Development of Spreadsheet Models for Forecasting Manpower Stocks and Flows

7/26/2019 Development of Spreadsheet Models for Forecasting Manpower Stocks and Flows

1/117

NAVAL

POSTGRADUATE

SCHOOL

Monterey,

California

THESIS

D E V E L O P M E N T

O FSPREADSHEET

M O D E L SFO R FORECASTINGM A N P O W E R

STOCKSA N D

FL OWS

by

Michael

G .

Earl

March1998

Thesis

Advisor: aul

R.Milch

Associate

Advisor:

ulie

Dougherty

Approvedfo r

public

release;distributionisunlimited.

] D T I C QUALITYIHSPECTfiD2

1 9 9 8 0 5 2 6 0 9 5


2/117

REPORTDOCUMENTATIONPAGE

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Approved

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No

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reporting

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fo r

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ofinformation

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pe rresponse,

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WashingtonDC 20503.

1.G E N C Y

USE

ON L Y(Leave blank)

2.

R E P O R T

DATE

March

1998

3.

E P OR T

TYPE

AND

DATES

C OV E R E D

Master's Thesis

4.ITLEA NDSUBTITLE

Developmentof SpreadsheetModelsfor

Forecasting

Manpower

Stocksand

Flows

6. AUTHOR(S)

Earl.

Michael

G.

7 .E RF O RM IN G

ORGANIZAT ION

NAME(S)

AND A D D R E SS( E S)

Naval

Postgraduate

School

Monterey,C A3943-5000

9 .P ON SOR I N G /MONITORING A G E N C YNAME(S)

AND

ADDRESS(ES)

5.UNDING N U MBE R S

8.

E R FOR MI N G

ORGANIZAT ION

E P O RT

N U MBE R

10.

SPONSORING/MONITORING

A G E N C Y

R E P OR T

N U MBE R

11 .

U P P L E M E N T A RY

N OT E S

The

viewsexpressed

in

this

thesis

are

those

of

theauthor

an ddono t

reflect

theofficialpolicyor

position

of the

Department

of

Defense

or

theU.S.

Government.

12a.ISTRIBUTION/AVAILABILITYSTATEM ENT

Approved

fo r

public

release;distribution

is

unlimited.

12b .DISTRIBUT IONC OD E

13.

ABSTRACT

(maximum

200

words)

The

computerizedmanpower

planning

models

developed

intinsthesis

weredesigned

to

be

usedby

students

takingth e

ManpowerPersonnel

Models

course,OS4701 ,

n

th eManpower

Systems

AnalysisCurriculumat

th e

Nava lPostgraduateSchool.he

purpose

ofth e

course

istointroducetudents

to

some

ofth e

basic

manpower

modelingconceptsand

these

models

are

th e

prime

instruments

toward

achieving

that

goal.

hemodels

constructed

using

MicrosoftExcel

include

a

MarkovChain

Model,

On e

GradeVacancymodel,

Multigrade

Vacancy

model

with

Non-InstantaneousFilling

of

Vacancies,

an d

a

Vacancymodel

with

Instantaneous

Filling

of

Vacancies.

Themodels

re

esigned

oe

unn

ersonal

omputers

with

Microsoft

Windows

5

operating

system.ser's

manuals

and

example

problems

are

included

for

eachmodel

in

th e

appendices.

14.

UBJECT

T E R M S

Excel,Modeling,Markov,Vacancy,Replace,

Manpower

Planning,Personnel

Flows,Spreadsheet

Modeling,

Manpower

Forecast,Stocks

17 .

E C U RIT Y

CLASSIFICAT ION

OF

R E P O R T

Unclassified

1 8.

ECURITY

CLASSIFICAT ION

O F

THIS

P AG E

Unclassified

NSN7540-01-280-5500

19 .

ECURITY

CLASSIFICATION

OFA B S T RA C T

Unclassified

15 .

U MBE R OF

P A G E S

118

16.

RICE C OD E

20 .

IMITATION

OF

ABSTRACT

U L

Standard

Form

298

(Rev.

2-89)

PrescribedbyA N S I

Std.239-18

298-102


3/117


4/117

Approved

fo rpublic

release;

distributionis

unlimited

D E V E L O P M E N T

O F

SPREADSHEETM O D E L S

FO R

F O R E C A S T I N G

M A N P O W E RSTOCKSAN DF L O W S

Michael

G.

Earl

Lieutenant,UnitedStates Navy

B.S. ,

UnitedStatesNaval

Academy,

993

Submitted

inpartial

fulfillmentof

th e

requirements

for

th e

degree

of

M A S T E RO F

SCIENCE

IN

M A N A G E M E N T

from

th e

N A V A L

P O S T G R A D U A T E

S C H O O L

March1998

Author:

Approved

by :

Paul

R .

Milch,

ThesisAdvisor

l^jJoOJlA

[ ^MAL

ReubenT.

Harris,

Chairman

Department

of

Systems

Management

1 1 1


5/117

IV


6/117

A B STRA C T

The

omputerized

anpowerlanning

odels

eveloped

n

his

hesis

ere

designed

tobe

used

by

studentstaking

th e

Manpower

Personnel

Models

course,OS4701 ,

in

th e

Manpower

ystems

Analysis

Curriculum

t

he

Nava l

Postgraduate

chool.

he

purpose

of

thecourseistointroducestudentsto

om e

ofth ebasicmanpowermodeling

concepts

nd

hese

modelsre

th e

rime

nstrumentsoward

chieving

hat

oal.he

modelsconstructed

using

Microsoft.

Excel

include

a

MarkovChain

Model ,

aOne

Grade

Vacancyodel, ultigradeacancyodelithon-Instantaneousilling

f

Vacancies,an d

aVacancymodelwithInstantaneous Filling

ofVacancies.

The

modelsre

esigned

o

e

un

nersonal

omputerswith

Microsoft

Windows95

operating

system.

ser's

manualsandexampleproblems

are

includedfor

each

modelin

the

appendices.


7/117

VI


8/117

THESISDISCLAIMER

Thereader/useriscautionedthatth ecomputermodelsdevelopedinthisresearch

m ay

not

have

been

exercised

forallpossible

cases.

hile

every

effort

was

madeto

ensure

th e

models

are

free

ofcomputational

an dlogic

errors,theycannotbe

consideredvalidated.

Anyapplicationof

these

models,

withoutadditional

verification,isat

th e

risk

of

th e

user.

vn


9/117

Vlll


10/117

TABLE

OF

CONTENTS

I.NT R OD U C T I ON

A.

A C K G R O U N D

B.

C O PE

OF

TH ETH E S I S

1 .

anpower

Modeling

Theory.

2.

preadsheets.

C .

V A I L A B I L I TY OF

M O D EL S

D.

O P Y R I G H T

NOTICE

n.

H E

BASICMARKOVCHAIN

M O D E L

A.N T R O D U C T I O N TO

TH E

MODEL

B.A S I C E Q U A T I O N SOFTH EM A R K O VC H A I NMODEL

1 .

quation

(1 )

2.

quation

(2 )

C .

N P U TOPTIONS

1 .

ength

of

Service

(LOS)

System

....

7

2.

ierarchicalSystem

3.eneral

System

D.

E C R U I T M E N T

OPTIONS

1 .ixedRecruitment.

2 .

dditiveIncreasesor

Decreases

in

Recruitment.

3 .ultiplicativeIncreasesorDecreasesinRecruitment.

4 .

dditive

Increasesor

Decreases

inSystem

Size

5 .

ultiplicative

IncreasesorDecreasesinSystemSize

6 .ixedSystem

Size

E.

TEADY S TA TE

1 .

teady

State

StockVector,

SSSV.

2 .teady

State

Distribution

Vector,

SSDV.

3 .

he

Zero

Vector. 0

F.

SER'S

M A N U A L AN DEX A M P L ES 1

IU .

ONE

G R A D E

V A C A N C Y

M O D E L

3

A.N T R O D U C T I O N TOTH E

MODEL

3

B.S S U M P T I O N SAN DNOTATION 3

1 .

otation

andDefinitions. 3

2.

elationshipBetween f(i)andG(i)

4

C .U B M O D E L S 4

/.

