STOCHASTIC CONTROL IN MANPOWER PLANNING

by

GRALl ABDALLAOUI MAAN

A Thesis submitted for the fulfilment of the

requirements of the University of London for the

degree of Doctor of Philosophy.

The London School of Economics and Political Science

1984

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ABSTRACT

Our concern is with control problems which arise in

connection wi th a discrete time Markov chain model for a

graded manpower system. In this model, the members of an

organisation are classified into distinct classes. As time

passes, they move from one class to another, or to the out-

side world, in a random way governed by fixed transition

probabilities. The emphasis is, then, placed on examining

means of reaching and then retaining the structure best

adapted to the aims of the organisation, with the assumption

that only the recruitment flows are subject to control.

Attainability and maintainability have received a

great deal of attention in recent years. However, much of

the work has been concerned with deterministic analysis, in

the sense that average values are used in place of random

variables. We adopt, instead, a stochastic approach to the

study of these forms of control.

We present some of the problems encountered when

evaluating probabilities related to the distribution of

stock numbers at different steps and we give a detailed

numerical comparison of different recruitment strategies.

An iterative method is developed to compute exact

values of the probabilities of attaining and maintaining a

structure in one step. It is designed for the special but

very important case of systems in which promotion is only

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possible to the next highest grade. Its efficiency makespossible the use of exact results in the comparison of therecruitment strategies, which was formerly accomplished bymeans of simulation techniques only.

As to the comparison itself, it emerges that thestrategy which, at each step, steers the system as far aspossible towards the goal is superior to all deterministicstrategies. Also, this strategy is shown to come close toproviding the highest level of control that is possible.

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ACKNOWLEDGEMENTS

I wis h to extend my deep thanks and grati tude to my

supervisor, Professor D.J. Bartholomew, for his invaluable

support, encouragement and guidance throughout the prepar-

ation of this work.

I am also most grateful to L'Institut National de

Statistique et d' Economie Appliquee, Rabat, Morocco for

supporting me. Thi s support has been both financial and

moral and without it the completion of this work would not

have been possible.

Addi tional financia1 support came from the Economic

Commission for Africa and for this I am also grateful.

For their patience and application in typing this

thesis, I would like to express my thanks to Mary Dobbyn and

Satta Jabati.

Finally, these acknowledgements would not be complete

without reference to the help and encouragement which I have

generously received from my family and friends.

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CONTENTS

TITLE

ABSTRACT

ACKNOWLEDGEMENTS

TABLE OF CONTENTS

LIST OF TABLES

LIST OF FIGURES

Page

1

2

4

5

8

10

CHAPTER I:1.11.2

1.2.11.2.21.2.3

1.2.41.2.5

1.3

1.3.11.3.21.4

INTRODUCTIONDefiniticns and notations

Maintainability in a deterministicenvironmentControl by promotionControl by recruitmentMaintainability under growth orcontractionPartial maintainabilityMaintainability and limit behaviourof n(t)Attainability in a deterministicenvironment

Control by recruitmentControl by promotion

Attainability in a partially stochasticenvironment

11

16

222326

27

29

29

323746

48

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Page

CHAPTER II:

2.12.22.2.12.2.2

2.2.2.1

2.2.2.2

2.2.2.32.2.2.42.32.4

PROBABILITY OF MAINTAINING OR ATrAININGA STRUCTURE IN ONE STEP

Notations and assumptionsExact methodsGeneral caseCase of upper-diagonal transitionmatricesEvaluation of the TrinomialdistributionEvaluation of the BinoIliialdistributionPrecision in the routine MAINTPerformance of the routine MAINTNormal approximationMaintainable and attainable regionsin a stochastic environment

52

525454

60

65

6767

7073

78

CHAPTER III: PROBABILITY OF MAINTAINING OR ATIAININGA STRUCTURE IN MANY STEPS 86

3.13.1.13.1.23.1.33.23.2.13.2. 2

3.33.3.1

3.3.2

Recruitment strategiesDeterministic and fixed strategiesAdaptive strategiesRounding R(T) to an integer vectorExact methodBasic formulae and limitationsCase of h=2Normal approximationRecruitment vectors that do notdepend on the flowsRecruitment vectors that dependon the flows

87

87899194

94

99101

101

104

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Page

CHAFI'ER IV:

4.14.24.34.4

4.5

4.5.14.5.2

MAINTAINABILITY AND ATTAINABILITY INA STOCHASTIC ENVIRONMENTNumerical approachRecruitment strategiesNumerical examplesComparison between the exact andsimulation methodsComparison of the recruitmentstrategiesCase of maintainabilityCase of attainability

107108

111

115

119

121122134

GENERAL CONCLUSIONS

REFERENCES

143

146

APPENDIX A: Resolution of some quadraticprogrammiig problems 150

APPENDIX B: Detailed results concerning thecomparison of different recruitmentstrategies 163

APPENDIX C: The Fortran routine MAINT 228APPENDIX D: The Fortran program GENERAL 237APPENDIX E: Fortran programs used in investigating

different recruitment strategies 240

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PageLIST OF TABLES

Table 1

Table 2

Table 3

Table 4

Table 5

Table 6

Table 7

Table 8

Table 9

Table 10

Number of combinations in Step Four

Execution times with routine GENERAL

Accuracy and execution time withroutine MAINT

Comparison of the multivariate Normalapproximation (NA) to P(O < £ < n/n)with the exact value (EV)

Number of evaluations of P(f(l)=v/n')

Computation times required to evaluateP(O < £(2) ~ mIn)

Comparison of the Normal approximation(NA) to P(O < £(2) ~ n/n) with theexact value (EV), in the case of therecruitment strategy F1

Comparison of the Normal approximation(NA) to P(O < £(2) < n/n) with theexact value (EV), in the case of therecruitment strategy F2

Systems to be investigated by exact methods

Systems to be investigated by simulation

58

59

71

76

97

100

103

106

117

118

Table 11.a The average structures maintained in asequence of 10,000 trials and the expectedstock vector at the 10th step 120

Table 11.b Variances of a sequence of 10,000 trialscompared to the exact variances of stockvectors at the 10th step 121

Table 12

Table 13

Table 14

Table 15

Table 16

Table 17

Table 18

Table 19

Tables 20-25

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The average bias of stock number n.(t)1

over a period of 10 steps, under theadaptive strategy

The average variances of stock numbersn.(t) over a period of 10 steps, under1

the adaptive strategy

The average mean square errors ofstock numbers n.(t) over a period of

110 steps, under the adaptive strategy

The average mean square errors ofstock numbers n.(t) over a period of

110 steps, under the fixed strategy

The average bias of stock numbersn.(t) over a period of 10 steps,

1under the goal strategy

The average variances of stocknumbers n. (t) over a period of 10

1steps, under the goal strategy

The average mean square errors ofstock numbers n. (t) over a period

1of 10 steps, under the goal strategy

The expected distances between thestock vectors at the 10th step andthe goal vector, under differentrecruitment strategies

Expected stock vector and mean squareerrors of the stock numbers at thethird step under different recruit-ment strategies (different cases)

Page

123

124

125

127

129

130

131

132

137-142

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LIST OF FIGURES

Page

Figure 1.1

Figure 1.2

Figure 2.1

Figure 2.2

Probabilities of maintainingdifferent structures by recruitment

Probabilities of attainingdifferent structures by recruitment

0.50 - Maintainable region

0.30 - Attainable region

80

81

83

84

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CHAPTER I

INTRODUCTION

During the last two decades a good deal of attentionhas been devoted to problems of control in manpower plan-ning. However, these problems are by no means so recent.Grinold and Marshall (1977), for example, argued that theconstruction of the pyramids of Egypt, the building of thegreat wall of China and the conduct of the military forceoperations of ancient Rome could not have been achievedwithout the assistance of manpower planners. But no recordsremain to show what their role was.

The earliest recorded work concerned with manpowersystems is generally attributed to actuaries and demo-graphers. They were the first to use mathematical techni-ques to forecast characteristics of distinct age groupswithin populations. In their original approaches, onlyaverages and ratios of such averag~s were used. Human popu-lations, their main concern, were large enough to ensure ahigh accuracy in the estimation of different flows and moresophisticated statistical approaches were not considerednecessary. Furthermore, forecasting was the sole purpose oftheir techniques, rather than control. They could do littleto influence the birth and death rates on which their pro-jections depended. These made the actuarial techniques

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inappropriate and insufficient to handle problems concerning

manpower systems in general and industrial organisation type

systems in particular. Indeed, such systems are not usually

as large as human populations and the ultimate function of

the modern manpower planner is as much control as forecast-

ing.

Some of the earliest work on the statistical approach

to manpower planning using modelling techniques can be

traced to the work of Seal (1945) and Vajda (1947) when the

consideration of stratefied populations with promotion from

grade to grade was first proposed. But Markov models, which

are the basis of most present-day work concerning graded

systems, were first used by Young and Almond (1961) and by

Gani (1963). Wastage and promotion flows were considered to

be governed by fixer transition probabilities and the pre-

diction of grade sizes was therefore the sole objective of

such models. In many organisations, however, the grade

sizes are not free to vary and promotion and recruitment can

only take place to fill vacancies as they occur. Models

designed for these systems were introduced first in

Bartholomew (1963) and much of their mathematical found-

ations were drawn from the renewal theory.

Applications and developments of the original Markov

models have been pursued since then but the emphasis on pre-

dicting stocks and flows remained dominant until the end of

the 1960s.

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Next, attention mov~d to aspects of control of amanpower system. Questions of how to steer the latter to-ward some desired direction became the main concern duringthe 1970s. In the basic model, the members of a system ororganisation are classified into distinct classes or groupsand, as time passes, individuals move from one class toanother as well as between each class and the outside world.The members of each class are homogeneous in the sense thateach member of a class has the same probability of a parti-cular move. The problem is, then, to devise means of reach-ing and then retaining the structure best adapted to theaims of the organisation. Control, therefore, has twoaspects which are known in the literature of manpower plan-ning as attainability and maintainability. Attainability isconcerned with whether or not a goal structure can bereached and, if so, by what means. Maintainability, how-ever, is concerned with remaining at the goal structure onceit has been attained.

Theoretical formulation and early developments ofthis model can be found in Bartholomew (1967). Since then,the related literature has been growing rapidly and researchhas taken different directions. Forbes (1970) investigatedthe control of manpower systems under expansion or contrac-tion. Morgan (1971) developed techniques for studyingcareer prospects in graded systems. Charnes, Cooper andNiehaus (1972) used goal programming techniques in theirmodel for pIanning the civi1ian rnanpower in the U.S . Navy

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Department. Davies (1973 and 1975) placed emphasis onattainability and maintainability in many steps. Heexamined and gave a geometrical description of families ofn-maintainable and n-attainable structures. Bartholomew(1973) advocated the use of linear and quadratic programmingto tackle problems of attainability in many steps. Vajda(1975 and 1978) gave a detailed discussion of the use oflinear programming methods and introduced weaker forms ofmaintainability in which it is sufficient to maintain thetotal size of some subset of grades. Grinold and Stanford(1974) developed several algorithms for calculating optimalcontrol policies and based their approach on a combined useof dynamic programming and generalised linear programming.Further developments as well as an account of application ofMarkov models across a wide range of social sciences can befound in Smith (1970), Bartholomew (1976, 1979 and 1982),Grinold and Marshall (1977).

Much of the early work on control, however, was con-cerned with deterministic analysis in the sense that averagevalues were used in place of random variables. Grinold andMarshall (1977) argued that the assumption in such ananalysis concerning the Markovian nature of the flows is notnecessary, and that the problem is essentially the same ifthe flow numbers are assumed to be proportional to thestocks from which they come. Their use of the term"fractional flow model" was therefore adopted in order toavoid the probabilistic connotation of the Markov chain.

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The first step towards exploring the consequences of control

in a stochastic environment and evaluating probabilities of

groups sizes being maintainable was taken in Bartholomew

(1975, 1977). Most calculations, however, were based on

simulation techniques and Mu1tinormal approximation. We

propose in the following two chapters to examine new

approaches for exact evaluation of the above probabilities.

Control can be exercised through three main aspects

of a manpower system: probabilities of loss from the

sys tern, promotion and demotion flows, recrui tment flows.

But, for practical reasons that will be discussed in the

remainder of this chapter and for other reasons inherent in

the methods used, our attention will be focussed on control

by recruitment.

A detailed comparison of different recrui trnent

strategies based on the new evaluation methods will be the

subject of chapter four.

We begin firstly with a brief review of the deter-

ministic theory.

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1.1 DEFINITIONS AND NOTATIONS

Let us consider a manpower system that consists of N

individuals of

(i=1,2, ••• ,k).

whom n.(T) are in class i at time T,1-

The vec tor n (T) = (nl(T), n2 (T), ... , nk (T) )

is referred to as the stock vector at T, and its elements as

stock numbers.

The partition of the system into classes, grades or

categories could be made in several ways and could follow

different criteria, but it is essential that each indivi-

dual, at each time, should belong to one and only one class.

The basis of a manpower classification is generally provided

by the nature of the system to be investigated and the pur-

pose of the investigation itself.

We assume that all flows take place at integral

points of time and the probability that an individual ~oves

from i to J. between one time point and the next is pij(i,j=1,2, ... ,k). If there are no gains or losses from or to

the outside world or if the losses are immediately made good

the system will be called a closed system. This assumption

is generally made when concentrating on the issue of

mobility of the individuals within the system. However,

losses are frequently observed and recruiting new indivi-

duals is often of primordial importance and has serious and

direct effects upon the evolution of a manpower system over

time. Thus, since our main concern is to investigate the

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evolution of a manpower system over time under differentrecruitment strategies, it is natural to suppose that thesystem is open and interacts with the outside world. Moreprecisely, we suppose that an individual in grade i mayleave the system with probability

kw. = 1 - 1: p .. > 01 j=1 1J

(i = 1,2, •.. ,k),

and new entrants at T are allocated to various grades pro-portionally to a recruitment vector

withk1:

j=1r. (T) = 1

Jand

rj(T) ~ 0 (j=1,2,...,k). The decision upon the total numberof recruits M(T) is supposed to be made at the end of aperiod of time, after flows and losses have taken place.

The implicit suggestion that membership of one classis the only factor which governs the flows and wastage prob-abilities could be regarded as very restrictive. However,since the partition of the system into classes is left tothe model-builder, this latter could include all the rele-vant factors which might influence different flows andlosses to define an appropriate classification and overcomethe restriction underlined above.

As to the way the stock numbers are related todifferent flows, let n ..(T) be the number of individuals who1Jmove from grade i to grade j during the period of time[T-l, T [, T>l. We have:

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kI niJ. (T) + M(T) x rJ.(T)

i=lj=1,2, .•• ,k

(1 .1)

This equation will give,

expectations:

after using conditional

kn .(T) = I n. (T-1 )p.. + M(T) x rJ.(T) (j=1 ,2, •..,k)

J i=1 1 1J

(1.2)

where the bar denotes the expected value.

Using matrix notation, the equation (1.2) becomes:

n(T) = n(T-1) P + M(T) x r(T) (1. 3 )

where P denotes the k x k matrix of transition probabili-

ties.

It can be shc~n that the stock vectors ~(T) and

n(T-1) will have the same total size if, and only if,

M(T) = n(T-1) * w' , where w denotes the row vector of

wastage probabilities and w' denotes its transpose. In this

case, equation (1.3) becomes:

n(T) = n(T-1)P + n(T-1) * w' * r(T) (1.4)

If we confine ourselves to the expected values of

different aggregates of a manpower system, equation (1.3)

provides a concise description of the state of the system

and the way in which it is changing. However, the same

equation holds in the case of p .. and w. being considered as1J 1.

fixed proportions. Thus, any analysis which relies only on

- 19 -

expected values of stock numbers and flows will be con-

sidered to operate in a deterministic environment and will

be called a deterministic analysis.

Recruitment policies and probabilities related to

different flows, as described above, might be understood to

be given and not subject to any control. Consequently, it

could be assumed that the only purpose of the model is to

make predictions about the future of the manpower system.

This is untrue; while prediction is an important feature of

a manpower model, its use in advising the best actions upon

different flows, to direct the evolution of a manpower

system toward a desired objective, is of the utmost impor-

tance. However, this kind of control could turn out to be

very difficult in practice; present environment and state of

the manpower system, social and economic considerations are

often the key factors which deny the management a meaningful

influence on all flows or even on some of them.

The wastage flows, for example, are subject mainly to

considerations, inherent in the iudividuals themselves and

are very difficul t to control. Obviously they could be

influenced directly through redundancies, but this action

could provoke strong reaction from within the organisation

and, in any case, should not normally be planned by effi-

cient management.

- 20 -

Another sensitive area of control is concerned withthe promotion flows. Although the latter result from directmanagement decisions, an action upon them could affectseriously the career prospects of the individuals within theorganisation and its implementation could be hampered byinternal factors such as hostile reaction from the indivi-duals concerned and lack of appropriate qualifications, orexternal factors such as legislation, competition and publicimage considerations. The environment of the organisationis therefore an important factor which makes any action onthe internal flows very difficult a!ld consequently infre-quent.

The recruitment flows also emanate from directmanagement decisions but present fewer difficulties than thepromotion flows. Recruitment could affect to some extentthe individuals already in an organisation but on a muchsmaller scale than a direct action upon the promotion flows.

Through all of our work, we will confine ourselves totypes of control which involve action upon only one type offlow at a time; control by recruitment arises when recruit-ment is the only area of direct action. If the probabili-ties of transition from one grade to another are the onlyelements for which choice is possible, the control is thenby promotion. In both types of control, we will suppose

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that the wastage probabilities are given and that redundan-cies must not occur. The objectives of such control couldbe very varied and too numerous to be covered in generalterms within the confines of one or few models. Thus, wewill concern ourselves only with cases in which the purposeof the control is to reach a structure m from a structure nin one or more steps; that is with attainability. If m andn are identical, the control will be said to be concernedwith maintainability.

This objective is generally p'lrsued when the struc-ture m presents some desirable features or when it satisfiessome temporary needs. If attaining a structure m in a fixednumber of steps turns out to be impossible, we will tryinstead to get as close as possible to m; a variety ofdefinitions for distance between two structures will beconsidered.

In the remainder of this chapter, a deterministicanalysis of attainability and maintainability will be made;control in a stochastic environment will be the subject ofthe next chapters.

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1.2 MAINTAINABILITY IN A DETERMINISTIC ENVIRONMENT

In this section, we will be concerned with theevolution of a manpower system in a single period of time,say [T, T+l[, T>O. Thus, since there is no confusion aboutthe time considered, the suffix T will be omitted from allour notations.

A stock vector or structure n is said to be main-tainable if there exist values of P, wand r such that:

n = nP + nw'r (1.5)

Because of equation (1.4), it is implied that theexpected stock vector, at the end of a single period oftime, could be made equal to n, if this latter was theinitial stock vector at the beginning of the same period.It is also implied that the total of recruits is equal onaverage to the total losses.

If we could act upon all the elements P~ wand r, theproblem would be trivial and all structures would be main-tainable. But as discussed earlier, this is almost imposs-ible and it would be reasonable to assume that only someflows can be controlled.

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1.2.1 Control by Promotion

Let us suppose that the wastage probabilities vectorwand the recruitment vector r are given.then to find a transition matrix P such that:

Our problem is

n = nP + nw'r (1.6a)

k1: p .. = 1 - w. i = 1,2, ...,k (1.6b)

j=l ~J ~

p .. > 0 i,j = 1,2, ...,k (1.6c)~J -

Bartholomew (1973) showed that a necessary conditionfor the existence of a matrix P which satisfies conditions(1.6) is that:

n - nw'r > 0 (1.7)

let us prove that (1.7) is also a sufficient condition forthe maintainability of n.

We will introduce first a new problem and show thatthe structure n will be maintainable if, and only if, thelatter problem has a solution. Our proof will be completedby showing that (1.7) is a sufficient condition for thesolvability of the new problem.

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The problem we intend to consider derives from condi-tions (1.6) by multiplying each member of equation (1.6b) byn. and introducing new variables x ..=n. p .. (i, j=l ,2, ... ,k).1 1J 1 1JThe concern is then to find values for all variables x ..1J

(1 . 8a )j = 1,2, ... ,k

such that the following constraints are all satisfied:kEx. . = n. - (nw') r .i=l 1J J JkI

j=lx ..

1J= n.

1n.w.

1. 1i = 1,2, ... ,k (1.8b)

x .. > 01J

i,j = 1,2, ... ,k (1 . 8c )

It is not difficult to check that if there existvalues p .. which satisfy conditions (1.6), there will be

1J"values x .. which satisfy conditions (1.8) and vice-versa;

1Jthis establishes the equivalence between maintainability ofn and solvability of the new problem. To prove that con-dition (1.7) is sufficient for the solvability of the latterproblem, let us suppose that condition (1.7) holds; that isn. - (nw')r. ~ 0, j = 1,2, ... ,k. Thus, since we have also:

J J

n. - n.w. = n. (1 - w. ) > 0 i=1,2, ... ,k1. 1. 1. 1. 1. -

k kand E (n . - nw 'r .) = I (n. - n.w. ),

j=l J J i=l 1. 1. 1.

the problem defined by constraints (1.8) could be seen as abalanced transportation problem and therefore admits atleast one solution.

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The presentation of the maintainability issue interms of a transportation problem introduces great ease andhigh flexibility into the analysis; optimisation of somelinear function related to the variables p.. can be easily1Jincorporated and a postoptimal analysis can be performed.It does also allow us to benefit from the simplicity offinding a solution for conditions (1.6). It does not,however, help in characterising the maintainable regionbetter than condition (1.7).

In the above formulation, it was assumed that thereare no conditions imposed upon p.. other than conditions1J(1.6) • But, situations in which many , are required top. . s

~Jbe zero arise in practice. In such cases, we will concernourselves first by finding values of x .. which comply with

1Jconstraints (1.8) and minimise the linear function:

kL = 1:

i=lk1:

j=lc .. x ..1.J 1.J

where c. .1.J = + ~ if P is required to be zeroij_ a constant otherwise.

Then we will conclude that the maintainabilityproblem, with the additional constraints on p .., admits a1.Jsolution if and only if the minimal value of L is finite.Moreover, if the additional constraints on p .. require all,1.Jexcept (2k-1) p.. to be zero, the latter problem will have1.Jeither no solution or a unique solution. The latter case

- 26 -

will occur if the minimal value of L is finite, and the 2k-lbasic variables of the corresponding solution will be thosewith finite coefficients cij.

The important class of hierarchical manpower systems,where the promotion is possible only into the next highergrade, is an example in which (2k-l)non-zero values.

1.2.2 Control by Recruitment

p .. only could take1.J

In this type of control we assume that the wastageprobabilities vector wand the transition matrix P are bothgiven and cannot be subject to any modification. Theproblem is to find a vector r such that:

r. > 01.

(i=1,2, .• ,k) andkI:

i=lr. = 1,

1.

and which satisfies equation (1.5).

Bartholomew (1973) showed that the vector r definedby:

r = n (I-P)/nw' (1.9 )

is the only solution to our problem if and only if n > nP.He showed also that the set of all maintainable structureswith total size N, or maintainable region, is the convexhull of the points N x e. x (I_P)-l/d. (i = 1,2, ...,k),

1. 1.

where d. is the sum of the elements of the ith row of the1.

- 27 -

matrix (I_P)-l and ei is the row vector with all entriesequal to zero except the ith element which is equal to 1;e. (I_P)-l is thus the ith row of the matrix (I_P)-l.1

More precisely, it was shown that any maintainablestructure n with total size N can be expressed as:

k r.d. N (I_P)-lL ~ ~ (1.10)n = e.k d. ~

i=l 1: r .d. ~

j=l J J

where r is the corresponding solution to equation (1.5).

Equations (1.9) and (1.10) establish a one to onecorrespondence between a maintainable structure n and arecruitment policy r. In particular, it can be easilychecked that policies which allow recruitment only into asingle grade correspond to the vertices of the maintainableregion. Consequently, the recruitment policy whichallocates new recruits to different grades in equalproportions will correspond to a structure well inside themaintainable region.

1.2.3 Maintainability under Growth or Contraction

The assumption that structures at successive pointsof time should have the same total size could be seen to berestrictive. Generally the size of a manpower system is

- 28 -

subject to variations either imposed by external factors orfreely decided upon in accordance with future planningrequirements.

Let us consider the latter case and assume that thesystem is expanding or contracting at a constant rate, thatis N(T+1) = (1+0) N(T), where N(T) is the total size of thesystem at time T and 0 a constant bigger than -1. Thesystem will be said to be under expansion if a is positive,under contraction if a is negative or of fixed total size ifa is zero.

If 0 is non-zero, it will be impossible to maintain astructure n, but one could instead try to maintain therelative sizes of the grades. This is what Forbes (1970)termed quasi-stationarity. More precisely, our concern willbe to find a vector r or a transition matrix P, according tothe type of control, such that:

(1+0) n = nP + (nw' + aN)r (1.11)where N is the total size of the structure n.

We note that the total number of recruits takes intoaccount the total losses and the required change in thetotal size.

It can be shown that if a structure n is quasi-stationary under an expansion rate a, it will be also quasi-stationary under a bigger a'; 0 and a' are not necessarily

- 29 -

positive. It was also shown in Bartholomew (1973), that theset of quasi-stationary structures, in the case of controlby recruitment, is the convex set with k vertices propor-tional respectively to:

-1e. (I (1+a) - P)1

1.2.4 Partial Maintainability

(i=1,2, •.. ,k).

The notion of maintainability as discussed abovecould be generalised in what Vajda (1978) called partial-maintainability; a structure is said to be partially main-tainable if the total stock numbers of a group of grades iskept constant. This notion introduces a more flexible con-trol in the context of manpower planning, but unlike main-tainability, it is not repetitive; a structure which ispartially maintainable after one step is not necessarilypartially maintainable after two or more steps.

