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Water Resources Development and Management

Optimization(Linear Programming) CVEN 5393

Feb 25, 2013Acknowledgements Dr. Yicheng Wang (Visiting Researcher, CADSWES during Fall 2009 early Spring 2010) for slides from his Optimization course during Fall 2009Introduction to Operations Research by Hillier and Lieberman, McGraw Hill

Todays LectureSimplex MethodRecap of algebraic formSimplex Method in Tabular form

Simplex Method for other formsEquality ConstraintsMinimization 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 Model4Simplex Method (Tabular Form)The Simplex Method in Tabular FormThe 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

Iteration2Step1: Choose the entering basic variable to be x1Step2: Choose the leaving basic variable to be x5

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 =36Tie Breaking in the Simplex MethodTie 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 VariableTie for the Leaving Basci Variable-DegeneracyIf 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 ZIf 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 SolutionIn 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 + 2x2C (2,6)E (4,3)Therefore, all optimal solutions are a weighted average of these two optimal CPF solutions.

Multiple Optimal SolutionAny 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 37.5Z=3/2b1: from 4 to 5Z: from 36 to 36Z=0b3: from 18 to 19Z: from 36 to 37Z=1

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)Sensitivity Analysis

Maximizeb, 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 ProgrammingSensitivity 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 ModelAugmented Form of the Model

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.

Augmented Form of the Artificial Problem

Initial Form of the Artificial ProblemThe Real Problem

The feasible region of the Real ProblemThe feasible region of the Artificial Problem

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)

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 as 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 theres 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 ProblemThe Artificial ProblemThe 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

Phase 1 Problem (The above example)Example:

Phase2 ProblemExample:Solving Phase 1 Problem

Preparing to Begin Phase 2

Solving Phase 2 Problem

How to identify the problem with no feasible solutonsThe 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.