Submodel

A

4

2.

ubmodel

B

5

3.

ubmodel

C.

5

D.

SER'S

M A N U A L AN DEX A M P L ES 6

IV .U L T I GR AD E

V A C A N C YM O D E L

WITHN O N

IN ST A N T A N E O USFILLING

O F

VAC ANC I E S

7

A.N T R O D U C T I O N

TO

TH E

MODEL

7

B.

O TA TI O NAN DDEFINITIONS 7

C .

A S I C E Q U A T I O N FOR C O M P U T I N G

V A C A N C I E S

8

IX


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D.L G O R I T H M FOR C O M P U T I N GV A C A N C I E S 9

E.N P U TOPTIONS 1

1 .

ierarchical

system ..21

2.

eneralsystem

1

E.

USER'SM A N U A L

AN D

E X A M PL E S 1

V .

ULT IG RA DEVACANCY

M O D E L

WITH

IN ST A N T A N E O USFILLING

O F

VAC ANC I E S 2 3

A.

N TR O D U C TI O N OF THEM O D E L 3

B.

OTATION,

DEFINITIONS AN D

E Q U A T I O N S

3

C .E S U L T S

:

6

E.SER'SM A N U A L AN DE X A M PL E S

..26

VI.X C E L

M OD E L I NG

F E A T URE SAN DP R O C E D U R E S 7

A.

E A T U R E S

AN DF U N C T I O N S 7

1 .

acros. 7

2 .

f,

Then,

Else

7

3 .

atrix

Multiplication 7

4 .

Offset.

8

5 .ranspose 8

6.

atrix

inverse 8

7 .

aximum

andminimum 8

8 .

ount.

8

9.mbedded/nested

functions. 9

B.

A K I N G

C H A N G E S

TOTH E

M O D E L S

9

A P P E N DIX

A .

O M P U T E R

S Y S T E M

H A R D W A R EAN D

S O F T W A R EREQUIREMENTS....

31

A.Y S T E M H A R D W A R ER E Q U I R E M E N T S 1

B.

O F T W A R E

R E Q U I R E M E N T S 1

A P P E N DIXB . USER'S

M A N U A L

FO RT H E

BASIC

M A R K O V

CHA IN

MODEL3

A:

L O A D I N G

TH E

M O D E L

3

B:

R U N N I N G

TH E

M O D E L

..

3

1 .

te p

(1 )

Start. 3

2.

te p(2 )

Choose

Model

Option

3

3.

tep

(3 )

Data

Input.

3

4 .te p

(4 )

Reinitializing. 6

5 .tep

(5 )

DisplayResults.

7

6 .

te p

(6 )

Combine

Results

ofth e

InitialForecast

and

Additional

Forecasts.

8

7 .

te p(7 )

Printing. 9

C .R R O R TR A PPI N G / W A R N I N G 9

1 .

rror

Displays.

9

2 .

onsequences

of

IgnoringWarnings. 0

A P P E N DIXC.

USER'S

M A N U A L

FO R

T H E

O N E

G R A D E

V A C A N C Y

MODEL1

A.

OADING

TH E

M O D E L

1

B.R U N N I N G TH EM O D E L

1

1 .

xplanationof

th e

Three

Submodels.

1

2 .

ata

Input.

2

3 .

isplaying

Results. 3

4 .singResults.

4

C .R R O R

TR A PPI N G / W A R N I N G5

x


12/117

1 .rror

Displays. 5

2.

onsequencesofIgnoringWarnings.

5

APPENDIX D.

USER'S

M A N U A L

FO RT H E

M U L T I G R A D EVACANCY

M O D E L

WITH

N O N -

IN ST A N T A N E O US

FILLING

O FV A CA N CIE S 7

A.OADING

TH E

M O D E L 7

B.

R U N N I N G

TH E

M O D E L

7

1 .

Step(1 )

Start. ..

4

7

2.te p

(2 )

Choose

Model

Option 7

3.te p

(3 )

Data

Input.

8

4 .te p(4 )Reinitializing.

9

5.

te p(5 )

Displaying

Results. 0

6.te p(6 )CombiningResultsofan InitialForecastandan

Additional

Forecast.

0

7 .

te p

(7 )

Printing.

1

C.

R R O R

TR A PPI N G / W A R N I N G 2

1 .

Error

Displays...:

2

2 .onsequencesof

IgnoringWarnings. 2

APPENDIX E. USER'S

M A N U A L

FO R

T H E

M U L T I G R A D EVACANCY

M O D E L

WITH

IN ST A N T A N E O US

FILLING

O F

V A CA N CIE S

3

A.

L O A D I N G

T H EM O D E L 3

B.

R U N N I N G

TH EM O D E L 3

1 .xplanationof

th e

Model.

3

2 .ata

Input.

3

3 .isplaying

Results.

4

4 .

sing

Results.

4

C .

R R O R

TR A PPI N G / W A R N I N G 5

1 .rrorDisplays.

5

2 .

onsequences

ofIgnoring

Warnings. 5

APPENDIX

F.

SA M P LE

P R O B L E M SUSIN G

T H E

M A RKO V .X LS M O D E L.57

E X A M PL E

1

E X A M PL E

2

E X A M PL E3

.57

.62

.67

APPENDIX

G .

SA M P LE

P R O B L E M S

USINGTH E

RE P LA CE .X LS MODEL

3

E X A M PL E1 :; 3

E X A M PL E

2:

.

6

E X A M PL E3: 9

A P P E N DIX

H .

SA M P LEP R O B L E M S

USINGT H E

V A CA N CY.X LS M O D E L

3

E X A M PL E

1 :

3

EX A M P L E

2:

7

E X A M PL E3: 1

APPENDIX

L

SA M P LE

P R O B L E M S

USIN G

T H E

IN ST A N T A N E O US.X LS

M O D E L

5

E X A M PL E

1

E X A M PL E2

E X A M PL E

3

.95

.97

.99

LIST

O F

RE F E RE N CE S

01

XI


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INITIALDISTRIBUTION

LIST 03

X ll


14/117

A C K N O W L E D G M E N T

Iwouldlike

tothank

Professor

Paul

R.

Milch

for

all

oftheattentionan dtime

that

he

gave

to

this

endeavor.

lso,

I

would

like

to

thank

Julie

Dougherty

for

her

assistance.

I

wouldlike

tothank

my

wife,Deedra Earl,fo rhe rsupportand

understanding,

and

most

importantly,my tw osons,DaltynandCurtis,fo rgiving

me

mymuch

needed

recess

timeaway

from

thecomputer.

xin


15/117

I.

NTRODUCTION

A.

BACKGROUND

In

eneral,

anpower

lanning

s

multi-disciplinary

ctivity

oncerned

with

matching

thesupplyofpersonnelwith

the jobsavailable(Ref.:

p.

V) .omputer

models

are

availabletohelp

personnel

managers

dotheir

jobs

more

efficientlybyaccelerating

the

calculations

se dn

redictingersonnel

lows

withutomation.

s

vailable

market

technologybecomesmoreser-friendly,

he

xpectations

of

usersorhat

moreser-

friendlysoftware

increase.

The

software

produced

in

this

thesis

is

designed

for

use

in

conjunction

with

the

courseOS4701,entitled"Manpower

and

Personnel

Models".his

is

a

required

course

in

th e

Manpower

ystemsAnalysis(MSA)urriculum

at

the

NavalPostgraduatechool.

M S A

raduates

re

equiredo

. . .have

he

bilityo

se

nd

nderstandomputer

systems

in

problem

solving..."

(Ref.

2:

p.41).

heyalso

mustbe

able

to

us eadvanced

quantitative

nalysisuc h

s

.. .

arkov

models

n

he

nalysis

fforce

tructure,

manpower

planning,

orecasting

an d

flow

models."

(Ref.

:

.

141).

eeping

the

course

models

up

todate

withregard

totechnologywillallow

thestudentstobettermeetthe

requirementsof

the

curriculum.

