Chapter 2
Linear and Integer Programming Models
2.1 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.
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Introduction to Linear Programming
The Importance of Linear Programming
Many real world problems lend themselves to linear programming modeling. Many real world problems can be approximated by linear models. There are well-known successful applications in: Manufacturing Marketing Finance (investment) Advertising Agriculture
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Introduction to Linear Programming
The Importance of Linear Programming
There are efficient solution techniques that solve linear programming models. The output generated from linear programming packages provides useful what if analysis.
Introduction to Linear Programming
Assumptions of the linear programming model
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 (the additivity assumption). The Continuity assumption: Variables can take on any value within a given feasible range.
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The Galaxy Industries Production Problem A Prototype Example
Galaxy manufactures two toy doll models:
Space Ray. Zapper.
Resources are limited to
1000 pounds of special plastic. 40 hours of production time per week.
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The Galaxy Industries Production Problem A Prototype Example
Marketing requirement Total production cannot exceed 700 dozens. Number of dozens of Space Rays cannot exceed number of dozens of Zappers by more than 350. Technological input
Space Rays requires 2 pounds of plastic and 3 minutes of labor per dozen. Zappers requires 1 pound of plastic and 4 minutes of labor per dozen.
The Galaxy Industries Production Problem A Prototype Example
The 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), while remaining within the marketing guidelines.
The current production plan consists of:
8(450) + 5(100) Space Rays = 450 dozen Zapper = 100 dozen Profit = $4100 per week
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Management is seeking a production schedule that will increase the companys profit.
A linear programming model can provide an insight and an intelligent solution to this problem.
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The Galaxy Linear Programming Model
Decisions variables:
X1 = Weekly production level of Space Rays (in dozens) X2 = Weekly production level of Zappers (in dozens).
Objective Function:
Weekly profit, to be maximized
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The Galaxy Linear Programming Model
Max 8X1 + 5X2 subject to 2X1 + 1X2 1000 3X1 + 4X2 2400 X1 + X2 700 X1 - X2 350 Xj> = 0, j = 1,2 (Weekly profit) (Plastic) (Production Time) (Total production) (Mix) (Nonnegativity)
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2.3 The Graphical Analysis of Linear Programming
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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Graphical Analysis the Feasible Region
X2
The non-negativity constraints
X1
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Graphical Analysis the Feasible Region
X2
1000
The Plastic constraint 2X1+X2 1000
Total production constraint: X1+X2 700 (redundant)
700 500
Infeasible
Production Time 3X1+4X2 2400
Feasible
500
700 X1
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Graphical Analysis the Feasible Region
X2
1000
The Plastic constraint 2X1+X2 1000 Total production constraint: X1+X2 700 (redundant)
700 500
Infeasible
Feasible
500
700
Production Time 3X1+4X22400
Production mix constraint: X1-X2 350
X1
Interior points. Boundary points. Extreme points.
There are three types of feasible points
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Solving Graphically for an Optimal Solution
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The search for an optimal solution
X2
1000
Start at some arbitrary profit, say profit = $2,000... Then increase the profit, if possible... ...and continue until it becomes infeasible
700 500
Profit =$4360
X1
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500
Summary of the optimal solution
Space Rays = 320 dozen Zappers = 360 dozen Profit = $4360
This solution utilizes all the plastic and all the production hours. Total production is only 680 (not 700). Space Rays production exceeds Zappers production by only 40 dozens.
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Extreme points and optimal solutions
If a linear programming problem has an optimal solution, an extreme point is optimal.
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Multiple optimal solutions
For multiple optimal solutions to exist, the objective function must be parallel to one of the constraints Any weighted average of optimal solutions is also an optimal solution.
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2.4 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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Sensitivity Analysis of Objective Function Coefficients.
1000 X2
500
X1
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500
800
1000
Sensitivity Analysis of Objective Function Coefficients. X
2
Range of optimality: [3.75, 10]
500
400
600
800
X1
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Reduced cost
Assuming there are no other changes to the input parameters, the reduced cost for a variable Xj that has a value of 0 at the optimal solution is: The negative of the objective coefficient increase of the variable Xj (-DCj) necessary for the variable to be positive in the optimal solution Alternatively, it is the change in the objective value per unit increase of Xj.
Complementary slackness
At the optimal solution, either the value of a variable is zero, or its reduced cost is 0.
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Sensitivity Analysis of Right-Hand Side Values
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 or fewer units will this per unit change be valid?
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Sensitivity Analysis of Right-Hand Side Values
Any change to the right hand side of a binding constraint will change the optimal solution. Any change to the 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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Shadow Prices
Assuming there are no other changes to the input parameters, the change to the objective function value per unit increase to a right hand side of a constraint is called the Shadow Price
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The Plastic constraint
Shadow Price graphical demonstration
X2
1000
When more plastic becomes available (the plastic constraint is relaxed), the right hand side of the plastic constraint increases.
Maximum profit = $4360
500
Maximum profit = $4363.4 Shadow price = 4363.40 4360.00 = 3.40
Production time constraint
500
X1
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Range of Feasibility
Assuming there are no other changes to the input parameters, the range of feasibility is
The range of values for a right hand side of a constraint, in which the shadow prices for the constraints remain unchanged. In the range of feasibility the objective function value changes as follows:
Change in objective value = [Shadow price][Change in the right hand side value]
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The Plastic constraint
Range of Feasibility
X2
1000
Increasing the amount of plastic is only effective until a new constraint becomes active.
Production mix constraint X1 + X2 700
A new active constraint
500
This is an infeasible solution
Production time constraint
X1
500
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The Plastic constraint
Range of Feasibility
X2
1000
Note how the profit increases as the amount of plastic increases.
