- Email: [email protected]
Military manpower planning models
0305-0548191f3.00+0.00
CompurersOps Res.Vol.18. No. 1, pp. 65-73.1991 Printed
in Great
Britain.
Copyright
All rights reserved
MILITARY
MANPOWER SAUL
PLANNING I.
C 1991 Pergamon
Press plc
MODELS
GASS*
College of Business and Management, University of Maryland, College Park, MD 20742, U.S.A.
(Received
September
1989; in revised form January
1990)
Scope and Purpose-Manpower planning is basic to all large-scale organizations. Mathematical models and related computer-based algorithms are now available that can aid personnel planners to analyze future goals and to establish tradeofls between these often conflicting goals. In this short, self-contained primer, we lirst review some general manpower concepts and then describe some recent basic modeling structures that have been of great value in analyzing extended planning horizon military manpower problems. Although manpower models have been addressed by a number of authors, we feel that our selection of material and its presentation will help make the subject more accessible and, hopefully, cause some readers to apply proven military manpower models to nonmilitary areas. Abstract-4 this paper we review elements of manpower planning with emphasis on the large-scale problems faced by the military. After a brief introduction to some basic concepts, we describe transition rate (Markov) models, network models and network-like goal-programming models. Specific applications are reviewed, as appropriate.
INTRODUCTION
The basic manpower (personnel) planning problem is the following: “Determine the number of personnel and their skills that best meets the future operational requirements of an enterprise.” Every large-scale employer encounters this problem. The automobile industry must determine how to make the transition from mainly human labor to a mix of humans and robotics, while the postal service has to integrate its labor force with automatic handling equipment and to deal with the impact of electronic mail. Universities must react to the ever-changing size of the student population and the demand for the “in” subjects such as computer science and business. And, the military, while recruiting and training its personnel to meet its short-term commitments, must plan now to have the proper force that can handle high-technology weapon systems and meet future political and military goals. Readers interested in the basics and details of the mathematics and statistics of manpower planning are referred to the books by Genold and Marshall [l], Bartholomew and Forbes [2] and Charnes et al. [3]; a number of applications (military and nonmilitary) are given in the latter publication. In what follows, we first describe some basic manpower concepts, and then concentrate on more recent large-scale military applications and their development. We hope that this short, self-contained primer will encourage and enable the interested reader to apply these manpower structures to nonmilitary areas. Our concern is in short-term planning (l-3 years) and long-term planning (3-20 years). Although of equal importance and just as challenging, we will not touch on the related problem of manpower scheduling to meet current operational requirements; e.g. the scheduling of telephone operators or nurses or bus drivers [4,5]. *Saul I. Gass received his B.S. in Education and M.A. in Mathematics from Boston University, Mass., and his Ph.D. in Engineering Science/Operations Research from the University of California, Berkeley. He is currently Professor of Management Science and Statistics at the College of Business and Management, University of Maryland. Dr Gass first served as a mathematician for the Aberdeen Bombing Mission, U.S. Air Force, and then transferred to Air Force Headquarters where he began his career in operations research with the Directorate of Management Analysis. He was Manager of the Project Mercury man-in-space program for IBM, and Manager of IBM’s Federal Civil Programs. He was Senior Vice-President of World Systems Laboratories and Vice-President of Mathematics. He has served as a consultant to the U.S. General Accounting Otlice, Congressional Budget Ofice, the National Bureau of Standards and other operations research and systems analysis organizations. Included in his many publications are the text Linear Programming, now in its 5th edition, the book An Illustrated Guide to Linear Programming and the recently published text Decision Making, Models and Algorirhms. Dr Gass is a Past-President of the Operations Research Society of America and of Omega Rho, the international operations research honorary society. 65
SAUL I. GASS
66
A person’s job status changes over time. An individual’s job profile consists of being hired, changing skills, changing jobs, being promoted or fired, quitting, retiring, dying. When studying large personnel systems, it is difficult to keep track of the progression of the work force by individuals (although individual records are kept). Instead, we lind it convenient and appropriate to aggregate individuals by class descriptors over planning timeperiods. For example, we may have the following classes: Descrip~or~~~la~~
Time hired (senio~ty, years of service). Skills (machinist, laborer, programmer). Function (data processing, accounting). Job title (supervisor, manager, rank, grade). Each person is a member of one and only one class, but a person can change classes based on transition assumptions. These distinctions are readily captured by the following notation and its extensions: X(g, s, y, t) = the number of individuals in grade 8, with skill s, with years of service y in planning period t. A combination (g, s, y, t) is called a state; an individual can be in only one state at time period t, with the initial state conditions given by X(g, s, y, 0). For each time period of the planning horizon, the planning problem solution determines how many persons are in each state based on the assumptions concerning the transitions from one state to another. The problems can be very large due to the dimensions of Q, s, y, t and other indicators. We discuss these types of problems in terms of transition rate, network flow and goal-programming models. TRANSITION
RATE
(MARKOV)
MODELS
Transition rate models are used to forecast personnel inventory levels based on known transition rates. The problem: “Given a work force described by class descriptors at the beginning of the planning period, what is the com~sition of the force at the end of the planning period?” Models for solving such problems are often termed Markov models due to the problem assumptions, as will be described below. To illustrate, we let X(i, t) = the number of persons in class i at the beginning of period c, Z(i, t) = the number of class i hires in time period t 4ji = the fraction of class i that moves to class j during time period t (transition rate). We assume that i
qji=l
for(i=l,...,n)
f=l and Figure 1 describes a network in which the nodes represent classes of personnel by time period and the arcs represents the flow or transition of persons from one class to another. Here we have three classes and transitions can be made only to the same class or to a higher index class. Note that the total flow into a node is equal to the total flow out of the node, i.e. every person is accounted for (conservation of flow). The usual Markov assumptions for transition rate models are: Each individual is governed by a Markov process, i.e. only the last state occupied determines the individual’s future. 2. The same Markov process applies to all individu~s. 3. All individuals behave independently. 1.
