University of Colorado at Boulder
Yicheng Wang, Phone:303-492-4228, Email:[email protected]
Optimization Techniques for
Civil and Environmental Engineering Systems
By
Yicheng Wang
University of Colorado at Boulder
Yicheng Wang, Phone:303-492-4228, Email:[email protected]
(2) Setting Up the Simplex Method
Original Form of the Model
Augmented Form of the Model
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Augmented Form of the Model
H
H
B
B
C
C
G
G
A
A
D
D
H(3,2)
H(3,2)
E
E
F
F
For example, H(3,2) is a solution for the original model, which yields the
augmented solution ( x1, x2, x3, x4, x5) = (3, 2 ,1 ,8, 5)
For example, G(4,6) is a corner-point infeasible solution, which yields the
corresponding basic solution ( x1, x2, x3, x4, x5) = (4, 5 ,0 ,0, -6)
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Yicheng Wang, Phone:303-492-4228, Email:[email protected]
The only difference between basic solutions and corner-point solutions is
whether the values of the slack variables are included
Relationship between Corner-Point Solutions and Basic Solutions
In the original model, we have
In the augmented model, we have
Corner-point solution
Basic solution
Corner-point feasible (CPF) solution
Basic Feasible (BF) solution
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Yicheng Wang, Phone:303-492-4228, Email:[email protected]
The corner-point solution (0,0) in
the original model corresponds to
the basic solution (0, 0, 4,12, 18) in
the augmented form, where x1 =0
and x2=0 are the nonbasic variables,
and x3=4, x4=12, and x5=18 are the
basic variables
H
B C G
D
A
E
F
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Yicheng Wang, Phone:303-492-4228, Email:[email protected]
Example:
The CPF solution (0,0) in the original model corresponds to the BF solution (0, 0,
4,12, 18) in the augmented form, where x1 =0 and x2=0 are the nonbasic variables,
and x3=4, x4=12, and x5=18 are the basic variables
Choose x1 and x4 to be the nonbasic variables that are set equal to 0. The three
equations then yield, respectively, x3=4, x2=6 , and x5=6 as the solution for the
three basic variables as shown below.
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Yicheng Wang, Phone:303-492-4228, Email:[email protected]
Example:
A(0,0) and B(0,6) are two CPF solutions
The corresponding BF solutions are
( x1, x2, x3, x4, x5) = (0, 0 ,4 ,12, 18) and
( x1, x2, x3, x4, x5) = (0, 6 ,4 ,0, 6)
A(0,0) and C(2,6) are two CPF solutions
The corresponding BF solutions are
( x1, x2, x3, x4, x5) = (0, 0 ,4 ,12, 18) and
( x1, x2, x3, x4, x5) = (2, 6 ,2 ,0, 0)
H
B
C
G
D
A
H(3,2)
E
F
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Yicheng Wang, Phone:303-492-4228, Email:[email protected]
University of Colorado at Boulder
Yicheng Wang, Phone:303-492-4228, Email:[email protected]
(3) The Algebra of the Simplex Method
Use the Wyndor Glass Co. Model to illustrate the algebraic
procedure
Initialization
Geometric interpretation
B
A
C
D
E
Algebraic interpretation
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Optimality Test
Geometric interpretation
A(0,0) is not optimal.
Algebraic interpretation
The objective function:
The rate of improvement of Z
by the nonbasic variable x1 is 3
B
A
The rate of improvement of Z by
the nonbasic variable x2 is 5
C
D
E
Conclusion: The initial BF solution
(0,0,4,12,18) is not optimal.
University of Colorado at Boulder
Yicheng Wang, Phone:303-492-4228, Email:[email protected]
Iteration1 Step1: Determining the Direction of Movement
Geometric interpretation
Move up from A(0,0)
to B(0,6)
B
A
C
D
E
Algebraic interpretation
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Yicheng Wang, Phone:303-492-4228, Email:[email protected]
Iteration1 Step2: Where to Stop
Geometric interpretation
Algebraic interpretation
Stop at B. Otherwise, it
will leave the feasible
region.
Step 2 determine how far
to increase the entering
basic variable x2.
B
A
C
D
E
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Thus x4 is the leaving basic variable for iteration 1 of the example.
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Iteration1 Step3: Solving for the New BF Solution
Geometric interpretation
The intersection of the
new pair of constraint
boundary: B(0,6)
Algebraic interpretation
Nonbasic variables
Basic variables
B
C
Nonbasic variables
A
D
E
Basic variables
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(0)
Initial BF Solution
Nonbasic variables: x1= 0
(1)
x2= 0
(2)
Basic variables: x3= 4
x4 =12
(3)
x5= 18
(0)
New BF Solution
Nonbasic variables: x1= 0
(1)
x4= 0
(2)
Basic variables: x3= ？
(3)
x2 =6
Basic variables: x3= 4
x2 =6
x5= 6
x5= ？
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Optimality Test
Geometric interpretation
B(0,6) is not optimal,
because moving from
B to C increases Z.
