1 1 Cap1 QUANTITATIVE METHODS FOR BUSINESS FOR BUSINESS Seventh Edition ANDERSON SWEENEY WILLIAMS SLIDES PREPARED BY JOHN LOUCKS © 1998 South-Western College

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QUANTITATIVE METHODSQUANTITATIVE METHODS FOR BUSINESSFOR BUSINESS

Seventh EditionSeventh Edition

ANDERSON SWEENEY ANDERSON SWEENEY WILLIAMSWILLIAMS

SLIDES PREPARED BY JOHN LOUCKS

© 1998 South-Western College Publishing© 1998 South-Western College Publishing

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Chapter 1Chapter 1IntroductionIntroduction

Problem Solving and Decision MakingProblem Solving and Decision Making Quantitative AnalysisQuantitative Analysis Models of Cost, Revenue, and ProfitModels of Cost, Revenue, and Profit The Management ScientistThe Management Scientist

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Body of KnowledgeBody of Knowledge

Quantitative approaches to decision making Quantitative approaches to decision making based on the scientific method.based on the scientific method.

Names for this body of knowledge include:Names for this body of knowledge include: Management ScienceManagement Science, , Operations ResearchOperations Research, , andand Decision Science Decision Science..

It had its early roots in World War II and is It had its early roots in World War II and is flourishing in business and industry with the aid flourishing in business and industry with the aid of computers in general and the microcomputer of computers in general and the microcomputer in particular.in particular.

Some of the primary Some of the primary applications areasapplications areas of this of this body of knowledge are forecasting, production body of knowledge are forecasting, production scheduling, inventory control, capital scheduling, inventory control, capital budgeting, and transportation.budgeting, and transportation.

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Problem Solving and Decision MakingProblem Solving and Decision Making

7 Steps of 7 Steps of Problem SolvingProblem Solving

(First 5 steps are the process of (First 5 steps are the process of decision makingdecision making))• Identify and define the problem.Identify and define the problem.• Determine the set of alternative solutions.Determine the set of alternative solutions.• Determine the criteria for evaluating the Determine the criteria for evaluating the

alternatives.alternatives.• Evaluate the alternatives.Evaluate the alternatives.• Choose an alternative.Choose an alternative.

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• Implement the chosen alternative.Implement the chosen alternative.• Evaluate the results.Evaluate the results.

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Quantitative Analysis and Decision MakingQuantitative Analysis and Decision Making

Potential Reasons for a Quantitative Analysis Potential Reasons for a Quantitative Analysis Approach to Decision MakingApproach to Decision Making• The problem is complex.The problem is complex.• The problem is very important.The problem is very important.• The problem is new.The problem is new.• The problem is repetitive.The problem is repetitive.

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Quantitative AnalysisQuantitative Analysis

Quantitative Analysis ProcessQuantitative Analysis Process• Model DevelopmentModel Development• Data PreparationData Preparation• Model SolutionModel Solution• Report GenerationReport Generation

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Model DevelopmentModel Development

ModelsModels are representations of real objects or are representations of real objects or situations.situations.

Three Three forms of modelsforms of models are iconic, analog, and are iconic, analog, and mathematical. mathematical. • Iconic modelsIconic models are physical replicas (scalar are physical replicas (scalar

representations) of real objects. representations) of real objects. • Analog modelsAnalog models are physical in form, but do not are physical in form, but do not

physically resemble the object being modeled.physically resemble the object being modeled.• Mathematical modelsMathematical models represent real world represent real world

problems through a system of mathematical problems through a system of mathematical formulas and expressions based on key formulas and expressions based on key assumptions, estimates, or statistical analyses.assumptions, estimates, or statistical analyses.

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Mathematical ModelsMathematical Models

Cost/benefit considerations must be made in Cost/benefit considerations must be made in selecting an appropriate mathematical model. selecting an appropriate mathematical model.

Frequently a less complicated (and perhaps Frequently a less complicated (and perhaps less precise) model is more appropriate than a less precise) model is more appropriate than a more complex and accurate one due to cost more complex and accurate one due to cost and ease of solution considerations.and ease of solution considerations.

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Mathematical ModelsMathematical Models

Mathematical models relate Mathematical models relate decision variablesdecision variables (or (or controllable inputs) with fixed or variable parameters controllable inputs) with fixed or variable parameters (or uncontrollable inputs). (or uncontrollable inputs).

