Kim Woods

DLSUBusiness & Economics Review 25.2 (2016), pp. 166-196

A DEA-based Performance Measurement Mathematical Model and Software Application System Applied to Public Hospitals in the Philippines R ic h a r d C. Li, J a z m in C. T a n g s o c , S o lo m o n L. S e e , V i c t o r J o h n M . C a n to r ,

M a r t h a L a u re n L. T a n , a n d R a c h e lle J o y S. Yu

De La Salle University, Manila, Philippines [email protected]

Evaluation o f performance is an important activity in identifying shortcomings in managerial efficiency and devising goals for improvement. However, measuring performance is not an easy task; more so in making sure that it captures a holistic view o f performance. This study identified four existing performance measurement issues that organizations face often. These issues were (a) existence o f missing data during data collection, (b) accounting undesirable or non-value adding outputs as opposed to desirable or marketable outputs, (c) inclusion o f exogenous or environmental factors that affects the organization performance, and (d) arriving with resource allocation decisions that will help improve organizational performance. Linear Programming (LP) and Data Envelopment Analy sis (DEA) were used to develop a performance measurement tool that addresses the aforementioned issues. Subsequently, this tool was used to develop the DEA-based performance measurement and reallocation software to aid managers in analyzing organizational performance. A case study on 14 NCR public hospitals was conducted to validate the logic and usefulness o f the software. Software results showed that there were two inefficient hospitals and corresponding decisions to increase performance were identified in terms o f inputs and outputs. An economic interpretation was then provided to realize the significance o f the performance measurement results.

JEL Classifications: C 61,C 88,I12

Keywords: Data envelopment analysis, Exogenous input, Missing data, Performance measurement, Reallocation decision, Undesirable output

Copyright © 2016 by De La Salle University

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Performance measurement is a fundamental building block o f any organization that aims to improve service and business performance. Historically, organizations have always measured performance through their financial performance ( G e rs c h e w s k i & X ia o , 2 0 1 5 ). H o w e v e r, trad itio n al perfo rm an ce m easures, based on cost accounting information, provide little to sup p o rt o rg an izatio n s on th eir p erfo rm an ce as a whole. This is because they do not map process perfo rm an ce w ith the consideration o f all in p u ts u sed to a tta in o rg a n iz a tio n a l goals (Colledani & Tolio, 2009). It is through perform ance measurement where organizations should be able to identify and track progress against organizational goals that are often not easily m easurable using financial indicators (Tung, Baird, & Schoch, 2011). Performance m easurem ent also plays an im portant role in identifying opportunities for improvement and comparing perform ance against standards (both internal and external) (de Lima, da Costa, & de Faria, 2009; Verbeeten & Boons, 2009). Clearly, m easuring perform ance in a holistic view is warranted for an organization to prosper.

Performance measurement is data intensive and requires the organization to have a data collection system in place. However, though p ro c e d u re s are d e v elo p ed in e n su rin g th a t important information needed for performance m easurem ent is collected, there are instances wherein data are missing. M issing data pertains to perform ance-related data that are unavailable due to several reasons that include, but are not lim ited to, adm inistrative fault in which the staff failed to collect or record the data, malfunctioning o f equipment that resulted to data corruption, and refusal o f respondent to answer the questions from a survey, among others (Zha, Song, Xu, & Yang, 2013). M issing data exists and are inevitable in all organizations, thus should be accounted for (Kao & Liu, 2000). In any case wherein data are missing, exclusion or elimination o f the cases or categories that contain

m issin g data p roduces a disto rted re su lt o f performance as some inefficient systems may be considered efficient in the absence o f considering a significant input or output in evaluation (Chen, Li, Xie, An, & Liang, 2014). M oreover, the impact o f missing data is detrimental not only through its potential hidden biases o f the results but also in its practical impact on the sample size available for analysis.

Aside from data collection inadequacies that result to missing data, organizations also need to decipher what kind o f data they need to collect and consider for perform ance m easurem ent. In the performance o f any operation, there are many aspects to look into in order to maximize efficien cy in clu d in g : re d u c in g co sts, cycle time, waste, material usage, and so forth while m axim izing throughput, quality, and so forth. There is m ultidimensionality in the goals that organizations want to achieve. In optim izing efficiency, organizations would want to use the least amount o f inputs to achieve the most amount o f output. However, it m ust be noted that in using inputs, it does not only produce desirable or marketable outputs. Undesirable outputs may also be produced in the process. These outputs are classified as w aste or non-value adding for the organization but are jo in tly produced (Seiford & Zhu, 2002). Hence, it makes sense for a perform ance measurement system to credit an organization for its provision o f desirable outputs and to penalize it for its production o f undesirable outputs. For instance, banks increase their num ber o f deposits and loans but incur overdue debts as well. Likewise, hospitals aim to maximize the total number o f patients served and treated but incur failed operations and diagnoses that result to deaths. These are some examples o f sim ultaneous occurrence o f desirable and undesirable outputs. Zanella, Cam anho, and Dias (2015) and Fare, Grosskopf, Lovell, and Pasurka (1989) claimed that ignoring undesirable outputs might produce misleading performance results. Thus, performance measurement should

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be able to simultaneously consider the decrease o f undesirable outputs and the increase o f the desirable outputs.

Furtherm ore, organizations m easure their perform ance in order to com pare them selves both with their own standards and the industry standards. Performance measurement results are used to benchmark if the current perform ance o f an o rg a n iz a tio n is lo w er, h ig h e r, or at p a r w ith o th e r o r g a n iz a tio n s . H o w e v e r, o rg an izatio n s tend to m easure perform ance and do b e n ch m a rk in g w ith o u t co n sid e rin g e x te rn a l fa c to rs th a t are n o n -c o n tro lla b le and yet have an effect on their perform ance. Inefficiencies in any organization should not be constrained only to m anagerial and operational inadequacies. Rather, the inefficiencies caused by its o p eratin g en v iro n m e n t m ust also be included in the analysis o f perform ance. These are called exogenous inputs that characterize the operating environm ent w ithin w hich the production or the organization is taking place or situated, respectively (Macpherson, Principe, & Shao, 2013). For instance, as described by A vkiran and R o w lan d s (2008), ed u catio n al a tta in m e n t o f p a re n ts co u ld be co n sid e re d as an exogenous input in m easuring literacy and n u m e ra c y in p rim a ry sc h o o ls fo r the reason that educated populations are likely to show higher rating on these m easures due to additional resources available to children. Smith and Street (2004) concluded that in whatever way operating environment is defined, usually some organizations operate in m ore adverse environments than others in the sense that the external circumstances make the achievement o f a given level o f attainment more or less difficult, thereby leading to im precise and unreliable assessment o f an organization’s inefficiency. The benefit o f accounting for these exogenous inputs in decision making lies on the idea that it makes the com parison betw een organizations m ore realistic by taking all influences (both internal or external) that contribute to performance rating

into account and also providing possible sources that can explain the performance behavior o f the organization.

Having said that performance measurement is im p o rtan t in any o rg an izatio n and given th a t m issing data, u n d esirab le o utputs, and exogenous inputs may conspire to distort the m easurem ent and analysis o f perform ance, a reliable perform ance m easurem ent system is therefore necessary to aid in effective decision­ m ak in g w hile co n sid e rin g all o f th e issues mentioned.

Data Envelopment Analysis (DEA) is used as a foundation in this paper to come up with a more reliable performance measurement system that addresses all the aforementioned issues. It is a performance measurement and benchmarking tool with the ability to simultaneously consider all inputs and outputs that may be o f interest in arriving at an overall performance or efficiency sco re (C h a rn e s, C ooper, & R h o d es, 1978; Sherman & Zhu, 2006). The model developed in this paper stems from the basic DEA model. The basic DEA model was modified to consider both exogenous inputs and undesirable outputs while a predictive LP model was form ulated alongside the modified DEA model in order to account for m issing data before com parative a n aly sis am ong o rg a n iz a tio n s or D e cisio n M ak in g U n its (D M U s) is p e rfo rm ed . The im proved perform ance m easurem ent system also includes the reallocation o f resources to organizations that can realize the most potential in terms o f output. An economic interpretation o f re allo catio n d ecisio n s m ade p ro v id es an indication o f progress towards specific defined organizational objectives and whether expected results are being achieved.