61
Military manpower planning models
X(3.2) Fig. 1. Transition rate model network representation.
Markov models are used to estimate new hires, separations, retirements, training requirements, shortages by class and steady-state inventories. If we let Q = (qji), the transition matrix, and X(r) be the inventory of personnel by class in time t, then X(t + 1) = QX(t), and, thus, X(t + 1) = QX(l), where t= 1,2,. . . . Under the Markov assumptions, the matrix product Q converges to a matrix P as t + co, where
1Pl **- Pl P=
and
:I P2
P2
. . .
. . .
. .
Pn
- **
P.
TPi=l,
= :I (P
* * . P)
Pi>O.
For t large, say t = T >>0, we have X(7’) = PX(0) = Q’X(O), where X(T) is called the steady-state inventory. Letting X(T) = X = X(x,), we can find X by solving the (n + 1) equations in n variables: X=QX and C
Xi =
Sy
where S is the size of the constant inventory. The reader is referred to Grinold and Marshall [l] for details and extensions of this type of model. Transition rate models can be used to forecast next period personnel inventories with some accuracy, given the initial class inventories and a correct set of qii. For large personnel systems, there is some difficulty in collecting the q,i, and the steady-state forecasts may not be accurate enough as the qii do change over time. However, there have been long-term applications that have proved of value to military planners. In the paper by Gass et al. [6], a Markov model is used to project the Sow of an initial US. Army enlisted force to a future force over a 20-year horizon. The model outputs include the resultant changes to the force size and its composition, and the associated costs. For example, the model was employed to evaluate the effect of the declining youth population in the 1990s. As the U.S. Army wanted to maintain a constant strength of 666,900 over that period, it had to determine those actions necessary to overcome the anticipated decrease in new recruits. Assuming that the army would continue to attract the same percentage of the eligible population, the army needed to look to other personnel actions that would maintain the desired strength levels. Using the Markov model, the army evaluated increases in retention (decreased separations) through
SAUL I. Gass
68
increased reenlistments. The study indicated that an increase of 16% in reenlistments during the 1990s was necessary to achieve the desired result. But this change would cause an increase in the number of midcareer personnel, i.e. an “older” force. To maintain the desired grade distribution within the older force, the model decreased promotions throughout the 1990s. NETWORK-FLOW
MODELS
Network representations of personnel systems were recognized as appropriate tools of analysis early in the development of operational research. Votaw and Orden 173 showed that the personnel assignment problem was equivalent to the Hitchcock-Koopmans transportation (network) problem. The personnel assignment problem is to assign ai persons in personnel categories i to bj jobs in job categories j so as to minimize the cost of the assignments or to maximize the value of the assignments. The mathematical model is given by minimize (maximize)
7 7
subject to
CXij=a,
(i=l,...,m)
Txij=bj
(j=l,...,n)
Cij
xijk"9
where Xii, a, and bj are integers. The cij can be interpreted as costs of assignment (minimization) or values (productivity) of the assignment (maximization). The mathematical structure of the assignment model requires I
j
this can always be made the case by addition of shortage or surplus nodes. A network representation for m = 4 and n = 5 is given in Fig. 2, where node 5 is a lo-worker surplus node (100 workers are to be assigned to 90 jobs). Note, again, that we assume conservation of flow through a node, i.e. what goes into a node must come out of the node. Nonconservation of flow applications are quite important, and models and algorithms for solving them have been developed; Klingman and Glover [8] describe applications of the generalized assignment problem in which the flow across an arc is multiplied by a factor. Network representations of personnel systems have been extended greatly beyond the basic assignment model; see, for example, Mulvey [9], where a personnel scheduling model for assigning faculty members to classes is described, and Klingman and Glover [8] for a generalized network description for scheduling undergraduate flight training for the U.S. Air Force. For these problems, the basic form of the network is that of the more general minimal-cost transshipment (flow) network
WORKERS
JOBS
Fig. 2. Assignment model networks.
models
Military manpower planning
model. Its mathematical
description assumes conservation
minimize
C,
subject to
~xij-~
69
of flow and is as follows:
cijxij
Xji=ai
;:Xij-~Xji=O
(i a source node) (i
5Xij-~xji= j
-bi
an intermediate node)
(iasinknode),
0 ~ Iij ~ Xij f Uij.
A typical military application of the minimum-cost network-flow model has the arcs representing flows of personnel, the source nodes representing initial personnel inventories, the sink nodes final inventories and the intermediate nodes established to maintain personnel balances and to force inventories to meet grade and skill goals. A simple network with two initial inventory source nodes, two final inventory sink nodes and seven intermediate transfer nodes is given in Fig. 3. A problem that arises in the use of networks in describing personnel systems is that the very nice mathematical and computational features of networks (see [lo]) are degraded when we try to use network structures to analyze variables (flows) that have many identifiers, especially over many time periods. A main drawback here is that multiple flows entering a node can only take on common identifiers when leaving that node. For example, if two flows X(1,2, 10, t) and X(1,2,5, t) enter a node, then the total flow out of that node, i.e. the sum of the two flows coming in, can be identified only by X(1,2, t). Further, ending year inventories of personnel are usually aggregated totals with few identifiers and these figures are initial inventories for the next year that require detailed identifiers. However, as we shall see, the use of a network structure is both an important application and explanatory tool for formulating and understanding large-scale personnel models. A typical manpower network-flow minimum-cost model is described by Gass [l l] and we next present elements of that application. The model is called an oflicer distribution plan (ODP). We use the simple network of Fig. 4 to discuss the model formulation and some of the basic conditions of the ODP. The purpose of the ODP is to provide for effective planning and control of the distribution process for commissioned and warrant officers, i.e. we want to determine the flow of officer personnel among army commands as a function of grade and skill. We need to differentiate between U.S.-based and overseas commands. Each command has an individual grade/skill authorization, as well as a total personnel inventory level. An overseas command must rotate a determined number of personnel back to the U.S.A. Under these and other conditions, the ODP determines the flow of personnel among the commands for each grade/skill combination. For discussion purposes, we assume 6 grades, 20 commands and up to 300 skills. In what follows, we describe a network optimization model that is based on standard linear-programming concepts, augmented by network goalprogramming procedures. The illustrative network in Fig. 4 is for a single grade/skill combination (g, s) and four army commands (1,2,3,4). Each command has a beginning inventory IN(g, s, i), i = 1,2, 3,4. Here we
D IN(Z
Fig. 3. Simple force inventory network.
