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Indian Institute of Management Bangalore Quantitative Methods II
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Question 1 The personnel director of a manufacturing company has collected the following data on the Salary earned by each machinist in the factory along with the average performance rating over the past 3 years, the years of service and the number of different machines each employee is certified to operate. Avg Years Performance of S. No. Salary Rating Service Certifications 1 48.20 3.50 9 6 2 55.30 5.30 20 6 3 53.70 5.10 18 7 4 61.80 5.80 33 7 5 56.40 4.20 31 8 6 52.50 6.00 13 6 7 54.00 6.80 25 6 8 55.70 5.50 30 4 9 45.10 3.10 5 6 10 67.90 7.20 47 8 11 53.20 4.50 25 5 12 46.80 4.90 11 6 13 58.30 8.00 23 8 14 59.10 6.50 35 7 15 57.80 6.60 39 5 16 48.60 3.70 21 4 17 49.20 6.20 7 6 18 63.00 7.00 40 7 19 53.00 4.00 35 6 20 50.90 4.50 23 4 21 55.40 5.90 33 5 22 51.80 5.60 27 4 23 60.20 4.80 34 8 24 50.10 3.90 15 5 Sum 1308.00 128.60 599.00 average 54.50 5.36 24.96 6.00 stdev 5.47 1.29 11.22 1.32 Covariance(Y,X(i)) 4.52 50.55 3.86 ∑(salary(i) - 54.50) = 689.26
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The personnel director would like to build a regression model to estimate the average salary an employee should expect to receive. a) If the regression model is to be built using only one independent variable, which variable should be used? Explain why.
b) What is the proportion of variation in salary that can be explained by this variable?
c) A regression model using average performance rating is fitted to the data. Given that the intercept for the model is 39.35, the slope is 2.83 and the R square value is 0.445, predict the salary range for an employee with 12 years of service, who has received average reviews of 4.5, and is certified to operate 4 pieces of machinery?
d) A regression model (Model 1) is built using Certifications and Years of Service as the independent variables. The Summary Output and Residual Output are given below. Complete the Summary Output by filling in ALL the missing entries. SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square 0.8557 Standard Error Observations ANOVA df Regression Residual Total Standard Error 2.0798 SS MS F Significance F 5.71299E10
Intercept
Coefficients 35.8489
t Stat P-value 17.2369 0.0000
Lower 95% 31.5237
Upper 95% 40.1740
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Certifications Years of Service RESIDUAL OUTPUT
Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Total
1.5489 0.3749
0.3388 0.0398
4.5723
0.0002
0.8444
2.2533
Predicted Salary 48.5165 52.6409 53.4399 59.0640 59.8630 50.0163 54.5156 53.2926 47.0168 65.8621 52.9668 49.2664 56.8635 59.8139 58.2159 49.9181 47.7667 61.6886 58.2650 50.6680 55.9663 52.1678 60.9878 49.2173 1308.0000
Residuals -0.3165 2.6591 0.2601 2.7360 -3.4630 2.4837 -0.5156 2.4074 -1.9168 2.0379 0.2332 -2.4664 1.4365 -0.7139 -0.4159 -1.3181 1.4333 1.3114 -5.2650 0.2320 -0.5663 -0.3678 -0.7878 0.8827 0.0000
Residual^2 0.1002 7.0708 0.0677 7.4855 11.9925 6.1687 0.2659 5.7956 3.6740 4.1531 0.0544 6.0833 2.0636 0.5097 0.1730 1.7374 2.0545 1.7197 27.7207 0.0538 0.3207 0.1353 0.6207 0.7791 90.7997
e) Carry out a Hypothesis test to determine if the variable Years of Service has the ability to explain variation in salary. State your Hypotheses and conclusion clearly. Support your conclusion adequately and show all work.
f) What does the Multiple R value imply?
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The summary of the regression model with all three independent variables is as follows: SUMMARY OUTPUT Regression Statistics Multiple R 0.9558 R Square 0.9135 Adjusted R Square Standard Error 1.7261 Observations 24 ANOVA df Regression Residual Total 3 20 23 Significance SS MS F F 629.6726 209.8909 70.4480 8.2824E-11 59.5874 2.9794 689.26 Standard Error 1.9490 0.3268 0.0364 0.2916 Upper 95% 36.9868 l.7395 0.4012 1.9074
Intercept Avg Performance Rating Years of Service Certifications
Coefficients 32.9212 1.0578 0.3252 1.2992
t Stat P-value 16.8911 0.0000 3.2367 8.9223 4.4555 0.0041 0.0000 0.0002
Lower 95% 28.8556 0.3761 0.2492 0.6909
g) Carry out a partial F test to decide whether adding the third variable is worthwhile. Which model should be used (Model 1 or Model 2)?
Question
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The data in table 1 is collected from 20 North American cities. The data consists of number of crimes per million per year, population, percentage of families with annual income of less than $5000 and percentage unemployed. Table 1: Number of crimes per 1 Percentage Percentage million of families of per Region Population AI < 5000 unemployed annum 1 587000 16.5 6.2 11.2 2 643000 20.5 6.4 13.4 3 635000 26.3 9.3 40.7 4 692000 16.5 5.3 5.3 5 1248000 19.2 7.3 24.8 6 643000 16.5 5.9 12.7 7 1964000 20.2 6.4 20.9 8 1531000 21.3 7.6 35.7 9 713000 17.2 4.9 8.7 10 749000 14.3 6.4 9.6 11 7895000 18.1 6 14.5 12 762000 23.1 7.4 26.9 13 2793000 19.1 5.8 15.7 14 741000 24.7 8.6 36.2 15 625000 18.6 6.5 18.1 16 854000 24.9 8.3 28.9 17 716000 17.9 6.7 14.9 18 921000 22.4 8.6 25.8 19 595000 20.2 8.4 21.7 20 3353000 16.9 6.7 25.7 Mean 1433000 19.72 6.935 20.57 standard deviation 1703726 3.242254 1.207117 9.881407 Model 1 The following regression model is constructed to analyze the number of crimes per million (Y) and the percentage unemployed (X1). Y = β0 + β1 X1 The Regression output from Excel is shown in Table 2.
Table 2 Regression Statistics Multiple R R Square Adjusted R Square
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Standard Error Observations ANOVA
5.0968524
Df Regression Residual Total
SS
MS
F
Significance F
Intercept Percentage of unemployed
Coefficients -28.5267
Standard Error 6.81371 0.96867
t Stat
P-value
Lower 95% -42.841798 5.04445402
Upper 95% -14.211619 9.11465453
What percentage of the variation in crime rate is explained by model 1? Using model 1 calculate the probability that the crime rate will be more than 21 per million per annum when the unemployment percentage is 6.4. How do you interpret the value of β0 in model 1? Comment on the negative value of β0, is the negative value justified? Regression output for the no-intercept model (model 2) is shown in table 3. (a) Calculate the percentage of variation explained by model 2. (b) Which model (between models 1 and 2) should be used to analyze the data in table 1, justify your answer using appropriate regression measures. Model 2: Y = β1 X1
Table 3 ANOVA Df Regression Residual Total Standard Error #N/A Upper 95% #N/A 3.544923 8 48.57609 6 SS MS F 193.4028 2 Significance F
Intercept Percentage of unemployed
Coefficients 0
t Stat #N/A
P-value #N/A
Lower 95% #N/A 2.61747
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SPSS output for Model 3 is shown in Table 3. Identify the explanatory variable that has the maximum impact on the crime rate. Model 3: Crime rate = β0 + β1 x population + β2 x percentage families with AI < $5000 + β3 x percentage unemployed
Table 3 Coefficients(a) Unstandardized Coefficients Model 1 B -36.765 Std. Error 7.011 .000 .562 1.530 Standardized Coefficients Beta t -5.244 1.199 2.123 3.084 Sig. .000 .248 .049 .007
(Constant) Population Percentage of families AI < 5000 Percentage of unemployed
a Dependent Variable: Number of crimes per 1 million per annum
Table 4 below shows the distance measures and influential statistics for the Model 3.
Region 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Mahalanobis distance 1.48755 1.67850 4.39730 2.48362 .58566 1.42796 1.00081 .34335 4.69735 5.49044 14.40156 1.87023 2.17375 2.42169 .46345 2.65850 .72160 2.16993 3.60786 Cook’s distance .00402 .05467 .01940 .03168 .01213 .00412 .00534 .09363 .01186 .01966 5.63672 .00093 .00010 .01737 .00243 .05081 .00427 .08234 .16107 Leverage value .07829 .08834 .23144 .13072 .03082 .07516 .05267 .01807 .24723 .28897 .75798 .09843 .11441 .12746 .02439 .13992 .03798 .11421 .18989 DFFIT DFBeta0 -.67020 -.70931 -1.5232 -2.0099 .17268 .74556 .11047 -1.5728 .96872 -1.1451 3.90214 -.13388 .03303 -1.2747 .38857 2.01915 -.50030 1.99450 .87133 DFBeta1 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 DFBeta2 .03777 -.1731 .05058 -.0247 -.0700 -.0222 .05116 .01718 .05522 .13913 -.03596 .02595 .00613 .05768 -.00289 -.15623 .03824 .09428 .30520 DFBeta3 -.0290 .52891 .09395 .31261 .20012 -.0250 -.1487 .2422 -.2727 -.2524 -.0443 -.0474 -.0213 .04279 -.0296 .1147 -.0571 -.6005 -1.0549
-.79828 .67834 -.69459 .28739 .20837 .21486 .73284 .54493 -.74943 -19.5900 .10766 .03710 .50967 .12335 -.90175 -.17789 -1.0673 -1.8043
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2.91888
.25223
.15363
2.08036
1.36347
.00000
-.36775
.84411
Identify the most influential observation in the data. What will be the model parameter values (beta values) if this observation is not included in the sample. Calculate the DFFIT for observation 1 (region 1) Question The data in the following table shows the exchange rate of Indian rupee for 1 US dollar for the past 10 trading days.
Date Exchange rate INR Vs 1 Dollar 2009-10-05 47.5198 2009-10-06 46.9203 2009-10-07 46.6898 2009-10-08 46.3199 2009-10-09 46.4102 2009-10-12 46.4897 2009-10-13 46.3449 2009-10-14 46.1152 2009-10-15 46.285 2009-10-16 46.2997
1. Use the method of moving averages with number of periods k = 4 and calculate the forecasted exchange rate for 2009-10-20. 2. Use the method of simple exponential smoothing and forecast the values for 200910-20. Use the value of α = 0.8. 3. Comment on the forecasting power of the moving average with 4 periods and simple exponential smoothing with α = 0.8 Show all work. For the data used in this problem what should be the appropriate values of k and α (small or large)? Justify your answer.
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Question The planning committee of a bank makes decisions at the beginning of every year on the amount of funds to allocate to loans and to government securities. Some of the loans are secured (backed by collateral) and some are unsecured. A list of various types of loans and their annual returns are shown in table 1. Table 1: Annual rate of return on investments Type of investment Annual rate of return Government Securities 9 Secured loans Residential mortgage 11 Commercial mortgage 12 Automobile 15 Home improvement 13 Unsecured loans Vacation 17 Student 10 In making its decision, the planning committee must satisfy certain legal requirements and bank policies. These can be summarized as the following set conditions. 1. The amount allocated to secured loans must be at least four times the amount allocated to unsecured loans. 2. Auto and Home loans should be no more than 20% of all secured loans. 3. Student loan should be no more than 30% of unsecured loan. 4. The amount allocated to government securities should be at least 10% but no more than 20% of the funds available. 5. The amount allocated to vacation loans must not exceed 10% of all loans. The bank has Rs 5 million available for loans and investments in the current year. The bank collects 10 percent of the principle for each of secured loans along with the interest at the end of the year (simple annual interest is calculated for the entire loan amount for the year). The entire principle and interest is collected on the unsecured loans at the end of the year. In addition to these, Rs 6 million is available at the beginning of the second year. This entire amount is available for allocation in the second year. Formulate a linear programming model that will enable the planning committee to determine the optimal allocation of funds if the objective is to maximize the annual return over the two years. Clearly state all the decision variables and the constraints.
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Question Sugarco can manufacture three types of candy bars. Each candy bar consists of sugar and chocolate. The composition of each type of candy bar and the profit earned from each candy bar is shown in the following table. Fifty ounces of sugar and 100 ounces of chocolate are available.
Amount of Sugar Bar 1 Bar 2 Bar 3 Used 1 1 1
Amount of Chocolate Used 2 3 1
Profit (in Rs per unit) 3 7 5
(a) Formulate a linear programming problem. Clearly state the decision variables.
(b) Solve the problem and find the optimal value of the objective function and the optimal values of the decision variables.
(c) If the sugar can be purchased at Rs 2 per ounce, comment whether Sugarco should buy one additional ounce of Sugar. What will be the optimal profit if 10 additional ounces of sugar is made available?
(d) What is the minimum profit for candy bar 1 for which the current optimal solution will remain as the optimal solution.
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(e) Suppose candy bar 1 requires only 0.5 ounces of sugar and 0.5 ounces of chocolate, will the current optimal solution remain as optimal solution?
(f) A new candy bar can be made using 3 ounces of sugar and 4 ounces of chocolate. What should be the minimum profit so that it is advisable to make this new candy bar?
Question Consider the following LP Formulation Max Profit = 300 X1 + 500 X2 + 250 X3 Subject to 4 X1 + 6 X2 + 2 X3 <= 100 (Saline) 2.5 X1 + X2 + 1.5 X3 <= 40 (Benzene) X1 – X2 = 0 (Balance) 2 X1 + X2 >= 10 (Demand) Xi >= 0 (Non-negativity) Answer Report Target Cell (Max) Cell Name $E$6 OBJ SUM Adjustable Cells Cell Name $B$7 Var X1 $C$7 Var X2 $D$7 Var X3 Constraints Cell Name $E$2 Saline SUM Benzene $E$3 SUM Balance $E$4 SUM Demand $E$5 SUM Sensitivity Report Adjustable Cells
Original Value 8562.5 Original Value 0 0 0 Cell Value 100 40 0 26.25
Final Value 8562.5 Final Value 8.75
Formula $E$2<=$G$2 $E$3<=$G$3 $E$4=$G$4 $E$5>=$G$5
Status
Slack
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Cell $B$7 $C$7 $D$7 Constraints Cell $E$2 $E$3 $E$4 $E$5
Name Var X1 Var X2 Var X3
Final Value 8.75
Reduced Cost 0 0 0
Objective Coefficient 300 500 250
Allowable Increase 450 450 92.86
Allowable Decrease 216.67 216.67 90
Name Saline SUM Benzene SUM Balance SUM Demand SUM
Final Value 100 40 0 26.25
Shadow Price 40.625 112.5 -143.75
Constraint R.H. Side 100 40 0 10
Allowable Increase 14.29 21.67 4.55
Allowable Decrease 28.89 5 10 1E+30
1. Which of the four constraints are binding? 2. What is the change in the objective function value if the minimum demand (RHS of DEMAND CONSTRAINT) is increased from 10 to 20? 3. A new constraint is added. The constraint is given below: 4 X1 + 8 X2 + 12 X3 <= 180 What is the impact of this constraint on the present optimal solution? Is this new constraint binding or not binding? 4. The objective function coefficient of X1 is increased from 300 to 400. What is the change in the objective function value? What is the change in the value of the variables in the optimal solution?
5. The objective function coefficient of X1 is decreased from 300 to 150 and that of X3 increased to 330 simultaneously. What is the change in the objective function value? What is the change in the value of the variables in the optimal solution? 6. The demand constraint is changed as follows: X1+ X2 <= 10.
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What are the values of X1 and X2 in the new optimal solution? 7. The objective function coefficients are changed simultaneously as follows: X1 450; X2 750; X3 375. Is the current optimal solution still optimal? What is the change in the value of the objective function in the optimal solution? 8. The availability of Saline and Benzene is changed to 50 and 20. What is value of the variables in the new optimal solution 9. A new variable X4 is introduced into the formulation. Each unit of X4 requires 4 units of Saline and 6 units of Benzene. The profit contribution is 840. Is it worthwhile producing X4? What is the minimum required value of the profit contribution of X4 for it to be worthwhile?
Question 1 The following table gives marks scored by 12 students in the QM-II exam along with their CAT score and the number of classes missed by these students. Three regression models are developed by treating the marks in the QM-II as response variable and CAT score and number of missed classes as explanatory variables. The data is shown in table 1 along with influential statistics data for model 2.
Student No QM-II Marks (Y) CAT Score (X1) No missed classes (X2) 1 7 6 1 6 0 1 0 2 1 1 2 2.3333 Maha distance Cook’s distance Leverage DFFit DFBeta0 DfBeta1 DfBeta2
1 85 65 0.30311 0.0058 0.02756 -0.232 2 66 50 3.52954 0.68655 0.32087 4.8320 3 58 55 2.24782 0.35484 0.20435 -2.930 4 90 65 0.30311 0.01639 0.02756 0.3910 5 65 55 2.24782 0.00044 0.20435 -0.103 6 87 70 1.28724 0.14518 0.11702 -1.564 7 94 65 0.30311 0.08488 0.02756 0.8899 8 98 70 1.28724 0.08424 0.11702 1.1916 9 81 66 0.52591 0.02677 0.04781 -0.543 10 91 70 1.36777 0.00196 0.12434 0.1850 11 76 50 6.94407 0.0347 0.63128 -1.444 12 74 55 1.65327 0.04416 0.1503 -0.931 Mean 80.416 61.33 Std. Dev. 12.688 7.773 2.4984 The following three models are used to analyse the relationship between the explanatory variables. Model 1: Y = β0 + β1 X1 Model 2: Y = β0 + β1X1 + β2X2 Model 3: ln(Y) = β0 + β1X1 + β2X2 The Excel output for model 1 is shown in Table 2: Table 2: Regression output for Model 1
0.04067 1.74349 3.83791 -0.06838 0.13572 4.23182 -0.15561 -3.22353 2.83831 -0.89955 -5.96439 -5.955
-0.0046 -0.0397 -0.0514 0.00786 -0.0018 -0.0828 0.01788 0.06307 -0.0486 0.0152 0.08586 0.08395
0.03049 0.72536 -0.6565 -0.0512 -0.0232 0.08405 -0.1166 -0.0640 -0.0863 0.0161 0.22721 0.20314
QM-II marks and the
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R gre e ssion S tatistic s Multiple R RS re qua Adjusted RS re qua S nda E ta rd rror 7 457 .4 3 5 O bserva tions ANO VA df S S MS F S ignific eF anc R ression eg R esidua l T l ota 1 7 .9 7 70 1 C ffic nts oe ie S tandard E rror t S tat P-value L owe 9 % Uppe 9 % r 5 r 5 Interc ept -0 4 1 .1 4 3 -4 .3 6 2 1 7 3 .1 4 7 78 C S ore AT c 0 871 .2 8 2 0 0789 .7 9 4 3 1
Using table 2, find the regression parameters for model 1. Comment whether the overall model can be accepted. What is the probability that a student with a CAT score of 65 will score more than 85 marks in QM_II? The Regression output for model 2 is shown in Table 3 Table 3: Regression output for model 2
SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error 5.29589207 Observations ANOVA df Regression Residual Total Coefficients Intercept CAT Score Missed Classes Standard Error 19.32753 0.289113 0.899491 t Stat P-value Lower 95% 1.512064 0.031836032 -4.984688867 Upper 95% SS MS 759.249 2 F Significance F
252.4183
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Write model 2 by determining the model parameters. Calculate the increase in the R-square value in model 2 from model 1. Comment whether this increase can be justified for the use of model 2 instead of model 1. Find the probability that a student with a CAT score of 65 and 2 missed classes will score more than 85 marks in QM-II. What is the correct classification of observation 11: is it an outlier, leverage or influential observation? Justify your classification.
The Regression output for model 3 is shown in table 4 Table 4: Regression output for model 3
SUMMARY OUTPUT Regression Statistics R Square Standard Error ANOVA df Regression Residual Total Coefficients 3.984976 0.007939 -0.04148 Standard Error t Stat P-value Lower 95% 3.426841632 -0.00041005 -0.067457281 Upper 95% SS 0.259066 MS 0.00457 F Significance F
Intercept CAT Score Missed Classes
Identify the significant independent variables in model 3 and the R-square value.
If the number of missed classes increases by 1, what will be the change in the marks scored in QM-II. Among the three models, which is the most preferred model? Justify your answer.
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Question
A company needs to lease warehouse storage space over the next 5 months. The space that will be required during each of these months is known. However, since these space requirements change from month to month, it may be most economical to lease only the amount needed each month on a month-by-month basis. On the other hand, the relative cost for leasing space for additional months is much less than for one month, so it may be less expensive to lease the maximum amount needed for the entire 5 months. Another option is the intermediate approach of changing the total amount of space leased by adding a new lease and/or having an old lease expire. The space requirement (in thousands of square feet) and the leasing costs (Rs.) for the various leasing periods (durations in months) are Month 1 2 3 4 5 Required Space (1000 sq ft) 15 10 20 5 25 Lease duration (months) 1 2 3 4 5 Cost (Rs.) per 1000 sq ft leased 2800 4500 6000 7300 8400
a) Assuming that two or more leases of different durations can begin at the
same time give a mathematical programming formulation to determine the leasing schedule that provides the required amounts of space at a minimum cost. Define all variables clearly and label all constraints. Consider the following parts independently. For each part make suitable modifications to the formulation in part a). State the modifications clearly. b) Only one kind of lease can be signed in a month. c) At most two changes can be made in the amount of space hired. d) Two or more leases cannot run simultaneously in any month.
e) It costs Rs. 2500 to process a lease.
f)
A 10% discount is available on the cost of leasing space that is in excess of 10000 sq ft. However, this discount is available only for lease of 5 month duration signed in the first month. cannot be signed in any subsequent month.
g) If in month 1 a one period (month) lease is signed then a one period lease
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Question
Nanjanguda Investments Company is considering four projects for investment. Project 1 will yield a net present value (NPV) of Rs. 250,000; project 2, an NPV of Rs. 180,000; project 3, an NPV of Rs. 120,000; and project 4, an NPV of Rs. 80,000. Each project requires a cash investment at the present time: project 1, Rs. 80,000; project 2, Rs. 50,000; project 3, Rs. 40,000; and project 4, Rs. 30,000. At present, a budget of Rs. 150,000 is available for investment. The following Binary Integer Program (BIP) has been formulated to help decide which projects should be chosen for investment: Let xi ( i = 1, 2, 3, 4) = 1 if investment is made in project I and 0 otherwise Max z = 25x1 + 16 x2 + 12 x3 + 8 x4 s.t. 8x1 + 5x2 + 4 x3 + 3x4 ≤ 15 xi = binary ( i = 1, 2, 3, 4 ) The above BIP has a single constraint and is referred to as a knapsack problem. The LP relaxation for this BIP and other subproblems encountered while finding the solution to the BIP can be solved quite easily by observing that projects will be chosen according to the return they earn for each unit of investment. For example, to solve the LP relaxation of the knapsack problem above, we will first select project 2 since it has the highest return (3.2) per unit investment. We can select all of project 2 (x2=1), and we will still have Rs. 100,000 left. We will next select project 1 since it has the second highest return per unit investment. We can afford to select all of project 2, and as a result Rs. 20,000 will be left. The next best selection is project 3. However, since it requires an investment of Rs. 40,000, we can take only half of project 3. Hence the optimal LP solution is: x1 = x2 = 1, x3 = ½, x4=0. In order to solve the Nanjanguda Investments Company’s decision problem the Branchand Bound method is used and the following results are obtained: LEVEL 0
LP0 x1 = x2 = 1 x3 = 1 =1/2 X3 x4 = 0
x3 = 0
LEVEL 1 LP1 X1 = 3/4 X2 = x3 = 1 x4 = 0 LP2 x1 = x2 = 1 x3 = 0
x4 = 2/3
Answer the following questions: a) The lower bound on the optimal objective function value to the BIP at level 0 is _______________. Give adequate explanations.
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b) The best upper bound on the optimal objective function value to the BIP at
level 1 is _______________ .Give adequate explanations.
c)
Complete the branch-and-bound (b-a-b) tree using the above information in order to find the optimal solution for the problem. Proceed down the node that has the most promising value. While branching, set the variable value on the left branch equal to 1 and on the right branch equal to zero. Solve the left branch problem before the right branch problem. Clearly • show all branches, branching variables and their values • number the subproblems created in the order that you solve them • give the optimal solutions/objective function values for each subproblem solved • state when a node is fathomed and why and • Give the optimal solution and optimal objective function value. Question Consider the following linear programming (Primal) problem (LP1): Max 2x1 + x2 + x3 + 4x5 - 3x5 s.t. x1 + 2x2 + x3 - x4 + 2x5 <= 8 2x1 + x2 + x3 + 2x4 + x5 <= 12 5x1 + x3 + x5 = 10 x1, x2, x3, x4, x5 ≥ 0 (1) (2) (3)
a) Write the dual linear programming problem (DLP1) for the above LP1 and define the dual variables
used. b) Determine if (1,1,3,2,2) is a feasible solution for the given primal problem LP1? Show all work c) Using the information obtained so far give a lower and an upper bound for the objective function value of the dual problem DLP1. Support your answer with adequate reasoning.
d) Use duality properties to determine if the following solutions are optimal for
the dual problem DLP1. Support your answer with adequate reasoning. Clearly state the properties being used and show all work. 1. (1,3,1) 2. (0,2,-0.4)
e) Give the optimal solution and the optimal objective function value for the
primal problem LP1. Support your answer with adequate reasoning.
f) Another non-negative variable x6 is added to the primal problem with coefficients 3, 4 and 1 in
constraints (1), (2) and (3) respectively and with an objective function coefficient of 7. Write the dual LP problem corresponding to this extended primal LP with 6 variables (call it LP2).
g) Use duality property(s) to determine the optimal solution for LP2. Support your answer
with adequate reasoning.
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h) A new constraint: 2x1 - 3x2 + 3x4 - 2x5 >= 15 is added to LP1. Determine the optimal
solution to this modified LP1. Support your answer with adequate reasoning.
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Question NS Consumer Electronics (NSCE) has recently added an LCD TV to its product portfolio. They have hired the Quickbuck Advertising Company (QAC) to promote the TV. QAC has come up with two possible options-(1) advertising on IPL and (2) advertizing on the popular TV serial “Bahu Bhi Saas Banegi” (also known as B2SB). They are trying to target Rural high income (HIR) households, Urban High income (HIU) households and Young unmarried urban men (YUM). They have estimated the following exposures: Exposure in ‘000/ min of Ad. Cost/minute of Ad. Advertisement in HIR HIU YUM Rs. ‘000 IPL 4 10 5 100 B2SB 8 5 4 60 NSCE would like to minimize the cost with the following constraints: The exposure to HIR must be at least 40,000 The exposure to HIU must be at least 60,000 The exposure to YUM must be at least 35,000 There must be at least two minutes of advertising on each of the shows (IPL and B2SB). 1. Find out how many minutes of advertising is to be purchased on IPL and B2SB so that the total cost of advertizing is minimum. What is the cost of advertising in the optimal solution? 2. The total budget given by NSCE is Rs. 600,000. Consider this as a hard constraint. Formulate the problem as a goal programming problem. Define all variables used.
3. Consider the constraints given above as the goals in the same order of priorities. For example “The exposure to HIR must be at least 40,000” is the first priority. If the budget constraint cannot be violated, find out what is the minimum possible deviation for each of the 4 goals, with respect to the solution obtained in question no. 1. In case more than one solution is possible, select the one which will minimize the goal deviations in the order of priority mentioned above.
4. Solve the goal programming problem formulated in question no. 2. Calculate the over achievement/under achievement for each of the goals. 5. How does the above solution change if goal no. 4 is given the top most priority and the budget constraint is a soft constraint and ranked 2 in priority?