The objective o f this research was to develop a perform ance m easurem ent tool in the form o f software that addresses the aforementioned issues. This software is aimed to help managers in p e rfo rm a n c e m e a s u re m e n t a n a ly s is by calculating efficiencies o f units and identifying

DEA-BASED PERFORMANCE MEASUREMENT MATHEMATICAL MODEL AND SOFTWARE LI, R.C., ET AL. 1 69

w hat levels o f inputs and outputs should be achieved to improve performance.

An overview o f the general methodology o f the study is discussed in Part II. Development o f a L in ea r P ro g ra m m in g (L P) m o d el and modification o f the basic DEA model to consider the aforementioned issues is then discussed in Part III. The developm ent o f the software based on the LP and DEA models and the selection o f inputs and outputs for analysis are discussed in Part IV and V, respectively. Discussion o f the results will follow in Part VI and conclusions based on all the insights gathered from the study will be presented in Part VII.

METHODOLOGY

S e v e ra l p h a s e s w ere u n d e rta k e n in the completion o f this research study as shown in Figure 1. The base methodology o f the study

involves development o f the mathematical model and software, the selection and preparation o f data to be exam ined, m easurem ent o f DM U efficiency using DEA analysis, reallocation o f resources among DMUs for the maximization o f overall system output using available resources, and the economic interpretation o f the results. The succeeding sub-sections discuss each o f the phases aforementioned.

A. Model Development

For the developm ent o f the model, existing performance measurement tools were reviewed to identify which tool best suits the objective o f the study. DEA was then selected as the foundation tool to be used because o f its advantages over other perform ance m easurem ent tools (e.g., single dimension performance indicators, ratio analysis, regression analysis) such as the ability to sim u lta n eo u sly c o n sid er m u ltip le inputs

Software Application ..S y s t e m .

Figure 1. Methodology flowchart.

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and outputs and it being a benchmarking tool. Limitations o f existing DEA models were then identified, which resulted to three key points to be addressed in this study.

First, existing D EA m odels do not allow missing cases in the data set. Second, existing DEA m odels account for undesirable outputs and exogenous inputs separately, which should be taken simultaneously. Third, existing DEA m o d els do n o t p ro v id e in sig h ts on how to reallocate excess inputs within the set o f DMUs in order to maximize overall system output.

Amodified additive DEAmodel was developed to address the aforementioned concerns and test d ata from case analyses used in a previous stu d y a b o u t lib raries (E loissenzadeh L o fti, Jahanshahloo, & Esmaeli, 2007) to verify if the model gives logical results. Sensitivity analysis was perform ed to investigate the robustness o f the model by considering the effects o f parameter adjustm ents on the m odel behavior. M odel behavior should showcase that inefficient DMUs can be identified if number o f DMUs used are w ith in 2 * (in p u t + o u tp u t factors); also th at improvement areas are identified for inefficient DMUs; and that expected increase in desirable output has a basis for movement, such that there is either increase in input combinations and/or in undesirable. Upon acceptance o f model behavior, software development was then commenced.

B. Software Development

T h e s o f tw a r e d e v e lo p m e n t s ta r te d by identifying user requirem ents and these were as follows: data setup in the software, needed functions based on the modified additive DEA model, and expected output (num erically and graphically) o f the software. To address the user requirem ents, the software developm ent was divided into phases: (1) data input interface, (2) missing data estimation, (3) DEA efficiency calculation, (4) reallocation o f resources, and (5) u sab ility ev alu atio n and en hancem ents.

User testing was conducted on the preliminary software application where comments received from a pool o f potential users w ere used to modify and enhance the software application.

C. Selection and Preparation of Data Set for Case Study

The preparation o f the data set began with the selection o f DMUs, inputs, and outputs involved in the study as well as the industry for the case study. The healthcare industry was chosen to be the industry for the case study o f this research. Typically, inputs co nsidered in analysis are critical resources that were used in the production o f the product or service o f a DMU. For this purpose, inputs may be, but not limited to, budget allocations, existing capital, buildings, or labor employed (Cantor, Tan, & Yu, 2008). In addition, i f there are factors that influence the outcome o f outputs and are not in the control o f the DMU, these factors should be considered in the analysis as exogenous inputs. Outputs to be considered in analyses, on the other hand, should be the major output, services or products that the company or organization is offering to consumers. I f there are waste or non-value adding output, these should minimize and can be considered as undesirable output in the analyses.

The to ta l n u m b e r o f in p u ts and o u tp u ts identified were then checked if compliant with the rule o f thumb o f DEA analysis that the number o f DM Us should be greater than or equal to two times the sum o f the number o f inputs and outputs. As DEA is data intensive, analysis cannot be conducted if there are missing data. As such, an LP model was developed that will estimate values for missing data.

D. Measurement of DMU Efficiency

Amodified additive DEAmodel was developed and was used to compute for efficiency scores o f DMUs under evaluation. Aside from the normal

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controllable inputs and desirable outputs that existing DEA models consider, the undesirable (bad) o u tp u ts, exogenous inputs, and p o in t estimates for missing data were also taken into consideration in the m odified additive DEA model.

DEA is a benchmarking tool where DMUs are evaluated relative to other business units by ranking. This is done in order to see the relationship between the set o f DMUs such that, for any two units, the first is either “ranked higher than”, “ranked lower than” or “ranked equal to” the second. Rankings make it possible to evaluate complex information according to certain criteria. Computation o f the DEA efficiency will generate input slacks and surplus to be used for resource reallocation decisions.

E. Reallocation of Resources

U pon efficien cy m easurem ent o f D M U s, resource distribution o f surplus inputs among DMUs was performed with the goal o f improving overall system output. The objective o f the r e a llo c a tio n m o d e l w as to m a x im iz e th e opportunity to increase the output o f the entire system whilst m axim izing the use o f existing resources within the system. It m ust be noted th at the re a llo c a tio n m odel only co n sid e rs reallocation o f the controllable inputs excluding exogenous inputs.

F. Economic Interpretation of Results

Results were analyzed and insights were drawn for possible enhancement o f system performance. Initially excess inputs were identified within the system o f units being assessed. These excesses were indications o f inefficiencies in the original allo c atio n and use o f reso u rces. E x cesses should not be interpreted as immediate removal o f a resource. Rather, it is an indication that measures should be done to reduce cost in that specific resource area. Efficient DM Us can also

be identified so that the inefficient DMUs may benchmark their operations against these DMUs.

When inefficiencies are addressed and savings are realized, these savings may be tapped to allocate to DMUs tow ards increasing overall o u tp u t o f th e system . G en erally , reso u rce reallocation can be looked into as a means to enhance overall system performance.

MODEL FORMULATION

There were three main phases that correspond to th e m a in f u n c t i o n a l i t i e s o f th e D E A benchmarking and reallocation tool. First was the estimation utility for cases o f missing data on the data set. Second was the facility for efficiency comparison alongside with the identification o f excess inputs or resources o f DMUs. Finally, the third functionality was a reallocation model w h e rein th e id e n tifie d ex ce ss in p u ts w ere reallocated to DMUs to maximize overall system output. Succeeding sub-sections discuss the details o f each o f the functionalities mentioned.

A. Estimation of Missing Data

Problem on m issing data arises frequently w hen an organization conducts perform ance measurement; and the most common method to deal with this problem is through the deletion o f categories or cases with missing data. However, such method can seriously affect the number o f cases left for analysis, which then can lead to bias and inaccurate findings. To address the problem o f deleting categories or cases w ith m issing data, the DEA-based performance measurement and reallocation software allows users to run an analysis with a data set containing missing data through data estimation. That is, the value o f the missing data is estimated from the data set instead o f deleting. An LP predictive model (see Appendix A) was recommended to estimate the value o f missing data. LP was utilized because

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it does not require assumption o f data normality. S ta tis tic a l re g re ss io n te c h n iq u e s h o ld tru e with assumptions o f data normality. LP-based estimates, however, can be used whether the data is normal or non-normal.

The LP p re d ic tiv e m odel uses p air-w ise com bination to find the best input or output category that can be used as basis in estimating missing data. Standard error (S ) is computed from the pair-wise combination and the category that gives the least Se becom es the basis for estimating the values o f missing data. It must be noted that different missing data from different categories may have different bases.

