cob 291 introduction to management science michael e. busing cis/om program, james madison...

COB 291

Introduction to Management ScienceMichael E. Busing

CIS/OM Program, James Madison University

What is Management Science?

What is Management Science?

field of study that uses computers, statistics and mathematics to solve business problems.

sometimes referred to as operations research or decision science.

formerly, the field was available to only those with advanced knowledge of mathematics and computer programming languages. PC’s and spreadsheets have made the tools available to a much larger audience.

Air New Zealand, “Optimized Crew Scheduling at Air New Zealand”

Air New Zealand, “Optimized Crew Scheduling at Air New Zealand”

The airline crew scheduling problem consists of the pairings problem involving the generation of minimum-cost pairings (sequences of duty periods) to cover all scheduled flights, and the rostering problem involving the assignment of pairings to individual crewmembers. Over the past fifteen years, eight application-specific optimization-based computer systems have been developed in collaboration with the University of Auckland to solve all aspects of the pairings and rostering processes for Air New Zealand's National and International operations. These systems have produced large savings, while also providing crew rosters that better respect crew preferences.

Federal Aviation Administration, “Ground Delay Program Enhancements (GDPE) under Collaborative Decision Making (CDM)”

Federal Aviation Administration, “Ground Delay Program Enhancements (GDPE) under Collaborative Decision Making (CDM)”

When airport arrival capacity is reduced, the demand placed by arriving aircraft may be too great. In these cases a ground delay program (GDP) is used to delay flights before departure at their origin airport, keeping traffic at an acceptable level for the impacted arrival airport. However, GDPs sometimes lacked current data and a common situational awareness. Working with the FAA and airline community, Metron, Inc. and Volpe National Transportation Systems Center improved the process by utilizing real-time data exchange between all users, new algorithms to assign flight arrival slots, and new software in place at FAA facilities and airlines.

Fingerhut Companies, Inc., “Mail Stream Optimization”

Fingerhut Companies, Inc., “Mail Stream Optimization”

Fingerhut mails up to 120 catalogs per year to each of its 6 million customers. With this dense mail plan and independent mailing decisions, many customers were receiving redundant and unproductive catalogs. To find and eliminate these unproductive catalogs, optimization models were developed to select the optimal sequence of catalogs, called a "mail stream", for each customer. With mail streams, Fingerhut is able to make mailing decisions at the customer level as well as increase profits. Today, this application runs weekly using current data to find the most profitable mail stream for each of its 6 million customers.

Ford, “Rightsizing and Management of Prototype Vehicle Testing at Ford Motor Company”

Ford, “Rightsizing and Management of Prototype Vehicle Testing at Ford Motor Company”

Prototype vehicles are used to verify new designs and represent a major annual investment at Ford Motor Company. Engineering managers studying in a Wayne State University master's degree program adapted a classroom set covering example and launched the development of the Prototype Optimization Model (POM). The POM is used for both operational and strategic planning. The modeling approach was lean and rapid and was designed to maintain the role of the experienced manager as the ultimate decision-maker.

IBM, “Matching Assets with Demand in Supply Chain Management at IBM Microelectronics”

IBM, “Matching Assets with Demand in Supply Chain Management at IBM Microelectronics”

The IBM Microelectronics Division is a leading-edge producer of semiconductor and packaged solutions supplying a wide range of customers inside and outside IBM. A critical component of customer responsiveness is matching assets with demand to correctly assess anticipated supplies linked with demand and provide manufacturing guidelines. A suite of tools was developed to handle matching in a division-wide "best can do", division wide ATP, and daily individual manufacturing location MRPs. The key modeling advance is the dynamic interweaving of linear programming, traditional MRP explosion and implosion-based heuristics and the ability to harness deep computing to solve large linear programming problems.

For More InformationFor More Information

http://www.lionhrtpub.com/ORMS.shtmlhttp://www.informs.org/http://dsi.gsu.edu/

http://www.lionhrtpub.com/ORMS.shtml




http://www.informs.org/

http://dsi.gsu.edu/

Application to InfoSec?Application to InfoSec?

