Tbs910 linear programming

Linear Programming ModelLinear Programming ModelLinear Programming ModelLinear Programming Model
TBS910 BUSINESS ANALYTICSTBS910 BUSINESS ANALYTICS
by
Prof.Stephen Ong
Visiting Professor, Shenzhen
University
Visiting Fellow, Sydney Business

Today’s OverviewToday’s Overview

7-3
Learning ObjectivesLearning Objectives
1.1. Understand the basic assumptions and properties ofUnderstand the basic assumptions and properties of
linear programming (LP).linear programming (LP).
2.2. Graphically solve any LP problem that has only twoGraphically solve any LP problem that has only two
variables by both the corner point and isoprofit linevariables by both the corner point and isoprofit line
methods.methods.
3.3. Understand special issues in LP such asUnderstand special issues in LP such as
infeasibility, unboundedness, redundancy, andinfeasibility, unboundedness, redundancy, and
alternative optimal solutions.alternative optimal solutions.
4.4. Understand the role of sensitivity analysis.Understand the role of sensitivity analysis.
5.5. Use Excel spreadsheets to solve LP problems.Use Excel spreadsheets to solve LP problems.
After this lecture, students will be able to:After this lecture, students will be able to:

7-4
OutlineOutline
7.17.1 IntroductionIntroduction
7.27.2 Requirements of a Linear ProgrammingRequirements of a Linear Programming
ProblemProblem
7.37.3 Formulating LP ProblemsFormulating LP Problems
7.47.4 Graphical Solution to an LP ProblemGraphical Solution to an LP Problem
7.57.5 Solving Flair Furniture’s LP ProblemSolving Flair Furniture’s LP Problem
using QM for Windows and Excelusing QM for Windows and Excel
7.67.6 Solving Minimization ProblemsSolving Minimization Problems
7.77.7 Four Special Cases in LPFour Special Cases in LP
7.87.8 Sensitivity AnalysisSensitivity Analysis

7-5
IntroductionIntroduction
 Many management decisions involveMany management decisions involve
trying to make the most effective use oftrying to make the most effective use of
limited resources.limited resources.
 Linear programmingLinear programming ((LPLP) is a widely used) is a widely used
mathematical modeling techniquemathematical modeling technique
designed to help managers in planningdesigned to help managers in planning
and decision making relative to resourceand decision making relative to resource
allocation.allocation.
 This belongs to the broader field ofThis belongs to the broader field of
mathematical programming.mathematical programming.
 In this sense,In this sense, programmingprogramming refers torefers to
modeling and solving a problemmodeling and solving a problem
mathematically.mathematically.

Requirements of a LinearRequirements of a Linear
Programming ProblemProgramming Problem
 All LP problems have 4 properties inAll LP problems have 4 properties in
common:common:
1.1. All problems seek toAll problems seek to maximizemaximize oror minimizeminimize somesome
quantity (thequantity (the objective functionobjective function).).
2.2. Restrictions orRestrictions or constraintsconstraints that limit the degreethat limit the degree
to which we can pursue our objective areto which we can pursue our objective are
present.present.
3.3. There must be alternative courses of action fromThere must be alternative courses of action from
which to choose.which to choose.
4.4. The objective and constraints in problems mustThe objective and constraints in problems must
be expressed in terms ofbe expressed in terms of linearlinear equations orequations or
inequalities.inequalities.

Basic Assumptions of LPBasic Assumptions of LP
 We assume conditions ofWe assume conditions of certaintycertainty exist andexist and
numbers in the objective and constraints arenumbers in the objective and constraints are
known with certainty and do not changeknown with certainty and do not change
during the period being studied.during the period being studied.
 We assumeWe assume proportionalityproportionality exists in theexists in the
objective and constraints.objective and constraints.
 We assumeWe assume additivityadditivity in that the total of allin that the total of all
activities equals the sum of the individualactivities equals the sum of the individual
activities.activities.
 We assumeWe assume divisibilitydivisibility in that solutions needin that solutions need
not be whole numbers.not be whole numbers.
 All answers or variables areAll answers or variables are nonnegative.nonnegative.

LP Properties andLP Properties and
AssumptionsAssumptions
PROPERTIES OF LINEAR PROGRAMSPROPERTIES OF LINEAR PROGRAMS
1. One objective function1. One objective function
2. One or more constraints2. One or more constraints
3. Alternative courses of action3. Alternative courses of action
4. Objective function and constraints are4. Objective function and constraints are
linear – proportionality and divisibilitylinear – proportionality and divisibility
5. Certainty5. Certainty
6. Divisibility6. Divisibility
7. Nonnegative variables7. Nonnegative variablesTable 7.1

Formulating LP ProblemsFormulating LP Problems
 Formulating a linear program involvesFormulating a linear program involves
developing a mathematical model to representdeveloping a mathematical model to represent
the managerial problem.the managerial problem.
 The steps in formulating a linear program are:The steps in formulating a linear program are:
1.1. Completely understand the managerialCompletely understand the managerial
problem being faced.problem being faced.
2.2. Identify the objective and the constraints.Identify the objective and the constraints.
3.3. Define the decision variables.Define the decision variables.
4.4. Use the decision variables to writeUse the decision variables to write
mathematical expressions for the objectivemathematical expressions for the objective
function and the constraints.function and the constraints.

Formulating LP ProblemsFormulating LP Problems
 One of the most common LP applications isOne of the most common LP applications is
thethe product mix problem.product mix problem.
 Two or more products are produced usingTwo or more products are produced using
limited resources such as personnel,limited resources such as personnel,
machines, and raw materials.machines, and raw materials.
 The profit that the firm seeks to maximize isThe profit that the firm seeks to maximize is
based on the profit contribution per unit ofbased on the profit contribution per unit of
each product.each product.
 The company would like to determine howThe company would like to determine how
many units of each product it should producemany units of each product it should produce
so as to maximize overall profit given itsso as to maximize overall profit given its
limited resources.limited resources.

Flair Furniture CompanyFlair Furniture Company
 The Flair Furniture Company produces inexpensiveThe Flair Furniture Company produces inexpensive
tables and chairs.tables and chairs.
 Processes are similar in that both require a certainProcesses are similar in that both require a certain
amount of hours of carpentry work and in the paintingamount of hours of carpentry work and in the painting
and varnishing department.and varnishing department.
 Each table takes 4 hours of carpentry and 2 hours ofEach table takes 4 hours of carpentry and 2 hours of
painting and varnishing.painting and varnishing.
 Each chair requires 3 of carpentry and 1 hour ofEach chair requires 3 of carpentry and 1 hour of
painting and varnishing.painting and varnishing.
 There are 240 hours of carpentry time available andThere are 240 hours of carpentry time available and
100 hours of painting and varnishing.100 hours of painting and varnishing.
 Each table yields a profit of $70 and each chair a profitEach table yields a profit of $70 and each chair a profit
of $50.of $50.

