linear programming.ppt

7/25/2019 Linear Programming.ppt

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Optimization Techniques: LinearProgramming


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Linear Programming

Linear programming is a mathematical

technique for solving constrainedmaximization and minimization problems,when there are many constraints and theobjective function to be optimized, as well

as the constraints faced, are linear (i.e.,can be represented by straight lines).

t is a technique for providing speci!cnumerical solutions of problems.

t bridges the gap between abstracteconomic theory and managerial decisionma"ing in practice.


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#he technique was developed by the

$ussian mathematician L.%. &antorovich in

' and extended by the *merican

mathematician +. . -antzig in '/.

0se of linear programming is expandingvery fast because of use of computer which

can quic"ly solve complex problems

involving the optimal use of many

resources, which are available to a !rm at a

particular time.


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Diference between TraditionalEconomic Analysis Vs. Linear

Programmingoth approaches show how economicagents (producers or consumers) reachoptimal choices, how they do their planning

or programming in order to attainmaximum utility, maximum pro!t,minimum cost, etc.

1either economic theory nor linearprogramming say anything about theimplementation of the optimal plan orsolution. #hey simply derive the optimal

solution in any particular situation.


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2owever,

n economic theory the optimal solution is

usually shown in qualitative abstractterms, diagrams, or general mathematicalsymbols. n contrast, linear programmingyields speci!c numerical solutions to theparticular optimization problems.

$elationships of economic theory areusually non3linear, expressed by curves

(not straight lines), while in linearprogramming all relationships betweenthe variables involved are assumed to belinear.


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Suppose a firm has the following quantities of factors ofproduction

L= 400 units of labour (hours

!= 300 units of capital (machine hours

S = "000 units of land (square feet

#he firm can produce either commodit$ % or commodit$ $with the following a&ailable processes (acti&ities

acti&it$ ' for % acti&it$ for $

Labour l%= 4 l$= "

)apital !%= " *$= "

Land S%= 2 S$= 5

#he production of one unit of % requires 4 hours of labour+" machine hour and 2 square feet of land, Similarl$+ theproduction of one unit of $ requires " hour of labour+ "machine hour and 5 square feet of land,


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Suppose % $ields a unit profit of ./ 2+ and commodit$ $

$ields a unit profit of ./ ", #he goal of the firm is to

choose the optimal product mi%+ that is+ the combination

that ma%imises its total profit,

#he total profit function can be written as

1 = 2 "

here+ 1 = total profit = quantit$ of commodit$ % (orle&el of 'cti&it$ ' = quantit$ of commodit$ $ (or le&el

of 'cti&it$ and 2 and " are the unit profits of the two

commodities,

Objective Function7 #he total profit function is called the

ob8ecti&e function as it e%presses the ob8ecti&e of the

firm, #his is the function+ which represents the goals of

the economic agent,


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Constraints

#echnical (or functional constraints+ and

/on:negati&it$ constraints,

#he technical constraints are set b$ the state of

technolog$ and the a&ailabilit$ of factors of

production, #here are man$ technicalconstraints as the factors of production

#he$ e%press the fact that the quantities of factors

which will be absorbed in the production of the

commodities cannot e%ceed the a&ailablequantities of these factors, #hus+ in our problem

100052;30011;40014 +++ YXYXYX


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here+ and are the le&els of commodities %

and $+ and integers on the left:hand side of the

equations are the technical coefficients ofproduction+ i,e,+ the factors inputs required for the

production of one unit of the products % and $,

#he figures on the right:hand side are the

resources that the firm has at its disposition,#he non:negati&it$ constraints e%press the

necessit$ that the le&els of production of the

commodities cannot be negati&e, #he le&el of

production of an$ one commodit$ can either be


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The linear programming problem may be stated as

a%imi


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Graphical Solution of the Linear Programming Problem

>raphical determination of the region of

feasible solutions+

>raphical determination of the ob8ecti&e

function+ and

?etermination of the optimal solution,


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Graphical determination of the region offeasible solutions

' solution will be feasible when it satisfies all the

constraints,

@ere we ha&e to satisf$ both non:negati&it$

constraints as well as technical constraints,

oundar$ set b$ the factor Labour7 #his isdefined b$ a straight line whose slope is the ratio

of the labour inputs in the production of the two

commodities,

@ence+ slope of the line = input of L in % A input of L in $

= 4A" = l%Al$


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Similarl$ slope of the line for capital will be

= input of ! in %A input of ! in $ = !%A!$= "A"="

'nd+ slope of the line for land will be

= input of S in % A input of S in $= S%A S$= 2A5

's ob8ecti&e function ma$ be represented b$ iso:profit lines i,e,+

Slope of the iso:profit line will be

YXYXZyx

+=+= 12

21

2====

ofyunitprofit

ofxunitprofit

X

Y

y

x


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Determination of the Optimal Solution

#he optimal solution will be found b$ the point of

tangenc$ of the frontier of the region of feasiblesolutions to the highest possible iso:profit cur&e,

#he optimal solution will be a point on the

frontier of the region of all feasible solutions,@ere it will be 1 = 2 " = 2(56 " ("-9 = 2;0


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The Duel Problem and Shadow Prices

#he basic problem whose solution is attempted b$

the linear programming technique is called the primalproblem,

#o each primal problem corresponds a dual problem+

which $ields additional information to the decision:

ma*er, #he nature of the dual problem depends on the

primal problem, .f the primal problem is a

ma%imi


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Shadow Prices

#hese are the imputed costs or opportunit$ costs

of the factors for a particular firm, #he$ are crucial indicators for the e%pansion of

the firm, #he$ show which factors are

bottlenec*s to the further e%pansion of the firm,

#he shadow prices of the resources can be

compared with their mar*et prices and help the

entrepreneur decide whether it is profitable to

hire additional units of these factors, .t decides how much the profit of the firm will be

increased if the firm emplo$s an additional unit of

this factor,


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Reference:

!outso$iannis+ ', (";-;+ ModernMicroeconomics+ acmillan Cress Ltd,+

London

linear programming.ppt

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