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Technical Note 2

Optimizing the Use

of Resources with

Linear Programming

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Linear Programming Basics

A Maximization Problem

A Minimization Problem

OBJECTIVES

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Linear Programming Essential

Conditions Is used in problems where we have

limited resources or constrained

resources that we wish to allocate

The model must have an explicitobjective (function)

Generally maximizing profit orminimizing costs subject to resource-based, or other, constraints

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Common Applications

Aggregate sales and operations planning

Service/manufacturing productivity analysis

Product planning Product routing

Vehicle/crew scheduling

Process control

Inventory control

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Linear Programming Essential

Conditions (Continued)

Limited Resources to allocate

Linearityis a requirement ofthe modelin both objective function and

constraints Homogeneityofproducts produced

(i.e., products must the identical) andall hours of labor used are assumedequally productive

Divisibilityassumes products andresources divisible (i.e., permitfractional values ifneed be)

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Objective Function

Maximize (or Minimize) Z = C1X1 + C2X2 + ... + CnXn

Cjis a constant that describes the

rate ofcontribution to costs orprofit of(Xj) units being produced

Z is the total cost or profit fromthe given number ofunits beingproduced

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Constraints

A11X1 + A12X2 + ... + A1nXneB1A21X1 + A22X2 + ... + A2nXnuB2

::

AM1X1 + AM2X2 + ... + AMnXn=BM Aijare resource requirements for each of

the related (Xj) decision variables Biare the available resource

requirements Note that the direction ofthe inequalities

can be all or a combination ofe, u, or =linear mathematical expressions

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Non-Negativity Requirement

X1,X2, , Xnu 0

All linear programming model

formulations require their decisionvariables to be non-negative While these non-negativity

requirements take the form ofa

constraint, they are considered amathematical requirement tocomplete the formulation ofan LPmodel

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An Example of a Maximization Problem

LawnGrow Manufacturing Company must determine the unitmix of its commercial riding mower products to beproduced next year. The company produces two productlines, the Max and the Multimax. The average profit is $400for each Max and $800 for each Multimax. Fabrication hours

and assembly hours are limited resources. There is amaximum of5,000 hours offabrication capacity availableper month (each Max requires 3 hours and each Multimaxrequires 5 hours). There is a maximum of3,000 hours ofassembly capacity available per month (each Max requires 1hour and each Multimax requires 4 hours). Question: Howmany units ofeach riding mower should be produced eachmonth in order to maximize profit?

Now lets formulate this problem as an LP model

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The Objective Function

Maximize Z = 400X + 800X

Where

Z = the monthly profit from Max and Multimax

X = the number of Max produced each month

X = the number of Multimax produced each month

1 2

1

2

Ifwe define the Max and Multimax products as the twodecision variables X1 and X2, and since we want tomaximize profit, we can state the objective function asfollows:

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Constraints

Max (X1) Multimax (X2)

Required Time/Unit Required Time/Unit Available Time/ onth

3 5 5,000 Fab

1 4 3,000 Assy

1 2

1 2

1 2

3 5 5,000 (Fab.)

X 4X 3,000 (Assy.)

X ,X 0 (Non-negativity)

e

e

u

Given the resource information below from the problem:

We can now state the constraints and non-negativityrequirements as:

Note that the inequalities are less-than-or-equal sincethe time resources represent the total availableresources for production

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Solution

Produce 715 Max and 571 Multimax per monthfor a profit of $742,800Produce 715 Max and 571 Multimax per monthfor a profit of $742,800

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An Example of a Minimization Problem

HiTech Metal Company is developing a plan for buyingscrap metal for its operations. HiTech receives scrapmetal from two sources, Hasbeen Industries andGentro Scrap in daily shipments using large trucks.Each truckload ofscrap from Hasbeen yields 1.5 tons

ofzinc and 1 ton of lead at a cost of$15,000. Eachtruckload ofscrap from Gentro yields 1 ton ofzinc and3 tons of lead at a cost of$18,000. HiTech requires atleast 6 tons ofzinc and at least 10 tons of lead per day.

Question: How many truckloads ofscrap should bepurchased per day from each source in order tominimize scrap metal costs to HiTech?

Now lets formulate this problem as an LP model

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The Objective Function

Minimize Z = 15,000 X1 + 18,000 X2WhereZ = daily scrap costX1 = truckloads from Hasbeen

X2 = truckloads from Gentro

Hasbeen

GENTRO

Ifwe define the Hasbeen truckloads and the Gentrotruckloads as the two decision variables X1 and X2, andsince we want to minimize cost, we can state theobjective function as follows:

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Constraints

1.5X1 + X2 > 6( inc/tons)

X1 + 3X2 > 10(Lead/tons)

X1, X2 > 0(Non-negativity)

Has een (X1) Gentro (X2)

Tons Tons Min Tons

1.5 1 6 Zinc

1 3 10 Lead

Given the demand information below from the problem:

We can now state the constraints and non-negativityrequirements as:

Note that theinequalities aregreater-than-or-

equal since thedemand informationrepresents theminimum necessaryfor production.

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Solution

Order 2.29 truckloads from Hasbeen and 2.57truckloads from Gentro for daily delivery. The dailycost will be $80,610.

Order 2.29 truckloads from Hasbeen and 2.57truckloads from Gentro for daily delivery. The dailycost will be $80,610.

Note: Do you see why in this solution that

integer linear programming methodologiescan have useful applications in industry?

Note: Do you see why in this solution that

integer linear programming methodologiescan have useful applications in industry?

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Question Bowl

Which ofthe following is consideredan essential condition for linearprogramming to be used in aproblem situation?

a. Limited resourcesb. Explicit objectivec. Divisibilityd. Linearitye. All ofthe above

A swer:e.Alloftheabove (Correct

a swerca i c l u e Homoge eity.)

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Question Bowl

The fact that a regular linear programmingmodel has to permit a solution that isfactional, refers to which ofthefollowing essential conditions?

a. Limited resourcesb. Explicit objectivec. Divisibilityd. Linearity

e. All ofthe above

A swer:c.Divisibility

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Question Bowl

Which ofthe following are means bywhich a linear programmingproblem can be solved?

a. Genetic algorithm

b. Microsoft Solver spreadsheetprogramc. Graphic LPd. All ofthe above

e. None ofthe aboveA swer: d.All ofthe above

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End of Technical

Note 2