B
A
Algebraic interpretation
The objective function:
The rate of improvement of Z
by the nonbasic varialle x1 is 3
The rate of improvement of Z by
the nonbasic varialle x4 is -5/2
C
D
E
Conclusion: The BF solution
(0,6,4,0,6) is not optimal.
University of Colorado at Boulder
Yicheng Wang, Phone:303-492-4228, Email:[email protected]
Iteration2
Step1: Determining the Direction of Movement
Choose x1 to be the entering basic variable
Step2: Where to Stop
The minimum ratio test
indicates that x5 is the
leaving basic variable
Step3: Solving for the New BF Solution
(0)
(1)
New BF Solution
(2)
Nonbasic variables: x1= 0, x4= 0
(3)
Basic variables: x3= 2, x2 =6, x1= 2
University of Colorado at Boulder
Yicheng Wang, Phone:303-492-4228, Email:[email protected]
Optimality Test
The objective function:
The coefficients of the nonbasic variables x4 and x5 are negative.
Increasing either x4 or x5 will decrease Z, so (x1, x2, x3, x4, x5) =
(2, 6, 2, 0, 0) must be optimal with Z = 36.
B
A
C
D
E
In terms of the original form of the
problem (no slack variables), the
optimal solution is (x1, x2) = (2, 6) ,
which yields Z = 3x1+5x2=36.
University of Colorado at Boulder
Yicheng Wang, Phone:303-492-4228, Email:[email protected]
(4) The Simplex Method in Tabular Form
The tabular form is more convenient form for performing the required calculations.
The logic for the tabular form is identical to that for the algebraic form.
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Yicheng Wang, Phone:303-492-4228, Email:[email protected]
Summary of the Simplex Method in Tabular Form
TABLE 4.3b The Initial Simplex Tableau
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TABLE 4.3b The Initial Simplex Tableau
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Iteration1
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University of Colorado at Boulder
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University of Colorado at Boulder
Yicheng Wang, Phone:303-492-4228, Email:[email protected]
Iteration2
Step1: Choose the entering basic variable to be x1
Step2: Choose the leaving basic variable to be x5
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Yicheng Wang, Phone:303-492-4228, Email:[email protected]
Step3: Solve for the new BF solution.
The new BF solution is (2,6,2,0,0) with Z =36
Optimality test: The solution (2,6,2,0,0) is optimal.
University of Colorado at Boulder
Yicheng Wang, Phone:303-492-4228, Email:[email protected]
(5) Tie Breaking in the Simplex Method
Tie for the Entering Basic Variable
The answer is that the selection between these contenders
may be made arbitrarily.
The optimal solution will be reached eventually, regardless
of the tied variable chosen.
University of Colorado at Boulder
Yicheng Wang, Phone:303-492-4228, Email:[email protected]
Tie for the Leaving Basci Variable-Degeneracy
If two or more basic variables tie for being the leaving basic
variable, choose any one of the tied basic variables to be the leaving
basic variable. One or more tied basic variables not chosen to be
the leaving basic variable will have a value of zero.
If a basic variable has a value of zero, it is called degenerate.
For a degenerate problem, a perpetual loop in computation is
theoretically possible, but it has rarely been known to occur in
practical problems. If a loop were to occur, one could always get
out of it by changing the choice of the leaving basic variable.
University of Colorado at Boulder
Yicheng Wang, Phone:303-492-4228, Email:[email protected]
No Leaving Basic Variable – Unbounded Z
If every coefficient in the pivot column of the simplex tableau is
either negative or zero, there is no leaving basic variable. This case
has an unbound objective function Z
If a problem has an unbounded objective function, the model
probably has been misformulated, or a computational mistake may
have occurred.
University of Colorado at Boulder
Yicheng Wang, Phone:303-492-4228, Email:[email protected]
Multiple Optimal Solution
In this example, Points C and D are two CPF Solutions, both of
which are optimal. So every point on the line segment CD is
optimal.
Therefore, all optimal
solutions are a weighted
average of these two optimal
CPF solutions.
Fig. 3.5 The Wyndor Glass Co.
problem would have multiple optimal
solutions if the objective function
were changed to Z = 3x1 + 2x2
C (2,6)
E (4,3)
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Yicheng Wang, Phone:303-492-4228, Email:[email protected]
Multiple Optimal Solution
Any linear programming problem with multiple optimal solutions
has at least two CPF solutions that are optimal. All optimal
solutions are a weighted average of these two optimal CPF
solutions.
Consequently, in augmented form, any linear programming
problem with multiple optimal solutions has at least two BF
solutions that are optimal. All optimal solutions are a weighted
average of these two optimal BF solutions.
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Multiple Optimal Solution
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These two are the only BF solutions that are optimal, and all other
optimal solutions are a convex combination of these .
