Linear Programming:  Sensitivity Analysis and Interpretation of Solution

 

Problems

(The source is Quantitative Methods for Business, by Andersen, Sweeney, and Williams.)

 

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)  What is the marginal rate of return on extra funds added to the portfolio?

h)  Suppose the risk index for the money market fund increases from its current value of 3 to 3.5.  How does the optimal solution , if at all?  What is the new total risk?

 

4.  Consider the following linear program.

 

Min z = 15x1 + 15x2 + 16x3

Subject to:

   1x1 +             1x3 ≤ 30

0.5x1 – 1x2 +   6x3 ≥ 15

   3x1 + 4x2 -    1x3 ≥ 20

               x1, x2, x3 ≥ 0

Solve the problem using Excel Solver.

a)  What is the optimal solution, and what is the value of the objective function?

b)  Which constraints are binding?

c)  What are the shadow prices?  Interpret each. 

d)  If you could change the right-hand side of one constraint by one unit, which one would you choose?  What would be the new value of the right-hand side?

 

5.  Supersport Footballs, Inc., has to determine the best number of All-Pro (A), College (C), and High School (H) models of footballs to produce in order to maximize profits.  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 All-Pro footballs.  The linear programming model of Supersport’s problem is shown here:

 

Max z = 3A + 5C + 4H

Subject to:

12A + 10C +   8H ≤ 18,000     Cutting and dying

15A + 15C + 12H ≤ 18,000     Sewing

  3A +   4C + 24H ≤   9,000     Inspection and modeling

  1A                       ≥   1,000     All-Pro model

                A, C, H ≥ 0

Solve the problem using Excel Solver.

 

a)  How many footballs of each type should Supersport produce 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 College ball is increased by $1.  How do you expect the solution to change?  What is the new value of the objective function (profit)?

 

6.  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.