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Educational models for manpower development
TECHNOLOGICAL
FORECASTING
AND SOCIAL
CHANGE
8,309-324
(1976)
309
Educational Models for Manpower Development WARREN L. BALINSKY
ABSTRACT In this paper a number of educational models are formulated and optimized. A single educational program multiperiod model is developed and optimized by means of dynamic programming. The type of results an educational admini,strator might obtain from this model and optimization are presented in an example which evaluates alternative future options. This single educational program model is extended to a multi-echelon model, allowing for the modifications to very realistic “compulsory-type” and “college-type” educational models. Linear programming generates the optimal decisions for the general multi-echelon multiperiod educational model.
1. Introduction and Historical Background To control the ever rising cost of education and to find improved and more effective methods of optimally allocating resources, the application of management techniques is becoming increasingly popular among individual practitioners and both private and public institutions. More often than not they build their models and structure their programs from huge statistical information, figures and graphs, various characteristics of the system, and impressive analysis of educational attainments of individuals at various levels in the system. But such efforts, though highly regarded for the validity of the results they produce, contribute very little, if at all, toward adequately clarifying the real issues involved, or even to a satisfactory policy formulation for the futurg. The main reason for such shortcomings is that most studies done in this area ignore the dynamics of the system concerning the issues in focus. A dynamic framework with unifying concepts of investigating the essential facts in quantitative terms and put together in a form readily understandable by policymakers is what is needed most for effective use of the results. The objective of this study is to consider the underlying relationship between the economic and educational sectors of a society by means of time dependent flow models. It is widely known that the sustenance of socioeconomic development is dependent upon a continuous and adequate supply of human resources of various skills and trainings. The spectrum of these resources is broad, ranging from people with basic literacy to those with advanced scientific and technical backgrounds. Nevertheless, the human-material resources are always scarce and the amount of these resources that can be allocated for educational purposes, for establishing and operating facilities, is limited. Thus it becomes relevant to explore and develop effective policies which will generate needed manpower DR. BALINSKY is Associate Professor in the Health Services Administration Program at the State University of New York at Stony Brook. Prior to this appointment he was Assistant Professor in the Management Systems Department of the School of Management, State University of New York at Buffalo. His research interests center on the application of management science techniques to problems in the areas of health management and educational management with special emphasis on the solution techniques of mathematical programming and statistics. 0 American
Elsevier Publishing
Company,
Inc., 1976
310
WARREN L. BALINSKY
skills through time in a quantity necessary to maintain an equilibrium between supply and demand of these skills. It is, therefore, imperative for the orderly planning of an educational structure to be consistent with the projected manpower needs and overall economic investment. This paper, in summary, is an attempt to look into this problem of systematic planning of educational structures for achieving manpower goals in an optimal fashion. There exists a small body of literature which has entertained the problem of orderly planning of an educational system. Bolt, Koltun and Levine [8] were the first to study the dynamics of this problem and develop a model which takes account of the characteristics of the system. Their model treats the educational system (higher education) as one sector and the rest of the economy as the other. A feedback model then represents the process of degree production. The model describes the flows of graduates between these two sectors by establishing that Ph.D. degree holders go back to the academic sector as faculty or administrators, while others are absorbed by industrial and governmental positions. The model makes use of two linear difference equations to represent flows in the economy. Simultaneous solutions of the difference equations are obtained for various parameter values. The idea of the parametric study is to provide policymakers with better insight to recognize the consequences of programs they might recommend. The model has a number of weaknesses. Some of the limitations recognized by the authors themselves are: “The model ignores the input of students into education,” and “The model is defined only for linear behavior.” Another weakness of the model is its handling of each level of education as an independent process, which is contrary to the fact. Bolt, Koltun and Levine’s model has been extended by R&man’s original model [ 191 and its extension by Reisman’s and Taft [20]. Their approach to the sectors is less aggregative than that of the Bolt, Koltun and Levine model. They consider the interactions between lower and higher levels of education, particularly the dependence of higher levels of education for their input on the output of lower levels. They also study flows as difference equations suitable for computer simulation, which give their model an advantage over the other model. These models are very helpful in understanding the feedback processes into higher education. But, as pointed out earlier, none of the studies in the area of manpower planning appearing in the literature so far has attempted to specifically obtain an optimal analytical solution to the problem. The development of this type of model as a policymaking tool is, therefore, in order. As indicated by Levine [ 141 , the analytical techniques for evaluating the effectiveness of a given set of policies is that of requiring the policies to produce a supply of trained manpower at each educational level within certain restrictive bounds. This is accounted for in the present study. Within the scope of this report, however, all the optimization techniques applicable to the problem are not explored. 2. A Single Educational Program-Multiperiod Model The objective of this paper is to formulate and discuss the optimization of progressively more realistic manpower planning models. The models are characteristically hierarchical and are built around the multilevel nature of the educational/training’ programs. First, a single educational program-multiperiod manpower planning model will be ’ Education and training may be used interchangeably in thib paper. well as formal education may be equally modelled using the developments
since job training, herein.
informal
as
EDUCATIONAL
MODELS
FOR MANPOWER
DEVELOPMENT
311
developed. A single educational program means that there is only one educational program in which individuals may or may not participate. For example, a training program for high school dropouts, a training program for high school graduates, or a master’s degree program for a particular specialty. As a qualification for a single level educational program, a program must have a provision for distinguishing between persons who successfully complete the program, like awarding a diploma, and persons not doing so. Under the provisions of this definition, on the job training-informal education, as well as formal education may be considered as a single level educational program. A multiperiod planning model takes into account the fact that a series of periodic events over time generates the process being modeled. The entire planning horizon consist of this whole series of events. The model assumes a planning horizon consisting of N periods of equal length. Further, it is assumed that the time duration of the educational process is equal to one of the N periods. Diagrammatically, this system is identical to the one shown in Fig. 1. In this model, assume that the values p1,p2, . , pN, representing the total population of eligible students for the future N periods, are known. Also, assume that on the basis of manpower studies, it is predicted that dI,l, cl,,?, . , dl~ and d,,, ,d2,2, , d2fl will be the manpower requirements of the economy for the future N periods on level 1 and level 2, respectively. These predictions take into account attrition (death, retirement, and all other flows out of the system) for the appropriate employment level. These manpower requirements shall be assumed to be required at the end of the time period. Likewise, although dropping out and graduating from the educational program may occur throughout the time period, these events are also assumed to occur at the end of the time period. This is merely a matter of convenience and is accomplished through proper aggregation.
__)
Available 2;;; 2pool
Employment Manpovler . Demand Level 2
Pool
d
Level 2
Attrition
W
Graduates
Educational
Students Selected
Total Eligible Population
Attrition (Non-Graduates)
-
L
Population Not Selected
Available
=a Labor Pool Level I
r
Manpower Demand Level I
FIG. 1. The single educational
Emppl;oyment
4
*
level system.
Level
I
Attrition
b
312
WARREN
L. BALINSKY
This model addresses itself to the question, what is the optimal number of students in each period, x;“, to be selected out of the total eligible population, pt, for entrance into the school system in that period given certain costs (educational costs, and penalty costs for manpower shortages or surpluses).2 The optimal numbers, XT, therefore minimize the costs to the system over the entire planning horizon. The sequence of events in this model shall be: (1) both available labor pools are checked for size, (2) manpower demand predictions are obtained for the planning period, (3) the optimization is performed and the optimal number of students, XT, are admitted into the educational system, (4) time elapses, graduation and dropouts increase the appropriate available labor pools, (5) the demanded manpower flows from the available labor pool to the employment labor pool, and (6) the sequence of events for one time period is concluded and the cycle begins again at step (1) for the next time period. From this sequence of events it follows that the “conservation” or “input-output” equations for the available labor pools are: 3 Si,,+i
=S,,,+(p,-x,)+X(x,)-d,,,
fort=
1.2..
,N.
(1)
and S 2,1+1 =S2,r+
[xr-X(x,)]
-d2,t
forf=
1,2,. ..,lV.
