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A generalized hierarchical model of resource allocation
Oq,llzG ~ The Int Jl of Mgmt Sci. Vol. tl. >4o 3, pp 2"9 2')1, 1'-/83
0305-O.a'S3:53 1~302-'-)-13S0300 0 Cop?right , 19~,3 Pergamon P:'es~ Lid
Pr:nted m Gr~:at Britain -Xil right~ re~,er'.ed
A Generalized Hierarchical Model of Resource Allocation DAVID T W H I T F O R D W A Y N E J DAVIS University
of Illinois at Urbana-Champaign,
USA
( ReceiL'ed Nocemher 1981: in revised./orm September 1982)
This study presents a three-level, linear version of the Generalized Hierarchical Model ( G H M ) and demonstrates how the model c a n be used in allocating resources for a hypothetical university. The G H M uses a composition approach to organizational modeling. By applying the model to a rather large university planning problem, it is demonstrated that the composition approach can serve as a planning tool that m a y be useful in allocating resources within hierarchical organizations.
INTRODUCTION Tttls PAPER has two objectives. First it describes a three-level, linear version of the Generalized Hierarchical Model (GHM) originally presented by Davis [2]. Second it demonstrates how the GHM might be used in allocating resources within a university. The GHM belongs in a class of multilevel composition models. See Sweeny et al. [18] for a di'scussion of the composition approach to organizational modeling. Composition models include those developed by Ruefli [16, 17], Freeland [6], Freeland & Baker [7] and Davis & Talavage [3]. Each of these models uses goal programming to solve resource allocation problems for multilevel organizations. Recently several studies have proposed the use of multicriteria or goal programming in university resource allocation (see [5,8,9, 10, 11, 13, 14, 20, 21]). Although enlightening, each has modeled a component of a university's overall resource allocation problem. That is, these studies have concentrated upon an individual department, a college or university administrative problem. None has investigated the composition problem created by forming an ensemble of subproblems. The individual subproblems of the university planning model presented herein closely resemble previous mul-
tiple criteria academic planning models. What differentiates the composition approach to university planning from previous studies is that it explicitly allows units within the organization to interact. By focusing solely on subproblems such as an academic department's problem, organizational characteristics are omitted. The advantages of the GHM's composition approach are: (1) the organizational structure is explicitly recognized; (2) the model converges rather quickly to a feasible solution; (3) individual subsidiary units have autonomy, yet they are sensitive to global organizational goals; (4) one can observe how organizational goals are transformed into goals for individual decision makers. A discussion of the GHM is given in the next section. The following two sections provide the formulation of the university resource allocation model and present the model's solutions characteristics, respectively. A final section gives concluding comments. 279
2XO
[thittbrd, Datts--Generahzed Hierarchical .!,lodet 0] Resource .4llocation
THE G E N E R A L I Z E D H I E R A R C H I C A L MODEL The version of the GHM discussed here is designed to capture the structure of the threelevel hierarchical organization similar to that depicted in Fig. 1. There is a central unit, whose responsibilities are to establish goals and allocate resources. In the university model, the central unit represents the university's highest administrators. Subordinate to the central unit are M management units (k = 1. . . . . M). A university's colleges would correspond to these management units. They coordinate their subordinate operating units (or departments) and allocate available resources, such that the programs and policies under a college's authority conform to organizational goals and priorities. In total there are N operating units. Define a set of integers, {r0, r t rM}, with r0 equal to zero and rM equal to N, such that the operating units r~_ ~+ I through r k will be subordinate to management unit k. The mathematical structure of the G H M is given in equations (1) to (13.0 in Table 1. Equations (1) to (4) define the central unit's problem, while equations (5.k) to (9.k) and (10.i) to (13.i) give the management and operating units' problems, respectively.
7.
oa
. . . . .
"G
"'
'
o~
v The m a n a g e m e n t unit's problem
Each management unit (k = 1. . . . . M) has two sets of goals. The first is a m~ column vector, Gk(t), which defines a set of external goals allocated by the central unit on iteration t. The second is a m~ column vector of internal goals, gk, which are held constant during the solution procedure. The internal goals can provide a certain amount of autonomy for each management unit. There are four appropriately dimensioned column vectors of deviations, Y { ( t ) , Yk- (t), ~,'~ (t) and y~- (t). These contain positive and negative deviations from Gk(t) and gk, respectively. Each deviation vector is premultiplied by the appropriate (ink x mk) or (rn~ x m~) identity matrix, denoted by/,,,~ or 1,6. Each management unit can select from a set of n, column vectors, IX,( 1) . . . . . Xi( t ) }, containing the proposals generated during iterations 1 through t by operating unit i (i =rk_ ~+ 1 . . . . . r~). B, and B: ( i = r k _ ~ + l . . . . . r~) are (ink X n,) and (m~. x n 3 matrices, respectively.
D
Omega, ~'ol. 11..Vo. 3
These matrices linearly relate the ith operating unit's proposals to the set of external and internal goals. In solving its problem on iteration t, each management unit generates a composite n: vector. X*(t). as a convex combination of the set of proposals previously generated by operating unit i. Mathematically X * ( t ) ( i = r k _ ~+ 1, . . . . r D, is defined by (14.i) in conjunction with
the organization's hierarchy. The central unit is "told" via the l"£-(t) and YE(t) deviation vectors how close manager k came to achieving the goals it was given on iteration t. G~(t). Operating units subordinate to management unit k receive an r/, vector of goals, ;,',(t + I) (i =rk_ ~+ I . . . . . r O. The ;:,(t + I) vector provides information to assist operating unit i in selecting a proposal, X,(t + 1), that can potentially improve goal achievement on iteration t + I. A discussion of the functional form of ;,',(t + 1) is given immediately following the description of the operating unit's problem.
(8.~) .V*(t)= V .V(r).;.,(r)
i=q..l+
for
I
G.
281
(14.i)
r=l
Using a goal programming framework X*(t) is obtained by selecting values for 2,(r) (i = rk_ t + 1 . . . . . r~ and r = 1. . . . . t) and the Y ; ( t ) , Y£-(t), y~r(t) and y~-(t) deviation vectors. This goal programming problem is controlled by the priority weights contained in the W~-, W~-. ,.~ and w~- row vectors. These weights are held constant throughout the solution process. Once a solution for the management unit's problem is obtained, the kth management unit passes coordinative information up and down
Tiw central ,rail's problem The central unit's rote is to select new goal vectors, G~(t + 1) (k = I . . . . . M). for its management units. In determining the new goal vectors the central unit must consider the constraints in equations (3) and (4). In (3) G, is a m~ column vector of overall organizational goals and P~ (k = 1. . . . . M) is a (m, x m D matrix that linearly relates Gk(t + I) to G~. Equation (2.k) allows the reassignment of positive and negative deviations among managers.
TAIILE I. SPECIFICArION OF THE G H M
The central unit's problem (I)
MinZ,,(t + I ) = i ~t
{W~[S~'( t + 1)]+ W~-[S~-(t + I)l I.
/,=1
(2.k) l,,,,G~(t + I) + I,,,S~'(t + l) - l,,,,S[(t + I) = Gk(t ) + I"~Y(t) -- Y~(t) for k = 1. . . . . ;'v[ where l,,,~ is the (m~ × m D identity matrix (3)
~.. P~G~(t + I) ~> G~
(41
G~(t+[)>_O.
S~(t+l)>O.
S~-(t+l)>0
The managen ent unit k's prohlem (k = I . . . . .
M)
(5.k) Min Z~(t) = I.V;[l".'(t)] + IVk-[yk-(t)] + w~-[y;(t)] + w~[y~-(t)] sc
~
(6.k) (7.k) ,=t~
(8.i)
i
-E-I
~" B , , V : ( r ) ) . , ( r ) - l ~ , Y ; ( t ) + I m ,
r£-(t)=G~(t )
y~ Bj(,(r)).,(r) - l,,,/vk'(t) + l~iv:~(t) = gk :=I
'~ ,;.,(r ) = I,
(9.k) Y ? ( t ) > O ,
,;.:(,2)> 0 for / = rk_ [ + I . . . . . r~
Y;(t)>O
The operating unit i's problem (i = I . . . . , N) (10.i)
Min~q,-[qS,-(t + I)]+ f2,-[qS,(t + I)] st
(l l . i ) [ f ~ , ] [ . l ( t ( t - t - l ) ] - / , d p - ( t + l ) + [ ~ c ~ , ( t + l ) = . / , ( t + l (12.i)
D,Y,(t+I)~F,
(13.i)
A;(t + l ) _ O ,
c/5,'(t+ll>O,
c~7(t4-1)>0
)
2,',2
~i71it/brd. Dut'is--Generalized Hierarchical 31odel o( Resource Agocatton
The reassigned deviations are contained in the nl, column vectors, S(-(t + 1) and S~-{t + 1). Note that the right hand side of(2.k) represents a goal vector which manager k could meet with no deviations, given the current values of A'7(t) (i = r~_t + I . . . . . q). The central unit's problem uses a goal programming formulation in which the sum of the weighted deviations, It'ES~(t + 1) and WE & - ( t + I), is minimized. Once the G~(t + 1) (k = I . . . . . M) goal vectors have been obtained, they are passed down to the appropriate management unit for consideration at the next iteration.