Linear ProgrammingQueuing Theory/SimulationProject managementForecastingDecision Analysis/Decision Trees

IdentifyProblem

Formulate andImplement

Model

AnalyzeModel

TestResults

ImplementSolution

unsatisfactory results

A Visual Model of the Problem-Solving Process

Linear ProgrammingLinear Programming

Problems characterized bylimited resources.decisions about how best to utilize the limited resources

available to an individual or a business.maximization or minimization of profits or costs.

Mathematical Programming (MP) is a field of management science that finds the optimal, or most efficient, way of using limited resources to achieve the objectives of an individual or a business. MP is sometimes referred to as optimization.

Optimization ExampleOptimization ExampleSeuss’s Sandwich Shop sells two types of sandwiches: green eggs and ham (GEH) and ham and cheese (HC). A green eggs and ham sandwich consists of 2 slices bread, 1 green egg, and 1 slice ham. It takes an employee 3 minutes to make one of these sandwiches. A ham and cheese sandwich consists of 2 slices bread, 2 slices ham and 1 slice cheese. It takes 2 minutes to make a ham and cheese sandwich. The sandwich shop presently has available 400 slices of bread, 80 slices cheese, 120 green eggs and 200 slices of ham. The shop also has one employee scheduled for 7 hours to make all of the sandwiches. If a green egg and ham sandwich sells for $5 and a ham and cheese sandwich sells for $4, how many of each type should be prepared to maximize revenue? (Assume that demand is great enough to ensure that all sandwiches made will be sold).

N u m 220 b e r 200 o f 180 H a 160 m a 140 n d 120 C h e e 100 s e 80 (H C) 60 t o 40 M a k 20 e 0 20 40 60 80 100 120 140 160 180 200 220 Number of Green Eggs and Ham Sandwiches (GEH) to Make

MAX 5 GEH + 4 HC Subject to 2 GEH + 2 HC < 400 (BREAD) HC < 80 (CHEESE) GEH < 120 (GREEN EGGS) GEH + 2 HC < 200 (HAM) 3 GEH + 2 HC < 420 (TIME) GEH, HC > 0

Post-Optimality AnalysisPost-Optimality Analysis

Range of Optimality – tells range that a decision variable’s coefficient can take on in the objective function without affecting current optimal solution (note that the objective function value WILL change).

Shadow/dual price – tells how receiving additional units of a resource affects the objective function value (for < constraints). Also tells how requiring more of something (for > constraints) affects the objective function value. Note that the changes in right hand side values are only good for a relevant range.


MAX 5 GEH + HC (REVISED) Subject to 2 GEH + 2 HC < 400 (BREAD) HC < 80 (CHEESE) GEH < 120 (GREEN EGGS) GEH + 2 HC < 200 (HAM) 3 GEH + 2 HC < 420 (TIME) GEH, HC > 0


MAX 5 GEH + 4 HC Subject to 2 GEH + 2 HC < 400 (BREAD) HC < 80 (CHEESE) GEH < 120 (GREEN EGGS) GEH + 2 HC < 220 (HAM REVISED) 3 GEH + 2 HC < 420 (TIME) GEH, HC > 0

Spreadsheet Solution to Linear Programming

Problems

Spreadsheet Solution to Linear Programming

ProblemsGreen Eggs and Ham Production Problem (Zeuss' Sandwich Shop)

Decision VariablesGreen Eggs and Ham Ham and Cheese

No. to produce: 0 0

Objective Max Total Profit

Unit Profits: 5 4 =SUMPRODUCT(B8:C8,B5:C5)

