Linear Programming

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Linear Programming
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Objectives

• Requirements for a linear programming model.
• Graphical representation of linear models.
• Linear programming results
• Unique optimal solution
• Alternate optimal solutions
• Unbounded models
• Infeasible models
• Extreme point principle.

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Objectives - continued

• Sensitivity analysis concepts
• Reduced costs
• Range of optimality--LIGHTLY
• Shadow prices
• Range of feasibility--LIGHTLY
• Complementary slackness
• Added constraints / variables
• Computer solution of linear programming models
• WINQSB
• EXCEL
• LINDO

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3.1 Introduction to Linear Programming

• A Linear Programming model seeks to maximize or
  minimize a linear function, subject to a set of
  linear constraints.
• The linear model consists of the following
  components
• A set of decision variables.
• An objective function.
• A set of constraints.
• SHOW FORMAT

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• The Importance of Linear Programming
• Many real static problems lend themselves to
  linear
• programming formulations.
• Many real problems can be approximated by linear
  models.
• The output generated by linear programs provides
• useful whats best and what-if information.

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Assumptions of Linear Programming (p. 48)

• The decision variables are continuous or
  divisible, meaning that 3.333 eggs or 4.266
  airplanes is an acceptable solution
• The parameters are known with certainty
• The objective function and constraints exhibit
  constant returns to scale (i.e., linearity)
• There are no interactions between decision
  variables

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Methodology of Linear Programming

• Determine and define the decision variables
• Formulate an objective function
• verbal characterization
• Mathematical characterization
• Formulate each constraint

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3.2 THE GALAXY INDUSTRY PRODUCTION PROBLEM -
A Prototype Example

• Galaxy manufactures two toy models
• Space Ray.
• Zapper.
• Purpose to maximize profits
• How By choice of product mix
• How many Space Rays?
• How many Zappers?
• A RESOURCE ALLOCATION PROBLEM

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Galaxy Resource Allocation

• Resources are limited to
• 1200 pounds of special plastic available per week
• 40 hours of production time per week.
• All LP Models have to be formulated in the
  context of a production period
• In this case, a week

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• Marketing requirement
• Total production cannot exceed 800 dozens.
• Number of dozens of Space Rays cannot exceed
  number of dozens of Zappers by more than 450.
• Technological input
• Space Rays require 2 pounds of plastic and
• 3 minutes of labor per dozen.
• Zappers require 1 pound of plastic and
• 4 minutes of labor per dozen.

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• Current production plan calls for
• Producing as much as possible of the more
  profitable product, Space Ray (8 profit per
  dozen).
• Use resources left over to produce Zappers (5
  profit
• per dozen).
• The current production plan consists of
• Space Rays 550 dozens
• Zapper 100 dozens
• Profit 4900 dollars per
  week

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Management is seeking a production schedule
that will increase the companys profit.
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• MODEL FORMULATION
• Decisions variables
• X1 Production level of Space Rays (in dozens
  per week).
• X2 Production level of Zappers (in dozens per
  week).
• Objective Function
• Weekly profit, to be maximized

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The Objective Function

• Each dozen Space Rays realizes 8 in profit.
• Total profit from Space Rays is 8X1.
• Each dozen Zappers realizes 5 in profit.
• Total profit from Zappers is 5X2.
• The total profit contributions of both is
• 8X1 5X2
• (The profit contributions are additive because of
  the linearity assumption)

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• we have a plastics resource constraint, a
  production time constraint, and two marketing
  constraints.
• PLASTIC each dozen units of Space Rays requires
  2 lbs of plastic each dozen units of Zapper
  requires 1 lb of plastic and within any given
  week, our plastic supplier can provide 1200 lbs.

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• The Linear Programming Model
• Max 8X1 5X2 (Weekly profit)
• subject to
• 2X1 1X2 lt 1200 (Plastic)
• 3X1 4X2 lt 2400 (Production Time)
• X1 X2 lt 800 (Total production)
• X1 - X2 lt 450 (Mix)
• Xjgt 0, j 1,2 (Nonnegativity)

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3.4 The Set of Feasible Solutions for
Linear Programs
The set of all points that satisfy all the
constraints of the model is called a
FEASIBLE REGION
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• Using a graphical presentation
• we can represent all the constraints,
• the objective function, and the three
• types of feasible points.

