Siting efficiency of long-term health care facilities

Socio-Econ. Plann. Sci. Vol. 32, No. 1, pp. 25-43, 1998 ~C 1998 Published by Elsevier Science Ltd. All rights reserved Printed in Great Britain PII: S0038-0121(97)00016-5 0038-0121/98 $19.00 + 0.00

Pergamon

Siting Efficiency of Long-term Health Care Facilities HOMEI~ F. E. S H R O F F Mountain States R&D International, Inc., Tucson, Arizona, U.S.A.

TH. R. G U L L E D G E t and K I N G S L E Y E. H A Y N E S The Institute of Public Policy, George Mason University, Fairfax, Virginia 22030-4444, U.S.A.

M O L L Y K. O ' N E I L L INOVA Health Systems, Springfield, Virginia, U.S.A. Abstract--This paper describes a multiple criteria location assessment model while providing an application using fractional programming. The model was developed for a major health care provider, and was subsequently used to support siting a decision for a long-term health care facility in Northern Virginia. The model incorporates efficiency measurement methodologies utilizing Data Envelopment Analysis (DEA) to estimate the relative siting efficiency of 26 potential sites. The study identified several Pareto-optimal sites as potential locations for the proposed long-term care facility. Management selected one of these sites for construction of the facility, and full implementation is now underway. © 1998 Published by Elsevier Science Ltd

INTRODUCTION Long-term care is an issue of increasing importance in the United States. As noted at the recent U.S. General Accounting Office/Kaiser Family Foundation long-term care forum, Long-term care is becoming increasingly important as the number of persons who need services grows and expenditures for services to assist them increase. Approximately 10 million Americans of all ages are chronically disabled and dependent on others for assistance in the basic tasks of daily living such as eating, toileting, moving around in the house, shopping, money management, and other activities that most Americans take for granted. The number of persons needing help with these things will increase substantially in the future [23]. Given this projected increase in demand for long-term services, there is an interest in constructing new long-term care facilities. This paper describes the long-term care facility siting decision of a major Northern Virginia health care provider. Data Envelopment Analysis (DEA) was used to analyze the multiple criteria siting problem. Results of the analysis were used by the health care provider to support the decision to build a new facility in the Northern Virginia region. The paper is organized along the same lines as our research team's presentation to upper management. First, the siting problem is described, and a brief introduction to the DEA methodology is provided. Next, the DEA formulation is discussed, including management's definition of relevant inputs and outputs. Finally, the solution and sensitivity analyses are presented, and the decision support implications discussed. t A u t h o r for correspondence.

25

26

Homed F. E. Shroff et al. THE LONG-TERM CARE FACILITY SITING PROBLEM

The current situation

For many years, I N O V A t has been, and remains, Northern Virginia's premiere health care organization. More specifically, I N O V A is a not-for-profit provider of health care at all levels of acuity within the Northern Virginia region. It presently operates three acute care hospitals, two 24-hour emergency care centers, six urgent care centers, and two long-term care facilities. The function of an I N O V A long-term care facility is to offer a continuum of care (whether for short or long-term accommodation) from Assisted Living to Intermediate, Skilled, and Sub-acute nursing care levels. Patients may also be admitted into a long-term care facility for rehabilitative therapy. In this regard, relocation to a long-term care facility from home or hospital enables on-going treatments to be continually monitored on a more personal basis and in a home-like atmosphere. In addition, I N O V A ' s long-term care facilities include an on-site or close-by 24-hour emergency room that provides immediate medical support and emergency treatment. Another function of I N O V A ' s long-term care facility is to provide full service living accommodations with complementary health care interventions for the elderly population of Northern Virginia. These 'home-for-adults' patients require minimal assistance or nursing care. Such assisted-living facilities are designed for residents with restricted mobility whose daily activities, such as cooking, cleaning, and necessity shopping are cumbersome. Given the changing health care regulatory environment and increasing long-term care demand, I N O V A forecasts the need for building a new long-term care facility. The two existing I N O V A long-term care facilities played a major role in providing management and the research team with an understanding of the siting requirements for the proposed facility. The siting decision thus required continuous involvement between the research team and the managerial and administrative staffs at both facilities. INOVA's

requirements ./'or a new f a c i l i t y

I N O V A ' s proposal for an additional long-term care facility in the Northern Virginia region is based on several considerations. The primary issues involve (1) an increased demand for long-term care facilities, and (2) the rising costs of daily hospital bed expenses for prolonged post-operative or trauma-induced patient recovery periods, i.e. those cases where hospital-level care is no longer requried. In short, if a patient's condition does not require hospital-level care, then it is less costly to house him/her in a long-term care facility. Further to this, I N O V A ' s management anticipates the lifting of those state regulatory policies that currently preclude the operation of additional long-term care beds. These economic and regulatory changes directly affect both the functional operation and financial management of all Virginia's health care organizations, including INOVA. The factors affecting I N O V A ' s decision to build a new facility are complex. They include health regulatory and zoning requirements as well as economic and political factors. Management knew that the decision must consider factors such as I N O V A ' s not-for-profit status, regulations regarding bed-type as well as acuity levels of care, and regulations with respect to pay-type +. Other factors that were considered include administrative staff philosophies regarding type and number of staff employed. Based on these observations, I N O V A ' s Planning and Marketing Department proposed a 120 bed facility§, but with several constraints imposed by management [20]. One such constraint was that the bed occupancy in the proposed facility be evenly distributed between private-pay and government-pay patients. Of course, if the only criterion was profit maximization, a preferred choice would fill all beds with private-pay patients since I N O V A incurs a monetary loss on each government-pay bed. But, given I N O V A ' s not-for-profit status and sense of social responsibility, tlNOVA is the organization's signature, representing its tradition in providing innovative health care. {'Pay-type," as used here, defines two categories of patients: "private-pay' and 'government-pay."The 'government pay' patients are those who have a portion of their costs off-set by government Medicare/Medicaid payments. {}The requirement that the facility contain 120 beds was based on operating and other regulatory requirements. It was independent of this study.