(2)
Equation (1) states that the number of persons in the available labor pool on level 1 at the beginning of the t+l time period, Sr,,+i , shall be composed of those persons available at the beginning of the t time period on level 1 (S,,,), plus the flow of the people generated by those not selected to enter the educational program in period t(pt -x,), plus the flow of people who do not complete the educational process in period r[X(x,)] , and reduced by the flow of people who leave the available labor pool to enter the employment pool in period t on level 1 (di,,). In a similar manner, Eq. (2) states that the number of persons in the available labor pool on level 2 at the beginning of the t+l time period, SZ,t+r , shall be composed of those persons available at the beginning of the t time period on level 2 (S,,,), plus the flow of people who graduate from the school system in period t[xt - X(x,)]. and reduced by the flow of people who leave the available labor pool to enter the employment pool in period t on level 2 (d,,,). These are the flow equations to be used in the minimization problem. The minimization that will be performed involves a summation of three terms. These three terms represent all the costs incurred by the system in a single period. The summation of these components yields the total cost incurred by the system over the entire planning horizon. The three terms for each period are: a processing cost and two penalty costs for having either a shortage or a surplus of manpower on each of the two 1eve1s.4 For this model, ct(xt) is the function representing the processing cost in period t, and gl,AS1,t+1 > d g2,t@2,t+l ) represent the penalty cost functions, on levels 1 and 2,
’ The difficulty of obtaining these costs in real life situations is full! appreciated by the author although a discussion of data acquisition is beyond the scope of this paper. The model presented herein, however, will indicate the benefits of mounting a program to obtain such data. 3 h(x,) indicates that the drop-out rate is 3 function of the number of persons admitted to the program in time period 1. ’ A shortage cost might be the cost of importing labor, loss in economic returns (GNP), etc.; surplus costs might bc the cost of unemployment, etc.
EDUCATIONAL
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313
DEVELOPMENT
respectively, in period t.5 It is assumed that these functions are known or can be obtained. The minimization in the single level-multiperiod case therefore involves the summation of discounted future costs or: Minimize 9x2,. . . , XN Equation
t
c”1 t=
[c&t)
(3) is subject to the manpower
+g*,t(S1,,+1)
constraints:
+g&2,t
+1>1.
(3)
0 < xt < pt, and Eqs. (1) and
(2). Therefore, the single educational program multiperiod manpower planning model is formulated as an n-variable optimization problem. Dynamic programming may be shown to be a general and efficient technique for solving this model.6 A two state variable dynamic programming algorithm can be converted to a one state variable algorithm. Also, upper and lower bounds are proven for the state space of the one state variable algorithm. The savings in terms of computer storage and time resulting from this conversion is quite significant. The type of results obtainable from this model and optimization are presented in the following example. A public school administrator must select the optimal number of students for admission to a special educational program (single educational level system as in Fig. 1). S/he has a four year planning horizon and the total eligible population each year is thirty-two. Also, on the basis of local manpower studies, it is predicted that demand on level one will be 12, 19, 5, and 9 for the future four years, whereas demand on level two will be 7, 11, 13 and 5 for the future four years. The initial supply at levels one and two are both zero. The administrator estimates his educational costs to consist of a $5,000 fixed charge and $1,000 per student variable charge. The penalty cost functions for having an over- or undersupply of trained manpower are $7,000 and $9,000 (per person) on levels one and two, respectively. The administrator’s assistant studies these costs and after evaluating several alternatives recommends accepting ten students per year for admission. An analysis of several alternatives are presented to support that recommendation (Table 1). The results obtained from the optimization model are summarized in Table 2. The optimal decision saves the administrator over $100,000 when compared to the assistant’s proposal. Furthermore. the administrator is guaranteed that no other alternative will cost less than the optimal presented. 3. The Multi-Echelon Multiperiod Education Model The model represented by Fig. 2 is a multi-echelon educational system. The fact that it is a multi-echelon system means that the process (educational/training) under investigation is hierarchial in nature. There may be 1,2,3 or any number of levels of education, where each level G+l) requires completion of the preceding level of education 0’) before commencing, in brief each level of education in the system must be adhered to in numerical order. Let j (j=1,2,. . . , M) represent the ith level of the system and let t (t=1,2, . . . , IV) represent the tth period of the planning horizon. As in the single level model, assume that the values p, , pz, . , pn;, representing the total population of eligible students for the ’ gj, t(Si, t+l) for i = 1 or 2 and all t are functions period. This is consistent with previous notation. 6 See [5] or [3].
of the available
labor
at the end of the time
WARREN
314 TABLE Recommendation Time period Population size Demand at level I Demand at level 2 Zero admission policy Total eligible adm. policy Level 2 demand adm. policy 30:‘6 eligible adm. policy
L. BALINSKY
1
of Assistant
Administrator
1 32 12 7 0
2 32 19 11 0
3 32 5 13 II
4 32 9 5 0
Total cost ($1
32
32
32
32
1.846,143
7
11
13
5
650,338
10
10
10
10
645,999
1,668,482
future N periods are known. Also assume that on the basis of some forecasting methods, it is predicted that dI,l.d,,,, , dldy,d2,1.d2,2.. , dZ4v,. ,dh~,~,d~,~, ~,VI,N are the manpower requirements dictated by the economy for the future IV periods on levels 1,2, . . fl, respectively. These predictions are aggregated so as to be required at the end of each time period. The model addresses itself to the question, what is the optimal number of students in each period XT,, , to be selected out of the total eligible population pt, for entrance into level 1 of the educational system in that period given certain costs (educational costs and penalty costs for having a surplus or shortage of manpower on any level)? These optimal numbers, x:,~, shall be the numbers which minimize the costs to the system over the entire planning horizon. The sequence of events in this model shall be (1) all available labor supplies are checked for size, (2) manpower demand predictions are obtained for all levels for the entire planning horizon, (3) the optimization is performed and the optimal number of students, XT,*, are admitted into the education system level 1, (4) time elapses, and graduates either increase their appropriate available labor pool or enter the next level in the educational system. Concurrently nongraduates increase their appropriate supply levels, (5) true manpower demands flow from the available labor pool to the employment labor pool, and (6) the sequence of events for one time period is concluded and the cycle begins again at step (1) for the next time period. It should be noted that for all available labor pools. except the first and the last, the input flows represent both graduates of the lower educational level and nongraduates of the higher educational level. For level 1, the inputs represent those nongraduates of educational level 1 and those persons who do not TABLE Optimization Time period Population size Demand at Iwe1 I Demand at level 2 Optimal admission policy
1
2 Results
32 12 I
2 32 19 I1
3 32 5 13
4 32 9 5
8
13
15
6
‘1.otal cost (S,
543,128
315
EDUCATIONAL MODELS FOR MANPOWER DFVELOPMENT AvaIlable Labor Pool Level
+
Emphyofnent Level
M+l
)
M+I
IEducational Available La~or$gl
Employment
m-
I
ass Section of Level J-l, Available Lolb,~~,PJo01
Manpower Demands
Available Labor Pool Level 3
Available
Level I -Graduate
Pool
__)
Level M
J System
Attritior
Emp;oyo;lent
*
+
Level J
>
Employment Pool Level 3
4
Employment Pool Level 2
__)
Leavers
Level I - Non-Graduates
FIG. 2. The multi-educational
level system.
enter tile educational system at all. For level M+ 1, inputs are comprised of graduates of educational level M. From this sequence of events it follows that the “conservation” or “input-output” equations for the available labor pools are:7
Sl
,t+1
=Sl,,t~,-x*,,)t~l(xl,,)-~l,t
t=lT
sj,*+l
=Sj,t
•t iji(xj,r)
+Lj-l
[xj-l,r-Xj-l(Xj-l,r>l
forj=2,3 ~,+1,r+1
=S.&f+l,t+
[xM,r--.&M,r)l
,...
,Mandt=
>-> . . . >
N
(4)
-dj,t
1,2,..
,N
(5)
-dilf+1,r t=12 , ,-..>
N
(6)
7 ;li(Xj,f) indicates that the defective rate is il function of the number of students in levelj in time period t.
316
WARREN
L. BALINSKY
where
forj=2,3
,...,
Mandt=
. . . . -2,-1,0,1,2
,...,
N
(7)
Equation (4) is identical in meaning to Eq. (1). Equation (5) states that the number of persons in the available labor pool on level j (for j = 2,3, . ,M) at the beginning of the t+l time period, Sj,t+l , shall be composed of those persons available at the beginning of the t time period on level j (Sj, t), plus the flow of nongraduates at the j level in the time period t [Xj(Xj,t)] , plus the flow of graduates who complete the G-1) educational level and who flow from the educational system to the economic system on the j level in period t, Lj_1 [xj_l,t -hjj-l(xj-l,t)], and reduced by the flow of persons who flow from the available labor supply as manpower demand in period t on level j (dj, t). Equation (6) is identical in meaning to Eq. (2) with available labor pool level M+l corresponding to available labor pool level 2 of the single level model. These equations relate the flows into and out of the available labor supply pools. An important consequence of this system’s structure links educational levels over time. The number of students in one of the j educational levels (for j = 2,3,. . , M) is equal to the number of graduates from level j-l which do not leave the educational system in the previous time period. Since Xi-1 ,r-r Xi-1(Xj-1 ,t-l) represents the graduates in time period r-1 from educational level j-l, and ~j-1(.) represents those graduates who flow from the educational system to the economy, then the quantity
flows into the next educational level in the next time period. This process yields Eq. (7) and is shown in Fig. 3. Equations (4-7) shall be used to define the constraint set of our minimization problem. It is now possible to write the expression for all costs in time period t (t = 13,“’ . . . 1IV), K(t), as: M+ 1
Xl
rc(t)
= c cj, kxj, f> + j=1
t=17 a-3 .
c gj, *
j=
I
. . 9 N,
(8)
where Cj,,(‘) is the function representing the educational cost on level j in period C, and gj,,(‘) is the function representing the penalty cost on level j in period t. It is assumed that these functions are known. Thcrefor-e. Eq. (8) includes all the costs in period t incurred by the educational system for the M levels (the first summation) and all the costs in period t incurred as a result of shortages or surpluses of manpower for thcM+l levels (the second summation). To obtain the present worth of the total cost for this system for the entire planning horizon. Eq. (8) must be appropriately discounted and summed for periods 1 to N. Thus
TC=
t= C
1
Or--’ 1 c cj,t(xj,r) + j=
1
c
j=
1
iTj,&S,,f+,)].