Frequently mathematical models seek to maximize Frequently mathematical models seek to maximize or minimize some or minimize some objective functionobjective function subject to subject to constraints.constraints.

If any of the uncontrollable inputs is subject to If any of the uncontrollable inputs is subject to variation the model is said to be variation the model is said to be stochasticstochastic; ; otherwise the model is said to be otherwise the model is said to be deterministicdeterministic. .

Generally, stochastic models are more difficult to Generally, stochastic models are more difficult to analyze. analyze.

The values of the decision variables that provide the The values of the decision variables that provide the mathematically-best output are referred to as the mathematically-best output are referred to as the optimal solutionoptimal solution for the model. for the model.

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Example: Project SchedulingExample: Project Scheduling

Consider a construction company building Consider a construction company building a 250-unit apartment complex. The project a 250-unit apartment complex. The project consists of hundreds of activities involving consists of hundreds of activities involving excavating, framing, wiring, plastering, painting, excavating, framing, wiring, plastering, painting, landscaping, and more. Some of the activities landscaping, and more. Some of the activities must be done sequentially and others can be must be done sequentially and others can be done simultaneously. Also, some of the done simultaneously. Also, some of the activities can be completed faster than normal activities can be completed faster than normal by purchasing additional resources (workers, by purchasing additional resources (workers, equipment, etc.). equipment, etc.).

What is the best schedule for the activities What is the best schedule for the activities and for which activities should additional and for which activities should additional resources be purchased?resources be purchased?

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Question: Question:

How could management science be used How could management science be used to solve this problem?to solve this problem?

Answer:Answer:

Management science can provide a Management science can provide a structured, quantitative approach for structured, quantitative approach for determining the minimum project completion determining the minimum project completion time based on the activities' normal times and time based on the activities' normal times and then based on the activities' expedited (reduced) then based on the activities' expedited (reduced) times.times.

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Question:Question:

What would be the uncontrollable What would be the uncontrollable inputs?inputs?

Answer:Answer:• Normal and expedited activity completion Normal and expedited activity completion

timestimes• Activity expediting costsActivity expediting costs• Funds available for expeditingFunds available for expediting• Precedence relationships of the activitiesPrecedence relationships of the activities

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Question:Question:

What would be the decision variables of the What would be the decision variables of the mathematical model? The objective function? mathematical model? The objective function? The constraints?The constraints?

Answer:Answer:• Decision variables: which activities to expedite Decision variables: which activities to expedite

and by how much, and when to start each and by how much, and when to start each activityactivity

• Objective function: minimize project Objective function: minimize project completion timecompletion time

• Constraints: do not violate any activity Constraints: do not violate any activity precedence relationships and do not expedite precedence relationships and do not expedite in excess of the funds available.in excess of the funds available.

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Question: Question:

Is the model deterministic or stochastic?Is the model deterministic or stochastic? Answer:Answer:

Stochastic. Activity completion times, Stochastic. Activity completion times, both normal and expedited, are uncertain and both normal and expedited, are uncertain and subject to variation. Activity expediting costs subject to variation. Activity expediting costs are uncertain. The number of activities and are uncertain. The number of activities and their precedence relationships might change their precedence relationships might change before the project is completed due to a before the project is completed due to a project design change. project design change.

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Question: Question:

Suggest assumptions that could be made Suggest assumptions that could be made to simplify the model.to simplify the model.

Answer:Answer:

Make the model deterministic by assuming Make the model deterministic by assuming normal and expedited activity times are known normal and expedited activity times are known with certainty and are constant. The same with certainty and are constant. The same assumption might be made about the other assumption might be made about the other stochastic, uncontrollable inputs.stochastic, uncontrollable inputs.

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Example: Austin Auto AuctionExample: Austin Auto Auction

An auctioneer has developed a simple An auctioneer has developed a simple mathematical model for deciding the starting bid mathematical model for deciding the starting bid he will require when auctioning a used he will require when auctioning a used automobile. automobile. Essentially, he sets the Essentially, he sets the starting bid at seventy percent of what he starting bid at seventy percent of what he predicts the final winning bid will (or should) be. predicts the final winning bid will (or should) be. He predicts the winning bid by starting with the He predicts the winning bid by starting with the car's original selling price and making two car's original selling price and making two deductions, one based on the car's age and the deductions, one based on the car's age and the other based on the car's mileage. other based on the car's mileage.

