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~,.
ti: PROBLEMS
.' ~ ff
1.
Recall the RMC problem (Chapter 2, Problem 21). Letting F
= tons of fuel additive
= tons of solvent base
S
leads to the formulation
Max 40F + 30S s.t.
%F + %S:$ 20 SS :$ 5 %F + %oS :S 21 ~S~O
Material 1 Material 2 Material 3
,
~
Use the graphical sensitivity analysis approach to determine the range of optimality for the ~ '
objectivefunctioncoefficients.
2. For Problem 1 use the graphical sensitivity approach to determine what happens if an additional3 tons of material 3 tIec,?me available. What is the corresponding dual price for the
,I
..
,
1
oons~a 3. Considerthe followinglinear program: s.t.
XI + X2:S 10 2x1 + X2 ~ 4 XI + 3X2 :S 24 2x1 + X2:$ 16 XI' X2.~ 0 a. Solve this problem using the graphical solution procedure.
~
I
~
b. Compute therangeofoptimality fortheobjective functioncoefficient XI' of
c.
d. e.
'
Computethe range of optimalityfor the objectivefunctioncoefficientof X2'
Suppos~ the obj~ve function coefficient of XI is increased from 2 to 2.5. What is the
'1
.
Suppose the objective function coefficient of X2 is decreased from 3 to 1. What is the 1
new Optimalsolution?
~
newoptimal olution'? s
4. Refer to Problem3. Computethe dual prices for co~ts
:5. ,Consider the following linear program:
.
.
1 and 2 and interpretthem.
,~
!
'
Min s.t.
XI + 2x1 + xI + xI,X2 a. b. 'c. d. 2x2 ~ 7 x2 ~ 5 6x2 ~ 11 ~0
Solve this problem using the graphical solution procedure. Compute the range of optimality for the objective function ooefficient of XI. Compute the range of optimality for the objective function coefficient of X2' Suppose the objective function coefficient of xI is increased to 1.5. Find the new optimal solution.
e.
Supposethe objectivefunctioncoefficientof X2is decreasedto VS.Find the new optimal solution.
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Chapter 3 6. 7.
Linear Programming: Sensitivity Analysis and Interpretation of Solution
131
Refer to Problem 5. Compute and interpret the dual prices for the constraints. Consider the following linear program:
Max s.t.
2xI + X2 ~ 3
5X2 ~ 4
I

XI
+ +
1
2xI  3X2 :S 6
3Xl
;
2X2 :S 35 ifX2:S 10 =S
%Xl
+
Xl,X2 ~ 0
a. b. c. d. e. 8.
Solve this problem using the graphical solution procedure. Compute the range of optimality for the objective function coefficient of XI' Compute the range of optimality for the objective function coefficient of X2' Suppose the objective function coefficient of Xl is decreased to 2. What is the new optimal solution? Suppose the objective function coefficient of X2 is increased to 10. What is the new optimal solution?
Refer to Problem 7. Suppose that the objective function coefficient of X2 is reduced to 3. a. Resolve using the graphical solution procedure. b. Compute the dual prices for constraints 2 and 3. Refer again to Problem 3. a. Suppose the objective function coefficient of XI is increased to 3 and the objective function coefficient of X2 is increased to 4. Find the new optimal solution. b. Suppose the objective function coefficient of Xl is increased to 3 and the objective function coefficient of X2 is decreased to 2. Find the new optimal solution.
9.
10. Refer again to Problem 7. a. Suppose the objective function coefficient of Xl is decreased to 4 and the objective function coefficient of X2 is increased to 10. Find the new optimal solution. b. Suppose the objective function coefficient of XI is decreased to 4 and the objective function coefficient of X2 is increased to 8. Find the new optimal solution. 11. Recall the Kelson Sporting Equipment problem (Chapter 2, Problem 22). Letting
R = number of regular gloves C = number of catcher's mitts
leads to the following formulation:
Max s.t.
5R + 8C R + %C :S 900 Cutting and sewing ~R + Vac :S 300 Finishing YsR + %C :s 100 Packaging and shipping R,C~ 0
The computer solution obtained using The Management Scientist is shown in Figure 3.13.
132
INTRODUCTIONO MANAGEMENT T SCIENCE
FIGURE 3.13
THE MANAGEMENT SCIENTIST SOLUTION FOR THE KELSON SPORTING EQUIPMENT PROBLEM
Function Value
Objective
=
3700.00150
Variable R C Constraint 1 2 3
OBJEcrIVE COEFFICIENT
Value 500.00150 149.99925 Slack/Surplus 174.99963 0.00000 0.00000
RANGES
Reduced Costs 0.00000 0.00000 Dual Prices 0.00000 2.99999 28.00006
Variable R C
RIGlIT' HAND
Lower Limit 4.00000 3.33330
CUrrent Value 5.00000 8.00000
Upper Limit 12.00012 10.00000
SIDE RANGES Lower Limit 725.00037 133.33200 75.00000 Current Value 900.00000 300.00000 100.00000 Upper Limit No Upper Limit 400.00000 134.99982
Constraint 
1 2 3
a.
