Setting Up the Simplex Method in Operations Research: Wyndor Glass Co. Example, Study notes of Systems Engineering

Download Setting Up the Simplex Method in Operations Research: Wyndor Glass Co. Example and more Study notes Systems Engineering in PDF only on Docsity!

ISE 330

Introduction to Operations Research:

Deterministic Models

www-scf.usc.edu/~ise330/

September 12, 2007

Lecture 5

Setting Up the Simplex Method

(Sec. 4.1 – 4.3)

Wyndor Glass Co. Example

Profit per batch

Plant^12

Product

Production TimeAvailable per Week,

Hours

Production Time per Batch,

Hours

Wyndor Glass Co. Example

Maximize Z = 3x

• 5x 1

2

Subject to:

x^1

≤^4

2x

3x

• 2x 1

and x^1

≥^ 0, x

x^1

x^2

x^1

2x

• Corner Point Feasible Solutions 3x

• 2x 1

Wyndor Glass Co. Example (0,6)

1. Choose (0,0) as initial CPF solution2. (0,0) NOT optimal – adjacent CPF solutions better 3. Consider two edges from (0,0)4. Move in direction with largest improvement of Z (x

) 2

5. Solve for the intersection of the new set of boundaries 6. Choose (0,6) as new CPF solution7. (0,6) NOT optimal – adjacent CPF solutions better 8. Consider two edges from (0,6)9. Move in direction with largest improvement of Z 10.Solve for the intersection of the new set of boundaries 11.Choose (2,6) as new CPF solution

x^1

3x

• 2x 1

x^1

x^2

2x

(0,0)

(4,3) (4,0) (2,6)

Setting up the Simplex Method

Original Form Maximize Z = 3x

• 5x 1

2

Subject to:

x^1

≤^4

2x

3x

• 2x 1

and x^1

≥^ 0, x

Augmented Form

Maximize ZSubject to:Z - 3x

• 5x 1

x^1

• x

2x

• x 2

3x

• 2x 1

• x 2

and

x^ ≥j^

0 for j = 1, 2, ..., 5

Simplex Method Definitions ‡^

Slack Variables

• Added to convert functional inequality constraints to

equivalent equality constraints ‡^

Augmented Solution

• Solution for the original variables that is

augmented by the slack variables^ „

(3,2)

→

(3,2,1,8,5)

‡^

Basic Solution

• Augmented corner-point solution

‡^

Basic Feasible (BF) Solution

• Augmented CPF solution

‡^

Degrees of Freedom

• Number of variables – number of equations

„^

5 – 3 = 2 „^

Number of variables that are set equal to zero

‡^

Nonbasic Variables

• Variables that are set equal to zero

‡^

Basic Variables

• Variables that are not set equal to zero

Adjacent BF Solutions ‡^

Two BF solutions are adjacent if all but one of their nonbasic variablesare the same ‡^

Consider (0,0) and (0,6) ‡^

Augmented solutions are (0,0,4,12,18) and (0,6,4,0,6) ‡^

Nonbasic variables are (x

,x 1

) and (x 2

,x 1

Homework (Due September 19, 2007) ‡

- 10 points

‡

- 10 points

‡

- 15 points

‡

- 10 points

‡

- 15 points

‡

- 15 points

‡

- 15 points

‡

- 10 points