Constraints:Used Available

Bread (slices) 2 2 =SUMPRODUCT(B12:C12,$B$5:$C$5) 400Cheese (slices) 0 1 =SUMPRODUCT(B13:C13,$B$5:$C$5) 80Eggs (units) 1 0 =SUMPRODUCT(B14:C14,$B$5:$C$5) 120Ham (slices) 1 2 =SUMPRODUCT(B15:C15,$B$5:$C$5) 200Time (minutes) 3 2 =SUMPRODUCT(B16:C16,$B$5:$C$5) 420

Microsoft Excel

Worksheet

Other Applications in LPOther Applications in LP1. (Operations planning) Diagnostic Corporation assembles two types of electronic

calculators. The DC1 Calculator provides basic math functions, while the DC2 also provides trigonometric computations needed by engineers. Due to a winter blizzard, incoming shipments of components have been delayed. Ron Beckman, Manager of the plant, has assembled his production staff to plan an appropriate response. Bob Driscoll, in charge of supplies, reports that only three items are in short supply and likely to run out before new shipments arrive. The items in short supply are diodes (16,000 available), digital displays (10,000 available), and resistors (18,000 on hand). The quantities of each of these components that are required by each calculator are shown below. Mr. Beckman states that he would like to plan production to maximize the profits that can be realized using the available supply of parts.

Number of Parts Required Profit Per Model Diodes Displays Resistors Calculator DC1 6 2 6 $10 DC2 4 3 1 $12

Other Applications in LPOther Applications in LP2. (Sales promotion) Riverside Auto wants to conduct an advertising campaign

where each person who comes to the lot to look at a car receives $1. To advertise this campaign, Riverside can buy time on two local TV stations. The advertising agency has provided Riverside with the following data:

Cost per Number of Number of Maximum

Station Ad Serious Buyers Freeloaders Spots

Ch. 47 $90 80 480 20Ch. 59 $100 100 360 20

Riverside wants to minimize costs, while giving away a maximum of $15,000, and ensuring that the ads attract at least 2400 serious buyers. Formulate Riverside’s problem as a linear program.

Other Applications in LPOther Applications in LP3. (Staff Scheduling) Palm General Hospital is concerned with the staffing of its

emergency department. A recent analysis indicated that a typical day may be divided into six periods with the following requirements for nurses:

Time Period Nurses Needed

7AM – 11AM 811AM – 3PM 63PM – 7PM 127PM – 11PM 611PM – 3AM 43AM – 7AM 2

Other Applications in LPOther Applications in LP4. (Agriculture) A farming organization operates 3 farms of comparable productivity. The output of each farm is limited both

by the usable acreage and by the amount of water available for irrigation. Data for the upcoming season are as follows:Farm Usable Acreage Water Available (Acre ft.) 1 400 1500 2 600 2000 3 300 900The organization is considering 3 crops for planting, which differ in their expected profit/acre and in their consumption of water. Furthermore, the total acreage that can be devoted to each of the crops is limited by the amount of appropriate harvesting equipment available:

Max Water Consumption Expected ProfitCrop Acreage in Acre ft./Acre Per Acre A 700 5 $400 B 800 4 $300 C 300 3 $100

In order to maintain a uniform workload among farms, the percentage of usable acreage planted must be the same at each farm. However, any combination of crops may be grown at any of the farms. Hw much of each crop should be planted at each farm to maximize expected profit?

Other Applications in LPOther Applications in LP5. (Blending) Suppose that an oil refinery wishes to blend 4 petroleum constituents into 3 grades of gasoline A, B, and C. The

availability and costs of the 4 constituents are as follows:Constituent Max Availability (bbls/day) Cost per Barrel W 3,000 $3 X 2,000 $6 Y 4,000 $4 Z 1,000 $5

To maintain the required quality for each grade of gasoline it is necessary to maintain the following maximum and/or minimum percentages of the constituents in each blend. Determine the mix of the 4 constituents that will maximize profit.Grade Specification Selling Price per Barrel A no more than 30% of W $5.50

at least 40% of Xno more than 50% of Y

B no more than 50% of W $4.50at least 10% of X

C no more than 70% of W $3.50

Queuing Theory/Simulation

Queuing Theory/Simulation

Probabilistic Models – must make decision today, but don’t know for sure what will happen.

