Many practical problems in operations research can be broadly formulated as linear programming problems, for which the simplex this is a general method and cannot be used for specific types of problems like,
(ii)transshipment models and
(iii) the assignment models.
The above models are also basically allocation models. We can adopt the simplex technique to solve them, but easier algorithms have been developed for solution of such problems. The following sections deal with the transportation problems and their streamlined procedures for solution.
In a transportation problem, we have certain origins, which may represent factories where we produce items and supply a required quantity of the products to a certain number of destinations. This must be done in such a way as to maximize the profit or minimize the cost. Thus we have the places of production as origins and the places of supply as destinations. Sometimes the origins and destinations are also termed as sources and sinks.
To illustrate a typical transportation model, suppose m factories supply certain items to n warehouses. Let factory i (i = 1, 2, …, m) produce ai units and the warehouse j (j = 1, 2, …, n) requires bj units. Suppose the
cost of transportation from factory i to warehouse j is cij. Let us define the decision variables xij being the amount transported from the factory i to the warehouse j. Our objective is to find the transportation pattern that will minimize the total transportation cost.
The model of a transportation problem can be represented in a concise tabular form with all the relevant parameters mentioned above. See table 1
Table 1 Origins (Factories) Destinations (Warehouses) Available
1 2 …….. n
…… … … …