Periodicity of the profile process in Markov manpower systems

~ 4- ~"

~i

EUROPEAN JOURNAL OF OPERATIONAL RESEARCH ELSEVIER

European Journal of Operational Research 85 (1995) 650-669

Theory and Methodology

Periodicity of the profile process in Markov manpower systems Ioannis I. Gerontidis Mathematics Department, University of Thessalonik~ 54006 Thessalonik~ Greece Received January 1992; revised november 1993

Abstract

This paper presents a new approach and provides a unified treatment to the problem of periodicity of the profile process in Markov manpower systems by specifying necessary and sufficient conditions for periodicity on the recruitment distribution and the wastage probabilities. It is shown among others that in the periodic case, recruitment control on the entire system is exercised only through the first cyclic subclass of the state space of the individual level chain. The approach adopted also provides tractable answers to the problems of asymptotic attainability and re-attainability of structures.

Keywords: Attainability; Control theory; Personnel; Planning; Reattainable structures

1. Introduction This paper is concerned with the periodic behaviour of the profile process in a Markov manpower system. A considerable amount of research has been devoted to the transient as well as the limiting behaviour of the expected grade or the relative structure, see for example Bartholomew (1973, 1982), Feichtinger (1976), Feichtinger and Mehlmann (1976) and Mehlmann (1977), under the assumption that the underlying individual level process is a time homogeneous aperiodic Markov chain. However as has been observed by Gerontidis (1991, 1994), there are certain cases where the individual member process may also be periodic and Woodward (1983), motivated by the work of Sykes (1969) in a study concerning the stochastic population models of Leslie type, provided an (informal) answer to the problem of when a Leslie matrix is periodic. The problem of controlling the profile process by recruitment or by promotion has also been a major area of research appearing in the works by Davies (1973, 1975), Bartholomew (1973), Grinold and Stanford (1976), Grinold and Marshall (1977) and Vajda (1978). The purpose of this paper is to present a new approach and to provide a unified treatment to the problem of periodicity, by specifying necessary and sufficient conditions on the recruitment distribution and the wastage probabilities of a Markov manpower system, for the relative structure to be periodic. In Section 2 we state the periodicity conditions and investigate their implications with respect to the behaviour of the singleton Markov replacement chain, the distribution of the completed length of service and the leaving process. Section 3 studies the periodicity of the profile process with particular attention 0377-2217/95/$09.50 © 1995 Elsevier Science B.V. All rights reserved SSDI 0 3 7 7 - 2 2 1 7 ( 9 4 ) 0 0 0 2 0 - D

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focused to the evolution of each cyclic subclass separately and introduces the concept of d-step stability by extending Feichtinger (1976) for the case of a vector growth factor. It is found that if the system has given input, the relative structure converges to a unique limit. In case that the system has given but changing total size and the proportionate growth of each subclass converges to some positive value, the profile process undergoes an asymptotically periodic behaviour. In case that all proportionate growths vanish asymptotically the profile process converges to the unique limiting distribution of the singleton replacement chain. The approach adopted also provides tractable answers to the problems of iasymptotic attainability and re-attainability of structures in manpower systems (Vajda, 1978). A further important conclusion of the present study is that recruitment control of the entire system is exercised only through the first cyclic subclass of the state space of the individual level chain, so that this subclass possesses a strong control element. In Section 4, the time-dependent case is considered for two types of environments, i.e. either convergent or cyclic. Finally in Section 5 two illustrative examples from the literature on educational manpower planning are given.

2. Periodicity conditions and their implications In this section we define the parameters of the manpower system and we provide the basic recurrence equations describing the evolution of the expected category (grade) and the relative structures across time. Consequently we provide necessary and sufficient conditions for periodicity on the recruitment distribution and the wastage probabilities and we examine their consequences with respect to the singleton Markov replacement chain, the completed length of service of an individual and ~he leaving process. The flows of a k-grade Markov manpower system may be divided into the number of new entrants R ( t ) allocated over the grades according to the distribution P0 = (/701. . . . , Pok), the transitions between the grades governed by the substochastic matrix P = (Pij) and the output flows from file system according to the wastage vector Pk +1 = (Pl,k + 1. . . . . Pk,k +1), (state k + 1 representing the world outside the system). Let ni(t) be the random number in grade i at time t and ~ ( t ) = [~l(t),..., ~k(t)] be the vector of expected grade sizes (bar denotes expectation of the corresponding random variable). Let also T ( t ) = ~(t)l' be the expected total membership in the system at time t, where 1 = (1,..., 1). Depending on assumptions about R(t), two types of models have appeared in the literature (Bartholomew, 1973, 1982). In the first model, R ( t ) may either be a sequence of given numbers or the realization of a known stochastic process (usually Poisson), giving rise to a system with given input and the evolution of the expected stock vector across time is given by (Bartholomew, 1982, p.52) ~ ( t + 1) = h ( t ) P + R ( t +

1)p 0.

(2.1)

In the second model, the grade sizes are allowed to vary around a given global total T(t), so that R ( t ) is now a random variable and the expectation R(t + 1) is a linear function of the existing expected stock vector h(t), i.e. R ( t + 1) = M ( t + 1) + ~ ( t )Pk+,, '

(2.2)

where M ( t + 1) = T(t + 1) - T(t). In this case R(t + 1) is determined by the need to replace losses and to cope with the change M ( t + 1) of the total size. Then we have a system with given total size and from (2.1) and (2.2) we get R(t + 1) = R ( t ) Q + M ( t + 1)p o

(2.3)

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where

Q = P +P~+lPo

(2.4)

is the stochastic transition matrix of the singleton Markov replacement chain (Keilson, 1979, p.41). The e l e m e n t qij = Pij + Pi,k + 1Poj is the total probability of transition (of any kind) out of grade i which results

in an addition to grade j, i.e. either by internal transfer or by the replacement of a leaver from i with a new member into j. Model (2.1) was proposed by Gani (1963) for the Australian university system, whereas model (2.3) was proposed by Young and Almond (1961). In case that the grade sizes are always growing, for practical convenience, our primary interest is concerned with the expected profile process q(t) = F~(t)/T(t). Expressing (2.3) in terms of q's we get

T(t) [ M ( t + l) ] q(t + 1) = q(t) T(t + 1) " Q + T(t) l'p 0 .

