Computational Methods for Management and Economics Carla Gomes

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Computational Methods forManagement and
EconomicsCarla Gomes

• Module 6a
• Introduction to Simplex
• (Textbook Hillier and Lieberman)

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Algebraic Model for Wyndor Glass Co.
Let D the number of doors to produce W the
number of windows to produce Maximize P 3 D
5 W subject to D 4 2W 12 3D 2W
 18 and D  0, W  0.
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Wyndor Glass
Edge of Feasible region
CPF
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LP Concepts

• Corner point feasible solution (CPF solution)
  intersection of n (or more n - number of
  variables) constraint boundaries
• For any LP with n decision variables two CPF
  solutions are adjacent to each other if they
  share (n-1) constraint boundaries
• Edge of feasible region intersection of the
  (n-1) constraint boundaries shared by two
  adjacent CPF solutions
• Optimality test for any LP problem that
  possesses at least one optimal solution, if a CPF
  solution has no adjacent CPF solutions that are
  better (as measure by Z) then it must be an
  optimal solution

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3D feasible region

Edge of feasible region between two CPFSs ? the
edge is the line that lies at the intersection of
the common constraint boundaries of the two CPFSs
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Corner Point Solutions

• Corner-point feasible solution special
  solution that plays a key role when the simplex
  method searches for an optimal solution.
  Relationship between optimal solutions and CPF
  solutions
• Any LP with feasible solutions and bounded
  feasible region ?
•  
• (1)  the problem must possess CPF solutions and
  at least one optimal solution
• (2)  the best CPF solution must be an optimal
  solution
• If the problem has exactly one optimal solution
  it must be a CFP solution
• If the problem has multiple optimal solutions, at
  least two must be CPF solutions

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Simplex Method

• Iterative procedure involving the following
  steps
• Initialization find initial CPF solution
• Optimality test
• if optimal stop
• if not optimal go to next iteration
• Iteration find a better CPF solution go to 2.

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Geometric View Point of Simplex Method

• Iterative procedure involving the following
  steps
• Initialization find initial CPF solution
• Whenever possible pick (0,0) as initial solution
• Optimality test
• (check value of Z of adjacent CPF solutions)
• Iteration find a better CPF solution go to 2.
• Consider edges that emanate from current CPF
  solution and pick the one that increases Z at a
  faster rate
• Stop at the first new constraint boundary

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Wyndor Glass
Let D the number of doors to produce W the
number of windows to produce Maximize P 3 D
5 W
1
Z30
2
Z36
Edge of Feasible region
3
Z27
Z0
0
CPF
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Simplex Key Concepts

• Concept 1 CPF solutions
• Simplex methods focuses only on CPF solutions
  (finite set)
• Concept 2 Flow of simplex method
• Iterative procedure
• Initialization find initial CPF solution
• Optimality test
• if optimal stop
• if not optimal go to next iteration
• Iteration find a better CPF solution go to 2.
• Concept 3 Initialization
• Whenever possible pick the origin otherwise
  special procedure

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Simplex Key Concepts (cont.)

• Concept 4 Path to optimal solution
• Simplex methods always chooses a CPF solution
  adjacent to the current one
• The entire path to the optimal solution is along
  the edges of the feasible solution
• Concept 5 Choice of new CPF solution
• From a CPF solution consider all edges emanating
  from it but it does not solve for each adjacent
  CPF solution it simply identifies the rate of
  improvement in Z along a given edge and than it
  picks the one with the largest improvement
• Concept 6 Optimality Test
• From a given CPF solution check if there is an
  edge that gives a positive rate of improvement in
  Z. If not, current CPF is optimal.

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Setting up Simplex Method for Algebraic Procedure

• Algebraic procedure solving systems of
  equations ? converting inequalities into
  equalities ? introduction of slack variables
• Slack variables
• x1 4 ?? x1 s1 4 s1 4 - x1
• x1 4 ?? x1 s1 4 and s1 0

Meaning of Slack variables 0 the solution lies
on the constraint boundary gt0 solution lies on
the feasible region lt0 solutions lies on the
infeasible region

• LP Augmented form standard form slack
  variables

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Augmented Form for Wyndor Glass
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Graphical Representation
Augmented solution solution for original
variables slack variables Basic solution ??
augmented corner point solution Basic feasible
solution ?? augmented corner point
feasible solution
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Simplex as an algebraic procedure

• System of functional constraints n (5) variables
  (5) and m (3) equations ? 2 degrees of freedom,
  (i.e., we can set those two variables to any
  arbitrary values) they are the nonbasic
  variables the other variables are the basic
  variables
• Simplex chooses to set the non-basic variables to
  ZERO.
• Simplex solves the simultaneous equations to set
  the values of the basic variables

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Properties of Basic Solutions