The

models

that

have

beeninuseinthecoursewerecreated

by

Ahmet

E.Gurdal

in1991and

are

thoroughly

explained

in

Reference

3.

hesemodels

are

technically

correct

but

lack

a

graphic

interface.

he

technology

used

to

create

the

models

in

the

course

has

been

overtakenbya

more

visual

based

interface,spreadsheets.

he

models

produced

for

this

thesisarebuilt

using

a

current

computer

softwaretool, graphicalinterfacedriven

spreadsheet.

oday's

usersfind

any

programs

thatdonot

have

graphicalinterfacestobe

ahindrance

tolearning.

preadsheets

are

becoming

one

of

the

standardsoftware

tools

used

today

by

anincreasingnumber

of

people

in

general,

andby

several

curriculaatthe

NavalPostgraduatechool

in

particular.

his

spreadsheet

technology,

ncelearned,

s

transferableotherMicrosoftWindows computer

ased

ystems.

anyofthe

techniques

ca n

also

betransferred

to

other

software

packages

as

well.

herefore,

hese

models

introduce

the

students

to

more

resources

for

analyzing

the

data

roduced

by

these

an d

other

models.

Themodels

produced

in

thisthesisaremoreuser-friendlythanthemodels

they

are

replacing.

he

improved

user

interface

means

the

student

doesnot

need

to

spend

as

much


16/117

timelearning

ho w

touse

the

model

and

can

concentrate

efforts

on

analysis

of

the

results.

With

aspreadsheetas

the

underlying

framework

fo rthesemodels,tudentscanuse

the

output

from

themodels

in

further

computationswithout

cumbersome

datainput

from

the

keyboard.

Since

the

primary

goal

of

this

thesis

is

to

produce

user-friendly

models

for

students

enrolled

in

thecourse,

OS4701,

every

effortis

made

toreduce

user

confusion

with

regard

toexpectedinput

values

an d

steps

needed

towork

the

models.

he

models

areMicrosoft

Windows

basedan dthereforehavegraphicaluserinterfaces.

ac h

model

is

thoroughly

documentedan dpackagedina

formatsimilartoallthe

models.

hemodels

are

currently

being

tested

in

the

course,

OS4701:

Thefollowingmodelswere

developed

tobe

used

onapersonal

computeras

part

of

this

thesis:

1 .

he

Basic

Markov

Chain

model,

MARKOV.XLS.

2.ne

Grade

Vacancymodel,REPLACE.XLS

3.Multigrade

Vacancy

model

with

non-instantaneous

filling

of

vacancies,

VACANCY.XLS.

4.acancy

model

with

instantaneous

filling

of vacancies,

INSTANTANEOUS.XLS.

Thesemodels

are

explained

in

Chapters

II,

III ,

IV,

and

V,

respectively.

ll

models

areprogrammed

in

MicrosoftExcel ,thespreadsheet

program

currently

accepted

as

the

U.S.

Navy's

standardspreadsheet.hecomputer

system

andsoftwarerequirementsfor

themodels

can

befoundinAppendixA.

ach

model

isprovided

with

a

user's

manual

in

Appendices

B,C,

D

an dE,respectively.

Each

model

is

furtherexplained

through

example

problems

presented

in

Appendices

F,

G,H

andI.

B.

SCOPEOF

THETHESIS

Therimaryntentof

this

hesisstoroduceser-friendlyMicrosoftExcel

models

for

use

in

thecourse,

OS4701.ser's

manualsare

also

providedto

help

students

makeeasyuseof

the

models.hesemodels

are

designed

forth esoleuseofth e

course.

Any

use

of

these

models

in

any

other

setting

is

not

recommended.

The

thesis

willexplain

the

basic

equations,

otationsandproceduresused

in

the

creation

of

the

models.talso

explains

some

of the

specific

Microsoft

Excel procedures

an d

functions

used

in

the

creation

of

the

models.


17/117

I t

is

notth eintent

of

thisthesis

toexplaineither

manpower

modeling

theory

or

th e

general

sefpreadsheetechniques.his

hesis

ssumeshe

eaderas

asic

understanding

of

personalcomputer

skills

and

isfamiliarwithvectors

and

matrices.

1.anpowerModeling

Theory.

Manpower

modeling theorym ayb elearnedforexample,fromReference

or

4,

or

acombination

of

both.S4701is

a

coursedevotedtothatgoal

at N P S .

2.preadsheets.

Spreadsheet

modelingtechniques

shouldbelearned

in

apecific

courseesigned

toteachspreadsheettechniquesorbyusinga

hands

onapproachand

spreadsheetuser's

manuals.here

are

numerous

spreadsheetuser'smanualsonth emarket.

sers

hould

find

th e

manual

that

best

suits

their

individual

learning

style,

urrent

level

of

competence

andtheir

learning

goals.

C .

VAILABILITYO FM O D E L S

Copies

of

th emodelspresentedinthisthesisar e

available

fromProfessorPaulR .

Milch

n

heperations

esearch

epartment

ndulie

ougherty

nhe

ystems

Management

Department

atth e

Naval

Postgraduate chool,Monterey,C A93943 .

he

models

re

lso

nstalled

nheearningesourceenteromputerabocated

n

Glasgow

203 .

D.

O P Y R I G H T

NOTICE

Themodelswritten

inthisthesis

must

beused

with

a

registeredcopyof Microsoft

Excel

.

he

modelscreatedby th eauthorareplaced

in th e

public

domain.


18/117


19/117

H .

H E

BASIC

MARKOVCHA IN

M O D E L

A .

N TRODUC TION

TOTH E

M O D E L

For

personnel

managers,

he

ability

to

accurately

model

their

personnel

system

in

such

way

so

orecast

he

vailable

workforce,

ccessioneeds,

nd

osses

ue

o

attrition

isvital.

his

abilityallows

personnel

managerstotestpersonnelpoliciesonth e

modelwithouthe

egative

epercussions

ssociated

withhe

rial

nd

rror

pproach

carried

outon

th ereal

system

of

personnel.Markov

Chain

modelm ay

be

used

for

these

purposes.

Thefirst

model

developedforthisthesis,M A RK OV .X L S ,

orecasts

stocksusing

internalflow

rates,attritionrates,an d

recruitment

flows.

asedon

an

initial

stockvector,

a

recruitment

proportionvector,

nd

atransitionratematrix,

th emodel

computesstocks

by

using

MarkovChainTheory

an d

a

varietyofplanning

scenarios.

Amore

etailedxplanationfhe

heory

nd

ssumptions

an

be

ound

n

Reference

1 ,pages

95-115.

B.

ASIC

E Q U A T I O N S

O FTH EM A R K O V

CHAIN

M O D E L

The

AR K OV .XLS

odel

ses

wo

orms

fhearkovhainquation,

described

in

Reference

4,

ages6-2.2,

o

redict

stock

sizes

undervarious"Recruitment

Options".

he

twoequations,an d

anexplanation

of

th enotationused,follow.

1.

Equation

(1).

n(t)

=

n(t-l)P

+R(t )r .

Thisquation

s

se d

o

redict

tock

izesn

he

arious

ategories

hile

controlling

he

umberfersonnelecruiteduring

he

orecastingeriod.he

definitions

of

th e

notationused

in

th e

equation

are:

a .

n(t).

n(t)

is

avector

of

th e

category

stocks

at

t ime

t.

his

isth e

predicted

stocks

vector.


20/117

b .

(t-l).

n(t-l)

isavectorof thecategory

stocks

at

thecurrenttime

t-1.

he nt=l,

this

isn(0),the

initialstock

vector.

c .

a nd

w.

P

is

thetransition

ratematrix

which

governs

the

internalpersonnelflows.

TheP

matrixan d

the

attritionratevector,w,arecodependent.The

relationship

between

themi s :on eminusthesum

of

therow

elements

inthe

P

matrix

equalsthe

corresponding

attrition

rateelement.

or

example,

if thesumof the

first

rowofa

P

matrix

is.8 ,then

the

value

of the

firstelementof

w

is. 2 .

d.

(t).

R(t)isthetotal

number

of

personnel

recruited,

duringthe

interval

t-1tot,

whosurvive

in

thesystem

until

time

t.

r

she

ecruitment

roportion

ector

whichetermines

ow

heR(t)

recruits

are

distributed

amongthe

categories.

herefore,

the

sum

of

itscomponentsmust

equal

one.

For

example,if r=(.85,.15,0),then

85percentofthe

new

recruitswill

enter

category

one,

5percent

of

thenewrecruitswillenter

category

two,

andno

recruits

will

enter

category

three.