1.2.5 Maintainability and Limit Behaviour of net)

The purpose of this paragraph is to prove, for fixedsize systems and under fixed control policies, that themaintainable region is the set of all possible limits ton(t) when t tends to the infinity. This result will beproved to hold for either type of control, by promotion orby recruitment, and independently of the initial structuren (0 ) •

- 30 -

In addition to the simplifications of the notation it

introduces, the assumption that the same control policy will

be used at each point of time allows the proof to be made at

the same time for both types of control; according to the

nature of the latter, one could consider the recruitment

vector r to be given and the transition matrix P subject to

control or vice-versa.

The proof that any limit to n(t) is maintainable

follows directly from the definition of a limit and from

equation (1.4).

To prove that any maintainable structure m is a limit

to n(t) we consider again equation (1.4) and note that at

each point of time t we have:

n (t ) = n(t-l)P + n(t-l)w'r

- (t-l) (P w'r)= n +

= n(O) (P + w'r)t

= n(O) Qt

where Q = p + w'r.

It can be shown easily that the matrix Q and hence

all matrices Qt (t~l) are stochastic.

Moreover, with

- 31 -

the sole assumption that all w.1.

(i = 1,2, .•• ,k) are strictly positive, we can use results

from the theory of Markov chains to show that when t tends

to the infinity, Qt will converge to a stochastic matrix 7lrwhose rows are all identical and equal to a vector

n=(n1,n2, .•. ,TIk) and that n is the unique solution to the

equation n=nQ.

Iosifiscu (1980) showed that such kind of convergence

holds for a stochastic matrix A whenever there exists a

na tural number to such that the ergodici ty coeffici~nt of

Ato ·1.S strictly positive; the ergodicity coefficient

a(B) of a stochastic matrix B is defined as:

a(B) = mini,j

1:h

min

In our case, the ergodicity coefficient of the matrix

Q is itself strictly positive and therefore we are justified

in using the results stated above.

kIndeed, since 1: ri=l and ri~O (i=1,2, ... ,k), there

i=1

should be at least one jo for which r. >0 and consequently,Jogiven that w. > 0 (i=1 ,2 ,...,k) , all Q .. = p .. + w. r .1 1Jo 1.Jo 1 Jo(i=1 ,2 ,..• ,k) will be strictly positive. Thus:

kI min

h=1(Q.. , Q .. ) > 0 (i, j=1 , 2 , ... , k )1.Jo JJ 0

and hence:k

a (Q) = min ( I mini,j h=l

for

- 32 -

The limit behaviour of Qt as stated above means thatlarge values of t, Q(~~ will be independent of the

1J

starting grade i. It implies also, that the stock vectorn(t) will converge always to a unique structure n* regard-less of the initial stock vector n(O); more precisely:

n* = lim n(t)t+ CID

= n(O)T1= ( ~ n. (O)n., j=1,2, .•• ,k)i=l 1 J

It follows also that n* is the only solution to theequation n* = n*Q = n* (P+w'r). Thus, according to the typeof control, we can make the limit n* to be equal to m byconsidering either the transition matrix P or the recruit-ment vector r which verifies the following equation:

m = mP + mw'r (1.12)

A solution to this equat50n exists since m is maintainable.

1.3 ATTAINABILITY IN A DETERMINISTIC ENVIRONMENT

The latter argument was about attainability in thelong run and about situations in which the goal vector ismaintainable. The control policies were in addition con-sidered to be fixed and repeated at different points oftime. These assumptions are unduly restrictive and oneought to investigate the attainability in much more generalterms, that is how to bring a manpower system from aninitial structure n to terminal structure m in a fixed

number of steps,

- 33 -

using either control by promotion or byrecruitment. Obviously, except in the case of very specialsituations, this cannot be achieved in exact terms but onlyon average. Other constraints about the intermediatetrajectory of the system could be also added. They aregenerally assumed to be linear and often of the type:

n(t)g' = a(t) (t = 1,2, .•. ,T)

where the vector g and all numbers a(t) are supposed to begiven. Many interpretations of this equation could be made;Bartholomew (1973) imposed constraints on the total size ofthe system at intermediary steps and therefore consideredthe vector g to be equal to vector e whose all its com-ponents are equal to one. Grinold and Stanford (1974) con-sidered other situations as well as the case where g.

~

(i = 1,2,...,k) represents the financial costs incurred byhaving an individual in grade i and regarded n(t)g'to be thetotal manpower cost at time t and a(t) as the correspondingbudget.

The numbers a (t) could be supposed to be given andindependent or related to a(O) in o~e way or another; themost used relation is the following:

a (t ) te ng'

or a(t) = e a(t-1)

- 34 -

If g, for example, is the k-dimensioned vector with

all its components equal to one, a(t)=N(t) and hence the

above relation would mean that the system is growing at rate

( 9-1) • We have expansion, no growth or contraction as e>l,

9=1 or 9<1.

A structure m is said to be attainable in T steps or

T-attainable from another structure n if there exist T

transition matrices pet) and T vectors u(t) such that:

n(t) = n (t-l) P (t) + u (t ) t =1 , 2 , ••• , T (1•13a )

n(t)g' = en(t-l)g'

neT) = mt=1,2, ... ,T-l (1.13b)

(1.13c)

kI p ..(t) = 1-w.

j=l 1J 1

n(O) = nu(t) > 0, n(t) > 0- -

t=l ,2,... ,T

t=l ,2 ,... ,T

(1 •13d )

(1 . 13e )

(1.13£)

where the wastage probabilities vector w, the vector g and

the constant e are supposed to be given.

The transition matrices P(t) (t = 1,2, ... ,T) will be

supposed to be given or subject to control depending on the

nature of the latter. As to the vectors u(t) (t=1,2, ... ,T),

we should notice first that their components u. (t) are the1

expected recruit numbers to grade i at time t and therefore:

(t =1 , 2 , ••• , T )

- 35 -

In addition, the expected total of recruits M(t)

at time t could be shown to be determined by the need toreplace losses and to provide for the increase or decrease

in overall size, that is:

t=1,2, ...,T

Thus, even if the control has to be exclusively by pro-

motion, we cannot assume that the vectors u(t) are fixed;

they could still be influenced by the type of promotion

policy pursued through the numbers M(t); the rec~uitment

vectors r(t) only can be assumed to be fixed. On the other

hand, when the transition matrices are given and the control

is by recruitment, the vectors u(t) could be influenced in

many ways; if there are no constraints fixing the inter-

mediate total sizes of the manpower system, u(t) could be

influenced by acting either on the vectors r(t) only, on the

numbers M(T) or on both of them. In this case, we will

as sume tha t control is about r (t) and M (t) (t = 1,2, ... ,T)

at the same time. This, however, is not possible if the

total system size trajectory is imposed; the numbers M(t)

cannot be subject to control and the latter can be exerted

on the vectors r(t) only.

The number of periods T need not necessarily be con-

sidered fixed. In some situations, the overall exercise of

the control is precisely concerned with finding the adequate

value of T according to some optimality criteria that have

to be defined. Bartholomew (1973) considered the case where

- 36 -

the objective is to find the minimal number of steps T* thatare necessary to attain the structure m from a given struc-ture n. He referred to this problem as free time control.For its solution, he advised a method based on successivetrials by increasing each time the value of T* until anadmissible value is reached; the first trial should be forT*=l. At the final trial some optimality criteria could beadded in order to differentiate between admissible controlstrategies. This method gives rise to two remarks. First,the full range of prescribed trials have to be done and inthe same indicated order. The fact that an T-attainablestructure from n is not necessarily T'-attainable from thesame structure if T and T' are different, prevents us fromreducing the number of trials. This leads to the secondremark concerning the association between a fixed and a freetime control problem; as ~mplied, the latter could be lookedupon as a collection of successive problems of the formertype. This relationship between the two types of problemsallows us henceforward to concentrate only on the case whereT is assumed to be given.

- 37 -

1.3.1 Control by Recruitment

Here, the problem is to find a set of vectors

u (1),u (2 ),••• ,u (T) and n(1),n(2 ),..• ,n(T-1) such that:

t=1 ,2 ,... ,T-1 (1.14a )

t=1,2, ... ,T-1 (1.14b)

m =n(T-1)P + u(T) (1.14c )

u (t) > 0 t=1 ,2 ,..• ,T (1.14d)-

net) > 0 t=1,2, ... ,T-1 (1.14e)-

nCO) = n (1.14f)

All transition matrices are supposed to be given and

not functions of time. The stationarity assumption about

the transition probabilities is not necessary but is made in

order to simplify the no_ation; the models that will be dis-

cussed and the methods for their solution will be shown not

to depend on that assumption.

The foregoing problem can obviously be solved by

linear programming techniques. But as noted by Bartholomew

(1973) the enumeration of the number of variables and con-

straints will show that, among all possible solutions, those

provided by the simplex method would be too extreme to be

acceptable in practice. He showed that, if all components

of the initial stock vector are non-zero, a basic solution

will have at most T+k-1 non-zero elements to be distributed

among the k*T components of the vectors u(t) (t=1,2, ... ,T).

- 38 -

This will leave almost every vector with only one non-zerocomponent. Such a recruitment strategy will be difficult to

implement in practice especially if the grade to which the

recruits will be allocated keeps changing. Thi s extreme

behaviour of a basic solution could be lessened by the

introduction of new constraints on the variables u. (t) to1

make sure that the solution will be within acceptable

limits. These latter, however, have to be set with extreme

care in order to avoid producing infeasibility.

Another way to select a solution which does not

require sharp and repeated changes in the recruitment

policies from one step to another, is to consider as the

objective the minimisation of a function of the type:

TD= L

t=2

linear programming will still be used since such a function

could be written as:

v.(t) + w.(t) )1 1

with

u.(t) - u.(t-l) = v.(t) - w.(t)1 1 1 1

and vi(t), wi(t) ~ 0 for i=1,2, ...,k and t=2,3, ... ,T.

- 39 -

A solution to the problem defined by conditions

(1.14) can also be worked out by a combined use of Dantzig-

Wolfe decomposition algorithm and dynamic programming;

Grinold and Stanford (1974) suggested a way of decomposition

which leads to subproblems without terminal conditions on

the structure n(t) and showed that these subproblems can be

solved effectively by the use of dynamic programming. They

showed that it is possible to define more general terminal

conditions than (1.14c) or to optimise in addition a linear

function related to economic or social considerations during

the whole period of T steps. Incidentally, this was the

main concern of that paper with different assumptions con-

cerning the finiteness of T and the nature of the transition

probabilities.

As to the foregoing problem, there was only a mention

that its solution can be worked out in the same way as

explained above; in their account they assumed always that

an initial solution is already known. We give below a brief

description of how the method above can be adapted to our

case.

Let AT(n) denote the set of all T-attainable struc-

tures from n. The attainability problem is then concerned

with finding a structure y such that:

- 40 -

( 1)

To find a solution to problem (1), if it exists, wepropose to solve another problem constructed from (1), thatis :

Min L = (a+b)e'(2) Subject to:

y + a - b = m

y E Ar(n)a, b E B

where e is a k-dimensioned vector with all its componentsequal to one and B a set of vectors defined by:

B = {x/O<x.<m.. i=1.2 ....,kl._ ~_ ~ J J J

If the minimal value of L is strictly positive, wewill conclude that the problem (1) has no solution,otherwise a solution to (1) exists and the corresponding

T{x(t)J u(t)lt=l will be also provided. The upper bounds

on the components of vectors a and b are not necessary butthey can be shown not to affect the feasibility of problem(1); they were added in order to avoid complications thatwould arise in the decomposition algorithm.the master problem as:

Let us define

- 41 -

( Minimi se L =III

Subject to:

( 3 )

h yj h' ,(aj bj)L-Aj + L).j - = m

j=l j=l

h2- Aj = 1

j=lh' ,

~A . = 1

Jj=l

(El)

(E2)

(E3)

A . > 0, j=1 ,2 ,... ,hJ -

l,

A • > 0, j=1 ,2 ,... ,h 'J -

where 1 2 h points in the attainable regiony , y , ..., y are

AT(n) and aj bj (j=1 ,2 ,... ,h ') are points in the set B, all,supposed to be known. hand h' are integers greater than orequal to one. Initially they are set to one but they could

could be obtained by selecting firstiterations.change in subsequent

vectors 1 1 1y , a and bpoint 1 fromany y ~(n)

The initial starting

and define after a~ and b~~ ~

(i = 1,2, ... ,k) by:

1 1 and b~ 0 if > 1a. = m. - y. = m. y.~ ~ ~ ~ ~ - ~

1 0 and b~ 1 if < 1or a. = = y. m. m. y.~ ~ ~ ~ ~ ~

let '" '"( , " )A, A denote an optimal solution of problem (3) and

'" '" '"u, v and v' denote the associated dual variables correspond-ing respectively to equation (El), (E2) and (E3); u is a

- 42 -

A A

k-dimensioned column vector, v and v' are scalars. We need

then to solve two subproblems:

A "

{

Max D4 = yu + v(4 )

Subject to: y E AT(n)

{

Max DS = (a-b)u - (a+b)e'+v'(S )

Subject to: a, b E B

Problem (4 ) can be solved directly by means of

dynamic programming as shown in Grinold and Stanford (1974).

The solution of problem ( S) is straightforward and can be

soown to be as follows:A

a. = m. , b. = 0 if u. > 1~ ~ ~ ~

A

a. = 0 , b. = 0 if -1 < u. < 1~ ~ - 1. -

A

a. = 0 , b. = m. if u. < -1 (i=1 ,2 ,... ,k) .~ ~ ~ 1.

*Let D4an optimal

denote the maximal value of D4solution of problem (4). Let

and y* denote

*DS denote the

b*) an optimal solution of

*DS are equal to zero, then:

maximal value of DS and (a*,

*problem (S). If both D4 and

(y =hr

j=l

h' A , • A

r A. aJ, b =j=l J

h'I

j=lA •

A '. bJ )J

is an optimal solution of problem (3). If on the other hand*D4 is positive and * h+1bigger than DS' we will put y = y*,

zero.

- 43 -

increase the value of h by one and try to solve the new*problem (3). If DS is positive and bigger than or equal to

* h'+l h'+lD4, we will put a =a* and b =b*, increase h' by oneand try to solve the new problem (3). This process can besoown to take a finite number of iterations and then willallow us to know if the optimal value of L is positive or

In the latter case, a solution to problem (1) existsand from this one ought to pursue the optimisation of alinear function in order to choose eventually an alternativeand better recruitment strategy.

This method has the merit of reducing the number ofconstraints in problem (1) from T*(k+1) equations to (k+2)equations in problem (3) and has a special appeal because ofthe great ease which it offers in solving the generated sub-problems. The master problem (3) itself does not need to besolved each time from seratch; the revised simplex methodprovides an efficient way of updating successive solutionsof this problem. Nevertheless, the amount of work assoc-iated with the process of decomposition itself makes any netadvantage on the linear programming doubtful especially ifthe number of constraints in problem (1) is not very large.

Whatever method of solution is adopted, the attain-ability problem could have a unique solution or could evenbe infeasible precisely because of reasons inherent in themodel assumptions and constraints such as equation (1.14c).In such a case, the issue of selecting a less extremerecruitment strategy is irrelevant and one ought to settle

- 44 -

for less restrictive conditions and, rather try instead to

direct the manpower system towards a structure m. Exact

attainability would cease to be the fundamental objective.

Bartholomew (1973) considered the attainability

problem as defined by conditions (1.14) but dropped al-

together the requirement that n(T) should be equal to m.

To control the evolution of the system, he regarded as

objective the minimisation of a distance D(n(T),m) between

the terminal structure n(T) and the goal structure m. This

new problem is always feasible and allows for more flexibi-

lity. It can be used to solve the original attainability

problem with the additional advantage of providing, in the

case of infeasibility of the latter problem, a solution with

the nearest distance between n(T) and m; it suffices, for

example, to put:

D(n(T),m) =kL

i =12(n.(T) - m.) •

~ ~

A distance between two structures does not have to be

of the latter type and the method of solution will depend

heavily on its nature.

as :

More generally, it can be defined

D(n(T),m) =kI

i=lw.

~n. (T)

~'

am.~

a > 0

where wi (i = 1,2, ... ,k) is a set of non-negative weights

chosen to reflect the importance of holding deviations for

grade i as small as possible. A large value of a will tend

to prevent a relatively large deviation at a single or group

- 45 -

of grades but would make the solving process very difficult

if not impracticable. It is then no surprise to find, in

the context of the foregoing analysis, that much of he

published work is concerned only with cases where a is not

bigger than 2.

Bartholomew (1973) considered both cases, a=l and

a=2. He advocated the use of linear programming in the case

of the former and quadratic programming in the case of the

latter. Moreover, in the special case where T is assumed to

be equal to one and where the total size of the organisation

is considered to be fixed, he advised a very simple way in

obtaining the solution for both cases a=l and a=2. He

argued hat models with T being assumed equal to one are of

the utmost importance in practice in spite of the apparently

excessive restriction; putting T=l means that, at each step,

we will consider the finishing structure as a new start and

try to move the system as far as possible towards the goal

m. He stressed that such a strategy is fully justified if

we remember the stochastic behaviour of different flows. He

pointed out that even in a deterministic environment, this

strategy would achieve a good control of the system;

numerical comparisons between strategies that concentrate

only on the step ahead and others which minimise the

- 46 -

distance between n(T) and m, for aT> 1, were made toenforce his point.

Meh1mann (1980) suggested different means of solutionbased on dynamic programming techniques. His model con-sidered a slightly different type of quadratic function inthe definition of D(n(T),m). It allows also the use of anytype of control: by promotion, by recruitment or by both ofthem. Unfortunately, the way the control variables aredefined might lead to an optimal solution with either therecruitment vectors r(t) or the transition matrices P(t)being infeasible. In addition, the method itself requires aconsiderable amount of calculations including inversion ofmany matrices of kxk dimension, and makes additionalassumptions about the regularity of the latter matrices.

But, as mentioned earlier, a more efficient use ofdynamic programming can be found in Grinold and Stanford(1974) if the issue is not about reducing a distance betweenn(T) and m but rather about minimising a linear costfunction related to the whole period of T steps.

1.3.2 Control by Promotion

In comparison with the recruitment control, itappears that much less work has been devoted to the pro-motion control. An explanation could find its roots in anearly discussion about the nature of control by promotion,

- 47 -

its undesirable features and the practical difficulties that

would face its implementation. Other reasons related to the

modelling process itself could also be invoked; modelling

could be very difficult given the considerable number of

variables under control or might turn out to be a duplic-

ation of models already discussed in the case of control by

recruitment.

The latter case is well illustrated by Bartholomew's

models; it was shown when writing conditions (1.14) as:

k k k kn.(t)= Ln •• (t-l)+r. { L n.(t-l)w.+ L n.(t)- L n.(t-l)}

J i=l 1J J i=l 1 1 i=l 1 i=l 1

(j = 1,2, ... ,k; t = 1,2, ... ,T-l) (l.15a)

kL

i=ln. (t)g.1 1

k= e L

i=1n. (t-l)g.1 1

(t = 1,2, ... , T-l) (1.15b)

km = L

i=l

k k kn ..(T-l) + r. { L n.(T-1)w. + L m. - L n.(T-l) }

1J J i=1 1 1 i=l 1 i=1 1

(j = 1,2, ... ,k) (1.15c)

n .. (t) > 0 (i ,j=1 ,2, ... ,k ; t=O,1, ... ,T-l) (1.15d)1J -

n(t) > 0 (t = 1,2, ... ,T-1) (l.15e)-

n(O) = n (1.15f)

that the attainability problem can still be solved by means

of mathematical programming. It was stressed that all

eventual solutions resul ting from this method would have

extreme characteristics which would severely limit its

practical value.

- 48 -

Therefore, the previous discussion con-cerning model assumptions, and especially terminal con-ditions, remains of fundamental relevance.

1.4 ATTAINABILITY IN A PARTIALLY STOCHASTIC ENVIRONMENT

As was pointed out earlier, the analysis in theprevious sections is concerned only with the average stocknumbers and can be used without any modification if all p ..~Jand w. are to be considered as fixed proportions and not as

~

probabilities. It does not however provide sufficientinformation about the actual path that the system wouldfollow since the deterministic assumption holds rarely andvariation in the flow numbers have generally to be assumed.This led Bartholomew (1975, 1977) to initiate an investi-gation concerning the behaviour of a manpower system in astochastic environment. He tackled the evaluation of theprobability of attaining a structure m from anotherstructure n and compared different control strategies.These considerations will form the subject of the nextchapters and a detailed discussion of the difficultiesinvolved will be entered into. Here, we shall give a briefdescription of a recent model which is easier to manipulatethan Bartholomew's and yet presents fewer limitations thanthe deterministic model by allowing for variation in someflows.

- 49 -

Davies (1982) considered a manpower system of fixed

size N. He assumed that, of the n. (T) individuals in grade1

i at time T (i = 1, 2 , . . . , k; T = 0, 1 , 2 , . . .) a fix ed pro p0r-

tion p .. are promoted to grade j, i<j, at step T+1; no1J

demotions are allowed. The numbers of grade i who are not

promoted at step T+1 are assumed either to stay in grade i

with probability p. or to leave the system with probability1

1-p.. All proportions p .. and probabilities are considered1 1J

to be given and only the recruitment vector r is assumed to

be subject to control. Finally, recruitment is assumed to

take place after the wastage numbers are known.

Under these conditions, a structure n(T+l) is said to

be attainable in one step from another structure n(T) if,

and only if,

n. (T+1) >~

i-1L

j=ln.(T)p .. + n .. (T+1) (i = 1,2, ... ,k)J J1 11

where n .. (T+1) is the number of members of grade i at time T11

who will stay at the same grade at time T+1. Thus n(T+1)

mayor may not be attainable from n(T) depending on the

values taken by the random variable n .. (T+1) (i=1,2, ... ,k).11

The probability P(n(T+1)/n(T)) of attaining n(T+l) in one

step from n(T) is thus of importance and can be expressed

as :

P (n (T+1 )/n (T ) )k

= L.. P(n .. (T+1)<n.(T+1)11 - 1i=l

i-1- L

j=ln.(T)p .. /n(T))J J1

where nii(T+1) are independent Binomial variables with para-

meters (n. (T)p .. ,1 11

p. )1

and p ..~1k

= 1 - Lj=i+1

p ...1J

- 50 -

Davies (1982) showed that, for an initial structuren, there is one and only one structure n* attainable from n

in one step with probability one; n*=nP and P={p ..}. How-1Jever, it was pointed out that for a given goal structure n*,there is no guarantee that all of the entries in n*p-1 will

be non-negative. It was shown also that the one-step prob-

abilities decrease on lines emanating from n*; that is, for

any structure n'(~n*) attainable with non-zero probability

in one step from n, we have:

o < P(n'/n) < P( (l-).)n' + ).n*/n) < 1 o < ). < 1

As to the probability P(n(T) /n(O» of attaining a

structure n(T) from a structure n(O) in T steps, its value

will depend on the recruitment vectors r(l), r(2), ... ,r(T)

decided upon at the points of time t=1,2, ... ,T. More

generally, the distribu~ion of the stock numbers at

different points of time will depend largely on the adopted

recruitment strategy. The purpose of this latter, could be

then motivated by maximising or minimising a function, not

necessarily linear, of the stock numbers according to an

objective decided upon by the management.

Davies (1983) aimed at the maximisation of

P(n(T)/n(O» and called the maximal value of the latter as

the "best T-step probability". From a previous result, the

best T-step probability of attaining npT from n is equal to

one by taking the system through the same path (n, nP, np2,T••• , nP ).

- 51 -

But in general, no method was advised to find a path

from nCO) to neT) with the highest probability of attain-

ability. Nevertheless, an indication about how the best T-

step probabilities vary with the goal structure were

obtained by numerical calculations. It was suggested that,

within the regions of structures attainable with non-zero

probability, the best T-step probabilities decrease on lines

emanating from the structure npT.

Although the model discussed in this section is only

partially stochastic, it gives a sounding of difficulties

that would arise with stochastic flows and introduces a

sample of new questions that have to be addressed. In the

following chapters, we will consider the model where all the

flows are stochastic, discuss different ways of evaluating

probabilities of attainability and give a comparison of

different recruitment strategies.

- 52 -

CHAPTER II

PROBABILITY OF MAINTAINING OR ATTAINING ASTRUCTURE IN ONE STEP

In the context of stochastic control of a graded man-

power system, the probability of maintaining or attaining a

structure in one step is of fundamental importance.

Unfortunately, its evaluation has presented many

difficulties and has been considered a major obstacle;

Bartholomew (1977) tried to overcome the problem by using a

Multinormal approximation and Davies (1982) was inclined to

consider partially stochastic systems. In this chapter we

will describe the computacional difficulties that occur in

the evaluation of the above probability. In the case of

systems with upper-diagonal transition matrices, a powerful

and fast approach for such evaluation will be presented.

The Normal approximation and the difficulties involved in

evaluating the mu1tivariable Normal integral will be also

discussed.

2.1 NOTATIONS AND ASSUMPTIONS

As in an early model, suppose the system consists of

N individuals of whom n. (T) are in grade i at time T,1

i = 1,2, .••,k. Time is assumed discrete and the probability

that an individual moves from i to j between one time point

- 53 -

and the next is Pij (i,j = 1,2,...,k). An individual mayalso leave the system from grade i with probability wi.F(T) will denote the kxk flow matrix with elements n..(T),~Jthese being the numbers moving from i to j in [T-l, T[, T~l.The stock numbers at time Tare given by the relation

kn (T) = f(T) + R (T), where f(T) = ( >: n.. (T), j= 1,2,...,k )i=l ~Jand R.(T) is the number of new entrants to grade i at time T

~

(i = 1,2,...k). Decision upon recruitment is supposed to bemade after flows and losses have taken place. Since controlis by recruitment, a structure m will be attainable in onestep from n(T) if, and only if, f(T) < m. Therefore thevector f(T) is of crucial importance and also its distri-bution which is needed to evaluate all probabilities ofmaintaining or attaining a structure.