The operating unit's problenl In this problem, f U and ~ 7 are q~ row vectors containing goal programming priority weights assigned to q, column vectors, ~bT(t + I) and 4~,-(t + 1), positive and negative deviations, respectively, from the goal vector, 7,(t + 1). In the university planning model the values of departmental priority weightings, f2,+ and ~,-, are equated to the augmented vectors, [W~lw~-] and [W~Tlwk-] of the appropriate governing college. X,(t + 1) is a t1~vector containing the unit's proposals for iteration t + I. /7, is a l, column vector of constraint constants and D, is a (l, x n,) matrix that linearly relates X,(t + I) to F~. Again the operating unit's decision is formulated as a goal programming problem which generates X,(t + I ) by minimizing the weighted sum of the deviations given in (10.i).
The GHM solution procedure The G H M follows an iterative solution procedure. The central unit's problem is not solved on the first iteration. Instead, external goals, G~(1) for each manager (k = 1. . . . . M) are set to zero. Initially, each operating unit provides its superior with a set of feasible operating decisions. The operating units do not consider equations (10.i) or (11.i) on the first iteration, and the operating units' proposals satisfy only the minimum specifications given in (12.i). Accordingly, this phase of the algorithm resembles a "zero-base' [15] budgeting system. Utilizing its subordinates" initial proposals, each management unit formulates and solves its problem. At this juncture two alternatives are available: (I) ask the central unit for a revised set of external goals and (II) ask the operating units to revise their proposals so that they conform more closely with the current external and inter-
hal goals of their superordinate manager. Task I is accomplished by informing the central unit on the success or lack thereof each management unit has had in achieving its goals on iteration t: that is manager k passes along the I"~7(t) and F , ( t ) deviation vectors to the central unit. Once the ]",,-(t) and YF(t) (k = 1. . . . . M) vectors are received, the central unit solves its problem and generates new goals for the management units on the next iteration. Task II involves the generation of the goal vector 7,(t + 1) defined in (15.i). In
' for
-L~J i=r~_t+l
.....
LT~J
{15.i)
r~
speci~'ing 7,(t + 1) (i = r k_ i + I . . . . . rk) manager k asks each operating unit to generate alternative proposals, X,(t + 1), which minimize the deviations that manager k is experiencing on iteration t. That is, each operating unit is asked to respond to its manager's total deviations. Once the coordinating goal vectors are received, the operating units solve their problems. The new proposals are then transmitted to the appropriate management unit and the process begins again. These information exchanges continue until convergence is achieved. Convergence is achieved when the central unit's objective function values on iterations t and t + 1 are equal. The analytical properties of the G H M have been investigated in detail; however, an extensive discussion of these properties will not be given here due to space limitations. For an in depth discussion see Davis and Whitford [4]. The authors have defined an overall problem for the three-level, linear version of the G H M . The G H M converges to a feasible solution to this overall problem. The G H M is one of several composition models [3, 6, 7, 16, 17] that can be applied to the same hierarchical problem. Hence it is possible to compare different solution characteristics of the composition approaches. The G H M differs from these other composition models is several ways. It uses a goal programming formulation at all levels of the hierarchy. Consequently one can observe how overall organizational goals are transformed into goals for individual decision makers. Unlike the majority of composition models, G H M does not rely upon simplex multipliers (often interpreted as transfer
Orne~,,a. ~bl. II, .'¢o. 3
prices/as a source of coordinative information. (The fact that the G H M does not use simplex multipliers clearly does not discredit their use by other composition models.) The G H M offers flexibility in its approach to organizational modeling. For example the strategy depicted in equation (15.i) is but one of many potential alternatives for generating goals to coordinate the operating units. A group of fractional allocation procedures in which only a portion of a manager's current deviations is assigned to each subordinate have been tested. The G H M presented in Table 1 uses a linear formulation for its objective functions, but other functional forms are possible. Currently a quadratic version of the G H M is being investigated. The model in Table I assumes a consistent ranking of goals throughout the hierarchy. For example the ~qi+ vector is equated to the composite vector, [W~-[w~]. This assumption can be relaxed to account for priority divergence within the organization. Unfortunately this type of modification makes the specification of an overall objective function for the organization difficult. By extending the G H M to incorporate these and other organizational characteristics, one quickly encounters the intractability of defining the G H M ' s mathematical properties. In this situation the G H M becomes a heuristic model of organizational decision making; however, it still remains within the general framework of a composition model.
SPECIFICATION OF T H E U N I V E R S I T Y MODEL The planning model presented in this study focuses upon university resource allocation over a three year horizon. As seen in Fig. 1, the university is composed of two major units: the College of Arts and Sciences (AS) and the College of Business Administration (BA). In turn each college is composed of several subordinate departments or institutes. A state-supported university located in the southern United States served as the structuring guide for this study. It should be noted that some of the model's characteristics do not conform to the actual university. Simplifications were made to keep the structure of the model within reasonable bounds. Accordingly the re-
283
suits of the research, although closely representative of an actual university, do not represent a "pure" application whose solution could be full',' implemented. The principal focus of the study was upon academic units. Administrative and support services, such as building maintenance and major additions to laboratory, library, classroom and office facilities, were not directly included in the model. However, sensitivity analysis could easily evaluate the potential trade-offs among these commitments.
The university lecel problem
A statement of the university level formulation requires the specification of the aggregate university goal vector, Go, the partitioned college goal vectors, Gk (k = 1,2), and the matrices which relate the colleges' programs and the university's goals. The university is assumed to have six performance goals in each planning period. These goals relate to the university's discretionary budget, graduate and undergraduate unfulfilled student demand, minimum levels of university-wide undergraduate core course offerings, and faculty composition. Panel A of Table 2 provides the values for each of the Go elements. The first three elements specify budgets of $8.0, $8.5 and $9.0 m during the next three years, respectively. The next six goals are related to unfulfilled undergraduate and graduate student demand. Each of the unfulfilled goal levels is constrained to be less than or equal to zero. The next three goals focus upon the freshman-sophomore core curriculum shared by AS and BA students. At the university modeled here, approximately 78 and 44°,o of the respective freshman and sophomore BA courses are provided by the AS faculty. Because AS decisions affect individual college and university programs, it seemed reasonable that the university should review these decisions to assure overall program quality. The final two goal sets involve proportional constraints related to faculty composition and accreditation. These goals are targeted at maintaining quality in teaching and research and impose the following conditions: (a) at least two-thirds of the university's full time faculty must have completed a PhD or other earned doctorate degree and (b) the percentage of full
284
~Vhitlbrd. Davis--Generalized Hierarchical Model o! Resource Allocatton
T~,BLE". COMPONENTSOF THE UNIVERSITYLEVELPROBLEM Goal
A. Elements o]" the G,3 cector
B. Unicersity decision cariables for period t
Planning ?ear 2
College AS(k =I) BAlk =2)
I 3 University budget 8.000.000 8.500.000 9.000,000 ceiling ($) Unfulfilled undergraduate 0 0 0 demand Unfulfilled graduate 0 0 0 demand Minimum uni~ersity-core 603 615 621 courses Terminal degree t:acultv 0 0 0 proportion Graduate student 0 0 0 instructor limits
time equivalent teaching assistants and instructors should not exceed 40% of the entire faculty. To understand how a zero goal can generate a proportional constraint, assume that M F is the number of faculty without a terminal (e.g. PhD) degree and D F represents the number of terminal degree faculty. Also assume that the university has a policy that requires at least two-thirds of its faculty to have a terminal degree. This implies
C. Unicersi O' letel constraint set
Gi.s.t -- G,..s.i < 8.000.000 (16.1) G~8.: + G,.s: < 8.500.000 (16.2)
Gt.s.,
Gz.s,
Gl.tt.,
G:tt~
G~B.3+ G:.8.3< 9,000.000 (16.3) GItL.,+G:.~: <0 V~ (16.4)
Gt.L~,.~
Gz.tI,~
GL.t~j.~-Gz.Lv.~
G,,~s.,
not applicable
G t.ro ~
G2.rD ;
Gt.FS.I > 603 GI.FS.," > 615 Gj.rs.3 >_621 GLro.,+G,_.ro.,>_O
G~.a~.,
Gz.a~.,
Gi.cc, + Gz.Gt., < 0 V,
•,
(16.5)
V¢
(16.61 (16.7) (16.8) (16.9) ( 16. I O)
inclusion of the G H M ' s goal programming deviation vectors. The college p r o b l e m s
Within the G H M framework the colleges review the proposals generated by their subordinate departments and recommend revised performance goals based upon current external university and internal college policies. I n t e r n a l goals. Because the majority of a university's course offerings are part of internal college programs, it was not necessary to superDF > 2/3 [MF + DF]. (17) impose external university constraints upon all departmental course offerings. However, there Rearranging (17) and inserting positive and was a need to specify that internal college negative deviation variables, d-~o and diD, yields programs were adequately staffed and funded. the following goal programming constraint: To accomplish this, three internal goal sets were -2/3MF+ l/3DF-d-/o+dro=O. (18) imposed; they controlled the colleges' doctoral programs and tenure decisions during the three The role of the university is to generate the right year planning horizon. These internal goals and hand side of this constraint which appears in the constraints are given in equations (19.1) colleges' formulation. (See equations (19.7)and through (19.3) in Table 3. (19.8).) E x t e r n a l goals. Equations (19.4) through In solving its problem, the university par- (19.9) specify the remainder of the AS and BA titions its goal vector by solving for the vari- problems (equation (19.9) is applicable only to ables defined in panel B of Table 2. These goal AS). These equations link the external goals, assignments are subject to the constraints ap- generated by the university, to the departmental pearing in panel C of Table 2 (i.e. equations operating proposals. They are targeted at meet(16.1 ) through (16.10)). It should be emphasized ing the university stipulated goals in each planthat these 'hard constraints' will not necessarily ning period for budget, unfulfilled underprevent budget overruns or other undesirable graduate and graduate demand, faculty goal violations. This possibility results from the composition and minimum core courses.
