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Partial maintainability and control in nonhomogeneous Markov manpower systems
European Journal of Operational Research 62 (1992) 241-251 North-Holland
241
Theory and Methodology
Partial maintainability and control in nonhomogeneous Markov manpower systems A.C. Georgiou Statistics and Operations Research Section, Department of Mathematics, University of Thessaloniki, 54006- ThessalonikL Greece Received August 1990; revised November 1990
Abstract: Concepts of recruitment control in nonhomogeneous Markov manpower systems are studied, in
order to determine subsets of structures in a graded population, which can be maintained in one step, using partial recruitment information necessary for each subset. Two necessary and sufficient conditions are derived, and the Simplex algorithm is used to describe the regions of the so-called partially maintainable structures. The results are illustrated by numerical examples and the consideration of some interesting special cases. Keywords: Personnel; planning; Markov processes; control theory
I. Introduction
Consider a population stratified into k classes (grades, states) according to some characteristic, and assume that individual transitions take place on discrete time intervals. The expected number of people in each grade changes over time through promotion rates, wastage and recruitment stream according to a discrete nonhomogeneous Markov chain with transition probability matrices P(t), t = 0, 1, 2 . . . . Let S = { 1, 2 . . . . . k} be the state space. Each member of the system may be in one and only one state at any given time. The population structure of the system at time t is denoted by the row vector N(t) = [Nl(t) . . . . . N~(t)] where N,(t) is the expected number of members in the i-th grade. The expected total number of members of the system at time t is denoted by T(t) whereas the vector q(t)= [ q l ( t ) , . . . , q g ( t ) ] = N(t)/T(t) is the relative population structure. A system defined in this way is called a nonhomogeneous Markov system, NHMS, (Vassiliou, 1982). Our interest is in controlling a subset of the population structure, q(t), using the recruitment flows connected with a given subset. The origins of the maintainability problem are in Bartholomew (1973,1975). Our problem here arises from Vajda (1975,1978a,1978b) and combines results from Vassiliou (1982) and Vassiliou and Tsantas (1984a,1984b). Vajda was concerned with homogeneous Markov systems and introduced the concept of partially re-attainable, partially maintainable and semi-stationary structures. Vassiliou and Tsantas (1984a) introduced the stochastic control in NHMS. 0377-2217/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
A.C. Georgiou / Partialmainta&abilityand controlin NHMS
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The goal of this paper is to use the stochastic control of NHMS in order to extend the concept of partial maintainability, and prove a basic theorem which characterizes the region of these structures (connected with a desired subset). It is clear that we are interested in subsets of states in hierarchically graded NHMS. A chosen group of grades from a system must maintain its structure and the tool is the partial control of the recruitment stream into the subset (independently from the effects of recruitment into the rest of the grades, as we shall see). The results of this work are illustrated in the final section by two examples concerning manpower systems. The first consists of three grades, while the second consists of four grades. The discussion of the illustration, mainly of the first example, is connected with the work done by Vassiliou and Tsantas (1984a).
2. Notation and formulae
We provide the following notation: : The subset of grades s ___S under consideration (and control). : The complement of s. Ns(t) : The population vector containing zeros in the grades of g and the corresponding N/(t)'s for s. s g
Ns(t) qs(t) #s(t) C~(t)
: Ns(t) + Ns(t) = N(t). : The relative structure vector constructed as N~(t).
: #~(t) + q~(t) = q(t). : A k × k matrix composed from the columns of P(t) which correspond to the grades belonging
in s. The rest of its columns are equal to zero. C.s(t) : e ( t ) = C~(t) + ~ ( t ) . Let us denote by w(t) the wastage vector and by po(t) the recruitment vector at time t. In a similar way, we define the partial wastage vector w~(t) and its complement ~s(t), the partial recruitment vector po~(t), the control parameter, and/50s(t). Also, A(t) = T(t + 1) - T(t) and e = [1, 1. . . . . 1]. Similarly are defined the vectors e s and ~s. Under the Markovian assumption the population structure at time t is given by:
N(t) = N ( t - 1 ) e ( t -
1) + N ( t -
1 ) w ' ( / - 1 ) P 0 ( t - 1) + z a ( t - 1 ) P 0 ( t - 1)
(2.1)
Substituting the vectors in (2.1) by their equivalent sums of partial vectors, we arrive at (2.2) in which the structure N~t) is expressed in terms of known parameters as well as the control partial vector pos(t). So, from (2.1) we obtain
N~(t) +/Vs(/) = [N~(t- 1) + N s ( t - 1)][Cs(t- 1) + C ' s ( t - 1)] + [N~(t- 1) +his(t- 1 ) ] [ w ' ( / - 1) + ~ ' ( t -
1)]
×[Pos(t-1)+ffo~(t-1)]+AT(t-1)[Pos(t-1)+~o,(t-1)].
(2.2)
By reordering we get that
N~(t) +Ns(t) = N ~ ( t - 1 ) C ~ ( t - 1) + , ~ ( t -
1 ) C ~ ( t - 1)
+ N , ( t - 1 ) w ' ( / - 1)Pos(t- 1) + N s ( t - 1 ) ~ ' ( / - 1)Pos(t- 1) + A T ( t - 1 ) P 0 , ( t - 1) + N s ( t - 1 ) C ' s ( t - 1) + N s ( t - 1 ) C s ( t - 1) + N s ( t - 1 ) w g ( t - 1)ffos(t - 1) + N ~ ( t - 1 ) ~ s ( t - 1).~0~(/- 1) + A T ( t -
1)ff0~(/- 1)
+ N s ( t - 1 ) ~ ' ( t - 1)Pos(t - 1) + . N ~ ( t - 1 ) w ' ( t - 1)Po~(t - 1) +Ns(t - 1 ) ~ ' ( t - 1 ) ~ o s ( t - 1) + , ~ ( t - 1 ) w ' ( t - 1 ) B o s ( t - 1).
(2.3)
A.C. Georgiou / Partialmaintainability and control in NHMS
243
The last four terms in (2.3) vanish due to the structure of the component vectors. So, (2.3) results in:
Ns(t ) = N s ( t -
1)Cs(t - 1) + )V~(t - 1 ) C ~ ( t - 1) + N s ( t -
+/Vs(t- 1)~'(t-
1 ) P o s ( t - 1) + A T ( t -
a ) w ' ( t - 1 ) P o s ( t - 1)
1 ) P o ~ ( t - 1),
N~(t) = N ~ ( t - 1 ) C s ( t - 1) + / V ~ ( t - 1 ) ~ ' s ( t - 1) + ~ / s ( t - 1 ) ~ ( t +N~(t- 1)~'(t-
1),~0s(t- 1) + A T ( t -
(2.4) 1)ffo~(t- 1)
1)ff0~(t- 1).
(2.5)
The recursive relation (2.4) provides the partial vector N~(t) in terms of the known partial parameters and the control partial vector po~(t - 1), i.e. the recruitment into the subset. Each term of (2.4) could be naturaly interpreted and all together sum up every possible outcome of the personnel flow into the subset's grades, Ns(t). The term N s ( t - 1)C~(t- 1) represents the transitions between the subset's grades, N s ( t - 1)C~(t- 1) corresponds to the transitions from the complement g into the grades of s, N~(t - 1)w'(t - 1)P0~(t - 1) is the expected recruitment stream into the subset due to wastage from s, while Ns(t - 1)~'(t - 1)p0~(t - 1) represents recruitment flow due to wastage from the complement g. Finally, A T ( t - 1)Pos(t - 1) is the recruitment due to expansion of the system (and therefore of the subset). Relation (2.4) could be easily rewritten in terms of the expected relative structures, i.e.:
qs(t) = [ q ~ ( t - 1) + O ~ ( t - 1)] × [ a ( t - 1 ) C s ( t - 1) + { b ( t -
1)e'+ a(t-
1 ) w ' ( t - 1 ) } p 0 s ( t - 1)]
(2.6)
where a(t - 1) = T(t - 1 ) / T ( t ) and b(t - 1) = IT(t) - T(t - 1)]/T(t), and
q s ( t ) e s = q ~ ( t ) e ' 4 1 and [q~(t) + ~ s ( t ) ] e ' = q ( t ) e ' =
1.