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X2
1200
Total production constraint X1X2lt800
Infeasible
600
Feasible
Production Time 3X14X2lt2400
X1
600
800
Interior points.

• There are three types of feasible points

Boundary points.
Extreme points.
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3.5 Solving Graphically for an Optimal
Solution
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We now demonstrate the search for an optimal
solution
Start at some arbitrary profit, say profit
2,000...
Then increase the profit, if possible...
X2
1200
...and continue until it becomes infeasible
Profit 5040
2,
4,
3,
800
600
X1
400
600
800
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X2
1200
Lets take a closer look at the optimal
point
800
Infeasible
600
Feasible region
Feasible region
X1
400
600
800
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Summary of the optimal solution

• Space Rays 480 dozens
• Zappers 240 dozens
• Profit 5040
• This solution utilizes all the plastic and all
  the production hours.
• Total production is only 720 (not 800).
• Space Rays production exceeds Zapper by only 240
  dozens (not 450).

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• Extreme points and optimal solutions
• If a linear programming problem has an optimal
  solution, it will occur at an extreme point.
• Multiple optimal solutions
• For multiple optimal solutions to exist, the
  objective function must be parallel to a
  constraint that defines the boundary of the
  feasible region.
• Any weighted average of optimal solutions is also
  an optimal solution.

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3.6 The Role of Sensitivity Analysis of the
Optimal Solution

• Is the optimal solution sensitive to changes in
  input parameters?
• Possible reasons for asking this question
• Parameter values used were only best estimates.
• Dynamic environment may cause changes.
• What-if analysis may provide economical and
  operational information.

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3.7 Sensitivity Analysis of Objective
Function Coefficients.

• Range of Optimality
• The optimal solution will remain unchanged as
  long as
• An objective function coefficient lies within its
  range of optimality
• There are no changes in any other input
  parameters.
• The value of the objective function will change
  if the coefficient multiplies a variable whose
  value is nonzero.

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The effects of changes in an objective function
coefficient on the optimal solution
X2
1200
800
Max 8x1 5x2
600
Max 4x1 5x2
Max 3.75x1 5x2
Max 2x1 5x2
X1
400
600
800
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The effects of changes in an objective function
coefficient on the optimal solution
X2
1200
Max8x1 5x2
10
Max 10 x1 5x2
Max 3.75 x1 5x2
3.75
800
600
Max8x1 5x2
Max 3.75x1 5x2
X1
400
600
800
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• Multiple changes
• The range of optimality is valid only when a
  single objective function coefficient changes.
• When more than one variable changes we turn to
  the
• 100 rule.
• This is beyond the scope of this course

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• Reduced costs
• The reduced cost for a variable at its lower
  bound (usually zero) yields
• The amount the profit coefficient must change
  before
• the variable can take on a value above its lower
  bound.
• Complementary slackness
• At the optimal solution, either a variable is at
  its lower bound or the reduced cost is 0.

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3.8 Sensitivity Analysis of Right-Hand Side
Values

• Any change in a right hand side of a binding
  constraint will change the optimal solution.
• Any change in a right-hand side of a non-binding
  constraint that is less than its slack or
  surplus, will cause no change in the optimal
  solution.

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• In sensitivity analysis of right-hand sides of
  constraints we are interested in the following
  questions
• Keeping all other factors the same, how much
  would the optimal value of the objective function
  (for example, the profit) change if the
  right-hand side of a constraint changed by one
  unit?
• For how many additional UNITS is this per unit
  change valid?
• For how many fewer UNITS is this per unit change
  valid?

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X2
1200
2x1 1x2 lt1200
2x1 1x2 lt1350
600
Feasible
Infeasible extreme points
X1
600
800
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• Correct Interpretation of shadow prices
• Sunk costs The shadow price is the value of an
  extra unit of the resource, since the cost of the
  resource is not included in the calculation of
  the objective function coefficient.
• Included costs The shadow price is the premium
  value above the existing unit value for the
  resource, since the cost of the resource is
  included in the calculation of the objective
  function coefficient.