Omega. Vol. 11..Vo. 3
The departmental problems Departmental decision t'ariables. The components of the departmental problems are the number, level, size and staffing of course offerings. Each department had seven levels of full time equivalent (FTE) staff and faculty positions as well as seven levels of course offerings. The staff levels were:
285
(2) junior-senior core. (3) junior-senior major-minor. (4) graduate (masters) core, (5) graduate (masters) major, (6) graduate (doctorate) core,
( 1) secretarial and research associate personnel, (7) graduate (doctorate) major courses. (2) graduate student research assistants, Associated with these course levels are seven unfulfilled course demand variables. Let 4%..... 7.k,j., and rk,.~., equal staff positions, course offerings and unfulfilled student demand associated with department j of college k at level i in planning period t. respectively. Finally, let/3kr ,, the budget of department j of college k during period t equal the sum of department's salaries, cry,.,, plus indirect costs, cJ~,.,. Departmental operating constraints and goals. Each department has five categories of operating constraints:
(3) graduate student teaching assistants, (4) instructors (nonterminal degree), (5) assistant professors, (6) associate (tenured) professors, (7) full (tenured) professors. The course levels were: (I) freshman-sophomore core,
TABLE 3.
(a) class size and enrollments,
SPECIFICATION OF THE
AS
AND
BA
INTERNAL AND EXTERNAL GOALS AND CONSTRAINTS
AS and BA college constraints and goals
Internal No. doctoral seminars offered in year t for deparment i No. internal or external promotions to associate professor in year t No. internal or external promotions to full professor in year t
- d i s . , + dos., = Target year t - d'de., + d5e., = Target year t - d ; e . , + d~.e., = Target year t
program level in t'or department i No. promotions in
',-',, (19.1) ~:
(19.2)
No. promotions in
z',
(19.3)
+ d i . , = External budget goal in year t - d £ : ~ . , + di:t. , = Unfulfilled undergraduate demand goal in year t - d ' ( c . , + d~c., = Unfulfilled graduate demand goal in year t
V.
(19.4)
,~:
(19.5)
V,
(19.6)
V,
(19.7)
v,
(19.8)
v,
(19.9)
External Total departmental budgets in year t
-d~.,
Total unfulfilled undergraduate student demand in year t Total unfulfilled graduate student demand in year t
-0.667
Total non-terminal degree faculty in year t
+ 0.333
0.6
Total FTE graduate student/ instructors in year t
0.4
Total terminal degree faculty in year t
E T°tal] professorial faculty in year t
The total number of AS freshman/sophomore core courses offered in year t
O~II/3-E
-d-~o., + dro., = Terminal degree faculty goal in year t
- d ~ u + d~t., = Graduate student/instructor goal in ,,'ear t
-d;s~.s. , + d~ses., = M i n i m u m freshman/ sophomore core courses goal in year t
"-S6
~Vhitti~rd. Davis--Generalized Hierarchical .!,lodel o! Re*nurce .4ll.t atl,m
lb) teaching loads and levels. Ic) secretarial and research support. (d) direct and indirect budgetary expenses, le) tenure obligations. These constraints are given in equations (20.1) through (20.16) in Table 4. Equations (20.1) through (20.5)incorporate university policy on average class size and staffing. For example, (20.1) states that the number of courses offered by department kj at level i in year t times the average class size for that level, plus any unfulfilled student demand in department k i for course level i in period t will equal the department's expected student demand for that course level in year t. In addition (20.2) through (20.5) specify that courses offered at various levels cannot exceed staff resources capable of teaching at that level. For example, only professorial faculty are allowed to teach doctoral courses: however, as the course offerings expand to include the freshman-sophomore level (20.5), all instruction staff resources are included. Equations (20.6) through (20.8) provide minimum and maximum levels of secretarial and departmental administrative support. Similarly (20.9) and (20.10) levy maximum and minimum levels for research assistant support. Equations (20. II) through (20.14) are related to budgetary items. They define departmental salaries, minimum and maximum indirect cost levels and a total departmental budget, respectively. Finally, (20.15) and (20.16) insure that tenure obligations are fulfilled. A final set of goal linkage equations is necessary to complete the departmental problems. Because these equations are departmental versions of each college's internal and external goal constraints (equations (19.1) through (19.9) in Table 3), they will not be detailed here. Instead of summing across a college's departments, these goal linkage constraints focus upon a single department. Not all constraints given in Table 3 are applicable to every department. As an example, a departmental version of (19.1) was applied only to those departments offering doctoral programs. Also, only AS departments incorporated a minimum freshman-sophomore core course constraint.
Goal deciation priority we(ghtings A procedure similar to the one used by Wacht & Whitford [19] was utilized to determine priorit',' weightings. The approach resembled the "Delphi technique" developed by the Rand Corporation mathematician, Olaf Helmer. After receiving a brief explanation of goal programming, individual administrators at various levels of the university were asked to provide anonymous rankings of undesirable deviations. The ranking scale ranged from 0 to 100: the higher the value, the less desirable the deviation. In the G H M , the colleges determine composite weighting vectors. Based upon rankings obtained from AS and BA administrators, these weights were computed by averaging within each college the responses assigned to a particular deviation. When a college's deviation was applicable to only one department (i.e. goals related to PhD seminars), no averaging of responses was necessary. The average deviation weights tbr AS and BA are given in Fig. 2. These rankings can be interpreted as the priorities expressed by a college's executive committee. These weighting vectors were held constant over the three year planning horizon. In a complex organization, it is likely that no weighting determination procedure can adequately capture true priorities. The possibility of priority measurement error points to the need for sensitivity analysis. Perhaps the greatest benefit of this type of study is the information obtained by observing the impacts that alternative goals and priorities can have upon the allocation of resources.
T H E MODEL'S SOLUTION The smallest of the university model's subproblems was at the university level: it had 69 variables and 36 constraints. The largest subproblems were in the BA departments which contained up to 375 variables and 141 constraints. The G H M ' s F O R T R A N code was written and tested on a Control Data Corporation CYBER-175 computer. The linear programming subroutines used by this code were developed by Marsten [12]. Although the current version of the model's code is solved on a single mainframe, the modular nature of the
Omen.a. ~.'ol. II..Vo. +;
T~,BLE
4.
DEP~,RTMENTAL
-
Tar,.z,et_ .class Zl ~izc tot i /a _ level i _i
F Maximum FTE ~'Z,
graduate teaching load per academic
_<
,,car
VT, ,_<
V" ,5, ,_< /
;
~" =_.
O,
=
l ......
%'} A N D
it =
l .....
