Simplex Method Meeting 5. Course: D0744 - Deterministic Optimization Year: 2009. What to learn?. Artificial variables Big-M method. The Facts. To start, we need a canonical form
Download Policy: Content on the Website is provided to you AS IS for your information and personal use and may not be sold / licensed / shared on other websites without getting consent from its author.While downloading, if for some reason you are not able to download a presentation, the publisher may have deleted the file from their server.
Simplex MethodMeeting 5
Course: D0744 - Deterministic Optimization
Year: 2009
What to learn?
Artificial variables
Big-M method
The 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.
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!
??????????????????
Compare!
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 0+0+0 5!
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 0+0+0 5
Idea: solve a first problem with
Min {a | constraint (I) + a 0 + other constraints }!
One Equality???
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.
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?
?
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?
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
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!
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!
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
Big-M Method
Combine both objectives :
(1)Min iai
(2)Max j cjxj
into a single one:
(3)Max – M iai + j cjxj
where M is a large number, larger than anything subtracted from it.
If one minimizes j cjxj
then the combined objective function is
Min M iai + j cjxj
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.
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.