A review of mathematical models in human resource planning

O~4EG4 The Int. Jl of Mgmt Sea., Vol. 4. No. 6. pp. 639 to t.,~5

03(}5-0,tg3 80 1101-1~39~)2.(~):0

C Pergamon Press Ltd 14'~0 Printed in Great 8rltam

A Review of Mathematical Models in Human Resource Planning WL

PRICE

A MARTEL KA LEWIS Universit6 Laval, Qu6bec, Canada (Received June 1979: in revised form March 1980)

To manage its manpower, the OtgHilation must be informed about its internal dynamics and about the dynamics of its environment. This involves the monitoring of internal personnel movements and the analysis of external supp4ies. The internal situation can largely be controlled through hirings, promotions, internal transfers, redundancies and retirement planning. The problem is precisely to plan and control these interrelated activities in order to achieve a stable organization capable of meeting its objectives. The influence of the enviroument, through the economic situation, legislation, competition and other factors complicates the problem further. To assist in the planning and eoatrol of these activities, the organization can have recourse to models that are either descriptive (Markov chains, renewal models) or normative (linear and goal programming, netwock method~ stochastic programming). Having reviewed a number of modeling approaches the authors are able to draw certain conclusions as to their applicability for solving various practical problems.

INTRODUCTION PROBLEMS IN human resource planning have been treated extensively in the literature on management science. It is the aim of the authors to outline those problems that affect the whole of the organization over the long term, and to classify and comment on available solution techniques. Such strategic problems are generally dynamic and involve parameters that may be uncertain t. The problem most often encountered in this literature is the planning and control of grade sizes. This is the problem of providing the quality and quantity of manpower that will permit the organization to fulfill its objectives. To manage its manpower, the organization must be informed about its internal dynamics Tactical problems such as shift scheduling, manpower leveling and the control of transfers are beyond the scope of this paper. The interested reader may consult, for example, references [15, 22, 24].

and about the. dynamics of its environment. This involves the monitoring of internal personnel movements, and the analysis of external supplies. Part of the internal dynamics is beyond the control of the planner since attrition, for example, results from individual decisions rather than organizational decisions. The internal situation can, however, largely be controlled through hirings, promotions, internal transfers, redundancies and retirement planning. The problem is precisely to plan and control these interrelated activities in order to achieve a stable organization capable of meetmg its objectives. The influence of the environment, through the economic situation, legislation, competition and other factors complicates the problem further. Before looking at how the organization can be modeled, various aspects of the overall planning problem will be examined using terminology familiar to human resource planners.

639

6.10

Price---A Re~'iew of Mathematical Models in Human Resource Plannin#

The control of promotions is a difficult area in many organizations. In some cases it is only necessary to ensure that promotions are not accorded at a rate that would cause the higher grades to become top-heavy, but in others it is also necessary to attempt to equalize promotion rates among different divisions or occupational classifications. Internal transfers are sometimes used as a device to broaden the experience of personnel or simply to meet the demands of a changing situation. If transfers are frequently used for these or other reasons, it can become difficult to evaluate the impact on grade sizes and on the promotion rates of the individual. Policies on promotions and transfers are directly linked to succession planning and career planning. Succession planning is the problem of ensuring that senior or key posts in the organization will always have suitable occupants and can be necessary either for individual positions or for an entire class of posts. Career planning attempts to guide new employees with high potential or with a needed skill in their choice of assignments within the organization. The aim of this guidance is to assist the individual in his personal development as well as to make the best use of his skills from the point of view of the organization. Both succession planning and career planning which in a sense are complements, will have an impact on hirin# and training policy. The determination of hiring criteria and the establishment of internal training programs are related to the classical 'make or buy' problem. This consideration is also related to the control of grade sizes, since access to some grades may be linked to qualifications. A further related question is that of whether to hire specialists (and broaden their horizons) or generalists (and give them specific training). Redundancies are sometimes necessary in any organization, however because of the disturbance that they cause both to the life of the individual and to the life of the organization, they must be planned carefully. It must be established, through careful calculation, that the personnel released are indeed no longer needed. More than one organization has given the 'golden handshake' to a part of its workforce, only to discover a short time later that they were needed and had to be re-hired. The establishment and operation of fair

