10 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, VOL. SMC-3, NO. 1, JANUARY 1973

incorporating widespread public participation in the devel- REFERENCESopment of goals and objectives of the New City design via [1] G. Breckenfeld, Coiumbia and the New Cities. New York:graphical analysis has been advocated. Certain common Washburn, 1971, p. 187.

,2] A Maslow, Psychology of Science. Chicago, Ill.: Regnery, 1969.questions about the proposal have been addressed [3] E. Howard, Garden Cities for Tomorrow, F. J. Osburn, Ed.This is a broad brush treatment of a topic early in its Cambridge, Mass.: M.I.T. Press, 1966.

[4] A. Downs, "Alternate forms of future urban growth in theformative stage. Obviously, we are far from a complete United States," AIP J., pp. 3-11, Jan. 1970.working proposal but preliminary calculations are en- [5] W. Alonso, "What are new towns for?" Urban Stud., Feb. 1970.

[6] D. Canty, Ed., The New City (Report by the National Committeecouraging. Work continues in interdisciplinary groups at on Urban Growth Policy). New York: Praeger, 1969.Oakland University and at Battelle Columbus Laboratories. [7] M. Apgar, IV, "New business from new towns?" Harvard Bus.

Rev., pp. 90-109, Jan./Feb. 1971.[8] J. N. Warfield and J. D. Hill, A Unified Systems EngineeringACKNOWLEDGMENT Concept. Seattle, Wash.: Battelle Memorial Inst., 1972.[9] J. Jacobs, The Death and Life of Great American Cities. New

Support by Battelle Memorial Institute and opportunity York: Random House, 1963.for criticisms by Institute members and other colleagues is [10] E. C. Banfield, The Unheavenly City. Boston, Mass.: Little,

Brown, 1970.hereby gratefully acknowledged. In particular, W. Turski, [11] J. D. Hill and J. N. Warfield, "Unified program planning,"A. Bilinski, R. White, R. Dayton, J. Warfield, J. D. Hill, IEEE Trans. Syst., Man, Cybern., pp. 610-621, Nov. 1972.

[12] L. Rodwin, Planning Urban Growth and Regional Development.have been most helpful. Cambridge, Mass.: M.I.T. Press, 1969.

Regional Health Care PlanningCARL G. LOVE AND GIORGIO TREBBI

Abstract-An approach is presented for planning a health care system region; the distribution of patients among the facilities in aon a regional basis. The major components of the process are embodied region; and, most importantly, the cost and effectiveness ofin four computerized models: a demographic model; a facilities-location the health care for the people of the region.model; a regional and local model of the interactions of physicians,patients, and facilities; and a model which uses dynamic programming Like all urban systems, a health care system iS not timeto determine the best strategy to follow in sequentially altering the health invariant and is fraught with uncertainty of both thecare system. Each of these models is described and representative results present and future. There are no accurate data availablefrom each are presented. which define the health needs of the current population or

INTRODUCTION the future needs of a projected population. Even if the needsof the population were known, there is a wide gap between

A REGIONAL health care delivery system is composed need and demand since the latter is greatly affected by suchof many diverse but interacting components. The factors as an individual's perception of need and ability to

population from which the patients are derived is a het- pay, and the accessibility and acceptability of the service.erogeneous group with corresponding variations in their Furthermore, medical and technical advances and changeshealth care needs. The medical professionals form a com- in the basic mechanisms of health care delivery and financingplex hierarchical structure that interacts with the other can have a profound impact on the required resources.professionals and service personnel. There is a wide dis- Regional health care planning is, therefore, a dificultparity in the type, quality, and quantity of health care task in such a complex environment, where change andservices throughout a typical region. These factors affect uncertainty is the rule rather than the exception. This paperthe number, location, and association of physicians in a presents an approach for regional planning which revolves

around a set of computerized models. The set of models isManuscript received May 18, 1972; revised August 21, 1972. This not all inclusive in that there are elements of the planning

work was supported by the Westinghouse Health Systems Department. process that we have not attempted to model. We have foundThe authors are with the Westinghouse Research Laboratories, tathscoperzdm elaeueflorvlain

