Step 1: Formulate the Problem

1
Step 1 Formulate the Problem Decision
Variables Objective Function (O.
F.) Constraints (S. T.) Sign
Constraints (URS) Step 2 Create the
Standard Form of LP Constraints ( s ,
- e , a ) Variables gt 0 Step 3
Create a Simplex Tableau Row 0 a version
of O.F. Row 1- .. constraint with
equality Variable gt 0 Initial bfs
IE 416, Chap 41, June 1999
2
RMC Inc. Problem, Summary
Mixture in
Product Raw Material Available
Fuel Solvent Material 1
20 tons 2/5
1/2 Material 2 5 tons
- 1/5 Material 3
21 tons 3/5 3/10
Profit /ton
40 30 Source An
Introduction to Management Science By Anderson,
Sweeney, Williams
IE 416, Chap 4, May 99
3
RMC Inc. Problem, Formulation X1 number of
tons of fuel, positive X2 number of tons of
solvent, positive O.F. S.T.
Material
1
Material 2

Material 3
IE 416, Chap 4, May 99
4
RMC Inc. Problem, Standard LP Form
IE 416, Chap 4, May 99
5
RMC Inc. Problem,Using Simplex Method
2 Ratio testing
1 Entering variable
Z X1 X2 S1 S2 S3 rhs
BV ratio 1 -40 -30 0 0
0 0 Z 0 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)
4 Pivot term
3 Pivot row
First iteration
IE 416, Chap 4, May 99
6
RMC Inc. Problem,Using Simplex Method, cont.
Z X1 X2 S1 S2 S3 rhs
BV ratio 1 0 -10 0 0
200/3 1400 Z 0 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
ratio 1 0 0 100/3 0 400/9
1600 Z 0 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
IE 416, Chap 4, May 99
7
Excess and Artificial Variables
8
Added Simplex Method Practical Variable
Application Application Slack
Equality of equation s gt 0 resource not
used BV for initial
s 0 binding constraint simplex
tableau Excess Equality of equation e gt
0 extra resource
required

e 0 binding constraint Artificial Added to gt
and No meaning
equations desire a 0
BV for initial a gt 0 no
solution simplex tableau
IE 416, Chap 41, Jan 99
9
Simplex Method (maximization) Entering
Variable (most -ve in Row 0) Ratio Testing
smallest ratio,
ratio (rhs) / (coefficient gt 0) Pivot
Term (entering pivot row) ERO
(next iteration, new bfs) Optimum
Criterion (no -ve in Row 0) Different
problems Effect on simplex method min
O.F. initial bfs big M method row 0
version multi-optimal LP entering
variable unbounded LP ratio test infeasible
LP optimum tableau URS decision variable
IE 416, Chap 42, July 98