Hello,

I having difficulties with the Task 3 (sensitive analysis) of this problem:

A company produces 3 types of cables: A, B, and C. In-house production costs per foot of cables A, B, and C are $6, $8, and $10, respectively. The production process requires 5 resources: Drawing, Annealing, Stranding, Extrusion, and Assembly. For each resource, the table below specifies the number of minutes of the resource needed to produce a foot of each type of cable. For example, to produce each foot of Cable C, we need 0.1 minutes of Drawing, 0.2 minutes of Annealing, 0.3 minutes of Stranding, 0.1 minutes of Extrusion, and 0.4 minutes of Assembly. The column Available hours, specifies the number of hours of each resource available during a production period.

For the next production period the firm is contractually obligated to produce 60,000 feet of A, 40,000 feet of B, and 120,000 feet of C. Due to limited resource availability, these demands cannot be met by in-house production alone. The company must procure cables from an outsourcing partner, at higher costs, to meet the demand. The costs per foot for purchasing cables A, B, and C from the outsourcing partner are $8, $10, and $15, respectively.

The production manager must decide how much of each type of cable to produce in-house and how much to purchase from the outsourcing partner to meet the demands at minimum cost.

Relevant data is summarized in the table below:

Cable Type	A	B	C	Available hours
Demand (ft)	60,000	40,000	120,000
Production Cost/ft	$6	$8	$10
Purchase Cost/ft	$8	$10	$15
Drawing (mins/ft)	0.1	0.2	0.1	400
Annealing (mins/ft)	0.1	0.2	0.2	600
Stranding (mins/ft)	0.1	0.3	0.3	800
Extrusion (mins/ft)	0.1	0.3	0.1	500
Assembly (mins/ft)	0.2	0.1	0.4	1000

Task 1. Formulate the problem as a Linear Program. (2 Points)

Define the decision variables and specify the objective function and constraints.

Decision Variables:

xA, xB, xC: Number of feet of Cables A, B, and C produced. yA, yB, yC: Number of feet of Cables A, B, and C purchased.

Objective Function:

The objective is to minimize total cost = cost of production + cost of purchase

Minimize Cost = 6 xA + 8 xB + 10 xC + 8 yA + 10 yB + 15 yC

Constraints:

Number of feet of Cable A produced + Number of feet of Cable A purchased should be no less than the demand for Cable A.

Demand_A: xA + yA >= 60000

Task 2. Solve the LP and report your optimal solutions. (4 Points)

Minimum cost attainable = $ 1,936,000

Decision variable values under optimal solutions:

Cable Type	A	B	C
Produce (feet)	48000	24000	120000
Procure (feet)	12000	16000	0

Resource use: Under the columns Used and Not Used, specify the number of minutes of each resource used, and left unused, under the optimal solution.

Resource	Available (minutes)	Used (minutes)	Not Used (minutes)
Drawing	24,000	21600	2400
Annealing	36,000	33600	2400
Stranding	48,000	48000	0
Extrusion	30,000	24000	6000
Assembly	60,000	60000	0

Task 3. Sensitivity analysis. (4 Points)

1. The column Cost per hour in the table below specifies the production costs per hour for available resources. For example, the first 400 hours of Drawing costs $60 per hour. Under the column Cost decrease per additional hour specify the decrease in total cost if an additional hour of the resource is available; availability of all other resources remain unchanged. Under the column Maximum amount for additional hour specify the maximum amount that the company should be willing to pay for an additional hour of each resource (beyond its current availability). Briefly explain your reasoning. (2 Points)

Resource	Current availability (hours)	Cost per hour	Cost decrease per additional hour	Maximum amount for additional hour
Drawing	400	$60
Annealing	600	$240
Stranding	800	$180
Extrusion	500	$120
Assembly	1000	$300

Explanation:

2. How would the minimum cost obtained in Task 2 be affected if the purchase cost per foot of Cable C was $20 (instead of $15). Briefly explain your reasoning. (1 Point)

3. The company has identified another potential supplier, New-Partner. New-Partner can supply at most 1000 feet of Cable C, but the purchase price is subject to negotiations. What is the maximum price that the company should be willing to offer New-Partner per foot of Cable C? Briefly explain your reasoning. (1 Point)