70
SAUL
I. Gus
Militarymanpbwerplanningmodels
71
do not distinguish between U.S. and overseas commands, and we assume that each command can ship to every other command. The nodes of the network are divided into five node types. Each command has node types l-4 associated with it. Node type 1 is a source node from which the initial inventories are split between personnel (flows) that stay at the command and those that are to be shipped to the other commands. The first flow (x) is from node type 1 to node type 4, and the second flow (2) is from node type 1 to node type 2. At a node type 2, the deployment of personnel from a command to the other commands takes place. These flows (y) go from node type 2 to node type 3. Note that a flow from node type 2 to node type 3 maintains the identity of the originating command. Each node type 3 sums all the personnel shipments going to the corresponding command. The flow out of node type 3 (Y) has lost the originating command identification and just represents the number of new personnel with grade g and skill s going to the corresponding command. At node type 4, the flow is the number of personnel that stayed at the command (x) plus the new shipment (Y ); these numbers are summed to X = x + Y. Note that X is constrained between a minimum fill goal FG and the strength maximum bound A for that command. Node type 5 illustrates how the totals for each command can also be summed so that they are constrained by some total strength upper bound. Note that the individual shipments x, 2, y, Y are also bounded by lower and upper bounds, as appropriate. As the grade/skill and total inventory targets are planning goals, a model constrained to exactly meet these goals, would, in general, be infeasible. Thus, it is appropriate to treat the targets in a goal-programming sense, i.e. we allow the final inventories to be above or below the targets and we attempt to mi’nimize the resultant weighted deviations. This is done by splitting the flows into the target nodes and forcing inventories to take place by use of negative and positive weights in the objective function. The final ODP network (which we do not illustrate here, see Ref. [l l]), is a modification of that shown in Fig. 4 and represents a minimum-cost network problem with 19,ooO nodes and 75,000 arcs. Networks of this and even larger dimensions can be readily solved using available specialized network computer codes. The reader is referred to Refs [12-151 for additional discussions on network goal-programming approaches to manpower planning; also, see Refs [8, 161 for modeling techniques and related applications using network technology.
MULTIYEAR MANPOWER PLANNING MODELS From the previous discussion, it should be clear that a network flow description of a manpower problem (or any other application) is a powerful descriptive device. One does not have to display complex equations to explain how flows are transformed, accumulated and controlled. Indeed, a network is worth 10,000 equations! However, as a tool for accomplishing detailed, time-dependent analyses, we find the network model too restrictive. As noted earlier, when two different flows X(g, s, t) enter a node, they can be accumulated (exit that node) only in terms of their similar indices. For example, in Fig. 4, the flows that originate from node type 4 are identified by command as they enter node 5 (the total inventory node). These flows are added and exit node 5 as the total force with no command identifiers. If we wanted to do a combined 2-year plan, i.e. start the second year with end-of-first-year inventories, we have no way of joining the first and second year networks as we cannot break the flow out of node 5 into the individual command end-of-year inventories. Each year’s network would have to be solved separately (suboptimized) and in succession. Thus, for multiyear, detailed manpower models we resort to explicit linear- and goal-programming formulations. To illustrate the power and scope of such multiyear planning models, we briefly discuss the manpower planning model of Gass et al. [6]. This model determines how an initial force inventory, indexed by grade (g = 1, . . . ,7) and skill categories (s = 1, . . . ,33) can be modified to meet desired future force targets over a lo-year planning horizon. The initial force is transformed into the succeeding years’ forces by separations, promotions and accessions; we assume known rates of separation and promotion, and limits on accessions. Yearly targets are given for separations and promotions (as functions of the previous year’s inventory for each grade and year-of-service combination), grade targets and total force targets. We next describe elements of the multiyear manpower planning goal-programming model, see Gass et al. [6] for full details. The goal-programming constraints are target conditions in that each represents a function of
72
SAUL
1. GASS
the variables (e.g. gains, losses, promotions, skill inventories) that is set equal, in a goal-programming sense, to a target (goal) value. A typical constraint can be written (functionally) as fCx(t, $7,s)l + m
89s)- m 9, s) = w, BY s),
