Lecture 3

Last Lecture Summary: Covered remaining of Sec. 10.2 and Sec. 10.3 Alternative optimal solution. No feasible solution. Unbounded solution. Applications of linear programming. Diet mix model and transportation model.

Today’s Main Topics We will start Chapter 11 Sec. 11.1: Simplex Preliminaries Overview of the Simplex Method Requirements of Simplex Method Basic feasible solutions Definitions of Feasible Solution; Basic Solution; and Basic Feasible Solution

The Simplex and Computer Solutions Methods Chapter 11 The Simplex and Computer Solutions Methods

Chapter Objectives Provide an understanding of the simplex method of solving LP problems Illustrate the ways in which special LP phenomena evidence themselves when using the simplex method Discuss computer solution methods for LP problems, with particular emphasis on the interpretation of computer results

Main Topics Simplex Preliminaries The Simplex Method Special Phenomena Computer Solution Methods The Dual Problem

Simplex Preliminaries WHY SIMPLEX METHOD??? ANSWER: Graphical solution methods are applicable to LP problems involving TWO variables The geometry of THREE-variable problems is very complicated, and Beyond Three-variables, there is no geometric frame of reference SO, NEED some NONGRAPHICAL Method...

The most popular nongraphical procedure is called The Simplex Method This is an algebraic procedure for solving system of equations where an objective function needs to be optimized. It is an iterative process, which identifies a feasible starting solution The procedure then searches to see whether there exists a better solution “Better” is measured by whether the value of the objective function can be improved

The Simplex Method Cont’d... If better solution is signaled, the search resumes The generation of each successive solution requires solving a system of linear equations The search continues until no further improvement is possible in the objective function

Requirements of the Simplex Method All constraints must be stated as equations. The right side of the constraint cannot be negative. All variables are restricted to nonnegative values.

Most linear programming problems contain constraints which are inequalities. Before we solve by the simplex method, these inequalities must be restated as equations. The transformation from inequalities to equations varies, depending on the nature of the constraints.

Transformation procedure for ≤ constraints For each “less than or equal to” constraint, a non-negative variable, called a slack variable, is added to the left side of the constraint. This variable serves the function of balancing the two sides of the equation. Note that the slack variables become additional variables in the problem and must be treated like any other variables. That means they are also subject to Requirement III; that is, they cannot assume negative values.

Example Consider the two constraints 2x1 + 3x2 ≤ 50 4x1 + 2x2 ≤ 60 where x1 and x2 are the number of units produced of products A and B, respectively. Assume that the two constraints represent the limited labor availability in two departments.

Transformation procedure for ≥ constraints For each “greater than or equal to” constraint, a non-negative variable E, called a surplus variable, is subtracted from the left side of the constraint. It keeps the two sides of equation in balance. In addition, a non-negative variable A, called an artificial variable, is added to the left of side of the constraint. The artificial variable has no real meaning in the problem; its only function is to provide a convenient starting point (initial guess) for the simplex.

Transformation procedure for = constraints For each “equal to” constraint, an artificial variable, is added to the left of side of the constraint.

Example Transform the following constraint set into the standard form required by the simplex method x1 + x2 ≤ 100 2x1 + 3x2 ≥ 40 x1 – x2 = 25 x1, x2 ≥ 0

Solution The transformed constraint set is x1 + x2 + S1 = 100 2x1 + 3x2 – E2 + A2 = 40 x1 – x2 + A3 = 25 x1, x2, S1, E2, A2, A3 ≥ 0 Note that each supplemental variable (slack, surplus, artificial) is assigned a subscript which corresponds to the constraint number. Also, the non-negativity restriction (requirement III) applies to all supplemental variables.

Requirement II of the simplex method states that the right side of any constraint equation not be negative. If a constraint has a negative right side, the constraint can be multiplied by – 1 to make the right side positive

Requirement III of the simplex method states that all variables be restricted to nonnegative values. There are specialize techniques for dealing with variables which can assume negative values; however, we will not examine these methods. The only point which should be repeated is that slack, surplus, and artificial variable are also restricted to being nonnegative.

Example An LP problem has 5 decision variables; 10 (≤) constraints; 8 (≥) constraints; and 2 (=) constraints. When this problem is restated to comply with requirement I of the simplex method, how many variables will there be and of what types?

Solution There will be 33 variables: 5 decision variables, 10 slack variables associated with the 10 (≤) constraints, 8 surplus variables associated with the 8 (≥) constraints, and 10 artificial variables associated with the (≥) and (=) constraints. 5 + 10 + 8 + 10 = 33

Basic Feasible Solutions Let’s state some definitions which are significant to our coming discussions. Assume the standard form of an LP problem which has m structural constraints and a total of n’ decision and supplemental variables.

FEASIBLE SOLUTION A feasible solution is any set of values for the n’ variables which satisfies both the structural and non-negativity constraints. BASIC SOLUTION A basic solution is any solution obtained by setting (n’–m) variables equal to 0 and solving the system of equations for the values of the remaining m variables. The m variables solved for are called basic variables. These variables are said to constitute a basis. The remaining (n’–m) variables, or those which have been assigned values of 0, are called non-basic variables. BASIC FEASIBLE SOLUTION A basic feasible solution is a basic solution which also satisfies the non-ngativity constraints.

It can be shown that the optimal solution to an LP problem is included in the set of basic feasible solutions. Thus, the optimal solution can be found by performing a search of the set of basic feasible solutions.

Review Covered Sec. 11.1: Simplex Preliminaries Requirements of Simplex Method Requirement I for different types of constraints Basic Feasible Solutions Next time, we’ll start Sec 11.2: The Simplex Method