Step 1: Formulate the ProblemDecision VariablesObjective Function (O. F.)Constraints (S. T.)Sign Constraints (URS)

Step 2: Create the Standard Form of LPConstraints = (+ s , - e , + a )Variables >= 0

Step 3: Create a Simplex TableauRow 0 : a version of O.F.Row 1- .. : constraint with equalityVariable >= 0Initial bfs

IE 416, Chap 4:1, June 1999

RMC Inc. Problem, Summary

Mixture in ProductRaw Material Available Fuel SolventMaterial 1 20 tons 2/5 1/2Material 2 5 tons - 1/5Material 3 21 tons 3/5 3/10 Profit $/ton 40 30

Source: An Introduction to Management ScienceBy: Anderson, Sweeney, Williams

IE 416, Chap 4, May 99

RMC Inc. Problem, Formulation

X1 = number of tons of fuel, positiveX2 = number of tons of solvent, positive

O.F.

S.T. Material 1

Material 2 Material 3

IE 416, Chap 4, May 99

2110

3

5

3

55

1

202

1

5

2

21

2

21

XX

X

XX

21 3040 XXZ

RMC Inc. Problem, Standard LP Form

IE 416, Chap 4, May 99

03040 21 XXZ

2110

3

5

3

55

1

202

1

5

2

321

22

121

SXX

SX

SXX

RMC Inc. Problem,Using Simplex Method

Z X1 X2 S1 S2 S3 rhs BV ratio1 -40 -30 0 0 0 0 Z0 2/5 1/2 1 0 0 20 S1 20/(2/5)0 0 1/5 0 1 0 5 S2 --0 3/5 3/10 0 0 1 21 S3 21/(3/5)

1: Entering variable

3: Pivot row4: Pivot term

2: Ratiotesting

IE 416, Chap 4, May 99

First iteration

RMC Inc. Problem,Using Simplex Method, cont.

IE 416, Chap 4, May 99

Z X1 X2 S1 S2 S3 rhs BV ratio1 0 -10 0 0 200/3 1400 Z0 0 3/10 1 0 -2/3 6 S1 6/(3/10)*0 0 1/5 0 1 0 5 S2 5/(1/5)0 1 1/2 0 0 5/3 35 X1 35/(1/2)

Z X1 X2 S1 S2 S3 rhs BV ratio1 0 0 100/3 0 400/9 1600 Z0 0 1 10/3 0 -20/9 20 X2 0 0 0 -2/3 1 4/9 1 S2 0 1 0 -5/3 0 25/9 25 X1

Excess and Artificial Variables

1038 21 XX

1038 1121 aeXX

2025 21 XX

2025 221 aXX

Added Simplex Method PracticalVariable Application ApplicationSlack Equality of equation s > 0 resource not used BV for initial s = 0 binding constraint

simplex tableau Excess Equality of equation e > 0 extra resource required e = 0 binding constraintArtificial Added to > and = No meaning equations desire a = 0 BV for initial a > 0 no solution simplex tableau

IE 416, Chap 4:1, Jan 99

Simplex Method: (maximization) Entering Variable (most -ve in Row 0) Ratio Testing [smallest ratio, ratio = (rhs) / (coefficient > 0)] Pivot Term (entering & pivot row) ERO (next iteration, new bfs) Optimum Criterion (no -ve in Row 0)

Different problems Effect on simplex methodmin O.F. initial bfsbig M method row 0 versionmulti-optimal LP entering variableunbounded LP ratio testinfeasible LP optimum tableauURS decision variable

IE 416, Chap 4:2, July 98