Pittsburgh, Pa. 15235. 'ta hs optrzdmdl r sflfreautn

LOVE AND TREBBI: REGIONAL HEALTH CARE PLANNING 11

DEMOGRAPHIC MODELDemographic Model

The first step in the planning process is to project thepopulation of the region, its characteristics, and its location.The demographic model produces yearly population pro-

Facilities Location Model jections by age-sex cohorts for various population groups.The natural processes of birth, death, and aging are modeledas shown in Fig. 2. For each simulated year, a percentage

Regional Simulation: of each cohort ages into the succeeding cohort; a specifiedDemand Models portion of each cohort (mortality rate) is assumed to die

each year; and new births are added to the first cohort bysumming the number of fertile females, usually those be-

Dynamic Planning tween 15 and 44, and multiplying that number by a fertilityrate. The basic mechanism is duplicated for each population

Fig. 1. Planning models. group for each geographic region and for each of the sexes.In addition, the migration into or out of a region is repre-sented by either specifying new housing starts or by using

the ramifications of change and uncertainty. We are not 'past migration levels. If new housing starts are used, theyusing the models to predict changes in medical practice, are classified according to type (single family, townhouse,for example, or to resolve future uncertainties, but rather apartments, low income, etc.) and the age distribution ofto determine the short-term and long-term effects of these the occupants specified for each type. The classification of aitems so as to be better able to plan given that we have an group of units is changed when, for example, single-familyuncertain future. These computerized models are tools units are converted to multiple-family units. The majorwhich complement the abilities of professional planners. advantage of disaggregating the population by age, sex,They provide a quick means of evaluating alternative plans, ethnic, and income categories is that it allows a morethereby allowing the user to direct his efforts toward defining precise definition of demand or need based on the popula-plans rather than spending his time doing tedious calcula- tion in each of these categories. A lumped population pro-tions. In fact, the models allow a much more thorough jection, a total number of people, does not contain sufficientevaluation because they provide a common framework and information to adequately plan a health care deliverya set of impartial criteria for evaluation. This latter charac- system.teristic will become increasingly important with the im- The model is designed such that a planning area can beposition of more federal, state, and local controls. divided into a number of regions. Such a subdivision allows

projections to be made for each region, thus permitting thePROBLEM DESCRIPTIONpopulation to be more precisely located than if the area

The planning problem that we wish to consider is the were treated as a whole. The model also contains an optionfollowing: determine for various points in time the location for computing the saturation population of each region.and type of health care resources required to best satisfy This feature is used in urban and suburban areas to deter-the health care needs of the population for a specified mine the target population if the area were to be fullygeographic region. A discussion of appropriate criteria for developed. Having obtained population projections, theevaluating solutions is included in a later section. next step is to use these projections and some coarse demandOur general approach to the regional planning problem is ratios to determine if new facilities need to be constructed

to do the following: use a demographic model to project to meet the health needs of the projected population.the population for defined regions by age, sex, and incomelevel; use a facilities-location model to aid in specifying the FACILITIESLOCATION MODELlocation of new facilities and the expansion of existing The problem of locating facilities or services throughoutfacilities; use a regional-demand model to determine the a geographic area is analyzed by dividing the area intodistribution of the demand among the physicians and regions, using population projections from the demographicfacilities that will result from the projected population; use model and morbidity rates to specify demand by region,a local-demand model to obtain a detailed specification of and then using linear programming to determine the bestthe required resources for each individual facility as a location and size of facilities as a function of time. Thefunction of time; and, finally, use dynamic programming criterion function for locating the facilities consists ofto determine the best strategies to follow in altering existing acquisition costs, operation costs, social costs for insuffi-facilities or constructing new facilities given the uncertainty cient capacity, travel costs, and preference costs. Thepresent in demand projections. Of course, the planning preference costs are an aggregation of factors which areprocess is never such a well-defined step-by-step procedure. used to account for a person's selection of a facility evenIn practice, the various models are utilized in an iterative though it results in higher direct cost to the individual formanner. They can also be used independently ofeach other. the same quality of care than another facility. ExistingA block diagram depicting the interrelations of the models facilities are treated as initial conditions; constraints canis shown in Fig. 1. be imposed on the location of new facilities, the amount of

