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WOOD 492 MODELLING FOR DECISION SUPPORT

WOOD 492 MODELLING FOR DECISION SUPPORT. Lecture 3 Basics of the Simplex Algorithm. Last Class. Introduction to Linear Programming Solving LPs with the graphical method. Example: Custom Cabinets company. x 1 =48, x 2 =12 Z=$2,520. Feasible Region. Why use a specialized algorithm?.

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WOOD 492 MODELLING FOR DECISION SUPPORT

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  1. WOOD 492 MODELLING FOR DECISION SUPPORT Lecture 3 Basics of the Simplex Algorithm

  2. Last Class • Introduction to Linear Programming • Solving LPs with the graphical method Wood 492 - Saba Vahid

  3. Example: Custom Cabinets company x1 =48, x2 =12 Z=$2,520 Feasible Region Wood 492 - Saba Vahid

  4. Why use a specialized algorithm? • Exhaustive search takes too long • Too many feasible solutions • We want to ask many “what if” questions • So we run the model over and over • We want to perform sensitivity analysis • What constraints are binding? • How much do the constraints cost us? • Which products are the most profitable? We can use Simplex Algorithm to solve LPs Wood 492 - Saba Vahid

  5. Terminology • Feasible solution • Solution where all constraints are satisfied • Many are possible • Optimal solution • Feasible solution with highest (or lowest) objective function value • Can be unique, but there are many cases where there are ties • Boundary equation • Constraint with inequality replaced by an equality • These define the feasible region • Corner-point solution • Where two or more constraints intersect Wood 492 - Saba Vahid

  6. Feasible Region Wood 492 - Saba Vahid

  7. Important properties of LP • An optimal solution is always at a feasible corner-point solution • If a feasible corner-point solution has an objective value higher than all the adjacent feasible corner-point solutions, then it is optimal • There is a finite number of feasible corner-point solutions for an LP These properties make it possible to use the simplex algorithm which is very efficient in practice Wood 492 - Saba Vahid

  8. (22,25) Z=$2130 (48,12) Z=$2520 Feasible Region (48,0) Z=$1920 (0,0) Z=$0 demo Wood 492 - Saba Vahid

  9. Simplex Algorithm • Has two steps: • Start-up: Find anyfeasible corner-point solution • Iterate: Move repeatedly to adjacent feasible corner-point solutions with the highest improvement in objective values, until no better values are achieved by moving to an adjacent feasible corner-point solution. The final corner-point solution is the optimal solution. (it is possible to have more than one optimal solution) • Excel Solver uses the Simplex algorithm for solving LPs Cabinet LP Example Wood 492 - Saba Vahid

  10. Assumptions of LP • For a system to be modelled with an LP, 4 assumptions must hold: • Proportionality: Contribution of each activity (decision variable) to the Obj. Fn. is proportional to its value (represented by its coefficient in the Obj. Fn.), e.g. Z=3x1+2x2 , when x1 is increased, its contribution to the Obj. (3x1) is always increased three-fold. • Additivity: Every function in an LP (Obj. Fn. Or the constraints) is the linear sum of individual contributions of the respective activities (decision variables) • Divisibility: Activities can be run at fractional level, i.e., decision variables can have any level (not just integer values). • Certainty: Parameter values (coefficients in the functions) are known with certainty Wood 492 - Saba Vahid

  11. Next Lecture • Assumptions of LP • More examples of LP matrixes and Solver • Overview of Lab 1 Problem • Quiz on Friday, Sept 14 Wood 492 - Saba Vahid

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