employment practices has become a major problem in many western countries today. The United Kingdom has passed some of the most stringent legislation regarding women and racial minorities, in Canada legislation and the regulations of the public service cover the hiring of linguistic as well as racial minorities, and in the United States the 'equal employment opportunity' legislation is well known. In some cases it is sufficient to be able to demonstrate non-discrimination in hiring to comply with the law, while in others it is, in addition, necessary to ensure that the composition of the workforce meets some specific targets by a given year. Compliance with fair employment legislation requires the monitoring of turnover and the establishment of programs to control or modify it. The management of turnover, then, is a problem that most organizations face. It should be noted that while some organizations seek to reduce turnover and induce more stability in the workforce, others seek to increase controlled turnover to reduce promotion blockages or to reduce the size of the workforce. In this context another factor which can be used to control the stock of personnel is the retirement plan of the organization. In order to plan properly for future contingencies, it is often necessary to make fore-

casts of the supply of specialized personnel. This usually involves demographic and labor mobility considerations [1]. The management scientist can design decision processes that will allow planners to make use of all the information that they possess and to help make known through the planning process the constraints under which the planners must operate and the objectives that they wish to meet. Since many of the problems that have been discussed are interrelated, the management scientist can show, through modeling, the effects of a 'solution' to one problem on the constraints of another. M O D E L I N G AS AN AID TO H U M A N RESOURCE PLANNING In assisting the planner, the management scientist is faced with the usual design problem of including sufficient characteristics of the organization in the model to attain the required

Omega, Vol. 8, No. 6

degree of realism without reaching a level of complexity such that the model is as difficult to understand as the organization itself or that computations become expensive and burdensome. While not forgetting the individuality of each member of an organization, the first step in modeling is usually to divide the workforce into subgroups that are homogeneous according to those characteristics to be treated in the model. Personnel can, for example, be classified according to their occupation and grade and each occupation/grade combination is called a state. The set of states defines a state-space and a vector of stocks is used to represent the numbers in each state. It is at the point where it becomes necessary to describe how personnel move between states that the model types diverge. Descriptive models can be constructed to imitate the behavior of the actual organization, and so permit the testing of various policies. This paper will cover the main types of descriptive models found in the literature: Markov models, fractional flow models and renewal models 2. Normative models prescribe a course of action from among a set of feasible solutions identified by the model itself, according to a criterion specified by the planner. Linear programming, goal-programming, stochastic programming and network methods can be used in the construction of normative manpower models. This brief article could not attempt to cover all the literature on the modeling of manpower systems, however, a number of books and articles of general interest must be mentioned. A series of books [7, 14,37,41] reporting the proceedings of NATO-sponsored meetings on human resource planning contains many illustrations of the use of diverse methods and models. In addition, Bartholomew [2], Bartholomew and Smith [5], Bartholomew and Forbes E3], Bartholomew and Morris [4], Charnes, Cooper and Niehaus [11], Grinold and Marshall [19], Niehaus [28] and Vajda [39] have published books on specific areas of z Simulation models, using specialized l a n g u a g e s such as GPSS, t)v~^,,.ioand so on could bc classified as descriptive models, however the simulation methodology usually leads to models well suited to the particular situation but without a great deal of generality. For this reason, they have been cxciuded from this review.

641

the mathematics of manpower planning. The RAND Corporation carried out and published an in-depth study [21] of various models while Lewis [22] and Piskor [31] compiled bibliographies on mathematical techniques for manpower planning. Other papers will be cited to illustrate the use of particular techniques and approaches to practical problems. MODELING THEORY AS USED IN PRACTICE

Descriptive models If the first step in the modeling process is the establishment of a state space, the second is the choice of a description of how people 'flow' between states. One must describe how promotions, transfers and so on occur. These flows, when calculated as proportions of people in each state lead to fractional flow models [19]. Fractional flow models become Markov chain models [2] if one assumes that the proportions are, indeed, probabilities. Where x(t) is the stocks vector that is observed at time t, n(t + 1) is the number of' entrants at time (t + 1), p is a vector showing how these entrants are distributed among the states of the system and P is the matrix of transition proportions among states, the expected stocks vector at time (t + l) is x(t + 1) and is given by the matrix equation: x(t + I) = x(t)P + n(t + l)p.