DataData
The company wants to determine the bestThe company wants to determine the best
combination of tables and chairs to producecombination of tables and chairs to produce
to reach the maximum profit.to reach the maximum profit.
HOURS REQUIREDHOURS REQUIRED
TO PRODUCE 1TO PRODUCE 1
UNITUNIT
DEPARTMENTDEPARTMENT
((TT))
TABLESTABLES
((CC))
CHAIRSCHAIRS
AVAILABLEAVAILABLE
HOURS THISHOURS THIS
WEEKWEEK
CarpentryCarpentry 44 33 240240
Painting andPainting and
varnishingvarnishing 22 11 100100
Profit per unitProfit per unit $70$70 $50$50
Table 7.2

 The objective is to:The objective is to:
Maximize profitMaximize profit
 The constraints are:The constraints are:
1.1. The hours of carpentry time used cannot exceedThe hours of carpentry time used cannot exceed
240 hours per week.240 hours per week.
2.2. The hours of painting and varnishing time usedThe hours of painting and varnishing time used
cannot exceed 100 hours per week.cannot exceed 100 hours per week.
 The decision variables representing the actualThe decision variables representing the actual
decisions we will make are:decisions we will make are:
TT = number of tables to be produced per week.= number of tables to be produced per week.
CC = number of chairs to be produced per week.= number of chairs to be produced per week.

7-14
 We create the LP objective function in terms ofWe create the LP objective function in terms of TT andand C:C:
Maximize profit = $70Maximize profit = $70TT + $50+ $50CC
 Develop mathematical relationships for the twoDevelop mathematical relationships for the two
constraints:constraints:
 For carpentry, total time used is:For carpentry, total time used is:
(4 hours per table)(Number of tables produced) +(4 hours per table)(Number of tables produced) +
(3 hours per chair)(Number of chairs produced).(3 hours per chair)(Number of chairs produced).
 We know that:We know that:
Carpentry time usedCarpentry time used ≤ Carpentry time available.≤ Carpentry time available.
44TT + 3+ 3CC ≤ 240≤ 240 (hours of carpentry time(hours of carpentry time))

 Similarly,Similarly,
Painting and varnishing time usedPainting and varnishing time used
≤ Painting and varnishing time available.≤ Painting and varnishing time available.
22 TT + 1+ 1CC ≤ 100≤ 100 (hours of painting and(hours of painting and
varnishing time)varnishing time)
This means that each tableThis means that each table
produced requires two hours ofproduced requires two hours of
painting and varnishing time.painting and varnishing time.
 Both of these constraints restrictBoth of these constraints restrict
production capacity and affect total profit.production capacity and affect total profit.

The values forThe values for TT andand CC must bemust be
nonnegative.nonnegative.TT ≥ 0≥ 0 (number of tables produced is greater than(number of tables produced is greater than
or equal to 0)or equal to 0)
CC ≥ 0≥ 0 (number of chairs produced is greater than(number of chairs produced is greater than
or equal to 0)or equal to 0)
The complete problem stated mathematically:The complete problem stated mathematically:
Maximize profit = $70Maximize profit = $70TT + $50+ $50CCsubject tosubject to
44TT + 3+ 3CC ≤240≤240 (carpentry constraint)(carpentry constraint)
22TT + 1+ 1CC ≤≤100100 (painting and(painting and
varnishing constraint)varnishing constraint)

Graphical Solution to an LPGraphical Solution to an LP
ProblemProblem
 The easiest way to solve a small LPThe easiest way to solve a small LP
problems is graphically.problems is graphically.
 The graphical method only works whenThe graphical method only works when
there are just two decision variables.there are just two decision variables.
 When there are more than two variables,When there are more than two variables,
a more complex approach is needed as ita more complex approach is needed as it
is not possible to plot the solution on ais not possible to plot the solution on a
two-dimensional graph.two-dimensional graph.
 The graphical method provides valuableThe graphical method provides valuable
insight into how other approaches work.insight into how other approaches work.

Graphical Representation of aGraphical Representation of a
ConstraintConstraint
100 –
–
80 –
–
60 –
–
40 –
–
20 –
–
–
C
| | | | | | | | | | | |
0 20 40 60 80 100 T
NumberofChairsNumberofChairs
Number of TablesNumber of Tables
This Axis Represents theThis Axis Represents the
ConstraintConstraint TT ≥ 0≥ 0
This Axis RepresentsThis Axis Represents
the Constraintthe Constraint CC ≥ 0≥ 0
Figure 7.1
Quadrant Containing All Positive ValuesQuadrant Containing All Positive Values

 The first step in solving the problem isThe first step in solving the problem is
to identify a set or region of feasibleto identify a set or region of feasible
solutions.solutions.
 To do this we plot each constraintTo do this we plot each constraint
equation on a graph.equation on a graph.
 We start by graphing the equalityWe start by graphing the equality
portion of the constraint equations:portion of the constraint equations:
44TT + 3+ 3CC = 240= 240
 We solve for the axis intercepts andWe solve for the axis intercepts and
draw the line.draw the line.