(2.7)
The construction of model (2.6) provides a means of finding and controlling the substructure q~(t - 1) in terms of the previous ones. It is not applicable if we just want to simulate the personnel movement, because this could be done better using the model provided by Vassiliou and Tsantas (1984a). Instead, the logic of (2.6) is the construction of a difference equation which uses the control parameter pos(t - 1) combined with the subset s. So, given a subset of grades, s c_ S, the question that arises is which is the acceptable partial recruitment policy for the subset in order to maintain its structure. The answer must be independent from the recruitment policy into the complement g in order to be meaningful.
3. Partial maintainability
First we provide a definition: Definition 1. Consider a nonhomogeneous Markov system. A relative structure q(t) is called partially maintainable in [t, t + 1), if for a given subset s of the grades space S, there exists an acceptable partial recruitment policy pos(t) so that the partial relative structure qs(t) is maintained, i.e. q~(t) = qs(t + 1) = q~.
We now provide, in the form of a theorem, the two necessary and sufficient conditions for a relative structure to be partially maintainable. Theorem 1. Let a N H M S be defined by the sequences {P(t)}t~=o, {w(t)~t~=o, {T(t)~,~ o and an arbitrarily chosen sequence {po~(t)}t~=o. A structure q(t) of this N H M S is partially maintainable subject to a chosen subset s c S of its grades (i.e. q~ =qs(t - 1) = q~(t)), in [t - 1, t), if and only if:
(i)
q~[l-a(t-
(ii)
~s(t- 1)[l-a(t-
1 ) C s ( t - 1)] > ~ ( t -
1 ) a ( t - 1 ) C ~ ( t - 1),
1)~'~(t- 1)]e' >~q~a(t- 1 ) C s ( t - 1)e'.
(3.1a) (3.1b)
A.C. Georgiou / Partial maintainability and control in NHMS
244
Proof. Let q~ = q~(t- 1)= q~(t). Then,
q~= [qs + ~ ( t -
1 ) ] [ a ( t - 1)Cs(t- 1) + { b ( t - 1)e' + a ( t - 1 ) w ' ( t - 1)}Pos(t- 1)],
which gives:
qs[l-a(t-
1 ) a ( t - 1)Cs(t- 1)
1)C~(t- 1)] - ~ ( t -
= [qs + ~ s ( t - 1)] [ b ( t - 1 ) e ' + a ( t - 1 ) w ' ( t - 1)]Pos(t- 1).
(3.2)
So, the partial recruitment policy that maintains qs(t - 1) (independently of po~(t - 1)) is given by: P ° ~ ( t - 1) =
q s [ l - a ( t - 1 ) C s ( t - 1)] - ~ ( t 1 ) a ( t - 1 ) C s ( t - 1) [q+~(t-1)][b(t-1)e'+a(t-1)w'(t-1)]
(3.3)
The partial recruitment policy defined by (3.3) is acceptable if: (i)
P o s ( t - 1) >/0,
(ii)
Pos(t - 1)e' <~1.
After finding po~(t - 1), the remaining portion 1 - p o ~ ( t - 1)e' =/~0s(t - 1)e' can be allocated in any random way in the coordinates of ffo~(t - 1), which correspond to g (since its coordinates corresponding to s are equal to zero by definition). Since Pos(t - 1) is a part of the stochastic vector po(t - 1), the first condition expresses this necessity in mathematical terms. The second one arises from the fact that
Pos(t- 1) +ffos(t- 1) = p 0 ( t - 1)
and
p o ( t - 1 ) e ' = 1 (stochastic vector). In order to have po~(t - 1) >/0, the numerator of (3.3) must be nonnegative (the denominator is always a positive scalar if the sequence T(t) is increasing or constant). So,
qs[l-a(t-
1)Cs(t- 1)] - ~ s ( t -
1 ) a ( / - 1)Cs(t- 1) >t O,
qs[l-a(t-
1)Cs(t- 1)] > ~ s ( t - 1 ) a ( t - 1)Cs(t- 1).
or
So, condition (3.1a) is necessary for the partial maintainability of q(t - 1). We want po~(t - 1)e' ~< 1 as well. Let A denote the numerator of (3.3) and B the denominator. Then:
A .e'=qs[l-a(t-
1)Cs(t- 1)]e'-~(t-
=qse'+~s(t- 1)e'- ~(t-
1 ) a ( t - 1 ) C s ( t - 1)e'
1 ) e ' - [q~ + ~ ( t -
1 ) ] a ( t - l ) C ~ ( t - 1)e'
-[q~+~s(t-1)]a(t-1)~(t-1)e'+[q~+~s(t-1)]a(t-1)C~(t-1)e',
(3.4)
and since
[qs +qs( t - 1 ) ] e ' - [qs +qs( t - 1 ) ] a ( t - 1 ) C s ( t - 1)e'
-[q~ + ~ ( t - 1)]a(/- 1)C~(/- 1)e' = [q~ + ~ s ( t - 1 ) ] [ l - a ( t -
1 ) P ( t - 1)]e',
we get that
A . e'= [qs + qs( t - 1)] [I - a(t - 1 ) P ( / - 1)] e ' - ~ ( t - 1)e' + [qs + ~ ( t -
1 ) ] a ( t - 1)C~(t- 1)e'.
(3.5)
A.C. Georgiou / Partial maintainability and control in NHMS
245
The denominator takes the following form: B = [qs + qs( t - 1)] [ b ( t - 1)e' - a ( t - 1 ) e ' - a ( t - 1 ) e ( t - 1 ) e ' ] , or
B = [q~ + ~ ( t -
1)] [ e ' - a ( t -
1)e(/-
1)e'] = [q~ + ~ s ( t - 1)] [ I - a ( t -
1 ) P ( t - 1)]e'.
(3.6) Since we want po~(t - 1)e' ~< 1, we have that A 'e' ~
1)e(/-
1)][I-a(t-
1)]e'-~(/-
1)e'
+ [q~ + ~ s ( t - 1 ) ] a ( / - 1)C~(/- 1)e'
~< [qs + ~ ( t -
1)P(t- 1)]e'
1)][I-a(t-
or
-~(t-
1 ) e ' + [qs +qs( t - 1 ) ] a ( t - 1 ) C s ( t - 1)e'~< 0
which results in q ~ a ( t - 1)~'~(t- 1)e' ~<~ ( / -
1)[l-C.s(t-
1)]e',
which is just condition (3.1b). So, condition 3.1b is also necessary for the system to be partially maintainable in s. On the other hand, assume that for an NHMS and for a subset s of its grades spaces S, conditions (3.1a) and (3.1b) hold. We are going to prove that they are sufficient to partially maintain the structure involved. First, from (3.1b) we have that [q~ + ~ s ( t - 1)] [ I - a ( t -
1 ) P ( t - 1 ) ] e ' - ~ s ( t - 1) + [q~ + ~ ( t -
<~ [qs + ~s(t - 1)] [I - a ( t -
1 ) ] a ( / - 1)~'~(/- 1)e'
1)P(t - 1)]e'.
Consequently:
[qs + #~(t- 1)] [I- a(t- 1)P(t- 1)]e'-~(t- 1) + [q~ + ~s(t- 1)]a(t- 1)Cs(/- 1)e' [qs +qs( t - 1)] [ I - a ( t -
1 ) P ( t - 1)]e'
~<1, { q s + ~ s ( t - 1) - [ q s + ~ s ( t - l ) ] a ( t -
l ) C ~ ( t - 1) - [ q s + ~ s ( t - 1 ) ] a ( t - a ) c ~ ( t - 1) - ~ s ( t - 1) + [q~ + ~ s ( t -
X {[q~ + ~ ( t -
1)] [ b ( t -
1)e'+ a(t-
1 ) ] a ( t - 1)C~(/- a)e'}
1 ) w ' ( t - 1)]}-1~< 1,
or
1 ) C s ( t - 1)] - ~ s ( t - 1 ) a ( t - 1 ) C s ( t - 1)]e' ~< 1. [qs + ~ ( t 1)] [ b ( t - 1 ) e ' + a ( t - 1 ) w ' ( t - 1)]
[qs[l-a(t-
(3.7)
The fraction (3.7) is the partial recruitment vector found in (3.3) which, as we have proved, maintains the structure qs(t - 1), post multiplied by e'. So, if (3.1b) holds, po~(t - 1)e' < 1 is true as well. Also, if (3.1a) holds, the vector po~(t - 1) is nonnegative since (3.1a) produces its numerator. Apparently, the last two conclusions showed that conditions (3.1) are sufficient for the partial maintainability of the structure of an NHMS and that they provide the desired acceptable partial recruitment policy (which is independent of the recruitment policy applied on the complement g). That completes the proof. []
A.C. Georgiou / Partial maintainability and control in NHMS
246
Remark. Conditions (3.1) could be interpreted as follows: the first one indicates that the expected relative number in each grade of qs must be greater than or equal to the total number of transitions in the grade at step [t - 1, t). The second one indicates that the aggregate internal movement into g must not in any case exceed the expected relative population that g's grades hold in the relative structure.
