Monge-Kantorovich Transportation Problem

Some homogeneous goods are stocked in storage centers in quantities , , , …, . The goods are required in corresponding consumption centers , , , …, . Knowing the unit transportation costs from each storage center to each consumption center, a plan that uses all the goods (offers) and satisfies all the consumers (requests) is needed.
denotes the quantity of goods transported from storage center to consumption center . The following is a mathematical model of the problem, where we try to minimize the total transportation cost :
,
, ,
, ,
is the total offer,
is the total request.
The transportation problem can be solved if . If , we introduce a fictional consumption center with the offer and transportation cost 0. if , we introduce a fictional storage center with the request and transportation cost 0.
The problem is shown for . The result shows a matrix whose row and column sums are the and . This matrix shows the quantity that each storage center should ship to each consumption center. If , we can obtain a 4×5 () matrix or a 5×4 () matrix, where the extra row or column corresponds to a fictional center.

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