Question

4. A local company restores cars and trucks for resale. Each vehicle must be processed in the refinishing/paint shop and the machine/body shop. Each car (on average) contributes $3000 to profit, and each truck contributes (on average) $2000 to profit. The refinishing/paint shop has 2400 work-hours available and the machine/body shop has 2500 work-hours available. A car requires 50 work-hours in the machine/body shop and 40 work-hours in the refinishing/paint shop. A truck requires 50 work-hours in the machine/body shop and 60 work-hours in the refinishing/paint shop. determine the sensativity of the optimal solution to a change in the profit for each truck.

Answer 1

Let x1 be the number of cars restored per day, and x2 be the number of trucks restored per day. The goal is to maximize 3000x+2000y subject to

MAX Z = 3000x1 + 2000x2
subject to
50x1 + 50x2 <= 2500
40x1 + 60x2 <= 2400
and x1,x2 >= 0

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

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

After introducing slack variables

MAX Z = 3000x1 + 2000x2 + 0S1 + 0S2
subject to
50x1 + 50x2 + S1 = 2500
40x1 + 60x2 + S2 = 2400
and x1,x2,S1,S2 >= 0

Iteration-1		Cj	3000	2000	0	0
B	CB	XB	x1	x2	S1	S2	MinRatio XBx1
S1	0	2500	(50)	50	1	0	250050=50→
S2	0	2400	40	60	0	1	240040=60
Z=0		Zj	0	0	0	0
		Cj-Zj	3000↑	2000	0	0

Positive maximum Cj-Zj is 3000 and its column index is 1. So, the entering variable is x1.

Minimum ratio is 50 and its row index is 1. So, the leaving basis variable is S1.

∴ The pivot element is 50.

Entering =x1, Departing =S1, Key Element =50

R1(new)=R1(old)÷50

R2(new)=R2(old) - 40R1(new)

Iteration-2		Cj	3000	2000	0	0
B	CB	XB	x1	x2	S1	S2	MinRatio
x1	3000	50	1	1	150	0
S2	0	400	0	20	-45	1
Z=150000		Zj	3000	3000	60	0
		Cj-Zj	0	-1000	-60	0

Since all Cj-Zj≤0

Hence, integer optimal solution is arrived with value of variables as :
x1=50,x2=0

Max Z=150000