 When Flair produces no tables, theWhen Flair produces no tables, the
carpentry constraint is:carpentry constraint is:
4(0) + 34(0) + 3CC = 240= 240
33CC = 240= 240
CC = 80= 80
 Similarly for no chairs:Similarly for no chairs:
44TT + 3(0) = 240+ 3(0) = 240
44TT = 240= 240
TT = 60= 60
 This line is shown on the following graph:This line is shown on the following graph:

100 –
–
80 –
–
60 –
–
40 –
–
20 –
–
–
C
| | | | | | | | | | | |
0 20 40 60 80 100 T
NumberofChairs
Number of Tables
(T = 0, C = 80)
Figure 7.2
(T = 60, C = 0)
Graph of carpentry constraint equationGraph of carpentry constraint equation

7-22
100 –
–
80 –
–
60 –
–
40 –
–
20 –
–
–
C
| | | | | | | | | | | |
0 20 40 60 80 100 T
Figure 7.3
 Any point on orAny point on or
below thebelow the
constraint plot willconstraint plot will
not violate thenot violate the
restriction.restriction.
 Any point aboveAny point above
the plot will violatethe plot will violate
the restriction.the restriction.
(30, 40)
(30, 20)(30, 20)
(70, 40)
Region that Satisfies the Carpentry ConstraintRegion that Satisfies the Carpentry Constraint

 The point (30, 40) lies on the plot and exactlyThe point (30, 40) lies on the plot and exactly
satisfies the constraintsatisfies the constraint
4(30) + 3(40) = 240.4(30) + 3(40) = 240.
 The point (30, 20) lies below the plot andThe point (30, 20) lies below the plot and
satisfies the constraintsatisfies the constraint
4(30) + 3(20) = 180.4(30) + 3(20) = 180.
 The point (70, 40) lies above the plot and doesThe point (70, 40) lies above the plot and does
not satisfy the constraintnot satisfy the constraint
4(70) + 3(40) = 400.4(70) + 3(40) = 400.

100 –
–
80 –
–
60 –
–
40 –
–
20 –
–
–
C
| | | | | | | | | | | |
0 20 40 60 80 100 T
NumberofChairs
Number of Tables
(T = 0, C = 100)
Figure 7.4
(T = 50, C = 0)
Region that Satisfies the PaintingRegion that Satisfies the Painting
and Varnishing Constraintand Varnishing Constraint

 To produce tables and chairs, bothTo produce tables and chairs, both
departments must be used.departments must be used.
 We need to find a solution that satisfies bothWe need to find a solution that satisfies both
constraintsconstraints simultaneously.simultaneously.
 A new graph shows both constraint plots.A new graph shows both constraint plots.
 TheThe feasible regionfeasible region (or(or area of feasiblearea of feasible
solutionssolutions) is where all constraints are) is where all constraints are
satisfied.satisfied.
 Any point inside this region is aAny point inside this region is a feasiblefeasible
solution.solution.
 Any point outside the region is anAny point outside the region is an infeasibleinfeasible
solution.solution.

100 –
–
80 –
–
60 –
–
40 –
–
20 –
–
–
C
| | | | | | | | | | | |
0 20 40 60 80 100 T
Number of Tables
Figure 7.5
Feasible Solution Region for the FlairFeasible Solution Region for the Flair
Furniture Company ProblemFurniture Company Problem
Painting/Varnishing ConstraintPainting/Varnishing Constraint
Carpentry ConstraintCarpentry Constraint
FeasibleFeasible
RegionRegion

 For the point (30, 20)For the point (30, 20)
CarpentryCarpentry
constraintconstraint
44TT + 3+ 3CC ≤ 240 hours available≤ 240 hours available
(4)(30) + (3)(20) = 180 hours used(4)(30) + (3)(20) = 180 hours used
PaintingPainting
(2)(30) + (1)(20) = 80 hours used(2)(30) + (1)(20) = 80 hours used


CarpentryCarpentry
(4)(70) + (3)(40) = 400 hours(4)(70) + (3)(40) = 400 hours
usedused
PaintingPainting
(2)(70) + (1)(40) = 180 hours(2)(70) + (1)(40) = 180 hours
usedused



CarpentryCarpentry
(4)(50) + (3)(5) = 215 hours(4)(50) + (3)(5) = 215 hours
usedused
PaintingPainting
(2)(50) + (1)(5) = 105 hours(2)(50) + (1)(5) = 105 hours
usedused



Isoprofit Line SolutionIsoprofit Line Solution
MethodMethod
 Once the feasible region has been graphed, weOnce the feasible region has been graphed, we
need to find the optimal solution from the manyneed to find the optimal solution from the many
possible solutions.possible solutions.
 The speediest way to do this is to use theThe speediest way to do this is to use the
isoprofit line method.isoprofit line method.
 Starting with a small but possible profit value,Starting with a small but possible profit value,
we graph the objective function.we graph the objective function.
 We move the objective function line in theWe move the objective function line in the
direction of increasing profit while maintainingdirection of increasing profit while maintaining
the slope.the slope.
 The last point it touches in the feasible region isThe last point it touches in the feasible region is
the optimal solution.the optimal solution.

MethodMethod
 For Flair Furniture, choose a profit of $2,100.For Flair Furniture, choose a profit of $2,100.
 The objective function is thenThe objective function is then
$2,100 = 70$2,100 = 70TT + 50+ 50CC
 Solving for the axis intercepts, we can draw theSolving for the axis intercepts, we can draw the
graph.graph.
 This is obviously not the best possible solution.This is obviously not the best possible solution.
 Further graphs can be created using larger profits.Further graphs can be created using larger profits.
 The further we move from the origin, the larger theThe further we move from the origin, the larger the
profit will be.profit will be.
 The highest profit ($4,100) will be generated whenThe highest profit ($4,100) will be generated when
the isoprofit line passes through the point (30, 40).the isoprofit line passes through the point (30, 40).

100 –
–
80 –
–
60 –
–
40 –
–
20 –
–
–
C
| | | | | | | | | | | |
0 20 40 60 80 100 T
Figure 7.6
Profit line of $2,100 Plotted for theProfit line of $2,100 Plotted for the
$2,100 = $70$2,100 = $70TT + $50+ $50CC
(30, 0)(30, 0)
(0, 42)(0, 42)
Isoprofit Line Solution MethodIsoprofit Line Solution Method

100 –
–
80 –
–
60 –
–
40 –
–
20 –
–
–
C
| | | | | | | | | | | |
0 20 40 60 80 100 T
NumberofChairs
Number of Tables
Figure 7.7
Four Isoprofit Lines Plotted for the FlairFour Isoprofit Lines Plotted for the Flair
Furniture CompanyFurniture Company
$2,100 = $70$2,100 = $70TT + $50+ $50CC
$2,800 = $70$2,800 = $70TT + $50+ $50CC
$3,500 = $70$3,500 = $70TT + $50+ $50CC
$4,200 = $70$4,200 = $70TT + $50+ $50CC
MethodMethod

7-33
100 –
–
80 –
–
60 –
–
40 –
–
20 –
–
–
C
| | | | | | | | | | | |
0 20 40 60 80 100 T
Figure 7.8
Optimal Solution to the FlairOptimal Solution to the Flair
Furniture problemFurniture problem
Optimal Solution PointOptimal Solution Point
((TT = 30,= 30, CC = 40)= 40)
Maximum Profit LineMaximum Profit Line
$4,100 = $70$4,100 = $70TT + $50+ $50CC
Isoprofit Line Solution MethodIsoprofit Line Solution Method

