Question

California Cabinets manufactures outdoor tables the company sells to local dealers throughout the Southwest. Because of an increase in demand for its products, California Cabinets is considering hiring subcontractors to complete its current backlog of orders. The following information is available for each subcontractor: Subcontractor 1 Subcontractor 2 Subcontractor 3 Hours required to complete wood tables 50 42 30 Hours required to complete steel tables 60 48 35 Hours available 40 30 35 Cost per hour $36 $42 $55 For example, Subcontractor 1 estimates it will take 50 hours to complete all wood tables and 60 hours to complete all steel tables. However, Subcontractor 1 only has 40 hours available. Therefore, Subcontractor 1 can only complete 40/50 = .80 or 80% of the wood tables if the company worked only on the wood tables or 40/60 = .67 or 67% if it worked only on the steel tables.

Formulate a linear programming model you can use to determine the percentage of wood tables and the percentage of steel tables that should be assigned to each of the three subcontractors to minimize the total cost of completing both projects.

Solve the model from Part 1. What percentage of the wood tables and what percentage of the steel tables should be assigned to each subcontractor?

Based on your model, what is the total cost of completing both projects?

Suppose Subcontractor 2 reduced its costs to $38 per hour. What effect would this change have on the optimal solution? Explain.

Answer 1

	SC-1	SC-2	SC-3
Hours required for [all] wood tables	50	42	30
Hours required for [all] steel tables	60	48	35
Hours available	40	30	35
Cost per hour	$36	$42	$55

Let Pwj be the percentage of the wooden tables given of subcontractor-j and Psj be the percentage of the steel tables given of subcontractor-j; j=1,2,3

Objective function: Minimize Z = total cost
Z = 36*(50Pw1 + 60Ps1) + 42*(42Pw2 + 48Ps2) + 55*(30Pw3 + 35Ps3)

Subject to,

Pw1 + Pw2 + Pw3 = 1
Ps1 + Ps2 + Ps3 = 1

50Pw1 + 60Ps1 <= 40
42Pw2 + 48Ps2 <= 30
30Pw3 + 35Ps3 <= 35

Pwj, Psj >= 0; j=1,2,3

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	SC-1	SC-2	SC-3	Total
% of wood tables	0.27	0.00	0.73	1
% of steel tables	0.00	0.63	0.38	1
	SC-1	SC-2	SC-3
Hours required for [all] wood tables	50	42	30
Hours required for [all] steel tables	60	48	35
Hours available	40	30	35
Cost per hour	$36	$42	$55
Hours consumed	13.54	30	35
Total cost	$3,672.50

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The coefficient of Ps2 suggests that up to an infinite increase in the coefficient, there will be no change in the optimal value 0.625. The capacity of 30 hours is already consumed. So, there cannot be any allocation to Pw2 as well. So, no change will be observed in the optimal solution.