4. Special cases
a) The first special case considers a subset s equal to the state space S. Then we arrive at the strict maintainability case in NHMS of Vassiliou and Tsantas (1984a), strict in the sense of maintaining the membership of all grades in the system. In fact, let us examine what becomes of conditions (3.1) if s = S. The first one will be
q[l-a(t-1)P(t-1)]
>10 since
g=~i, q =q and ~ ( t - 1 ) = 0 ,
(4.1)
that is, the necessary and sufficient condition of maintainability of Vassiliou and Tsantas (1984a). Furthermore, (3.1b) is meaningless since it appears to be an identity (0 = 0), because C~(t - 1) will be a zero matrix, as well as the vector ~ ( t - 1). Instead of (3.1b) (which is involved with the necessity of pos(t - 1)e' ~< 1), Vassiliou and Tsantas proved that in strict maintainability in NHMS the condition pos(t - 1)e' = 1 is always true when q(t - 1) is maintained in [ t - 1, t). In partial maintainability this is not always true and (3.1b) is necessary and sufficient for Pos(t - 1)e' ~< 1 to hold. b) Consider a system consisting of k grades. In this case, partial maintainability of a structure with k - 1 grades implies strict maintainability. For a homogeneous Markov system with three grades we arrive at the partial re-attainable structures of Vajda (1978a). Vajda preserved the total membership of a set of two grades, a case which is identical to the partial maintainability of a structure with one grade. If s consists of any (k - 1)-tuple of states, then q~(t - 1)= q~ = q~(t) results in ~s(t - 1)= ~ = ~(t), since the last coordinate left for ~ represents the membership of the remaining state. So (3.1a) gives
qs[l-a(t-
1 ) C ~ ( / - 1)] > ~ a ( t -
1 ) C ~ ( t - 1),
(4.2a)
but also
~ s [ I - a ( t - 1 ) C ~ ( t - 1)] >~q~a(t- 1)tff~(t- 1)
(4.2b)
By addition we get
q~+~-q~a(t-1)[Cs(t-1)+C~(t-1)]-~sa(t-1)[Cs(t-1)+C~(t-1)]
>~0,
(4.3)
or
q - q s a ( t - 1 ) P ( t - 1) - ~ s a ( t - 1 ) P ( t -
1) >/0,
(4.4)
or
q[l-a(t-
1)P(t-
1)] >/0,
(4.5)
which is the condition of strict maintainability in NHMS (Vassiliou and Tsantas, 1984a). Condition (3.1b) becomes
q s a ( t - 1 ) C , ( t - 1 ) e ' < 0 , [ I - a ( t - 1 ) C s ( t - 1)]e', and also
~ s a ( t - 1)Cs(t- 1)e'~< q s [ l - a ( t -
1 ) C ~ ( t - 1)]e'.
A.C. Georgiou / Partial maintainability and control in NHMS
247
After some manipulation we get a)e(t-
q[l-a(t-
1)]e'>~ 0,
a straightforward result from (4.5).
5. The region of the partially maintainable structures It is obvious that the conditions proved in Theorem 1 provide a descriptive means of the region of the partially maintainable structures. That is, the set of all structures q ( t - 1) (for a given time t) for which the partial vector qs(t - 1) could be maintained in [t - 1, t). This region could be defined by linear constraints and its vertices are the feasible solutions. Let an NHMS be given with S = { 1 , 2 , 3 . . . . . k} its state space. If s is a subset of S, s = {s~, s 2, s 3 . . . . . sin} , m <~k, and s i ~ S, we denote the region of all partially maintainable structures in [t - 1, t) by Bsq The description of B~ is twofold. We might search for all q ( t - 1)'s independently from the expected relative population in the subset, or, we may have qse' = d,, 0 < ds ~< 1, with d s a given constant. In the last case we denote B~ with B~. The conditions of the linear problem - without objective function - for B t follow: Problem (5.1): q ~ - [qs + ~ ( t [~s(t-
1)]a(t-
1) - [qs + ~ s ( t -
[qs+~s(t-1)]e'= qs >~O,
1 ) C ~ ( t - 1) >I0, 1)]a(t-
1 ) C s ( t - 1)]e' >~0,
(5.1a) (5.1b) (5.1c)
l,
~ s ( t - 1) >/0.
(5.1d)
We could use the Simplex algorithm (see for example Chvfital, 1980) to find the vertices of the feasible region of (5.1), i.e. B~'. Unknowns are the coordinates of qs and ~s(t - 1). The conditions of the linear problem - without objective function - for Bias are as shown in the following Problem (5.2): q,-
[qs+~s(t - 1)]a(t-
1 ) C s ( t - 1) >/0,
(5.2a)
1 - d s - [[qs +qs( t - l ) ] a ( t - l ) C , ( t - 1)]e'>~ 0,
(5.2b)
qs e' = d , ,
(5.2c)
qs >>"O,
~ s ( t - 1) >/0.
(5.20)
Again we find the vertices of the feasible region of (5.2) which is B,~, and the unknowns are the coordinates of q, and ~i(t - 1). A third case could be considered as a special case of (5.2), assuming that ~,(t - 1) is given. Then we may again find the region of all partially maintainable structures. Since ~ , ( t - 1) is constant (known), d, = 1 - ~ , ( t - 1)e'. Then, the linear constraints are the same as (5.2) with one basic alteration. The coordinates of ~,(t - 1) are given and the unknowns are only the coordinates of qs. Clearly, the convex hull of the maintainable structures defined by Vassiliou and Tsantas (1984a) will be a subregion of the convex hull B,t defined by (5.1). This result is illustrated in the next section.
6. Illustrations a) The first example comes from a system consisting of three grades, because we would like to provide a graphic representation of the results. Such a system is in a way trivial, because partial maintainability
248
A.C. Georgiou / Partialmaintainability and control in NHMS
can be applied only on subsets consisting of one grade (or two, which is an identical case), as we mentioned in the special cases. Consider the following transition matrix: [0.17188 P ( 0 ) = |0.00000 [ 0.00000
0.81875 0.29873 0.00000
0.00000] 0.68644 and 0.32303
The linear problems (5.1) for B s=(l), I i For Bs={l }.
T(0) --8970,
T(1) = 9000.
i 1 Bs=(2) and Bs:(3}, respectively,
are:
- 0.8160q~ + 0.0182q 2 + 0.6808q3 >~ 0, ql +q2 +q3 = 1, qi/> 0, i = 1, 2, 3, where qs = (ql, 0, 0) and ~s(0) = (0, q2, 1 }. For Bs={2
q3)-
- 0 . 8 1 6 0 q l + 0.7023q 2 >/0, -0.8288qj - 0.6842q 2 + 0.67089q3 >~ 0, q~ + q2 + q3 = 1, qi>/0,
i=1,2,3
where
qs=(O, q 2 , 0 ) and ~ s ( O ) = ( q t , O , q2).
1
For B s = (3}- 0 . 6 8 4 1 q 2 + 0.6808q 3 >~ 0, - 0.0128qt - 0.7023q 2 >t 0, ql+qz+q3= qg>/0,
1,
i=1,2,3
where
q s = ( 0 , 0 , q3) and ~ ( O ) = ( q ~ , q 2 , 0 ) .
The vertices are found to be as follows:
B~={l}: z,, = (0, 1, 0),
z,2 = (0, 0, 1),
z,3 = (0.0218, 0.9782, 0.0000),
zl4 = (0.4548, 0.0000, 0.5452). Bsl={2}: z21 = (0.4625 0.5375, 0),
z22 = (0, 0, 1),
z23 = (0.0000, 0.4988, 0.5012),
zz4 = (0.4522, 0.5478, 0.0000).