In the following sections we will be concerned with asingle period [T-l, T[, T~l. Since there is no conf~sionwhich T is being considered, we will omit the suffix in allour notations.

The stock vector n at the beginning of the period isassumed to be known; the vector to be considered at the endof the same period will be denoted by m.

- 54 -

2.2 EXACT METHODS

We will present two methods to evaluate the exactvalue of the probability of attaining the structure m.Whilst the first method deals with the general case, thesecond concentrates on a particular and important class ofsystems in which the transition matrix is upper-diagonal.

2.2.1 General Case

The probability P( 0 ~ f ~ min) of attaining thestructure m from a structure n in one step could be evalu-ated as follows:

v . ( 1 ), n2.J J

= v. ( 2 ) , ..• , nk .=V . ( k ) ,J J J

j = 1,2, .•• ,k)

(2 .1)where the summation is over all possible vectors v(1),v(2),...,v(k) such that:

kI: v(i) < m

i=l(2.2a)

kI: v.(i)

j=l J< n.

~ i = 1,2, ... ,k ( 2 • 2b)

v.(i) > 0J - i,j = 1,2, ... ,k (2.2c)

(n. ;P·l'P· 2'··· ,P·k)~ ~ ~ ~

and their distribution can be evaluated directly. Thus it

The vectors (n.., j=1,2, ...,k) (i=1,2,...,k) are in-~Jdependent Multinomials with parameters

would be useful to express P(O ~ f ~ min) as:

P(O < f < mIn)

- 55 -

= ~ ~ P(n .. = v.(i), j = 1,2, .•• ,kli =1 1. J J

(2.3)

However, the process of enumeration of all possible sets ofk vectors v(l), v(2), ... , v(k) which comply with conditions

k(2 .2 ), and the eva 1ua tion of n P (n. . = v. (i ), j=l, 2 ,... ,k )

i=l 1. J J

for each set, would be too long; the enumeration in itselfwould be very complex, and some probabilities would have tobe evaluated many times unless some sophisticated, and in-evitably time consuming, storage procedures were introduced.Moreover, a direct evaluation of P(n .. = v.(i), j=1,2, ... ,k)

1.J Jfor a particular i and vector v( i) would be very lengthygiven the large number of such evaluations that would beneeded. In order to avoid the problems discussed above, wepropose to proceed as follows:

Step One:

For each grade i, start by consideringv (i )= ( 0 ,0 ,.. .,0 ) and ass ume that P (n. .= 0 , j=1 ,2 ,...,k ) is1.Jequal to a constant C. Then, enumerate all possible vectorsv(i) which comply with conditions (2.2b) and (2.2c). Inthis enumeration, a new vector v(i) would be obtained froman old vector v'(i) byThis would allow the use

,adding one to an element v. (i).Jo

of a recursi ve relation of theMultinomial distribution to evaluate the probabilityP(n ..=v.(i), j=1,2, ... ,k); more precisely we will have:1.J J

P(n .. =v . (i) , j=1,2, •. ,k)1J J

- 56 -

k ,p.. (n. - I v. (i»1Jo 1 j=l J ,

= -- --,----- x pen .. =v.(i),j=I,2, .• ,k)w. (v. (i) + 1) 1J J1 Jo

(2.4)

The constant C is deduced from the fact that all probabili-

ties should add up to one.

Step Two:

For the remaining steps and for all grades i, ignore~c1allLvectors v(i), enumerated in the latter step, which donot satisfy the condition (2.2a).

Step Three:

Let i = 2 and s(l) = v(l)

Step Four:

iLet P( I nhj = s.(i), j = 1,2, ... ,k) = 0 for all

h=l J

integer vectors s(i) , such that 0 < s (i ) < ffi.- -

Consider all combinations of vectors v(i) and s(i-1)

and calculate s(i) = s(i-1) + v(i). If for particular

i-1P ( I nhJ. =

h=l

o 0s. (i.-l),j=1,2,... ,k) x P(n ..=v.(i), j=1,2, ... ,k)

J 1J J

to P(iI nhJ. =

h=lo

s.(i),J

j=l ,2 ,... ,k ). This will lead to a

- 57 -

simultaneous evaluation of all probabilities

iP ( I nh J. = s. ( i ), j = 1, 2 , . • . , k), s ( i) < m•

h=l J

Step Five:

If i is less than k, put i = i+l and go to step four;

otherwise, end the procedure by putting:

P(O < f < min)s (k )~m

kP ( I

h=l

= s. (k), j=l, 2 , .•• , k ) (2. 5 )J

This procedure would involve too many operations for

any practical use, especially if the number of grades is

bigger than three or if ~he total size of the structures n

and m are high. To have an idea about the kind of combin-

ations that are involved, we evaluated the total number of

combinations of vectors v( i) and s(i-i) , (i = 2,3, .•. ,k)

that have to be considered in step four of the above pro-

cedure; the values of such numbers, obtained by enumeration,

are given in table 1.

- 58 -

Table 1Number of Combinations in Step Four

Number of Structure to Total Size Number ofgrades be maintained Combinations

3 (3,3,3) 9 1 480(5,5,5) 15 13 272(8,8,8) 24 127 710(7,7,10) 24 145 272(10,10,10) 30 3 453 312(20,20,20) 60 16 810 332

4 (3,3,3,3) 12 14 910(6,6,6,6) 24 798 210(15,15,15,15) 60 389 491 488

5 (3,3,3,3,3) 15 124 208(5,5,5,5,5) 25 4 114 908(12,12,12,12,12) 60 4 690 850 528

However, if the transition matrix has some entriesi-l

equal to zero, many P( E nhJ· = SJ.(i-l), j = 1,2, ... ,k)h=l

would also be equal to zero. Therefore the number of com-

binations to be considered in Step Four could be reduced,

without affecting the result, by considering only s(i-l) for

which the latter probability is non-zero.

In the case of k=3, a Fortran routine "General" based

on the above procedure and incorporating the latest modifi-

cation, has been developed to calculate the exact value of

P( 0 < f < mIn).

- 59 -

This routine has been used to evaluate P(O ~ f ~ n/n)for three grades structures n in table 1. For each of these

structures,

sidered:

the following transition matrices were con-

(0.7 0.1 0.1 ) ( 0.4 0.2 0.2)P1 = 0.1 0.7 0.1 P2= 0.2 0.4 0.2 (6a)

0.1 0.1 0.7 0.2 0.2 0.4

= (0.8 0.1 ( 0.5 0.3, "P 0.8 0.1 ), P = 0.5 0.3 ) (6b)3 0.8 3 0.5

The execution time required on the CDC 7600 to carry

out the evaluation of P(O ~ f ~ n/n), for each example, is

given in table 2.

Table 2Execution Times with routine "General"

Execution time, in CP secondsStructure Total

n size Matrix Pl Matrix P2 Matrix p'3

Matrix p"3

(3,3,3) 9 0.01 0.01 <0.01 <0.01(5,5,5) 15 0.05 0.05 0.04 0.04(8,8,8) 24 0.48 0.48 0.37 0.37(7,7,10) 24 0.53 0.53 0.41 0.41(10,10,10) 30 1.49 1.49 1.15 1.15(20 ,20 ,20 ) 60 61.55 61.54 46.63 46.62

- 60 -

As would be expected, the execution times do not

depend on the magnitude of the transition matrix entries

but, rather, on how many and which entries are zero. How-

ever, the execution times increase sharply as the stock

numbers increase. Also, given the strong relation between

the execution times in table 2 and the number of combin-

ations in table 1, we expect that the execution times will

increase sharply if we increase the number of grades. Thus,

it would not be suitable to use the proposed approach in the

case of systems with large total size or number of grades.

2.2.2 Case of Upper-diagonal Transition Matrices

In this section, we will suppose that:

p .. = 0~J i,j = 1,2,.o.,k and j f:. i,i+1

This new assumption reduces considerably the previous diffi-

culties and allows for the introduction of an iterative

method to evaluate P(O < f ~ min);

let us denote by P(O; m· h· j/n) the probability that, ,

"0 < nll ~ ml, 0 ~ n12 + n22 ~ m2,-

... , 0 < nh_l h + nh h ~ mh, n = j/n(O)=n",, , h,h+l

where n.. are the numbers moving from grade i to J.~J '

m = (ml,m2, .•. ,mk) is a fixed and known vector and h < k-l.

- 61 -

Thus,

P(O; m; h; j In)

= L P(O < nll ~ m1, 0 < n12 + n22 < m2,- - -i £ I .

JIn)..., 0 < nh_l h + i ~ mh, nh h =i,nh,h+l =j- , ,

... ,

= j In)

=L L-iEI tET

j i

P(O; m- h-l; tin) P(nh h=i, n =j In), , h,h+l

(2.7)

Where Ij = { i 10< i < Min (nh-j, mh) }

and Ti = { t I 0 < t < Min (mh-i, nh_1) }.

We shall refer to the equation (2.7) as the iterative

equation, since it implies an iterative way to evaluate

P(O; m; k-l; j/n). This latter probability is of particular

interest since it can be related to the probability of

attaining m from n in one step as follows:

- 62 -

P(O < f ~ m In)

= P(O < nll~ m1, 0 ~ n12 + n22 < m2,••., 0 < nk_1 k + nk k ~ mk In), ,

= ~P(O ~ n11 ~ m1, a ~ n12 + n22~ m2,jEJ

/n)... , a < nk_1 k + nk k ~ mk, nk_1 k = j- , , ,

= Lp(a ~ n11 ~ m1, a ~ n12 + n22 < m2,-jEJ

/n)... , a < nk,k ~ mk- j, nk_1 k = j- ,

= L pro; m" k-l- j In) p(a < nk k ~ mk - j /n), , -jEJ ,

where J = { j / a < j < Min (nk_1, mk) }

( 2 • 8 )

Equations (2.7) and (2.8) could be used to evaluate

p(a < f < m In) as follows;

Step One:

Put h = 1.

pea; m; h; j In) = p(a ~ n11 ~ ro1, n12 = j In) can

be deduced from the Trinomial distribution with parameters

(n1; Pl1' P12)· If m1 ~ n1, pea; m; 1; j In) can be deduced

from the Binomial distribution with parameters (n1; P12).

- 63 -

Step Two:

h = h+l. If h = k go to Step Four.

Step Three:

Evaluate PCO; m-, h-, j In) using equation (2.7) and

the Trinomial distribution with parameters

Go to Step Two.

Step Four:

Evaluate P(O ~ f ~ mIn) using equation (2.8) and

the Binomial distribution with parameters (nk; Pk k).,

This algorithm could be modified to evaluate probabi-

lities of the kind P (a ~ f < b In), where a and bare

non-negative integer vectors. It would be sufficient to

replace P(O; fil; h; j In) by pea; b; h; j In) and equations

(2.7), (2.8) respectively by (2.9), (2.10) where:

pea; b- h- j In), ,

= P(a1 ~ n11 ~ b1, a2 ~ n12 + n22 ~ b2,.. ., 8h ~ nh_1 h + nh h ~ bh, n = j In), , h,h+l

Pea; b; h-l; tIn) P(nh,h=i, nh,h+l=j In)

(2.9)

- 64 -

and Pea < f < b In)- -

= Lp(a; b; k-l ; j In) P(ak-j < nk k < bk-j In) (2.10)- -,jEJ'

,I . = { i I Max (0 , ah - nh_1) < i < Min (nh-j, bh) }

J - -

,T. = { t I Max (0 , ah - i ) < t < Min (nh_1, bh-i) }1 - -

,J = { j I Max (0 , ak - nk) < j < Min (nk_1, bk) }- -

The probability P(O ~ f ~ n In) is a special case of

Pea ~ f ~ b In) where a = 0 and b = m. The probability of

making the first move from n to m, before recruitment has

taken place is P(f = mIn), which is a special case

of P(a < f < b In) where ~ = b = m.

A Fortran routine "MAINT" , based on the latter

algorithm, has been developed to evaluate P(a < f < bin).

But before we discuss the performance of this routine, we

shall present first the methods by which it evaluates

Binomial and Trinomial distributions.

2.2.2.1

- 65 -

Evaluation of the Trinomial distribution

We started by calculating the probability density

function at the mode (k1' k2) of the Trinomial distribution;

we used a procedure in Finucan (1964 ) to find the mode.

Then using the following recursive relations:

n-i-j P11° P (i+1 , j ) = x - P(i,j) 0 < i < n-1- -(i+1) P3 0 < j < n-i-1

i P32° P (i-1, j) = x - P(i,j) 1 < i < n- -n-i-j+l Pi 0 < j < n-i

n-i-j P23° P (i, j+ 1) = x P(i,j) 0 < i < n- -(j + 1) P3 0 < j < n-i-l

j P34° P (i, j-l) = x P(i,j) 0 < i < n- -n-i-J+l P2 1 < j < n-i

We calculated the probabilities P(u,v) at the remain-

ing points, where P3 = 1 - Pi - P2 and

P(u, v) =n! u

Pu!v! (n-u-v)! 1v

P2

n-u-vP

3

However, there would be (n+l)(n+2)/2 possible (u,v)

and their complete enumeration could be very lengthy and

unnecessary, since most P(u,v) are nearly zero. One way to

cut down significantly the amount of calculations is to

choose an enumeration process which considers the more

- 66 -

likely outcomes first and stops when the probability ofoccurrence of all the remaining (u,v) is less than a fixedprecision E; P(u,v) for all remaining (u,v) are set to zero.

The enumeration process of (u,v) in the routine MAINT

is as follows:

986

5

14

13

12 4

11 10

15

where the cells are numbered according to the order in which

probabilities of their occurrence were calculated. The cell

number 1 corresponds to a mode of the Trinomial

distribution.

2.2.2.2

- 67 -

Evaluation of the Binomial distribution

A similar approach to the latter method was adopted;

the procedure starts by calculating P(m) for m = np and then

using alternately the following recursive equations:

P(j+l) = p x (n-j) P(j)(l-p) (j+l)

P(j-l) = .!=£ x j P (j)p (n-j+l)

m < j < n-l

1 < j < m

(n) w n-wto evaluate the other P(w)'s, where pew) = w P (l-p) ·

The procedure stops when the probability of occurence of all

remaining w is less than a fixed precision E; pew) for all

remaining ware set to zero.

2.2.2.3 Precision in tne routine MAINT

The process of enumeration of (u,v) in the case of

the Trinomial distribution and w in the case of Binomial

distribution, especially if E is not equal to zero, could

lead to an under-evaluation of Pea < f < bIn).

to find an upper bound a to this error;

We propose

- 68 -

let us denote by:

SChl = ~ PCa; b; h; x/nlxe:Xh

(2.11)

where h ~ 1 and Xh = {j/O ~ j ~ Min Cnh, bh+1l}, and by~CZl

the tDlder-evaluation error in the calculation of an

expression Z.

From equation (2.10) we have:

D. [P(a < f < bin)]

~LD.[P Ca;b;k-l;j/nl) PCak-j <

je:J

P(a;b;k-1;j/n)~[P(ak-j <

~LA[PCa;b;k-l;j/nl) + EL PCa;b;k-l;j/nlje:J je:J

< lJ. [S(k-1)] + e: (2.12)

We will suppose thatl1[S(h)] < Qh and attempt to find

a relation between Qh and Qh-1 for h > 2.

(2.9) and (2.11) we have:

From equations

n = j In)h,h+l

- 69 -

S(h) =L. L.. L P(a;b;h-l;t In) penh h=i,nh h+l=j In)j E X

hi E I '. t E T ! "J 1

This would lead to:

11 [S(h)]

< ~ ~ 6[~ P(a;b;h-1;t In)] P(nh h=i,j E Xh i E I j t £ T i '

+L L Z- P(a;b;h-1;t In) 6[P(nh,h=i,nh,h+1 = j In)]j E Xh i E I '. t E T!

J 1

<

Moreover since,S(1) = ~ P(a;b;1;j/nl

jEX1

(2.13)

Where I'! = {i181 < i < Min (n1 - j , b1 )}, thenJ

~ [S(l)] = L- L ~ [P(n11 = i,n12 = j /n)]~ EjEX1 iEX'!

J

Thus a1 < E

From relations (2.12), (2.13) and 2.14) we havea = ~ [ Pea ~ f ~ bin)] ~ k E

(2.14)

(2.15)

2.2.2.4- 70 -

Performance of the routine MAINT

In comparison to the routine GENERAL, the routine

MAINT achieves a remarkable reduction in the execution time;

this latter routine has been used to evaluate P(O ~ f ~ n/n)

for all structures in table 2 but only for transition

matrices P3 and P3. The error a in MAINT has been set to

zero and the computa tions have been made on the CDC 7600.

Under such conditions, the execution times needed for all

above examples were at most equal to 0.01 CP seconds.

Table 3 shows the effect of increasing the value of a on the

execution times needed to evaluate P(O < f ~ n/n), using the

rout ine MA INT,

matrices.

in the case of upper diagonal transi tion

- Matrix P' where:k, 0.8, , 0.1, w! 0.1 1 < i < k-lp. . = Pi ,i +1 = =~,~ ~ - -

, 0.8, w' 0.2 (2.16a)Pk k = =, k

- Matrix P" where:k

p'.'. 0.5, " 0.3, w'.' 0.2 1 < i < k-l= Pi ,i+1 = =1,~ ~ - -

p" = 0.5, w" = 0.5 (2.16b)k,k k

P' and pIt have already been defined by relation (2.6b) . The3 3computations have been made on the CDC 7600.

- 71 -

Table 3Accuracy and Execution tiae vi til routiDe JlAINT

Execution times, in CP secondsNumber

ofgrades

Structure n

3 (100,100,100 ) 0.34 0.42 0.02 0.10 0.02 0.05

(200,200,200 ) 2.56 3.20 0.10 0.44 0.06 0.24

(500,500,500 ) 41.60 51.21 0.72 2.81 0.37 1.47

(1000,1000,1000 ) 441.96 486.16 2.90 12·33 1.45 6.44

( 5000 , 5000 , 5000 ) >1200 >1200 87.43 369.82 44.11 193.18

4 (100 , 100 , 100 , 100 ) 0.68 0.83 0.05 0.22 0.03 0.12

(200,200,200,200 ) 5.12 6.42 0.20 0.91 0.12 0.51

(500,500, SOO,500) 83.58 102.46 1.52 5.77 0.80 3.12

(1000,1000,1000,1000 ) 883.60 970. 51 6.00 25. 47 3.05 13.47

( 5000 , 5000 , 5000 , 5000 ) > 1200 > 1200 180.46 760.98 93.43 406.61

5 (100,100,100,100,100) 1.02 1.25 0.07 0·33 0.05 0.18

(200,200,200,200,200) 7.68 9.63 0.31 1.41 0.18 0.79

(500,500,500,500,500 ) 125.68 153. SO 2.31 8.82 1.24 4.81

(1000 , 1000 , 1000 , 1000 , 1000 ) >1200 > 1200 9.19 38.69 4.82 20.97

(5000,5000,5000,5000,5000 ) >1200 > 1200 277.42 1165.39 144.61 631.70

30 (1000 , 1000 •. • 1000,1000 ) >1200 > 1200 101.33 427.60 59 ·71 259. 54

- 72 -

Table 3 shows that a substantial reduction in theexecution times can be achieved by fixing a, even at a lowlevel, so long as this is not at zero. We noticed that theexecution time depends on the magnitude of the transitionmatrix entries. this dependency increases when a, and thus£, becomes non-zero. This could be explained by the way theTrinomial distributions are evaluated; the enumeration inthe case of Trinomial variable with parameters (h; P1' P2)and for a fixed E, is shorter if P1' P2 and (1-Pl-P2) arevery different, rather than if they are similar; In theformer case, fewer outcomes (u,v) would have P(u, v) non-negligible and all of these outcomes would be concentratedaround the mode (k1, k2). Table 3 also shows that theexecution time increase linearly with the number of grades

problem in evaluating< n/n). Potential

is not a realprobabilities of the type P(Oproblems may arise only in thehigh stock numbers. However,

case ofthethatshows

systems with very3

f

table

<

this latterand that

execution times are reasonably low and yet the stock numbersare too high for most practical situations.

Unfortunately, the routine MAINT deals only with aparticular type of transition matrices and we still have tofind another method to cope with the general case; theNormal approximation seems to be the most suitable.

- 73 -

2.3 RORMAL APPROXIMATION

Bartholomew (1977) showed, by means of simulation andin the case of some numerical examples, that the Multinormaldistribution does approximate reasonably well to the

kdi stributi on of f = (I n.. j = 1,2 ..•k )1J,

i=l

It was suggested that

P (a < f < bin) ~ P (a. - 0.5 < Y. < b. + 0.5, j = 1,2, ... k)J J J

(2.17)

where Y = (Y1'Y2' ....Yk) has the multivariate Normal distri-bution with mean vector ~ and variances-covariances matrix Awi th:

k

~ . =Lnh PhjJ h=l

and

kA •. =Lnh Phi (I-Phi)11

h=l

j = 1, 2, ... k

i=l, 2, ... k

A ..1J i ,j = 1,2, ... k; i~j

- 74 -

However, the evaluation of the multivariate Normalintegral is not easy in itself. Milton (1972) developed aFortran program using a multidimensional iterated Simpson'squadrature adapted to the Multinormal integral. Unfortu-nately, from our own experience, this program failed veryoften in evaluating integrals of five or more variables. Inthe case of fewer variables, its failure was less frequentbut still occasional. Bohrer and Schervish (1981),Schervish (1984) discuss these shortcomings in some detail.In 1981 the NAG Iibrary contained new Fortran routines toevaluate multi-dimensional definite integrals for generalfunctions. We found routine D01FCF to be suitable for ourneeds. This routine is based on automatic adaptiveprocedures which involve subdivision of the region ofintegration into subregions, concentrating the divisions inthose parts of the region where the integrand is worstbehaved, and apply numerical integration rules separately toeach subregion.

The routine D01FCF was used successfully to evaluatethe right hand side of relation (2.17), for values of k upto the value eight. A potential relative error 8, equal to1%, was accepted in such evaluations and all computationswere made on the CDC7600. The evaluations took a fractionof a second for k < 5, a few seconds for 6 < k < 7 and justover 60 seconds for k = 8. These times increased sharplywhen a higher accuracy was required; they reached 120 CPseconds when we considered the case of k equal to five andset B equal to 0.1%. However, since a relative error of 1%

- 75 -

is already a very small error, and given the purpose of anapproximation method, we see no need to require higheraccuracy than S = 0.01, especially if it is costly in time.

As to the quality of the Multinormal approximation, table 4

confirms the positive conclusions in Bartholomew (1977); it

gives, in the case of several numerical examples, the exact

and approximate values of the probability of maintaining a

structure. The approximate values were obtained by the use

of routine 00lFCF with B = 0.01. The exac t va 1ue s were

obtained by the use of the routine MAINT in the case of

upper diagonal transition matrices Pk and Pk and the routine

General in the case of matrices P1 and P2. These matrices

are defined by relations (2.6) and (2.16).

This table shows that whilst the quality of the

approximation improves with the size of stock numbers, it

does not seem to be affected by the number of grades. In

most cases the relative error in the approximation is small,

except in the case of matrices Pk where it reaches 13%.

This could be explained by the combination of big

differences among the elements of these matrices and low

stock numbers.

In the case of upper diagonal transition matrices, if

n1 is equal to zero, the variance-covariance matrices would

be singular. We would then modify the equation (2.17) asfollows:

- 76 -

Table 4ea.pariSOD of the ..utiYariate No~ approrlJlation

(0) to p(O < f < DID) ridl die exact Talue (D)

k n EV NA NA~X 100 EV NA NA~ xl00

3 (5,5,5) 0.364 0.362 -1 0.479 0.482 1(10,10,10) 0.389 0.385 -1 0.577 0.581 1(20,20,20) 0.478 0.476 -0.4 0.729 0.745 2

"pi Pk k

k EV NA NA-EV 100 EV NA NA-EV xl00n EVx EV

3 (5,5,5) 0.667 0.606 -9 0.686 0.681 -1(10,10,10) 0.678 0.649 -4 0.759 0.771 2(20,20,20 ) 0.747 o ~')~ -3 0.860 0.870 1./ -

4 (5,5,5,5) o. 533 0.477 -11 0.559 0.556 -1(10,10,10,10) 0.548 0.522 -5 0.656 0.657 0.2(20,20,20,20) 0.639 0.631 -1 0.797 0.800 0.4

5 (5,5,5,5,5) 0.425 0.376 -12 0.455 0.454 -0.2(10,10,10,10,10) 0.443 0.420 -5 0.566 0.570 1(20, 20, 20, 20, 20) 0.549 0.537 -2 0.737 0.741 1

6 (5,5,5,5,5,5) 0.339 0.296 -13 0.370 0.370 0(10,10,10,10,10) 0.358 0.337 -6 0.489 0.496 1(20, 20, 20, 20, 20) 0.468 0.458 -2 0.682 0.687 1

P (a < f < bIn) =

- 77 -

P (a. - 0.5 < Y. < b. + 0.5, j = 2, .•. k)J J J

where Y'=(Y2, ••• ,Yk) has a multivariate Normal distribution.

Finally, in the case of upper diagonal transition

matrices, there is no gain in computing time by using the

Multinormal approximation unless the stock numbers are very

high (exceeding a thousand).

quicker in most cases.