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Question Consider a balanced transportation problem with 3 sources and 4 destinations. The sensitivity report of the solution is given below.
Name Source 1 Destination $B$10 1 Source 1 Destination $C$10 2 Source 1 Destination $D$10 3 Source 1 Destination $E$10 4 Source 2 Destination $B$11 1 Source 2 Destination $C$11 2 Source 2 Destination $D$11 3 Source 2 Destination $E$11 4 Source 3 Destination $B$12 1 Source 3 Destination $C$12 2 Source 3 Destination $D$12 3 Source 3 Destination $E$12 4
Cell
Final Reduced Objective Allowable Allowable Value Cost Coefficient Increase Decrease 0 2000 7000 0 6000 0 0 1000 0 1000 0 3000 1 0 0 1 0 4 4 0 1 0 2 0 4 2 1 4 1 4 3 1 3 1 2 2 1 2 1 1 2 1 1 3 1 2 1 1
Cell $B$15 $B$16 $B$17 $B$18 $B$19 $B$20 $B$21
Name Source 1 Constraint Source 2 Constraint Source 3 Constraint Dest 1 Constraint Dest 2 Constraint Dest 3 Constraint Dest 4 Constraint
Final Value 9000 7000 4000 6000 3000 7000 4000
Shadow Constraint Allowable Allowable Price R.H. Side Increase Decrease 0 9000 1E+30 0 -2 7000 2000 0 -1 4000 2000 0 3 6000 0 2000 2 3000 0 2000 1 7000 0 7000 3 4000 0 2000
Since this is a balanced transportation problem, any increase or decrease of the requirement at any destination has to be matched with a corresponding increase or decrease at one of the sources.
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1. If, for some reason, Route 3 to 1 (Source 3 to Destination 1) is not available, what will be the change in the solution?
2. If the requirement at destination 3 is increased by one unit, and the corresponding increase is at source 2, what is the change in the value of the objective function?
3. If the requirement at Destination 4 is to be increased by 1 unit, which is the best source from which the extra unit is to be supplied?
4. Which is the costliest combination for any increase in the requirement at any of the destinations by one unit? Which the source involved with this combination?
5. What will be the change in the solution if the objective function coefficient of the route from source 1 to destination 1 is changed from 4 to 3?
6. What will be the change in the solution if the objective function coefficient of the route from source 2 to destination 2 is changed from 4 to 5?
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Question NLNN Kumar who just completed a short term course at the International Institute of Management and Accounting (IIMA), is trying his newly acquired knowledge with the data he received from his company. He is working in NS Software Services (popularly known as NS3), and received the data from the software testing division of the company. He had presumed that the number of bugs per 10 kilolines of code can be explained by two explanatory variables namely, the months of experience of the team (calculated as the average no. of months of total experience for the team) and size of the team (number of team members). He fitted the regression in Excel and got the following output. Summary Statistics: n= 10 ∑X1i = 150 ∑X2i = 831 ∑X1iX2i = 13990 ∑X21i = 2580 ∑X22i = 76117 SUMMARY OUTPUT Regression Statistics Multiple R 0.981583 R Square 0.963504 Adjusted R Square Standard Error Observations 10 ANOVA Significance df SS MS F F Regression 2 70.7897 35.39485 Residual 7 2.681371 0.383053 Total 9 73.47107 a) Calculate the adjusted R2 for this model. Explain how adjusted R2 differs from R2 and what is its significance?
b) Test the following null hypothesis. Use α = 0.01. State the conclusion of the result of the hypothesis test clearly
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H0: β1 = β2 = 0 H1 : At least one β is not equal to zero
Kumar also got the following output regarding the regression coefficients and the corresponding confidence intervals Standard PLower Upper Coefficients Error t Stat value 95% 95% Intercept 4.3036 2.2625 1.9022 0.0989 -1.0462 9.6535 Teamsize 0.8559 0.7481 1.1442 0.2902 -0.9130 2.6249 Months of Exp -0.0849 0.1613 -0.5261 0.6151 -0.4664 0.2966 c) Test the null hypothesis that the team size has no impact on number of bugs and also that months of experience has no impact on the number of bugs. State your conclusion of the hypothesis test clearly. Use α = 0.05 d) Comment on the results of Question nos. 1, 2 and 3, by explaining any consistency or inconsistency among your answers. e) Calculate the Variance Inflation Factor (VIF) of the regression coefficients and comment on the value obtained.
Question A regression analysis is performed on the annual food expenditure (AFE) with annual income (AI) and family size (FS) as explanatory variable. The following two models are used to establish the relationship between Annual Food Expenditure and Annual Income and Family Size. Model 1: AFE = B0 + B1 × AI Model 2: AFE = β0 + β1 × AI + β2 × FS Regression output for Model 1 is shown below:
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SUMMARY OUTPUT (Model 1)
Regression Statistics Multiple R R Square Adjusted R Square 0.77684 Standard Error 1.907748 Observations 20 ANOVA Df SS MS F Significance F Regression 244.3584 Residual Total 309.8695 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 1.043354 0.934366 1.116644 0.278831 -0.91968 3.006385 AI 0.149707 0.01827
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Model 2: Summary Output Regression Statistics Multiple R 0.894788 R Square Adjusted R Square Standard Error 1.906239 Observations 20 ANOVA Df Regression Residual Total 17 19 SS MS 124.0479 F Significance F
309.8695 Upper 95% 2.781999 0.183532
Intercept AI FS
Coefficients Standard Error t Stat P-value 0.396888 1.130483 0.351079 0.729843 0.140701 0.020301 6.930686 2.43E-06 0.391196 0.385735
Lower 95% -1.98822 0.097869
Table 1 shows the predicted value of AFE, distance measures, Mahalanobis Distance (Mah_D), Cook’s Distance (Cook_D), leverage values and DFBeta Values.
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Table 1:
S.No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 AFE 5.2 5.1 5.6 7.2 11.3 8.1 7.8 5.8 5.1 18 4.9 11.8 5.2 4.8 7.9 6.4 15.2 13.7 5.1 2.9 AI 28 26 32 24 54 59 44 30 40 82 42 58 28 20 42 47 112 85 31 26 FS 3 3 2 1 4 2 3 2 1 3 3 4 1 5 3 1 4 5 2 2 Predicted Y 5.5101 5.2287 5.6817 4.1649 9.55951 9.48062 7.76131 5.4003 6.41611 13.10793 7.47991 10.12231 4.7277 5.16688 7.47991 7.40102 17.72015 14.31243 5.541 4.8375 Cook_D 0.00107 0.00021 0.00005 0.17839 0.0371 0.02388 0.00001 0.00132 0.03423 0.60735 0.03918 0.03436 0.00413 0.01847 0.00104 0.02333 0.90717 0.01769 0.00156 0.03617 Leverage 0.04832 0.05784 0.02292 0.10186 0.05656 0.05855 0.00479 0.02686 0.10414 0.13414 0.00705 0.05629 0.09698 0.40093 0.00705 0.1234 0.40778 0.22229 0.02477 0.03746 DFB0 -0.02595 -0.01164 -0.01169 0.7918 -0.11727 -0.09611 0.00138 0.05971 -0.27917 -0.39432 -0.1063 -0.13264 0.11672 0.02602 0.01731 -0.19511 0.98484 0.17584 -0.06449 -0.31501 DFB1 0.00078 0.00036 0.00008 -0.00299 -0.00051 -0.00339 -0.00002 -0.00048 -0.00152 0.02313 0.00186 0.00036 -0.00021 0.00338 -0.0003 -0.00215 -0.02936 -0.00195 0.00047 0.00329 DFB2 -0.0099 -0.00442 0.0014 -0.17658 0.08809 0.0641 0.00056 -0.00608 0.10024 -0.13265 -0.04264 0.07781 -0.02941 -0.07899 0.00694 0.08601 0.04395 -0.04792 0.00715 0.02183
a) What percentage of variations in Annual food expenditure (AFE) is explained by Model 1? b) In Model 1, calculate F statistic and comment what inference can be obtained using the F statistic value of model 1. c) In model 1, what is the 95% confidence interval for mean value of AFE when the annual income is 30? d) Comment whether model 2 (addition of new variable, family size) adds any value in predicting the annual food expenditure compared to model 1. e) In model 2, check whether the beta coefficient for FS is different from zero. f) Identify if there are any potential outliers in the model.
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g) Calculate the change in the predicted value of AFE for the sample point 1, when it is removed from the sample in estimating the parameter values. h) Identify the sample point that has highest influence on the regression model 2. What will be the change in the Beta co-efficient values if this sample point is removed from the sample?
Question The quarterly revenue of H&B, a catalog company, from 2005 till 2008 quarter 3 is shown in the following table (Table 1) along with trend to moving average values calculated for the multiplicative model Yt = Tt × St × Et (where Yt is the value at time t, Tt is the trend component in the value, St is the seasonal component and Et is the error component). The regression output for trend component is shown in table 2. Table 1: Quarterly revenue data and ratio to moving average S.No Year, Quarter Revenue (in Millions) Moving Average Ratio to Moving Average 1 2005,Q1 72 2 2005,Q2 68 3 2005,Q3 80 72.5 1.1034 4 2005,Q4 70 73.5 0.9523 5 2006,Q1 76 74 1.0270 6 2006,Q2 70 74.5 0.9395 7 2006,Q3 82 75.5 1.0860 8 2006,Q4 74 75 0.9866 9 2007,Q1 74 74 1 10 2007,Q2 66 74.5 0.8859 11 2007,Q3 84 76 1.1052 12 2007,Q4 80 76.5 1.0457 13 2008,Q1 76 78.5 0.9681 14 2008,Q2 74 78.5 0.9426 15 2008,Q3 84
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Table 2: Regression output for Trend SUMMARY OUTPUT Regression Statistics Multiple R 0.892401 R Square 0.79638 Adjusted R Square 0.776018 Standard Error 0.882523 Observations 12 ANOVA df SS MS Regression 1 30.46154 30.46154 Residual 10 7.788462 0.778846 Total 11 38.25 Standard Error t Stat 0.677061 105.3478 0.0738 6.253888
F 39.11111
Intercept t
Coefficients 71.32692 0.461538
P-value 1.46E-16 9.46E-05
a) Calculate the seasonality index for Quarters 1, 2, 3 and 4 b) Estimate the trend component for the revenue data shown in table 1 c) Forecast the revenue for 2008 quarter 4 and 2009 quarter 2.
Question NS International Instruments, located in Bangalore, is planning to fill 4 positions in the company. It is not necessary that all the four positions need to be filled. They have shortlisted 8 potential candidates. Their names and other details are given below: Details Ambika Bhama Chandana Divya Eileen Farah Geetha Hema Degree MCA MCA MBA MBA BE BE B Com B Com Local Yes Yes Yes Yes Non-Local Yes Yes Yes Yes Age (years) 25 27 26 29 24 23 21 22 Contribution 6 5 4 6.5 4.5 7 5.5 3.5 to the company
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The contribution to the company is in Rs. Lakhs per annum. If the company selects any non-locals, it will have to arrange for a special coaching in Kannada, at a cost of Rs. 50,000 (irrespective of the number of non-locals). The company wants to position itself as a young company, and hence the average age of the selected persons should be less than 25 years. a) Formulate this problem as a mixed integer problem to maximize the contribution to the company. Modify the above formulation to incorporate the following requirements (Treat each of the following questions independently). b) Of the persons selected, there must be at least one Post graduate and at least one graduate. c) There must be as many locals as non locals d) It turns out that Farah and Geetha are sisters and insists that either both should be offered or neither should be offered. e) If two MCAs are selected, one BE must be selected.
Question Mehak Perfumeries, located in Mysore, manufactures three types of perfumes: Anand, Bela and Chameli. The monthly demand for each perfume is 4000 fluid ounces. The market price for each fluid ounce of the perfume is: Rs. 600 for Anand, Rs. 750 for Bela and Rs. 1000 for Chameli. Production of Anand and Bela requires the natural ingredient, ING. This ingredient can be purchased at a price of Rs 230 per fluid ounce. A chemical process (Process 1) can convert one fluid ounce of ING into one fluid ounce of Anand. Similarly, another chemical process (Process 2) can convert one fluid ounce of ING into one fluid ounce of Bela. Anand can be further processed to produce Bela and Chameli using a chemical process (Process 3) that requires one fluid ounce of Anand to produce 0.6 fluid ounces of Bela and 0.4 fluid ounces of Chameli at a cost of Rs 160 per fluid ounce. Similarly another chemical process (Process 4) can further process Bela and convert one fluid ounce of Bela to 0.8 fluid ounces of Chameli at a cost of Rs 120 per fluid ounce. In order to decide what quantities of Anand, Bela and Chameli should be sold to maximize monthly profits a linear programming model is formulated and solved as follows:
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Decision Variables: Let ING be the fluid ounces of the natural ingredient purchased; INGA and INGB be the fluid ounces of the natural ingredient used to produce Anand and Bela; AS, BS and CS be the fluid ounces of Anand, Bela and Chameli sold; AP and BP be the fluid ounces of Anand and Bela processed further. Model: Maximize 600AS + 750BS + 1000CS – 230ING – 160AP – 120BP s.t. ING – INGA – INGB = 0 INGA – AS – AP = 0 BS + BP – INGB – 0.6AP = 0 0.4AP + 0.8BP – CS = 0 AS <= 4000 BS <= 4000 CS <= 4000 AS, BS, CS, AP, BP, ING, INGA, INGB are non-negative.
SOLUTION OUTPUT Adjustable Cells Cell $B$12 $C$12 $D$12 $E$12 $F$12 $G$12 $H$12 $I$12 Constraints Cell $J$13 $J$14 $J$15 $J$16 Name Final Value -1.82E-12 4.55E-13 3.64E-12 -4.55E-13 Shadow Price Constraint Allowable Allowable R.H. Side Increase Decrease 0 1E+30 13000 -230 0 1E+30 4000 230 0 9000 1E+30 -437.5 0 1E+30 4000 Name AS BS CS AP BP ING INGA INGB Final Value 4000 4000 4000 0 5000 13000 4000 9000 Reduced Objective Allowable Allowable Cost Coefficient Increase Decrease 370 600 1E+30 370 520 750 1E+30 520 562.5 1000 1E+30 562.5 -353 -160 353 1E+30 0 -120 1E+30 450 0 -230 320.91 370 0 0 353.00 370 0 0 3529.99 450
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Use the solution output above to answer the following questions. State your answers precisely and give adequate explanations. a) What is the optimal production plan for Mehak perfumeries? State your answers precisely. b) If the price of ING goes up by Rs 120, what will be the impact on the optimal production plan? Will it change? What will be the impact on the optimal profit? Assume that all else remains the same. Support your answers with precise reasons. c) Determine the shadow price associated with the first constraint of the model.
d) Suppose that a new process can convert each ounce of Anand to 0.25 fluid ounces of Bela and 0.75 fluid ounces of Chameli at a cost of Rs 100. Will it be worthwhile using this process? Why or Why not? Assume that all else remains the same. e) Suppose that the demand for Anand goes up by 10%, for Bela goes up by 15%, while that for Chameli goes down by 20%. What will be the impact on the optimal profit? Assume that all else remains the same. What impact will this have on the optimal production plan? Will it change? f) Suppose that the availability of ING were limited to 15000 fluid ounces. How will this impact the optimal production plan and the optimal profit? What would the shadow price of this constraint be when added to the formulation? Give the range within which this shadow price will remain valid. g) If it was decided that Anand was to be processed further to produce Bela and Chameli will this have any impact on the profit? If so, what would the impact be?
Question Hassan Electric Winders (HEW) Inc., located in Hassan, manufactures voltage converters and stabilizers. Processed tungsten is a key ingredient required for this purpose. An important step in the manufacturing of converters is the melting of tungsten which can be done using several methods. The cost involved in using each method varies, and so does the quality of the tungsten processed.
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At HEW two methods are used, Method 1 and Method 2. If Method 1 is used to obtain processed tungsten, the cost on a per converter basis is Rs 250, while using Method 2 the cost turns out to be Rs 325 per converter. Using the two methods, the quality of tungsten obtained after melting and, therefore, the quality of the converters obtained is as given in Table 1 below. For instance, if Method 1 is used for melting tungsten, 25% of the converters produced are of Grade 2. Note that Grade 1 indicates poor (but acceptable) quality, while Grade 4 indicates excellent quality. HEW can improve the quality of the converters obtained by “refiring” the melted tungsten of a particular grade. The refiring process will cost Rs 120 on a per converter basis. The results of the refiring process are as given in Table 2. For instance, when defective grade melted tungsten is refired, 30% gets upgraded to Grade 1, 15% gets upgraded to Grade 2, 10% to Grade 3 and 20% to Grade 4. Assume no loss of material. HEW has sufficient furnace capacity to melt and/or refire tungsten for at most 15,000 converters per month. HEW’s monthly demands for converters of various grades are: 2000 for grade 4, 2000 for grade 3, 3000 for grade 2 and 4000 grade 1.
Table 1. Percentage Yield from the two methods used for melting
Grade of Transistor Produced Defective Grade 1 Grade 2 Grade 3 Grade 4 20 35 25 10 10
Method 1 (Percentage of converters)
Method 2 (Percentage of converters)
15 25 25 15 20
Table 2. Percentage Yield from the refiring process
Refired Grade of Converters Defective Grade 1 Grade 2 Grade 3 Grade 4 a)
Defective 25 30 15 10 20
Grade 1 0 20 20 30 30
Grade 2 0 0 35 35 30
Grade 3 0 0 0 40 60
Develop an appropriate linear programming model for HEW that will help them determine how to meet the monthly demand for the various grades of converters at a minimum cost. Define the decision variables needed to develop your model precisely. State the purpose and meaning of each constraint and label the constraints appropriately.
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b)
Now, suppose that the melting and refiring process also results in a loss of tungsten. Specifically, if Method 1 is used, 6% of the tungsten is lost in the melting process. For Method 2, the rate of loss is 5%, while the loss rate in the refiring process is 2%. Assume that after accounting for the losses, the yield rates are the same as given in Table 1 and Table 2. Modify the model described in a) to account for the losses. Define clearly any additional variables, if used. Describe clearly the appropriate modifications made in the objective function and the constraints in the model.
Question Quitmeyer electronics incorporated (QEI) manufactures the following five microcomputer peripheral devices: internal modems, external modems, CD drives, hard disk drives and memory expansion boards. Each of these products requires time in hours on two types of electronic testing devices as shown on the following table (Table 1). Table 1. Device Internal External CD Drives Hard disk Memory Modem Modem drives expansion boards Test device 7 3 6 18 17 1 Test device 2 5 2 15 17 2 Test device 1 is available for 120 hours per week whereas test device 2 is available for 80 hours per week. Table 2 summarizes the revenues and material costs for each product. Table 2. Device Internal External CD Drives Hard disk Memory Modem Modem drives expansion boards Revenue per 200 120 130 430 260 unit sold Material and 50 20 80 180 60 testing cost per unit a) Formulate a linear programming model to maximize QEI’s total weekly profit. b) Use the graphical method to find the maximum profit that Quitmeyer can make. Show all work on the graph paper and also state the maximum profit below.
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c) Calculate the value of shadow price for the resources, test device 1 and 2.
d) If one unit of CD driver is manufactured, what will be the change in the objective function value of the profit maximization problem? Write all the steps involved in calculation of the new profit.
e) What should be the minimum revenue for CD driver so that it is advisable to make CD drivers? f) Find the optimal product mix
Question Doosra Life Insurance (DLI) is an insurance company based in Chennai. The absenteeism among employees is a major problem at DLI and the company is interested in establishing the relationship between number of absent days and duration of the employment at the company and a measure of attitude toward the company (part of recently concluded clinical interview with the company psychologist). The attitude scale ranged from 1 (extremely favorable) to 13 (extremely unfavorable). The data corresponding to 12 clerks are shown in Table 1. Table 1. Absent data for 12 clerks
C lerk 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 S um Avera e g S ev td.D D ysAbsent, Y a Attitude, X 1 Y rswithc pa X ea om ny, 2 1 1 1 0 2 1 1 2 2 4 3 2 3 5 4 2 5 6 5 6 5 6 7 4 9 1 0 8 1 3 1 1 7 1 5 1 1 9 1 6 1 2 1 0 7 5 7 5 5 9 6 5 .2 6 5 .2 4 1666 .9 6 6 6 7 5 7948 .6 4 0 8 5 3 3 1 6 1 .9 4 1 5 2 3 1625 .1 7 4 8 5
C enteredX 1 C enteredX 2 -5 5 .2 -3 2 .9 -4 5 .2 -3 2 .9 -4 5 .2 -2 2 .9 -3 5 .2 -2 2 .9 -1 5 .2 -0 2 .9 -1 5 .2 1 8 .0 -0 5 .2 0 8 .0 0 5 .7 -0 2 .9 3 5 .7 3 8 .0 4 5 .7 2 8 .0 4 5 .7 4 8 .0 5 5 .7 5 8 .0 0 0 0 0 3 3161 .9 4 1 5 2 3 1625 .1 7 4 8 5
Correlation matrix
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Y Y X1 X2 CX1 CX2 1 0.94978 0.890216 0.94978 0.890216
X1 0.94978 1 0.950585 1 0.950585
X2 CX1 CX2 0.890216 0.94978 0.890216 0.950585 1 0.950585 1 0.950585 1 0.950585 1 0.950585 1 0.950585 1
CX1: Centered attitude variable; CX2: Centered years with company The values of various sums of squares are given below:
∑ ( x1 − x1)( y − y) = 739.53; ∑ ( x2 − x2)( y − y) = 573.53; ∑ (x1 - x1) 2 = 665.30 ∑ (x2 - x2) 2 = 414.82
The SPSS output corresponding to the above data is shown below:
Model Summary(b) Adjusted R Std. Error of the R R Square Square Estimate .951(a) .904 a Predictors: (Constant), Years with company, Attitude b Dependent Variable: Days Absent ANOVA(b) Model 1 Sum of Squares 320.149
-
−
−
−
−
-
Model 1
Df
Mean Square
F
Regression Residual Total
Sig. .000(a)
354.250 a Predictors: (Constant), Years with company, Attitude b Dependent Variable: Days Absent Coefficients(a) Model 1 (Constant) Attitude Years with company a Dependent Variable: Days Absent Unstandardized Coefficients B -2.263 1.550 -.239 Std. Error 1.096 .481 .606 t Sig. Collinearity Statistics Tolerance VIF 10.375 10.375
a) Comment on the overall fitness of the regression model. Identify the explanatory variables that are significant. Use α = 0.05.
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b) Mr Jim Bob, the statistician at DLI after noticing that the variance inflation factor (VIF) is 10.375, removes all the data corresponding to the explanatory variable, “years with company” and runs a regression with only attitude as the explanatory variable. What will be new coefficient value for the attitude? What is the impact of this removal on standard error of estimate of the model? c) If the decision is to remove an explanatory variable from the analysis, which explanatory variable should be removed? Justify your answer d) Jason, the senior statistician at DLI advices Jim Bob to center the data (deviation from mean) on, “number of years” about its mean and run the regression between number of absent days and attitude and centered number of years. What will be the impact of this strategy on the overall regression model e) What inference can be obtained from the following scatter plot between standardized predicted value and standardized residual of the regression model?
Regression Standardized Residual
S at r lo c t ep t
D p n e tV r b :D y A s n e e d n a ia le a s b e t
1 .5
1 .0
0 .5
0 .0
- .5 0
- .0 1
- .5 1 - .5 1 - .0 1 - .5 0 0 .0 0 .5 1 .0 1 .5
R ge s nSa d r iz dPe ic dV lu e r s io t n ad e r d te a e
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Question 3: Binary Integer Programming Consider the following binary integer programming (BIP) problem: Max 5x1 + 3x2 + 4x3 + 2x4 s.t. 4x1 + x2 + x3 + 2x4 <= 3 (1) x1 + 3x2 + x3 + 3x4 <= 5 (2) 2x1 + x2 - 2x3 + 4x4 >= 4 (3) x1, x2, x3, x4 binary a) State the LP relaxation (LP0) for the problem.
b) Solve the BIP using the branch-and-bound algorithm. Clearly • show all branches, branching variables and their values. You may branch on the first variable which has a fractional value • number the sub-problems created in the order that you solve them • give their optimal solutions/values and • state when a node is fathomed and why You may use the following information if required: - Optimal solution to LP0 is: (0.035, 0.55, 0.31, 1). - Optimal solution to LP0 with an additional constraint x1=0 is: (0, 0.57, 0.28, 1).
c) When solving integer programming problems preprocessing of the given formulation is usually carried out before using the b-a-b algorithm. Some of the reasons for doing this is to: i) identify variables that can be fixed at one of their possible values because the other value is precluded by the constraints. ii) Tightening the constraints to reduce the feasible region of the LP relaxation without eliminating any feasible solution to the BIP problem. At times this can result in substantial reduction in the effort and time required to solve the problem. Preprocess the given BIP and i) Identify two variables whose values can be fixed. Support your answers with adequate explanations. ii) Eliminate the variables that have been fixed in i) by substituting for their values in the given BIP. Call the resulting modified 2 variable problem BIP1. Solve the LP relaxation for BIP-1. Does it give an integer solution?
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If not, solve it using the b-a-b method. iii) Incorporating the tighter form of the constraint (1) for BIP-1 solve its LPrelaxation. Does it give an integer solution? Comment on the effect of preprocessing done in i)
Question: Bectel Telecom needs to install towers to cover an upcoming area in Bangalore that has been divided into 12 regions as follows:
5 2 1 11 8 6 10 4 3 7 9 12 11
a) A tower located in a given region can provide communication in that region as well as to any of the adjacent regions (regions with a common boundary). Bectel would like to use a minimum number of towers to cover the entire area. Give an appropriate formulation that will help Bectel in making the decision. Instead of proceeding as above Bectel decides to look at the population in each region which is as follows: Region 1 2 3 Population (100s) 50 70 55
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4 5 6 7 8 9 10 11 12
95 45 150 60 200 225 90 175 120
b) Assume that only one tower can be installed in the entire area. The region in which the tower is located as well as any adjacent regions can be served by the tower. Where the tower should be located to maximize the population served? Question Tiny Twinkler Toasters (3T) produces two different models of toasters. The number of toasters that it can sell depends on the price of the two models. The demand functions for these two models are given below: Model 1: Q1 = 500 – 0.1 P1 + 0.15 P2 Model 2: Q2 = 350 + 0.1 P1 – 0.2 P2 Where Qi is the quantity of Model i and Pi is the price of model i. The cost of production is given by the following cost functions: C1 = 150 + 0.8 Q1 if Q1 ≤ 100 C1 = 170 + 0.6 Q1 if Q1 ≥ 100 C2 = 200 + 0.9 Q2 Where C1 and C2 are the total cost of producing the respective models. The amount of working capital available for the decision period is 8000. The total cost of production should be met from this amount. The profit is defined as the difference between the total revenue and total cost. Formulate the above problem to solve the number of toasters for each model to be produced in order to maximize the profit. Question Nidhi would like to invest Rs. 100,000 and is considering investment options A, B, C, D and E. She has a 3 year horizon in mind and would like to maximize the cash on hand at the end of the third year. The cash flows associated with these investments now and over the next 3 years are as given below: Cash flow (Rs.) in Year 0 1 2 3 -1 +0.5 +1 0
A
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B C D E
0 -1 -1 0
-1 +1.2 0 0
+0.50 0 0 -1
+1 0 +1.9 +1.5
For example, a rupee invested in investment B requires no cash outflow right now (year 0), a cash outflow of Rs. 1 in year 1, a return of Rs. 0.50 in year 2 and a return of Rs. 1 in year 3. Any returns can be immediately reinvested. Nidhi would not like to invest more than Rs. 75,000 in a single investment. In addition to these investment options Nidhi can get a return of 8% per year if she leaves her money in the bank. Nidhi’s investment decision problem has been formulated as an LP. The formulation and the solution are as given below: Max B + 1.9D + 1.5E +1.08F2 Subject to A + C + D + F0 = 100,000 0.5A + 1.2C + 1.08F0 – B – F1 = 0 A + 0.5B + 1.08F1 – E – F2 = 0 A < = 75000 B < = 75000 C < = 75000 D < = 75000 E < = 75000 All vars >= 0 Adjustable Cells Final Name Value A 60000 B 30000 C 0 D 40000 E 75000 F0 0 F1 0 F2 0
Reduced Objective Cost Coefficient 0 0 0 1 -0.0280 0 0.0000 1.9 0.0000 1.5 -0.2152 0 -0.3504 0 -0.0400 1.0800
Allowable Increase 0.0500 0.0292 0.0280 0.4750 1E+30 0.2152 0.3504 0.0400
Allowable Decrease 0.0583 0.2844 1E+30 0.0500 0.38 1E+30 1E+30 1E+30
Constraints Cell $J$3 $J$4 Final Value 100000 0 Shadow Price 1.9 -1.56 Constraint R.H. Side 100000 0 Allowable Increase 35000 37500 Allowable Decrease 40000 56250
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$J$5 $J$6 $J$7 $J$8 $J$9 $J$10
0 60000 30000 0 40000 75000
-1.12
0 75000 75000 75000 75000 75000
18750 1E+30 1E+30 1E+30 1E+30 18750
43750 15000 45000 75000 35000 43750
a) Define the variables used in the given formulation and give the optimal investment policy for Nidhi. b) If Nidhi has Rs. 15000 more on hand at the beginning (year 0), how much will she have on hand at the end of year 3? c) If Nidhi could borrow Rs. 15000 at the beginning of year 1 for which she would be charged an annual interest rate of 15% should she borrow the money? d) If Nidhi had an additional Rs. 20000 at the beginning of year 1 and Rs. 15000 at the beginning of year 2, how much will she have at the end of year 3? e) If investment D yielded 1.80 at time 3 would Nidhi’s investment plan change? f) Nidhi is considering a new investment option G where investing 1 rupee in year 0 generates Rs. 1.10 in year 1, Rs. 0.20 in year 2 and Rs. 0.10 in year 3. Should Nidhi invest in this option? g) Write the dual of the given problem. Define all the variables clearly. h) Should Nidhi consider the option of investing more than Rs. 75000/- in any one of the investments option A,B,C,D,E? If so, in which one(s)? Up to how much? Exactly how will this help her?