B~ = {3}: z31 = (0, 0, 1),
z32 = (1, 0, 0),
Z33 = (0.4548, 0.0000, 0.5452).
Using Theorem 2.1 from Vassiliou and Tsantas (1984a) we find the vertices of the strict maintainability region: Bl: z 1 = (0.2999, 0.3485, 0.3516),
z 2 = (0.0000, 0.4988, 0.5012),
z3 = (0.0000, 0.0000, 1.0000). ~ ~ } and B 1. As expected, B~ is denoted clearly In Figure 1 we can see all four regions Bs=~l }, Bs={2}, B={3 as the intersection of the three regions of partial maintainability in [0, 1). In Table 1 we provide some structures and test the conditions (3.1) of Theorem 1, for each region B~. If the conditions hold, the partial recruitment policy is found, and according to Theorem 1, partial maintainability of the given qs(0) is certain and independent from the distribution of the recruitment in
~, (~0s(0)).
A.C. Georgiou / Partial maintainability and control in N H M S
249
Z12"Z22"Zsl
Z•/
BB~~z23-
X , ( O ) / * / ' ............ ~ ' " ~ x ,
B1 /B.'.,o, Z
/. >" . .
¢
a20,o,o)
.
..... >~:: ......
: :
.
Fig. 1. Regions
.
~ .
xX,(O)
B'\ " ~( \ (o,to) Z . ,.",
.
Z2~ Z24
1 B,=(l l,
zBs
Z18Xa(O)
l Bs=(2 P
B~=13} and B 1
T h e s t r u c t u r e s of T a b l e 1 a r e c l e a r l y m a r k e d w i t h stars. O b v i o u s l y , x~(0) lies o n Bt={3}, w h i l e x2(0) a n d x3(0) b e l o n g to r e g i o n B,=0}. b) T h e s e c o n d e x a m p l e c o n s i d e r s a s y s t e m w i t h f o u r states. L e t s = {1, 3}, i.e. g r a d e s 1 a n d 3 a r e s u b j e c t to c o n t r o l . T h e t r a n s i t i o n m a t r i x P ( 0 ) is:
10
6
P(0) = / ~
55
55
kN
N
~ I' a n d T ( 0 ) =
1900,
T(1)=
2000.
' WI2 4
~
~j
T h e l i n e a r p r o b l e m follows:
Bs1= { 1,3}" 0.7ql - 0.1q2 - 0.2q3 - 0.3q4 >/0, -0.1ql
- 0.3q2 + 0.6q3 - 0.2q4 >~ 0,
-0.5q]
+ 0 . 4 q 2 - 0 . 3 q 3 + 0.7q 4 >/0,
ql+q2+q3 qi>/O,
= 1, i=1,2,3,4
Table 1 Population structures and corresponding partial input policy when the partial maintainability conditions hold Structure
Ci
Cii
Pos(O)
Yes Yes Yes
No Yes Yes
(0.5015, 0.0000, 0.0000) (0.0000, 0.0000, 0.0000)
No Yes Yes
Yes No No
Yes No No
Yes Yes Yes
s = {1}:
x1(0) = (0.5000, 0.0000, 0.5000) x2(0) = (0.0000, 0.5000, 0.5000) x3(0) = (0.0109, 0.9891, 0.0000) s = {2}:
x1(0) = (0.5000, 0.0000, 0.5000) x2(O) = (0.0000, 0.5000, 0.5000) x3(O) = (0.0109, 0.9891, 0.0000) s = {31:
xl(O) = (0.5000, 0.0000, 0.5000) x2(O) = (0.0000, 0.5000, 0.5000)
x3(O) = (0.0109, 0.9891, 0.0000)
(0.0000, 0.0000, 0.9817)
A.C. Georgiou / Partial maintainability and control in NHMS
250
Table 2 Population structures and corresponding partial input policy when the partial maintainability conditions hold s = {1, 3}Structure
Ci
Cii
xl(0) = (0.2000 x2(0) = (0.2242 x3(0) = (0.5000 x4(0) = (0.3100 xs(0) = (0.0000 x6(0) = (1.0000 x7(0) = (0.0000 xs(0) = (0.3000 Xg(0) = (0.3000 xl0(0) = (0.3000 Xll(0) = (0.2000 xl 2(0) = (0.2000, xl3(0) = (0.2000, x14(0) = (0.2000, x15(0) = (0.2000,
N Y Y Y N N N Y Y N Y Y Y Y Y
Y Y N Y Y Y Y Y Y Y Y Y Y Y Y
0.2000, 0.2000, 0.4000) 0.4920, 0.2838, 0.0000) 0.0000, 0.5000, 0.0000) 0.0000, 0.2600, 0.4300) 0.5000, 0.0000, 0.5000) 0.0000, 0.0000, 0.0000) 0.0000, 0.0000, 1.0000) 0.0000, 0.2500, 0.4500) 0.2800, 0.2500, 0.1700) 0.3220, 0.2500, 0.1280) 0.4750, 0.3250, 0.0000) 0.3663, 0.3250, 0.1087) 0.3499, 0.3250, 0.1251) 0.3475, 0.3250, 0.1275) 0.3471, 0.3250, 0.1279)
Po,(O) (0.6761, 0.0000, 0.0034, 0.0000) (0.2517, 0.0000, 0.2727, 0.0000)
(0.1724, 0.0000, 0.2069, 0.0000) (0.7864, 0.0000, 0.0194, 0.0000) (0.3607, (0.0622, (0.0261, (0.0210, (0.0201,
0.0000, 0.4662, 0.0000) 0.0000, 0.4686, 0.0000) 0.0000, 0.4737, 0.0000) 0.0000, 0.4744, 0.0000) 0.0000, 0.4746, 0.0000)
where qs=(q,,O,
q3,0)
and
~s(0) = ( 0 ,
qz,0,
q4).
The vertices o f BI={1,3I are: z I = (0.1626, 0.5400, 0.2974, 0.0000),
z a = (0.2789, 0.0000, 0.2155, 0.5056),
z 3 = (0.2550, 0.4680, 0.2770, 0.0000),
z4
z s = (0.1740, 0.4460, 0.3800, 0.0000),
z 6 = (0.2617, 0.0000, 0.3851, 0.3532).
=
(0.4182, 0.0000, 0.1981, 0.3837),
z~ and z2 maintain grades 1 and 3 using
pos(O) = (o, o, o, o),
z3 and z 4 maintain the subset using
Pos(O) =
z 5 and z 6 maintain the subset using
Vo (O) = (o, o, 1, o).
(1, 0, 0, 0),
In Table 2 we provide a collection of structures and test the conditions of Theorem 1. Again, when both hold, the partial recruitment policy necessary for maintaining the grades of subset {1,3} is given. Note that structure xts(0) belongs to the strictly maintainable region as well, with recruitment policy:
P0(0) =J'0s(0) =P0s(0) = [0.0201, 0.0000, 0.4746, 0.0000] + [0.0000, 0.5053, 0.0000, 0.0000] = [0.0201, 0.5053, 0.4746, 0.0000]
Acknowledgement The author wishes to thank Prof. P.-C.G. Vassiliou for his suggestions on the subject
References Bartholomew, D.J. (1973), Stochastic Models for Social Processes, 2nd ed., Wiley, Chichester, UK. Bartholomew, D.J. (1975), "A stochastic control problem in the social sciences", Bulletin International Statistical Institute 46, 670-680.
A.C Georgiou / Partial maintainability and control in NHMS
251
Chv~ital, V. (1983), Linear Programming, Freeman, New York. Vajda, S. (1975), "Mathematical aspects of manpower planning", Operations Research Quarterly 26, 527-542. Vajda, S. (1978a), Mathematics of Manpower Planning, Wiley, Chichester, UK. Vajda, S. (1978b), "Maintainability and preservation of graded population structure", Studies in the Management Sciences 8, 219-230. Vassiliou, P.-C.G. (1982), "Asymptotic behavior of Markov systems", Journal of Applied Probability 19, 815-857. Vassiliou, P.-C.G., and Tsantas, N. (1984a), "Stochastic control in nonhomogeneous Markov systems", International Journal of Computer Mathematics 16, 139-155. Vassiliou, P.-C.G., and Tsantas, N. (1984b), "Maintainability of structures in nonhomogeneous Markov Systems under cyclic behavior and input control", SIAM Journal on Applied Mathematics 44, 1014-1022.