The routine MAINT would be

- 78 -

2.4 MAINTAINABLE AND ATTAINABLE REGIONS IN A STOCHASTIC

ENVIROItMENT

In our di scussion of the deterministic theory, we

referred to results which Bartholomew (1973) obtained in the

case of control by recruitment concerning maintainable

regions. We saw that these latter are convex and that the

position of a structure within these regions gives a clear

indication concerning the recruitment policy that should be

adopted for maintaining that structure. Davies (1973) made

a detailed investigation concerning the attainable and main-

tainable regions. He considered, in particular, the set of

all structures that are attainable in T steps from a given

structure, T > 1. He found that all regions for T=1,2 ....

are convex but pointed out that in general no inclusion

relationship exists between them except in the case of the

initial structure being maintainable in one or many steps.

In the above analysis, all structures can be divided into

two distinct groups: maintainable (or attainable) struc-

tures and the others. In a stochastic environment, such

classification is not possible; the maintainability (or

attainability) of a structure mayor may not hold depending

on the actual values taken by the random flows. Davies

(1982) faced with the same problem in the case of a part-

ially stochastic model, considered a classification which

distinguishes between structures depending on whether their

probabilities of attainability are zero or not. He proved

that the attainable structures with non-zero probabilities

form a convex set and that the attainable region as defined

- 79 -

in the deterministic theory is included in the latter set.

In the case of a completely stochastic environment, however,all structures have a non-zero probability of maintain-ability (or attainability) and hence the latter classifi-

cation is inappropriate. Nevertheless, this difficulty can

be easily removed by considering regions where all struc-

tures have a probability of maintainability (or attain-

ability) greater than a given constant a, 0 < a < 1. We

will refer to the newly defined regions as a-maintainable

(or a-attainable) regions. As in Davies (1983) we will try

to investigate the properties of these latter, especially

their nature and their relationship with the corresponding

regions in the deterministic theory. No theoretical results

are available for such investigation but much insight can be

gained from a recourse to g""-aphical representation. This

however will not permit us to look into systems with a

number of grades greater than three. In addition, so as to

reduce the computing time in the evaluation of all required

probabilities, it is desirable to consider only systems with

upper-diagonal transition matrices. Finally, we will focus

our discussion on the case of T being fixed at one. Greater

values of T, as will be argued in the next chapter, would

make the evaluation of the needed probabilities very lengthy

and would require the examination of an extensi ve range ofrecruitment strategies.

Two graphical representations are possible. the

first is in three dimensional space where each structure is

represented in X-Y plane by its barycentric co-ordinates and

- 80 -

•

•z~.,)...••

I.".-3L

¥L

1:~."~L

."•

."ik _

t: I!I::0 •••

! ~~~.~ ~II3 ...c _

iL

A.

:."-.-

- 81 -

N

- 82 -

the probabilities are regarded as the heights of the surface

at these points. Gino-Surf, a library of subroutines fordisplaying three-dimensional surfaces was used for producing

an isometric projection of this surface. Figures 1.1 and

1.2 show a typical projection as produced by Gino-Surf. In

each figure, the four pictures correspond to the same pro-

jection but viewed from different angles. Figure 1.1 shows

that the structures with higher probabilities of maintain-

ability are top heavy and suggests that a-maintainable

regions are convex for any level a. The latter property

holds also in the case of a-attainable regions as can be

deduced from figure 1.2. This representation, however, does

not allow for an accurate description of the probabilities

nor does it deal directly with the a-maintainable (or a-

attainable) regions; its main function is primarily

pictorial. In the second type of plotting, each structure

is also identified by its barycentric co-ordinates but, in

addition, its probability of maintainability or attain-

ability is printed at these co-ordinates.

To identify the borders of an a-maintainable or a-

attainable region, we used a Fortan program based on the

assumption that these regions are convex. Beginning with a

list of all eligible structures, the program finds their

convex hull and finally draws the edges of the region by

joining the determined vertices with straight lines. Should

the convexity assumptions happen to be false, we would have

within the boundary of the region a structure with probabi-

Iity less than or equal to a. But from a wide range of

- 83 -

c 0,12,0)28

07(a, a.12) (12,0,0)

.59 - Moinlolnable rPQlon wilh N = 12 and

(.70 .20 .00)

p = •ee •se . 1 e.ee .0e .ge

Figure 2.1

( a, 9,12>

- 84 -

.Je - AllQlnabl. rpqion Fr~ ( 1, 1, 1) wIlh

(

•7e .20 •Be )P = .se .S9 .,e.ee .09 .90

Figure 2.2

00(12,0,0)

- 85 -

examples which were considered, no single case was found to

contradict the above assumption. Figures 2.1 and 2.2 show atypical graphical representation as produced by the programdiscussed above. The dashed lines are the edges of the

maintainable (or attainable) region as defined in the deter-

ministic theory.

It was observed, in the case of maintability, that

the maintainable regions are generally included in a-main-

tainable regions of a close to 0.5. In the case of attain-

ability, however, no similar conclusion could be drawn given

the strong dependency of the probabilities on the initial

stock vectors. Nevertheless, in both cases it is clear that

the probabilities within the maintainable and attainable

regions are higher than those outside these regions.

- 86 -

CHAPTER III

PROBABILITY OF MAINTAINING OR

ATTAINING A STRUCTURE IN MANY STEPS

In this chapter, we will be concerned with the evalu-

ation of P(O ~ f(h) ~ mIn), the probability of attaining a

structure m from a structure n in h steps, h ~ 2, where n is

kthe stock vector at time T=O and f(h)=( L n ..(h) ,j=1,2, ...,k),

i=1 1Jn ..(h) being the flow numbers from i to j in [h-1, h [.

1J

In the case of h being equal to one, we knew the

distribution of the vectors (n..(h), j = 1,2, ... ,k),1J

(i = 1,2, ... k) . However, we were unable to describe analy-

tically the distribution of their sum. For h ~ 2, we do not

know even the distribution of the vectors themselves; this

distribution depends on the stock vector at T = h-1, which

in its turn depends on all flows in the previous periods.

They all depend on recrui trnent vectors decided upon at the

end of preceding periods.

As in the case of h = 1, we will examine in turn two

methods in evaluating P(O < f(h) < m):

- 87 -

Exact approachMultinorrnal approximation

But firstly, we will review different types of recruitment

policies and strategies.

3.1 RECRUITMENTSTRATEGIES

A recruitment policy is a set of rules to obtain the

recruitment vector at a given time T. A recruitment

strategy over a period is a sequence of such policies. The

recruitment vector at each time is assumed to be non-

negative, otherwise we will be dealing with a highly

undesirable situation in which redundancies have to take

place.

Instead of introducing hypothetical recrui tment

strategies, we will concentrate on a few types which are the

most relevant to our work and those which will be used most

extensively in the control context.

3.1.1 Deterministic and Fixed Strategies

Firstly, generalising from the notion of a

T*-maintainable path in Davies (1975), let us suppose that

there is a sequence of non-negative vectors m(T) and R(T)

such that:

- 88 -

m(T) = m(T-l) P + R(T), (1 < T < T*) (3.1 )

where m(O)=n and m(T*) is a fixed goal vector that we wish

to attain in a fixed number of steps T*. (T* is notnecessarily equal to h). Then { n, m(l), m(2), ..., m(T*) }

will be said to form a T*-attainable path from n to m(T*).

The recruitment strategy defined by (3.1) will be called a

deterministic strategy. This definition supposes that m(T*)

could be attained from n in T* steps and a T*-attainable

path from n to m(T*) is known.

Under a deterministic strategy the vectors m(T),

(T = 1,2 ,...,T*) will be successive 1y attaine d but on1y on

average. In particular, if n belongs to the maintainable

region and m(T*) = n, { n, n, ... , n } will be a

T*-maintainable path for n anc the corresponding recruitment

strategy will be defined by:

R (T) = n ( I - P), (T = 1, 2 , ••. , T*) (3.2)

The recruitment strategy defined by (3.2), and

denoted by F1 in Bartholomew (1977), will keep the total

size of the system constant only on average. However, if

the losses are known before the recruitment vectors have to

be chosen, we can improve F1 by making recrui tment equal

losses and allocating the new recrui ts in the same

proportions as in (3.2). This strategy, denoted by F2 in

- 89 -

Bartholomew (1977), will ensure at each time that the total

size of the system is kept constant and the structure n is

maintained on average.

Both strategies F1 and F2 allocate recruits in fixed

proportions, and therefore are described as "fixed".

3.1.2 Adaptive Strategies

An adaptive strategy should try at each time T, to

readjust the evolution of the system and use R(T) which

minimises the distance between n(T) and a fixed vector m(T);

the vectors m(T) could belong to a T*-attainable path from n

to m(T*), or they could be all equal to a fixed vector m.

If we consider a squared distance function, we would

choose R(T) which minimises:

kD(m(T), n(T» = L

j=lwhich is given by:

( m.(T) - f.(T) - R.(T) )2,J J J

R .(T) = Max ( 0, m. (T) - f. (T ) ), (j=1 ,2 ,...,k ).J J J

This strategy assumes that f(T) is known at time T,

when a decision upon R(T) has to be made. Bartholomew(1977) introduced this strategy in the case of m(T)=n and

odenoted it by 52.size of

distancesame

oBut since 52

the system, a policy 5~D( n(T), m(T) )

tends to increase the total

which aims to minimise theo

as in 52' but with anadditional constraint:

- 90 -

kt

j=lR.(T) =

J

kt

j=ln. -

J

kE

j=lf .(T )

J

was introduced. It ensures that the total size of the

system remains fixed.

At time T, the recruitment vector R(T) is then a

solution to an integer quadratic programming problem. An

optimal solution should try to make the differences

The following r.llgorithrn achieves this

(n. - f.(T) - R.(T) )J J J

other as possible.

aim:

(j = 1,2, ... ,k), as close to each

Put a. = n. - f.(T) and S =J J J

kE

j=la.

J

Set R.(T) = 0J

j = 1,2, ... ,k

S·,

If S is equal to zero, stop.

Find the maximum a . , then put:J

R.(T) = R.(T) + 1J J

a. = a. - 1J J

S = S - 1

Go to step 3.

This algorithm can be improved in the case of a large

its proof and the improvement are fully discussed in

Appendix A.

- 91 -

3.1.3 Rounding ReT) to an Integer Vector

Some definitions of R(T) in the above examples, may

lead to a vector which is not integer. One way of

overcoming this difficulty is to round up or down the

elements of R(T) with probabilities chosen to ensure that

the expectation of the new random vector R*(T) is equal to

R(T). These transformations could be done in many ways and

the results obtained could be dependent on them.

In the case of k=3, we were able to find a general

way to obtain an R* vector. Two cases are considered:

(i) Case 1:

we define:

3S = L

j=lR.(T) is integer;

J

and a

[R(T)] = ( [Rj(T)] , j =

= R(T) - [Rj(T)]

1,2,3 )

Since

S i~ integer,

3a . < 1 (j=1,2,3), we have L a . < 3, and since

J j=l J3 3L a . = S - L [R.(T)] is integer. Therefore

j=l J j=l J3L a

J. can take only three possible values 0, 1 or 2.

j=l

3If L

j=la. = 0:

J

we put R*(T) = R(T) with probability equal to one.

If3I

j=1a. = 1:

J

- 92 -

We can check that E(R*(T))

with probability a1with probability a2with probability a3

= [R(T)] +

we put:

R*(T) t·(1,0,0)

(0,1,0)(0,0,1)

3= R(T) and I

j=1R* . (T) = S

J

If

we put:

3I

j=1a. = 2:

J

R* ( T) = [R ( T) ]

We can check that E(R*(T))

{

(0,1,1) with+ (1,0,1) with

(1,1,0) with3

= R(T) and Ij=l

probability (1-a1)probability (1-a2)probability (1-a3)

R* .(T) = S.J

(ii ) Case 2 : S is not integer;we define: S' = [S ] + 1

S" = [S]and put p = S - [S ]

Thus S = pS' + (l-p) S"S'

we define: R'(T) = R(T)S

S"R"(T) = R(T)

S

- 93 -

R'(T) and R"(T) could be dealt with as in the previous case.

Thus:

R*(T) = {R'*(T) with probability pR"*(T) with probability (l-p)

we can check that E(R*(T»3

= R(T) and Ij=l

R* . (T) = S.J

These transformations have the advantage of being general,

simple and the most natural way of rounding a real vector

R(T) to an integer vector R*(T) such that:

E(R*(T» = R(T)

and3I

j=lR* . (T) =

J

3E

j=lR.(T)

J

Unfortunately, for k bigger than three, no similar methods

could be found; to illustrate the difficulties involved, letus consider the following example when k=4. Let us define

4a = R(T) - [R(T) ] and suppose that I a .=2. Thus the vector

j=l J

R*(T) should have the following distribution:

- 94 -

(1,1,0,0) with probability 81(1,0,1,0) with probability 82

R*(T) = [R(T) ] + (1,0,0,1) with probability 83(0,1,1,0) with probability 84(0,1,0,1) with probability 85(0,0,1,1) with probability 86

Such that:

81 > 0, 82 :::.0, 83 ~ 0, 84 ~ 0, 85 > 0, 86 > 0

81 + 82 + 83 + 84 + 85 + 86

81 + 82 + 83

81 + 84 + 85

82 + 84 + 86

83 + B5 + 86

=

=

=

=

=

1

Obviously, a solution for such a problem could be found by

means of linear programming techniques, but this process of

rounding R(T) in itself will make the calculations very

lengthy.

3. 2 EXACT METHOD

3.2.1 Basic Formulae and Limitations

Unlike the case of h = 1, we were unable to find a

direct and exact method to evaluate the probability

P(O < f(h) < min), of attaining a structure m from an

- 95 -

initial structure n in h steps, h > 2. But indirectly, this

probability could be evaluated by the use of the following

recursive relation:

P(O ~f(hl~m/nl = ~ P(f(ll=v/nlP(O~f(hl~m/n(ll=v+RlvEA

= L P(f(ll=v/nlP(O::f(h-l)::m/n' lvEA

(3.4)

where A is the set of all structures that could be reached

at time T=l, before recruitment has taken place, R is the

recruitment vector at the end of the first step of the con-

sidered period, and n'=v+R. In this relation, we have

(3.5)

supposed that once the outcome of f(l) is known, we will be

able to provide a unique recruitment vector R according to a

given recruitment strategy. However, in some situations, as

shown in section (3.1.3), R could be a random vector. In

this case, relation (3.4) could be modified as follows:

p(O ~ f(h) ~ min)

= L.p (f(1)=v In) LP (R=r I f (1)=v) P (O~f (h-1 )~m In' )vEA rEE

where E is the set of all possible outcomes of R given that

To see how this recursive approach could be used, let

us start with a simple case and suppose that under the

recruitment strategy, the total size N of the system will be

kept constant and only relation (3.4) will be needed. In

- 96 -

this case, the calculations will be made backwards; we will

start by evaluating P(O ~ f(l) ~ mIn') for all possible

integer vectors n' whose total size is equal to N·,P(O ~ f(l) ~ mIn') for a particular n' , will be used when-

ever n(h-1) is equal to n'. Since there would be many paths

{n,n(1), ••• ,n(h-2),n'} from n to n' in (h-1> steps, it is

better to evaluate P(O~f(l) ~m/n') once and to store it

for subsequent use. For the same reason, using the relation

(3.4), we will evaluate successively for t=2, ... ,h-l, the

probabilities P(O ~ f(t) ~ min') for all possible n'. The

probabilities P(O ~ f(h-l) ~ mIn') will be then used to

evaluate P(O ~ f(h) ~ mIn). We should point out that in

using the relation (3.4) to evaluate P(O ~ f(t) ~ min'), the

vector R stands for R(h-t+l). The evaluation of

P(f(l)=v/n') and P(O ~ f(l) ~ mIn') could be done by one of

the two exact methods discussed in the previous chapter.

A serious difficulty in this approach, is the high

number of evaluations of probabilities of type P(f(l)=v/n');

if h~3, these probabilities should be evaluated, at least

once, for all possible vectors n', whose total size is equal

to N, and for all possible outcomes v of f (1) from each of

these vectors n'. More precisely, if we denote by M1 the

number of vec tors n' and by M2 the number of vectors v

mentioned above, we will have:

and

Thus,

- 97 -

there will be a need to evaluate M3=M1 xM2probabilities of the kind P(f(1)=v/n').

Table 5 gives the values of M1, M2 and M3 in the case

of k=3 and some values of N.

Table 5Number of evaluations of P(f(l)=v/n')

N M1 M2 M3 M4

10 66 286 18 876 8 29415 136 816 110 976 47 32818 190 1 330 252 700 106 66620 231 1 771 409 101 171 78724 325 2 925 950 625 396 04525 351 3 2-'5 1 149 876 478 29650 1 326 23 426 31 062 876 12 673 466

M4 represents the total number of evaluations that

would be required in the case of an upper-diagonal trans-

ition matrix; in this latter case, the number of outcomes

for f(1) depends on the initial stock vectors n'. The

process of enumeration suggests that there would be:

, , ,possible vectors v from n'=(n1,n2,n3).

- 98 -

In the case of systems of 3 grades and with upper-

diagonal transition matrices, the computation times needed

to complete the evaluation of all required probabilities

P(f(l)=v/n') are approximately as follows:

N TIME

18 14524 84025 1100

All the calculations were made in the CDC 7600

computer and times are expressed in CP seconds.

Additionally, i~ the transition matrix is not upper-

diagonal and the routine MAINT is not used, the execution

times will be considerably higher than those given above.

Nevertheless, since the total size is assumed to be constant

anrl the possible n' at each step will be always the same,

these evaluations have to made only once and the probabili-

ties could be used as many times as requi red. If , in

addition, the vectors R(T) are easily obtainable, the

evaluation of P(O < f(h') ~ mIn) would be slightly longer

than P(O < f(h) < mIn) for h' > h > 3.

If the total size under a recruitment strategy is not

kept constant, the above difficulties would be dramatically

amplified. It would be difficult to predict the outcomes

- 99 -

of n(h-l), ... ,n(2). Also, for each step, we will have a setof new vectors n' for which the probabilities P(f(l)=v/n')

have to be evaluated at this step.

3.2.2 Case of h=2

This is a special case, since the issue of the number

of evaluations, as discussed above, is relatively much less

serious than in the case of h ~ 3 and the total size problem

is irrelevant. In the case of h=2, equation (3.4) becomes:

P (0 < f (2) < mIn) =L P (f(1)=v In) P (0 < f (1) < mIn')v£A

Since P(O ~ f(1) ~ mIn') could be evaluated directly

without a further use c relation (3.4), we only need to

evaluate P(f(1 )=v/n) for all possible vectors v, but only

from the initial structure n. However, even in this special

case, the number of vectors v could still be high and would

affect the computation time of P(O ~ f(2) ~ mIn). Table 6

gives the time required to compute the probability of main-

taining a structure n in two steps. All calculations were

made on the CDC 7600 and the times are expressed in CP

seconds. The transition matrices P' and3 p) are upper-

diagonal and therefore the routine MAINT was used to evalu-

ate the probabilities P(f(l)=v/n) and P(O ~ f(1) < n/n').

- 100 -

Table 6Ca-putation times required to evaluate P(O < £(2) < n/n)

Recruitment StrategyTransition Matrix Structure n

F1 F2 Sl2

( 5, 5, 5) 0.83 1.29 0.66

(0.8 0.1 (10,10,10) 10.81 24.80 12.29P' = 0.8 0.1 ) (15,15,15) 97.01 162 .47 79 .253

0.8 (20 ,20 ,20 ) 269 .83 648.30 315. 55

( 5, 5, 5) 2.29 1.43 0.67

(0.5 0.3

)(10,10,10) 13.01 27.92 12.61

P" = 0.5 0.3 (15,15,15) 293.93 181.91 80 .5230.5 (20 ,20 ,20 ) 326 .29 720.43 316.48

The computation times become very high if the

recruitment vector, obtained by the rules of a recruitment

policy, is real and has to be rounded to an integer vector.

Thus, these times are relatively much higher in the case of

F2 than S~; the rules of the policy S~ lead always to an

integer vector, while those of F2 lead, for most outcomes of

f (1), to a real vector. As to the policy F1, since the

recruitment vector is fixed in advance and does not depend

on the outcomes of f(1), either of the equations (3.4) or

(3.5) should always be used depending on whether R=n-nP is

integer or not. If R is not integer, the computation times

will still depend on the distribution of R obtained from the

rounding process.

- 101 -

The computation times in table 6 are already high,

even for small structures, and increase sharply with stock

numbers. They would be much higher if the transition

matrices were not upper-diagonal or if the number of grades

was larger than three. There is obviously a need to look

for an alternative method;

could be such a method.

3.3 NORMAL APPROXIMATION

the Multinormal approximation

As in the case of h=1, we will assume that the Multi-

normal distribution would approximate to the distribution

kof f (h)= ( Ln. . (h), j =1 ,2 ,... ,k) . But, un like the case of

i =1 1Jh=l, the mean vector and the variance-covariance matrix of

the distribution are not immediately available and will

depend on the recrui tment strategy over the period r O,h [.

Thus, since it would be difficult, if not impossible, to

investigate the Multinormal approximation issue in general,

we will review in turn different recruitment strategies;

these latter will be differentiated according to the

dependence of recruitment vectors R(T) on the previous

flows.

3.3.1 Recruitment Vectors that Do Not Depend on the Flows

In this case, the recruitment vectors R(T) ,

(T=1,2, ... ,h) could be any vectors decided upon without any

knowledge of the previous flows. They could be random or

- 102 -

constant, equal or different. If a vector R(T) is random,

its distribution should be given in advance. Under these

assumptions, we can use the following recursive equations toevaluate the mean vectors n(T) and the variance-covariance

matrices V(T) successively for the stock vectors n(T),

(T=l,2,... ,h):

(3.6a)

V(T)=P'V(T-l)P+[n(T-l)PJd-P' [j1(T-1)JdP+V(R(T» (3.6b)

n(O)=n,v(O)=O (3.6c)

where [-Jd denotes a diagonal matrix with the elements of

the vector contained within it on the principal diagonal,

R(T) and V(R(T» respectively the mean vector and the

variance-covariance matrix of R(T).

The relations (3.6a) and (3.6b) were quoted in

Bartholomew (1977) for slightly more restrictive assulOptions

than those considered in this section, but could be shown to

hold for the latter assumptions.

The mean vector and the variance-covariance matrix of

f(h) will be identical to those of n(h) if we put R(h)=O.

Having now the possibility of evaluating the required

moments of f(h) we will try to assess the quality of the

multivariate Normal approximation to the probability of

attaining a structure in many steps. We will restrict our

comparison to the case of two steps and will consider the

strategy F1, i.e. R(l)=n(O)(I-P). If R(l) is not integer,

- 103 -

we will use the procedure in section (3.1.3) to round it to

an integer vector. This definition of R(I) satisfies the

assumptions of this section and therefore the relations(3.6) will be valid for use. The results from table 7

suggest that the multivariate Normal approximation toP(O ~ f(h) < n/n) is still good for h=2 and we think it

would be also good for h>3.

Table 7Comparison of the Normal approximation (NA) to

P(O < f(2) < n/n) with the exact value (EV),in the case of the recruitment strategy F1

Structure n,

Transition Matrix P3 "Transition Matrix P3

EV NA NAE~Vxl00 EV NA NAE~Vxl00

( 5, 5, 5) 0.477 0.470 -1.5 0.639 0.631 -1(10,10,10) O. 537 O. 531 -1 O. 716 0.717 0(15,15,15) 0.579 O. 573 -1 0.773 0.788 2( 20 , 20 , 20 ) 0.626 0.630 1 0.819 O. 826 1

For all the examples in table 7, the computation

times of NA were equal to a fraction of a second. And since

the computation of NA requires the evaluation of the mean

vector and the variance-covariance matrix of f(h), and only

one multivariate Normal integral, independently of h, we

expect the latter times to increase just slightly with h.

- 104 -

They would increase much more significantly with the numberof grades but compared with those which would be requiredusing the exact methods, they would still be negligible.

3.3.2 Recruitment Vectors that Depend on the Flows

Under such an assumption, there is no general method

to evaluate the mean vector and the variance-covariance

rna t ri x 0f f (h ) • In some cases, this evaluation cannot be

made unless the distribution of f(h) itself is known, and

thus any approximation is irrelevant. An example of these

strategies is the adaptive strategy Sl. Under this latter,2

the recrui tment vector R (h) would depend on the flows to

each grade by the end of the period [h-1,h[, but there is no

analytical expression of R(h). This makes the task of

evaluating the mean vector and the variance-covariance

matrix of f(h) difficult, if not impossible. Another

example is the fixed strategy F2 under which the recruitment

vectors R(T) depend only on the total losses during the

periods [T-1,T[, (T=1,2, ... ,h). In the case of this

example, results from Bartholomew (1975) are of interest; it

was shown that if:

kR (T) = { r n. k 1 (T) }r, (T=l ,2, ••• , h )

i =1 1, +

Where r is a fixed vector, then:

(3.7)

ii(T) = ii(T-l)Q

- 105 -

,(T=1,2, .•• ,h) (.J.8a)

VeT) = Q'V(T-l)Q+[n(T-l)Q]d-Q'[n(T-l)]dQ-n(T-l)w'{[r]d-r'r},(T=1,2, ... ,h) (3.8b)

V(O) = O,n(O)=n (3.8c)

where, in addition to the notations used in equations (3.6),

w is the wastage vector and Q=P+w'r. A useful generalis-

ation of these results is to replace r by reT) in equation

( 3. 7 ) ; if r (T) , (T=1,2, ... ,h) are still fixed in advance,

but not necessarily equal, we can prove that we need only to

replace r by reT) in (3.8a) and in the definition of Q. In

the case of strategy F2, we have:

kr(T)=r={l/ 1:

i=ln.w

~}n(I-P) , (T~l)

But for the purpose of using equations (3.8), we will put

r(h)=O; the mean vector and the variance-covariance matrix

of f(h) will be respectively equal to n(h) and V(h). How-

ever, this development does not take into account the fact

that R(T), as defined in equation (3.7), might need to be

rounded to an integer vector. Fortunately, even with this

omission, the results in table 8 show that the multivariate

Normal distribution, with mean vector and variance-

covariance matrix as defined by equation (3.8), does

approximate quite well the distribution of f(h).

