Part 3 Linear Programming 3 4 Transportation Problem

Part 3 Linear Programming 3. 4 Transportation Problem

The Transportation Model

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.

Constraint Structure

Problem Structure

Model Parameters

Transformation of Standard Form of Transportation Problem into the Primal Form

Asymmetric Form of Duality

Dual Transportation Problem

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 -u 1, -u 2, …, -um for the product at the m origins and then receive v 1, v 2, …, 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 D 1 D 2 12 O 1 O 2 O 3 Amount required x 11 D 3 3 x 12 7 x 21 x 13 x 22 8 D 4 8 4 4 x 14 6 x 23 7 Amount Available 9 x 24 3 6 x 31 x 32 x 33 x 34 b 1=4 b 2=8 b 3=11 b 4=6 a 1=7 a 2=10 a 3=12

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 12 4 3 8 4 3 7 7 4 5 9 3 6 10 5 8 7 6 4 6 8 12 6 11 6

Triangular Matrix • Definition: A nonsingular square 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? 1. Find a row with exactly one nonzero entry. 2. 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 v 1=12 u 1 = 0 u 2 = 1 u 3 = -2 12 4 v 2=3 v 3=5 v 4=8 12 3 3 3 5 7 4 VIOLATION 5 4 6 5 6 9 7 3 OK 6 3 6 6 OK 13 10 8 1 VIOLATION 4 6 8 8 8 4 VIOLATION 9 10 OK 11 7 6 12

Cycle of Change v 1 v 2 v 3 v 4 u 1 -1 c 11 +1 c 12 x 11 x 12 0 u 2 +1 c 21 -1 c 22 c 23 0 x 22 x 23 0 u 3 c 31 0 c 24 c 33 x 33 b 2 c 14 0 c 32 0 b 1 c 13 b 3 c 34 x 34 b 4 a 1 a 2 a 3

Selection of the New Basic Variable

Step 4: Find a basic feasible solution of the dual problem – Loop identification

Step 4: Move 4 unit around loop 1 v 1=6 v 2=3 v 3=5 v 4=8 u 1 = 0 6 0 12 3 7 3 5 0 8 8 0 4 u 2 = 1 7 4 1 4 6 5 6 9 0 9 u 3 = -2 4 0 8 1 0 7 3 6 6 6 4 6 8 11 7 10 12

Repeat Step 3 Violation: Cell 14

Repeat Step 4: Move 5 unit around the loop v 1=6 v 2=3 v 3=1 v 4=4 12 3 5 0 8 8 5 4 u 1 = 0 6 0 u 2 = 1 7 4 6 4 6 0 6 9 0 9 u 3 = 2 4 0 8 1 0 7 3 11 3 6 1 6 4 8 11 NO VIOLATION!!! 7 6 10 12

Solution

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).

Example - Given Data

Temperature Partition

Definitions

Transportation Formulation

Cost Coefficients

Additional Constraints

Solution