Question: The year is 1250 B.C. and Egyptian Pharaoh Ramses II has commissioned a granite causeway (elevated road) to be built from the gates of the temple at Karnak down to the east bank of the Nile River. The causeway will commemorate the Pharaoh's triumphs at the battle of Kadesh, so he insists it be completed before the scheduled visit of several princes from that region. This means that Imhotep, the Chief architect of Public Works of Ramses II, has exactly six months to finish the project. The project is fairly simple: Imhotep needs to transport a total of 5,000 carefully carved blocks from the granite
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quarries of Lower Egypt, downstream to the current site at Karnak. The stonemasons at Karnak do not need all 5,000 blocks at once; rather, the monthly requirements of the granite blocks are shown in table 1: Table: Monthly requirements of granite blocks Month Blocks Required 1 500 2 500 3 1,000 4 1,000 5 1,750 6 250
The requirements of the stonemasons must be met. Imhotep has a choice of two granite quarries from which to purchase his blocks: the quarry at Deir El Medinah, which can produce a maximum of 600 blocks a month for the next six months, or the quarry near Fayum, which can produce a maximum of 1100 blocks a month during months 1, 2, 3 and 6. Unfortunately, the annual flooding of the Nile River, which will happen during months 4 and 5, makes the Fayum quarry inaccessible during that time. Hence production during those months is zero (at Fayum). In addition, while the stonemasons at Karnak do not mind having more blocks than they need in a given month, they cannot have more than 1,200 blocks "left over" at the end of a month, due to space considerations. Imhotep needs to decide how many blocks to have delivered to Karnak each month. He also needs to decide on which quarry to order those blocks from - he is free to order blocks from both quarries in the same month, if he wishes. His objective is to minimise the cost of the entire project. The labour costs involved in moving a single block of granite can vary, according to the month. In months 1 through 3, Imhotep must rely primarily on the Pharaoh's military to do the transporting: they will charge the modern day equivalent of Rs 15000 per block moved from either quarry to Karnak. In months 4, 5 and 6, Imhotep does not have to contract with the military; the flooding of the Nile will leave all local farmers idle, they will move as many blocks as he wants for the equivalent of Rs 7500 per block. In addition to the transportation costs, Imhotep must consider the actual purchase price of the blocks: the quarry at Deir El Medinah charges Rs 200,000 per block it produces, while the Fayum quarry charges Rs 225,000 per block it produces. Finally, it costs Rs 1000 per block "left over" at the end of the month in inventory carrying costs. a) Assuming that blocks produced during a particular month can be transported and used during the same month, formulate Imhotep's problem as a linear programming model. Clearly write all the decision variables.
b) If 2% of all the blocks purchased from Deir El Medinah are damaged during transportation and thus cannot be used for construction. Change the formulation in Q1 to include this new information
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Question : Mr Kunjithapatham has been recently appointed as the head of newly created transportation security department by the Indian government after the 26/11 terrorists attack on Mumbai. The department is responsible for providing security at 12 international airports in India. Even prior to 2008, airline passengers had become familiar with the two basic types of systems used to check each passenger at a security checkpoint. One is a portal that can detect concealed weapons as the passenger walks through. The other is a screening system that scans the passenger’s carry-on luggage. Various proposals have been made for advanced security technology that would improve these two systems. Mr Kunjithapatham’s task force now needs to make recommendations on which direction to go for the next generation of these systems. The task force has been told that the functional requirement for the new portal system is that it must be able to detect even one ounce of explosives and hazardous liquids as well as metallic weapons being concealed by a passenger. The technology needed to do this includes quadrupole resonance and magnetic sensors. There are various ways to design the portal with this technology that would satisfactorily meet the functional requirement. However, the designs would differ greatly in the frequency with which false alarms would occur as well as in the purchase cost and maintenance cost for the portal. The frequency of false alarms is a key consideration since it substantially affects the efficiency with which the passengers can be processed. Even more importantly, a high frequency of false alarms greatly decreases the alertness of the security personnel for detecting the relatively rare terrorists who are actually concealing destructive devices. The security portals have to be imported either from USA or Germany. The most basic version of the portal system that satisfactorily meets the functional requirement has an estimated purchase price of $100,000 and, on the average, would incur an annual maintenance cost of $15,000. The drawback of this version is that it would generate a false alarm for approximately 10 percent of the passengers. This false alarm rate can be reduced by using more expensive versions of the system. Each additional $1000 in the cost of the portal system would lower the false alarm rate 0.1 percent and also would increase the annual maintenance cost by $200. The most expensive version would cost $180,000, so it would have a false alarm rate of only 2 percent of the customers as well as an annual maintenance cost of $31,000. Regarding the new screening system for carry-on luggage, the functional requirement is that it must clearly reveal suspicious objects as small as the smallest Swiss army knife. The technology needed to do this combines X-ray imaging, a thermal neutron scanner, and computer tomography imaging (which compares the density and other physical properties of any suspicious objects with known high-risk materials). It is estimated that the most basic version that satisfactorily meets this functional requirement would cost $60,000 plus an annual maintenance cost of $9,000. As with the most basic portal system, the drawback of this version is that it doesn’t sufficiently discriminate between suspicious objects that actually are destructive devices and those that are harmless. Thus, this version would generate false alarms for approximately 16 percent of the customers. In addition to wasting time and delaying passengers, such a high false alarm rate would make it very difficult for
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the screening operator to pay sufficient attention when the far more unusual true alarms occur. However, more expensive versions of the screening system would be considerably more discriminating. In particular, each additional $1,000 in the cost of the system would enable a reduction of 0.1 percent in the false alarm rate, while also increasing the annual maintenance cost by $300. Thus, the most expensive version, costing $160,000, would decrease the false alarm rate to 6 percent and incur an annual maintenance cost of $45,000. The task force has been given two budgetary guidelines. The false alarms due to security portal and the screening systems are independent. First budgetary guideline: Plan on a total expenditure of $200 ,000 for both the portal system and the screening system for carry-on luggage at each security checkpoint. Second budgetary guideline: Plan on holding down the average total maintenance costs for the two systems at each security checkpoint to no more than $70,000. These budgetary guidelines prohibit using the most expensive versions of both the portal system and the screening system for carry-on baggage. Therefore, the task force needs to determine which financially feasible combination of versions for the two systems will maximize the effectiveness with which passengers can be screened. Doing this requires first obtaining input from the top management of the Transportation Security Administration regarding what is the measure of effectiveness should be and then what management’s goals and priorities are for achieving substantial effectiveness and meeting the budgetary guidelines. Fortunately, Mr Kunjithapatham already has had extensive discussions with home department to obtain its guidance on these matters. These discussions led to the adoption of a clear policy that was approved all the way up to the home minister. The policy establishes the following order of priorities. Priority1: The total false alarm rate for both systems should not exceed 0.1 per passenger. Priority 2: Meet the first budgetary guideline. Priority 2: Meet the second budgetary guideline. Now that it has obtained all the needed managerial input, the task force is ready to begin its analysis. (a) Formulate the above problem as a preemptive goal programming problem. Clearly state all the decision variables involved in formulating the Goal programming problem
(b) Solve the preemptive goal programming formulation using graphical method.
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Question : A consumer advocacy group would like to evaluate the efficiency of four airline companies with respect to several criteria that are important to consumers. The four airlines are Tom, Dick, Harry and Mary. The output measures are identified as average delay in minutes, number of luggage lost per 10,000 checked-in bags, waiting time for check-in (in minutes) and quality of service (measured through questionnaire on a 100 point scale where higher the score better the quality of service). The input measures are labour costs per day, number of flights per day and overhead costs per day. The following table gives the details of inputs and outputs for the airlines. Airline Inputs Labour cost per day (in lacs) Overhead cost per day (in lacs) Number of flights per day Outputs Average delay in minutes Lost baggage per 10,000 bags Waiting time for check-in (minutes) Quality of service Tom 800 50 320 15 10 10 85 1000 60 416 20 15 20 80 Dick 600 40 280 7 5 5 95 Harry 750 28 312 9 7 8 92 Mary
a) Formulate an input-oriented DEA model to check the efficiency of the airline Tom.
b)
If the efficiency score of Mary is 1, what conclusions can be derived about efficiency of other three airlines?
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Question A textbook publisher wants to determine the relationship between the number of book copies sold (Y), the number of pages in the book X1, the number of analytical examples X2, the number of end-chapter problems X3, the ratio of favourable to unfavouarble reviews X4 and the nature of the support package D1. The data corresponding to a sample with 10 books are shown in table 1.
Table 1: Data, standardized and studentized residual
Sample point 1 2 3 4 5 6 7 8 9 10 Mean Stdev Copies sold 7 11 14 17 29 39 52 6 3 2 18 16.76637 Examples X2 0 0 0 15 15 30 15 92 0 0 16.7 28.38251 Problems X3 0 0 0 100 100 900 500 900 10 0 251 374.1791 Favorable to unfavourable reviews X4 5.1 4.6 6 2 3.1 0.2 0.1 10.3 5 12 4.84 3.91101 Support Package, D1 0 0 0 0 1 1 1 0 0 0 0.3 0.483046 Standardized residual -0.33812 0.24162 1.02037 0.18403 -0.3777 -0.51353 0.89123 0.05372 -1.24189 0.08027 Studentized residual -0.3762 0.28327 1.26043 0.26654 -1.89154 -1.49315 1.67628 0.3558 -1.4828 0.1961
Pages X1 390 472 512 408 610 830 310 910 211 172 482.5 243.7518
The table 2 gives various distance measures and DFBeta values of the data in table 1.
Sample 1 2 3 4 5 6 7 Mahalanobis Distance 0.82976 1.55226 2.20176 3.80992 7.74117 7.03542 5.55594 Cooks Distance 0.00561 0.00501 0.13924 0.013 14.36016 2.76978 1.18843 Leverage Values 0.0922 0.17247 0.24464 0.42332 0.86013 0.78171 0.61733 DFB0 -0.45514 0.03699 -2.85942 2.16623 21.2656 11.62836 15.52249 DFB1 0.00069 0.0016 0.01126 0.00134 0.04056 0.02758 0.03353 DFB2 0.01041 0.01338 0.10902 0.03225 0.81685 0.59576 0.16079 DFB3 0.00002 0.00012 0.00045 0.00145 0.11267 0.04313 0.00618 DFB4 0.02448 0.02565 0.18246 -0.2254 1.12292 0.37244 0.64421 DFB5 0.63484 0.61483 1.95958 1.04778 59.8744 3.7449 6.30033
49 0.00237 0.00129 0.00194 1.49047 2.50934 0.92671
8 9 10
7.8948 1.78686 6.59212
0.90428 0.15596 0.03184
0.8772 0.19854
-1.12667 -7.55361 -0.62546
-0.0011 0.01047 0.00184
0.21159 0.03711 0.02833
0.19135 0.28319 0.32631
Table 3. ANOVA table Sum of Squares 2307.621
ANOVA(b)
Model 1
df
Mean Square
F
Sig.
Regression Residual Total
2530.000 a Predictors: (Constant), Support Package, D1, Examples X2, Favorable to unfavourable reviews X4, Pages X1, Problems X3 b Dependent Variable: Copies sold
(a) Calculate the co-efficient of determination of the model and comment whether the overall model can be accepted? (b) Calculate the leverage value for sample number 10. (c) Based on table 1 and 2, comment on the possibility of heteroskedasticity in the data (d) Identify all the outliers in the data and state the reason(s) why these sample points are outliers? (e) Calculate the DFFit value for sample point 1 Question Good Day Dairies (GDD) is entering the milk distribution business. To begin with they will purchase raw milk to resell as milk or cream. They can purchase milk from two regions: Anekhalli (A) and Banekhalli (B). Prices, butterfat content, and separation properties of milk differ between the two regions. GDD processes the raw milk to produce cream and milk to desired specifications for sale to the customers. There is a one time contract payment of Rs.50,000 to be made if GDD decides to buy any amount of raw milk from a region. Purchase Price: The purchase prices of raw milk from the first region is Rs. 7 per litre and from region 2 is Rs. 6 per litre.
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Milk from region A has 15% butterfat and can be separated at a cost of Rs. 0.50 per litre. When separated it yields two ‘milks’, one with 31% butterfat and another with 8% butterfat. The volume of milk is conserved in the separation processing. Milk from region B has 10% butterfat and can be separated at a cost of Rs. 0.70 per litre. When separated it yields two types of ‘milks’, one with 33% butterfat and another with 5% butterfat. The volume of milk is conserved in the separation process. Production Process: After the milk is purchased and collected at the plant, it is either mixed directly or separated and then mixed. Mixing is done at no cost, and its purpose is to produce cream and milk to specifications. Demand and Selling Price: The demand and selling price are as follows: Minimum required per cent of butterfat Cream Milk 30 6 Maximum demand in litres 2500 20000 Selling Price in Rs / litre 28 11
a) Draw a flow diagram for the decision process that will aid you in formulating the problem of GDD maximizing its profit as a linear/integer programming problem. b) Formulate a mathematical programming problem that when solved will help GDD maximize its profits. Assume that the disposal of any extra milk or cream does not cost anything. Define all variables clearly and explain each of the constraints. Variables without proper definitions and constraints without proper labels and explanations will not be considered. Note: Use variables starting with ’X’ and ‘Y’ to define all decisions pertaining to Region A and Region B respectively. You are free to use any other notation for all additional variables used.
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c) Consider the situation where the purchase prices of raw milk from the 2 regions are lower if larger quantities of milk are purchased and are as indicated in the following diagrams: Region A Cost (Rs.) 5 4 7 5 Cost (Rs.) Region B
5000 Vol (Litres)
7000 Vol (Litre)
For example, the purchase of 7000 litres of raw milk from region A would cost Rs. (7*5000 + 5*2000). There is no upper bound on the amount of milk availability. Define any new variables required, if any, to formulate the decision problem under this cost structure. Give the appropriate formulation. Question The Brilliant Investments Company is considering four investments. Investment 1 will yield a net present value (NPV) of $16,000; investment 2, an NPV of $22,000; investment 3, an NPV of $12,000; and investment 4, an NPV of $8,000. Each investment requires a certain cash outflow at the present time: investment1, $5000; investment 2, $7000; investment 3, $4000; and investment 4, $3000. At present, $14,000 are available for investment. Let xj ( j = 1, 2, 3, 4) = 1 if investment j is made 0 otherwise
Then the following Binary Integer Program (BIP) can help decide which investments should be chosen: Max z = 16x1 + 22 x2 + 12 x3 + 8 x4 s.t. 5x1 + 7x2 + 4 x3 + 3x4 ≤ 14 xj = binary ( j = 1, 2, 3, 4 )
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A linear programming problem with a single constraint is referred to as a Knapsack Problem. The above BIP is a knapsack problem in which each variable must equal 0 or 1. When knapsack problems are solved by branch-and-bound, two aspects of the method greatly simplify. Since each variable must equal 0 or 1, branching on xj will yield xj = 0 and xj = 1 branches. Also, the LP relaxation for this and other subproblems can be solved in the following way. Observe that cj / aj (where cj and aj are the coefficients of xj in the objective function and constraint, respectively) may be interpreted as the benefit item j earns for each unit of the resource used by item j. Thus, the best items have the largest values of cj / aj and the worst items have the smallest values of c j / aj . To solve any sub-problem encountered while solving a binary knapsack problem, compute all the ratios cj/ aj . Select the best item first and fill the knapsack with as much of this item as possible. Then select and place as much as possible of the second-best item in the knapsack. Continue in this fashion until the knapsack is full. For example, to solve the LP relaxation of the knapsack problem above, we will first select investment 1 since it has the highest ratio of 3.2. We can select all of investment 1 (x1=1), and we will still have $9000 left. We will next select investment 2 since it has the highest ratio amongst the ones remaining (ratio of 22/7). We can afford to select all of investment 2, and as a result $2000 will be left. The next best selection is investment 3. However, here since investment 3 requires $4000, we can take in only half of investment 3. Hence the optimal LP solution is: x1 = x2 = 1, x3 = ½. In the process of solving the Brilliant Investments Company’s decision problem, using the Branch-and Bound method, the following steps were taken:
LEVEL 0
LP0 z = 44 x1 = x2 = 1 x3 =1/2 x4 = 0 x3 = 0 LP1 z = 43 1/3 x1 = x2 = 1 x3 = 0 x4 = 2/3 x3 = 1 LP2 z = 43 5/7 x1 = x3 = 1 x2 = 5/7 x4 = 0
LEVEL 1
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Answer the following questions: c) The value of the upper bound on the optimal solution to the BIP at level 0 is _______________ and the value of the lower bound is ___________________. Explain why.
d) The value of the upper bound on the optimal solution to the BIP at level 1 with x3 = 0 is _______________ and the value of the lower bound is ___________________. Explain why.
c) Complete the branch-and-bound (b-a-b) tree using the above information in order to find the optimal solution for the problem. Proceed down the node that has the most promising value. Clearly • show all branches, branching variables and their values • number the sub-problems created in the order that you solve them • give their optimal solutions/values and • state when a node is fathomed and why.
Question Jalahalli Machine Tools (JMT) is in the business of manufacturing and selling three products: Product 1, Product 2 and Product 3. The relevant data is provided in the table below: Product 1 Rs 1500 0.75 hrs 1.5 hrs 2 units Product 2 Rs 800 0.5 hrs 0.8 hrs 1 units Product 3 Rs 900 1.00 hrs 1.2 hrs 1.5 units
Selling Price Labour Required Machine Time Required Raw material required
Each week, up to 400 units of raw material can be purchased at a cost of Rs 150 per unit. The company employs four workers, who work 40 hours per week. (Their salaries are considered a fixed cost.) Workers are paid Rs 600 per hour to work overtime. Each week, 320 hours of machine time are available. In the absence of advertising, 50 units of product 1, 60 units of product 2 and 40 units of product 3 will be demanded each week. Advertising can be used to increase demand for product 1 and 2. It is found that for product 1, Rs 10 spent on advertising increases the demand by a unit. For product 2, Rs 6.67 spent on advertising increases the demand by one
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unit. At most, Rs 10,000 can be spent on advertising. In order to figure out what to produce and sell, JMT modeled and solved the following LP: Let, Pi = number of units of product i produced each week, i = 1, 2, 3. OT = number of hours of overtime labour used each week. RM = number of units of raw materials purchased each week. Ai = money spent of advertising for product i each week, i = 1, 2.
Max 1500P1 + 800P2 + 900P3 – 600OT – 150RM – A1 – A2 S.T. P1 – 0.1A1 <= 50 P2 – 0.15A2 <= 60 P3 <= 40 0.75P1 + 0.5P2 + P3 – OT <= 160 2P1 + P2 + 1.5P3 – RM <= 0 RM <= 400 A1 + A2 <= 10000 1.5P1 + 0.8P2 + 1.2P3 <= 320 All variables are nonnegative. The LINDO output to the above problem is given below: LP OPTIMUM FOUND AT STEP 5
OBJECTIVE FUNCTION VALUE 1) 242766.7 REDUCED COST 0.000000 0.000000 386.666656 213.333328 0.000000 0.000000 0.000000
VARIABLE VALUE P1 160.000000 P2 80.000000 P3 0.000000 OT 0.000000 RM 400.000000 A1 1100.000000 A2 133.333328
ROW SLACK OR SURPLUS DUAL PRICES 2) 0.000000 10.000000
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3) 4) 5) 6) 7) 8) 9)
0.000000 40.000000 0.000000 0.000000 0.000000 8766.666992 16.000000 5
6.666667 0.000000 386.666656 600.000000 450.000000 0.000000 0.000000
NO. ITERATIONS=
RANGES IN WHICH THE BASIS IS UNCHANGED: VARIABLE P1 P2 P3 OT RM A1 A2 OBJ COEFFICIENT RANGES CURRENT ALLOWABLE ALLOWABLE COEF INCREASE DECREASE 1500.000000 96.666664 53.333336 800.000000 26.666668 48.333332 900.000000 386.666656 INFINITY -600.000000 213.333344 INFINITY -150.000000 INFINITY 450.000000 -1.000000 1.000000 5.333333 -1.000000 1.000000 7.250000
ROW 2 3 4 5 6 7 8 9
RIGHTHAND SIDE RANGES CURRENT ALLOWABLE RHS INCREASE 50.000000 110.000000 60.000000 20.000000 40.000000 INFINITY 160.000000 27.500000 0.000000 6.666667 400.000000 6.666667 10000.000000 INFINITY 320.000000 INFINITY
ALLOWABLE DECREASE 876.666687 1315.000122 40.000000 2.500000 55.000000 55.000000 8766.666992 16.000000
Using the information provided above, answer the following questions: a. This week, the supplier from whom the raw material is purchased informs JMT that in addition to the 400 units that are available at the current price, an additional 5 units may be purchased at a price of Rs 500 per unit. Would it be worthwhile for JMT to purchase the additional 5 units? Why or Why not?
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b. The workers union (comprising of the four employees) would like to earn some additional wages by working overtime. It seems that the current rate of Rs 600 per hour for overtime is too high to interest the management. The union would then like to know the highest wage rate that they can negotiate, below which the management would be interested using overtime hours? What is that wage rate? Explain your answer. c. Supposing that Ravi Nachappa approached JMT and expressed interest in working for them. He is as skilled as the four workers currently working for JMT. Ravi makes two offers. He is either willing to work full-time (40 hours a week) for a wage of Rs 300 per hour, or part-time (for 20 hours a week) for a wage of Rs 320 per hour. Of course, JMT can choose one of these two offers, or choose not to hire Ravi in the first place. What option must JMT choose? Why? State your reasons precisely.
d. The LP solution suggests that currently it is not worthwhile making any of Product 3. One of the workers feels that it is because the production process for making this product is not efficient. He feels that by utilizing a smaller quantity of labour hours and also a lesser amount of machine time, the same product can be made with as good a quality. If JMT were to focus on the labour utilization of Product 3, what must its utilization be reduced to before it becomes profitable to make any of Product 3? Note that currently each unit of product 3 utilizes 1 hour of labour. Show your work clearly. Question Nilgiri’s, a popular grocery chain would like to develop a model that helps to predict the stores’ monthly revenues depending on the area it is located and the fact that it may or may not be located in a shopping centre. A stores’ location is identified by the median annual income in that area. That is because it is believed to have a bearing on weekly revenues. Below is a display of data obtained from a random sample of 15 locations where Nilgiri stores are currently located. Store Number 1 2 3 4 5 6 Weekly Revenues (in lakhs of Rs) 12.00 9.90 11.50 8.20 12.20 11.60 Median Value of Annual Income (lakhs) 2.25 1.70 1.53 1.32 2.37 1.87 Store Location (Shopping Center?) Yes No Yes No Yes Yes
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7 8 9 10 11 12 13 14 15
12.50 11.50 11.50 9.70 11.70 8.60 11.50 9.40 11.20
2.45 1.25 2.15 1.70 2.23 1.47 1.97 1.67 2.10
Yes Yes Yes No No No Yes No No
Given the above data, the average weekly revenues are Rs. 10.87 lakhs and the standard deviation of the weekly revenues is 1.35. The average weekly revenues for stores located in a shopping centre are Rs. 11.79 lakhs. The average of the median value of annual income across all 15 areas is Rs. 1.87 lakhs and the standard deviation of the median annual income is 0.38 Let y = Weekly Revenues of the store in lakhs of rupees; x1 = Median value of annual incomes of the area in lakhs of rupees; x2 = 1 if the store is located in a shopping centre = 0 if the store is not located in a shopping centre Correlation matrix:
Y X1 X2 Y 1 0.703 0.757 X1 1.00 0.321 X2
1
The regression outputs for the models fitted are as given below: Model B: y i = β 0 + β 1 x1i + β 2 x 2i + ε i , for i = 1, 2, ..., 15.
SUMMARY OUTPUT Regression Statistics Multiple R 0.899 R Square 0.809 Adjusted R Square 0.777 Standard Error 0.636 Observations 15.000 ANOVA df Regression 2.000 Residual 12.000 Total 14.000 Coefficients
SS 20.517 4.857 25.373 Standard Error
MS 10.258 0.405 t Stat
F 25.347
Significance F 0.000
P-value
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Intercept x1 x2 Model C:
6.683 1.798 1.544
0.849 0.468 0.348
7.868 3.844 4.442
0.000 0.002 0.001 for i = 1, 2, ..., 15.
y i = β 0 + β1 x1i + β 2 x 2i + β 3 ( x1i x2i ) + ε i ,
SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations ANOVA Regression Residual Total Intercept x1 x2 x1*x2
0.988 0.977 0.971 0.230 15.000
SS 3.000 24.792 11.000 0.581 14.000 25.373 Coefficients Standard Error 2.986 0.513 3.921 0.290 7.397 0.663 -3.212 0.357
df
MS 8.264 0.053 t Stat 5.823 13.511 11.165 -8.997
F Significance F 156.463 0.000
P-value 0.000 0.000 0.000 0.000
If, to begin with, a model A was fitted to explain the variation in weekly revenues using the location of the stores (X2) as the only explanatory variable answer the following questions: (Parts b-e are worth 2 points each. All other parts are worth 4 points each). Show all work for each part and give adequate explanations. a) b) c) d) e) Specify the model A along with the values of the beta estimates. What is the R Square value for the Model A? What is the amount of the total variation in the response variable that will be explained by the explanatory variable (x2) used in Model A? What is the standard error for Model A? If using the given data, two separate regression models were developed to explain variations in weekly revenues using median value of income; one for stores located in shopping centres and the other for stores not located in shopping centres. What would be the expected difference in the y-intercept values for the two regression models?