It is usually true using statistical regression that approximately 68% o f the estimated values will be w ithin one 5 , and approximately 95% o f estimated values will be within two Se (Winston, 2004). Validation o f the LP predictive model was conducted through the data set retrieved from a DEA study on libraries (Hoissenzadeh Lofti et al., 2007) and it was found that approximately 83% o f the estimated values were within one S .

e

B. Efficiency Comparison Using the Modified Additive DEA Model With Undesirable Output and Exogenous Input

A modified additive DEAmodel (see Appendix B) was developed to consider simultaneously u n d e sira b le o u tp u ts and e x o g en o u s in p u ts aside from the controllable inputs and desirable outputs. The modified additive D EA m odel was used to compute for efficiency scores o f DMUs. The efficiency scores were used as a means to identify whether a DMU was relatively efficient or relatively inefficient and from these, identify the rankings o f all DMUs. In addition, excess re so u rces w ere id e n tifie d from D M U s th at contributed to the system ’s inefficiencies.

U ndesirable output. Undesirable outputs were treated similar as a resource input such as the model developed by Korhonen and Luptacik

(2004) and further examined by Yang and Pollitt (2009). It was considered as an input in a manner that for a DMU to be efficient, it should be able to produce more output with the least amount o f undesirable outcomes.

Exogenous input. The impact o f exogenous inputs or environm ental factors in efficiency m e a s u r e m e n t w a s c o n s i d e r e d t h r o u g h benchmarking selection. The most representative model within this option was the one proposed by Banker and M orey (1986). In the Banker and Morey (1986) model, the comparison may include units that operate in a similar or more unfavorable environm ent com pared w ith the assessed unit. However, this assumes a positive impact on the desirable output. A ccording to Hua, Bian, and Liang (2007), such treatm ent was not as applicable w hen both inputs and o u tp u ts w ere sim u ltan eo u sly c o n sid e re d in perform ance assessm ent o f DM Us. Positive impact o f the exogenous input cannot be assumed since its impacts to the inputs and undesirable output indicators cannot be alw ays affirm ed to be positive. Instead, to be able to consider exogenous inputs, this requires that reference co m p ariso n s u tiliz e the sam e lev els o f the transform ed exogenous inputs as th at o f the assessed business units.

C. Reallocation M odel

The third functionality o f the D EA -based p erfo rm an ce m easu rem en t and re a llo ca tio n software was a reallocation model (see Appendix C). The excess resources identified during efficiency comparisons o f DMUs were allocated to other DMUs needing these excess resources w ith the goal o f m axim izing overall system output under the assum ption o f m aintaining current relative operational efficiency. To provide for the reallocation decision functionality, a forecast model in the form o f an inverse DEA model is utilized. Inverse to the DEA, the input reallocation problem determines how much an

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input should be allocated given that outputs are increased and efficiency remains the same (Wei, Zhang, & Zhang, 2000). Another type o f the inverse DEA problem was the forecasting problem which determines how much an output should increase given that inputs are increased and efficiency rem ains the same (Wei et al., 2000). With the forecast m odel, it can now determine how additional input can affect output changes.

The objective o f the reallocation model was to reallocate resources that can increase the overall output o f the whole system. The DEA-based p e rfo rm an ce m e asu rem en t and re a llo ca tio n s o f tw a r e w a s d e v e lo p e d to i n c o r p o r a t e reallocation o f excess inputs to enhance output production o f the entire system. The process o f selection on reallo catio n depends on the influence o f that reallocation to the changes in all output indicators. The software chooses to allocate resources that have the highest increase in percentage output change. The increase in output production w as presented through an index that aggregates the percentage increase in output production o f each output indicator (e.g., the model will choose to allocate to a DM U if

percentage increase in an output or a combination thereof is greater than percentage increase in other output w hen allocated to another unit).

SOFTWARE DEVELOPM ENT

With considerations to the functionalities and mathematical foundations o f the tool, the software prototype was developed using Microsoft.NET framework 4.0 with C# as its prim ary language. The software was designed and built following the architectural design shown in Figure 2. The software has three core modules:

1. D a ta R e a d e r M o d u le - T he m o d u le responsible for reading inputted data from Microsoft Excel, data analysis and storing as a .NET data table.

2. Solver M odule - The module responsible for estimating missing data from the input data set and execution o f DEA analysis.

3. V isu a liz a tio n M o d u le - The m o d u le responsible for generating the charts and table outputs from the input data sets and the resulting output.

Figure 2. Architectural design.

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Users o f the software would place the data they wanted to be analyzed in table form on a single Excel spread sheet. The software will then allow the users to specify the excel file where their data is located for DEA analysis. The specified file is then processed by the Data Reader module that reads the data from the excel spread sheet and stores the data in memory as a .NET System.Data. DataTable. The DataTable is then processed by the Solver module to estimate missing data and then performs the DEA analysis. For the Solver module to perform the D EA analysis, the model constraints discussed above are retrieved from the Constraints Rule Database. Finally, the results o f the LP Solver is stored, to another DataTable and is processed by the Visualizer module to format the data on screen and generate charts and graphs for the user to easily visualize the results.

A. Data Reader Module

The Data Reader module is responsible for reading the data set from the user (currently only supports MS Excel files) and converts it into a System.Data.DataTable that the Solver module takes as input to perform the DEA analysis. This m odule m akes use o f M icrosoft Excel COM Interop to read excel files (.xls and .xlsx files) into the software. M icrosoft Excel COM Interop makes use o f Microsoft Excel’s shared libraries to natively read and write excel files. This method provides a fast and direct way o f reading Excel files but im poses a restriction that M icrosoft Excel has to be installed on the machine where it is going to be used.

Once the data is read from the excel file, it is stored in memory as a System.Data.DataTable. The DataTable class stores the read file as a series o f columns and rows. The module further annotates the colum ns in the D ataTable for analysis, by asking the user to identify which fields are considered as Input, Exogenous Input, Desirable Output, and Undesirable Output. The annotated DataTable is then finally passed to

the Visualizer module, to display on-screen the data read from the Excel file, and to the Solver module, to perform the DEA analysis.

B. Solver Module

T h e S o lv e r m o d u le is r e s p o n s ib le fo r estimating missing data from the input data set as well as performing the analysis by solving the modified additive DEA model discussed above. The Solver module forwards the DataTable to the Data Estimator sub-module that first checks for missing data from the input DataTable. This is done by checking every cell in the table if they are empty or not. For all empty cells found in the DataTable, the data estimation algorithm described in the section above is run to complete the DataTable.

O nce a co m plete D ataT able is obtained, the data is forw arded to the LP Solver sub- module. The LP Solver sub-module makes use o f M icrosoft Solver Foundation (MSF) to solve the m odified additive DEA m odel using the Simplex method. This sub-module takes as input the DataTable as well as the linear constraints stored in the Constraints Rule Database written in Optimization M odelling Language (OML). Decision variables from the model are then read and stored into a DataTable and passed to the Visualizer Module for display.

C. Visualizer Module

The Visualizer module converts the DataTable resu lts o f the D EA analysis into individual G ridV iew s and g e n erate s ch arts fo r b e tte r visualization o f the user. This m odule splits the resulting DataTable into smaller tables and displays them into individual GridViews located in different tabs. The DataTable is divided into two smaller tables which contains its calculated efficiency and reallocation. Finally, these data are then converted into bar charts using Microsoft Charting Controls.

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D. Verification of Software Results

C om parison o f results from the softw are developed was conducted to verify if output resu lts w ere lo g ical and valid. To te st the resulting values, software results were verified against reference models ran in General Algebraic M odeling System (GAMS). Verification runs resu lted to the sam e output in the softw are d e v elo p ed and on GA M S. C o m p ariso n o f stan d ard e rro r show s th at the LP estim ates resulted in lower standard error as compared to using regression analysis. Short discussion on the details o f the verification can be found on Appendices D and E.

E. Usability Evaluation and Enhancement

The usability o f the software interface was evaluated using the Nielsen (1993) heuristics as it was considered to be one o f the most dependable usability heuristics. Using Nielsen heuristics for software usability ensures that users can easily use and understand the software. The usability evaluation focused on two things: (a) data entry and (b) presentation o f results.

