Presentation is loading. Please wait.

Presentation is loading. Please wait.

1 Dantzig-Wolfe Decomposition. 2 Outline  block structure of a problem  representing a point by extreme points  Revised Simplex to the extreme point.

Similar presentations


Presentation on theme: "1 Dantzig-Wolfe Decomposition. 2 Outline  block structure of a problem  representing a point by extreme points  Revised Simplex to the extreme point."— Presentation transcript:

1 1 Dantzig-Wolfe Decomposition

2 2 Outline  block structure of a problem  representing a point by extreme points  Revised Simplex to the extreme point representation  an example

3 General Problem  min s.t.s.t.  B 1 x 1 = b 1,  B 2 x 2 = b 2,  …  B K x K = b K,  0  x k, k = 1, 2, …, K.  common in network- based problems  distribution of K types of products  B k x k = b k : constraints related to the flow of the kth type of products  constraints of common resources for the K products: 3

4 4 General Problem  min s.t.s.t.  B 1 x 1 = b 1,  B 2 x 2 = b 2, ……  B K x K = b K,  0  x k, k = 1, 2, …, K.

5 5 General Problem  min  s.t.  B 1 x 1 = b 1,  B 2 x 2 = b 2,  …  B K x K = b K,  0  x k, k = 1, 2, …, K. Possible to avoid solving a large problem?

6 6 A Numerical Example  Problem P2:  min–3x 1 – 2x 2 – 2x 3 – 4x 4,  s.t. x 1 + x 2 + 2x 3 + x 4  10,  x 1 + 2x 2  8,  x 2  3,  x 3 + 3x 4  6,  x 3  4,  x i  0.

7 7 A Numerical Example  feasible region: {(x 1, x 2 ): x 1 + 2x 2  8, x 2  3}  extreme points:  a feasible point: convex combination of the extreme points

8 8 A Numerical Example  feasible region: {(x 3, x 4 ): x 3 + 3x 4  6, x 3  4}  extreme points:  a feasible point: convex combination of the extreme points

9 9 A Numerical Example  Problem P3: problem in terms of extreme points  min -3(8   13 ) – 2(3   14 ) – 2(4   23 ) – 4(   24 ),  s.t. (8   13 ) + (3   14 ) + 2(4   23 ) + (   24 )  10,   11 +  12 +  13 +  14 = 1,   21 +  22 +  23 +  24 = 1,   ij  0,i = 1, 2; j = 1, 2, 3, 4. a subspace B k x k = b k is represented by a single constraint  n  kn = 1

10 10 A Numerical Example  Problem P3: problem in terms of extreme points  min –24  12 – 12  13 – 6  14 – 8  22 – (32/3)  23 – 8  24,  s.t. 8       24  10,   11 +  12 +  13 +  14 = 1,   21 +  22 +  23 +  24 = 1,   ij  0, i = 1, 2; j = 1, 2, 3, 4. The previous representation suits DW Decomposition more.

11 11 Practical?  impractical approach  impossible to generate all the extreme points of a block B k x k = b k  Dantzig-Wolfe Decomposition: check all extremely points without explicitly generating them

12 12 General Problem  K blocks, B k x k = b k, k = 1, …, K  N k extreme points in the kth block dual variable  0  dual variable  1  dual variable  2  dual variable  K  …

13 13 Reduced Cost for a Non-Basic Variable  kn  reduced cost of  kn

14 14 Most Negative Reduced Cost Check each extreme point for each block, which is equivalent to solving a linear program. Result: Solving K+1 small linear programs. where

15 15 General Approach  1  Form the master program (MP) by the  representation  2  Get a feasible solution of the MP; find the corresponding   3  Solve the subproblems to check the reduced costs of  kn  3.1  stop if the MP is optimal;  3.2  else carry a standard revised simplex iteration (i.e., identifying the entering and leaving variables, stopping for an unbounded problem, and determining B -1 otherwise)  3.3  go back to 2  if the problem is not unbounded

16 16 An Example of the Dantzig-Wolfe Approach An Example of the Dantzig-Wolfe Approach  min –3y 1 – 2y 2 – 2y 3 – 4y 4,  s.t. y 1 + y 2 + 2y 3 + y 4  10,  y 1 + 2y 2  8,  y 2  3,  y 3 + 3y 4  6,  y 3  4,  y i  0.

17 17 An Example of the Dantzig-Wolfe Approach An Example of the Dantzig-Wolfe Approach  problem in general form with

18 18 An Example of the Dantzig-Wolfe Approach An Example of the Dantzig-Wolfe Approach  let be the extreme points of B 1 x 1 = b 1  any point in B 1 x 1 = b 1 :  similarly, any point in B 2 x 2 = b 2 :

19 19 An Example of the Dantzig-Wolfe Approach An Example of the Dantzig-Wolfe Approach  in terms of the extreme points

20 20 An Example of the Dantzig-Wolfe Approach An Example of the Dantzig-Wolfe Approach  to solve the problem, introduce the slack variable x 5 and artificial variables x 6 and x 7

21 21 An Example of the Dantzig-Wolfe Approach An Example of the Dantzig-Wolfe Approach  initial basic variable x B = (x 5, x 6, x 7 ) T, B = I, (c B ) T = (0, M, M), b = (10, 1, 1) T   = (c B ) T B = (0, M, M)  for the kth subproblem, the reduced costs:


Download ppt "1 Dantzig-Wolfe Decomposition. 2 Outline  block structure of a problem  representing a point by extreme points  Revised Simplex to the extreme point."

Similar presentations


Ads by Google