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f)
Perform a Partial F test to decide if the addition of explanatory variables x1 and x1*x2 (Model C) add significant additional information when x2 is already present. Use the appropriate model to answer the following: Does the average increase in weekly revenues of stores for every Rs. I lakh increase in median income depend on the location of the stores? How much is the increase? Use the appropriate model to answer the following: Does Nilgiris always get higher/lower revenues for stores located in shopping centres versus those not located in shopping centres for all levels of median income? If yes, for which location and by how much? If not, how exactly does it differ with respect to levels of median income? Use model C to give a point estimate and a 95% prediction interval for the actual weekly revenues for a store that is located in a shopping centre in an area where the median value of annual income is Rs. 2 lakhs. Tinv(.05,11) = 2.201, Tinv(.05,12) = 2.179, Tinv(.05,13) = 2.160
g)
h)
i)
Question The following table (table 1) lists the total compensation (salary + bonuses) paid to CEO’s of 30 fortune 500 companies. Tables 1 and 2 give various distance measures and BFFIT and DFBeta values of the sample points. It is believed that there is a linear relationship between the total compensation and the following explanatory variables: 1. 2. 3. 4. 5. Tenure – Number of years as CEO ( = 0 if less than 6 months) Age – Age of the CEO Sales – Total sales revenue of the firm Profits – Profit of the firm Assets – Total Assets of the firm
Compensat ion Studentized Residual Cook's Distance
Table 1: Values of Response and explanatory variables and distance measures
Firm tenure age sales profits Assets Leverage Values
1 2 3 4 5 6 7 8 9 10 11 12
6600 68500 24000 29600 83700 33300 131350 6830 13450 16920 134000 13700
7 0 11 6 18 6 15 5 3 2 16 5
61 51 63 60 63 57 60 61 57 60 63 61
161315.0 144416.0 139208.0 100697.0 100469.0 81667.0 76431.0 57813.0 56154.0 53588.0 50777.0 47678.0
2956.0 22071.0 4430.0 6370.0 9296.0 6328.0 5807.0 5372.0 1120.0 6398.0 5165.0 1704.0
257389.0 237545.0 49271.0 92630.0 355935.0 86100.0 668641.0 59920.0 36672.0 59550.0 617679.0 42754.0
-3.62 -0.17 2.603 0.916 0.561 1.017 -0.16 -1.44 0.589 -0.64 0.759 0.149
2.60327 0.01799 0.59574 0.01602 0.01147 0.01229 0.00277 0.03988 0.00423 0.01245 0.04397 0.00035
0.51108 0.75329 0.31199 0.06937 0.14631 0.03316 0.34573 0.06981 0.03483 0.11915 0.2807 0.05153
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13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Mean S.D.
6800 9000 7000 67500 8000 47000 80000 8000 23000 6000 72000 6000 7500 14500 6600 7100 13500 30000
32715 36578.67
7 4 0 2 4 32 5 3 4 15 17 5 4 2 2 2 9 9
7.333 6.91
58 59 56 60 60 74 63 56 52 57 55 56 57 56 52 49 57 59
58.4 4.67
47061.0 41322.0 37154.0 35853.0 33674.0 33296.0 32379.0 31707.0 31565.9 31260.0 31131.0 30951.0 30678.0 30219.0 30147.0 29398.0 28777.0 28203.3
55499.64 37532.061
2945.0 1048.0 3780.0 1259.0 568.0 3765.6 3782.0 578.0 2965.3 703.0 3276.0 935.0 594.0 1614.0 970.0 -962.0 4023.0 410.8
3642.39 4217.05
33673.0 37675.0 30966.0 299804.0 14166.0 194398.0 365875.0 28570.0 55143.9 29350.0 317590.0 15666.0 23638.0 13465.0 26720.0 28728.0 45066.0 6700.1
137709.3 177455.4
-0.69 -0.19 -0.82 0.806 0.086 -1.29 0.104 -0.19 0.313 -0.82 -0.06 -0.29 -0.2 0.523 -0.38 0.035 -0.89 2.225
0.00399 0.00043 0.01162 0.02996 0.00013 0.53962 0.0007 0.00042 0.00218 0.02834 0.00023 0.00097 0.00044 0.00345 0.00322 0.00007 0.01381 0.06802
0.01472 0.03339 0.06062 0.18358 0.05863 0.62671 0.24643 0.03311 0.08456 0.16695 0.26245 0.03005 0.02911 0.03707 0.08244 0.21991 0.06052 0.04284
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Table 2: Mahalanobis distance and DFFIT and DFBeta value of the sample points Firm
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Mahalanobis
DFFIT
-30155.9 -3013.54 11489.19 1027.563 1149.984 724.0253 -820.414 -1624.67 430.2569 -1103.62 2976.441 137.1235 -350.59 -136.272 -837.026 2042.017 86.14769 -15117.3 355.3213 -133.827 405.9233 -1908.32 -210.569 -199.043 -133.267 394.5543 -488.99 106.4118 -911.846 1823.341
DFB0
1771.836 -1676.74 -8486.12 -2767.38 -67.6465 1757.23 -843.842 9748.579 293.9678 5103.653 -1043.12 -853.204 -433.195 535.6712 1283.115 -5054.45 -412.874 21594.88 -1164.24 -306.256 2777.848 -7568.23 -875.639 -849.33 -104.536 228.6693 -3082.53 543.1996 -2604.81 1221.323
DFB1
90.02119 3.38324 142.1058 -17.8337 49.60728 17.36036 -2.72151 90.61141 -17.1551 66.84446 -13.0592 -8.96651 -11.3315 9.18545 61.07498 -127.692 -5.35156 -338.52 -18.4274 4.392 13.97231 -140.255 -10.929 -3.21383 4.18633 -20.1834 -6.79856 2.41693 -47.7577 75.88662
DFB2
84.39044 33.23607 92.00443 47.01162 -7.89183 -31.2681 16.65544 -192.081 0.52404 -102.793 19.56062 17.3027 1.91654 -12.3811 -39.4801 108.3992 8.93141 -371.146 23.33959 2.48231 -45.8936 135.2919 15.87497 11.24521 -1.27187 5.43253 49.53844 -9.29264 36.88248 3.2469
DFB3
-0.2482 0.00124 0.10742 0.01125 0.00351 0.00617 -0.00082 0.00954 0.00392 0.00778 -0.00732 -0.00024 0.00101 0.00027 0.00788 -0.00637 -0.0004 0.03042 -0.00203 0.00044 -0.0015 -0.00127 0.00025 0.0008 0.00066 -0.0032 0.00035 0.00025 0.01001 -0.00799
DFB4
1.60971 -0.15021 -0.42897 0.01078 0.04511 0.04052 0.01785 -0.1465 -0.04885 -0.09905 -0.01313 -0.002 -0.01085 0.00653 -0.05617 -0.05553 -0.00181 -0.2895 0.00562 0.00858 0.00413 0.03923 0.0017 0.00839 0.00699 0.00164 0.02088 -0.0058 -0.08531 -0.05056
DFB5
0.00942 0.00044 0.00888 -0.0013 0.00014 0.00134 0.00146 0.00211 -0.0002 0.00073 0.00556 0.00013 0.00105 0.0001 0.00034 0.00382 0.00007 0.00855 0.00051 0.00004 0.00011 0.00219 0.00011 0.00026 0.00011 0.00029 0.00001 0.00002 0.00168 0.00394
14.821 21.845 9.0477 2.0118 4.2431 0.9615 10.026 2.0244 1.01 3.4552 8.1404 1.4944 0.4268 0.9682 1.7579 5.3237 1.7004 18.175 7.1466 0.9602 2.4523 4.8415 7.6109 0.8714 0.8443 1.0749 2.3906 6.3773 1.7551 1.2422
Table 3: Coefficients(a) Model 1 (Constant) tenure age Unstandardized Coefficients B 8090.913 269.073 16.825 Std. Error 31304.542 403.708 571.963 t .258 .667 .029 Sig. .798 .511 .977
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sales profits assets
-.202 2.080
.068 .624 .013
-2.972 3.332 13.946
.007 .003 .000
.184 a Dependent Variable: Compensation
a). Table 3 gives the multiple regression output of coefficients obtained using SPSS. Identify the explanatory variable that has the highest relative impact on the compensation paid to the CEOs.
b). Two MBA students, Tom and Jerry, draw a sample of 7 firms each from the original list of 30 firms and estimate the regression coefficients. The firms in Tom’s and Jerry’s sample are shown in Table 4. Who is likely to get a better estimate and why? Write all the steps clearly
Table 4: Tom and Jerry’s Sample Sample Point 1 2 Firms in Tom’s 1 2 sample Firms in Jerry’s 24 25 Sample 3 3 26 4 4 27 5 5 28 6 6 29 7 7 30
c). Which firm has the highest influence on the regression coefficients and why? d). If all the sample points in which the actual values lie outside 3-sigma level from the predicted values are removed from the data, what will be the new values of regression coefficients? e). Figure 1 shows the plot between un-standardized residual and age. What would be your inference from the plot? Suggest a necessary action if the plots indicates any problem
Figure 1. Residual plot
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30000 20000 Residuals 10000 20 40 60 80 0 -100000 -20000 -30000 ag e
f). Identify all the outliers in the sample, state why these sample points are outliers? g). If the standardized residual for sample 1 is -2.44, what inference you can make about the standard deviation of the residuals? Question Data collected on the monthly demand for Gold Jewellery in 2007 (measured in Tons) is shown in the following table. Table 1 Month January February March April May June July August September Demand 100 90 80 150 240 320 300 280 220
a). What is the 4-month Moving Average forecast for period 5? b). What is the forecast for period 3 using the exponential smoothing methods with a smoothing factor of 0.4? c). What conclusion can you draw with respect to the length of moving average versus smoothing effect for this time series data?
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d). Assume that the 12 month moving average centred on July is 250. What is the seasonal effect of the month July on demand? Explain. e). Trend line based on the data in table 5 is given by: Demandt = 40.44 + 27.50 × t. Calculate the demand for gold in July 2008. f.) Calculate the forecasting power of 4 periods moving average method for the data in table 1 Question A company manufactures tables and chairs. Each table requires 4 hours of assembly time, 2 hours of finishing time, 20 square feet of plywood, and 16 running feet of wood beam (section 2" X 3"). Each chair requires 2 hours of assembly time, 4 hours of finishing time, 5 square feet of plywood, and 8 running feet of wood beam. The profit per table is Rs 80 and per chair is Rs 60. Constraints that must be satisfied include the following: A. 600 hours of assembly time are available. B. 450 hours of finishing time are available. C. 1000 square feet of plywood are available. D. At least as many chairs as tables must be produced. E. The amount of wood beam available is uncertain, but estimated to be at least 10,000 feet. The LP formulation for the above problem such that profit can be maximized is given below: Max 80T + 60C s.t. 4T + 2C ≤ 600 2T + 4C ≤ 450 20T + 5C ≤ 1000 - T + C ≥ 0 16T + 8C ≤ 10000 T ≥ 0, C ≥ 0 Define the decision variables used above and label the constraints.
Show your work on the given graph paper to answer the following questions in the given space:
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1. 2.
Find the optimal solution, using the graphical technique. Mark the feasible region, optimal point, and the value of the objective function clearly. How much would you be willing to pay for: a) each extra hour of assembly time? b) each extra square foot of plywood? 3. If the amount of available plywood doubled, what would be the new solution? 4. The company wants to get out of the table-making business, and make only chairs. What should be the minimum profit per chair (assuming the same profit per table) to make this possible? 5. Management requires that tables and chairs should be sold in sets. Two alternatives are being considered: a) Set of one table and 4 chairs. b) Set of one table and 6 chairs. Which alternative do you recommend? Why? In each case show the feasible region. 6. Management has mandated that at least 50 tables should be produced. How will this effect the solution? Question A company operates for five seasons, and makes two products, Product 1 and Product 2. It manufactures the products in the first four seasons and sells them in seasons 2 through 5. Products made in a season can be sold from the next season onwards. One unit of product 1 requires 5 person hours in the preparatory shop and 3 person hours in the finishing shop. One unit of product 2 requires 6 person hours in the preparatory shop and 1 person hour in the finishing shop. During each season, the company has at most 12,000 hours in the preparatory shop and 3000 hours in the finishing shop. The company wants to formulate a linear program to decide its production and sales plans. It wants to maximize revenues. The expected selling price per unit of each product for each season is given below: Product 1 Product 2 Season 2 Rs.20 Rs.45 Season 3 Rs.25 Rs.40 Season 4 Rs.30 Rs.40 Season 5 Rs.15 Rs.30
(a) Define the variables needed to formulate the problem as an LP.
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(b) Formulate the problem as an LP. Label all constraints clearly. inventory carrying cost is zero for both products.
Assume that the
(c) If a unit is available for sale during a season but is not sold, the manufacturer has to pay Rs. 2 per unit as charges for carrying inventory to the next period. How will the formulation change? Define any new variables clearly, if used. Selling the products requires some marketing efforts, and 0.1 hours and 0.2 hours are needed to sell 10 units of products 1 and 2 respectively. This can be hired from outside at the following rates per hour: Season 2 3 4 5 Rate Rs.2 Rs.4 Rs.1 Rs.10. (d) How does the problem formulation change to ensure sufficient marketing effort? Define any new variables clearly, if used. Suppose the number of marketing hours that can be hired at the given rates is limited. Any hours over and above that have to be hired at a much higher rate per person hour as follows (Assume that an unlimited number of these hours are available): Season Maximum hours at low rates Higher rates 2 400 Rs.10 3 300 Rs.10 4 600 Rs.10 5 1000 Rs.10
(e) How would the formulation change to take this into account? Define any new variables clearly if used. Question Saloni Malhotra has recently been named the CEO of the Vittal Iron and Steel (VIS) Company. The company makes steel, heavy machinery and trucks for export. Saloni has decided that, in view of the government policies, the company’s performance (and hers) would be best served by maximizing the net dollar value of the exports by the company for the coming year. (The net dollar value of exports is defined as exports less the cost of all materials imported by the company). For the coming year it is assessed that the company can sell all it can produce of these items at the existing prices (in equivalent dollar amounts) of $900/unit for steel, $2500/unit for machinery and $3000/unit for trucks. In order to produce one unit of steel with the existing technology, it takes 0.05 units of machinery, 0.08 units of trucks, two units of ore imported for $100/unit, and other imported materials costing $100. In addition it takes 0.5 man-years of labour to produce each unit of steel. The steel mills have a maximum usable capacity of 300,00 units/year.
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To produce one unit of machinery it requires 0.75 units of steel, 0.12 units of trucks, and 5 man-years of labour. In addition, $150 of materials must be imported for each unit of machinery produced. The practical capacity of the machinery plants is 50,000 units/year. In order to produce one unit of trucks, it takes one unit of steel, 0.10 units of machinery, three man-years of labour, and $500 worth of imported materials. Existing production capacity for trucks is 550,000 units/year. The total manpower available for production of steel, machinery and trucks is 1,200,000 men/year. To plan for the coming year Saloni Malhotra has asked his OR consultant, Ms. Mahima Mukherjee, for inputs. Mahima has formulated a linear program to arrive at the best strategy to suggest. The model and the SOLVER solution are as given below: Max 900 SE + 2500 ME + 3000 TE - 300SP - 150MP - 500TP s.t. .75 MP + TP - SP + SE = 0 0.05SP + .10TP - MP + ME = 0 .08SP + .12MP - TP + TE = 0 .5SP + 5MP + 3TP <= 1200 SP <= 300 MP <= 50 TP <= 550 end
Microsoft Excel 11.0 Sensitivity Report Worksheet: [VIS Company.xls]Sheet1 Report Created: 12/6/2007 5:16:01 PM Adjustable Cells Final Reduced Objective Cell Name Value Cost Coefficient $B$2 SP 300 0 -300 $C$2 MP 50 0 -150 $D$2 TP 262.5 0 -500 $E$2 SE 0 -1350.00 900 $F$2 ME 8.75 0 2500 $G$2 TE 232.5 0 3000 Constraints Final Shadow Constraint Cell Name Value Price R.H. Side $H$3 0 2250 0 $H$4 0 2500 0 $H$5 0 3000 0 $H$6 1187.5 0 1200 $H$7 300 1585 300 $H$8 50 302.5 50 $H$9 262.5 0 550
Allowable Increase 1E+30 1E+30 403.33 1350 10566.67 347.70 Allowable Increase 4.17 9.4E+15 1E+30 1E+30 3.57 4.54 1E+30
Allowable Decrease 1585 302.5 1350 1E+30 281.40 1350 Allowable Decrease 232.5 8.75 232.5 12.5 252.72 8.14 287.5
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Use the output to answer the following questions precisely. Support each answer with adequate work and explanations. 1. Define the variables used in the formulation. What is the optimal production and export strategy for VIS Company? What is the net dollar value of exports under this plan? 2. The optimal solution suggests that 50,000 units of machinery should be produced. How are these units to be utilized during the year? 3. What would happen to the value of net exports if the price of steel increased to $1500/unit and the company chose to export one unit of it. 4. There is a chance that Saloni Malhotra may have $400,000 to spend on expanding capacity. If this investment can buy 250 units of steel capacity, 1000 units of machinery capacity, or 2000 units of truck capacity, what would the best investment be? Why? 5. If the price of the imported materials needed to produce one unit of trucks were to increase by $400, what would be the optimal export mix for the company? What would be the dollar value of their net exports? 6. The R&D group at the company has come to Saloni with a new product. Product X, that can be produced for export with 1.5 man-years of labour and 0.5 units of machinery for each unit produced. What must Product X sell for to make it attractive for production? [2 points] 7.If Saloni decides to stockpile (inventory) 25,000 units of steel during the coming year, what effect will this have on the related constraint equation(s), and what impact will it have on net dollar exports? Question Consider the following linear programming problem: Min 4x1 + 6x2 + 7x3 Subject to: + x5 + 5x6 + 2x6 >= 5 + x6 >= 3 + x6 >= 3 0 (1) (2) (3)
- x1 + 2x2 + x3 x1 - 2x3 + x5 x2 + 2x4 x1, x2, x3, x4, x5, x6 ≥
i) Write the dual linear programming problem for the above LP
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j) Use duality properties to determine if the following solutions are optimal for the dual problem. Clearly explain why or why not. Determine the primal optimal solution and the optimal objective function value. Show all work. 1. 2. (2.5,1,0) (2,1,0)
k) Introduce a new non-negative variable x7 in the primal problem with coefficients 1, 5 and 1 in constraints (1), (2) and (3) respectively. The objective function coefficient of x7 is 1. 1. Will this variable be included in the new primal optimal solution? Why or why not? 2. Write the dual LP problem corresponding to this extended primal LP with 7 variables. Question The RMC Corporation blends three raw materials in order to produce two products: a fuel additive and a solvent base. Each ton of fuel additive is a mixture of 2/5 ton of material 1 and 3/5 ton of material 3. A ton of solvent base is a mixture of 1/2 ton of material 1, 1/5 ton of material 2, and 3/10 of material 3. RMC’s production is constrained by a limited availability of the three raw materials. For the current production period RMC has the following quantities of each raw material: material 1, 20 tons; material 2, 5 tons; material 3, 21 tons. Management would like to achieve the following P1 priority level goals: Goal 1: Produce at least 30 tons of fuel additive. Goal 2: Produce at least 15 tons of solvent base. Assume there are no other goals in the problem. a. Is it possible for management to achieve both P1 level goals, given the constraints on the amounts of each material available? Explain . Treating the amounts of each material available as constraints, formulate a goal programming model to determine the optimal product mix. Assume both of the P1 priority level goals are equally important to management. Define all variables and label all constraints.
b.
c.
The feasible region defined by the material constraints is given by ABCDE where:
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A = (0,0), B = (0,25), C = (75/4,25), D = (25,20), E = (35,0) Which of these corner points will be optimal solution to the formulation in part b? Explain clearly why. What is the optimal objective function value? d. If goal 1 is thrice as important as goal 2, give the new formulation. What is the optimal product mix and the optimal objective function value?
Reconsider the RMC data presented above. Assume the two P1 priority level goals remain the same and both goals are equally important to management. Suppose management has learned that additional amounts of material 3 can be obtained from another RMC plant. Although management would like to obtain a solution that satisfies their production goals using the 21 tons of material 3 currently available, they are willing to consider using additional amounts of material 3 from the other plant. Introducing this new objective or goal as a P2 priority level goal, the problem goals can now be restated as follows: Priority Level 1 Goals Goal 1: produce at least 30 tons of fuel additive. Goal 2: produce at least 15 tons of solvent base. Priority Level 2 Goals Goal 3: Use at most 21 tons of material 3. e. Treating the amounts of materials 1 and 2 available as problem constraints, formulate a goal programming model for this problem.
Question Soft Solution (SS) is a software solution provider and has recently started using Six Sigma to improve their business processes. Currently the management is analyzing 20 different projects and would like identify the best project bases on their relative efficiency scores. The inputs and outputs for the 20 projects are given in the following table.
Project Number Expected Project Cost Expected Project Duration in days Number of Black and Percentage Increase in Customer Satisfaction Impact on Business Strategy Financial Impact Expected Increase in Sigma Quality Expected Percentage Increase in Productivity Inputs Outputs
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Green 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 212 199 214 280 263 203 196 215 281 233 263 198 220 284 214 235 200 217 198 227 70 63 88 77 72 70 61 79 71 66 84 60 80 79 87 80 63 75 63 70 Belts 10 3 5 8 11 11 3 5 8 6 10 3 4 6 7 5 4 11 3 5 11 29 28 29 19 21 31 22 17 10 27 32 15 23 19 27 33 13 31 10 4 8 4 6 2 5 9 5 3 6 8 8 7 4 7 2 8 5 8 6 331 342 333 303 240 306 345 264 239 338 310 341 308 325 314 236 339 313 343 317 0.24 0.77 0.33 0.48 0.41 0.52 0.78 0.27 0.66 0.3 0.88 0.51 0.31 0.31 0.54 0.34 0.83 0.74 0.87 0.46 20 23 10 10 11 17 21 6 19 16 19 22 7 7 5 17 23 7 22 17
(a) A DEA model for calculating the efficiency of project 1 has the following constraints in addition to few other constraints. yij is the output i for project j and uj is the weights allocated to project j. 1. ∑ y1 j u j ≥ 11 ,
j =1
20
20
2. ∑ y2 j u j ≥ 4 , 3.
j =1
20
∑ y3 j u j ≥ 331 4.
j =1
20
∑y
j =1
20
4j
u j ≥ 0.24
5.
∑y
j =1
5j
u j ≥ 20
Complete the DEA model by identifying the objective function and remaining constraints. (b) The following table gives the efficiency scores of projects and the corresponding benchmark projects (also known as peer projects which are on the efficiency frontier) along with the weights allocated by the DEA model in question (a). Calculate the relative efficiency scores of projects 4 and 5. Weights and the corresponding peer projects
Project Number
1 2
Relative Efficiency Score θ
1.000
Weight (uj)
0.9594 1
First Peer Project Number
7 2
Weight(uj)
2nd Peer Project Number
72
3 4 5
0.9652 0.8458 0.1933
7 12 7
0.0585 0.5081
17 12
Question The XYZ Company produces two different types of electronic components; say component A and component B, for a major customer. The customer notifies the XYZ Company each quarter as to what the monthly requirement for components will be during the next three months. The following order is received by XYZ Company for the next 3 month's period. April May June ----------------------------------------------Component A Component B 1000 3000 5000 1000 500 3000 -----------------------------------------------
The production control department must now develop a 3-month production plan for the components taking all applicable costs into account. The production costs of components A and B are as given below: Month April May June Production cost per unit Component A Component B -----------------------------------------------------200 150 220 160 230 165 -----------------------------------------------------
The inventory costs are calculated on the month end inventory level for each month. There should not be any short supply of components in any month. In any given month the XYZ company cannot supply to the customer more than what is required for that month. Any inventory left at the end of June will be a waste and does not involve any extra cost of disposal. The inventory carrying costs are 1.5% of the cost of the product kept in stock at the end of each month. If the production level of a component changes from one month to the next month, then it involves an additional cost. For a component, if the production level is more than the production level in the previous month, then a cost of Rs. 50 per unit increase is incurred. If the production level in a month is less than the production level in the previous month, then a cost of Rs. 20 per unit decrease is incurred. This is applicable for changes in the production levels for each component. For example, if the production of component A in May is 53.4 units higher than the production in April then a cost of 50*53.4 is incurred for increasing the production level in May. At the beginning of April, the production level for the two components can be set at any levels without incurring any cost.
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Moreover, there is a limitation on the storage capacity. The Company has 10,000 sq.ft. of storage capacity and extra storage capacity cannot be hired. Components A and B require 2 sq.ft. and 3 sq.ft. of storage capacities per unit respectively. The objective is to arrive at a production plan for the next 3 months that will minimize the total cost. For planning purposes the number of units to be produced of the two different types of electronic components can be fractional. a) Formulate this decision problem as a mathematical programming model. Clearly define all the variables that you introduce and label all constraints The variable cost involved when the production level of a component changes from one month to the next month increases with the degree of change. For a component, if the production level is more than the production level in the previous month, then a cost of Rs. 50 per unit increase is incurred as long as the increase is up to 200 units. Beyond 200 units the variable cost incurred is Rs. 60 per unit. If the production level in a month is less than the production level in the previous month, then a cost of Rs. 20 per unit decrease is incurred as long as the decrease is up to 200 units. Beyond 200 units the variable cost incurred is Rs. 25 per unit. This is applicable for changes in the production levels for each component. Modify the formulation in part a) to incorporate this information. Taking the base case as in part a) consider the following information: There is a fixed setup cost of Rs. 100 involved whenever there is a change in the production level. Modifiy your formulation in part a) to incorporate this information.
b)
c)
Question Consider the following Integer Linear Programming (ILP) problem: Max 17 X1 + 20 X2 Subject to: 4 X1 + 5 X2 <= 18 6 X1 + 3 X2 <= 22 X1, X2 are non-negative integers On solving the LP relaxation of the problem the following optimal solution is obtained: X1 = 3.1111, X2 = 1.1111 , Objective Function Value = 75.1111 a) Using this information can you say anything about the optimal solution and/or objective function value of the ILP?
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b)
Solve the ILP using the branch-and-bound (b-a-b) method branching on the variable X1 first. Draw the b-a-b tree and clearly indicate the branching variables, solution at each node and the reason for fathoming a node. c) Now solve the ILP again branching on X2 first. Draw the b-a-b tree and clearly indicate the branching variables, solution at each node and the reason for fathoming a node. d) Compare the number of LPs required to solve the ILP in b) and c). Comment on the difference in the two, if any.
Question Ramlakhan owns a term life policy that has a surrender value of Rs. 60,000 and death benefit of Rs. 250,000. Ram is considering investing in a different policy that has a higher death benefit of Rs. 500,000. In order to do this he would like to cash the current life policy and invest the Rs. 60,000 elsewhere such that the interest earned per year is sufficient to make the first 6 annual premium payments for the new policy. Of course, the interest income he earns will be taxable at a rate of 30% but he would have to pay taxes only on the interest amount in excess of what he uses towards making the premium payments for the new policy every year. At the end of each year the excess amount of interest remaining, if any, after paying taxes is invested along with the principal investment. The premium on the new policy for the next 6 years are: Year 1 2 3 4 5 6 Premium 4230 4570 4890 5160 5300 5580 Ramlakhan would like to determine the minimum annual rate of return that will be required on his investment to make this possible. This will help him decide on whether he can afford to make the switch. Give a mathematical formulation that will help Ramlakhan determine the rate of return. Do not attempt to solve the problem
Indian Institute of Management Bangalore Quantitative Methods II
Please do not print this unless you really need to.