The initial software interface with respect to data entry was found to have usability problems. L ab els and d ialo g u e boxes w ere co n fu sin g for the user and the software did not provide feedback or status for the actions done on the software. This m eans that the term inologies initially used confused the user on w hat to do and did not give the user an idea on what was happening while extracting the data file. For the presentation o f results, heavy user’s mental load was an initial problem because a lot o f numbers were presented but were not actually significant for interpretation. Graphs and tables were not appropriate for the results presented, which adds to the confusion as to how the different sets o f results were related. In addition, coded labels for the different variables were not easily identifiable to a specific variable and were consistently in

places that required the user to m em orize the labels.

Enhancem ents were done to ensure users w ould n o t have a d iffic u lt tim e to use the software and interpret the results. In the data entry, commonly used terms were used; visibility o f the system status were also done as the user ex tracts the file and w hile an aly sis w as in progress. Graphs with the corresponding variable nam es w ere used instead o f coded variables and tabulated results were shown. Graphical representation o f the results were also added to make it easy to differentiate values among the input and output variables.

SELECTION OF INPUTS AND OUTPUTS

A. A Case Study on NCR Hospitals

This study considered the analysis o f efficiency o f h o sp itals located in the N atio n al C apital Region (NCR) o f the Philippines. Categorization o f hospitals was done in choosing the final set o f hospitals to be included in the study following the D ep artm en t o f H e a lth ’s (2012) hospital classification. Department o f Health classified hospitals in the Philippines based on ownership (g o v ern m en t or p riv a te ), scope o f serv ices (general or specialized), and functional capacity (level 1, 2 or 3). For the purpose o f this study, hospitals that are government-owned, offering general services, and have Level 3 functional capacity were chosen.

Government hospitals are hospitals that are owned by the Philippine government and receive g o v e rn m e n t fu n d in g . M ea n w h ile , g e n eral hospitals provide medical and surgical care to the sick and injured, maternity care and shall have as minimum the following clinical services: medicine, pediatrics, obstetrics and gynecology, surgery and anesthesia, em ergency services, o u tpatient, and an cillary services. G eneral hospitals are further classified in three levels

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o f functional capacity. Functional capacity o f Level 3 means that a hospital has the necessary equipm ent and m anpow er for all the clinical services aforem entioned w ith the presence o f te ach in g w ith accred ited residency train in g program in major clinical services.

A total o f 14 government, general, and Level 3 hospitals were chosen to form the final data set for this study. Some hospitals w ere not included because o f inadequate information in the statistical reports gathered from the Department o f Health.

B. Input Selection

M easuring efficiency o f hospitals requires appropriate selection o f inputs to be considered in the evaluation. H ospital input categories generally fall into three broad sub-categories: capital investment, labor, and other operating expenses (O ’N eill, Rauner, H eidenberger, & Kraus, 2008). Capital investment usually refers to beds, equipment, and different facilities that can be directly used for clinical services. The num ber o f fully staffed hospital beds is most often used as a proxy for hospital size and capital investment. O ’Niell et al. (2008) made a comparison and taxonomy o f hospital efficiency studies and showed that 55 out o f 79 research studies included the number o f beds as an input category. Some o f these research studies were from Ballestero and M aldonado (2004), Chem and Wan (2000), and Grosskopf, Margaritis, and Valdmanis (2004), among others.

M o re o v er, a b o u t tw o -th ird s o f h o sp ita l operating costs is due to payroll expenses that usually refer to labor costs (Sahin & Ozcan, 2 0 0 0 ). H o sp ita l c lin ic a l s ta f f c o n sists o f physicians, nurses, and other health/m edical personnel. Som m ersguter-Reichm ann (2000) d e fin e d n u m b e r o f p e rso n n e l as a g en eral labor input category. Finally, other than labor costs, non-labor costs are also being incurred by h o sp itals d u rin g o p eratio n s. N o n -lab o r

costs include m edical supply, food, drug and pharmaceutical, material costs, among others.

Hosseinzadeh Lofti et al. (2007) purported that influences o f external conditions affect the performance o f each organization that is regarded as the exogenous inputs that are identifiable b u t u n c o n tro lla b le by th e o rg a n iz a tio n in consideration. For this study, the population o f a city where a hospital is situated was considered as exogenous input. Thus, the corresponding group o f inputs that describe the health care services offered in the hospitals under consideration were: (a) authorized bed capacity, (b) total personnel, (c) total expenditure, and (d) population.

C. Output Selection

The Agency on H ealth Care R esearch and Quality (AHRQ, 2011) o f the United States o f America distinguished two types o f outputs in healthcare, namely: health services which refer to the products and services that healthcare units provide to constituents (e.g., visits, admissions, drugs, etc.), and health outcomes w hich refer to resulting output o f the services availed (e.g., preventable deaths, functional status, and blood pressure control). Selection o f outputs was considered to reflect the general range o f hospital activities (Al-Shammari, 1999). In the case o f this study, the corresponding group o f outputs that describe the health care services offered in the hospitals in assessment were: (a) total patients adm inistered (health services), (b) laboratory services (health services), and (c) net death rate (health outcome).

N um ber o f patients (A l-Sham m ari, 1999; Katharaki, 2008; Steinmann, Dittrich, Karman, & Z w e ife l, 2 0 0 4 ) and la b o ra to ry serv ice s (Katharaki, 2008) were selected as criteria for efficiency assessment o f DMUs sim ilar to the study conducted by M aria K atharaki on the management o f Greek hospitals’ gynecological and obstetrics unit. Meanwhile, outputs used in efficiency m easures were usually a mix o f

DEA-BASED PERFORMANCE MEASUREMENT MATHEMATICAL MODEL AND SOFTWARE LI, R.C., ET AL. 177

hospital services such as discharges, visits, and procedures (AHRQ, 2011). In the case o f this study, the mix o f hospital services was represented through the total patients administered and the laboratory services.

N et death rate, on the other hand, refers to a health outcome measure for quality o f hospital system-level performance. Although there were debates on the drawbacks on the use o f mortality as a quality m easure, it was one o f the m ost widely used (Kroch & Duan, 2008). Mortality rate was considered to be a simple measure as it is easily observable by counting deaths from discharges. In addition, a docum ent review o f hospital perform ance reports indicated net death rate to be a common measure among N CR tertiary hospitals (Katharaki, 2008).

DISCUSSION

T h e c a s e s tu d y u s e s th e D E A - b a s e d p erfo rm an ce m e asu rem en t and re a llo ca tio n softw are to assess 14 N C R hospital units in the perspective o f a managing body. The case study uses seven efficiency factors. Inputs used in analysis were authorized bed capacity, total personnel, and total expenditure. Exogenous input used is the population o f the local government unit being served. Outputs considered are total p a tie n ts served and the lab o rato ry serv ices offered. Undesirable output considered is the net death rate o f the hospital unit.

A. Missing Data

Ideally, DEA analysis should only be carried out with all available data on hand and complete. However, even at the best o f efforts to gather data, it is a reality that there will be cases o f data that will not be available. Performance data o f each o f the hospitals in this case analysis was gathered from D O H perform ance reports. A thorough docum ent review o f reports was done

to gather the m ost com plete inform ation on hospital performance. Unfortunately, there are still instances o f missing data as shown in Table 1. It was observed that 83% o f efficiency factors data is available; however nine o f the 14 hospitals have at least one efficiency factor missing. In traditional DEA software, the efficiency analysis can now only be carried out for the five remaining hospitals w ith com plete data. With ju s t five remaining hospital qualified for analysis, only 36% o f total data available will be utilized.

With the DEA-based performance measurement and reallocation software developed, it allows the flexibility o f carrying out efficiency analysis despite some cases o f m issing data. The LP predictive model optimizes the best predictor in the data set to estimate values for missing data. Using the software with the LP predictive model developed, it estimated data for: total personnel, total patients, laboratory services, and net death rate (see Table 2). As such, DEA efficiency analysis can now be perform ed and the data set with estimated values was used.

B. Insights on Efficiency

The case an aly sis u sin g the D E A -b ased p erfo rm an ce m easu rem en t and re a llo ca tio n so ftw a re in F ig u re 3 d e te rm in e d th a t tw o hospitals are below par o f its peers. These are East Avenue M edical Center and Tondo Medical Center.