where f[_Y(t, g, s)] is a function of decision variables such as separations or promotions or inventories in time period t, grade g and skill s; P(t, g, s) and N(t, g, s) are the goal-programming under- and overachievement deviation variables for the target constraint, respectively; and T(t, g, s) is the target goal for the function. Each P(t, g, s) and N(t, g, s) appears in the linear objective function multiplied by a weight WP(t, g, s) and WN(t, g, s), respectively. These weights are meant to indicate the importance of meeting the associated target and reflect the decision maker’s explicit and implicit tradeoffs in selecting a particular solution to implement (or to use as a basis for further planning). It should be clear that the element that makes this type of problem difficult is that there is no true single optimizing solution and the solution selection process is one of compromise and satisficing. The analysis problem reduces to determining values of the thousands of weights WP(t, g, s) and WN(t, g, s) such that the solution produced in optimizing (here minimizing) the objective function subject to the target constraints would be a compromise solution acceptable to the decision maker. (See Ref. [17] for a discussion of weight determination.) Typical constraints include the following: (i) Grade-skill target goal constrainrs INVX(t, g, s) + TARP@, g, s) - TARN@, g, s) = TARG(t, g, s). For each grade g and skill combination s in year t, if the total INVX(t, g, s) does not equal the desired end-strength target, TARG(t, g, s), for that grade and skill, then a shortage is noted by TARP(t, g, s), or a surplus by TARN@, g, s). (ii) Promotion goal constraints PROX(t, g, s) + PROP@, g, s) - PRON(t, g, s) =RPRO(tl,g- l,s)INVX(t-l,g1,s) The desired number of promotions to grade g during year t is the rate of promotion RPRO(r - 1, g - 1, s) times the year t beginning inventory INVX(r - 1, g - I, s) in grade g - 1 for all s. These numbers are treated as goals. If the actual number of promotions PROX(t, g, s) is less than the corresponding goal, then a shortage is noted by adding PROP& g, s); if the actual number of promotions is greater than the goal, then a surplus is noted by subtracting PRONft, g, s). An application of this model had 9060 constraints and 28,730 variables; 6950 of the equations were goal constraints, and it was solved in 3.72 min. The objective function consisted of a weighted function of the 13,900 goal-programming deviation variables. A related model that dealt with 7 grades, 20 years-of-service categories over a 20-year time horizon is also detailed in Ref. [6]. Other multiperiod/goal-programming models are described elsewhere [ 18-2 11.
SUMMARY
As the preceding discussion demonstrates, many important technical and application developments have been made in the general field of military manpower planning. The basic problems are amenable to operational research and computer-based techniques, and, we feel, can be applied to civilian organizations. However, there will be a continuing need for new developments, especially those that deal with long-range planning where future organizational goals (military and nonmilitary) are ever-changing and, in some sense, indeterminate. Operational researchers must develop methods that will yield consistent and realistic solutions so that manpower planners can be confident that their model-based decisions reelect the planners’ version of future manpower requirements.
Military manpower planning models
73
REFERENCES 1. R. C. Grinold and K. T. Marshall, Manpower Plunning Models. North-Holland, Amsterdam (1977). 2. D. J. Bartholomew and A. F. Forbes, Statistical Techniquesfor Manpower Planning. Wiley, New York (1979). 3. A. Chames, W. W. Cooper and R. J. Niehaus, Management Science Approaches to Manpower Planning and Organization Design. North-Holland, Amsterdam (1978). 4. M. Segal, The operator scheduling problem: a network flow approach. Ops Res. 22 (1974). 5. J. M. Tien and A. Kamiyama, On manpower scheduling algorithms. SIAM Rev. 275-287 (1982). 6. S. I. Gass, R. W. Collins, C. W. Meinhardt, D. M. Lemon and M. D. Gillette, The army manpower long-range planning system. Ops Res. 36, 5-17 (1988). 7. D. F. Votaw Jr and A. Orden, The Personnel Assignment Problem.In Linear Inequalities and Programming (Edited by A. Orden and L. Goldstein). DLS/Comptroller, Headquarters U.S. Air Force, Washington, D.C. (1952). 8. D. Klingman and F. Glover, Tutorial: networks. Working Paper CBDA 118, Center for Business Decision Analysis, Univ. of Texas, Austin (1984). 9. J. H. Mulvey, Strategies in modeling: a personnel scheduling example. Interjaces 9, 66-76 (1979). 10. P. A. Jensen and J. W. Barnes, Network Flow Programming. Wiley, New York (1980). 11. S. I. Gass, On the development of large-scale personnel planning models. In Svstem Mode&g and Optimization (Edited by P. Throft-Christensen). Springer, Berlin (1983). 12. W. L. Price, Solving goal-programming manpower models using advanced network codes. J. Opl Res. Sot. 29,123 1-1239 (1978). 13. W. L. Price and M. Gravel, Network manpower models: including attrition calculations. Working Paper 79-09, Lava1 Univ., Quebec (1979). 14. D. Klingman, M. Mead and N. V. Phillips, Network optimization models for manpower planning. In Operational Research ‘84 (Edited by Brams). Elsevier, Amsterdam (1984). 15. D. Klingman and N. V. Phillips, Topological and computational aspects of preemptive multicriteria military personnel assignment problems. Mgmt Sci. 30, 1362-1375 (1984). 16. F. Glover and D. Klingman, Network applications in industry and government. AIIE Trans. 363-376 (1977). 17. S. I. Gass, A process for determining priorities and weights for large-scale linear goal-programming models. J. Opl Res. Sot. 37, (1986). 18. W. L. Price and W. G. Priskor, The use of goal programming in manpower planning. INFOR 10, 3 (1972). - __. 19. ci. Mltra and K. C. Chai, A multicriteria stochastic programming model for manpower planning. In Proceedings of the Third Symposium on Operations Research (Edited by W. Oettli and F. Steffens). Athenaum, Hain, Scriptor & Hanstein, Berlin (1978). 20. E. S. Bres, D. Bums, A. Chames and W. W. Cooper, A goal programming model for planning officer accessions. Research Report CSC 335, Center for Cybernetic Studies, Univ. of Texas, Austin (1980). 21. R. J. Niehaus, Computer Assisted Human Resources Planning. Wiley, New York (1979).
