1. 2 chapter 8 strategic allocation of resources

1 1 2002 South-Western/Thomson Learning 2002 South-Western/Thomson Learning TMTM

Slides preparedSlides preparedby John Loucksby John Loucks

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Chapter 8Chapter 8Chapter 8Chapter 8

Strategic AllocationStrategic Allocation

of Resourcesof Resources

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OverviewOverviewOverviewOverview

IntroductionIntroduction Recognizing Linear Programming (LP) ProblemsRecognizing Linear Programming (LP) Problems Formulating LP ProblemsFormulating LP Problems Solving LP ProblemsSolving LP Problems Real LP ProblemsReal LP Problems Interpreting Computer Solutions of LP ProblemsInterpreting Computer Solutions of LP Problems Wrap-Up: What World-Class Companies DoWrap-Up: What World-Class Companies Do

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IntroductionIntroductionIntroductionIntroduction

Today many of the resources needed as inputs to Today many of the resources needed as inputs to operations are in limited supply.operations are in limited supply.

Operations managers must understand the impact of Operations managers must understand the impact of this situation on meeting their objectives.this situation on meeting their objectives.

Linear programming (LP) is one way that operations Linear programming (LP) is one way that operations managers can determine how best to allocate their managers can determine how best to allocate their scarce resourcesscarce resources..

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Linear Programming (LP) in OMLinear Programming (LP) in OMLinear Programming (LP) in OMLinear Programming (LP) in OM

There are five common types of decisions in which There are five common types of decisions in which LP may play a roleLP may play a role Product mixProduct mix Ingredient mixIngredient mix TransportationTransportation Production planProduction plan AssignmentAssignment

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LP Problems in OM: Product MixLP Problems in OM: Product MixLP Problems in OM: Product MixLP Problems in OM: Product Mix

ObjectiveObjectiveTo select the mix of products or services that results To select the mix of products or services that results in maximum profits for the planning periodin maximum profits for the planning period

Decision VariablesDecision VariablesHow much to produce and market of each product or How much to produce and market of each product or service for the planning periodservice for the planning period

ConstraintsConstraintsMaximum amount of each product or service Maximum amount of each product or service demanded; Minimum amount of product or service demanded; Minimum amount of product or service policy will allow; Maximum amount of resources policy will allow; Maximum amount of resources availableavailable

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LP Problems in OM: Ingredient MixLP Problems in OM: Ingredient MixLP Problems in OM: Ingredient MixLP Problems in OM: Ingredient Mix

ObjectiveObjectiveTo select the mix of ingredients going into products To select the mix of ingredients going into products that results in minimum operating costs for the that results in minimum operating costs for the planning periodplanning period

Decision VariablesDecision VariablesHow much of each ingredient to use in the planning How much of each ingredient to use in the planning periodperiod

ConstraintsConstraintsAmount of products demanded; Relationship Amount of products demanded; Relationship between ingredients and products; Maximum amount between ingredients and products; Maximum amount of ingredients and production capacity availableof ingredients and production capacity available

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LP Problems in OM: TransportationLP Problems in OM: TransportationLP Problems in OM: TransportationLP Problems in OM: Transportation

ObjectiveObjectiveTo select the distribution plan from sources to To select the distribution plan from sources to destinations that results in minimum shipping costs destinations that results in minimum shipping costs for the planning periodfor the planning period

Decision VariablesDecision VariablesHow much product to ship from each source to each How much product to ship from each source to each destination for the planning perioddestination for the planning period

ConstraintsConstraintsMinimum or exact amount of products needed at each Minimum or exact amount of products needed at each destination; Maximum or exact amount of products destination; Maximum or exact amount of products available at each sourceavailable at each source

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LP Problems in OM: Production PlanLP Problems in OM: Production PlanLP Problems in OM: Production PlanLP Problems in OM: Production Plan

ObjectiveObjectiveTo select the mix of products or services that results To select the mix of products or services that results in maximum profits for the planning periodin maximum profits for the planning period