Americans are reported to spend 37 billion hours a year waiting in line!

E-mail often waits in a queue (i.e., line) on the Internet at intermediate computing centers before sent to final destination.

Subassemblies often wait in a line at machining centers to have the next operations performed.

Queuing theory represents the body of knowledge dealing with waiting lines.

Queuing TheoryQueuing Theory

Conceived in the early 1900s when Danish telephone engineer, A.K. Erlang, began studying the congestion and waiting times occurring in the completion of telephone calls.

Since that time, a number of quantitative models have been developed to help business people understand waiting lines and make better decisions about how to manage them.


Any time there is a finite capacity for service, you have a queuing system.

Channels and Stages:Channel: How many servers available for initial

operation step?Stage: How many servers must an individual entity

see before service is completed?

Queuing TheoryChannels and Stages


Single Channel (# of servers available at each stage) / Single Stage (how many servers the entity must see before service is complete.

InputSource

Service Facility



Multiple Channel (# of servers available at each stage) / Single Stage (how many servers the entity must see before service is complete.

InputSource Service Facility

Note: Every entity joins the same line and waits for theNext available server.



Question: What is the difference between the following?Multiple Channel/Single Stage

Multiple – Single Channel Single Stage Systems

InputSource

InputSourceInput

Source

Service facility

Service facility

Service facility

Other FormsQueuing Systems

Other FormsQueuing Systems

Single Channel/Multiple Stage

Multiple Channel/Multiple Stage

InputSource

InputSource


Managers use queuing theory to answer:How many servers should we have?How long should a customer wait, on average?

Queue Discipline:Infinite Calling PopulationInfinite Queue CapacityNo Balking or RenegingFirst Come First ServedRequires that service rate is greater than or equal to arrival rate.In multiple server case, all servers are of equal capability.

Queuing TheoryProcesses

Queuing TheoryProcesses

Input process can be either deterministic (D), general (G), or it can follow a Poisson Process (M). Poisson Process says that I know, on average, how many customers arrive per unit of time. Average arrival RATE is represented by .

Service process can be either deterministic, general or follow a Poisson Process. Average service RATE is represented by .

Number of servers is represented by K.

Queuing TheoryKendall’s NotationQueuing Theory

Kendall’s NotationThere are infinite numbers of possible queuing systems:

3 Possible Arrival Processes X 3 Possible Service Processes X Infinite Possible Number of Servers

To make sense of this, we use standard notation – Kendall’s Notation.

Input Process Service Process # of Servers Queue Capacity

M / M / 1 / M / M / 2 / M / M / K / M / G / K / G / M / K / G / G / K /

Queuing TheoryM/M/1 ExampleQueuing Theory

M/M/1 ExampleAverage Arrival Rate =45 customers per hourAverage Service Rate =60 customers per hourQuestions of interest:

Average time a customer spends waiting in line?Average number of customers waiting in line?Average time a customer spends in the system?Average number of customers in the system?Probability that the system is empty and idle?

Little’s Flow Equationsfor M/M/1/

Little’s Flow Equationsfor M/M/1/

WL

1WW

LW

L

PP

-1P

q

qq

q

0i

0

i

Little’s Flow Equationsfor M/M/1/Example

Little’s Flow Equationsfor M/M/1/Example

customers 3(0.067)45 W L

minutes 4 hours 067.060

10.05

1WW

minutes 3hours 05.045

2.25

LW

customers 25.2456060

45 L

1875.025.060

45PPP

25.060

45-1 -1P

q

qq

q

1

1 0i

0

i

Little’s Flow Equationsfor M/M/n/

Little’s Flow Equationsfor M/M/n/

WL

1WW

LW

q

qq

20q

k-i

0

0

i

k

1-k

0i

0

)(k1)!-(kPL

k i ifk!k

P

k i if P!

P

table)use(or

k-1k!

!