CompurersOps Res.Vol.18. No. 1, pp. 65-73.1991 Printed
in Great
Britain.
Copyright
All rights reserved
MILITARY
MANPOWER SAUL
PLANNING I.
C 1991 Pergamon
Press plc
MODELS
GASS*
College of Business and Management, University of Maryland, College Park, MD 20742, U.S.A.
(Received
September
1989; in revised form January
1990)
Scope and Purpose-Manpower planning is basic to all large-scale organizations. Mathematical models and related computer-based algorithms are now available that can aid personnel planners to analyze future goals and to establish tradeofls between these often conflicting goals. In this short, self-contained primer, we lirst review some general manpower concepts and then describe some recent basic modeling structures that have been of great value in analyzing extended planning horizon military manpower problems. Although manpower models have been addressed by a number of authors, we feel that our selection of material and its presentation will help make the subject more accessible and, hopefully, cause some readers to apply proven military manpower models to nonmilitary areas. Abstract-4 this paper we review elements of manpower planning with emphasis on the large-scale problems faced by the military. After a brief introduction to some basic concepts, we describe transition rate (Markov) models, network models and network-like goal-programming models. Specific applications are reviewed, as appropriate.
INTRODUCTION
The basic manpower (personnel) planning problem is the following: “Determine the number of personnel and their skills that best meets the future operational requirements of an enterprise.” Every large-scale employer encounters this problem. The automobile industry must determine how to make the transition from mainly human labor to a mix of humans and robotics, while the postal service has to integrate its labor force with automatic handling equipment and to deal with the impact of electronic mail. Universities must react to the ever-changing size of the student population and the demand for the “in” subjects such as computer science and business. And, the military, while recruiting and training its personnel to meet its short-term commitments, must plan now to have the proper force that can handle high-technology weapon systems and meet future political and military goals. Readers interested in the basics and details of the mathematics and statistics of manpower planning are referred to the books by Genold and Marshall [l], Bartholomew and Forbes [2] and Charnes et al. [3]; a number of applications (military and nonmilitary) are given in the latter publication. In what follows, we first describe some basic manpower concepts, and then concentrate on more recent large-scale military applications and their development. We hope that this short, self-contained primer will encourage and enable the interested reader to apply these manpower structures to nonmilitary areas. Our concern is in short-term planning (l-3 years) and long-term planning (3-20 years). Although of equal importance and just as challenging, we will not touch on the related problem of manpower scheduling to meet current operational requirements; e.g. the scheduling of telephone operators or nurses or bus drivers [4,5]. *Saul I. Gass received his B.S. in Education and M.A. in Mathematics from Boston University, Mass., and his Ph.D. in Engineering Science/Operations Research from the University of California, Berkeley. He is currently Professor of Management Science and Statistics at the College of Business and Management, University of Maryland. Dr Gass first served as a mathematician for the Aberdeen Bombing Mission, U.S. Air Force, and then transferred to Air Force Headquarters where he began his career in operations research with the Directorate of Management Analysis. He was Manager of the Project Mercury man-in-space program for IBM, and Manager of IBM’s Federal Civil Programs. He was Senior Vice-President of World Systems Laboratories and Vice-President of Mathematics. He has served as a consultant to the U.S. General Accounting Otlice, Congressional Budget Ofice, the National Bureau of Standards and other operations research and systems analysis organizations. Included in his many publications are the text Linear Programming, now in its 5th edition, the book An Illustrated Guide to Linear Programming and the recently published text Decision Making, Models and Algorirhms. Dr Gass is a Past-President of the Operations Research Society of America and of Omega Rho, the international operations research honorary society. 65
SAUL I. GASS
66
A person’s job status changes over time. An individual’s job profile consists of being hired, changing skills, changing jobs, being promoted or fired, quitting, retiring, dying. When studying large personnel systems, it is difficult to keep track of the progression of the work force by individuals (although individual records are kept). Instead, we lind it convenient and appropriate to aggregate individuals by class descriptors over planning timeperiods. For example, we may have the following classes: Descrip~or~~~la~~
Time hired (senio~ty, years of service). Skills (machinist, laborer, programmer). Function (data processing, accounting). Job title (supervisor, manager, rank, grade). Each person is a member of one and only one class, but a person can change classes based on transition assumptions. These distinctions are readily captured by the following notation and its extensions: X(g, s, y, t) = the number of individuals in grade 8, with skill s, with years of service y in planning period t. A combination (g, s, y, t) is called a state; an individual can be in only one state at time period t, with the initial state conditions given by X(g, s, y, 0). For each time period of the planning horizon, the planning problem solution determines how many persons are in each state based on the assumptions concerning the transitions from one state to another. The problems can be very large due to the dimensions of Q, s, y, t and other indicators. We discuss these types of problems in terms of transition rate, network flow and goal-programming models. TRANSITION
RATE
(MARKOV)
MODELS
Transition rate models are used to forecast personnel inventory levels based on known transition rates. The problem: “Given a work force described by class descriptors at the beginning of the planning period, what is the com~sition of the force at the end of the planning period?” Models for solving such problems are often termed Markov models due to the problem assumptions, as will be described below. To illustrate, we let X(i, t) = the number of persons in class i at the beginning of period c, Z(i, t) = the number of class i hires in time period t 4ji = the fraction of class i that moves to class j during time period t (transition rate). We assume that i
qji=l
for(i=l,...,n)
f=l and Figure 1 describes a network in which the nodes represent classes of personnel by time period and the arcs represents the flow or transition of persons from one class to another. Here we have three classes and transitions can be made only to the same class or to a higher index class. Note that the total flow into a node is equal to the total flow out of the node, i.e. every person is accounted for (conservation of flow). The usual Markov assumptions for transition rate models are: Each individual is governed by a Markov process, i.e. only the last state occupied determines the individual’s future. 2. The same Markov process applies to all individu~s. 3. All individuals behave independently. 1.
