Methodology

\(\text { Minimise } z=\sum_{i=1}^{m} \sum_{j=1}^{n} C_{i j} X_{i j}\)

\(\sum_{j=1}^{n} X_{i} \leq a_{i}, i=1,2,3 \ldots m\)

(Demand constraint)

\(\sum_{j=1}^{n} X_{i j} \geq b_{j}(j=1,2,3 \ldots n)\)

(Supply constraint)

\(X_{i j} \geq 0(i=1,2,3 \ldots m, j=1,2,3 \ldots n)\)

This is a linear program with m, n decision variables, m+n functional constraints, and m, n non-negative constraints.

m=Number of sources, n= Number of destinations, ai= Capacity of ith source (in tons, pounds, litres, etc.), bj =Demand of jth destination (in tons, pounds, litres, etc.)

cij = cost coefficients of material shipping (unit shipping cost) between ith source

and jth destination (in $ or as a distance in kilometres, miles, etc.), xij= amount of material shipped between ith source and jth destination (in tons, pounds, litres etc.)

A necessary and sufficient condition for the existence of a feasible

\(\sum_{i=1}^{m} a_{i}=\sum_{j=1}^{n} b_{j}\)

Remark. The set of constraints

\(\sum_{j=1}^{n} X_{i j}=b_{j} \text { and } \sum_{i=1}^{m} X_{i j}=a_{i}\)


represents m+n equations in non-negative variables. Each variable appears in exactly two constraints, one is associated with the origin and the other is associated with the destination.

Unbalanced Transportation Problem

 If \( \sum_{i=1}^{m} a_{i} \neq \sum_{j=1}^{n} b_{j} \)


The transportation problem is known as an unbalanced transportation problem. There are two cases:
Case (1)

\(\sum_{i=1}^{m} a_{i}>\sum_{j=1}^{n} b_{j}\)

Case (2)

\(\sum_{i=1}^{m} a_{i}<\sum_{j=1}^{n} b_{j}\)


Introduce a dummy origin in the transportation table; the cost associated with this origin is set equal to zero. The availability at this origin is:

\(\sum_{i=1}^{m} a_{i}<\sum_{j=1}^{n} b_{j}\)