Tondo M edical C enter’s efficiency is rated at 93.40%, and East Avenue M edical C enter’s efficiency is rated at 91.84%. The efficiency ratings indicated that they are only perform ing at a rate to what is considered as efficient based on other hospital peers’ input-output performance.

C. Excess Inputs/Resources Identification

Aside from the ratings that indicate relative efficiency; the softw are was also capable o f identifying excess inputs. Excess inputs were

17 8 BUSINESS & ECONOMICS REVIEW V O L 2 5 NO. 2

Table 1. Initial Data Set for Case Analysis o f NCR Hospitals

DMU Authorized Total

Bed Capacity Personnel Total Expenditure Population Total Patients

Laboratory Services

N et Death Rate

Amang Rodriguez Memorial Medical Center 300.00 I111I11IIS11I 424,150,.00 84,005..00 566,,159..00 0.0380 Dr. Jose N. Rodriguez Memorial Hospital 200.00 97.00 54,980,664.4500 744,500..00 70,007..00 453,,005..00 M m East Avenue Medical Center 600.00 1,025.00 345,215,,00 155,396 .00 villi' in 0.0332 Gat Andres Bonifacio Memorial Medical Center 150.00 i l l l l i l i i 330,434..20 89,842..00 244.,754..00 0.0290 Jose R. Reyes Memorial Medical Center 450.00 1,074.00 909,947,310.53 330,434..20 224,044 .00 1,154,,747 .00 0.0530 Justice Jose Abad Santos General Hospital 150.00 454.00 49,112,110.56 330,434..20 62,795,,00 70,,716 .00 Hitt Las Pinas General Hospital & Satellite Trauma Center 150.00 292.00 137,627,154.70 552,573..00 49,456..00 128.,625 .00 0.0300 Mandaluyong City Medical Center 150.00 547.00 328,699 .00 72,672..00 64.,849 .00 0.2950 Pasay City General Hospital 150.00 196,434 .50 54,925 .00 173,,849 .00 0.0200 Philppine General Hospital 1,346.00 3,653.00 2,248,344,940.00 330,434 .20 525,741 .00 1,341,,067 .00 0.0453 Diosdado Macapagal Memorial Medical Center 2,000.00 m s u m 744,500..00 36,384 .00 54.,392 .00 0.0193 Quezon City General Hospital 250.00 424.00 297,597,919.78 345,215..00 85,355 .00 277,,814 .00 0.0130 Quirino Memorial Medical Center 350.00 192.00 269,836,926.00 345,215 .00 / | Ml Tondo Medical Center 200.00 419.00 249,406,489.13 330,434 .20 78,775 .00 209,,112 .00 0.0326

Table 2. Final Data Set with Estimates fo r Case Analysis o f NCR Hospitals

DMU Authorized

Bed Capacity Total

Personnel Total Expenditure Population Total Patients

Laboratory N et Death Services Rate

Amang Rodriguez Memorial Medical Center 300.00 518.4033 272,796,295.2854 424,150.00 84,005.00 566,159.00 0.0380 Dr. Jose N. Rodriguez Memorial Hospital 200.00 97.00 54,980,664.4500 744,500.00 70,007.00 453,005.00 0.0294 East Avenue Medical Center 600.00 1,025.00 592,073,857.6638 345,215.00 155,396.00 409,626.37 0.0332 Gat Andres Bonifacio Memorial Medical Center 150.00 559.8232 298,900,750.0710 330,434.20 89,842.00 244,754.00 0.0290 Jose R. Reyes Memorial Medical Center 450.00 1,074.00 909,947,310.53 330,434.20 224,044.00 1,154,747.00 0.0530 Justice Jose Abad Santos General Hospital 150.00 454.00 49,112,110.56 330,434.20 62,795.00 70,716.00 0.0294 Las Pinas General Hospital & Satellite Trauma Center 150.00 292.00 137,627,154.70 552,573.00 49,456.00 128,625.00 0.0300 Mandaluyong City Medical Center 150.00 547.00 222,112,418.9643 328,699.00 72,672.00 64,849.00 0.2950 Pasay City General Hospital 150.00 312.0491 142,743,606.3754 196,434.50 54,925.00 173,849.00 0.0200 Philippine General Hospital 1,346.00 3,653.00 2,248,344,940.00 330,434.20 525,741.00 1,341,067.00 0.0453 Diosdado Macapagal Memorial Medical Center 2,000.00 180.4806 598,223,836.5998 744,500.00 36,384.00 54,392.00 0.0193 Quezon City General Hospital 250.00 424.00 297,597,919.78 345,215.00 85,355.00 277,814.00 0.0130 Quirino Memorial Medical Center 350.00 192.00 269,836,926.00 345,215.00 83,343.2800 220,681.3083 0.0310 Tondo Medical Center 200.00 419.00 249,406,489.13 330,434.20 78,775.00 209,112.00 0.0326

identified in comparison to other hospitals, that is, a select hospital w ith consideration o f its efficiency is using more than what it needs to produce its current level o f output.

W ith the help o f the softw are (see Table 3), excess resources have been identified for authorized bed capacity (133.84), personnel (total 103.84), and total expenditure (5,040,625).

T he to o l w as able to id e n tify th a t E a st Avenue Medical Center have excess resources in authorized bed capacity by approximately 134 excess beds (see Figure 4). This is indicative that the hospital has a larger hospital size in terms o f

capital investment (O ’Neill et al., 2008) relative to its output perform ance. Sim ilarly, Tondo M edical C enter is identified to have excess in total expenditure amounting to as much as 5 m illion pesos. M eanw hile, M andaluyong Medical Center is considered relatively efficient but has an excess o f 103.84 personnel. Excess personnel at the efficient level indicate that the hospital can afford to lessen human capital and still expect to achieve the same level o f relative efficiency.

Id en tify in g excess reso u rces is usefu l to indicate which aspects or area a hospital unit

DEA-BASED PERFORMANCE MEASUREMENT MATHEMATICAL MODEL AND SOFTWARE LI, R.C., ET AL. 179

Efficiency Scores East Avenue Medical Center

Tondo Medical Center

Quirino Memorial Medical Center

Quezon City General Hospital

Diosdado Macapagal Memorial Medical Center

Philippine General Hospital

Pasay City General Hospital

Mandaluyong City Medical Center

Las Pinas General Hospital & Satellite Trauma Center

Justice Jose Abad Santos General Hospital

Jose R. Reyes Memorial Medical Center

Gat Andres Bonifacio Memorial Medical Center

Dr. Jose N. Rodriguez Memorial Hospital

Amang Rodriguez Memorial Medical Center

1 91.84%

- 3 93.40%

■100.00%

1100.00%

■100.00%

■100.00%

■100.00%

■100.00%

■100.00%

■100.00%

■100.00%

■100.00%

■100.00%

■100.00%

Figure 3. Efficiency results and excess inputs.

is w eak at. This gives insights for hospitals on where to put focus to improve operational perform ance. H ow ever, as a m anaging body there are strateg ic decisions to be m ade on how best to utilize limited resources that would m axim ize the o verall system (all ho sp itals) to tal output perform ance. G iven th a t units are already identified with excess resources, a managing body has to make a decision whether other hospitals may need the additional budget or resources to produce more output.

The softw are developed has an additional capability to provide analysis on reallocation o f excess inputs. The objective o f the reallocation decision is to enhance the percentage increase in the output indicators as a system (all hospitals considered), namely: total patients administered and laboratory services. The increase in output production is indicative through an index that aggregates the percentage increase in output production o f each output category.

Reallocation analysis provides insights in two areas: (1) target operational improvements for inefficient DMUs, and (2) reallocation excess inputs and expected im provem ents in system output performance.