What
is the optimal
solution, and what
is the value of the total profit contribution?
b. Which
constraintsare binding?
c. d. 12.
What are the dual prices for the resources? Interpret each. If overtime can be scheduled in one of the departments, where would you recommend doing so?
Refer to the computer solution of the Kelson Sporting Equipment problem in Figure 3.13 (see Problem ll). a. Compute the ranges of optimality for the objective function coefficients. b. Interpret the ranges in part (a). c. Interpret the range of feasibility for the righthand sides. d. How much will the value of the optimal solution improve if 20 extra hours of packaging and shipping time are made available? Investment Advisors, Inc., is a brokerage firm that manages stock portfolios for a number of clients. A particular portfolio consists of U shares of V.S. Oil and H shares of Huber Steel. The annual return for V.S. Oil is $3 per share and the annual return for Huber Steel is $5 per share. V.S. oil sells for $25 per share and Huber Steel sells for $50 per share. The portfolio has $80,000 to be invested. The portfolio risk index (0.50 per share of V.S. Oil and 0.25 per share of Huber Steel) has a maximum of 700. In addition, the portfolio is lim
13.
Chapter 3
Linear Programming: Sensitivity Analysis and Interpretation of Solution
133
ited to a maximum of 1000 shares ofU.S. Oil. The linear programming formation that will maximum the total annual return of the portfolio is as follows:
Max s.t.
3U +
SH
Maximize total annual return
Funds available Risk maximum U.S. Oil maximum
25 U + SOH::5 80,000 O.SOU + 0.2SH ::5 700 1U ::5 1000 U,H~ 0
The computer solution of this problem is shown in Figure 3.14. a. What is the optimal solution, and what is the value of the total annual return? b. Which constraints are binding? What is your interpretation of these constraints in terms of the problem? c. What are the dual prices for the constraints? Interpret each. d. Would it be beneficial to increase the maximum amount invested in U.S. Oil? Why or why not?
FIGURE 3.14
THE MANAGEMENT SCIENTIST SOLUTION FOR THE INVESTMENT ADVISORS PROBLEM
Function Value
Objective
=
Value

8400.000
Reduced Costs 
Variable

U H
Constraint 
800.000 1200.000
Slack/Surplus 
0.000 0.000
Dual Prices 
1 2 3
OBJECTIVE COEFFICIENT RANGES
0.000 0.000 200.000
0.093 1.333 0.000
Variable

Lower Limit 
Current Value 
Upper Limit 
U H
RIGHT HAND SIDE RANGES Constraint 1 2 3
2.500 1. 500
3.000 5.000
10.000 6.000
Lower Limit 65000.000 400.000 800.000
Current Value 80000.000 700.000 1000.000
Upper Limit 140000.000 775.000 No Upper Limit
134
INTRODUCTION
TO MANAGEMENT
SCIENCE
14.
Refer to Figure 3.14, which shows the computer solution of Problem 13. a. How much would the estimated pershare return for D.S. Oil have to increase before it would be beneficial to increase the investment in this stock? b. How much would the estimated pershare return for Huber Steel have to decrease before it would be beneficial to reduce the investment in this stock? c. How much would the total annual return be reduced if the D.S. Oil maximum were reduced to 900 shares? Recall the Tom's, Inc., problem (Chapter 2, Problem 26). Letting W = jars of Western Foods Salsa M = jars of Mexico City Salsa leads to the formulation:
15.
Max s.t.
I W + 1.25M 5W + 3W+ 2W + . 7M ~ 4480 Whole tomatoes Tomato sauce IM~2080 2M ~ 1600 Tomato paste
W, M ===0
The Management Scientist solution is shown in Figure 3.15.
FIGURE 3.15 TIIE MANAGEMENTSCIENTISTSOLUTIONFORTIIE TOM'S, INC.,PROBLEM
Objective Function Value
=
Value

860.000
Reduced Costs 
Variable

W M
Constraint 
560.000 240.000
Slack/Surplus 
0.000 0.000
Dual Prices 
1 2 3
OBJECTIVE COEFFICIENT RANGES
0.000 160.000 0.000
0.125 0.000 0.188
Variable

Lower Limit 
Current Value 
Upper Limit 
W M
RIGHT HAND SIDE RANGES
0.893 1.000
1.000 1.250
1.250 1.400
Constraint 1 2 3
Lower Limit 4320.000 1920.000 1280.000
Current Value 4480.000 2080.000 1600.000
Upper Limit 5600.000 No Upper Limit 1640.000
Chapter 3 3. b. c. d. 16.
linear Programming: SensitivityAnalysisand Interpretationof Solution What is the optimal solution, and what are the optimal production quantities? Specify the range of optimality for the objective function coefficients. What are the dual prices for each constraint? Interpret each. Identify the range of feasibility for each of the righthandside values.