Question 1 The personnel director of a manufacturing company has collected the following data on the Salary earned by each machinist in the factory along with the average performance rating over the past 3 years, the years of service and the number of different machines each employee is certified to operate. Avg Years Performance of S. No. Salary Rating Service Certifications 1 48.20 3.50 9 6 2 55.30 5.30 20 6 3 53.70 5.10 18 7 4 61.80 5.80 33 7 5 56.40 4.20 31 8 6 52.50 6.00 13 6 7 54.00 6.80 25 6 8 55.70 5.50 30 4 9 45.10 3.10 5 6 10 67.90 7.20 47 8 11 53.20 4.50 25 5 12 46.80 4.90 11 6 13 58.30 8.00 23 8 14 59.10 6.50 35 7 15 57.80 6.60 39 5 16 48.60 3.70 21 4 17 49.20 6.20 7 6 18 63.00 7.00 40 7 19 53.00 4.00 35 6 20 50.90 4.50 23 4 21 55.40 5.90 33 5 22 51.80 5.60 27 4 23 60.20 4.80 34 8 24 50.10 3.90 15 5 Sum 1308.00 128.60 599.00 average 54.50 5.36 24.96 6.00 stdev 5.47 1.29 11.22 1.32 Covariance(Y,X(i)) 4.52 50.55 3.86 ∑(salary(i) - 54.50) = 689.26
2
The personnel director would like to build a regression model to estimate the average salary an employee should expect to receive. a) If the regression model is to be built using only one independent variable, which variable should be used? Explain why.
b) What is the proportion of variation in salary that can be explained by this variable?
c) A regression model using average performance rating is fitted to the data. Given that the intercept for the model is 39.35, the slope is 2.83 and the R square value is 0.445, predict the salary range for an employee with 12 years of service, who has received average reviews of 4.5, and is certified to operate 4 pieces of machinery?
d) A regression model (Model 1) is built using Certifications and Years of Service as the independent variables. The Summary Output and Residual Output are given below. Complete the Summary Output by filling in ALL the missing entries. SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square 0.8557 Standard Error Observations ANOVA df Regression Residual Total Standard Error 2.0798 SS MS F Significance F 5.71299E10
Intercept
Coefficients 35.8489
t Stat P-value 17.2369 0.0000
Lower 95% 31.5237
Upper 95% 40.1740
3
Certifications Years of Service RESIDUAL OUTPUT
Observation 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Total
1.5489 0.3749
0.3388 0.0398
4.5723
0.0002
0.8444
2.2533
Predicted Salary 48.5165 52.6409 53.4399 59.0640 59.8630 50.0163 54.5156 53.2926 47.0168 65.8621 52.9668 49.2664 56.8635 59.8139 58.2159 49.9181 47.7667 61.6886 58.2650 50.6680 55.9663 52.1678 60.9878 49.2173 1308.0000
Residuals -0.3165 2.6591 0.2601 2.7360 -3.4630 2.4837 -0.5156 2.4074 -1.9168 2.0379 0.2332 -2.4664 1.4365 -0.7139 -0.4159 -1.3181 1.4333 1.3114 -5.2650 0.2320 -0.5663 -0.3678 -0.7878 0.8827 0.0000
Residual^2 0.1002 7.0708 0.0677 7.4855 11.9925 6.1687 0.2659 5.7956 3.6740 4.1531 0.0544 6.0833 2.0636 0.5097 0.1730 1.7374 2.0545 1.7197 27.7207 0.0538 0.3207 0.1353 0.6207 0.7791 90.7997
e) Carry out a Hypothesis test to determine if the variable Years of Service has the ability to explain variation in salary. State your Hypotheses and conclusion clearly. Support your conclusion adequately and show all work.
f) What does the Multiple R value imply?
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The summary of the regression model with all three independent variables is as follows: SUMMARY OUTPUT Regression Statistics Multiple R 0.9558 R Square 0.9135 Adjusted R Square Standard Error 1.7261 Observations 24 ANOVA df Regression Residual Total 3 20 23 Significance SS MS F F 629.6726 209.8909 70.4480 8.2824E-11 59.5874 2.9794 689.26 Standard Error 1.9490 0.3268 0.0364 0.2916 Upper 95% 36.9868 l.7395 0.4012 1.9074
Intercept Avg Performance Rating Years of Service Certifications
Coefficients 32.9212 1.0578 0.3252 1.2992
t Stat P-value 16.8911 0.0000 3.2367 8.9223 4.4555 0.0041 0.0000 0.0002
Lower 95% 28.8556 0.3761 0.2492 0.6909
g) Carry out a partial F test to decide whether adding the third variable is worthwhile. Which model should be used (Model 1 or Model 2)?
Question
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The data in table 1 is collected from 20 North American cities. The data consists of number of crimes per million per year, population, percentage of families with annual income of less than $5000 and percentage unemployed. Table 1: Number of crimes per 1 Percentage Percentage million of families of per Region Population AI < 5000 unemployed annum 1 587000 16.5 6.2 11.2 2 643000 20.5 6.4 13.4 3 635000 26.3 9.3 40.7 4 692000 16.5 5.3 5.3 5 1248000 19.2 7.3 24.8 6 643000 16.5 5.9 12.7 7 1964000 20.2 6.4 20.9 8 1531000 21.3 7.6 35.7 9 713000 17.2 4.9 8.7 10 749000 14.3 6.4 9.6 11 7895000 18.1 6 14.5 12 762000 23.1 7.4 26.9 13 2793000 19.1 5.8 15.7 14 741000 24.7 8.6 36.2 15 625000 18.6 6.5 18.1 16 854000 24.9 8.3 28.9 17 716000 17.9 6.7 14.9 18 921000 22.4 8.6 25.8 19 595000 20.2 8.4 21.7 20 3353000 16.9 6.7 25.7 Mean 1433000 19.72 6.935 20.57 standard deviation 1703726 3.242254 1.207117 9.881407 Model 1 The following regression model is constructed to analyze the number of crimes per million (Y) and the percentage unemployed (X1). Y = β0 + β1 X1 The Regression output from Excel is shown in Table 2.
Table 2 Regression Statistics Multiple R R Square Adjusted R Square
6
Standard Error Observations ANOVA
5.0968524
Df Regression Residual Total
SS
MS
F
Significance F
Intercept Percentage of unemployed
Coefficients -28.5267
Standard Error 6.81371 0.96867
t Stat
P-value
Lower 95% -42.841798 5.04445402
Upper 95% -14.211619 9.11465453
What percentage of the variation in crime rate is explained by model 1? Using model 1 calculate the probability that the crime rate will be more than 21 per million per annum when the unemployment percentage is 6.4. How do you interpret the value of β0 in model 1? Comment on the negative value of β0, is the negative value justified? Regression output for the no-intercept model (model 2) is shown in table 3. (a) Calculate the percentage of variation explained by model 2. (b) Which model (between models 1 and 2) should be used to analyze the data in table 1, justify your answer using appropriate regression measures. Model 2: Y = β1 X1
Table 3 ANOVA Df Regression Residual Total Standard Error #N/A Upper 95% #N/A 3.544923 8 48.57609 6 SS MS F 193.4028 2 Significance F
Intercept Percentage of unemployed
Coefficients 0
t Stat #N/A
P-value #N/A
Lower 95% #N/A 2.61747
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SPSS output for Model 3 is shown in Table 3. Identify the explanatory variable that has the maximum impact on the crime rate. Model 3: Crime rate = β0 + β1 x population + β2 x percentage families with AI < $5000 + β3 x percentage unemployed
Table 3 Coefficients(a) Unstandardized Coefficients Model 1 B -36.765 Std. Error 7.011 .000 .562 1.530 Standardized Coefficients Beta t -5.244 1.199 2.123 3.084 Sig. .000 .248 .049 .007
(Constant) Population Percentage of families AI < 5000 Percentage of unemployed
a Dependent Variable: Number of crimes per 1 million per annum
Table 4 below shows the distance measures and influential statistics for the Model 3.
Region 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 Mahalanobis distance 1.48755 1.67850 4.39730 2.48362 .58566 1.42796 1.00081 .34335 4.69735 5.49044 14.40156 1.87023 2.17375 2.42169 .46345 2.65850 .72160 2.16993 3.60786 Cook’s distance .00402 .05467 .01940 .03168 .01213 .00412 .00534 .09363 .01186 .01966 5.63672 .00093 .00010 .01737 .00243 .05081 .00427 .08234 .16107 Leverage value .07829 .08834 .23144 .13072 .03082 .07516 .05267 .01807 .24723 .28897 .75798 .09843 .11441 .12746 .02439 .13992 .03798 .11421 .18989 DFFIT DFBeta0 -.67020 -.70931 -1.5232 -2.0099 .17268 .74556 .11047 -1.5728 .96872 -1.1451 3.90214 -.13388 .03303 -1.2747 .38857 2.01915 -.50030 1.99450 .87133 DFBeta1 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 .00000 DFBeta2 .03777 -.1731 .05058 -.0247 -.0700 -.0222 .05116 .01718 .05522 .13913 -.03596 .02595 .00613 .05768 -.00289 -.15623 .03824 .09428 .30520 DFBeta3 -.0290 .52891 .09395 .31261 .20012 -.0250 -.1487 .2422 -.2727 -.2524 -.0443 -.0474 -.0213 .04279 -.0296 .1147 -.0571 -.6005 -1.0549
-.79828 .67834 -.69459 .28739 .20837 .21486 .73284 .54493 -.74943 -19.5900 .10766 .03710 .50967 .12335 -.90175 -.17789 -1.0673 -1.8043
8
20
2.91888
.25223
.15363
2.08036
1.36347
.00000
-.36775
.84411
Identify the most influential observation in the data. What will be the model parameter values (beta values) if this observation is not included in the sample. Calculate the DFFIT for observation 1 (region 1) Question The data in the following table shows the exchange rate of Indian rupee for 1 US dollar for the past 10 trading days.
Date Exchange rate INR Vs 1 Dollar 2009-10-05 47.5198 2009-10-06 46.9203 2009-10-07 46.6898 2009-10-08 46.3199 2009-10-09 46.4102 2009-10-12 46.4897 2009-10-13 46.3449 2009-10-14 46.1152 2009-10-15 46.285 2009-10-16 46.2997
1. Use the method of moving averages with number of periods k = 4 and calculate the forecasted exchange rate for 2009-10-20. 2. Use the method of simple exponential smoothing and forecast the values for 200910-20. Use the value of α = 0.8. 3. Comment on the forecasting power of the moving average with 4 periods and simple exponential smoothing with α = 0.8 Show all work. For the data used in this problem what should be the appropriate values of k and α (small or large)? Justify your answer.
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Question The planning committee of a bank makes decisions at the beginning of every year on the amount of funds to allocate to loans and to government securities. Some of the loans are secured (backed by collateral) and some are unsecured. A list of various types of loans and their annual returns are shown in table 1. Table 1: Annual rate of return on investments Type of investment Annual rate of return Government Securities 9 Secured loans Residential mortgage 11 Commercial mortgage 12 Automobile 15 Home improvement 13 Unsecured loans Vacation 17 Student 10 In making its decision, the planning committee must satisfy certain legal requirements and bank policies. These can be summarized as the following set conditions. 1. The amount allocated to secured loans must be at least four times the amount allocated to unsecured loans. 2. Auto and Home loans should be no more than 20% of all secured loans. 3. Student loan should be no more than 30% of unsecured loan. 4. The amount allocated to government securities should be at least 10% but no more than 20% of the funds available. 5. The amount allocated to vacation loans must not exceed 10% of all loans. The bank has Rs 5 million available for loans and investments in the current year. The bank collects 10 percent of the principle for each of secured loans along with the interest at the end of the year (simple annual interest is calculated for the entire loan amount for the year). The entire principle and interest is collected on the unsecured loans at the end of the year. In addition to these, Rs 6 million is available at the beginning of the second year. This entire amount is available for allocation in the second year. Formulate a linear programming model that will enable the planning committee to determine the optimal allocation of funds if the objective is to maximize the annual return over the two years. Clearly state all the decision variables and the constraints.
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Question Sugarco can manufacture three types of candy bars. Each candy bar consists of sugar and chocolate. The composition of each type of candy bar and the profit earned from each candy bar is shown in the following table. Fifty ounces of sugar and 100 ounces of chocolate are available.
Amount of Sugar Bar 1 Bar 2 Bar 3 Used 1 1 1
Amount of Chocolate Used 2 3 1
Profit (in Rs per unit) 3 7 5
(a) Formulate a linear programming problem. Clearly state the decision variables.
(b) Solve the problem and find the optimal value of the objective function and the optimal values of the decision variables.
(c) If the sugar can be purchased at Rs 2 per ounce, comment whether Sugarco should buy one additional ounce of Sugar. What will be the optimal profit if 10 additional ounces of sugar is made available?
(d) What is the minimum profit for candy bar 1 for which the current optimal solution will remain as the optimal solution.
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(e) Suppose candy bar 1 requires only 0.5 ounces of sugar and 0.5 ounces of chocolate, will the current optimal solution remain as optimal solution?
(f) A new candy bar can be made using 3 ounces of sugar and 4 ounces of chocolate. What should be the minimum profit so that it is advisable to make this new candy bar?
Question Consider the following LP Formulation Max Profit = 300 X1 + 500 X2 + 250 X3 Subject to 4 X1 + 6 X2 + 2 X3 <= 100 (Saline) 2.5 X1 + X2 + 1.5 X3 <= 40 (Benzene) X1 – X2 = 0 (Balance) 2 X1 + X2 >= 10 (Demand) Xi >= 0 (Non-negativity) Answer Report Target Cell (Max) Cell Name $E$6 OBJ SUM Adjustable Cells Cell Name $B$7 Var X1 $C$7 Var X2 $D$7 Var X3 Constraints Cell Name $E$2 Saline SUM Benzene $E$3 SUM Balance $E$4 SUM Demand $E$5 SUM Sensitivity Report Adjustable Cells
Original Value 8562.5 Original Value 0 0 0 Cell Value 100 40 0 26.25
Final Value 8562.5 Final Value 8.75
Formula $E$2<=$G$2 $E$3<=$G$3 $E$4=$G$4 $E$5>=$G$5
Status
Slack
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Cell $B$7 $C$7 $D$7 Constraints Cell $E$2 $E$3 $E$4 $E$5
Name Var X1 Var X2 Var X3
Final Value 8.75
Reduced Cost 0 0 0
Objective Coefficient 300 500 250
Allowable Increase 450 450 92.86
Allowable Decrease 216.67 216.67 90
Name Saline SUM Benzene SUM Balance SUM Demand SUM
Final Value 100 40 0 26.25
Shadow Price 40.625 112.5 -143.75
Constraint R.H. Side 100 40 0 10
Allowable Increase 14.29 21.67 4.55
Allowable Decrease 28.89 5 10 1E+30
1. Which of the four constraints are binding? 2. What is the change in the objective function value if the minimum demand (RHS of DEMAND CONSTRAINT) is increased from 10 to 20? 3. A new constraint is added. The constraint is given below: 4 X1 + 8 X2 + 12 X3 <= 180 What is the impact of this constraint on the present optimal solution? Is this new constraint binding or not binding? 4. The objective function coefficient of X1 is increased from 300 to 400. What is the change in the objective function value? What is the change in the value of the variables in the optimal solution?
5. The objective function coefficient of X1 is decreased from 300 to 150 and that of X3 increased to 330 simultaneously. What is the change in the objective function value? What is the change in the value of the variables in the optimal solution? 6. The demand constraint is changed as follows: X1+ X2 <= 10.
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What are the values of X1 and X2 in the new optimal solution? 7. The objective function coefficients are changed simultaneously as follows: X1 450; X2 750; X3 375. Is the current optimal solution still optimal? What is the change in the value of the objective function in the optimal solution? 8. The availability of Saline and Benzene is changed to 50 and 20. What is value of the variables in the new optimal solution 9. A new variable X4 is introduced into the formulation. Each unit of X4 requires 4 units of Saline and 6 units of Benzene. The profit contribution is 840. Is it worthwhile producing X4? What is the minimum required value of the profit contribution of X4 for it to be worthwhile?
Question 1 The following table gives marks scored by 12 students in the QM-II exam along with their CAT score and the number of classes missed by these students. Three regression models are developed by treating the marks in the QM-II as response variable and CAT score and number of missed classes as explanatory variables. The data is shown in table 1 along with influential statistics data for model 2.
Student No QM-II Marks (Y) CAT Score (X1) No missed classes (X2) 1 7 6 1 6 0 1 0 2 1 1 2 2.3333 Maha distance Cook’s distance Leverage DFFit DFBeta0 DfBeta1 DfBeta2
1 85 65 0.30311 0.0058 0.02756 -0.232 2 66 50 3.52954 0.68655 0.32087 4.8320 3 58 55 2.24782 0.35484 0.20435 -2.930 4 90 65 0.30311 0.01639 0.02756 0.3910 5 65 55 2.24782 0.00044 0.20435 -0.103 6 87 70 1.28724 0.14518 0.11702 -1.564 7 94 65 0.30311 0.08488 0.02756 0.8899 8 98 70 1.28724 0.08424 0.11702 1.1916 9 81 66 0.52591 0.02677 0.04781 -0.543 10 91 70 1.36777 0.00196 0.12434 0.1850 11 76 50 6.94407 0.0347 0.63128 -1.444 12 74 55 1.65327 0.04416 0.1503 -0.931 Mean 80.416 61.33 Std. Dev. 12.688 7.773 2.4984 The following three models are used to analyse the relationship between the explanatory variables. Model 1: Y = β0 + β1 X1 Model 2: Y = β0 + β1X1 + β2X2 Model 3: ln(Y) = β0 + β1X1 + β2X2 The Excel output for model 1 is shown in Table 2: Table 2: Regression output for Model 1
0.04067 1.74349 3.83791 -0.06838 0.13572 4.23182 -0.15561 -3.22353 2.83831 -0.89955 -5.96439 -5.955
-0.0046 -0.0397 -0.0514 0.00786 -0.0018 -0.0828 0.01788 0.06307 -0.0486 0.0152 0.08586 0.08395
0.03049 0.72536 -0.6565 -0.0512 -0.0232 0.08405 -0.1166 -0.0640 -0.0863 0.0161 0.22721 0.20314
QM-II marks and the
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R gre e ssion S tatistic s Multiple R RS re qua Adjusted RS re qua S nda E ta rd rror 7 457 .4 3 5 O bserva tions ANO VA df S S MS F S ignific eF anc R ression eg R esidua l T l ota 1 7 .9 7 70 1 C ffic nts oe ie S tandard E rror t S tat P-value L owe 9 % Uppe 9 % r 5 r 5 Interc ept -0 4 1 .1 4 3 -4 .3 6 2 1 7 3 .1 4 7 78 C S ore AT c 0 871 .2 8 2 0 0789 .7 9 4 3 1
Using table 2, find the regression parameters for model 1. Comment whether the overall model can be accepted. What is the probability that a student with a CAT score of 65 will score more than 85 marks in QM_II? The Regression output for model 2 is shown in Table 3 Table 3: Regression output for model 2
SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error 5.29589207 Observations ANOVA df Regression Residual Total Coefficients Intercept CAT Score Missed Classes Standard Error 19.32753 0.289113 0.899491 t Stat P-value Lower 95% 1.512064 0.031836032 -4.984688867 Upper 95% SS MS 759.249 2 F Significance F
252.4183
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Write model 2 by determining the model parameters. Calculate the increase in the R-square value in model 2 from model 1. Comment whether this increase can be justified for the use of model 2 instead of model 1. Find the probability that a student with a CAT score of 65 and 2 missed classes will score more than 85 marks in QM-II. What is the correct classification of observation 11: is it an outlier, leverage or influential observation? Justify your classification.
The Regression output for model 3 is shown in table 4 Table 4: Regression output for model 3
SUMMARY OUTPUT Regression Statistics R Square Standard Error ANOVA df Regression Residual Total Coefficients 3.984976 0.007939 -0.04148 Standard Error t Stat P-value Lower 95% 3.426841632 -0.00041005 -0.067457281 Upper 95% SS 0.259066 MS 0.00457 F Significance F
Intercept CAT Score Missed Classes
Identify the significant independent variables in model 3 and the R-square value.
If the number of missed classes increases by 1, what will be the change in the marks scored in QM-II. Among the three models, which is the most preferred model? Justify your answer.
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Question
A company needs to lease warehouse storage space over the next 5 months. The space that will be required during each of these months is known. However, since these space requirements change from month to month, it may be most economical to lease only the amount needed each month on a month-by-month basis. On the other hand, the relative cost for leasing space for additional months is much less than for one month, so it may be less expensive to lease the maximum amount needed for the entire 5 months. Another option is the intermediate approach of changing the total amount of space leased by adding a new lease and/or having an old lease expire. The space requirement (in thousands of square feet) and the leasing costs (Rs.) for the various leasing periods (durations in months) are Month 1 2 3 4 5 Required Space (1000 sq ft) 15 10 20 5 25 Lease duration (months) 1 2 3 4 5 Cost (Rs.) per 1000 sq ft leased 2800 4500 6000 7300 8400
a) Assuming that two or more leases of different durations can begin at the
same time give a mathematical programming formulation to determine the leasing schedule that provides the required amounts of space at a minimum cost. Define all variables clearly and label all constraints. Consider the following parts independently. For each part make suitable modifications to the formulation in part a). State the modifications clearly. b) Only one kind of lease can be signed in a month. c) At most two changes can be made in the amount of space hired. d) Two or more leases cannot run simultaneously in any month.
e) It costs Rs. 2500 to process a lease.
f)
A 10% discount is available on the cost of leasing space that is in excess of 10000 sq ft. However, this discount is available only for lease of 5 month duration signed in the first month. cannot be signed in any subsequent month.
g) If in month 1 a one period (month) lease is signed then a one period lease
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Question
Nanjanguda Investments Company is considering four projects for investment. Project 1 will yield a net present value (NPV) of Rs. 250,000; project 2, an NPV of Rs. 180,000; project 3, an NPV of Rs. 120,000; and project 4, an NPV of Rs. 80,000. Each project requires a cash investment at the present time: project 1, Rs. 80,000; project 2, Rs. 50,000; project 3, Rs. 40,000; and project 4, Rs. 30,000. At present, a budget of Rs. 150,000 is available for investment. The following Binary Integer Program (BIP) has been formulated to help decide which projects should be chosen for investment: Let xi ( i = 1, 2, 3, 4) = 1 if investment is made in project I and 0 otherwise Max z = 25x1 + 16 x2 + 12 x3 + 8 x4 s.t. 8x1 + 5x2 + 4 x3 + 3x4 ≤ 15 xi = binary ( i = 1, 2, 3, 4 ) The above BIP has a single constraint and is referred to as a knapsack problem. The LP relaxation for this BIP and other subproblems encountered while finding the solution to the BIP can be solved quite easily by observing that projects will be chosen according to the return they earn for each unit of investment. For example, to solve the LP relaxation of the knapsack problem above, we will first select project 2 since it has the highest return (3.2) per unit investment. We can select all of project 2 (x2=1), and we will still have Rs. 100,000 left. We will next select project 1 since it has the second highest return per unit investment. We can afford to select all of project 2, and as a result Rs. 20,000 will be left. The next best selection is project 3. However, since it requires an investment of Rs. 40,000, we can take only half of project 3. Hence the optimal LP solution is: x1 = x2 = 1, x3 = ½, x4=0. In order to solve the Nanjanguda Investments Company’s decision problem the Branchand Bound method is used and the following results are obtained: LEVEL 0
LP0 x1 = x2 = 1 x3 = 1 =1/2 X3 x4 = 0
x3 = 0
LEVEL 1 LP1 X1 = 3/4 X2 = x3 = 1 x4 = 0 LP2 x1 = x2 = 1 x3 = 0
x4 = 2/3
Answer the following questions: a) The lower bound on the optimal objective function value to the BIP at level 0 is _______________. Give adequate explanations.
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b) The best upper bound on the optimal objective function value to the BIP at
level 1 is _______________ .Give adequate explanations.
c)
Complete the branch-and-bound (b-a-b) tree using the above information in order to find the optimal solution for the problem. Proceed down the node that has the most promising value. While branching, set the variable value on the left branch equal to 1 and on the right branch equal to zero. Solve the left branch problem before the right branch problem. Clearly • show all branches, branching variables and their values • number the subproblems created in the order that you solve them • give the optimal solutions/objective function values for each subproblem solved • state when a node is fathomed and why and • Give the optimal solution and optimal objective function value. Question Consider the following linear programming (Primal) problem (LP1): Max 2x1 + x2 + x3 + 4x5 - 3x5 s.t. x1 + 2x2 + x3 - x4 + 2x5 <= 8 2x1 + x2 + x3 + 2x4 + x5 <= 12 5x1 + x3 + x5 = 10 x1, x2, x3, x4, x5 ≥ 0 (1) (2) (3)
a) Write the dual linear programming problem (DLP1) for the above LP1 and define the dual variables
used. b) Determine if (1,1,3,2,2) is a feasible solution for the given primal problem LP1? Show all work c) Using the information obtained so far give a lower and an upper bound for the objective function value of the dual problem DLP1. Support your answer with adequate reasoning.
d) Use duality properties to determine if the following solutions are optimal for
the dual problem DLP1. Support your answer with adequate reasoning. Clearly state the properties being used and show all work. 1. (1,3,1) 2. (0,2,-0.4)
e) Give the optimal solution and the optimal objective function value for the
primal problem LP1. Support your answer with adequate reasoning.
f) Another non-negative variable x6 is added to the primal problem with coefficients 3, 4 and 1 in
constraints (1), (2) and (3) respectively and with an objective function coefficient of 7. Write the dual LP problem corresponding to this extended primal LP with 6 variables (call it LP2).
g) Use duality property(s) to determine the optimal solution for LP2. Support your answer
with adequate reasoning.
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h) A new constraint: 2x1 - 3x2 + 3x4 - 2x5 >= 15 is added to LP1. Determine the optimal
solution to this modified LP1. Support your answer with adequate reasoning.
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Question NS Consumer Electronics (NSCE) has recently added an LCD TV to its product portfolio. They have hired the Quickbuck Advertising Company (QAC) to promote the TV. QAC has come up with two possible options-(1) advertising on IPL and (2) advertizing on the popular TV serial “Bahu Bhi Saas Banegi” (also known as B2SB). They are trying to target Rural high income (HIR) households, Urban High income (HIU) households and Young unmarried urban men (YUM). They have estimated the following exposures: Exposure in ‘000/ min of Ad. Cost/minute of Ad. Advertisement in HIR HIU YUM Rs. ‘000 IPL 4 10 5 100 B2SB 8 5 4 60 NSCE would like to minimize the cost with the following constraints: The exposure to HIR must be at least 40,000 The exposure to HIU must be at least 60,000 The exposure to YUM must be at least 35,000 There must be at least two minutes of advertising on each of the shows (IPL and B2SB). 1. Find out how many minutes of advertising is to be purchased on IPL and B2SB so that the total cost of advertizing is minimum. What is the cost of advertising in the optimal solution? 2. The total budget given by NSCE is Rs. 600,000. Consider this as a hard constraint. Formulate the problem as a goal programming problem. Define all variables used.
3. Consider the constraints given above as the goals in the same order of priorities. For example “The exposure to HIR must be at least 40,000” is the first priority. If the budget constraint cannot be violated, find out what is the minimum possible deviation for each of the 4 goals, with respect to the solution obtained in question no. 1. In case more than one solution is possible, select the one which will minimize the goal deviations in the order of priority mentioned above.
4. Solve the goal programming problem formulated in question no. 2. Calculate the over achievement/under achievement for each of the goals. 5. How does the above solution change if goal no. 4 is given the top most priority and the budget constraint is a soft constraint and ranked 2 in priority?
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Question Consider a balanced transportation problem with 3 sources and 4 destinations. The sensitivity report of the solution is given below.