500
Production time constraint
X1
500
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Range of Feasibility
X2
Infeasible solution
1000
Less plastic becomes available (the plastic constraint is more restrictive).
The profit decreases
500
A new active constraint
X1 500
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The 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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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 coefficients.
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2.5 Using Excel Solver to Find an Optimal Solution and Analyze Results
To see the input screen in Excel click Galaxy.xls Click Solver to obtain the following dialog box.
This cell contains Set Target cell $D$6 the value of the Equal To: objective function By Changing cells These cells contain $B$4:$C$4 the decision variables To enter constraints click All the constraints have the same direction, thus are included in one Excel constraint.
$D$7:$D$10
$F$7:$F$10
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Using Excel Solver
To see the input screen in Excel click Galaxy.xls Click Solver to obtain the following dialog box.
This cell contains Set Target cell $D$6 the value of the Equal To: objective function By Changing cells These cells contain $B$4:$C$4 the decision variables Click on Options and check Linear Programming and Non-negative. $D$7:$D$10<=$F$7:$F$10
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Using Excel Solver
To see the input screen in Excel click Galaxy.xls Click Solver to obtain the following dialog box.
Set Target cell $D$6 Equal To: By Changing cells $B$4:$C$4 $D$7:$D$10<=$F$7:$F$10
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Using Excel Solver Optimal Solution
GALAXY INDUSTRIES
Dozens Profit Plastic Prod. Time Total Mix Space Rays 320 8 2 3 1 1 Zappers 360 5 1 4 1 -1 Total 4360 1000 2400 680 -40 Limit <= <= <= <= 1000 2400 700 350
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Using Excel Solver Optimal Solution
GALAXY INDUSTRIES
Dozens Profit Plastic Prod. Time Total Mix Space Rays 320 8 2 3 1 1 Zappers 360 5 1 4 1 -1 Total 4360 1000 2400 680 -40 Limit <= <= <= <= 1000 2400 700 350
Solver is ready to provide reports to analyze the optimal solution.
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Using Excel Solver Answer Report
Microsoft Excel 9.0 Answer Report Worksheet: [Galaxy.xls]Galaxy Report Created: 11/12/2001 8:02:06 PM
Target Cell (Max) Cell Name $D$6 Profit Total
Original Value 4360
Final Value 4360
Adjustable Cells Cell Name Original Value $B$4 Dozens Space Rays 320 $C$4 Dozens Zappers 360
Final Value 320 360
Constraints Cell Name $D$7 Plastic Total $D$8 Prod. Time Total $D$9 Total Total $D$10 Mix Total
Cell Value 1000 2400 680 -40
Formula $D$7<=$F$7 $D$8<=$F$8 $D$9<=$F$9 $D$10<=$F$10
Status Slack Binding 0 Binding 0 Not Binding 20 Not Binding 390
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Using Excel Solver Sensitivity Report
Microsoft Excel Sensitivity Report Worksheet: [Galaxy.xls]Sheet1 Report Created:
Adjustable Cells Final Reduced Objective Allowable Allowable Cell Name Value Cost Coefficient Increase Decrease $B$4 Dozens Space Rays 320 0 8 2 4.25 $C$4 Dozens Zappers 360 0 5 5.666666667 1 Constraints Cell $D$7 $D$8 $D$9 $D$10 Name Plastic Total Prod. Time Total Total Total Mix Total Final Shadow Constraint Allowable Allowable Value Price R.H. Side Increase Decrease 1000 3.4 1000 100 400 2400 0.4 2400 100 650 680 0 700 1E+30 20 -40 0 350 1E+30 390
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2.7 Models Without Unique Optimal Solutions
Infeasibility: Occurs when a model has no feasible point.
Unboundness: Occurs when the objective can become
infinitely large (max), or infinitely small (min). Alternate solution: Occurs when more than one point optimizes the objective function
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Infeasible Model
No point, simultaneously, lies both above line 1 and
2
below lines 2 and 3 .
1
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Solver Infeasible Model
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Unbounded solution
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Solver Unbounded solution
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Solver An Alternate Optimal Solution
Solver does not alert the user to the existence of alternate optimal solutions. Many times alternate optimal solutions exist when the allowable increase or allowable decrease is equal to zero. In these cases, we can find alternate optimal solutions using Solver by the following procedure:
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Solver An Alternate Optimal Solution
Observe that for some variable Xj the Allowable increase = 0, or Allowable decrease = 0. Add a constraint of the form: Objective function = Current optimal value. If Allowable increase = 0, change the objective to Maximize Xj If Allowable decrease = 0, change the objective to Minimize Xj
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2.8 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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Cost Minimization Diet Problem
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 Cost per 2 oz. Subject to 20X1 + 50X2 100 Vitamin A 25X1 + 25X2 100 Vitamin D % Vitamin A provided per 2 oz. 50X1 + 10X2 100 Iron % required X1, X2 0
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The Diet Problem - Graphical solution
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The Iron constraint
Feasible Region
Vitamin D constraint
Vitamin A constraint
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Cost Minimization Diet Problem
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 requirement 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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Copyright 2002John Wiley & Sons, Inc. All rights reserved. Reproduction or translation of this work beyond that named in Section 117 of the United States Copyright Act without the express written consent of the copyright owner is unlawful. Requests for further information should be addressed to the Permissions Department, John Wiley & Sons, Inc. Adopters of the textbook are granted permission to make back-up copies for their own use only, to make copies for distribution to students of the course the textbook is used in, and to modify this material to best suit their instructional needs. Under no circumstances can copies be made for resale. The Publisher assumes no responsibility for errors, omissions, or damages, caused by the use of these programs or from the use of the information contained herein.
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