**LEARNING OBJECTIVES **

- Formulating a transportation problem
- Finding basic feasible solution of a transportation problem
- Determining minimum transportation costs
- Making an unbalanced transportation problem a balanced one
- Understanding the maximization objective function related to transportation problems
- Solving transportation problems using different methods

**CHAPTER OUTLINE **

**1.**Introduction**2.**Terminology used in transportation model**3.**Solution of transportation problem- North-West corner rule
- Row-Minima method
- Column-Minima method
- Least-Cost method
- VAM
**4.**Performing optimality test- Stepping stone method
- MODI method or UV method
**5.**Degeneracy in the transportation problem**6.**Review and Discussion Questions

**INTRODUCTION **

The transportation model is basically a Linear Program. Simplex algorithm can be used to solve any LP model but since it is very laborious and time consuming, there is a need to look for a model with simplified calculations. The name of the model is based on the fact that it was first used for transportation problems. The model deals with the determination of minimum cost for transporting one commodity from a number of sources for example manufacturing units to a number of destinations e.g. clearing and forwarding (C and F) agents. The name can be misleading because it is not that only transportation and destination problems can be solved. It can be extended to be used for facility and location (plant location, machine assignments) planning as also for many production planning and control (PPC) problems.

Studies were conducted in 1940′s to find the optimal solution to the problem of distributing product from several sources to numerous locations. These problems were essentially confined to number of shipping sources and number of destinations with specific capacities and requirements. Most of the problems require minimizing the transportation cost. Since these problems have the essential feature of Linear Programming, they can be solved by LP simplex method. But as stated earlier, simplex method could be very complicated.

**DEFINITION **

The transportation model seeks the determination of a transporation plan of a single commodity from a number of sources to a number of destinations. The model must have the following information.

(a) Amount of demand at each destination

(b) Availability at each source

(c) The unit transportation cost of commodity from each source to each destination

Since we are concerned with only one commodity, the destination can get the commodity from any of the sources. The objective of the problems is to find out the amount (quantity) to be transported from each of the sources to each destination so that the total transportation cost is minimum.

Let us put the definition in mathematical terms

Let m = number of sources origin (say plant location)

n = number of destinations

a_{i} = number of units of the commodity available at source (i = 1,2,3 …. m)

b_{j} = number of units required at destination (j = 1,2,3, … n)

C_{ij} = unit transportation cost for transporting the commodity from source i to destination j.

It is clear that the objective in this type of situation is to determine the number of units to be transported from source i (ai) to destination j(b_{j}) so that the total transportation cost is minimized.

Let X_{ij} = the number of units to be transported from source i to destination j.

The above objective function is to be minimized with the constraints given above.

It may be seen that the equation representing the constraints as well as that of objective function are linear equations in x_{ij}. Hence, essentially it is a Linear Problem (LPP).

The model makes an assumption to simplify problem, the transportation cost on a given route is directly proportional to the number of units transported. It must be noted that the constraint of availability of commodity at Source must specify that all the transportation from the source cannot exceed the supply.

In real life situations, supply may not equal demand or exceed it. In such situations, the transportation model needs to be balanced.

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