12 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, JANUARY 1973

Fig.2. Fucto aldigae ffebckefcsffml pouato thog expce births.

e Cohort bCohort facilits | |Cost p paient Coho t dCohort r

I_0-5 6-11 _ 84~3-950-614_65 +|

l Years Years Yearer ars Years

stucio avriu pitsi Exetie;adteemnd ma x lt temnl vleo nrmna aaiyaddabe~~~~~~ ~ ~ ~subivde by tye inain antuptet n y tim tli eink

lvt l rs ofldependen

ThFpanigntrvlis n otonsubintrvals andb tef Theobjective function ithoent Cohs

demandsforeachregionisattrebec atedas a stantthrughuf ti NT1os perpn Year

eacsubtionateval.rBecausedentmeandcaothe predictd with objeatives Femm {alue (Finale()Rnalesca i)

certainty, sverval ivdemanddlevelsareprtojtedvfor each The (1v+iDst

certaint, severa demand etedsaepoetdfrec

decision time point. The excess capacity, the unfilled demand + rei2(tn)R2k(t)+ b21(ti)R211(ti)from each region, and the manner in which patientsdistribute themselves throughout the region are explicit + ti2(ti)R221(ti) +k31(t.)Z1 (ti) + 3(ti)Yr'(ti)parts of the problem formulation. + n2(ti)Z2n(t) + 2(tI)Y2e(tI)]

In order to illustrate the problem formulation and methodof solution, we will consider a problem with only two + p2(t )[41p(t1)R112(ti) + 012(ti)R121(ti)regions and two possible levels of demand. Let + 021(ti)R12'(ti) + 422(ti)R222(ti)

pf(th) probability of demand level I at time ti + fl2(ti)ZR2(ti) + 5 Z(t1)Y12(ti) + /32(ti)Z22(ti)Rkdi(ti) number of patients from region k who receive

service in region] when demand level I is present + fl2(ti)Z22(ti)] + (ti) -C1(t _)]at time t h

Zkr(ti) amount of unused capacity in region k with + 21(ti)[C2(ti)- C2(ti)]R- x2(ti)[C(t)demand level present at time t - Cl(ti1)] - x2(ti)[C2(ti)- C2(tiZ2)]} (1)

Yk'(ti) number of patients in region k who do notreceive service when demand level I is present where NT is the number of intervals and DR is the rateat time ti used to convert all costs to present values. In addition, two

ZkI(ti) sum of travel cost, operation cost, and preference types of constraints must be imposed. If the unfilledcost per patient for patients originating in demand is explicitly considered, then for each time pointregion k and obtaining service in region j at the sum of the unfilled demand from a region and thetime tr number of patients receiving service in the facilities in all

Ck(tI) capacity in region k at time ti the regions must be equal to the total demand from thatAk(tI) cost per unit additional capacity in region k at region. Also, the number of patients receiving care at one

time ti facility plus the unused capacity must equal the capacity offlk(ti) cost per unit of unused capacity in region k at the facility. The above constitutes a standard linear pro-

time ti gramming problem where the Rkj1(tL), Zk'(ti), Yk'(ti), and

LOVE AND TREBBI: REGIONAL HEALTH CARE PLANNING 13

TABLE I TABLE IVDEMAND PROJECTIONS: NOMINAL DEMAND (1000 VISITS PER YEAR) DISTRIBUTION OF DEMAND (1000 VISITS PER YEAR) FROM REGION 3,

DEMAND LEVEL 2Region

(+1) RegionTime 1 2 3 4 Variations -

Time 1 2 3 40 800 80 50 20 --1 816 96 75 40 1 1 75 0 0 02 816 125 113 80 2 2 113 0 ,0 0

3 808 162 157 140 3 3 158 0 0 0

4 800 195 205 210 4 4 136 0 191 0

5 784 234 246 294 5 5 138 0 107 0

6 745 269 283 368 6 6 0 0 283 0

7 707 309 325 459 7 7 0 0 325 0

8 672 355 374 574 8 8 129 0 245 0

9 638 409 430 632 9 9 0 0 430 0

10 606 470 494 663 10 10 80 0 415 0

TABLE II TABLE VPERMITTED CONSTRUCTION TIMES MISMATCH OF DEMAND AND CAPACITY (1000 VISITS PER YEAR)

