1 1 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. INTRODUCTION TO MANAGEMENT SCIENCE, 13e Anderson Sweeney Williams Martin

2 2 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Chapter 2 An Introduction to Linear Programming n Linear Programming Problem n Problem Formulation n A Simple Maximization Problem n Graphical Solution Procedure n Extreme Points and the Optimal Solution n Computer Solutions n A Simple Minimization Problem n Special Cases

3 3 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Linear Programming n Linear programming has nothing to do with computer programming. n The use of the word “programming” here means “choosing a course of action.” or planning n Linear programming involves choosing a course of action when the mathematical model of the problem contains only linear functions. n Maximize or Minimize subject to some constraints

4 4 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Linear Programming (LP) Problem n The maximization or minimization of some quantity is the objective in all linear programming problems. n All LP problems have constraints that limit the degree to which the objective can be pursued. n A feasible solution satisfies all the problem's constraints. n An optimal solution is a feasible solution that results in the largest possible objective function value when maximizing (or smallest when minimizing). n A graphical solution method can be used to solve a linear program with two variables.

5 5 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Linear Programming (LP) Problem n Linearity Assumption Objective function and constraints are all linear Objective function and constraints are all linear n What is Linearity? Proportionality Proportionality Additivity Additivity Divisibility Divisibility n Practical Assumption Linearity Linearity Deterministic : parameter values are fixed not stochastic Deterministic : parameter values are fixed not stochastic

6 6 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Problem Formulation

7 7 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Guidelines for Model Formulation n Understand the problem thoroughly. n Describe the objective. n Describe each constraint. n Define the decision variables. n Write the objective in terms of the decision variables. n Write the constraints in terms of the decision variables.

8 8 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 1: Graphical Solution n First Constraint Graphed

9 9 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 1: Graphical Solution n Other Constraints

10 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 1: Graphical Solution n Combined-Constraint Graph Showing Feasible Region

11 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 1: Graphical Solution n Objective Function Lines

12 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 1: Graphical Solution n Optimal Solution

13 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Summary of the Graphical Solution Procedure for Maximization Problems n Prepare a graph of the feasible solutions for each of the constraints. n Determine the feasible region that satisfies all the constraints simultaneously. n Draw an objective function line. n Move parallel objective function lines toward larger objective function values without entirely leaving the feasible region. n Any feasible solution on the objective function line with the largest value is an optimal solution.

14 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slack Variables

15 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. n Example 1 in Standard Form Slack Variables (for < constraints) s 1, s 2, s 3, and s 4 are s lack variables

16 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Slack and Surplus Variables n A linear program in which all the variables are non- negative and all the constraints are equalities is said to be in standard form. n Standard form is attained by adding slack variables to "less than or equal to" constraints, and by subtracting surplus variables from "greater than or equal to" constraints. n Slack and surplus variables represent the difference between the left and right sides of the constraints. n Slack and surplus variables have objective function coefficients equal to 0.

17 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Extreme Points and the Optimal Solution n If

18 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Extreme Points and the Optimal Solution n The corners or vertices of the feasible region are referred to as the extreme points. n An optimal solution to an LP problem can be found at an extreme point of the feasible region. n When looking for the optimal solution, you do not have to evaluate all feasible solution points. n You have to consider only the extreme points of the feasible region. n But, if the number of decision variables is 50 and the number of constraints is 100 the number of extreme points is the number of extreme points is

19 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Computer Solutions n Management Scientist

20 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Computer Solutions n LP problems involving 1000s of variables and 1000s of constraints are now routinely solved with computer packages. n Linear programming solvers are now part of many spreadsheet packages, such as Microsoft Excel. n Leading commercial packages include CPLEX, LINGO, MOSEK, Xpress-MP, and Premium Solver for Excel. n Using Excel n Using LINGO

21 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Reduced Cost n The reduced cost for a decision variable whose value is 0 in the optimal solution is: the amount the variable's objective function coefficient would have to improve (increase for maximization problems, decrease for minimization problems) before this variable could assume a positive value. n The reduced cost for a decision variable whose value is > 0 in the optimal solution is 0. (see. p.115)

22 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 2: A Simple Minimization Problem n LP Formulation

23 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 2: Graphical Solution n Constraints Graphed

24 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 2: Graphical Solution n Optimal Solution

25 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Summary of the Graphical Solution Procedure for Minimization Problems n Prepare a graph of the feasible solutions for each of the constraints. n Determine the feasible region that satisfies all the constraints simultaneously. n Draw an objective function line. n Move parallel objective function lines toward smaller objective function values without entirely leaving the feasible region. n Any feasible solution on the objective function line with the smallest value is an optimal solution.

26 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Surplus Variables n Example 2 in Standard Form s 1 and s 2 are surplus variables

27 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example 2: Spreadsheet Solution n Interpretation of Computer Output n Excel output n LINGO output

28 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Feasible Region n The feasible region for a two-variable LP problem can be nonexistent, a single point, a line, a polygon, or an unbounded area. n Any linear program falls in one of four categories: is infeasible is infeasible has a unique optimal solution has a unique optimal solution has alternative optimal solutions has alternative optimal solutions has an objective function that can be increased without bound has an objective function that can be increased without bound n A feasible region may be unbounded and yet there may be optimal solutions. This is common in minimization problems and is possible in maximization problems.

29 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Special Cases n Alternative Optimal Solutions In the graphical method, if the objective function line is parallel to a boundary constraint in the direction of optimization, there are alternate optimal solutions, with all points on this line segment being optimal.

30 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Special Cases n Infeasibility No solution to the LP problem satisfies all the constraints, including the non-negativity conditions. No solution to the LP problem satisfies all the constraints, including the non-negativity conditions. Graphically, this means a feasible region does not exist. Graphically, this means a feasible region does not exist. Causes include: Causes include: A formulation error has been made.A formulation error has been made. Management’s expectations are too high.Management’s expectations are too high. Too many restrictions have been placed on the problem (i.e. the problem is over-constrained).Too many restrictions have been placed on the problem (i.e. the problem is over-constrained).

31 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Infeasible Problem n Consider the following LP problem.

32 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Infeasible Problem

33 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Example: Unbounded Solution n Consider the following LP problem. Max 20 X + 10 Y s.t. X > 2 Y > 5 Y > 5 X, Y > 0 X, Y > 0

34 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. Special Cases n Unbounded The solution to a maximization LP problem is unbounded if the value of the solution may be made indefinitely large without violating any of the constraints. The solution to a maximization LP problem is unbounded if the value of the solution may be made indefinitely large without violating any of the constraints. For real problems, this is the result of improper formulation. (Quite likely, a constraint has been inadvertently omitted.) For real problems, this is the result of improper formulation. (Quite likely, a constraint has been inadvertently omitted.)

35 Slide © 2008 Thomson South-Western. All Rights Reserved © 2011 Cengage Learning. All Rights Reserved. May not be scanned, copied or duplicated, or posted to a publicly accessible website, in whole or in part. End of Chapter 2