(I)

Repeated application of this equation, or equivalent mathematical procedure, allows forecasting of the stocks vector for later points in time. These models may be thought of as 'push' models, since the state vector at a particular point in time is determined by the number of people 'pushed" from the previous state vector into the various states by the Markov matrix. Grinold and Marshall [19, ch. 2] show how a simple variant of equation (l) was used in the forecasting of the size of the student body at the Berkeley campus of the University of California and of the distribution of the enrollment in the four years of study. Bartholomew and Forbes [3, ch. 6] show how this model can be used to study the effect of changing promotion and career patterns on the age distribution

642

Price--A Review of Mathematical Models in Human Resource Pl~Inniny

mated externally and xtlt), x21r) and x3(t) are supposed known. The set of equations can then easily be solved by starting with (2) and proceeding to (4). More complex versions of this model can be constructed to account for hiring at all levels. multiple job classifications leading to a number of vertical career streams and other practical considerations. Versions of this type of model that permit the evaluation of the x-variables in continuous time rather than at fixed points are described in Bartholomew [2]. Bartholomew and Forbes [3] show how renewal models can be used to study career patterns and contrast these results with those that can be obtained from Markov models as previously mentioned. Piskor and Dudding 1-32] describe the incorporation of a renewal model in a conversational program in use for the planning of grade sizes, hiring and transfers in the Canadian Public Service, and Butler [8] has studied the problem of equalizing promotion rates in different classifications of the U K Civil Service. Stewman [38] compares the performance of Markov and renewal models in a number of cases. Grinold and Marshall [19] have described a class of longitudinal flow models in which it is supposed that entrants to the system belong to one of K 'chains'. Each chain describes the car(2) eer progression of its members. For example, in x3(t + 1) = x3(t) + f23(t + 1) - wj(t); a two year college, students who don't fail any xz(t + 1) = x2(t) - f z 3 ( t + 11 + f l z ( t + 1) -- w2(t): (31 courses are in a chain 'Year 1-Year 2', and xt(t + I) = xt(t) - - f t z ( t + 11 + h(t) - wl(t). (4) students who fail may be in chains 'Year 1-Year l-Year 2' or 'Year 1-Year 2-Year 2'. In these equations, only the f-variables and Sufficient chains are enumerated to allow a h(t) are unknown, since the wastage is esti- reasonably complete description of the flows in the organization. Where the system has N classes, and K chains have been enumerated, a series of N x K matrices P(u) are specified to STATE3 ] show the distribution of individuals in the K chains over the N classes in the uth time period % (losses) t fz3 (Promotions) in which they are counted. If the number of new entrants is 9(0 in period t, the total stock STATE2 in class i at time t is given by:

within grades of a hierarchy. They have applied this type of approach in their work with the UK Civil Service (See also Bartholomew's paper in [14, pp 81-90]. Price [34] shows how a fractional flow model can be used to accurately forecast the total turnover in a large organization. Heneman and Sandver [20] have recently reviewed a number of applications of this model type. In many situations it is more appropriate to construct 'pull' models that determine the flows in the system as a function of a desired stocks vector, rather than to observe the stocks vector that results from assumptions on the proportions of stocks that will flow from one state of the system to another. Renewal models can be used to describe this situation. Figure 1 shows a simplified hierarchical structure that will be used to illustrate the nature of such models. Let the desired number in each state be x t, x2 and x3 respectively and let this number be fixed for each time period. Let new hires in a given time period be h(t) (for simplicity, hiring is supposed to take place at the lowest level only) and let promotions between levels be denoted f12(t) and f23(t). Wastage can be estimated and is noted wl(t), w2(t), and w3(t). In order to maintain the desired stock levels, the flows must then satisfy the equations:

*z

(losses)

