Planning horizons for manpower planning : a theoretical analysis

Planning horizons for manpower planning : a theoretical

analysis

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Nuttle, L. W., & Wijngaard, J. (1981). Planning horizons for manpower planning : a theoretical analysis. OR Spektrum, 3(3), 153-160.

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OR Spektrum (1981) 3:153-160

ORSpekt m

9 Springe~-Verlag 198t

Planning Horizons for Manpower Planning

A Theoretical Analysis

L. W. Nuttle I and J. Wijngaard 2

1 North Carolina State University at Raleigh, Department of Industrial Engineering, Raleigh, NC, USA and 2 Eindhoven University of Teehnology, Department of Industrial Engineering, Eindhoven, The Netherlands Received April 11, 1980 / Accepted in revised form February 24, 1981

Summary. The purpose of manpower planning is to get a better matching between manpower requirement andman- power availability. The difficult part of manpower plan- ning is to get reliable forecasts for future manpower re- quirement. It is important, therefore, to know what infor- mation one needs ab out the future to make good decisions now. How detailed should our knowledge of future manpower requirement be and of how far in the future? The last point is directly related to the problem of the planning horizon. This problem is investigated in this paper for a hierarchical manpower system with two grades, recruitment at the bottom and a promotion policy formulated in the grade-age (number of years in grade one). There is a goal on the total content of the system and a goal on the content of the second level. These goals may be interpreted as the future require- ments. The penalties for deviations from the goals are assumed to be proportional to these deviations. The only way to control the system is by recruitment.

For the case where all employees have the same career pattern one can get rather general results since this problem is almost equivalent to the case with only one grade. For the more general case it is only possible to get planning horizon results if conditions are added on the penalty functions and the goal patterns. The general two-level case appears to be equivalent to the production-smoothing problem without inventory with a lower bound on the difference between the hiring in two subsequent periods.

Zusammenfassung. Ziel der Personalplanung ist ein besserer Ausgleich zwischen Personalbedarf und -verftig- barkeit. Zul~issige Vorhersagen fOr den zukiinftigen Be- darf zu erhalten, ist der schwierige Anteil der Personal- planung. Daher ist es also wichtig, den Informationsbe- daft tiber die Zukunft zu kennen, um in der Gegenwart gute Entscheidungen zu treffen. Wie genau sollte nun

unsere Kenntnis des zukiinftigen Personalbedarfs sein und wie welt in die Zukunft sollte sie reichen? Die letzte Frage h~ingt direkt mit dem Planungshorizont zusammen. In dieser Arbeit wird diese for ein hierarchisches Personal- planungssystem mit zwei Laufbahnstufen untersucht: Neueinstellung und eine Bef6rderungspolitik, die yon der Anzahl der Jahre der Betriebszugeh6rigkeit abh~ingt. Ziel der Einstellungspolitik ist es, den Stellenplan for beide Stufen m6glichst gut zu erf'tillen. Abweichunge n yore Stellenplan werden mit dazu proportionalen Strafkosten belegt. Die alleinige Steuerungsm6glichkeit fiir das Per- sonalplanungssystem besteht in der Nnderung der Ein- stellungspolitik. Ftir den Fall, daf~ alle Beschgftigten dieselben Karrierevoraussetzungen besitzen, k6nnen sehr allgemeine Ergebnisse erhalten werden, da das Problem dem mit nur einer Laufbahnstufe nahezu/iquivalent ist. Ftir den allgemeineren Fall kann man Ergebnisse fur den Planungshorizont nur dann erreichen, wenn zus~itzliche Bedingungen an die Strafkosten und Ziele gestellt wer- den. Der allgemeinste Zweistufenfall ist formal gquiva- lent zu einem Produktionsgl~ittungsproblem ohne Lager- haltung und mit einer unteren Schranke fiir die Differenz zwischen den Kosten zweier aufeinanderfolgenden Perio- den.

1. Introduction

The problem of medium and long term manpower planning is to get a good match of the future require- ment of personnel and the future availability of per- sonnel.

The future requirement of personnel in the various categories is determined by the organization activity plans. The future availablity of personnel is determined by the actual population, together with the policy with respect to recruitment, promotion, job rotation, and 0171-6468/81/0003/0153/$01.60

154 H.L.W. Nuttle and J. Wijngaard: Planning Horizons for Manpower Planning so on. The problem is to matchrequirement and avail-

ability as closely as possible. The difficulty with man- power planning is that all decisions made to adjust availability and requirement to each other have a long lasting impact on the organization. It is not possible to adjust from year to year because of, for instance, the following points.