The age deduction is $800 per year and The age deduction is $800 per year and the mileage deduction is $.025 per mile.the mileage deduction is $.025 per mile.

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Question: Question:

Develop the mathematical model that will give Develop the mathematical model that will give the starting bid (the starting bid (BB) for a car in terms of the car's ) for a car in terms of the car's original price (original price (PP), current age (), current age (AA) and mileage () and mileage (MM). ).

Answer:Answer:

The expected winning bid can be expressed The expected winning bid can be expressed as:as:

PP - 800( - 800(AA) - .025() - .025(MM))

The entire model is:The entire model is:

BB = .7(expected winning bid) or = .7(expected winning bid) or

BB = .7( = .7(PP - 800( - 800(AA) - .025() - .025(MM)) or)) or

BB = .7( = .7(PP) - 560() - 560(AA) - .0175() - .0175(MM) )

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Question: Question:

Suppose a four-year old car with 60,000 Suppose a four-year old car with 60,000 miles on the odometer is up for auction. If its miles on the odometer is up for auction. If its original price was $12,500, what starting bid original price was $12,500, what starting bid should the auctioneer require? should the auctioneer require?

Answer:Answer:

BB = .7(12,500) - 560(4) - .0175(60,000) = = .7(12,500) - 560(4) - .0175(60,000) = $5460.$5460.

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Question:Question:

The model is based on what assumptions? The model is based on what assumptions? Answer:Answer:

The model assumes that the only factors The model assumes that the only factors influencing the value of a used car are the influencing the value of a used car are the original price, age, and mileage (not condition, original price, age, and mileage (not condition, rarity, or other factors). rarity, or other factors).

Also, it is assumed that age and mileage Also, it is assumed that age and mileage devalue a car in a linear manner and without devalue a car in a linear manner and without limit. (Note, the starting bid for a very old car limit. (Note, the starting bid for a very old car might be negative!)might be negative!)

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Example: Iron Works, Inc.Example: Iron Works, Inc.

Iron Works, Inc. (IWI) manufactures two Iron Works, Inc. (IWI) manufactures two products made from steel and just received this products made from steel and just received this month's allocation of month's allocation of bb pounds of steel. It takes pounds of steel. It takes aa11 pounds of steel to make a unit of product 1 and it pounds of steel to make a unit of product 1 and it takes takes aa22 pounds of steel to make a unit of product 2. pounds of steel to make a unit of product 2.

Let Let xx11 and and xx22 denote this month's production denote this month's production level of product 1 and product 2, respectively. level of product 1 and product 2, respectively. Denote by Denote by pp11 and and pp22 the unit profits for products 1 the unit profits for products 1 and 2, respectively. and 2, respectively.

The manufacturer has a contract calling for at The manufacturer has a contract calling for at least least mm units of product 1 this month. The firm's units of product 1 this month. The firm's facilities are such that at most facilities are such that at most uu units of product 2 units of product 2 may be produced monthly. may be produced monthly.

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Mathematical ModelMathematical Model• The total monthly profit = The total monthly profit =

(profit per unit of product 1) (profit per unit of product 1)

x (monthly production of product 1) x (monthly production of product 1)

+ (profit per unit of product 2) + (profit per unit of product 2)


= = pp11xx11 + + pp22xx22

We want to maximize total monthly profit:We want to maximize total monthly profit:

Max Max pp11xx11 + + pp22xx22

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Mathematical Model (continued)Mathematical Model (continued)• The total amount of steel used during monthly The total amount of steel used during monthly

production = production =

(steel required per unit of product 1) (steel required per unit of product 1)

x (monthly production of product 1)x (monthly production of product 1)

+ (steel required per unit of product 2) + (steel required per unit of product 2)


