Assignment #4: Case Problem “Stateline Shipping and Transport Company”

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Assignment #4: Case Problem “Stateline Shipping and Transport Company”  

1) This transportation model problem consists of 18 decision variables, representing the number of barrels of wastes product transported from each of the 6 plants to each of the 3 waste disposal sites: [pic]= Number of Barrels transported per week from plant ‘i’ to the j-th waste disposal site, where i = 1, 2, 3, 4, 5, 6 and j = A, B, C. The objective function is to minimize the total transportation cost for all shipments. So the objective function is the sum of the individual shipping costs from each plant to each waste disposal site: Minimize Z = 12[pic]+ 15[pic]+ 17[pic]+ 14[pic]+ 9[pic]+ 10[pic]+ 13[pic]+ 20[pic] +11[pic]

+17[pic] +16[pic] +19[pic] +7[pic] +14[pic] +12[pic] +22[pic] +16[pic] +18[pic] Let’s assume the constraints are the number of barrels of wastes available per week at each plant and the number of barrels of wastes contained at each waste disposal site. Therefore there are 9 constraints- one for each plant supply and one for each waste disposal site’s demand. The six supply constraints are:

[pic]+ [pic]+ [pic] = 35
[pic]+ [pic]+ [pic] = 26
[pic]+ [pic] +[pic] = 42
[pic] + [pic] +[pic] = 53
[pic] +[pic] +[pic] = 29
[pic] + [pic] +[pic] = 38
For example, let’s say the supply constraint [pic]+ [pic]+ [pic] = 35 represents the number of barrels transported from the plant Kingsport to all the three waste disposal sites. The amount transported from Kingsport is limited to the 35 barrels available. The three demand constraints are:

[pic]+ [pic]+ [pic]+[pic]+ [pic]+[pic] ≤ 65
[pic]+ [pic]+ [pic]+ [pic]+[pic] +[pic] ≤ 80
[pic]+ [pic]+[pic]+ [pic]+[pic] +[pic] ≤ 105
Here the demand constraint [pic]+ [pic]+ [pic]+[pic]+ [pic]+[pic] ≤ 65 represents the number of barrels transported to the waste disposal site Whitewater from all the six plants. The barrel of wastes that can accommodate in the waste disposal site Whitewater is limited to 65 barrels. The demand constraints are ≤ inequalities because the total demand (65+80+105) = 250 exceeds the total supply (26+42+53+29+38) = 223. The linear programming model for the transportation problem is summarized as follows: Minimize Z = 12[pic]+ 15[pic]+ 17[pic]+ 14[pic]+ 9[pic]+ 10[pic]+ 13[pic]+ 20[pic] +11[pic]

+17[pic] +16[pic] +19[pic] +7[pic] +14[pic] +12[pic] +22[pic] +16[pic] +18[pic] Subject to
[pic]+ [pic]+ [pic] = 35
[pic]+ [pic]+ [pic] = 26
[pic]+ [pic] +[pic] = 42
[pic] + [pic] +[pic] = 53
[pic] +[pic] +[pic] = 29
[pic] + [pic] +[pic] = 38
[pic]+ [pic]+ [pic]+[pic]+ [pic]+[pic] ≤ 65
[pic]+ [pic]+ [pic]+ [pic]+[pic] +[pic] ≤ 80
[pic]+ [pic]+[pic]+ [pic]+[pic] +[pic] ≤ 105
[pic]

3) If each of the plant and waste disposal sites is considered as intermediate shipping points, the transportation model becomes a transshipment model. [pic]= Number of barrels of waste shipped from plant ‘i’ to plant ‘j’,

where i = 1, 2, 3, 4, 5, 6 and j = 1, 2, 3, 4, 5, 6
[pic]= Number of barrels of waste shipped from waste disposal site ‘k’ to waste disposal site ‘l’, where k = A, B, C and l = A, B, C.

The new objective function of the transshipment model is
Minimize Z = 12[pic]+ 15[pic]+ 17[pic]+ 14[pic]+ 9[pic]+ 10[pic]+ 13[pic]+ 20[pic] +11[pic]
+ 17[pic] +16[pic] +19[pic] +7[pic] +14[pic] +12[pic] +22[pic] +16[pic] +18[pic]
+ 6[pic]+ 4[pic]+ 9[pic]+ 7[pic]+ 8[pic]+ 6[pic]+ 11[pic]+ 10[pic] +12[pic]+ 7[pic]
+ 5[pic]+ 11[pic]+ 3[pic]+ 7[pic]+ 15[pic]+ 9[pic]+ 10[pic]+ 3[pic] + 3[pic]+ 16[pic]
+ 7[pic]+ 12[pic]+ 7[pic]+ 3[pic]+ 14[pic]+ 8[pic]+ 7[pic]+ 15[pic] + 16[pic]+ 14[pic]
+ 12[pic]+ 10[pic]+ 12[pic]+ 15[pic]+ 10[pic]+ 15[pic]
The number of barrels of wastes available at plant ‘i’ for shipment to one or more of the waste disposal sites is given by
(Number of barrels already available at plant ‘i’) + (Number of barrels shipped to plant ‘i’ from other plants) – (Number of barrels shipped from plant ‘i’ to other plants)...
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