Section 4.3

The Simplex Method: Nonstandard Problems

Ex.

A nonstandard maximization problem: Maximize

P

= 8

x

+ 3

y

Subject to 2

x

8

x

 0,

y

2

y

 0 First change the inequalities to less than or equal to.

2

x

8

x y

2

x

 0,

y

 0 Now proceed with the simplex method

Introduce

slack variables

to make equations out of the inequalities and set the objective function = 0: 2

x x



x y



v

 8   2 

P

 0 The initial tableau and notice

v

= –2 (not feasible):

x

2 1  8

y u

1 1  1 0  3 0

v

0 1 0

P

0 0 1 constant 8  2 0 We need to pivot to a feasible solution

x

2 1  8

y u

1 1  1 0  3 0

v

0 1 0

P

0 0 1 constant 8  2 0 Ratios 8 2 Locate any negative number in the constant column ( –2). Now go to the first negative to the left of that constant (–1). This determines the pivot column. The pivot row is found by examining the positive ratios. So –1 is our pivot. Create unit column



R

2

x

2  1  8

y

1 1  3

u

1 0 0

v

0  1 0

P

0 0 1 constant 8 2 0

R

1

R

3 

R

2  3

R

2

x y

 3 1 0 1  11 0

u

1 0 0

v

1  1  3

P

0 0 1 constant 6 2 6 New pivot since it is the only positive ratio Note: now we have a feasible solution proceed with simplex

1 3

R

1

R

2

R

3 

R

1  11

R

1

x y u v

1  1 0 1/3 1/3 1  11 0 0 0  1  3

P

0 0 1 constant 2 2 6

x

1 0 0

y

0 1

u

1/3 1/3 0 11/ 3

v

1/3  2 / 3 2 / 3

P

0 0 1 constant 2 4 28 This is the final tableau:

x

= 2,

y

= 4,

P

= 28

The Simplex Method for Nonstandard Problems

1. If necessary, rewrite as a maximization problem.

2. If necessary, rewrite inequalities as less or equal to.

3. Introduce slack variables and write simplex tableau.

4. If no negative constants (upper column) use simplex method, otherwise go to step 5.

5. Pick a negative entry in a row with a negative constant (this is the pivot column). Compute positive ratios to determine pivot row. Then pivot and return to step 4.