(2.5)

If T(t)/T(t -1)>_maxi~,j=~Pij k for all t (Feichtinger and Mehlmann, 1976), the matrix [ l - { T ( t 1)/T(t)}P]l'p o contains one column vector with positive elements only and by Theorem 2 in Isaacson and Madsen (1974) it is aperiodic. Therefore

S(t)

T(t-1) Q+ T-~-l'po

T ( t - l) P + ( TI ( t - 1 )T ( t i

P l'p0 ,

t = 1, 2 . . . . . is a sequence of aperiodic stochastic matrices, irrespectively of whether P is periodic or aperiodic and thus the study of periodicity of q(t) cannot be based on (2.5), which is suitable only for the aperiodic case. The correct approach is to focus attention on the evolution of the equivalence classes of communicating states of the state space. Consequently in the rest of this section we determine the necessary and sufficient conditions on the recruitment distribution and the wastage probabilities for the singleton Markov replacement chain to be periodic. This makes it possible (Section 3 and 4) to study the evolution of the profile process associated to each cyclically moving subclass separately and to obtain tractable asymptotic results.

(i) Quasi-periodic chains Since the steady state behaviour of a periodic Markov chain is characterized by the equivalence classes of communicating states and their corresponding stationary vectors, we assume that the state space C of the periodic chain is decomposed into d mutually exclusive cyclically moving subclasses Co, C 1. . . . . C d_ ~, where C r has k~ states such that E rd-1 f 0 k r = k and C - C Ot2 C 1 u ... u Ca_ 1 (Iosifescu, 1980). Let, after rearrangement,

p=

Co

C1

C2

Co

0

Po

0

C1

0

0

P1

• "" "'"

Cd-

0

• Ca-2 Ca-1

1

0

(2.6) 0

0

0

...

Pd-2

Ud_ 1 0

0

""

0

where Pr = (Pr,iy) is k r X kr+ 1 stochastic, for r = 0, 1 , . . . , d - 2, and Ud_ 1 is k d _ 1 X k o substochastic. Thus P is an irreducible quasi-periodic substochastic matrix (cyclic of period d). The chain is quasi-periodic (Whittle, 1975) in the sense that a single step transition from C r always leads to Cr+ 1 for r -- 0, 1,

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2 , . . . , d - 2, whereas C d_ 1 leads either to C o or to the absorbing state k + 1. It is apparent from (2.6) that Co P k + l = (0o, • " ,

Ca - 2

Ca - 1

0d-2,

Pk+l ),

d-1

(2.7)

where p dk+l - 1, = l ~ - l - - U d - l l ~ and 0 r, 1 r are l × k 1 . . . . . d - 1. Assume further that Co

p0 = (p0,

c1

O1 . . . .

r vectors of zero and ones respectively, r = 0 ,

ca- 1

, 0d_l).

(2.8)

For a vector a = (a i) denote by Diag(a) the matrix with elements [6ija~], where 6/y is the K r o n e c k e r delta. In case that a i is a square matrix instead of a scalar, then D i a g ( . ) assumes the block diagonal form. When P is raised to the d-th power, p d has the form p d = Diag

CO

C1

( Wo,

W 1....

Ca- 1 ,

Wd_ 1

),

where ~ V r = e r " " e d - 2 U d - l e O " " er-1

(2.9)

is k r × k r substochastic, r = 0, 1. . . . . d - 1. Consider also the vectors (2.10)

P~ = P ° P o " ' ' P r - 1 ,

mapping the external environment to C r so that p~ may be considered as an initial distribution on C r and Pk'+l = P r " "

(2.11)

Pd-2pd-71',

mapping Cr to the outside world, so that P~,+I may be considered as the wastage vector from Cr, r = 0, 1.....

d-1.

(ii) Periodic r e p l a c e m e n t chains

F r o m (2.4) and (2.6) to (2.8), we get that after rearrangement Q has the canonical form

Q=

Co

C1

C2

...

Co

0

Po

0

...

0

C1

O

O

P1

".

0

"

Ca- 1

"

,

Ca_ =

0

0

0

...

Pd-2

Cd-1

ed-1

0

0

...

0

(2.12)

-a-Pk _d-l,_0 where P a - 1 = Ud-1 -+ l 1% is kd_ 1 X k o stochastic. Woodward (1983) motivated by the work of Sykes (1969), provided conditions for periodicity of the transition matrix of the linear (Leslie) I model of population dynamics applied to a closed (with regard to migration) population in relation to mortality an fertility. Since a Leslie matrix is a special case of (2.4) (Feichtinger and Mehimann, 1976), conditions (2.7) and (2.8) provide a formal generalization of T h e o r e m 3 in Woodward (1983). W e :prove the following result.

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LL Gerontidis ~European Journal of Operational Research 85 (1995) 650-669

L e m m a 2.1. Conditions (2.7) and (2.8) are necessary and sufficient for the transition matrix Q of the singleton Markov replacement chain to be periodic. Proof. It is apparent from (2.4) that Q is periodic of the form (2.12), if and only if both P and P~+lP0 are periodic of the same form. Periodicity of p:,+ lP0 is ensured if d equals the greatest common divisor (g.c.d.) of {i[ Pi,k+l > 0 and k} and if P0 has the form (2.8). For if recruitment and (or) wastage are allowed to (from) any other state the periodic behaviour of the replacement process is violated. This comes from the fact that some power of Q will contain a column with positive elements only and by Theorem 2 in Isaacson and Madsen (1974), Q will be aperiodic. [] By raising Q to the d-th power, CO

C1

Ca - I

Qd = Oiag(A0, A l , . . . ,

Ad_l),

where A r . .er. .

e d _ 2 ( U d _ 1 + P kd+- Il ,P o )0P o

"'" e r - 1 _~ W r + P kr,+ l P Or

(2.13)

is k r × k r stochastic irreducible and aperiodic, for r = O, 1 , . . . , d - 1. Definition 2.1 (Boverman et al., 1977). An irreducible stochastic matrix Q is called strongly ergodic if its eigenvalue 1 is simple. If Q has d > 1 eigenvalues of modulus 1 and A r is strongly ergodic, d -- 0, 1. . . . , d - 1, then Q is called periodic strongly ergodic.

Thus Q is periodic strongly ergodic and Qnd converges to Co

C1

Cd- i

Q~-I = Oiag(l~1r 0 l~¢r 1. . . . , l~_lard_ 1), where ¢rr and 1 r are the 1 × k r left and right eigenvectors of A~, respectively, corresponding to the eigenvalue 1. From (2.13) we get that p ~ ( l r --

~:r) -1

qTr "~ PO( Ir -- W r ) _ l l r r

,

(2.14)

where I, is the k, x k r identity matrix. Consider the sequence of kr-simplexes .~k,.~

{,Wrl,Wr>Or,

,Wrltr =

1},

of the k:dimensional Euclidean space R k, and denote by 8"r(d)the region in R k, generated by the equilibrium distribution 7,, as p0° varies over the stochastic k0-simplex ~ko. It can be proved (see Theorem 2.2 in Gerontidis, 1994) that 8"r(d) is the convex hull of the normalized rows of

( / r - - Wr) -1,

r=0, 1,...,d-

1.