Siting efficiency of long-term health care facilities

27

as expressed in its mission, such an allocation is not acceptable. Hence, a 50/50 payment mix was determined by I N O V A ' s management to be a reasonable expectation. It was thus imposed as a constraint. 1NOVA's management also imposed restrictions on the beds allocated to different acuity care levels. The restrictions include the following: • Category One: Sub-acute Care patients. The patients in this category include symptomatic HIVs, head and spinal chord injuries, and ventilator-dependencies. The total number of beds allocated by I N O V A ' s management to this category was 12. • Category Two: Skilled Care patients. The total number of beds allocated by I N O V A ' s management to this category was 36. • Category Three: Intermediate Care patients. These are elderly patients needing assistance, or patients who have recently suffered a stroke or post cardiovascular arrest. The total number of beds allocated by I N O V A ' s management to this category was 72. In summary, the total number of bed-types allocated to each of the categories, as well as the expected payment distribution, were defined by 1NOVA's planning and long-term care staffs. Both were subsequently set as constraints on the siting analysis. These self-imposed constraints meet I N O V A ' s present social responsibility criteria for servicing patients and fulfilling current demand levels within the Northern Virginia region. PROBLEM SPECIFICS The analysis team was asked by I N O V A ' s management to address the following question: "Given all relevant impacting variables, tell us the preferred area within Northern Virginia for building a new long-term care facility?" To handle the multiple criteria nature of the problem, we selected D a t a Envelopment Analysis (DEA) as the decision support methodologyt. This choice was made for two reasons. There is some literature to support the use of D E A for this type of analysis++, and the concept has intuitive appeal. While the latter point may seem trivial, it weighed heavily in our decision since we had to ultimately present our business case to top management.

Data envelopment analysis The D a t a Envelopment Analysis (DEA) methodology provides a non-parametric approach for measuring relative efficiencies of decision making 'entities' [8J. Such entities, generally referred to as Decision Making Units (DMUs), encompass both private and public sector organizations, agencies, etc. This methodology has been applied to measuring efficiency in general§ and to siting efficiency in particular [11]. The application of D E A here has some characteristics of the Desai and Storbeck approach, but the critical spatial characteristics are not present. The problem of this study is more appropriately described as 'locational benchmarking.' (See K a o and Yang [14] for an example of this concept). That is, spatial efficiency analysis explicitly considers some measure of distance in the analysis, while locational benchmarking examines the relative efficiency of geographical regions. For the hospital studies addressed by Desai and Storbeck, spatial variables were extremely important, especially as they relate to access time for emergency services. They are of minimal importance for this study, however. While there are cases where long-term care patients are transferred to hospitals, our surveys and interviews indicate that travel time and other speed of access spatial variables are not as important for siting long-term care facilities.

Production, relative efficiency, and data envelopment analysis A production function is a technical relationship that describes the maximal outputs attainable from a given set of inputs. The degree to which the actual output of a production unit approaches tWhile the relationship between DEA and multiple criteria decision analysis is well known by many researchers, there has been little literature written on the subject. A good explanation of the relationship is provided by Belton [5]. ~See, for example, Desai and Storbeck [1 I]. §We do not provide a complete review of the DEA literature here. The interested reader is directed to the bibliography constructed by Seiford [19], which contains many applications of DEA in diverse problem solving settings.

28

Homed F. E. Shroff et al.

a m a x i m u m level depends upon its efficiency. Since it is unlikely that all units achieve maximal output, efficiency becomes an important evaluation criteria. Efficiency may be increased by minimizing inputs while holding output rates constant, or, by holding input rates constant while increasing outputs, or, by a combination of both. For example, any machine, process, business, public sector program, or, in this case, a potential 'site,' utilizes a number of inputs to produce 'desirable' outputs. Examples of inputs are raw materials, capital equipment, personnel, information, real estate, distance, and so forth. Examples of outputs are production goods, energy distribution, client satisfaction, public access, and market penetration. Efficiency strives to achieve maximal outputs with minimal inputs. F r o m such an input-output productivity analysis, a site [i.e. a Decision Making Unit (DMU)] can be designated as efficient, if it is 'relatively' more efficient than the other D M U s (sites) with which it is being compared; that is, efficiency can me measured relative to all other D M U s (sites) in a realized input-output data set. The Pareto-optimal D M U s (sites) are located on the efficient frontier'L The name D E A defines a set of mathematical programming procedures applied to observational data that are used to establish efficiency frontiers via an envelope function encompassing all observed units. This is contrary to parametric frontier estimation techniques that use statistical methods to estimate the efficiency frontier. In a traditional application of DEA, the inputs and outputs are discretionary variables to be manipulated by a D M U in an attempt to create a certain level of efficient production. This application treats the choice of sites (DMUs) and the related input and output characteristics as discretionary only in the situation of be/bre the long-term care facility is built, where alternative sites with different input/output characteristics are part of the choice set. Once a facility is built, these locational characteristics become nondiscretionary. To state this differently, the locational siting approach views these decisions as 'long-run,' in which all inputs and outputs are variable. The D E A m e t h o d o l o g y

The fractional programming formulation for the D E A problem is presented below~. The formulation is motivated by the classical engineering-science definition of productivity: v productivity = = x where y is the realized output rate and x is the realized input rate. D E A extends the classical definition to multiple outputs and inputs without requiring the researcher to a priori assign weights to each input and output. For example, suppose there are n D M U s to be analyzed, each being subscripted with a j . Each D M U uses m inputs to produce s outputs. With this notation, xij is the rate of input i applied to DMUj, and yr, is the rate of output r produced by DMU~. The productivity, h, for DMUi is then:

• r=[

blryU

h, -

,

(1)

m

~ UiXij i=l

where x!j > 0, for i = 1,... m , and, Y,:i> 0 for r = 1..... s; and .j is the subscript defining the j t h DMU. The difficulty with directly applying the above definition involves determination of the weights to assign to each input and output (i.e. the ur and t,i values). The D E A methodology employs

tA comparison of various efficiency measurement methodologies is provided by Lovell and Schmidt [15]. +In the more recent literature, it is common to go directly to the transformed dual linear programming formulation of the DEA problem. We elected to present the fractional form, mainly because of the ease of presenting the concept to management. Good introductions to DEA and its models are provided Charnes and Cooper [9] and Banker et al. [2].

