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:
•
•
•
•
•
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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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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 “what’s best” and “what-if” information.
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Assumptions of Linear
Programming
• 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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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).
WinQSB report is at the end.
• The current production plan consists of:
Space Rays = 550 dozens
Zapper
= 100 dozens
Profit
= 4900 dollars per week
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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 < = 1200 (Plastic)
3X1 + 4X2 < = 2400 (Production Time)
X1 + X2 < = 800
(Total production)
X1 - X2 < = 450
(Mix)
Xj> = 0, j = 1,2 (Nonnegativity)
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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
The plastic constraint:
The
Plastic constraint
2X1+X2<=1200
Total production constraint:
X1+X2<=800
Infeasible
600
Production
Feasible
Time
3X1+4X2<=2400
Production mix
constraint:
X1-X2<=450
600
800
Interior points.
Boundary points.
Extreme points.
X1
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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
800
Profit
4,
Profit
= $=$5040
2,
3,
000
600
X1
400
600
800
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1200
X2
Let’s take a closer look at
the optimal point
Infeasible
800
600
Feasible
Feasible
region
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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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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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
1200
X2
800
600
X1
400
600
800
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The effects of changes in an objective function coefficient
on the optimal solution
1200
X2
Range of optimality
800
600
400
600
800
X1
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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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Sensitivity Analysis of
Right-Hand Side Values
• Any change in a right hand side of a binding
constraint will change the optimal solution.
• Small change in a right-hand side of a nonbinding 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
The Plastic constraint
The new Plastic constraint
Maximum profit = 5040
Production mix constraint
600
Production time
constraint
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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Other Post Optimality Changes
• 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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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 1 and
below lines 2
and 3 .
2
3
1
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Unbounded solution

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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
Cost per 2 oz.
Minimize 0.60X1 + 0.50X2
Subject to
20X1 + 50X2  100 Vitamin A
% Vitamin A
25X1 + 25X2  100 Vitamin D
provided per 2 oz.
50X1 + 10X2  100 Iron
X1, X2  0
% required
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The Graphical solution
5
4
The Iron constraint
Feasible Region
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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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
Variables are
restricted to >= 0
Click to solve
No upper bound
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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 with10 products
• max 10x1 + 12 x2 + 15 x3 + 5 x4 + 8 x5 + 17x6
+ 3 x7 + 9x8 + 11x10
• s.t.
• 2x1 + x2 + 3x3 + x4 + 2x5 + 3x6 + x7 + 3x8 +
2x9 + x10 <= 100
• all xi >= 0
• How many basis variables?
• How many products should we be making?
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