Name Source 1 Destination $B$10 1 Source 1 Destination $C$10 2 Source 1 Destination $D$10 3 Source 1 Destination $E$10 4 Source 2 Destination $B$11 1 Source 2 Destination $C$11 2 Source 2 Destination $D$11 3 Source 2 Destination $E$11 4 Source 3 Destination $B$12 1 Source 3 Destination $C$12 2 Source 3 Destination $D$12 3 Source 3 Destination $E$12 4
Cell
Final Reduced Objective Allowable Allowable Value Cost Coefficient Increase Decrease 0 2000 7000 0 6000 0 0 1000 0 1000 0 3000 1 0 0 1 0 4 4 0 1 0 2 0 4 2 1 4 1 4 3 1 3 1 2 2 1 2 1 1 2 1 1 3 1 2 1 1
Cell $B$15 $B$16 $B$17 $B$18 $B$19 $B$20 $B$21
Name Source 1 Constraint Source 2 Constraint Source 3 Constraint Dest 1 Constraint Dest 2 Constraint Dest 3 Constraint Dest 4 Constraint
Final Value 9000 7000 4000 6000 3000 7000 4000
Shadow Constraint Allowable Allowable Price R.H. Side Increase Decrease 0 9000 1E+30 0 -2 7000 2000 0 -1 4000 2000 0 3 6000 0 2000 2 3000 0 2000 1 7000 0 7000 3 4000 0 2000
Since this is a balanced transportation problem, any increase or decrease of the requirement at any destination has to be matched with a corresponding increase or decrease at one of the sources.
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1. If, for some reason, Route 3 to 1 (Source 3 to Destination 1) is not available, what will be the change in the solution?
2. If the requirement at destination 3 is increased by one unit, and the corresponding increase is at source 2, what is the change in the value of the objective function?
3. If the requirement at Destination 4 is to be increased by 1 unit, which is the best source from which the extra unit is to be supplied?
4. Which is the costliest combination for any increase in the requirement at any of the destinations by one unit? Which the source involved with this combination?
5. What will be the change in the solution if the objective function coefficient of the route from source 1 to destination 1 is changed from 4 to 3?
6. What will be the change in the solution if the objective function coefficient of the route from source 2 to destination 2 is changed from 4 to 5?
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Question NLNN Kumar who just completed a short term course at the International Institute of Management and Accounting (IIMA), is trying his newly acquired knowledge with the data he received from his company. He is working in NS Software Services (popularly known as NS3), and received the data from the software testing division of the company. He had presumed that the number of bugs per 10 kilolines of code can be explained by two explanatory variables namely, the months of experience of the team (calculated as the average no. of months of total experience for the team) and size of the team (number of team members). He fitted the regression in Excel and got the following output. Summary Statistics: n= 10 ∑X1i = 150 ∑X2i = 831 ∑X1iX2i = 13990 ∑X21i = 2580 ∑X22i = 76117 SUMMARY OUTPUT Regression Statistics Multiple R 0.981583 R Square 0.963504 Adjusted R Square Standard Error Observations 10 ANOVA Significance df SS MS F F Regression 2 70.7897 35.39485 Residual 7 2.681371 0.383053 Total 9 73.47107 a) Calculate the adjusted R2 for this model. Explain how adjusted R2 differs from R2 and what is its significance?
b) Test the following null hypothesis. Use α = 0.01. State the conclusion of the result of the hypothesis test clearly
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H0: β1 = β2 = 0 H1 : At least one β is not equal to zero
Kumar also got the following output regarding the regression coefficients and the corresponding confidence intervals Standard PLower Upper Coefficients Error t Stat value 95% 95% Intercept 4.3036 2.2625 1.9022 0.0989 -1.0462 9.6535 Teamsize 0.8559 0.7481 1.1442 0.2902 -0.9130 2.6249 Months of Exp -0.0849 0.1613 -0.5261 0.6151 -0.4664 0.2966 c) Test the null hypothesis that the team size has no impact on number of bugs and also that months of experience has no impact on the number of bugs. State your conclusion of the hypothesis test clearly. Use α = 0.05 d) Comment on the results of Question nos. 1, 2 and 3, by explaining any consistency or inconsistency among your answers. e) Calculate the Variance Inflation Factor (VIF) of the regression coefficients and comment on the value obtained.
Question A regression analysis is performed on the annual food expenditure (AFE) with annual income (AI) and family size (FS) as explanatory variable. The following two models are used to establish the relationship between Annual Food Expenditure and Annual Income and Family Size. Model 1: AFE = B0 + B1 × AI Model 2: AFE = β0 + β1 × AI + β2 × FS Regression output for Model 1 is shown below:
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SUMMARY OUTPUT (Model 1)
Regression Statistics Multiple R R Square Adjusted R Square 0.77684 Standard Error 1.907748 Observations 20 ANOVA Df SS MS F Significance F Regression 244.3584 Residual Total 309.8695 Coefficients Standard Error t Stat P-value Lower 95% Upper 95% Intercept 1.043354 0.934366 1.116644 0.278831 -0.91968 3.006385 AI 0.149707 0.01827
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Model 2: Summary Output Regression Statistics Multiple R 0.894788 R Square Adjusted R Square Standard Error 1.906239 Observations 20 ANOVA Df Regression Residual Total 17 19 SS MS 124.0479 F Significance F
309.8695 Upper 95% 2.781999 0.183532
Intercept AI FS
Coefficients Standard Error t Stat P-value 0.396888 1.130483 0.351079 0.729843 0.140701 0.020301 6.930686 2.43E-06 0.391196 0.385735
Lower 95% -1.98822 0.097869
Table 1 shows the predicted value of AFE, distance measures, Mahalanobis Distance (Mah_D), Cook’s Distance (Cook_D), leverage values and DFBeta Values.
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Table 1:
S.No 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 AFE 5.2 5.1 5.6 7.2 11.3 8.1 7.8 5.8 5.1 18 4.9 11.8 5.2 4.8 7.9 6.4 15.2 13.7 5.1 2.9 AI 28 26 32 24 54 59 44 30 40 82 42 58 28 20 42 47 112 85 31 26 FS 3 3 2 1 4 2 3 2 1 3 3 4 1 5 3 1 4 5 2 2 Predicted Y 5.5101 5.2287 5.6817 4.1649 9.55951 9.48062 7.76131 5.4003 6.41611 13.10793 7.47991 10.12231 4.7277 5.16688 7.47991 7.40102 17.72015 14.31243 5.541 4.8375 Cook_D 0.00107 0.00021 0.00005 0.17839 0.0371 0.02388 0.00001 0.00132 0.03423 0.60735 0.03918 0.03436 0.00413 0.01847 0.00104 0.02333 0.90717 0.01769 0.00156 0.03617 Leverage 0.04832 0.05784 0.02292 0.10186 0.05656 0.05855 0.00479 0.02686 0.10414 0.13414 0.00705 0.05629 0.09698 0.40093 0.00705 0.1234 0.40778 0.22229 0.02477 0.03746 DFB0 -0.02595 -0.01164 -0.01169 0.7918 -0.11727 -0.09611 0.00138 0.05971 -0.27917 -0.39432 -0.1063 -0.13264 0.11672 0.02602 0.01731 -0.19511 0.98484 0.17584 -0.06449 -0.31501 DFB1 0.00078 0.00036 0.00008 -0.00299 -0.00051 -0.00339 -0.00002 -0.00048 -0.00152 0.02313 0.00186 0.00036 -0.00021 0.00338 -0.0003 -0.00215 -0.02936 -0.00195 0.00047 0.00329 DFB2 -0.0099 -0.00442 0.0014 -0.17658 0.08809 0.0641 0.00056 -0.00608 0.10024 -0.13265 -0.04264 0.07781 -0.02941 -0.07899 0.00694 0.08601 0.04395 -0.04792 0.00715 0.02183
a) What percentage of variations in Annual food expenditure (AFE) is explained by Model 1? b) In Model 1, calculate F statistic and comment what inference can be obtained using the F statistic value of model 1. c) In model 1, what is the 95% confidence interval for mean value of AFE when the annual income is 30? d) Comment whether model 2 (addition of new variable, family size) adds any value in predicting the annual food expenditure compared to model 1. e) In model 2, check whether the beta coefficient for FS is different from zero. f) Identify if there are any potential outliers in the model.
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g) Calculate the change in the predicted value of AFE for the sample point 1, when it is removed from the sample in estimating the parameter values. h) Identify the sample point that has highest influence on the regression model 2. What will be the change in the Beta co-efficient values if this sample point is removed from the sample?
Question The quarterly revenue of H&B, a catalog company, from 2005 till 2008 quarter 3 is shown in the following table (Table 1) along with trend to moving average values calculated for the multiplicative model Yt = Tt × St × Et (where Yt is the value at time t, Tt is the trend component in the value, St is the seasonal component and Et is the error component). The regression output for trend component is shown in table 2. Table 1: Quarterly revenue data and ratio to moving average S.No Year, Quarter Revenue (in Millions) Moving Average Ratio to Moving Average 1 2005,Q1 72 2 2005,Q2 68 3 2005,Q3 80 72.5 1.1034 4 2005,Q4 70 73.5 0.9523 5 2006,Q1 76 74 1.0270 6 2006,Q2 70 74.5 0.9395 7 2006,Q3 82 75.5 1.0860 8 2006,Q4 74 75 0.9866 9 2007,Q1 74 74 1 10 2007,Q2 66 74.5 0.8859 11 2007,Q3 84 76 1.1052 12 2007,Q4 80 76.5 1.0457 13 2008,Q1 76 78.5 0.9681 14 2008,Q2 74 78.5 0.9426 15 2008,Q3 84
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Table 2: Regression output for Trend SUMMARY OUTPUT Regression Statistics Multiple R 0.892401 R Square 0.79638 Adjusted R Square 0.776018 Standard Error 0.882523 Observations 12 ANOVA df SS MS Regression 1 30.46154 30.46154 Residual 10 7.788462 0.778846 Total 11 38.25 Standard Error t Stat 0.677061 105.3478 0.0738 6.253888
F 39.11111
Intercept t
Coefficients 71.32692 0.461538
P-value 1.46E-16 9.46E-05
a) Calculate the seasonality index for Quarters 1, 2, 3 and 4 b) Estimate the trend component for the revenue data shown in table 1 c) Forecast the revenue for 2008 quarter 4 and 2009 quarter 2.
Question NS International Instruments, located in Bangalore, is planning to fill 4 positions in the company. It is not necessary that all the four positions need to be filled. They have shortlisted 8 potential candidates. Their names and other details are given below: Details Ambika Bhama Chandana Divya Eileen Farah Geetha Hema Degree MCA MCA MBA MBA BE BE B Com B Com Local Yes Yes Yes Yes Non-Local Yes Yes Yes Yes Age (years) 25 27 26 29 24 23 21 22 Contribution 6 5 4 6.5 4.5 7 5.5 3.5 to the company
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The contribution to the company is in Rs. Lakhs per annum. If the company selects any non-locals, it will have to arrange for a special coaching in Kannada, at a cost of Rs. 50,000 (irrespective of the number of non-locals). The company wants to position itself as a young company, and hence the average age of the selected persons should be less than 25 years. a) Formulate this problem as a mixed integer problem to maximize the contribution to the company. Modify the above formulation to incorporate the following requirements (Treat each of the following questions independently). b) Of the persons selected, there must be at least one Post graduate and at least one graduate. c) There must be as many locals as non locals d) It turns out that Farah and Geetha are sisters and insists that either both should be offered or neither should be offered. e) If two MCAs are selected, one BE must be selected.
Question Mehak Perfumeries, located in Mysore, manufactures three types of perfumes: Anand, Bela and Chameli. The monthly demand for each perfume is 4000 fluid ounces. The market price for each fluid ounce of the perfume is: Rs. 600 for Anand, Rs. 750 for Bela and Rs. 1000 for Chameli. Production of Anand and Bela requires the natural ingredient, ING. This ingredient can be purchased at a price of Rs 230 per fluid ounce. A chemical process (Process 1) can convert one fluid ounce of ING into one fluid ounce of Anand. Similarly, another chemical process (Process 2) can convert one fluid ounce of ING into one fluid ounce of Bela. Anand can be further processed to produce Bela and Chameli using a chemical process (Process 3) that requires one fluid ounce of Anand to produce 0.6 fluid ounces of Bela and 0.4 fluid ounces of Chameli at a cost of Rs 160 per fluid ounce. Similarly another chemical process (Process 4) can further process Bela and convert one fluid ounce of Bela to 0.8 fluid ounces of Chameli at a cost of Rs 120 per fluid ounce. In order to decide what quantities of Anand, Bela and Chameli should be sold to maximize monthly profits a linear programming model is formulated and solved as follows:
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Decision Variables: Let ING be the fluid ounces of the natural ingredient purchased; INGA and INGB be the fluid ounces of the natural ingredient used to produce Anand and Bela; AS, BS and CS be the fluid ounces of Anand, Bela and Chameli sold; AP and BP be the fluid ounces of Anand and Bela processed further. Model: Maximize 600AS + 750BS + 1000CS – 230ING – 160AP – 120BP s.t. ING – INGA – INGB = 0 INGA – AS – AP = 0 BS + BP – INGB – 0.6AP = 0 0.4AP + 0.8BP – CS = 0 AS <= 4000 BS <= 4000 CS <= 4000 AS, BS, CS, AP, BP, ING, INGA, INGB are non-negative.
SOLUTION OUTPUT Adjustable Cells Cell $B$12 $C$12 $D$12 $E$12 $F$12 $G$12 $H$12 $I$12 Constraints Cell $J$13 $J$14 $J$15 $J$16 Name Final Value -1.82E-12 4.55E-13 3.64E-12 -4.55E-13 Shadow Price Constraint Allowable Allowable R.H. Side Increase Decrease 0 1E+30 13000 -230 0 1E+30 4000 230 0 9000 1E+30 -437.5 0 1E+30 4000 Name AS BS CS AP BP ING INGA INGB Final Value 4000 4000 4000 0 5000 13000 4000 9000 Reduced Objective Allowable Allowable Cost Coefficient Increase Decrease 370 600 1E+30 370 520 750 1E+30 520 562.5 1000 1E+30 562.5 -353 -160 353 1E+30 0 -120 1E+30 450 0 -230 320.91 370 0 0 353.00 370 0 0 3529.99 450
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Use the solution output above to answer the following questions. State your answers precisely and give adequate explanations. a) What is the optimal production plan for Mehak perfumeries? State your answers precisely. b) If the price of ING goes up by Rs 120, what will be the impact on the optimal production plan? Will it change? What will be the impact on the optimal profit? Assume that all else remains the same. Support your answers with precise reasons. c) Determine the shadow price associated with the first constraint of the model.
d) Suppose that a new process can convert each ounce of Anand to 0.25 fluid ounces of Bela and 0.75 fluid ounces of Chameli at a cost of Rs 100. Will it be worthwhile using this process? Why or Why not? Assume that all else remains the same. e) Suppose that the demand for Anand goes up by 10%, for Bela goes up by 15%, while that for Chameli goes down by 20%. What will be the impact on the optimal profit? Assume that all else remains the same. What impact will this have on the optimal production plan? Will it change? f) Suppose that the availability of ING were limited to 15000 fluid ounces. How will this impact the optimal production plan and the optimal profit? What would the shadow price of this constraint be when added to the formulation? Give the range within which this shadow price will remain valid. g) If it was decided that Anand was to be processed further to produce Bela and Chameli will this have any impact on the profit? If so, what would the impact be?
Question Hassan Electric Winders (HEW) Inc., located in Hassan, manufactures voltage converters and stabilizers. Processed tungsten is a key ingredient required for this purpose. An important step in the manufacturing of converters is the melting of tungsten which can be done using several methods. The cost involved in using each method varies, and so does the quality of the tungsten processed.
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At HEW two methods are used, Method 1 and Method 2. If Method 1 is used to obtain processed tungsten, the cost on a per converter basis is Rs 250, while using Method 2 the cost turns out to be Rs 325 per converter. Using the two methods, the quality of tungsten obtained after melting and, therefore, the quality of the converters obtained is as given in Table 1 below. For instance, if Method 1 is used for melting tungsten, 25% of the converters produced are of Grade 2. Note that Grade 1 indicates poor (but acceptable) quality, while Grade 4 indicates excellent quality. HEW can improve the quality of the converters obtained by “refiring” the melted tungsten of a particular grade. The refiring process will cost Rs 120 on a per converter basis. The results of the refiring process are as given in Table 2. For instance, when defective grade melted tungsten is refired, 30% gets upgraded to Grade 1, 15% gets upgraded to Grade 2, 10% to Grade 3 and 20% to Grade 4. Assume no loss of material. HEW has sufficient furnace capacity to melt and/or refire tungsten for at most 15,000 converters per month. HEW’s monthly demands for converters of various grades are: 2000 for grade 4, 2000 for grade 3, 3000 for grade 2 and 4000 grade 1.
Table 1. Percentage Yield from the two methods used for melting
Grade of Transistor Produced Defective Grade 1 Grade 2 Grade 3 Grade 4 20 35 25 10 10
Method 1 (Percentage of converters)
Method 2 (Percentage of converters)
15 25 25 15 20
Table 2. Percentage Yield from the refiring process
Refired Grade of Converters Defective Grade 1 Grade 2 Grade 3 Grade 4 a)
Defective 25 30 15 10 20
Grade 1 0 20 20 30 30
Grade 2 0 0 35 35 30
Grade 3 0 0 0 40 60
Develop an appropriate linear programming model for HEW that will help them determine how to meet the monthly demand for the various grades of converters at a minimum cost. Define the decision variables needed to develop your model precisely. State the purpose and meaning of each constraint and label the constraints appropriately.
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b)
Now, suppose that the melting and refiring process also results in a loss of tungsten. Specifically, if Method 1 is used, 6% of the tungsten is lost in the melting process. For Method 2, the rate of loss is 5%, while the loss rate in the refiring process is 2%. Assume that after accounting for the losses, the yield rates are the same as given in Table 1 and Table 2. Modify the model described in a) to account for the losses. Define clearly any additional variables, if used. Describe clearly the appropriate modifications made in the objective function and the constraints in the model.
Question Quitmeyer electronics incorporated (QEI) manufactures the following five microcomputer peripheral devices: internal modems, external modems, CD drives, hard disk drives and memory expansion boards. Each of these products requires time in hours on two types of electronic testing devices as shown on the following table (Table 1). Table 1. Device Internal External CD Drives Hard disk Memory Modem Modem drives expansion boards Test device 7 3 6 18 17 1 Test device 2 5 2 15 17 2 Test device 1 is available for 120 hours per week whereas test device 2 is available for 80 hours per week. Table 2 summarizes the revenues and material costs for each product. Table 2. Device Internal External CD Drives Hard disk Memory Modem Modem drives expansion boards Revenue per 200 120 130 430 260 unit sold Material and 50 20 80 180 60 testing cost per unit a) Formulate a linear programming model to maximize QEI’s total weekly profit. b) Use the graphical method to find the maximum profit that Quitmeyer can make. Show all work on the graph paper and also state the maximum profit below.
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c) Calculate the value of shadow price for the resources, test device 1 and 2.
d) If one unit of CD driver is manufactured, what will be the change in the objective function value of the profit maximization problem? Write all the steps involved in calculation of the new profit.
e) What should be the minimum revenue for CD driver so that it is advisable to make CD drivers? f) Find the optimal product mix
Question Doosra Life Insurance (DLI) is an insurance company based in Chennai. The absenteeism among employees is a major problem at DLI and the company is interested in establishing the relationship between number of absent days and duration of the employment at the company and a measure of attitude toward the company (part of recently concluded clinical interview with the company psychologist). The attitude scale ranged from 1 (extremely favorable) to 13 (extremely unfavorable). The data corresponding to 12 clerks are shown in Table 1. Table 1. Absent data for 12 clerks
C lerk 1 2 3 4 5 6 7 8 9 1 0 1 1 1 2 S um Avera e g S ev td.D D ysAbsent, Y a Attitude, X 1 Y rswithc pa X ea om ny, 2 1 1 1 0 2 1 1 2 2 4 3 2 3 5 4 2 5 6 5 6 5 6 7 4 9 1 0 8 1 3 1 1 7 1 5 1 1 9 1 6 1 2 1 0 7 5 7 5 5 9 6 5 .2 6 5 .2 4 1666 .9 6 6 6 7 5 7948 .6 4 0 8 5 3 3 1 6 1 .9 4 1 5 2 3 1625 .1 7 4 8 5
C enteredX 1 C enteredX 2 -5 5 .2 -3 2 .9 -4 5 .2 -3 2 .9 -4 5 .2 -2 2 .9 -3 5 .2 -2 2 .9 -1 5 .2 -0 2 .9 -1 5 .2 1 8 .0 -0 5 .2 0 8 .0 0 5 .7 -0 2 .9 3 5 .7 3 8 .0 4 5 .7 2 8 .0 4 5 .7 4 8 .0 5 5 .7 5 8 .0 0 0 0 0 3 3161 .9 4 1 5 2 3 1625 .1 7 4 8 5
Correlation matrix
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Y Y X1 X2 CX1 CX2 1 0.94978 0.890216 0.94978 0.890216
X1 0.94978 1 0.950585 1 0.950585
X2 CX1 CX2 0.890216 0.94978 0.890216 0.950585 1 0.950585 1 0.950585 1 0.950585 1 0.950585 1 0.950585 1
CX1: Centered attitude variable; CX2: Centered years with company The values of various sums of squares are given below:
∑ ( x1 − x1)( y − y) = 739.53; ∑ ( x2 − x2)( y − y) = 573.53; ∑ (x1 - x1) 2 = 665.30 ∑ (x2 - x2) 2 = 414.82
The SPSS output corresponding to the above data is shown below:
Model Summary(b) Adjusted R Std. Error of the R R Square Square Estimate .951(a) .904 a Predictors: (Constant), Years with company, Attitude b Dependent Variable: Days Absent ANOVA(b) Model 1 Sum of Squares 320.149
-
−
−
−
−
-
Model 1
Df
Mean Square
F
Regression Residual Total
Sig. .000(a)
354.250 a Predictors: (Constant), Years with company, Attitude b Dependent Variable: Days Absent Coefficients(a) Model 1 (Constant) Attitude Years with company a Dependent Variable: Days Absent Unstandardized Coefficients B -2.263 1.550 -.239 Std. Error 1.096 .481 .606 t Sig. Collinearity Statistics Tolerance VIF 10.375 10.375
a) Comment on the overall fitness of the regression model. Identify the explanatory variables that are significant. Use α = 0.05.
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b) Mr Jim Bob, the statistician at DLI after noticing that the variance inflation factor (VIF) is 10.375, removes all the data corresponding to the explanatory variable, “years with company” and runs a regression with only attitude as the explanatory variable. What will be new coefficient value for the attitude? What is the impact of this removal on standard error of estimate of the model? c) If the decision is to remove an explanatory variable from the analysis, which explanatory variable should be removed? Justify your answer d) Jason, the senior statistician at DLI advices Jim Bob to center the data (deviation from mean) on, “number of years” about its mean and run the regression between number of absent days and attitude and centered number of years. What will be the impact of this strategy on the overall regression model e) What inference can be obtained from the following scatter plot between standardized predicted value and standardized residual of the regression model?
Regression Standardized Residual
S at r lo c t ep t
D p n e tV r b :D y A s n e e d n a ia le a s b e t
1 .5
1 .0
0 .5
0 .0
- .5 0
- .0 1
- .5 1 - .5 1 - .0 1 - .5 0 0 .0 0 .5 1 .0 1 .5
R ge s nSa d r iz dPe ic dV lu e r s io t n ad e r d te a e
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Question 3: Binary Integer Programming Consider the following binary integer programming (BIP) problem: Max 5x1 + 3x2 + 4x3 + 2x4 s.t. 4x1 + x2 + x3 + 2x4 <= 3 (1) x1 + 3x2 + x3 + 3x4 <= 5 (2) 2x1 + x2 - 2x3 + 4x4 >= 4 (3) x1, x2, x3, x4 binary a) State the LP relaxation (LP0) for the problem.
b) Solve the BIP using the branch-and-bound algorithm. Clearly • show all branches, branching variables and their values. You may branch on the first variable which has a fractional value • number the sub-problems created in the order that you solve them • give their optimal solutions/values and • state when a node is fathomed and why You may use the following information if required: - Optimal solution to LP0 is: (0.035, 0.55, 0.31, 1). - Optimal solution to LP0 with an additional constraint x1=0 is: (0, 0.57, 0.28, 1).
c) When solving integer programming problems preprocessing of the given formulation is usually carried out before using the b-a-b algorithm. Some of the reasons for doing this is to: i) identify variables that can be fixed at one of their possible values because the other value is precluded by the constraints. ii) Tightening the constraints to reduce the feasible region of the LP relaxation without eliminating any feasible solution to the BIP problem. At times this can result in substantial reduction in the effort and time required to solve the problem. Preprocess the given BIP and i) Identify two variables whose values can be fixed. Support your answers with adequate explanations. ii) Eliminate the variables that have been fixed in i) by substituting for their values in the given BIP. Call the resulting modified 2 variable problem BIP1. Solve the LP relaxation for BIP-1. Does it give an integer solution?
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If not, solve it using the b-a-b method. iii) Incorporating the tighter form of the constraint (1) for BIP-1 solve its LPrelaxation. Does it give an integer solution? Comment on the effect of preprocessing done in i)
Question: Bectel Telecom needs to install towers to cover an upcoming area in Bangalore that has been divided into 12 regions as follows:
5 2 1 11 8 6 10 4 3 7 9 12 11
a) A tower located in a given region can provide communication in that region as well as to any of the adjacent regions (regions with a common boundary). Bectel would like to use a minimum number of towers to cover the entire area. Give an appropriate formulation that will help Bectel in making the decision. Instead of proceeding as above Bectel decides to look at the population in each region which is as follows: Region 1 2 3 Population (100s) 50 70 55
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4 5 6 7 8 9 10 11 12
95 45 150 60 200 225 90 175 120
b) Assume that only one tower can be installed in the entire area. The region in which the tower is located as well as any adjacent regions can be served by the tower. Where the tower should be located to maximize the population served? Question Tiny Twinkler Toasters (3T) produces two different models of toasters. The number of toasters that it can sell depends on the price of the two models. The demand functions for these two models are given below: Model 1: Q1 = 500 – 0.1 P1 + 0.15 P2 Model 2: Q2 = 350 + 0.1 P1 – 0.2 P2 Where Qi is the quantity of Model i and Pi is the price of model i. The cost of production is given by the following cost functions: C1 = 150 + 0.8 Q1 if Q1 ≤ 100 C1 = 170 + 0.6 Q1 if Q1 ≥ 100 C2 = 200 + 0.9 Q2 Where C1 and C2 are the total cost of producing the respective models. The amount of working capital available for the decision period is 8000. The total cost of production should be met from this amount. The profit is defined as the difference between the total revenue and total cost. Formulate the above problem to solve the number of toasters for each model to be produced in order to maximize the profit. Question Nidhi would like to invest Rs. 100,000 and is considering investment options A, B, C, D and E. She has a 3 year horizon in mind and would like to maximize the cash on hand at the end of the third year. The cash flows associated with these investments now and over the next 3 years are as given below: Cash flow (Rs.) in Year 0 1 2 3 -1 +0.5 +1 0
A
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B C D E
0 -1 -1 0
-1 +1.2 0 0
+0.50 0 0 -1
+1 0 +1.9 +1.5
For example, a rupee invested in investment B requires no cash outflow right now (year 0), a cash outflow of Rs. 1 in year 1, a return of Rs. 0.50 in year 2 and a return of Rs. 1 in year 3. Any returns can be immediately reinvested. Nidhi would not like to invest more than Rs. 75,000 in a single investment. In addition to these investment options Nidhi can get a return of 8% per year if she leaves her money in the bank. Nidhi’s investment decision problem has been formulated as an LP. The formulation and the solution are as given below: Max B + 1.9D + 1.5E +1.08F2 Subject to A + C + D + F0 = 100,000 0.5A + 1.2C + 1.08F0 – B – F1 = 0 A + 0.5B + 1.08F1 – E – F2 = 0 A < = 75000 B < = 75000 C < = 75000 D < = 75000 E < = 75000 All vars >= 0 Adjustable Cells Final Name Value A 60000 B 30000 C 0 D 40000 E 75000 F0 0 F1 0 F2 0
Reduced Objective Cost Coefficient 0 0 0 1 -0.0280 0 0.0000 1.9 0.0000 1.5 -0.2152 0 -0.3504 0 -0.0400 1.0800
Allowable Increase 0.0500 0.0292 0.0280 0.4750 1E+30 0.2152 0.3504 0.0400
Allowable Decrease 0.0583 0.2844 1E+30 0.0500 0.38 1E+30 1E+30 1E+30
Constraints Cell $J$3 $J$4 Final Value 100000 0 Shadow Price 1.9 -1.56 Constraint R.H. Side 100000 0 Allowable Increase 35000 37500 Allowable Decrease 40000 56250
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$J$5 $J$6 $J$7 $J$8 $J$9 $J$10
0 60000 30000 0 40000 75000
-1.12
0 75000 75000 75000 75000 75000
18750 1E+30 1E+30 1E+30 1E+30 18750
43750 15000 45000 75000 35000 43750
a) Define the variables used in the given formulation and give the optimal investment policy for Nidhi. b) If Nidhi has Rs. 15000 more on hand at the beginning (year 0), how much will she have on hand at the end of year 3? c) If Nidhi could borrow Rs. 15000 at the beginning of year 1 for which she would be charged an annual interest rate of 15% should she borrow the money? d) If Nidhi had an additional Rs. 20000 at the beginning of year 1 and Rs. 15000 at the beginning of year 2, how much will she have at the end of year 3? e) If investment D yielded 1.80 at time 3 would Nidhi’s investment plan change? f) Nidhi is considering a new investment option G where investing 1 rupee in year 0 generates Rs. 1.10 in year 1, Rs. 0.20 in year 2 and Rs. 0.10 in year 3. Should Nidhi invest in this option? g) Write the dual of the given problem. Define all the variables clearly. h) Should Nidhi consider the option of investing more than Rs. 75000/- in any one of the investments option A,B,C,D,E? If so, in which one(s)? Up to how much? Exactly how will this help her?
