Question

A linear programming problem has been formulated as follows: Maximize 20 X1 + 10 X2 Xy+ 2 X, s 100 2 X1 + X2 s 100 2 X1 + 2 X2 2 100 X120, X220 Which of the following represents the maximum value of the OF to this problem? Select one: O a. 950 O b. 1200 O c. 2000 O d. 1000 O e. 1350

Answer 1

1. To draw constraint X1 + 2x2 < 100 - (1) Treat it as X1 + 2x2 = 100 When X1 - then X2 = ? a (0) + 2x2 = 100 2x2 = 100 100 X2 50 2 When x2 = 0 then x1 ? 3 X1 + 2(0) = 100 X1 100 X1 0 100 X2 50 0 2. To draw constraint 2x1 + x2 < 100 → (2) Treat it as 2x1 + x2 = 100 When X1 o then X2 = ? 2(0) + X2 = 100 X2 = 100 When x2 = 0 then x1 ? 2X1 + (0) = 100 2x1 = 100 100 → X1 50 X1 0 50 X2 1000

3. To draw constraint 2x1 + 2x2 100 (3) Treat it as 2x1 + 2x2 = 100 When x1 = 0 then x2 = ? ► 2(0) + 2x2 = 100 = 2x2 = 100 100 = = X2 :50 2 When x2 o then x1 = ? 2x1 + 2(0) = 100 2x1 = 100 100 = = X1 50 2 X10 50 X2 500

2100 2x+2100 10 so 80 70 co 140 30 10 ht 10 20 30 40 90 60 70 80 SO 100 The value of the objective function at each of these extreme points is as follows: Extreme Point Coordinates Objective function value Lines through Extreme Point (x2-x2) Z = 20x1 + 10x2 1 - A(0,50) 20(0) + 10(50) = 500 3 2 B(50,0) X1 + 2x2 < 100 2x1 + 2x2 > 100 2x1 + x2 < 100 2X1 + 2x2 > 100 X1 + 2x2 < 100 2x1 + x2 < 100 20(50) + 10(0) = 1000 3 1 C(33.33, 33.33) 20(33.33) + 10(33.33) = 1000 2 + The maximum value of the objective function Z = 1000 occurs at 2 extreme points. Hence, problem has multiple optimal solutions and max 2 = 1000.

option d is correct.

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