Chapter 9

 Dual of minimum cost problem seeks highest. production value given resource constraints[r]

MANAGERIAL ECONOMICS

MANAGERIAL ECONOMICS

12

12thth Edition Edition

By By

Linear Programming

Linear Programming

Chapter 9 Chapter 9 OVERVIEW OVERVIEW

 Basic Assumptions

 Production Planning for a Single Product  Production Planning for Multiple Products  Graphic Specification and Solution

 Algebraic Specification and Solution  Dual in Linear Programming

 Dual Specification

Chapter 9 Chapter 9

KEY CONCEPTS KEY CONCEPTS

 linear programming  optimal solution

 relative distance method

 feasible space

 objective function  corner point

 slack variables

 simplex solution method

 primal  dual

Basic Assumptions  Inequality Constraints

 Resource constraints limit usage to ≤ some

fixed amount.

 Output quantity or quality constraints limit

production to ≥ some fixed amount.

 Linear Assumptions

 Constant output prices.  Constant input prices.

Production Planning for a Single Product

 Production Processes

 Equal distances along the same process ray indicate

equal output quantities

 Equal distances along different process rays indicate

different output quantities

 Production Isoquants

 Linear segments represent input combinations used

to produce a given level of output

 Least-cost input combination is on feasible isocost

line closest to origin

 Maximum output with limited resources is on feasible

Production Planning for Multiple Products

 Objective Function Specification

 Maximize profits, revenue or quantity subject to ≤

resource constraints

 Minimize cost subject to ≥ output constraints

 Constraint Equation Specification

 Resource use cannot exceed availability

 Output quantity/quality constraints must be met

 Nonegativity Requirement

Graphic Specification and Solution  Solving the LP Problem

 Analytic expression is the first step.

 Graphic representation builds intuition.

 Graphing the LP Problem

 The LP feasible space graph shows

possibilities.

 The LP objective function depicts most

Algebraic Specification and Solution

 Slack Variables

 Slack variables convert ≥ or ≤ constraints into equalities  Zero slack implies full employment

 Positive slack implies excess capacity  Algebraic Solution

 Corner point with highest value is maximum  Corner point with lowest value is minimum  Slack Variables at the Solution Point

 Binding constraints imply no slack  Nonbinding constraints imply slack

 Computer-based solution methods work best for

Dual in Linear Programming

 Duality Concept

 Pairs of symmetrical LP problems are called

the primal and dual.

 Every primal has a dual and vice versa.  Primal and dual solutions are related.

 Shadow Prices

 Shadow prices are opportunity costs.

 Remember: costs of constrained resources

Dual Specification and Solution

 Dual Objective Function

 Dual of profit maximization problem seeks minimum

cost solution

 Dual of minimum cost problem seeks highest

production value given resource constraints

 Dual Constraints

 Binding constraints imply no slack  Nonbinding constraints imply slack

 Dual Slack Variables