- It is difficult to fire people or to move people from one location to another.

- People have (implicit or explicit) career rights; career possibilities have to remain stable therefore.

- For many functions one needs people with experience in the organization and these people are not directly available at the labor market.

The difficulty of this long lasting impact of personnel decisions is even more severe because of the fact that it is not possible to get good forecasts for future require- ment. It is difficult to plan more than five years ahead while the decisions have certainly a longer impact than five years. It is extremely important, therefore, to know how much of the future one has to know to be able to make good decisions now. So the question is in the first place, how far in the future one needs to have informa- tion about the requirement. In the second place, how detailed should this information be. The first point is the problem of choosing the proper planning horizon. The second point has to do with the level of aggregation. Although these points are related (longer planning horizon, higher level of aggregation) we will consider here only the planning horizon problem.

In thinking about planning horizons there are two pos- sible points of view, the deterministic and the stochastic. In the deterministic approach one assumes that it is possible to acquire perfect information about future data (in this case future personnel requirement), but that it is difficult to get this information and that it is im- portant, therefore, to know how much information one needs to make a good first-period decision (see, for in- stance, Lundin and Morton [5] and Morton [6] for this ap- proach in production planning). In the other approach one assumes that the forecasts have a given (un)reliability and one considers the quality of the planning as function of the horizon (see Baker and Peterson [2] for an ex- ample of this approach).

In this paper we use the deterministic approach. The problem we consider is a two4evel hierarchical system where the only control possibility is recruitment (only at the lower level) and where promotion depends on grade-age, that is the time spent in the last grade. The system is rather typical for formal organizations (see, for instance, van der Beeket al. [3]). The only purpose of the decisions is to minimize the deviation of future (expected) availability from future requirement (goals). It will turn out that it is not possible for the general two-level case to get good planning horizon results with-

out making extra assumptions about goal patterns and penalty functions. The special, but interesting, case where all employees have the same career pattern is easier since in this case the system is almost equivalent to a one-level system.

The model is described in more detail in Sect. 2. Section 3 gives a transformation of the problem which brings it somewhat closer to the production planning type problems. In Sect. 4 the one-level case is considered and in Sect. 5 the two-level case. Subsection 5.1 gives the special case; Subsect. 5.2 the general case. Finally, Sect. 6 gives a discussion of the differences between this problem and a related production-smoothing problem considered in Aronson et al. [1] and explains why it is not possible to get planning horizon results for the more-level case without making extra assumptions on goal patterns and penalty functions.

2. The Model

Consider a linear hierarchical system with two grades. The promotion policy is such that people with less than l years of service in grade 1 (grade-age < l) cannot be pro- moted to grade 2. The probability to be promoted for people with l years of service in grade 1, or more is p. Recruitment is only in grade 1. Promotion as well as re- cruitment are assumed to take place once a year at a certain fixed date. The probability to leave the system (turnover) is a, independent of grade and grade-age. That means that the system may be described by a Markov- type model with states (1, 1), (1, 2) . . . (1, l) and (2), where (1, i) indicates the category of people in grade 1 with grade-age i (see Fig. t). People recruited in grade 1 now are assumed to be in state (1, 1) during this year. Next year they enter state (1,2).

C~

t ~

t

Suppose that the current content is given by the num- bers (..011 , (.012 . . . &)ll and COz, let

z(t)

H. L. W. Nuttle and J. Wijngaard: Planning Horizons for Manpower Planning 155

merit in year t and x11(t), xx2(t) ... x l l ( t ) a n d x 2 ( t ) the expected content of the different categories in year t; then the following equations are satisfied

x,i(O) = cow x2(0) = co2 and for t ~> 0

X l l ( t + 1) = z ( t + 1)

X 1 / ( t + 1) = (1 - a)xl]_ 1 (t), ] = 2 ... I - 1 x l l ( t + 1) = (1 - a)x11_a(t) + (1 - a)(1 - p ) x l t ( t ) x 2 ( t + 1) = (1 - a)PXll(t) + (1 - a)x2(t)

We assume that there are goals on the future expected content o f the system, in the first place a goal g l ( t ) o n

t

the content o f the whole system at time t, ~ x l i ( t ) +

]=1

x 2 ( t ) , and in the second place a goal g2(t) on the con- tent o f the second level, x 2 ( t ) . The costs o f deviations from these goals are assumed to be given b y

l

Cll3tlgl(t) - ( x 2 ( t ) + ]~ Xx](t))[ and j = l

c213t[g2(t) - x2(t)] ,

where t3 is a discount factor, 0 < 13 ~< 1.

