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Artificial Variables, 2-Phase and Big M Methods

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!

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Artificial Variables, 2-Phase and Big M Methods

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    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. Bevco’s 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 department’s 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

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