INTRODUCTION TO LINEAR PROGRAMMING

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INTRODUCTION TO LINEAR PROGRAMMING

• CONTENTS
• Introduction to Linear Programming
• Applications of Linear Programming

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A Typical Linear Programming Problem

• Linear Programming Formulation
• Minimize c1x1 c2x2 c3x3 . cnxn
• subject to
• a11x1 a12x2 a13x3 . a1nxn ? b1
• a21x1 a22x2 a23x3 . a2nxn ? b2
• am1x1 am2x2 am3x3 . amnxn ? bm
• x1, x2, x3 , ., xn ? 0
• or,
• Minimize ?j1, n cjxj
• subject to
• ?j1, n aijxj ? bi for all i 1, , m
• xj ? 0 for all j 1, , N

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Matrix Notation

• Maximize cx
• subject to
• Ax b
• x ? 0

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Features of a Linear Programming (LP) Problem

• Decision Variables
• We minimize (or maximize) a linear function of
  decision variables, called objective function.
• The decision variables must satisfy a set of
  constraints.
• Decision variables have sign restrictions.

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An Example of a LP

• Giapettos woodcarving manufactures two types of
  wooden toys soldiers and trains
• Constraints
• 100 finishing hour per week available
• 80 carpentry hours per week available
• produce no more than 40 soldiers per week
• Objective maximize profit

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An Example of a LP (cont.)

• Linear Programming formulation

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Assumptions of Linear Programming

• Proportionality Assumption
• Contribution of a variable is proportional to its
  value.
• Additivity Assumptions
• Contributions of variables are independent.
• Divisibility Assumption
• Decision variables can take fractional values.
• Certainty Assumption
• Each parameter is known with certainty.

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Linear Programming Modeling and Examples

• Stages of an application
• Problem formulation
• Mathematical model
• Deriving a solution
• Model testing and analysis
• Implementation

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Capital Budgeting Problem

• Five different investment opportunities are
  available for investment.
• Fraction of investments can be bought.
• Money available for investment
• Time 0 40 million
• Time 1 20 million
• Maximize the NPV of all investments.

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Transportation Problem

• The Brazilian coffee company processes coffee
  beans into coffee at m plants. The production
  capacity at plant i is ai.
• The coffee is shipped every week to n warehouses
  in major cities for retail, distribution, and
  exporting. The demand at warehouse j is bj.
• The unit shipping cost from plant i to warehouse
  j is cij.
• It is desired to find the production-shipping
  pattern xij from plant i to warehouse j, i 1,
  .. , m, j 1, , n, that minimizes the overall
  shipping cost.

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Static Workforce Scheduling

• Number of full time employees on different days
  of the week are given below.
• Each employee must work five consecutive days and
  then receive two days off.
• The schedule must meet the requirements by
  minimizing the total number of full time
  employees.

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Feasible Region

• Feasible Region Set of all points satisfying all
  the constraints and all the sign restrictions
• Example
• Max. z 3x1 2x2
• subject to
• 2x1 x2 100
• x1 x2 80
• x1 40
• x1 ³ 0
• x2 ³ 0

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Example 1

• Maximize z 50x1 100x2
• subject to
• 7x1 2x2 ³ 28
• 2x1 12x2 ³ 24
• x1, x2 ³ 0

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Example 2

• Maximize z 3x1 2x2
• subject to
• 1/40x1 1/60x2 1
• 1/50x1 1/50x2 1
• x1 ³ 30
• x2 ³ 20
• x1, x2 ³ 0

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Example 3

• Max. z 3x1 2x2
• subject to
• 1/40 x1 1/60x2 1
• 1/50 x1 1/50x2 1
• x1, x2 ³ 0