PROBLEM DESCRIPTION: International Textile Company
International Textile Company, Ltd., is a Hong Kong-based firm that distributes textiles worldwide. The company is owned by the Lao family. International Textile has mills in the Bahamas, Hong Kong, Korea, Nigeria, and Venezuela, each weaving fabrics out of two or more raw fibers: cotton, polyester, and/or silk. The mills service eight distribution centers located near the customers' geographical centers of activity.
Because transportation costs historically have been less than 10% of the total expenses, management has paid little attention to extracting savings through judicious routing of shipments. However, Ching Lao is returning from the United States, where he has just completed his bachelor's degree in marketing. He believes that each year he can save the company hundreds of thousands of dollars-perhaps millions-just by better routing of fabrics from mills to distribution centers. One glaring example of poor routing is the current assignment of fabric output to the Mexico City distribution center from Nigeria instead of from Venezuela, less than a third the distance. Similarly, the Manila center now gets most of its textiles from Nigeria and Venezuela, although the mills in Hong Kong itself are much closer. Of course, the cost of shipping a bolt of cloth does not depend on distance alone. Table 1 provides the actual shipping costs supplied to Lao from company headquarters.
Table 1. Shipping cost data ($ per bolt of cloth)
|
Mill
|
Distribution Center
|
|
Los Angeles
|
Chicago
|
London
|
Mexico
|
City
|
Manila
|
Rome
|
Tokyo
|
New York
|
|
Bahamas
|
2
|
2
|
3
|
3
|
7
|
4
|
7
|
1
|
|
Hong Kong
|
6
|
7
|
8
|
10
|
2
|
9
|
4
|
8
|
|
Korea
|
5
|
6
|
8
|
11
|
4
|
9
|
1
|
7
|
|
Nigeria
|
14
|
12
|
6
|
9
|
11
|
7
|
5
|
10
|
|
Venezuela
|
4
|
3
|
5
|
1
|
9
|
6
|
11
|
4
|
Distribution center demands are seasonal, so a new shipment plan must be made each month. Table 2 provides the fabric requirements for the month of March. International Textile's mills have varying capacities for producing the various types of cloth, where Table 3 provides the quantities that apply during March.
Table 2. Fabric demands for the month of March (in bolts)
|
Fabric
|
Distribution Center
|
|
Los Angeles
|
Chicago
|
London
|
Mexico City
|
Manila
|
Rome
|
Tokyo
|
New York
|
|
Cotton
|
500
|
800
|
900
|
900
|
800
|
100
|
200
|
700
|
|
Polyester
|
1000
|
2000
|
3000
|
1500
|
400
|
700
|
900
|
2500
|
|
Silk
|
100
|
100
|
200
|
50
|
400
|
200
|
700
|
200
|
Table 3. March production quantities (in bolts)
|
Mill
|
Production Capacity
|
|
Cotton
|
Polyester
|
Silk
|
|
Bahamas
|
1000
|
3000
|
0
|
|
Hong Kong
|
2000
|
2500
|
1000
|
|
Korea
|
1000
|
3500
|
500
|
|
Nigeria
|
2000
|
0
|
0
|
|
Venezuela
|
1000
|
2000
|
0
|
Lao wants to schedule production and shipments in such a way that the most costly customers are shorted when there is insufficient capacity, and the least-efficient plants operate at
Managerial Report
a) Using your own wordings, describe the problem at hand and clearly list what the objective of the analysis is.
b) For each type of fabric (e.g. cotton, polyester cloth and silk), formulate the problem as a linear program that determines the optimal shipping schedule of that fabric separately. Clearly define the decision variables and provide a verbal description of the objective function as well as each of the constraints.
c) Solve the models developed in part (b) using Excel solver.
d) Solve the models developed in part (b) using LINGO.
e) For each type of fabric, what is the optimal shipment schedule obtained, and what is the associated optimal objective function value?
f) For each type of fabric, is the optimal solution obtained degenerate? Why?
g) For each type of fabric, is the optimal solution obtained unique? Why?
h) For a specific type of fabric of your choice, discuss the range of optimality for the objective function coefficients and provide its interpretation.
i) For a specific type of fabric of your choice, discuss the range of feasibility for the constraints right hand sides and provide its interpretation.
j) For a specific type of fabric of your choice, present the shadow prices for the constraints along with their interpretations.
k) What are the resources that exist in this problem? Comment on the utilization of these resources (i.e., which of the resources have been fully utilized and which haven't).
l) If the demand for the cotton fabric at the Mexico City distribution center is increased from 900 to 1000, is the shadow price associated with that constraint still valid to assess what impact this increase in the right hand side would have on the objective function value?
m) Assume that the mill located in Nigeria breaks down at the beginning of March and it won't be fixed till the end of the month indicating that no production occurs at that facility during March. How would such unexpected event impact the company's shipping schedule?
n) If the cost associated with shipping one bolt of Polyester fabric from the mill located in Korea to the distribution center in London is changed to $10, would the optimal solution change? Why?
o) What is the maximum that International Textile Company should be willing to pay in order to increase the available production capacity at the Venezuela's production mill by one bolt during the month of March (assuming such an increase is possible)?
p) The company will be opening a silk-making department in the Nigeria mill. Although it will not be completed for several months, a current capacity of 1,000 bolts for that fabric might be used during March for an added one time cost of $2,000. Find the new optimal shipment schedule and the total cost for that fabric. Should the Nigeria mill process silk in March?
q) Suppose International Textile Company is thinking about signing an agreement that allows the option of paying a $500 penalty for each distribution center that gets only 90% of its demand fulfilled (i.e. demand is not fully satisfied at that distribution center). Should the company proceed ahead with signing that agreement? Why or Why not?
r) Lao learns that changes might have to be made to the March plans. If a new customer is obtained, the cotton demand in Manila and in Mexico City will increase by 10% at each location. Meanwhile, a big New York customer might cut back, which would reduce polyester demand by 10% in both New York and Chicago. Find the contingent optimal schedules and total costs (a) for cotton and (b) for polyester.
s) International Textile loses a profit of $10 for each bolt of cotton it falls short of meeting the distribution center's demand. For polyester, the loss is $20 per bolt; for silk, it is a whopping $50 per bolt. By running the mills on overtime, the company can produce additional bolts at the additional costs shown in Table 4 below. Using only the original data from Tables 1 through 3 and the information in Table 4, determine new production schedules to maximize overall profit for (a) cotton, (b) polyester, and (c) silk. Which fabrics and locations involve overtime production, and what are the overtime quantities?
Table 4. Overtime production costs
|
Mill
|
Cost
|
per Bolt ($)
|
|
Cotton
|
Polyester
|
Silk
|
|
Bahamas
|
10
|
10
|
N.A.
|
|
Hong Kong
|
15
|
12
|
25
|
|
Korea
|
5
|
8
|
22
|
|
Nigeria
|
6
|
N.A.
|
N.A.
|
|
Venezuela
|
7
|
6
|
N.A.
|
t) Formulate the original problem (with the original data figures) as a single linear program that includes all three fabrics in the same model. Solve the resulting model using the software and report the optimal solution obtained.
u) How does the solution found in part (t) differ from that found in parts (c) and (d)? What does this imply?
v) Without making any calculations, offer Lao other suggestions for reducing costs of transportation.