Decision VariablesDecision VariablesHow much to produce on straight-time labor and How much to produce on straight-time labor and overtime labor during each month of the yearovertime labor during each month of the year

ConstraintsConstraintsAmount of products demanded in each month; Amount of products demanded in each month; Maximum labor and machine capacity available in Maximum labor and machine capacity available in each month; Maximum inventory space available in each month; Maximum inventory space available in each montheach month

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LP Problems in OM: AssignmentLP Problems in OM: AssignmentLP Problems in OM: AssignmentLP Problems in OM: Assignment

ObjectiveObjectiveTo assign projects to teams so that the total cost for To assign projects to teams so that the total cost for all projects is minimized during the planning periodall projects is minimized during the planning period

Decision VariablesDecision VariablesTo which team is each project assignedTo which team is each project assigned

ConstraintsConstraintsEach project must be assigned to a team; Each team Each project must be assigned to a team; Each team must be assigned a projectmust be assigned a project

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Recognizing LP ProblemsRecognizing LP ProblemsRecognizing LP ProblemsRecognizing LP Problems

Characteristics of LP Problems in OMCharacteristics of LP Problems in OM A well-defined single objective must be stated.A well-defined single objective must be stated. There must be alternative courses of action.There must be alternative courses of action. The total achievement of the objective must be The total achievement of the objective must be

constrained by scarce resources or other restraints.constrained by scarce resources or other restraints. The objective and each of the constraints must be The objective and each of the constraints must be

expressed as linear mathematical functions.expressed as linear mathematical functions.

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Steps in Formulating LP ProblemsSteps in Formulating LP ProblemsSteps in Formulating LP ProblemsSteps in Formulating LP Problems

1.1. Define the objectiveDefine the objective..

2.2. Define the decision variables. Define the decision variables.

3.3. Write the mathematical function for the objective. Write the mathematical function for the objective.

4.4. Write a 1- or 2-word description of each constraint. Write a 1- or 2-word description of each constraint.

5.5. Write the right-hand side (RHS) of each constraint. Write the right-hand side (RHS) of each constraint.

6.6. Write Write <<, =, or , =, or >> for each constraint. for each constraint.

7.7. Write the decision variables on LHS of each constraint. Write the decision variables on LHS of each constraint.

8.8. Write the coefficient for each decision variable in each Write the coefficient for each decision variable in each constraint.constraint.

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Cycle Trends is introducing two new lightweight bicycle frames, the Deluxe and the Professional, to be made from aluminum and steel alloys. The anticipated unit profits are $10 for the Deluxe and $15 for the Professional.

The number of pounds of each alloy needed per frame is summarized on the next slide. A supplier delivers 100 pounds of the aluminum alloy and 80 pounds of the steel alloy weekly. How many Deluxe and Professional frames should Cycle Trends produce each week?

Example: LP FormulationExample: LP FormulationExample: LP FormulationExample: LP Formulation

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Aluminum AlloyAluminum Alloy Steel AlloySteel Alloy

DeluxeDeluxe 2 3 2 3

Professional Professional 4 4 22


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Define the objective Maximize total weekly profit

Define the decision variables x1 = number of Deluxe frames produced weekly

x2 = number of Professional frames produced weekly

Write the mathematical objective function Max Z = 10x1 + 15x2

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Write a one- or two-word description of each constraint Aluminum available Steel available

Write the right-hand side of each constraint 100 80

Write <, =, > for each constraint < 100 < 80

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Write all the decision variables on the left-hand side Write all the decision variables on the left-hand side of each constraintof each constraint x1 x2 < 100

x1 x2 < 80

Write the coefficient for each decision in each constraint

+ 2x1 + 4x2 < 100

+ 3x1 + 2x2 < 80

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LP Problems in GeneralLP Problems in GeneralLP Problems in GeneralLP Problems in General

Units of each term in a constraint must be the same as Units of each term in a constraint must be the same as the RHSthe RHS

Units of each term in the objective function must be Units of each term in the objective function must be the same as Zthe same as Z

