Chapter 9

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MANAGERIAL ECONOMICS

MANAGERIAL ECONOMICS

12

12

thth

Edition

_Edition

By

By

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Linear Programming

Linear Programming

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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

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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

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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.

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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

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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

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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

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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

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Dual in Linear Programming



Duality Concept



Pairs of symmetrical LP problems are called

the primal and dual.



Every primal has a dual and

<i>vice versa</i>

.



Primal and dual solutions are related.



Shadow Prices



Shadow prices are opportunity costs.



Remember: costs of constrained resources

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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

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