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Part 3 Linear Programming. 3.4 Transportation Problem. The Transportation Model. Theorem.
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Part 3 Linear Programming 3.4 Transportation Problem
Theorem A transportation problem always has a solution, but there is exactly one redundant equality constraint. When any one of the equality constraints is dropped, the remaining system of n+m-1 equality constraints is linearly independent.
Transformation of Standard Form of Transportation Problem into the Primal Form
Interpretation of the Dual Transportation Problem Let us imagine an entrepreneur who, feeling that he can ship more efficiently, come to the manufacturer with the offer to buy his product at origins and sell it at the destinations. The entrepreneur must pay -u1, -u2, …, -um for the product at the m origins and then receive v1, v2, …, vn at the n destinations. To be competitive with the usual transportation modes, his prices must satisfy ui+vj<=cij for all ij, since ui+vj represents the net amount the manufacturer must pay to sell a unit of product at origin i and but it back again at the destination j.
Example Amount Available a1=7 a2=10 a3=12 D1 D2 D3 D4 O1 O2 O3 Amount required b1=4 b2=8 b3=11 b4=6
Solution Procedure • Step 1: Set up the solution table. • Step 2: “Northwest Corner Rule” – when a cell is selected for assignment, the maximum possible value must be assigned in order to have a basic feasible solution for the primal problem.
Northwest Corner Rule 7 10 12 4 8 11 6
Triangular Matrix • Definition: A nonsingularsquare matrix M is said to be triangular if by a permutation of its rows and columns it can be put in the form of a lower triangular matrix. • Clearly a nonsingular lower triangular matrix is triangular according to the above definition. A nonsingular upper triangular matrix is also triangular, since by reversing the order of its rows and columns it becomes lower triangular.
How to determine if a given matrix is triangular? • Find a row with exactly one nonzero entry. • Form a submatrix of the matrix used in Step 1 by crossing out the row found in Step 1 and the column corresponding to the nonzero entry in that row. Return to step 1 with this submatrix. If this procedure can be continued until all rows have been eliminated, then the matrix is triangular.
The importance of triangularity is the associated method of back substitution in solving
Basis Triangularity • Basis Triangularity Theorem: Every basis of the transportation problem is triangular.
Step 3: Find a basic feasible solution of the dual problem – initial guess Due to one of the constraints in the primal problem is redundant!
Step 3 v1=12 v2=3 v3=5 v4=8 u1 = 0 u2 = 1 u3 = -2 7 10 12 4 8 11 6
Cycle of Change a1 a2 a3 v1 v2 v3 v4 u1 u2 u3 b1 b2 b3 b4
Step 4: Find a basic feasible solution of the dual problem – Loop identification
Step 4: Move 4 unit around loop 1 v1=6 v2=3 v3=5 v4=8 u1 = 0 u2 = 1 u3 = -2 7 10 12 4 8 11 6
Repeat Step 3 Violation: Cell 14
Repeat Step 4: Move 5 unit around the loop v1=6 v2=3 v3=1 v4=4 u1 = 0 u2 = 1 u3 = 2 7 10 12 4 8 11 6 NO VIOLATION!!!
Application – Minimum Utility Consumption Rates and Pinch Points Cerda, J., and Westerberg, A. W., “Synthesizing Heat Exchanger Networks Having Restricted Stream/Stream Matches Using Transportation Formulation,” Chemical Engineering Science, 38, 10, pp. 1723 – 1740 (1983).