7-34
 A second approach to solving LP problemsA second approach to solving LP problems
employs theemploys the corner point method.corner point method.
 It involves looking at the profit at everyIt involves looking at the profit at every
corner point of the feasible region.corner point of the feasible region.
 The mathematical theory behind LP is thatThe mathematical theory behind LP is that
the optimal solution must lie at one of thethe optimal solution must lie at one of the
corner pointscorner points, or, or extreme pointextreme point, in the, in the
feasible region.feasible region.
 For Flair Furniture, the feasible region is aFor Flair Furniture, the feasible region is a
four-sided polygon with four corner pointsfour-sided polygon with four corner points
labeled 1, 2, 3, and 4 on the graph.labeled 1, 2, 3, and 4 on the graph.
Corner Point SolutionCorner Point Solution
MethodMethod

100 –
–
80 –
–
60 –
–
40 –
–
20 –
–
–
C
| | | | | | | | | | | |
0 20 40 60 80 100 T
Figure 7.9
Four Corner Points of theFour Corner Points of the
Feasible RegionFeasible Region
1
2
3
4
Corner Point Solution MethodCorner Point Solution Method

 To find the coordinates for Point accurately we haveTo find the coordinates for Point accurately we have
to solve for the intersection of the two constraint lines.to solve for the intersection of the two constraint lines.
 Using theUsing the simultaneous equations methodsimultaneous equations method, we multiply, we multiply
the painting equation by –2 and add it to the carpentrythe painting equation by –2 and add it to the carpentry
equationequation
44TT + 3+ 3CC == 240240 (carpentry line)(carpentry line)
–– 44TT – 2– 2CC ==––200200 (painting line)(painting line)
CC == 4040
 Substituting 40 forSubstituting 40 for CC in either of the originalin either of the original
equations allows us to determine the value ofequations allows us to determine the value of T.T.
44TT + (3)(40) =+ (3)(40) =240240 (carpentry line)(carpentry line)
44TT + 120 =+ 120 = 240240
TT == 3030
3

7-37
3
1
2
4
Point : (Point : (TT = 0,= 0, CC = 0)= 0) Profit = $70(0) + $50(0) = $0Profit = $70(0) + $50(0) = $0
Point : (Point : (TT = 0,= 0, CC = 80)= 80) Profit = $70(0) + $50(80) = $4,000Profit = $70(0) + $50(80) = $4,000
Because Point returns the highest profit, this is theBecause Point returns the highest profit, this is the
optimal solution.optimal solution.
3

Slack and SurplusSlack and Surplus
 SlackSlack is the amount of a resource that isis the amount of a resource that is
not used. For a less-than-or-equalnot used. For a less-than-or-equal
constraint:constraint:
 SlackSlack = Amount of resource available –= Amount of resource available –
amount of resource used.amount of resource used.
 Surplus is used with a greater-than-or-Surplus is used with a greater-than-or-
equal constraint to indicate the amount byequal constraint to indicate the amount by
which the right hand side of the constraintwhich the right hand side of the constraint
is exceeded.is exceeded.
 SurplusSurplus = Actual amount – minimum amount.= Actual amount – minimum amount.

Summary of GraphicalSummary of Graphical
Solution MethodsSolution Methods
ISOPROFIT METHODISOPROFIT METHOD
1.1. Graph all constraints and find the feasible region.Graph all constraints and find the feasible region.
2.2. Select a specific profit (or cost) line and graph it to find the slope.Select a specific profit (or cost) line and graph it to find the slope.
3.3. Move the objective function line in the direction of increasingMove the objective function line in the direction of increasing
profit (or decreasing cost) while maintaining the slope. The lastprofit (or decreasing cost) while maintaining the slope. The last
point it touches in the feasible region is the optimal solution.point it touches in the feasible region is the optimal solution.
4.4. Find the values of the decision variables at this last point andFind the values of the decision variables at this last point and
compute the profit (or cost).compute the profit (or cost).
CORNER POINT METHODCORNER POINT METHOD
1.1. Graph all constraints and find the feasible region.Graph all constraints and find the feasible region.
2.2. Find the corner points of the feasible reason.Find the corner points of the feasible reason.
3.3. Compute the profit (or cost) at each of the feasible corner points.Compute the profit (or cost) at each of the feasible corner points.
4.4. Select the corner point with the best value of the objectiveSelect the corner point with the best value of the objective
function found in Step 3. This is the optimal solution.function found in Step 3. This is the optimal solution.
Table 7.4

Copyright ©2012 Pearson
Education, Inc. publishing as
Prentice Hall7-40
Solving Flair Furniture’s LP ProblemSolving Flair Furniture’s LP Problem
Using QM for Windows and ExcelUsing QM for Windows and Excel
 Most organizations have access to softwareMost organizations have access to software
to solve big LP problems.to solve big LP problems.
 While there are differences between softwareWhile there are differences between software
implementations, the approach each takesimplementations, the approach each takes
towards handling LP is basically the same.towards handling LP is basically the same.
 Once you are experienced in dealing withOnce you are experienced in dealing with
computerized LP algorithms, you can easilycomputerized LP algorithms, you can easily
adjust to minor changes.adjust to minor changes.

Using QM for WindowsUsing QM for Windows
 First select the Linear ProgrammingFirst select the Linear Programming
module.module.
 Specify the number of constraints (non-Specify the number of constraints (non-
negativity is assumed).negativity is assumed).
 Specify the number of decision variables.Specify the number of decision variables.
 Specify whether the objective is to beSpecify whether the objective is to be
maximized or minimized.maximized or minimized.
 For the Flair Furniture problem there areFor the Flair Furniture problem there are
two constraints, two decision variables,two constraints, two decision variables,
and the objective is to maximize profit.and the objective is to maximize profit.