2. Equation

(2).

n(t)

=

n(t-l)Q

+

M(t)r .

Thisquation

s

sedoredicttock

izes

n

he

ariousategorieswhile

controllingthesize

of

th e

system

duringthe

forecasting

period.

he

definitionsof

the

notation

used

in

theequationare:

a,

(t),

n(t-l),

andr .

n(t),n(t-l),

and

r

are

the

same

as

previously

explained.


21/117

Q

is

amatrix

similartothe

matrix

P

used

in

equation(1).

he

0 _matrix

is

derivedby

the

followingequation:

Q _

=

P+w'r

where

w'

s

heolumn

ector

ersionftheowector

w.

Fo r morehorough

explanationof

thisequationseeReference

4,

pages

12-13.

c . M(t).

M(t)represents

the

change

insystemsize

during

the

interval,

[t-1

to

t) .

f

N(t)is

thesystem

size

attime

t,

and N(t-l)

is

the

system

size

at

time

t-1,

then:

M(t)=N(t)-N(t-l).

C.

INPUT

OPTIONS

To

make

userinput

easier,

as

wellas

foster

abetterunderstanding

of

the

course

material

hrough

isualization,

hree

nput

ptions

rerovidedorseith

MARKOV.XLS.

hehreeptionswhichepresent

ifferent

cenarios

n

manpower

planningare,lengthof

service,

hierarchical,

andgeneral.

1.

ength

of

Service

(LOS)

System.

A

system

s

called

L OS

ystem

i f ,

uringny

ne

eriod,

n

ndividualn

category

must

either

leave

the

system

ormove

to

the

next

higher

category.

nexception

is

madein

the

last

category.

ere

duringaperiod,anindividual

musteither

stay

within

thatcategoryor

leave

the

system.

herefore,th e

P

matrix

of

an

L OS

system

ha s

positive

elements

only

inthe

cellsimmediately

above

the

maindiagonal

and

possiblyin

thelast

elementof

the

maindiagonal.

2.

ierarchical

System.

Asystemiscalled

hierarchical

i f ,

during

any

oneperiod,

the

onlypersonnel

flows

are

promotion

to

the

nexthighercategory,

attrition,or

remaining

in

the

original

category.


22/117

Therefore,

there

are

nodouble

promotions

or

demotionsallowedinahierarchical

system.

The

Pmatrix

of

a

hierarchicalsystem

can

onlyhave

positiveelements

in

th e

main

diagonal

and

immediately

above

it.

3.

General

System.

The

general

systemcovers

all

possible

situations.

herearenorestrictionsonth e

placementof

th epositive

elements

in

th etransitionmatrix.

D.

REC RUITM EN T

OPTION S

Once

th einput

option

has

been

chosen,

th e

model

then

allows

th e

user

to

specify

"RecruitmentOptions".

he

irsthreeRecruitmentOptions"rese dopecifyhe

mannern

which

ccessions

re

rought

nto

he

ystem.heemaining

hree

ontrol

systemsizean d

allowrecruitmenttoconform

accordingly.

1.

ixedRecruitment.

Theumberofpersonnelnteringheystemsixedthe

nitial

ecruitment

level,R .

2.

dditive

Increases

or

Decreases

in

Recruitment.

This

ption

ncreasesr

ecreases

he

nitial

ecruitment

y

etmount

ach

period.

or

example,f

th e

value

-2 0

is

ntered

n

thisption,

hen

n

ach

eriod

he

recruitment

level

isreduced

by

20

until

recruitment

goes

tozero.

3.

ultiplicative

Increasesor

Decreases

in

Recruitment.

This

optionincreases

or

decreasesth einitial

recruitmentby

a

se t

percentageeach

period.or

example,f

th e

value

15s

ntered

nhisption,hen

n

acheriodhe

recruitment

level

is

increased

y

15percent.

4.

dditive

Increases

or

Decreases

inSystemSize.

Thisptionncreases

r

decreases

th eizeofth eystemy et

mount

ac h

period.orexample,

f

th e

value

200

s

ntered

in

this

ption,

hen

in

achperiodthe

system

size

is

increased

by

200 .


23/117

5.

ultiplicative

Increases

or

Decreases

in

System

Size.

This

option

increases

or

decreases

thesystem

size

by

a

se t

percentageeach

period.

For

example,

if

th e

value

-.05

isentered

inthis

option,then

in

each

period

th e

system

size

is

decreased

y

five

percent

until

th e

system

size

goes

to

zero.

6.

ixedSystem

Size.

Thisoptionholds

th esizeof

th e

system

fixedat

th e

level

of

th einitial

system

size.

E.

STEADYSTATE

The

equations

used

to

deriveth e

various

steadystate

vectors

thatresult

fromth e

six"Recrui tmentOptions"are

beyondth e

scope

ofthis

thesis.

detailed

explanation

of

th esteadystate

equations

for

each

"Recrui tment

Option"

an

be

found

n

Reference

:

pages17-45.

1.

teady

State

Stock

Vector,SSSV.

When

omputing

uccessive

tock

ectors,t

ay

eoticed

nderertain

conditions,

hatth e

values

of

th ecomponents

of

n(t)

remain

th e

am e

beyond

certain

valueoft.ystemsthatreachthispoint

are

saidtobeinsteadystate.hesevaluesare

called

teady

tatetocks

an d

their

vector,

(t),

s

alled

th e

teady

tatetockvector.

The

S S S V

exists

in

caseof

Recruitment

Options:

a .

ecruitmentOptions

(1).

Fixedrecruitment.

b .

ecruitment

Options

(6).

Fixedsystemsize.

2.

teadyStateDistribution

Vector,

SSDV .

When

comput ing

successive

stock

vectors,

t

m ay

lso

e

oticed

under

ertain

conditions,

hatheistribution

of

th e

omponents

ofn(t)emain

he

am e

eyond

certainvalue

oft.

n

somescenariositm ay

not

be

possible

for

the

stock

sizestoreach

steady

state

yetth e

stocks

m ay

reachsteadystatein

their

relative

sizes

to

eachother.

n


24/117

this

ase,we

ay

hat

th e

ystemeaches

teadytate

n

istribution".heesulting

vector

of

percentagesis

called

the

steady

state

distribution

vector.S S DValwaysexists

when

S S Vxists.

n

ummary,heS DVxistsnder

he

ollowingRecruitment

Optionsand

values.

a .

ecruitment

Option

(1).

Fixed

recruitment.

b .

ecruitment

Option

(2).

Additive

ncrease

r

ecreasen

ecruitment

ith

ositivedditive

increase.

c.

ecruitment

Option(3).

Multiplicative

increase or

decrease

in

recruitment

with a

positive

multiplicative

increase.

d.

ecruitment

Option

(4).

Additivencrease

r

ecrease

nystem

iz eithositive

dditive

increase.

e.ecruitmentOption

(5).

Multiplicativeincreaseor

decrease

in

system

sizewith

apositive

multiplicative

increase.

ecruitmentOption(6).

Fixedsystem

size

3.

TheZeroVector.

The

zero

vector

is

th e

value

of

both

th e

S S S V

and

S S D V

in

case

of

he

following

Recrui tment

Options

an d

values:

10


25/117

a.

ecruitment

Option(2).

Additive

ncrease

r

ecrease

n

ecruitment

ith

egativedditive

increase.

b.

ecruitment

Option(3).

Multiplicative increaseor

decrease

inrecruitmentwithanegative

multiplicative

increase.

c.

ecruitment

Option(4).

Additive

ncrease

r

ecrease

n

ystem

ize

withegativedditive

increase.

d.

ecruitment

Option(5).

Multiplicative

ncreaserdecrease

in

system

size

ith

a

negative

multiplicative

increase.

F.

USER'S

MANUAL

AN D

EXAMPLES

The

user'sanual

or

heMAR K OV .XLS

odel

s

ocated

n

ppendixB.

Appendix F

contains

threeexampleproblemsusingth eMAR K OV .XLS

model.

1 1


26/117

12


27/117

in.N E

G R A D E


A.

