2 linear programming

5/28/2018 2 Linear Programming

1/18

2008 Prentice-Hall, Inc.

Chapter 7

To accompanyQuant i tat ive Analysis for Management, Tenth Edit io n,

by Render, Stair, and HannaPower Point slides created by Jeff Heyl

Linear Prog ramm ing Models:Graph ical and Compu ter

Methods

2009 Prentice-Hall, Inc.


2/18 2009 Prentice-Hall, Inc. 72

In t roduct ion

Many management decisions involve trying tomake the most effective use of limited resources Machinery, labor, money, time, warehouse space, raw

materials

Linear programm ing(LP) is a widely usedmathematical modeling technique designed tohelp managers in planning and decision makingrelative to resource allocation

Belongs to the broader field of mathematicalprogramming

In this sense, programmingrefers to modeling andsolving a problem mathematically



Requ irements of a Linear

Programm ing Problem

LP has been applied in many areas over the past50 years

All LP problems have 4 properties in common1. All problems seek to maximizeor minimizesome

quantity (the object ive func t ion)2. The presence of restrictions or constra in tsthat limit the

degree to which we can pursue our objective

3. There must be alternative courses of action to choosefrom

4. The objective and constraints in problems must beexpressed in terms of l inearequations or inequali t ies



LP Propert ies and Assumpt ions

PROPERTIES OF LINEAR PROGRAMS

1. One objective function

2. One or more constraints

3. Alternative courses of action

4. Objective function and constraints are linear

ASSUMPTIONS OF LP

1. Certainty

2. Proportionality

3. Additivity

4. Divisibility

5. Nonnegative variables

Table 7.1



Basic Assumpt ions of LP

We assume conditions of certaintyexist andnumbers in the objective and constraints areknown with certainty and do not change duringthe period being studied

We assume proport ional i tyexists in the objectiveand constraints

We assume addit iv i tyin that the total of allactivities equals the sum of the individual

activities We assume div is ib i l i tyin that solutions need not

be whole numbers

All answers or variables are nonnegat ive



Formulat ing LP Prob lems

Formulating a linear program involves developinga mathematical model to represent the managerialproblem

The steps in formulating a linear program are

1. Completely understand the managerialproblem being faced

2. Identify the objective and constraints

3. Define the decision variables

4. Use the decision variables to writemathematical expressions for the objectivefunction and the constraints



Formulat ing LP Prob lems

One of the most common LP applications is theproduct m ix prob lem

Two or more products are produced usinglimited resources such as personnel, machines,and raw materials

The profit that the firm seeks to maximize isbased on the profit contribution per unit of eachproduct

The company would like to determine howmany units of each product it should produceso as to maximize overall profit given its limitedresources



Flair Furn i ture Company

The Flair Furniture Company producesinexpensive tables and chairs

Processes are similar in that both require a certainamount of hours of carpentry work and in the

painting and varnishing department Each table takes 4 hours of carpentry and 2 hours

of painting and varnishing

Each chair requires 3 of carpentry and 1 hour ofpainting and varnishing

There are 240 hours of carpentry time availableand 100 hours of painting and varnishing

Each table yields a profit of $70 and each chair aprofit of $50




The company wants to determine the bestcombination of tables and chairs to produce toreach the maximum profit

HOURS REQUIRED TO

PRODUCE 1 UNIT

DEPARTMENT(T)

TABLES(C)

CHAIRSAVAILABLE HOURSTHIS WEEK

Carpentry 4 3 240

Painting and varnishing 2 1 100

Profit per unit $70 $50

Table 7.2




The objective is to

Maximize profit

The constraints are

1. The hours of carpentry time used cannotexceed 240 hours per week

2. The hours of painting and varnishing timeused cannot exceed 100 hours per week

The decision variables representing the actual

decisions we will make areT= number of tables to be produced per week

C= number of chairs to be produced per week




We create the LP objective function in terms of Tand C

Maximize profit = $70T+ $50C

Develop mathematical relationships for the twoconstraints

For carpentry, total time used is(4 hours per table)(Number of tables produced)

+ (3 hours per chair)(Number of chairs produced)

We know thatCarpentry time used Carpentry time available

4T+ 3C 240 (hours of carpentry time)




Similarly

Painting and varnishing time used Painting and varnishing time available

2 T+ 1C 100 (hours of painting and varnishing time)

This means that each table producedrequires two hours of painting andvarnishing time

Both of these constraints restrict productioncapacity and affect total profit




The values for Tand Cmust be nonnegative

T 0 (number of tables produced is greaterthan or equal to 0)

C 0 (number of chairs produced is greater

than or equal to 0)

The complete problem stated mathematically

Maximize profit = $70T+ $50C

subject to4T+ 3C240 (carpentry constraint)2T+ 1C100 (painting and varnishing constraint)

T, C0 (nonnegativity constraint)



Cycle Trends is introducing two new lightweight bicycleframes, the Deluxe and the Professional, to be made from

aluminum and steel alloys. The anticipated unit profits are $10

for the Deluxe and $15 for the Professional.

The number of pounds of each alloy needed per frame is

summarized on the table. A supplier delivers 100 pounds of the

aluminum alloy and 80 pounds of the steel alloy weekly. How

many Deluxe and Professional frames should Cycle Trends

produce each week?

Pounds of each alloy needed per frame

1- Example: LP Formulation

Aluminum Alloy Steel Alloy

Deluxe 2 3

Professional 4 2



Montana Wood Products manufacturers two-high quality products, tables and chairs. Its profit is

$15 per chair and $21 per table. Weekly production is

constrained by available labor and wood. Each chair

requires 4 labor hours and 8 board feet of wood whileeach table requires 3 labor hours and 12 board feet of

wood. Available wood is 2400 board feet and available

labor is 920 hours. Management also requires at least

40 tables and at least 4 chairs be produced for every

table produced. To maximize profits, how many chairs

and tables should be produced?




The Sureset Concrete Company produces

concrete. Two ingredients in concrete are sand (costs

$6 per ton) and gravel (costs $8 per ton). Sand and

gravel together must make up exactly 75% of theweight of the concrete. Also, no more than 40% of the

concrete can be sand and at least 30% of the concrete

be gravel. Each day 2000 tons of concrete are

produced. To minimize costs, how many tons of graveland sand should be purchased each day?



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2009 Prentice-Hall, Inc. 717

A company produces two products that are processed on twoassembly lines. Assembly line 1 has 100 available hours, andassembly line 2 has 42 available hours. Each product requires

10 hours of processing time on line 1, while on line 2 product1 requires 7 hours and product 2 requires 3 hours. The profitfor product 1 is $6 per unit, and the profit for product 2 is $4per unit.

Formulate a linear programming model for this problem.

4- Example: LP Formu lat ion


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2009 Prentice-Hall, Inc. 7 18

A California grower has a 50-acre farm on which to plantstrawberries and tomatoes. The grower has available 300 hours oflabor per week and 800 tons of fertilizer, and he has contracted forshipping space for a maximum of 26 acres' worth of strawberries

and 37 acres' worth of tomatoes. An acre of strawberries requires10 hours of labor and 8 tons of fertilizer, whereas an acre oftomatoes requires 3 hours of labor and 20 tons of fertilizer. Theprofit from an acre of strawberries is $400, and the profit from anacre of tomatoes is $300. The farmer wants to know the number of

acres of strawberries and tomatoes to plant to maximize profit.

Formulate a linear programming model for this problem.

5- Example: LP Formu lat ion

2 linear programming

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