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

© 2015 McGraw-Hill Education. All rights reserved.


Frederick S. Hillier Gerald J. Lieberman

Chapter 4Solving Linear Programming Problems: The Simplex Method


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

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The Essence of the Simplex Method

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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 found4


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• First step: convert functional inequality constraints into equality constraints– Done by introducing slack variables

– Resulting form known as augmented form

– Example: constraint is equivalent to

4.2 Setting Up the Simplex Method


Setting Up the Simplex Method

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• 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

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• 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

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4.3 The Algebra of the Simplex Method

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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

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The Simplex Method in Tabular Form

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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 test12


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

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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

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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

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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

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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

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4.7 Postoptimality Analysis

• Simplex method role


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Postoptimality Analysis

• 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


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• 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 ith slack variable in row 0 of the final simplex tableau


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• 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



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• 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 8.2


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

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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

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The Interior-Point Approach to Solving Linear Programming Problems

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The Interior-Point Approach to Solving Linear Programming Problems

• Disadvantage– Limited capability for performing a

postoptimality analysis• Approach: switch over to simplex method

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4.10 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

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

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Transcript of © 2015 McGraw-Hill Education. All rights reserved. Chapter 4 Solving Linear Programming Problems:...