1. Outline Simplex algorithm The initial basic feasible solution
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2. An example of simplex algorithm
Linear program in standard form
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3. An example of simplex algorithm
Putting the linear program into slack form(29.1)
The system of constraints (29.62)-(29.64) has 3 equations and 6 variables. Any setting of the variables defines
values for ; therefore an infinite number of solutions to this system of equations.
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4. Basic solution
Set all the (nonbasic) variables on the right- hand to 0 and then compute the values of the (basic) variables on the left-hand side. In this example,the basic solution is
If a basic solution is also feasible, we call a basic feasible solution.
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5. Our goal
Our goal in each iteration is to reformulate the linear program so that the basic solution has a greater objective value.
We select a nonbasic variable x whose coefficient in the objective function is positive and we increase the value of x as much as possible without violating any of the constraints
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6. Example
To continue the example let us think about increasing the value of As we increase the values of all decrease. Because we have a nonnegativity constraint for each variable we cannot allow the any of them to become negative. If increases above 30, then becomes negative,while become negative when increases above 12,and 9 respectively
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7. An example of simplex algorithm
Switch the role of and
Rewrite other equations
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8. An example of simplex algorithm
After rewrite the linear program
New basic solution
This operation is pivot. Choose a nonbasic variable-entering
variable,and a basic variable-leaving variable, exchanges roles.
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9. An example of simplex algorithm
After entering and leaving
Basic solution (33/4, 0, 3/2, 69/4, 0, 0)
Objective value = 111/4
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10. An example of simplex algorithm
After entering and leaving
Basic solution (8,4,0,18,0,0)
Objective value = 28
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11. Basic solutions
The objective function variable P is always selected a basic variables and never selected as a nonbasic varaible
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12. Selecting Basic and Nonbasic variables
Determine the number of basic variables. These numbers do nıt change during simplex process
Basic varaibles: A variable can be selected as a basic variable only if it correspondes to a column in the tableu that has exactly one nonzero element(usually 1)
Nonbasic variables: Remaining variables
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13. Example
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14. Example
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15. Example
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16. Initial simplex tableu
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17. Pivot operation Entering variable 1 2 1 0 0 32 3 4 0 1 0 84
32:2=16 small
84:4=21
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18. Simplex tableu
1/ /
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19. Simplex tableu
1/ /
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20. Simplex tableu
1/ /
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21. Simplex tableu
1/ /
16:1/2= :1=20 pivot 1
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22. Simplex tableu
/2 -1/
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23. Pivoting
The procedure PIVOT takes an input a slack form,given by the tuple (N,B,A,b,c,v) the index l of the living variable and the index e of the entering variable
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24. Copyright © The McGraw-Hill Companies, Inc
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

25. The formal simplex algorithm
Assume that we have a procedure INITIALIZE-SIMPLEX(A,b,c) that takes as inpur a linear program in standart form,that is an m x n matrix A, an m dimensional vector b and n-dimensinal vector c. If the problem infeasible it returns a message that the program infeasible and then terminates.Otherwise it returns a slack form for which the initial basic solution is feasible.
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26. Copyright © The McGraw-Hill Companies, Inc
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

27. The initial basic feasible solution
In this section we first describe how test if a linear program is feasible, and it is how to produce a slack form for which the basic solution is feasible. We conclude fundamental theorem of linear programming, which says that the SIMPLEX procedure always produces the correct result.
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28. Example Consider the following linear program 18/03/2009
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29. Example
If we were convert this linear program to slack form the basic solution would set
and This solution violates constraint
(29.107) and it is not a feasible solution. Thus
INITIALIZE-SIMPLEX cannot just returns the obvious slack form. By inspection it is not clear whether this linear program even has any feasible solutions
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30. Auxiliary linear program
In order to determine whether it does we can formulate an auxilary linear program . For this auxiliary program we will be able to find a slack form for which the basic solution is feasible. Furtermore, the solution of auxilary linear program will determine whether the initial linear program is feasible and if so it will provide a feasible solution with which we can initialize SIMPLEX
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31. Auxiliary linear program
Lemma.
Let L be a linear program in standart form . Let Laux be the following linear program with n+1 variables:
Then L is feasible if and only if the optimal objective value of Laux is 0
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32. Auxiliary linear program
Proof of lemma. Suppose that L has a feasible solution
Then the solution combined with x is a feasible solution Laux with objective value 0. Since is a constraint of Laux this solution be optimal for Laux.Conversely, suppose that the optimal objective value of Laux is 0. Then remaining variables satisfy L
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33. Example.
We now demonstrate the operation INITIALIZE-SIMPLEX on the example.
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34. Copyright © The McGraw-Hill Companies, Inc
Copyright © The McGraw-Hill Companies, Inc. Permission required for reproduction or display.

35. Example We write this linear program in slack form,obtaining
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36. Example
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37. Example
Set x0=0 and simplifying
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38. Example.
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39. Fundamental theorem of linear programming
Any linear program L, given in standard form, either:
Has an optimal solution with a finite objective value
Is infeasible
Is unbounded
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40. Background George Dantzig born 8.11.1914, Portland
invented "Simplex Method of Optimisation" in 1947
this grew out of his work with the USAF
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The Simplex Algorithm