241
Theory and Methodology
Partial maintainability and control in nonhomogeneous Markov manpower systems A.C. Georgiou Statistics and Operations Research Section, Department of Mathematics, University of Thessaloniki, 54006- ThessalonikL Greece Received August 1990; revised November 1990
Abstract: Concepts of recruitment control in nonhomogeneous Markov manpower systems are studied, in
order to determine subsets of structures in a graded population, which can be maintained in one step, using partial recruitment information necessary for each subset. Two necessary and sufficient conditions are derived, and the Simplex algorithm is used to describe the regions of the so-called partially maintainable structures. The results are illustrated by numerical examples and the consideration of some interesting special cases. Keywords: Personnel; planning; Markov processes; control theory
I. Introduction
Consider a population stratified into k classes (grades, states) according to some characteristic, and assume that individual transitions take place on discrete time intervals. The expected number of people in each grade changes over time through promotion rates, wastage and recruitment stream according to a discrete nonhomogeneous Markov chain with transition probability matrices P(t), t = 0, 1, 2 . . . . Let S = { 1, 2 . . . . . k} be the state space. Each member of the system may be in one and only one state at any given time. The population structure of the system at time t is denoted by the row vector N(t) = [Nl(t) . . . . . N~(t)] where N,(t) is the expected number of members in the i-th grade. The expected total number of members of the system at time t is denoted by T(t) whereas the vector q(t)= [ q l ( t ) , . . . , q g ( t ) ] = N(t)/T(t) is the relative population structure. A system defined in this way is called a nonhomogeneous Markov system, NHMS, (Vassiliou, 1982). Our interest is in controlling a subset of the population structure, q(t), using the recruitment flows connected with a given subset. The origins of the maintainability problem are in Bartholomew (1973,1975). Our problem here arises from Vajda (1975,1978a,1978b) and combines results from Vassiliou (1982) and Vassiliou and Tsantas (1984a,1984b). Vajda was concerned with homogeneous Markov systems and introduced the concept of partially re-attainable, partially maintainable and semi-stationary structures. Vassiliou and Tsantas (1984a) introduced the stochastic control in NHMS. 0377-2217/92/$05.00 © 1992 - Elsevier Science Publishers B.V. All rights reserved
A.C. Georgiou / Partialmainta&abilityand controlin NHMS
242
The goal of this paper is to use the stochastic control of NHMS in order to extend the concept of partial maintainability, and prove a basic theorem which characterizes the region of these structures (connected with a desired subset). It is clear that we are interested in subsets of states in hierarchically graded NHMS. A chosen group of grades from a system must maintain its structure and the tool is the partial control of the recruitment stream into the subset (independently from the effects of recruitment into the rest of the grades, as we shall see). The results of this work are illustrated in the final section by two examples concerning manpower systems. The first consists of three grades, while the second consists of four grades. The discussion of the illustration, mainly of the first example, is connected with the work done by Vassiliou and Tsantas (1984a).
2. Notation and formulae
We provide the following notation: : The subset of grades s ___S under consideration (and control). : The complement of s. Ns(t) : The population vector containing zeros in the grades of g and the corresponding N/(t)'s for s. s g
Ns(t) qs(t) #s(t) C~(t)
: Ns(t) + Ns(t) = N(t). : The relative structure vector constructed as N~(t).
: #~(t) + q~(t) = q(t). : A k × k matrix composed from the columns of P(t) which correspond to the grades belonging
in s. The rest of its columns are equal to zero. C.s(t) : e ( t ) = C~(t) + ~ ( t ) . Let us denote by w(t) the wastage vector and by po(t) the recruitment vector at time t. In a similar way, we define the partial wastage vector w~(t) and its complement ~s(t), the partial recruitment vector po~(t), the control parameter, and/50s(t). Also, A(t) = T(t + 1) - T(t) and e = [1, 1. . . . . 1]. Similarly are defined the vectors e s and ~s. Under the Markovian assumption the population structure at time t is given by:
N(t) = N ( t - 1 ) e ( t -
1) + N ( t -
1 ) w ' ( / - 1 ) P 0 ( t - 1) + z a ( t - 1 ) P 0 ( t - 1)
(2.1)
Substituting the vectors in (2.1) by their equivalent sums of partial vectors, we arrive at (2.2) in which the structure N~t) is expressed in terms of known parameters as well as the control partial vector pos(t). So, from (2.1) we obtain
N~(t) +/Vs(/) = [N~(t- 1) + N s ( t - 1)][Cs(t- 1) + C ' s ( t - 1)] + [N~(t- 1) +his(t- 1 ) ] [ w ' ( / - 1) + ~ ' ( t -
1)]
×[Pos(t-1)+ffo~(t-1)]+AT(t-1)[Pos(t-1)+~o,(t-1)].
(2.2)
By reordering we get that
N~(t) +Ns(t) = N ~ ( t - 1 ) C ~ ( t - 1) + , ~ ( t -
1 ) C ~ ( t - 1)
+ N , ( t - 1 ) w ' ( / - 1)Pos(t- 1) + N s ( t - 1 ) ~ ' ( / - 1)Pos(t- 1) + A T ( t - 1 ) P 0 , ( t - 1) + N s ( t - 1 ) C ' s ( t - 1) + N s ( t - 1 ) C s ( t - 1) + N s ( t - 1 ) w g ( t - 1)ffos(t - 1) + N ~ ( t - 1 ) ~ s ( t - 1).~0~(/- 1) + A T ( t -
1)ff0~(/- 1)
+ N s ( t - 1 ) ~ ' ( t - 1)Pos(t - 1) + . N ~ ( t - 1 ) w ' ( t - 1)Po~(t - 1) +Ns(t - 1 ) ~ ' ( t - 1 ) ~ o s ( t - 1) + , ~ ( t - 1 ) w ' ( t - 1 ) B o s ( t - 1).
(2.3)
A.C. Georgiou / Partialmaintainability and control in NHMS
243
The last four terms in (2.3) vanish due to the structure of the component vectors. So, (2.3) results in:
Ns(t ) = N s ( t -
1)Cs(t - 1) + )V~(t - 1 ) C ~ ( t - 1) + N s ( t -
+/Vs(t- 1)~'(t-
1 ) P o s ( t - 1) + A T ( t -
a ) w ' ( t - 1 ) P o s ( t - 1)
1 ) P o ~ ( t - 1),
N~(t) = N ~ ( t - 1 ) C s ( t - 1) + / V ~ ( t - 1 ) ~ ' s ( t - 1) + ~ / s ( t - 1 ) ~ ( t +N~(t- 1)~'(t-
1),~0s(t- 1) + A T ( t -
(2.4) 1)ffo~(t- 1)
1)ff0~(t- 1).
(2.5)
The recursive relation (2.4) provides the partial vector N~(t) in terms of the known partial parameters and the control partial vector po~(t - 1), i.e. the recruitment into the subset. Each term of (2.4) could be naturaly interpreted and all together sum up every possible outcome of the personnel flow into the subset's grades, Ns(t). The term N s ( t - 1)C~(t- 1) represents the transitions between the subset's grades, N s ( t - 1)C~(t- 1) corresponds to the transitions from the complement g into the grades of s, N~(t - 1)w'(t - 1)P0~(t - 1) is the expected recruitment stream into the subset due to wastage from s, while Ns(t - 1)~'(t - 1)p0~(t - 1) represents recruitment flow due to wastage from the complement g. Finally, A T ( t - 1)Pos(t - 1) is the recruitment due to expansion of the system (and therefore of the subset). Relation (2.4) could be easily rewritten in terms of the expected relative structures, i.e.:
qs(t) = [ q ~ ( t - 1) + O ~ ( t - 1)] × [ a ( t - 1 ) C s ( t - 1) + { b ( t -
1)e'+ a(t-
1 ) w ' ( t - 1 ) } p 0 s ( t - 1)]
(2.6)
where a(t - 1) = T(t - 1 ) / T ( t ) and b(t - 1) = IT(t) - T(t - 1)]/T(t), and
q s ( t ) e s = q ~ ( t ) e ' 4 1 and [q~(t) + ~ s ( t ) ] e ' = q ( t ) e ' =
1.