135
Recall the Innis Investments problem (Chapter 2, Problem 37). Letting
S = units purchased in the stock fund M = units purchased in the money market fund
leads to the following formulation: Min s.t.
8S +
3M
Funds available Annual income Units in money market
50S + 100M::5 1,200,000 5S + 4M 2= 60,000 M 2= 3000 S, M 2= 0 The computer solution is shown in Figure 3.16.
FIGURE 3.16
THE MANAGEMENT SCIENTIST SOLUTION FOR THE INNIS INVESTMENTS PROBLEM
Function Value
Objective
=
Value
62000.000 Reduced Costs
Variable
S M
Constraint 
4000.000 10000.000
Slack/Surplus 
0.000 0.000
Dual Prices 
1 2 3
OBJECTIVE COEFFICIENT RANGES
0.000 0.000 7000.000
0.057 2.167 0.000
Variable


Lower Limit

Current
Value
No Upper
Upper
Limit Limit
S M
3.750 No Lower Limit
8.000 3.000
6.400
RIGHT HAND SIDE RANGES
Constraint 1 2 3 Lower Limit 780000.000 48000.000 No Lower Limit Current Value 1200000.000 60000.000 3000.000 Upper Limit 1500000.000 102000.000 10000.000
136
INTRODUCTIONO MANAGEMENT T SCIENCE
a. b. c. d. e. f. 17.
What is the optimal solution, and what is the minimum total risk? Specify the range of optimality for the objective function coefficients. ~ow much annual income will be earned by the portfolio? What is the rate of return for the portfolio? What is the dual price for the funds available constraint? What is the marginal rate of return on extra funds added to the portfolio?
Refer to Problem 16 and the computer solution shown in Figure 3.16. a. Suppose the risk index for the stock fund (the value of Cs) increases from its current value of 8 to 12. How does the optimal solution change, if at all? b. Suppose the risk index for the money market fund (the value of CM)increases from its current value of 3 to 3.5. How does the optimal solution change, if at all? c. Suppose Cs increases to 12 and CM increases to 3.3. How does the optimal solution change, if at all? Quality Air Conditioning manufactures three home air conditioners: an economy model, a standard model, and a deluxe model. The profits per unit are $63, $95, and $135, respectively. The production requirements per unit are as follows:
18.
Number of Fans Economy Standard Deluxe
Number of Cooling Coils 1 2 4
Manufacturing Time (hours) 8 12 14
For the coming production period, the company has 200 fan motors, 320 cooling coils and 2400 hours of manufacturing time available. How many economy models (E), standard models (S), and deluxe models (D) should the company produce in order to maximize profit? The linear programming model for the problem is as follows: M~)\. (\'~E
* 9SS * BSO
Fan motors Cooling coils Manufacturing time
s.t. lE + IS + ID:S 200 lE + 2S + 4D:S 320 8E + l2S + 14D :S 2400 E,S,D;::: 0
The computer solution using The Management Scientist is shown in Figure 3.17. a. What is the optimal solution, and what is the value of the objective function? b. Which constraints are binding? c. Which constraint shows extra capacity? How much? d. If the profit for the deluxe model were increased to $150 per unit, would the optimal solution change? Use the information in Figure 3.17 to answer this question. 19. Refer to the computer solution of Problem 18 in Figure 3.17. a. Identify the range of optimality for each objective function coefficient. b. Suppose the profit for the economy model is increased by $6 per unit, the profit for the standard model is decreased by $2 per unit, and the profit for the deluxe model is increased by $4 per unit. What will the new optimal solution be? c. Identify the range of feasibility for the righthandside values. d. If the number of fan motors available for production is increased by 100, will the dual price for that constraint change? Explain.
Chapter3
Linear Programming:
Sensitivity Analysis and Interpretation
of Solution
137
FIGURE 3.17
THE MANAGEMENT SCIENTIST SOLUTION FOR THE QUALITY AIR CONDmONING PROBLEM
Function Value
Objective
=
16440.000
Variable E S D Constraint 1 2 3
OBJECTIVE
Value 80.000 120.000 0.000 Slack/Surplus 0.000 0.000 320.000
Reduced Costs 0.000 0.000 24.000 Dual Prices 31.000 32.000 0.000
COEFFICIENT'RANGES
Variable E S D
Lower Limit 47.500 87.000 No Lower Limit
Current Value 63.000 95.000 135.000
Upper Limit 75.000 126.000 159.000
RIGHT HAND SIDE RANGES Constraint Lower Limit Current Value Upper Limit 
1 2 3
160.000 200.000 2080.000
200.000 320.000 2400.000
280.000 400.000 No Upper Limit
20. Digital ontrols,Inc. C
(DCI),manufacturestwo models of a radar gun used by police to
monitor the speed of automobiles. Model A has an accuracy of plus or minus 1 mile per hour, whereas the smaller model B has an accuracy of plus or minus 3 miles per hour. For the next week, the company has orders for 100 units of model A and 150 units of model B. Although DCI purchases all the electronic components used in both models, the plastic cases for both models are manufactured at a DCI plant in Newark, New Jersey. Each model A case requires 4 minutes of injectionmolding time and 6 minutes of assembly time. Each model B case requires 3 minutes of injectionmolding time and 8 minutes of assembly time. For next week, the Newark plant has 600 minutes of injectionmolding time available and 1080 minutes of assembly time available. The manufacturing cost is $10 per case for model A and $6 per case for model B. Depending upon demand and the time available at the Newark plant, DCI occasionally purchases cases for one or both models from an outside supplier in order to fill customer orders that could not be filled otherwise. The purchase cost is $14 for each model A case and $9 for each model B case. Management wants to develop a minimumcost plan that will determine how many cases of each model should be produced at the Newark plant and how many cases of each model should
138
INTRODUOION TO MANAGEMENT SCIENCE
be purchased. The following decision variables were used to formulate a linear programming model for this problem:
AM = number of cases of model A manufactured BM = number of cases of model B manufactured AP = number of cases of model A purchased BP = number of cases of model B purchased
The linear programming model that can be used to solve this problem is as follows: Min s.t.
