INTRODUCTION TO LINEAR PROGRAMMING

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INTRODUCTION TO LINEAR PROGRAMMING

• CONTENTS
• Introduction to Linear Programming
• Applications of Linear Programming
• Reference Chapter 1 in BJS book.

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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 - xni bi for all i 1, ,
  m
• xj ? 0 for all j 1, , n

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Matrix Notation

• Minimize cx
• subject to
• Ax b
• x ? 0
• where

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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
• Maximize z 3x1 2x2 (Obj. Func.)
• subject to
• 2x1 x2 ? 100 (Finishing constraint)
• x1 x2 ? 80 (Carpentry constraint)
• x1 ? 40 (Bound on soldiers)
• x1 ? 0 (Sign restriction)
• x2 ? 0 (Sign restriction)

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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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Multi-Period Financial Models

• Determine investment strategy for the next three
  years
• Money available for investment at time 0
  100,000
• Investments available A, B, C, D E
• No more than 75,000 in one invest
• Uninvested cash earns 8 interest
• Cash flow of these investments

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Cutting Stock Problem

• A manufacturer of metal sheets produces rolls of
  standard fixed width w and of standard length l.
• A large order is placed by a customer who needs
  sheets of width w and varying lengths. He needs
  bi sheets of length li, i 1, , m.
• The manufacturer would like to cut standard rolls
  in such a way as to satisfy the order and to
  minimize the waste.
• Since scrap pieces are useless to the
  manufacturer, the objective is to minimize the
  number of rolls needed to satisfy the order.

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Multi-Period Workforce Scheduling

• Requirement of skilled repair time (in hours) is
  given below.
• At the beginning of the period, 50 skilled
  technicians are available.
• Each technician is paid 2,000 and works up to
  160 hrs per month.
• Each month 5 of the technicians leave.
• A new technician needs one month of training, is
  paid 1,000 per month, and requires 50 hours of
  supervision of a trained technician.
• Meet the service requirement at minimum cost.

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Solution Capital Budgeting Problem

• Decision Variables
• xi fraction of investment i purchased
• Formulation
• Maximize z 13x1 16x2 16x3 14x4 39x5
• subject to
• 11x1 53x2 5x3 5x4 29x5 40
• 3x1 6x2 5x3 x4 34x5 20
• x1 1
• x2 1
• x3 1
• x4 1
• x5 1
• x1, x2, x3, x4, x5 ³ 0

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Solution Transportation Problem

• Decision Variables
• xij amount shipped from plant i to warehouse j
• Formulation
• Minimize z
• subject to
• ai, i 1, , m
• ? bj, j 1, , n
• xij ? 0, i 1, , m, j 1,
  , n

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Solution Static Workforce Scheduling

• LP Formulation
• Min. z x1 x2 x3 x4 x5 x6 x7
• subject to
• x1 x4 x5 x6 x7 ³ 17
• x1 x2 x5 x6 x7 ³ 13
• x1 x2 x3 x6 x7 ³ 15
• x1 x2 x3 x4 x7 ³ 19
• x1 x2 x3 x4 x5 ³ 14
• x2 x3 x4 x5 x6 ³ 16
• x3 x4 x5 x6 x7 ³ 11
• x1, x2, x3, x4, x5, x6, x7 ³ 0

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Solution Multiperiod Financial Model

• Decision Variables
• A, B, C, D, E Dollars invested in the
  investments A, B, C, D, and E
• St Dollars invested in money market fund at time
  t (t 0, 1, 2)
• Formulation
• Maximize z B 1.9D 1.5E 1.08S2
• subject to
• A C D S0 100,000
• 0.5A 1.2C 1.08S0 B S1
• A 0.5B 1.08S1 E S2
• A 75,000
• B 75,000
• C 75,000
• D 75,000
• E 75,000
• A, B, C, D, E, S0, S1, S2 ³ 0

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Solution Multiperiod Workforce Scheduling

• Decision Variables
• xt number of technicians trained in period t
• yt number of experienced technicians in period t
• Formulation
• Minimize z 1000(x1 x2 x3 x4 x5)
  2000(y1 y2 y3 y4 y5)
• subject to
• 160y1 - 50 x1 ³ 6000 y1 50
• 160y2 - 50 x2 ³ 7000 0.95y1 x1 y2
• 160y3 - 50 x3 ³ 8000 0.95y2 x2 y3
• 160y4 - 50 x4 ³ 9500 0.95y3 x3 y4
• 160y5 - 50 x5 ³ 11000 0.95y4 x4 y5
• xt, yt ³ 0, t 1, 2, 3, , 5

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Cutting Stock Problem (contd.)

• Given a standard sheet of length l, there are
  many ways of cutting it. Each such way is called
  a cutting pattern.
• The jth cutting pattern is characterized by the
  column vector aj, where the ith component,
  namely, aij, is a nonnegative integer denoting
  the number of sheets of length li in the jth
  pattern.
• Note that the vector aj represents a cutting
  pattern if and only if ?i1,n aijli ? l and each
  aij is a nonnegative number.

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Cutting Stock Problem (contd.)

• Formulation
• Minimize ?i1,n xi
• subject to
• ?i1,n aij xi ? bi i 1, , m
• xi ? 0 j 1, , n
• xi integer j 1, , n

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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

Feasible region in this example is unbounded.
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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

This linear program does not have any feasible
solutions.
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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

This linear program has multiple or alternative
optimal solutions.