1P

k

i

i

i

i

i

NOTE: These aresame as for M/M/1/

Values for P0 for Multiple Channel Waiting Lines with Poisson Arrivals

and Exponential Service Times

Values for P0 for Multiple Channel Waiting Lines with Poisson Arrivals

and Exponential Service Times

2 channels 3 channels 4 channels 5 channels0.15 0.8605 0.8607 0.8607 0.86070.2 0.8182 0.8187 0.8187 0.8187

0.25 0.7778 0.7788 0.7788 0.77880.3 0.7391 0.7407 0.7408 0.7408

0.35 0.7021 0.7046 0.7047 0.70470.4 0.6667 0.6701 0.6703 0.6703

0.45 0.6327 0.6373 0.6376 0.63760.5 0.6000 0.6061 0.6065 0.6065

0.55 0.5686 0.5763 0.5769 0.57690.6 0.5385 0.5479 0.5487 0.5488

0.65 0.5094 0.5209 0.5219 0.52200.7 0.4815 0.4952 0.4965 0.4966

0.75 0.4545 0.4706 0.4722 0.47240.8 0.4286 0.4472 0.4491 0.4493

0.85 0.4035 0.4248 0.4271 0.42740.9 0.3793 0.4035 0.4062 0.4065

0.95 0.3559 0.3831 0.3863 0.38671 0.3333 0.3636 0.3673 0.3678

1.2 0.2500 0.2941 0.3002 0.30111.4 0.1765 0.2360 0.2449 0.24631.6 0.1111 0.1872 0.1993 0.20141.8 0.0526 0.1460 0.1616 0.16462 0.1111 0.1304 0.1343

2.2 0.0815 0.1046 0.10942.4 0.0562 0.0831 0.08892.6 0.0345 0.0651 0.07212.8 0.0160 0.0521 0.05813 0.0377 0.0466

3.2 0.0273 0.03723.4 0.0186 0.02933.6 0.0113 0.02283.8 0.0051 0.01744 0.0130

4.2 0.00934.4 0.00634.6 0.00384.8 0.0017

P0


M/M/2 ExampleAverage Arrival Rate =45 customers per hourAverage Service Rate =60 customers per hourNumber of Channels, K, = 2

4545.0

)60(245

-12!

6045

!16045

!06045

1P

k-1k!

!

1P

2100

k

1-k

0i

0

i

i

NOTE: =45/60=0.75 – this value is on lookup table for k=2 0.4545


M/M/2 Example

customers 0.87304)(45)(0.019W L

minutes 1.16or hours 0194.060

10.0027

1WW

minutes 0.16or hours 0027.045

0.1227

LW

customers 1227.0)45(2(60)1)!-(2

)60(452

6045

4545.0)(k1)!-(k

PL

q

qq

220q

k

Cost/Service TradeoffsCost/Service Tradeoffs

To Recap Service Statistics:

So, is it worth the extra cost to add an additional server? Suppose servers earn $15 per hour, but customer cost of waiting (loss of goodwill, etc.) has been estimated to be $25 per hour.

Service Statistic k=1, k=2, Probability No Entities in System 0.25 0.4545Average Number of Entities Waiting in Queue 2.25 customers 0.1227 customersAverage Time Spent in Queue 3.00 minutes 0.16 minutesAverage Time Spent in System 4.00 minutes 1.16 minutesAverage Number of Entities in System 3 customers 0.8730 customers

Cost/Service Tradeoffs (cont’d)

Cost/Service Tradeoffs (cont’d)

Customer waiting cost can either be associated with queue time or total time in system. We’ll go with “total time in system.”

Therefore, our per-hour total (system) cost is represented by

TC=(Cs X K)+ (CW X L)where Cs = hourly cost for server and

CW = hourly waiting cost TCk=1 =(15 X 1) + (25 X 3) = $90TCk=2 =(15 X 2) + (25 X 0.8730) = $51.83

SimulationSimulation

Assumptions of Queuing - RevisitedInfinite Calling PopulationInfinite Queue CapacityNo Balking or RenegingFirst Come First ServedRequires that service rate is greater than or equal to arrival rate.In multiple server case, all servers are of equal capability.