(iii) The distribution of the completed length of service For a single individual denote by X t the grade he entered at its t-th transition within the system and let T be the random variable defined by T = i n f { t > 0: X t = k + 1}.

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Then T is the completed length of service of an individual or otherwise the (random) time he spends in the system. This has probability distribution

F ( t ) = l - P o P ' l ',

t=O, 1 . . . . .

which is said to be of phase-type (PH) with representation (P0, P) of order k (Neuts, 1981), with (discrete) density

f ( s d + r + 1) = F ( s d + r + 1) - F ( s d +r) [0 =pojDsd+rpk+lt

if r = 0 , 1 . . . . . d - 2 ,

= ~ nd-lws

kP0

.d-lt

if r = d -

d-1/'k+l

1,

and hazard function (age specific failure rate) 0

q~(sd+r+ 1)

= l~ Po

if r = 0 , 1 , . . . , d -

_d-lws

~

~

PO

2,

_d-lt

d-lPk+l

if r = d - 1 ,

VVd-lJtd-1

for s = 0, 1. . . . . . The hazard function enables one to identify times in an individual's service: when the risk of leaving is particularly high or low, thus providing a concise description of the leaving process (Bartholomew, 1973, p.183). L e t / . ~ be the mean of F(t). Then we have that (Lemma 3.1 in Gerontidis, 1994)

(2.15)

=

where lrd_ 1 is given by (2.14).

(iv) The expected number of losses in (0, t] Let H(t) be the expected number of successive replacements for a single individual realization line in the time interval (0, t ]. The probability that a replacement occurs at time t is h(t) =poQt-lp'k+l,

t = 1, 2, 3 . . . . .

(2.16)

and this is the renewal density of the associated (discrete) renewal process. By taking H(0) -- 0, we have t-1

H ( t ) = ~_, poQip],+,.

(2.17)

i=O

Since for s = O, 1, 2 . . . . .

h(sd + r) =

0

d_lA s d-V Po d-lPk+a

if r = 0 , 1 . . . . , d - 2 , if r = d -

(2.18)

1,

from (2.17) and (2.18) we get d-1

H(sd+r)

=

E

s-1

m=0 i=0

for r = 0 , 1. . . . . d - 2 .

H(sd) = H ( s d + X )

r

E P OI~I m fZl i dP. ,k + l -F

s-1

" ~poQSd+'P~+I i=0

= E n dPo -lAi i=0

Thus .....

H[(s+l)d-ll,

s=0,

1,2 .....

d - - l P. dk-+l tl '

(2.19)

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LL Gerontidis / European Journal of Operational Research 85 (1995) 650-669

An expression for the asymptotic replacement rate may be obtained from (2.15) and (2.18) as 0 s--,~limh ( s d + r ) =

if r = 0, 1 , . . . , d -

2,

~'~a_l/,,k+l"d-l'=d/ixl if r = d - 1 ,

(2.20)

and thus from (2.19) and (2.20) asymptotically,

H ( s d + r ) =sd/l~l,

r=0, 1.... ,d-1.

For a system with a fixed number of T members say, the recruitment and wastage processes are identical since each loss is directly responsible for one recruitment. The expected number of losses in (0, t] is the superposition of T identical renewal process as defined above and thus, fik+i(t) = T" H(t). If the system is expanding, the recruitment and wastage process are not the same since by (2.2) recruitment exceeds wastage because of the need to feel vacancies created by expansion. In this case we have that sd+r

~k+~(sd + r) = E T(sd + r - i)H(i). i=O

This is the discrete time analogue of the result derived in Bartholomew (1973, p.231) for the continuous time renewal model and thus pointing out the importance of the renewal function in determining expected wastage.

3. Periodicity of the profile process In order to study the periodicity of q we introduce some additional notation. Rewrite ~(t) in canonical form as CO

~(t)=

C1

Cd - 1

[T°(t), ~'(t), ..., ~d-l(t)],

and let Tr(t)=~r(t)l'r be the expected total size of Cr, r = 0 , 1 . . . . . d - 1 , d - 1 Tr(t) • We next define a version of the canonical form of q(t), as T(t) = Y'-r=0 qd(t)=

Co [q0(t)

at time t such that

C1 Ca- 1 ql(t) .... , qa-l(t)],

where qr(t)= ~r(t)/Tr(t), r = 0, 1 . . . . . d - 1. The symbol qa(t) may be considered as a generalized profile consisting of the d profile processes associated with Cr, r = 0, 1. . . . . d - 1, such that d-1

qa(t) 1'= E qr(t)l"=d. r=O

For A = [aij] a real matrix and b = (b i) a real vector, we define their norms as II A II = maxiEj=l k laijl and II b II = maxilbil. In the rest of this paper the convergence results will be assumed with respect to these norms.

(i) The case of independent input In this version of the model the number R(t) of recruits entering the system in the time interval (t - 1, t) may either be a given number or a Poisson random variable independent of the existing stock vector n(t). In the latter case R(t) represents the parameter of the associated non-homogeneous Poisson process. In this section we shall be concerned only with the expectation R(t) without making any distributional assumptions, so that distinction between the above mentioned cases is irrelevant.