(9)
EDUCATIONAL
MODELS
FOR MANPOWER
J EDUCATIONAL -k
‘J-l+I
J-l
LEVEL
(XJ_,,t_,
317
DEVELOPMENT ENTRANTS
)-
-LJ_,CxJ-~,t-i-XJ-I(XJ-,,+-I)3
A
GRADUATES EDUCATIONAL L
cx
J-l
LEAVING SYSTEM
J-l,t-I-XJ-I(XJ-l,t-,”
EDUCATIONAL LEVEL
J-l
I
I
*
NON - GRADUATES
1
J-i
FIG. 3. Flows generated
from thej-1
(x
J-l,+?
educational
level.
is Eq. (9) that will be minimized, resulting in the selection ofx;,, for f = 1,2, . . . ,N, equations. The subject to the availability restrictions on each x~,~ and the input-output minimization problem now becomes It
{Xl,lJ1,2,.
.
r. e’-’ ,Xl,N}
tt1)I . t)+ 2Zgj,dsj, j=cj,dxj, M
A’
Minimize
t=1
c
M+l
j= 1
1
Subject to the following constraints: 0
S 1,t+1 sj, t+ 1
t=1,2,...,N
GP*
GX1,t
=
s1,t + 6,
-x1,t)
+ Xl(Xl,,)
-d1,t
t= 1,2,. . . ,N
=Sj,t+Xj(xj,t)+Lj-l[Xj-l,t-Aj-l(Xj-l,t)-dj,t
fori=2,3,... sM+l,ttl
= sM+l,t
+ [XM,t-XM(XM,tr)l
-dMtl,t
,Mand t=
t= 1,2,. . . ,N
152,. . . ,N
(10)
WARREN
318
L.BALINSKY
and
xi t =
xi-1
,/-I
-
Ai-1 (Xi-IJ-1)
- I+1
forj=2,3,...,Mand
[q-l
J-1
-
+I
(Xi-1,t-1 ,I
t=....-2,-1,0,1,2,...,N
Under particular conditions, the multi-echelon model leads to some very nice closed form optimal decision rules.’ Under one set of conditions. the optimal policy is to admit each period that number of eligible persons into the educational system such that the manpower demands at the highest level are exactly satistied. Also, under another set of conditions, the optimal policy is to admit the total number of eligible persons into the educational system each period. Specifically, the conditions which guarantee that the optimal decision rule meets exactly the manpower demands on the highest level each period are: 1. No shortages of manpower are permitted. 2. All functions are linear. 3. The cost of educating and storing manpower is strictly decreasing with time. Therefore, one wishes to defer education or training. 4. The yearly supply of eligible population is sufficiently large such that manpower demands on level M+l can be met on a yearly basis, that is without storing on level M+l. 5. The cumulative demands on level M+l (appropriately adjusted) must generate enough nongraduates and graduates who seek employment at all the lower levels such that no shortages occur. Thus one prefers deferring education (training); however, to avoid manpower shortages, if level M+l demand is met exactly (each period) all lower level demands are met and this generates the least cost solution. The nice properties that this solution exhibits are: (1) it is independent of initial available labor pool supplies, demand estimates and all parameters related to all but the highest level in the system, (2) the solution can be written in an analytically closed form which is very rare when treating a complex system, and (3) it justifies mathematically what is an intuitively correct and often practiced decision rule. The conditions guaranteeing an optimal decision rule of admitting the total eligible population into the educational system each period are: 1. No shortages of manpower are permitted. 2. All functions are linear. 3. The cost of educating and storing manpower is strictly less than the cost of not educating regardless of the time period. Therefore, one wishes to educate the total eligible population every time period. 4. Enough nongraduates must be generated from educating the entire population each period to satisfy level 1 manpower demands when the only flow into level 1 manpower supply is via the nongraduate route. In this case, one prefers to educate the total eligible population each time period, and this course of action guarantees no manpower shortages and the least cost solution. Again, this solution exhibits nice properties in that it: (1) is independent of all system :)See [4]or [3]
EDUCATIONAL
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319
DEVELOPMENT
parameters and demand forecasts, (2) can be written in an analytically closed form and (3) again justifies mathematically what is an intuitively correct and often practiced decision rule. 4. Educational Program Duration Modifications and Other Extensions This section shall deal with modification of the previously developed models in order to render models more specific and at the same time more realistic. The first modification to be explored is the situation in which each educational level does not require a single planning time period for completion. Two situations shall be explored: the first is when some educational levels require fractional time periods, the second is when some educational levels require multiple time periods. To illustrate the first case, suppose a 4-level educational system in which each educational level required 8 months is to be analyzed. If the population estimates and manpower requirements are given as aggregates over g-month time periods, then certainly the multi-educational level planning models as stated can be used.’ However, if the population estimates and manpower requirements are given as yearly aggregates, then we proceed as follows. Taking two-thirds of the yearly population estimates and two-thirds of the yearly manpower requirements allows us to use the multi-educational level models with planning periods of 8 months. Suppose one is confronted with a 3 level educational system in which the 1st level requires 6 months, the 2nd level requires 3 months, and the 3rd level requires 6 months. Taking the common denominator between 6 months, 3 months and 6 months, one should develop a system with planning periods of 3 months. All yearly demand estimates must of course be adjusted to quarterly estimates. But now it takes 2 time periods to graduate from the 1st educational level, 1 to graduate from the 2nd level and 2 from the 3rd. To continue with this example, we shall investigate the second modification mentioned before. That is, the case when some educational levels require multiple time periods. In general, the technique to modify the multi-educational level model to account for multiples of some time period (to complete the requirements at some given level) is to increase the number of levels in the system. Continuing with the example of the previous paragraph we are confronted with a 3 level system in which the 1st level requires 2 time periods, the 2nd level 1 time period and the 3rd level 2 time periods. Adding together the number of periods required to graduate from the system, we obtain 5, therefore we shall require a 5 level educational system. Observe Fig. 4: it is a 5 level educational system in which levels 1 and 4 have only graduates (no dropouts). The flow equations for this system follow:
s 1,t+1
=s1,,
+ (Pt -X1,t)
S3,t+ I = S3,r + X363,t)
+ X2@2,t)
+ J52 b2,t
-d1,,
- hz(xz,t>l
r =
1,2,.
- d3,t
. ,N
(11)
t= 12> ,..., 4,
s4,t+
1 = S4,t + X5 (X5,t)
+ L3
LX3,t - A3 CX3,dl
-d4,t
t= 1,2,. . . ,N (13)
Scj,r+1 ’ One merely
assumes
=
S6,r + [x5,r
n period
- b(x5,r)l
of 8 months
-d6,r
t=1,2,...,N
(14)
WARREN
320
-c r------
* * * N
Available Labor Pool Level 6
.