Siting efficiency of long-term health care facilities

29

mathematical p r o g r a m m i n g techniques to optimally assign the weights, subject to the following imposed restrictions: Condition 1:

Condition 2:

The weights for each D M U are assigned subject to the constraint that no other D M U has an efficiency greater than 1.0 if it uses the same weights. This implies that efficient D M U s will have a ratio value o f 1.0. The derived weights u, and vl are not negative.

These conditions imply the following constraint set:

• ury,i r=l

- -

m E UiX~j

_< 1 for j = 1,2..,n

(2)

i=1

where, v, > 0 for i = 1..... m; ur_> 0 for r = 1..... s, and n is the total n u m b e r o f D M U s t . Each D M U ' s productivity ratio must be maximized subject to the constraints in eqn (2), hence, the concept o f relative efficiency. The fractional objective function is the ratio o f the total weighted o u t p u t divided by the total weighted input; i.e.

• Ur.}',k r=l

Maximize h , -

,

(3)

m

E ~.!iXik i=1

where k is the subscript defining the current D M U under investigation. The n fractional p r o g r a m s are then formulated, one for each D M U , in the realized dataset. The solution to the fractional p r o g r a m s defines an efficient frontier, and, hence, a ratio-definition of efficiency. The complete formulation is: s

E u,-yrk r=l

Maximize h k -

(4) m

~ UiXik i=1

subject to s

E U~yrj r=l

• UiX(i

_< 1 for j = 1,2..,n

i=[

where vi > 0 for i = 1...... m;, ur > 0 for r = 1..... s,; and n is the total n u m b e r o f D M U s . The analysis is summarized as follows: 1. Collect data on the o u t p u t and input variables; i.e. xq and yrj.

t F o r many, the variables v~and ur should be constrained to be positive. This is handled mathematically by requiring ur,r~ >_ E where E > 0 is a small non-Archimedean infinitesimal, smaller than any positive real number. Computationally, this is handled in D E A computer codes by a two-stage optimization. For a proof that this is equivalent to introducing such a non-Archimedean infinitesimal, see Arnold et al. [1].

Homed F. E. Shroff et al.

30

2. Solve n mathematical programs, one for each D M U , with decision variables vl and Ur. 3. The solution to each program defines the efficiency of one D M U relative to the other DMUs. 4. The relatively efficient D M U s have an optimal objective function value of 1.0. Charnes and Cooper [10] have shown that the fractional program defined in (4) may be transformed to an equivalent linear programming problem. Hence, the efficient D M U s are determined by solving a sequence of linear programs. These transformations, summarized by Charnes and Cooper [9], are well known and thus not repeated here. We elected to present our models in fractional form because the concepts were easily understood by both I N O V A ' s corporate management and our I N O V A partners o,1 the research team.

The Northern Virginia region The Northern Virginia region is part of the ~Bo-Washt' megalopolis corridor. The region is comprised of four counties and the City of Alexandria. The counties are Arlington, Loudoun, Fairfax and Prince William (see Fig. 1)$. Presently, only five cities in the U.S. have populations that exceed that of Northern Virginia. In the 1980s, Northern Virginia was described as the 'Mecca' of suburban building and mall markets in the world. Garreau [13], in his definition of a functional edge city, defines the Northern Virginia area as " p a r t of America's life on the new frontier." Some of the factors responsible for the "hyper-growth' activity within the Northern Virginia area are economic restructuring toward an information-based economy and the availability of computational-intensive and communications technologies, as well as the area's attraction to private sector information technology-based service firms. Increases in the elderly and middle-aged population, increased life expectancy, 'echo-effect' of baby-boomers, high fertility rates amongst minority groups and growing in-migration and immigration of elderly citizens continue to alter the region's demographic profile into the next century. Five of the nine jurisdictions in the United States with the highest income per capita are in Northern Virginia. The region also contains the highest number of graduate degrees per capita in the United States. As a consequence, the region is known for its retail business growth and high per capita retail spending levels. The region will probably continue to experience unprecedented growth for a decade, and should generate a high demand for personal services such as health care.

T H E DEA A P P L I C A T I O N

D M U definition Our strategy, similar to that used by Plane and Hendrick [18], involved dividing the Northern Virginia region into a set of demand points. The D M U s , each a demand point, were defined as local government planning districts. Fairfax County and the City of Alexandria employ planning districts as a measure for identifying homogeneous pockets of demographics, income, industry and service within a geographic region. The remaining three counties, Arlington, Loudoun and Prince William, use census tracts as their planning areas. In addition, I N O V A and other organizations compile economic and demographic data by post office zip codes across the counties. In order to accommodate all these units (i.e. by zip code, census tract and planning district), the data were aggregated to the planning district level. The aggregation rules were the same as those used by Fairfax County and the City of Alexandria in defining their planning districts. The result was a consistent data set that spanned 26 planning districts, where each planning district represented a potential site or D M U (see Fig. 2). I N O V A representatives, especially those associated with the Planning and Marketing Department, participated in all data decisions. t"Bo-Wash" is a name that was proposed by Garreau [13]. SThe Potomac River is the boundary between Northern Virginia, and Maryland and the District of Columbia.

