MMIS 671
Homework 1. Constrained Optimization Problems
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:
Objective Function:
Constraints:
Task 2. Solve the LP and report your optimal solutions. (4 Points)
Minimum cost attainable = $ ___________________
Decision variable values under optimal solutions:
|
Cable Type |
A |
B |
C |
|
Produce (feet) |
|||
|
Procure (feet) |
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 |
||
|
Annealing |
36,000 |
||
|
Stranding |
48,000 |
||
|
Extrusion |
30,000 |
||
|
Assembly |
60,000 |
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)