3)
(20.1)
J
| . | S" ' / +":' /
(20.2)
j
load per academic
< .%'--
(k
7 / ]
I
Maximum FTE graduate teaching load per academic .sear Maximum FTE undergraduate teaching load per academic year
I
CONSTRAINTS
]
Maximum FTE TA teaching /
OPERATING
{-Expected student demand for le',el : .... = i. department k, L n per od t
2S"
x7 ,Oa.... ,~
(20.3)
(20.4)
"Maximum FTE- l undergraduate [ , teaching ] +,O~:.+.,+ load p~ [ ,=~at/'+'~..... academic
yea r
)'ear Ratio equating I .Minimum FTE ] FTE TAs to secretarial ,/J~, _> equal 1 professorial facuhy ratio l:aculty secretarial requirement I Ratio equating Maximum FTE ] FTE TAs to secretarial ,,0a.k,_< equal I professorial t'acuhy ratio faculty secretarial requirement
[
(20.5)
] J
'/q,.3., + L '#~.....
(20.6)
'/%.3., + L ¢'k.....
(20.7)
,=4
t=4
+].,,..t: > 2
(20.8)
blaximum FTE RA ] f ) Elculty ratio ] ,L-'a ..... L [Minirnum FTE RA ] f ,b~ : g facuhy ratio _ '];a..... t=a
(20.9)
~/q
:. _>
]
(20.10)
[- Appropriate level i salary, ] a. ,= ~V [ofdepartmentjofcoIIege k ] ,]q..... '~' L in period t ]
(20.11)
s7 IMinimum indirect cost-{ ¢'~' ' = "7~ FTE t'acuhy member j ~ .....
(20.12)
[ Maximum indirect cost ] total salary ratio
~e"a '--<
[~
.
=a
4
.+
.
u,^
,
[ No. tenured associate ] '~.... >- a professors in period t ] [ NO. tenured full ] '); "'-> professors m period t
o-~,
(20.13) (20.14) (20.15) (20.16)
258
Whtt]i~rd, Duv~s--Generahzed Hierarchical Model o/ Resource .4/Iota'ion
Z 0
r~ ~J Z ~J 0 0 Z uJ Z
t~
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GHM's structure and solution process makes implementation via a network of interactive minicomputers feasible. It is in this network setting that the GHM offers the distinct advantage of merging intbrmation and decision support systems into the resource allocation process.
2S9
these alternatives created trade-offs in maintaining faculty morale, educational quality and instructional effectiveness. Further, accreditation standards limited several courses of action. Although numerous policy alternatives ~vere tested with varying degrees of success, the compromise version of the model reported here focused upon five a r e a s .
Solution characteristics qf the #titial formulation The initial university formulation converged to a final solution in twelve iterations. Although the model generated no unfulfilled graduate or undergraduate student demand in any planning year, this was achieved at a considerable price. Most dramatic were budget deficits of approximately SI.0, $1.1 and $1.2m in years one through three, respectively. With few exceptions, the departments relied upon teaching assistants and temporary instructors rather than assistant professors. The model was not totally successful at meeting the colleges' internal goals. Although minimum support was generated for doctoral programs, the results of the promotion goals were inconsistent in that no promotions were incorporated in the BA solution. This result can be attributed to the algorithm's 'perception' that BA was generating the university's deficits. As seen in Fig. 2, the AS penalties associated with budget overruns were significantly higher than BA (100 vs 75). This higher level of priority weighting may be interpreted as 'political clout'. Analysis of the colleges' budgets revealed that all deficits occurred in BA. From a behavioral viewpoint, a scenario in which a bureaucratic organization with relatively more political power receives a disproportionately large share of available funding is plausible. But from the university administration's viewpoint, this type of situation may not be desirable. In the light of these and other characteristics it was necessary to revise the original formulation. Problem revision and sensitirit)' analysis To eliminate budget deficits several alternative strategies were available. Although ad hoc in nature, each strategy involved realistic, but politically unpopular compromises. These included increasing average class sizes and faculty teaching loads, decreasing faculty raises and indirect expenses and allowing 'less expensive' graduate student teaching assistants and instructors to teach upper level courses. Each of
The first policy change involved an increase in average class size. Even though these increases could potentially lower educational quality, the Carnegie Commission has endorsed this approach [1]. The second change attempted to counterbalance the potential diminution of educational quality precipitated by class size increases. In the original formulation, temporary instructors were allowed to teach at all levels except doctoral seminars (see equations (20.2) through (20.5)). In the revised version, temporary instructors were allowed to teach only freshman-sophomore and junior-senior core courses. It was anticipated that these changes would force the model to incorporate additional assistant professors into the final solution. Based upon the size of the university's deficit, it was unlikely that campus-wide increases in class size could totally offset funding shortfalls. As a result the third change in university policy focused upon limiting the size of departmental indirect costs. The fourth revision provided parity in the budget priorities of the AS and BA colleges. Both colleges' budget overrun penalty weights were increased and equated so that no college had an advantage in bargaining power. The final changes eliminated the internal promotion goals and lowered the minimum number of departmental doctoral seminars to the number of seminars observed in the oriNnal solutions. This level of doctoral program support was felt to be more than adequate. Because promotions can be incorporated into the formulation within the department's constraints (see equations (20.15) and (20.16)) omission of promotion constraints does not severely limit the formulation's applicability.
Results of the revised formulation The modified version of the model reached a final solution after four iterations and, unlike the previous versions, all university and internal college goals were achieved. One of the most surprising aspects of the revised formulation
290
[{71it/ord. Dat'is--Generalized Hierarchical .$[odel o~ Resource .4/location
was its rapid convergence. Not only did the changes in budget priorities improve computational time and expense, the,,' generated small budget surpluses for BA in years one and two and AS for year three. As one might expect, these restrictive budgets had a direct effect on departmental research support and indirect cost allocations, but none of these support levels was felt to be below minimum acceptable levels. One disappointing aspect of the revised formulation was the model's inability to incorporate significant levels of assistant professors in departments with declining student demand. Based upon the complexity and interactions of departmental, college and university policies, the timely analysis of the major issues facing all levels of a university's administration can be extremely difficult. However, the computational ease engendered by the use of the GHM prorides a convenient and straightforward solution procedure that allows testing and sensitivity analysis of an unlimited number of alternative strategies and policies. Although the GHM cannot guarantee an "optimal" allocation of an organization's resources, it can, if properly employed, significantly improve upon the rationality and reasonableness of resource allocation decisions in many decentralized, hierarchical organizations.
ACK NOW L EDG EM ENTS This study draws heavil? upon Professor Whitford's doctoral dissertation submitted in 1980 at Georgia State University, which was prepared under the super',ision of Prolessors Richard F Wacht, Chairman, Dileep R Mehta. David L Sjoquist and Roger O Miller. Vice-President for Financial Affairs at GSU. These individuals provided many helpful insights and comments. The authors v,ould like to acknowledge the computational support and assistance provided by Mr Stanley P Kerr of the Computing Services Office at the University of Illinois at Urbana-Champaign and b,v Professor Roy E Marsten of the Universit? of Arizona. The helpful comments of two anonymous referees are also acknowledged.
REFERENCES I. CARNEGIE COMMISSION ON" HIGHER EDUCATION (1972)
2. 3. 4.
5.
6. 7. 8.
SUMMARY AND CONCLUSIONS This study has presented a three-level, linear version of the GHM and has shown how this model could be used in allocating resources for a hypothetical university. The GHM uses a composition approach to organizational modeling similar to the composition models developed by Ruefli [16, 17], Freeland [6], Freeland & Baker [7] and Davis & Talavage [3]. Unlike these composition models the GHM uses a goal programming format at each level of the organizational hierarchy. By applying the GHM to a large university planning problem, it was demonstrated that the composition approach can serve as a viable planning tool. Sensitivity analysis allowed one to observe and respond to the interactions of organizational goals, priorities and policy guidelines. In summary it appears that the GHM may be useful in allocating resources within hierarchical organizations.
9.
lO.
11. 12.
13.
14. 15. 16. 17.