QM for Windows Linear ProgrammingQM for Windows Linear Programming
Computer screen for Input of DataComputer screen for Input of Data
Program 7.1A

QM for Windows Data Input for FlairQM for Windows Data Input for Flair
Furniture ProblemFurniture Problem
Program 7.1B

QM for Windows Output for Flair Furniture Problem
Program 7.1C

QM for Windows Graphical Output for FlairQM for Windows Graphical Output for Flair
Program 7.1D

Using Excel’s Solver Command toUsing Excel’s Solver Command to
Solve LP ProblemsSolve LP Problems
The Solver tool in Excel can beThe Solver tool in Excel can be
used to find solutions to:used to find solutions to:
LP problems.LP problems.
Integer programming problems.Integer programming problems.
Noninteger programmingNoninteger programming
problems.problems.
Solver is limited to 200 variables andSolver is limited to 200 variables and
100 constraints.100 constraints.

Using Solver to Solve the FlairUsing Solver to Solve the Flair
 Recall the model for Flair Furniture is:Recall the model for Flair Furniture is:
Maximize profit =Maximize profit =$70$70TT ++$50$50CC
Subject toSubject to 44TT ++ 33CC ≤ 240≤ 240
22TT ++ 11CC ≤ 100≤ 100
To use Solver, it is necessary toTo use Solver, it is necessary to
enter formulas based on theenter formulas based on the
initial model.initial model.

7-48
Using Solver to Solve theUsing Solver to Solve the
Flair Furniture ProblemFlair Furniture Problem
1.1. Enter the variable names, theEnter the variable names, the
coefficients for the objective functioncoefficients for the objective function
and constraints, and the right-hand-sideand constraints, and the right-hand-side
values for each of the constraints.values for each of the constraints.
2.2.Designate specific cells for the values ofDesignate specific cells for the values of
the decision variables.the decision variables.
3.3.Write a formula to calculate the value ofWrite a formula to calculate the value of
the objective function.the objective function.
4.4.Write a formula to compute the left-handWrite a formula to compute the left-hand
sides of each of the constraints.sides of each of the constraints.

Program 7.2A
Excel Data Input for the Flair Furniture ExampleExcel Data Input for the Flair Furniture Example

Program 7.2B
Formulas for the Flair Furniture ExampleFormulas for the Flair Furniture Example

Program 7.2C
Excel Spreadsheet for the Flair Furniture ExampleExcel Spreadsheet for the Flair Furniture Example

7-52
 Once the model has been entered, the followingOnce the model has been entered, the following
steps can be used to solve the problem.steps can be used to solve the problem.
In Excel 2010, selectIn Excel 2010, select Data – Solver.Data – Solver.
If Solver does not appear in the indicatedIf Solver does not appear in the indicated
place, see Appendix F for instructions onplace, see Appendix F for instructions on
how to activate this add-in.how to activate this add-in.
1.1. In the Set Objective box, enter the cell address for theIn the Set Objective box, enter the cell address for the
total profit.total profit.
2.2. In the By Changing Cells box, enter the cell addressesIn the By Changing Cells box, enter the cell addresses
for the variable values.for the variable values.
3.3. ClickClick MaxMax for a maximization problem andfor a maximization problem and MinMin for afor a
minimization problem.minimization problem.

7-53
44. Check the box for. Check the box for Make UnconstrainedMake Unconstrained
Variables Non-negativeVariables Non-negative..
55. Click the. Click the Select Solving MethodSelect Solving Method buttonbutton
and selectand select Simplex LPSimplex LP from the menu thatfrom the menu that
appears.appears.
6.6.ClickClick AddAdd to add the constraints.to add the constraints.
7.7.In the dialog box that appears, enter theIn the dialog box that appears, enter the
cell references for the left-hand-side values,cell references for the left-hand-side values,
the type of equation, and the right-hand-sidethe type of equation, and the right-hand-side
values.values.
8.8.ClickClick SolveSolve..

7-54
Starting Solver
Figure 7.2D

7-55
Figure 7.2E
SolverSolver
ParametersParameters
Dialog BoxDialog Box

7-56
Figure 7.2F
Solver Add Constraint Dialog BoxSolver Add Constraint Dialog Box

7-57
Figure 7.2G
Solver Results Dialog BoxSolver Results Dialog Box

Figure 7.2H
Solution Found by SolverSolution Found by Solver

7-59
Solving Minimization ProblemsSolving Minimization Problems
 Many LP problems involve minimizing anMany LP problems involve minimizing an
objective such as cost instead of maximizingobjective such as cost instead of maximizing
a profit function.a profit function.
 Minimization problems can be solvedMinimization problems can be solved
graphically by first setting up the feasiblegraphically by first setting up the feasible
solution region and then using either thesolution region and then using either the
corner point method or an isocost linecorner point method or an isocost line
approach (which is analogous to the isoprofitapproach (which is analogous to the isoprofit
approach in maximization problems) to findapproach in maximization problems) to find
the values of the decision variables (e.g.,the values of the decision variables (e.g., XX11
andand XX22) that yield the minimum cost.) that yield the minimum cost.

7-60
The Holiday Meal Turkey Ranch is consideringThe Holiday Meal Turkey Ranch is considering
buying two different brands of turkey feed andbuying two different brands of turkey feed and
blending them to provide a good, low-cost diet for itsblending them to provide a good, low-cost diet for its
turkeysturkeys
Minimize cost (in cents) = 2Minimize cost (in cents) = 2XX11 + 3+ 3XX22
subject to:subject to:
55XX11 + 10+ 10XX22 ≥≥ 90 ounces90 ounces (ingredient constraint A)(ingredient constraint A)
44XX11 + 3+ 3XX22 ≥≥ 48 ounces48 ounces (ingredient constraint B)(ingredient constraint B)
0.50.5XX11 ≥≥ 1.5 ounces1.5 ounces (ingredient constraint C)(ingredient constraint C)
XX11 ≥≥ 00 (nonnegativity constraint)(nonnegativity constraint)
XX22 ≥≥ 00 (nonnegativity constraint)(nonnegativity constraint)
Holiday Meal Turkey RanchHoliday Meal Turkey Ranch
XX11 = number of pounds of brand 1 feed purchased= number of pounds of brand 1 feed purchased
XX22 = number of pounds of brand 2 feed purchased= number of pounds of brand 2 feed purchased
Let

INGREDIENTINGREDIENT
COMPOSITION OFCOMPOSITION OF
EACH POUND OFEACH POUND OF
FEED (OZ.)FEED (OZ.)
MINIMUMMINIMUM
MONTHLYMONTHLY
REQUIREMENTREQUIREMENT
PER TURKEYPER TURKEY
(OZ.)(OZ.)
BRAND 1BRAND 1
FEEDFEED
BRAND 2BRAND 2
FEEDFEED
AA 55 1010 9090
BB 44 33 4848
CC 0.50.5 00 1.51.5
Cost perCost per
poundpound
2 cents2 cents 3 cents3 cents
Holiday Meal Turkey Ranch dataHoliday Meal Turkey Ranch data
Table 7.5

 Use the corner point method.Use the corner point method.
 First construct the feasibleFirst construct the feasible
solution region.solution region.
 The optimal solution will lie atThe optimal solution will lie at
one of the corners as it wouldone of the corners as it would
in a maximization problem.in a maximization problem.