N TRODUC TION

T O

TH EM O D E L

The

implest

of

all

acancy

models

s

he

model

with

nly

ne

ategory.

n

model

of

thistype,here

an

be

nlytwo

flows:ttrition

and

ecruitment.

he

model,

R E P L A C E . X L S ,

omputes

replacement

ates

nd

numbers

of recruits

eededased

n

either

ttritionorurvivalehavior,

engthof

serviceofpersonnel

tim eero,

nd

numberofjobscreated

ineachfutureperiod.

Amore

etailed

xplanationfhe

heory

nd

ssumptionsan

eound

n

Reference1,pages

139-145,

andReference4,pages

62-71.

B.SS UM P TION SA NDN O T A T I O N

Since

there

areno

ategories

in

a

one

grade

vacancy

model,he

only

flows

that

existare

eitherattrition

out

of

th e

system,

orrecruitment

into

th e

system.

or

that

reason,

recruits

re

ften

eferred

oseplacementsn

this

model.ttritionshoughtofas

occurringniformlyhroughoutheeriodndecruitmentshought

of

asccurring

instantly

at

th e

en d

of th e

period.

1.otationandDefinitions.

The

following

notationan ddefinitions

are

introduced:

f(i)

isth eattritionrateamong

personnel

withi

years

of

service.

T )m ay

be

interpretedasth eprobability

that

anemployee,withatleast butlessthani+1

eriods

of

service,

will

leave

th e

system

beforecompleting

i

+

eriods

ofservice.

(i)m ayalsobe

interpreteds

he

roportion

of

employees,

with

at

east

ut

ess

han+1eriods

of

service,

leaving th e

systembefore

completing

i

+ periods

ofservice.

b .

G(i).

G(i )

isth e

probability

thatan

employee

willsurvive

in

th e

system

to

i

years

of service

orth eproportionof

such

employees.

13


28/117

2. Relationship

Between

f(i)

an d

G(i).

The

relationships

betweentheattrition

rates,

f ,

and

thesurvivorrates,G,re

the

following:

a .

(i)=G(i)-G(i+l).

Giventh e

G

rates,

the

above

formula

computes

the

corresponding

f

rates.

b .(i+l)=G(i)-f(i).

Giventhe

frates,

theabove

formula

computesthecorresponding

G

rates

using

the

additionalfact

that

G(0)=

1 .

C. SUBMODELS

Threespecific

submodelsare

discussed

in

thissection.he

differences

amongthe

threesubmodels

hingeon

theassumptionsmadein

thedefinitionsof

eachmodel.here

is

a

atural

rogression

of

thought

ranscendinghe

hree

ubmodels

nwhichhe

irst

submodelmakesthreestringent

assumptions

and

the

remainingtw osubmodelsrelax

some

of

theassumptions

in

order

to

make

the

modelmorewidely

applicable.

1 .

SubmodelA.

a.

odel

Assumptions:

(1)ystem

size

is

fixed,

at

size

N ,

at

the

en dofeach

timeinterval.

(2)

Attimezero,

all

employees

have

zero

yearsof

service.

(3)llrecruitsenteringthesystemstartwithzeroyearsofservice

at

theirtime

of

entry.

b .eplacementRates

a nd

Number

ofRecruits.

The

eplacement

ates,

(i),

re

omputed

sing

he

ormulas

ound

n

Reference

4:

page

64 .he

number

of

recruits,R(i),

is

computed

as:

R(i)=

Nh(i)

where:

1 4


29/117

(1)

(i )

=th erate

twhich

ecruits

re

ireduring

heeriod

[i ,

i+1).

(2 )R(i )

=

th eexpectednumberofreplacements

or

recruits,uring

the

period[i ,

i+1).

c. Steady

State.

Theformulasforcomputingth e

steadystatereplacementratesandsteady

state

numberofrecruits

for

submodel

A

arefound

in

Reference

4:

pages67-69.

2.

Submodel

B.

a.

ssumptions.

Submodelakes

he

am essumptions

s

ubmodel

ith

he

exception

of

assumption

2).

nubmodelB,

ssumption2 )selaxedollow

distributionof

length

of

serviceamong

all

personnel

at

t imezero.

b.

eplacement

Ratesand

Number

of

Recruits.

Theeplacementates, '(i),re

omputedsingheormulas

oundn

Reference

4:page

66 .

he

numberof

recruits,R'( i) ,arecomputed

as:

R'(i) =

Nh'(i)

c.

teadyState.

The

steadystatereplacement

rates

for

SubmodelB

are

th e

sameas tothose

computed

in

submodel

A.

3 .

Submodel

C .

a.

Assumptions.

Submodelakes

he

am essumptions

submodel

ith

he

exceptionof

Assumption

1).ere,heriginalystemizem ayehangedy

he

creationof

M(i)

newbillets

at

future

periodsi

=

1,2, .. . .

15


30/117

b.

eplacement

Rates

and

Number

ofRecruits.

The

replacementrates,"(i)

and

R"(i ) ,are

computed

using

theformulas

found

in

Reference

4:

pages

70-71.

c .

teady

State.

The

steadystate

numberofrecruitsdoesnotexistforsubmodelC.

The

steady

state

replacement

rates

arebeyond

thescope

of

thisthesis.

D. U S E R ' S

M A N U A L

AN D

E X A MP L E S

Theser'smanual

or

heREPLACE.XLS

modelsocated

n

Appendix .

AppendixG containsthree

example

problemsusingtheREPLACE.XLSmodel.

16


31/117

IV .

U L T I G R A D E


WITHNONI N S T A N T A N E O U S

FILLINGO FV A CA N CD3 S

A.N T R O D U C T I O N

TOT H E

M O D E L

The

urposeof

a

multigrade

acancy

model

soorecasttocksof

vacancies,

numbersofpersonnelfilling

jobs,an d

flowsofpersonnel.

nliketh esinglegrademodel,

this

odel

llows

or

he

ovementetween

ategories

ithin

heystem.

V A C A N C Y . X L S

computes

vacancies,vailable

jobs,

nd

th e

numberofpeople,n

each

category,

based

on

input

criteriafurtherdescribedin

thischapter.

Amore

etailed

xplanation

fth e

heory

nd

ssumptionsaneoundn

Reference1 ,pages152-156,an dReference4,pages71-81.

B.

OTATION

A N D

DEFINITIONS

The

following

notation

an d

definitions

are

usedin

the


model:

1.

(t).

n(t)

isth e vectorof

th e

number

of

jobs

in

each

categoryattime

t.

2.(t).

v(t)

isth evectorof

th enumberof

vacanciesineachcategoryatt ime t.

3.

and

W.

wis

th evectorof personnelattritionratesin

each

category

during

an y

oneperiod.

W

is

a

square

matrix

withth eattritionrates

n

its

main

iagonalnd

eros

verywhere

else.

4.

.

TheSmatrixisth e

transitionratematrix

which

governs

th e

internal

vacancy

flows

amongth ecategories.

17


32/117

5.(t).

e(t)sth evectorofth enumberofpersonnel

n

ach

ategoryat

time

.(t)s

computed

using

th e

formula:

e(t)=

n(t)-v(t).

6.

lpha.

Alpha

is

th e

growth

rateof jobsineachcategoryduringanyperiod.lpha

can

be

positive,zero,

or

negative.positive

alpha would

imply

an

increase

in

th e

number

of

jobs

of

all

categoriesatthat

samerate.

zero

value

ofalpha

wouldindicatenogrowth

in

th e

number

of

jobsin

allcategories.

negative

alpha

would

imply

adecreasein

the

number

of

jobs

of

all

categories

at

that

same

rate.

C.

BASICEQUA TION

FO RC O M P U T I N GV A C A N C I E S

The

basic

equation

used

in

predicting

vacancies

is:

v(t)=

v(t-l)S

+

e(t-l)W

+n(t)-n(t-l).

The

equationis

explainedthrougheach

of

th e threecomponents.

1-

v(t-l)S.

Thistermisthenumber,ndlocation,of

vacanciesoccurringduetoth einternal

movementof

vacancies

over

a

period.

2.

(t-l)W.

Thisermsheumberofvacanciesreated

ver

eriodueo

ersonnel

attritions

from

th e

system

in

all

categories.

3 .

n(t)-n(t-l)).

