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CHAPTER 7 Transportation Problem
7.1 Introduction The transportation problem is one of the subclasses of LPPs in which the objective is to transport various quantities of a single homogeneous commodity that are initially stored at various origins to different destinations in such a way that the transportation cost is minimum. To achieve this objective we must know the amount and location of available supplies and the quantities demanded. In addition we must known the costs that result from transporting one unit of commodity from various origins to various destinations. Let m be the number of sources n be the number of destinations ai be the supply at the source i bj be the demand at the destination j cij be the cost of transportation per unit from source i to destination j Xij be the number of units to be transported from the source i to the destination j. The Linear programming model representing the transportation problem is given by Minimize Z = ∑∑ C ij X ij
i =1 j=1 m n
subject to the constraints
∑X
j=1
n
ij
≤ a ij ,
i = 1, 2, 3....m
(Row Sum) and
∑X
i =1
m
ij
≥ b ij ,
j = 1, 2, 3....n
(Column Sum)
X ij ≥ 0 for all i and j
The objective function minimizes the total cost of transportation (z) between various sources and destinations. The constraint i in the first set of constraints ensures that the total units transported from the source i is less than or equal to its supply. The constraint j in the second set of constraints ensures that the total units transported to the destination j is greater than or equal to its demand.
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Fig. 7.1 Schematic Diagram of Simple Transportation Example 3.1 Consider the following transportation problem (Table 3.3) involving 3 sources and 3 destinations. Develop a linear programming (LP) model for this problem and solve it. The given transportation problem is said to be balanced if
∑ai = ∑bj
i =1 j=1
m
n
ie. if the total supply is equal to the total demand....