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**Unformatted text preview: **Chapter 4
Solving Linear
Programming Problems:
The Simplex Method
Dr. David R. Nowicki
Professor
Fall 2018 2 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 3 The Essence of the Simplex Method 4 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 5 4.2 Setting Up the Simplex Method
• First step: convert functional inequality constraints
into equality constraints
– Done by introducing slack variables
– Resulting form known as augmented form
– Example: constraint 1 ≤ 4 is equivalent to
1 + 3 = 4 and 3 ≥ 0 6 Setting Up the Simplex Method 7 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 8 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 9 4.3 The Algebra of the Simplex Method 0 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 1 The Simplex Method in Tabular Form 2 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 3 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 4 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 5 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 6 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 7 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 8 4.7 Postoptimality Analysis
• Simplex method role 9 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 0 Postoptimality Analysis
• 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 1 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 2 Postoptimality Analysis 3 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 8.2 4 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 5 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 6 The Interior-Point Approach to Solving
Linear Programming Problems 7 The Interior-Point Approach to Solving
Linear Programming Problems
• Disadvantage
– Limited capability for performing a postoptimality
analysis
• Approach: switch over to simplex method 8 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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- Spring '16
- Operations Research, Linear Programming, Optimization, It, CPF