I f~z (Promotions) u=O

~

STATEI

(losses)

1 h (NewHires)

FtG. 1. A simple hierarchy.

it= I

Longitudinal flow models require more data (to estimate the series of matrices P(u)) than the fractional flow models, however if such data are available, it can be expected that they will more accurately reflect the evolution of the sys-

Omega, Vol. 8, No. 6

tem. Grinold and Marshall report applications of these original models within the US Navy and Marine Corps and to data drawn from student registrations at the Berkely campus of the University of California [19, ch. 3],

Normative models The previous manpower planning models are descriptive in nature and are typically used to forecast manpower requirements or to study the effects of various policies on the manpower systems. Normative models take the next step. They start from forecast requirements and suggest a set of personnel management decisions that will attain required goals in a manner that is optimal according to some stated objective function. Many normative techniques have been used in manpower planning models; however, the most popular has certainly been linear programming and its extensions. In such a model, a set of basic accounting equations, similar to (2), (3) and (4) are established to describe the relationship between stocks and flows. In this type of model, however, the x-variables are not fixed, but still remain to be determined. Other constraints can be placed on the stocks and flows to describe, for example, labor market restrictions on the numbers that can be hired (this may require a supply forecasting model), budgetary limitations, upper and lower bounds on the stocks in various states, the desired ratio of the stock in one state to the stock in another, and so on. An objective function is then specified. For example, one may wish to minimise the sum of hiring, training, shortage and salary costs over a number of years. Mathematically, the constraints may be expressed in the matrix equation: Ax = b, (x /> O)

(5)

where A is the matrix of the coefficient of constraints, x is a column vector of decision variables (stocks, promotions, hiring) and b is the vector of data concerning bounds on stocks, ratios between stocks in various states, and so on. The objective function can be expressed as: Minimise ex

(6)

where c is a row vector of costs. The concept of goal programming is due to Charnes & Cooper [9] who presented, with Niehaus, the first manpower planning applications of the technique.

643

Purkiss [36] describes a linear programming model that was used to help derive training budgets for manpower in the British steel industry, while Morgan [27] and Ciough, Dudding and Price [13] use this framework in studies of the Royal Air Force and the Canadian forces respectively. One of the particular problems addressed in [ 13] was the control of training costs associated with the supply of specialized personnel (pilots). In their text, Grinold and Marshall [19] show how optimizations similar to these can be based on their longitudinal models. A major objection to these approaches is that in personnel planning, objectives are multiple and only rarely would one wish to drive an entire manpower system according to a single narrow objective. Where it is necessary to take into account more than one objective, the goal-programming ~ modeling technique can be used. In goalprogramming, each objective (costs, promotion rates, hiring quotas, etc.) is included as a constraint which should be satisfied within a range satisfactory to the decision-maker, that is to say a 'goal'. The objective function takes the form of a minimization of the weighted sum of the deviations from the stated goals. The weights express the importance of various constraints and sub-objectives according to priorities expressed by the one or several decisionmakers involved. Empirically, the results obtained with these models have been good, despite the lack of any deep theoretical justification for the way in which the objective function weights are obtained. Their determination, to a largeextent, has been justified by management in terms of the structure of the given organization and society within that organization. Mathematically, the programs obtained have the form: rain w ' y * + w - y -

(7)

subject to Ax + iy + - iy- = b

where y + and y- are column vectors of discrepancies from the stated goals and where w ÷ and w- are the corresponding row vectors of penalty weights. Charnes, Cooper and Niehaus [11] describe the results of a number of years of application

