(1) Original problem
This is a Transportation problem for cost minimization. The problem is that the company has six collection centres from where company wants to collect waste and then transported to three disposable centres at a minimum cost. The different costs from collection centres to disposable sites are given in a table as shown in the following table.

White water Los Canos Duras Availability (bbl)
Kingsport $12.00 $15.00 $17.00 35
Danville $14.00 $9.00 $10.00 26
Macon $13.00 $20.00 $11.00 42
Selma $17.00 $16.00 $19.00 53
Columbus $7.00 $14.00 $12.00 29
Allentown $22.00 $16.00 $18.00 38
Capacity (barrels) 65 80 105

Model in the mathematical form

Let Xij i=1,2,3,4,5,6; j=1,2,3 denote the quantity of waste transported from i-th plant to j-th waste disposal centre. The objective function Z and constraints can be expressed as

Minimize Z=12X11+15X12+17X13+14X21+9X22+10X23+13X31+20X32+11X33+17X41+16X42+19X43+7X51+14X52+12X53+22X61+16X62+18X63
Subject to
X11+X12+X13=35
X21+X22+X23=26
X31+X32+X33=42
X41+X42+X43=53
X51+X52+X53=29
X61+X62+X63=38
X11+X21+X31+X41+X51+X61 65
X12+X22+X32+X42+X52+X62 80
X13+X23+X33+X43+X53+X63 105
Xij 0, i=1,2,3,4,5,6; j=1,2,3.

(2) The original Problem

The above problem without considering the intermediates sites are solved using excel solver and the final solution are given as follows

White water Los Canos Duras
Kingsport 35
Danville 26
Macon 42
Selma 1 52
Columbus 29
Allentown 28 10
Dummy 27

Therefore, the optimal transportation procedure is to transport
(1) 35 bbl from Kingport, 1 bbl from Selma and 29 bbl from Columbus to the White water waste disposal plant
(2) 52 bbl from Selma and 28 bbl from Allentown to Los Canos waste disposal plant
(3) 26 bbl from Danville, 42 bbl from Macon and 10 bbl from Allentown to the Duras waste disposal plant

The minimum cost=35*12+26*10+42*11+1*17+52*16+29*7+28*16+10*18
=$2822.

(3) Modified Problem

Since the company having another option to drop a load at a plant or disposal sites and then picked up and carried on the final destination without extra cost, we can find certain roots in this model, it is cheaper to drop and pick up loads at intermediate points rather than ship them directly. So dropping the waste at certain intermediate position as described below will give certain cost advantage to the shipping company. So by considering all such possibilities some intermediate points are identified to drop the waste to minimize cost of transportation. All such roots with the lowest cost are given in the following table. Among the new roots, the roots with minimum cost are marked in green.

Sl. No. Plant From Intermediate Position Disposal centre Cost Minimum cost
1 Kingsport ------------- Duras $17 $15 Kingsport Danville Duras $15
2 Macon -------------- Los Canos $20 $19 Macon Selma Los Canos $19
3 Selma ------------- White water $17 $10 Selma Columbus White water $10
4 Selma ------------- Duras $19 $14 Selma Macon Duras $14
5 Allentown ------------ White water $22 $20 Allentown Kingsport White water $20
6 Allentown ------------ Duras $18 $17 Allentown Danville Duras $17

Therefore, the original transportation problem can be modified using the intermediate points as follows. The reduced costs from the original model is marked in brown colour.

White water Los Canos Duras Availability (bbl)
Kingsport $12.00 $15.00 $15.00 35
Danville $14.00 $9.00 $10.00 26
Macon $13.00 $19.00 $11.00 42
Selma $10.00 $16.00 $14.00 53
Columbus $7.00 $14.00 $12.00 29
Allentown $20.00 $16.00 $17.00 38
Capacity (barrels) 65 80 105

Then the objective function Z of the modified problem and constraints are written as
Minimize Z=12X11+15X12+15X13+14X21+9X22+10X23+13X31+19X32+11X33+10X41+16X42+14X43+7X51+14X52+12X53+20X61+16X62+17X63
Subject to
X11+X12+X13=35
X21+X22+X23=26
X31+X32+X33=42
X41+X42+X43=53
X51+X52+X53=29
X61+X62+X63=38
X11+X21+X31+X41+X51+X61 65
X12+X22+X32+X42+X52+X62 80
X13+X23+X33+X43+X53+X63 105
Xij 0, i=1,2,3,4,5,6; j=1,2,3.

(4) The modified transportation problem can be solved using excel solver and the following solutions were obtained

The new solution is given below.

White water Los Canos Duras
Kingsport 16 19
Danville 26
Macon 42
Shelma 36 17
Columbus 29
Allentown 38
Dummy 27

The minimum cost=16*15+19*15+26*9+42*11+36*10+17*14+29*7+38*16
=$2630.

Therefore, the new optimal transportation procedure using the intermediate sites is to transport
(1) 36 bbl from Shelma and 29 bbl from Columbus to the White water waste disposal plant
(2) 16 bbl from Kingsport, 26 bbl from Danville and 38 bbl from Allentown to Los Canos waste disposal plant
(3) 19 bbl from Kingsport, 42 bbl from Macon, 17 bbl from Shelma and 10 bbl from Allentown to the Duras waste disposal plant.
To get this cost advantage company must use the following roots for transportation. Waste from Kingsport to Duras (19 bbl) must take to Danville first and then from take Danville to Duras. Waste from Shelma to White water (36 bbl) must be transported and dropped at Columbus first and from there it must be shipped to White water. Waste from Shelma to Duras ($17 bbl) must be transported and dropped at Macon first and from there it must be shipped to Duras.

(5) Conclusion

Using the original problem the minimum cost of the company was estimated as $2822. But, using certain intermediate points to drop the waste to get cost advantage the original problem was modified and again solve using solver. The result showed that the new model caused a considerable reduction in transportation costs. The optimum cost of the Stateline Shipping Company reduced to $2630. To get this cost advantage company must use the following roots for transportation. Waste from Kingsport to Duras must take to Macon first and then from take Macon to Duras. Waste from Macon Los Canos should be transported to Selma first and then from Shelma to Los Canos. Waste from Shelma to White water must be transported and dropped at Columbus first and from there it must be shipped to White water. Waste from Shelma to Duras must be transported and dropped at Macon first and from there it must be shipped to Duras. Waste from Allentown to White water must be transported and dropped at Kingsport first and from there it must be shipped to White water. Finally waste from Allentown to Duras must be transported and dropped at Danville first and from there it must be shipped to Duras.

Limitations
The main limitation is that, in the above calculations, we have considered the cost aspect only. Handling of very dangerous industrial wastes cannot be considered based on cost aspect alone. In our problem, it should be noted that this waste contains hazardous elements and hence transportation of waste to many intermediate positions will increase the risk of the company. Another important limitation of this study is we have not considered the time required to transport waste from one station to another. If the waste from the sites do not reach at the disposable sites at proper time, dispose of wastes may become difficult. If we use intermediate sites to drop waste for cost advantage this delay become more serious. Also note that transporting and dumping of waste from one plant to another plant which is not meant for waste disposable may become legal problems.