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Linear Transportation Problem Example

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Transportation problem

There are three warehouses at different cities: Detroit, Pittsburgh and Buffalo. They have 250, 130 and 235 tons of paper accordingly. There are four publishers in Boston, New York, Chicago and Indianapolis. They ordered 75, 230, 240 and 70 tons of paper to publish new books.
There are the following costs in dollars of transportation of one ton of paper:

From \ To	Boston (BS)	New York (NY)	Chicago (CH)	Indianapolis (IN)
Detroit (DT)	15	20	16	21
Pittsburgh (PT)	25	13	5	11
Buffalo (BF)	15	15	7	17

We denote the cost of transportation from one city to another by C_ prefix, for example cost from Buffalo to Chicago is C_BF_CH.

We have to find a plan that all orders will be performed and the transportation costs will be minimized. The decision variables are amounts of tons transported from one city to another denoted by prefix X_:
X_DT_BS= tons from Detroit to Boston,
X_PT_BS= tons from Pittsburgh to Boston,
X_BF_NY= tons from Buffalo to New York and etc.

The objective function is sum of multiplications of transported tons and their costs:
F = C_DT_BS*X_DT_BS + C_DT_NY*C_DT_NY + C_DT_CH*C_DT_CH + �

The constraints are defined in terms of the transported paper amount from one warehouse or to one publisher.

For example:
Amount transported from Detroit cannot be more than 250 tons:
X_DT_BS + X_DT_NY + X_DT_CH + X_DT_IN <= 250

The problem has been solved successfully with the following plan:

From \ To	Boston	New York	Chicago	Indianapolis	Sent
Detroit	75	175	-	-	250
Pittsburgh	-	21.34	35.66	70	130
Buffalo (BF)	-	33.66	201.34	-	235
Received:	75	230	240	70

The Total Cost of Transportation is $7780

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Linear programming examples

There are several examples of linear programming intended to make the users of GIPALS familiar with it. These examples are included in GIPALS installation and can be found in ..\GIPALS\Examples folder.

Transport problem

Production (profit maximization) problem

Dietary optimization

Alloy mixture optimization

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