Avoilable Labor Pool Level 4
w
Employment Pool Level 4
;-
Employment Pool Level 3
*
Employment Pool Level I
L. BALINSKY
EmpzY+ Level 6
Educotionol
Avoiloble Lof;vreep~l
Educational Level 2
d +
I,
Level
-J I L7~___1
I
I
Total Eligible Population
Available *Labor Pool * Level I -
FIG. 4. A modified
S-level system.
for j = 3,4 and t = . . . , -2,-1,0,1,2, x2,t
=x1,t-I
x5,t = x4,t-1
. . . ,N
(15)
for I = . . , -2,-l
,0,1,2,
. ,N
(16)
for t = . . , -2,-l
,0,1,2,
. . ,N
(17)
The system represents a modified 5 level educational system which accounts for the differences in educational duration at the various levels. One can see both from Fig. 4 and flow equations (16) and (I 7) that educational levels 1 and 4 are really artificial educational levels in that all students pass through these levels after one time period. These levels delay the educational process in a manner which allows the proper time durations to be accounted for in the model. Since the original problem was concerned with only 3 educational levels, there are 4 available manpower pools. and 4 employment pools; the pools 2 and 5 corresponding to artificial educational levels 1 and 4 have been appropriately discarded. One can think of levels 1 and 2 as a single level (level l*), level 3 as a single level (level 2*) and levels 4 and 5 as a single level (level 3*) of the original 3 level
EDUCATIONAL
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DEVELOPMENT
321
system and therefore, by way of illustration, some methods for modifying the multilevel models to account for differences in educational duration have been presented. Figure 5 represents a very realistic multilevel modification. Suppose, one would like to model the situation in which there is a minimum educational level which the entire population must attain. For example, if each educational level represented a 4 year period of study, levels 1 and 2 of Fig. 5 could represent the elementary school system, levels 3, 4, and 5 high school, college and graduate school, respectively. The flow equations for this system would be: S3,t+1
= S3,t
+ (X&t
-x3,2
+ h3b3,t)
Si, t+ 1 = Sj,t + Ai(xj, t>+ Lj- 1[xi- 1,t -
Ai-1(xi- 1,t>l - di,t j=4,5andt=1,2
S 6,r+
1 = S6,t
+ [Xs,t
-
~5@5,t>1
t=
- d6,r
A
Educational Level
1,2,. . . N
Level
4
*
3
+
-
Available L$ov$gol
L
L-l Totol Population
,...,
Empboyyent
Available
Labor Pool Level 4
.z t _-I
__)
Employment Pool Level 3
I I I
FIG. 5. A system with compulsory
(18)
t=1,2,...,N
-d3,1
educational
levels.
N
(19) (20)
322
WARREN L. BALINSKY
Xi>t = Xj-l,t-1
- Aj-1
(Xj-1,1-I
)-
Lj-1
[Xj-1 ,r--l
ij-1
-
Cxj-l ,I--1 )I
j = 4,5 and t = . . . ) -2,-1,0,1,2, t =
x1,t = Pt X2,t
,-2,-l
,O,I ,2,
t= . . . ) -2,-1,0,1,2,.
= x1,t-1
(21)
. . ,N
,N
(22) ,N
(23)
where the decision level for this model is level 3. That is, the decision problem is how many students to optimally admit to the third educational level from an available amount ~2,~ on the second level. Essentially, this model shifts the entire population up 2 levels and adjusts for the educational delay involved in doing this. (One could renumber the system, such that the total population and educational levels 1 and 2 become the total population *, and higher educational levels 3, 4, 5 become l*, 2*. 3”. By doing this, the system in Fig. 5, would correspond to the system in Fig. 2 in which there is a time adjustment to account for the total population * becoming eligible for level I*.) Figure 6 represents a modification similar- to that represented by Fig. 4. This moditication of the multilevel system incorporates the possibility of a student dropping out of an educational program after having completed a few educational levels, but having not completed the entir-e program.” That is, if levels 1, 3, 3. and 4 constitute a program (such as a four year college), any student who drops out before completing the entire 4 level program essentially gets no credit for attending the program and flows into available labor pool 1, as if he had not attended this program at all. The flow equations corresponding to this modification are: s,,t+ 1 = s1,t
+
(/It
-x1.r)
+ Xl(XlJ)
+
X*(x-2.t)
+ b(X3.t)
+
X4b4,t)
t=1,2,....1v S 2,t+ 1
= s2,t
+
Xi.r = Xj-
1 ,t_
b4,t
1
-
-
Xj_
h4(x4,t)l 1
-d2,1
t=1? ,-.
. , IV
(Xi_1) j = 2,3,4 and t = . . . , -2,-1,0,1,2, 1,
-d1,t
(24) (25)
t-
. . ,N
(26)
For the purpose of clar-ity, each modification in this section has been presented as a special case of the multi-echelon multiperiod educational model. However, combining these modifications is quite simple and the number of further special cases is only bounded by the limits of one’s imagination.
5. Conclusions A number of educational planning models are developed and optimized. Beginning with a single educational program-multiperiod model, and then after presenting a general multi-echelon multiperiod educational model, a very realistic “compulsory type” educational system and a “college type” educational system are presented. By means of sample r-esults for the single educational program-multiperiod dynamic programming, model are demonstrated; whereas, linear programming generates the optimal decisions presented for the general multi-echelon multiperiod educational model.
EDUCATIONAL
MODELS
FOR MANPOWER
4
DEVELOPMENT
323
Available Labor Pool __)I Level 2
Employment Pool
Level
2
Educational Level 4
Educational Level I 4
& --c Available Y Labor Pool --Z Level I
>
FIG. 6. A 4-level “college-type”
system.
Employment Pool Level I
All of the models presented are but a few of the many that can be generated by modifying the flows, the levels, etc. Elsewhere [3], there exists a general scheme which presents all of the manpower planning models in a systematic fashion. The scheme is a method of unification, through generalization, of all such models. The most general model in the scheme contains all the components which are currently perceived as being germane to these models. Most models in the literature, however, are of lesser complexity and can be shown to be subcases of this general model. These subcases may be obtained merely by dropping or simplifying certain terms in the general model. In this manner, it is possible to delineate the entire field of manpower flow models in a systematic fashion. The classification scheme, LASTFENKCO, is first presented and then used to classify the models in the literature. This scheme can also be used to indicate those models that have yet to be developed. References 1. Adehnan, I., A Linear Programming Model of Educational Planning: A Case Study of Argentina, Paper delivered at the Joint Session of the Econometric Society and the American Economic Association, New York, 1965. 2. Balinsky, Warren, An Annotated Bibliography: Applications of Operations Research to Educa-
324
WARREN
L. BALINSKY
tional Systems. Technical Memorandum No. 96, Case Western Reserve University, November 1967. 3. Balinsky, Warren, Some Manpower Planning Models Based on Levels of Educational Attainment. Technical Memorandum ifl85, Case Western Reserve University, Cleveland, Ohio, June 1970, also available from University Microfilms. 4. Balinsky, Warren, Some Optimal Decision Rules for a Multi-Echelon Multi-Period Production System. Working Paper No. 99, State University of New York at Buffalo, February 1971. 5. Balinsky, Warren, Some Dynamic Programming Models for a Single Process-Multiperiod Inventory and Production Systems. Working Paper No. 102, State University of New York at Buffalo, March 1971. 6. Bellman, R., and Dreyfus, S.E., Applied Dynamic Programming, Princeton University Press, Princeton, NJ, 1962. 7. Bellman, R., and Dreyfus, S.E. Adaptive Control Processes, Princeton University Press, Princeton, NJ, 1961. 8. Bolt, R.H., W.L. Koltun, and O.H. Levine, Doctoral Feedback into Higher Education, Science 148,918 (1965). 9. Bowles, Samuel, Planning Educational Systems for Economic Growth, Harvard University Press, Cambridge, Massachusetts, 1961. 10. Correa, Hector, Quantitative Methodologies of Educational Planning, International Institute for Educational Planning, UNESCO, France, 1967. 11. Hadley, G., Nonlinear and Dynamic Programming, Addison-Wesley, Reading, Massachusetts, 1964. 12. Hadley, G., and T. Whitin, Analysis of Inventory Systems, Prentice-Hall, Englewood Cliffs, New Jersey, 1963. 13. Harbison, I., and C. Meyers, Education, Manpower and Economic Growth, McGraw Hill, New York, 1964. 14. Levine, Sumner, Economic Growth and the Development of Educational Facilities: A Systems Analysis, Socio-Economic Planning Science 1, 21-32 (1967). 15. Moonan, W.J., and M.H. Covher, A Computer Program for the Determination of an Optimal Advancement Policy for Petty Officers: A Dynamic Programming Approach. AD 635850, September 1966. 16. Nemhauser, G.L., Introduction to Dynamic Programming, John Wiley & Sons, New York, 1966. 17. O.E.C.D., MathematicalModels in Educational Planning, Paris, 1967. 18. O.E.C.D., Planning Education for Economic and Social Development, Paris, 1962. 19. Reisman, Arnold, A Population Plow 1:eedback Model, Science 153, (3731) 89-91 (1966). 20. Reisman, Arnold, and Martin I. Taft, On the Generation of Doctorates and Their Feedback Into Higher Education, Socio-Economic Planning Science 2, (2, 3, 4) (April 1969). 21. Scarf, H.E., D. Gilford, and M. Shelley, Multistage Inventory Models and Techniques, Stanford University Press, Stanford, California, 1963. 22. Tinbergen, Jan, and H.C. Bos, A Planning Model for the Educational Requirements of Economic Development,
The Residual Factor and Economic Growth, O.E.C.D.,
Received I7 June 1974; revised 2 June 1975
Paris,
1964.