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• Range of feasibility
• The set of right - hand side values for which the
  same set of constraints determines the optimal
  extreme point.
• The range over-which the same variables remain in
  solution (which is another way of saying that the
  same extreme point is the optimal extreme point)
• Within the range of feasibility, shadow prices
  remain constant however, the optimal objective
  function value and decision variable values will
  change if the corresponding constraint is binding

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3.9 Other Post Optimality ChangesSKIP THIS

• Addition of a constraint.
• Deletion of a constraint.
• Addition of a variable.
• Deletion of a variable.
• Changes in the left - hand side technology
  coefficients.

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3.10 Models Without Optimal Solutions

• Infeasibility Occurs when a model has no
  feasible point.
• Unboundedness Occurs when the objective
  can become infinitely large.

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• Infeasibility

No point, simultaneously, lies both above line
and below lines and .
1
2
3
2
1
3
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• Unbounded solution

The feasible region
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3.11 Navy Sea Ration

• A cost minimization diet problem
• Mix two sea ration products Texfoods,
  Calration.
• Minimize the total cost of the mix.
• Meet the minimum requirements of
• Vitamin A, Vitamin D, and Iron.

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• Decision variables
• X1 (X2) -- The number of two-ounce portions of
  Texfoods (Calration)
  product used in a serving.
• The Model
• Minimize 0.60X1 0.50X2
• Subject to
• 20X1 50X2 100 Vitamin A
• 25X1 25X2 100 Vitamin D
• 50X1 10X2 100 Iron
• X1, X2 0

Cost per 2 oz.
Vitamin A provided per 2 oz.
required
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• The Graphical solution

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The Iron constraint
Feasible Region
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Vitamin D constraint
2
Vitamin A constraint
2
4
5
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• Summary of the optimal solution
• Texfood product 1.5 portions ( 3 ounces)
• Calration product 2.5 portions ( 5 ounces)
• Cost 2.15 per serving.
• The minimum requirements for Vitamin D and iron
  are met with no surplus.
• The mixture provides 155 of the requirement for
  Vitamin A.

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3.12 Computer Solution of Linear Programs
With Any Number of Decision Variables

• Linear programming software packages solve large
  linear models.
• Most of the software packages use the algebraic
  technique called the Simplex algorithm.
• The input to any package includes
• The objective function criterion (Max or Min).
• The type of each constraint .
• The actual coefficients for the problem.

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• The typical output generated from linear
  programming software includes
• Optimal value of the objective function.
• Optimal values of the decision variables.
• Reduced cost for each objective function
  coefficient.
• Ranges of optimality for objective function
  coefficients.
• The amount of slack or surplus in each
  constraint.
• Shadow (or dual) prices for the constraints.
• Ranges of feasibility for right-hand side values.

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Variable and constraint name can be changed here
WINQSB Input Data for the Galaxy Industries
Problem
Click to solve
Variables are restricted to gt 0
No upper bound
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Simplex Algorithm Basics

• Starting at a feasible extreme point, the
  algorithm proceeds from extreme point to extreme
  point until one is found that is better than all
  neighboring extreme points
• The transition from one extreme point to the next
  is called an iteration.
• The algorithm chooses which extreme point to go
  to next based on the fastest rate of improvement

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An iteration

• One non-basic variable enters the basis and
  becomes a basic variable
• One basis variable exits the basis and becomes
  non-basic
• The basis variables re-solved in terms of the
  non-basis variables
• The non-basis variables are fixed, usually at
  their lower bounds, which is zero, usually

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Basis and non-basis variables

• The basis variable values are free to take on
  values other than their lower bounds
• The non-basis variables are fixed at their lower
  bounds (0)
• THERE ARE ALWAYS AS MANY BASIS VARIABLES AS THERE
  ARE CONSTRAINTS, ALWAYS

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Another problem--10 products

• max 10x1 12 x2 15 x3 5 x4 8 x5 17x6 3
  x7 9x8 5x9 11x10
• s.t.
• 2x1 x2 3x3 x4 2x5 3x6 x7 3x8 2x9
  x10 lt 100
• all xi gt 0
• How many basis variables?
• How many products should we be making?

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