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Home6 F. E. Shroff et al. NORTHERN VIRGINIA COUNTIES PLANNING DISTRICTS

Alexandria Arlington Co

D @ m

Fairfax Co Loudoun Co Prince William Co

PDAL3 / PDAL1

Co Alexandria \ PDAX2

Mt. Vernon

+ Lower Potomac

Fig. 2. Planning districts in Northern Virginia. Long-term

care i n p u t s

The long-term care inputs selected by I N O V A ' s management as being most important for the siting decision are as follows: Xl = x2 = x3 = x4 = x5 = x6 =

Desirability of location; Land acquisition costs; Annual average household income; Population density of potential elderly (Intermediate Care) residents; Population density of children of potential elderly (Intermediate Care) residents; Availability of professional staff; e.g. registered nurses and licensed practitioner nurses (RNs, LPNs); x7 = Availability of other professional staff; e.g. dietitians and therapists (occupational, speech and physical);

Siting efficiency of long-term health care facilities

33

xa = Availability of semi-skilled staff; e.g. certified nurses, nurses aides and orderlies (CNAs); and x9 = Availability o f physicians (practitioners, internists, primary care physicians, and psychiatrists). These inputs are discussed in more detail below. Desirability of location (x0:. INOVA's long-term care administrative staff stated that there existed a high rate of turnover a m o n g all types of nursing staff within their existing long-term care facilities. I N O V A understands the factors that influence staff turnover. Some of these are locational, while others are seasonal. A separate analysis attempted to identify factors that influence turnover rates. It was not a part of this study. This led to an investigation o f the 'desirability' of a potential employment location for different classifications of long-term care employees. The results of a questionnaire survey distributed among the nursing staff within INOVA's existing facilities allowed for the construction of a desirability index. Amenities such as easy access to mass transit and highways; and/or close proximity to services such as banks, dry-cleaning, day care, etc. added to the attractiveness of a particular location. Land acquisition costs (x2):. For the proposed 120 bed facility, the I N O V A planning staff suggested that five one-acre lots be examined. In estimating the cost of these lots, much uncertainty was encountered. First, the average costs in a particular planning district may grossly under-represent or over-represent a particular parcel of land, even within the same block. Second, due to complexity and peculiarities of multiple zoning categories, site locations are negotiable between the buyer and the seller, as well as with neighborhood groups and the county zoning office. Hence, even though INOVA's Planning and Marketing Department provided data on the costs of particular land parcels, it was clear that a variable describing average land costs should be treated with caution and subjected to extensive sensitivity analyses. Household income (x3):. Average household income was selected as a measure of planning district income. This is preferred over average family income since such a measure only includes blood-related household members and, therefore, understates the income of non-traditional households. Such households include single adults living together and other lessors or renters living in the same household. This input is an indicator o f private-pay potential within a planning district. Table 1. Initial DEA results for Northern Virginia Planning districts Fairfax County Annandale Bailey's Crossroads Bull Run Fairfax City Jefferson Lincoluia Lower Potomac McLean Mount Vernon Pohick Rose Hill Springfield Upper Potomac Vienna Arlington County SW Arlington Central Arlington North Arlington SE Arlington City of Alexandria Seminary Potomac Old Towne Loudoun County Central Loudoun East Loudoun Prince William County East Prince William Central Prince William North Prince William

Efficiency scores (%) 0.66 1.00 0.72 0.59 0.44 0.22 0.13 1.00 1.00 1.00 1.00 0.94 1.00 0.82 0.75 1.00 0.85 0.75 0.76 0.98 1.00 0.46 0.76 0.78 1.00 0.86

Home6 F. E. Shroff et al.

34

Population densiO' of elderly (potential intermediate care residents) (x4) and their children (xs). Data on the concentration and distribution of the elderly as well as their children were examined. By I N O V A ' s definition the elderly population consists of age cohorts 85 years or older with children falling within the range of 45 to 64 years of age. The 85 + population also includes patients with Alzheimer's, Parkinson's or other related diseases. These residents are candidates for self-admission into a nursing facility or admission on the recommendation of their physician or children. I N O V A ' s market research department has determined that it is quite c o m m o n for older children (i.e. 45 to 64 years of age) to move their elderly parents to Northern Virginia and admit them to long-term care facilities. Hence, the population density of these children was considered important. Availability of staff" and physicians (x7 to x~). The long-term care staff was divided into three categories, as follows:

NORTHERN VIRGINIA COUNTIES PROPOSED EFFICIENT SITES FOR INOVA LONGTERM CARE INITIAL RESULTS

r "/

i,.•.'".,,

/ ....

;:>

. .....

+,7%111/k .

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EFFICIENCY

D

',.., "-.,

EFFICIENT PD > 0.94

NORTHERN VIRGINIA

Co

".,.. ""

i

, 'IU" .. --i-=

Fig. 3. Initial D E A results for N o r t h e r n Virginia.

Alexandria

Siting efficiency of long-term health care facilities

35

1. Professional Staff: This category includes registered nurses (RNs) and licensed practitioner nurses (LPNs). They are part of the professional team and, in some cases, are a substitute for physician care when physicians are not available 2. Other Professional Staff: This category is comprised of dietitians and therapists. Three types of therapists were considered: physical, occupational, and speech. 3. Semi-Skilled Staff: This category contains certified nursing assistants (CNAs), nurses aides, and orderlies. Four types of physicians were considered in defining physician availability. They were INOVA-specified family practice or general practitioners, internists or internal medics, primary-care physicians, and psychiatrists. These categories cover the physician types that service long-term care facilities.