FOR DEMAND LEVEL 3Region

RegionTime 1 2 3 4

Time 1 2 3 4 Total

2 X 1 0 0 0 -37 -373 X 2 0 0 0 0 I4 X X X 3 0 0 -5 -144 -1495 X 4 92 0 0 0 926 X X X 5 0 0 0 -78 -727 X X 6 0 0 0 -100 -100

8 x X X 7 46 0 0 0 469 X 8 0 0 0 -158 -15810 x x x 9 0 0 0 -64 -64

10 0 -28 -36 -159 -223

A positive number indicates excess capac-TABLE III ity; a negative number indicates insufficient

MINIMUM COST CONFIGURATION (1000 VISITS PER YEAR) capacity.(p = 0.1, 0.8, 0.1)

Region permitted construction times are as shown in Table II, then

10 o3 the minimum cost capacity for each region is given in Table

2 1156 0 0 0 III. The distribution of the demand from each region for3 1156 0 0 04 1156 0 401 0 each time point is determined as part ofthe minimum cost5 1156 0 401 tie i s fcs6 1156 0 507 A) configuration as a function of time. The distribution of7 1156 0 507 3128 1156 0 507 312 demand for region 3 demand level 2 is given in Table IV.9 1156 0 507 571 frdmns i

10 1156 0 507 571 The mismatch of demand and capacity for demand level 3is shown in Table V. These excesses and deficiencies ofcapacity are those resulting after the demand has optimally

CQ(t1) are the unknown variables. The reader is referred to distributed itself among the facilities in the four regions.[1] for a more complete discussion. This approach for locating new facilities can be easily

extended to the problem with multiple demand types [1].Representative Example

Consider the problem of planning outpatient clinics. REGIONAL-DEMAND MODELSuppose the geographic region is divided into four regions, This section describes a regional simulation of the inter-where region 1 represents an urban core and the other actions among the population (patients), physicians, andthree regions represent growing subregions. By using a health care facilities. The geographic area is divided intodemographic model and specifying an average number of regions and the existing facilities and practicing physiciansvisits per thousand of population, one can obtain prob- are located in the appropriate region. Population projectionsabilistic demand projections as shown in Table I. Note from the demographic model are specified for each regionthat a nominal projection is made and then the percentage and for each future year. The regional-demand modelvariation, plus and minus, is specified probabilistically. projects the health care demand using the population pro-Thus three demand levels are specified for each time point. jections, distributes the patients among the facilities, andWe assume our short-term projections have less spread models the location process of physicians. The user mustthan the long-term projections. This is just one of many specify the capacity, by type, of existing facilities and anypossible demand models that could be used. additions to existing facilities or the opening of newSuppose that initially there is an outpatient facility in facilities. The facilities-location model described in the

region I with a capacity of 1 000 000 outpatient visits per previous section generates the best location and size ofyear. For the demand projections shown in Table I, we additional capacity for future time points. In succeedingwish to determine where and how much capacity should be paragraphs, a description of the health care demand processadded in the region for the next ten years. If the probabilities is presented and then followed by an actual planningassigned to the demand levels are (0.1, 0.8, 0.1), and the application.

14 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, JANUARY 1973

Health Care Demand Process where di,k is the distance between regions i and k; De is aThe health care demand process evolves through two parameter that controls where accessibility starts to de-

important stages. The potential patient, his perception of crease sharply; and ac is a parameter related to the rate ofhealth care need, and his decision of contacting a physician degradation of accessibility with increasing distance. Dedominate the first stage. The second stage deals with the assumes the meaning of "convenient distance," i.e., aphysician-patient interaction. The choice by a patient of a distance a patient considers convenient to travel. As such,particular physician depends on the physician's availability it depends on patient class. This function has been derivedand accessibility to the patient. The availability of a from an analysis of the origin of patients admitted tophysician to a patient is affected by the requests for the hospitals [3], [4]. Similar results have been obtained inphysician's services by other patients in the region. Acces- marketing and transportation studies [5], [6].sibility is related to geographical distances and to the The degree of availability of physicians in a given regionexisting transportation system. is directly related to the demand for services placed onThe second stage of the health care process is dominated them. When the demand is low, physician availability is