(2.7)
The construction of model (2.6) provides a means of finding and controlling the substructure q~(t - 1) in terms of the previous ones. It is not applicable if we just want to simulate the personnel movement, because this could be done better using the model provided by Vassiliou and Tsantas (1984a). Instead, the logic of (2.6) is the construction of a difference equation which uses the control parameter pos(t - 1) combined with the subset s. So, given a subset of grades, s c_ S, the question that arises is which is the acceptable partial recruitment policy for the subset in order to maintain its structure. The answer must be independent from the recruitment policy into the complement g in order to be meaningful.
3. Partial maintainability
First we provide a definition: Definition 1. Consider a nonhomogeneous Markov system. A relative structure q(t) is called partially maintainable in [t, t + 1), if for a given subset s of the grades space S, there exists an acceptable partial recruitment policy pos(t) so that the partial relative structure qs(t) is maintained, i.e. q~(t) = qs(t + 1) = q~.
We now provide, in the form of a theorem, the two necessary and sufficient conditions for a relative structure to be partially maintainable. Theorem 1. Let a N H M S be defined by the sequences {P(t)}t~=o, {w(t)~t~=o, {T(t)~,~ o and an arbitrarily chosen sequence {po~(t)}t~=o. A structure q(t) of this N H M S is partially maintainable subject to a chosen subset s c S of its grades (i.e. q~ =qs(t - 1) = q~(t)), in [t - 1, t), if and only if:
(i)
q~[l-a(t-
(ii)
~s(t- 1)[l-a(t-
1 ) C s ( t - 1)] > ~ ( t -
1 ) a ( t - 1 ) C ~ ( t - 1),
1)~'~(t- 1)]e' >~q~a(t- 1 ) C s ( t - 1)e'.
(3.1a) (3.1b)
A.C. Georgiou / Partial maintainability and control in NHMS
244
Proof. Let q~ = q~(t- 1)= q~(t). Then,
q~= [qs + ~ ( t -
1 ) ] [ a ( t - 1)Cs(t- 1) + { b ( t - 1)e' + a ( t - 1 ) w ' ( t - 1)}Pos(t- 1)],
which gives:
qs[l-a(t-
1 ) a ( t - 1)Cs(t- 1)
1)C~(t- 1)] - ~ ( t -
= [qs + ~ s ( t - 1)] [ b ( t - 1 ) e ' + a ( t - 1 ) w ' ( t - 1)]Pos(t- 1).
(3.2)
So, the partial recruitment policy that maintains qs(t - 1) (independently of po~(t - 1)) is given by: P ° ~ ( t - 1) =
q s [ l - a ( t - 1 ) C s ( t - 1)] - ~ ( t 1 ) a ( t - 1 ) C s ( t - 1) [q+~(t-1)][b(t-1)e'+a(t-1)w'(t-1)]
(3.3)
The partial recruitment policy defined by (3.3) is acceptable if: (i)
P o s ( t - 1) >/0,
(ii)
Pos(t - 1)e' <~1.
After finding po~(t - 1), the remaining portion 1 - p o ~ ( t - 1)e' =/~0s(t - 1)e' can be allocated in any random way in the coordinates of ffo~(t - 1), which correspond to g (since its coordinates corresponding to s are equal to zero by definition). Since Pos(t - 1) is a part of the stochastic vector po(t - 1), the first condition expresses this necessity in mathematical terms. The second one arises from the fact that
Pos(t- 1) +ffos(t- 1) = p 0 ( t - 1)
and
p o ( t - 1 ) e ' = 1 (stochastic vector). In order to have po~(t - 1) >/0, the numerator of (3.3) must be nonnegative (the denominator is always a positive scalar if the sequence T(t) is increasing or constant). So,
qs[l-a(t-
1)Cs(t- 1)] - ~ s ( t -
1 ) a ( / - 1)Cs(t- 1) >t O,
qs[l-a(t-
1)Cs(t- 1)] > ~ s ( t - 1 ) a ( t - 1)Cs(t- 1).
or
So, condition (3.1a) is necessary for the partial maintainability of q(t - 1). We want po~(t - 1)e' ~< 1 as well. Let A denote the numerator of (3.3) and B the denominator. Then:
A .e'=qs[l-a(t-
1)Cs(t- 1)]e'-~(t-
=qse'+~s(t- 1)e'- ~(t-
1 ) a ( t - 1 ) C s ( t - 1)e'
1 ) e ' - [q~ + ~ ( t -
1 ) ] a ( t - l ) C ~ ( t - 1)e'
-[q~+~s(t-1)]a(t-1)~(t-1)e'+[q~+~s(t-1)]a(t-1)C~(t-1)e',
(3.4)
and since
[qs +qs( t - 1 ) ] e ' - [qs +qs( t - 1 ) ] a ( t - 1 ) C s ( t - 1)e'
-[q~ + ~ ( t - 1)]a(/- 1)C~(/- 1)e' = [q~ + ~ s ( t - 1 ) ] [ l - a ( t -
1 ) P ( t - 1)]e',
we get that
A . e'= [qs + qs( t - 1)] [I - a(t - 1 ) P ( / - 1)] e ' - ~ ( t - 1)e' + [qs + ~ ( t -
1 ) ] a ( t - 1)C~(t- 1)e'.
(3.5)
A.C. Georgiou / Partial maintainability and control in NHMS
245
The denominator takes the following form: B = [qs + qs( t - 1)] [ b ( t - 1)e' - a ( t - 1 ) e ' - a ( t - 1 ) e ( t - 1 ) e ' ] , or
B = [q~ + ~ ( t -
1)] [ e ' - a ( t -
1)e(/-
1)e'] = [q~ + ~ s ( t - 1)] [ I - a ( t -
1 ) P ( t - 1)]e'.
(3.6) Since we want po~(t - 1)e' ~< 1, we have that A 'e' ~
1)e(/-
1)][I-a(t-
1)]e'-~(/-
1)e'
+ [q~ + ~ s ( t - 1 ) ] a ( / - 1)C~(/- 1)e'
~< [qs + ~ ( t -
1)P(t- 1)]e'
1)][I-a(t-
or
-~(t-
1 ) e ' + [qs +qs( t - 1 ) ] a ( t - 1 ) C s ( t - 1)e'~< 0
which results in q ~ a ( t - 1)~'~(t- 1)e' ~<~ ( / -
1)[l-C.s(t-
1)]e',
which is just condition (3.1b). So, condition 3.1b is also necessary for the system to be partially maintainable in s. On the other hand, assume that for an NHMS and for a subset s of its grades spaces S, conditions (3.1a) and (3.1b) hold. We are going to prove that they are sufficient to partially maintain the structure involved. First, from (3.1b) we have that [q~ + ~ s ( t - 1)] [ I - a ( t -
1 ) P ( t - 1 ) ] e ' - ~ s ( t - 1) + [q~ + ~ ( t -
<~ [qs + ~s(t - 1)] [I - a ( t -
1 ) ] a ( / - 1)~'~(/- 1)e'
1)P(t - 1)]e'.
Consequently:
[qs + #~(t- 1)] [I- a(t- 1)P(t- 1)]e'-~(t- 1) + [q~ + ~s(t- 1)]a(t- 1)Cs(/- 1)e' [qs +qs( t - 1)] [ I - a ( t -
1 ) P ( t - 1)]e'
~<1, { q s + ~ s ( t - 1) - [ q s + ~ s ( t - l ) ] a ( t -
l ) C ~ ( t - 1) - [ q s + ~ s ( t - 1 ) ] a ( t - a ) c ~ ( t - 1) - ~ s ( t - 1) + [q~ + ~ s ( t -
X {[q~ + ~ ( t -
1)] [ b ( t -
1)e'+ a(t-
1 ) ] a ( t - 1)C~(/- a)e'}
1 ) w ' ( t - 1)]}-1~< 1,
or
1 ) C s ( t - 1)] - ~ s ( t - 1 ) a ( t - 1 ) C s ( t - 1)]e' ~< 1. [qs + ~ ( t 1)] [ b ( t - 1 ) e ' + a ( t - 1 ) w ' ( t - 1)]
[qs[l-a(t-
(3.7)
The fraction (3.7) is the partial recruitment vector found in (3.3) which, as we have proved, maintains the structure qs(t - 1), post multiplied by e'. So, if (3.1b) holds, po~(t - 1)e' < 1 is true as well. Also, if (3.1a) holds, the vector po~(t - 1) is nonnegative since (3.1a) produces its numerator. Apparently, the last two conclusions showed that conditions (3.1) are sufficient for the partial maintainability of the structure of an NHMS and that they provide the desired acceptable partial recruitment policy (which is independent of the recruitment policy applied on the complement g). That completes the proof. []
A.C. Georgiou / Partial maintainability and control in NHMS
246
Remark. Conditions (3.1) could be interpreted as follows: the first one indicates that the expected relative number in each grade of qs must be greater than or equal to the total number of transitions in the grade at step [t - 1, t). The second one indicates that the aggregate internal movement into g must not in any case exceed the expected relative population that g's grades hold in the relative structure.