The only way to control the system and keep it close to the goals is by choosing the appropriate recruitment. The problem we will consider is to choose recruitment such that the total cost o f deviations over a finite horizon T is minimized. The recruitment from period 1 on in- fluences the total content of the system from period 1 on, but it influences the content o f the second grade only from period l + 1 on. That means that one has to consider costs of deviations from the goal for the total content from period 1 on and costs o f deviations from the goal for the content o f the second grade from period l + 1 on. So, the relative weights o f the penalties for de- viations depend on the horizon. To circumvent this we choose the following objective function

T l C(T) := N clt3t[gl(t) - (X2(t)+ ~ Xlj(t))] t = l ]=1 T + l S, c2t3tlg2(t)- x2(t)l t = l + l

So the problem is a simple goal-programming problem (see Charnes, et al. [4] and the references given there for applications of goal-programming techniques in more complex problems). However, here we are not interested in efficient algorithms or nice interactive computer packages, but only in the planning horizon, the length o f time over which we need information about the goals to

make a good first-period decision. A number N is called a planning-horizon here if the problem of minimizing

C(T) gives the same first-period recruitment for all

T >~ N.

The goals chosen here are somewhat uncommon. More c o m m o n is to have a goal on each o f the grades. Both choices are more or less arbitrary. Important is that one has some way to express the preference for certain grade contents. Since this study is n o t a direct application, but a way to get some insight in the amount of information required in this type o f problem, it is not t o o important which way o f representing this preference is chosen.

3. Transformation of the Problem

First, it is possible to reduce the problem to a problem with turnover equal to 0. Define

t

Xlj( t): = (1 - a)-t X lj( t )

t

x2(,): = (1 - a ) - t x 2 ( t ) z ' ( t ) : = (1 - a ) - t z ( t )

Then the x' develops as in an equivalent system without turnover with recrmtment z . If we define ' '

gj(t): = g l ( t ) ' ( 1 --a) - t g;(t): = g 2 ( t ) " ( 1 _ ~ ) - t

then the costs in period t are given by I

c1" (1 - a)tl3tl~a(t ) + ~ x'li(t) - g ~ ( t ) l

]=1

c2 " (1 - a)t(3t[x;(t) - g~(t)

and

So of the same type as before. It is important to notice that if the original goals were (about) constant then the revised goals are (about) increasing at rate 1/(1 - a). Therefore, the case where the revised goals are increasing is more or less normal.

Second, we may assume without loss o f generality that co = 0, start with an empty system. It is always possible to subtract from the goals the content due to the starting population. However, after such a t r a n s f o r m a t i o n t h e goals are not necessarily positive.

The third transformation is the most important one. It reduces the problem to an equivalent problem without grade-age. Let

l

y l ( t ) : --- :~

x~j(t) +

]=1

156 H. L. W. Nuttle and J. Wijngaard: Planning Horizons for Manpower Planning

All people in the system at time t are at time t + l - 1 either in state (1, l) or in state (2). So x i l ( t + l - 1) = y l ( t ) - y 2 ( t + l - 1). The input at time t + l from grade

1 to grade 2 is p ( y l ( t ) - y 2 ( t + l - 1)). Hence y 2 ( t + l) = y 2 ( t + l - 1) + p ( y l ( t ) - y 2 ( t + t - 1)) =(1 - p ) y 2 ( t + l - 1)+ p y x ( t ) = p t - 1 ~, ( 1 - p ) i y l ( t - - i ) i=0 y 2 ( t + l)

Now define y ~ ( t ) : - - - , then P

choose Yl ( ' ) such that the expression

the problem is to T {Cl(1

--

a)tflyl(t)

-g~ (t)l

is minimized under the conditions Yl (1)/> 0, Yl (t + 1) ~> y l ( t ) , t = 1 , 2 . . . . (since the total content at time t + 1 can never be less than the total content at time t). The relationship between Yl and y ~ is given b y

t Pk-1

t Pk-2

t p,

t p,

yl(. ) (input)

4. The One-Level Case

In this section we consider the case in which there is a single grade. According to Sect. 3 the problem is then to

T Minimize ~ { d l y ( t ) - h(t)l(1 -

oot~ t}

So we may view Yl (') and y ~ ( ' ) as input and content o f a system where each period a fraction p o f the old popu- lation leaves (see Fig. 2).

t p

...