lOAM + 6BM + l4AP + 9BP + lAP + IBM + lBP 4AM + 3BM MM + 8BM lAM +
=
=
:5
:5
100 Demand for model A 150 Demand for model B
600 Injectionmolding time 1080 Assembly time
AM,BM,AP, BP ~ 0
The computer solution developed using The Management Scientist is shown in Figure 3.18. a. What is the optimal solution, and what is the optimal value of the objective function? b. Which constraints are binding? c. What are the dual prices? Interpret each. d. If you could change the righthand side of one constraint by one unit, which one would you choose? Why? 21. Refer to the computer solution of Problem 20 in Figure 3.18. a. Interpret the ranges of optimality for the objective function coefficients. b. Suppose that the manufacturing cost increases to $11.20 per case for model A. What is the new optimal solution? c. Suppose that the manufacturing cost increases to $11.20 per case for model A and the manufacturing cost for model B decreases to $5 per unit. Would the optimal solution change? Use the 100 percent rule and discuss. Tucker Inc. produces highquality suits and sport coats for men. Each suit requires 1.2 hours of cutting time and 0.7 hours of sewing time, uses 6 yards of material, and provides a profit contribution of $190. Each sport coat requires 0.8 hours of cutting time and 0.6 hours of sewing time, uses 4 yards of material, and provides a profit contribution of $150. For the coming week, 200 hours of cutting time, 180 hours of sewing time, and 1200 yards of fabric are available. Additional cutting and sewing time can be obtained by scheduling overtime for these operations. Each hour of overtime for the cutting operation increases the hourly cost by $15, and each hour of overtime for the sewing operation increases the hourly cost by $10. A maximum of 100 hours of overtime can be scheduled. Marketing requirements specify a minimum production of 100 suits and 75 sport coats. Let
22.
s = number of suits produced se = number of sport coats produced D 1 = hours of overtime for the cutting operation D2 = hours of overtime for the sewing operation
Chapter 3
linear Programming: SensitivityAnalysisand Interpretationof Solution
139
FIGURE.18 3
THE MANAGEMENT SCIENTIST SOLUTION FOR THE DIGITAL CONTROLS, INC., PROBLEM
Function Value
Objective
=
2170.000 Reduced Costs
Variable AM BM AP BP Constraint 1 2 3 4
OBJECTIVE COEFFICIENT
Value 100.000 60.000 0.000 90.000 Slack/Surplus 0.000 0.000 20.000 0.000
RANGES
0.000 0.000 1.750 0.000 Dual Prices 12.250 9.000 0.000 0.375
Variable AM BM AP BP
Lower Limit No Lower Limit 3.667 12.250 6.000
Current Value 10.000 6.000 14.000 9.000
Upper Limit 11.750 9.000 No Upper Limit 11.333
RIGHT HAND SIDE RANGES Constraint Lower Limit Current Value Upper Limit

1 2 3 4
0.000 60.000 580.000 600.000
100.000 150.000 600.000 1080.000
111.429 No Upper Limit No Upper Limit 1133.333
The computer solution developed using The Management Scientist is shown in Figure 3.19. 8. What is the optimal solution, and what is the total profit? What is the plan for the use of overtime? b. A price increase for suits is being considered that would result in a profit contribution of $210 per suit. If this price increase is undertaken, how will the optimal solution change? c. Discuss the need for additional material during the coming week. If a.rush order for material can be placed at the usual price plus an extra $8 per yard for handling, would you recommend the company consider placing a rush order for material? What is the maximum price Tucker would be willing to pay for an additional yard of material? How many additional yards of material should Tucker consider ordering? d. Suppose the minimum production requirement for suits is lowered to 75. Would this change help or hurt profit? Explain.
