University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
Optimization Techniques forCivil and Environmental Engineering Systems
By
Yicheng Wang
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
(2) Setting Up the Simplex Method
Original Form of the Model
Augmented Form of the Model
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
F
G
H
C
D
B
A EH(3,2)
Augmented Form of the Model
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)
F
G
H
C
D
B
A EH(3,2)
University of Colorado at BoulderYicheng 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
Corner-point solution
Corner-point feasible (CPF) solution
In the augmented model, we have
Basic solution
Basic Feasible (BF) solution
University of Colorado at BoulderYicheng 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
F
G
H
C
D
B
A E
University of Colorado at BoulderYicheng 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.
University of Colorado at BoulderYicheng 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)
F
G
H
C
D
B
A EH(3,2)
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)
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
University of Colorado at BoulderYicheng 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 Algebraic interpretation
C
D
B
A E
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
Optimality Test
Geometric interpretation Algebraic interpretation
C
D
B
A E
A(0,0) is not optimal.
Conclusion: The initial BF solution (0,0,4,12,18) is not optimal.
The objective function:
The rate of improvement of Z by the nonbasic variable x1 is 3
The rate of improvement of Z by the nonbasic variable x2 is 5
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
Iteration1 Step1: Determining the Direction of Movement
Geometric interpretation Algebraic interpretation
C
D
B
A E
Move up from A(0,0) to B(0,6)
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
Iteration1 Step2: Where to Stop
Geometric interpretation Algebraic interpretation
C
D
B
A E
Stop at B. Otherwise, it will leave the feasible region.
Step 2 determine how far to increase the entering basic variable x2.
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
Thus x4 is the leaving basic variable for iteration 1 of the example.
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
Iteration1 Step3: Solving for the New BF Solution
Geometric interpretation Algebraic interpretation
C
D
B
A E
The intersection of the new pair of constraint boundary: B(0,6) Nonbasic variables
Basic variables
Nonbasic variables
Basic variables
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
(0)
(1)
(2)
(3)
Nonbasic variables: x1= 0
x2= 0Basic variables: x3= 4
x4 =12
x5= 18
Initial BF Solution
Nonbasic variables: x1= 0
x4= 0Basic variables: x3= ?
x2 =6
x5= ?
New BF Solution(0)
(1)
(2)
(3)
Basic variables: x3= 4
x2 =6
x5= 6
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
Optimality Test
Geometric interpretation Algebraic interpretation
C
D
B
A E
B(0,6) is not optimal, because moving from B to C increases Z.
Conclusion: The BF solution (0,6,4,0,6) is not optimal.
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
University of Colorado at BoulderYicheng 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)
(2)
(3)
Nonbasic variables: x1= 0, x4= 0
Basic variables: x3= 2, x2 =6, x1= 2
New BF Solution
University of Colorado at BoulderYicheng 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.
C
D
B
A 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 BoulderYicheng 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.
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
Summary of the Simplex Method in Tabular Form
TABLE 4.3b The Initial Simplex Tableau
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
TABLE 4.3b The Initial Simplex Tableau
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
Iteration1
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
University of Colorado at BoulderYicheng 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
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
Step3: Solve for the new BF solution.
Optimality test: The solution (2,6,2,0,0) is optimal.The new BF solution is (2,6,2,0,0) with Z =36
University of Colorado at BoulderYicheng 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.
Tie for the Entering Basic Variable
University of Colorado at BoulderYicheng 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 BoulderYicheng 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 BoulderYicheng 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.
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)
Therefore, all optimal solutions are a weighted average of these two optimal CPF solutions.
University of Colorado at BoulderYicheng 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.
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
Multiple Optimal Solution
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
These two are the only BF solutions that are optimal, and all other optimal solutions are a convex combination of these .
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
(6) Adapting to Other Model Forms
Original Form of the Model Augmented Form of the Model
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
Artificial-Variable TechniqueThe 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.
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
Augmented Form of the Artificial Problem
Initial Form of the Artificial Problem The Real Problem
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
The feasible region of the Real Problem The feasible region of the Artificial Problem
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
Converting Equation (0) to Proper FormThe 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)
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
Application of the Simplex MethodThe 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.
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
Solution to the Artificial Problem
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
Functional Constraints in ≥ Form
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
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
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
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.
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
The Real Problem The Artificial Problem
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
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.
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
Summary of the Two-Phase Method
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
Phase 1 Problem (The above example)
Example:
Phase2 Problem
Example:
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
Solving Phase 1 Problem
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
Preparing to Begin Phase 2
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
Solving Phase 2 Problem
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
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 BoulderYicheng 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.
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
(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 ?
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
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
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
indicates that adding 1 more hour of production time in Plant 2 for the twonew 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)
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
(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.
University of Colorado at BoulderYicheng 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)
University of Colorado at BoulderYicheng Wang, Phone:303-492-4228, Email:[email protected]
(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.