Content is yz*(. )

Input is yl[- )

For cases with more grades one can get the same type of transformation. If there are k grades in t h e original system with 11, ..., /k-1 the maximal grade-ages and Pl . . . Ptc-1 the promotion probabilities from the highest grade-age to the next grade, then the system is equivalent to a system as depicted in Fig. 3. The total content in the original system corresponds to the input in the transformed system, the total content o f the grades 2 . . . . , k in the original system corresponds to the con- tent of grade 1 in the transformed system, and so on.

under the conditions y ( 1 ) ~> 0 and y ( t + 1) >~y(t) for

t ~> 1. Since the goals h(.) are transformed to take into account the contribution due to the starting content, it is possible that h ( t ) < 0. The one-level case is interesting

in its own fight, but also because o f its similarity with the two-level case with promotion probability p = 1 (see Subsect. 5.1).

tn the single4evel case it is possible to give an explicit expression for the optimal first-period decision. This can be used to derive planning horizon results. Let y (tl T) be the optimal contentin period t for the T-period problem. Define sgnt(x): = - 1 for x ~< h ( t ) , sgnt(x): = +1 for x >

tl

h ( t ) . Let s n ( x ) : = ~ {sgnt(x)(1 - a)t~ t} and let nT

t = l

be the supremum o f all x such that s t l ( x ) <<. 0 for alt

t~ ~ T. Notice that n r <~h(1).

An integer T * is called a planning horizon if y ( 1 IT) = y ( l l T * ) for all T ~> T*, independent o f the h ( t ) for t >

T*. T* is called a weak planning horizon if y ( 1 IT) = y ( l l T * ) is only true under certain conditions on the

h ( t ) f o r t > T*.

L e m m a 4.1. y (1 IT) = max (0, n T ) f o r all T >~ 1.

Proof. Suppose y ( 1 IT) > max (0, nT) for some T. Let

H. L. W. Nuttle and J. Wijngaard: Planning Horizons for Manpower Planning 157

lution y ' ( . ) b y y'(t) = y ( t i T ) - e for all t ~< 71 and y'(t) = y (tl T) for all t > {2- Then, b y the definition o f st1, the so- lution y'(.) is better than the solution y ( ' I T ) , which yields a contradiction.

Suppose now that y (1 IT) < max (0, nT), so 0 < y (1 [7) <nT-. Let t' be the first t such t h a t y ( t l T ) >1 max (0, n r ) (if such a t exists; otherwise t': = T + 1). Construct the solution y'(-) b y y ' ( t ) = max (0, n r ) for t < t' and y ' ( t ) = y ( t l T ) for t ~> t'. By the definition o f n T the solution y ' ( - ) is cheaper than the solution y (. IT), which yields a

contradiction again.

The following corollaries are immediate consequences

of L e m m a 4.1. El

Corollary 4.1. The optimal first-period decision y ( 1 IT)

is non-increasing in T. That means, if y(1 IT*) = 0 then

T* is a planning horizon.

Corollary 4.2. I f h ( t ) >~ h(1) for all t ~ to then to is a (weak) planning horizon.

A special case were the conditions o f Corollary 4.2 are satisfied for to = 1 is the case where h ( t ) is increasing. The optimal policy is myopic in this case, y ( I [ T ) = m a x (0, h(1)). In Sect. 3 we mentioned already that the case o f increasing (revised) goals is the normal case.

The discount factor/~ and the turnover rate a can also help in getting planning horizons.

Corollary 4.3. I f for t* >~ 1 the value st,(h(1)) <~

--(1 -- ~ ) t * + l ~ t * + l

then t* is a planning horizon.

1 - ( 1 - a ) / 3

- ( 1 - o 0 t * + l / ~ t * + l

Proof st* (h(1)) ~< implies that

1 - ( 1 -

a)~

st(h(1)) <~ 0 for all t ~> t*. Since nT ~<h(1) this implies that n T is constant for T ~> t*.

T

Minimize 2; {d 1 [Yl ( t ) -- h 1 (t)l + d2 lY2 ( t )

t = l

- h2(l)l)(1 - o t ) t ~ t ,

t - 1

where Y2(t) = 2; (1 - p)iy~(t - i) andYx(-) has to

i=0

satisfy the conditions y 1 ( 1 ) / > 0 , y x ( t + 1) >~yl(t) for t ~ > l .

The constants d 1 and d 2 correspond to the constants c 1 and e2p(1 - a)1~ t in the original model, while h~(t) and h2 ( t ) correspond to g~ ( t ) and g ; ( t + l)/p (see Sect. 3).

In Subsect. 5.1 the case p = 1 is considered. The re- suits for this case are very similar to the one-level results. Although the case p = 1 is special it is certainly not un- interesting. In m a n y formal organizations the promotion restrictions, especially in the lower grades, are so tight that one may approximate it b y p = 1. One may also think of a situation where the lower level is a training- type level.

