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CHAPTER 2, , PRELIMINARIES, In this chapter, we present few important existing basic concepts and well known, results in transportation theory, optimal theory, interval and fuzzy set theory and, rough set theory which are used in the thesis., Now, we present some terms in transportation theory which are used in the thesis., , 2.1, , Transportation and optimal theory, , 2.1.1, , Transportation problem with equality constraints, , Consider the following transportation problem with equality constraints, m, , (P), , Minimize Z , i 1, , n, , , j 1, , cij xij, , Subject to, n, , xij ai , i 1, 2,..., m, , (2.1), , j 1, , m, , xij b j , j 1, 2,..., n, , (2.2), , xij 0 , for all i and j and integers,, , (2.3), , i 1, , Where m = the number of supply points;, n = the number of demand points;, xij = the number of units shipped from supply point i to demand point j;, cij = the cost of shipping one unit from supply point i to the demand point j;, , 14
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ai = the supply at supply point i and, b j = the demand at demand point j., , Now, the above problem can be put in the following transportation table, Table 2.1 Transportation problem with equality constraints, , Supply, , Demand, , 2.1.2, , c11, , c12, , , , c1 n, , a1, , c21, , c22, , , , c2 n, , a2, , , , , , , , , , , , cm1, , cm 2, , , , cm n, , am, , b1, , b2, , , , bn, , Balanced transportation problem, , A transportation problem is said to be balanced if the total supply from all sources, equals to the total demand in the destinations, that is,, , 2.1.3, , m, , n, , i 1, , j 1, , ai b j ., , Unbalanced transportation problem, , A transportation problem is said to be unbalanced if the total supply from all sources, is not equal to the total demand in the destinations, that is,, 2.1.4, , m, , n, , i 1, , j 1, , ai b j ., , Feasible solution, , Any set of non-negative allocations of a transportation problem which satisfies the, row sum or column sum (if the problem is unbalanced) or both (if the problem is, balanced) is called a feasible solution., 2.1.5, , Basic feasible solution, , Any feasible solution is called basic feasible solution if the number of non-negative, allocations is equal to m n 1 ., , 15
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2.1.6, , Optimal solution, , A feasible solution which minimizes the total shipping cost is said to be an optimal, solution., 2.1.7, , Non-degenerate solution, , Any feasible solution to a transportation problem containing m-sources and ndestinations is said to be non-degenerate if it contains m n 1 occupied cells and, each allocation in independent positions., 2.1.8, , Degenerate solution, , If the number of occupied cells in any feasible solution is less than m n 1 then the, solution is called a degenerate solution (i.e., The number of allotted cells m n 1 )., , 2.2, , Interval and fuzzy set theory, , We need the following definitions of the basic arithmetic operators and partial, ordering on closed bounded intervals and closed bounded fuzzy intervals and also, the, basic definitions of fuzzy set, triangular fuzzy number which can be found in George, J. Klir and Bo Yuan (2008), Ebrahimnejad (2015)., 2.2.1, , Interval theory, , Let D denote the set of all closed bounded intervals on the real line R. That is,, D [a, b] : a b, a and b are in R ., , Definition 2.2.1: Let A [a, b] and B [c, d ] be in D. Then,, (i) A B [a c, b d ], (ii) AB [a d , b c], (iii) kA [ka, kb] if k is a positive real number, (iv) kA [kb, ka] if k is a negative real number and, (v) A B [ p, q] where p min{ac, ad , bc, bd} and q max{ac, ad , bc, bd}., Definition 2.2.2: Let A [a, b] and B [c, d ] be in D. Then,, (i) A B if and only if a c and b d and, (ii) A B if and only if a c and b d ., , 16