644

Price--A Review of Mathematical Models in Human Resource Planniny

of goal-programming models to the civilian and military personnel of the US Navy. They treat a variety of problems, related to the control of grade sizes. In a further paper, Charnes, Cooper, Lewis and Niehaus [10] model the US Navy system with a view to planning for fair employment practices ('Equal Employment Opportunities'). Price and Piskor 1-35] describe a successful application of goal programming to the planning of hirings and promotions in the Canadian Armed Forces. The latest generation of normative manpower models uses network algorithms to perform the optimizations within the models. The main advantage of the network codes is their speed, however, the network model itself is often more graphic and easier to describe to non-technical people than other normative models. Gorham [17] describes an early application of network technology to the planning of hiring, training and re-training of personnel, and Price 1"33] showed how the military problem of [35] can be re-cast into network terms. Billionnet [6] has applied this technique of modeling for the definition of personnel policies at the Rdgie Nationale des Usines Renault, and has incorporated the model in an interactive computer program for policy testing. Other applications of network technology are certain to follow in the coming years. Planning involves a consideration of the future and since the future is rarely known with certainty the use of deterministic models such as the ones described in the previous paragraphs may not be appropriate. In manpower planning problems, the supply of qualified personnel is often uncertain and, for later years, the demand for personnel, the available budgets and so on may very well be random. Probabilistic programming models have been developed by Charnes, Cooper and Niehaus and Sholtz [12] for the US Navy and by Martel and Price [25] for the Canadian Armed Forces. D~/namic programming techniques also provide a natural way of obtaining integer solutions to manpower planning problems (e.g. Nuttle [-29]). Markov programming is an extension of the approach which permits the optimization of manpower policies directly from the Markov chain models already discussed (e.g. Grinold [18], Wagner [40]). Bartholomew [2] also presents optimal control

theory models which exploit this basic structure. Although these techniques are conceptually appealing, they lead to mathematical programs which are difficult to solve. CONCLUSIONS Mathematical models for manpower planning have been appearing regularly in the literature since the early and mid-fifties and a number of books and anthologies of research papers have been published in recent years. An examination of this literature shows that the majority of the papers have been written by practitioners or by academics who are reporting empirical studies. Thus the persistance of such articles in the literature, their number, and their source show tile extent to which the techniques described have found considerable practical application. A pertinent question is to ask which type of model is most appropriate in which situation. In practice, a number of factors will have to be taken into account, not the least of which are the availability of data, the familiarity of the modeler with one or another of the techniques and the availability of software and computing power. Notwithstanding these considerations a number of general remarks can be made. The fractional-flow or Markov models would seem to be most appropriate for systems in which personnel movements between states are generated largely by the individuals and as such are not specifically controlled. It is then reasonable to speak of the 'proportion' of individuals who will move from one state to another, or the 'probability' that an individual will make the transition. The model can then be used as a forecasting tool to describe the future behavior of the systems. Renewal-type models are most appropriate where grade size is closely controlled within the organization and where promotion and hiring decisions are made only to fill vacant positions. In many situations, this type of model can be used to examine various policies (for example, concerning different growth rates) and to evaluate the results of their application on system parameters such as promotion rates, length of stay in grade and so on. In organizations where costs are an overriding factor or where conflicting objectives must be resolved optimization models (linear pro-

Omega, VoL 8. No. 6

gramming, goal-programming, etc.) are possibly the best approach. They require, however, more care in their formulation and more sophisticated software and computing facilities. Optimization models are also useful in situations where complex constraints must be taken into account. REFERENCES 1. AGGARWALSP (1969) Manpower Supply: Concepts and Methodology. Meenakshi Prakashan, 2. BARTHOLOMEWDJ (1973)Stochastic Models for Social Processes. 2nd Ed. John Wiley, New York, USA. 3. BARTHOLOMEW DJ & FOgBES AF (1979) Statistical Techniques for Manpower Planning. John Wiley. New York, USA. 4. BARTHOLOMEW DJ & MORRIS BR (1971) Aspects of Manpower Planning. Elsevier, New York, USA. 5. BAaTHOCOMEWDJ & SMITI-I AR (1971) Manpower and Management Science. English Universities Press. London, UK. 6. BILLIONNETA (1978) Mod61e adaptant les politiques de personnel aux besoins pr6visionnels. RAIRO Recherche Operationnelle i 2( l ), 41-58. 7. BRYANT DT & NIEHAUS RJ (1978) Manpower Planning and Organization Design. Plenum Press, New York, USA. 8. BUTLER AD (1974) A system for equalizing promotion rates between grades within similar hierarchies. In [..37]. pp. 205-218. 9. C8^RNES A & COOPER WW (1961) Management

