# Linear Programming

1. The Scrod Manufacturing Co. produces two key items â€“ 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 â€“ 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 â€œfeasible solutionâ€ 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 intersection point 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.
3. 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. 24Ã—1 + 12Ã—2 + 36Ã—3

Subject to: .75Ã—1 + .75Ã—2 + 1.5Ã—3 < 300 (manufacturing)

.8Ã—1 + .4Ã—2 + .4Ã—3 < 200 (testing)

x1 + x2 + x3 < 500 (canning)

x1, 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 Â½ and Â¾ 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

Max 24Ã—1 + 12Ã—2 + 36Ã—3
Subject to
.75Ã—1 + .75Ã—2 + 1.5Ã—3 <300
.8Ã—1 + .4Ã—2 + .4Ã—3 <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

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