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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