Input measurement The absolute input levels are not appropriate measures to be used in this type of D E A problem. For example, five doctors available to service 500 beds is much different than five doctors available to service 100 beds. Hence, all the inputs, with the exception of desirability of location and land acquisition costs, were measured on a per-bed basis. That is, with the exception of xt and x2, all inputs (x~ to x9) were defined relative to the total number of beds in the planning district. This measurement procedure also indirectly considers competitive pressures within the districts, as will be explained below. Let ' a ' be a particular input (e.g. total number of LPNs in the district). After correcting for the number of beds, input j is thus: a x,= ~ =

input value number of beds in planning district"

The denominator of this ratio includes all I N O V A and competitor hospital and long-term care beds. Also included in the denominator are the 120 beds for I N O V A ' s new proposed facility. The following simple example is used to explain how the ratio implicitly considers competitive effects, and why the ratio must be inverted before it can be used as input data for the siting analysis. Assume there are two planning districts, PDI and PD2. Also assume that there are 10 LPNs (licensed practitioner nurses) available in each district. Further, assume that there are 50 beds present in PD~ and 100 beds in PD2. The input ratios for the two districts are thus: a PDl:Xjl- b a PD2:xj2- b -

10LPNs 50 beds -

0.2LPNs b e ~ ' and

10LPNs 100 beds -

0.1LPNs bed

F r o m I N O V A ' s point of view, xj~ is clearly preferred over xj> Although the same number of LPNs are available in both planning districts, the LPNs in PDI have to 'cover' a smaller number of beds. Therefore, more LPNs per bed are available for employment in PDI as compared to PD,. According to the 'classical' engineering-science definition of productivity, smaller values of the denominator (X) are preferred when the value of the numerator (Y) is fixed. That is, a 'best' productive unit is one that attains a fixed output with the least input. Under these circumstances, the ratio a/b must be inverted. Hence, all inputs that were measured on a per-bed basis were inverted prior to analyzing the output/input ratios.

Long-term care outputs I N O V A has a long-held reputation as the premiere health care provider in Northern Virginia. Hence, the I N O V A staff suggested that 'quality' long-term care should be their final output. While quality is of primary importance, the concern of this analysis was locating the facility so that I N O V A would be able to fill the given bed types, given the values of the inputs noted earlier. Hence, for the siting decision, the outputs should be defined in terms of the proposed bed types, relative to the number of competing beds in the planning district. Once the new facility is constructed, then quality of care is indeed the primary objective.

36

Home6 F. E. Shroff et al. Table 2. DEA results for Northern Virginia:Land acquisitioncosts excluded Planningdistricts Efficiencyscores Fairfax County Annandale 0.36 Bailey's Crossroads 0.91 Bull Run 0.61 Fairfax City 0.52 Jefferson 0.40 Lincolnia 0.22 Lower Potomac 0.11 McLean 1.00 Mount Vernon 0.95 Pohick 1.00 Rose Hill 1.00 Springfield 0.94 Upper Potomac 1.00 Vienna 0.79 ArlingtonCounty SW Arlington 0.75 Central Arlington 1.00 North Arlington 0.78 SE Arlington 0.38 City of Alexandria Seminary 0.65 Potomac 0.84 Old Towne 1.00 Loudoun County Central Loudoun 0.39 East Loudoun 0.76 Prince WilliamCounty East PrinceWilliam 0.47 Central Prince William 1.00 North Prince William 0.31

The i n t e r m e d i a t e o u t p u t s were determined by I N O V A ' s staff as the following: Yz = Sub-acute care bed o c c u p a n c y for private-pay patients; Y2 = Skilled care bed o c c u p a n c y for private-pay patients; a n d Y3 = I n t e r m e d i a t e care bed o c c u p a n c y for private-pay patients Since the ratio o f private-pay to g o v e r n m e n t - p a y was pre-determined by I N O V A to be a 50/50 split, it was n o t necessary to explicitly consider the g o v e r n m e n t - p a y beds in the analysis. This was the case since, for a 120 bed facility, the c o m p l e m e n t of the private-pay beds are the g o v e r n m e n t - p a y beds. The initial D E A solution The initial solution for all 26 N o r t h e r n Virginia p l a n n i n g districts are s u m m a r i z e d in Table 1 while the efficient p l a n n i n g districts are indicated in Fig. 3. The efficient p l a n n i n g districts, using radial efficiency,? are Old T o w n e Alexandria; Central A r l i n g t o n C o u n t y ; C e n t r a l Prince W i l l i a m C o u n t y ; a n d Fairfax C o u n t y p l a n n i n g districts that include: Bailey's Crossroads, M c L e a n , Mt. V e r n o n , Pohick, Rose Hill, a n d U p p e r P o t o m a c . O n e x a m i n i n g these p l a n n i n g districts, we recognized that high income a n d p o p u l a t i o n , a l o n g with the availability of e m p l o y m e n t , was the driving force b e h i n d the relative siting efficiencies in these p l a n n i n g districts. Extensive sensitivity analyses led to an e x p l a n a t i o n as well as an a p p r e c i a t i o n o f the power of D E A .

EXTENDED DEA ANALYSES Once the initial analyses were completed, sensitivity analyses a n d a d d i t i o n a l f o r m u l a t i o n s were executed. The extended analyses c o n c e n t r a t e d o n two inputs, land acquisition costs a n d desirability of location, as discussed earlier. Inconsistencies were also e x a m i n e d a n d subjected to further analyses. F o r example, i f a p a r t i c u l a r p l a n n i n g district was considered efficient in the analysis, b u t its efficiency score was counter-intuitive, then that p l a n n i n g district was subjected to further analyses. Finally, Eigenvalue tRadial efficiencyis defined in Chapter three of [12] for the linear programs defined earlier. It is also called 'weak efficiency" because the non-zero slack components are ignored [7].