Units between constraints do not have to be the sameUnits between constraints do not have to be the same LP problem can have a mixture of constraint typesLP problem can have a mixture of constraint types

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Solving LP ProblemsSolving LP ProblemsSolving LP ProblemsSolving LP Problems

Graphical Solution ApproachGraphical Solution Approach - used mainly as a - used mainly as a teaching toolteaching tool

Simplex MethodSimplex Method - most common analytic tool - most common analytic tool Transportation MethodTransportation Method - one of the earliest methods - one of the earliest methods Assignment MethodAssignment Method - occasionally used in OM - occasionally used in OM

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Graphical Solution ApproachGraphical Solution ApproachGraphical Solution ApproachGraphical Solution Approach

1.1. Formulate the objective and constraint functions. Formulate the objective and constraint functions.

2.2. Draw the graph with one variable on the horizontal Draw the graph with one variable on the horizontal axis and one on the vertical axis.axis and one on the vertical axis.

3.3. Plot each of the constraints as if they were lines or Plot each of the constraints as if they were lines or equalities.equalities.

4.4. Outline the feasible solution space. Outline the feasible solution space.

. . . more. . . more

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Graphic Solution ApproachGraphic Solution ApproachGraphic Solution ApproachGraphic Solution Approach

5.5. Circle the potential solution points -- intersections on Circle the potential solution points -- intersections on the inner (minimization) or outer (maximization) the inner (minimization) or outer (maximization) perimeter of the feasible solution space.perimeter of the feasible solution space.

6.6. Substitute each of the potential solution point values Substitute each of the potential solution point values of the two decision variables into the objective of the two decision variables into the objective function and solve for Z.function and solve for Z.

7.7. Select the solution point that optimizes Z. Select the solution point that optimizes Z.

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Example: Graphical Solution ApproachExample: Graphical Solution Approach

4040

3535

3030

2525

2020

1515

1010

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5 10 15 20 25 30 35 40 45 505 10 15 20 25 30 35 40 45 50

xx22

x1

MAX 10xMAX 10x11 + 15x + 15x22 (Profit) (Profit)

2x2x11 + 4x + 4x2 2 << 100 (Aluminum) 100 (Aluminum)

Optimal Solution PointOptimal Solution Pointxx11 = 15, x = 15, x22 = 17.5, Z = $412.50 = 17.5, Z = $412.50

3x3x11 + 2x + 2x2 2 << 80 (Steel) 80 (Steel)

FeasibleFeasibleSolutionSolutionSpaceSpace

Potential Solution PointPotential Solution Pointxx11 = 0, x = 0, x22 = 25, Z = $375.00 = 25, Z = $375.00

Potential Solution PointPotential Solution Pointxx11 = 26.67, x = 26.67, x22 = 0, Z = $266.67 = 0, Z = $266.67

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Simplex MethodSimplex MethodSimplex MethodSimplex Method

Examples of standard LP computer programs that useExamples of standard LP computer programs that usethe simplex method are:the simplex method are: IBM’s IBM’s Optimization Solutions LibraryOptimization Solutions Library POM Software LibraryPOM Software Library GAMSGAMS MPL for WindowsMPL for Windows SolverSolver -- available within spreadsheet packages such -- available within spreadsheet packages such

as Microsoftas MicrosoftExcel, Lotus 1-2-3, and Quattro ProExcel, Lotus 1-2-3, and Quattro Pro What’s Best!What’s Best! and and Premium SolverPremium Solver -- add-ons to -- add-ons to

common spreadsheet packagescommon spreadsheet packages

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Example: Excel/Solver SolutionExample: Excel/Solver SolutionExample: Excel/Solver SolutionExample: Excel/Solver Solution

Partial Spreadsheet Showing Problem Data

A B C D1 Amount2 Material Deluxe Profess. Available3 Aluminum 2 4 1004 Steel 3 2 805 Profit/Bike 10 15

Material Requirements

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Partial Spreadsheet Showing Formulas

