1. Artificial Variables, 2-Phase and Big M Methods
For the example, go to p. 16.

2. 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.
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3. 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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4. 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 5!
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5. 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 5
Idea: ! solve a first problem with
Min {a | constraint (E) + (a  0) + other constraints }!
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6. 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.
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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 for x=y=0, it is -5 !
How do we force s  0? ?
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8. how? 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?
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9. 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
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10. 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!
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11. 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!
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12. 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
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13. 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 with no “cheating”… the original problem is infeasible
Phase 2: solve original LP
Start from the phase 1 BFS, and maximize the original objective function.
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14. 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
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15. The simplex method algorithm requires a starting bfs.
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.
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16. 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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17. The Big M Method
Let x1 = number of ounces of orange soda in a bottle of Oranj
x2 = number of ounces of orange juice in a bottle of Oranj x1 x2

18. The Big M Method
Letting x1 = number of ounces of orange soda in a bottle of Oranj
x2 = number of ounces of orange juice in a bottle of Oranj
The LP is:
min z = 2x x2
st 0.5x x2 ≤ 4 (sugar constraint)
x x2 ≥ 20 (Vitamin C constraint)
x x2 = 10 (10 oz in 1 bottle)
x1, x2  0
The LP in standard form is shown on the next slide.
© Brooks/Cole

19. The Big M Method Row 1:-z + 2x1 + 3x2 = 0
Row 2: x x2 + s = 4
Row 3: x x s2 = 20
Row 4: x x = 10
The LP in standard form has z and s1 which could be used for BVs but row 2 would violate sign restrictions and row 3 no readily apparent basic variable.
In order to use the simplex method, a bfs is needed. To remedy the predicament, artificial variables are created. The variables will be labeled according to the row in which they are used as seen below. The basic variables are in red:
Row 1:-z + 2x x = 0
Row 2: x x2 + s = 4
Row 3: x x s2 + a = 20
Row 4: x x a3 = 10
© Brooks/Cole

20. The Big M Method
If all artificial variables in the optimal solution equal zero, the solution is optimal. If some artificial variables are positive in the optimal solution, the problem is infeasible.
The Bevco example continued:
Initial Tableau
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21. 4.10 – The Big M Method

22. The Big M Method