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Linear Programing

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Quantitative Module

LEARNING OBJECTIVESWhen you complete this module you

should be able to

IDENTIFY OR DEFINE:

Objective function

Constraints

Feasible region

Iso-profit/iso-cost methods

Corner-point solution

Shadow price

DESCRIBE OR EXPLAIN:

How to formulate linear models

Graphical method of linear programm

How to interpret sensitivity analysis

BLinear Programming

Module OutlineREQUIREMENTS OF A LINEAR

PROGRAMMING PROBLEMFORMULATING LINEAR

PROGRAMMING PROBLEMS

Shader Electronics Example

GRAPHICAL SOLUTION TO A LINEAR

PROGRAMMING PROBLEM

Graphical Representation of Constraints

Iso-Profit Line Solution Method

Corner-Point Solution Method

SENSITIVITY ANALYSIS

Sensitivity Report

Changes in the Resources

or Right-Hand-Side ValuesChanges in the Objective Function

Coefficient

SOLVING MINIMIZATION PROBLEMS

LINEAR PROGRAMMING APPLICATIONS

Production-Mix Example

Diet Problem Example

Production Scheduling Example

Labor Scheduling Example

THE SIMPLEX METHOD OF LP

SUMMARY

KEY TERMS

USING SOFTWARE TO SOLVE LP PROBLEMS

SOLVED PROBLEMS

INTERNET AND STUDENT CD-ROM EXERCISES

DISCUSSION QUESTIONS

ACTIVE MODEL EXERCISE

PROBLEMS

INTERNET HOMEWORK PROBLEMS

CASE STUDY: GOLDING LANDSCAPING AND PLANTS, INC.

ADDITIONAL CASE STUDIES

BIBLIOGRAPHY

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692 MO D U L E B L I N E A R PR O G R A M M I N G

Linear Programming(LP)A mathematicaltechnique designed tohelp operations managersplan and make decisionsrelative to the trade-offsnecessary to allocateresources.

The storm front closed in quickly on Chicagos OHare Airport, shutting it down without warning. The heavy

thunderstorms, lightning, and poor visibility sent American Airlines passengers and ground crew scurrying.

Because American Airlines uses linear programming (LP) to schedule flights, hotels, crews, and refueling, LP has a

direct impact on profitability. As the president of AAs Decision Technology Group says, Finding fast solutions to

LP problems is essential. If we get a major weather disruption at one of the hubs, such as Dallas or Chicago, then a

lot of flights may get canceled, which means we have a lot of crews and airplanes in the wrong places. What we

need is a way to put that whole operation back together again. LP is the tool that helps airlines such as American

unsnarl and cope with this weather mess.

Many operations management decisions involve trying to make the most effective use of an organi-

zations resources. Resources typically include machinery (such as planes, in the case of an airline)

labor (such as pilots), money, time, and raw materials (such as jet fuel). These resources may be

used to produce products (such as machines, furniture, food, or clothing) or services (such as airline

schedules, advertising policies, or investment decisions). Linear programming (LP) is a widely

used mathematical technique designed to help operations managers plan and make the decisions

necessary to allocate resources.

A few examples of problems in which LP has been successfully applied in operations manage-

ment are

1. Scheduling school buses to minimize the total distance traveled when carrying students.

2. Allocating police patrol units to high crime areas to minimize response time to 911

calls.

3. Scheduling tellers at banks so that needs are met during each hour of the day while

minimizing the total cost of labor.

4. Selecting the product mix in a factory to make best use of machine- and labor-hours available while maximizing the firms profit.

5. Picking blends of raw materials in feed mills to produce finished feed combinations atminimum cost.

6. Determining the distribution system that will minimize total shipping cost from several

warehouses to various market locations.

7. Developing a production schedule that will satisfy future demands for a firms product and

at the same time minimize total production and inventory costs.

8. Allocating space for a tenant mix in a new shopping mall so as to maximize revenues to the

leasing company. (See the OM in Action box Using LP to Select Tenants in a Shopping

Mall.)

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REQUIREMENTS OF A LINEAR PROGRAMMING PROBLEM

All LP problems have four properties in common:

1. LP problems seek to maximize or minimize some quantity (usually profit or cost). We re

to this property as the objective function of an LP problem. The major objective of a ty

cal firm is to maximize dollar profits in the long run. In the case of a trucking or airline d

tribution system, the objective might be to minimize shipping costs.

