ALGORITHM FOR TRANSPORTATION AND ASSIGNMENT PROBLEMS GASPAR ASAMPANA 1. TRANSPORTATION PROBLEM 1.1. INTRODUCTION A transportation problem involves the shipment of goods from various origins or sources of supply to a set of destinations (e.g. retail outlets, cities), each destination demanding a specified level of the commodity. For a typical transportation problem, there are: 1. m origins with the ith having a supply of s i units, i  1... m 2. n destinations with the jth demanding d j units, j  1,... n 3. cost c ij for shipping a unit of the commodity from origin i to destination j. The cost of distributing units from any origin to any particular location s i is proportional to the number of units distributed. The objective function is to minimize total transportation cost or to maximize the total profit contribution from the allocation. 1.2. CLASSICAL TRANSPORTATION MODEL The transportation problem can be formulated into a LP problem. Let x ij , i  1... m, j  1... n be the number of units transported from origin i to destination j. The LP problem is as follows Minimize :  ii m  j1 n c ij x ij Subject to :  j1 n x ij  s i , i  1..., m 1  i1 m x ij  d j , j  1,..., n x ij  0, i  1,... m, j  1,... n. A transportation problem is said to be balanced if  i1 m s i   j1 n d j , All constraints in any balanced transportation problem must be binding. For a balanced TP, the original problem is equivalent to 1