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3.1. A Markov manpower system with given quasi-periodic matrix P and input sequence R(t) has an asymptotically attainable structure qd(oo) = [q°(oo), q~(oo). . . . . qa-~(oo)], under recruitment control, if there exists a recruitment policy P0 = (pO, 0~. . . . ,0d_~), such that lim, _~qd(t)= qd(oo). Definition

We now determine the asymptotic form of qd(t) as t ---, 0% and the set .~'a(oo) of possible q's that can be attained asymptotically from any initial structure, as pO varies over .~ko. Let o.)1 be the maximum modulus of the eigenvalues of P (there are d such eigenvalues). Then h I = ~Ol a is the dominant eigenvalue of pd which is of multiplicity d (Keilson and Wishart, 1964) and also h~ is t h e dominant eigenvalue of W~, r = 0, 1 , . . . , d - 1, of multiplicity one. The following result (extending Mehlmann, 1977) shows that although the ~(t) process may be asymptotically periodic (see Eq. (3.4) below), the profile process converges to a unique limit. T h e o r e m 3.1. Let for a quasi-periodic Markov manpower system the parameters P, Po and R(t) be given. Let also A(t ) be a function with the following properties: (1) A(1) = A is positive and finite. (2) {A(t)/A(t - 1)} > o)1, for all t >_2. (3) liras ~ { A ( s d + h + 1)/A(sh + h]} = bh = 1/c h, exists and is finite for h = 1, 2,..., d. (4) lim,_.®{R(t)/A(t)} = 1. Then, (a) The profile process qa( t ) converges to a unique limit Co C~ Cd- 1 qa(°°) = [ pO(blo _ Wo) -1 p ~ ( b l l _ Wl ) -1 p g - l ( b l d _ l -- W d _ l ) -1

p°(blo - Wo)-ll{~

p~(bl I - W,)-11~ ' . . . . p g - l ( b l d _ l

i_

-- W d _ l ) - 1 1 ' d-1

where b= 1-[bi=l

ci = I / c > A

i=1

(3.1)

1.

i=1

(b) The set sJa(~) of asymptotically attainable structures qa(oo), as pO varies over the stochastic ko_simplex ~ko is the product space

where s~r(~) is the convex hull of the normalized rows of (b~ - ~)-' Proof.

,

r=O,

1,...,d-1.

(a): Iterating (2.1) we get

~ ( s d + h + l)

psd+h+l d-1 s-1 ~ [ ( s _ m ) d + h + l _ r ] pmd E P o Pr E A ( s d + h + l) A ( s d + h + l) = n ( O ) A ( s d + h + l ) + r=0 m=0 h R(sd+h+l-r)

+ E

r=0

A ( s d + h + l)

poe re'd.

(3.2)

By property (2) the first and third parts in the right of (3.2) vanish in the limit. It follows from property (4) and the fact that pd is substochastic, that for given h the series s-1 R [ ( s - m ) d + h

E m=O

+ 1--r] pmd

A( sd + h + 1)

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is convergent for r = 0, 1 . . . . , d - 1. Thus

~ ( s d + h + l) d-1 s--1 ~ [ ( s _ m ) d + h + v ( h + 1) = lim = ~ po Pr lim s-)~A(sd+h+l) r=0 s--)°°m=O A(sd+h+l)

l _ r ] p,,a

exists and is finite. Taking limits in (2.1), together with the properties (3) and (4) we have that

v ( h + 1) = v ( h ) P c h +Po,

(3.3)

Iterating (3.3) d - 1 times, we get h-1

r

d-1

r

v(h) =v(h)edc + E pOprHch-i + E poPrHCd+h-i, r=0 i=1 r=h i=l where c = I-ld=lci, l-I°ffilCi = 1 and Cmd+h = Ch. Thus [h-1 r 8-1 r v ( h ) = [ ~_, poPrX-[ch_i + ~_~ poPrI-ica+h_i](l--cpd) r=0 i=1 r=h i=1 ]

-1

and from v(h) = [v°(h), v l ( h ) , . . . , v d- l(h)] together with (2.10) we get r

vr(h)=

1--[Ch_iP~(Ir--CW~) -1 i=lr

if r < h ,

I-Ica+h_iP~(Ir--CW~) -1

ifr>_h,

(3.4)

i=1

for r = 0, 1 , . . . , d - 1 ; h = 1, 2 . . . . ,d. Since q r ( h ) = v r ( h ) / v r ( h ) l ", (3.1) follows from (3.4) and is independent of h. (b): From (3.1) we also have that

k, qr(o0) =

E i=1

p;ifi r er (bIr - Wr)-1 kr fi r E p~jfl r j=l

where e~ is the 1 × kr row vector having 1 in the i-th position and zeros elsewhere and f i r is the sum of elements in the i-th row of (bl r - W r ) -1. Thus for varying pO, qr(~) can be expressed as a convex combination of the normalized rows of (bl r - Wr)-1, r = O, 1 , . . . , d - 1. [] Remark 3.1. T h e o r e m 3.1 is applicable in a series of special cases. For example properties (3) and (4) imply that R(sd + r) =A(Hr£11bi)b s. If in particular b 1 . . . . . bd = 1 = b, then R(t) = A for all t and qr(oo) -- ~rr, which is the limiting distribution (2.14) of the singleton Markov replacement chain. Thus in the case of constant input, qr(~) is the same as the distribution obtained if the system is immediately replacing the losses incurred in order to keep the total size constant (see also Theorem 3.3(b), below). We next investigate the transient behaviour of ~(t) with respect to stability. Since P maps C o to C 1, C 1 to C 2. . . . , Ca-1 to Co, whereas pa maps C o to C o , . . . , C a_ 1 to C a_ 1, one step stability in the sense of Feichtinger (1976) does not apply. We can introduce instead a weaker form of stability, i.e. d-step stability with respect to vector growth factor x = (x 0, Xl . . . . . xa_ 1) and thus extending Feichtinger (1976). Denote by p a = ( p O , p~ . . . . ,pa-~), where p~, r = 0 , 1 , . . . , d - 1 , as in (2.10) and R d ( s ) = [R(sd), R(sd - 1 ) , . . . , R{(s - 1)d + 1}]. After d steps we have

~[(s + 1)d] = ~(sd)P d +pg Diag{Rd(s + 1)I},

s = 0, 1 . . . . .

which provides the evolution of ~(t) over time intervals of one period length.

(3.5)

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659

Definition 3.2. The expected structure ~(sd) is d-step stable with growth factor x = (x0, x l , . . . , Xd_~) for xr>A1, r = 0 , 1. . . . . d - 1, if

~ [ ( s + 1)d] = ~ ( s d ) D i a g ( x / ) ,

s = 0 , 1, 2 . . . . .

(3.6)

where xl = (xolo, x l i 1..... Xd_lld_I). If ~(sd) is d-step stable from (3.5) and (3.6), we have that

~(sd) =pod Diag{Rd(s + 1) I} [Diag(x/) - pd]-1

(3.7)

The following result provides necessary and sufficient conditions on the initial establishment and the input function for the structure ~(sd) to be d-step stable. Theorem 3.2. The expected structure ~(sd) is d-step stable with growth factor x = (x0, x 1. . . . ,Xd_l), for Xr > A 1, r = 0 , 1. . . . . d - 1 , if and only if

n(0) = pod Diag{Rd(1)} [Diag(x/) - pd] -1

(3.8)

and the input has the form

R(sd-r)=RX r,

r=0,1

s=l,2

.... ,d-l,

.....