2. The presence of restrictions, orconstraints, limits the degree to which we can pursue

objective. For example, deciding how many units of each product in a firms product lin

manufacture is restricted by available labor and machinery. We want, therefore, to maximor minimize a quantity (the objective function) subject to limited resources (the constrain

3. There must be alternative courses of action to choose from. For example, if a comp

produces three different products, management may use LP to decide how to alloc

among them its limited production resources (of labor, machinery, and so on). If there w

no alternatives to select from, we would not need LP.

4. The objective and constraints in linear programming problems must be expressed in ter

of linear equations or inequalities.

FORMULATING LINEAR PROGRAMMING PROBLEMS

One of the most common linear programming applications is theproduct-mix problem. Two or m

products are usually produced using limited resources. The company would like to determine h

many units of each product it should produce to maximize overall profit given its limited resourLets look at an example.

Shader Electronics ExampleThe Shader Electronics Company produces two products: (1) the Shader Walkman, a porta

CD/DVD player, and (2) the Shader Watch-TV, a wristwatch-size internet-connected color tele

sion. The production process for each product is similar in that both require a certain number

hours of electronic work and a certain number of labor-hours in the assembly department. E

Walkman takes 4 hours of electronic work and 2 hours in the assembly shop. Each Watch-

requires 3 hours in electronics and 1 hour in assembly. During the current production period,

FO R M U LAT I N G L I N E A R PR O G R A M M I N G PR O B L E M S 6

Objective functionA mathematicalexpression in linearprogramming thatmaximizes or minimizes

some quantity (oftenprofit or cost, but anygoal may be used).

ConstraintsRestrictions that limitthe degree to whicha manager can pursuean objective.

Using LP to Select Tenantsin a Shopping Mall

Homart Development Company is one of the largest

shopping-center developers in the U.S. When starting anew center, Homart produces a tentative floor plan, orfootprint, for the mall. This plan outlines sizes, shapes,and spaces for large department stores. Leasing agree-ments are reached with the two or three major depart-ment stores that will become anchor stores in the mall.The anchor stores are able to negotiate highly favorableoccupancy agreements. Homarts profits come primarilyfrom the rent paid by the nonanchor tenantsthe smallerstores that lease space along the aisles of the mall. Thedecision as to allocating space to potential tenants is,therefore, crucial to the success of the investment.

The tenant mix describes the desired stores in the mallby their size, general location, and type of merchandise

or service provided. For example, the mix might specify

two small jewelry stores in a central section of the maand a medium-size shoe store and a large restaurant one of the side aisles. In the past, Homart developed plan for tenant mix using rules of thumb develope

over years of experience in mall development.Now, to improve its bottom line in an increasing

competitive marketplace, Homart treats the tenant-mproblem as an LP model. First, the model assumes thatenants can be classified into categories according tthe type of merchandise or service they providSecond, the model assumes that for each store typstore sizes can be estimated by distinct category. Fexample, a small jewelry store is said to contain abo700 square feet and a large one about 2,200 squafeet. The tenant-mix model is a powerful tool for enhaning Homarts mall planning and leasing activities.

Sources: Chain Store Age(March 2000): 191192; Business World(Mar

18, 2002): 1; and Interfaces(MarchApril 1988): 19.

O M I N A C T I O N

Active Model B.1

This example is furtherillustrated in Activemodel B.1 on theCD-ROM and in theexercise on page 713.

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694 MO D U L E B L I N E A R PR O G R A M M I N G

TABLE B.1

Shader ElectronicsCompany ProblemData

HOURSREQUIRED TOPRODUCE1 UNIT

WALKMANS WATCH-TVS

DEPARTMENT (X1) (X2) AVAILABLEHOURSTHISWEEK

Electronic 4 3 240

Assembly 2 1 100

Profit per unit $7 $5

hours of electronic time are available, and 100 hours of assembly department time are available

Each Walkman sold yields a profit of $7; each Watch-TV produced may be sold for a $5 profit.

Shaders problem is to determine the best possible combination of Walkmans and Wat