Ans.

Transportation Problem & its basic assumption This model studies the minimization of the cost of transporting a commodity from a number of sources to several destinations. The supply at each source and the demand at each destination are known. The transportation problem involves m sources, each of which has available. i (i = 1, 2, â€Ś..,m) units of homogeneous product and n destinations, each of which requires bj (j = 1, 2â€Ś., n) units of products. Here a i and bj are positive integers. The cost cij of transporting one unit of the product from the ith source to the jth destination is given for each i and j . The objective is to develop an integral transportation schedule that meets all demands from the inventory at a minimum total transportation cost.It is assumed that the total supply and the total demand are equal.i.e.

Condition (1)The condition (1) is guaranteed by creating either a fictitious destination with a demand equal to the surplus if total demand is less than the total supply or a (dummy) source with a supply equal to the shortage if total demand exceeds total supply. The cost of transportation from the fictitious destination to all sources and from all destinations to the fictitious sources are assumed to be zero so that total cost of transportation will remain the same.

Formulation of Transportation Problem The standard mathematical model for the transportation problem is as follows. Let xij be number of units of the homogenous product to be transported from source i to the destination j Then objective is to

Theorem: A necessary and sufficient condition for the existence of a feasible solution to the transportation problem (2) is that

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