Siting efficiency of long-term health care facilities

37

diagnostic measures were used to identify collinear inputs, while statistical methods were used to search for extreme observations and to evaluate their effects on the efficiency measurements. As previously discussed, we were apprehensive a b o u t including average land costs in the model. This hesitation was due to variability in costs within the planning districts. Hence, we executed the model with land costs excluded, finding that the efficient districts changed slightly. The results o f this analysis are presented in Table 2 and Fig. 4. The scores for Bailey's Crossroads and M o u n t Vernon were reduced from 1.0 to 0.91 and 0.95, respectively, indicating that land costs were a factor in these previously efficient districts. A l t h o u g h the scores o f the inefficient D M U s changed, the efficient districts from the initial analysis remained the same. This indicates that income, demographics, and e m p l o y m e n t are the 'drivers' o f the relative locational efficiencies o f these planning districts. The lack o f sensitivity seems plausible.

NORTHERN VIRGINIA COI)NTIES

PROPOSED EFFICIENT SITES FOR INOVA LONGTERM CARE WITHOUT LAND ACQUISTION COSTS

l1 PDPW2

t

EFFICIENTPD

::i! I

EFFICIENCY> 0.94

D

INEFFICIENTPD

~!~iii~iilii!i~i ii~~i~i~!i~!~,~i~!)il!i!~ii¸i!~il! i¸¸

~i~ili!~ii~il~ii ~ii' ~!i! il !' ~ i~

Co Alexandria

Fig. 4. DEA results Northern Virginia planning districts: Land acquisition costs excluded.

38

Home6 F. E. Shroff

et al.

Table 3. DEA results for Northern Virginia: impact of a facility in central Prince William Planning districts Fairfax County Annandale Bailey's Crossroads Bull Run Fairfax City Jefferson Lincolnia Lower Potomac McLean Mount Vernon Pohick Rose Hill Springfield Upper Potomac Vienna Arlington County SW Arlington Central Arlington North Arlington SE Arlington City of Alexandria Seminary Potomac Old Towne Loudoun County Central Loudoun East Loudoun Prince William County East Prince William Central Prince William North Prince William

Efficiency scores 0.36 0.91 0.61 0.52 0.40 0.23 0.1 t 1.00 0.95 1.00 1.00 0.94 1.00 0.79 0.75 1.00 0.78 0.38 0.65 0.84 1.00 0.39 0.84 0.47 0.55 0.31

Once the land cost is incurred, the real drivers of efficiency should be the demographics, income, and employment availability in the districtL Since the 'desirability of location' input was not calculated on a per-bed basis, and to justify the emerging hypothesis that the efficiency sensitive inputs were income, population, and employee availability, the model was solved after excluding the 'desirability of location' input. The resulting efficient planning districts set remained unchanged, but the scores of the inefficient planning districts did change. Planning districts that were assigned a high value for desirability of location experienced a significant drop in their efficiency scores, whereas districts with low desirability scores remained unchanged. This sensitivity analysis revealed that the efficient planning districts were not responsive to changes in this input, and that the changes in their scores agreed with our expectations. Counter-intuitive results Central Prince William County was efficient in the original analysis, as well as in the sensitivity analyses. Interestingly, this result did not agree with our expectations. Central Prince William County is low in population density, income, and employee availability when compared to districts in Fairfax or Arlington counties. Sensitivity analyses led to an explanation as to why Central Prince William County was efficient. This district contained no existing beds, either in long-term care or hospital facilities. The lack of any long-term or hospital beds (zero supply) created what appeared to be a high demand for beds in the district. However, when an additional 120 hypothetical beds were assigned to the district, the efficiency score dropped to 0.55. This sensitivity analysis provides a clear indication that the initial driving force for efficiency within Central Prince William County was the complete lack of beds, and that the additional 120 beds saturated this demand. Hence, the size of the anticipated long-term care unit was a determining factor in this particular case. As previously, this analysis led to an appreciation of the power of D E A (see Table 3 and Fig. 5). t i n the end, this truly b e c a m e a non-issue. I N O V A already o w n e d land in one o f the efficient districts, and land cost was not a determining factor in the location decision. H o w e v e r , this fact was only revealed to us after presenting results to top m a n a g e m e n t .

Siting efficiency of long-term health care facilities I

39

NORTHERN VIRGINIA COUNTIES PROPOSED EFFICIENT SITES FOR INOVA LONGTERM CARE W I T H 120 B E D S I N P W 2 W I T H O U T L A N D A C Q U I S T I O N C O S T S

EFFICIENT PD

®

EFFICIENCY > 0.94

INEFFICIENT PD (

:"

k' ,,.,,

Co ¸;

¸ i iiiiii!i!i

Alexandria

Fig. 5. DEA results for Northern Virginia: impact of a facility in Central Prince William.

Collinearity diagnostics There were no indications of severe collinearity problems in any of the analyses; however, we decided to apply diagnostic procedures to increase our confidence in the solutionS'. Each output was thus regressed on the input set, and the resulting eigenvalues examined:~. The analysis indicated a potential problem with two input variables: professional staff availability, and 'other' staff availability. When either input was deleted from the program, the efficient set remained unchanged, an indication that the solution is stable. The conclusion of these sensitivity analyses is that if there is a collinearity problem in the INOVA formulation, it does not appear to change the DEA results.

tCollinearity can be a problem in DEA models [16]. One approach for dealing with it is presented by Olesen and Petersen [17], They advocate input aggregation using column aggregation techniques. :~The diagnostic procedure, discussed by Belsley et al. [6], uses the work of Silvey [21] to identify near dependencies among input columns.