(3.9)

Proof. Rewrite (3.9) in vector form as

s) = Rx s, where xS=(x~, x~,...,x~_l) , s = l , equivalent to

2. . . . .

It can be proved by induction that (3.8) and (3.9) are

~(sd) = Rp d D i a g ( x l ) ~+ l [ D i a g ( x l ) - pd] -1

(3.10)

Let (3.8) and (3.9) both hold. Rearranging (3.10) together with (3.5) we get d-step stability. Suppose now that ~(sd) is d-step stable. Then (3.7) holds and for s = 0 we get (3.8). From (3.5) and (3.7) we also have that p0a Diag{Rd(S + 2) I} [Diag(xl) - pal]-1 =pd Diag{Rd(s + 1) I} [Diag(xl) - pd]-1 D i a g ( x / ) . Summation over the individual subvector elements gives

Rd(S+2) =Rd(s+I ) Diag(xl), from which (3.9) follows.

s = O , 1. . . . .

[]

Definition 3.3. The expected relative structure qr(t) is asymptotically stable with growth factor Xr, for Xr > A~, if for any initial structure n~(0) we have

p (Xr1 limqr(t)

=

-- W,)--1

Xrlr-- l rr)-ll'r

,

r=0,1

..... d-1.

(3.11)

Remark 3.2. Condition (3.9) splits R(t) into d geometric subsequences. If we drop the assumption (3.8) on the initial structure but maintain exponential recruitment we have R[(s + 1)d + r]/R(sd + r) = x, and Theorem 3.1 when applied to (3.5) implies asymptotic stability of qr(~), r -- 0, 1. . . . . d - 1.

LL Gerontidis~European Journal of OperationalResearch 85 (1995) 650-669

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Remark 3.3. From (3.7) we have that if nr(sd) is d-step stable, then qr(sd) =

p ~ ( X r i r - Wr) - 1 -1 , = q r [ ( s + l ) d ] , pS( Xrlr -- l~rr) 1 r

s=0,

1 ....

so that qr(sd) is re-attainable after d steps (see Definition 3.4, below) and from (3.11) it is also asymptotically stable for r = 0, 1. . . . . d - 1.

(ii) The case of given total size A model more relevant to the manpower planning context is to allow the grade sizes to vary within a given global T(t). In this case the number of recruits R(t + 1) entering the system in the time interval (t, t + 1) is determined by the number of vacancies occuring in the system and it is thus a random variable. Its expectation depends linearly on the expected stock vector ~(t) as in (2.2). The total membership T(t) may either be a sequence of given numbers or the expectation of a known stochastic process. The distinction is of importance only if we consider the distribution of the grade sizes (Bartholomew, 1982, p.72). Iterating (2.3) h - 1 times, we get the evolution of ~(sd + h) in terms of ~(sd). From (2.3) together with (2.8) and (2.12), the stock vector ~'(sd + h) corresponding to C~ after h steps is given by

LI. Gerontidis / European Journal of Operational Research 85 (1995) 650-669 Lemma

661

3.1. The following relations hold: Zr(Sd -~-h) - Zd_h+r(Sd ) 0

(a)

M(sd-r+h)=

(b)

M [ ( s + 1)d - r] = T~[(s + 1)d] - Tr(Sd),

(C)

Zr(sd+h)=(zd-h+r[(s-I-1)d] rd_h+r(Sd )

if r > h ,

if r < h ,

d-1

T ( s a dr-h) = E Td-h+r[(S -[- 1)el

+ E

Td-h+r(Sd),

r=h

r=0

for r=O, 1. . . . . d - l ;

r
if r > h,

h-1 (d)

if

h = 1. . . . . d.

Proof. To obtain (a) postmultiply (3.12) by 1'r, which for h = d provides (b). Replacing r by d - h + r, for r < h in (b) we get

M ( s d - r + h) = Ta_h+~[(s + 1)d] - Ta_h+r(sd), which together with (a) implies (c). Relation (d) is now straightforward.

[]

Let Mr(sd) = Tr(sd) - Tr[(S - 1)d] be the increase (expansion) of the total size attributable to C~ at time sd. From Lemma 3.1(b) we get that M(sd-

r) = Tr(sd ) - Tr[(S - 1)d] = M r ( s d ).

Then

M(sd -r)

Tr(sa)

Zr(Sd ) -- Zr[(S - 1)d] =

Tr(sa)

= 1 - y,(sd),

is the proportionate growth of C r at time sd, with

Yr(Sd) =

T,[(s- 1)d] Zr(Sd )

,

r=0,

1. . . . . d -

1.

Combining (3.13) together with the results (a) and (c) of Lemma 3.1 we arrive at qr( sd q- h) = qd-h+r( Sd) ×

[ Y d - h + r ( s d ) P d - h + r ' ' ' P o ' ' ' Pr-l + {1--Yd_h+~(sd)}l'rP~]

if r < h ,

Pa-h+r "'" Po "'" Pr-1

if r > h,

(3.14)

for r=O, 1. . . . , d - 1 and h = 1 . . . . . d. Eq. (3.14) is the basic recurrence relation on which the subsequent analysis is based. It determines the evolution of qr(sd + h), during the (s + 1)st cycle, given that qr(sd) is known. The following lemma provides asymptotic results for non-homogeneous Markov chains (Isaacson and Madsen, 1976). L e m m a 3.2. Let Q(t), t = O, 1 , . . . , be a sequence of stochastic matrices with lim t _.~Q(t ) = Q, where Q is strongly ergodic with normalized left eigenvector u corresponding to eigenvalue 1. Then lira t _~®I-[[=s Q(i) = l'u, uniformly in s. []

/./. Gerontidis~European Journal of Operational Research 85 (1995) 650-669

662

The following theorem is the main result of this section. It extends Feichtinger and Mehlmann (1976) and provides a characterization of the asymptotic periodicity of qd(t), by examining each subclass separately.

Theorem 3.3. Let for a manpower system with given P and Po, its total size T( t ) be determined in advance. (a) Let also Tr(Sd) be such that Yr(Sd) > m.aX(Wrl'r) i f o r s = 0 , 1,...,

(3.15)

I

with lim y , ( s d ) =y~ = 1/z~

(3.16)

S -...~ oo

and the matrix t

r

Ur = y r ~ r r -.{-( I r - Y r W r ) l r P o

be strongly ergodic with normalized left eigenvector u r corresponding to the eigenvalue 1. Then q r ( h ) = lim qr(sd + h) $ ...e ct~

-~ /Ud_h+r{Yd_h+red_h+r ' ' " eo " " er_l d- (1 --Yd_h+r)ltrP~}

Ud-h+rPd-h+~ " " Po "'" Pr-1

if r < h ,

if r > h,

(3.17)

where p ~ ( Z r I r - ~ r ) -1

ur

p~(zrlr_W~)_ll"

r = O , 1,.