= = aa11xx11 + a + a22xx22

This quantity must be less than or equal to This quantity must be less than or equal to the the allocated allocated bb pounds of steel: pounds of steel:

aa11xx11 + + aa22xx2 2 << bb

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Mathematical Model (continued)Mathematical Model (continued)• The monthly production level of product 1 The monthly production level of product 1

must be greater than or equal to must be greater than or equal to mm::

xx11 >> mm

• The monthly production level of product 2 The monthly production level of product 2 must be less than or equal to must be less than or equal to uu::

xx22 << uu

• However, the production level for product 2 However, the production level for product 2 cannot be negative: cannot be negative:

xx22 >> 0 0

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Mathematical Model SummaryMathematical Model Summary

Max Max pp11xx11 + + pp22xx22

s.t. s.t. aa11xx11 + + aa22xx22 << bb

xx11 >> mm

xx22 << uu

xx22 >> 0 0

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Question:Question:

Suppose Suppose bb = 2000, = 2000, aa11 = 2, = 2, aa22 = 3, = 3, mm = 60, = 60, uu = 720, = 720, pp11 = 100, = 100, pp22 = 200. Rewrite the model with = 200. Rewrite the model with these specific values for the uncontrollable inputs.these specific values for the uncontrollable inputs.

Answer:Answer:

Substituting, the model is:Substituting, the model is:

Max 100Max 100xx11 + 200 + 200xx22

s.t. 2s.t. 2xx11 + 3 + 3xx22 << 2000 2000

xx11 >> 60 60

xx22 << 720 720

xx22 >> 0 0

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Question:Question:

The optimal solution to the current model The optimal solution to the current model is is xx11 = 60 and = 60 and xx22 = 626 = 626 2/32/3. If the product were . If the product were engines, explain why this is not a true optimal engines, explain why this is not a true optimal solution for the "real-life" problem. solution for the "real-life" problem.

Answer:Answer:

One cannot produce and sell 2/3 of an One cannot produce and sell 2/3 of an engine. Thus the problem is further restricted by engine. Thus the problem is further restricted by the fact that both the fact that both xx11 and and xx22 must be integers. must be integers. They could remain fractions if it is assumed They could remain fractions if it is assumed these fractions are work in progress to be these fractions are work in progress to be completed the next month.completed the next month.

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Example: Ponderosa Development Example: Ponderosa Development CorporationCorporation

Ponderosa Development Corporation (PDC) is a Ponderosa Development Corporation (PDC) is a small real estate developer operating in the Rivertree small real estate developer operating in the Rivertree Valley. It has seven permanent employees whose Valley. It has seven permanent employees whose monthly salaries are given in the table on the next monthly salaries are given in the table on the next slide.slide.

PDC leases a building for $2,000 per month. PDC leases a building for $2,000 per month. The cost of supplies, utilities, and leased equipment The cost of supplies, utilities, and leased equipment runs another $3,000 per month.runs another $3,000 per month.

PDC builds only one style house in the valley. PDC builds only one style house in the valley. Land for each house costs $55,000 and lumber, Land for each house costs $55,000 and lumber, supplies, etc. run another $28,000 per house. Total supplies, etc. run another $28,000 per house. Total labor costs are figured at $20,000 per house. The one labor costs are figured at $20,000 per house. The one sales representative of PDC is paid a commission of sales representative of PDC is paid a commission of $2,000 on the sale of each house. The selling price of $2,000 on the sale of each house. The selling price of the house is $115,000.the house is $115,000.

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EmployeeEmployee Monthly SalaryMonthly Salary

President President $10,000$10,000

VP, Development 6,000VP, Development 6,000

VP, Marketing VP, Marketing 4,500 4,500

Project Manager Project Manager 5,500 5,500

Controller Controller 4,000 4,000

Office Manager Office Manager 3,000 3,000

Receptionist Receptionist 2,000 2,000

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Question:Question:

Identify all costs and denote the marginal Identify all costs and denote the marginal cost and marginal revenue for each house.cost and marginal revenue for each house.

Answer:Answer:

The monthly salaries total $35,000 and The monthly salaries total $35,000 and monthly office lease and supply costs total monthly office lease and supply costs total another $5,000. This $40,000 is a monthly fixed another $5,000. This $40,000 is a monthly fixed cost. cost.

The total cost of land, material, labor, and The total cost of land, material, labor, and sales commission per house, $105,000, is the sales commission per house, $105,000, is the marginal cost for a house. marginal cost for a house.

The selling price of $115,000 is the The selling price of $115,000 is the marginal revenue per house.marginal revenue per house.