- 106 -

Table 8Ca-parison of the Normal approximation (HA)

to P(O < f(2) < n/n) with the exact value (EV),in the case of the recruitment strategy F2

,Transition Matrix P3Structure n

EV NA NA-EV 100EV x

"Transition Matrix P3

EV NA NA-EV 100EV x

( 5, 5, 5) 0.47 5 0.484 2 0.642 0.643 0(10,10,10) O. 531 O. 533 0 o . 725 0.732 1(15,15,15) 0.585 0.585 0 0.785 0.794 1( 20, 20, 20) O. 632 0.633 0 0.831 O. 841 1

As to the computation times,

those concerning table 7 still apply.

the same remarks as

- 107 -

CHAPTER IV

MAINTAINABILITY AND ATTAINABILITY IN ASTOCHASTIC ENVIRONMENT

The stock numbers in a graded manpower system change

over time as a result of wastage, promotion and recruitment

flows. By acting upon the latter, we would direct the

evolution of the system towards some desired situations.

More precisely, the objective of control by recruitment, in

a stochastic environment, would be minimising or n~ximising

the expected value of some function of economic or social

interest. Generally, this function would depend on the

flows from and to di fferent grades and on the importance

given to each grade and to different steps. One particular

function could be seen as a generalisation of deterministic

theory; in the latter theory, there has been extensive work,

by Bartholomew (1973), Grinold and Stanford (1974), Davies

(19 75) and Va jda (19 78 ), on IDri intaining or attaining a

structure in one or many steps. In a stochastic environ-

ment, such an aim would be impossible to achieve but one

would try, instead, to get as close as possible to the goal

structure in a specified number of steps. Our objective is

then to minimise the expected distance between the structure

reached in h steps and a target structure. We will refer to

the problem as the maintainability problem if the initial

and target structures are identical and as an attainability

problem otherwise.

- 108 -

Given the complexi ty of the stochastic environment,

and the lack of analytical methods to evaluate even simple

probabilities that would be needed, we have to rely onnumerical methods to assess different recruitment strate-

gies.

In the following sections, after describing the basis

of our numerical approach, we will concentrate on maintain-

ability and attainability in a stochastic environment. We

will describe the recruitment strategies and numerical

examples that we will consider and present the main con-

clusions shown from such cases.

4.1 NUMERICAL APPROACH

The aim of this section is to explain how we propose

to evaluate the expected value V(n,h) of an objective

function over a period of h steps, starting at vector n.

Let us suppose that at the end of each step, after wastage

and promotion flows have taken place, we know what recruit-

ment policy we should apply and we also know how to deter-

mine the recrui tment vector R. At this stage it is not

necessary to specify whether R depends on wastage and pro-

motion flows or not. Our approach will be shown to deal

with both cases in the same way.

Let us denote by A(v,R,T) a measure of the immediate

consequences in taking R as the recruitment vector at time

T, given that f(T)=v, l<T<h. We will refer to the function

A as the return function.

- 109 -

This function does not take into

account any future consequences to the choice of R. For

example, in a manpower context, we would he interested in

the total sa1ary hi11 and the recrui ting costs over the

period of h steps. Thus, if at time T, f(T)=v and R(T)=R we

have:

kA(v,R,T)= 1:

i=lb.R. +

1. 1.

k1:

i=lc. (v. +R. )

1. 1. 1.

where b. and c. are respectively the average salary and1. 1.

recruiting cost in grade i; the costs b. and c. could depend1. 1.

on T. In this example, it is clear that whilst the function

A takes into account the costs incurred at time T, it does

not consider the effects of the decision upon R on future

costs.

Thus, in general, if the return functions are

additive, V(n,h) can be evaluated by using the following

iterative equation:

where R is the recruitment vector at the end of the first

period.

In addition, if we are interested in the minimal

value V*(n,h) of V(n,h) with respect to

{R(1),R(2), ... ,R(h)}, we would change the equation (4.1) to:

- 110 -

V*(n,h)= L:P(f(l)=v /n(O)=n)Min A(v,R,1)+V*(v+R,h-1)v

(4.2)

The equation (4.1) could be seen as a generalisation of the

equation (3.4) in which:

A - 0

V(n',T) _ P(O~f(T)~m/n') ,1~T~h

V(n',O) = 1 if n'=m

o if n':/=m

where n' is a possible stock vector at time T, 0 < T < h.

As in the discussion of quation (3.4), if R is a random

vector, equation (4.1) should be modified as follows:

v (n ,h) =L P (f (1)=v /n (0 )=n ) xv

)L p(R=r/f(1)=V)X{A(V"R,1)+V(V+R,h-1)}} (4.3)( rEE

where E is the set of all possible outcomes of vector R,

given f(l)=v.

The equations (4.1), (4.2) and (4.3) are very general

in the sense that they could deal with a very wide range of

objective functions and recruitment strategies. In the case

of the attainability problem, for example, V(m, h) will

represent the expected distance between n(h) and a goal

•

- 111 -

structure m, under a specified recruitment strategy, and

V*(m,h) will represent the minimal value of such distance

over an acceptable set of recrui tment strategies; in both

cases, n is considered to be the initial vector. These

values could be calculated respectively by using equation

(4.1) and equation (4.2). We need, then, to put A=O and

V(n',O)=V*(n' ,O)=D(n',m), where n' is a possible outcome of

n(h) and D is a measure of the distance between two vectors.

However, since our approach is mainly a generalisation of

equations (3.4) and (3.5), and for the same reasons given in

section (3.2.1), it would be impracticable to use this

approach for general recrui tment strategies or for systems

with either a high number of grades or high stock numbers;

the amount of calculations would be too high. Therefore, we

will consider only recruitment strategies which will not

change the total size of the system. Also, we will concen-

tra te only on s truc tures wi th total sizes not greater than

twenty four.

4.2 RECRUITMENT STRATEGIES

Besides very rare and special situations, there could

be no certittrle of attaining exactly a structure m from a

structure n in one step or, indeed, in many steps. Thus for

a fixed number h of steps, we would be interested to know

how a recruitment strategy could bring us close to the

target structure m.

- 112 -

Recruitment policies at the end of each step could be

divided into two groups according to the timing of thedecision on the number of recruits to different grades;

policies which depend on internal transfer and losses,

during an observed period, and policies which make no use of

such flows. We will consider only policies from the former

group; with this choice, we have the possibility of

controlling exactly the total size of the system and taking

into account the limitations of the total size discussed

earlier. Many strategies could still be considered but we

retain only three relevant strategies:

Fixed strategy F: This, at each step, allocates new

recruits to different grades proportionally to the

fixed vector a=m(I-P). This is the same strategy as

F2 from the previous chapter, where m was equal to n.

The suffix "2" is dropped since it is the only fixed

strategy that we will consider and therefore there is

no need for such precision. This strategy cannot be

used if m is not maintainable.

Goal strategy Ah: This tries to get as close as

possible to the target structure m in a fixed number

h of steps, h>2.

Adaptive strategy A1 This is a particular case of

the latter strategy and tries, at each step, to get

as close as possible to the target structure m. It1is the same strategy as 52 from the previous

chapter.

- 113 -

The distance D(u,v) between two vectors u and v could

be measured in many ways, but we consider the squared

distance function:k

D(u,v)= Ii=l

2(u. - v. )1 1

to be the most

appropriate. In the case of the fixed strategy, it may be

necessary to round up or down the entries of a recrui tment

vector to integer values. There are many ways in which this

could be done, but we will use the same techniques intro-

duced in the previous chapter.

In the case of the fixed strategy, the choice of r as

the vector with entries proportional to those of m(I-P) was

moti vated by two considerations. Firstly, as shown in

chapter one, the expected stock vector will converge in the

long run to the goal vect~r m, independently of the initial

struc ture. Secondly, if we consider the entries of r as

probabilities and not as fixed proportions, such a recruit-

ment vector will ensure, in comparison to other vectors,

the best expected distance D(n(~),m) between n(~) and ffi.

A proof of this proposition is given below.

Given the new interpretation concerning the elements

of the recruitment vector and the fixed total size assump-

tion, a transition from grade i to grade j can take place

either within the system or by loss from grade i and

replacement to grade j with total probability p ..+w.r.. It1J 1 J

can then be srown that Qt represents the probabil ities of

transition from one grade to another in t steps, with

Q=P+w'r. The matrix Qt was already proved to converge as

- 114 -

t+- to a stochastic matrix 1T= en', where n is the only

solution to the equation n=nQ and e the row vector with all

its elements equal to one. Thus, independently of nCO), thestock vector n(m) has a Multinomial distribution with para-

meters (N,n), where N is the system total size. The limit-

ing values of the expected stock numbers at grade i and the

variance of the latter numbers are therefore as follows:

0i (m) = Nni

Vi ( CD ) = Nn. (1- n. )1 1

i =1 , 2 , ... , k

These results hold for any recruitment vector r, but

in the case of r*=m(I-P)/rnw' we have in addition:

n*i(CD) - m - Nn*- i - 1

V*(~) = Nn* (l-n*)111

It follows then:

i =1 , 2 , .•. , k

k kD* = D'{ n * (C1D) , m) = N r II ~ (1- II *) + N( 1- 1:

i =1 1 1 i =1For any other recruitment vector r, we have:

kN2 k

II~)2D(n(CID) ,m) N r n. (l-n.) + 1: ( n .i=l 1 1 i=l 1 1

kn~) N2 k ..:, ) 2N(1- L + 1: ( n . n .

i=l 1i =1 1 1

k 2 k 2 2k *2= N(l- L (n.-n~) - 1: n~ ) + N 1: (n.-n.). 1 1 1 . 1 1 1·--1 1 11= 1=

Henc e,

k= D* + N(N-1) r

i =1

D* < D(n(QO) ,m)

(n._n~)21 1

- 115 -

4.3 truMERICAL EXAMPLES

To see how the recrui tment strategies affect the

evolution of a system, we have to rely on numerical com-

parisons. In the choice of the transition matrices, we took

into account two factors:

The relative magnitudes of the off-diagonal elements

in comparison to the diagonal elements.

The wastage rates at all grades.

In his examples, Bartholomew (1977) considered only the

first factor and used two transition matrices:

0.20.8 0.1

0.9

and

0.40.6 0.3

0.8

Both matrices present low wastage rates, but, the

ratios between the diagonal and off-diagonal elements are

much bigger in the case of Pl than in the case of P2. Thus,

in order to allow a comparison wi th Bartholomew's work and

to take into account the second factor, we considered in

addition to P1 and P2 the following matrices:

- 116 -

0.16

0.62 0.80.70

and

0.31

0.47 0.230.63

The entries of P3 and P4 are in the same proportions

as those of P1 and P2 respectively but the wastage rates in

the case of the former matrices are much higher than those

of the latter. As to the choice of the goal vector m, we

considered in the case of each transition matrix and each

fixed total size, two vectors, one that could be maintained

by recruiting only at the lowest level and the other that

could be maintained by equal recrui tment at each level.

When, for a given transition matrix and a fixed total size,

there were no such vectors with integer entries, we con-

sidered neighbouring vec~ors. Finally, given the limitation

of our numerical approach, we considered only systems with

total sizes equal to 12 and 24.

above are listed in table 9.

The examples dis(.ussed

- 117 -

Table 9Syste.s to be investigated by exact methods

Transition Total Goal Vector m-mPMatrix m Size m

12 ( 4, 4, 4) (1.20 ,0, 0)

(0.70.2

0.1 0)( 1 , 3, 8) (0 .30,0 .40 ,0 .50)

P1= 0.8 24 ( 8, 8, 8) (2.40,0, 0)0.9 ( 3, 7,14) (0.90,0.80,0.70)

12 ( 7, 3, 2 ) (3.22,0.02,0 .36 )(O.~ 0.16 ( 3, 4, 5 ) (1.38 ,1 .04 ,1 .18 )

P3= 0.62 0.08 ) 24 (15, 7 , 2 ) (6.90,0.26,0.04)0.70 ( 5 , 9,10) (2 .30,2 .62,2 .28 )

12 ( 3, 3, 6) (1.50,0.00 ,0.30 )

(0.50.4

0.3 )

( 1 , 3, 8 ) (0 .50 ,0 .80 ,0 .70 )P2= 0.6 ,-4 ( 6, 7,11) (0.30,0.40,0.10)

0.8 ( 3, 6,15) (1.50 ,1.20 ,1.20 )

12 ( 5, 4, 3) (3.05,0 .57,0 .28 )0.39 0.31 ( 2 , 4, 6 ) (1.22 ,1 .50 ,1 .48 )

P4= 0.47 0.23 ) 24 (11, 8, 5 ) (6.71,0.83,0.16)0.60 ( 4, 8,12) (2 .44 ,3.00 ,2 .96 )

We notice that all matrices are upper-diagonal and

the number of grades in all systems is equal to three. For

other systems which do not fall within such specifications

we have to rely on simulation methods.

latter type are listed in table 10.

Examples of this

- 118 -

Table 10Syste.s to be investigated by simulation

Transition Total Goal Vector m-mPMatrix P Size m

12 ( 6, 3, 3) (2.4,0, 0)

(0 . 4 O. 2 O. 2 ) ( 4, 4, 4) (0.8,0.8,0.8)PS= 0.2 0.4 0.2 24 (12, 6, 6 ) (4.8,0, 0)

0.2 0.2 0.4 ( 8, 8, 8 ) (1.6,1.6,1.6)

0.6 0.3 12 ( 3, 3, 3, 3 ) (1.2,0, 0, o )0.7 0.2 ( 1 , 2 , 3, 6 ) (0.4,0.3,0.2,0.3)

P6= 0.8 0.1 48 (12,12,12,12) (4.8,0, 0, 0)0.9 ( 3, 7,13,25) (1.2,1.2,1.2,1.2)

The first four e~'1mples in table 9 are treated by

simulation and exact methods in order to allow a comparison

of both methods.

All the examples in table 9 and table 10 will be used

to survey the effects of different recruitment strategies

upon the evolution of a graded system. This evolution will

be observed over a quite long period in order to examine the

behaviour of the system at its steady state. From our cal-

culations, we found that a period of 10 steps meets that

objective.

- 119 -

4.4 COMPARISON BETWEEN THE EXACT AND SIMULATION METHOD

The simulation cannot be applied in the case of the

goal strategy; this latter requires a prior knowledge, at

any step t, of smallest average distances, between all

possible stock vectors and the goal vector in (h-t) steps.

The evaluation of such distances by simulation would be too

lengthy and of a very poor accuracy. By contrast, the use

of simulation method in the case of fixed and adapti ve

strategies is much easier and prese~ts more flexibility than

our exact method. This is more true if we consider systems

with general transition matrices or with a number of grades

bigger than three.

As to the quality jf the results obtained by means of

simulation, we conducted a comparison of the latter results

with those obtained by the exact method. This comparison

was done in the case of two sets of examples; the first set

consists of the first examples in table 9. In this case, we

compa red the resul ts obtai ned by 10, 000 simula tions of a

period of 10 steps. We were able in both methods, and in

the case of fixed and adaptive strategies, to have inform-

ation on the whereabouts of the system at each step. The

resul ts of such comparisons are given in table F.1

and A.1

(i=1,2 ,3,4) in Appendix B. They show the high quality of

the simulation method. As to the second set, we considered

four examples from Bartholomew (1977); the author observed

the evolution of the systems over a sequence of 10,000

steps, using simulation.

- 120 -

His resul ts are therefore con-cerned with the whereabouts of the systems only when theyreach their steady state behaviour. We found, as shown intables 11.a and 11.b, the results obtained by the exactmethod, at the 10th step, to be very close to Bartholomew's.

Table 11.aThe average structures maintained in a sequenceof 10,000 trials and the expected stock vector

at the 10th step (initial and goal vectors are equal)

Transition Goal Recrui t- Average Expected StockMatrix Vector rnent Structures Vector

Strategy

(8 ,8, 8 ) F * (8.0,8.0,8.0)0.7 0.2 ~ A (6.16,8.21,9.63) (6.5,8.24,9.26)

0.8 0.1 (2,5,11) F * (2.0,5.0,11.0)0.9 A1 (1.75,4.95,11.30) (1.84,4.94,11.22)

0.5 0.4 (7,7,11) F * (7.0,7.0,11.0)0.6 0.3 A1 (5.58,7.11,12.31) (5.8,7.06,12.14)

0.8 (3,6,16) F * (3.0,6.0,16.0)A1 (2.78,5.96,16.27) (2.86,5.94,16.20)

* Results for fixed strategies were not given inBartholomew (1977).

- 121 -

Table 11.bVariances of a sequence of 10,000 trials compared tothe exact variances of stock vectors at the 10th step

(Initial and goal vectors are equal)

Transition Goal Recruit- Variances from Vari(\.lIlc..~ ~}rc:>~Matrix Vector ment simulation to''LU ~d...

Strategy method

(8,8, 8) F (5.30,5.07,5.10) (5.33,5.23,5.19)O.7 O.2 A1 (2.03,2.34,3.31) (1.92,2.71,3.09)

0.8 0.1 (2,5,11) F (1.25,2.97,3.68) (1.29,2.93,3.35)0.9 A1 (0.31,0.59,0.71) (O.16~0.58,0.64)

0.5 0.4 (7,7,11) F (5.06,5.16,6.23) (5.03,5.04,6.14)0.6 0.3 A1 (1.66,1.98,3.08) (1.60,2.36,3.21)

0.8 (3,6,16) F (1.53,3.33,4.23) (1.63,3.51,4.23)A1 (0.29,0.52,0.76) (0.16,0.53,0.71)

The comparison above, in the case of both sets of

examples, shows that would be safe to rely on simulation

whenever the use of exact methods is not practicable.

4.5 COMPARISON OF THE RECRUITMENT STRATEGIES

Let us denote by E (n.(t)-m. )2 the expec ted di stance~ ~

between the number of individuals in the i th grade of the

stock vector n(t) reached after t steps and the number of

those in the ith grade of the structure m to be maintained.

Then:

E (n. (t )-m. ) 21 ~

E (n. (t )-n. (t ) )2~ ~

= E (n. (t )-n. (t ) )2~ ~

is the variance of+ - 2(n. (t)-m.), where

~ ~

n. (t) and n. (t) is the~ ~

expected value of n. (t).1

If under

The difference

grade.

- 122 -

(n.(t)-m.) will be called bias at the1 1

a recruitment strategy n. (t)=m.,1 1

(i=1,2, .•. ,k), the strategy will be said to be unbiased. Atthe tth step, the expected distance E(n. (t)_m.)2 between

1 1

the stock vector net) and the structure m is, thus, affected

by two factors: variability and degree of bias of the

variables n.(t), i=1,2, ... ,k.1

Following on from this dis-

cussion, let us now assess the performance of each recruit-

ment strategy.

4.5.1 Case of Maintainability

In this case the initial and goal vectors are iden-

tical. The results concerning the evolution of the twenty

four examples discus~ d previously are given in tables 12 to

19 and in greater detail in Appendix B. They show that:

(a) Although the adaptive strategy is biased at all

steps, the bias in most cases is not significant and

does not show big fluctuations from one step to

another. Furthermore, the bias tends to become

relatively smaller when the total size of the system

increases. In the case of systems with high wastage

rates, the mentioned decrease in the bias was

observed even in absolute terms. These observations

hold better when the structure to be maintained is in

the middle of the maintainable region than when it is

at the border of this latter.

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•-B••••1

- 126 -

(b) The variances of the stock numbers under the adaptivestrategy, increase with the number of steps. This

increase is more significant at the beginning of theperiod of observation and tends to disappear after

six or seven steps when the systems start to present

a steady state behaviour. These variances are small

in general but are remarkably much smaller in the

case of structures at the centre of the maintainable

region. They increase almost in the same proportions

as the total size. However, in the case of systems

at the middle of the maintainable region or systems

with very high wastage rates, the variances increase

in lesser proportions than the total size.

(c ) Although the fixed strategy F is unbiased, the

variances of the stock numbers n. (t )~

(i=1,2,3),

(t=1,2, ...,10) are very high and are bigger th~n the

mean square errors

strategy Ai.

of n. (t )~

under the adaptive

127

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(d)

- 128 -

Under the fixed strategy F, the variances in all

numerical examples increase in the same proportions

as the total size, regardless of the position of the

structure to be maintained in the maintainable region

and of the wastage rates. This observation could be

explained by the relation (3.8b) and (3.Bc). As tothe evolution of these variances over time, the same

pattern as the adaptive strategy was observed: an

increase in the first steps and a steady state

behaviour later.

(e) In the early steps, the goal strategy takes the stock

vectors n(t) very far from the goal structure m,

presumably to where it would be easy to attain m,

before bringing it back to the latter structure at

the 10th step. This behaviour introduces a very high

bias at all steps except at the final step.

(f) The va riances of the stock numbers under the goal

strategy do not behave in t:he same way as under the

fixed and adaptive strategies: high fluctuations are

observed from one step to another and no kind of

limit is reached. On average, they are smaller than

those under the fixed strategy but are higher than

those under the adaptive strategy. As to the other

strategies, the goal strategy presents less variabi-

lity in the case of structures in the middle of the

maintainable region than in the case of those at the

border of the latter region.

129

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i-'

- 133 -

From the observations above, it is clear that there

is no advantage in looking many steps ahead (goal strategy),

rather than concentrating only on the next step (adapti ve

strategy). Indeed, as shown in table 19, expected distances

between the stock vector at the 10th step and the goal

vector m, under the adaptive strategy Al, are not very far

from those under the goal strategy AIO. Moreover, unlike

the goal strategy, the adaptive strategy does not show bad

behaviour in the intermediate steps (t=1,2, ... ,9) but always

keeps n(t) very close to the goal vector m. It al so gives

better results than the fixed strategy and its implement-

ation would not require more elaborate efforts.

These conclus_ Jns should not be considered to be

specific to the observed numerical examples; indeed, these

latter represent a very wide range of type of systems and

take into consideration all relevant factors. It would be

safe then to assume that for a graded system from the real

world, the adaptative strategy would behave better than the

goal and fixed strategies and would exert a tight control on

its evolution at all steps, especially if the goal vector is

not at the border of the maintainable region; the control of

a graded manJX'wer system in a stochastic environment could

thus be well achieved by simple and practical strategies.

- 134 -

4.5.2 Case of Attainability

In this section, we propose to examine the behaviour

of different recrui tment strategies when the initial andgoal vectors are not equal. This situation would occur ifwe were initially at a given structure and want to move to

another one which is, according to some economic or social

criteria, considered to be more desirable. We will, how-

ever, be concerned only with the periods of transition of

short durations and in particular with three-step periods.

Should a problem of attainability with a long period arise,

one ought to rely on the conclusions from the previous

section. Indeed, we observed in the case of long periods

and under different recruitment strategies that the initial

structure has littJ~ effect upon the behaviour of the

system, especially at the final steps. This can be ex-

plained easily by the Markovian property of the different

flows.

In addition to the three strategies that were con-

sidered in the previous section, two more strategies will be

introduced. In that section, we considered only one deter-

ministic strategy, that is F, under which the new recruits

are always allocated to different grades proportionally to

the elements of the vector m(I-P), where m is the goal

structure. The initial and goal vectors were identical and

equal to m, and consequently the strategy F was proved to

maintain on average the structure m. In the present con-

text, the ini tial and goal vectors are not equal and the

- 135 -

latter property no longer holds. There is therefore a need

to consider a new deterministic strategy which allows for

attaining the goal vector. Such a requirement can be shown

to be met by considering the strategy which allocates thenew recruits to different grades proportionally to (m(l)-nP)

in the first step, to (m(2)-m(1)P) in the second step and to

(m-m(2)P) in the third and final step, {n,m(1),m(2),m}

being a three-step attainable path from n to m. We will

reserve the qualification "deterministic" for this strategy

and will denote it by D3. There would be as many strategies

of type D3 as attainable paths from n to m and henceforward

the attainable path will always be given whenever the

deterministic strategy is used.

In conjunction with an attainable path

{n , m ( 1) ,m ( 2 ) ,m } from n to m, another stra tegy can be

defined. It is similar to the adaptive strategy Al but

tries, at each step t (t=1,2,3) to get as close as possible

to m( t. ) , m( 3) being equal to m. Such strategy will be,

denoted by A1

It is clear that the fixed strategy F cannot be used

if m is,

and A1

goal and

not maintainable and that the same applies for D3if m is not attainable in three steps from n. The

adaptive strategies, however, can be used in all

situations and do not require any prior knowledge concerning

the maintainability or attainability of m. Thus, in order

to compare these five strategies wi thout restricting un-

- 136 -

necessarily the numerical investigation to particular

s1 tuations, we considered six different examples, which

include all possible combinations with regard to the main-

tainability and the three-step attainability of the goal

vector m.

Tables 20 to 25 give a comparison of different

recruitment strategies in the case of the six examples.

They show that at the third step, the adaptive strategy

exerts a tight control on the evolution of the systems

considered. Indeed, in tables 20, 22 and 23 the strategies

Al and A3 are almost identical and the differences between

the two in the other tables are not very important. They

are well compensated for by the better behaviour of the

system under Al at the termediate steps.

