Transportation And Assignment Problem
Chapter 6.
1.0 TRANSPORTATION PROBLEM
The transportation problem is a special class of the linear programming problem. It deals with the situation in which a commodity is transported from Sources to Destinations. The objective is to determine the amount of commodity to be transported from each source to each destination so that the total transportation cost is minimum.
EXAMPLE 1.1
A soft drink manufacturing firm has m plants located in m different cities. The total production is absorbed by n retail shops in n different cities. We want to determine the transportation schedule that minimizes the total cost of transporting soft drinks from various plants to various retail shops. First we will formulate this as a linear programming problem.
MATHEMATICAL FORMULATION
Let us consider the m-plant locations (origins) as O1 , O2 , …., Om and the n-retail shops (destination) as D1 , D2 , ….., Dn respectively. Let ai 0, i= 1,2, ….m , be the amount available at the ith plant Oi . Let the amount required at the jth shop Dj be bj 0, j= 1,2,….n.
Let the cost of transporting one unit of soft drink form ith origin to jth destination be Cij , i= 1,2, ….m, j=1,2,….n. If xij 0 be the amount of soft drink to be transported from ith origin to jth destination , then the problem is to determine xij so as to Minimize
Subject to the constraint and xij 0 , for all i and j.
This lPP is called a Transportation Problem.
THEOREM 1.1
A necessary and sufficient condition for the existence of a feasible solution to the transportation problem is that
Remark. The set of constraints
Represents m+n equations in mn non-negative variables. Each variable xij appears in exactly two constraints, one is associated with the origin and the other is associated with the destination.
Note. If we are putting in the matrix from, the elements of A are either 0 or 1.
THE TRANSPORTATYION TABLE: D1 D2 ……
1.0 TRANSPORTATION PROBLEM
The transportation problem is a special class of the linear programming problem. It deals with the situation in which a commodity is transported from Sources to Destinations. The objective is to determine the amount of commodity to be transported from each source to each destination so that the total transportation cost is minimum.
EXAMPLE 1.1
A soft drink manufacturing firm has m plants located in m different cities. The total production is absorbed by n retail shops in n different cities. We want to determine the transportation schedule that minimizes the total cost of transporting soft drinks from various plants to various retail shops. First we will formulate this as a linear programming problem.
MATHEMATICAL FORMULATION
Let us consider the m-plant locations (origins) as O1 , O2 , …., Om and the n-retail shops (destination) as D1 , D2 , ….., Dn respectively. Let ai 0, i= 1,2, ….m , be the amount available at the ith plant Oi . Let the amount required at the jth shop Dj be bj 0, j= 1,2,….n.
Let the cost of transporting one unit of soft drink form ith origin to jth destination be Cij , i= 1,2, ….m, j=1,2,….n. If xij 0 be the amount of soft drink to be transported from ith origin to jth destination , then the problem is to determine xij so as to Minimize
Subject to the constraint and xij 0 , for all i and j.
This lPP is called a Transportation Problem.
THEOREM 1.1
A necessary and sufficient condition for the existence of a feasible solution to the transportation problem is that
Remark. The set of constraints
Represents m+n equations in mn non-negative variables. Each variable xij appears in exactly two constraints, one is associated with the origin and the other is associated with the destination.
Note. If we are putting in the matrix from, the elements of A are either 0 or 1.
THE TRANSPORTATYION TABLE: D1 D2 ……