A B C D67 Deluxe Professional8 Bikes Made9

10 =B5*C10+C5*D101112 Constraints Amount Used Amount Avail.13 Aluminum =B3*B8+C3*C8 <= 10014 Steel =B4*B8+C4*C8 <= 80

Decision Variables

Maximized Total Profit

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Partial Spreadsheet Showing Solution

A B C D67 Deluxe Professional8 Bikes Made 15 17.5009

10 412.5001112 Constraints Amount Used Amount Avail.13 Aluminum 100 <= 10014 Steel 80 <= 80

Decision Variables

Maximized Total Profit

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Transportation MethodTransportation MethodTransportation MethodTransportation Method

This method can solve a special form of LP problem, This method can solve a special form of LP problem, including the classical including the classical transportation problemtransportation problem, with , with these typical characteristics:these typical characteristics: mm sources and sources and nn destinations destinations number of variables is m x nnumber of variables is m x n number of constraints is m + n (constraints are for number of constraints is m + n (constraints are for

source capacity and destination demand)source capacity and destination demand) costs appear only in objective function (objective costs appear only in objective function (objective

is to minimize total cost of shipping)is to minimize total cost of shipping) coefficients of decision variables in the constraints coefficients of decision variables in the constraints

are either 0 or 1are either 0 or 1

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Example: Transportation LPExample: Transportation LPExample: Transportation LPExample: Transportation LP

National Packaging has plants in Tulsa, National Packaging has plants in Tulsa, Memphis, and Detroit that ship to the firm’s Memphis, and Detroit that ship to the firm’s warehouses in San Diego, Norfolk, and Pensacola. warehouses in San Diego, Norfolk, and Pensacola. The three warehouses require at least 4,000, 2,500, The three warehouses require at least 4,000, 2,500, and 2,500 pounds of cardboard per week, and 2,500 pounds of cardboard per week, respectively. The plants each have 3,000 pounds of respectively. The plants each have 3,000 pounds of cardboard per week available for shipment. cardboard per week available for shipment.

The shipping cost per pound from each plant to The shipping cost per pound from each plant to each warehouse is shown on the next slide. each warehouse is shown on the next slide. Formulate and solve an LP to determine the shipping Formulate and solve an LP to determine the shipping arrangements (source, destination, and quantity) that arrangements (source, destination, and quantity) that will minimize the total weekly shipping cost.will minimize the total weekly shipping cost.

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DestinationDestination

SourceSource San DiegoSan Diego NorfolkNorfolk PensacolaPensacola

Tulsa $12 Tulsa $12 $ 6 $ 5$ 6 $ 5

Memphis 20 11Memphis 20 11 9 9

DetroitDetroit 30 26 28 30 26 28

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Define the objectiveDefine the objective

Minimize the total weekly shipping costMinimize the total weekly shipping cost Define the decision variablesDefine the decision variables

There are m x n = 3 x 3 = 9 decision variables.There are m x n = 3 x 3 = 9 decision variables.

xxijij represents the number of pounds of cardboard represents the number of pounds of cardboard

to be shipped from plant to be shipped from plant ii to warehouse to warehouse jj..

San DiegoSan Diego NorfolkNorfolk PensacolaPensacola Tulsa Tulsa xx1111 xx1212 xx1313

MemphisMemphis xx2121 xx2222 xx2323

DetroitDetroit xx3131 xx3232 xx3333

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Write the mathematical function for the objectiveWrite the mathematical function for the objective

Min Z = 12Min Z = 12xx1111 + 6 + 6xx1212 + 5 + 5xx1313 + 20 + 20xx2121 + 11 + 11xx2222

+ 9+ 9xx2323 + 30 + 30xx3131 + 26 + 26xx3232 + 28 + 28xx3333

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Write the constraintsWrite the constraints

(1) (1) xx1111 + + xx1212 + + xx1313 << 3000 (Plant 1 capacity in pounds) 3000 (Plant 1 capacity in pounds)



