2 Linear Programming
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Transcript of 2 Linear Programming
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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.
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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
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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
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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
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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
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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
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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
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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
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Flair Furn i ture Company
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
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Flair Furn i ture Company
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
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Flair Furn i ture Company
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)
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Flair Furn i ture Company
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
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Flair Furn i ture Company
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)
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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
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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?
2- Example: LP Formulation
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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?
3- Example: LP Formulation
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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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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