linear programming

Question

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).

Answer 1

1. LP Formulation

Decision Variables: Let x_A, x_B, x_C = feet of cable A, B, C produced in-house Let y_A, y_B, y_C = feet of cable A, B, C procured externally

Objective Function: Minimize Z = 6x_A + 12x_B + 10x_C + 8y_A + 15y_B + 12y_C

Constraints: Demand (exact): x_A + y_A = 60,000; x_B + y_B = 80,000; x_C + y_C = 90,000

Machine time: 0.50x_A + 0.50x_B + 0.60x_C ≤ 48,000 minutes (800 hrs)

Finishing time: 0.60x_A + 0.20x_B + 0.40x_C ≤ 24,000 minutes (400 hrs)

Non-negativity: all variables ≥ 0

2. Optimal Solution

Minimum cost attainable = $2,490,000

Cable Type A B C
Produce (ft)	10,000	80,000	5,000
Procure (ft)	50,000	0	85,000

Resources used: Machine time: 0.5(10,000) + 0.5(80,000) + 0.6(5,000) = 48,000 minutes (fully used) Finishing time: 0.6(10,000) + 0.2(80,000) + 0.4(5,000) = 24,000 minutes (fully used)

Both constraints are binding. Cable B is produced entirely in-house because its savings ($3/ft) are the highest per unit of both resources. The remaining capacity is split between A and C at the intersection of the two binding resource constraints.

3. Sensitivity Analysis

(i) Production cost for cable A increases from $6 to $7. Cost will increase. Reasoning: The optimal solution produces 10,000 ft of A in-house. Raising the per-foot production cost for a variable currently at a positive value must raise total cost (or leave it unchanged only if the basis shifts to fully remove A, which still yields a higher total than the original). In this case the savings margin for A drops from $2 to $1, and the optimum shifts to x_A = 0, x_C ≈ 13,333 ft, giving a new cost of about $2,493,333.

(ii) Purchase cost for cable B increases from $15 to $16. Cost will remain unchanged. Reasoning: The current solution procures zero feet of B (y_B = 0), so the procurement price for B has no effect on total cost. Raising it only makes in-house production of B even more attractive, but B is already at its demand ceiling, so nothing shifts.

4. Memorandum to the COO

To: Chief Operations Officer From: Production Manager Re: Expansion of Finishing Time Capacity

The current production plan uses all 400 available finishing-time hours and all 800 machine-time hours, which forces us to procure 50,000 ft of cable A and 85,000 ft of cable C externally at a premium. Sensitivity analysis of the optimal LP solution shows that the shadow price of finishing time is $1.25 per minute, or $75 per hour. This means each additional hour of finishing capacity reduces total procurement-and-production cost by $75, as long as machine time remains the co-binding constraint. I recommend pursuing additional finishing capacity, and the firm should be willing to pay up to $75 per hour above current costs to secure it. Beyond a certain expansion point the machine-time constraint will become the sole bottleneck and the marginal value of additional finishing time will drop to zero, so any arrangement should be structured on an incremental-hour basis rather than a long fixed commitment.