The More Eff~'ctice L"se of Resources: an hnperative /br Higher Education. McGraw-Hill, New York. DAVIS WJ (1978) A generalized decomposition procedure and its application to engineering design. Jl Mech. Des. 100(4), 739-746. DAVISWJ & TALAVAGEJ (1977) Three-level models for hierarchical coordination. Omega 5(6), 709-722. DAvis WJ & WmTFORD DT (1981) The generalized hierarchical model: a new approach to resource altocation within multilevel organizations. BEBR Working Paper 797, University of Illinois at Urbana-Champaign (August). DYER JF & MULVEY JM (1976) An integrated optimization/information system for academic departmental planning. Mgmt Sci. 22(12), 1332-134 I. FREELANDJR (1976) A note on goal decomposition in a decentralized organization.Mgmt Sci. 23(I), 100--102. FREELANDJR & BAKERJR (1975) Goal partitioning in a hierarchical organization. Omega 3(6), 673-688. GEOFFRION AM, DYER JF & FIENBERG A (1972) An interactive approach for multi-criterion optimization with an application to the operation of an academic department. Mgmt Sci. 19(4), 357-368. HARWOODGB & LAWLESSRW (1975) Optimizing organizational goals in assigning faculty teaching schedules. Dec. Sci. 6(3), 513-524. HOPKINS DSP, LARRECHE JC & MASSY WF (1977) Constrained optimization of a university administrator's preference function. Mgmt Sci, 24(4). 365-377. LEE SM & CLAYTON ER (1972) A goal programming model for academic resource allocation. Mgmt Sci. 18(8), B395-B408. MARSTEN RE (1978) XMP: a structured library of ~ubroutines for experimental mathematical programming. Technical Report 351, Department of MIS, University of Arizona. MASSY WF (1978) Reflections on the application of a decision science model to higher education. Dec'. Sci. 9(2), 362-369. McNAMARAJF (1973) Mathematical programming applications in educational planning. Socio-econ. Plann. Sci. 7(1). 19-35. PHYRR P (1970) Zero-based budgeting. Hart. Bus. Rer. 48(6), 111-121. RUEELI TW (1971) A generalized goal decomposition model. Mgmt Sci. 17(8), B505-518. RUEELI TW (1971) PPBS--an analytic approach. In BYRNE F R e t al. Studies in Budgeting, Vol. II of Budgeting of Interrelated ,4cticities. North-Holland, Amsterdam.
Omeea. V~)I. 1 I..Vo. 3
18. SWEEN~" DJ, WINKOFSKY EP, ROY P & BAKER NR 11978) Composition ~s decomposition: two approaches to modeling organizational decision processes..1,Igmt Sci. 24~ 14). 149[-[499. 19. W~,fnx RF & WHI~ORD DT 11976) A goal programming model for capita[ investment analysis in nonprofit hospitals. Fin. Mgmt 5(2). 37-47. 20. W~.LTERS A, MANGOLD J & HAR,-XN EGP (1976) A comprehensive planning model for long-range academic strategies. 3,Igmt Sci. 22(7), 727-738.
29[
21. WEHRUNG DA, HOPKINS DSP & MASS'~" WF (1978) Interactive preference optimization t'or university administrators..$fgrnt Sci. 2416). 599-611. ~DDRESS FOR CORRESPONDENCE: Pro/essor
David T Whit/ord. College ql' Commerce and Business Admini.~'trt~ti(m, (.'nt£'ersit.v of Illinois at Urbana-Champa~en. 3.10 Commerce Building ( West), 1206South Sixth Street. Champa~en. I L 6182O, USA.
0305-O.a'S3:53 1~302-'-)-13S0300 0 Cop?right , 19~,3 Pergamon P:'es~ Lid
Pr:nted m Gr~:at Britain -Xil right~ re~,er'.ed
A Generalized Hierarchical Model of Resource Allocation DAVID T W H I T F O R D W A Y N E J DAVIS University
of Illinois at Urbana-Champaign,
USA
( ReceiL'ed Nocemher 1981: in revised./orm September 1982)
This study presents a three-level, linear version of the Generalized Hierarchical Model ( G H M ) and demonstrates how the model c a n be used in allocating resources for a hypothetical university. The G H M uses a composition approach to organizational modeling. By applying the model to a rather large university planning problem, it is demonstrated that the composition approach can serve as a planning tool that m a y be useful in allocating resources within hierarchical organizations.
INTRODUCTION Tttls PAPER has two objectives. First it describes a three-level, linear version of the Generalized Hierarchical Model (GHM) originally presented by Davis [2]. Second it demonstrates how the GHM might be used in allocating resources within a university. The GHM belongs in a class of multilevel composition models. See Sweeny et al. [18] for a di'scussion of the composition approach to organizational modeling. Composition models include those developed by Ruefli [16, 17], Freeland [6], Freeland & Baker [7] and Davis & Talavage [3]. Each of these models uses goal programming to solve resource allocation problems for multilevel organizations. Recently several studies have proposed the use of multicriteria or goal programming in university resource allocation (see [5,8,9, 10, 11, 13, 14, 20, 21]). Although enlightening, each has modeled a component of a university's overall resource allocation problem. That is, these studies have concentrated upon an individual department, a college or university administrative problem. None has investigated the composition problem created by forming an ensemble of subproblems. The individual subproblems of the university planning model presented herein closely resemble previous mul-
tiple criteria academic planning models. What differentiates the composition approach to university planning from previous studies is that it explicitly allows units within the organization to interact. By focusing solely on subproblems such as an academic department's problem, organizational characteristics are omitted. The advantages of the GHM's composition approach are: (1) the organizational structure is explicitly recognized; (2) the model converges rather quickly to a feasible solution; (3) individual subsidiary units have autonomy, yet they are sensitive to global organizational goals; (4) one can observe how organizational goals are transformed into goals for individual decision makers. A discussion of the GHM is given in the next section. The following two sections provide the formulation of the university resource allocation model and present the model's solutions characteristics, respectively. A final section gives concluding comments. 279
2XO
[thittbrd, Datts--Generahzed Hierarchical .!,lodet 0] Resource .4llocation
THE G E N E R A L I Z E D H I E R A R C H I C A L MODEL The version of the GHM discussed here is designed to capture the structure of the threelevel hierarchical organization similar to that depicted in Fig. 1. There is a central unit, whose responsibilities are to establish goals and allocate resources. In the university model, the central unit represents the university's highest administrators. Subordinate to the central unit are M management units (k = 1. . . . . M). A university's colleges would correspond to these management units. They coordinate their subordinate operating units (or departments) and allocate available resources, such that the programs and policies under a college's authority conform to organizational goals and priorities. In total there are N operating units. Define a set of integers, {r0, r t rM}, with r0 equal to zero and rM equal to N, such that the operating units r~_ ~+ I through r k will be subordinate to management unit k. The mathematical structure of the G H M is given in equations (1) to (13.0 in Table 1. Equations (1) to (4) define the central unit's problem, while equations (5.k) to (9.k) and (10.i) to (13.i) give the management and operating units' problems, respectively.
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Each management unit (k = 1. . . . . M) has two sets of goals. The first is a m~ column vector, Gk(t), which defines a set of external goals allocated by the central unit on iteration t. The second is a m~ column vector of internal goals, gk, which are held constant during the solution procedure. The internal goals can provide a certain amount of autonomy for each management unit. There are four appropriately dimensioned column vectors of deviations, Y { ( t ) , Yk- (t), ~,'~ (t) and y~- (t). These contain positive and negative deviations from Gk(t) and gk, respectively. Each deviation vector is premultiplied by the appropriate (ink x mk) or (rn~ x m~) identity matrix, denoted by/,,,~ or 1,6. Each management unit can select from a set of n, column vectors, IX,( 1) . . . . . Xi( t ) }, containing the proposals generated during iterations 1 through t by operating unit i (i =rk_ ~+ 1 . . . . . r~). B, and B: ( i = r k _ ~ + l . . . . . r~) are (ink X n,) and (m~. x n 3 matrices, respectively.
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These matrices linearly relate the ith operating unit's proposals to the set of external and internal goals. In solving its problem on iteration t, each management unit generates a composite n: vector. X*(t). as a convex combination of the set of proposals previously generated by operating unit i. Mathematically X * ( t ) ( i = r k _ ~+ 1, . . . . r D, is defined by (14.i) in conjunction with
the organization's hierarchy. The central unit is "told" via the l"£-(t) and YE(t) deviation vectors how close manager k came to achieving the goals it was given on iteration t. G~(t). Operating units subordinate to management unit k receive an r/, vector of goals, ;,',(t + I) (i =rk_ ~+ I . . . . . r O. The ;:,(t + I) vector provides information to assist operating unit i in selecting a proposal, X,(t + 1), that can potentially improve goal achievement on iteration t + I. A discussion of the functional form of ;,',(t + 1) is given immediately following the description of the operating unit's problem.
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Using a goal programming framework X*(t) is obtained by selecting values for 2,(r) (i = rk_ t + 1 . . . . . r~ and r = 1. . . . . t) and the Y ; ( t ) , Y£-(t), y~r(t) and y~-(t) deviation vectors. This goal programming problem is controlled by the priority weights contained in the W~-, W~-. ,.~ and w~- row vectors. These weights are held constant throughout the solution process. Once a solution for the management unit's problem is obtained, the kth management unit passes coordinative information up and down
Tiw central ,rail's problem The central unit's rote is to select new goal vectors, G~(t + 1) (k = I . . . . . M). for its management units. In determining the new goal vectors the central unit must consider the constraints in equations (3) and (4). In (3) G, is a m~ column vector of overall organizational goals and P~ (k = 1. . . . . M) is a (m, x m D matrix that linearly relates Gk(t + I) to G~. Equation (2.k) allows the reassignment of positive and negative deviations among managers.
