Linear Programming: Sensitivity Analysis and Interpretation of Solution
Problems
(The source is Quantitative Methods for Business, by Andersen, Sweeney, Williams, Camm, and Martin.)
1. Investment Advisors, Inc., is a brokerage firm that manages stock portfolios for a number of clients. A particular portfolio consists of U shares of U.S. Oil and H shares of Huber Steel. The annual return for U.S. Oil is $3 per share and the annual return for Huber Steel is $5 per share. U.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 U.S. Oil and 0.25 per share for Huber Steel) has a maximum of 700. In addition, the portfolio is limited to a maximum of 1000 shares of U.S. Oil. The linear programming formulation that will maximize the total annual return of the portfolio is as follows:
Max z = 3U + 5H
Subject to:
25U + 50H ≤ 80,000 Funds available
0.50U + 0.25H ≤ 700 Risk maximum
1U ≤ 1000 U.S. Oil maximum
U, H ≥ 0
Solve the problem using Excel Solver.
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 shadow 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?
2. Refer to the printout of Problem 1.
a) How much would the return for U.S. Oil have to increase before it would be beneficial to increase the investment in this stock?
b) How much would the 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 U.S. Oil maximum were reduced to 900 shares?
3. Innis Investments manages funds for a number of companies and wealthy clients. The investment strategy is tailored to each client’s needs. For a new client, Innis has been authorized to invest up to $1.2 million in two investment funds: a stock fund and a money market fund. Each unit of the stock fund costs $50 and provides an annual rate of return of 10%; each unit of the money market fund costs $100 and provides an annual rate of return of 4%.
The client wants to minimize risk subject to the requirement that the annual income from the investment be at least $60,000. According to Innis’s risk measurement system, each unit invested in the stock fund has a risk index of 8, and each unit invested in the money market fund has a risk index of 3; the higher risk index associated with the stock fund simply indicates that it is the riskier investment. Innis’s client has also specified that at least $300,000 be invested in the money market fund.
Letting S = units purchased in the stock fund
M = units purchased in the money market fund
leads to the following formulation:
Min z = 8S + 3M
Subject to:
50S + 100M ≤ 1,200,000 Funds available
5S + 4M > 60,000 Annual income
M ≥ 3,000 Units in money market
S, M ≥ 0
Solve the problem using Excel Solver.
a) What is the optimal solution, and what is the minimum risk?
b) What does s3 = 7,000 represent, a slack or a surplus? Explain what it means for this problem.
c) Specify the objective coefficient ranges.
d) How much annual income will be earned by the portfolio?
e) What is the rate of return for the portfolio?
f) What is the shadow price for the funds available constraint? Explain what it means for this problem.
g) Suppose the risk index for the money market fund increases from its current value of 3 to 3.5. How does the optimal solution change, if at all? What is the new total risk?
4. A company has to determine the best number of three models of a product to produce in order to maximize profits. The models are, an economy model, a standard model, and a deluxe model. Constraints include production capacity limitations (time available in minutes) in each of three departments (cutting and dyeing, sewing, and inspection and packaging) as well as constraint that requires the production of at least 1000 economy models. The linear programming model is shown here:
Max z = 3x_{1} + 5x_{2} + 4.5x_{3}
Subject to:
12x_{1} + 10x_{2} + 8 x_{3} ≤ 18,000 Cutting and dying
15x_{1} + 15x_{2} + 12 x_{3} ≤ 18,000 Sewing
3x_{1} + 4x_{2} + 2x_{3} ≤ 9,000 Inspection and modeling
1x_{1} ≥ 1,000 Economy model
x_{1}, x_{2}, x_{3} ≥ 0
Solve the problem using Excel Solver.
a) How many units of each model should be produced to maximize the total profit contribution?
b) Which constraints are binding?
c) Interpret slack and/or surplus in each constraint.
d) Overtime rates in the sewing department are $12 per hour. Would you recommend that the company consider using overtime in that department? Explain.
e) What is the shadow price for the fourth constraint? Interpret its value for management.
f) Suppose that the profit contribution of the economy model is increased by $1. How do you expect the solution to change? What is the new value of the objective function (profit)?
g) The profit contribution for the standard model is $5 per unit. How much would this profit contribution have to change to make it worthwhile to produce some units of standard model?
5. A company manufactures two products, identified as model A and model B. Each model has its lowest possible production cost when produced on the new production line. However, the new production line does not have the capacity to handle the total production of both models. As a result, at least some of the production must be routed to a higher-cost, old production line. The minimum production requirement for next month for model A is 50,000 units; and for model B, it is 70,000 units. The production line capacities in units per month are 80,000 and 60,000 for the new line and the old line, respectively. The production cost for model A produced on the new line is $30/unit; on the old line, it is $50/unit. The production cost for model B produced on the new line is $25/unit; on the old line, it is $40/unit.
Let AN = Units of model A produced on the new production line
AO = Units of model A produced on the old production line
BN = Units of model B produced on the new production line
BO = Units of model B produced on the old production line
The objective of the company is to determine the minimum cost production plan. The linear programming model has been formulated below.
Min z = 30AN + 50AO + 25BN + 40BO
Subject to:
AN + AO + > 50,000 Minimum production for model A
BN + BO ≥ 70,000 Minimum production for model B
AN + BN < 80,000 Capacity of the new production line
AO + BO < 60,000 Capacity of the old production line
AN, AO, BN, BO ≥ 0
Solve the problem using Excel Solver.
a) What is the optimal solution and what is the total production cost associated with this solution?
b) Which constraints are binding? Explain.
c) Would you recommend increasing the capacity of the old production line? Explain.
d) Would an increase in capacity for the new production line be desirable? Explain.
e) Suppose that the minimum production requirement for model B is reduced from 70,000 units to 60,000 units. What effect would this change have on the total production cost?
6. The Porsche Club
of
Max z = 200DRB + 280DRW
Subject to:
20DRB + 25DRW ≤ 40,000 Steel available
40DRB + 100DRW ≤ 120,000 Manufacturing minutes
60DRB + 40DRW ≤ 96,000 Assembly minutes
DRB, DRW ≥ 0
Solve the problem using Excel Solver.
a) What are the optimal solution and the total profit contribution?
b) If the available manufacturing time is increased by 500 hours, will the shadow price for the manufacturing time constraint change? Explain.
c) Should Deegan consider using overtime to increase the available assembly time? Why or why not?
d) Because of increased competition, Deegan is considering reducing the price of model DRB such that the new contribution to profit is $175 per unit. How would this change in price affect the optimal solution? Explain. What is the new total profit contribution?
e) Another supplier offered to provide Deegan Industries with an additional 500 pounds of the steel alloy at $2 per pound. Should Deegan purchase the additional pounds of the steel alloy? Explain.
7. 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 are as follows:
Category |
Product 1 |
Product 2 |
Product 3 |
Profit/unit |
$30 |
$50 |
$20 |
Machine 1 time/unit |
0.5 |
2.0 |
0.75 |
Machine 2 time/unit |
1.0 |
1.0 |
0.5 |
Two 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 time. A maximum of 100 labor-hours 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) Formulate a linear programming model that can be used to determine the number of units of each product to produce to maximize the total profit contribution.
b) Solve the problem with Excel Solver. What is the optimal solution, and what is the projected weekly profit associated with your solution?
c) How many hours of production time will be scheduled on each machine?
d) What is the value of an additional hour of labor?
e) 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 production mix assuming the extra hours are made available.