Question

UESTION 3 (TOTAL 10 MARKS Stelle Office Supplies must fill an order for 2000 modular office dividers. Each divider consists of a frame, a set of legs, and a panel. SOS has limited production and finishing time available and is considering the purchase of some of the components. Let x1, X2, and x3 be the number of frames, leg sets, and panels to make, and x4, X5, and x6 be the number of each to buy. The model reflects the costs to be minimized, the amount of production time, the amount of assembly time, and the need for 2000 of each component Min 20x1 14x2 + 15x +28x4 20x5 25x6 30x1 +40x2 + 25x3 < 180000 15x1 + 10x2 30x3 90000 X1 +X4 -2000 X2 +x5 -2000 x3 +x6-2000 all xi > 0 The final tableau is X2 X3 X6 S1 S2 -17.5 .333 -25 0 0-17.5 -31.67 0 1 20 -14-15 -28 0 Basis св 25 .5 333 0 033666.67 X6 833 6666.67 0 0 201 2000 14 0 2000 150 .333 00 033 1333.33 X3 20-14 15 -25 -17.33 -25 0 68667 -3 -2.67 0 0-.33 Calculate the range of optimality for the number of panels to buy. (3 marks) b. Calculate the range of feasibility for the assembly time. (3 marks) How much less expensive would it have to be to buy frames before you would consider it? a. c. (1 mark) d. What would the change in total cost be if the cost to make a panel increased by $3.00? (2 marks) e. What would you be willing to pay for more production time? (1 mark)

Answer 1

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AS FOR GIVEN DATA...

Stelle Office Supplies must fill an order for 2000 modular office dividers. Each divider consists of a frame, a set of legs, and a panel. SOS has limited production and finishing time available and is considering the purchase of some of the components. Letxj,x2. andx3 be the number of frames, leg sets, and panels to make, and x4xz, andx be the mumber of each to buy. The model reflects the costs to be minimized, the amount of production time, the amount of assembly time, and the need for 2000 of each component.

EXPLANATION ::-

Note the value of Z in the optimal tableau is wrong. It must be  -25 x 666.67 + 0 x 6666.67 - 20 x 2000 - 14 x 2000 - 15 x 1333.33 = -104667

a)

. Calculate the range of optimality for all of the objective function coefficients.

SOL ::-

		x1	x2	x3	x4	x5	x6	s1	s2
Basis	cB	-20	-14	-15	-28	-20	-25	0	0
x6	-25	0	0	0	0.5	0.333	1	0	-0.033
s1	0	0	0	0	-17.5	-31.67	0	1	-0.833
x1	-20	1	0	0	1	0	0	0	0
x2	-14	0	1	0	0	1	0	0	0
x3	-15	0	0	1	-0.5	-0.333	0	0	0.033
	Zj	-20	-14	-15	-25	-17.33	-25	0	0.33
	Cj- Zj	0	0	0	-3	-2.67	0	0	-0.33
	(Cj - Zj)/x6	-	-	-	-6	-8.018	0	-	10
	(Cj - Zj)/x1	0	-	-	-3	-	-	-	-
	(Cj - Zj)/x2	-	0	-	-	-2.67	-	-	-
	(Cj - Zj)/x3	-	-	0	6	8.018	-	-	-10

	Allowable Increase	Allowable decrease
x1	INFINITY	3
x2	INFINITY	2.67
x3	6	10
x4	3	INFINITY
x5	2.67	INFINITY
x6	10	6

b)

. Calculate the range of feasibility for the first two right-hand sides.

SOL ::-

		x1	x2	x3	x4	x5	x6	s1	s2
Basis	cB	-20	-14	-15	-28	-20	-25	0	0.000	RHS	RHS/s1	RHS/s2
x6	-25	0	0	0	0.5	0.333	1	0	-0.033	666.67	-	-20202
s1	0	0	0	0	-17.5	-31.67	0	1	-0.833	6666.67	6666.67	-8000
x1	-20	1	0	0	1	0	0	0	0.000	2000	-	-
x2	-14	0	1	0	0	1	0	0	0.000	2000	-	-
x3	-15	0	0	1	-0.5	-0.333	0	0	0.033	1333.33	-	40000
	Zj	-20	-14	-15	-25	-17.33	-25	0	0.33	-104667
	Cj - Zj	0	0	0	-3	-2.67	0	0	-0.33

	Dual Price	Allowable Increase	Allowable decrease
b1	0	INFINITY	6666.67
b2	-0.33	8000	40000

c.)

How much less expensive would it have to be to buy frames before you would consider it?

SOL::-

Present cost of x4 = 28

Need to reduce by 3 (note the sensitivity of x4 in part (a)) i.e. when the cost becomes 25 or less, it is beneficial to produce instead of buying.

d.)

How much more expensive would legs have to be to make before you would change your

SOL ::-

Present cost of x2 = 14

Need to increase by 2.67 (note the sensitivity of x2 in part (a)) i.e. when the cost becomes 16.67 or more, it is beneficial to buy instead of producing.

e)

. What would the total cost be if the cost to make a panel increased by 3.00?

SOL ::-

The Same optimal solution will be found earlier as the change (3 units) is in the range of optimality of x3 and the total cost becomes 20 x 2000 + 14 x 2000 + 18 x 1333.33 + 28 x 0 + 20 x 0 + 25 x 666.67 = 108667.

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