Question

Find the optimal solution for the following problem. (Round your answers to 3 decimal places.) 20 points Maximize C = 15x + 11y + 4z subject to 8x + 10 + 172 s 62 15x + 18y + 11z s 124 and X 20, y 20. eBook a. What is the optimal value of x? Print References b. What is the optimal value of y? c. What is the optimal value of z?

Answer 1

MAXIMIZE C= 15x+11y+4z

Subject To,

            8x+10y+17z ≤ 62

            15x+18y+11z ≤ 124

            x, y, z ≥ 0

The problem is converted to canonical form by adding slack, surplus and artificial variables as appropriate

1. As the constraint-1 is of type '≤' we should add slack variable S₁.

2. As the constraint-2 is of type '≤' we should add slack variable S₂.

Re-writing the problem introducing the slack variables we get-

MAXIMISE C= 15x+11y+4z+0S₁+0S₂

Subject To,

            8x+10y+17z+ S₁ = 62

            15x+18y+11z+ S₂ = 124

            x, y, z, S₁, S₂ ≥ 0

BV	Basic Table
	Zj	15	11	4	0	0	Solution	Replacement Ratio
	CB	x	y	z	s1	s2
s1	0	8	10	17	1	0	62	(62/8) = 7.75
s2	0	15	18	11	0	1	124	(124/15) = 8.267
Cj		0	0	0	0	0	0
Cj-Zj		-15	-11	-4	0	0

Maximization requires All C_j-Z_j ≥ 0.

Since All C_j-Z_j in Basic Table is less than or equal to zero, we will form the Iteration- 1 follows.

In Iteration-1:

x₁ is entering variable and

s₁ is departing variable

Key Element 8

BV	Iteration 1
	Zj	15	11	4	0	0	Solution	Replacement Ratio
	CB	x	y	z	s1	s2
x	15	1	1.25	2.125	0.125	0	7.75
s2	0	0	-0.75	-20.875	-1.875	1	7.75
Cj		15	18.75	31.875	1.875	0	116.25
Cj-Zj		0	7.75	27.875	1.875	0

Since all Zj-Cj ≥ 0

Hence, optimal solution is arrived with value of variables as:
x = 7.75, y = 0, z = 0

Max C = 15x + 11y + 4z = (15x7.75) + (11x0)+ (4x0) = 116.25

You can also see the below images of the table

Basic Table:

Basic Table Solution 15 Replacement Ratio CB Si Si у 10 18 1 0 | z 17 11 0 -4 62 124 (62/8) = 7.75 (124/15) = 8.267 S2 15 1 Ci Cj-Z; -15 -11 0 | 0

Iteration 1:

BV Iteration 1 Solution 4 0 lacement Ratio Zi СВ 15 2 s1 S2 15 X 1 0 15 11 y 1.25 -0.75 18.75 7.75 S2 0 2.125 -20.875 31.875 27.875 0.125 -1.875 1.875 1.875 0 1 0 7.75 | 7.75 116.25 Cj-Zj

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