4. Special cases
a) The first special case considers a subset s equal to the state space S. Then we arrive at the strict maintainability case in NHMS of Vassiliou and Tsantas (1984a), strict in the sense of maintaining the membership of all grades in the system. In fact, let us examine what becomes of conditions (3.1) if s = S. The first one will be
q[l-a(t-1)P(t-1)]
>10 since
g=~i, q =q and ~ ( t - 1 ) = 0 ,
(4.1)
that is, the necessary and sufficient condition of maintainability of Vassiliou and Tsantas (1984a). Furthermore, (3.1b) is meaningless since it appears to be an identity (0 = 0), because C~(t - 1) will be a zero matrix, as well as the vector ~ ( t - 1). Instead of (3.1b) (which is involved with the necessity of pos(t - 1)e' ~< 1), Vassiliou and Tsantas proved that in strict maintainability in NHMS the condition pos(t - 1)e' = 1 is always true when q(t - 1) is maintained in [ t - 1, t). In partial maintainability this is not always true and (3.1b) is necessary and sufficient for Pos(t - 1)e' ~< 1 to hold. b) Consider a system consisting of k grades. In this case, partial maintainability of a structure with k - 1 grades implies strict maintainability. For a homogeneous Markov system with three grades we arrive at the partial re-attainable structures of Vajda (1978a). Vajda preserved the total membership of a set of two grades, a case which is identical to the partial maintainability of a structure with one grade. If s consists of any (k - 1)-tuple of states, then q~(t - 1)= q~ = q~(t) results in ~s(t - 1)= ~ = ~(t), since the last coordinate left for ~ represents the membership of the remaining state. So (3.1a) gives
qs[l-a(t-
1 ) C ~ ( / - 1)] > ~ a ( t -
1 ) C ~ ( t - 1),
(4.2a)
but also
~ s [ I - a ( t - 1 ) C ~ ( t - 1)] >~q~a(t- 1)tff~(t- 1)
(4.2b)
By addition we get
q~+~-q~a(t-1)[Cs(t-1)+C~(t-1)]-~sa(t-1)[Cs(t-1)+C~(t-1)]
>~0,
(4.3)
or
q - q s a ( t - 1 ) P ( t - 1) - ~ s a ( t - 1 ) P ( t -
1) >/0,
(4.4)
or
q[l-a(t-
1)P(t-
1)] >/0,
(4.5)
which is the condition of strict maintainability in NHMS (Vassiliou and Tsantas, 1984a). Condition (3.1b) becomes
q s a ( t - 1 ) C , ( t - 1 ) e ' < 0 , [ I - a ( t - 1 ) C s ( t - 1)]e', and also
~ s a ( t - 1)Cs(t- 1)e'~< q s [ l - a ( t -
1 ) C ~ ( t - 1)]e'.
A.C. Georgiou / Partial maintainability and control in NHMS
247
After some manipulation we get a)e(t-
q[l-a(t-
1)]e'>~ 0,
a straightforward result from (4.5).
5. The region of the partially maintainable structures It is obvious that the conditions proved in Theorem 1 provide a descriptive means of the region of the partially maintainable structures. That is, the set of all structures q ( t - 1) (for a given time t) for which the partial vector qs(t - 1) could be maintained in [t - 1, t). This region could be defined by linear constraints and its vertices are the feasible solutions. Let an NHMS be given with S = { 1 , 2 , 3 . . . . . k} its state space. If s is a subset of S, s = {s~, s 2, s 3 . . . . . sin} , m <~k, and s i ~ S, we denote the region of all partially maintainable structures in [t - 1, t) by Bsq The description of B~ is twofold. We might search for all q ( t - 1)'s independently from the expected relative population in the subset, or, we may have qse' = d,, 0 < ds ~< 1, with d s a given constant. In the last case we denote B~ with B~. The conditions of the linear problem - without objective function - for B t follow: Problem (5.1): q ~ - [qs + ~ ( t [~s(t-
1)]a(t-
1) - [qs + ~ s ( t -
[qs+~s(t-1)]e'= qs >~O,
1 ) C ~ ( t - 1) >I0, 1)]a(t-
1 ) C s ( t - 1)]e' >~0,
(5.1a) (5.1b) (5.1c)
l,
~ s ( t - 1) >/0.
(5.1d)
We could use the Simplex algorithm (see for example Chvfital, 1980) to find the vertices of the feasible region of (5.1), i.e. B~'. Unknowns are the coordinates of qs and ~s(t - 1). The conditions of the linear problem - without objective function - for Bias are as shown in the following Problem (5.2): q,-
[qs+~s(t - 1)]a(t-
1 ) C s ( t - 1) >/0,
(5.2a)
1 - d s - [[qs +qs( t - l ) ] a ( t - l ) C , ( t - 1)]e'>~ 0,
(5.2b)
qs e' = d , ,
(5.2c)
qs >>"O,
~ s ( t - 1) >/0.
(5.20)
Again we find the vertices of the feasible region of (5.2) which is B,~, and the unknowns are the coordinates of q, and ~i(t - 1). A third case could be considered as a special case of (5.2), assuming that ~,(t - 1) is given. Then we may again find the region of all partially maintainable structures. Since ~ , ( t - 1) is constant (known), d, = 1 - ~ , ( t - 1)e'. Then, the linear constraints are the same as (5.2) with one basic alteration. The coordinates of ~,(t - 1) are given and the unknowns are only the coordinates of qs. Clearly, the convex hull of the maintainable structures defined by Vassiliou and Tsantas (1984a) will be a subregion of the convex hull B,t defined by (5.1). This result is illustrated in the next section.
6. Illustrations a) The first example comes from a system consisting of three grades, because we would like to provide a graphic representation of the results. Such a system is in a way trivial, because partial maintainability
248
A.C. Georgiou / Partialmaintainability and control in NHMS
can be applied only on subsets consisting of one grade (or two, which is an identical case), as we mentioned in the special cases. Consider the following transition matrix: [0.17188 P ( 0 ) = |0.00000 [ 0.00000
0.81875 0.29873 0.00000
0.00000] 0.68644 and 0.32303
The linear problems (5.1) for B s=(l), I i For Bs={l }.
T(0) --8970,
T(1) = 9000.
i 1 Bs=(2) and Bs:(3}, respectively,
are:
- 0.8160q~ + 0.0182q 2 + 0.6808q3 >~ 0, ql +q2 +q3 = 1, qi/> 0, i = 1, 2, 3, where qs = (ql, 0, 0) and ~s(0) = (0, q2, 1 }. For Bs={2
q3)-
- 0 . 8 1 6 0 q l + 0.7023q 2 >/0, -0.8288qj - 0.6842q 2 + 0.67089q3 >~ 0, q~ + q2 + q3 = 1, qi>/0,
i=1,2,3
where
qs=(O, q 2 , 0 ) and ~ s ( O ) = ( q t , O , q2).
1
For B s = (3}- 0 . 6 8 4 1 q 2 + 0.6808q 3 >~ 0, - 0.0128qt - 0.7023q 2 >t 0, ql+qz+q3= qg>/0,
1,
i=1,2,3
where
q s = ( 0 , 0 , q3) and ~ ( O ) = ( q ~ , q 2 , 0 ) .
The vertices are found to be as follows:
B~={l}: z,, = (0, 1, 0),
z,2 = (0, 0, 1),
z,3 = (0.0218, 0.9782, 0.0000),
zl4 = (0.4548, 0.0000, 0.5452). Bsl={2}: z21 = (0.4625 0.5375, 0),
z22 = (0, 0, 1),
z23 = (0.0000, 0.4988, 0.5012),
zz4 = (0.4522, 0.5478, 0.0000).
