Chapter 3 New Alternate Methods of Transportation Problem
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CHAPTER 3 NEW ALTERNATE METHODS OF TRANSPORTATION PROBLEM
3.1 Introduction The transportation problem and cycle canceling methods are classical in optimization. The usual attributions are to the 1940's and later. However, Tolsto (1930) was a pioneer in operations research and hence wrote a book on transportation planning which was published by the National Commissariat of Transportation of the Soviet Union, an article called Methods of ending the minimal total kilometrage in cargo-transportation planning in space, in which he studied the transportation problem and described a number of solution approaches, including the, now well-known, idea that an optimum solution does not have any negative-cost cycle in its residual graph. He might have been the first to observe that the cycle condition is necessary for optimality. Moreover, he assumed, but did not explicitly state or prove, the fact that checking the cycle condition is also sufficient for optimality. The transportation problem is concerned with finding an optimal distribution plan for a single commodity. A given supply of the commodity is available at a number of sources, there is a specified demand for the commodity at each of a number of destinations, and the transportation cost between each source-destination pair is known. In the simplest case, the unit transportation cost is constant. The problem is to find the optimal distribution plan for transporting the products from sources to destinations that minimizes the total transportation cost. This can be seen in Figure 1.
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Here sources indicated the place from where transportation will begin, destinations indicates the place where the product has to be arrived and cij indicated the transportation cost transporting from source to destination and Sink denotes the destination. There are various types of transportation models and the simplest of them was first presented by Hitchcock (1941). It was further developed by Koopmans (1949) and Dantzig
CHAPTER 3 NEW ALTERNATE METHODS OF TRANSPORTATION PROBLEM
3.1 Introduction The transportation problem and cycle canceling methods are classical in optimization. The usual attributions are to the 1940's and later. However, Tolsto (1930) was a pioneer in operations research and hence wrote a book on transportation planning which was published by the National Commissariat of Transportation of the Soviet Union, an article called Methods of ending the minimal total kilometrage in cargo-transportation planning in space, in which he studied the transportation problem and described a number of solution approaches, including the, now well-known, idea that an optimum solution does not have any negative-cost cycle in its residual graph. He might have been the first to observe that the cycle condition is necessary for optimality. Moreover, he assumed, but did not explicitly state or prove, the fact that checking the cycle condition is also sufficient for optimality. The transportation problem is concerned with finding an optimal distribution plan for a single commodity. A given supply of the commodity is available at a number of sources, there is a specified demand for the commodity at each of a number of destinations, and the transportation cost between each source-destination pair is known. In the simplest case, the unit transportation cost is constant. The problem is to find the optimal distribution plan for transporting the products from sources to destinations that minimizes the total transportation cost. This can be seen in Figure 1.
21
Here sources indicated the place from where transportation will begin, destinations indicates the place where the product has to be arrived and cij indicated the transportation cost transporting from source to destination and Sink denotes the destination. There are various types of transportation models and the simplest of them was first presented by Hitchcock (1941). It was further developed by Koopmans (1949) and Dantzig