by the physician's behavior. Once a patient is in the high. But, as the demand approaches the capacity of thephysician's office, he transfers to the physician a consider- physicians, long waiting lists make it difficult to obtainable amount of his decision independence. It is mainly the prompt appointments, service tends to degrade, andphysician that decides the treatment, eventual successive physicians are actually not available to all the patientsvisits, specialist consultations, and the need of a hospital requesting their services. A function that describes thisadmission. In this last eventuality, the physician's admitting relationship between availability and demand may be ex-privileges and the availability to the patient of the required pressed analytically byhospital services play a dominant role in the ultimate choice Fof the hospital. This process is represented in the form of AVk,e = e,k5)an interactance model similar to those that have been 1 + (WklPk)(successfully applied to marketing analyses and other health where Fe,k is a factor introduced to account for the fact thatcare planning. The interactance model used here places physicians may be unequally available to all populationemphasis on the phenomenon of saturation of the available classes; fl is a parameter which controls the reduction rateresources and introduces the capability of handling socio- of availability with an increasing demand; Pk representseconomic variables [2]. the capability of physicians in region k, expressed by the

In the demand model, the entire area under study is number of conditions the physicians can reasonably attenddivided into N regions. In each region the distribution of in one year; and Wk represents the demand, in number ofthe population by age and sex is assumed to be given, conditions per year, placed on the physicians in region k.Generally, the population will be further classified according The variables x k and Wk are related by the equationto any socioeconomic, ethnic, or religious attribute (hence- I,k,eforth, simply called "population class") that is relevant to N NCthe health care demand process. Wk = Y Si,eXi,k,e5 k = 1,2,- *,N (6)

Let Si,e be the number of conditions manifested in a yearby the population in region i, population class e. Let xi,k,e where NC is the number of population classes. In conditionsbe the probability that such conditions will be attended by of equilibrium, (2)-(6) form a system of simultaneousa physician in region k. The model assumes that equations that can be solved with respect to the variables

Xi,k,e and Wk, i = I N, k = I N, e = I NC,Xi,k,e = Ri * ACCk,i,e - AVk,e (2) which represent, respectively, the distribution of attended

conditions among regions and the total number of attendedwhere Ri is a constant of proportionality that ensures that conditions by region of service. The distribution of admis-

N sions among the various hospitals in the area lends itselfE Xi,k,e = 1 (3) to a similar mathematical treatment.k=1 Functions similar in structure to those discussed for

ACCk,i,e represents the accessibility of physicians in region assigning patients to physicians [see (4), (5)] are used tok as perceived by a patient in region i, class e; and AVk,e express the hospital accessibility to patients and physiciansrepresents the availability of the resources in region k to and specialty availability. In the same way, a set of equationspatients of class e. can be derived whose solution, in conditions of equilibrium

Accessibility is a function of distance-usually travel and for each specialty considered, provides the distributiontime distance. As the distance between the patient and the of patients among the various facilities in the area underphysician increases, accessibility decreases, but not propor- study.tionally. An analytical representation of such a function Apiaino h einlDmn oeiS givenl by the expression

This section discusses an application of the regional-ACCk,i,e 1 (4) demand model. The area studied is portrayed in Fig. 3,

'' I + (di,k/DeY2 which also indicates the various regions and the average

LOVE AND TREBBI: REGIONAL HEALTH CARE PLANNING 15

LEGEND

\ \)!t .,,^f,, Lc.. b.fe' . |- - |0-15 Mi n>, 5 ) *- .' ~~~~~~~~~~~15-20 Min|t>>ef bu ell 00 0 \0 j ~~~20-25 Min

30-35 Min.t~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~.Fig. 3. Regional map showing average admission travel time at year 0.

