Stateline Shipping and Transport Company

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1. Formulation of the Problem
The Stateline Shipping and Transport Company wanted to transport industrial wastes from the 6 plants to the 3 waste disposable sites. The problem can be represented in as a Transportation table as shown below. Our problem is to find roots to disposable sites, such that the cost of transportation is minimized..

White waterLos CanosDurasAvailability (bbl)
Kingsport$12.00$15.00$17.0035
Danville$14.00$9.00$10.0026
Macon$13.00$20.00$11.0042
Selma$17.00$16.00$19.0053
Columbus$7.00$14.00$12.0029
Allentown$22.00$16.00$18.0038
Capacity (barrels)6580105

Mathematical Formulation
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. Then the objective function Z representing the cost and different constraints of the problem can be written 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 transportation problem described above can be solved mathematically using a computer package. The solution to the problem using excel solver are given in the following table.

WhitewaterLos CanosDuras
Kingsport35---------
Danville--------26
Macon---------42
Selma152-----
Columbus29----------
Allentown------2810
Dummy---------27

The above table can be explained, in the following way
(1)35 bbl from Kingport, to the Whitewater waste disposal plant. (2)1 bbl from Selma to the Whitewater waste disposal plant. (3)29 bbl from Columbus to the Whitewater waste disposal plant. (4)52 bbl from Selma to Los Canos waste disposal plant.

(5)28 bbl from Allentown to Los Canos waste disposal plant. (6)26 bbl from Danville to the Duras waste disposal plant.
(7)42 bbl from Macon to the Duras waste disposal plant.
(8)10 bbl from Allentown to the Duras waste disposal plant.

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

3. Modified Problem Using Intermediate Sites

A closer look at the problem reveals that, there are certain other roots exists in the model and using these roots along with some intermediate positions in these roots to drop the waste and then picked up to the final destination will reduce transportation cost, instead of transport them directly to a waste disposal centre. So dropping the waste at certain intermediate position as described below will give considerable cost advantage to the shipping company. So by considering all such possibilities some intermediate places are found to drop the waste to minimize transportation costs. All such roots with the lowest cost are given in bold letter in the following table.

Sl. No.Plant FromIntermediate PositionDisposal centreCostMinimum cost 1Kingsport--DirectDuras$17$15
KingsportDanvilleDuras$15
2Macon--DirectLos Canos$20$19
MaconSelmaLos Canos$19
3Selma--DirectWhite water$17$10
SelmaColumbusWhite water$10
4Selma--DirectDuras$19$14
SelmaMaconDuras$14
5Allentown--DirectWhite water$22$20
AllentownKingsportWhite water$20
6Allentown--DirectDuras$18$17
AllentownDanvilleDuras$17

Since, we found 6 new roots, through which cost is less compared to the original transportation problem; the new transportation table with reduced costs are given below. The reduced costs are given in bold letter.

White waterLos CanosDurasAvailability (bbl)
Kingsport$12.00$15.00$15.0035
Danville$14.00$9.00$10.0026
Macon$13.00$19.00$11.0042
Selma$10.00$16.00$14.0053
Columbus$7.00$14.00$12.0029
Allentown$20.00$16.00$17.0038
Capacity (barrels)6580105

The...
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