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© 2015 McGraw-Hill Education. All rights reserved. Chapter 4 Solving Linear Programming Problems: The Simplex Method.

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1 © 2015 McGraw-Hill Education. All rights reserved. Chapter 4 Solving Linear Programming Problems: The Simplex Method

2 © 2015 McGraw-Hill Education. All rights reserved. 4.1 The Essence of the Simplex Method Algebraic procedure –Underlying concepts are geometric Revisit Wyndor example –Figure 4.1 shows constraint boundary lines Points of intersection are corner-point solutions Five points on corners of feasible region are CPF solutions Adjacent CPF solutions –Share a constraint boundary 2

3 © 2015 McGraw-Hill Education. All rights reserved. The Essence of the Simplex Method 3

4 © 2015 McGraw-Hill Education. All rights reserved. The Essence of the Simplex Method Optimality test –If a CPF solution has no adjacent CPF solution that is better (as measured by Z): It must be an optimal solution Solving the example with the simplex method –Choose an initial CPF solution (0,0) and decide if it is optimal –Move to a better adjacent CPF solution –Iterate until an optimal solution is found 4

5 © 2015 McGraw-Hill Education. All rights reserved Setting Up the Simplex Method

6 © 2015 McGraw-Hill Education. All rights reserved. Setting Up the Simplex Method 6

7 © 2015 McGraw-Hill Education. All rights reserved. Setting Up the Simplex Method Augmented solution –Solution for the original decision variables augmented by the slack variables Basic solution –Augmented corner-point solution Basic feasible (BF) solution –Augmented CPF solution 7

8 © 2015 McGraw-Hill Education. All rights reserved. Setting Up the Simplex Method Properties of a basic solution –Each variable designated basic or nonbasic –Number of basic variables equals number of functional constraints –The nonbasic variables are set equal to zero –Values of basic variables obtained as simultaneous solution of system of equations –If basic variables satisfy nonnegativity constraints, basic solution is a BF solution 8

9 © 2015 McGraw-Hill Education. All rights reserved. 4.3 The Algebra of the Simplex Method 9

10 © 2015 McGraw-Hill Education. All rights reserved. 4.4 The Simplex Method in Tabular Form Tabular form –Records only the essential information: Coefficients of the variables Constants on the right-hand sides of the equations The basic variable appearing in each equation –Example shown in Table 4.3 on next slide 10

11 © 2015 McGraw-Hill Education. All rights reserved. The Simplex Method in Tabular Form 11

12 © 2015 McGraw-Hill Education. All rights reserved. The Simplex Method in Tabular Form Summary of the simplex method –Initialization Introduce slack variables –Optimality test Optimal if and only if every coefficient in row 0 is nonnegative –Iterate (if necessary) to obtain the next BF solution Determine entering and leaving basic variables Minimum ratio test 12

13 © 2015 McGraw-Hill Education. All rights reserved. 4.5 Tie Breaking in the Simplex Method Tie for the entering basic variable –Decision may be made arbitrarily Tie for the leaving basic variable –Matters theoretically but rarely in practice –Choose arbitrarily Condition of no leaving basic variable –Z is unbounded –Indicates a mistake has been made 13

14 © 2015 McGraw-Hill Education. All rights reserved. Tie Breaking in the Simplex Method Multiple optimal solutions –Simplex method stops after one optimal BF solution is found –Often other optimal solutions exist and would be meaningful choices –Method exists to detect and find other optimal BF solutions 14

15 © 2015 McGraw-Hill Education. All rights reserved. 4.6 Adapting to Other Model Forms Simplex method adjustments –Needed when problem is not in standard form –Made during initialization step Artificial-variable technique –Dummy variable introduced into each constraint that needs one –Becomes initial basic variable for that equation 15

16 © 2015 McGraw-Hill Education. All rights reserved. Adapting to Other Model Forms Types of nonstandard forms –Equality constraints –Negative right-hand sides –Functional constraints in greater-than-or- equal-to form –Minimizing Z Solving the radiation therapy problem –Text reviews two methods: Big M and two- phase 16

17 © 2015 McGraw-Hill Education. All rights reserved. Adapting to Other Model Forms No feasible solutions –Constructing an artificial feasible solution may lead to a false optimal solution –Artificial-variable technique provides a way to indicate whether this is the case Variables are allowed to be negative –Example: negative value indicates a decrease in production rate –Negative values may have a bound or no bound 17

18 © 2015 McGraw-Hill Education. All rights reserved. 4.7 Postoptimality Analysis 18 Simplex method role

19 © 2015 McGraw-Hill Education. All rights reserved. Postoptimality Analysis 19 Reoptimization –Alternative to solving the problem again with small changes –Involves deducing how changes in the model get carried along to the final simplex tableau –Optimal solution for the revised model: Will be much closer to the prior optimal solution than to an initial BF solution constructed the usual way

20 © 2015 McGraw-Hill Education. All rights reserved. Postoptimality Analysis 20 Shadow price –Measures the marginal value of resource i –The rate at which Z would increase if more of the resource could be made available –Given by the coefficient of the i th slack variable in row 0 of the final simplex tableau

21 © 2015 McGraw-Hill Education. All rights reserved. Postoptimality Analysis Sensitivity analysis –Purpose: to identify the sensitive parameters These must be estimated with special care –Can be done graphically if there are just two variables –Can be performed in Microsoft Excel 21

22 © 2015 McGraw-Hill Education. All rights reserved. Postoptimality Analysis 22

23 © 2015 McGraw-Hill Education. All rights reserved. Postoptimality Analysis Parametric linear programming –Study of how the optimal solution changes as many of the parameters change simultaneously over some range –Used for investigation of trade-offs in parameter values –Technique presented in Section

24 © 2015 McGraw-Hill Education. All rights reserved. 4.8 Computer Implementation Simplex method ideally suited for execution on a computer Computer code for the simplex method –Widely available for all modern systems –Follows the revised simplex method Main factor determining time to solution –Number of functional constraints Rule of thumb: number of iterations required equals twice the number of functional constraints 24

25 © 2015 McGraw-Hill Education. All rights reserved. 4.9 The Interior-Point Approach to Solving Linear Programming Problems Alternative to the simplex method developed in the 1980s –Far more complicated Uses an iterative approach starting with a feasible trial solution –Trial solutions are interior points Inside the boundary of the feasible region Advantage: large problems do not require many more iterations than small problems 25

26 © 2015 McGraw-Hill Education. All rights reserved. The Interior-Point Approach to Solving Linear Programming Problems 26

27 © 2015 McGraw-Hill Education. All rights reserved. The Interior-Point Approach to Solving Linear Programming Problems Disadvantage –Limited capability for performing a postoptimality analysis Approach: switch over to simplex method 27

28 © 2015 McGraw-Hill Education. All rights reserved Conclusions Simplex method –Efficient and reliable approach for solving linear programming problems –Algebraic procedure –Efficiently performs postoptimality analysis –Moves from current BF solution to a better BF solution –Best performed by computer except for the very simplest problems 28


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