(3.18)

,d-l,

for h = 1 .... , d, i.e. asymptotically qd( t ) splits into d convergent subsequences with limits qd(h)=[q°(h),ql(h)

..... qd-l(h)],

h = l . . . . . d.

(b) I f T ( t ) converges to some fixed value, i.e. lim t _~=T(t) = T, then

p6( z Ir-

qr(o0)

-1

p~(Zrlr__ Wr )-llr,

~rr,

r = O , 1..... d

1,

(3.19)

i.e. qa(t) converges to lim qd( t) = ( Tro, lrl,...,~ra_l),

t--*~

which is the limiting distribution (2.14) of the singleton Markov replacement chain. Proof. (a): For h = d (3.14) gives q r [ ( s + 1 ) d ] = q r ( s d ) [ Y r ( S d ) A r + {1 - yr(sd)}l'rP~],

(3.20)

where Ar as in (2.13). Iterating (3.20), we get $

sd) = qr(o) I-I U,( i),

(3.21)

i=1

where

U,(i) = y r [ ( i - 1)d] W~+ {Ir - y ~ [ ( i -

1)d]Wr}l"p~.

(3.22)

l.L Gerontidis ~European Journal o f Operational Research 85 (1995) 650-669

663

Taking limits in (3.22) together with (3.16) we have that U r ~-- lim U r ( i ) = [YrWr + (I~ - - Y r W r ) l ~ p ~ ]

(3.23)

i ---~ o o

is strongly ergodic with normalized left eigenvector u r corresponding to the eigenvalue 1. Conditions (3.15) and (3.16) imply the existence of the non-negative matrix ( I r -YrWr) - I and from (3.23) we derive (3.18). F r o m L e m m a 3.2 we get lim f i Ur(i ) = l'Ur,

s-*~ i=1

and (3.21) provides lim q r [ ( s + l ) d ] = u r,

r=0,1,...,d-1.

(3.24)

s ---~ t:¢

Taking limits in (3.14) together with (3.24) we arrive at (3.17). (b): If lim t _.~T(t) ffi T, then lira t _.®M(t) = lira t _.~[T(t) - T(t - 1)] = 0 and also lim M [ ( s + 1)d - r] -- lira {Tr[(S + 1)d] - Tr(Sd)} = O. S --~ oo

$ ---~ O0

Therefore, lim ( M r ( s d ) / T r ( s d ) ) =

0 = 1--Yr,

s ---~ o o

i.e. y~ = 1 and case (a) in our t h e o r e m is valid with y~ = z, = 1 for r = 0, 1. . . . . d - 1. Expression (3.18) becomes

p~(l~

-

Wr)--1

ur = p ~ ( I r -- Wr)

(3.25)

--1 ,¢ ~'"/'rr' lr

i.e. the equilibrium distribution (2.14) of the singleton Markov replacement chain and from (3,17), we get lim

qr(sa +h) ~ ' ' l r d _ h + r e d _ h + r

'''

e 0 "" Pr-l"

(3.26)

s ---~ ¢~

Since ~ r = T r o P o "'" P,-1,

r= 1..... d-

1

(3.27)

( T h e o r e m 4.2 in Keilson and Wishart, 1964), (3.25) together with (3.26) and (3.27) imply (3.19), which is independent of h. [] R e m a r k 3.4. If C r expands at a constant rate in time intervals of one period length, i.e. Tr[(s + 1)d] = Tr(Sd)z r, for s = 0, 1. . . . . and z, > A1, r = 0, 1 . . . . , d - 1 from L e m m a 3.1(c)-(d), we get that

Tr(sa +h)

=

/Td_h+r(Sd)Zd_h+r Ta_h+r(sd )

if r < h , if r > h,

and h-1

T(sd+h)=

d-1

~ T d - h + r \¢0~zS+l / d - h + r @ E Td - h + r ( O ) Z d - h + r

rffi0

rfh

for h = 1 , . . . , d, i.e. T ( t ) is a sum of d geometric sequences.

LI. Gerontidis~European Journalof OperationalResearch 85 (1995) 650-669

664

Definition 3.4. The structure qa(sd) = [q°(sd), ql(sd),..., qa-l(sd)] is re-attainable after d steps (Vajda, 1978) or d-step maintainable (Davies, 1973), at the s-th cycle under recruitment control, if there exists a recruitment vector P0 = (p0, 01 . . . . . 0d_ 1) such that

qa(sd)=qa[(s+ l)d]=qa,

s=0,1,....

Let Ra(sd) be the set of qa structures for the entire system that can be re-attained at the s-th cycle as pO varies over ~ k o and let Rr(Sd) be the set of qr re-attainable structures for r = 0, 1 . . . . . d - 1. By looking at time intervals of one period length, (3.20) provides qr[(s + 1)d] = qr(sd)[y~(sd)Wr + {I r - y,(sd)W,}l~p~]. If qr(sd) is re-attainable at the s-th cycle, then

(3.28)

qr = q,[ Yr( Sd)W ~ __ {ir _ Yr( Sd)Wr}l;p~] .

Condition (3.15) ensures the existence of the non-negative matrix [I,- y,(sd)Wr] -I and from (3.28) we get that p~[ I, - y r ( s d ) ~ ] -1

q'= p~[1,-y,(sd)W,]-ll"

r=0,

1 ..... d-

1.

We have thus proved the following result: Theorem 3.4. A s p ° caries over ~ko, the set Ra(sd) of d-step re-attainable structures qa = (qO, ql,..., qa-~) at the s-th cycle is the product space

Rd(sd) -Ro(sd) ×n~(sd) ×

... ×Rd_~(sd),

where Rr(sd) is the convex hull of the normalized rows of

[Z,-yr(sd)W,] -1, r = 0 , 1 , . . . , d - 1 .