140
INTRODUaJON TO MANAGEMENT SCIENCE
FIGURE 3.19
THE MANAGEMENT SCIENTIST SOLUTION FOR THE TUCKER INC. PROBLEM
Ftmction Value
Objective
=
40900.000
Variable S SC D1 D2 Constraint 1 2 3 4 5 6
OBJECrIVE COEFFICIENT
Value 100.000 150.000 40.000 0.000 Slack/Surplus 0.000 20.000 0.000 60.000 0.000 75.000
RANGES
Reduced Costs 0.000 0.000 0.000 10.000 Dual Prices 15.000 0.000 34.500 0.000 35.000 0.000
S SC D1 D2
RIGHT
Variable
Lower Limit No Lower Limit 126.667 187.500 No Lower Limit
CUrrent Value 190.000 150.000 15.000 10.000
Upper Limit 225.000 No Upper Limit 0.000 0.000
HAND SIDE RANGES
Lower Limit CUrrent Value
Constraint 
200.000 180.000 1200.000 100.000 100.000 75.000
1 2 3 4 5 6
140.000 160.000 1000.000 40.000 0.000 No Lower Limit
240.000 No Upper Limit 1333.333 No Upper Limit 150.000 150.000
Upper
Limit
23.
Round Tree Manor is a hotel that has two types of rooms with three rental classes: Super Saver, Deluxe, and Business. The profit per night for each type of room and rental class is as follows:
Room
Type I TypeD
Rental Class Deluxe Super Saver $30 $35 $20 $30
Business
$40
Chapter3
linear Programming: Sensitivity Analysis and Interpretation of Solution
141
Type I rooms do not have Internet access and are not available for the Business rental class. Round Tree's management makes a forecast of the demand by rental class for each night in the future. A linear programming model developed to maximize profit is used to determine how many reservations to accept for each rental class. The demand forecast for a particular night is 130 rentals in the Super Saver class, 60 rentals in the Deluxe class, and 50 rentals in the Business class. Round Tree 100 has Type I rooms and 120 Type 11rooms. a. Use linear programming to determine how many reservations to accept in each rental class and how the reservations should be allocated to room types. Is the demand by any rental class not satisfied? Explain. b. How many reservations can be accommodated in each rental class? c. Management is considering offering a free breakfast to anyone upgrading from a Super Saver reservation to Deluxe class. If the cost of the breakfast to Round Tree is $5, should this incentive be offered? . d. With a little work, an unused office area could be converted to a rental room. If the conversion cost is the same for both types of rooms, would you recommend converting the office to a Type I or a Type 11room? Why? e. Could the linear programming model be modified to plan for the allocation of rental demand for the next night? What information would be needed and how would the model change? 24. Adirondack Savings Bank (ASB) has $1 million in new funds that must be allocated to home loans, personal loans, and automobile loans. The annual rates of return for the three types of loans are 7% for home loans, 12% for personal loans, and 9% for automobile loans. The bank's planning committee decided that at least 40% of the new funds must be allocated to home loans. In addition, the planning committee specified that the amount allocated to personal loans cannot exceed 60% of the amount allocated to automobile loans. a. Formulate a linear programming model that can be used to determine the amount of funds ASB should allocate to each type of loan in order to maximize the total annual return for the new funds. b. How much should be allocated to each type of loan? What is the total annual return? What is the annual percentage return? c. If the interest rate on home loans increased to 9%, would the amount allocated to each type of loan change? Explain. d. Suppose the total amount of new funds available was increased by $10,000. What effect would this change have on the total annual return? Explain. e. Assume that ASB has the original $1 million in new funds available and that the planning committee agreed to relax by 1% the requirement that at least 40% of the new funds must be allocated to home loans. How much would the annual return change? How much would the annual percentage return change? Better Products, Inc., manufactures three products on two machines. In a typical week, 40 hours are available on each machine. The profit contribution and production time in hours per unit follow:
25.
Category Profit/unit Machine 1time/unit Machine 2 time/unit
Product 1 $30 0.5 1.0
Product 2 $50 2.0 1.0
Product 3 $20 0.75 0.5
1\vo operators are required for machine 1; thus, 2 hours of labor must be scheduled for each hour of machine 1 time. Only one operator is required for machine 2. A maximum of
142
INTRODUCTIONO MANAGEMENT T SCIENCE
100 laborhours is available for assignment to the machines during the coming week. Other production requirements are that product 1 cannot account for more than 50% of the units produced and that product 3 must account for at least 20% of the units produced. a. How many units of each product should be produced to maximize the total profit contribution? What is the projected weekly profit associated with your solution? b. How many hours of production time will be scheduled on each machine? c. What is the value of an additional hour of labor? d. Assume that labor capacity can be increased to 120 hours. Would you be interested in using the additional 20 hours available for this resource? Develop the optimal product mix assuming the extra hours are made available. 26. Industrial Designs has been awarded a contract to design a label for a new wine produced by Lake View Winery. The company estimates that 150 hours will be required to complete the project. Three of the firm's graphics designers are available for assignment to this project: Lisa, a senior designer and team leader; David, a senior designer; and Sarah, a junior designer. Because Lisa has worked on several projects for Lake View Winery, management has specified that Lisa must be assigned at least 40% of the total number of hoUIS that are assigned to the two senior designers. To provide labeldesigning experience, Sarah. must be assigned at least 15% of the total project time. However, the number of hours assigned to ~arah must not exceed 25% of the total number of hours that are assigned to the two senior designers. Due to other project commitments, Lisa has a maximum of 50 hoUIS available to work on this project. Hourly wage rates are $30 for Lisa, $25 for David, aid $18 for Sarah. a. Formulate a linear program that can be used to determine the number of hours each graphic designer should be assigned to the project in order to minimize total cost. b. How many hours should each graphic designer be assigned to the project? What is the total cost? c. Suppose Lisa could be assigned more than 50 hours. What effect would this change have on the optimal solution? Explain. d. If Sarah were not required to work a minimum number of hours on this project, would the optimal solution change? Explain. Vollmer Manufacturing makes three components for sale to refrigeration companies. The components are processed on two machines: a shaper and a grinder. The times (in minutes) required on each machine are as follows:
27.