FORECASTING
AND SOCIAL
CHANGE
8,309-324
(1976)
309
Educational Models for Manpower Development WARREN L. BALINSKY
ABSTRACT In this paper a number of educational models are formulated and optimized. A single educational program multiperiod model is developed and optimized by means of dynamic programming. The type of results an educational admini,strator might obtain from this model and optimization are presented in an example which evaluates alternative future options. This single educational program model is extended to a multi-echelon model, allowing for the modifications to very realistic “compulsory-type” and “college-type” educational models. Linear programming generates the optimal decisions for the general multi-echelon multiperiod educational model.
1. Introduction and Historical Background To control the ever rising cost of education and to find improved and more effective methods of optimally allocating resources, the application of management techniques is becoming increasingly popular among individual practitioners and both private and public institutions. More often than not they build their models and structure their programs from huge statistical information, figures and graphs, various characteristics of the system, and impressive analysis of educational attainments of individuals at various levels in the system. But such efforts, though highly regarded for the validity of the results they produce, contribute very little, if at all, toward adequately clarifying the real issues involved, or even to a satisfactory policy formulation for the futurg. The main reason for such shortcomings is that most studies done in this area ignore the dynamics of the system concerning the issues in focus. A dynamic framework with unifying concepts of investigating the essential facts in quantitative terms and put together in a form readily understandable by policymakers is what is needed most for effective use of the results. The objective of this study is to consider the underlying relationship between the economic and educational sectors of a society by means of time dependent flow models. It is widely known that the sustenance of socioeconomic development is dependent upon a continuous and adequate supply of human resources of various skills and trainings. The spectrum of these resources is broad, ranging from people with basic literacy to those with advanced scientific and technical backgrounds. Nevertheless, the human-material resources are always scarce and the amount of these resources that can be allocated for educational purposes, for establishing and operating facilities, is limited. Thus it becomes relevant to explore and develop effective policies which will generate needed manpower DR. BALINSKY is Associate Professor in the Health Services Administration Program at the State University of New York at Stony Brook. Prior to this appointment he was Assistant Professor in the Management Systems Department of the School of Management, State University of New York at Buffalo. His research interests center on the application of management science techniques to problems in the areas of health management and educational management with special emphasis on the solution techniques of mathematical programming and statistics. 0 American
Elsevier Publishing
Company,
Inc., 1976
310
WARREN L. BALINSKY
skills through time in a quantity necessary to maintain an equilibrium between supply and demand of these skills. It is, therefore, imperative for the orderly planning of an educational structure to be consistent with the projected manpower needs and overall economic investment. This paper, in summary, is an attempt to look into this problem of systematic planning of educational structures for achieving manpower goals in an optimal fashion. There exists a small body of literature which has entertained the problem of orderly planning of an educational system. Bolt, Koltun and Levine [8] were the first to study the dynamics of this problem and develop a model which takes account of the characteristics of the system. Their model treats the educational system (higher education) as one sector and the rest of the economy as the other. A feedback model then represents the process of degree production. The model describes the flows of graduates between these two sectors by establishing that Ph.D. degree holders go back to the academic sector as faculty or administrators, while others are absorbed by industrial and governmental positions. The model makes use of two linear difference equations to represent flows in the economy. Simultaneous solutions of the difference equations are obtained for various parameter values. The idea of the parametric study is to provide policymakers with better insight to recognize the consequences of programs they might recommend. The model has a number of weaknesses. Some of the limitations recognized by the authors themselves are: “The model ignores the input of students into education,” and “The model is defined only for linear behavior.” Another weakness of the model is its handling of each level of education as an independent process, which is contrary to the fact. Bolt, Koltun and Levine’s model has been extended by R&man’s original model [ 191 and its extension by Reisman’s and Taft [20]. Their approach to the sectors is less aggregative than that of the Bolt, Koltun and Levine model. They consider the interactions between lower and higher levels of education, particularly the dependence of higher levels of education for their input on the output of lower levels. They also study flows as difference equations suitable for computer simulation, which give their model an advantage over the other model. These models are very helpful in understanding the feedback processes into higher education. But, as pointed out earlier, none of the studies in the area of manpower planning appearing in the literature so far has attempted to specifically obtain an optimal analytical solution to the problem. The development of this type of model as a policymaking tool is, therefore, in order. As indicated by Levine [ 141 , the analytical techniques for evaluating the effectiveness of a given set of policies is that of requiring the policies to produce a supply of trained manpower at each educational level within certain restrictive bounds. This is accounted for in the present study. Within the scope of this report, however, all the optimization techniques applicable to the problem are not explored. 2. A Single Educational Program-Multiperiod Model The objective of this paper is to formulate and discuss the optimization of progressively more realistic manpower planning models. The models are characteristically hierarchical and are built around the multilevel nature of the educational/training’ programs. First, a single educational program-multiperiod manpower planning model will be ’ Education and training may be used interchangeably in thib paper. well as formal education may be equally modelled using the developments
since job training, herein.
informal
as
EDUCATIONAL
MODELS
FOR MANPOWER
DEVELOPMENT
311
developed. A single educational program means that there is only one educational program in which individuals may or may not participate. For example, a training program for high school dropouts, a training program for high school graduates, or a master’s degree program for a particular specialty. As a qualification for a single level educational program, a program must have a provision for distinguishing between persons who successfully complete the program, like awarding a diploma, and persons not doing so. Under the provisions of this definition, on the job training-informal education, as well as formal education may be considered as a single level educational program. A multiperiod planning model takes into account the fact that a series of periodic events over time generates the process being modeled. The entire planning horizon consist of this whole series of events. The model assumes a planning horizon consisting of N periods of equal length. Further, it is assumed that the time duration of the educational process is equal to one of the N periods. Diagrammatically, this system is identical to the one shown in Fig. 1. In this model, assume that the values p1,p2, . , pN, representing the total population of eligible students for the future N periods, are known. Also, assume that on the basis of manpower studies, it is predicted that dI,l, cl,,?, . , dl~ and d,,, ,d2,2, , d2fl will be the manpower requirements of the economy for the future N periods on level 1 and level 2, respectively. These predictions take into account attrition (death, retirement, and all other flows out of the system) for the appropriate employment level. These manpower requirements shall be assumed to be required at the end of the time period. Likewise, although dropping out and graduating from the educational program may occur throughout the time period, these events are also assumed to occur at the end of the time period. This is merely a matter of convenience and is accomplished through proper aggregation.
__)
Available 2;;; 2pool
Employment Manpovler . Demand Level 2
Pool
d
Level 2
Attrition
W
Graduates
Educational
Students Selected
Total Eligible Population
Attrition (Non-Graduates)
-
L
Population Not Selected
Available
=a Labor Pool Level I
r
Manpower Demand Level I
FIG. 1. The single educational
Emppl;oyment
4
*
level system.
Level
I
Attrition
b
312
WARREN
L. BALINSKY
This model addresses itself to the question, what is the optimal number of students in each period, x;“, to be selected out of the total eligible population, pt, for entrance into the school system in that period given certain costs (educational costs, and penalty costs for manpower shortages or surpluses).2 The optimal numbers, XT, therefore minimize the costs to the system over the entire planning horizon. The sequence of events in this model shall be: (1) both available labor pools are checked for size, (2) manpower demand predictions are obtained for the planning period, (3) the optimization is performed and the optimal number of students, XT, are admitted into the educational system, (4) time elapses, graduation and dropouts increase the appropriate available labor pools, (5) the demanded manpower flows from the available labor pool to the employment labor pool, and (6) the sequence of events for one time period is concluded and the cycle begins again at step (1) for the next time period. From this sequence of events it follows that the “conservation” or “input-output” equations for the available labor pools are: 3 Si,,+i
=S,,,+(p,-x,)+X(x,)-d,,,
fort=
1.2..
,N.
(1)
and S 2,1+1 =S2,r+
[xr-X(x,)]
-d2,t
forf=
1,2,. ..,lV.