B~ = {3}: z31 = (0, 0, 1),
z32 = (1, 0, 0),
Z33 = (0.4548, 0.0000, 0.5452).
Using Theorem 2.1 from Vassiliou and Tsantas (1984a) we find the vertices of the strict maintainability region: Bl: z 1 = (0.2999, 0.3485, 0.3516),
z 2 = (0.0000, 0.4988, 0.5012),
z3 = (0.0000, 0.0000, 1.0000). ~ ~ } and B 1. As expected, B~ is denoted clearly In Figure 1 we can see all four regions Bs=~l }, Bs={2}, B={3 as the intersection of the three regions of partial maintainability in [0, 1). In Table 1 we provide some structures and test the conditions (3.1) of Theorem 1, for each region B~. If the conditions hold, the partial recruitment policy is found, and according to Theorem 1, partial maintainability of the given qs(0) is certain and independent from the distribution of the recruitment in
~, (~0s(0)).
A.C. Georgiou / Partial maintainability and control in N H M S
249
Z12"Z22"Zsl
Z•/
BB~~z23-
X , ( O ) / * / ' ............ ~ ' " ~ x ,
B1 /B.'.,o, Z
/. >" . .
¢
a20,o,o)
.
..... >~:: ......
: :
.
Fig. 1. Regions
.
~ .
xX,(O)
B'\ " ~( \ (o,to) Z . ,.",
.
Z2~ Z24
1 B,=(l l,
zBs
Z18Xa(O)
l Bs=(2 P
B~=13} and B 1
T h e s t r u c t u r e s of T a b l e 1 a r e c l e a r l y m a r k e d w i t h stars. O b v i o u s l y , x~(0) lies o n Bt={3}, w h i l e x2(0) a n d x3(0) b e l o n g to r e g i o n B,=0}. b) T h e s e c o n d e x a m p l e c o n s i d e r s a s y s t e m w i t h f o u r states. L e t s = {1, 3}, i.e. g r a d e s 1 a n d 3 a r e s u b j e c t to c o n t r o l . T h e t r a n s i t i o n m a t r i x P ( 0 ) is:
10
6
P(0) = / ~
55
55
kN
N
~ I' a n d T ( 0 ) =
1900,
T(1)=
2000.
' WI2 4
~
~j
T h e l i n e a r p r o b l e m follows:
Bs1= { 1,3}" 0.7ql - 0.1q2 - 0.2q3 - 0.3q4 >/0, -0.1ql
- 0.3q2 + 0.6q3 - 0.2q4 >~ 0,
-0.5q]
+ 0 . 4 q 2 - 0 . 3 q 3 + 0.7q 4 >/0,
ql+q2+q3 qi>/O,
= 1, i=1,2,3,4
Table 1 Population structures and corresponding partial input policy when the partial maintainability conditions hold Structure
Ci
Cii
Pos(O)
Yes Yes Yes
No Yes Yes
(0.5015, 0.0000, 0.0000) (0.0000, 0.0000, 0.0000)
No Yes Yes
Yes No No
Yes No No
Yes Yes Yes
s = {1}:
x1(0) = (0.5000, 0.0000, 0.5000) x2(0) = (0.0000, 0.5000, 0.5000) x3(0) = (0.0109, 0.9891, 0.0000) s = {2}:
x1(0) = (0.5000, 0.0000, 0.5000) x2(O) = (0.0000, 0.5000, 0.5000) x3(O) = (0.0109, 0.9891, 0.0000) s = {31:
xl(O) = (0.5000, 0.0000, 0.5000) x2(O) = (0.0000, 0.5000, 0.5000)
x3(O) = (0.0109, 0.9891, 0.0000)
(0.0000, 0.0000, 0.9817)
A.C. Georgiou / Partial maintainability and control in NHMS
250
Table 2 Population structures and corresponding partial input policy when the partial maintainability conditions hold s = {1, 3}Structure
Ci
Cii
xl(0) = (0.2000 x2(0) = (0.2242 x3(0) = (0.5000 x4(0) = (0.3100 xs(0) = (0.0000 x6(0) = (1.0000 x7(0) = (0.0000 xs(0) = (0.3000 Xg(0) = (0.3000 xl0(0) = (0.3000 Xll(0) = (0.2000 xl 2(0) = (0.2000, xl3(0) = (0.2000, x14(0) = (0.2000, x15(0) = (0.2000,
N Y Y Y N N N Y Y N Y Y Y Y Y
Y Y N Y Y Y Y Y Y Y Y Y Y Y Y
0.2000, 0.2000, 0.4000) 0.4920, 0.2838, 0.0000) 0.0000, 0.5000, 0.0000) 0.0000, 0.2600, 0.4300) 0.5000, 0.0000, 0.5000) 0.0000, 0.0000, 0.0000) 0.0000, 0.0000, 1.0000) 0.0000, 0.2500, 0.4500) 0.2800, 0.2500, 0.1700) 0.3220, 0.2500, 0.1280) 0.4750, 0.3250, 0.0000) 0.3663, 0.3250, 0.1087) 0.3499, 0.3250, 0.1251) 0.3475, 0.3250, 0.1275) 0.3471, 0.3250, 0.1279)
Po,(O) (0.6761, 0.0000, 0.0034, 0.0000) (0.2517, 0.0000, 0.2727, 0.0000)
(0.1724, 0.0000, 0.2069, 0.0000) (0.7864, 0.0000, 0.0194, 0.0000) (0.3607, (0.0622, (0.0261, (0.0210, (0.0201,
0.0000, 0.4662, 0.0000) 0.0000, 0.4686, 0.0000) 0.0000, 0.4737, 0.0000) 0.0000, 0.4744, 0.0000) 0.0000, 0.4746, 0.0000)
where qs=(q,,O,
q3,0)
and
~s(0) = ( 0 ,
qz,0,
q4).
The vertices o f BI={1,3I are: z I = (0.1626, 0.5400, 0.2974, 0.0000),
z a = (0.2789, 0.0000, 0.2155, 0.5056),
z 3 = (0.2550, 0.4680, 0.2770, 0.0000),
z4
z s = (0.1740, 0.4460, 0.3800, 0.0000),
z 6 = (0.2617, 0.0000, 0.3851, 0.3532).
=
(0.4182, 0.0000, 0.1981, 0.3837),
z~ and z2 maintain grades 1 and 3 using
pos(O) = (o, o, o, o),
z3 and z 4 maintain the subset using
Pos(O) =
z 5 and z 6 maintain the subset using
Vo (O) = (o, o, 1, o).
(1, 0, 0, 0),
In Table 2 we provide a collection of structures and test the conditions of Theorem 1. Again, when both hold, the partial recruitment policy necessary for maintaining the grades of subset {1,3} is given. Note that structure xts(0) belongs to the strictly maintainable region as well, with recruitment policy:
P0(0) =J'0s(0) =P0s(0) = [0.0201, 0.0000, 0.4746, 0.0000] + [0.0000, 0.5053, 0.0000, 0.0000] = [0.0201, 0.5053, 0.4746, 0.0000]
Acknowledgement The author wishes to thank Prof. P.-C.G. Vassiliou for his suggestions on the subject
References Bartholomew, D.J. (1973), Stochastic Models for Social Processes, 2nd ed., Wiley, Chichester, UK. Bartholomew, D.J. (1975), "A stochastic control problem in the social sciences", Bulletin International Statistical Institute 46, 670-680.
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Chv~ital, V. (1983), Linear Programming, Freeman, New York. Vajda, S. (1975), "Mathematical aspects of manpower planning", Operations Research Quarterly 26, 527-542. Vajda, S. (1978a), Mathematics of Manpower Planning, Wiley, Chichester, UK. Vajda, S. (1978b), "Maintainability and preservation of graded population structure", Studies in the Management Sciences 8, 219-230. Vassiliou, P.-C.G. (1982), "Asymptotic behavior of Markov systems", Journal of Applied Probability 19, 815-857. Vassiliou, P.-C.G., and Tsantas, N. (1984a), "Stochastic control in nonhomogeneous Markov systems", International Journal of Computer Mathematics 16, 139-155. Vassiliou, P.-C.G., and Tsantas, N. (1984b), "Maintainability of structures in nonhomogeneous Markov Systems under cyclic behavior and input control", SIAM Journal on Applied Mathematics 44, 1014-1022.