61
Military manpower planning models
X(3.2) Fig. 1. Transition rate model network representation.
Markov models are used to estimate new hires, separations, retirements, training requirements, shortages by class and steady-state inventories. If we let Q = (qji), the transition matrix, and X(r) be the inventory of personnel by class in time t, then X(t + 1) = QX(t), and, thus, X(t + 1) = QX(l), where t= 1,2,. . . . Under the Markov assumptions, the matrix product Q converges to a matrix P as t + co, where
1Pl **- Pl P=
and
:I P2
P2
. . .
. . .
. .
Pn
- **
P.
TPi=l,
= :I (P
* * . P)
Pi>O.
For t large, say t = T >>0, we have X(7’) = PX(0) = Q’X(O), where X(T) is called the steady-state inventory. Letting X(T) = X = X(x,), we can find X by solving the (n + 1) equations in n variables: X=QX and C
Xi =
Sy
where S is the size of the constant inventory. The reader is referred to Grinold and Marshall [l] for details and extensions of this type of model. Transition rate models can be used to forecast next period personnel inventories with some accuracy, given the initial class inventories and a correct set of qii. For large personnel systems, there is some difficulty in collecting the q,i, and the steady-state forecasts may not be accurate enough as the qii do change over time. However, there have been long-term applications that have proved of value to military planners. In the paper by Gass et al. [6], a Markov model is used to project the Sow of an initial US. Army enlisted force to a future force over a 20-year horizon. The model outputs include the resultant changes to the force size and its composition, and the associated costs. For example, the model was employed to evaluate the effect of the declining youth population in the 1990s. As the U.S. Army wanted to maintain a constant strength of 666,900 over that period, it had to determine those actions necessary to overcome the anticipated decrease in new recruits. Assuming that the army would continue to attract the same percentage of the eligible population, the army needed to look to other personnel actions that would maintain the desired strength levels. Using the Markov model, the army evaluated increases in retention (decreased separations) through
SAUL I. Gass
68
increased reenlistments. The study indicated that an increase of 16% in reenlistments during the 1990s was necessary to achieve the desired result. But this change would cause an increase in the number of midcareer personnel, i.e. an “older” force. To maintain the desired grade distribution within the older force, the model decreased promotions throughout the 1990s. NETWORK-FLOW
MODELS
Network representations of personnel systems were recognized as appropriate tools of analysis early in the development of operational research. Votaw and Orden 173 showed that the personnel assignment problem was equivalent to the Hitchcock-Koopmans transportation (network) problem. The personnel assignment problem is to assign ai persons in personnel categories i to bj jobs in job categories j so as to minimize the cost of the assignments or to maximize the value of the assignments. The mathematical model is given by minimize (maximize)
7 7
subject to
CXij=a,
(i=l,...,m)
Txij=bj
(j=l,...,n)
Cij
xijk"9
where Xii, a, and bj are integers. The cij can be interpreted as costs of assignment (minimization) or values (productivity) of the assignment (maximization). The mathematical structure of the assignment model requires I
j
this can always be made the case by addition of shortage or surplus nodes. A network representation for m = 4 and n = 5 is given in Fig. 2, where node 5 is a lo-worker surplus node (100 workers are to be assigned to 90 jobs). Note, again, that we assume conservation of flow through a node, i.e. what goes into a node must come out of the node. Nonconservation of flow applications are quite important, and models and algorithms for solving them have been developed; Klingman and Glover [8] describe applications of the generalized assignment problem in which the flow across an arc is multiplied by a factor. Network representations of personnel systems have been extended greatly beyond the basic assignment model; see, for example, Mulvey [9], where a personnel scheduling model for assigning faculty members to classes is described, and Klingman and Glover [8] for a generalized network description for scheduling undergraduate flight training for the U.S. Air Force. For these problems, the basic form of the network is that of the more general minimal-cost transshipment (flow) network
WORKERS
JOBS
Fig. 2. Assignment model networks.
models
Military manpower planning
model. Its mathematical
description assumes conservation
minimize
C,
subject to
~xij-~
69
of flow and is as follows:
cijxij
Xji=ai
;:Xij-~Xji=O
(i a source node) (i
5Xij-~xji= j
-bi
an intermediate node)
(iasinknode),
0 ~ Iij ~ Xij f Uij.
A typical military application of the minimum-cost network-flow model has the arcs representing flows of personnel, the source nodes representing initial personnel inventories, the sink nodes final inventories and the intermediate nodes established to maintain personnel balances and to force inventories to meet grade and skill goals. A simple network with two initial inventory source nodes, two final inventory sink nodes and seven intermediate transfer nodes is given in Fig. 3. A problem that arises in the use of networks in describing personnel systems is that the very nice mathematical and computational features of networks (see [lo]) are degraded when we try to use network structures to analyze variables (flows) that have many identifiers, especially over many time periods. A main drawback here is that multiple flows entering a node can only take on common identifiers when leaving that node. For example, if two flows X(1,2, 10, t) and X(1,2,5, t) enter a node, then the total flow out of that node, i.e. the sum of the two flows coming in, can be identified only by X(1,2, t). Further, ending year inventories of personnel are usually aggregated totals with few identifiers and these figures are initial inventories for the next year that require detailed identifiers. However, as we shall see, the use of a network structure is both an important application and explanatory tool for formulating and understanding large-scale personnel models. A typical manpower network-flow minimum-cost model is described by Gass [l l] and we next present elements of that application. The model is called an oflicer distribution plan (ODP). We use the simple network of Fig. 4 to discuss the model formulation and some of the basic conditions of the ODP. The purpose of the ODP is to provide for effective planning and control of the distribution process for commissioned and warrant officers, i.e. we want to determine the flow of officer personnel among army commands as a function of grade and skill. We need to differentiate between U.S.-based and overseas commands. Each command has an individual grade/skill authorization, as well as a total personnel inventory level. An overseas command must rotate a determined number of personnel back to the U.S.A. Under these and other conditions, the ODP determines the flow of personnel among the commands for each grade/skill combination. For discussion purposes, we assume 6 grades, 20 commands and up to 300 skills. In what follows, we describe a network optimization model that is based on standard linear-programming concepts, augmented by network goalprogramming procedures. The illustrative network in Fig. 4 is for a single grade/skill combination (g, s) and four army commands (1,2,3,4). Each command has a beginning inventory IN(g, s, i), i = 1,2, 3,4. Here we
D IN(Z
Fig. 3. Simple force inventory network.