D. Targets on Operational Improvements

Summary results o f the reallocation analysis presented in Table 4 showed that the inefficient units could still increase their level o f output given current levels o f relative efficiency. For East Avenue Medical Center to retain its current level o f relative efficiency, even with 22% less capital investment (bed capacity), it has to focus on improving operations with targets in increasing the number o f laboratory services by 51.24%. This is also a cue for East Avenue Medical Center to review their capital expenditures and target to minimize unnecessary capital spending, while focusing on improvements to increase laboratory

180 BUSINESS & ECONOMICS REVIEW VOL. 25 NO. 2

Table 3. Excess Inputs

DMU Efficiency Score Excess Authorized

Bed Capacity Excess Total

Personnel Excess Total Expenditure

Amang Rodriguez Memorial Medical Center 100.00% 0.00 0.00 0.00 Dr. Jose N . Rodriguez Memorial Hospital 100.00% 0.00 0.00 0.00 East Avenue Medical Center 91.84% 133.84 0.00 0.00 Gat Andres Bonifacio Memorial Medical Center 100.00% 0.00 0.00 0.00 Jose R. Reyes Memorial Medical Center 100.00% 0.00 0.00 0.00 Justice Jose Abad Santos General Hospital 100.00% 0.00 0.00 0.00 Las Pinas General Hospital & Satellite Trauma Center 100.00% 0.00 0.00 0.00 Mandaluyong City Medical Center 100.00% 0.00 103.84 0.00 Pasay City General Hospital 100.00% 0.00 0.00 0.00 Philippine General Hospital 100.00% 0.00 0.00 0.00 Diosdado Macapagal Memorial Medical Center 100.00% 0.00 0.00 0.00 Quezon City General Hospital 100.00% 0.00 0.00 0.00 Quirino Memorial Medical Center 100.00% 0.00 0.00 0.00 Tondo Medical Center 93.40% 0.00 0.00 5,040,625.27

Tondo Medical Center mmm 200.00 Quirino Memorial Medical Center mmm 350.00

Quezon City General Hospital ■ ■ ■ 250.00

Diosdado M acapagal Memorial Medical Center

Philippine General Hospital

Pasay City General Hospital ■ ■ 150.00

M andaluyong City M edical Center ■ 1 150.00

Las Pinas General Hospital & Satellite Trauma Center p a 150.00

Justice Jose Abad Santos General Hospital ■ ■ 150.00

Jose R. Reyes Memorial Medical Center ■ M a n 450.00

Gat Andres Bonifacio Memorial Medical Center ■ ■ 150.00

East Avenue M edical Center Wmmmmm, 600.00 Dr. Jose N. Rodriguez Memorial Hospital n 200.00

Amang Rodriguez Memorial Medical Center ^mm 300.00

□ Desired Authorized Bed Capacity ■ Current Authorized Bed Capacity

Figure 4. Identification of excess authorized bed capacity.

DEA-BASED PERFORMANCE MEASUREMENT MATHEMATICAL MODEL AND SOFTWARE LI, R.C., ET A L 181

Table 4. Summary o f Reallocation o f Inputs and Expected Change in Output

DM U _ „ , Reallocated _ . Change in Change in Reallocated Reallocated Total Change m v ®

„ , ,, . Total Laboratory N et Death Bed Capacity , Expenditure Total Patients

_______________ Personnel_____ Services Rate

209,786.08 0.01

Amang Rodriguez Memorial Medical Center Dr. Jose N. Rodriguez Memorial Hospital East Avenue Medical Center Gat Andres Bonifecb Memorial Medical Center Jose R. Reyes Memorial Medical Center Justice Jose Abad Santos General Hospital 1.07 Las Pinas General Hospital & Satellite Trauma Center 51.43 Mandaluyong City Medical Center 81.34 Pasay City General Hospital Phi%>pirte General Hospital Diosdado Macapagal Memorial Medical Center Quezon City General Hospital Quirino Memorial Medical Center Tondo Medical Center

- 5,040,625.77 - 10,641.23 - - - 26,158.79 286,696.31 - ■ 1,218.82 332,519.51 -

- : 33,623.00 398,613.00 0.01 103.84 _ 3,500.26 97,182.57 _

- - - 112,045.91 -

services. However, East Avenue Medical Center should take caution or precautionary measures in increasing the laboratory services w ith less capital investm ent as it w ill also potentially affect health outcome by potentially increasing net death rate by 19%.

M e a n w h ile , T o n d o M ed ic a l C e n te r has excess expenditure o f about 2% o f their total allocation. While budget for total expenditure can be reduced, it should focus on improving its laboratory services from 53.58% to 93.4% efficiency.

Efficiency results also showed that Diosdado M ac a p ag a l M e m o ria l M ed ic a l C e n te r w as e f f ic ie n t r e la tiv e to its p e e rs. H o w e v e r, reallocation results as seen in the summary results in Table 4 showed that health services outcome can still be maximized given the current resources o f the hospital. Diosdado Macapagal M emorial M edical Center can further study their internal operations to focus on increasing its health services p articu larly on laboratory services. However, the hospital should take precautionary measures against increasing its health services w hile m ain tain in g cu rren t resources as this will also potentially affect health outcome by potentially increasing net death rate up to 52%.

While there is room for improvement for the two inefficient units, there is also reason to believe

that output performance o f efficient hospital units can be expected to increase given that excess budgets are reallocated to these hospital units. R eallocation analysis has identified areas for improvement for the inefficient units as well as identified candidates for potential reallocation o f resources and targeted increase in output performance.

E. Resource Reallocation Decision

D ecision v ariab les on the reallo catio n o f excess inputs have certain economic impact on the hospitals. Economic impact on the hospitals can result to im proved services that translate to serving more patients and increasing their satisfaction level. Additional income generated from the increase in patients can be used to buy additional input resources such as equipm ent to further strengthen the service capabilities. In addition, excess inputs can be translated to realignm ent o f capital investm ent w ithin the same hospital.

The software in Figure 5 identified hospitals as candidates for reallocation o f capital investments (authorized bed capacity); and these are: Justice Jose Abad Santos M edical Center, Las Pinas General Hospital & Satellite Trauma Center, and M andaluyong M edical Center. By allocating

182 BUSINESS & ECONOMICS REVIEW VOL. 25 NO. 2

T ondo Medical Center

Quirino Memorial Medical Center

Quezon City General Hospital

Diosdado Macapagal Memorial Medical Center

Philippine General Hospital

Pasay City General Hospital

Mandaluyong City Medical Center

Las Pinas General Hospital & Satellite Trauma Center

Justice Jose Abad Santos General Hospital

Jose R. Reyes Memorial Medical Center

Gat Andres Bonifacio Memorial Medical Center

East Avenue Medical Center

Dr. Jose N. Rodriguez Memorial Hospital

Amang Rodriguez Memorial Medical Center

2,000.00

0 Desired Authorized Bed Capacity ■ Current Authorized Bed Capacity

Figure 5. Identification o f reallocation o f authorized bed capacity.

capital investm ent to these h o sp itals, using the same relative efficiency, increase in both health services o f total patients adm inistered and laboratory services can be expected. An increase by 34.29% o f capital investm ent in Las Pinas General Hospital & Satellite Trauma C enter can potentially increase adm inistered patients by 42% and laboratory services by as much as three times its current performance (see Table 4); while an increase by 54% o f capital investment in Mandaluyong Medical Center can potentially increase patients administered by 2% and laboratory services by as much as six times its current output performance. Reallocation for bed capacity should be in line with the approved bed capacity issued w ith the hospital license.

The re a llo ca tio n can d id ate id e n tified by the software as shown in Figure 6 for excess

personnel is Quirino Memorial M edical Center. An increase o f 54% in personnel can potentially increase health services by 4% and 44% for patients adm inistered and laboratory services respectively. The reallocated excess personnel should have the same position and skill needed to render the services.

For other excess inputs that can be reallocated to hospitals such as medical equipment, there is a need to have a change in accountability o f the assigned resource. The additional services generated from the reallocated resources lead to revenues th a t can be used for additional operatio n al funds in the form o f additional incentive to personnel or increase in medical supplies/equipment. In the case o f Justice Jose Abad Santos Medical Center, to have an increase in capital investment by 1 % and total expenditure

DEA-BASED PERFORMANCE MEASUREMENT MATHEMATICAL MODEL AND SOFTWARE LI, R.C., ET AL. 183

Tondo Medical Center

Quirino Memorial Medical Center

Quezon City General Hospital

Diosdado Macapagal Memorial Medical Center

Philippine General Hospital

Pasay City General Hospital

Mandaluyong City Medical Center

Las Pinas General Hospital & Satellite Trauma Center

Justice Jose Abad Santos General Hospital

Jose R. Reyes Memorial Medical Center

Gat Andres Bonifacio Memorial Medical Center

East Avenue Medical Center

Dr. Jose N. Rodriguez Memorial Hospital

Amang Rodriguez Memorial Medical Center

0 Desired Total Personnel ■ Current T otal Personnel

Figure 6. Identification o f reallocation o f total personnel

by 10%, at its current efficiency level, it can potentially increase laboratory services by 15%. The increase in laboratory services can generate income that can help the hospital to further equip the laboratory w ith the latest medical equipment.

