University of Colorado at Boulder Yicheng Wang, Phone: , Optimization Techniques for Civil and Environmental Engineering Systems By Yicheng Wang

University of Colorado at Boulder Yicheng Wang, Phone: , (2) Setting Up the Simplex Method Original Form of the Model Augmented Form of the Model

University of Colorado at Boulder Yicheng Wang, Phone: , F G H C D B AE H(3,2) Augmented Form of the Model For example, H(3,2) is a solution for the original model, which yields the augmented solution ( x 1, x 2, x 3, x 4, x 5 ) = (3, 2,1,8, 5) For example, G(4,6) is a corner-point infeasible solution, which yields the corresponding basic solution ( x 1, x 2, x 3, x 4, x 5 ) = (4, 5,0,0, -6) F G H C D B AE H(3,2)

University of Colorado at Boulder Yicheng Wang, Phone: , 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 Boulder Yicheng Wang, Phone: , 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 x 1 =0 and x 2 =0 are the nonbasic variables, and x 3 =4, x 4 =12, and x 5 =18 are the basic variables F G H C D B AE

University of Colorado at Boulder Yicheng Wang, Phone: , 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 x 1 =0 and x 2 =0 are the nonbasic variables, and x 3 =4, x 4 =12, and x 5 =18 are the basic variables Choose x 1 and x 4 to be the nonbasic variables that are set equal to 0. The three equations then yield, respectively, x 3 =4, x 2 =6, and x 5 =6 as the solution for the three basic variables as shown below.

University of Colorado at Boulder Yicheng Wang, Phone: , Example: A(0,0) and B(0,6) are two CPF solutions The corresponding BF solutions are ( x 1, x 2, x 3, x 4, x 5 ) = (0, 0,4,12, 18) and ( x 1, x 2, x 3, x 4, x 5 ) = (0, 6,4,0, 6) F G H C D B AE H(3,2) A(0,0) and C(2,6) are two CPF solutions The corresponding BF solutions are ( x 1, x 2, x 3, x 4, x 5 ) = (0, 0,4,12, 18) and ( x 1, x 2, x 3, x 4, x 5 ) = (2, 6,2,0, 0)

University of Colorado at Boulder Yicheng Wang, Phone: ,

University of Colorado at Boulder Yicheng Wang, Phone: , (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 AE

University of Colorado at Boulder Yicheng Wang, Phone: , Optimality Test Geometric interpretation Algebraic interpretation C D B AE 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 x 1 is 3 The rate of improvement of Z by the nonbasic variable x 2 is 5

University of Colorado at Boulder Yicheng Wang, Phone: , Iteration1 Step1: Determining the Direction of Movement Geometric interpretation Algebraic interpretation C D B AE Move up from A(0,0) to B(0,6)

University of Colorado at Boulder Yicheng Wang, Phone: , Iteration1 Step2: Where to Stop Geometric interpretation Algebraic interpretation C D B AE Stop at B. Otherwise, it will leave the feasible region. Step 2 determine how far to increase the entering basic variable x 2.

University of Colorado at Boulder Yicheng Wang, Phone: , Thus x 4 is the leaving basic variable for iteration 1 of the example.

University of Colorado at Boulder Yicheng Wang, Phone: , Iteration1 Step3: Solving for the New BF Solution Geometric interpretation Algebraic interpretation C D B AE The intersection of the new pair of constraint boundary: B(0,6) Nonbasic variables Basic variables Nonbasic variables Basic variables

University of Colorado at Boulder Yicheng Wang, Phone: , (0) (1) (2) (3) Nonbasic variables: x 1 = 0 x 2 = 0 Basic variables: x 3 = 4 x 4 =12 x 5 = 18 Initial BF Solution Nonbasic variables: x 1 = 0 x 4 = 0 Basic variables: x 3 = ？ x 2 =6 x 5 = ？ New BF Solution (0) (1) (2) (3) Basic variables: x 3 = 4 x 2 =6 x 5 = 6

University of Colorado at Boulder Yicheng Wang, Phone: , Optimality Test Geometric interpretation Algebraic interpretation C D B AE 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 x 1 is 3 The rate of improvement of Z by the nonbasic varialle x 4 is -5/2

University of Colorado at Boulder Yicheng Wang, Phone: , Iteration2 Step1: Determining the Direction of Movement Choose x 1 to be the entering basic variable Step2: Where to Stop The minimum ratio test indicates that x 5 is the leaving basic variable Step3: Solving for the New BF Solution (0) (1) (2) (3) Nonbasic variables: x 1 = 0, x 4 = 0 Basic variables: x 3 = 2, x 2 =6, x 1 = 2 New BF Solution

University of Colorado at Boulder Yicheng Wang, Phone: , Optimality Test The objective function: The coefficients of the nonbasic variables x 4 and x 5 are negative. Increasing either x 4 or x 5 will decrease Z, so (x 1, x 2, x 3, x 4, x 5 ) = (2, 6, 2, 0, 0) must be optimal with Z = 36. C D B AE In terms of the original form of the problem (no slack variables), the optimal solution is (x 1, x 2 ) = (2, 6), which yields Z = 3x 1 +5x 2 =36.