The introduction of the strategy Ai did not bring

any major improvement over Al. Its requirement of prior

knowledge and the existence of an attainable path denies

the strategy Ai any real advantage on Al. Furthermore,

results from tables 20 and 22 show that Al can provide

better control than Ai.

As to the fixed and deterministic strategies, they

both perfonn badly. Their main problem lies in the high

variations of the stock numbers and makes the quality of

their control doubtfUl, especially if the total system size

happens to be high. In the latter case, the variations will

be higher. Nevertheless, if much importance is attached to

- 137 -

the unbiasedness of a strategy, D3 is obviously of interest.Indeed, D3 can be shown to allow for the attainability onaverage of m(t) at each step t (t=1,2,3). Results in tables20, 21 and, much more markedly, in table 22 show that D3exerts tighter control than F.

Table 20Expected stock vector and mean square errors of the

stock numbers at the third step, under differentrecruitment strategies

Re crui tment Expec ted stock vector Mean square errorsstrategy

Grade Grade Grade Total Grade Grade Grade Total1 2 3 1 2 3

A3 1.76 3.17 7.07 12.00 0.38 0.80 0.73 1.91A1 1.71 3.19 7.10 12.00 0.38 0.85 0.78 2.01A' 1.61 3.29 7.10 12. 00 0.51 1.01 0.93 2.451F 1.98 3.6L 6.40 12.00 1.26 2.58 2.76 6.60D3 2.00 3.00 7.00 12 .00 1.50 2.01 2.59 6.10

with:Initial vector nGoal vector mTransition matrix

= (3, 5, 4)

= (2 , 3, 7 )

= 0.5 0.40.6 0.3

0.8Three-step attainable path from n to m:

m(O) = nm (1) = (2 , 5, 5)m(2) = (1, 4, 7 )m (3) = m

Observation: m is maintainable and attainable in three stepsfrom n.

- 138 -

Table 21Expected stock vector and mean square errors of the

stock numbers at the third step, under differentrecruitment strategies

Recruitment Expected stock vector Mean square errorsstrategy

Grade Grade Grade Total Grade Grade Grade Total1 2 3 1 2 3

A3 1.91 2.70 7.40 12.00 0.31 0.68 1.03 2.02A1 1.66 2.61 7.73 12.00 0.42 0.81 1.75 2.99A' 1.89 2.59 7.59 12.00 0.31 0.76 1.24 2.311F 2.00 1.93 8.07 12.00 1.19 2.51 3.27 6.96D3 2.00 3.00 7.00 12.00 1.45 1.92 2.79 6.15

with:

Initial vector n = (0, 0, 12)

Goal vector m = (2, 3, 7)

Transition matrix 0.4 )0.6 0.3

0.8

Three-step attainable path from n to m:

m(O) = nm (1) = (2 , 0, 10)m (2) = (3, 1, 8)m ( 3) = m

Observation: m is maintainable and attainable in three stepsfrom n.

- 139 -

Table 22Expected stock vector and mean square errors of the

stock numbers at the third step, under differentrecruitment strategies

Recruitment Expected stock vector Mean square errorsstrategy

Grade Grade Grade Total Grade Grade Grade Total1 2 3 1 2 3

A3 1.14 3.01 7.85 12.00 0.34 0.50 0.53 1.36A1 1.13 3.08 7.79 12.00 0.34 0.54 0.58 1.46A' 1.22 3.32 7.46 12. 00 0.47 0.83 0.83 2.131F 1.53 3.95 6.52 12.00 1.37 2.95 4.32 8.64D3 1.00 3.00 8.00 12.00 0.83 1.73 1.77 4.33

with:

Initial vector n

Goal vector m

Transition matrix

= (6, 0, 6)

= (1, 3, 8)

= (0.5 0.40.6 0.3 )

0.8

Three-step attainable path from n to m:

m(O) = nm ( 1) = ( 3, 3, 6 )m(2) = (2, 3, 7 )m (3) = m

Observation: m is maintainable and attainable in three stepsfrom n.

- 140 -

Table 23Expected stock vector and mean square errors of the

stock numbers at the third step, under differentrecruitment strategies

Recruitment Expected stock vector Mean square errorsstrategy

Grade Grade Grade Total Grade Grade Grade Total1 2 3 1 2 3

A3 1.85 4.49 5.65 12.00 0.92 4.46 3.46 8.84A1 1.85 4.50 5.65 12.00 0.92 4.47 3.47 8.86A'1F 2.72 5.45 3.83 12.00 2.37 8.86 12.44 23.67D3

with:

Initial vector n = (12 , 0, 0)

Goal vector m = ( 2 , 3, 7 )

Transition matrix = (0. 5 0.40.3 )0.60.8

Observation: m is maintainable but it is not attainable inthree steps from n.

- 141 -

Table 24Expected stock vector and mean square errors of the

stock numbers at the third step, under differentrecruitment strategies

Recruitment Expected stock vector Mean square errorsstrategy

Grade Grade Grade Total Grade Grade Grade Total1 2 3 1 2 3

A3 2.07 5.62 4.31 12.00 0.63 1.48 1.31 3.41

Ai 1.84 5.31 4.85 12.00 0.61 1.68 2.32 4.61

A' 2.15 5.50 4.34 12. 00 0.67 1.54 1.31 3.511F

D3 2.00 6.00 4.00 12.00 1.62 2.87 2.40 6.89

with:

Initial vector n

Goal vector m

Transition matrix

= (10, 0, 2 )

= ( 2, 6, 4)

=(0.5 0.4 )0.6 0.30.8

Three-step attainable path from n to m:

m(O) = nm (1) = (6 , 4, 2 )m(2) = (4, 5, 3 )m (3) - m

Observation: m is not maintainable but it is attainable inthree steps from n.

- 142 -

Table 25Expected stock vector and mean square errors of the

stock numbers at the third step, under differentrecruitment strategies

Recruitmentstrategy

Expected stock vector Mean square errors

Grade Grade Grade Total Grade Grade Grade Total1 2 3 1 2 3

2.06 5.81 4.13 12.00 0.65 1.42 1.19 3.261.92 5.49 4.60 12.00 0.66 1.55 1.92 4.13

A'1F

with:

Initial vector n (11 , 1 , 0)

Goal vector m = ( 2 , 6, 4)

Transition matrix = (0.5 0.40.6 0.3 )

0.8

Observation: m is neither maintainable nor attainable inthree steps from n.

- 143 -

GENERAL CONCLUSIONS

In addition to other questions that arise inconnection with attainability and maintainability in

stochastic or deterministic environments, we addressed twomajor and important issues: the evaluation of

probabilities related to the distribution of stock numbers

at different steps, and a detailed comparison of a range

of recruitment strategies on exact results basis.

So far as the computational aspects of our study

are concerned, we believe that some progress has been

achieved. An exact and efficient method has been

developed to evaluate probabilities of maintaining or

attaining a structure in one step. It was designed for a

special but very important case of systems in which

promotion is only possible to the next higher grade. Its

generalisation to cover a wide range of systems and,

especially, to deal with the maintainability and

attainability in many steps, is still open and much

needed. Nevertheless, as shown in chapters three and

four, its efficiency has made possible the use of exact

results in the comparison of different recruitment

strategies and brought to completion the computation of

the distribution of stock numbers which was formerly

accomplished by means of simulation techniques only. The

approach followed in the latter comparison imposed

restrictions in the choice of numerical examples. Systems

with a high number of grades or large total size, and

- 144 -

strategies ·which allow for a variation in the total sizeof the system were not considered. These restrictions,

however, were not so severe as to prevent us from coveringa wide range of situations and from considering different

factors which would have influenced the results of the

numerical investigation.

From our comparison, the most practical and

important conclusion to emerge is that the adaptive

strategy is generally superior to all deterministic

strategies and does achieve a tight control on the

evolution of a manpower system. The assessment concerning

the quality of the adaptive strategy was reinforced by the

comparison of the latter with the newly defined goal

strategy. This latter, however, cannot be recommended for

implementation in practice given the complexity of its

rules and, especially, its markedly bad behaviour in the

ear:y steps prior to the fixed time horizon. The fixed

strategy in the context of maintainability and the

deterministic strategy in the context of attainability, on

the other hand, were proved to be unbiased. This could be

weighted against the high variability they cause if much

importance is attached to the unbiassedness.

The foregoing analysis focussed on models in which

the objective is to bring the system as close as possible

to a fixed goal structure. This objective is by no means

- 145 -

the concern of every organisation but, as explained in

chapter four, the approach can be modified easily to copewith other situations.

We made recourse to simulation techniques in the

case of very limited number of examples. This was

motivated by two main reasons. Firstly, having developed

methods to obtain exact results, there was a good

opportunity to test the quality of these techniques. We

have agreed that these techniques offer a high flexibility

with regard to the characteristics of the systems to be

chosen. Indeed, they have been used successfully to

compare different strategies in the case of systems that

do not fall within the confines of limits imposed by the

use of exact methods. This was precisely the second

motivation for their use.

The determination of the number of simulations to

be used in such techniques was based on experimentEtion

only; we considered , successively, different numbers

ranging from 10,000 to 100, 000 and observed that 10, 000

simulations leads to satisfactory results. Theoretically

sound methods for such determination have yet to be

developed.

- 146 -

REFERENCES

Bartholomew, D.J. (1963). "A multi-stage renewal process".

J.R. Statist. Soc. B, 25, pp. 150-168.

Bartholomew, D.J. (1967). Stochastic Models for Social

Processes (1st ed.), Wiley, London.

Bartholomew, D.J. (1973). Stochastic Models for Social

Processes (2nd ed.), Wiley, London.

Bartholomew, D.J. (1975). "A Stochastic control problem in

the social science". Bull. Int. Statist. lnst. 46,

pp. 670-680.

Bartholomew, D.J. (Ed.) (1976). Manpower Planning, Penguin

Modern Management Readings, Penguin Books,

Harmondsworth, Middlesex, England.

Bartholomew, D.J. (1977). "Maintaining a grade or age

structure in a stochastic environment". Adv. Appl.

Pro b ., 9, pp. 1-1 7.

Bartholomew, D.J.,

Techniques

Chichester.

and A.F. Forbes (1979).

for Manpower Planning,

Statistical

John Wiley,

- 147 -

Bartholomew, D.J. (1979). "The control of a grade structure

in a stochastic environment using promotion control".

Adv. Appl. Prob., 11, pp. 603-615.

Bartholomew, D.J. (1982). Stochastic Models for Social

Processes (3rd ed.), Wiley, Chichester.

Bohrer, R.E. and M.J. Schervish (1981). "An error-bounded

algorithm for normal probabilities of rectangular

regions". Technometrics, 23, pp. 297-300.

Charnes, A., W.W. Cooper and R.J. Niehaus (1972). Studies

in Manpower Planning, Office of Civilian Manpower

Management, Department of the Navy, Washington D.C.

Davies, G.S. (1973). "Structural control in a graded

manpower system". Man. Sci., 20, pp. 76-84.

Davies, G.S. (1975). "Maintainability of structures in

Markov chains in models under recrui tment control".

J. Appl. Prob., 12, pp. 376-382.

Davies, G.S. (1982). "Control of grade sizes in partially

stochastic Markov manpower model". J. App. Prob., 19,

pp. 439-443.

Davies, G.S. (1983). "A note on the geometric/probabilistic

relationship in a Markov manpower Model ". J. Appl.

Prob., 20, pp. 423-428.

- 148 -

Finucan, H.M. (1964). "The mode of a multinomialdistribution". Biometrika, 51, pp. 513-517.

Forbes, A.F. (1971). "Promotions and recruitment policiesfor the control of quasi-stationary hierarchicalsystems". In A.R. Smith (1971), pp. 401-414.

Gani, J. (1963). "Fonnulae for projecting enrolments anddegrees awarded in universities". J.R. Statist.Soc., A, 126, pp. 400-409.

Grinold, R.C. and K.T. Marshall (1977).Models, North-Holland, New York.

Manpower Planning

Grinold, R.C. and R.E. S~anford (1974).a graded manpower system". Man.1216.

"Optimal control ofSci., 20, pp. 1201-

Iosifiscu, M. (1980). Finite Markov Processes and theirApplications, Wiley, Chichester.

Mehlmann, A. (1980). "An approach to optimal recruitmentand transition strategies for manpower systems usingdynamic programming". J. Operate Res. Soc., 31, pp.1009-1015.

Milton, R.C. (1972). "Computer evaluation of themultivariate normal integral". Technometrics, 14, pp.881-889.

- 149 -

Morgan, R.W. (1971). "The use of a steady state model to

obtain the recruitment and promotion policies of an

expanding organisation". In D.J. Bartholomew and A.R.

Smith (1971), pp. 283-291.

Schervish, M.J. (1984). Algorithm AS 195. "Multivariate

normal probabilities with error bound". Appl.

Statist., 33, pp. 81-87.

Seal, H.L. (1945). "The mathematics of a population

composed of k stationary strata each recrui ted from

the stra tum below and supported at the lowest

level by a uniform annual number of entrants".

Bi ome t ri k a, 33 , pp. 2 26- 2 30 •

Smith, A.R. (Ed.) (1971). Models for Manpower Systems,

English Universities Press, London.

Vajda, s. (1947). "The stratified semi-stationary

population". Biometrika, 34, pp. 243-254.

Vajda, S. (1975). "Mathematical aspects of manpower

planning". Operate Res. Quart., 26, pp. 527-542.

Vajda, S. (1978). Mathematics of Manpower Planning, Wiley,

Chichester.

Young, A. and G. Almond (1961). "Predicting distributions

of s t a f f". Comp. J., 3, pp. 246- 250 .

- 150 -

APPENDIX A

RESOLl1I'ION OF SOME QUADRATIC

PROGRAMMING PROBLEMS

A.l INTRODUCTION

Consider the problem

Minimise f(x)

Subject to:

k 2= L (a . - x .)j=l J J

kL x. = S

j=l J

x. > 0J - j=l ,2,... ,k

For its solution we will consider two cases:

x. are realsJ

x. are integersJ

In the latter case we will refer to the problem by

P(a,S), where a=(a1, a2, ..., ak) and S an integer.

A.2 CASE 1: x. ARE REALSJ

Let F =k 2 kL (a. - x.) + A( S - 1: x.)

j=l J J j=l J

- 151 -

aF= - 2 (a . - x .) - ).

ax . J JJ

a 2 F= l~ if i 1= j

ax. ax . if i = j~ J

Therefore, according to Kuhn-Tucker theory, any point

which satisfies the first-order

necessary conditions is minimum. Moreover, since n, the set

of feasible points, is convex and f is convex on n, any

relative minimum of f is a global minimum.

The necessary conditions for a minimum are:

aF= - 2 (a . - x~) - ). > 0 if x-k = 0

J J - Jax .J

aF= - 2 (a . - x~) - ). = 0 if x~ > 0

ax . J J JJ

k1: x~ = S

j=l J

In order to find a solution to this system we will lookfirst at some properties of any solution.

- 152 -

Proposition 1

*If x.Jo = o *and x. > 0, then a.J+ Jo <

Proof: From Kuhn-Tucker conditions we have*- 2 (a . - x. )

J+ J+- 2 a.Jo

= ).

> ).

(1 )

(2 )

Hence *- 2a. > - 2a. + 2x. > - 2a.Jo J+ J+ J+

i.e. a. < a.Jo J+

Therefore, if we suppose, without loss of generality that

> ak, there should be an integer L such

x*; > 0 j < LJ -

x* = 0 j > Lj

Proposition 2

L(S - La. ) + LaLj=l J

L+l( S - La. ) + (L+l)aL+1j=l J

(iii) L is the largest p such that

p(S - La.) + pa

j=l J P > 0

- 153 -

Proof of (1)By definition of L we have:

*2 (x . - a.) = AJ J

- 2 a. > AJ -

j=l , 2 , ••• , L

j=L+l, .•• ,k

( 3 )

(4 )

from which we can deduce:

L*

L2( E x. E a .) = L A ( 5 )

j=l J j=l J

L-l*

L-12 ( E x. E a .) = (L-l) A (6)

j=11

j=1 J.J

* *But given that x. = 0 for j > L and xL > 0 we haveJ

L*

L-1 *E x. = S and 1: x. < Sj=l J i=1 J

Thus, the previous system becomes:

L2 ( S - L a. ) = L A ( 7 )

j=1 J

L-12 ( S - L a. > (L-1) A ( 8 )

• "1 JJ=..l

Substituting in (8), using (7), we obtain:

L-12 ( S - E

j=la.

J

L> (L-l) x 1 x 2 ( S - L

L j=la . )

J

- 154 -

which leads to:

( S -LI

j=la. )

J+

Proof of (ii):From (4) and (7) we deduce that:

L+l2 ( S - I

j=la.

J> (L+l) A (9 )

Substituting into (9), using (7) we obtain:L+l 2 L

2 ( S - I a. > (L+l) - ( S - I a.J - j=l Jj=l L

which leads to:

( s -L+l

Lj=l

a. ) + (L+ 1)J

< 0

Proof of (iii):

Let t(p) = ( s -PI

j=la. ) + pa .

J pThen:

t(p+l) - t(p)p+l p

= ( S - I a. ) + (P+ 1)a 1 ~ ( S ~ La. ) - pa .j=l J p+ j=l J P

= - a 1 + (p+l)a 1 - pap+ p+ p= pea 1 a) < 0 (we assumed that a. 1 < a.).p+ p ~+ ~

And sin ce t (L) > 0 > t (L+1 ), L=Max {p £ IN / t (p) > O}.

This last property will allow us to find L easily; knowingL, the minimum to our problem will be:

- lSS -

* ).x. = - + a. j < LJ J -2

* 0x. = j > LJ

L5 - I a.

j=1 J). = 2

L

A.3 CASE 2: x. ARE INTEGERSJ

Our approach in finding a solution when S is integer,

rests on two fundamental propositions:

Proposition 3

Let 51 and 52 be two integers such that

and

If m*51 is an optimal

an optimal solution to

is an optimal solution

Proof:

solution to P(a, 51) and m*52 is5 5 5 SP(a-m* 1, 52) then m* = m* l+m* 2

to P ( a, 5).

Let m = (m1, m2, ... , mk) be an integer vector such that

kI

i=lm. = 5 and m. > 0,1 1

i=1,2, ... ,k.

- 156 -

the definition of 52 have:Given m* , we

k 2 k *51 *51 2I (a. - m. ) = I [ (a. - m. ) - (m. - m. ) ]

.11 1 i=1 1 1 1 11=

k *51 *52 2> I [ (a. - m ) - m ]- 1i=l i i

5incek 21: (a. - m.) >.11 1.1=

k1:

i=la. -1

5(*1m.

1

5 2* 2, ]+ m. ,1.

for any m and sincek S1: (m.* 1

i =1 1.

S+ m.* 2) = S

1.

51 52m* = m* + m* is an optimal solution to P(a, 5).

Proposition 4:

If 5 = 1 and J. an integer such that a. > a. for allJ - 1.

. 12k th ,\..5 J. h J.. th t .th1.= , ,..." ·en m'" = e, were e 1.S e vec or W1.one at the jth position and zeros elsewhere.

Proof

Consider any solution m. Since S=l there should be a j'

such that mot = 1 and m. = 0 for ilj'. In this case weJ 1.

wi 11 have:

- 157 -

k 2 k 2E (a. - m. ) = E a. - 2a ., + 11=1 1 1 i=1 1 J

k k 5 22> I a. - 2a . + 1 = I (ai - m*. )- 1 J i=1 1i=1

The combination of these two propositions suggests aniterative approach which solves successively the problemsP(a,1), P(a,2), ..., Pea,S) taking always 51=1. Moreprecisely the iterative method is as follows:

1° Set m. = 0 j=1 ,2 ,... ,kJ

2° If S is zero, go to 43° Find the maximum a .. Put :

Jm. = m. + 1

J Ja. = '.:' 1

J jS = 5-1Go to 2

4° The solution of pea,S) is m*S= (m1 ' m 2 ' • • • , mk ) •

STOP

However, when S is large, this method would be very long.Fortunately, the two following propositions will allow usto introduce an improvement;

- 158 -

Proposition 5

If: m* is an optimal solution to Pea,S),b is an integer vector such that 0 < b < m*,

kd = a-b and S' = S - I

i=lb. and1

n* is an optimal solution to P(d,S')

Then n*+b is an optimal solution to pea,S).

Proofk * kI (n. + b. ) = 5' + I b. = S.

i=1 1 1 i=1 1

* *Since n. > 0 and b. > 0 then n. + b. > o.1 - 1 - 1 1 -

MoreoverkI

j=1* 2(a. - m.) =

1 ~

kI

i=1[ (a . -b.) - (m~ - b.) ] 2

~ 1. ~ ~

i .e.kI

i=1* 2(a. - m.) =

~ ~

kI

i=1d.

~*(m.~

k*Thus, since m~ b. > 0 and I (m. b. ) = S'

~ ~ - ~ ~i=1and given the definition of n* we have,

k * 2I (a. - m) >

i =1 ~ i

k * 2I (d. - n. )

. 1 ~ 11==

kI

i=1a.

~*(n.~

- 159 -

Therefore n* + b is an optimal solution to P(a,S).

Note that in this proof we did not use the fact that thedifferent vectors are integers and therefore this

proposition would apply to the real case.

Proposition 6

Let us consider the problem P(a,S). If x* is the real

optimal solution and m* is its integer optimalthen:

1 •SO.1.ut10n,

Proof

*ffi. ~1

*[x. ]1

i=1,2, ... ,k

As we saw in the previous section, there exists an integer*L such that x = 0 for any j > L.j

*ffi. > 0 = [x*].1 - 1

Therefore if *i > L , [x.] = 0 an.d by1

definition of m*,

Suppose that there exists an integer j ~ L such that* *m. < [x .]. Then we should have at least one other p ~ LJ J

such that:

* * L * L *m > [x ], otherwi se 1: m. < 1: [x. ] < SP P i=l 1 1 -i=l

*Thus, since mj

- 160 -

*< x. - 1 we haveJ

*a. - m.J J

* ~> a. - x. + 1 = - - + 1, where ~ is the same- J J 2

as defined in the previous section,

Moreover, since * *m > [x ]p p we have *m* > x which givesp p

*a - mp p *< a - x =p p 2

* * (10)Therefore a - m < a. - m. - 1P P J J

*,

*Let m. = m. + 1 and rn = m - 1 • We have:J J P P

(a . ' 2 (a ' ,2- m.) + - m )J J P P

* 1)2 + * _ 1)2= (a . - m. - (a - mJ J P P

(a. - * 2 * 2 - 2{(a j * * - 1}= m .) + (a m ) m . ) (a m )J J P P J P P

* 2 * 2< (a. - m) + (a - m) because of inequation (10).J j P P

This result contradicts the definition of m*. Thus, there* *is no j ~ L such that m. < [x.].J J

- 161 -

These two propositions suggest the following procedure:1° Release the constraint of integrity of the

variables and solve the problem as in Case 1.

* j=1 ,2, .•• , k2° Define a. = a. - [x . ]J J J

L *5' = 5 - 1: [x . ]j=1 J

3° 501ve the problem P (a', 5 ') using the previousiterati ve method .

4° Then, if n* is an optimal solution of P (a ', 5 ' ) ,

* * *m. = n. + [x.], i=1 ,2 ,... ,k is an optimal solution~ ~ ~

to P (a, S ) •

Proposition 7

(i) S' < L

(ii) There is an optimal solution n* of P (a ', S') such*that n. = 0~

Proof of (i)

i=L+l, L+2, ... ,k

S' = S - L *1: [x.] <

j=1 J

L5 - 1:

j=l*(x. - 1) = 5-S + L = LJ

- 162 -

Proof of (iJ)

Let i > L and j < L-

* * AWe have a. - [x . ] > a. - x. = - - > a.J J - J J - 12

, ,which is equivalent to a. > a ..

J - J

Since S' is at most equal to L and since there are L,

coefficients a. bigger than a., there should be an optimalJ 1

solution with n. = 0 for any i > L because of propositions1

3 and 4.

- 163 -

APPENDIX B

DETAILED RESULTS CONCERNING TIlE EVOLtrrlON OF24 EXAMPLES UNDER DIFFERENT RECRUITMENT STRATEGIES

In thi s Appendix, tables are self-explanatory and

thei r numbers consist of a letter X and a serial number,

where:F

X = AG

in the case of fixed strategyin the case of adaptive strategy

in the case of goal strategy.

The examples considered are those gi ven in tables 9

and 10; their serial numbers are as follows:

Structures StructuresTransition matrix Total maintained by Serial main tained by Serial

Size recrui tment Number equal recruit- Numberat the bottom ment at each

level

0.70 0.20 12 (4,4,4) 1 (1,3,8) 2p - 0.80 0.10 (8,8,8) (3,7,14)1- 24 3 4

0.900.54 0.16 12 (7,3,2) S (3,4,S) 6

p - 0.62 0.08 (IS,7,2) (S,9,10) 83- 24 70.70

0.50 0.40 12 (3,3,6) 9 (1,3,8) 10P2= 0.60 0.30 24 (6,7,11) 11 (3,6,1S) 12

0.800.39 0.31 12 (S,4,3) 13 (2,4,6) 14

P = 0.47 0.23 (11,8,S) (4,8,12)4 24 IS 160.60

0.40 0.20 0.20 12 (6,3,3) 17 (4,4,4) 18PS= 0.20 0.40 0.20 24 (12,6,6) 19 (8,8,8) 200.20 0.20 0.40

0.6 0.3 12 (3,3,3,3) 21 (1,2,3,6) 22P6= 0.7 0.2 48 (12,12,12,12) (3,7,13,25)23 240.8 0.1

0.9

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- 228 -

)PENDIX C: THE ROUTINE "MAINT"

The purpose of this routine is to evaluate the prob-

abilities PR = P(a ~ f(l) ~ b/n(O) = n), in which a, band nare given non-negative vectors.

