A basic graphical intro to linear programming

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Unit – IUnit – I
Linear ProgrammingLinear Programming

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OutlineOutline
 Linear Programming ProblemLinear Programming Problem
 Linear Production FunctionsLinear Production Functions
 Iso Quants & Endowments pointsIso Quants & Endowments points
 Process Mixing and Full employmentProcess Mixing and Full employment

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Linear ProgrammingLinear Programming
Linear programmingLinear programming has nothing to do with computerhas nothing to do with computer
programming.programming.
The use of the word “programming” here meansThe use of the word “programming” here means
“choosing a course of action.”“choosing a course of action.”
Linear programming involves choosing a course ofLinear programming involves choosing a course of
action when the mathematical model of the problemaction when the mathematical model of the problem
contains only linear functions.contains only linear functions.
Calculus method of optimization cannot be used toCalculus method of optimization cannot be used to
solve linear problems as first and second derivativesolve linear problems as first and second derivative
would be constant and zero respectivelywould be constant and zero respectively
Linear programming is a special tool wherein weLinear programming is a special tool wherein we
consider the maximization as well as minimization ofconsider the maximization as well as minimization of
linear objective functions subject to a number oflinear objective functions subject to a number of
constraints.constraints.

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Linear Programming (LP)Linear Programming (LP)
ProblemProblem
 All LP problems haveAll LP problems have constraintsconstraints that limit the degreethat limit the degree
to which the objective can be pursued.to which the objective can be pursued.
 AA feasible solutionfeasible solution satisfies all the problem'ssatisfies all the problem's
constraints.constraints.
 AnAn optimal solutionoptimal solution is a feasible solution that results inis a feasible solution that results in
the largest possible objective function value whenthe largest possible objective function value when
maximizing (or smallest when minimizing).maximizing (or smallest when minimizing).
 AA graphical solution methodgraphical solution method can be used to solve acan be used to solve a
linear program with two variables.linear program with two variables.

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Linear Production functionsLinear Production functions
 Linear production activity is a process in whichLinear production activity is a process in which
one or more outputs are produced by using oneone or more outputs are produced by using one
or more inputs in fixed proportions.or more inputs in fixed proportions.
 Thus, the relevant linear production functionThus, the relevant linear production function
exhibits constant returns to scale.exhibits constant returns to scale.
 In such cases, a rise in all inputs by constantIn such cases, a rise in all inputs by constant
proportion necessarily leads to a rise in all theproportion necessarily leads to a rise in all the
outputs by the same constant proportionoutputs by the same constant proportion

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Linear Production functionsLinear Production functions
Linear function with two
inputs and one output
Iso-quants

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Linear Production functionsLinear Production functions
Iso-quants
01 – Expansion path
Case – 1: 3 units of L and 12 units of K

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Linear Production functionsLinear Production functions
Case – 2: 3 units of L and 14 units of K
How many units could be produced?

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Linear Production functionsLinear Production functions
Case – 3: 6 units of L and 12 units of K
How many units could be produced?

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Linear Production functionsLinear Production functions
Iso-quants
In general, if al and ak are the input requirements of L and K
respectively, then x = min[(L/al), (K/ak)]
M, N & C – Endowment Points

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Linear Production functionsLinear Production functions
Multiple Process – Process mixing and Full Employment

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Linear Production functionsLinear Production functions
Multiple Process – Process mixing and Full Employment

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Linear Production functionsLinear Production functions
Case value: 70 units of L and 100 units of K
How many units could be produced?
Adopting Process 1 alone, we have
X = min[(70/1), (100/4)]
X = 25
Adopting Process 2 alone, we have
X = min[(70/2), (100/2)]
X = 35
Adopting Process 3 alone, we have
X = min[(70/4), (100/1)]
X = 17.5

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Linear Production functionsLinear Production functions
Case value: 70 units of L and 100 units of K

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Linear Production functionsLinear Production functions
Case value: 70 units of L and 100 units of K
Process 1 → L=10 & K=40 → X1=10
Process 2 → L=60 & K=60 → X2=30
Total X = 40

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Linear Production functionsLinear Production functions
Case value: 70 units of L and 100 units of K
Process 1 → L=22 & K=88 → X1=22
Process 3 → L=48 & K=12 → X2=12
Total X = 34

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Linear Production functionsLinear Production functions
Process Mixing and Full Employment
1) For efficient process mix one must choose the
closest two production processes on either side of
the given endowment point
2) The process mix is purely technical in the sense
that there is no influence of output and input
prices in this analysis