220 likes | 409 Views
Compare!. 2x 3y ? 5 ? 2x 3y s = 5, s ? 0 (s basic)2x 3y = 5 ? ??????? Infeasible if x=y=0!2x 3y ? 5 ? 2x 3y - s = 5, s ? 0 (??????) Infeasible if x=y=0!
E N D
1. Artificial Variables, 2-Phase and Big M Methods Facts:
To start, we need a canonical form
If we have a ? constraint with a nonnegative right-hand side, it will contain an obvious basic variable (which?) after introducing a slack var.
If we have an equality constraint, it contains no obvious basic variable
If we have a ? constraint with a nonnegative right-hand side, it contains no obvious basic variable even after introducing a surplus var. 1
2. Compare! 2x + 3y ? 5 ? 2x + 3y + s = 5, s ? 0 (s basic)
2x + 3y = 5 ? ??????? Infeasible if x=y=0!
2x + 3y ? 5 ? 2x + 3y - s = 5, s ? 0 (??????)
Infeasible if x=y=0!
??????????????????
2
3. One Equality??? 2x + 3y = 5 ? 2x + 3y + a = 5, a = 0 (E)
(a is basic, but it should be 0!)
How do we force a = 0? This is of course not feasible if x=y=0, as 0+0+0 ?5!
3
4. One Equality??? 2x + 3y = 5 ? 2x + 3y + a = 5, a = 0 (E)
(a basic, but it should be 0!)
How do we force a = 0? This is of course not feasible if x=y=0, as 0+0+0 ?5
Idea: solve a first problem with
Min {a | constraint (E) + a ? 0 + other constraints }! 4
5. Artificial Variables Notice: In an equality constraint, the extra variable is called an artificial variable.
For instance, in
2x + 3y + a = 5, a = 0 (E)
a is an artificial variable.
5
6. One Inequality ? ??? 2x + 3y ? 5 ? 2x + 3y - s = 5, s ? 0 (I)
s could be the basic variable,
but it should be ? 0
and for x=y=0, it is -5 !
How do we force s ? 0?
? 6
7. One Inequality ? ??? 2x + 3y ? 5 ? 2x + 3y - s = 5, s ? 0 (I)
s could be the basic variable,
but it should be ? 0
and it is -5 for x=y=0!
How do we force s ? 0?
By making it 0!
how?
7
8. One Inequality ? ??? 2x + 3y ? 5 ? 2x + 3y - s = 5, s ? 0 (E)
s could be basic, but it should be ? 0
and it is -5 for x=y=0!
How do we force s ? 0?
By making it 0! But we have to start with a canonical form
so
treat is as an equality constraint!
2x + 3y - s + a = 5, s ? 0, a ? 0 and Min a 8
9. Artificial Variables Notice: In a ? inequality constraint, the extra variable a is called an artificial variable.
For instance, in
2x + 3y s + a = 5, s ? 0, a ? 0 (E)
a is an artificial variable.
In a sense, we allow temporarily a small amount of cheating, but in the end we cannot allow it!
9
10. What if we have many such = and ? constraints? 7x - 3y s1 + a1 = 6, s1,a1 ? 0 (I)
2x + 3y + a2 = 5, a2 ? 0 (II)
a1 and a2 are artificial variables, s1 is a surplus variable.
One minimizes their sum:
Min {a1+a2 | a1, a2 ? 0, (I), (II), other constraints}
i.e., one minimizes the total amount of cheating! 10
11. Then What? We have two objectives:
Get a feasible canonical form
Maximize our original problem
Two methods:
?2-phase method (phase 1, then phase 2)
?big M method 11
12. 2-Phase Method Phase I: find a BFS
Minimize the sum of the artificial variables
If min = 0, we have found a BFS
If min > 0, then we cannot find a solution without cheating
the original problem is infeasible
Phase 2: solve original LP
Start from the phase 1 BFS, and maximize the original objective function.
12
13. Big-M Method Combine both objectives :
(1) Min ?i ai
(2) Max ?j cj xj
into a single one:
(3) Max M ?i ai + ?j cj xj
where M is a large number, larger than anything subtracted from it.
If one minimizes ?j cj xj
then the combined objective function is
Min M ?i ai + ?j cj xj 13
14. The Big M Method 14
15. Example Bevco manufactures an orange-flavored soft drink called Oranj by combining orange soda and orange juice. Each orange soda contains 0.5 oz of sugar and 1 mg of vitamin C. Each ounce of orange juice contains 0.25 oz of sugar and 3 mg of vitamin C. It costs Bevco 2˘ to produce an ounce of orange soda and 3˘ to produce an ounce of orange juice. Bevcos marketing department has decided that each 10-oz bottle of Oranj must contain at least 30 mg of vitamin C and at most 4 oz of sugar. Use linear programming to determine how Bevco can meet the marketing departments requirements at minimum cost.
15
16. The Big M Method 16
17. The Big M Method 17
18. The Big M Method 18
19. The Big M Method 19
20. 4.10 The Big M Method 20
21. The Big M Method 21