admission travel time by region of patient origin. In theexample chosen, the model was used to establish the 120 Utilization _ YearOm__ Year 6feasibility of replacing in six years the facility in region 3 ICOwith a new inpatient facility in region 7. The analysis was eofurther complicated by already proposed expansions of the 60hospitals in regions 4, 5, and 11, which were expected to beoperational within the planning period.The model produces data describing the evolution of the 20

regional health care system. Such data are provided for ° 1 3 4 5 7 11every two years in the planning period. Included is informa- Regiontion about the distribution of outpatient visits, the expected Fig. 4. Hospital utilization by region for year 0 and year 6.number of physicians, the distribution of hospital admis-sions, and patient visit and admission travel time. 91-percent occupancy rate and is expected to serve mainlyOf particular relevance to this example is the description the northeastern population.

of the situation after six years of simulation, when the new The availability of a new facility in region 7 also has anfacility becomes available in region 7. At that time, all the impact upon the patient admission travel time. At time 0planned expansions have already taken place and their there is a remarkable variation between the average travelimpact may be evaluated. The comparison of the hospitals time of patients living in the western and eastern regions.in regions 1, 4, 5, and II at the beginning of the simulation, In particular, admission travel time is very high for patientstime 0, and after six years, time 6, leads to several observa- coming from regions 6 and 9 (Fig. 3). The construction oftions. Fig. 4 shows that, assuming constant lengths of stay the new facility improves the situation by achieving a(LOS), the occupancy rate in all the existing hospitals general reduction of the average travel time for the entiredecreases. This has the effect of alleviating the pressure on population and, more importantly, smoothes the differencesthe hospitals in regions 4 and 5, perhaps ensuring a less existing among regions (Fig. 5).congested operation. However, the other hospitals will be The simulation covers a ten-year interval, so that it isunderoccupied. The new facility is expected to operate at a possible to project the demand on the new facility at the

16 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, JANUARY 1973

, : . LEGEND

m ,,,B,,0-5MinZJ15-20 Min

20-25 Min

25-30 Min

Fig. 5. Average admission travel time by region of patient origin, year 6.

time of its opening and for the next four years. The model of facilities that should be built and the manner in whichpredicts that the occupancy rate will grow from 91 to 98 they should be altered with time to best meet a changingpercent at the end of the ten years. This suggests that the demand. Furthermore, we wish to consider the tradeoffsfacility will eventually need to be expanded. between acquisition and operation costs over the entire lifeThe actual planning of the new facility requires more cycle of the facility. We also wish to consider the projec-

detailed information about the health care demand the tions from the demand model as only nominal projectionsfacility will serve. The resources required depend not only and add some uncertainty to them. Thus we will consideron the number of inpatient admissions and outpatient visits, the planning of a facility or a group of facilities as a dynamicbut also on the mix of diseases and on the characteristics programming problem and explicitly include the uncer-of the patients admitted. Another model, the local-demand tainty associated with demand projections. The reader ismodel, is used to produce this type of information. In each referred to [7] for a more complete discussion.region and for each point in time, the demand is projected Consider the planning of a health care system which hasin terms of medical and surgical admissions by diagnostic light care, moderate care, intensive care, and outpatientcategory, the number of attended conditions and ambu- capabilities. This system is modeled by two state variableslatory visits by age, sex, and diagnostic category, and in- which specify the system configuration and demand. Bothpatient and outpatient X-ray services and laboratory state variables are decomposed into the aforementionedprocedures. types of care. The demand is specified probabilistically as

shown in Table I.DYNAMIC PLANNING The objective function is the minimization of the ex-

Up to this point we have discussed a demographic model pected present value of the sum of acquisition, operating,which can be used to obtain population projections, a model and penalty costs. The acquisition costs arc a function offor locating facilities in a geographic region, and regional- time, the existing configuration, and the configuration toand local-demand models which yield projections of the which the system is to change. (This transition is determineddemand by type and the corresponding resources for each by the decision variable.) The operating costs are a functionfacility located in the region. In this section, we consider of time, the system configuration, and the patient demand.the problem of deciding upon the physical configuration The penalty costs are a function of time, the configuration,

LOVE AND TREBBI: REGIONAL HEALTH CARE PLANNING 17

and the patient demand. Penalty costs are incurred when a TABLE VI. . , , .~~~~~~~~~~~~~~CONFIGURATION CAPACITIESconfiguration does not possess enough total capacity or CONFIGURATION_CAPACITIES

the correct type of capacity (the mix of intensive care, Configuration Intensive Moderate Light Total Beds OptimumCare Care Care Visits/Year

moderate care, light care, and outpatient capacities) for the 01D1211 15 234 204 453 250000existing demand. 02E12M 15 234 204 453 320000