4. The time-dependent case

In this section we relax the stationarity assumption by allowing the parameters of the manpower system to be given functions of time. Time dependence is introduced via the sequences po(t), P(t) and Pk+l(t), t ----0, 1, 2 . . . . , from which we construct the sequence of replacement matrices

Q(t) = P ( t ) +p~+l(t)po(t),

t - 1, 2 . . . . ,

(4.1)

having the canonical form (2.12). Eq. (3.14) now becomes qr( sd -F h) = qd-h+r( Sd)

([ya_h+r(Sd)Pa_h+r(sd + 1 ) ' "

Pr-l(sd + h - 1)

×l +{1-ya_h+r(sd)}l'p~(sd+h-r-1)] /

[Pa_h+r(sd + 1) " " P,_l(Sd + h - 1)

if r
(4.2)

if r > h,

where

p~(sd+h-r-1)=p°(sd+h-r-1)Po(sd+h-r)... for s = 0, 1, 2 , . . . ; r = 0, 1 , . . . , d - 1 ;

Pr_l(sd+h-1),

r
h = 1 , . . . , d . It is apparent from (4.2) that the evolution of

qr(sd + h) across time depends on the behaviour of the environment, i.e. the behaviour of the sequences

LI. Gerontidis/ European Journal of Operational Research 85 (1995) 650-669

665

po(t), P ( t ) and p~÷~(t), t = 0, 1, 2 , . . . , which may be either convergent or cyclic. Therefore we consider two different types of environments. (i) Convergent environment

In the convergent case we assume that lim

P(t)=P,

lim P k ÷ l ( t ) = P k + l

t ---~ c¢

and lim P o ( t ) = P 0

t --~ oo

t -'~ oo

elementwise, and that Q ( t ) = P ( t ) + pk+l(t)'po(t) is a sequence of d-periodic stochastic matrices having the same cyclic decomposition C ~ C O U • • • U Ca_ r Let also lim, _.~Q(t) = Q be a d-periodic strongly matrix of the form (2.12) with left eigenvector ¢r = (~r0, ~rl,..., ~ra_ 1) corresponding to the eigenvalue 1. Since time-dependence was implicitly involved via the y-parameter in the derivation of T h e o r e m 3.3, the above assumptions provide the necessary facts for T h e o r e m 3.3 to hold also in the case of a convergent environment. (ii) Cyclic environment

Another type of behaviour is to assume that the sequences P ( t ) , Pk÷ ~(t) and Po(t) repeat themselves in a cyclic fashion, i.e. there exists a d I > 0 such that e(sdl +r ) =e(r),

Po(Sdl + r ) = P 0 ( r ) ,

Pk+l(Sdl + r ) = P k + l ( r ) ,

and thus r=0,1 ..... dl-1,

Q(Sdl+r)=Q(r),

for any s - 0, 1, 2 , . . . . It is not restrictive to assume that d = d 1, where d is the period of the cyclic decomposition C - C O u • • • u Ca_ 1 and d~ is the cycle of the sequence Q(t), t = 1, 2 . . . . . FOr if d ¢ d~, then d* = d . d ~ is a common period for the overall (combined) behaviour. In a cyclic environment Eq. (4.2) becomes

ar( sd + h) =qd-h+r( Sd) [ Y d - h + r ( s d ) P d - h + r ( O ) ' ' " e r - l ( h - 1)

×

I

if r < h ,

+{1-Yd_h+r(sd)}l'rP~(d+h-r-1)]

Pa_h+r(O) ' ' ' Pr_~(h -- 1)

for r = 0, 1. . . . . d Q(0, d -

if r >_h,

1; h = 1. . . . . d. Define

1) = Q ( 0 ) Q ( 1 ) " -

Q(d-

1)

to be the cycle matrix of the form

Q(0, d - 1) = Diag[Ao(0, d - 1), A~(0, d - 1) ....

,Ad_~(O, d -

1)],

where At(0 , d -

1) = P r ( 0 ) ' " xPo(d= W~(0, d -

Pd_2(d-

r- 2)[Ua_l(d- r-

r)...P~_~(d-

1) + p ~ + a ( d - r -

1) + p ~ + ~ ( d -

r-

1 ) ' p o ° ( d - r - 1)]

1) 1)'p~(d-r-

1)

is the stochastic and aperiodic, with p ~ + l ( d - r - 1)' = Pr(0) " " " P d _ 2 ( d - r - 2)Pka~_~(d -- r -- 1)',

(4.3)

666

I.L Gerontidis ~European Journal of Operational Research 85 (1995) 650-669

and p ~ ( d - r - 1) = p ° ( d -

r - 1 ) P 0 ( d - r) . . . P r _ l ( d - 1),

such that p ~ ( d - r - 1 ) l ' = l , for r = O , 1. . . . . d - 1 . It is apparent from (4.3) that if ~rra-1 is the normalized left eigenvector of At(0, d - 1) corresponding to the eigenvalue 1, then I1"/-1= p 6 ( d - r -

1 ) [ I r - ~r(O, d - 1)]-1

p~(d-r-1)[rr-Wr(O,d-1)]-lr

'

r=0,1

.... ,d-1.

Therefore Trd-l=

(,/l. d - l ,

1,1-d - 1 . . . . . s/eft--?)

is the left eigenvector of Q(0, d - 1) corresponding to the eigenvalue 1 and ~.h _~ ~trd- l O ( O, h )

is a vector of the form =

. . . . .

with

~rrh = Tl'dd~-rl-h-led +r-h -1(0) Pal+r-h( 1 ) ' ' " Pr- l( h )

being the normalized left eigenvector of Ar(h + 1, d + h) corresponding to C r, r = 0, 1,..., d - 1, at the h-th cycle, h = 0, 1. . . . . d - 2 (Lemma 4.3 in Gerontidis, 1994). The following result highlights the effects of the cyclic environment on the asymptotic behaviour of qd(t). T h e o r e m 4.1. Let for a non-stationary manpower system the sequences P(t), Pk +l( t ) and p0(t), t = 1,

2 . . . . . be such that

e(sd+r)

=P(r),

Pk+l(sd+r)=Pk+l(r),

Po(sd+r)=Po(r),

for any s ffi O, 1, 2 . . . . and r = O, 1 , . . . , d - 1. Let also Q(t) be a sequence o f periodic matrices having the same cyclic decomposition C - C o u • .. u Ca_ 1 and Q(O, d - 1) be a d-periodic strongly ergodic matrix with left eigenvector ~lrd - l = ("/l"d-l, 'Trld - 1 . . . . , ~ra-11) corresponding to the eigenvalue 1. (a) Let also Tr(sd) be such that y,(sd) > maxi[Wr(O, d - 1)l'r] i, for S = O, 1 , . . . , with linl Yr( s d ) = Yr S ..-#oo

and the matrix

U,(0, d - 1) =YrWr(0, d - 1)

+ [Ir--Yrl~rr(O, d -

1)]l'rp~(d-r-

1)

be strongly ergodic with normalized left eigenvector UZr- 1 corresponding to the eigenvalue 1. Then, U dd-1 - h + r { Y d - h + r Pd - h + r ( 0 ) ° ' " e r - l ( h -

q r ( h ) = lim q r ( s d + h ) = $ ---~00

+ ( 1 - - Y d _ h + r ) l r r P ~ ( d - - r + h -- 1)} t d-1 d - h + r p d - h + r kt OI~ ' " P r _ l ( h - 1 )

1) if r < h , ifr>h,

where d-1 Ur

p~(d-r1) [ l , - y , W , ( O , d - 1)] -~ p~(d_r_l)[lr_y,W~(O,d_l)]_ll,r,

r = O , 1 .... , d - l ,

h=l,...,d.

