Linear Programming

1
Linear Programming

• Introduction

2
What is Linear Programming?

• A Linear Programming model seeks to maximize or
  minimize a linear function, subject to a set of
  linear constraints.

3
What are linear functions?

• y mxb is the equation of a straight line
• e.g. y -4/3 x 6
• Multiplying by 3 and rearranging 4x 3y 18

A linear function consists of the sum of
positive, negative or 0 constants times
variables e.g. 5X1 - 4X2 0X3 6X4 is a
linear function in 4 variables.
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What are Linear Constraints?

• Linear constraints have the form
• ltLinear Functiongt lthas some relation togt lta
  constantgt
• The relation is one of the following
• ?, , ? ---- they all contain the equal to
  part
• Examples
• 4X1 5X2 - 6X3 2X5 ? 34
• 2X1 - 5X2 1X4 ? 47
• - 2X2 8X3 9X4 2X5 67
• X1 ? 0
• X5 ? 0

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Example of a Linear Program

• MAX 4X1 7X3 - 6X4
• s.t. 2X1 3X2 - 2X4 20
• - 2X2 9X3 7X4 ? 10
• -2X1 3X2 4X3 8X4 ? 35
• X2 ? 5
• All Xs ? 0

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Another Example

• MIN 6X1 8X2 11X3 10X4 5X5 14X6
• S.T. X1 X2 X3
  ? 20
• X4 X5 X6 ? 30
• X1 X4
  12
• X2
  X5 15
• X3
  X6 22
• All Xs ?
  0

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Components of a Linear Programming Model

• A linear programming model consists of
• A set of decision variables
• A (linear) objective function
• A set of (linear) constraints

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Why are Linear Programs Important?

• Many real world problems lend themselves to
  linear programming modeling.
• Other real world problems can be approximated by
  linear models.
• There are well-known successful applications in
• Manufacturing, Marketing, Finance (investment),
  Advertising, Agriculture, Energy, etc.
• There are efficient solution techniques and
  software programs that solve linear programming
  models.
• The output generated from linear programming
  packages provides useful what if analysis.

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Linear Programming Assumptions

• The parameter values are known with certainty.
• The objective function and constraints exhibit
  constant returns to scale.
• There are no interactions between the decision
  variables (additivity assumption).
• Continuity of the decision variables means they
  can take on any value within a given feasible
  range.
• Integer programming models can only take on
  integer values within a given feasible range.

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Example

• Galaxy Industries manufactures two toy gun
  models
• Space Rays Each dozen nets an 8 profit and
• Requires 2 lbs. of plastic 3 minutes of
  production time
• Zappers Each dozen nets a 5 profit and
• Requires 1 lb. of plastic 4 minutes of
  production time
• Weekly resource limits
• 1000 pounds of plastic 40 hours of production
  time
• Weekly production limits
• Maximum 700 dozen total units
• Space Rays cannot exceed Zappers by more than 350
  dozen

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Current Production

• Current reasoning calls for a production plan
  that
• Produces as much as possible of the more
  profitable product, Space Ray (8 profit per
  dozen).
• Uses any left over resources to produce Zappers
  (5 profit per dozen), while remaining within the
  marketing guidelines of 700 total dozen produced
  and Space Rays Zappers
  350.
• Using a simple spreadsheet, letting the (cell for
  production of Zappers) (cell for production of
  Space Rays 350), trial and error gives the
  following good solution that uses all the
  available weekly plastic
• Space Rays 450 dozen Zappers 100 dozen
• Profit 8(450) 5(100) 4100
• This is a good solution Can we do better?

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The Mathematical Model

• Recall a mathematical model consists of
• Set of decision variables
• Objective function
• Constraints
• Decision Variables
• (Include both a measurement unit (dozens) and a
  time unit (week))
• X1 dozens of Space Rays produced weekly
• X2 dozens of Zappers produced weekly

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2. OBJECTIVE FUNCTION

• Objective is to maximize the total weekly profit.
• How much profit will be made each week?

How much profit will be made weekly from Space
Rays?
How much profit will be made weekly from Zappers?
5 per dozen
8 per dozen
Make X2 dozen Zappers per week
Make X1 dozen Space Rays per week
5X2
8X1

MAX 8X1 5X2
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3. Constraints -- PLASTIC

• At most 1000 pounds of plastic available weekly.
• How much will be used?

How much plastic will be used weekly making Space
Rays?
How much plastic will be used weekly
making Zappers?
1 lb per dozen
2 lbs per dozen
Make X2 dozen Zappers per week
Make X1 dozen Space Rays per week
1X2
2X1

2X1 1X2 ? 1000
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Constraints -- Production Time

• At most 40 hours 40x60 2400 minutes available
  weekly. How much will be used?

How many minutes will be used weekly making Space
Rays?
How many minutes will be used weekly
making Zappers?
4 min per dozen
3 min per dozen
Make X2 dozen Zappers per week
Make X1 dozen Space Rays per week
4X2
3X1

3X1 4X2 ? 2400
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Constraints -- Max Production

• At most 700 dozen total units can be produced
  weekly. How many will be produced?

How many dozen Space Rays are produced weekly?
How many dozen Zappers are Produced weekly?
Make X1 dozen Space Rays per week
Make X2 dozen Zappers per week
X1

X2
X1 X2 ? 700
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Constraints -- Product Mix

• Space Rays can be at most 350 dozen units greater
  than Zappers each week. How many more dozen
  units of Space Rays will be produced weekly?

How many dozen Space Rays are produced weekly?
How many dozen Zappers are Produced weekly?
Make X2 dozen Zappers per week
Make X1 dozen Space Rays per week
X2
-
X1
X1 - X2 ? 350
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Constraints -- Nonnegativity

• Cannot produce a negative amount of Space Rays or
  Zappers

X1 ? 0 X2 ? 0
or
All Xs ? 0
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The Complete Galaxy IndustriesLinear Programming
Model

• MAX 8X1 5X2
• s.t. 2X1 1X2 1000 (Plastic)
• 3X1 4X2 2400 (Prod. Time)
• X1 X2 700 (Total Prod.)
• X1 - X2 350 (Mix)
• All Xs 0

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Review

• A linear program seeks to maximize or minimize a
  linear objective subject to linear constraints.
• Many problems are or can be approximated by
  linear programming models.
• Linear programs possess the features of
• Certainty, Constant Returns to Scale, Additivity
  and Continuity
• There exists efficient algorithms for solving
  linear programs that provide many sensitivity
  analyses as a by-product.