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Ponderosa Development Corporation (PDC)Ponderosa Development Corporation (PDC)

Question:Question:

Write the monthly cost function Write the monthly cost function cc((xx), ), revenue function revenue function rr((xx), and profit function ), and profit function pp((xx).).

Answer:Answer:

cc((xx) = variable cost + fixed cost = ) = variable cost + fixed cost = 105,000105,000xx + 40,000 + 40,000

rr((xx) = 115,000) = 115,000xx

pp((xx) = ) = rr((xx) - ) - cc((xx) = 10,000) = 10,000xx - 40,000 - 40,000

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Ponderosa Development Corporation (PDC)Ponderosa Development Corporation (PDC)

Question:Question:

What is the breakeven point for monthly sales What is the breakeven point for monthly sales of the houses?of the houses?

Answer:Answer:

rr((xx) = ) = cc((xx) or 115,000) or 115,000xx = 105,000 = 105,000xx + + 40,000 40,000

Solving, Solving, xx = 4. = 4. Question:Question:

What is the monthly profit if 12 houses per What is the monthly profit if 12 houses per month are built and sold?month are built and sold?

Answer:Answer:

pp(12) = 10,000(12) - 40,000 = $80,000 monthly (12) = 10,000(12) - 40,000 = $80,000 monthly profitprofit

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The Management ScientistThe Management Scientist

A software package called A software package called The Management The Management ScientistScientist has been prepared by the authors to has been prepared by the authors to accompany the textbook.accompany the textbook.

Version 4.0 is available for Windows and Version 4.0 is available for Windows and Windows 95 systems.Windows 95 systems.

This software can be used to solve problems in This software can be used to solve problems in the text as well as small-scale problems the text as well as small-scale problems encountered in practice.encountered in practice.

The Management Scientist contains 12 modules The Management Scientist contains 12 modules or programs.or programs.

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Module Selection ScreenModule Selection Screen

Select A ModuleSelect A Module o Linear Programmingo Linear Programming o PERT/CPM o PERT/CPM

o Transportationo Transportation o Inventory Models o Inventory Models

o Assignmento Assignment o Waiting Lines o Waiting Lines

o Integer Linear Programming o Decision Analysiso Integer Linear Programming o Decision Analysis

o Shortest Routeo Shortest Route o Forecasting o Forecasting

o Minimal Spanning Treeo Minimal Spanning Tree o Markov Processes o Markov Processes

Select A ModuleSelect A Module o Linear Programmingo Linear Programming o PERT/CPM o PERT/CPM

o Transportationo Transportation o Inventory Models o Inventory Models

o Assignmento Assignment o Waiting Lines o Waiting Lines

o Integer Linear Programming o Decision Analysiso Integer Linear Programming o Decision Analysis

o Shortest Routeo Shortest Route o Forecasting o Forecasting

o Minimal Spanning Treeo Minimal Spanning Tree o Markov Processes o Markov Processes

OKOKOKOK

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File MenuFile Menu• NewNew - - to begin a new problemto begin a new problem• OpenOpen - to retrieve a saved problem - to retrieve a saved problem• SaveSave - to save the current problem - to save the current problem• Change ModulesChange Modules - to return to Module - to return to Module

Selection Selection Screen Screen• ExitExit - to exit - to exit The Management ScientistThe Management Scientist

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Advice About Data InputAdvice About Data Input• Do not enter commas (,) in your numbers.Do not enter commas (,) in your numbers.• Do not enter the dollar sign ($).Do not enter the dollar sign ($).• Do not enter the percent sign (%). For Do not enter the percent sign (%). For

example, enter .25 rather than 25%.example, enter .25 rather than 25%.• Do not enter fractions. For example, enter .25 Do not enter fractions. For example, enter .25

rather than 1/4.rather than 1/4.• Attempt to scale extremely large input data. Attempt to scale extremely large input data.

For example, $2,500,000 might be scaled to For example, $2,500,000 might be scaled to 2.52.5

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The End of Chapter 1The End of Chapter 1

Documents

1 1 Cap1 QUANTITATIVE METHODS FOR BUSINESS FOR BUSINESS Seventh Edition ANDERSON SWEENEY WILLIAMS SLIDES PREPARED BY JOHN LOUCKS © 1998 South-Western College