Question: The year is 1250 B.C. and Egyptian Pharaoh Ramses II has commissioned a granite causeway (elevated road) to be built from the gates of the temple at Karnak down to the east bank of the Nile River. The causeway will commemorate the Pharaoh's triumphs at the battle of Kadesh, so he insists it be completed before the scheduled visit of several princes from that region. This means that Imhotep, the Chief architect of Public Works of Ramses II, has exactly six months to finish the project. The project is fairly simple: Imhotep needs to transport a total of 5,000 carefully carved blocks from the granite
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quarries of Lower Egypt, downstream to the current site at Karnak. The stonemasons at Karnak do not need all 5,000 blocks at once; rather, the monthly requirements of the granite blocks are shown in table 1: Table: Monthly requirements of granite blocks Month Blocks Required 1 500 2 500 3 1,000 4 1,000 5 1,750 6 250
The requirements of the stonemasons must be met. Imhotep has a choice of two granite quarries from which to purchase his blocks: the quarry at Deir El Medinah, which can produce a maximum of 600 blocks a month for the next six months, or the quarry near Fayum, which can produce a maximum of 1100 blocks a month during months 1, 2, 3 and 6. Unfortunately, the annual flooding of the Nile River, which will happen during months 4 and 5, makes the Fayum quarry inaccessible during that time. Hence production during those months is zero (at Fayum). In addition, while the stonemasons at Karnak do not mind having more blocks than they need in a given month, they cannot have more than 1,200 blocks "left over" at the end of a month, due to space considerations. Imhotep needs to decide how many blocks to have delivered to Karnak each month. He also needs to decide on which quarry to order those blocks from - he is free to order blocks from both quarries in the same month, if he wishes. His objective is to minimise the cost of the entire project. The labour costs involved in moving a single block of granite can vary, according to the month. In months 1 through 3, Imhotep must rely primarily on the Pharaoh's military to do the transporting: they will charge the modern day equivalent of Rs 15000 per block moved from either quarry to Karnak. In months 4, 5 and 6, Imhotep does not have to contract with the military; the flooding of the Nile will leave all local farmers idle, they will move as many blocks as he wants for the equivalent of Rs 7500 per block. In addition to the transportation costs, Imhotep must consider the actual purchase price of the blocks: the quarry at Deir El Medinah charges Rs 200,000 per block it produces, while the Fayum quarry charges Rs 225,000 per block it produces. Finally, it costs Rs 1000 per block "left over" at the end of the month in inventory carrying costs. a) Assuming that blocks produced during a particular month can be transported and used during the same month, formulate Imhotep's problem as a linear programming model. Clearly write all the decision variables.
b) If 2% of all the blocks purchased from Deir El Medinah are damaged during transportation and thus cannot be used for construction. Change the formulation in Q1 to include this new information
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Question : Mr Kunjithapatham has been recently appointed as the head of newly created transportation security department by the Indian government after the 26/11 terrorists attack on Mumbai. The department is responsible for providing security at 12 international airports in India. Even prior to 2008, airline passengers had become familiar with the two basic types of systems used to check each passenger at a security checkpoint. One is a portal that can detect concealed weapons as the passenger walks through. The other is a screening system that scans the passenger’s carry-on luggage. Various proposals have been made for advanced security technology that would improve these two systems. Mr Kunjithapatham’s task force now needs to make recommendations on which direction to go for the next generation of these systems. The task force has been told that the functional requirement for the new portal system is that it must be able to detect even one ounce of explosives and hazardous liquids as well as metallic weapons being concealed by a passenger. The technology needed to do this includes quadrupole resonance and magnetic sensors. There are various ways to design the portal with this technology that would satisfactorily meet the functional requirement. However, the designs would differ greatly in the frequency with which false alarms would occur as well as in the purchase cost and maintenance cost for the portal. The frequency of false alarms is a key consideration since it substantially affects the efficiency with which the passengers can be processed. Even more importantly, a high frequency of false alarms greatly decreases the alertness of the security personnel for detecting the relatively rare terrorists who are actually concealing destructive devices. The security portals have to be imported either from USA or Germany. The most basic version of the portal system that satisfactorily meets the functional requirement has an estimated purchase price of $100,000 and, on the average, would incur an annual maintenance cost of $15,000. The drawback of this version is that it would generate a false alarm for approximately 10 percent of the passengers. This false alarm rate can be reduced by using more expensive versions of the system. Each additional $1000 in the cost of the portal system would lower the false alarm rate 0.1 percent and also would increase the annual maintenance cost by $200. The most expensive version would cost $180,000, so it would have a false alarm rate of only 2 percent of the customers as well as an annual maintenance cost of $31,000. Regarding the new screening system for carry-on luggage, the functional requirement is that it must clearly reveal suspicious objects as small as the smallest Swiss army knife. The technology needed to do this combines X-ray imaging, a thermal neutron scanner, and computer tomography imaging (which compares the density and other physical properties of any suspicious objects with known high-risk materials). It is estimated that the most basic version that satisfactorily meets this functional requirement would cost $60,000 plus an annual maintenance cost of $9,000. As with the most basic portal system, the drawback of this version is that it doesn’t sufficiently discriminate between suspicious objects that actually are destructive devices and those that are harmless. Thus, this version would generate false alarms for approximately 16 percent of the customers. In addition to wasting time and delaying passengers, such a high false alarm rate would make it very difficult for
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the screening operator to pay sufficient attention when the far more unusual true alarms occur. However, more expensive versions of the screening system would be considerably more discriminating. In particular, each additional $1,000 in the cost of the system would enable a reduction of 0.1 percent in the false alarm rate, while also increasing the annual maintenance cost by $300. Thus, the most expensive version, costing $160,000, would decrease the false alarm rate to 6 percent and incur an annual maintenance cost of $45,000. The task force has been given two budgetary guidelines. The false alarms due to security portal and the screening systems are independent. First budgetary guideline: Plan on a total expenditure of $200 ,000 for both the portal system and the screening system for carry-on luggage at each security checkpoint. Second budgetary guideline: Plan on holding down the average total maintenance costs for the two systems at each security checkpoint to no more than $70,000. These budgetary guidelines prohibit using the most expensive versions of both the portal system and the screening system for carry-on baggage. Therefore, the task force needs to determine which financially feasible combination of versions for the two systems will maximize the effectiveness with which passengers can be screened. Doing this requires first obtaining input from the top management of the Transportation Security Administration regarding what is the measure of effectiveness should be and then what management’s goals and priorities are for achieving substantial effectiveness and meeting the budgetary guidelines. Fortunately, Mr Kunjithapatham already has had extensive discussions with home department to obtain its guidance on these matters. These discussions led to the adoption of a clear policy that was approved all the way up to the home minister. The policy establishes the following order of priorities. Priority1: The total false alarm rate for both systems should not exceed 0.1 per passenger. Priority 2: Meet the first budgetary guideline. Priority 2: Meet the second budgetary guideline. Now that it has obtained all the needed managerial input, the task force is ready to begin its analysis. (a) Formulate the above problem as a preemptive goal programming problem. Clearly state all the decision variables involved in formulating the Goal programming problem
(b) Solve the preemptive goal programming formulation using graphical method.
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Question : A consumer advocacy group would like to evaluate the efficiency of four airline companies with respect to several criteria that are important to consumers. The four airlines are Tom, Dick, Harry and Mary. The output measures are identified as average delay in minutes, number of luggage lost per 10,000 checked-in bags, waiting time for check-in (in minutes) and quality of service (measured through questionnaire on a 100 point scale where higher the score better the quality of service). The input measures are labour costs per day, number of flights per day and overhead costs per day. The following table gives the details of inputs and outputs for the airlines. Airline Inputs Labour cost per day (in lacs) Overhead cost per day (in lacs) Number of flights per day Outputs Average delay in minutes Lost baggage per 10,000 bags Waiting time for check-in (minutes) Quality of service Tom 800 50 320 15 10 10 85 1000 60 416 20 15 20 80 Dick 600 40 280 7 5 5 95 Harry 750 28 312 9 7 8 92 Mary
a) Formulate an input-oriented DEA model to check the efficiency of the airline Tom.
b)
If the efficiency score of Mary is 1, what conclusions can be derived about efficiency of other three airlines?
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Question A textbook publisher wants to determine the relationship between the number of book copies sold (Y), the number of pages in the book X1, the number of analytical examples X2, the number of end-chapter problems X3, the ratio of favourable to unfavouarble reviews X4 and the nature of the support package D1. The data corresponding to a sample with 10 books are shown in table 1.
Table 1: Data, standardized and studentized residual
Sample point 1 2 3 4 5 6 7 8 9 10 Mean Stdev Copies sold 7 11 14 17 29 39 52 6 3 2 18 16.76637 Examples X2 0 0 0 15 15 30 15 92 0 0 16.7 28.38251 Problems X3 0 0 0 100 100 900 500 900 10 0 251 374.1791 Favorable to unfavourable reviews X4 5.1 4.6 6 2 3.1 0.2 0.1 10.3 5 12 4.84 3.91101 Support Package, D1 0 0 0 0 1 1 1 0 0 0 0.3 0.483046 Standardized residual -0.33812 0.24162 1.02037 0.18403 -0.3777 -0.51353 0.89123 0.05372 -1.24189 0.08027 Studentized residual -0.3762 0.28327 1.26043 0.26654 -1.89154 -1.49315 1.67628 0.3558 -1.4828 0.1961
Pages X1 390 472 512 408 610 830 310 910 211 172 482.5 243.7518
The table 2 gives various distance measures and DFBeta values of the data in table 1.
Sample 1 2 3 4 5 6 7 Mahalanobis Distance 0.82976 1.55226 2.20176 3.80992 7.74117 7.03542 5.55594 Cooks Distance 0.00561 0.00501 0.13924 0.013 14.36016 2.76978 1.18843 Leverage Values 0.0922 0.17247 0.24464 0.42332 0.86013 0.78171 0.61733 DFB0 -0.45514 0.03699 -2.85942 2.16623 21.2656 11.62836 15.52249 DFB1 0.00069 0.0016 0.01126 0.00134 0.04056 0.02758 0.03353 DFB2 0.01041 0.01338 0.10902 0.03225 0.81685 0.59576 0.16079 DFB3 0.00002 0.00012 0.00045 0.00145 0.11267 0.04313 0.00618 DFB4 0.02448 0.02565 0.18246 -0.2254 1.12292 0.37244 0.64421 DFB5 0.63484 0.61483 1.95958 1.04778 59.8744 3.7449 6.30033
49 0.00237 0.00129 0.00194 1.49047 2.50934 0.92671
8 9 10
7.8948 1.78686 6.59212
0.90428 0.15596 0.03184
0.8772 0.19854
-1.12667 -7.55361 -0.62546
-0.0011 0.01047 0.00184
0.21159 0.03711 0.02833
0.19135 0.28319 0.32631
Table 3. ANOVA table Sum of Squares 2307.621
ANOVA(b)
Model 1
df
Mean Square
F
Sig.
Regression Residual Total
2530.000 a Predictors: (Constant), Support Package, D1, Examples X2, Favorable to unfavourable reviews X4, Pages X1, Problems X3 b Dependent Variable: Copies sold
(a) Calculate the co-efficient of determination of the model and comment whether the overall model can be accepted? (b) Calculate the leverage value for sample number 10. (c) Based on table 1 and 2, comment on the possibility of heteroskedasticity in the data (d) Identify all the outliers in the data and state the reason(s) why these sample points are outliers? (e) Calculate the DFFit value for sample point 1 Question Good Day Dairies (GDD) is entering the milk distribution business. To begin with they will purchase raw milk to resell as milk or cream. They can purchase milk from two regions: Anekhalli (A) and Banekhalli (B). Prices, butterfat content, and separation properties of milk differ between the two regions. GDD processes the raw milk to produce cream and milk to desired specifications for sale to the customers. There is a one time contract payment of Rs.50,000 to be made if GDD decides to buy any amount of raw milk from a region. Purchase Price: The purchase prices of raw milk from the first region is Rs. 7 per litre and from region 2 is Rs. 6 per litre.
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Milk from region A has 15% butterfat and can be separated at a cost of Rs. 0.50 per litre. When separated it yields two ‘milks’, one with 31% butterfat and another with 8% butterfat. The volume of milk is conserved in the separation processing. Milk from region B has 10% butterfat and can be separated at a cost of Rs. 0.70 per litre. When separated it yields two types of ‘milks’, one with 33% butterfat and another with 5% butterfat. The volume of milk is conserved in the separation process. Production Process: After the milk is purchased and collected at the plant, it is either mixed directly or separated and then mixed. Mixing is done at no cost, and its purpose is to produce cream and milk to specifications. Demand and Selling Price: The demand and selling price are as follows: Minimum required per cent of butterfat Cream Milk 30 6 Maximum demand in litres 2500 20000 Selling Price in Rs / litre 28 11
a) Draw a flow diagram for the decision process that will aid you in formulating the problem of GDD maximizing its profit as a linear/integer programming problem. b) Formulate a mathematical programming problem that when solved will help GDD maximize its profits. Assume that the disposal of any extra milk or cream does not cost anything. Define all variables clearly and explain each of the constraints. Variables without proper definitions and constraints without proper labels and explanations will not be considered. Note: Use variables starting with ’X’ and ‘Y’ to define all decisions pertaining to Region A and Region B respectively. You are free to use any other notation for all additional variables used.
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c) Consider the situation where the purchase prices of raw milk from the 2 regions are lower if larger quantities of milk are purchased and are as indicated in the following diagrams: Region A Cost (Rs.) 5 4 7 5 Cost (Rs.) Region B
5000 Vol (Litres)
7000 Vol (Litre)
For example, the purchase of 7000 litres of raw milk from region A would cost Rs. (7*5000 + 5*2000). There is no upper bound on the amount of milk availability. Define any new variables required, if any, to formulate the decision problem under this cost structure. Give the appropriate formulation. Question The Brilliant Investments Company is considering four investments. Investment 1 will yield a net present value (NPV) of $16,000; investment 2, an NPV of $22,000; investment 3, an NPV of $12,000; and investment 4, an NPV of $8,000. Each investment requires a certain cash outflow at the present time: investment1, $5000; investment 2, $7000; investment 3, $4000; and investment 4, $3000. At present, $14,000 are available for investment. Let xj ( j = 1, 2, 3, 4) = 1 if investment j is made 0 otherwise
Then the following Binary Integer Program (BIP) can help decide which investments should be chosen: Max z = 16x1 + 22 x2 + 12 x3 + 8 x4 s.t. 5x1 + 7x2 + 4 x3 + 3x4 ≤ 14 xj = binary ( j = 1, 2, 3, 4 )
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A linear programming problem with a single constraint is referred to as a Knapsack Problem. The above BIP is a knapsack problem in which each variable must equal 0 or 1. When knapsack problems are solved by branch-and-bound, two aspects of the method greatly simplify. Since each variable must equal 0 or 1, branching on xj will yield xj = 0 and xj = 1 branches. Also, the LP relaxation for this and other subproblems can be solved in the following way. Observe that cj / aj (where cj and aj are the coefficients of xj in the objective function and constraint, respectively) may be interpreted as the benefit item j earns for each unit of the resource used by item j. Thus, the best items have the largest values of cj / aj and the worst items have the smallest values of c j / aj . To solve any sub-problem encountered while solving a binary knapsack problem, compute all the ratios cj/ aj . Select the best item first and fill the knapsack with as much of this item as possible. Then select and place as much as possible of the second-best item in the knapsack. Continue in this fashion until the knapsack is full. For example, to solve the LP relaxation of the knapsack problem above, we will first select investment 1 since it has the highest ratio of 3.2. We can select all of investment 1 (x1=1), and we will still have $9000 left. We will next select investment 2 since it has the highest ratio amongst the ones remaining (ratio of 22/7). We can afford to select all of investment 2, and as a result $2000 will be left. The next best selection is investment 3. However, here since investment 3 requires $4000, we can take in only half of investment 3. Hence the optimal LP solution is: x1 = x2 = 1, x3 = ½. In the process of solving the Brilliant Investments Company’s decision problem, using the Branch-and Bound method, the following steps were taken:
LEVEL 0
LP0 z = 44 x1 = x2 = 1 x3 =1/2 x4 = 0 x3 = 0 LP1 z = 43 1/3 x1 = x2 = 1 x3 = 0 x4 = 2/3 x3 = 1 LP2 z = 43 5/7 x1 = x3 = 1 x2 = 5/7 x4 = 0
LEVEL 1
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Answer the following questions: c) The value of the upper bound on the optimal solution to the BIP at level 0 is _______________ and the value of the lower bound is ___________________. Explain why.
d) The value of the upper bound on the optimal solution to the BIP at level 1 with x3 = 0 is _______________ and the value of the lower bound is ___________________. Explain why.
c) Complete the branch-and-bound (b-a-b) tree using the above information in order to find the optimal solution for the problem. Proceed down the node that has the most promising value. Clearly • show all branches, branching variables and their values • number the sub-problems created in the order that you solve them • give their optimal solutions/values and • state when a node is fathomed and why.
Question Jalahalli Machine Tools (JMT) is in the business of manufacturing and selling three products: Product 1, Product 2 and Product 3. The relevant data is provided in the table below: Product 1 Rs 1500 0.75 hrs 1.5 hrs 2 units Product 2 Rs 800 0.5 hrs 0.8 hrs 1 units Product 3 Rs 900 1.00 hrs 1.2 hrs 1.5 units
Selling Price Labour Required Machine Time Required Raw material required
Each week, up to 400 units of raw material can be purchased at a cost of Rs 150 per unit. The company employs four workers, who work 40 hours per week. (Their salaries are considered a fixed cost.) Workers are paid Rs 600 per hour to work overtime. Each week, 320 hours of machine time are available. In the absence of advertising, 50 units of product 1, 60 units of product 2 and 40 units of product 3 will be demanded each week. Advertising can be used to increase demand for product 1 and 2. It is found that for product 1, Rs 10 spent on advertising increases the demand by a unit. For product 2, Rs 6.67 spent on advertising increases the demand by one
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unit. At most, Rs 10,000 can be spent on advertising. In order to figure out what to produce and sell, JMT modeled and solved the following LP: Let, Pi = number of units of product i produced each week, i = 1, 2, 3. OT = number of hours of overtime labour used each week. RM = number of units of raw materials purchased each week. Ai = money spent of advertising for product i each week, i = 1, 2.
Max 1500P1 + 800P2 + 900P3 – 600OT – 150RM – A1 – A2 S.T. P1 – 0.1A1 <= 50 P2 – 0.15A2 <= 60 P3 <= 40 0.75P1 + 0.5P2 + P3 – OT <= 160 2P1 + P2 + 1.5P3 – RM <= 0 RM <= 400 A1 + A2 <= 10000 1.5P1 + 0.8P2 + 1.2P3 <= 320 All variables are nonnegative. The LINDO output to the above problem is given below: LP OPTIMUM FOUND AT STEP 5
OBJECTIVE FUNCTION VALUE 1) 242766.7 REDUCED COST 0.000000 0.000000 386.666656 213.333328 0.000000 0.000000 0.000000
VARIABLE VALUE P1 160.000000 P2 80.000000 P3 0.000000 OT 0.000000 RM 400.000000 A1 1100.000000 A2 133.333328
ROW SLACK OR SURPLUS DUAL PRICES 2) 0.000000 10.000000
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3) 4) 5) 6) 7) 8) 9)
0.000000 40.000000 0.000000 0.000000 0.000000 8766.666992 16.000000 5
6.666667 0.000000 386.666656 600.000000 450.000000 0.000000 0.000000
NO. ITERATIONS=
RANGES IN WHICH THE BASIS IS UNCHANGED: VARIABLE P1 P2 P3 OT RM A1 A2 OBJ COEFFICIENT RANGES CURRENT ALLOWABLE ALLOWABLE COEF INCREASE DECREASE 1500.000000 96.666664 53.333336 800.000000 26.666668 48.333332 900.000000 386.666656 INFINITY -600.000000 213.333344 INFINITY -150.000000 INFINITY 450.000000 -1.000000 1.000000 5.333333 -1.000000 1.000000 7.250000
ROW 2 3 4 5 6 7 8 9
RIGHTHAND SIDE RANGES CURRENT ALLOWABLE RHS INCREASE 50.000000 110.000000 60.000000 20.000000 40.000000 INFINITY 160.000000 27.500000 0.000000 6.666667 400.000000 6.666667 10000.000000 INFINITY 320.000000 INFINITY
ALLOWABLE DECREASE 876.666687 1315.000122 40.000000 2.500000 55.000000 55.000000 8766.666992 16.000000
Using the information provided above, answer the following questions: a. This week, the supplier from whom the raw material is purchased informs JMT that in addition to the 400 units that are available at the current price, an additional 5 units may be purchased at a price of Rs 500 per unit. Would it be worthwhile for JMT to purchase the additional 5 units? Why or Why not?
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b. The workers union (comprising of the four employees) would like to earn some additional wages by working overtime. It seems that the current rate of Rs 600 per hour for overtime is too high to interest the management. The union would then like to know the highest wage rate that they can negotiate, below which the management would be interested using overtime hours? What is that wage rate? Explain your answer. c. Supposing that Ravi Nachappa approached JMT and expressed interest in working for them. He is as skilled as the four workers currently working for JMT. Ravi makes two offers. He is either willing to work full-time (40 hours a week) for a wage of Rs 300 per hour, or part-time (for 20 hours a week) for a wage of Rs 320 per hour. Of course, JMT can choose one of these two offers, or choose not to hire Ravi in the first place. What option must JMT choose? Why? State your reasons precisely.
d. The LP solution suggests that currently it is not worthwhile making any of Product 3. One of the workers feels that it is because the production process for making this product is not efficient. He feels that by utilizing a smaller quantity of labour hours and also a lesser amount of machine time, the same product can be made with as good a quality. If JMT were to focus on the labour utilization of Product 3, what must its utilization be reduced to before it becomes profitable to make any of Product 3? Note that currently each unit of product 3 utilizes 1 hour of labour. Show your work clearly. Question Nilgiri’s, a popular grocery chain would like to develop a model that helps to predict the stores’ monthly revenues depending on the area it is located and the fact that it may or may not be located in a shopping centre. A stores’ location is identified by the median annual income in that area. That is because it is believed to have a bearing on weekly revenues. Below is a display of data obtained from a random sample of 15 locations where Nilgiri stores are currently located. Store Number 1 2 3 4 5 6 Weekly Revenues (in lakhs of Rs) 12.00 9.90 11.50 8.20 12.20 11.60 Median Value of Annual Income (lakhs) 2.25 1.70 1.53 1.32 2.37 1.87 Store Location (Shopping Center?) Yes No Yes No Yes Yes
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7 8 9 10 11 12 13 14 15
12.50 11.50 11.50 9.70 11.70 8.60 11.50 9.40 11.20
2.45 1.25 2.15 1.70 2.23 1.47 1.97 1.67 2.10
Yes Yes Yes No No No Yes No No
Given the above data, the average weekly revenues are Rs. 10.87 lakhs and the standard deviation of the weekly revenues is 1.35. The average weekly revenues for stores located in a shopping centre are Rs. 11.79 lakhs. The average of the median value of annual income across all 15 areas is Rs. 1.87 lakhs and the standard deviation of the median annual income is 0.38 Let y = Weekly Revenues of the store in lakhs of rupees; x1 = Median value of annual incomes of the area in lakhs of rupees; x2 = 1 if the store is located in a shopping centre = 0 if the store is not located in a shopping centre Correlation matrix:
Y X1 X2 Y 1 0.703 0.757 X1 1.00 0.321 X2
1
The regression outputs for the models fitted are as given below: Model B: y i = β 0 + β 1 x1i + β 2 x 2i + ε i , for i = 1, 2, ..., 15.
SUMMARY OUTPUT Regression Statistics Multiple R 0.899 R Square 0.809 Adjusted R Square 0.777 Standard Error 0.636 Observations 15.000 ANOVA df Regression 2.000 Residual 12.000 Total 14.000 Coefficients
SS 20.517 4.857 25.373 Standard Error
MS 10.258 0.405 t Stat
F 25.347
Significance F 0.000
P-value
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Intercept x1 x2 Model C:
6.683 1.798 1.544
0.849 0.468 0.348
7.868 3.844 4.442
0.000 0.002 0.001 for i = 1, 2, ..., 15.
y i = β 0 + β1 x1i + β 2 x 2i + β 3 ( x1i x2i ) + ε i ,
SUMMARY OUTPUT Regression Statistics Multiple R R Square Adjusted R Square Standard Error Observations ANOVA Regression Residual Total Intercept x1 x2 x1*x2
0.988 0.977 0.971 0.230 15.000
SS 3.000 24.792 11.000 0.581 14.000 25.373 Coefficients Standard Error 2.986 0.513 3.921 0.290 7.397 0.663 -3.212 0.357
df
MS 8.264 0.053 t Stat 5.823 13.511 11.165 -8.997
F Significance F 156.463 0.000
P-value 0.000 0.000 0.000 0.000
If, to begin with, a model A was fitted to explain the variation in weekly revenues using the location of the stores (X2) as the only explanatory variable answer the following questions: (Parts b-e are worth 2 points each. All other parts are worth 4 points each). Show all work for each part and give adequate explanations. a) b) c) d) e) Specify the model A along with the values of the beta estimates. What is the R Square value for the Model A? What is the amount of the total variation in the response variable that will be explained by the explanatory variable (x2) used in Model A? What is the standard error for Model A? If using the given data, two separate regression models were developed to explain variations in weekly revenues using median value of income; one for stores located in shopping centres and the other for stores not located in shopping centres. What would be the expected difference in the y-intercept values for the two regression models?
