
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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