Models and Industrial Applications of Linear Programming. John Wiley, New York, USA. 10. CHARNES A, COOPER WW, LEWIS KA & NtEHAUS RJ (1975) A multi-objective model for planning equal employment opportunities. In Multi-Objective Programming (Ed. ZELENY M). Springer-Verlag, Berlin, Germany. I 1. CHARNESA. COOPER WW ,g' NIEHAU$ RJ (1972) Studies in Manpower Planning. US Navy Office of Civilian Manpower Management, Washington DC, USA. 12. CHARNES A, COOPER WW, NIEHAUS RJ & SHOLTZ D (1974) Multi level models for career management and resource planning. In [14], pp. 91-112. 13. CLOUGH DJ, DUDDING RC & PRICE WL (1971) Mathematical programming models of a quasi-independent subsystem of the Canadian forces manpower systems. In 1"37], pp. 299-316. 14. CLOUGH DJ, LEWIS CG & OLIVER AL (1974) Manpower Planning Models. English Universities Press, London, UK. 15. ELBERT NF & KEHOE WJ (1976) How to bridge fact and theory in manpower planning. Personnel 53(6), 31-39. 16. FERRISR, HULTZ J & KARNEY D (1978) Modeling and Solving large Scale Personnel Assignment Problems. Pap. Pres. at ORSA/TIMS meet., Los Angeles, USA. 17. GORHAM W (1963) An application of a network flow model to personnel planning. IEEE Trans. Engng Mgmt EMIO(3), 113-123.

O.~.L86--D

645

18. GRINOLD RC (1976) Manpower planning with uncertain requirements. Ops Res. 24(3), 387-399. 19. GRINOLD RC & MARSHALLKT (1977) Manpower Planning Models. North-Holland. New York, USA. 20. HENEMAN [II HG & SANDVER MG (1977) Markov analysis in human resource administration: applications and limitations. Acad. Mgmt Rev. 2(4L 21. JAQUETTEDL, NELSON GR & SMITH RJ (1977) An ana-

lytic reriew of personnel models in the Department of Defense. Rand Corporation rep. R-1920-ARPA. 22. LEWIS DG (Ed.) (1969) Manpower Plannin#: a Biblioyraphy. Elsevier, New York, USA. 23. L~wts KA. MAR~L A & PRICE WL (1977) Problems of manpower planning in decentralized organizations. In Proc XIII An Meet. Southeastern Chap. TIMS (Ed. HEBERT JE). 24. MARTEL A & AL-NUAIMI A (1973) Tactical manpower planning via programming under uncertainty. Opl Res. Q. 24; 571-585. 25. M~dt'I-ELA & PRXCl-'-WL (1978) A normative model for manpower planning under risk. In [7], pp. 291-306. 26. MILLER HE (1976) Personnel scheduling in public systems: a survey. Socio--Econ. Plann. Scis 10(6), 27. MORGAN RW (1971) Manpower planning in the royal air force: an exercise in linear programming. In [37]. pp. 317-326. 28. NIEHAUSRJ (1979) Computer Assisted Human Resource Planning. Wiley-lnterscience, New York, USA. 29. NUTTLE H (1969) Application of dynamic programming to employment planning. JI Ind. Engn# 20, , 30. PATZ A (1970) Linear programming applied to manpower management. Ind. Mgmt Rev. 11(2). 31. P1SKOR WG (1976) A Bibliography of Quantitative Techniques for Manpower Planning. Sloan School of Management, MIT, Cambridge, MA, USA. 32. PISKOR WG ~ DUDDING RC (1978) A computer assisted manpower planning model. In 17], pp. 145-154. 33. PRICE WL (1978) Solving goal-programming 'models using advanced network codes. JI Ops Res. Soc. 20(12), 1231-1239. 34. PRICE WL (1978) Measuring labor turnover for manpower planning. TIMS Stud. Mgmt Scis 8, 61-73. 35. l~lCt WL & Ptst
Facultd des Sciences de I'administration, Cit~ Universitaire, Qudbec lOe, Canada GIK 7P4.