LI. Gerontidis / European Journal of Operational Research 85 (1995) 650-669

(b)

667

If lira t _.~T(t)= T, then qa(sd +h) =(~-oh-~, ~-1h-~ . . . . . ~.h_-?), h = 1 . . . . . d.

lim

$ ---~ Oo

5. Two illustrative examples In this section we provide two examples from the literature on educational manpower planning illustrating the above theoretical results. Consider a university system offering a three year undergradul ate course of study, where each academic year is divided into two semesters. By taking the time interval to be one semester and the state of an individual the semester he occupies, we have lthat C {1, 2, 3, 4, 5, 6}. Let also a student pass the first semester independently of his performance !and at the end of the second semester be judged on his performance during the first year. He either passes the year with probability pEa or he fails with probability P27 in which case he repeats the year. The same rule applies for the second year except in case he fails he is replaced by a new student. At the end Of the third year he either receives his degree or he fails completely. In either case he is replaced by a new student. This type of behaviour is adequately modelled by P0 -- (1, 0, 0, 0, 0, 0),

p=

1

2

3

4

5

6

1

0

1

0

0

0

0-

2

0

0

PEa

0

0

0

3

0

0

0

1

0

0

4

0

0

0

0

P45

0

5

0

0

0

0

0

1

6

0

0

0

0

0

0

and thus P7 = ( 0 , P27, O, P47, O, 1). Since g.c.d. {il Pi7 > 0 and 6 } = 2 , we have d = 2 rearrangement has the canonical form

Q

1

3

5

2 [1

1

0

0

0

3

0

0

0

5

0

0

0

2

P27

PEa

0

0

4

7

0

P45

0

0

0

0

6

o]

and Q after

6

t00

0 1 0 0 0

0 0

with cyclically moving subclasses C o - {1, 3, 5} and C 1 = {2, 4, 6}. If the system keeps the total size fixed, assuming numerical values pEa = 0.7, P45 = 0.8, p~ = (0.3, 0.2, 1.0) and pO = (1, 0, 0), we get that Q is periodic strongly ergodic of period 2 and Q2 has left eigenvector ~ -- (~'o, ~'~) such that "8"0 = 'B"1 = ( 0 . 4 4 2 ,

0.310, 0.248).

Thus from Theorem 3.3(b), q2(oo) = [q°(oo), ql(ao)] = [(0.442, 0.310, 0.248), (0.442, 0.310, 0.248)]. In case that To{2(s + 1)} -- 1.25. To(2S) and Tl{2(s + 1)} = 1.1. Tl(2s), i.e. C O and C 1 expand at constant rates z o = 1.25 and zl = 1.1 respectively at time intervals of length 2, from Theorem 3.3(a) we get q2(1) = [q°(1), q~(1)] = [(0.480, 0.302, 0.218), (0.522, 0.291, 0.187)]

1.L Gerontidis / European Journal of Operational Research 85 (1995) 650-669

668

and q2(2) = [q°(2), q'(2)] = [(0.521, 0.292, 0.187), (0.480, 0.302, 0.218)]. We now provide an example for the cyclic case. Suppose that the university is offering a two year post-graduate course and and each academic year is divided into three terms. By taking the time interval to be one term, we have C = {1, 2, 3, 4, 5, 6}. Let also a student pass the first two terms independently of his performance and at the end of third term be judged on his performance during the first year. He either passes the year with probability P34, or he repeats the year with probability P37- The same rule applies for the second year of study except that in the last term he either receives his degree or fails completely. In both cases he is replaced by a new student. In this case p o ( t ) = (1, 0, 0, 0, 0, 0) and P7(t) = [0, 0, P37(t), 0, 0, 1] SO that d = g.c.d. {il Pi7(t) > 0 and 6} = 3, leading to C o - {1, 2}, C1 = {2, 4}, C 2 = {3, 6} and 1

4

0

0

2

l O o 4

5

[10

o° o° 63 [0°

3

6

0

0

00

o°°o °1] 0000 0000

Assuming that the system undergoes a cyclic behaviour of period 3, i.e. for m = 1, 2 . . . . . P ( 3 m + r) = P ( r ) and p kd-1 + l ( 3 m + r ) - - - - p ~ ( r ) , r = 0, 1, 2, let the actual sequence be given by Pa4(3m)= 0.25, P34(3m + 1) = 0.15 and Pa4(3m + 2) = 0.05, m = 0, 1 , . . . . Then 1 1 - 0.05

4

2

5

3

6

0.95

0

0

0

0

4

1

0

0

0

0

0

2

0

0

0.15

0.85

0

0

5

0

0

1

0

0

0

3

0

0

0

0

0.25

0.75

6

0

0

0

0

1

0

Q(0, 2) =

is 3-periodic strongly ergodic with ~-o2 = (0.513, 0.487),

~r 2 = (0.541, 0.459) and ~r22= (0.571, 0.429),

and ~r~ -1 --~'o2, ~rln-I =~r 2 and ~r2h-1 =~r~, for h = 1, 2. Let ~{z~, z 2} be the convex hull of two points If the system keeps the total size fixed, the sets dr2(Oo), r = O, 1, 2, of asymptotically attainable profiles qr(OO), r = O, 1, 2, obtained as po° varies over the .~2 are convex sets with vertices

zl,z 2 ~2.

=

~¢°~(°°)

~ z~ = (0.541 0.459)

z°

( 0.5

0.5)

'

and ~,[z2--(0.571 0.429)]

~¢~(oo)=

~

(0.5

0.5)

"

z~

( 0.5

0.5 ) j '

I.I. Gerontidis / European Journal of Operational Research 85 (1995) 650-669

669

Acknowledgement T h e a u t h o r w o u l d like to t h a n k t h e r e f e r e e s f o r u s e f u l c o m m e n t s .

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