Note that the above assumptions are fairly restrictive! Simulation offers us flexibility that Queuing Theory does not.

SimulationSimulation

Simulation is a quick way to model long periods of time.Simulation requires that we generate a stream of

numbers that are random and that have no relationship to each other. The RANDOM NUMBER GENERATOR IS KEY.

In MS Excel, I can use =rand() to generate a random number greater than 0.00 but less than 1.00.

Can humans generate random numbers?

Simulation ExampleSimulation Example

HouseSales-person

answer0.70

no answer0.30

M0.20

F0.80

no purchase 0.75

no purchase0.85

Purchase0.25

purchase0.15

0.10

0.40

0.300.20

1

234

12

3

0.60

0.30

0.10

Simulation Example (cont’d)

Simulation Example (cont’d)We now need to construct probability distributions associated

with each event in our simulation example

Answer? Probability RNUMGender Probability RNUM Y 0.7 0.00-0.69 M 0.2 0.00-0.19 N 0.3 0.70-0.99 F 0.8 0.20-0.99

Female Sale? Probability RNUM Male Sale? Probability RNUM Y 0.15 0.00-0.14 Y 0.25 0.00-0.24 N 0.85 0.15-0.99 N 0.75 0.25-0.99

Number (F)? Probability RNUM Number (M)? Probability RNUM 1 0.60 0.00-0.59 1 0.10 0.00-0.09 2 0.30 0.60-0.89 2 0.40 0.10-0.49 3 0.10 0.90-0.99 3 0.30 0.50-0.79

4 0.20 0.80-0.99



Microsoft Excel

Worksheet

ForecastingForecasting

A forecast is an estimate of future demandForecasts contain errorForecasts can be created by subjective means

by estimates from informal sourcesOR forecasts can be determined mathematically

by using historical dataOR forecasts can be based on both subjective

and mathematical techniques.

Forecast RangesForecast Ranges

Long Range (i.e., greater than one year)production capacityautomation needs

Quantitative Methods - L.S. Regression ExampleQuantitative Methods -

L.S. Regression ExamplePerfect Lawns, Inc., intends to use sales of lawn

fertilizer to predict lawn mower sales. The store manager feels that there is probably a six-week lag between fertilizer sales and mower sales. The pertinent data are shown below. =>


L.S. Regression ExamplePeriod Fertilizer Sales Number of Mowers Sold

(Tons) (Six-Week Lag)1 1.7 112 1.4 93 1.9 114 2.1 135 2.3 146 1.7 107 1.6 98 2 139 1.4 910 2.2 1611 1.5 1012 1.7 10

A) Use the least squares method to obtain a linear regression line for the data.


L.S. Regression ExamplePeriod Fertilizer Sales Number of Mowers Sold (X) (Y) X2 Y2

(Tons) (X) (Six-Week Lag) (Y)

1 1.7 11 18.7 2.89 121

2 1.4 9 12.6 1.96 81

3 1.9 11 20.9 3.61 121

4 2.1 13 27.3 4.41 169

5 2.3 14 32.2 5.29 196

6 1.7 10 17.0 2.89 100

7 1.6 9 14.4 2.56 81

8 2 13 26.0 4.00 169

9 1.4 9 12.6 1.96 81

10 2.2 16 35.2 4.84 256

11 1.5 10 15.0 2.25 100

12 1.7 10 17.0 2.89 100


L.S. Regression ExamplePeriod Fertilizer Sales Number of Mowers Sold (X) (Y) X2 Y2

(Tons) (X) (Six-Week Lag) (Y)1 1.7 11 18.7 2.89 121

2 1.4 9 12.6 1.96 813 1.9 11 20.9 3.61 1214 2.1 13 27.3 4.41 1695 2.3 14 32.2 5.29 1966 1.7 10 17.0 2.89 1007 1.6 9 14.4 2.56 818 2 13 26.0 4.00 1699 1.4 9 12.6 1.96 8110 2.2 16 35.2 4.84 25611 1.5 10 15.0 2.25 10012 1.7 10 17.0 2.89 100

SUM 21.5 135 248.9 39.55 1575


L.S. Regression Example

b

n X Y X Y

n X X

( )( )2

2



b

( )( . ) ( . )( )

( )( . ) ( . )

.

