L14_10 - ESI 6314 Deterministic Methods in Operations...

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1/24 ESI 6314 Deterministic Methods in Operations Research Lecture Notes University of Florida Department of Industrial and Systems Engineering / REEF

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2/24 The transportation simplex method Previously, we have considered three diﬀerent methods for ﬁnding a bfs for a transportation problem: 1 Northwest Corner (NWC) method 2 Minimum Cost (MC) method 3 Vogel’s method Recall that in these methods we use the fact that in a balanced transportation problem we may omit any of the m + n constraints, since it will be automatically satisﬁed if all the remaining constraints are satisﬁed. We may arbitrarily assume that that the constraint that we drop is the ﬁrst supply constraint. As a result, we have m + n - 1 constraints and, therefore, m + n - 1 basic variables. How do we know if a given set of m + n - 1 variables can form a basis?
3/24 The transportation simplex method The matrix B must be nonsingular! To easily test whether B is nonsingular and thus the corresponding variables of the transportation problem can form a bfs, we will use the concept of a loop deﬁned for the transportation table. Deﬁnition An ordered sequence of at least four diﬀerent cells is called a loop if 1 Any two consecutive cells lie in either the same row or same column 2 No three consecutive cells lie in the same row or column 3 The last cell in the sequence has a row or column in common with the ﬁrst cell in the sequence

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4/24 The transportation simplex method Theorem In a balanced transportation problem with m supply points and n demand points, the cells corresponding to a set of m + n - 1 variables contain no loop if and only if the m + n - 1 variables yield a bfs.
5/24 The transportation simplex method Based on the transportation tableau, the following steps should be performed. 1 Determine (by a criterion to be developed shortly) the variable that should enter the basis. 2 Find the loop (it can be shown that there is only one loop) involving the entering variable and some of the basic variables. 3 Counting the cells in the loop (starting with 0 corresponding to the entering variable), label them as even cells or odd cells. 4 Find an odd cell whose variable assumes the smallest value θ .

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L14_10 - ESI 6314 Deterministic Methods in Operations...

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