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LP Formulation and Simplex Method Example

This example illustrates the formulation of a linear programming problem and the application of the Simplex method to find the optimal solution. It includes the problem description, decision variables, objective function, constraints, sign constraints, and the step-by-step process of creating a simplex tableau. The example also covers scenarios such as multi-optimal solutions, unbounded problems, and infeasible problems.

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LP Formulation and Simplex Method Example

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  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 tons2/5 1/2 Material 2 5 tons- 1/5 Material 3 21 tons3/5 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 0Z 0 2/5 1/2 1 0 0 20S1 20/(2/5) 0 0 1/5 0 1 0 5S2 -- 0 3/5 3/10 0 0 1 21S3 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 1400Z 0 0 3/10 1 0 -2/3 6S1 6/(3/10)* 0 0 1/5 0 1 0 5S2 5/(1/5) 0 1 1/2 0 0 5/3 35X1 35/(1/2) Z X1 X2 S1 S2 S3 rhs BV ratio 1 0 0 100/3 0 400/9 1600Z 0 0 1 10/3 0 -20/9 20X2 0 0 0 -2/3 1 4/9 1S2 0 1 0 -5/3 0 25/9 25X1 IE 416, Chap 4, May 99

  7. Excess and Artificial Variables

  8. Added Simplex Method Practical Variable Application Application Slack Equality of equations > 0 resource not used BV for initials = 0 binding constraint simplex tableau Excess Equality of equatione > 0 extra resource required e = 0 binding constraint Artificial Added to > and =No meaning equationsdesire a = 0 BV for initiala > 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 method row 0 version multi-optimal LP entering variable unbounded LP ratio test infeasible LP optimum tableau URS decision variable IE 416, Chap 4:2, July 98

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