Machine Component 1 2 3 Shaper 6 4 4 Grinder 4 5 2
The shaper is available for 120 hours, and the grinder is available for 110 hours. No more than 200 units of component 3 can be sold, but up to 1000 units of each of the other components can be sold. The company already has orders for 600 units of component 1 that must be satisfied. The profit contributions for components 1, 2, and 3 are $8, $6, and $9, respectively. a. Formulate a linear programming model and solve for the recommended production quantities. b. What are the ranges of optimality for the profit contributions of the three components? Interpret these ranges for company management.
Chapter 3
linear Programming: Sensitivity Analysis and Interpretation of Solution
143
c. d. e. 28.
What are the ranges of feasibility for the righthand sides? Interpret these ranges for company management. If more time could be made available on the grinder, how much would it be worth? If more units of component 3 can be sold by reducing the sales price by $4, should the company reduce the price?
National Insurance Associates carries an investment portfolio of stocks, bonds, and other investment alternatives. Currently $200,000 of funds are available and must be considered for new investment opportunities. The four stock options National is considering and the relevant financial data are as follows:
Stock Price per share Annual rate of return Risk measure per dollar invested
A $100 0.12 0.10
B $50 0.08 0.07
C $80 0.06 0.05
D $40 0.10 0.08
The risk measure indicates the relative uncertainty associated with the stock in terms of its realizing the projected annual return; higher values indicate greater risk. The risk measures are provided by the firm's top financial advisor. National's top management has stipulated the following investment guidelines: tile annual rate of return for the portfolio must be at least 9% and no one stock can account for more than 50% of the total dollar investment. a. Use linear programming to develop an investment portfolio that minimizes risk. b. If the firm ignores risk and uses a maximum returnoninvestment strategy, what is the investment portfolio? c. What is the dollar difference between the portfolios in parts (a) and (b)? Why might the company prefer the solution developed in part (a)? 29. Georgia Cabinets manufactures kitchen cabinets that are sold to local dealers throughout the Southeast. The company has a large backlog of orders for oak and cherry cabinets and has decided to contract with three smaller cabinetmakers to do the final finishing operation. For the three cabinetmakers, the number of hours required to complete all the oak cabinets, the number of hours required to complete all the cherry cabinets, the number of hours available for the final finishing operation, and the cost per hour to perform the work are shown here.
Cabinetmaker Hours required to complete all the oak cabinets Hours required to complete all the cherry cabinets Hours available Cost per hour 50
1
Cabinetmaker 42
2
Cabinetmaker 30
3
"
60
48
35
40 $36
30 $42
35 $55
For example, Cabinetmaker 1 estimates it will take 50 hours to complete all the oak cabine!S and 60 hours to complete all the cherry cabinets. However, Cabinetmaker I only has
144
INTRODualON TO MANAGEMENT SCIENCE
40 hours available for the final finishing operation. Thus, Cabinetmaker I can only complete 40/50 = 0.80 or 80% of the oak cabinets if it worked only on oak cabinets. Similarly, Cabinetmaker 1can only complete 40/60 = 0.67 or 67% of the cherry cabinets if it worked only on cherry cabinets. a. Formulate a linear programming model that can be used to determine the percentage of the oak cabinets and the percentage of the cherry cabinets that should be assigned to each of the three cabinetmakers in order to minimize the total cost of completing both projects. b. Solve the model formulated in part (a). What percentage of the oak cabinets and what percentage of the cherry cabinets should be assigned to each cabinetmaker? What is the total cost of completing both projects? c. If Cabinetmaker 1 has additional hours available, would the optimal solution change? Explain. d. If Cabinetmaker 2 has additional hours available, would the optimal solution change? Explain. e. Suppose Cabinetmaker 2 reduced its cost to $38 per hour. What effect would this change have on the optimal solution? Explain. 30. Benson Eiectronics ma,nufactures three components used to produce cellular telephones and other communication devices. In a given production period, demand for the three components may exceed Benson's manufacturing capacity. In this case, the company meets demand by purchasing the components from another manufacturer at an increased cost per unit. Benson's manufacturing cost per unit and purchasing cost per unit for the three components are as follows:
Source Manufacture Purchase
Component 1 $4.50 $6.50
Component 2 $5.00 $8.80
Component 3 $2.75 $7.00
Manufacturing times in minutes per unit for Benson's three departments are as follows:
Department Production Assembly Testingand packaging
Component 1 2 1 1.5
Component 2 3 1.5 2
Component 3 4 3 5
For instance, each unit of component 1 that Benson manufactures requires 2 minutes of production time, 1 minute of assembly time, and 1.5 minutes of testing and packaging time. For the next production period, Benson has capacities of 360 hours in the production department, 250 hours in the assembly department, and 300 hours in the testing and packaging department. a. Formulate a linear programming model that can be used to determine how many units of each component to manufacture and how many units of each component to purchase. Assume that component demands that must be satisfied are 6000 units for component 1,4000 units for component 2, and 3500 units for component 3. The objective is to minimize the total manufacturing and purchasing costs. b. What is the optimal solution? How many units of each component should be manufactured and how many units of each component should be purchased?