TAIILE I. SPECIFICArION OF THE G H M
The central unit's problem (I)
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Min~q,-[qS,-(t + I)]+ f2,-[qS,(t + I)] st
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The reassigned deviations are contained in the nl, column vectors, S(-(t + 1) and S~-{t + 1). Note that the right hand side of(2.k) represents a goal vector which manager k could meet with no deviations, given the current values of A'7(t) (i = r~_t + I . . . . . q). The central unit's problem uses a goal programming formulation in which the sum of the weighted deviations, It'ES~(t + 1) and WE & - ( t + I), is minimized. Once the G~(t + 1) (k = I . . . . . M) goal vectors have been obtained, they are passed down to the appropriate management unit for consideration at the next iteration.
The operating unit's problenl In this problem, f U and ~ 7 are q~ row vectors containing goal programming priority weights assigned to q, column vectors, ~bT(t + I) and 4~,-(t + 1), positive and negative deviations, respectively, from the goal vector, 7,(t + 1). In the university planning model the values of departmental priority weightings, f2,+ and ~,-, are equated to the augmented vectors, [W~lw~-] and [W~Tlwk-] of the appropriate governing college. X,(t + 1) is a t1~vector containing the unit's proposals for iteration t + I. /7, is a l, column vector of constraint constants and D, is a (l, x n,) matrix that linearly relates X,(t + I) to F~. Again the operating unit's decision is formulated as a goal programming problem which generates X,(t + I ) by minimizing the weighted sum of the deviations given in (10.i).
The GHM solution procedure The G H M follows an iterative solution procedure. The central unit's problem is not solved on the first iteration. Instead, external goals, G~(1) for each manager (k = 1. . . . . M) are set to zero. Initially, each operating unit provides its superior with a set of feasible operating decisions. The operating units do not consider equations (10.i) or (11.i) on the first iteration, and the operating units' proposals satisfy only the minimum specifications given in (12.i). Accordingly, this phase of the algorithm resembles a "zero-base' [15] budgeting system. Utilizing its subordinates" initial proposals, each management unit formulates and solves its problem. At this juncture two alternatives are available: (I) ask the central unit for a revised set of external goals and (II) ask the operating units to revise their proposals so that they conform more closely with the current external and inter-
hal goals of their superordinate manager. Task I is accomplished by informing the central unit on the success or lack thereof each management unit has had in achieving its goals on iteration t: that is manager k passes along the I"~7(t) and F , ( t ) deviation vectors to the central unit. Once the ]",,-(t) and YF(t) (k = 1. . . . . M) vectors are received, the central unit solves its problem and generates new goals for the management units on the next iteration. Task II involves the generation of the goal vector 7,(t + 1) defined in (15.i). In
' for
-L~J i=r~_t+l
.....
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{15.i)
r~
speci~'ing 7,(t + 1) (i = r k_ i + I . . . . . rk) manager k asks each operating unit to generate alternative proposals, X,(t + 1), which minimize the deviations that manager k is experiencing on iteration t. That is, each operating unit is asked to respond to its manager's total deviations. Once the coordinating goal vectors are received, the operating units solve their problems. The new proposals are then transmitted to the appropriate management unit and the process begins again. These information exchanges continue until convergence is achieved. Convergence is achieved when the central unit's objective function values on iterations t and t + 1 are equal. The analytical properties of the G H M have been investigated in detail; however, an extensive discussion of these properties will not be given here due to space limitations. For an in depth discussion see Davis and Whitford [4]. The authors have defined an overall problem for the three-level, linear version of the G H M . The G H M converges to a feasible solution to this overall problem. The G H M is one of several composition models [3, 6, 7, 16, 17] that can be applied to the same hierarchical problem. Hence it is possible to compare different solution characteristics of the composition approaches. The G H M differs from these other composition models is several ways. It uses a goal programming formulation at all levels of the hierarchy. Consequently one can observe how overall organizational goals are transformed into goals for individual decision makers. Unlike the majority of composition models, G H M does not rely upon simplex multipliers (often interpreted as transfer
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prices/as a source of coordinative information. (The fact that the G H M does not use simplex multipliers clearly does not discredit their use by other composition models.) The G H M offers flexibility in its approach to organizational modeling. For example the strategy depicted in equation (15.i) is but one of many potential alternatives for generating goals to coordinate the operating units. A group of fractional allocation procedures in which only a portion of a manager's current deviations is assigned to each subordinate have been tested. The G H M presented in Table 1 uses a linear formulation for its objective functions, but other functional forms are possible. Currently a quadratic version of the G H M is being investigated. The model in Table I assumes a consistent ranking of goals throughout the hierarchy. For example the ~qi+ vector is equated to the composite vector, [W~-[w~]. This assumption can be relaxed to account for priority divergence within the organization. Unfortunately this type of modification makes the specification of an overall objective function for the organization difficult. By extending the G H M to incorporate these and other organizational characteristics, one quickly encounters the intractability of defining the G H M ' s mathematical properties. In this situation the G H M becomes a heuristic model of organizational decision making; however, it still remains within the general framework of a composition model.
SPECIFICATION OF T H E U N I V E R S I T Y MODEL The planning model presented in this study focuses upon university resource allocation over a three year horizon. As seen in Fig. 1, the university is composed of two major units: the College of Arts and Sciences (AS) and the College of Business Administration (BA). In turn each college is composed of several subordinate departments or institutes. A state-supported university located in the southern United States served as the structuring guide for this study. It should be noted that some of the model's characteristics do not conform to the actual university. Simplifications were made to keep the structure of the model within reasonable bounds. Accordingly the re-
283
suits of the research, although closely representative of an actual university, do not represent a "pure" application whose solution could be full',' implemented. The principal focus of the study was upon academic units. Administrative and support services, such as building maintenance and major additions to laboratory, library, classroom and office facilities, were not directly included in the model. However, sensitivity analysis could easily evaluate the potential trade-offs among these commitments.
The university lecel problem
A statement of the university level formulation requires the specification of the aggregate university goal vector, Go, the partitioned college goal vectors, Gk (k = 1,2), and the matrices which relate the colleges' programs and the university's goals. The university is assumed to have six performance goals in each planning period. These goals relate to the university's discretionary budget, graduate and undergraduate unfulfilled student demand, minimum levels of university-wide undergraduate core course offerings, and faculty composition. Panel A of Table 2 provides the values for each of the Go elements. The first three elements specify budgets of $8.0, $8.5 and $9.0 m during the next three years, respectively. The next six goals are related to unfulfilled undergraduate and graduate student demand. Each of the unfulfilled goal levels is constrained to be less than or equal to zero. The next three goals focus upon the freshman-sophomore core curriculum shared by AS and BA students. At the university modeled here, approximately 78 and 44°,o of the respective freshman and sophomore BA courses are provided by the AS faculty. Because AS decisions affect individual college and university programs, it seemed reasonable that the university should review these decisions to assure overall program quality. The final two goal sets involve proportional constraints related to faculty composition and accreditation. These goals are targeted at maintaining quality in teaching and research and impose the following conditions: (a) at least two-thirds of the university's full time faculty must have completed a PhD or other earned doctorate degree and (b) the percentage of full
284
~Vhitlbrd. Davis--Generalized Hierarchical Model o! Resource Allocatton
T~,BLE". COMPONENTSOF THE UNIVERSITYLEVELPROBLEM Goal
A. Elements o]" the G,3 cector
B. Unicersity decision cariables for period t
Planning ?ear 2
College AS(k =I) BAlk =2)
I 3 University budget 8.000.000 8.500.000 9.000,000 ceiling ($) Unfulfilled undergraduate 0 0 0 demand Unfulfilled graduate 0 0 0 demand Minimum uni~ersity-core 603 615 621 courses Terminal degree t:acultv 0 0 0 proportion Graduate student 0 0 0 instructor limits
time equivalent teaching assistants and instructors should not exceed 40% of the entire faculty. To understand how a zero goal can generate a proportional constraint, assume that M F is the number of faculty without a terminal (e.g. PhD) degree and D F represents the number of terminal degree faculty. Also assume that the university has a policy that requires at least two-thirds of its faculty to have a terminal degree. This implies
C. Unicersi O' letel constraint set
Gi.s.t -- G,..s.i < 8.000.000 (16.1) G~8.: + G,.s: < 8.500.000 (16.2)
Gt.s.,
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Gl.tt.,
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not applicable
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Gi.cc, + Gz.Gt., < 0 V,
•,
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V¢
(16.61 (16.7) (16.8) (16.9) ( 16. I O)
inclusion of the G H M ' s goal programming deviation vectors. The college p r o b l e m s
Within the G H M framework the colleges review the proposals generated by their subordinate departments and recommend revised performance goals based upon current external university and internal college policies. I n t e r n a l goals. Because the majority of a university's course offerings are part of internal college programs, it was not necessary to superDF > 2/3 [MF + DF]. (17) impose external university constraints upon all departmental course offerings. However, there Rearranging (17) and inserting positive and was a need to specify that internal college negative deviation variables, d-~o and diD, yields programs were adequately staffed and funded. the following goal programming constraint: To accomplish this, three internal goal sets were -2/3MF+ l/3DF-d-/o+dro=O. (18) imposed; they controlled the colleges' doctoral programs and tenure decisions during the three The role of the university is to generate the right year planning horizon. These internal goals and hand side of this constraint which appears in the constraints are given in equations (19.1) colleges' formulation. (See equations (19.7)and through (19.3) in Table 3. (19.8).) E x t e r n a l goals. Equations (19.4) through In solving its problem, the university par- (19.9) specify the remainder of the AS and BA titions its goal vector by solving for the vari- problems (equation (19.9) is applicable only to ables defined in panel B of Table 2. These goal AS). These equations link the external goals, assignments are subject to the constraints ap- generated by the university, to the departmental pearing in panel C of Table 2 (i.e. equations operating proposals. They are targeted at meet(16.1 ) through (16.10)). It should be emphasized ing the university stipulated goals in each planthat these 'hard constraints' will not necessarily ning period for budget, unfulfilled underprevent budget overruns or other undesirable graduate and graduate demand, faculty goal violations. This possibility results from the composition and minimum core courses.