70
SAUL
I. Gus
Militarymanpbwerplanningmodels
71
do not distinguish between U.S. and overseas commands, and we assume that each command can ship to every other command. The nodes of the network are divided into five node types. Each command has node types l-4 associated with it. Node type 1 is a source node from which the initial inventories are split between personnel (flows) that stay at the command and those that are to be shipped to the other commands. The first flow (x) is from node type 1 to node type 4, and the second flow (2) is from node type 1 to node type 2. At a node type 2, the deployment of personnel from a command to the other commands takes place. These flows (y) go from node type 2 to node type 3. Note that a flow from node type 2 to node type 3 maintains the identity of the originating command. Each node type 3 sums all the personnel shipments going to the corresponding command. The flow out of node type 3 (Y) has lost the originating command identification and just represents the number of new personnel with grade g and skill s going to the corresponding command. At node type 4, the flow is the number of personnel that stayed at the command (x) plus the new shipment (Y ); these numbers are summed to X = x + Y. Note that X is constrained between a minimum fill goal FG and the strength maximum bound A for that command. Node type 5 illustrates how the totals for each command can also be summed so that they are constrained by some total strength upper bound. Note that the individual shipments x, 2, y, Y are also bounded by lower and upper bounds, as appropriate. As the grade/skill and total inventory targets are planning goals, a model constrained to exactly meet these goals, would, in general, be infeasible. Thus, it is appropriate to treat the targets in a goal-programming sense, i.e. we allow the final inventories to be above or below the targets and we attempt to mi’nimize the resultant weighted deviations. This is done by splitting the flows into the target nodes and forcing inventories to take place by use of negative and positive weights in the objective function. The final ODP network (which we do not illustrate here, see Ref. [l l]), is a modification of that shown in Fig. 4 and represents a minimum-cost network problem with 19,ooO nodes and 75,000 arcs. Networks of this and even larger dimensions can be readily solved using available specialized network computer codes. The reader is referred to Refs [12-151 for additional discussions on network goal-programming approaches to manpower planning; also, see Refs [8, 161 for modeling techniques and related applications using network technology.
MULTIYEAR MANPOWER PLANNING MODELS From the previous discussion, it should be clear that a network flow description of a manpower problem (or any other application) is a powerful descriptive device. One does not have to display complex equations to explain how flows are transformed, accumulated and controlled. Indeed, a network is worth 10,000 equations! However, as a tool for accomplishing detailed, time-dependent analyses, we find the network model too restrictive. As noted earlier, when two different flows X(g, s, t) enter a node, they can be accumulated (exit that node) only in terms of their similar indices. For example, in Fig. 4, the flows that originate from node type 4 are identified by command as they enter node 5 (the total inventory node). These flows are added and exit node 5 as the total force with no command identifiers. If we wanted to do a combined 2-year plan, i.e. start the second year with end-of-first-year inventories, we have no way of joining the first and second year networks as we cannot break the flow out of node 5 into the individual command end-of-year inventories. Each year’s network would have to be solved separately (suboptimized) and in succession. Thus, for multiyear, detailed manpower models we resort to explicit linear- and goal-programming formulations. To illustrate the power and scope of such multiyear planning models, we briefly discuss the manpower planning model of Gass et al. [6]. This model determines how an initial force inventory, indexed by grade (g = 1, . . . ,7) and skill categories (s = 1, . . . ,33) can be modified to meet desired future force targets over a lo-year planning horizon. The initial force is transformed into the succeeding years’ forces by separations, promotions and accessions; we assume known rates of separation and promotion, and limits on accessions. Yearly targets are given for separations and promotions (as functions of the previous year’s inventory for each grade and year-of-service combination), grade targets and total force targets. We next describe elements of the multiyear manpower planning goal-programming model, see Gass et al. [6] for full details. The goal-programming constraints are target conditions in that each represents a function of
72
SAUL
1. GASS
the variables (e.g. gains, losses, promotions, skill inventories) that is set equal, in a goal-programming sense, to a target (goal) value. A typical constraint can be written (functionally) as fCx(t, $7,s)l + m
89s)- m 9, s) = w, BY s),