(2)
Equation (1) states that the number of persons in the available labor pool on level 1 at the beginning of the t+l time period, Sr,,+i , shall be composed of those persons available at the beginning of the t time period on level 1 (S,,,), plus the flow of the people generated by those not selected to enter the educational program in period t(pt -x,), plus the flow of people who do not complete the educational process in period r[X(x,)] , and reduced by the flow of people who leave the available labor pool to enter the employment pool in period t on level 1 (di,,). In a similar manner, Eq. (2) states that the number of persons in the available labor pool on level 2 at the beginning of the t+l time period, SZ,t+r , shall be composed of those persons available at the beginning of the t time period on level 2 (S,,,), plus the flow of people who graduate from the school system in period t[xt - X(x,)]. and reduced by the flow of people who leave the available labor pool to enter the employment pool in period t on level 2 (d,,,). These are the flow equations to be used in the minimization problem. The minimization that will be performed involves a summation of three terms. These three terms represent all the costs incurred by the system in a single period. The summation of these components yields the total cost incurred by the system over the entire planning horizon. The three terms for each period are: a processing cost and two penalty costs for having either a shortage or a surplus of manpower on each of the two 1eve1s.4 For this model, ct(xt) is the function representing the processing cost in period t, and gl,AS1,t+1 > d g2,t@2,t+l ) represent the penalty cost functions, on levels 1 and 2,
’ The difficulty of obtaining these costs in real life situations is full! appreciated by the author although a discussion of data acquisition is beyond the scope of this paper. The model presented herein, however, will indicate the benefits of mounting a program to obtain such data. 3 h(x,) indicates that the drop-out rate is 3 function of the number of persons admitted to the program in time period 1. ’ A shortage cost might be the cost of importing labor, loss in economic returns (GNP), etc.; surplus costs might bc the cost of unemployment, etc.
EDUCATIONAL
MODELS
FOR MANPOWER
313
DEVELOPMENT
respectively, in period t.5 It is assumed that these functions are known or can be obtained. The minimization in the single level-multiperiod case therefore involves the summation of discounted future costs or: Minimize 9x2,. . . , XN Equation
t
c”1 t=
[c&t)
(3) is subject to the manpower
+g*,t(S1,,+1)
constraints:
+g&2,t
+1>1.
(3)
0 < xt < pt, and Eqs. (1) and
(2). Therefore, the single educational program multiperiod manpower planning model is formulated as an n-variable optimization problem. Dynamic programming may be shown to be a general and efficient technique for solving this model.6 A two state variable dynamic programming algorithm can be converted to a one state variable algorithm. Also, upper and lower bounds are proven for the state space of the one state variable algorithm. The savings in terms of computer storage and time resulting from this conversion is quite significant. The type of results obtainable from this model and optimization are presented in the following example. A public school administrator must select the optimal number of students for admission to a special educational program (single educational level system as in Fig. 1). S/he has a four year planning horizon and the total eligible population each year is thirty-two. Also, on the basis of local manpower studies, it is predicted that demand on level one will be 12, 19, 5, and 9 for the future four years, whereas demand on level two will be 7, 11, 13 and 5 for the future four years. The initial supply at levels one and two are both zero. The administrator estimates his educational costs to consist of a $5,000 fixed charge and $1,000 per student variable charge. The penalty cost functions for having an over- or undersupply of trained manpower are $7,000 and $9,000 (per person) on levels one and two, respectively. The administrator’s assistant studies these costs and after evaluating several alternatives recommends accepting ten students per year for admission. An analysis of several alternatives are presented to support that recommendation (Table 1). The results obtained from the optimization model are summarized in Table 2. The optimal decision saves the administrator over $100,000 when compared to the assistant’s proposal. Furthermore. the administrator is guaranteed that no other alternative will cost less than the optimal presented. 3. The Multi-Echelon Multiperiod Education Model The model represented by Fig. 2 is a multi-echelon educational system. The fact that it is a multi-echelon system means that the process (educational/training) under investigation is hierarchial in nature. There may be 1,2,3 or any number of levels of education, where each level G+l) requires completion of the preceding level of education 0’) before commencing, in brief each level of education in the system must be adhered to in numerical order. Let j (j=1,2,. . . , M) represent the ith level of the system and let t (t=1,2, . . . , IV) represent the tth period of the planning horizon. As in the single level model, assume that the values p, , pz, . , pn;, representing the total population of eligible students for the ’ gj, t(Si, t+l) for i = 1 or 2 and all t are functions period. This is consistent with previous notation. 6 See [5] or [3].
of the available
labor
at the end of the time
WARREN
314 TABLE Recommendation Time period Population size Demand at level I Demand at level 2 Zero admission policy Total eligible adm. policy Level 2 demand adm. policy 30:‘6 eligible adm. policy
L. BALINSKY
1
of Assistant
Administrator
1 32 12 7 0
2 32 19 11 0
3 32 5 13 II
4 32 9 5 0
Total cost ($1
32
32
32
32
1.846,143
7
11
13
5
650,338
10
10
10
10
645,999
1,668,482
future N periods are known. Also assume that on the basis of some forecasting methods, it is predicted that dI,l.d,,,, , dldy,d2,1.d2,2.. , dZ4v,. ,dh~,~,d~,~, ~,VI,N are the manpower requirements dictated by the economy for the future IV periods on levels 1,2, . . fl, respectively. These predictions are aggregated so as to be required at the end of each time period. The model addresses itself to the question, what is the optimal number of students in each period XT,, , to be selected out of the total eligible population pt, for entrance into level 1 of the educational system in that period given certain costs (educational costs and penalty costs for having a surplus or shortage of manpower on any level)? These optimal numbers, x:,~, shall be the numbers which minimize the costs to the system over the entire planning horizon. The sequence of events in this model shall be (1) all available labor supplies are checked for size, (2) manpower demand predictions are obtained for all levels for the entire planning horizon, (3) the optimization is performed and the optimal number of students, XT,*, are admitted into the education system level 1, (4) time elapses, and graduates either increase their appropriate available labor pool or enter the next level in the educational system. Concurrently nongraduates increase their appropriate supply levels, (5) true manpower demands flow from the available labor pool to the employment labor pool, and (6) the sequence of events for one time period is concluded and the cycle begins again at step (1) for the next time period. It should be noted that for all available labor pools. except the first and the last, the input flows represent both graduates of the lower educational level and nongraduates of the higher educational level. For level 1, the inputs represent those nongraduates of educational level 1 and those persons who do not TABLE Optimization Time period Population size Demand at Iwe1 I Demand at level 2 Optimal admission policy
1
2 Results
32 12 I
2 32 19 I1
3 32 5 13
4 32 9 5
8
13
15
6
‘1.otal cost (S,
543,128
315
EDUCATIONAL MODELS FOR MANPOWER DFVELOPMENT AvaIlable Labor Pool Level
+
Emphyofnent Level
M+l
)
M+I
IEducational Available La~or$gl
Employment
m-
I
ass Section of Level J-l, Available Lolb,~~,PJo01
Manpower Demands
Available Labor Pool Level 3
Available
Level I -Graduate
Pool
__)
Level M
J System
Attritior
Emp;oyo;lent
*
+
Level J
>
Employment Pool Level 3
4
Employment Pool Level 2
__)
Leavers
Level I - Non-Graduates
FIG. 2. The multi-educational
level system.
enter tile educational system at all. For level M+ 1, inputs are comprised of graduates of educational level M. From this sequence of events it follows that the “conservation” or “input-output” equations for the available labor pools are:7
Sl
,t+1
=Sl,,t~,-x*,,)t~l(xl,,)-~l,t
t=lT
sj,*+l
=Sj,t
•t iji(xj,r)
+Lj-l
[xj-l,r-Xj-l(Xj-l,r>l
forj=2,3 ~,+1,r+1
=S.&f+l,t+
[xM,r--.&M,r)l
,...
,Mandt=
>-> . . . >
N
(4)
-dj,t
1,2,..
,N
(5)
-dilf+1,r t=12 , ,-..>
N
(6)
7 ;li(Xj,f) indicates that the defective rate is il function of the number of students in levelj in time period t.
316
WARREN
L. BALINSKY
where
forj=2,3
,...,
Mandt=
. . . . -2,-1,0,1,2
,...,
N
(7)
Equation (4) is identical in meaning to Eq. (1). Equation (5) states that the number of persons in the available labor pool on level j (for j = 2,3, . ,M) at the beginning of the t+l time period, Sj,t+l , shall be composed of those persons available at the beginning of the t time period on level j (Sj, t), plus the flow of nongraduates at the j level in the time period t [Xj(Xj,t)] , plus the flow of graduates who complete the G-1) educational level and who flow from the educational system to the economic system on the j level in period t, Lj_1 [xj_l,t -hjj-l(xj-l,t)], and reduced by the flow of persons who flow from the available labor supply as manpower demand in period t on level j (dj, t). Equation (6) is identical in meaning to Eq. (2) with available labor pool level M+l corresponding to available labor pool level 2 of the single level model. These equations relate the flows into and out of the available labor supply pools. An important consequence of this system’s structure links educational levels over time. The number of students in one of the j educational levels (for j = 2,3,. . , M) is equal to the number of graduates from level j-l which do not leave the educational system in the previous time period. Since Xi-1 ,r-r Xi-1(Xj-1 ,t-l) represents the graduates in time period r-1 from educational level j-l, and ~j-1(.) represents those graduates who flow from the educational system to the economy, then the quantity
flows into the next educational level in the next time period. This process yields Eq. (7) and is shown in Fig. 3. Equations (4-7) shall be used to define the constraint set of our minimization problem. It is now possible to write the expression for all costs in time period t (t = 13,“’ . . . 1IV), K(t), as: M+ 1
Xl
rc(t)
= c cj, kxj, f> + j=1
t=17 a-3 .
c gj, *
j=
I
. . 9 N,
(8)
where Cj,,(‘) is the function representing the educational cost on level j in period C, and gj,,(‘) is the function representing the penalty cost on level j in period t. It is assumed that these functions are known. Thcrefor-e. Eq. (8) includes all the costs in period t incurred by the educational system for the M levels (the first summation) and all the costs in period t incurred as a result of shortages or surpluses of manpower for thcM+l levels (the second summation). To obtain the present worth of the total cost for this system for the entire planning horizon. Eq. (8) must be appropriately discounted and summed for periods 1 to N. Thus
TC=
t= C
1
Or--’ 1 c cj,t(xj,r) + j=
1
c
j=
1
iTj,&S,,f+,)].