In Subsect. 5.2 the case p < 1 is investigated. In this case it is only possible to give good results under extra conditions on the constants d 1 and d z and on the goal patterns. We will emphasize the case o f increasing goals.

5.1. The Case p = 1

When promotion is assured (p = 1), grade 2 is effectively the highest grade-age in a single grade system; but with a separate goal on its content. In this case y l ( t ) = y2(t) and the problem reduces to

Minimize

T

2; ( d l lYl ( t ) - h I ( t ) [ + d 2 [ y l ( t ) t = l

- h 2 ( t ) [ ) (1 - a)tl3 t

For instance, if (1 - a)/3 ~< 89 then s 1 (h (1)) : - (1 - a)/t - (1 - a)2132 <~ - ( 1 - a)el32

( 1 - a ) ~ 1 - ( 1 - a)13

and Corollary 4.3 implies that 1 is a planning horizon,

n T = h(1).

5. The Two-Level Case

In this section we consider the two-level case. According to Sect. 3 we m a y consider a problem o f the following type

under the conditions y l ( 1 ) / > 0 a n d y l ( t + 1) >~yi(t) for t / > 1. That means that this problem is almost equi- valent to the one-level case. As in that case there is only one variable which can be controlled. The difference is the shape o f the penalty function.

Let Xo(t ) be the largest value o f x for which dl Ix - hi(t)[ + d2 ix - h2(t)l is minimal. Let rt(x ) be the left- hand derivative o f this function in x. Define, as in Sect.

tl

4, for all t 1 ~> 1, sq(x): = E rt(x)(1 - ot)t~ t and let

t = l

n r be the supremum o f allx such that s q ( x ) ~ 0 for all t 1 ~< T. Observe that nT ~< X0(1). The following lemma and corollaries correspond to Lemma 4.1 and Corollaries 4.1,4.2 and 413, and have similar proofs.

15 8 H.L.W. Nuttle and J. Wijngaard: Planning Horizons for Manpower Planning Lemma 5.1. For each T there is an optimal solution

Yl ( t i T ) with Yl (1 IT) = m a x (0, nT-).

Corollary 5.1. I f Yl ( l i T * ) = 0 then T* is a planning horizon.

Corollary 5.2. I f h l ( t ) and h2 ( t ) are such that xo(t) >~

Xo(1) for all t >7 t o then t o is a (weak)planning horizon.

A special case where these conditions are satisfied for t o = 1 is the case where h 1 (.) and h2 (') are increasing; this implies that Xo(') is also increasing. In this case y 1(1 IT) = max (0, Xo(1)).

Corollary 5.3. I f for certain t* >~ 1 the value

- ( 1 - a ) t * + 13t*+ 1

st,(Xo(1)) ~ ( d l + d:l),

1 - ( 1 - a)13

then t* is a planning horizon.

Since the transformations applied in Sect. 3 play a more significant role in this case than in the one-grade case we have to check what a planning horizon T in this transformed model means in the original problem. The main part o f the transformation was a shift in the time- axis for Y2. That means that a planning horizon T for the transformed problem implies that in the original problem one needs to know the goals gl(1) . . . & ( T )

and g2(1) . . . . g2(T + l). In case p = 1 the maximum time

spent in the lowest grade is l. The result shows that 1 contributes directly to the length o f the proper planning horizon.

Reviewing this subsection shows that the results can be generalized to the case with more than two levels and more general penalty functions.

tion Yl (tl T) satisfies

Yl ( t ! T) = max (hi ( t ) , y ~ ( t - 1 t T)) for all t / > to. Proof. We will give the p r o o f for the case a = 0, 3 = 1 ; the p r o o f for the general case is similar.

Let t I ~> t o and suppose y l ( q l T ) > max (ha.(q), Y l ( q - liT)). Then the optimal solution can be im-

proved by the revised solution y~ (.) defined b y y~ ( t ) =

Y1(tIT), t r tl a n d y ~ ( t l ) = Y l ( q IT) - e. The total cost

for deviations from goal h 1 (-) is ed 1 less for solution y~ (-)

than for Yl(" ] T), while the cost for deviations from goal h2(-) is at most e(d 2 + d2(1 - p ) + d2(1 - p)2 + . . . ) =

ed2- more for y ] (') than for Yl (' I T). This yields a contra- P

diction.