Omega. Vol. 11..Vo. 3
The departmental problems Departmental decision t'ariables. The components of the departmental problems are the number, level, size and staffing of course offerings. Each department had seven levels of full time equivalent (FTE) staff and faculty positions as well as seven levels of course offerings. The staff levels were:
285
(2) junior-senior core. (3) junior-senior major-minor. (4) graduate (masters) core, (5) graduate (masters) major, (6) graduate (doctorate) core,
( 1) secretarial and research associate personnel, (7) graduate (doctorate) major courses. (2) graduate student research assistants, Associated with these course levels are seven unfulfilled course demand variables. Let 4%..... 7.k,j., and rk,.~., equal staff positions, course offerings and unfulfilled student demand associated with department j of college k at level i in planning period t. respectively. Finally, let/3kr ,, the budget of department j of college k during period t equal the sum of department's salaries, cry,.,, plus indirect costs, cJ~,.,. Departmental operating constraints and goals. Each department has five categories of operating constraints:
(3) graduate student teaching assistants, (4) instructors (nonterminal degree), (5) assistant professors, (6) associate (tenured) professors, (7) full (tenured) professors. The course levels were: (I) freshman-sophomore core,
TABLE 3.
(a) class size and enrollments,
SPECIFICATION OF THE
AS
AND
BA
INTERNAL AND EXTERNAL GOALS AND CONSTRAINTS
AS and BA college constraints and goals
Internal No. doctoral seminars offered in year t for deparment i No. internal or external promotions to associate professor in year t No. internal or external promotions to full professor in year t
- d i s . , + dos., = Target year t - d'de., + d5e., = Target year t - d ; e . , + d~.e., = Target year t
program level in t'or department i No. promotions in
',-',, (19.1) ~:
(19.2)
No. promotions in
z',
(19.3)
+ d i . , = External budget goal in year t - d £ : ~ . , + di:t. , = Unfulfilled undergraduate demand goal in year t - d ' ( c . , + d~c., = Unfulfilled graduate demand goal in year t
V.
(19.4)
,~:
(19.5)
V,
(19.6)
V,
(19.7)
v,
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v,
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-d~.,
Total unfulfilled undergraduate student demand in year t Total unfulfilled graduate student demand in year t
-0.667
Total non-terminal degree faculty in year t
+ 0.333
0.6
Total FTE graduate student/ instructors in year t
0.4
Total terminal degree faculty in year t
E T°tal] professorial faculty in year t
The total number of AS freshman/sophomore core courses offered in year t
O~II/3-E
-d-~o., + dro., = Terminal degree faculty goal in year t
- d ~ u + d~t., = Graduate student/instructor goal in ,,'ear t
-d;s~.s. , + d~ses., = M i n i m u m freshman/ sophomore core courses goal in year t
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lb) teaching loads and levels. Ic) secretarial and research support. (d) direct and indirect budgetary expenses, le) tenure obligations. These constraints are given in equations (20.1) through (20.16) in Table 4. Equations (20.1) through (20.5)incorporate university policy on average class size and staffing. For example, (20.1) states that the number of courses offered by department kj at level i in year t times the average class size for that level, plus any unfulfilled student demand in department k i for course level i in period t will equal the department's expected student demand for that course level in year t. In addition (20.2) through (20.5) specify that courses offered at various levels cannot exceed staff resources capable of teaching at that level. For example, only professorial faculty are allowed to teach doctoral courses: however, as the course offerings expand to include the freshman-sophomore level (20.5), all instruction staff resources are included. Equations (20.6) through (20.8) provide minimum and maximum levels of secretarial and departmental administrative support. Similarly (20.9) and (20.10) levy maximum and minimum levels for research assistant support. Equations (20. II) through (20.14) are related to budgetary items. They define departmental salaries, minimum and maximum indirect cost levels and a total departmental budget, respectively. Finally, (20.15) and (20.16) insure that tenure obligations are fulfilled. A final set of goal linkage equations is necessary to complete the departmental problems. Because these equations are departmental versions of each college's internal and external goal constraints (equations (19.1) through (19.9) in Table 3), they will not be detailed here. Instead of summing across a college's departments, these goal linkage constraints focus upon a single department. Not all constraints given in Table 3 are applicable to every department. As an example, a departmental version of (19.1) was applied only to those departments offering doctoral programs. Also, only AS departments incorporated a minimum freshman-sophomore core course constraint.
Goal deciation priority we(ghtings A procedure similar to the one used by Wacht & Whitford [19] was utilized to determine priorit',' weightings. The approach resembled the "Delphi technique" developed by the Rand Corporation mathematician, Olaf Helmer. After receiving a brief explanation of goal programming, individual administrators at various levels of the university were asked to provide anonymous rankings of undesirable deviations. The ranking scale ranged from 0 to 100: the higher the value, the less desirable the deviation. In the G H M , the colleges determine composite weighting vectors. Based upon rankings obtained from AS and BA administrators, these weights were computed by averaging within each college the responses assigned to a particular deviation. When a college's deviation was applicable to only one department (i.e. goals related to PhD seminars), no averaging of responses was necessary. The average deviation weights tbr AS and BA are given in Fig. 2. These rankings can be interpreted as the priorities expressed by a college's executive committee. These weighting vectors were held constant over the three year planning horizon. In a complex organization, it is likely that no weighting determination procedure can adequately capture true priorities. The possibility of priority measurement error points to the need for sensitivity analysis. Perhaps the greatest benefit of this type of study is the information obtained by observing the impacts that alternative goals and priorities can have upon the allocation of resources.
T H E MODEL'S SOLUTION The smallest of the university model's subproblems was at the university level: it had 69 variables and 36 constraints. The largest subproblems were in the BA departments which contained up to 375 variables and 141 constraints. The G H M ' s F O R T R A N code was written and tested on a Control Data Corporation CYBER-175 computer. The linear programming subroutines used by this code were developed by Marsten [12]. Although the current version of the model's code is solved on a single mainframe, the modular nature of the
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(20.11)
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o-~,
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258
Whtt]i~rd, Duv~s--Generahzed Hierarchical Model o/ Resource .4/Iota'ion
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GHM's structure and solution process makes implementation via a network of interactive minicomputers feasible. It is in this network setting that the GHM offers the distinct advantage of merging intbrmation and decision support systems into the resource allocation process.
2S9
these alternatives created trade-offs in maintaining faculty morale, educational quality and instructional effectiveness. Further, accreditation standards limited several courses of action. Although numerous policy alternatives ~vere tested with varying degrees of success, the compromise version of the model reported here focused upon five a r e a s .