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f)
Perform a Partial F test to decide if the addition of explanatory variables x1 and x1*x2 (Model C) add significant additional information when x2 is already present. Use the appropriate model to answer the following: Does the average increase in weekly revenues of stores for every Rs. I lakh increase in median income depend on the location of the stores? How much is the increase? Use the appropriate model to answer the following: Does Nilgiris always get higher/lower revenues for stores located in shopping centres versus those not located in shopping centres for all levels of median income? If yes, for which location and by how much? If not, how exactly does it differ with respect to levels of median income? Use model C to give a point estimate and a 95% prediction interval for the actual weekly revenues for a store that is located in a shopping centre in an area where the median value of annual income is Rs. 2 lakhs. Tinv(.05,11) = 2.201, Tinv(.05,12) = 2.179, Tinv(.05,13) = 2.160
g)
h)
i)
Question The following table (table 1) lists the total compensation (salary + bonuses) paid to CEO’s of 30 fortune 500 companies. Tables 1 and 2 give various distance measures and BFFIT and DFBeta values of the sample points. It is believed that there is a linear relationship between the total compensation and the following explanatory variables: 1. 2. 3. 4. 5. Tenure – Number of years as CEO ( = 0 if less than 6 months) Age – Age of the CEO Sales – Total sales revenue of the firm Profits – Profit of the firm Assets – Total Assets of the firm
Compensat ion Studentized Residual Cook's Distance
Table 1: Values of Response and explanatory variables and distance measures
Firm tenure age sales profits Assets Leverage Values
1 2 3 4 5 6 7 8 9 10 11 12
6600 68500 24000 29600 83700 33300 131350 6830 13450 16920 134000 13700
7 0 11 6 18 6 15 5 3 2 16 5
61 51 63 60 63 57 60 61 57 60 63 61
161315.0 144416.0 139208.0 100697.0 100469.0 81667.0 76431.0 57813.0 56154.0 53588.0 50777.0 47678.0
2956.0 22071.0 4430.0 6370.0 9296.0 6328.0 5807.0 5372.0 1120.0 6398.0 5165.0 1704.0
257389.0 237545.0 49271.0 92630.0 355935.0 86100.0 668641.0 59920.0 36672.0 59550.0 617679.0 42754.0
-3.62 -0.17 2.603 0.916 0.561 1.017 -0.16 -1.44 0.589 -0.64 0.759 0.149
2.60327 0.01799 0.59574 0.01602 0.01147 0.01229 0.00277 0.03988 0.00423 0.01245 0.04397 0.00035
0.51108 0.75329 0.31199 0.06937 0.14631 0.03316 0.34573 0.06981 0.03483 0.11915 0.2807 0.05153
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13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Mean S.D.
6800 9000 7000 67500 8000 47000 80000 8000 23000 6000 72000 6000 7500 14500 6600 7100 13500 30000
32715 36578.67
7 4 0 2 4 32 5 3 4 15 17 5 4 2 2 2 9 9
7.333 6.91
58 59 56 60 60 74 63 56 52 57 55 56 57 56 52 49 57 59
58.4 4.67
47061.0 41322.0 37154.0 35853.0 33674.0 33296.0 32379.0 31707.0 31565.9 31260.0 31131.0 30951.0 30678.0 30219.0 30147.0 29398.0 28777.0 28203.3
55499.64 37532.061
2945.0 1048.0 3780.0 1259.0 568.0 3765.6 3782.0 578.0 2965.3 703.0 3276.0 935.0 594.0 1614.0 970.0 -962.0 4023.0 410.8
3642.39 4217.05
33673.0 37675.0 30966.0 299804.0 14166.0 194398.0 365875.0 28570.0 55143.9 29350.0 317590.0 15666.0 23638.0 13465.0 26720.0 28728.0 45066.0 6700.1
137709.3 177455.4
-0.69 -0.19 -0.82 0.806 0.086 -1.29 0.104 -0.19 0.313 -0.82 -0.06 -0.29 -0.2 0.523 -0.38 0.035 -0.89 2.225
0.00399 0.00043 0.01162 0.02996 0.00013 0.53962 0.0007 0.00042 0.00218 0.02834 0.00023 0.00097 0.00044 0.00345 0.00322 0.00007 0.01381 0.06802
0.01472 0.03339 0.06062 0.18358 0.05863 0.62671 0.24643 0.03311 0.08456 0.16695 0.26245 0.03005 0.02911 0.03707 0.08244 0.21991 0.06052 0.04284
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Table 2: Mahalanobis distance and DFFIT and DFBeta value of the sample points Firm
1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30
Mahalanobis
DFFIT
-30155.9 -3013.54 11489.19 1027.563 1149.984 724.0253 -820.414 -1624.67 430.2569 -1103.62 2976.441 137.1235 -350.59 -136.272 -837.026 2042.017 86.14769 -15117.3 355.3213 -133.827 405.9233 -1908.32 -210.569 -199.043 -133.267 394.5543 -488.99 106.4118 -911.846 1823.341
DFB0
1771.836 -1676.74 -8486.12 -2767.38 -67.6465 1757.23 -843.842 9748.579 293.9678 5103.653 -1043.12 -853.204 -433.195 535.6712 1283.115 -5054.45 -412.874 21594.88 -1164.24 -306.256 2777.848 -7568.23 -875.639 -849.33 -104.536 228.6693 -3082.53 543.1996 -2604.81 1221.323
DFB1
90.02119 3.38324 142.1058 -17.8337 49.60728 17.36036 -2.72151 90.61141 -17.1551 66.84446 -13.0592 -8.96651 -11.3315 9.18545 61.07498 -127.692 -5.35156 -338.52 -18.4274 4.392 13.97231 -140.255 -10.929 -3.21383 4.18633 -20.1834 -6.79856 2.41693 -47.7577 75.88662
DFB2
84.39044 33.23607 92.00443 47.01162 -7.89183 -31.2681 16.65544 -192.081 0.52404 -102.793 19.56062 17.3027 1.91654 -12.3811 -39.4801 108.3992 8.93141 -371.146 23.33959 2.48231 -45.8936 135.2919 15.87497 11.24521 -1.27187 5.43253 49.53844 -9.29264 36.88248 3.2469
DFB3
-0.2482 0.00124 0.10742 0.01125 0.00351 0.00617 -0.00082 0.00954 0.00392 0.00778 -0.00732 -0.00024 0.00101 0.00027 0.00788 -0.00637 -0.0004 0.03042 -0.00203 0.00044 -0.0015 -0.00127 0.00025 0.0008 0.00066 -0.0032 0.00035 0.00025 0.01001 -0.00799
DFB4
1.60971 -0.15021 -0.42897 0.01078 0.04511 0.04052 0.01785 -0.1465 -0.04885 -0.09905 -0.01313 -0.002 -0.01085 0.00653 -0.05617 -0.05553 -0.00181 -0.2895 0.00562 0.00858 0.00413 0.03923 0.0017 0.00839 0.00699 0.00164 0.02088 -0.0058 -0.08531 -0.05056
DFB5
0.00942 0.00044 0.00888 -0.0013 0.00014 0.00134 0.00146 0.00211 -0.0002 0.00073 0.00556 0.00013 0.00105 0.0001 0.00034 0.00382 0.00007 0.00855 0.00051 0.00004 0.00011 0.00219 0.00011 0.00026 0.00011 0.00029 0.00001 0.00002 0.00168 0.00394
14.821 21.845 9.0477 2.0118 4.2431 0.9615 10.026 2.0244 1.01 3.4552 8.1404 1.4944 0.4268 0.9682 1.7579 5.3237 1.7004 18.175 7.1466 0.9602 2.4523 4.8415 7.6109 0.8714 0.8443 1.0749 2.3906 6.3773 1.7551 1.2422
Table 3: Coefficients(a) Model 1 (Constant) tenure age Unstandardized Coefficients B 8090.913 269.073 16.825 Std. Error 31304.542 403.708 571.963 t .258 .667 .029 Sig. .798 .511 .977
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sales profits assets
-.202 2.080
.068 .624 .013
-2.972 3.332 13.946
.007 .003 .000
.184 a Dependent Variable: Compensation
a). Table 3 gives the multiple regression output of coefficients obtained using SPSS. Identify the explanatory variable that has the highest relative impact on the compensation paid to the CEOs.
b). Two MBA students, Tom and Jerry, draw a sample of 7 firms each from the original list of 30 firms and estimate the regression coefficients. The firms in Tom’s and Jerry’s sample are shown in Table 4. Who is likely to get a better estimate and why? Write all the steps clearly
Table 4: Tom and Jerry’s Sample Sample Point 1 2 Firms in Tom’s 1 2 sample Firms in Jerry’s 24 25 Sample 3 3 26 4 4 27 5 5 28 6 6 29 7 7 30
c). Which firm has the highest influence on the regression coefficients and why? d). If all the sample points in which the actual values lie outside 3-sigma level from the predicted values are removed from the data, what will be the new values of regression coefficients? e). Figure 1 shows the plot between un-standardized residual and age. What would be your inference from the plot? Suggest a necessary action if the plots indicates any problem
Figure 1. Residual plot
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30000 20000 Residuals 10000 20 40 60 80 0 -100000 -20000 -30000 ag e
f). Identify all the outliers in the sample, state why these sample points are outliers? g). If the standardized residual for sample 1 is -2.44, what inference you can make about the standard deviation of the residuals? Question Data collected on the monthly demand for Gold Jewellery in 2007 (measured in Tons) is shown in the following table. Table 1 Month January February March April May June July August September Demand 100 90 80 150 240 320 300 280 220
a). What is the 4-month Moving Average forecast for period 5? b). What is the forecast for period 3 using the exponential smoothing methods with a smoothing factor of 0.4? c). What conclusion can you draw with respect to the length of moving average versus smoothing effect for this time series data?
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d). Assume that the 12 month moving average centred on July is 250. What is the seasonal effect of the month July on demand? Explain. e). Trend line based on the data in table 5 is given by: Demandt = 40.44 + 27.50 × t. Calculate the demand for gold in July 2008. f.) Calculate the forecasting power of 4 periods moving average method for the data in table 1 Question A company manufactures tables and chairs. Each table requires 4 hours of assembly time, 2 hours of finishing time, 20 square feet of plywood, and 16 running feet of wood beam (section 2" X 3"). Each chair requires 2 hours of assembly time, 4 hours of finishing time, 5 square feet of plywood, and 8 running feet of wood beam. The profit per table is Rs 80 and per chair is Rs 60. Constraints that must be satisfied include the following: A. 600 hours of assembly time are available. B. 450 hours of finishing time are available. C. 1000 square feet of plywood are available. D. At least as many chairs as tables must be produced. E. The amount of wood beam available is uncertain, but estimated to be at least 10,000 feet. The LP formulation for the above problem such that profit can be maximized is given below: Max 80T + 60C s.t. 4T + 2C ≤ 600 2T + 4C ≤ 450 20T + 5C ≤ 1000 - T + C ≥ 0 16T + 8C ≤ 10000 T ≥ 0, C ≥ 0 Define the decision variables used above and label the constraints.
Show your work on the given graph paper to answer the following questions in the given space:
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1. 2.
Find the optimal solution, using the graphical technique. Mark the feasible region, optimal point, and the value of the objective function clearly. How much would you be willing to pay for: a) each extra hour of assembly time? b) each extra square foot of plywood? 3. If the amount of available plywood doubled, what would be the new solution? 4. The company wants to get out of the table-making business, and make only chairs. What should be the minimum profit per chair (assuming the same profit per table) to make this possible? 5. Management requires that tables and chairs should be sold in sets. Two alternatives are being considered: a) Set of one table and 4 chairs. b) Set of one table and 6 chairs. Which alternative do you recommend? Why? In each case show the feasible region. 6. Management has mandated that at least 50 tables should be produced. How will this effect the solution? Question A company operates for five seasons, and makes two products, Product 1 and Product 2. It manufactures the products in the first four seasons and sells them in seasons 2 through 5. Products made in a season can be sold from the next season onwards. One unit of product 1 requires 5 person hours in the preparatory shop and 3 person hours in the finishing shop. One unit of product 2 requires 6 person hours in the preparatory shop and 1 person hour in the finishing shop. During each season, the company has at most 12,000 hours in the preparatory shop and 3000 hours in the finishing shop. The company wants to formulate a linear program to decide its production and sales plans. It wants to maximize revenues. The expected selling price per unit of each product for each season is given below: Product 1 Product 2 Season 2 Rs.20 Rs.45 Season 3 Rs.25 Rs.40 Season 4 Rs.30 Rs.40 Season 5 Rs.15 Rs.30
(a) Define the variables needed to formulate the problem as an LP.
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(b) Formulate the problem as an LP. Label all constraints clearly. inventory carrying cost is zero for both products.
Assume that the
(c) If a unit is available for sale during a season but is not sold, the manufacturer has to pay Rs. 2 per unit as charges for carrying inventory to the next period. How will the formulation change? Define any new variables clearly, if used. Selling the products requires some marketing efforts, and 0.1 hours and 0.2 hours are needed to sell 10 units of products 1 and 2 respectively. This can be hired from outside at the following rates per hour: Season 2 3 4 5 Rate Rs.2 Rs.4 Rs.1 Rs.10. (d) How does the problem formulation change to ensure sufficient marketing effort? Define any new variables clearly, if used. Suppose the number of marketing hours that can be hired at the given rates is limited. Any hours over and above that have to be hired at a much higher rate per person hour as follows (Assume that an unlimited number of these hours are available): Season Maximum hours at low rates Higher rates 2 400 Rs.10 3 300 Rs.10 4 600 Rs.10 5 1000 Rs.10
(e) How would the formulation change to take this into account? Define any new variables clearly if used. Question Saloni Malhotra has recently been named the CEO of the Vittal Iron and Steel (VIS) Company. The company makes steel, heavy machinery and trucks for export. Saloni has decided that, in view of the government policies, the company’s performance (and hers) would be best served by maximizing the net dollar value of the exports by the company for the coming year. (The net dollar value of exports is defined as exports less the cost of all materials imported by the company). For the coming year it is assessed that the company can sell all it can produce of these items at the existing prices (in equivalent dollar amounts) of $900/unit for steel, $2500/unit for machinery and $3000/unit for trucks. In order to produce one unit of steel with the existing technology, it takes 0.05 units of machinery, 0.08 units of trucks, two units of ore imported for $100/unit, and other imported materials costing $100. In addition it takes 0.5 man-years of labour to produce each unit of steel. The steel mills have a maximum usable capacity of 300,00 units/year.
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To produce one unit of machinery it requires 0.75 units of steel, 0.12 units of trucks, and 5 man-years of labour. In addition, $150 of materials must be imported for each unit of machinery produced. The practical capacity of the machinery plants is 50,000 units/year. In order to produce one unit of trucks, it takes one unit of steel, 0.10 units of machinery, three man-years of labour, and $500 worth of imported materials. Existing production capacity for trucks is 550,000 units/year. The total manpower available for production of steel, machinery and trucks is 1,200,000 men/year. To plan for the coming year Saloni Malhotra has asked his OR consultant, Ms. Mahima Mukherjee, for inputs. Mahima has formulated a linear program to arrive at the best strategy to suggest. The model and the SOLVER solution are as given below: Max 900 SE + 2500 ME + 3000 TE - 300SP - 150MP - 500TP s.t. .75 MP + TP - SP + SE = 0 0.05SP + .10TP - MP + ME = 0 .08SP + .12MP - TP + TE = 0 .5SP + 5MP + 3TP <= 1200 SP <= 300 MP <= 50 TP <= 550 end
Microsoft Excel 11.0 Sensitivity Report Worksheet: [VIS Company.xls]Sheet1 Report Created: 12/6/2007 5:16:01 PM Adjustable Cells Final Reduced Objective Cell Name Value Cost Coefficient $B$2 SP 300 0 -300 $C$2 MP 50 0 -150 $D$2 TP 262.5 0 -500 $E$2 SE 0 -1350.00 900 $F$2 ME 8.75 0 2500 $G$2 TE 232.5 0 3000 Constraints Final Shadow Constraint Cell Name Value Price R.H. Side $H$3 0 2250 0 $H$4 0 2500 0 $H$5 0 3000 0 $H$6 1187.5 0 1200 $H$7 300 1585 300 $H$8 50 302.5 50 $H$9 262.5 0 550
Allowable Increase 1E+30 1E+30 403.33 1350 10566.67 347.70 Allowable Increase 4.17 9.4E+15 1E+30 1E+30 3.57 4.54 1E+30
Allowable Decrease 1585 302.5 1350 1E+30 281.40 1350 Allowable Decrease 232.5 8.75 232.5 12.5 252.72 8.14 287.5
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Use the output to answer the following questions precisely. Support each answer with adequate work and explanations. 1. Define the variables used in the formulation. What is the optimal production and export strategy for VIS Company? What is the net dollar value of exports under this plan? 2. The optimal solution suggests that 50,000 units of machinery should be produced. How are these units to be utilized during the year? 3. What would happen to the value of net exports if the price of steel increased to $1500/unit and the company chose to export one unit of it. 4. There is a chance that Saloni Malhotra may have $400,000 to spend on expanding capacity. If this investment can buy 250 units of steel capacity, 1000 units of machinery capacity, or 2000 units of truck capacity, what would the best investment be? Why? 5. If the price of the imported materials needed to produce one unit of trucks were to increase by $400, what would be the optimal export mix for the company? What would be the dollar value of their net exports? 6. The R&D group at the company has come to Saloni with a new product. Product X, that can be produced for export with 1.5 man-years of labour and 0.5 units of machinery for each unit produced. What must Product X sell for to make it attractive for production? [2 points] 7.If Saloni decides to stockpile (inventory) 25,000 units of steel during the coming year, what effect will this have on the related constraint equation(s), and what impact will it have on net dollar exports? Question Consider the following linear programming problem: Min 4x1 + 6x2 + 7x3 Subject to: + x5 + 5x6 + 2x6 >= 5 + x6 >= 3 + x6 >= 3 0 (1) (2) (3)
- x1 + 2x2 + x3 x1 - 2x3 + x5 x2 + 2x4 x1, x2, x3, x4, x5, x6 ≥
i) Write the dual linear programming problem for the above LP
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j) Use duality properties to determine if the following solutions are optimal for the dual problem. Clearly explain why or why not. Determine the primal optimal solution and the optimal objective function value. Show all work. 1. 2. (2.5,1,0) (2,1,0)
k) Introduce a new non-negative variable x7 in the primal problem with coefficients 1, 5 and 1 in constraints (1), (2) and (3) respectively. The objective function coefficient of x7 is 1. 1. Will this variable be included in the new primal optimal solution? Why or why not? 2. Write the dual LP problem corresponding to this extended primal LP with 7 variables. Question The RMC Corporation blends three raw materials in order to produce two products: a fuel additive and a solvent base. Each ton of fuel additive is a mixture of 2/5 ton of material 1 and 3/5 ton of material 3. A ton of solvent base is a mixture of 1/2 ton of material 1, 1/5 ton of material 2, and 3/10 of material 3. RMC’s production is constrained by a limited availability of the three raw materials. For the current production period RMC has the following quantities of each raw material: material 1, 20 tons; material 2, 5 tons; material 3, 21 tons. Management would like to achieve the following P1 priority level goals: Goal 1: Produce at least 30 tons of fuel additive. Goal 2: Produce at least 15 tons of solvent base. Assume there are no other goals in the problem. a. Is it possible for management to achieve both P1 level goals, given the constraints on the amounts of each material available? Explain . Treating the amounts of each material available as constraints, formulate a goal programming model to determine the optimal product mix. Assume both of the P1 priority level goals are equally important to management. Define all variables and label all constraints.
b.
c.
The feasible region defined by the material constraints is given by ABCDE where:
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A = (0,0), B = (0,25), C = (75/4,25), D = (25,20), E = (35,0) Which of these corner points will be optimal solution to the formulation in part b? Explain clearly why. What is the optimal objective function value? d. If goal 1 is thrice as important as goal 2, give the new formulation. What is the optimal product mix and the optimal objective function value?
Reconsider the RMC data presented above. Assume the two P1 priority level goals remain the same and both goals are equally important to management. Suppose management has learned that additional amounts of material 3 can be obtained from another RMC plant. Although management would like to obtain a solution that satisfies their production goals using the 21 tons of material 3 currently available, they are willing to consider using additional amounts of material 3 from the other plant. Introducing this new objective or goal as a P2 priority level goal, the problem goals can now be restated as follows: Priority Level 1 Goals Goal 1: produce at least 30 tons of fuel additive. Goal 2: produce at least 15 tons of solvent base. Priority Level 2 Goals Goal 3: Use at most 21 tons of material 3. e. Treating the amounts of materials 1 and 2 available as problem constraints, formulate a goal programming model for this problem.
Question Soft Solution (SS) is a software solution provider and has recently started using Six Sigma to improve their business processes. Currently the management is analyzing 20 different projects and would like identify the best project bases on their relative efficiency scores. The inputs and outputs for the 20 projects are given in the following table.
Project Number Expected Project Cost Expected Project Duration in days Number of Black and Percentage Increase in Customer Satisfaction Impact on Business Strategy Financial Impact Expected Increase in Sigma Quality Expected Percentage Increase in Productivity Inputs Outputs
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Green 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 212 199 214 280 263 203 196 215 281 233 263 198 220 284 214 235 200 217 198 227 70 63 88 77 72 70 61 79 71 66 84 60 80 79 87 80 63 75 63 70 Belts 10 3 5 8 11 11 3 5 8 6 10 3 4 6 7 5 4 11 3 5 11 29 28 29 19 21 31 22 17 10 27 32 15 23 19 27 33 13 31 10 4 8 4 6 2 5 9 5 3 6 8 8 7 4 7 2 8 5 8 6 331 342 333 303 240 306 345 264 239 338 310 341 308 325 314 236 339 313 343 317 0.24 0.77 0.33 0.48 0.41 0.52 0.78 0.27 0.66 0.3 0.88 0.51 0.31 0.31 0.54 0.34 0.83 0.74 0.87 0.46 20 23 10 10 11 17 21 6 19 16 19 22 7 7 5 17 23 7 22 17
(a) A DEA model for calculating the efficiency of project 1 has the following constraints in addition to few other constraints. yij is the output i for project j and uj is the weights allocated to project j. 1. ∑ y1 j u j ≥ 11 ,
j =1
20
20
2. ∑ y2 j u j ≥ 4 , 3.
j =1
20
∑ y3 j u j ≥ 331 4.
j =1
20
∑y
j =1
20
4j
u j ≥ 0.24
5.
∑y
j =1
5j
u j ≥ 20
Complete the DEA model by identifying the objective function and remaining constraints. (b) The following table gives the efficiency scores of projects and the corresponding benchmark projects (also known as peer projects which are on the efficiency frontier) along with the weights allocated by the DEA model in question (a). Calculate the relative efficiency scores of projects 4 and 5. Weights and the corresponding peer projects
Project Number
1 2
Relative Efficiency Score θ
1.000
Weight (uj)
0.9594 1
First Peer Project Number
7 2
Weight(uj)
2nd Peer Project Number
72
3 4 5
0.9652 0.8458 0.1933
7 12 7
0.0585 0.5081
17 12
Question The XYZ Company produces two different types of electronic components; say component A and component B, for a major customer. The customer notifies the XYZ Company each quarter as to what the monthly requirement for components will be during the next three months. The following order is received by XYZ Company for the next 3 month's period. April May June ----------------------------------------------Component A Component B 1000 3000 5000 1000 500 3000 -----------------------------------------------
The production control department must now develop a 3-month production plan for the components taking all applicable costs into account. The production costs of components A and B are as given below: Month April May June Production cost per unit Component A Component B -----------------------------------------------------200 150 220 160 230 165 -----------------------------------------------------
The inventory costs are calculated on the month end inventory level for each month. There should not be any short supply of components in any month. In any given month the XYZ company cannot supply to the customer more than what is required for that month. Any inventory left at the end of June will be a waste and does not involve any extra cost of disposal. The inventory carrying costs are 1.5% of the cost of the product kept in stock at the end of each month. If the production level of a component changes from one month to the next month, then it involves an additional cost. For a component, if the production level is more than the production level in the previous month, then a cost of Rs. 50 per unit increase is incurred. If the production level in a month is less than the production level in the previous month, then a cost of Rs. 20 per unit decrease is incurred. This is applicable for changes in the production levels for each component. For example, if the production of component A in May is 53.4 units higher than the production in April then a cost of 50*53.4 is incurred for increasing the production level in May. At the beginning of April, the production level for the two components can be set at any levels without incurring any cost.
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Moreover, there is a limitation on the storage capacity. The Company has 10,000 sq.ft. of storage capacity and extra storage capacity cannot be hired. Components A and B require 2 sq.ft. and 3 sq.ft. of storage capacities per unit respectively. The objective is to arrive at a production plan for the next 3 months that will minimize the total cost. For planning purposes the number of units to be produced of the two different types of electronic components can be fractional. a) Formulate this decision problem as a mathematical programming model. Clearly define all the variables that you introduce and label all constraints The variable cost involved when the production level of a component changes from one month to the next month increases with the degree of change. For a component, if the production level is more than the production level in the previous month, then a cost of Rs. 50 per unit increase is incurred as long as the increase is up to 200 units. Beyond 200 units the variable cost incurred is Rs. 60 per unit. If the production level in a month is less than the production level in the previous month, then a cost of Rs. 20 per unit decrease is incurred as long as the decrease is up to 200 units. Beyond 200 units the variable cost incurred is Rs. 25 per unit. This is applicable for changes in the production levels for each component. Modify the formulation in part a) to incorporate this information. Taking the base case as in part a) consider the following information: There is a fixed setup cost of Rs. 100 involved whenever there is a change in the production level. Modifiy your formulation in part a) to incorporate this information.
b)
c)
Question Consider the following Integer Linear Programming (ILP) problem: Max 17 X1 + 20 X2 Subject to: 4 X1 + 5 X2 <= 18 6 X1 + 3 X2 <= 22 X1, X2 are non-negative integers On solving the LP relaxation of the problem the following optimal solution is obtained: X1 = 3.1111, X2 = 1.1111 , Objective Function Value = 75.1111 a) Using this information can you say anything about the optimal solution and/or objective function value of the ILP?
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b)
Solve the ILP using the branch-and-bound (b-a-b) method branching on the variable X1 first. Draw the b-a-b tree and clearly indicate the branching variables, solution at each node and the reason for fathoming a node. c) Now solve the ILP again branching on X2 first. Draw the b-a-b tree and clearly indicate the branching variables, solution at each node and the reason for fathoming a node. d) Compare the number of LPs required to solve the ILP in b) and c). Comment on the difference in the two, if any.
Question Ramlakhan owns a term life policy that has a surrender value of Rs. 60,000 and death benefit of Rs. 250,000. Ram is considering investing in a different policy that has a higher death benefit of Rs. 500,000. In order to do this he would like to cash the current life policy and invest the Rs. 60,000 elsewhere such that the interest earned per year is sufficient to make the first 6 annual premium payments for the new policy. Of course, the interest income he earns will be taxable at a rate of 30% but he would have to pay taxes only on the interest amount in excess of what he uses towards making the premium payments for the new policy every year. At the end of each year the excess amount of interest remaining, if any, after paying taxes is invested along with the principal investment. The premium on the new policy for the next 6 years are: Year 1 2 3 4 5 6 Premium 4230 4570 4890 5160 5300 5580 Ramlakhan would like to determine the minimum annual rate of return that will be required on his investment to make this possible. This will help him decide on whether he can afford to make the switch. Give a mathematical formulation that will help Ramlakhan determine the rate of return. Do not attempt to solve the problem