Chapter
3
linear Programming: Sensitivity Analysis and Interpretation of Solution
145
c. d.
Which departments are limiting Benson's manufacturing quantities? Use the dual price to determine the value of an extra hour in each of these departments. Suppose that Benson had to obtain one additional unit of component 2. Discuss what the dual price for the component 2 constraint tells us about the cost to obtain the additional unit.
31.
Golf Shafts, Inc. (GSI), produces graphite shafts for several manufacturers of golf clubs. Two GSI manufacturing facilities, one located in San Diego and the other in Tampa, have the capability to produce shafts in varying degrees of stiffness, ranging from regular models used primarily by average golfers to extra stiff models used primarily by lowhandicap and professional golfers. GSI just received a contract for the production of 200,000 regular shafts and 75,000 stiff shafts. Both plants are currently producing shafts for previous orders, which means that neither plant has sufficient capacity by itself to fill the new order. The San Diego plant can produce up to a total of 120,000 shafts and the Tampa plant can produce up to a total of 180,000 shafts. Because of equipment differences at each of the plants and differing labor costs, the perunit production costs vary as shown here:
Regular shaft Stiff shaft
San Diego Cost $5.25 $5.45
Tampa Cost $4.95 $5.70
a. b. c.
d.
Formulate a linear programming model to determine how GSI should schedule production for the new order in order to minimize the total production cost. Solve the model that you developed in part (a). Suppose that some of the previous orders at the Tampa plant could be rescheduled in order to free up additional capacity for the new order. Would this option be worthwhile? Explain. Suppose that the cost to produce a stiff shaft in Tampa had been incorrectly computed, and that the correct cost is $5.30 per shaft. What effect, if any, would this change have on the optimal solution developed in part (b)? What effect would it have on total production cost?
32.
"
The Pfeiffer Company manages approximately $15 million for clients. For each client, Pfeiffer chooses a mix of three investment vehicles: a growth stock fund, an income fund, and a money market fund. Each client has different investment objectives and different tolerances for risk. To accommodate these differences, Pfeiffer places_limitson the percentage of each portfolio that may be invested in the three funds and assigns a portfolio risk index to each client. Here's how the system works for Dennis Hartmann, one of Pfeiffer's clients. Based on an evaluation of Hartmann's risk tolerance, Pfeiffer has assigned Hartmann's portfolio a risk index of 0.05. Furthermore, to maintain diversity, the fraction of Hartmann's portfolio invested in the growth and income funds must be at least 10% for each, and at least 20% must be in the money market fund. The risk ratings for the growth, income, and money market funds are 0.10, 0.05, and 0.0 I, respectively. A portfolio risk index is computed as a weighted average of the risk ratings for the three funds where the weights are the fraction of the portfolio invested in each of the funds. Hartmann has given Pfeiffer $300,000 to manage. Pfeiffer is currently forecasting a yield of 20% on the growth fund, 10% on the income fund, and 6% on the money market fund. a. Develop a linear programming model to determine the best mix of investments for Hartmann's portfolio. b. Solve the model you developed in part (a). c. How much may the yields on the three funds vary before it will be necessary for Pfeiffer to modify Hartmann's portfolio?
146
INTRODUCTIONO MANAGEMENT T SCIENCE
d. e. f. g.