University of Colorado at Boulder Yicheng Wang, Phone: , (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 Boulder Yicheng Wang, Phone: , Summary of the Simplex Method in Tabular Form TABLE 4.3b The Initial Simplex Tableau

University of Colorado at Boulder Yicheng Wang, Phone: , TABLE 4.3b The Initial Simplex Tableau

University of Colorado at Boulder Yicheng Wang, Phone: , Iteration1

University of Colorado at Boulder Yicheng Wang, Phone: ,

University of Colorado at Boulder Yicheng Wang, Phone: ,

University of Colorado at Boulder Yicheng Wang, Phone: ,

University of Colorado at Boulder Yicheng Wang, Phone: , Iteration2 Step1: Choose the entering basic variable to be x 1 Step2: Choose the leaving basic variable to be x 5

University of Colorado at Boulder Yicheng Wang, Phone: , 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 Boulder Yicheng Wang, Phone: , (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 Boulder Yicheng Wang, Phone: , 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: , 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: , 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 = 3x 1 + 2x 2 C (2,6) E (4,3) Therefore, all optimal solutions are a weighted average of these two optimal CPF solutions.

University of Colorado at Boulder Yicheng Wang, Phone: , 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 Boulder Yicheng Wang, Phone: , Multiple Optimal Solution

University of Colorado at Boulder Yicheng Wang, Phone: ,

University of Colorado at Boulder Yicheng Wang, Phone: , 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 Boulder Yicheng Wang, Phone: , (6) Adapting to Other Model Forms Original Form of the Model Augmented Form of the Model

University of Colorado at Boulder Yicheng Wang, Phone: , 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.

University of Colorado at Boulder Yicheng Wang, Phone: , Augmented Form of the Artificial Problem Initial Form of the Artificial Problem The Real Problem

University of Colorado at Boulder Yicheng Wang, Phone: , The feasible region of the Real ProblemThe feasible region of the Artificial Problem

University of Colorado at Boulder Yicheng Wang, Phone: , 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)

University of Colorado at Boulder Yicheng Wang, Phone: , Application of the Simplex Method The new Eq. (0) gives Z in terms of just the nonbasic variables (x 1, x 2 ) 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 Boulder Yicheng Wang, Phone: , Solution to the Artificial Problem

University of Colorado at Boulder Yicheng Wang, Phone: , Functional Constraints in ≥ Form

University of Colorado at Boulder Yicheng Wang, Phone: , 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 Boulder Yicheng Wang, Phone: , 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 Boulder Yicheng Wang, Phone: ,

University of Colorado at Boulder Yicheng Wang, Phone: , The Real ProblemThe Artificial Problem

University of Colorado at Boulder Yicheng Wang, Phone: , 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 Boulder Yicheng Wang, Phone: , Summary of the Two-Phase Method

University of Colorado at Boulder Yicheng Wang, Phone: , Phase 1 Problem (The above example) Example: Phase2 Problem Example:

University of Colorado at Boulder Yicheng Wang, Phone: , Solving Phase 1 Problem

University of Colorado at Boulder Yicheng Wang, Phone: , Preparing to Begin Phase 2

University of Colorado at Boulder Yicheng Wang, Phone: , Solving Phase 2 Problem

University of Colorado at Boulder Yicheng Wang, Phone: , 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: , 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 Boulder Yicheng Wang, Phone: , (7) Shadow Prices (0) (1) (2) (3) Resource b i = production time available in Plant i for the new products. How will the objective function value change if any b i is increased by 1 ?

University of Colorado at Boulder Yicheng Wang, Phone: , b 2 : from 12 to 13 Z: from 36 to 37.5 △ Z=3/2 b 1 : from 4 to 5 Z: from 36 to 36 △ Z=0 b 3 : from 18 to 19 Z: from 36 to 37 △ Z=1

University of Colorado at Boulder Yicheng Wang, Phone: ,

University of Colorado at Boulder Yicheng Wang, Phone: , 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)

University of Colorado at Boulder Yicheng Wang, Phone: , (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 Boulder Yicheng Wang, Phone: , How are the sensitive parameters identified? In the case of b i, the shadow price is used to determine if a parameter is a sensitive one. For example, if > 0, the optimal solution changes with the b i. However, if = 0, the optimal solution is not sensitive to at least small changes in b i. For c 2 =5, we have c 1 =3 can be changed to any other value from 0 to 7.5 without affecting the optimal solution (2,6)

University of Colorado at Boulder Yicheng Wang, Phone: , (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.