Feasible Region for the HolidayFeasible Region for the Holiday
Meal Turkey Ranch ProblemMeal Turkey Ranch Problem
–
20 –
15 –
10 –
5 –
0 –
X2
| | | | | |
5 10 15 20 25 X1
PoundsofBrand2
Pounds of Brand 1
Ingredient C ConstraintIngredient C Constraint
Ingredient B ConstraintIngredient B Constraint
Ingredient A ConstraintIngredient A Constraint
a
b
c
Figure 7.10

 Solve for the values of the three corner points.Solve for the values of the three corner points.
 PointPoint aa is the intersection of ingredientis the intersection of ingredient
constraints C and B.constraints C and B.
44XX11 + 3+ 3XX22 = 48= 48
XX11 = 3= 3
 Substituting 3 in the first equation, we findSubstituting 3 in the first equation, we find XX22
= 12.= 12.
 Solving for pointSolving for point bb with basic algebra we findwith basic algebra we find
XX11 = 8.4 and= 8.4 and XX22 = 4.8.= 4.8.
 Solving for pointSolving for point cc we findwe find XX11 = 18 and= 18 and XX22 = 0.= 0.

Substituting these value back into theSubstituting these value back into the
objective function we findobjective function we find
CostCost = 2= 2XX11 + 3+ 3XX22
Cost at pointCost at point aa = 2(3) + 3(12) = 42= 2(3) + 3(12) = 42
Cost at pointCost at point bb = 2(8.4) + 3(4.8) = 31.2= 2(8.4) + 3(4.8) = 31.2
Cost at pointCost at point cc = 2(18) + 3(0) = 36= 2(18) + 3(0) = 36
The lowest cost solution is to purchase 8.4The lowest cost solution is to purchase 8.4
pounds of brand 1 feed and 4.8 pounds ofpounds of brand 1 feed and 4.8 pounds of
brand 2 feed for a total cost of 31.2 cents perbrand 2 feed for a total cost of 31.2 cents per
turkey.turkey.

Graphical Solution to the Holiday Meal TurkeyGraphical Solution to the Holiday Meal Turkey
Ranch Problem Using the Isocost ApproachRanch Problem Using the Isocost Approach
–
20 –
15 –
10 –
5 –
0 –
X2
| | | | | |
5 10 15 20 25 X1
PoundsofBrand2
Pounds of Brand 1
Figure 7.11
5454¢ = 2
¢ = 2XX
11 + 3
+ 3XX
22 Isocost Line
Isocost Line
Direction of Decreasing Cost
Direction of Decreasing Cost
31.2¢ = 2X
1 + 3X
2
(X1 = 8.4, X2 = 4.8)

7-67
Solving the Holiday Meal Turkey Ranch ProblemSolving the Holiday Meal Turkey Ranch Problem
Program 7.3

Program 7.4A
Excel 2010 Spreadsheet for the Holiday MealExcel 2010 Spreadsheet for the Holiday Meal
Turkey Ranch problemTurkey Ranch problem

7-69
Program 7.4B
Excel 2010 Solution to the Holiday MealExcel 2010 Solution to the Holiday Meal
Turkey Ranch ProblemTurkey Ranch Problem

7-70
Four Special Cases in LPFour Special Cases in LP
 Four special cases and difficultiesFour special cases and difficulties
arise at times when using thearise at times when using the
graphical approach to solving LPgraphical approach to solving LP
problems.problems.
 No feasible solutionNo feasible solution
 UnboundednessUnboundedness
 RedundancyRedundancy
 Alternate Optimal SolutionsAlternate Optimal Solutions

7-71
No feasible solutionNo feasible solution
 This exists when there is no solution toThis exists when there is no solution to
the problem that satisfies all thethe problem that satisfies all the
constraint equations.constraint equations.
 No feasible solution region exists.No feasible solution region exists.
 This is a common occurrence in the realThis is a common occurrence in the real
world.world.
 Generally one or more constraints areGenerally one or more constraints are
relaxed until a solution is found.relaxed until a solution is found.

A problem with no feasible solutionA problem with no feasible solution
8 –
–
6 –
–
4 –
–
2 –
–
0 –
X2
| | | | | | | | | |
2 4 6 8 X1
Region Satisfying First Two ConstraintsRegion Satisfying First Two Constraints
Figure 7.12
RegionRegion
SatisfyingSatisfying
ThirdThird

7-73
UnboundednessUnboundedness
 Sometimes a linear program will not haveSometimes a linear program will not have
a finite solution.a finite solution.
 In a maximization problem, one or moreIn a maximization problem, one or more
solution variables, and the profit, can besolution variables, and the profit, can be
made infinitely large without violatingmade infinitely large without violating
any constraints.any constraints.
 In a graphical solution, the feasibleIn a graphical solution, the feasible
region will be open ended.region will be open ended.
 This usually means the problem hasThis usually means the problem has
been formulated improperly.been formulated improperly.

A Feasible Region That is UnboundedA Feasible Region That is Unbounded
to the Rightto the Right
15 –
10 –
5 –
0 –
X2
| | | | |
5 10 15 X1
Figure 7.13
XX11 ≥ 5≥ 5
XX22 ≤ 10≤ 10
XX11 + 2+ 2XX22 ≥ 15≥ 15

RedundancyRedundancy
 A redundant constraint is one that doesA redundant constraint is one that does
not affect the feasible solution region.not affect the feasible solution region.
 One or more constraints may be binding.One or more constraints may be binding.
 This is a very common occurrence in theThis is a very common occurrence in the
real world.real world.
 It causes no particular problems, butIt causes no particular problems, but
eliminating redundant constraintseliminating redundant constraints
simplifies the model.simplifies the model.