..

12 248 9 215 135

12 39 55 215

84 3

12 356 826

2



a

Y

n

b X

n

135

12

6 826 215

120 98

( . )( . ).



Y X

Y

e

e

0 98 6826

0 98 6826 2 12 67

. . ( )

. . ( ) .

Predict lawn mower sales for the first week in August, given fertilier

sales six weeks earlier of two tons.

lawn mowers

Time Series AnalysisTime Series Analysis

A time series is a set of numbers where the order or sequence of the numbers is important, e.g., historical demand

Analysis of the time series identifies patternsOnce the patterns are identified, they can be

used to develop a forecast

Time Series ModelsTime Series Models

Simple moving averageWeighted moving averageExponential smoothing (exponentially weighted

moving average)Exponential smoothing with random fluctuationsExponential smoothing with random and trendExponential smoothing with random and seasonal

component

Time Series Models Simple Moving Average


Sample Data (3-period moving average) t Dt Ft Dt-Ft | Dt-Ft |

Quarter Actual Demand Forecast Error Error 1 100 2 110 3 110 4 ? (100+110+110)/3=106.67




Quarter Actual Demand Forecast Error Error

1 100 2 110 3 110 4 80 (100+110+110)/3=106.67 80-106.67=-26.67 26.67




Quarter Actual Demand Forecast Error Error 1 100 2 110 3 110 4 80 (100+110+110)/3=106.67 80-106.67=-26.67 26.67 5 ? (110+110+80)/3 = 100.00




Quarter Actual Demand Forecast Error Error 1 100 2 110 3 110 4 80 (100+110+110)/3=106.67 80-106.67=-26.67 26.67 5 100 (110+110+80)/3 = 100.00 0 0

Time Series Models Exponential smoothing (exponentially weighted moving

average)


average)

S D S

F St t t

t t

( )1 1

1


average)


average)

Wheret=time period

St=smoothed average at end of period t

Dt=actual demand in period t

=smoothing constant (0<<1) Ft=forecast for period t


average)


average)

Sample Data (alpha = 0.2) t Dt St Ft Dt-Ft

Quarter Actual Demand Smoothed Average Forecast Error 0 100


average)


average)

Sample Data (alpha=0.2) t Dt St Ft Dt-Ft

Quarter Actual Demand Smoothed Average Forecast Error 0 100 1 ? 100


average)


average)


Quarter Actual Demand Smoothed Average Forecast Error 0 100 1 100 100


average)


average)


Quarter Actual Demand Smoothed Average Forecast Error 0 100 1 100 100 100-100=0


average)


average)


Quarter Actual Demand Smoothed Average Forecast Error 0 100 1 100 .2(100)+.8(100)=100 100 100-100=0


average)


average)



2 ? 100


average)


average)



2 110 100 110-100=10


average)


average)



2 110 .2(110)+.8(100)=102 100 110-100=10


average)


average)



2 110 .2(110)+.8(100)=102 100 110-100=10

3 ? 102


average)


average)



2 110 .2(110)+.8(100)=102 100 110-100=10

3 110 102 110-102=8

Make forecasts for periods 4-12.

Time Series Models Forecast Error

Time Series Models Forecast Error

2 error measures:Bias

tells direction (i.e., over or under forecast)

Mean Absolute Deviation

tells magnitude of forecast error

( )D F

n

t t

D F

n

t t

Characteristics of Good Forecasts

Characteristics of Good Forecasts

StabilityResponsivenessData Storage Requirements

BESM Example Cont’dBESM Example Cont’d

Microsoft Excel

Worksheet

BESM - ExpandedBESM - Expanded

The Basic Exponential Smoothing Model (BESM) is nothing more than a cumulative weighted average of all past demand (and the initial smoothed average).