(9)
EDUCATIONAL
MODELS
FOR MANPOWER
J EDUCATIONAL -k
‘J-l+I
J-l
LEVEL
(XJ_,,t_,
317
DEVELOPMENT ENTRANTS
)-
-LJ_,CxJ-~,t-i-XJ-I(XJ-,,+-I)3
A
GRADUATES EDUCATIONAL L
cx
J-l
LEAVING SYSTEM
J-l,t-I-XJ-I(XJ-l,t-,”
EDUCATIONAL LEVEL
J-l
I
I
*
NON - GRADUATES
1
J-i
FIG. 3. Flows generated
from thej-1
(x
J-l,+?
educational
level.
is Eq. (9) that will be minimized, resulting in the selection ofx;,, for f = 1,2, . . . ,N, equations. The subject to the availability restrictions on each x~,~ and the input-output minimization problem now becomes It
{Xl,lJ1,2,.
.
r. e’-’ ,Xl,N}
tt1)I . t)+ 2Zgj,dsj, j=cj,dxj, M
A’
Minimize
t=1
c
M+l
j= 1
1
Subject to the following constraints: 0
S 1,t+1 sj, t+ 1
t=1,2,...,N
GP*
GX1,t
=
s1,t + 6,
-x1,t)
+ Xl(Xl,,)
-d1,t
t= 1,2,. . . ,N
=Sj,t+Xj(xj,t)+Lj-l[Xj-l,t-Aj-l(Xj-l,t)-dj,t
fori=2,3,... sM+l,ttl
= sM+l,t
+ [XM,t-XM(XM,tr)l
-dMtl,t
,Mand t=
t= 1,2,. . . ,N
152,. . . ,N
(10)
WARREN
318
L.BALINSKY
and
xi t =
xi-1
,/-I
-
Ai-1 (Xi-IJ-1)
- I+1
forj=2,3,...,Mand
[q-l
J-1
-
+I
(Xi-1,t-1 ,I
t=....-2,-1,0,1,2,...,N
Under particular conditions, the multi-echelon model leads to some very nice closed form optimal decision rules.’ Under one set of conditions. the optimal policy is to admit each period that number of eligible persons into the educational system such that the manpower demands at the highest level are exactly satistied. Also, under another set of conditions, the optimal policy is to admit the total number of eligible persons into the educational system each period. Specifically, the conditions which guarantee that the optimal decision rule meets exactly the manpower demands on the highest level each period are: 1. No shortages of manpower are permitted. 2. All functions are linear. 3. The cost of educating and storing manpower is strictly decreasing with time. Therefore, one wishes to defer education or training. 4. The yearly supply of eligible population is sufficiently large such that manpower demands on level M+l can be met on a yearly basis, that is without storing on level M+l. 5. The cumulative demands on level M+l (appropriately adjusted) must generate enough nongraduates and graduates who seek employment at all the lower levels such that no shortages occur. Thus one prefers deferring education (training); however, to avoid manpower shortages, if level M+l demand is met exactly (each period) all lower level demands are met and this generates the least cost solution. The nice properties that this solution exhibits are: (1) it is independent of initial available labor pool supplies, demand estimates and all parameters related to all but the highest level in the system, (2) the solution can be written in an analytically closed form which is very rare when treating a complex system, and (3) it justifies mathematically what is an intuitively correct and often practiced decision rule. The conditions guaranteeing an optimal decision rule of admitting the total eligible population into the educational system each period are: 1. No shortages of manpower are permitted. 2. All functions are linear. 3. The cost of educating and storing manpower is strictly less than the cost of not educating regardless of the time period. Therefore, one wishes to educate the total eligible population every time period. 4. Enough nongraduates must be generated from educating the entire population each period to satisfy level 1 manpower demands when the only flow into level 1 manpower supply is via the nongraduate route. In this case, one prefers to educate the total eligible population each time period, and this course of action guarantees no manpower shortages and the least cost solution. Again, this solution exhibits nice properties in that it: (1) is independent of all system :)See [4]or [3]
EDUCATIONAL
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FOR MANPOWER
319
DEVELOPMENT
parameters and demand forecasts, (2) can be written in an analytically closed form and (3) again justifies mathematically what is an intuitively correct and often practiced decision rule. 4. Educational Program Duration Modifications and Other Extensions This section shall deal with modification of the previously developed models in order to render models more specific and at the same time more realistic. The first modification to be explored is the situation in which each educational level does not require a single planning time period for completion. Two situations shall be explored: the first is when some educational levels require fractional time periods, the second is when some educational levels require multiple time periods. To illustrate the first case, suppose a 4-level educational system in which each educational level required 8 months is to be analyzed. If the population estimates and manpower requirements are given as aggregates over g-month time periods, then certainly the multi-educational level planning models as stated can be used.’ However, if the population estimates and manpower requirements are given as yearly aggregates, then we proceed as follows. Taking two-thirds of the yearly population estimates and two-thirds of the yearly manpower requirements allows us to use the multi-educational level models with planning periods of 8 months. Suppose one is confronted with a 3 level educational system in which the 1st level requires 6 months, the 2nd level requires 3 months, and the 3rd level requires 6 months. Taking the common denominator between 6 months, 3 months and 6 months, one should develop a system with planning periods of 3 months. All yearly demand estimates must of course be adjusted to quarterly estimates. But now it takes 2 time periods to graduate from the 1st educational level, 1 to graduate from the 2nd level and 2 from the 3rd. To continue with this example, we shall investigate the second modification mentioned before. That is, the case when some educational levels require multiple time periods. In general, the technique to modify the multi-educational level model to account for multiples of some time period (to complete the requirements at some given level) is to increase the number of levels in the system. Continuing with the example of the previous paragraph we are confronted with a 3 level system in which the 1st level requires 2 time periods, the 2nd level 1 time period and the 3rd level 2 time periods. Adding together the number of periods required to graduate from the system, we obtain 5, therefore we shall require a 5 level educational system. Observe Fig. 4: it is a 5 level educational system in which levels 1 and 4 have only graduates (no dropouts). The flow equations for this system follow:
s 1,t+1
=s1,,
+ (Pt -X1,t)
S3,t+ I = S3,r + X363,t)
+ X2@2,t)
+ J52 b2,t
-d1,,
- hz(xz,t>l
r =
1,2,.
- d3,t
. ,N
(11)
t= 12> ,..., 4,
s4,t+
1 = S4,t + X5 (X5,t)
+ L3
LX3,t - A3 CX3,dl
-d4,t
t= 1,2,. . . ,N (13)
Scj,r+1 ’ One merely
assumes
=
S6,r + [x5,r
n period
- b(x5,r)l
of 8 months
-d6,r
t=1,2,...,N
(14)
WARREN
320
-c r------
* * * N
Available Labor Pool Level 6
.