This

term

is

the

number

of

vacancies

created,or

eliminated,by

th e

creationof

new

jobs,

orth eelimination

of

existing

jobs.

18


33/117

D .

A L GORITH M

FO RC O M P U T I N GV A C A N C I E S

The

modeluses

a

ten

step

lgorithmtocompute

vacancies

successively

att imes

t

=

1 ,2,

...he

stepslisted

incorporateth e

restrictions

imposed

onth e

system

by

reality.

1.

ompute

K.

Ksheumberof

vacancies

hat

re

reated

nneeriods esultof

th e

internalmovementof vacancies.

K

=

v(t-l)S.

2.

omputen(t-l).

Thisis

th evector

of

th enumbers

of jobs

in

each

categoryat

time

t-1.

D(t-l)

=(l+a)

t

1

n

0).

3.

ompute

e(t-l).

Thisis

thevectorofth e

numbersof

personnel

ineach

category

att ime t-1.

e(t-l)

=

n(t-l)-v(t-l).

4 .

ompute

X.

Since acancy

an

move

nneirection

nly

when

ersonmovesnhe

opposite

direction,the

number

of

vacancies

created

by

internal

vacancy

movements

cannot

exceed

th e

available

numberof

peoplemoving

in

th e

opposite

direction.

X

is

th e

vector

that

ensuresthisruleisnotbroken.

X

=Min(K,_e(t-l)).

5.ompute

Y .

Y

is

the

vector

of

vacancies

created

bypeople

leaving

th e

system.

Y =

e(t-l)W.

19


34/117

6.

Compute

Z.

Z

is

jobs

during

th e

vectorofvacancies

th e

period

(t-1,

t] .

createdoreliminatedbyth ecreationoreliminationof

Z

=

an(t-1).

7.

ComputeU.

This

stepsums

upth ethree

up

th e

results

ofsteps

4,

5,

an d

6.

waysvacanciesmay

be

created.

TocomputeU ,ad d

U

=X +Y+Z.

8.

ComputeQ.

Q

is

th e

vectorthatensures

any

category.

thatnegative

numbers

of

vacanciesarenotcreatedin

Q =M ax

(U

,0).

9.

Compute

n(t).

This

is

th e

vector

of

the

numbersof jobsineachcategoryatt ime

t.

n(t)

-

(l+a)n(t-l).

10.

Computev(t).

Thenumber

of

vacancies

cannotexceed

th e

number

of

jobsin

necessitates

choosing

the

smallervalue

between

components

of

Q

and

an y

category.

This

n(t).

v(t)

=

Min(Q,n(t)).

2 0


35/117

E.

INPUTOPTIONS

Tomakeuserinput

easier,

swellas

foster

a

better

understandingofthe

course

materialthrough

visualization,wo

input

options

are

providedfor

theV A C A N C Y . X L S

model.

hetw o

options

arehierarchical

an d

general.

1.

ierarchical

system.

Aystemsalled

ierarchical

f,

uring

ny

neeriod,he

nly

lowsre :

vacancydemotionstoth e

next

lower

category,

vacancy

attrition,

rvacancies

remaining

unfillednheam e

ategory.

hese

orrespondoersonnelromotions,

ersonnel

attrition,

andpersonneltransfer

within

th e

same

category,

respectively.

hereareneither

doublevacancydemotionsnor

promotions

of

vacancies

allowedinahierarchical

system.

For

this

reason,

th e

Smatrixof

a

hierarchical

system

ca n

only

have

positive

elementsin

th emaindiagonal

and

immediately

below

it.

2.

eneralsystem.

The

general

system

covers

allpossible

vacancy

movement

situations.

here

are

no

restrictions

on th e

placement

of

th e

positiveelementsin the

transition

matrix.

E.

USER'S

M A N U A L A N D

E X A M P L E S

The

user's

manualor

he


model

s

ocated

n

Appendix

D.

Appendix

H

contains

three

example

problems

using

th e


model.

21


36/117

22


37/117

V .ULTIGRADEV A C A N C YM O D E LWITHI N S T A N T A N E O U SFILLINGO F

V A C A N C I E S

A .N T R O D U C T I O NO F

TH E

M O D E L

The

urposefth e

N S T A N T A N E O U S . X L S

odel

s

o

orecast

ersonnel

movementsduring

a

period.

his

model

operates

under

th eassumption

that

all

vacancies

are

illednstantaneously.hilehisonceptm ay

tretch

eality,

t

maye ood

approximationof

systemsthat

fill

vacancies

intime

periods

thataresmall

fractions

oftheir

accounting

periods.inceall

vacanciesar e

filled

instantly,

teady

stateis

achievedduring

one

period.

herefore,

th eonly

resultsof

th e

model

are

steady

state

results.

his

model

isnot

dependent

on

time.

A

more

etailedxplanation

f

heheory

nd

ssumptions

an

e

oundn

Reference

1 ,

pages

146-152,

and

Reference

4,

pages

81-86.

B .

OTATION,

DEFINITIONSA N D

EQUATIONS

Theollowingotation,efinitions

nd

quationsrese d

n

he

I N S T A N T A N E O U S . X L S

model:

1.

.

n

is

the vector

of th e

initialnumber

of jobs

in

each

category.

2.

.

w

is

th e

vector

ofpersonnel

attrition

rates

in

each

category.

3.

.

m

is

th e

vectorof

th e

umberofjobs

eing

reated

nach

ategory

uring

period.

negativecomponentof

m

impliesthat

jobs

are

beingeliminated

in that

category.

For

example,

m

=

(5 ,

3,

)

impliesthat

five

jobs

are

created

in

category

one,

threejobs

are

eliminated

in

category

tw o

an d

th e

number

of

jobs

in

category

three

remains

th e

same

duringth eperiod.

23


38/117

3..

This

vectoris

given

by

th eformula:

u=

(njwi+

m i,

n

2

w

2

+

m

2

,

..,

n

k

w

k

+

m

k

).

Each

componentof

this

vector

is

th enumber

ofvacanciescreatedinacategoryby

the

attrition

of

personnel

and

th e

creation/eliminationof

jobs

in

that

category.

The

vector

isderived

by

taking

th e

greater

value

in

each

category

between

th e

zero

vectoran du.heformula

is:

g

=

Max(0 ,u) .

This

formula

assures

that,

ineachcategory,

th e

number

of jobs

eliminated

does

not

exceed

th enumber

of

jobs

vacated

by

attriting

personnel.

6.

S .

The

S

matrix

is

the

transition

ratematrix

which

governs

th einternalvacancyflows.

The

o

atrix

is

the

matrix

augmentedwithn

dditional

olumn,

he

"0

U

" '

olumn,

consistingof vacancyttritionates.

he-transposeof

S

0

,writtens

0

\sheam e

matrix

withhe

ows

ndolumns

nterchanged.

he

0

nd

0

'

atrices

ave

he

following

appearances

when

constructed.

o

Sl O

S2 0

S 3 0

S k O

S l l

S 2 1

S 3 1

S k i

S l 2

ik

S 2 2

2k

S 32

3

k

Sk2

kk

2 4


39/117

S o

o

S lO

S20 S30

Sll

S2 1

S3 1

Sl2

S 22

S 32

S lk

S

2

k

S 3 k

Sk O

S k i

Sk2

S k k

Note

that

theSo'isa

matrix

with

k+1

rows

an dk

columns.

7.

D.

The

D

matrix

is

the

inverse

of

the

I

-

S

matrix:

where

Iis

the

identity

matrix

of

the

samesize

asS .n

equation

form:

D

=

( I -S)

1

.

8.

f.

f

is

avector

produced

by

the

matrix

multiplicationof

q

and

D.n

equation

form:

f=(a)(D)-

This

ector

must

e

onverted

o matrix

with

+1ow s

nd olumns

o

conformtothesizeofS

0

'.

ac h

of

thek+1ow sarethesame.

his

isaccomplished

by

establishing

thematrix

F

to

have

k+1rows,

each

on e

identical

to

thevectorf .

he

matrix

Fas

the

form:

F

=

fl

f

2

f

3

fl

f

2

f

3

fl

f

2

f

3

fl

fk

fk

fk

25


40/117


41/117

V I . EXCEL

MODELING

FEATURES

AND

PROCEDURES

A .