Now suppose Y l ( q I T ) < hl(ta) for some t I ~> t 0. Let

fi + k be the first period t where y l ( t l T ) >1 h t ( t ) Of

such a period exists; otherWise t a + k: = T + 1). This im- plies t h a t y a ( q + kiT) >Yl(ta + k - liT). Define the re-

vised solution y~ (.) b y Yi(t): = y l ( q l T ) + e

y~(t 1 + k): = y l ( t l + kiT) - e(1 _ p ) k

y ~ ( t ) : = y l ( t l T ) , t g = q , t l +k

This is possible for e small enough since y l ( t l + kiT)

> Yx(tl + k - l i T ) . Now y ; ( . ) differs from y E ( ' I T ) only in the periods t I . . . q + k - 1. The reduction in costs o f deviations from the goal h 1 (.) is at least (d 1 -

(1 - p ) k d l ) e while the increase in costs of deviations

from the goal h 2 ( ' ) is at most e(d= + d2(1 - p ) + ... +

d 2 ( l _ p ) k - 1 ) = e 9 d2(1 - (1 - p)k)/p. So in total

y ' ( - ) is better than y (-IT), which yields a contradiction. o In case hx(. ) is increasing the conditions o f L e m m a 5.2 are satisfied for t o = 1. We have the following corol- lary.

5.1. The Case p < l

In this case one has to add conditions on the ratio of d 1 and d2. If dl/d 2 is large (i.e., t.he goal on total system content is significantly more important than that on grade 2) and hl(" ) can be followed (i.e., h 1 (.) is non- decreasing) then hi(- ) should be followed indeed. I f

d2/dl is large and h 2 (-) can be followed then h2 (')should

be followed indeed. However, there is a gap between the two regions; a gap which is widening with decreasing p. In L e m m a 5.2 we consider the case where h I is the most important goal.

L e m m a 5.2. Suppose that dl > d2/P and hi ( t ) is non- decreasing for t ~ to. Then for each T the optimal solu-

Corollary 5.4. I f d 1 > d2/p and hi(. ) is increasing then the optimal first-period decision is Yl (t ] T) = max (0,

h 1 (1)) and 1 is therefore a (weak) planning horizon.

For to > 1 the problem is more difficult here than in the one-level case or the two-level case with p = 1. In the first part of the p r o o f we do not use the fact that hi (') is non-decreasing. That means that y l ( t l T ) <~ max ( h i ( t ) ,

Yl (t - i IT)) for all t. Let t* ~> t o be such that h 1 ( t * ) / >

hl(s ) for all s < t o and also hi(t* ) >~ O. Then for all

problems with T / > t* the optimal y l ( t * l T ) = hi(t* ).

However, this does not necessarily mean that t* is a planning horizon, since y2(t*) is still free and will

depend in general on the behavior o f h 2 ( ' ) a n d h 1 (-)be- yond t*.

In the next lemma we consider the case where h2(') is the most important goal.

H. L. W. Nuttle and J. Wijngaard: Planning Horizons for Manpower Planning 159

Lemma 5.3. Let d e > d1(2 - p) and let h2(t + 1) -

(1 - p)h2( t ) be non-decreasing from t = t o - 1 on, Then for each T the optimal solution Yl (" l T), Y2 (" l T) has the following properties.

(a) Let t I >1 t o and let yf(.),ykz(.) be a feasible solution such that y f ( t ) = y t ( t l T) for t <~ tt - 1 , y g ( t ) ~< h2(t)for t >i tl and y f ( t ) = he(t ) for t ~> fi + k. Then Ye (tl IT) ) y ~ ( t a).

(b) I f ( 1 - p ) y 2 ( t l - l l T ) + Y l ( t l - lIT)~< h2(tl) for some tl >>- to then ya ( q IT) ~< h2(tl).

(c) If(1 - p ) y 2 ( t l - 1 IT) + y l ( t l -- 1 I T ) > ~ h 2 ( t l ) f o r

some t 1 >1 t o then Yl (tl IT) = y l ( t l - lIT).

Proof. We will give the p r o o f for the case where a = 0, /3 = 1 ; the p r o o f for the general case is similar.

(a) Let yaX(.), y f ( - ) be as stated. Suppose y 2 ( q l T ) <

y((tl).

exists; otherwise t 1 + t: = T + 1). T h e n y e ( t l T ) <y2k(t)

for t I ~< t < tl + l. Define the revised solution y[ (.), y ; (.) b y

y ; ( t + i ) : = y l ( t l l T ) + e , i = 0 , 1 , . . . , I - 1

y[(fi +l): = Y l ( q + l I T ) - e ( ( 1 - p ) + (1 _ p ) 2

+ . . . + (1 - p ) Z }

y ] ( t ) : = y l ( t l T ) , t<~q a n d t > q +l.