CONCLUSION

The case an aly sis w as able to sho w case functionalities and application o f the DEA-based p e rfo rm an ce m easu rem en t and re a llo ca tio n softw are. T hrough the softw are application system, DEA analysis was carried out despite instances o f m issing data in some o f the hospital units. Also the tool was able to identify that A m ang Rodriguez M emorial M edical Center, East Avenue M edical Center, and Tondo Medical

C enter w ere perform ing relatively below its peers. Through the use o f the software excess input analysis, targets for output perform ance and candidates for resource reallocation were identified.

To im prove p erform ance o f E ast Avenue M edical C enter, it is sug g ested th at capital expenditure is reviewed as this was identified to be in excess. Internal operations o f East Avenue Medical Center should be reviewed with focus on increasing health services, particularly in laboratory services. The medical center should also take caution that focusing on increasing health services should not affect service outcome. Similar to East Avenue Medical Center, Tondo Medical C enter’s internal operations should also be im proved to increase health services with particular focus on laboratory services.

1 8 4 BUSINESS & ECONOMICS REVIEW VOL. 25 NO. 2

I f savings from capital expenditure (from East Avenue Medical Center), personnel expenditure (from M andaluyong Medical Center), and total e x p e n d itu re (from T ondo M ed ic a l C en ter) are realized these excess resources or budget can be re-allocated to other m edical centers. To m a x im iz e in c re a s e in h e a lth s e rv ic e s , excess budget for capital investm ents should be re a llo ca te d to Ju stice Jose A bad Santos M edical Center, Las Pinas G eneral H ospital & Satellite Trauma Center, and M andaluyong M edical Center. Excess personnel resources on personnel can be m aximized with Quirino M em orial M edical Center. Excess budget on total expenditure should be best re-allocated to Mandaluyong M edical Center.

ACKNOWLEDGMENT

I would like to extend my utmost gratitude on some people who have helped on this study in one way or another: (a) to Engr. Cecille Matienzo, Division C hief o f the Center for Medical Device R e g u la tio n and R a d ia tio n , F o o d and D rug Administration, Departm ent o f Health for her valuable inputs in the economic interpretation o f the results; (b) to Dr. Nicolas Lutero III, CESO III and D irector IV o f the B ureau o f H ealth Facilities and Services, Department o f Health and his staff for allowing me to gather data from hospital statistical reports and for entertaining my questions about hospital performance indicators and procedures on performance measurement; and la stly (c) to th e U n iv e rs ity R e se a rc h Coordination Office o f De La Salle University for funding this research.

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DEA-BASED PERFORMANCE MEASUREMENT MATHEMATICAL MODEL AND SOFTWARE LI, R.C., ET AL. 187

APPENDIX A: M odel Formulation o f Linear Programming Predictive M odel f o r M issing Data

Indices i = Data set category

Parameters Y. = Actual value o f param eter to be estimated in data set i X. = Predictor parameter for data set i

Variables Positive slope coefficient Negative slope coefficient Positive y intercept coefficient Negative y intercept coefficient Estimate error for data set i

Linear program m ing data estimation m odel

n

i = 1

Subject to:

Yt - [(ffi - A 2) * X i + B1 - B2] + <5; > 0, Vi

Yi — [(Tb — A2) * Xi + B-l — B2] — 8i > 0, Vi

A2, Bi , B2, Si , Vi

Objective function. The objective o f the model is to find the best combination o f coefficients to minimize the sum o f estimate error from what is estimated by the model, and the actual value o f the param eter to be estimated (this is denoted by the variable 5).

n

M m £ S, (!) i = l

Constraints. Estimate errors are obtained and calculated through error constraints. These error constraints limit that the error should be the difference o f the predicted and actual value. Two error types are possible, one would be that the predicted value is greater than the actual value (negative error),

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and the other type would be that the predicted value is less than the actual value (positive error). To differentiate positive and negative errors two constraints are introduced for the two possible error types.

(a) Negative error constraint. The inequality constrains that if an estimated value is less than the actual value, the estimate error should be greater than zero (2).

(b) Positive error constraint. This constraint limits that if an estimated value is less than the actual value, the estimate error should be less than zero (3).

Utilizing the LP predictive model, a series o f steps is done to generate estimate values for missing data. Having a data set with missing data, initially use all data that are complete, that is, those cases that are complete across all inputs and outputs. The LP predictive model is then run for each pair wise combination between and among inputs and outputs. The coefficients, sum error, and standard residuals o f each pair wise comparison were recorded. After all runs were done, the results o f the sum errors are compared. For every input and output, the linear function w ith the least sum o f errors was noted; these linear functions will be used to estimate missing data.

The LP predictive model then generates a single point estimate o f the missing data, however, since the values are ju st an estimate there is no guarantee that the calculated value will represent the “real” value o f the missing data. An interval estimate is deemed better to represent the range o f values in which the missing value can be expected. From each o f the pair wise combination comparisons, the standard error o f the linear function was calculated using the following formula given in (5):

Yi - [ ( A 1 - A 2) * X i + B1 - B 2] + 8l > 0 , Vi (2)

Yi - [ ( A 1 - A 2) * X i + B1 - B 2] - 8 i > 0 , Vi (3)

(c) Nonnegativity constraint.

A\, A2, B-y, B2, Si, Vi (4)

SSE (5)

It is usually true that approximately 68% o f the estimated values o f y will be w ithin Se, and approximately 95% o f estimated y will be w ithin 2Se (Winston, 2004). In the library data set used from a DEA study (Hoissenzadeh Loftiet al., 2007), it was found that approximately 83% are within 5 .

DEA-BASED PERFORMANCE MEASUREMENT MATHEMATICAL MODEL AND SOFTWARE LI, R.C., ET AL. 189

APPENDIX B: Model Formulation fo r DEA with considerations to exogenous input and undesirable output

Model indices j = Decision Making Unit (DMU) (j = 1,2,3 ... n) k = DMU in assessment {k = 1, 2, 3 ... n) i = Type of input (/ = 1, 2, 3 ... m) r = Type of output (r= 1, 2, 3 ... s)

Decision variables 0k = Proportion of input use in output production Ajk = Proportion of input and output benchmarked from DMU for DMU

Model parameters E = Exogenous input of DMU j

X.. = Input i used by DMU j

T. = Output r of DMU j

Urj = Undesirable output r of DMU j

Modified DEA model

M in z = 0,k

Subject to:

n

* Xij) < 9k *Xik, VkVi (Input constraint) j =i n

' T s xjk * ^ij) ~ * Eik> VkVi (Exogenous input constraint) 7=1 n

Y j ( Ajk * Yrj) > Yrk, V/cVr (Output constraint) 7=1

n

^ ( a;7c * Urj) < urk, 7=1

V/cVr (Undesirable output constraint)

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n

/ . 7̂'fr “ V/c 7=1

Xjk > o, v y v / c

dk > 0 , VtV/c

(Variable re tu rn s - to - scale co n strain t)

(N onnegativity co n strain t)

(N onnegativity co n strain t)

The DEA model is similar to that o f Cham es et al. (1978), however to accommodate variations in input and output, additional constraints were introduced. Such that o f exogenous input and undesirable output:

Exogenous input. The impacts o f the exogenous inputs are reflected by the way o f reference units selection. The most representative model within this option is the one proposed by Banker and Morey (1986). In the Banker and M orey (1986) model, the reference set may include units that operate in the similar or more unfavorable environment compared w ith the assessed unit (Banker & Morey, 1986). Furthermore, it assumes a positive impact on the desirable output, such that the effect o f the exogenous variable is deducted from the objective function o f the primal model. However, according to Hua et al. (2007), such treatm ent is not as applicable when both inputs and outputs (both good and bad) are simultaneously considered in the objective function. Positive impact o f the exogenous cannot be assumed to the objective function since the impacts to the inputs and bad outputs cannot be affirmed to be positive. Instead, to be able to consider exogenous inputs, this requires that reference units utilize the same levels o f the transformed exogenous inputs as that o f the assessed DM U on average, which is depicted on the equality constraint:

n

* E t j ) = 6 « * E ^> VfcVi (6) 7=1

Undesirable output. Undesirable outputs can be treated as an input, such as the model developed by Korhonen and Luptacik (2004) and further examined by Yang and Pollitt (2007). The undesirable outputs are constrained that a DMU being evaluated should only benchm ark itself to units that are perform ing better than itself with the undesirable output concerned. That is, the benchmarked undesirable outputs o f DM Us in the reference set should have less undesirable output than that o f the DMU in evaluation.

