Linear Programming

SUPPLEMENT OUTLINE Introduction, 6S-2

Solving LP Models Using Excel, 6S-17

SUPPLEMENT TO

CHAPTER

LEARNING OBJECTIVES

Linear Programming Model, 6S-2

Formulating Some Other Types of Constraints, 6S-3

Sensitivity Analysis, 6S-19

Objective Function Coefficient Changes, 6S-20 Changes in the Right-Hand Side (RHS) Value of a Constraint, 6S-21

After completing this supplement, you should be able to:

1 Describe the type of problem

Graphical Solution Method, 6S-4

Outline of Graphical Solution Method, 6S-4 Plotting Constraints, 6S-6 Identifying the Feasible Solution Space, 6S-9 Plotting an Objective Function Line, 6S-9 Redundant Constraints, 6S-13 Solutions and Corner Points, 6S-13 Minimization, 6S-14 Slack and Surplus, 6S-15

The Simplex Method, 6S-16 Computer Solutions, 6S-16

Key Terms, 6S-23 Solved Problems, 6S-23 Discussion and Review Questions, 6S-26 Problems, 6S-26 Mini-Case: Airline Fuel Management, 6S-31 Mini-Case: Son Ltd., 6S-32 Selected Bibliography and Further Reading, 6S-32

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that would lend itself to solution using linear programming. Formulate a linear programming model from a description of a problem. Solve simple linear programming problems using the graphical method. Interpret computer solutions of linear programming problems. Do sensitivity analysis on the solution of a linear programming problem.

6S-1

6S-2

SUPPLEMENT TO CHAPTER SIX LINEAR PROGRAMMING

INTRODUCTION

Linear programming is a powerful quantitative tool used by operations managers and other managers to obtain optimal solutions to problems that involve restrictions or limitations, such as the available materials, budgets, and labour and machine time. These problems are referred to as constrained optimization problems. There are numerous examples of linear programming applications to such problems, including: • Establishing locations for emergency equipment and personnel that will minimize response time • Determining optimal schedules for airlines for planes, pilots, and ground personnel • Developing financial plans • Determining optimal blends of animal feed mixes • Determining optimal diet plans • Identifying the best set of worker–job assignments • Developing optimal production schedules • Developing shipping plans that will minimize shipping costs • Identifying the optimal mix of products in a factory

LINEAR PROGRAMMING MODEL

A Linear programming model is a mathematical representation of a constrained optimization problem. Three components provide the structure of a linear programming model: 1. Decision variables. 2. Objective. 3. Constraints. decision variables Amounts of either inputs or outputs.

objective function Mathematical statement of profit (or cost, etc.). constraints Limitations that restrict the available alternatives.

feasible solution space The set of all feasible combinations of decision variables as defined by the constraints. parameters Numerical constants.

Decision variables represent choices available to the decision maker in terms of amounts of either inputs or outputs. Each unknown quantity is assigned a decision variable (e.g., x1, x2). A Linear programming model requires a single goal or objective. The two general types of objective are maximization and minimization. A maximization objective might involve profit, revenue, etc. Conversely, a minimization objective might involve cost, time, or distance. The objective function is a mathematical expression of decision variables. Constraints are limitations that restrict the alternatives available to decision makers. The three types of constraints are less than or equal to ( ), greater than or equal to ( ), and simply equal to ( ). A constraint implies an upper limit on the amount of some resource (e.g., machine hours, labour hours, materials) available for use. A constraint specifies a minimum that must be achieved in the final solution (e.g., must contain at least 10 percent real...