Avoilable Labor Pool Level 4
w
Employment Pool Level 4
;-
Employment Pool Level 3
*
Employment Pool Level I
L. BALINSKY
EmpzY+ Level 6
Educotionol
Avoiloble Lof;vreep~l
Educational Level 2
d +
I,
Level
-J I L7~___1
I
I
Total Eligible Population
Available *Labor Pool * Level I -
FIG. 4. A modified
S-level system.
for j = 3,4 and t = . . . , -2,-1,0,1,2, x2,t
=x1,t-I
x5,t = x4,t-1
. . . ,N
(15)
for I = . . , -2,-l
,0,1,2,
. ,N
(16)
for t = . . , -2,-l
,0,1,2,
. . ,N
(17)
The system represents a modified 5 level educational system which accounts for the differences in educational duration at the various levels. One can see both from Fig. 4 and flow equations (16) and (I 7) that educational levels 1 and 4 are really artificial educational levels in that all students pass through these levels after one time period. These levels delay the educational process in a manner which allows the proper time durations to be accounted for in the model. Since the original problem was concerned with only 3 educational levels, there are 4 available manpower pools. and 4 employment pools; the pools 2 and 5 corresponding to artificial educational levels 1 and 4 have been appropriately discarded. One can think of levels 1 and 2 as a single level (level l*), level 3 as a single level (level 2*) and levels 4 and 5 as a single level (level 3*) of the original 3 level
EDUCATIONAL
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FOR MANPOWER
DEVELOPMENT
321
system and therefore, by way of illustration, some methods for modifying the multilevel models to account for differences in educational duration have been presented. Figure 5 represents a very realistic multilevel modification. Suppose, one would like to model the situation in which there is a minimum educational level which the entire population must attain. For example, if each educational level represented a 4 year period of study, levels 1 and 2 of Fig. 5 could represent the elementary school system, levels 3, 4, and 5 high school, college and graduate school, respectively. The flow equations for this system would be: S3,t+1
= S3,t
+ (X&t
-x3,2
+ h3b3,t)
Si, t+ 1 = Sj,t + Ai(xj, t>+ Lj- 1[xi- 1,t -
Ai-1(xi- 1,t>l - di,t j=4,5andt=1,2
S 6,r+
1 = S6,t
+ [Xs,t
-
~5@5,t>1
t=
- d6,r
A
Educational Level
1,2,. . . N
Level
4
*
3
+
-
Available L$ov$gol
L
L-l Totol Population
,...,
Empboyyent
Available
Labor Pool Level 4
.z t _-I
__)
Employment Pool Level 3
I I I
FIG. 5. A system with compulsory
(18)
t=1,2,...,N
-d3,1
educational
levels.
N
(19) (20)
322
WARREN L. BALINSKY
Xi>t = Xj-l,t-1
- Aj-1
(Xj-1,1-I
)-
Lj-1
[Xj-1 ,r--l
ij-1
-
Cxj-l ,I--1 )I
j = 4,5 and t = . . . ) -2,-1,0,1,2, t =
x1,t = Pt X2,t
,-2,-l
,O,I ,2,
t= . . . ) -2,-1,0,1,2,.
= x1,t-1
(21)
. . ,N
,N
(22) ,N
(23)
where the decision level for this model is level 3. That is, the decision problem is how many students to optimally admit to the third educational level from an available amount ~2,~ on the second level. Essentially, this model shifts the entire population up 2 levels and adjusts for the educational delay involved in doing this. (One could renumber the system, such that the total population and educational levels 1 and 2 become the total population *, and higher educational levels 3, 4, 5 become l*, 2*. 3”. By doing this, the system in Fig. 5, would correspond to the system in Fig. 2 in which there is a time adjustment to account for the total population * becoming eligible for level I*.) Figure 6 represents a modification similar- to that represented by Fig. 4. This moditication of the multilevel system incorporates the possibility of a student dropping out of an educational program after having completed a few educational levels, but having not completed the entir-e program.” That is, if levels 1, 3, 3. and 4 constitute a program (such as a four year college), any student who drops out before completing the entire 4 level program essentially gets no credit for attending the program and flows into available labor pool 1, as if he had not attended this program at all. The flow equations corresponding to this modification are: s,,t+ 1 = s1,t
+
(/It
-x1.r)
+ Xl(XlJ)
+
X*(x-2.t)
+ b(X3.t)
+
X4b4,t)
t=1,2,....1v S 2,t+ 1
= s2,t
+
Xi.r = Xj-
1 ,t_
b4,t
1
-
-
Xj_
h4(x4,t)l 1
-d2,1
t=1? ,-.
. , IV
(Xi_1) j = 2,3,4 and t = . . . , -2,-1,0,1,2, 1,
-d1,t
(24) (25)
t-
. . ,N
(26)
For the purpose of clar-ity, each modification in this section has been presented as a special case of the multi-echelon multiperiod educational model. However, combining these modifications is quite simple and the number of further special cases is only bounded by the limits of one’s imagination.
5. Conclusions A number of educational planning models are developed and optimized. Beginning with a single educational program-multiperiod model, and then after presenting a general multi-echelon multiperiod educational model, a very realistic “compulsory type” educational system and a “college type” educational system are presented. By means of sample r-esults for the single educational program-multiperiod dynamic programming, model are demonstrated; whereas, linear programming generates the optimal decisions presented for the general multi-echelon multiperiod educational model.
EDUCATIONAL
MODELS
FOR MANPOWER
4
DEVELOPMENT
323
Available Labor Pool __)I Level 2
Employment Pool
Level
2
Educational Level 4
Educational Level I 4
& --c Available Y Labor Pool --Z Level I
>
FIG. 6. A 4-level “college-type”
system.
Employment Pool Level I
All of the models presented are but a few of the many that can be generated by modifying the flows, the levels, etc. Elsewhere [3], there exists a general scheme which presents all of the manpower planning models in a systematic fashion. The scheme is a method of unification, through generalization, of all such models. The most general model in the scheme contains all the components which are currently perceived as being germane to these models. Most models in the literature, however, are of lesser complexity and can be shown to be subcases of this general model. These subcases may be obtained merely by dropping or simplifying certain terms in the general model. In this manner, it is possible to delineate the entire field of manpower flow models in a systematic fashion. The classification scheme, LASTFENKCO, is first presented and then used to classify the models in the literature. This scheme can also be used to indicate those models that have yet to be developed. References 1. Adehnan, I., A Linear Programming Model of Educational Planning: A Case Study of Argentina, Paper delivered at the Joint Session of the Econometric Society and the American Economic Association, New York, 1965. 2. Balinsky, Warren, An Annotated Bibliography: Applications of Operations Research to Educa-
324
WARREN
L. BALINSKY
tional Systems. Technical Memorandum No. 96, Case Western Reserve University, November 1967. 3. Balinsky, Warren, Some Manpower Planning Models Based on Levels of Educational Attainment. Technical Memorandum ifl85, Case Western Reserve University, Cleveland, Ohio, June 1970, also available from University Microfilms. 4. Balinsky, Warren, Some Optimal Decision Rules for a Multi-Echelon Multi-Period Production System. Working Paper No. 99, State University of New York at Buffalo, February 1971. 5. Balinsky, Warren, Some Dynamic Programming Models for a Single Process-Multiperiod Inventory and Production Systems. Working Paper No. 102, State University of New York at Buffalo, March 1971. 6. Bellman, R., and Dreyfus, S.E., Applied Dynamic Programming, Princeton University Press, Princeton, NJ, 1962. 7. Bellman, R., and Dreyfus, S.E. Adaptive Control Processes, Princeton University Press, Princeton, NJ, 1961. 8. Bolt, R.H., W.L. Koltun, and O.H. Levine, Doctoral Feedback into Higher Education, Science 148,918 (1965). 9. Bowles, Samuel, Planning Educational Systems for Economic Growth, Harvard University Press, Cambridge, Massachusetts, 1961. 10. Correa, Hector, Quantitative Methodologies of Educational Planning, International Institute for Educational Planning, UNESCO, France, 1967. 11. Hadley, G., Nonlinear and Dynamic Programming, Addison-Wesley, Reading, Massachusetts, 1964. 12. Hadley, G., and T. Whitin, Analysis of Inventory Systems, Prentice-Hall, Englewood Cliffs, New Jersey, 1963. 13. Harbison, I., and C. Meyers, Education, Manpower and Economic Growth, McGraw Hill, New York, 1964. 14. Levine, Sumner, Economic Growth and the Development of Educational Facilities: A Systems Analysis, Socio-Economic Planning Science 1, 21-32 (1967). 15. Moonan, W.J., and M.H. Covher, A Computer Program for the Determination of an Optimal Advancement Policy for Petty Officers: A Dynamic Programming Approach. AD 635850, September 1966. 16. Nemhauser, G.L., Introduction to Dynamic Programming, John Wiley & Sons, New York, 1966. 17. O.E.C.D., MathematicalModels in Educational Planning, Paris, 1967. 18. O.E.C.D., Planning Education for Economic and Social Development, Paris, 1962. 19. Reisman, Arnold, A Population Plow 1:eedback Model, Science 153, (3731) 89-91 (1966). 20. Reisman, Arnold, and Martin I. Taft, On the Generation of Doctorates and Their Feedback Into Higher Education, Socio-Economic Planning Science 2, (2, 3, 4) (April 1969). 21. Scarf, H.E., D. Gilford, and M. Shelley, Multistage Inventory Models and Techniques, Stanford University Press, Stanford, California, 1963. 22. Tinbergen, Jan, and H.C. Bos, A Planning Model for the Educational Requirements of Economic Development,
The Residual Factor and Economic Growth, O.E.C.D.,
Received I7 June 1974; revised 2 June 1975
Paris,
1964.