Mat 540 Stateline shipping

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 1) The model for the transportation problem consists of 18 decision variables, representing the number of barrels of wastes transported from each of the 6 plants to each of the 3 waste disposal sites: = 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 of the manager is to minimize the total transportation cost for all shipments. Thus the objective function is the sum of the individual shipping costs from each plant to each waste disposal site: Minimize Z = 12+ 15+ 17+ 14+ 9+ 10+ 13+ 20 +11

+17 +16 +19 +7 +14 +12 +22 +16 +18
The constraints in the model are the number of barrels of wastes available per week at each plant and the number of barrels of wastes accommodated at each waste disposal site. There are 9 constraints- one for each plant supply and one for each waste disposal site’s demand. The six supply constraints are:

+ + = 35
+ + = 26
+ + = 42
+ + = 53
+ + = 29
+ + = 38
As an example, here the supply constraint + + = 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:

+ + ++ + ≤ 65
+ + + + + ≤ 80
+ ++ + + ≤ 105
Here the demand constraint + + ++ + ≤ 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+ 15+ 17+ 14+ 9+ 10+ 13+ 20 +11

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

2) Because the transportation model is formulated as a linear programming model, it can be solved with Excel Solver. The spreadsheet solution is shown in the following table. Plant
Waste Disposal Sites
Supply
Waste Shipped

Whitewater
Los Canos
Duras

Kingsport
35.0
0.0
0.0
35.0
35.0
Danville
0.0
0.0
26.0
26.0
26.0
Macon
0.0
0.0
42.0
42.0
42.0
Selma
1.0
52.0
0.0
53.0
53.0
Columbus
29.0
0.0
0.0
29.0
29.0
Allentown
0.0
28.0
10.0
38.0
38.0
Waste Received
65.0
80.0
105.0

223.0
Waste Disposed
65.0
80.0
78.0
223.0

Cost=
$ 2,822

Thus the optimum solution of the transportation problem is given in the following table. From
To
Shipment unit
Cost per unit
Shipment cost
Kingsport
Whitewater
35
$12
$420
Danville
Duras
26
$10
$260
Macon
Duras
42
$11
$462
Selma
Whitewater
1
$17
$17
Selma
Los Canos
52
$16
$832
Columbus
Whitewater
29
$7
$203
Allentown
Los Canos
28
$16
$448
Allentown
Duras
10
$18
$180
Total Cost
$2822

The total transportation cost for the optimal route is $2822. 3) If each of the plant and waste disposal sites is considered as intermediate shipping points, the transportation model becomes a transshipment model. The additional decision variables included in the revised model are = 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
= 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+ 15+ 17+ 14+ 9+ 10+ 13+ 20 +11
+ 17 +16 +19 +7 +14 +12 +22 +16 +18
+ 6+ 4+ 9+ 7+ 8+ 6+ 11+ 10 +12+ 7
+ 5+ 11+ 3+ 7+ 15+ 9+ 10+ 3 + 3+ 16
+ 7+ 12+ 7+ 3+ 14+ 8+ 7+ 15 + 16+ 14
+ 12+ 10+ 12+ 15+ 10+ 15
The number of barrels of wastes available at...
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