7-76
Problem with a Redundant ConstraintProblem with a Redundant Constraint
30 –
25 –
20 –
15 –
10 –
5 –
0 –
X2
| | | | | |
5 10 15 20 25 30 X1
Figure 7.14
RedundantRedundant
FeasibleFeasible
RegionRegion
X1 ≤ 25
22XX11 ++ XX22 ≤ 30≤ 30
XX11 ++ XX22 ≤ 20≤ 20

Alternate Optimal SolutionsAlternate Optimal Solutions
 Occasionally two or more optimalOccasionally two or more optimal
solutions may exist.solutions may exist.
 Graphically this occurs when theGraphically this occurs when the
objective function’s isoprofit orobjective function’s isoprofit or
isocost line runs perfectly parallel toisocost line runs perfectly parallel to
one of the constraints.one of the constraints.
 This actually allows management greatThis actually allows management great
flexibility in deciding whichflexibility in deciding which
combination to select as the profit iscombination to select as the profit is
the same at each alternate solution.the same at each alternate solution.

7-78
Example of Alternate Optimal SolutionsExample of Alternate Optimal Solutions
8 –
7 –
6 –
5 –
4 –
3 –
2 –
1 –
0 –
X2
| | | | | | | |
1 2 3 4 5 6 7 8 X1
Figure 7.15 FeasibleFeasible
RegionRegion
Isoprofit Line for $8Isoprofit Line for $8
Optimal Solution Consists of AllOptimal Solution Consists of All
Combinations ofCombinations of XX11 andand XX22 Along theAlong the ABAB
SegmentSegment
Isoprofit Line for $12 OverlaysIsoprofit Line for $12 Overlays
Line SegmentLine Segment ABAB
B
A

7-79
Sensitivity AnalysisSensitivity Analysis
 Optimal solutions to LP problems thus far haveOptimal solutions to LP problems thus far have
been found under what are calledbeen found under what are called deterministicdeterministic
assumptions.assumptions.
 This means that we assume complete certaintyThis means that we assume complete certainty
in the data and relationships of a problem.in the data and relationships of a problem.
 But in the real world, conditions are dynamicBut in the real world, conditions are dynamic
and changing.and changing.
 We can analyze howWe can analyze how sensitivesensitive a deterministica deterministic
solution is to changes in the assumptions of thesolution is to changes in the assumptions of the
model.model.
 This is calledThis is called sensitivity analysissensitivity analysis,, postoptimalitypostoptimality
analysisanalysis,, parametric programmingparametric programming, or, or optimalityoptimality
analysis.analysis.

7-80
Sensitivity AnalysisSensitivity Analysis
 Sensitivity analysis often involves a series ofSensitivity analysis often involves a series of
what-if? questions concerning constraints,what-if? questions concerning constraints,
variable coefficients, and the objective function.variable coefficients, and the objective function.
 One way to do this is the trial-and-error methodOne way to do this is the trial-and-error method
where values are changed and the entire modelwhere values are changed and the entire model
is resolved.is resolved.
 The preferred way is to use an analytic post-The preferred way is to use an analytic post-
optimality analysis.optimality analysis.
 After a problem has been solved, we determine aAfter a problem has been solved, we determine a
range of changes in problem parameters that willrange of changes in problem parameters that will
not affect the optimal solution or change thenot affect the optimal solution or change the
variables in the solution.variables in the solution.

7-81
 The High Note Sound Company manufactures quality CDThe High Note Sound Company manufactures quality CD
players and stereo receivers.players and stereo receivers.
 Products require a certain amount of skilled artisanshipProducts require a certain amount of skilled artisanship
which is in limited supply.which is in limited supply.
 The firm has formulated the following product mix LPThe firm has formulated the following product mix LP
model.model.
High Note Sound CompanyHigh Note Sound Company
Maximize profit =Maximize profit = $50X$50X11 ++
$120X$120X22
Subject toSubject to 2X2X11 + 4X+ 4X22 ≤ 80≤ 80
(hours of(hours of
electrician’s timeelectrician’s time
available)available)
3X3X11 + 1X+ 1X22 ≤ 60≤ 60
(hours of audio(hours of audio

The High Note Sound Company Graphical SolutionThe High Note Sound Company Graphical Solution
b = (16, 12)
a = (0, 20)
Isoprofit Line: $2,400 = 50Isoprofit Line: $2,400 = 50XX11 + 120+ 120XX22
60 –
–
40 –
–
20 –
10 –
0 –
X2
| | | | | |
10 20 30 40 50 60 X1
(receivers)
(CD players)c = (20, 0)
Figure 7.16

7-83
Changes in theChanges in the
Objective Function CoefficientObjective Function Coefficient
 In real-life problems, contribution rates in theIn real-life problems, contribution rates in the
objective functions fluctuate periodically.objective functions fluctuate periodically.
 Graphically, this means that although the feasibleGraphically, this means that although the feasible
solution region remains exactly the same, thesolution region remains exactly the same, the
slope of the isoprofit or isocost line will change.slope of the isoprofit or isocost line will change.
 We can often make modest increases orWe can often make modest increases or
decreases in the objective function coefficient ofdecreases in the objective function coefficient of
any variable without changing the current optimalany variable without changing the current optimal
corner point.corner point.
 We need to know how much an objective functionWe need to know how much an objective function
coefficient can change before the optimal solutioncoefficient can change before the optimal solution
would be at a different corner point.would be at a different corner point.

7-84
Objective Function CoefficientObjective Function Coefficient
Changes in the Receiver Contribution CoefficientsChanges in the Receiver Contribution Coefficients
b
a
Profit Line for 50Profit Line for 50XX11 + 80+ 80XX22
(Passes through Point(Passes through Point bb))
40 –
30 –
20 –
10 –
0 –
X2
| | | | | |
10 20 30 40 50 60 X1
c
Figure 7.17
Old Profit Line for 50Old Profit Line for 50XX11 + 120+ 120XX22
(Passes through Point(Passes through Point aa))
Profit Line for 50Profit Line for 50XX11 + 150+ 150XX22
(Passes through Point(Passes through Point aa))

7-85
QM for Windows and Changes inQM for Windows and Changes in
Objective Function CoefficientsObjective Function Coefficients
Input and Sensitivity Analysis for High Note SoundInput and Sensitivity Analysis for High Note Sound
Data Using QM For WindowsData Using QM For Windows
Program 7.5B
Program 7.5A

Excel Solver and Changes inExcel Solver and Changes in
Excel 2010 Spreadsheet for High Note Sound CompanyExcel 2010 Spreadsheet for High Note Sound Company
Program 7.6A

7-87
Excel 2010 Solution and Solver Results
Window for High Note Sound Company
Figure 7.6B

7-88
Excel 2010 Sensitivity Report for High NoteExcel 2010 Sensitivity Report for High Note
Sound CompanySound Company
Program 7.6C

7-89
Technological CoefficientsTechnological Coefficients
 Changes in theChanges in the technological coefficientstechnological coefficients
often reflect changes in the state ofoften reflect changes in the state of
technology.technology.
 If the amount of resources needed toIf the amount of resources needed to
produce a product changes, coefficients inproduce a product changes, coefficients in
the constraint equations will change.the constraint equations will change.
 This does not change the objectiveThis does not change the objective
function, but it can produce a significantfunction, but it can produce a significant
change in the shape of the feasible region.change in the shape of the feasible region.
 This may cause a change in the optimalThis may cause a change in the optimal
solution.solution.

