Question

Exercise 2

Linear Programming

1.         The Scrod Manufacturing Co. produces two key items – special-purpose Widgets (W) and more generally useful Frami (F). Management wishes to determine that mix of W & F which will maximize total Profits (P).

Data                                                                      W      F

            Unit profit contributions                     $ 30   $ 20

            Demand estimates (unit/week)               250      500

            Average processing rates – each product requires processing on both machines (units/hour)

                                    Machine #1                        2          4   

                                    Machine #2                        3          3

The two products compete for processing time using the same limited plant capacity. Only 160 hours are available on each of two machines (1 and 2) during each week, barring unexpected equipment breakdowns. Management has established a desired minimum production level of 200 units per week (total output: W + F) in order to maintain distribution outlets.

As a newly hired management analyst for Scrod, you have been asked to analyze the available options and recommend an appropriate product mix. Your boss has suggested that you structure a model of the underlying constrained optimization problem and test to be sure that a feasible solution exists before proceeding to analyze the alternatives. (You do not have to solve this problem; just set it up and make sure that a feasible solution exists. You should try this both with and without the demand estimates included as constraints).

2.         The Ace Manufacturing Company produces two lines of its product, the super and the regular. Resource requirements for production are given in the table. There are 1,600 hours of assembly worker hours available per week, 700 hours of paint time, and 1200 hours of inspection time. Regular customers will demand at least 150 units of the regular line and at least 90 of the super line.

                                            Profit         Assembly           Paint         Inspection

Product Line               Contribution       time (hr.)     time (hr.)        time (hr.)

Regular                                50                  1.2                     .8               1.5

Super                                   75                   1.6                     .5                 .7

a)     Formulate an LP model which the Ace Company could use to determine the optimal product mix on a weekly basis. Use two decision variables (units of regular and units of super). Suggest any feasible solution and explain what “feasible solution” means.

b)    Find the optimal solution by using the graphical solution technique. What is the value of the objective function? What are the values of all variables?

c)     By how many units can the demand for the super product increase before the optimal solution changes? Explain. For the regular product?

d)    How much would it be worth to the Ace Company if it could obtain an additional hour of paint time? Of assembly time? Of inspection time? Explain fully. Show all calculations.

e)     Find the upper and lower bounds for assembly time by identifying the corner points on either end of the line and substituting these points into the assembly equation. What do these bounds mean? Explain.

f)     Solve this problem with LINDO or POM and verify that your answers are correct.

Matchpoint Company produces 3 types of tennis balls: Heavy Duty, Regular, and Extra Duty, with a profit contribution of $24, $12, and $36 per gross (12 dozen), respectively.

   The linear programming formulation is:

   Max.                             24x₁ + 12x₂ + 36x₃

   Subject to:                   .75x₁ + .75x₂ + 1.5x₃           <    300 (manufacturing)

                                           .8x₁ + .4x₂ + .4x₃                     <    200 (testing)

x₁ + x₂ + x₃                      <    500 (canning)

x_1, x₂, x₃                       >       0

where x₁, x₂, x₃ refer to Heavy Duty, Regular, and Extra Duty balls (in gross). The LINDO solution is on the following page.

a)     How many balls of each type will Matchpoint product?

b)    Which constraints are limiting and which are not? Explain.

c)     How much would you be willing to pay for an extra man-hour of testing capacity? For how many additional man-hours of testing capacity is this marginal value valid? Why?

d)    By how much would the profit contribution of Regular balls have to increase to make it profitable for Matchpoint to start producing Regular balls?

e)     By how much would the profit contribution of Heavy Duty balls have to decrease before Matchpoint would find it profitable to change its production plan?

f)     Matchpoint is considering producing a low-pressure ball, suited for high altitudes, called the Special Duty. Each gross of Special Duty balls would require 1 ½ and ¾ man-hours of manufacturing and testing, respectively, and would give a profit contribution of $33 per gross. Special Duty balls would be packed in the same type of cans as the other balls.

Should Matchpoint produce any of the Special duty balls? Explain; provide support for your answer.

Max 24x1 + 12x2 + 36x3

Subject to

.75x1 + .75x2 + 1.5x3 <300

.8x1 + .4x2 + .4x3 <200

x1 + x2 + x3 < 500

end

LP OPTIMUM FOUND AT STEP      2

OBJECTIVE FUNCTION VALUE

   1)      8400.000

VARIABLE        VALUE         REDUCED COST

        X1              200.000000          0.000000

        X2                  0.000000          8.000000

        X3              100.000000          0.000000

   ROW   SLACK OR SURPLUS     DUAL PRICES

        2)                   0.000000                    21.333334

        3)                   0.000000                    10.000000

        4)               200.000000                      0.000000

NO. ITERATIONS=       2

RANGES IN WHICH THE BASIS IS UNCHANGED:

   OBJ COEFFICIENT RANGES

VARIABLE         CURRENT        ALLOWABLE        ALLOWABLE

                                   COEF              INCREASE              DECREASE

       X1                    24.000000           48.000000                   6.000000

       X2                    12.000000             8.000001                   INFINITY

       X3                    36.000000           12.000000                 24.000000

RIGHTHAND SIDE RANGES

      ROW         CURRENT        ALLOWABLE        ALLOWABLE

                             RHS                INCREASE              DECREASE

        2               300.000000         450.000000               112.500000

        3               200.000000         120.000000               120.000000

        4               500.000000         INFINITY                 200.000000

Answer 1

1) LP model

Max P = 30W + 20F

s.t.

W/2 + F/4 <= 160

W/3 + F/3 <= 160

W + F >= 200

W, F >= 0

Additional demand constraints:

W <= 250

F <= 500