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Yicheng Wang, Phone:303-492-4228, Email:[email protected]
(6) Adapting to Other Model Forms
Original Form of the Model
Augmented Form of the Model
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Artificial-Variable Technique
The purpose of artificial-variable technique is to obtain an initial BF solution.
The procedure is to construct an artificial problem that has the same optimal solution as the
real problem by making two modifications of the real problem.
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Yicheng Wang, Phone:303-492-4228, Email:[email protected]
Augmented Form of the Artificial Problem
Initial Form of the Artificial Problem
The Real Problem
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The feasible region of the Real Problem
The feasible region of the Artificial Problem
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Converting Equation (0) to Proper Form
The system of equations after the artificial problem is augmented is
To algebraically eliminate
from Eq. (0), we need to subtract from Eq. (0)
the product, M times Eq. (3)
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Application of the Simplex Method
The new Eq. (0) gives Z in terms of just the nonbasic variables (x1, x2)
The coefficient can be expressed as a linear function aM+b, where a is
called multiplicative factor and b is called additive term.
When multiplicative factors a’s are not equal, use just multiplicative
factors to conduct the optimality test and choose the entering basic
variable.
When multiplicative factors are equal, use the additive term to conduct
the optimality test and choose the entering basic variable.
M only appears in Eq. (0), so there’s no need to take into account M
when conducting the minimum ratio test for the leaving basic variable.
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Yicheng Wang, Phone:303-492-4228, Email:[email protected]
Solution to the Artificial Problem
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Functional Constraints in ≥ Form
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The Big M method is applied to solve the following artificial problem
(in augmented form)
The minimization problem is converted to the maximization problem by
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Solving the Example
The simplex method is applied to solve the following example.
The following operation shows how Row 0 in the simplex tableau is
obtained.
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Yicheng Wang, Phone:303-492-4228, Email:[email protected]
University of Colorado at Boulder
Yicheng Wang, Phone:303-492-4228, Email:[email protected]
The Real Problem
The Artificial Problem
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The Two-Phase Method
Since the first two coefficients are negligible compared to M, the two-phase
method is able to drop M by using the following two objectives.
The optimal solution of Phase 1 is a BF solution for the real problem, which
is used as the initial BF solution.
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Yicheng Wang, Phone:303-492-4228, Email:[email protected]
Summary of the Two-Phase Method
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Example:
Phase 1 Problem (The above example)
Phase2 Problem
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Solving Phase 1 Problem
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Preparing to Begin Phase 2
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Solving Phase 2 Problem
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How to identify the problem with no feasible solutons
The artificial-variable technique and two-phase method are used to find the
initial BF solution for the real problem.
If a problem has no feasible solutions, there is no way to find an initial BF
solution.
The artificial-variable technique or two-phrase method can provide the
information to identify the problems with no feasible solutions.
University of Colorado at Boulder
Yicheng Wang, Phone:303-492-4228, Email:[email protected]
To illustrate, let us change the first constraint in the last example as follows.
The solution to the revised example is shown as follows.
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(7) Shadow Prices
(0)
(1)
(2)
(3)
Resource bi = production time available in Plant i for the new
products.
How will the objective function value change if any bi is
increased by 1 ?
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b2: from 12 to 13
Z: from 36 to 37.5
△Z=3/2
b1: from 4 to 5
Z: from 36 to 36
△Z=0
b3: from 18 to 19
Z: from 36 to 37
△Z=1
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University of Colorado at Boulder
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indicates that adding 1 more hour of production time in Plant 2 for the two
new products would increase the total profit by $1,500.
The constraint on resource 1 is
not binding on the optimal
solution, so there is a surplus of
this resource. Such resources are
called free goods
The constraints on resources 2 and
3 are binding constraints. Such
resources are called scarce goods.
H(0,9)
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(8) Sensitivity Analysis
Maximize
b, c, and a are parameters whose values will not be known exactly
until the alternative given by linear programming is implemented in
the future.
The main purpose of sensitivity analysis is to identify the sensitive
parameters.
A parameter is called a sensitive parameter if the optimal solution
changes with the parameter.
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Yicheng Wang, Phone:303-492-4228, Email:[email protected]
How are the sensitive parameters identified?
In the case of bi , the shadow price is used to determine if a parameter
is a sensitive one.
For example, if
> 0 , the optimal solution changes with the bi.
However, if
= 0 , the optimal solution is not sensitive to at least
small changes in bi.
For c2 =5, we have
c1 =3 can be changed to any other value
from 0 to 7.5 without affecting the
optimal solution (2,6)
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(8) Parametric Linear Programming
Sensitivity analysis involves changing one parameter at a time in the
original model to check its effect on the optimal solution.
By contrast, parametric linear programming involves the
systematic study of how the optimal solution changes as many of
the parameters change simultaneously over some range.