7-90
Technological CoefficientsTechnological Coefficients
Change in the Technological Coefficients for theChange in the Technological Coefficients for the
(a) Original Problem
3X1 + 1X2 ≤ 60
2X1 + 4X2 ≤ 80
Optimal
Solution
X2
60 –
40 –
20 –
–
| | |
0 20 40 X1
StereoReceivers
CD Players
(b) Change in Circled
Coefficient
2 X1 + 1X2 ≤ 60
2X1 + 4X2 ≤ 80
Still
Optimal
3X1 + 1X2 ≤ 60
2X1 + 5 X2 ≤ 80
Optimal
Solutiona
d
e
60 –
40 –
20 –
–
| | |
0 20 40
X2
X1
16
60 –
40 –
20 –
–
| | |
0 20 40
X2
X1
|
30
(c) Change in Circled
Coefficient
a
b
c
f
g
c
Figure 7.18

7-91
Changes in Resources orChanges in Resources or
Right-Hand-Side ValuesRight-Hand-Side Values
 The right-hand-side values of theThe right-hand-side values of the
constraints often represent resourcesconstraints often represent resources
available to the firm.available to the firm.
 If additional resources were available, aIf additional resources were available, a
higher total profit could be realized.higher total profit could be realized.
 Sensitivity analysis about resources willSensitivity analysis about resources will
help answer questions about how muchhelp answer questions about how much
should be paid for additional resourcesshould be paid for additional resources
and how much more of a resource wouldand how much more of a resource would
be useful.be useful.

Changes in Resources or Right-Changes in Resources or Right-
Hand-Side ValuesHand-Side Values
 If the right-hand side of a constraint is changed,If the right-hand side of a constraint is changed,
the feasible region will change (unless thethe feasible region will change (unless the
constraint is redundant).constraint is redundant).
 Often the optimal solution will change.Often the optimal solution will change.
 The amount of change in the objective functionThe amount of change in the objective function
value that results from a unit change in one of thevalue that results from a unit change in one of the
resources available is called theresources available is called the dual pricedual price oror dualdual
valuevalue ..
 The dual price for a constraint is the improvementThe dual price for a constraint is the improvement
in the objective function value that results from ain the objective function value that results from a
one-unit increase in the right-hand side of theone-unit increase in the right-hand side of the
constraint.constraint.

7-93
Changes in Resources orChanges in Resources or
 However, the amount of possible increase inHowever, the amount of possible increase in
the right-hand side of a resource is limited.the right-hand side of a resource is limited.
 If the number of hours increased beyond theIf the number of hours increased beyond the
upper bound, then the objective functionupper bound, then the objective function
would no longer increase by the dual price.would no longer increase by the dual price.
 There would simply be excess (There would simply be excess (slackslack) hours) hours
of a resource or the objective function mayof a resource or the objective function may
change by an amount different from the dualchange by an amount different from the dual
price.price.
 The dual price is relevant only within limits.The dual price is relevant only within limits.

7-94
Changes in the Electricians’ Time ResourceChanges in the Electricians’ Time Resource
for the High Note Sound Companyfor the High Note Sound Company
60 –
40 –
20 –
–
25 –
| | |
0 20 40 60
|
50 X1
X2 (a)
a
b
c
Constraint Representing 60 Hours of AudioConstraint Representing 60 Hours of Audio
Technician’s Time ResourceTechnician’s Time Resource
Changed Constraint RepresentingChanged Constraint Representing 100100
Hours of Electrician’s Time ResourceHours of Electrician’s Time Resource
Figure 7.19

7-95
Changes in the Electricians’ Time ResourceChanges in the Electricians’ Time Resource
for the High Note Sound Companyfor the High Note Sound Company
60 –
40 –
20 –
–
15 –
| | |
0 20 40 60
|
30 X1
X2 (b)
a
b
c
Constraint Representing 60 Hours of AudioConstraint Representing 60 Hours of Audio
Technician’s Time ResourceTechnician’s Time Resource
Changed Constraint RepresentingChanged Constraint Representing 6060
Hours of Electrician’s Time ResourceHours of Electrician’s Time Resource
Figure 7.19

Changes in the Electricians’ TimeChanges in the Electricians’ Time
Resource for the High Note SoundResource for the High Note Sound
CompanyCompany
60 –
40 –
20 –
–
| | | | | |
0 20 40 60 80 100 120
X1
X2 (c)
Constraint RepresentingConstraint Representing
60 Hours of Audio Technician’s60 Hours of Audio Technician’s
Time ResourceTime Resource
Changed Constraint RepresentingChanged Constraint Representing
240240 Hours of Electrician’s TimeHours of Electrician’s Time
ResourceResource
Figure 7.19

7-97
QM for Windows and ChangesQM for Windows and Changes
in Right-Hand-Side Valuesin Right-Hand-Side Values
Sensitivity Analysis for High Note Sound CompanySensitivity Analysis for High Note Sound Company
Program 7.5B

7-98
Excel 2010 Sensitivity Analysis for High NoteExcel 2010 Sensitivity Analysis for High Note
Sound CompanySound Company
Program 7.6C

TutorialTutorial
Lab Practical : SpreadsheetLab Practical : Spreadsheet
1 - 99

Further ReadingFurther Reading
 Render, B., Stair Jr.,R.M. & Hanna, M.E.
(2013) Quantitative Analysis for
Management, Pearson, 11th
Edition
 Waters, Donald (2007) Quantitative
Methods for Business, Prentice Hall, 4th
Edition.
 Anderson D, Sweeney D, & Williams T.
(2006) Quantitative Methods For
Business Thompson Higher Education,
10th Ed.

Tbs910 linear programming

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