n

» u r j ) < UTk, V k V r 0 7=1

The expected output from the model would be the relative efficiency o f each DMU (i.e., 6 k, V/c)

and excess inputs from DMU, that is, ( 9 k * x i k ) ~ Y I j = i ( ^ j k * X i j ) -

DEA-BASED PERFORMANCE MEASUREMENT MATHEMATICAL MODEL AND SOFTWARE LI, R.C., ET AL. 191

APPENDIX C: Model Formulation o f the DEA Resource Reallocation Model

Inverse to the DEA, the input allocation problem determines how much an input should be allocated given that outputs are increased and efficiency remains the same (Wei et al., 2000). Another type of the inverse DEA problem is the forecasting problem; which determines how much an output should increase given that inputs are increased and efficiency remains the same (Wei et al., 2000). With the forecast model, decision makers can now determine how additional input can affect output changes. This model is further enhanced to incorporate re-allocation of excess inputs to enhance output production of the entire system.

Reallocation model indices j = Decision Making Unit (DMU) (/' = 1, 2, 3 ... n) k = DMU in assessment {k = 1, 2, 3 ... n) i = Type of input (/ = 1 ,2 ,3 ... m) r = Type of output (r = 1,2, 3 ... m)

Decision variables

A.jk = Proportion of input and output benchmarked from DMU j for DMU k A.k = Increase in input i for DMU k

Crk = Change in desirable output r for DMU k Nrk = Change in undesirable output r for DMU k

Model parameters

0k = Proportion of input use in output production of DMU k E = Exogenous input of DMU j

X = Input i used by DMU j Yrj = Output r of DMU j

Urj = Undesirable output r of DMU j S. = Excess input i

Reallocation Model

n s

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Subject to:

n

X ( y * X i j ) ^ &k * ( x i k + A i k ) ,

7=1

V /cV i

n

* E i j ) = O k * E ik> ; = i

V /cV i

n

^ i^-jk * Yr j ) > Yr k + Cr k ,

7=1

V /cV i

n

^ \ ^ j k * U r j ) ^ U r k + N r k , 7=1

V /cV r

I '

M 3

II h-* V/c

n n

X X Aik = s ° k = l i = l

V i

Aj k ^ o, V/V/c

A i k ^ 0- ViV/c

Cr k > 0, VrV/c

N r k > 0, ViV/c

(Input constraint)

(Exogenous input constraint)

(Output constraint)

(Undesirable output constraint)

(Variable retu rn — to - scale constraint)

(Resource allocation limit constraint)

(Nonnegativity constraint)

(Nonnegativity constraint)

(Nonnegativity constraint)

(Nonnegativity constraint)

DEA-BASED PERFORMANCE MEASUREMENT MATHEMATICAL MODEL AND SOFTWARE LI, R.C., ET AL. 193

APPENDIX D: Software Development Verification - Data Estimation

Table D .l Data set for data estimation

DMU XI X2 El Y1 Y2 U1 1 163,523.00 26.00 49,196.00 5,561.00 105,321.00 63.53 2 338,671.00 30.00 78,533.00 18,106.00 314,682.00 90.47 3 281,655.00 51.00 176,381.00 16,498.00 542,349.00 108.23 4 400,993.00 78.00 189,397.00 90,810.00 847,872.00 228.79 5 363,116.00 69.00 192,235.00 52,279.00 158,704.00 69.87 6 541,658.00 114.00 194,091.00 66,139.00 1,438,746.00 223.87 7 508,141.00 61.00 228,535.00 35,295.00 839,597.00 166.96 8 338,804.00 74.00 238,691.00 33,188.00 540,821.00 142.08 9 511,467.00 84.00 267,385.00 65,391.00 1,562,274.00 192.23 10 393,815.00 68.00 277,402.00 41,197.00 978,117.00 152.81 11 509,682.00 96.00 330,609.00 47,032.00 930,437.00 236.04 12 527,457.00 92.00 332,609.00 56,064.00 1,345,185.00 236.52 13 601,594.00 127.00 356,504.00 69,536.00 1,164,801.00 156.48 14 528,799.00 96.00 365,844.00 37,467.00 1,348,588.00 259.86 15 394,158.00 77.00 389,894.00 57,727.00 1,100,779.00 137.08

For LP predictive model, additional verification was done where the results were compared to regression analysis results. Table D.l above contains data set used for LP estimation verification. Table D.2 shows a sample comparison of standard error results of predicting target X : by predictor data (X2, E, Yp Yy Y3). Comparison of standard error shows that the LP estimates resulted in lower standard error as compared to using regression analysis. Best predictor identified is the same between the two methods that is X , is best used to estimate for missing X f data.

Table D.2 Comparison o f standard error estimates

Predictor Target Standard Error

LP Estimate Regression£ X, 56,361.00 65,118.06 E x, 79,156.64 246,792.90 Y, X, 93,898.55 101,941.67 L X, 59,855.47 82,825.22_£_ 62,056.26 69,455.33

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To verify if the results are accurate and acceptable, the returned results o f the software are compared to that o f results from a hard-coded model ran in GAMS. Comparing software results to reference model ran in GAMS; both results are similar and comparable (see Table D.3).

Table D.3 Comparison o f results between GAMS and DEA software

System Predictor Target Slope Intercept

GAMS *2 x , 3521.68 154340.0

DEA X, X, 3521.68 154340.9

Software 2 1

DEA-BASED PERFORMANCE MEASUREMENT MATHEMATICAL MODEL AND SOFTWARE LI, R.C., ET AL. 195

APPENDIX E: Software Development Verification - Efficiency Results and Reallocation

Using the same data set; results are compared using GAMS and using the software developed. For the test data run (see Table E .l), the same DMUs are identified under the efficient frontier; both from GAMs and from the software developed. Resulting efficiency scores index are also the same. Table E.2 presents the sample run efficiency result using the test data.

Table E.l Data set for test run

DMU XI X2 El Y1 Y2 U1 1 67,126,924.17 371.06 324,356.00 106,879.70 11,355.03 1,186.00 2 45,874,108.45 249.00 231,717.00 96,195.81 10,170.13 1,116.00 3 106,201,374.50 490.88 555,272.00 219,551.56 21,986.98 2,055.00 4 206,524,724.17 496.00 1,445,209.00 409,966.00 9,828.00 4,151.00 5 62,964,402.70 262.21 445,510.00 123,555.00 9,663.52 2,325.00 6 244,920,515.58 313.00 557,297.00 232,901.70 17,054.98 2,652.00 7 16,485,119.10 66.00 60,378.00 17,136.00 1,402.00 232.00 8 102,579,154.32 774.00 630,161.00 261,366.68 26,680.83 1,433.00 9 247,710,131.71 905.00 2,468,417.00 1,008,935.00 58,466.00 10,762.00 10 731,442,498.98 760.05 389,478.00 818,226.29 48,316.02 2,659.00

Table E.2 Efficiency results o f test run

DMU DEA Software GAMS 1 0.7896 0.7896 5 0.7773 0.7773 6 0.9477 0.9477

Likewise, re-allocation decisions are similarly compared both from the output o f the DEA software developed and reference model run in GAMS. For the test data run; the same re-allocation decisions were derived from both output. Table E.3 presents the sample run allocation result using the test data.

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Table E.3 Reallocation results o f test run

Allocation DEA Software GAMS DMU4 x, 90921182.92 x , 9.092118E+7

DMU8 10.18 X2 10.18

Output DEA Software GAMS

Y, 34411.48 Y, 34410.53

DMU4 y2 4727.50 y2 4727.37

U, U,414.14 414.12

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