MMIS 671 – Constrained Optimization Problems

A small business produces 3 types of cables: A, B, and C. The cost of in-house production is estimated to be $6, $12, and $10 per foot of A, B, and C respectively. The manufacturing process requires machining and finishing. The machining and finishing time needed to produce a foot of each type of cable are presented in the table below.
For the coming production period the firm is contractually obligated to produce 60,000 feet of A, 80,000 feet of B, and 90,000 feet of C. As only 800 hours of machine time and 400 hours of finishing time are available, these demands cannot be met by in-house production alone. The firm has the option of procuring these cables from an external supplier in order to meet the demand. The cost prices (from the external supplier) per foot of cable are as follows: $8 for A, $15 for B, and $12 for C.
The production manager has to decide how much of each type of cable to produce in-house and how much to purchase from the external supplier in order to meet the demands exactly and minimize total cost. The data is summarized below:
Cable Type
A
B
C
Demand (ft)
60,000
80,000
90,000
Production Cost/ft
$6
$12
$10
Procurement Cost/ft
$8
$15
$12
Machine time needed (mins/ft)
0.50
0.50
0.60
Finishing time needed (mins/ft)
0.60
0.20
0.40
1. Formulate the problem as a Linear Program:
Decision Variables:
Objective Function:
Constraints:
2. Solve the LP and report your optimal solutions:
Minimum cost attainable = $ ___________________
Decision variable values under optimal solutions:
Cable Type
A
B
C
Produce (ft)
Procure (ft)
Resources used:
Machine time (minutes)
Finishing time (minutes)
3. Sensitivity analysis:
Explain in each of the following cases whether you expect the costs to increase, decrease, or remain unchanged when the following parameters are changed (one at a time). Assume that all other parameters remain at their original values. Please be as brief and precise as possible in your explanations. 
(i) The production cost for a foot of cable A increases to from $6 to $7.
Cost will: increase, decrease, remain unchanged.
Reasoning: 
(ii) The purchase cost for a foot of cable B increases from $15 to $16.
Cost will: increase, decrease, remain unchanged.
Reasoning: 
4. Memorandum based on sensitivity analysis:
Given that the current production capacity is insufficient to meet the demand through in-house production alone, the Chief Operations Officer (COO) wants to know whether the firm should consider increasing the availability of finishing time. Write a short (one paragraph) memorandum to the COO with your recommendation. The memorandum should also specify at most how much the firm should be willing to pay per hour to increase the availability of finishing time (beyond the current availability of 400 hours).