If Hartmann were more risk tolerant. how much of a yield increase could he expect? For instance. what if his portfolio risk index is increased to 0.06? If Pfeiffer revised the yield estimate for the growth fund downward to 0.10. how would you recommend modifying Hartmann's portfolio? What information must Pfeiffer maintain on each client in order to use this system to manage client portfolios? On a weekly basis Pfeiffer revises the yield estimates for the three funds. Suppose Pfeiffer has 50 clients. Describe how you would envision Pfeiffer making weekly
modtflcattOt\ ttlead.\di..et\t ~mQti..Q m.d ~\.\.<:Y::.~t\.w& \Q\'l..\.fu~d." w.~"h1,~d 'hWI.~"i\~ *'~
the three investment funds. 33. La Jolla Beverage Products is considering producing a wine cooler that would be a blend of a white wine. a rose wine. and fruit juice. To meet taste specifications. the wine cooler must consist of at least 50% white wine. at least 20% and no more than 30% rose. and exactly 20% fruit juice. La Jolla purchases the wine from local wineries and the fruit juice from a processing plant in San Francisco. For the current production period. 10.000 gallons of white wine and 8000 gallons of rose wine can be purchased; there is no limit on the amount of fruit juice that can be ordered. The costs for the wine are $1.00 per gallon for the white and $1.50 per gallon for the rose; the fruit juice can be purchased for $0.50 per gallon. La Jolla Beverage Products can sell all of the wine cooler they can produce for $2.50 per gallon. a. Is the cost of the wine and fruitjuice a sunk cost or a relevant cost in this situation? Explain. b. Formulate a linear program to determine the blend of the. fuIe.e.ingre.die.nt'& will that maximize the total profit contribution. Solve the linear program to determine the number of gallons of each ingredient La Jolla should purchase and the total profit contribution they will realize from this blend. c. If La Jolla could obtain additional amounts of the white wine. should they do so? If so. how much should they be willing to pay for each additional gallon. and how many additional gallons would they want to purchase? d. If La Jolla Beverage Products could obtain additional amounts of the rose wine. should they do so? If so. how much should they be willing to pay for each additional gallon. and how many additional gallons would they want to purchase? e. Interpret the dual price for the constraint corresponding to the requirement that the wine cooler must contain at least 50% white wine. What is your advice to management given this dual price? f. Interpret the dual price for the constraint corresponding to the requirement that the wine cooler must contain exactly 20% fruit juice. What is your advice to management given this dual price? The program manager for Channel 10 would like to determine the best way to allocate the time for the 11:0011 :30 evening news broadcast. Specifically. she would like to determine the number of minutes of broadcast time to devote to local news. national news. weather. and sports. Over the 30minute broadcast, 10 minutes are set aside for advertising. The station's broadcast policy states that at least 15% of the time available should be devoted to local news coverage; the time devoted to local news or national news must be at least 50% of the total broadcast time; the time devoted to the weather segment must be less than or equal to the'time devoted to the sports segment; the time devoted to the sports segment should be no longer than the total time spent on the local and national news; and at least 20% of the time should be devoted to the weather segment. The production costs per minute are $300 for local news. $200 for national news. $100 for weather. and $100 for sports. a. Formulate and solve a linear program that can determine how the 20 available minutes should be used to minimize the total cost of producing the program. b. Interpret the dual price for the constraint corresponding to the available time. What advice would you give the station manager given this dual price?
34.
Chapter
3
linear Programming: Sensitivity Analysis and Interpretation of Solution
147
c.
d.
e.
Interpret the dual price for the constraint corresponding to the requirement that at least 15% of the available time should be devoted to local coverage. What advice would you give the station manager given this dual price? Interpret the dual price for the constraint corresponding to the requirement that the time devoted to the local and the national news must be at least 50% of the total broadcast time. What advice would you give the station manager given this dual price? Interpret the dual price for the constraint corresponding to the requirement that the time devoted to the weather segment must be less than or equal to the time devoted to the sports segment. What advice would you give the station manager given this dual price?
35.
Gulf Coast Electronics is ready to award contracts for printing their annual report. For the past several years, the fourcolor annual report has been printed by Johnson Printing and Lakeside Litho. A new firm, Benson Printing, has inquired into the possibility of doing a portion of the printing. The quality and service level provided by Lakeside Litho has been extremely high; in fact, only 0.5% of their reports have had to be discarded because of quality problems. Johnson Printing has also had a high quality level historically, producing an average of only I % unacceptable reports. Because Gulf Coast Electronics lacks any experience with Benson Printing, they estimate their defective rate to be 10%. Gulf Coast would like to determine how many reports should be printed by each firm to obtain 75,000 acceptablequality reports. To ensure that Benson Printing will receive some of the contract, management specified that the number of reports awarded to Benson Printing must be at least 10% of the volume given to Johnson Printing. In addition, the total volume assigned to Benson Printing, Johnson Printing, and Lakeside Litho should not exceed 30,000, 50,000, and 50,000 copies, respectively. Because of their longterm relationship with Lakeside Litho, management also specified that at least 30,000 reports should be awarded to Lakeside Litho. The cost per copy is $2.45 for Benson Printing, $2.50 for Johnson Printing, and $2.75 for Lakeside Litho. a. Formulate and solve a linear program for determining how many copies should be assigned to each printing firm to minimize the total cost of obtaining 75,000 acceptablequality reports. b. Suppose that the quality level for Benson Printing is much better than estimated. What effect, if any, would this quality level have? c. Suppose that management is willing to reconsider their requirement that Lakeside Litho be awarded at least 30,000 reports. What effect, if any, would this consideration have?
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