The routine is written in standard Fortran IV. It

can be called in any other routine by the use of the state-

ment "CALL MAINT (PR)". The input parameters should, then,

be passed to it by the use of the two following statements:

COMMON/Cl/NLOW(30), NHIGH(30), NSTART(30), P(30,31),N

COMMON/C2/SLMT

where:

N

P(i,j)

=

=

number of grades

transition probabilities between

grades i and j (i , j= 1 , 2, ••• , N)

NLOW(i) = a.~

NHIGH(i) = b.~

NSTART(i) = n.~

SLMT = l-a /N,

(i =1,2, ... ,N)

(i =1,2, ... ,N)

(1 =1,2, •.• ,N)

a being the absolute error

acceptable to the user.

The routine, as listed, can be used for any hier-

archical system with a maximum of 30 grades and a maximum of

1000 members at any grade. If the need arises to alter such

limits, all the 30s and 1000s in the dimension statements

have to be modified in accordance with the new requirements.

- 229 -

LISTING

SUBROUTINE MAINT(PR)IMPLICIT DOUBLE PRECISION (A-H,O-Z)COMMON/Cl/NLOW(30),NHIGH(30),NSTART(30),P(30,31),NCOHMON/C2/SLMTCOMMON/C3/MTOT,INDEX,SUM,KANUSE,NOTUSECOMMON/C4/B(2,lOOl)KANUSE=lNOTUSE=2NMIN1=N-lIF(NLOW(l).GT.O.OR.NHIGH(l).LT.NSTART(l» GO TO 3CALL BINOM(NSTART(1),P(l,2),O)IDEBUT=2GO TO 8

3 IDEBUT=l8 DO 5 ID=IDEBUT,NMINI

KEEP=OINTERM=KANUSEKANUSE=NOTUSENOTUSE=INTERHNZ=NSTART(ID)NZZ=NZ+lDO 7 J=l,NZZ

7 B(KANUSE,J)=O.OCALL ENUMER(NZ,P(ID ID),P(ID,ID+l),ID,KEEP)

5 CONTINUEINTERM=KANUSEKANUSE=NOTUSENOTUSE=INTERMCALL BINOM(NSTART(N),P(N,N),I)PR=O.OJUP=MI~O(NSTART(N-l),NHIGH(N»+lDO 6 JX=l,JUPJ=JX-IJUN=MINO(NHIGH(N)-J,NSTART(N»+lJDEUX=MINO(NLOW(N)-J-l,NSTART(N»+lPRl=B(KANUSE,JUN)PR2=O.OIF(JDEUX.GE.l) PR2=B(KANUSE,JDEUX)

6 PR=PR+B(NOTUSE,JX)*(PRI-PR2)RETURNEND

C++++++++++++C++++++++++++C++++++++++++C++++++++++++C++++++++++++CSUBROUTINE BINOM(N,P,KCUMUL)IMPLICIT DOUBLE PRECISION (A-H,O-Z)COMMON/C2/GRANDCOMMON/C3/MTOT,INDEX,SUM,KANUSE,NOTUSECOMMON/C4/B(2,lOOl)PP=PNR=N+ 1DO 273 I=l,NR

- 230 -

273 B(KANUSE,I)=O.IF(N.EQ.O) GO TO 210IND=OIF(PP.LE ••5) GO TO 201IND=lPP=I.-PP

201 M=INT(DFLOAT(N)*PP)+1Z=I.L=N- ~1Q=l.-PPLL=LDO 203 I=l,MJ=M-I+lZ=Z*(DFLOAT(L+J)/DFLOAT(J»*PP

3 IF(LL.EQ.O.OR.Z.LT.I.) GO TO 203Z=Z*QLL=LL-lGO TO 3

203 CONTINUEIF(LL.NE.O) Z=Z*(Q**LL)B(KANUSE,M+l)=ZSUN=ZYG=Q/PPY D=P P / QZG=ZZD=ZJ=O

10 J=J+lI=M-JIF(I)20,30,30

30 ZG=ZG*YG*DFLOAT(I+l)/DFLOAT(K-I)B(KANUSE,I+l)=ZGSUM=SUM+ZGIF(SUM.GT.GRAND) GO TO 209

20 I=M+JIF(I-N) 40,40,50

40 ZD=ZD*YD*DFLOAT(NR-I)/DFLOAT(I)B(KANUSE,I+l)=ZDSUM=SUM+ZDIF(SUM.GT.GRAND) GO TO 209

50 IF(I.GT.N.AND.J.GT.M) GO TO 209GO TO 10

209 CONTINUEIF(IND.NE.l) GO TO 220NN=INT(O.S*DFLOAT(N»+lDO 207 J=l,NNI=NN-JZ=B(KANUSE,N-I+I)B(KANUSE,N-I+I)=B(KANUSE,I+I)B(KANUSE,I+l)=Z

207 CONTINUEGO TO 220

- 231 -

210 IF(N.EQ.O) B(KANUSE,l)=l.IF(N.EQ.O) RETURNB(KANUSE,I)=l.-PB(KANUSE,2)=P

220 IF(KCUMUL.EQ.O) RETURNDO 300 I=2,NRB(KANUSE,I)=B(KANUSE,I)+B(KA~USE,I-1)

300 CONTINUERETURNEND

C++++++++++++C++++++++++++C++++++++++++C++++++++++++C++++++++++++CSUBROUTINE ENUMER(N,Pl,P2,ID,KEEP)IMPLICIT DOUBLE PRECISION (A-H,O-Z)DIMENSION AX(3),MODE(3),P(3)COHMON/C2/SLMTCOMMON/C3/MTOT,INDEX,SUM,KANUSE,NOTUSEP(l)=PlP(2)=P2P(3)=1.-P(1)-P(2)SUM=O.MTOT=(N+1)*(N+2)/2INDEX=OIF(N.EQ.O) GO TO 96NSUM=ODO 41 1=1,3AX(I)=DFLOAT(N+1)*P(I)MODE(I)=INT(AX(I»

41 NSUM=NSUM+MODE(I)ISG=OJM=1IF(NSUM-N) 42,43,44

42 XM=(1.-AX(1)+DFLOAT(MODE(l»)/P(l)ISG=lDO 46 1=2,3X=(l.-AX(I)+DFLOAT(MODE(I»)/P(I)IF(X.GE.XM) GO TO 46XM=XJM=I

46 CONTIt\UEGO TO 43

44 XM=(l.+AX(1)-DFLOAT(MODE(1»)/P(l)ISG=-lDO 47 1=2,3X=(l.+AX(I)-DFLOAT(MODE(I»)/P(I)IF(X.GE.XM) GO TO 47XM=XJM=I

47 CONTINUE43 MODE(JM)=MODE(JM)+ISG

Kl=MODE(1)

- 232 -

K2=MODE(2)KJ=MODE(3)IF(Kl+K2.NE.N) GO TO 49IF(K2.EQ.0) Kl=Kl-lIF(K2.NE.0) K2=K2-1K3=K3+1

49 CONTINUED=TRINOM(N,Kl,K2,KJ,P)IS=KlJS=K2JOLD=-1IOLD=-1IF(JS.NE.O) GO TO 170IOLD=ISOLDI=D

170 IF(IS.NE.O) GO TO 171JOLD=JSOLDJ=D

171 CONTINUEINC=1ISGN=1JJ=(N-KI-K2-1)/2+K2II=N-I-JJIF(IS.EQ.II.AND.JS.EQ.JJ) OLD1J=DJLMT=K2+INC*ISGNILMT=KlGO TO 97

96 IS=OJS=OD= 1 •

97 CONTINUECALL MAKIP(KEEP,ID,IS,JS,D)IF(KEEP.EQ.-l) RETURN

3 JS=JS+ISGNIF(JS*ISGN.GT.JLMT*ISGN) GO TO 1IF(JS.GE.O) GO TO 4ILMT=ILMT-INCISGN=lCOTO 2

4 CONTINUED=RECENT(D,2,ISGN,IS,JS,N,P)CALL MAKIP(KEEP,ID,IS,JS,D)IF(KEEP.EQ.-l) RETURNIF(IS.EQ.II.AND.JS.EQ.JJ) OLDIJ=DIF(IS.NE.O) GO TO 71JOLD=JSOLDJ=D

71 CONTINUEIF(IS+JS.NE.N) GO TO 3

8 INC=INC+lJLMT=JLMT-INCIF(JJ.EQ.-l.AND.IOLD.EQ.O.AND.JOLD.EQ.N) GO TO 50

- 233 -

1F(JJ.EQ.-l) GO TO 21S=11+lJS=JJ1LMT=11+ID=RECENT(OLDIJ,l,l,1S,JS,N,P)CALL MANIP(KEEP,1D,IS,JS,D)IF(KEEP.EQ.-1) RETURN11=1 SJJ=JS-1OLDIJ=D1SGN=-lGOTO 3

1 JS=JS-1SGN1LMT=ILMT+ISGN*INCGOTO 6

7 1S=IS-ISGN1SGN=-1SGN1NC=1NC+lJLMT=JLMT+ISGN*1NCGO TO 3

6 IS=1S+1SGN1F(IS*1SGN.GT.ILMT*ISGN) GO TO 7D=RECENT(D,l,ISGN,IS,JS,N,P)CALL MA~IP(KEEP,ID,1S,JS,D)IF(KEEP.EQ.-1) RETUF~1F(1S.EQ.11.AND.JS.E~.JJ) OLDIJ=D1F(JS.NE.O) GO TO 7210LD=1SOLOI=O

72 CONTINUEIF(1S.EQ.ILMT) GO TO 61F(1S.NE.O.ANO.IS+JS.NE.N) GO TO 61F(1S+JS.EQ.~) GO TO 81NC=1NC+1JLMT=JLMT+1NC1LMT=ILMT+1NCIF(JOLO.EQ.-1) GO TO 3GO TO 14

2 CONTINUE1F(10LD.EQ.-l) GO TO 31NC=1NC+IJLMT=JLMT+1NC1F(IOLD.EQ.O) GO TO 1415=10LD-1JS=O10LO=1SD=RECENT(OLDI,1,-I,IS,JS,N,P)CALL MANIP(KEEP,1D,1S,JS,n)1F(KEEP.EQ.-1) RETURNILMT=I5OL01=OGO TO 3

- 234 -

14 IF(JOLD.EQ.N) GO TO 8J8=JOLD+l15=0JOLD=JSD=RECENT(OLDJ,2,1,IS.JS.N.P)CALL MANIP(KEEP,ID,I5,JS,D)IF(KEEP.EQ.-l) RETURNOLDJ=DILMT=ILMT+INCISGN=lIF(JS.NE.N) GO TO 6GO TO 8

50 ISGN=lILMT=lJS=O15=11GO TO 6END

C++++++++++++C++++++++++++C++++++++++++C++++++++++++C++++++++++++CSUBROUTINE MANIP(KEEP,ID,IX,JX,DX)IMPLICIT DOUBLE PRECISION (A-H,O-Z)COMMO~/CI/NLOW(30),NHIGH(30),NSTART(30),P(30,31),NCOM}10~/C2 / SLMTCOMMON/C3/MTOT,INDEX,3UM.KANUSE,NOTUSECOMMON/C4/B(2,1001)INTEGER T,TMIN,TMAX,ALMI,BLMIJ=JX+lIF(IX.GT.NHIGH(ID).OR.JX.GT.NHIGH(ID+l» GO TO 90IF(ID.NE.l) GO TO 10IF(IX.LT.NLOW(ID» GO TO 90B(KANUSE,J)=B(KANUSE,J)+DXGO TO 90

10 NLMl=NSTART(ID-l)IF(IX.LT.NLOW(ID)-NLMl) GO TO 90A L M I = N L 0 \-1( I D ) - 1XBLMI=NHIGH(ID)-IXTMIN=MAXO(O.ALMI)+1TMAX=MINO(NLMI,BLMI)+1DO 1 T=TMIN,TNAX

1 B(KANUSE,J)=H(KANUSE.J)+B(NOTUSE,T)*DX90 INDEX=INDEX+l

SUH=SUM+DXIF(SUM.LT.SLMT.AND.INDEX.LT.MTOT) RETURNKEEP=-1RETURNEND

C++++++++++++C++++++++++++C++++++++++++C++++++++++++C++++++++++++C

- 235 -

DOUBLE PRECISION FUNCTION RECENT(X,IV,ISGNX,ISX,JSX,N,P)IMPLICIT DOUBLE PRECISION (A-H,O-Z)DIMENSION P(3)IF(IV.EQ.2) GO TO 2IF(ISGNX.GT.O) GO TO 3RECENT=X*DFLOAT(ISX+I)*P(3)/(DFLOAT(N-ISX-JSX)*P(1»RETURN

3 RECENT=X*DFLOAT(N-ISX-JSX+I)*P(1)/(P(3)*DFLOAT(ISX»RETURN

2 IF(ISGNX.GT.O) GO TO 5RECENT=X*DFLOAT(JSX+I)*P(3)/(DFLOAT(N-ISX-JSX)*P(2»RETURN

5 RECENT=X*DFLOAT(N-ISX-JSX+I)*P(Z)/(P(3)*DFLOAT(JSX»RETURNEND

C++++++++++++C++++++++++++C++++++++++++C++++++++++++C++++++++++++CDOUBLE PRECISION FUNCTION TRINOM(N,KI,K2,K3,P)IMPLICIT DOUBLE PRECISION (A-H,O-Z)DIME~SION K(3),P(3),Q(3)K(1)=K1K(Z)=K2K(3)=K3Q(l)=P(l)Q(2)=P(2)Q(3)=P(3)DO 2 1=1,211=1+1DO 2 J=II,3IF(K(I)-K(J» 2,2,3

3 IK=K(I)K(I)=K(J)K(J)=IKX=Q(I)Q(I)=Q(J)Q(J)=X

2 CONTINUEM=ND= 1.L L=K (3)IF(K(2).EQ.O) GO TO 8KKZ=K(2)DO 4 I=1,KK2J=K(2)-ID=D*Q(Z)*(DFLOAT(M-J)/DFLOAT(K(Z)-J»

5 IF(D.LE.l •• OR.LL.EQ.O) GO TO 4D=D*Q(3)LL=LL-1GO TO 5

4 CONTINUEM=M-K(2)

- 236 -

IF(K(l).EQ.O) GO TO 8KK1-K(1)DO 6 I=1,KK1J~I-1D=D*Q(l)*(DFLOAT(M-J)/DFLOAT(K(l)-J»

7 IF(D.LT.l ••OR.LL.EQ.O) GO TO 6D=D*Q(3)LL=LL-1GO TO 7

6 CONTINUE8 IF(LL.NE.O) D=D*(Q(3)**LL)

TRINOH=DRETURNEND

c++++++++++++c++++++++++++c++++++++++++c++++++++++++c++++++++++++C

- 237 -

APPENDIX D: THE ROUTINE "GENERAL"

This routine is designed to obtain the exact value ofthe probability PR = P(Oa ~ f(l) ~ m/n(O) = n) for a three-

grade system with general transition matrix, m and n being

two fixed non-negative vectors.

The routine GENERAL can be called in any other

routine by the use of the statement "CALL GENERAL (PR)".

The input parameters should, then, be passed to it by the

use of the following statement:

COMMON/GNI NHIGH(3), NSTART(3), P(3,4)

where:

P(i,j) = transition probability between

NHIGH(i)

NSTART(i)=

=

grades i and j

m.1

n.1

(i,j=1,2,3)

(i =1,2,3)(i =1,2,3)

The routine, as listed, can be used for any three-

grade system in which:

(m1+1) (m2+1) (m3+1)

(ni+1) (ni+2) (ni+3) 16< 10000

< 3000 (i =1,2,3)

If the system does not fall within such limits, all

the 10000s and 3000s in the dimension statements have to be

replaced with the values of the left-hand sides of the above

inequalities.

- 238 -

LISTING

SUBROUTINE GENERAL(PR)COMMON/GN/ NSTART(3).NHIGH(3),P(3,4)DIMENSION TAB(lOOOO).MV(3)DIMENSION PP(3000.3),IVEC(3000.3,3),KKK(3)N=3CALL QUADR(PP,IVEC,KKK)MY=(NHIGH(3)+1)MX=MY*(NHIGH(2)+1)KK4=MX*(NHIGH(I)+1)DO 1 I=l,KK4TAB(I)=O.O

1 CONTINUEKKl=KKK(l)KK2=KKK(2)KK3=KKK(3)DO 2 Kl=l,KKlIF(PP(Kl,l).EQ.O.O) GO TO 2DO 3 K2=1,KK2DO 5 J=1,3MV(J)=IVEC(Kl,J,l)+IVEC(K2,J,2)

5 CONTINUEDO 6 J=1.3IF(MV(J).GT.NHIGH(J» GO TO 3

6 CONTINUELL=MV(1)*MX+MV(2)*MY+MV(3)+1TAB(LL)=TAB(LL)+PP(Kl.l)*PP(K2,2)

3 CONTINUE2 CONTINUE

PR=O.ODO 8 LL=1.KK4IF(TAB(LL).EQ.O.O) GO TO 8K4=LL-lL2=MOD(K4,MX)L3=MOD(L2.MY)Ll=K4/NXL2=L2/MYDO 7 K3=1,KK3MV(1)=IVEC(K3,l.3)+LlMV(2)=IVEC(K3,2,3)+L2MV(3)=IVEC(K3,3,3)+L3DO 9 J=l,3IF(MV(J).GT.NHIGH(J» GO TO 7

9 CONTINUEPR=PR+TAB(LL)*PP(K3,3)

7 CONTINUE8 CONTINUE

RETURNEND

C++++++++++++C++++++++++++C++++++++++++C++++++++++++C++++++++++++C

- 239 -

SUBROUTINE QUADR(PP,IVEC,KKK)COMMON/GN/ NHIGH(3),NSTART(3),P(3,4)DIMENSION Q(4)DIMENSION PP(3000,3),IVEC(3000,3,3) ,KKK(3)DO 10 IX=I,3Q(4)=1.0DO 4 J=I,3Q(J)=P(IX,J)Q(4)=Q(4)-Q(J)

4 CONTINUENl=NSTART(IX)+1SUM=O.OKK=OP 1= 1 • 0LI=NSTART(IX)DO 1 I=I,NlPJ=PIJLMT=NI-I+lLJ=NI-IDO 2 J=1,JLHTPK=PJKLMT=JLMT-J+lDO 3 K=l,KLMTL=NI-I-J-K+2KK=KK+lPP(KK,IX)=PKSUM=SUM+PKIVEC(KK,I,IX)=I-1IVEC(KK,2,IX)=J-lIVEC(KK,3,IX)=K-1PK=(Q(3)*PK*L)/(Q(4)*K)

3 CONTINUEPJ=(PJ*Q(2)*LJ)/(Q(4)*J)LJ=LJ-l

2 CONTINUEPI=(PI*Q(I)*LI)/(Q(4)*I)L1=LI-l

1 CO~T1NUEDO 5 I=l,KKPP(I,IX)=PP(I,IX)/SUM

5 CONTINUEKKK(IX)=KK

10 CONTINUERETURNEND

C++++++++++++C++++++++++++C++++++++++++C++++++++++++C++++++++++++c

- 240 -

APPENDIX E: FORTRAN PROGRAMS USED IN INVESTIGATING"DIFFERENT RECRUITMENT STRATEGIES

E.l GENERAL

The examination of the behaviour of a system of total

size N, under a given recruitment strategy, is achieved in

three stages:

1° Firstly, the probabilities P(f(l)=v/n(O)=n) for all

structures n of total size N and all possible structure

v of total size less than or equal to N, are evaluated.

This requires lengthy computations and is achieved by

the program FLOWBR (flows before recruitment).

2° Secondly, the program FLOWAR (flows after recruitment)

uses the above probabilities and follows the rules of

the chosen recruitment policy to evaluate the probabili-

ties P(n(l)=n'/n(O)=n) for all possible vectors n' and n

of total size N.

3° Finally, the program TABLES uses the probabilities eval-

uated by the program FLOWAR at different steps to pro-

duce, for a given recruitment strategy, tables identical

to those in Appendix B.

The probabilities evaluated by the program FLOWAR

will be written in a file assigned to the logical unit 9.

However, if the recruitment policy is "GOAL" and the horizon

time is ITER, ITER files assigned to the logical units

9,18, ••. ,9*ITER are generated.

- 241 -

The logical unit 9*t (t=1,2, ...,ITER) will hold theprobabilities related to the recruitment policy at the step

(ITER-t+1).

The files generated by the program FLOWBR and FLOWAR

can be saved for subsequent use. Our three-stage approach

allows the maximum flexibility in combining a wide range of

recruitment policies and benefits largely from filing faci-

lities offered by the Fortran.

E. 2 Program FLOWBR

E.2.1 Parameters

Input of the parameters is from logical unit 5

(cards); output is tJ logical unit 15.

format of each input card are given below.

Description and

Card Parameters name and description Format

1 P(l,l) , P(1,2) Transition probabilitiesfrom grade 1 2F10.5

2 P(2,2) , P(2,3) Transition probabilitiesfrom grade 2 2Fl0.5

3 P(3,3), P(3,4) Transition probabilitiesfrom grade 3 2F10.5

4 NTOT System total size 15

- 242 -

E.2.2 Auxiliary Routines

MAINT

E.3 Program FLOWAR

E.3.1 Parameters

Input of the parameters is from logical unit 5

(cards); output is to logical unit(s) 9 (,18, ... ).

Description and format of each input card are given

below:

Card Parameters name and description Format

1 P(l,l), P(1,2) Transition probabilitiesLrom grade 1 2Fl0.5

2 P(2,2), P(2,3) Transition probabilitiesfrom grade 2 2Fl0.5

3 P(3,3), P(3,4) Transition probabilitiesfrom grade 3 2Fl0.5

4 NPO Recruitment policy A45 ITER Number of steps (=1 if

NPO is Fixed or ADAPTIVE) 15

6 NGOAL GOAL vector m 315

7 MR Vector with integerentries proportional tom-mP (for fixed policyonly) 315

- 243 -

E.3.2 Auxiliary Routines

INITFINITGMAJITREXARECEXAFIXEXAGLAFFECTVERFT

E.3.3 Observation

This program cannot be run unless the probabilities

P(f(l)=v/n(O)=n) are already available in a file assigned to

logical unit 15.

E.4 Program TABLES

E.4.1 Parameters

Input of the parameters is from logical unit 5

(cards); output to logical unit 6 (printer).

Description and format of each input card are given

below:

Card

1

2

3

4

5

6

7

8

9

- 244 -

Parameters name and descriptionP(l,l), P(1,2) Transition probabilities

from grade 1P(2,2), P(2,3) Transition probabilities

from grade 2

P(3,3), P(3,4) Transition probabilitiesfrom grade 3

ITER Number of steps

NSTART Initial vector n

NGOAL GOAL vector m

ITAPE Logical units assignedto files holding theprobabilitiesP(n(t+l)=n'/n(t)=n);(ITAPE(t) correspondsto step t)

NPOLI Recruitment strategy

MR Vector with integerentries proportional tom-mP (for fixed policyonly)

Format

2Fl0.5

2Fl0.5

2Fl0.5

IS315

315

ITER*I5

A4

315

E.4.2 Auxiliary Routines

ATNCRl

MAINT

VERFT

PRTTAB

SUBPRT

- 245 -

E.4.3 Observation

This program cannot be run unless the probabilitiesP(n(t+l)=n'/n(t)=n) are already in files assigned to thelogical units ITAPE(t) (t=1,2, ...,ITER).

E.5 RESTRICTIONS

These programs, as lis ted, can be used for any

three-grade hierarchical system with a total size less than

or equal to 25. The observation period to consider for any

recruitment strategy should not exceed 10 steps. These

limits are those discussed in chapter four.