FEATURES

AND

FUNCTIONS

I t

is

not

th e

intent

of

this

thesis

to

explain

all

of

th e

spreadsheet

techniques

used

in

th e

models

createdfor

this

thesis.he

features

and

functions

described

belowar e

chosen

because

theyareeither

used

often

in

th emodels

or

they

are

rarely

taken

advantage

of

by

th e

casual

spreadsheet

user.

1.

acros.

Macros

re

xcel

eatures

hatllow

he

ser

o

erformredetermined

sequence

of

taskswith

oneclick

of

a

button.

he

advantage

ofmacros

is

that

th e

userca n

define

th esequence

of tasks

to

fi t

specifically

tohisownneeds.eforecreatinga

macro,

it

isadvised

thatthe

user

first

test

th e

desired

process

onan

example

of

known

result

to

make

sureth e

correct

result

occurs,

t

is

alsosuggested

thatth euserwrite

downth esteps

onapieceofscratch

papertobe

followed

uring

th e

reationof

th emacro.efer

to

either the

help

function

oran

Excel

user's

manualfor

moredetailedinformationonhowto

create

a

macro.

2.

f,Then,

Else.

The

Ifthen"

tatement

is

a

basicdecision

statement

n

computer

programming.

Excel

s

imited

o

n

If,

hen,

lse"

tatement.

ince

Excel

s

ot

rogramming

language,itisincapableofperforming"If ,then,do"functions.

hislimitationin

Excel

is

thecauseofth e

burdensome

amount

of

if

statements

usedintheformattingportionsof

the

M A RK OV .X L S

model.

3.

atrix

Multiplication.

Matrix

multiplication

s

he

ackbone

of

th e

mathematicssednhe

Markov

Chain

quations.xcel

as

uilt

nunction

esigned

pecificallyo

andle

matrix

multiplication.

he

unction

oes

ot

ermitnvalidrray

ntries

whichakeshe

functionmore

user-friendlyto

th e

userwho

understands

th e

rules

ofmatrix

multiplication.

Before

omputing

matrix

multiplication,

he

sermust

irst

ighlight

heestination

vectoror

matrix.

hehighlighted

areamust

be

th e

propersize

accordingtoth erules

of

matrixmultiplication.Afterentering

th e

formula

in

the

formulabar,

Excelrequiresthree

27


42/117

keyboard

eys

o

eressed

imultaneously. The

hree

eysre ,ontrol,hift,

nd

Enter.

4.ffset.

The"offset"functionallowsth euserto

select

avalueor

an

array

of

values,or

display

or

further

computation,based

ona

volatile

input

parameter.

Offset"

an

usea

variable

such

asth e

forecast

number

of

years

and

return

th e

valueofth e

result

forth eyear

forecast.hisfunction

is

used

often

when

follow

on

computationsarerequiredwhichare

dependent

on

volatileinput

parameters.

5.ranspose.

The transpose function

simply

converts

columns

to

rows

and

rows

tocolumns.

This

function

can

be

used

on

matrices

as

well

as

vectors.

lthough

this

does

not

sound

like

afunction

advancedenoughto

mention,

ts

t ime

saving

quality

is

fantastic

because,

e.g.

withoutthis

function,

interchangingof

rows

and

columns

ofa

20X

20

matrixwould

requireat

least

400

individualcellentries.

6.

atrix

inverse.

Theatrix

nverse

unctionperates

nder

imilar

ules

s

he

atrix

multiplication

function

described

above.

he

matrix

inverse

returns

the

inverse

ofa

square

matrix.

he

user

must

be

aware

of

th e

rules

for

taking

th e

inverse

of

a

matrix

in

order

to

highlight

th e

appropriatedestination

area.

7.

aximumand

minimum.

These

twofunctionsar e

used

an y

t ime

th e

user

wants

toknoweither

th emaximum

or

minimum

value

of

an

array.

lonethese

functions

do

little

morethanreturn

a

value.

The

ower

fhese

unctions

omes

henmbedding

hemn

ther

unctionsr

displaying

ranges.

8.

ount.

Thecount unctionimply

ounts

th e

umberofentriesmade

in

anrrayf

cells. Toma k e

his

unctionmoreseful,t

s

ftense dnonjunctionwithother

28


43/117

functions.

Twopecific

ases

reo

ommon

that

oftware

esigners

reated

pecial

count

functions

for

them:

a.

ount

blank.

"Count

blank"

ountsa l lof

theblankcellsinan

array.

cell

with

the

value"0"

ntered

n

t

s

ot

ounted

s lank

ell.

his

s

ne

ofthe

ew

Excel

functions

thatdifferentiatesa

zero

from

an

empty

cell.

b.

ount

if.

The

"count

if

function

allows

the

user

the

ability

to

select

thecriteria

by

which

the

count

function

isconstrained.

he

user

can

usea

host

of

functions

to

create

the

criteria

uchshemaximum

function

xplained

bove.

t

hould

eotedha this

function

can

be

recreatedby

the

userby embedding

a

count

function

inan

if

statement.

9.

mbedded/nested

functions.

Excelallows

theuser

to

embed

functions

within

functions

many

layers

deep.he

term

"many"is

used

because

there

are

limitations

whichvary

from

functionto

function,

ye t

it

is

arare

case

when

this

limit

is

reached

in

practice.he

best

way

to

visualize

the

processofembedding

unctions

so

ecallhe

omplexitiesurrounding

he

seof

parentheses

in

simple

addition

and

multiplication.

he

usermustkeeptrackoftheorder

of

the

functions

to

be

performed.

or

many

people,

thestepsused

in

formulating

a

series

of

embedded

functions

maygetextremelyconfusing.

or

this

reason,

a

sketchof

a

flow

diagram

is

highlyrecommendedforany

userattemptingtoembedseveralfunctions.

his

flow

diagram

should

be

done

before

attempting

toenterformulas

into

the

formula

bar.

B. MAKINGCHANGESTOTHEMODELS

All

of the

modelscreated

in

this

thesis

arepasswordprotected.his

is

doneso

the

user

oes

ot

elete

ecessary

ellnformation

y

ccident.

hemodels

reoth

workbook

nd

worksheetrotected.

f

a

worksheet

s

rotected,

tmeans

hat

ll

protectedcells

in

the

worksheet

cannot

bechanged

bytheuser

without

unprotecting

the

worksheet

first.ftheworkbookis

protected,

itmeansthatallhiddensheets

cannot

be

shown,thearrangementof

displayed

sheetscannotbealtered

an d

sheets

cannot

beadded

or

deleted

without

unprotecting

the

workbook

first.

Theonly

sheet

not

protected

inany

29


44/117

ofthe

models

is

the

"Student

Worksheet". Only

an

experienced

user

should

attempt

to

makealterationstoamodel.

Tomakechanges

toany

of

the

models

in

this

thesis,thepasswordthatprotects

the

workbookandworksheetsfromtampering

must

be

known.

he

password

sednl l

workbook

an d

worksheet

protections

is

the

ame.

o

unprotect

either

a

workbook

or

worksheet,clickonthe"Tools"dropdownmenulocated

at

thetopofthescreen,elect

"Protection",

nd

hooseitherthe

heet

rtheworkbook,

epending

nheyp eof

changes

beingmade.

he n

enter

the

password

xactlyasgivenbelow.hepassword

used

in

all

of

the

models

is

CURTIS .

t

is

important

torememberthat

the

passwordis

case

sensitive.hispassword

is

anallcapital

letter,

sixletterword,withno

punctuation

or

spacing.

30


45/117

APPENDIX

A.

O M P U T E R

SY STEM

H A R D W A R EA N D

SOFTWARE

R E Q U I R E M E N T S

Thereare

certain

minimum

levels

of

computingpoweran dtechnologyrequiredto

operateth emodels

iscussed

inthisthesis.

hese

equirementsre

eparatednto

wo

categories.

he

two

ategories

re

ystemardwareequirements

ndmodel

oftware

requirements.

A.

YSTEMH A R D W A R ER E Q U I R E M E N T S

MAR K OV .XLS ,E P L A C E . X L S ,

A C A N C Y . X L S

ndN S T A N T A N E O U S .

