Simplex Method Meeting 5 Course: D Deterministic Optimization Year: 2009

Bina Nusantara University 3 What to learn? Artificial variables Big-M method

Bina Nusantara University 4 The Facts 1.To start, we need a canonical form 2.If we have a  constraint with a nonnegative right-hand side, it will contain an obvious basic variable (which?) after introducing a slack var. 3.If we have an equality constraint, it contains no obvious basic variable 4.If we have a  constraint with a nonnegative right-hand side, it contains no obvious basic variable even after introducing a surplus var.

Bina Nusantara University 5 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! ??????????????????

Bina Nusantara University 6 One Equality??? 2x + 3y = 5  2x + 3y + a = 5, a = 0 (I) (s basic, but it should be 0!) How do we force a = 0? This is of course not feasible if x=y=0, as  5!

Bina Nusantara University 7 One Equality??? 2x + 3y = 5  2x + 3y + a = 5, a = 0 (I) (a basic, but it should be 0!) How do we force a = 0? This is of course not feasible if x=y=0, as  5 Idea: solve a first problem with Min {a | constraint (I) + a  0 + other constraints }!

Bina Nusantara University 8 Artificial Variables Notice: In an equality constraint, the extra variable is called an artificial variable. For instance, in 2x + 3y + a = 5, a = 0 (I) a is an artificial variable.

Bina Nusantara University 9 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? ?

Bina Nusantara University 10 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?

Bina Nusantara University 11 2x + 3y  5  2x + 3y - s = 5, s  0 (I) 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

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Bina Nusantara University 13 Artificial Variables Notice: In a  inequality constraint, the extra variable is called an artificial variable. For instance, in 2x + 3y – s + a = 5, s  0, a  0 (I) a is an artificial variable. In a sense, we allow temporarily a small amount of cheating, but in the end we cannot allow it!

Bina Nusantara University 14 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!

Bina Nusantara University 15 Then What? We have two objectives: Get a “feasible” canonical form Maximize our original problem Two methods:  big M method  phase 1, then phase 2

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Bina Nusantara University 17 Big-M Method Combine both objectives : (1)Min  i a i (2)Max  j c j x j into a single one: (3)Max – M  i a i +  j c j x j where M is a large number, larger than anything subtracted from it. If one minimizes  j c j x j then the combined objective function is Min M  i a i +  j c j x j

Bina Nusantara University 18 The Big M Method The simplex method algorithm requires a starting bfs. Previous problems have found starting bfs by using the slack variables as our basic variables. If an LP have ≥ or = constraints, however, a starting bfs may not be readily apparent. In such a case, the Big M method may be used to solve the problem. Consider the following problem.

Bina Nusantara University 19 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.

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