Solution characteristics qf the #titial formulation The initial university formulation converged to a final solution in twelve iterations. Although the model generated no unfulfilled graduate or undergraduate student demand in any planning year, this was achieved at a considerable price. Most dramatic were budget deficits of approximately SI.0, $1.1 and $1.2m in years one through three, respectively. With few exceptions, the departments relied upon teaching assistants and temporary instructors rather than assistant professors. The model was not totally successful at meeting the colleges' internal goals. Although minimum support was generated for doctoral programs, the results of the promotion goals were inconsistent in that no promotions were incorporated in the BA solution. This result can be attributed to the algorithm's 'perception' that BA was generating the university's deficits. As seen in Fig. 2, the AS penalties associated with budget overruns were significantly higher than BA (100 vs 75). This higher level of priority weighting may be interpreted as 'political clout'. Analysis of the colleges' budgets revealed that all deficits occurred in BA. From a behavioral viewpoint, a scenario in which a bureaucratic organization with relatively more political power receives a disproportionately large share of available funding is plausible. But from the university administration's viewpoint, this type of situation may not be desirable. In the light of these and other characteristics it was necessary to revise the original formulation. Problem revision and sensitirit)' analysis To eliminate budget deficits several alternative strategies were available. Although ad hoc in nature, each strategy involved realistic, but politically unpopular compromises. These included increasing average class sizes and faculty teaching loads, decreasing faculty raises and indirect expenses and allowing 'less expensive' graduate student teaching assistants and instructors to teach upper level courses. Each of
The first policy change involved an increase in average class size. Even though these increases could potentially lower educational quality, the Carnegie Commission has endorsed this approach [1]. The second change attempted to counterbalance the potential diminution of educational quality precipitated by class size increases. In the original formulation, temporary instructors were allowed to teach at all levels except doctoral seminars (see equations (20.2) through (20.5)). In the revised version, temporary instructors were allowed to teach only freshman-sophomore and junior-senior core courses. It was anticipated that these changes would force the model to incorporate additional assistant professors into the final solution. Based upon the size of the university's deficit, it was unlikely that campus-wide increases in class size could totally offset funding shortfalls. As a result the third change in university policy focused upon limiting the size of departmental indirect costs. The fourth revision provided parity in the budget priorities of the AS and BA colleges. Both colleges' budget overrun penalty weights were increased and equated so that no college had an advantage in bargaining power. The final changes eliminated the internal promotion goals and lowered the minimum number of departmental doctoral seminars to the number of seminars observed in the oriNnal solutions. This level of doctoral program support was felt to be more than adequate. Because promotions can be incorporated into the formulation within the department's constraints (see equations (20.15) and (20.16)) omission of promotion constraints does not severely limit the formulation's applicability.
Results of the revised formulation The modified version of the model reached a final solution after four iterations and, unlike the previous versions, all university and internal college goals were achieved. One of the most surprising aspects of the revised formulation
290
[{71it/ord. Dat'is--Generalized Hierarchical .$[odel o~ Resource .4/location
was its rapid convergence. Not only did the changes in budget priorities improve computational time and expense, the,,' generated small budget surpluses for BA in years one and two and AS for year three. As one might expect, these restrictive budgets had a direct effect on departmental research support and indirect cost allocations, but none of these support levels was felt to be below minimum acceptable levels. One disappointing aspect of the revised formulation was the model's inability to incorporate significant levels of assistant professors in departments with declining student demand. Based upon the complexity and interactions of departmental, college and university policies, the timely analysis of the major issues facing all levels of a university's administration can be extremely difficult. However, the computational ease engendered by the use of the GHM prorides a convenient and straightforward solution procedure that allows testing and sensitivity analysis of an unlimited number of alternative strategies and policies. Although the GHM cannot guarantee an "optimal" allocation of an organization's resources, it can, if properly employed, significantly improve upon the rationality and reasonableness of resource allocation decisions in many decentralized, hierarchical organizations.
ACK NOW L EDG EM ENTS This study draws heavil? upon Professor Whitford's doctoral dissertation submitted in 1980 at Georgia State University, which was prepared under the super',ision of Prolessors Richard F Wacht, Chairman, Dileep R Mehta. David L Sjoquist and Roger O Miller. Vice-President for Financial Affairs at GSU. These individuals provided many helpful insights and comments. The authors v,ould like to acknowledge the computational support and assistance provided by Mr Stanley P Kerr of the Computing Services Office at the University of Illinois at Urbana-Champaign and b,v Professor Roy E Marsten of the Universit? of Arizona. The helpful comments of two anonymous referees are also acknowledged.
REFERENCES I. CARNEGIE COMMISSION ON" HIGHER EDUCATION (1972)
2. 3. 4.
5.
6. 7. 8.
SUMMARY AND CONCLUSIONS This study has presented a three-level, linear version of the GHM and has shown how this model could be used in allocating resources for a hypothetical university. The GHM uses a composition approach to organizational modeling similar to the composition models developed by Ruefli [16, 17], Freeland [6], Freeland & Baker [7] and Davis & Talavage [3]. Unlike these composition models the GHM uses a goal programming format at each level of the organizational hierarchy. By applying the GHM to a large university planning problem, it was demonstrated that the composition approach can serve as a viable planning tool. Sensitivity analysis allowed one to observe and respond to the interactions of organizational goals, priorities and policy guidelines. In summary it appears that the GHM may be useful in allocating resources within hierarchical organizations.
9.
lO.
11. 12.
13.
14. 15. 16. 17.
The More Eff~'ctice L"se of Resources: an hnperative /br Higher Education. McGraw-Hill, New York. DAVIS WJ (1978) A generalized decomposition procedure and its application to engineering design. Jl Mech. Des. 100(4), 739-746. DAVISWJ & TALAVAGEJ (1977) Three-level models for hierarchical coordination. Omega 5(6), 709-722. DAvis WJ & WmTFORD DT (1981) The generalized hierarchical model: a new approach to resource altocation within multilevel organizations. BEBR Working Paper 797, University of Illinois at Urbana-Champaign (August). DYER JF & MULVEY JM (1976) An integrated optimization/information system for academic departmental planning. Mgmt Sci. 22(12), 1332-134 I. FREELANDJR (1976) A note on goal decomposition in a decentralized organization.Mgmt Sci. 23(I), 100--102. FREELANDJR & BAKERJR (1975) Goal partitioning in a hierarchical organization. Omega 3(6), 673-688. GEOFFRION AM, DYER JF & FIENBERG A (1972) An interactive approach for multi-criterion optimization with an application to the operation of an academic department. Mgmt Sci. 19(4), 357-368. HARWOODGB & LAWLESSRW (1975) Optimizing organizational goals in assigning faculty teaching schedules. Dec. Sci. 6(3), 513-524. HOPKINS DSP, LARRECHE JC & MASSY WF (1977) Constrained optimization of a university administrator's preference function. Mgmt Sci, 24(4). 365-377. LEE SM & CLAYTON ER (1972) A goal programming model for academic resource allocation. Mgmt Sci. 18(8), B395-B408. MARSTEN RE (1978) XMP: a structured library of ~ubroutines for experimental mathematical programming. Technical Report 351, Department of MIS, University of Arizona. MASSY WF (1978) Reflections on the application of a decision science model to higher education. Dec'. Sci. 9(2), 362-369. McNAMARAJF (1973) Mathematical programming applications in educational planning. Socio-econ. Plann. Sci. 7(1). 19-35. PHYRR P (1970) Zero-based budgeting. Hart. Bus. Rer. 48(6), 111-121. RUEELI TW (1971) A generalized goal decomposition model. Mgmt Sci. 17(8), B505-518. RUEELI TW (1971) PPBS--an analytic approach. In BYRNE F R e t al. Studies in Budgeting, Vol. II of Budgeting of Interrelated ,4cticities. North-Holland, Amsterdam.
Omeea. V~)I. 1 I..Vo. 3
18. SWEEN~" DJ, WINKOFSKY EP, ROY P & BAKER NR 11978) Composition ~s decomposition: two approaches to modeling organizational decision processes..1,Igmt Sci. 24~ 14). 149[-[499. 19. W~,fnx RF & WHI~ORD DT 11976) A goal programming model for capita[ investment analysis in nonprofit hospitals. Fin. Mgmt 5(2). 37-47. 20. W~.LTERS A, MANGOLD J & HAR,-XN EGP (1976) A comprehensive planning model for long-range academic strategies. 3,Igmt Sci. 22(7), 727-738.
29[
21. WEHRUNG DA, HOPKINS DSP & MASS'~" WF (1978) Interactive preference optimization t'or university administrators..$fgrnt Sci. 2416). 599-611. ~DDRESS FOR CORRESPONDENCE: Pro/essor
David T Whit/ord. College ql' Commerce and Business Admini.~'trt~ti(m, (.'nt£'ersit.v of Illinois at Urbana-Champa~en. 3.10 Commerce Building ( West), 1206South Sixth Street. Champa~en. I L 6182O, USA.