Proof:

Decision Analysis/Decision Trees

Decision Analysis/Decision Trees

Decision Analysis (also called Bayesian Statistics) allows the decision maker to make a choice today (based on probability of future events) even though the future is unclear. The problems are generally multi-stage.

We can use Decision Trees to structure the above problems. All decision analysis includes:

Decision Alternatives (must pick one and stick with it)State(s) of Nature (after the decision is made, something

happens – something which the decision maker does not control).

Outcomes or payoffs.

Decision Analysis ExampleDecision Analysis ExampleA famous news anchor, who is thinking seriously about running for the US Senate, estimates the probability of being elected as 60%. While campaign expenditures are not a problem, the anchor feels that the gain or loss of prestige needs to be evaluated. The anchor estimates that, if prestige can be quantified, a win would be worth 100 “prestige points” and a loss would be represented by negative 50 “prestige points.”

There is no shortage of political advisors who would love to have this anchor for a client. One in particular offers to study the electorate and give an opinion of whether or not the race could be won. If this analyst returns a positive opinion, the anchor’s chance of winning rises to 92%. On the other hand, a negative opinion decreases the anchor’s winning chance to 29%. There is a 49% probability that the analyst will return a positive opinion.

If the analyst charges $15,000 for his opinion, what would the dollar amount of a “prestige point” have to be in order for the anchor to consider hiring the analyst.

Draw TreeStructure

hire

don’t hire

run

don’t run

win

lose

“+”

“-”

run

don’t run

don’t run

run

win

lose

win

lose

Label the Tree

hire

don’t hire

run

don’t run

win

lose

“+”

“-”

run

don’t run

don’t run

run

win

lose

win

lose

Add Probabilities

.49

.51

.60

.40

.92

.08

.29

.71

P(win|neg)

hire

don’t hire

run

don’t run

win

lose

“+”

“-”

run

don’t run

don’t run

run

win

lose

win

lose

Add Values

.49

.51

.60

.40

.92

.08

.29

.71

100

-50

0

100

-50

0

100

-50

0

43.12

40

40

0

88

88

hire

don’t hire

run

don’t run

win

lose

“+”

“-”

run

don’t run

don’t run

run

win

lose

win

lose

Fold Back

.49

.51

.60

.40

.92

.08

.29

.71

100

-50

0

100

-50

0

100

-50

0

-6.5

43.12

40

40

0

88

88

hire

don’t hire

run

don’t run

win

lose

“+”

“-”

run

don’t run

don’t run

run

win

lose

win

lose

Trim UndesirableDecisions Away

.49

.51

.60

.40

.92

.08

.29

.71

100

-50

0

100

-50

0

100

-50

0

-6.5

Decision Analysis ExampleDecision Analysis ExampleEVSI = 43.12 – 40 = 3.12

What is analyst charging per point?$15,000/3.12 = $4,807.69

Perfect Information: The analyst that is never wrong:If the analyst knows that you will win, you will and he’ll tell you to RUN

winrun100 If the analyst knows you will lose, you will and he’ll tell you to NOT RUN

losenot run0

Perfect Analyst: 0.6(100)+0.4(0)=60 NOTE: This is EVWPI

EVPI = 60-40=20NOTE: My analyst was only worth 3.12 points

Decision Analysis ExampleDecision Analysis Example

What is the EFFICIENCY of my analyst?EVSI / EVPI = 3.12 / 20 = 0.156

What if I thought 1 “prestige point” was worth $100? How much would I then be worth to pay ANY analyst?

$2000 the PERFECT analyst can only get me 20 points above and beyond what I can do on my own.

cob 291 introduction to management science michael e. busing cis/om program, james madison...

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