Review of Operation Reasearch
There is a types on linear programming problem that may be solved using a simplified version of the simplex technique called transportation method. Because of it’s major application in solving problems involving several product sources and several destinations of production. This types of problem is frequently called a transportation problem. It gets it’s name from it’s application to problem involving transportation of products from several destinations.
The transportation model is a special class of linear programming. The model assumes that the shipping cost is proportional to the number of units shipped on given route. In general, the transportation model can be extended to other areas of operation including among others inventory control, employment scheduling and personal assignment. The objective of such problem are either minimizing the cost of shipping m units to N destination or maximize the profit of shipping M units to N destinations.
According to Hiller and Lieberman (2000) introduction to operations research (OR) can be traced back many decades, when early attempts were made to use a scientific approach in the management of organizations. The beginning of the activities (OR) has generally been attributed to the military services early in world war II.
Because of the war efforts, there was an urgent need to allocate scarce resource to the various military operation and to activities within each operation in an effective manner therefore, the British and then the U.S military manager called upon a larger number of scientists to apply a scientist approach to dealing with this and other strategic and tactical problems. In effects, oresearch. This brought the use of operation research within the easy reach of much number of people. Today literally, millions of individual have ready access to operation research software.
Let use assume there are m sources supplying N destinations. Sources capacities destination requirement, and cost material shipping from each source to each destinations are given constantly. The transportation. Problem and usually it appears in a transportation table.
The general problem is represented by the Network below
Network Representation of A Transportation problem
There are M source and N destinations, Each represented by a node. The area represents the routes linking the sources and destination Arc (ii) for instance joins source I to destination J carries two pieces of information. The transportation cost per units, cij the amount shipped xij the amount of supply at source i is ai and amount of demand at destination j is by the total transportation cost while satisfying all the supply and demand restrictions.
A general transportation model with M sources and n destination. However, because the transportation model is always balance i.e sum of supply equals sum of demand one of these equation is redundant. Here has M+N-I independents constraints equations which mean that the starting base solution consists of M+N-I basic variables.
Often there is an excess supply or demand in such situation for the transportation method to work, a dumping warehouse or factory must be added procedurally. This involves inserting an extra row (for an additional factory) or an extra column (for additional warehouse). The amount of supply or demand required by the “dummy” equals the difference between the row and column total
Examples” supposing total factory supply equals (=) 51 and total depot demand = 52 this involves inserting an extra row-an additional factory the amount of supply by the difference between the row and column total. In the case above
Dummy = 52-51=1
The cost figures in each of the dummy row would be set at zero. So any units sent these would not incur a transportation cost theoretically, this adjustment is equivalent to the simplex procedure of inserting a slack variable in a constraint inequality to convert it to an equation and as in the simplex; the cost of the dummy would be zero in objective function.