Part 3 Linear Programming

1
Part 3 Linear Programming

• 3.4 Transportation Problem

2
(No Transcript)
3
The Transportation Model
4
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
  nm-1 equality constraints is linearly
  independent.

5
Constraint Structure
6
Problem Structure
7
Model Parameters
8
Transformation of Standard Form of Transportation
Problem into the Primal Form
9
Asymmetric Form of Duality
10
Dual Transportation Problem
11
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 uivjltcij for all ij, since uivj
  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.

12
Example
Amount Available a17 a210 a312
D1 D2 D3
D4
12 x11 3 x12 8 x13 4 x14
7 x21 4 x22 6 x23 9 x24
8 x31 7 x32 3 x33 6 x34
O1 O2 O3 Amount required
b14 b28 b311
b46
13
(No Transcript)
14
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.

15
Northwest Corner Rule
12 4 3 3 8 4
7 4 5 6 5 9
8 7 3 6 6 6
7 10 12
4 8 11
6
16
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.

17
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.

18
The importance of triangularity is the associated
method of back substitution in solving
19
Basis Triangularity

• Basis Triangularity Theorem Every basis of the
  transportation problem is triangular.

20
(No Transcript)
21
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!
22
Step 3
v112 v23
v35 v48
12 12 4 3 3 3 5 8 OK 8 4 VIOLATION
13 7 VIOLATION 4 4 5 6 6 5 9 9 OK
10 8 VIOLATION 1 7 OK 3 3 6 6 6 6
u1 0 u2 1 u3 -2
7 10 12
4 8 11
6
23
Cycle of Change
a1 a2 a3
v1 v2
v3 v4
-1 c11 x11 1 c12 x12 c13 0 c14 0
1 c21 0 -1 c22 x22 c23 x23 c24 0
c31 0 c32 0 c33 x33 c34 x34
u1 u2 u3
b1 b2
b3 b4
24
Selection of the New Basic Variable
25
Step 4 Find a basic feasible solution of the
dual problem Loop identification
26
Step 4 Move 4 unit around loop 1
v16 v23
v35 v48
6 12 0 3 3 7 5 8 0 8 4 0
7 7 4 4 4 1 6 6 5 9 9 0
4 8 0 1 7 0 3 3 6 6 6 6
u1 0 u2 1 u3 -2
7 10 12
4 8 11
6
27
Repeat Step 3
Violation Cell 14
28
Repeat Step 4 Move 5 unit around the loop
v16 v23
v31 v44
6 12 0 3 3 2 5 8 0 8 4 5
7 7 4 4 4 6 6 6 0 9 9 0
4 8 0 1 7 0 3 3 11 6 6 1
7 10 12
u1 0 u2 1 u3 2
4 8 11
6
NO VIOLATION!!!
29
Solution
30
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).

31
Example - Given Data
32
Temperature Partition
33
Definitions
34
Transportation Formulation
35
Cost Coefficients
36
Additional Constraints
37
Solution