Since y ( ( t ) <~ h2(t) for t ~> tl and y e ( t l T ) < y g ( t ) f o r

q ~ t < t I + l, we have also y~(t) < y g ( t ) K ha(t) for

t I ~< t < tl + I (and e sufficiently small). For t >~ tl + l we have y~(t) = y2(tl T). The reduction in cost o f devia-

tions from the goal he(- ) is equal to

such a period exists; otherwise q + l: = T + 1). Define the revised solution y~ (.), y ; (-) b y

y [ ( t l ) : = yx(tl IT) - e

Y~(tl + l): = Y l ( q + liT) + e(1 _ p ) l

y ' l ( t ) : = y l ( t l T ) for all t 4: tx, t x + l

The feasibility of Y l ( h ) : = Y l ( q [T) - e follows from

(1 - p)ya(tl - llT)+ y l ( t I - llT)<-h2(fi)and y2(tl IT) = (1 - P)Ye (tl -- 1[ T) + y 1 (tl IT) > he (tl), which implies

that Yl (tl IT) ) Yl (tl - 11 T) and therefore that Yl (tl IT) can be reduced indeed. That Yl (tx + I I T ) can be increased without increasing y x ( t l T ) for t > tl + l, follows from y l ( t l +l+ llT)>~h2(q +l+ 1 ) - ( 1 - p ) h 2 ( t 1 + l ) = h e ( r I + l + 1 ) - ( 1 - P ) Y 2 ( q +l)>~h2(q + l ) - ( 1 - p)h2( q + l - 1 ) > y 2 ( t l + l I T ) - yz(tl + l - 1 ] T ) = Yl (tl + II T). For y ; (.) we have y ; ( t ) = y 2 ( t l T ) , t < t l

y ; ( t + i) =Y2(t + ilT) - e(1 - p ) i ,

y ; ( t ) = y 2 ( t [ T ) , t>~tl +l

i = 0 , 1 . . . l - 1

The reduction in cost o f deviations from the goal h2(- ) is

l - 1 1 - ( l - p ) 1

d2e ~ (1 - p)i = d2 e . The increase in

i=0 P

cost of deviations from the goal hx(- ) is at most die +

dxe( 1 _ p ) l . From d 2 > dx(2 - p ) it follows that the total cost o f y[(-), y ; ( - ) is less than the total cost o f Yl (" i T), Y2 ( ' I T ) , which contradicts the optimality o f the last solution.

l - 1 d 2 Z, e ( l + ( 1 - p ) + . . . + ( 1 - p ) i) i=0 1-1 1 - ( 1 _ p ) i + l =d2e i=0 P _ d 2 e ( 1 1 - p l - p (l_p-p)t)

The increase in cost o f deviations from the goal h 1(.) is at most dlle + dfle(1 - p) 1 - (1 - p ) l . It is easy to

P

prove by induction, using d 2 > d 1 (2 - p ) , that the total cost of y i ( ' ) , y ; ( ' ) is less than the total cost o f y l ( - [ T ) , Y2 (" [ T), which contradicts the optimality o f this last so- lution.

(b) Let (1 - P)Y2(tl - l i T ) + y l ( t l - l I T ) ~ < h 2 ( q )

for some t I ~> t o and suppose Y2 (tl I T) > h 2 (tl). Ob serve first that by part a and the non-decreasingness o f h2(t + 1)

- (1 - p)h2(t ) from t = t o - 1 on this implies that

y2(tl T) ~> h 2 ( t ) for all t ~> tl. Choose l such that t I + l is the first period t after tl such thaty2(tlT) = h2(t) Of

('c) The p r o o f o f c is similar to the p r o o f o f b.

Application of this lemma to the case with t o = 1 (with h2(0): = 0) gives also myopic-type results. In the first place, it follows from c that h2(l ) ~< 0 implies Yl ( l I T ) = 0. So in this case the (weak) planning horizon is indeed equal to 1. Further, it follows from a and b that if there is a feasible policy y~(.),y~(.)with y~(t)=

h 2 ( t ) for all t ~> 1 then this is the optimal policy. Since h2(2) - (1 - p ) h 2 ( 1 ) ~ > h2(1) such a policyy~(.),y~(.) exists if h2(1) ~> 0. So in this case the (weak) planning horizon is equal to 2 ( a n d y 1 (1]T) = h2(1)).