where f[_Y(t, g, s)] is a function of decision variables such as separations or promotions or inventories in time period t, grade g and skill s; P(t, g, s) and N(t, g, s) are the goal-programming under- and overachievement deviation variables for the target constraint, respectively; and T(t, g, s) is the target goal for the function. Each P(t, g, s) and N(t, g, s) appears in the linear objective function multiplied by a weight WP(t, g, s) and WN(t, g, s), respectively. These weights are meant to indicate the importance of meeting the associated target and reflect the decision maker’s explicit and implicit tradeoffs in selecting a particular solution to implement (or to use as a basis for further planning). It should be clear that the element that makes this type of problem difficult is that there is no true single optimizing solution and the solution selection process is one of compromise and satisficing. The analysis problem reduces to determining values of the thousands of weights WP(t, g, s) and WN(t, g, s) such that the solution produced in optimizing (here minimizing) the objective function subject to the target constraints would be a compromise solution acceptable to the decision maker. (See Ref. [17] for a discussion of weight determination.) Typical constraints include the following: (i) Grade-skill target goal constrainrs INVX(t, g, s) + TARP@, g, s) - TARN@, g, s) = TARG(t, g, s). For each grade g and skill combination s in year t, if the total INVX(t, g, s) does not equal the desired end-strength target, TARG(t, g, s), for that grade and skill, then a shortage is noted by TARP(t, g, s), or a surplus by TARN@, g, s). (ii) Promotion goal constraints PROX(t, g, s) + PROP@, g, s) - PRON(t, g, s) =RPRO(tl,g- l,s)INVX(t-l,g1,s) The desired number of promotions to grade g during year t is the rate of promotion RPRO(r - 1, g - 1, s) times the year t beginning inventory INVX(r - 1, g - I, s) in grade g - 1 for all s. These numbers are treated as goals. If the actual number of promotions PROX(t, g, s) is less than the corresponding goal, then a shortage is noted by adding PROP& g, s); if the actual number of promotions is greater than the goal, then a surplus is noted by subtracting PRONft, g, s). An application of this model had 9060 constraints and 28,730 variables; 6950 of the equations were goal constraints, and it was solved in 3.72 min. The objective function consisted of a weighted function of the 13,900 goal-programming deviation variables. A related model that dealt with 7 grades, 20 years-of-service categories over a 20-year time horizon is also detailed in Ref. [6]. Other multiperiod/goal-programming models are described elsewhere [ 18-2 11.
SUMMARY
As the preceding discussion demonstrates, many important technical and application developments have been made in the general field of military manpower planning. The basic problems are amenable to operational research and computer-based techniques, and, we feel, can be applied to civilian organizations. However, there will be a continuing need for new developments, especially those that deal with long-range planning where future organizational goals (military and nonmilitary) are ever-changing and, in some sense, indeterminate. Operational researchers must develop methods that will yield consistent and realistic solutions so that manpower planners can be confident that their model-based decisions reelect the planners’ version of future manpower requirements.
Military manpower planning models
73
REFERENCES 1. R. C. Grinold and K. T. Marshall, Manpower Plunning Models. North-Holland, Amsterdam (1977). 2. D. J. Bartholomew and A. F. Forbes, Statistical Techniquesfor Manpower Planning. Wiley, New York (1979). 3. A. Chames, W. W. Cooper and R. J. Niehaus, Management Science Approaches to Manpower Planning and Organization Design. North-Holland, Amsterdam (1978). 4. M. Segal, The operator scheduling problem: a network flow approach. Ops Res. 22 (1974). 5. J. M. Tien and A. Kamiyama, On manpower scheduling algorithms. SIAM Rev. 275-287 (1982). 6. S. I. Gass, R. W. Collins, C. W. Meinhardt, D. M. Lemon and M. D. Gillette, The army manpower long-range planning system. Ops Res. 36, 5-17 (1988). 7. D. F. Votaw Jr and A. Orden, The Personnel Assignment Problem.In Linear Inequalities and Programming (Edited by A. Orden and L. Goldstein). DLS/Comptroller, Headquarters U.S. Air Force, Washington, D.C. (1952). 8. D. Klingman and F. Glover, Tutorial: networks. Working Paper CBDA 118, Center for Business Decision Analysis, Univ. of Texas, Austin (1984). 9. J. H. Mulvey, Strategies in modeling: a personnel scheduling example. Interjaces 9, 66-76 (1979). 10. P. A. Jensen and J. W. Barnes, Network Flow Programming. Wiley, New York (1980). 11. S. I. Gass, On the development of large-scale personnel planning models. In Svstem Mode&g and Optimization (Edited by P. Throft-Christensen). Springer, Berlin (1983). 12. W. L. Price, Solving goal-programming manpower models using advanced network codes. J. Opl Res. Sot. 29,123 1-1239 (1978). 13. W. L. Price and M. Gravel, Network manpower models: including attrition calculations. Working Paper 79-09, Lava1 Univ., Quebec (1979). 14. D. Klingman, M. Mead and N. V. Phillips, Network optimization models for manpower planning. In Operational Research ‘84 (Edited by Brams). Elsevier, Amsterdam (1984). 15. D. Klingman and N. V. Phillips, Topological and computational aspects of preemptive multicriteria military personnel assignment problems. Mgmt Sci. 30, 1362-1375 (1984). 16. F. Glover and D. Klingman, Network applications in industry and government. AIIE Trans. 363-376 (1977). 17. S. I. Gass, A process for determining priorities and weights for large-scale linear goal-programming models. J. Opl Res. Sot. 37, (1986). 18. W. L. Price and W. G. Priskor, The use of goal programming in manpower planning. INFOR 10, 3 (1972). - __. 19. ci. Mltra and K. C. Chai, A multicriteria stochastic programming model for manpower planning. In Proceedings of the Third Symposium on Operations Research (Edited by W. Oettli and F. Steffens). Athenaum, Hain, Scriptor & Hanstein, Berlin (1978). 20. E. S. Bres, D. Bums, A. Chames and W. W. Cooper, A goal programming model for planning officer accessions. Research Report CSC 335, Center for Cybernetic Studies, Univ. of Texas, Austin (1980). 21. R. J. Niehaus, Computer Assisted Human Resources Planning. Wiley, New York (1979).