Minimizing the Cost of Transportation

Read this article. The study employed a transportation model to find the minimum cost of transporting manufactured goods from factories to warehouses. As you read, think of some factors that can greatly increase transportation and logistics time.


\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}

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}