40

Home6 F. E. Shroff et al. Table 4. DEA results for Northern Virginia with central Arlington omitted Planning districts

Efficiency scores

Fairfax County Annandale Bailey's Crossroads Bull Run Fairfax City Jefferson Lincolnia Lower Potomac McLean Mount Vernon Pohick Rose Hill Springfield Upper Potomac Vienna Arlington County SW Arlington North Arlington SE Arlington City of Alexandria Seminary Potomac Old Towne Loudoun County Central Loudoun East Loudoun Prince William County East Prince William Central Prince William North Prince William

0.36 0.98 0.61 0,65 0.64 0.23 0.1 I 1.00 1.00 1.00 1.00 1.00 1.00 0.84 0.97 0.83 0.43 0,82 1.00 1.00 0.40 0.95 0.51 0.55 0.36

Identification o f extreme observations

One problem with deterministic frontier estimation methods is that an extreme data point can define the frontier and 'dominate' the analysis. To search for extreme observations here, the outputs were regressed against the inputs for each planning district, and the residuals examined. With the exception of Central Arlington County, there was no indication of a potential problem. A closer examination of Central Arlington County revealed that the long-term care bed need projection for this district was indeed high; i.e. this is a very wealthy district with many elderly citizens. When the DEA results were computed with the Central Arlington County D M U omitted, the efficiency scores for some districts increased slightly?. The planning districts in close proximity to Central Arlington County experienced the largest increases, again agreeing with our expectations. The results of these analyses are presented in Table 4 and Fig. 6. Final D E A solution

In our study the efficient planning districts may be grouped into two primary geographic areas. One is comprised of planning districts Pohick, Rose Hill, Springfield, and Old Towne Alexandria, while the other consists of Upper Potomac, McLean and Central Arlington (see Table 3 and Fig. 5). Also, it should be noted that the Mt. Vernon and Springfield planning districts continued to have an efficiency score greater than 0.94 in all the analyses. IMPLEMENTATION At the time this study was initiated, there was a moratorium on long-term care facility construction in the state of Virginia. Because of the tremendous growth in Northern Virginia throughout the 1980s, INOVA knew there was substantial existing demand for additional long-term care facilities. In anticipation of the lifting of the moratorium, INOVA's planning staff had been collecting demographic and economic data for a number of years. However, because of t L a n d acquisition costs were omitted from these analyses as well. However, the additional 120 hypothetical long-term care beds in Central Prince William County were retained in the analyses. Additional runs were also conducted that did not include the hypothetical beds. The results remained essentially the same with the exception of Central Prince William County which, of course, became efficient.

Siting efficiency of long-term health care facilities

41

N O R T H E R N VIRGINIA COUNTIES PROPOSED EFFICIENT SITES FOR INOVA LONGTERM CARE WITHOUT PD: ARLINGTON 2

EFFICIENTPD ~

EFFICIENCY> 0,94

~

NORTHERNVIRGINIA

Co :Alexandria

Fig. 6. DEA results for Northern Virginia with Central Arlington omitted.

the perceived difficulty in formulating and solving a multiple criteria location model, a formal model had not been constructed. The staff had many years experience in health care planning and management, and hence had some preconceived 'hunches' about where a facility might be located. Formal decision support was lacking, however. The results of this study were presented to INOVA's planning and marketing executives. The efficient planning districts identified in the study supported INOVA's a priori prediction that the Pohick and Rose Hill/Springfield areas were probably good sites to consider. The executives' subjective predictions were based mainly on a good understanding of the population and income data. Of course, these 'hunches' were not presented to the study team until after the analysis and formal presentation was completed. A parcel of land with the proper zoning requirements was identified and secured in the Pohick planning district. Full implementation of the recommendations of this study are currently underway, with INOVA management acknowledging the importance of DEA in supporting their final decision.

42

Home6 F. E. Shroff et al. CONCLUSIONS

The analysis o u t l i n e d here p r e s e n t e d I N O V A d e c i s i o n - m a k e r s with a choice o f several efficient l o c a t i o n sites for their l o n g - t e r m care facility. It also p r o v i d e d an ability to e x a m i n e less efficient sites m o r e closely. The e x t e n d e d sensitivity analyses identified the m o r e sensitive inputs a n d thus c r e a t e d the f o u n d a t i o n s for extensive t r a d e - o f f discussions on the efficient sites within the N o r t h e r n Virginia area. The efficiency-sensitive inputs were revealed relative to c u r r e n t I N O V A a n d c o m p e t i t o r beds in the p l a n n i n g districts. These inputs are income, d e m o g r a p h i c s , a n d e m p l o y e e availability. L a n d a c q u i s i t i o n costs a n d desirability o f l o c a t i o n are a p p a r e n t l y insensitive inputs. These results thus p r o v i d e an i n d i c a t i o n o f those i n p u t s t h a t should be m o r e closely e x a m i n e d by I N O V A ' s p l a n n i n g staff. Since c o m p l e t i o n o f this work, it has c o m e to o u r a t t e n t i o n that, given the choice is between discrete values, the t r e a t m e n t should f o r m a l l y be m a d e by reference to efficiency d o m i n a n c e [22], r a t h e r t h a n simple efficiency, as o u t l i n e d here. T h e difference is that, in the s t a n d a r d D E A model, one a s s u m e s a c o n t i n u i t y across the v a r i o u s D M U points; but, in some a p p l i c a t i o n s , the D M U s are real a n d discrete. In those cases, a m i x e d - i n t e g e r p r o g r a m m i n g a p p r o a c h is available [3, 4] to restrict the efficiency e v a l u a t i o n s to c o m p a r i s o n s o f the a c t u a l D M U s . W e feel t h a t this is a m a j o r a d v a n c e , b u t also note, that given the spatial a u t o c o r r e l a t i o n across D M U s within a m e t r o p o l i t a n context, as in the present example, treating D M U sites as o b s e r v a t i o n s f r o m a c o n t i n u o u s surface is n o t an u n w a r r a n t e d a s s u m p t i o n . Certainly, efficiency d o m i n a n c e w o u l d seem a v a l u a b l e a l t e r n a t i v e and, p e r h a p s , an efficient s o l u t i o n strategy. F r o m o u r analysis, D E A has p r o v e d a vital a n d p o w e r f u l tool in siting I N O V A ' s l o n g - t e r m care facility. T h e decision s u p p o r t o f the results thus indicate the p o w e r o f D E A in an a c t u a l decision m a k i n g setting, a n d the p o t e n t i a l usefulness o f i n t e g r a t i n g concepts o f p r o d u c t i v i t y into l o c a t i o n analysis. F u r t h e r , the sensitivity analyses indicate the r o b u s t n e s s o f the results, the stability o f the findings, a n d p o i n t to the wider use o f this m e t h o d o l o g y to s u p p o r t siting decisions in general. Acknowledgements--The authors wish to express their appreciation to Professor W.W. Cooper for reviewing and providing advice on this manuscript. Professor Cooper's advice has helped with the clarity and articulation of the ideas of the paper. All errors in analysis or interpretation are the responsibility of the authors.

