Water Resources Development and Management
Optimization
(Linear Programming)
CVEN 5393
Feb 25, 2013
Acknowledgements
• Dr. Yicheng Wang (Visiting
Researcher, CADSWES during Fall
2009 – early Spring 2010) for
slides from his Optimization
course during Fall 2009
• Introduction to Operations
Research by Hillier and Lieberman,
McGraw Hill
Today’s Lecture
• Simplex Method
– Recap of algebraic form
– Simplex Method in Tabular form
• Simplex Method for other forms
– Equality Constraints
– Minimization Problems
(Big M and Twophase methods)
• Sensitivity / Shadow Prices
• R-resources / demonstration
(2) Setting Up the Simplex Method
Original Form of the Model
Augmented Form of the Model
SIMPLEX METHOD
(TABULAR FORM)
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.
Summary of the Simplex Method in Tabular Form
TABLE 4.3b The Initial Simplex Tableau
TABLE 4.3b The Initial Simplex Tableau
Iteration1
Iteration2
Step1: Choose the entering basic variable to be x1
Step2: Choose the leaving basic variable to be x5
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.
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 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.
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.
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)
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.
Multiple Optimal Solution
These two are the only BF solutions that are optimal, and all other
optimal solutions are a convex combination of these .
SENSITIVITY
(SHADOW PRICES & SIGNIFICANCE)
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 ?
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
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)
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.
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)
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.
ADAPTING SIMPLEX TO OTHER FORMS
(MINIMIZATION, EQUALITY AND
GREATER THAN EQUAL CONSTRAINTS, )
Adapting to Other Model Forms
Original Form of the Model
Augmented Form of the Model
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.
Augmented Form of the Artificial Problem
Initial Form of the Artificial Problem
The Real Problem
The feasible region of the Real Problem
The feasible region of the Artificial Problem
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)
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.
Solution to the Artificial Problem
Functional Constraints in ≥ Form
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
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.
The Real Problem
The Artificial Problem
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.
Summary of the Two-Phase Method
Example:
Phase 1 Problem (The above example)
Phase2 Problem
Solving Phase 1 Problem
Preparing to Begin Phase 2
Solving Phase 2 Problem
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.
To illustrate, let us change the first constraint in the last example as follows.
The solution to the revised example is shown as follows.