(4) (4) xx1111 + + xx2121 + + xx3131 >> 4000 (Wareh. 1 demand in pounds) 4000 (Wareh. 1 demand in pounds)



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Assignment MethodAssignment MethodAssignment MethodAssignment Method

This method can solve a special form of LP problem, This method can solve a special form of LP problem, including the classical including the classical assignment problemassignment problem, with , with these typical characteristics:these typical characteristics: is a special case of a transportation problemis a special case of a transportation problem the right-hand sides of constraints are all 1the right-hand sides of constraints are all 1 the signs of the constraints are = rather than the signs of the constraints are = rather than << or or >> the value of all decision variables is either 0 or 1the value of all decision variables is either 0 or 1

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Example: Assignment LPExample: Assignment LPExample: Assignment LPExample: Assignment LP

A sprinkler system installation company has A sprinkler system installation company has three residential projects to complete and three work three residential projects to complete and three work teams available and capable of completing the teams available and capable of completing the projects. The work teams are not equally efficient at projects. The work teams are not equally efficient at completing a particular project. completing a particular project.

Shown on the next slide are the estimated Shown on the next slide are the estimated labor-hours required for each team to complete each labor-hours required for each team to complete each project. How should the work teams be assigned in project. How should the work teams be assigned in order to minimize total labor-hours?order to minimize total labor-hours?

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Example: Assignment LPExample: Assignment LPExample: Assignment LPExample: Assignment LP

ProjectsProjects

Work TeamWork Team AA BB CC

Alice, Ted Alice, Ted 28 30 18 28 30 18

Gary, MarvGary, Marv 35 32 2035 32 20

Tina, Sam Tina, Sam 25 25 1425 25 14

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Real LP ProblemsReal LP ProblemsReal LP ProblemsReal LP Problems

Real-world LP problems often involve:Real-world LP problems often involve: Hundreds or thousands of constraintsHundreds or thousands of constraints Large quantities of dataLarge quantities of data Many products and/or servicesMany products and/or services Many time periodsMany time periods Numerous decision alternativesNumerous decision alternatives … … and other complicationsand other complications

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Computer Solutions of LP ProblemsComputer Solutions of LP ProblemsComputer Solutions of LP ProblemsComputer Solutions of LP Problems

Computer output typically contains:Computer output typically contains: Original-problem formulationOriginal-problem formulation Basis variables (variables in the final solution) and Basis variables (variables in the final solution) and

their valuestheir values Slack-variable values -- slack represents the amount Slack-variable values -- slack represents the amount

of a scarce resource that is not used by the decision of a scarce resource that is not used by the decision variablesvariables

Shadow prices -- which indicate the impact on Z if Shadow prices -- which indicate the impact on Z if the constraints’ right-hand sides are changedthe constraints’ right-hand sides are changed

… … moremore

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Computer Solutions of LP ProblemsComputer Solutions of LP ProblemsComputer Solutions of LP ProblemsComputer Solutions of LP Problems

Computer output typically contains:Computer output typically contains: Ranges of right-hand side values for which the Ranges of right-hand side values for which the

shadow prices are validshadow prices are valid For each nonbasic variable: amount Z is reduced (in a For each nonbasic variable: amount Z is reduced (in a

MAX problem) or increased (in a MIN problem) for MAX problem) or increased (in a MIN problem) for one unit of x in the solutionone unit of x in the solution

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Wrap-Up: World-Class PracticeWrap-Up: World-Class PracticeWrap-Up: World-Class PracticeWrap-Up: World-Class Practice

Managers at all levels use LP to solve complex Managers at all levels use LP to solve complex problems and aid in decision makingproblems and aid in decision making Some companies establish formal operations Some companies establish formal operations

research, management science, or operations research, management science, or operations analysis departmentsanalysis departments

Other companies employ consultantsOther companies employ consultants LP is applied to long, medium, and short range LP is applied to long, medium, and short range

decision makingdecision making

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End of Chapter 8End of Chapter 8

1. 2 chapter 8 strategic allocation of resources

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