• A basic solution is composed of
• Non-basic variables
• number of non-basic variables equals (total
  number of variables - number of functional
  constraints)
• They are set to ZERO
• Basic variables
• number of basic variables equals number of
  functional constraints
• Their values results from solving the system of
  functional constraints (non-basic variables set
  to 0)
• They form the
• Basic Feasible solution it is a basic solution
  that satisfies the non-negativity constraints
• Adjacent basic feasible solutions all but one
  of their basic (non-basic) variables are the same
  ? moving from one basic feasible solution to an
  adjacent one involves switching one variable from
  non-basic to basic and one variable from basic to
  non-basic (check graph)

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Getting ready for the Simplex

• Standard form
• lt constraints
• Non-negativity constraints on all variables
• Positive right hand sides
• (if these assumptions are not valid ? additional
  adjustments need to be done)
• Transform the objective function and constraints
  into equalities (introduction of slack variables)

Whats wrong with this format?
maximize 3x1 2x2 - x3 x4 x1 2x2
x3 - x4 ? 5 -2x1 - 4x2 x3 x4 ? -1
x1 ? 0, x2 ? 0
not equality
not equality
not equality/negative
x3 may be negative
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Simplex Procedure

• Initialization origin whenever possible
  (decision variables 0)
• (okay if standard form with positive
  RHSs basic feasible solution (BFS) each
  equation has a basic variable with coefficient 1
  (slack variable RHS) and the variable does not
  appear in any other eq the decision variables
  are the non-basic variables set to 0)
• Optimality test is current BFS optimal?
• (the coefficients of the objective function
  of the non-basic variables gives the rate of
  improvement in Z )

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Simplex Procedure (cont.)

• Iteration move to a better adjacent BFS
• a)Variable entering the basis
• Consider non-basic variables (Graphically -
  Consider edges emanating from current CPF
  solution)
• Pick the variable that increases Z at a faster
  rate
• b)Variable leaving the basis
• One of the basic variables will become non-basic
  write all the basic variables as a function of
  the entering variable the most stringent value
  (i.e., smallest) will be the value for the new
  entering variable the basic variable associated
  with the most stringent constraint will become
  non-basic, leaving the basis.
• (Graphically where to stop? As much as
  possible without leaving the feasible region)
• c)Solving for the new BFS
• Objective of this step convert the system of
  equations into a more convenient form (1) to
  perform optimality test and (2) to perform next
  iteration if necessary
• New basic variable should have coefficient 1 in
  the equation of the leaving variable and 0 in all
  the other equations, including the objective
  function
• Valid operations
• Multiplication (or division) of an equation by a
  non-zero constant
• Addition (or subtraction) of a multiple of an
  equation to (from) another eqution

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Simplex Procedure Wyndor Glass
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Simplex Method in Tabular Form

• Tabular form more compact form it records
  only the essential information namely
• Coefficients of variables
• The constants on the right hand sides
• Basic variable in each equation
• Note only xj vars are basic and non-basic we
  can think of Z as the basic var. of objective
  function.

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Simplex Method in Tabular Form

• Assumption Standard form
• max only lt functional constraints all vars
  have non-negativity constraints rhs are positive
• Initialization
• Introduce slack variables
• Decision variables ? non-basic variables (set to
  0)
• Slack variables ? basic variables (set to
  corresponding rhs)
• Transform the objective function and the
  constraints into equality constraints
• Optimality Test
• Current solution is optimal iff all the
  coefficients of objective function are
  non-negative

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Simplex Method in Tabular Form (cont.)

• Iteration Move to a better BFS
• Step1 Entering Variable non-basic variable with
  the most negative coefficient in the objective
  function. Mark that column as the pivot column.
• Step2 Leaving basic variable apply the minimum
  ratio test
• Consider in the pivot column only the coeffcients
  that are strictly positive
• Divide each of theses coefficients into the rhs
  entry for the same row
• Identify the row with the smallest of these
  ratios
• The basic variable for that row is the leaving
  variable mark that row as the pivot row
• The number in the intersection of the pivot row
  with the pivot column is the pivot number

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Simplex Method in Tabular Form (cont.)

• Iteration Move to a better BFS
• Step3 Solve for the new BFS by using elementary
  row operations to construct a new simplex tableau
  in proper form
• Divide pivot row by the pivot number
• for each row (including objective function) that
  has a negative coefficient in the pivot column,
  add to this row the product of the absolute value
  of this coefficient and the new pivot row.
• for each row that has a positive coefficient in
  the pivot column, add to this row the product the
  new pivot row.multiplied by the negative of the
  coefficient.

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Wyndor Glass1st Simplex Tableau
Optimal? Entering Variable?
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What variable will enter the basis? Why? What
variable will leave the basis? Why? What
transformations do we need to perform to the
tableau to get the new basic variable into the
right format?
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What operations did we perform?
1 divide the pivot row by 2 2 multiplied the
new pivot row by (-2) and added it to eq. 3 3
multiplied the new pivot row by (5) and added it
to eq. 0
What are the new basic / non-basic variables?
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Optimal? Entering Variable?
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What operations did we perform?
1 divide the pivot row by 2 multiplied
the new pivot row by ( ) and added it to eq.
1 3 multiplied the new pivot row by ( ) and
added it to eq. 0
What are the new basic / non-basic variables?
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Optimal?
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Simplex Method in Tabular Form