- 246 -

E.6 LISTINGS

PROGRAM FLOWBRIMPLICIT DOUBLE PRECISION (A-II,O-Z)REAL P(3,4),PRECICHARACTER NCTRL*4COMMON/Cl/NLOW(30),NHIGH(30),NSTART(30),PP(30,31),NCOMMON/C2/SLMTDI~ENSION PRMOVE(3000)DATA NCTRL/'PFLW'/NSTEP=lN=3DO 2 I=l,NDO 3 J=l,NP(I,J)=O.O

3 CONTINUEREAD(S,200) P(I,I),P(I,I+l)

2 CONTINUEPRECI=O.OSLMT=l.-PRECI/DFLOAT(N)NN=N+lDO 1 I=l,NDO 1 J=l,NNPP(I,J)=P(I,J)

1 CONTINUEREAD(S,lOO) NTDTIENLT=NTOT+lDO 9 IEN=l,IENLTNSTART(l)=IEN-lJENLT=IENLT-IEN+lDO 9 JEN=l,JENLTNSTART(2)=JEN-lNSTART(3)=NTOT-NSTART(l)-NSTART(2)KK=ONl=NSTART(l)+lDO 4 I=l,NlN2=NSTART(l)-I+2N3=NSTART(2)+NSTART(3)+1NHIGH(l)=I-lNLOW(l)=NHIGH(l)DO 6 J=1,N2NHIGH(2)=J-lNLOW(2)=NHIGH(2)DO 6 K=1,N3NHIGH(3)=K-lNLOW(3)=NHIGH(3)CALL MAINT(PR)KK=KK+lPRMOVE(KK)=PR

6 CONTINUEIF(NSTART(2).EQ.O) GO TO 4N2=N2+1N4=NSTART(1)+NSTART(2)+2-1N3=NTOT-I+3

- 247 -

DO 7 J=N2,N4NS=N3-JNHIGH(2)=J-lNLOW(2)-=NHIGH(2)DO 7 K=l,NSNHIGH(3)=K-lNLOW(3)=NHIGH(3)CALL MAINT(PR)KK=KK+lPRHOVE(KK)=PR

7 CONTINUE4 CONTINUE

WRITE(lS) (NSTART(J),J=l,N),«P(I,J),J=l,N),I=l,N),+ NSTEP,KK,NCTRL

WRITE(lS) (PRMOVE(I),I=l,KK)9 CONTINUE

100 FORMAT(16IS)200 FORMAT(8FIO.5)

STOPEND

C++++++++++++C++++++++++++C++++++++++++C++++++++++++C++++++++++++c

- 248 -

PROGRAM FLO~ARIMPLICIT DOUBLE PRECISION (A-H,O-Z)REAL P(3,4)COMMON/AD/NREC(B,3),PREC(B),NR,MTOTCOMMON/AD5/MR(3),NTRECOMMON/GNRL1/PRMOVE(3000),ATN(351)COMMON/GNRL2/N,MT,NTOTCOMMON/GNRL3/COMP(351),VALITR(351,lO),IFROM(351,3)COMMON/GNRL4/P,PRECICOMMON/GNRL5/NGOAL(3)CHARACTER*4 NPON=3DO 1 I=l,NDO 2 J=l,NP(I,J)=O.O

2 CONTINUEREAD(5,200) P(I,I),P(I,I+l)

1 CONTINUEREAD(5,400) NPOREAD(5,lOO) ITERREAD(5,lOO) (NGOAL(I),I=l,N)NTOT=ODO 41 I=l,NNTOT=NTOT+NGOAL(I)

41 CONTINUECALL INITGMTOT=NTOTIF(NPO.EQ. 'GOAL' .OR.NPO.EQ. 'ADAP') GO TO 4IF(NPO. EQ. 'FIXE') GO TO 3WRITE(6,300) NPOSTOP

3 CALL INITF4 CONTINUE

DO 20 ITR=l,ITERCALL MAJITR(ITR,NPO)

20 CONTINUESTOP

100 FORMAT(1615)200 FORMAT(BFIO.S)300 FORMAT(lX,'NO ROUTINE FOR THE RECRUITME~T POLICY' ,A4)400 FORMAT(A4)

END

C++++++++++++C++++++++++++C++++++++++++C++++~+++++++C++++++++++++CSUBROUTINE INITFCOMMON/AD5/MR(3),NTRECOMMON/GNRL2/N,MT,NTOTREAD(S,lOO) (MR(J) ,J=l ,N)NTRE=ODO 7 I=l,NNTRE=NTRE+MR(I)

- 249 -

7 CONTINUE100 FORHAT(16I5)

RETURNEND

C++++++++++++C++++++++++++C++++++++++++C++++++++++++C++++++++++++CSUBROUTINE INITGIMPLICIT DOUBLE PRECISION (A-H,O-Z)COMMON/GNRL2/N,K,NTOTCOMMON/GNRL3/COMP(351),VALITR(351,lO),IFROM(351,3)COMMON/GNRL5/NGOAL(3)K=ONTOTT=NTOT+lDO 1 I=l,NTOTTNJ=NTOTT-I+lDO 1 J=l,NJK=K+lIFROH(K,l)=I-lIFROM(K,2)=J-lIFROM(K,3)=NTOTT-I-J+lCOMP(K)=O.ODO 2 L=l,NCOMP(K)=COMP(K)+(NGOAL(L)-IFROM(K,L»*(NGOAL(L)-IFROt1(K,L»

2 CONTINUE1 CONTINUE

RETURNEKD

C++++++++++++C++++++++++++C++++++++++++C++++++++++++C++++++++++++CSUBROUTINE MAJITR(ITR,NPO)IMPLICIT DOUBLE PRECISION (A-H,O-Z)REAL P(3,4),PRECICHARACTER *4 NPOCOHMON/GNRLl/PRHOVE(3000),ATK(351)COMMON/GNRLZ/N,MT,NTOTCO NH 0 N / G N R L 3 / CO rIP ( 3 5 1 ) , V A LIT 1\ ( 351 , 1 0) , I FRO H ( 3 5 1 , 3 )COM~ON/GKRL4/P,PRECICOMMON/GNRL5/NGOAL(3)COM}10N/~1AJ 1/NFL 0W (3 ) , L LL ,K KCOM~10N/AD5/MR(3) ,NTRED I ~1ENS ION K S ( 3 )ITP=9*ITRNN=N+lDO 3 I=l,MTVALITR(I,ITR)=O.O

3 C or~T I NU EREWIND 15DO 9 LX=I,MTLLL=LXDO 1 J=l,N

- 250 -

NS(J)=IFROM(LLL,J)1 CONTINUE

DO 5 I=l,MTATN(I)=O.O

5 CONTINUECAL L VERFT (N S ,P ,1 ,N ,15 ,ME P S ,'PFLW' ,NFLO W )READ(lS) (PRMOVE(L),L=l,MEPS)KK=ONI-NS(l)+1DO 4 1=I,NlN2=NS(I)-I+2N 3 =N S(2 )+N S(3 )+ 1NFLOW(l)=I-IDO 6 J=1,N2NFLOW(2)=J-lDO 6 K=I,N3NFLOW(3)=K-lKK=KK+lCALL EXAREC(ITR,NPO)

6 CONTINUEIF(NS(2).EQ.O) GO TO 4N2=N2+1N4=NS(I)+NS(2)-I+2N3=NTOT-I+3DO 7 J=N2,N4N5=N3-JNFLO\~(2) =J-lDO 7 K= 1 ,N 5NFLOW(3)=K-lKK=KK+lCALL EXAREC(ITR,NPO)

7 CONTI~UE4 CONTINUE

WRITE(ITP)(NS(I) ,1=1 ,N) ,«P(I,J) ,J=1 ,N) ,1=1 ,N) ,ITR,MT,NPOIF(NPO.EQ.' FIXE') WRITE(ITP) (MR(J) ,J=1 ,N)IF(NPO.NE.'FIXE') WRITE(ITP) (~GOAL(J),J=l,N)WRITE(ITP) (ATN(I),I=I,MT)

9 CONTINUEREWI~D ITPIF(NPO.NE.'GOAL') RETURNDO 2 I=I,MTCOMP(I)=VALITR(I,ITR)

2 CONTINUERETURNEND

C++++++++++++C++++++++++++C++++++++++++C++++++++++++C++++++++++++CSUBROUTINE EXAREC(ITR,NPO)CHARACTER*4 NPOIF(NPO.NE.'FIXE') GO TO 1CALL EXAFIX(ITR)

- 251 -

RETURN1 IF(NPO.NE.'GOAL'.AND.NPO.NE.'ADAP') GO TO 2

CALL EXAGL(ITR)RETURN

2 WHITE(6,100) NPO100 FORMAT(2X, 'NO ROUTINE FOR THE RECRUITMENT POLICY' ,A4)

STOPEND

C++++++++++++C++++++++++++C++++++++++++C++++++++++++C++++++++++++CSUBROUTINE EXAFIX(ITR)IMPLICIT DOUBLE PRECISION (A-H,O-Z)COMMO~/MAJI/NFLOW(3),LLL,KKCOMMON/AD/NREC(B,3),PREC(B),NR,MTOTCOMMON/GNRLl/PRMOVE(3000),ATN(351)COMMON/GNRL2/N,MT,NTOTDI~ENSION MX(3)KR=NTOTNT=N TO T+ 1DO 1 I=l,NKR=KR-NFLOW(I)

1 CONTINUEX=l.ONR=OCALL AFFECT(KR,X)DO 2 I=l,NRDO 3 J=l,NMX(J)=NFLOW(J)+NREC(I,J)

3 CONTINUEIJ=NT*MX(I)+MX(2)+I-MX(I)*(MX(I)-I)/2ATN(IJ)=ATN(IJ)+PRMOVE(KK)*PREC(I)

2 CONTINUERETURNEND

C++++++++++++C++++++++++++C++++++++++++C++++++++++++C++++++++++++CSUBROUTINE EXAGL(ITR)IMPLICIT DOUBLE PRECISION (A-H,O-Z)COMMON/GNRLI/PRMOVE(3000),ATN(351)C 0 ~1M 0 N / G N R L 2 / N , M T , NT 0 TCOMMON/GNRL3/COMP(351),VALITR(351,IO),IFROM(351,3)COMMON/MAJI/NFLOW(3),LLL,KKPMIN=N*NTOT*NTOTNTOTT=NTOT+III~NFLOW(I)+1I~T=NTOTT-NFLOW(3)IFI=INT-NFLOW(2)JI=NFLOW(2)+1DO 2 I=II,IFIIS=NTOTT*(I-l)-(I-l)*(I-2)/2

- 252 -

JF=1NT-1+IDO 2 J=JI,JFL-1S+J1F(PMIN-COMP(L» 2,3,3

3 PM1N=COMP(L)1J=L

2 CONTINUEVAL1TR(LLL,1TR)=VALITR(LLL,ITR)+PRMOVE(KK)*COMP(IJ)ATN(IJ)=ATN(IJ)+PRMOVE(KK)RETURNEND

C++++++++++++C++++++++++++C++++++++++++C++++++++++++C++++++++++++CSUBROUTINE AFFECT(KR,P)IMPLICIT DOUBLE PRECISION (A-H,O-Z)COMMON/AD/NREC(B,3),PREC(B),NR,NTOTCOMMON/ADS/ MR(3),NTREDIMENSION MX(3)NI=NRII=NI+1IF=NI+3DO 1 J=1,3KY=KR*MR(J)M=KY/NTREDO 2 I=II,IFNREC(I,J)=M

2 CONTINUEMX(J)=MOD(KY,NTRE)

1 CONTINUEMSUM=ODO 3 J=1,3MSUM=MSUM+MX(J)

3 CONTINUEt-1SUM=MSUM/NTREIF(MSUM-1) 4,S,6

4 NR= 1I=NI+1PREC(I)=1.0GO TO 10

5 NR=3K=ODO 7 I=II,IFK=K+lNREC(I,K)=NREC(I,K)+1PREC(I)=DFLOAT(MX(K»/DFLOAT(NTRE)

7 CONTINUEGO TO 10

6 NR=3K=ODO B I=II,IFDO 9 J=1,3

- 253 -

NREC(I,J)=NREC(I,J)+l9 CONTINUE

K-K+lNREC(I,K)=NREC(I,K)-lPREC(I)=l.-DFLOAT(MX(K»/DFLOAT(NTRE)

8 CONTINUE10 NR=NI+NR

DO 11 I=II,NRPREC(I)=PREC(I)*P

11 CONTINUERETURNEND

c++++++++++++c++++++++++++c++++++++++++c++++++++++++c++++++++++++c

- 254 -

PROGRAM TABLESIMPLICIT DOUBLE PRECISION (A-H,O-Z)REAL P(3,4),PD(3,4),PRECI,EPSDIMENSION TAB(351,lO),PR(3S1),NSTART(3),MV(3)DIMENSION NPO(lO),ITAPE(IO)COMMON/PRT/AV(IO,3),AAV(lO,3),DDIS(IO,3)DIMENSION MR(3),NGOAL(3),NS(3)COMMON/AD7/IFROM(3S1,3),ATN1(3SI)DIMENSION LET(12),PRATN(IO),KSTEP(IO)DI~ENSION NVEC(lO,3),NDUMY(IO)CHARACTER NPOLI*4,LET*3,NCTX*4DATA LET/'F ','A ','G I','G 2','G 3','G 4',

+ 'G S','G 6','G 7','G B','G 9','GI0'/

N= 3NN=N+lDO 45 I=l,NDO 46 J=I,NP(I,J)=O.O

46 CONTINUEREAD(S,200) P(I,I),P(I,I+I)

4S CONTINUEPRECI=O.OOREAD(S, 100) ITERREAD(S,IOO) (NSTAR~ J),J=I,N)READ(S,100) (NGOAL(J) ,J=I,N)READ(5,100) (ITAPE(I), I=l,ITER)

IDX=OREAD(5,SOO) NPOLI,IFXIF(IFX.EQ.O) GO TO 18IF(NPOLI.NE.'GOAL') GO TO 51READ(S,lOO) (KSTEP(I),I=I,ITER)GO TO 23

51 CONTINUEDO 19 1=I,ITERREAD(S,IOO) (NVEC(I,J),J=l,N)KSTEP(I)=l

19 CONTINUEGO TO 20

18 IF(NPOLI.EQ. 'GOAL') IDX=lDO 31 1=1, ITERKSTEP(I)=(ITER-I)*IDX+I

31 CONTINUEIF(NPOLI.NE.'FIXE') GO TO 23READ(S,IOO) (MR(I),I=I,N)DO 22 1=I,ITERDO 22 J=I,NNVEC(I,J)=MR(J)

22 CONTINUEGO TO 20

23 DO 24 1=1,1TERDO 24 J=I,N

- 255 -

NVEC(I,J)=NGOAL(J)24 CONTINUE20 IF(NPOLI.EQ. 'FIXE') IDX=l

IF(NPOLI.EQ.'ADAP') 1DX=21F(NPOL1.EQ. 'GOAL') IDX=3DO 35 1=l,1TERPRATN(I)=O.O

35 CONTINUECALL ATNCRl(NGOAL,P,PRECI,N,NTOT,MT)ITR=lNT=NTOT+lIJ=NT*NSTART(I)+NSTART(2)-NSTART(I)*(NSTART(I)-1)/2IDG=IJ+lITP=ITAPE(I)IF(IJ.EQ.O) GO TO 3DO 4 1=I,IJREAD(ITP) (NS(J) ,J=1 ,N) ,«PD(L,J) ,J=1 ,N) ,L=1 ,N),

+ NSTEP,MT,EPS,NCTXREAD(ITP) (NDUMY(J),J=I,N)READ(ITP) (PR(J),J=l,MT)

4 CONTINUE3 CONTINUE

NSTEP=KSTEP(ITR)DO 71 J=I,NNDUMY(J)=~VEC(I,J)

71 CONTINUECALL VERFT(NSTART,P,NSTEP,N,ITP,MT,NPOLI,NDUMY)READ(ITP) (PR(I),I=I,MT)DO 10 I=I,MTTAB(I,I)=PR(I)DO 10 1TR=2,ITERTAB(I,ITR)=O.O

10 CONTINUEPRATN(I)=ATNl(IDG)DO 11 ITR=2,ITERITP=ITAPE(ITR)NSTEP=KSTEP(ITR)~O 72 J=I,NNDUMY(J)=NVEC(ITR,J)

72 CONTINUEREWIND ITP1T=ITR-IDO 8 1=I,MTDO 1 J=I,NNS(J)=IFROM(I,J)

1 CONTINUEPRATN(ITR)=PRATN(ITR)+TAB(I,IT)*ATNI(I)CALL VERFT(NS,P,NSTEP,N,ITP,MT,NPOLI,NDUMY)READ(ITP) (PR(L),L=I,MT)DO 9 L=I,MTTAB(L,ITR)=TAB(L,ITR)+TAB(I,IT)*PR(L)

9 CONTlt;UE

- 256 -

8 CONTINUE11 CONTINUE

DO 14 I=I,ITERDO 14 J=I,NAV(I,J)=O.OAAV(I,J)=O.OOOIS(I,J)=O.O

14 CONTINUEDO 15 IJ=I,MTDO 21 J=I,NMV(J)=IFROM(IJ,J)

21 CONTINUEDO 15 I=I,ITERDO 15 J=I,NAV(I,J)=AV(I,J)+TAB(IJ,I)*MV(J)AAV(I,J)=AAV(I,J)+TAB(IJ,I)*MV(J)*MV(J)X=FLOAT(MV(J)-NGOAL(J»DDIS(I,J)=DDIS(I,J)+TAB(IJ,I)*X*X

15 CONTINUEDO 41 I=I,ITERDO 41 J=I,NAAV(I,J)=AAV(I,J)-AV(I,J)*AV(I,J)

41 CONTINUECALL PRTTAB(~,P,NSTART,NGOAL,NPOLI,IDX,ITER)WRITE(6,1400)WRITE(6,400) (I,I=I,ITER)WRITE(6,150) (PRATN(J),J=I,ITER)IF(IFX.EQ.O) STOPWRITE(6,1500)DO 16 I=I,ITERID=IDX+KSTEP(I)-1WRITE(6,1300) I,LET(ID),(NVEC(I,J),J=I,N)

16 CONTINUElOa FORHAT(16Is)ISO FORMAT(4X,10(F8.4,3X»200 FORMAT(8FI0.5)400 FORMAT(/,4X,10(3X,IHP,I2,5X»500 FORMAT(A4,6X,Il)

1300 FORMAT(6X,Il,5X,A3,3X,'(',3(I3,IX),')')1400 FORMAT(//,3X,'PROBABILITY PI OF ATTINING THE VECTOR',

+ ' M FROM THE VECTOR N IN I STEPS')1500 FORMAT(//,3X,'DETAILED INFORMATION ABOUT THE RECRUI',

+ 'TMENT STRATEGY' ,//,5X, 'STEP' ,lX, 'POLICY' ,lX,+ 'CORRESPONDING VECTOR')

STOPEND

C++++++++++++C++++++++++++C++++++++++++C++++++++++++C++++++++++++c

---- , /',' ADAPTIVE'

- 257 -

SUBROUTINE ATNCRl(NGOAL,P,PRECI,NNN,NTOT,K)IMPLICIT DOUBLE PRECISION (A-II,O-Z)REAL P(3,4),PRECICOMMON/Cl/NLOW(30),NHIGH(30),NSTART(30),PP(30,31),NCOMMON/C2/SLMTDIMENSION NGOAL(3)COMMON/AD7/ IFROM(351,3),ATN(351)N=NNNSLMT=l.-PRECI/FLOAT(N)Nl=N+lNTOT=ODO 1 I=l,NNTOT=NTOT+NGOAL(I)NHIGH(I)=NGOAL(I)NLOW(I)=NGOAL(I)NLOW(I)=ODO 1 J=I,NlPP(I,J)=P(I,J)

1 CONTINUEK=ONI=NTOT+lDO 2 I=l,NINSTART(l)=I-lNJ=NI-I+lDO 2 J=l,NJNSTART(2)=J-!NSTART(3)=NI-I-J+lCALL MAINT(PR)K=K+lDO 3 L=l,N

3 IFROM(K,L)=NSTART(L)ATN(K)=PR

2 CONTINUERETURNEND

C++++++++++++C++++++++++++C++++++++++++C++++++++++++C++++++++++++CSUBROUTINE PRTTAB(K,P,N,M,NPOLI,L,ITER)IMPLICIT DOUBLE PRECISION (A-H,O-Z)REAL P(3,4)DIMENSION N(3),M(3),V(3),LET(3),STRAT(3)~TIRET(3)CHARACTER STRAT*lO,LET*lCHARACTER TIRET*lO,NPOLI*4DATA TIRET/' ----- ',' -------- ','DATA LET/'F' ,'A' ,'G'/,STRAT/' FIXED

+ ' GOAL '/DO 2 J=I,KV(J)=M(J)DO 2 I=l,KV(J)=V(J)-M(I)*P(I,J)

2 CONTINUE

- 258 -

WRITE(6,lOO) LET(L),ITER,STRAT(L)WRITE(6,600) TIRET(L)WRITE(6,300) (P(l,J),J=l,K)W RITE (6 ,200) (N (I ) ,1=1 ,K) ,(M (I ) ,1=1 ,K ) ,(p (2 ,J ) ,J= 1 ,K )WRITE(6,SOO) (V(I),I=l,K),(P(3,J),J=I,K)CALL SUBPRT(ITER)

I00 FOR HA T (1HI, I / I ,2X, ,TAB L E " A I ,2X, ': EVOL UTI 0N 0 FA' ,+ 'GRADED SYSTEM OVER A PERIOD OF ',12,' STEPS,',+ 'UNDER THE' ,AIG,'RECRUITMENT STRATEGY.')

200 FOR ~IA T (3 X, ,INIT IAL VEe TOR N= ( ,,2 (I2 " ,'), I2 ,') ;',+ 'GOAL VECTOR M=(',2(I2,' ,'),12,') ,+ 'TRANSITION MATRIX P=' ,3(2X,FS.3»

300 FORMAT(83X,3(2X,FS.3»500 FOR MA T (3X, ,V ECTOR M-M * P = ( ,,2 (F5 •2 " ,'),

+ FS.2,' )',41X,3(2X,FS.3»600 FORMAT(7BX,AIO,/)

RETU Rr~END

C++++++++++++C++++++++++++C++++++++++++C++++++++++++C++++++++++++cSUBROUTINE SUBPRT(ITER)IMPLICIT DOUBLE PRECISION (A-H,O-Z)COMMO~/pRT/XAV(10,3),XAAV(10,3),XDDIS(10,3)DIMENSION TAB(12),AVfI2)WRITE(6,400)WRITE(6,IOOO) «I,I=I,3),J=I,3)DO 2 J=I,12AV(J)=O.O

2 CONTI~UEDO 1 I=I,ITERDO 6 J=I,3JJ=J+4JJJ=J+8TAB(J)=XAV(I,J)TAB(JJ)=XAAV(I,J)TAB(JJJ)=XDDIS(I,J)

6 cor,TINUETAB(4)=TAB(1)+TAB(2)+TAB(3)TAB(8)=TAB(S)+TAB(6)+TAB(7)TAB(12)=TAB(9)+TAB(10)+TAB(II)DO 3 J=I,12AV(J)=AV(J)+TAB(J)

3 CO~TINUEWRIT£(6,lI00) I,(TAB(J),J=I,12)

1 COKTIKUEWRIT£(6,1200)DO 4 J=I,12AV(J)=AV(J)/FLOAT(ITER)

4 COt; TIN U £WRITE(6,1300) (AV(J),J=1,12)\.JRITE(6,1200)

- 259 -

400 FORMAT(IOX,'... RESULTS OBTAINED BY EXACT METHODS',+ ' ••• ' )

lIDO FORMAT(2X,IHI,3X,I2,4X,lHI,3(2X,4(lHI,F7.2,lX),IHI»1200 FORMAT(2X,II(IH-),3(2X,37(IH-»)1300 FORMAT(2X,llHI AVERAGE I,3(2X,4(lHI,F7.2,IX),IHI»1000 FORMAT(II,22X,'EXPECTED STOCK VECTOR' ,15X,

+ 'VARIA~CES OF THE STOCK VECTORS' ,9X,+ 'M.S.E. OF THE STOCK VECTORS',I,+ 2X,ll(lH-),3(2X,37(IH-»,I,+ 2X, I HI, 9X, I HI, 3 ( 2 X, 1HI, lOX, 'G RA DE' , 1 1X, 1HI, 8 X , 1HI) , I ,+ 2X,IIHI STEP I,3(2X,lHI,26(lH-),10HI TOTAL 1),1,+ 2X, IHI,9X, IHI,3(2X, 3(lHI,3X, II ,4X) ,IHI,8X, IHI),I,+ 2X,11(IH-),3(2X,37(IH-»)

RETURNEt-;D

C++++++++++++C++++++++++++C++++++++++++C++++++++++++C++++++++++++CSUBROUTINE VERFT(KG,P,ITX,N,ITP,MT,NCT,MR)DI~ENSIO~ MRR(3),MR(3)CHARACTER *4 ~CT,NCTT

DIMENSION NG(3),ND(3)REAL P(3, 4) ,PD(3, 4)SMALL=1.E-6READ(ITP) (~D(J) ,J=1 ,N) ,( (PD(I ,J) ,J=1 ,N) ,1=1 ,N),

+ NT,MT,NCTTIF(NCT.NE.'PFLW') READ(ITP) (MRR(I),I=l,N)IFLG=3IF(NT.NE.ITX) GO TO 3DO 1 1= 1 , NIFLG=lIF(ND(I).NE.~G(I» GO TO 3IFLG=2DO 1 J=l,NIF(ABS(PD(I,J)-P(I,J».GT.SMALL) GO TO 3

1 CONTINUEIFLG=7IF(NCT.EQ. 'ADAP' .AND.NCTT.EQ. 'GOAL') GO TO 5IF(NCTT.NE.NCT) GO TO 3I F ( NCT. EQ. 'P F L \~ ') RET URN

5 IFLG=8DO 4 I=l,NIF(MR(I).NE.MRR(I» GO TO 3

4 CONTINUERETURN

3 WRITE(6,100) ITPWRITE(6,200) IFLG

200 FORMAT(lHO,'CONFLICT IN ARGUMENT ',12)WRITE(6,lOOO)IARG=1WRITE(6,1100) IARG, (NG( 1),1=1, N), (ND( 1),1=1, N)I ARC= 2

- 260 -

WRITE(6,1200) IARG,(P(l,J),J=l,Nl),(PD(l,J),J=l,Nl)DO 2 I=2,NWRITE(6,1300) (P(I,J),J=1,N1),(PD(I,J),J=1,N1)

2 CONTINUEIARG=3WRITE(6,1500) IARG,ITX,NTIARG=7WRITE(6,1600) IARG,NCT,NCTTIARG=8IF(NCT.NE.'PFLW') WRITE(6,lI00) IARG,(MR(I),I=l,N),

+ (MRR(I),I=l,K)1000 FORMAT(3X,'ARGUMENT',22X, 'SUPPLIED' ,42X, 'READ IN' ,I)1100 FORMAT(6X,I2,4X,3(I3,2X),35X,3(I3,2X»1200 FORMAT(6X,I2,4X,4FIO.3,10X,4F10.3)1300 FORMAT(12X,4FIO.3,lOX,4FIO.3)1500 FORMAT(6X,I2,4X,I3,47X,I3)1600 FORMAT(6X,I2,4X,A4,46X,A4)

STOP100 FORMAT(lHO,'THE DATA IN FILE' ,I3,2X,'IS INAPROPRIATE')

END

C+++++++++++TC++++++++++++C++++++++++++C++++++++++++C++++++++++++C