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 >= 0 Step 3: Create a Simplex Tableau Row 0 : a version of O.F. Row 1-.. : constraint with equality Variable >= 0 Initial bfs IE 416, Chap 4:1, 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 X 1 = number of tons of fuel, positive X 2 = 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 Z X 1 X 2 S 1 S 2 S 3 rhs BV ratio 1 -40 -30 0 0 0 0 Z 0 2/5 1/2 1 0 0 20 S 1 20/(2/5) 0 0 1/5 0 1 0 5 S 2 -- 0 3/5 3/10 0 0 1 21 S 3 21/(3/5) 1: Entering variable 3: Pivot row 4: Pivot term 2: Ratio testing IE 416, Chap 4, May 99 First iteration

6. RMC Inc. Problem,Using Simplex Method, cont. IE 416, Chap 4, May 99 Z X 1 X 2 S 1 S 2 S 3 rhs BV ratio 1 0 -10 0 0 200/3 1400 Z 0 0 3/10 1 0 -2/3 6 S 1 6/(3/10)* 0 0 1/5 0 1 0 5 S 2 5/(1/5) 0 1 1/2 0 0 5/3 35 X 1 35/(1/2) Z X 1 X 2 S 1 S 2 S 3 rhs BV ratio 1 0 0 100/3 0 400/9 1600 Z 0 0 1 10/3 0 -20/9 20 X 2 0 0 0 -2/3 1 4/9 1 S 2 0 1 0 -5/3 0 25/9 25 X 1

7. Excess and Artificial Variables

8. Added Simplex Method Practical Variable Application Application Slack 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 constraint Artificial 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

9. 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 problemsEffect on simplex method min O.F.initial bfs big M methodrow 0 version multi-optimal LPentering variable unbounded LPratio test infeasible LPoptimum tableau URSdecision variable IE 416, Chap 4:2, July 98