In order to reduce the quantity of input data, all costs 03F12M 15 234 204 453 37000004E15M 30 268 272 570 320000

are defined at time 0 and then escalated at specified annual 05F15M 30 268 272 570 370000rates. If E denotes the expected value to be taken over all 06E150 30 268 272 570 320000possible demand levels, then the objective function is given 07F150 30 268 272 570 370000

08F180 30 340 306 676 370000by 09G180 30 340 306 676 440000

1OH180 30 340 306 676 500000objective (t) llFlS0 30 350 306 686 370000

NN-1 12G150 30 350 306 686 440000

min E E [acquisition cost (k,c(k - 1),c(k))ea(k) 13H150 30 350 306 686 500000Q(t) k =t 14F180 30 384 374 788 370000

15GI80 30 384 374 788 440000

+ operating cost (k,c(k),p(k))e0(k) 16H180 30 384 374 788 500000

1 7II80 30 384 374 788 600000

+ penalty cost (k,c(k),p(k))ep(k)]g(t,k) 18F210 30 384 374 788 37000019G210 30 384 374 788 44000029H210 30 384 374 788 4400000

- S(N,c(N),p(N))g(t,N) , t = 0,l, ,N - 1 20I210 30 4 4 788 50000021G240 45 452 408 905 440000

22H240 45 452 408 905 500000

where c(t) and p(t) are the state variables which define the 23I240 45 452 408 905 600000system configuration and demand at time t; g(t,k) discounts 24G270 45 486 476 1007 440000k-money back to time t; and e0(k), eo(k), and ep(k) escalate 251270 45 486 476 1007 600000the acquisition, operating, and penalty costs from time zeroto time t. S(N,c,p) specifies the terminal value of the con-figurations (in t = N dollars) and is included to accountfor the relative merit of ending the planning period in one 26

configuration as opposed to others. Q(t) defines the set of 4allowable configuration transitions at planning time t to the 25 < 60end of the planning period. Acquisition budget constraintsare considered when defining Q(t). This optimizationproblem is solved using dynamic programming. 0 32 597

>~~ \ t4 DV 1~039Dynamic Planning Example

Total Expected Cost = 215.31 M $A planning period of 25 years is considered. Every five

years a decision is made with the purpose of minimizing (a)the total expected cost over the remainder of the planning 3 4

interval. The total cost is composed of acquisition, operating,and penalty costs as previously discussed. A four-year delay -7

is assumed between the making of a decision and the bene- _Y 30< 597ficial occupancy of the alternative specified by that decision.This delay accounts for the time required for budget ap-proval and construction. Table VI gives the capacity of the Total Expected Cost = 303.64 M$different configurations considered in this example. (b)An optimum strategy is represented by an oriented graph Fig. 6. (a) Optimum expansion strategy without budget constraints.

with essentially a tree structure. An example may help its (b) Optimum expansion strategy with budget constraints.interpretation. Consider the branch between configurations8 and 22 in Fig. 6(a). The label 2D < 560 indicates that if straints on the total cost. A cut in the budget for construc-the facility in the second interval has configuration 8 and tion necessarily results in an increase of operating andthe demand for beds is less than 560, then the next optimum penalty costs and may have disastrous effects over a longdecision is to expand the facility from 8 to 22. period of time. This point is illustrated by the following

In a real situation the planner operates and makes his example.decisions and choices within certain given budget con- The unconstrained optimum strategy, in this case, isstraints. It is interesting to investigate what are actually the shown in Fig. 6(a). The total expected cost is 285.31 millionconsequences of this limitation on the choice of the op- dollars. The initial alternative selected is configuration 8, totimum expansion strategy and the life-cycle cost. Many which a construction cost of 18.254 million dollars is asso-times the way in which budget restrictions are imposed does ciated. It is next assumed that a budget cut of 18 percent onnot recognize the long-range implications of such con- the initial construction is imposed. The allocation of funds

