1. Linear Programming – Simplex Method

2. Linear Programming - Review
Graphical Method:
What is the feasible region?
Where was optimal solution found?
What is primary limitation of graphical method?
Conversion to Standard Form: -

3. Linear Programming – Review
Solving Systems of Linear Equations:
What is a basic solution?
How did we obtain a basic solution?
What is a basic feasible solution?
Relationship between graphical and algebraic
representation of the feasible region:
corner point basic solution

4. Linear Programming – Review
Fundamental insight – the optimal solution to a linear
program, if it exists, is also a basic feasible solution.
Naïve approach – solve for all basic solutions and find
the feasible solution with the largest value (maximization
problem).
What is the problem with this approach? – there are
possible basic solutions, where m is the number of
constraints and n is the number of variables.

5. Linear Programming – Simplex Algorithm
Step 1 Convert the LP to standard form.
Step 2 Obtain a bfs (if possible) from the standard form.
Step 3 Determine whether the current bfs is optimal.
Step 4 If the current bfs is not optimal, then determine which nonbasic basic variable should become a basic variable and which basic variable should become a nonbasic variable to find a new bfs with a better objective function value. (pivot operation)
Step 5 Use EROs to find the new bfs with the better objective function value. Go back to step 3.
Operations Research, Wayne L. Winston

6. Linear Programming – Simplex Method
Review Simplex Handouts

7. Linear Programming – Simplex Method
Minimization Problems:
Min Z = cx  (-) Max Z = -cx
Ex. Min 2x1 – 3x2 + x3
s.t x1 + 2x < 5
2x x3 > 10
x1, x2, x3 > 0
(-)Max -2x1 + 3x2 - x3
s.t x1 + 2x < 5
2x x3 > 10
x1, x2, x3 > 0

8. Linear Programming – Simplex Method: Computational Problems
Breaking Ties in Selection of Non-Basic Variable – if tie for
non-basic variable with largest relative profit ( ), arbitrarily
select incoming variable.
Ties in Minimum Ratio Rule (Degeneracy) – if more than one
basic variable have same minimum ratio, select either variable
to leave the basis. This will result in a basic variable taking on
a value of 0. When this occurs, the solution is referred to as
a degenerate basic feasible solution.
When this occurs, you may transition through more than one
simplex tableau with the same objective (Z) value.

9. Linear Programming – Simplex Method: Computational Problems
Unbounded Solutions – if when performing the minimum ratio
rule, none of the ratios are positive, then the solution is
unbounded (e.g Max Z = or Min = ).

10. Simplex Method – Finding an Initial Basic Feasible Solution
Min Z = -3x1 + x2 + x3
s.t x1 – 2x2 + x3 <= 11
-4x1 + x2 +2x3 >= 3
2x x3 = -1
x1, x2, x3 >= 0
Standard Form:
(-) Max Z = 3x1 - x2 - x3
s.t x1 – 2x2 + x3 + x = 11
-4x1 + x2 +2x x5 = 3
-2x x = 1
x1, x2, x3, x4, x5 >= 0

11. Simplex Method – Finding an Initial Basic Feasible Solution
(-) Max Z = 3x1 - x2 - x3
s.t x1 – 2x2 + x3 + x = 11
-4x1 + x2 +2x x5 = 3
-2x x = 1
x1, x2, x3, x4, x5 >= 0
Only x4 is basic.
Introduce artificial variables.
s.t x1 – 2x2 + x3 + x = 11
-4x1 + x2 +2x x5 + x = 3
-2x x x7 = 1
x1, x2, x3, x4, x5, x6, x7 >= 0

12. Simplex Method – Solve Using Big-M Method
Let M be an arbitrarily large number, then:
(-) Max Z = 3x1 - x2 - x3 + 0x4 + 0x5 – Mx6 – Mx7
s.t x1 – 2x2 + x3 + x = 11
-4x1 + x2 +2x x x = 3
-2x x x7 = 1
x1, x2, x3, x4, x5, x6, x7 >= 0
Note: If the simplex algorithm terminates with one of the artificial variables as a basic variable, then the original problem has no feasible solution.