To evaluate the usefulness o f L e m m a 5.3 observe that

h2(t + 1) - (1 - p)h2(t) >~ h2(t) - (1 - p)h2(t - 1)

can also be written h2(t + 1) - h 2 ( t ) ~> (1 - p)(h2(t ) - h2(t - 1)). So convexity of h2(. ) is sufficient for all p,

but for the case where p is close to 1 the condition is much weaker. According to Sect. 3 one may expect in most cases h2(') increasing about geometrically. That means that the condition is not too severe.

160 H.L.W. Nuttle and J. Wijngaard: Planning Horizons for Manpower Planning 6. Production-Smoothhag Problem

In this section we discuss the relationship o f this problem to the production-smoothing problem without inven- tory investigated by Aronson et al. [1]. First we apply one more transformation. Define

)3a(t): = y a ( t ) - h a ( t )

t - 1

f i 2 ( t ) : = y 2 ( t ) - ~ ( I - p ) i h a ( t - i )

i=o

The problem formulated in 371 and 372 is then

Minimize

T

{dl IPl (t)l 4 d2 lfl2 ( t ) - ( h 2 ( t )

t=l

out adding extra information on the goal patterns. To clarify this we give an example where an increase in he (t') for a certain t ' implies a decrease in y 2 ( t ) for certain

t < t ' .

Example. L e t T = 3, p = 1/2, a = 0,/3 = 1 , d a = 1 , d 2 = 2

and let the goals be h 1 (1) = h 1 (2) = h I (3) = 2 and h2(1 )

= 0, h2(2) = 3, h2(3) = 389 Then the optimal solution is

Yt (1) = Yl (2) = Yl (3) = 2 which yields Y2 (1) = 2,3,2 (2) = 3, y2(3) = 389 However, if h2(3 ) = 489 instead o f 389

then the optimal solution isy I (1) = 0 , y 1 ( 2 ) = 3 ,Yl ( 3 ) =

3 which yieMs y2(1) = 0 , y 2 ( 2 ) = 3,y2(3) = 489 So an in-

crease in h2(3) causes a decrease in y2(1). It is clear that

this is due to the fact that an increase in YI (2) can also

imply an increase o f y 1 ( 3 ) .

t - I

- E (1 - p ) i h l ( t - i))1}(1 - a ) t [ J t,

i=0 t - a

where)32(t ) = 2 (1 - p ) i f i a ( t - i) and)71(. ) has tosat- i=0

isfy the conditions

~1 (1) ~> -hi (1),

i l l ( t + 1 ) > ~ f i a ( t ) - h l ( t + 1) + ha(t ) for t ~ > l .

Since the possibility to derive planning horizons as in [1 ] relies heavily on the monotonicity of the decisions as function of the goals one may not expect to be able to get these types of planning horizons here.

It is also possible o f course to explain the difficulty to derive planning horizons for this problem by mentioning that for p < 1 the state space is two-dimensional while the state space in the production-smoothing problem is one-dimensional.

The problem considered in [ 1 ] is of the following type

Minimize T (alxa(t)l + b l x 2 ( t ) - d(t)l), t = l t - 1 w h e r e x 2 ( t ) = ~ x l ( t - i ). i=0

The most essential difference is that in the problem considered here )31 (.) is not free while in the production- smoothing problem xa(') is free. In [1] the following planning horizon result is given. If for certain T = T* the optimal policy is such that xl (tx) > 0 and x 1 (t 2 ) < 0 for certain q =P t2 smaller than T* then the optimal policy on the interval [1, min ( q , t2) - 1] i,s not changed by a further increase o f T.

In the problem under consideration here it is n o t pos- sible to get such type o f planning horizon results with-

R e f e r e n c e s

1. Aronson JE, Morton TE, Thompson GE (1978) A forward algorithm and planning horizon procedure for the production- smoothing problem without inventory. Carnegie-Mellon, Re- port WP 20-78-79

2. Baker KR, Peterson DW (1979) An analytic framework for evaluating rolling schedules. Manag Sci 25:341- 351

3. van der Beek E, Verhoeven CJ, Wessels J (1977) Some appli- cations of the manpower planning system FORMASY. Manp P1 Rep No 6, Eindhoven University of Technology

4. Charnes A, Cooper WW, Lewis KA, Niehaus RJ (1978) Equal employment opportunity planning and ~taffing models. In: Bryant D, Niehaus RN (eds) Manpower planning and organiza- tion design. Plenum Press, New York, pp 367-382

5. Lundin R, Morton TE (1975) Planning horizons for the dynamic lot size model: Zabel vs. protective procedures and computational results. Operat Res 23:711-734

6. Morton TE (1978) Universal planning horizons for generalized convex production scheduling. Operat Res 26:1046-1058