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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