18 IEEE TRANSACTIONS ON SYSTEMS, MAN, AND CYBERNETICS, JANUARY 1973

per interval is expected to be 14.980, 0, 3.750, unlimited, used to validate this aspect of the model are the ease ofunlimited, 0. The corresponding optimum expansion strategy reproducing the initial status of the system modeled and theappears in Fig. 6(b). The final configurations are not too reasonableness of the results obtained. At the beginning ofdifferent from those found in the unconstrained case. the simulation the parameters of the model are chosen soThis means that the total construction costs in the two cases that the model reproduces two successive time points ofare essentially the same. There are only minor differences the system. The parameters are then kept constant for alldue to the escalation of construction costs and the discount the future time intervals. This procedure has given satis-rate, whose effects vary with the time sequence of the facility factory results on applications to date and is thereforeexpansion. The total expected cost is 303.64 million dollars, gaining increasing acceptance by health care planners.18.33 million dollars higher than the expected cost for thestrategy computed with no budget constraints. This relevant SUMMARYincrease in total cost, about 6 percent, is due to the penalty The planning of a regional health care system is a com-cost that must be paid for not having the necessary funds

pplex process which requires the projection of future popula-at the time when the level of the demand requires an tions and their health care needs, and a determination ofexpansion of the health facility. It is more a consequence of the most appropriate means of meeting these needs. Alack of timing than a lack of adequate total funds since, as large amount of data is required in this process because itpreviously discussed, the total life-cycle cost for construc- is necessary to consider a planning horizon of at least tention is essentially the same. years in order to include the future ramifications of present

MODEL VALIDATION actions.Four computerized models have been presented which

Validation will only be discussed for the demographic allow the user to systematically develop a plan for anand regional-demand models since the other two models efficient regional health care system. These models are ofare primarily optimization algorithms. The internal logic greatest use in planning a regional health care system inof the demographic model is basically the well-known aging which several health care facilities exist, interact, and com-and birth-death processes of a population. The accuracy of pete. They display, in a very comprehensive fashion, thethe proJections of the demographic modellsdirectly related impact of different courses of action as the system evolvesto the accuracy of the estimates for fertility rates, morbidity in time. Such quantitative tools are helpful for evaluatingrates, and changes in the housing stock or migration trends. area-wide plans and constitute a means of generating andThis model has been used extensively for both health care srting on ar onsithe decns of gen cies

and educationplanning. ~~~~supporting, on a regional basis, the decisions of the agenciesandeucatin planing responsible for health care delivery.The construction of the regional-demand model was

preceded by an extensive study of the factors influencing REFERENCESthe health care demand process. Information was drawn

[1] C. G. Love, "Specification of regional health care system con-from the existing literature and, where necessary, from the figurations," in Proc. 3rd Annu. Pittsburgh Conf: Modeling andexpertise of consultants. Furthermore, the use of a gravita- Simulation, Apr. 1972.

[2] G. Trebbi, "Planning of regional inpatient facilities," in Proc. 3rdtional model, incorporated in the regional-demand model, Annu. Pittsburgh Conf. Modeling and Simulation, Apr. 1972.is a widely accepted and tested technique. The logical and [3] Medical Facilities Planning Group, "Clinical teaching and health

care systems: models and evaluation," Stanford Univ. School ofinternal correctness of the model has been verified by several Medicine, Stanford, Calif., 1969.simple runs, in which the results could be manually checked, [4] D. L. Drosness and J. W. Lubin, "Planning can be based on patient

travel," Modern Hospital, no. 106, 1966.and by testing separately the various modules that con- [5] T. R. Lakshmanan and W. G. Hansen, "A retail market potentialstitute the model. model," AIP J., May 1965.

The validation oftepei[6] Dep. Highways, Commonwealth of Pennsylvania, Pittsburgh AreaThe validation of the predictive capablity of the model Transportation Study, vol. 1, 1961.is a much more difficult step. A complete set of historic data [7] C. G. Love, R. A. Mathias, and G. Trebbi, "Minimization of life

cycle costs for a health care system," in Proc. 1970 IEEE Syst. Sci.is very expensive and difficult to obtain. Hence, the criteria Cybern. Conf.