XL Saredesignedfor

useonpersonalcomputers.heremustbeatleastfivemegabytes

( MB)of

memory

emaining

nth eardrive

n

rder

o

oadll

hreemodels.

he

computer

mustru n

at

a

speed

ofat

least

00

M Hz.he

computer

must

have

a

minimum

of

eight

M B

of

random

access

memory.

he

system

must

be

equipped

with

a

mouse.

At

his

ime,urrent

echnologysuc hreaterhan

he

inimumystem

requirements

escribedbove.heodels

un

aster

nigher

apacity

ersonal

computers.

or

this

reason,

it

is

recommended

that

th e

user

exceed

theminimum

system

requirements.

B .

O F T W A R ER E Q U I R E M E N T S

The

omputermusteperatingnheMicrosoftWindows5,

r

igher,

operating

system.

he

computer

musthaveMicrosoftExcel version7. 0forWindows

95,

or

higher.

tthis

time

thereisaversion

of

Excel moreadvancedthanExcel

7.0

or

Windows5

.

here

s

o

ncreasederformance

ained

y

sing

ewer

software

packages

toru n

th e

models.f

a

newer

versionofExcelisth emostcommonly

used

version,tmight

e

asiertounthemodeln

th eewer

version.smentioned

above,thiswillnot

enhanceperformance;it

will

make

loading

th e

model

more

convenient.

31


46/117

32


47/117

APPENDIX

B. USER'S

MANUAL

FO R

THE

BASIC

MA RK O VCHAIN

MODEL

A.

OADING

THEMODEL

In

order

to

use

this

Excel

based

model,

the

user

must

first

open

the

Excel

program

that

meets

thecompatibilityrequirements

statedin

AppendixA.

nce

Excel

is

open,

the

usershouldpenthefile"MARKOV.XLS"

ocated

intheubdirectoryof

therivein

whichthemodel

is

contained.

orusersinthe

Learning

Resource

Center

lab,Glasgow

203,

this

subdirectory

an d

itslocation

will

be

providedby

the

instructor.Users

who

load

the

application

on

their

homecomputer

willfind

the

fileinthe

ubdirectory

to

which

the

user had

copiedit

earlier.

B.

UNNING

THE

MODEL

1.

Step

(1)Start.

Once

the

model

is

displayed

onthe

screen,select

the

"Start"sheetbyclicking

on

the

tab

at thebottomof

the

spreadsheet.

simplified

step

bystepguide

to

the

MARKOV

model

is

presentedon thissheet.

cell

fo rthe

user'sname

is

alsoprovided

here.

or

the

inexperienced

ser,

tsecommended

hat

he

Start"

heet

e

rintednd

the

instructions

on

it

be

followed.

2.

tep

(2)

Choose

Model

Option.

Next,the

user

should

select

the

appropriate

tab

at

the

bottom

ofthe

spreadsheet,

correspondingto

theMarkov

modelthatbest

represents

the

problem.

he

three

options

are:

Length

ofService

(LOS),

Hierarchical(Hier),an d

General

(Gen).

oran

explanation

of

theconstraintsand

guidelines

forthesethree

options,referto

Chapter

IIof

this

thesis.

3.

tep

(3)

DataInput.

a.

Initial

Stock

a nd

Recruitment

Proportion

Vectors.

All

threeof

the

above

options

require

theinput

of

an

initialstock

vector

an darecruitmentproportion

vector.

heuserislimitedtonomorethan

20

categories

an d

the

recruitment

proportionvector

must

sum

toone.

33


48/117

b .

ransition

Matrix.

Each

optionrequiresspecificinput

to

derive

the

transition

rate

matrix.

he

following

explanations

detail

the

specific

input

requirements

for

eachoption:

(1)

OS

option.

he

user

must

input

either

the

attrition

rates

or

thecontinuationrates

in

rder

forthepreadsheettoevelophentiretransitionate

matrix.

(2)

ierarchicalption.

he

usermustnputothhe

ttrition

ratesandthepromotion

rates

in

order

for

the

spreadsheetto

develop

the

entiretransition

ratematrix.

(3)eneraloption.he

user

mustinput

all

positiveelements

in

the

transition

rate

matrix.

c .

ecruitment

Options.

Oncethetransitionratematrixisestablished,theuser

must

chooseon eof

the

ix"RecruitmentOptions".

f

theuserdoesnotmake

an

ntryintheRecruitment

Optionsection,

then "Recruitment

Option1"

is

automatically

selected

by

default.n

order

to

makeanentryin

the

Recruitment

Option

ells,

he

user

must

nput

he

ppropriate

valueinthe

cell

corresponding

to

the

optionselected.nexplanationof each

ofthe

six

possibleoptions

follows:

(1)ixedrecruitment.

his

is

the

defaultsetting

and

requires

no

entry

in

the

Recruitment

Option

section.

owever,

this

option

requires

an

input

in

th e

cell

for

initial

recruitment.

(2)

dditive

ncrease

r

ecrease

n

ecruitment.his

ption

requires

anentryintw o

cells.he

first

entry

is

avalue

in

the

Recruitment

Optionblock

equaltotheamountbywhich

the

userwantstotal

recruitmenttoincrease/decrease

each

year.he

second

entry

is

the

initialrecruitment.

or

example,ifthe

user

wants

to

start

with

aninitialrecruitment

of

1000andwantstodecreaserecruitmentby50ac h

year,

thenhese r

must

nter

-50"

n

Recruitment

Option

"

nd1000"

n

he

nitial

recruitmentcell.

(3)ultiplicativeincreaseordecrease

in

recruitment.his

option

also

requiresentries

in

twocells.

he

first

entry

isa

value

n

theRecruitment

Option

block

equal

tothe

percentage

the

user

wants

totalrecruitment

toincrease/decrease

each

year.

The

valid

valuesthatcanbeentered

in

theRecruitmentOptioncell

range

from

-1

to

34


49/117

1 ,

notincludingzero.hesecondentry

isth e

initialrecruitment.or

example,

if

th euser

wants

to

startwithan

initial

recruitmentof

1000

an d

wantstoincreaserecruitmentby5

percenteachyear,thenthe

user

mustenter" .05"

in

"RecruitmentOption

3"

and

" 1000"

in

th einitial

recruitmentcell.

(4 )dditiveincrease

or

decrease

in

totalsystemsize.hisoption

requiresavaluein

"Recruitment

Option

4"

equalto

th e

numberbywhichth euser

wants

th e

totalsystem

size

to

change

each

year.

hereis

no

input

of

initial

recruitment

here

as

that

isnot

a

value

to

bechosen

byth e

user.For

example,

ifth e

userwants

toincrease

th e

totalsystemsize

by00per

year,

then

th eusermustenter" 100"

in

"Recruitment

Option

4".

(5 )

ultiplicativeincreaseordecrease

in

totalystem

ize.

his

option

equires

umbernheangeof-1o,

otncludingero,o

e

ut

nto

"Recruitment

Option

".

hi s

umber

epresents

he

ercentage

hange

o

he

otal

system

size

each

year.

here

isnoinput

ofinitial

recruitment

here

as

that

is

not

a

value

to

bechosen

by

th euser.

or

example,if

th euser

wantstodecrease

th e

total

system

sizeby

three

percent

eachyear,then

th e

user

must

enter

"- .03"

in"Recruitment

Option

5".

(6)

ixed

total

system

size.

his

option

requires

th enumber

"1"

to

beentered

in"RecruitmentOption

6" .

his

holds

th e

system

size

to

th e

su m

of

th e

initial

stocks

for

the

baseyear.

here

is

noinputofinitial

recruitment

hereas

thatisnota

value

tobe

chosen

bythe

user.

d.

nitialRecruitment

Initial

recruitment

is

only

neededfor

RecruitmentOptions

(1) ,(2) ,an d

(3).

As

explainedabove,

there

is

no

input

of

initial

recruitmentin

RecruitmentOptions

(4) ,

(5 ) ,

an d(6)

asit

isnota

value

to

bechosen

by

th e

user.

e.aseYear.

The

base

year,

e.g.

998,

an

beentered

Documents

Development of Spreadsheet Models for Forecasting Manpower Stocks and Flows