REFERENCES 1. Arnold, V., Bardhan, I., Cooper, W. W. and Gallegos, A., Primal and dual optimality in DEA computer codes. In Operations Research: Methods, Models, and Applications, ed. J. Aronson and S. Zionts. Quorum Books, New York~ 1996. 2. Banker, R., Charnes, A., Cooper, W. W., Swarts, J. and Thomas, D. A., An introduction to data envelopment analysis with some of its models and their uses. Research in Governmental and Nonprofit Accounting, 1989, 5, 125-163. 3. Bardhan, I., Bowlin, W. F., Cooper, W. W. and Sueyoshi, T,, Models and measures for efficiency dominance in DEA, Part I: Additive models and MED measures, Journal of the Operations Research Society of Japan, forthcoming. 4. Bardhan, I., Bowlin, W. F., Cooper, W. W. and Sueyoshi, T., Models and measures for efficiency dominance in DEA, Part II: Free disposal hull (FDH) and Russell measure (RM) approaches, Journal of the Operations Research Socie O' of Japan, forthcoming. 5. Belton, V., An integrating of data envelopment analysis with multiple criteria decision analysis. In Multipk, Criteria Decision Making, ed. A. Goicoechea, L. Duckstein, and S. Zionts. Springer-Verlag, New York, 1992, pp. 71-79. 6. Belsley, D. A., Kuh, E. and Welch, R. E., Regression Diagnostics: Identifying Influential Data and Sources of Collinearity. John Wiley, New York, 1980. 7. Charnes, A., Cooper, W. W., Sears, M. and Zlobec, S., Efficiency evaluations in perturbed data envelopment analysis. In Parametric Optimization and Related Topics II, Mathematical Forschung, Band 62, ed. J. Guddat, H. Jongen, B. Kummer, and F. Nozicka. Akademie Verlag, Berlin, 1991. 8. Charnes, A., Cooper, W. W. and Rhodes, E., Measuring the efficiency of decision-making units,. European Journal of Operational Research, 1978, 2, 429~444. 9. Charnes, A. and Cooper, W. W., Preface to topics in data envelopment analysis. Annals of Operations Research, 1985, 2, 59-94. 10. Charnes, A. and Cooper, W. W., Programming with linear fractional functionals. Naval Research Logistics Quarterly, 1962, 9, 181-186. l l. Desai, A. and Storbeck, J., A data envelopment analysis for spatial efficiency. Computers, Em, ironment and Urban Systems, 1990, 14, 1,$5-156. 12. F/ire, R., Grosskopff, S. and Lovell, C. A. K., The Measurement of Productive Efficiency. Kluwer, Boston, 1985. 13. Garreau, J., Edge City: Life on the New Frontier. Doubleday, New York, 1991. 14. Kao, Chiang and Yong Chi Yang, Reorganization of forest districts via efficiency measurement. European Journal of Operational Research, 1992, 58, 35(~362. 15. Lovell, C. A. Knox and Schmidt, P., A comparison of alternative approaches to the measurement of productive

Siting efficiency of long-term health care facilities

16, 17. 18. 19. 20. 21. 22. 23.

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efficiency. In Applications o]" Modern Production Theory: F~ff4ciencyand ProductiviO', ed. A. Dogramaci and R. Ffire. Kluwer, Boston, 1988, pp. 3-32. Olesen, O. B. and Petersen, N. C., Collinearity in data envelopment analysis: An extended facet approach, Working Paper, Department of Management, Odense University, Odense, Denmark, 1991. Olesen, O. B. and Petersen, N. C., Indicators of ill-conditioned data sets and model misspecification in data envelopment analysis: Aggregation as a potential remedy. Working Paper, Department of Management, Odense University, Odense, Denmark, 1993. Plane~ D. R. and Hendrick, T. E., Mathematical programming and the location of fire companies for the Denver Fire Department. Operations Research, 1977, 25, 563 578. Seiford, L., A bibliography of data envelopment analysis. Working Paper, Department of Industrial Engineering and Operations Research, The University of Massachusetts, Amherst, Massachusetts, 1991. Shroff, H. F. E., Siting efficiencies of long-term care facilities: The Northern Virginia health system. Unpublished Ph.D. Dissertation, Boston University, 1992. Silvey, S., Multicollinearity and imprecise estimation. Journal of the Royal Statistical SocieO', 1969, 31 B, 539 552. Tulkens, H., On FDH efficiency analysis: Some methodological issues and applications to retail banking, courts, and urban transit. Journal of Productivity Analysis, 1993, 1/2, 183-210. U.S. General Accounting Office, Long-Term Care Forum, GAO/HRD-93-1-SP, U.S. GAO, Washington, 1993.