Data envelopment analysis (DEA), as a linear programming technique, serves several benefits in comparison with the other techniques for performance measurement. DEA requires no preference articulation on priority of inputs/ outputs, and can estimate the production function as a non-parametric approach. DEA determines the efficient DMUs among others, and suggests the projection of efficiency for the inefficient DMUs. Moreover, Return to Scale (RTS) issues can be determined using an especial variants of DEA models. There are several successful applications of DEA models in the healthcare systems (Chowdhury and Zelenyuk, 2015). Thanassoulisa, Portelab and Graveneyc (2016) identify the usefulness of the length of stay of each episode and explores the differences between hospitals and between the care teams within the hospitals.

Thus, in this study DEA models are developed in order to measure the efficiency scores, RTS and Damage to Scale (DTS) issues of Iranian public health organizations in Tehran. The main objectives of this research are:

The development of DEA models to measure the efficiency scores of public health organization with multiple inputs/outputs.

The structure of this paper is as follows. In Section 2, literature of past works is reviewed. In Section 3 the proposed models are developed. In Section 3, new models for assessing Tehran’s health systems are presented, decision-making departments, factors would be defined. the solution algorithm that is applied for finding the application of scale in radial and non-radial position is presented. The justification of the application of such models will be discussed in Section 3. In Section 4, the case study and data analysis are presented. The results and findings are presented in Section 5. On the other hand, in Section 5, the Weak Natural Disposability (WND) and weak managerial disposability (WMD) models’ results are presented in both radial and non-radial status. The models are used for the evaluation of public health organization of Medicinal Universities using desirable and undesirable outputs. And also, the economics of scale of public health organization has been defined using radial, non-radial models and the obtained results were compared using the collected data. Finally, the paper will be summarized in Section 6 with concluding remarks and future research directions.

Although hospital efficiency analysis has attracted a large number of studies (e.g., see Goldstein et. al., 2002; Hollingsworth, 2003; O’Neill et al., 2008; Garcia- Lacalle and Martin, 2010; Rosko and Mutter, 2011; Mitropoulosa et al., 2015 and Thanassoulisa et al., 2016), there are less research whose focal point is on analyzing the determinants of hospital efficiency (e.g., Grosskopf et al., 2004; Lee et al., 2008; Blank and Valdmanis, 2010; Tsekouras et al., 2010; Cristian and Fannin, 2013; Ding, 2014). DEA is a method which evaluates service providers and defines a rate as the ratio of perceived performance to its closed potential. Or it shows the amount of the desirable efficiency of resources. In a juxtapose with the above results, Gok and Sezen (2013) showed that efficiency of small hospitals is relatively more, and satisfaction is higher compared to medium and large hospitals.

The theoretical improvement of the DEA approach commences with the influencing study of Charnes, Cooper and Rhodes (1979) towards the efficiency evaluationof DMUs. Mentionedinthefirstapplicationof DEAinthehealthcaresector was the works by Nunamaker (1983). Since then, DEA was vastly used for technical efficiency of public health organization in US and other parts of the world. Sherman (1984) was the first scholar who applied DEA for technical efficiency evaluation of public health organization in the US. Sherman (1984) analyzed factors like budget, day beds, number of full-time physicians, +65 years old patients, -65 years old patients, instructed nurses and interns as inputs and outputs. Butler and Li (2005) evaluated the benefits of RTS in DEA for rural public health organization of Michigan State. They used the BCC model and considered all the costs except salaries, the number of hospital beds, number of services and total number of employees as inputs, and the number of days of care for a patient, number of surgery operations, number of emergency rooms and number of hospitalized patients as outputs. Al-Shammari (1999) investigated the efficiency of 15 public health organization. He considered the number of beds’ active days, number of physicians, and number of personnel as input and number of hospitalized days, number of emergency surgeries, and number of general surgeries as output.

Tsai and Mar (2002) surveyed the scale efficiency in five departments of public health organizations in Britain using the variable RTS assumption that considered sum of operational costs, number of hospitalized, and emergency patients as main criteria Nayar and Ozcan (2008) investigated efficiency of Virginia public health organization using DEA. The results demonstrated that in most cases, working on quality resulted in an increase in hospital costs and hence efficiency was decreased. Joses et al., (2008) measured the efficiency of 54 public health organization in Kenya - although they found that there was a big gap between scientific and real- life evaluation of health care departments in Africa’s central Sahara. The obtained results showed that %26 of public health organization were inefficient. The projection of inefficient DMUs towards efficient frontier was calculated.

Kawaguchi, Tone and Tsutsui (2014) applied DEA for evaluating the efficiency of governmental public health organization. They applied both static and dynamic DEA models and categorized public health organization into two namely: managerial and operational departments. The obtained results demonstrated that there was a slight difference between static and dynamic models.

There are several approaches and perspective when measuring efficiency scores using DEA models. In this section, some commonly used approaches are briefly presented.

Assumeas / input vector, as desirable output vector, and as undesirable output vector. Therefore, the possibility production set (PPS) in a weak consumption environment would be defined as (1).

(1)

Where, is assessed under weak disposability approach if it seeks to increase desirable outputs and decrease inputs while the undesirable outputs are assumed to be constant (Sueyoshi, and Goto, 2011a; Sueyoshi and Goto, 2012a).

In a strong disposability approach, attempts to increase both desirable and undesirable outputs while the inputs are assumed to be decreased (Sueyoshi, and Goto, 2011b; Sueyoshi and Goto, 2012b). In this way, the PPS is defined as (2).

(2)

The natural and managerial consumption approach was presented first in a study by Sueyoshi and Goto (2012c) and later in many other researches (Cooper, Seiford and Zhu, 2011; Sueyoshi and Goto,2012d; Sueyoshi and Goto,2012e; Sueyoshi and Goto,2012f ). Under such conditions, the influence of the input vector on the increase and decrease of inputs is discussed. In the natural disposability approach, with an increase in input vectors, we attempt to decrease undesirable output vectors and increase desirable vectors. In this way, the PPS is defined as (3).

(3)

But in some cases, a DMU tries to develop available resources assuming an increase in inputs leads to an increase in desirable outputs and a decrease in undesirable outputs. In this way, the preliminary hypotheses of classic CCR and BCC models are not supported. In this way, the PPS would be shown as (4).

(4)

In this study, based on the requirement of case study, a new condition for PPS is proposed in which a combination of natural and managerial approaches under weak disposability condition is used. Assume that DMU is not responsible for any changes made in undesirable outputs. Moreover, DMU tries to increase desirable outputs through increasing inputs, therefore the two types of PPS would be shown as follows.

- A PPS under natural consumption approach considering weak disposability can be shown as (5).

(5)

And a PPS under managerial consumption approach considering weak disposability can be shown as (6).

(6)

Thus, in this study, two new types of PPS are used to develop the radial and non- radial DEA models. The developed models under such circumstances are used to determine the return and DTSs at some public health organization in Tehran.

In this subsection, the proposed models of this study, under several assumptions, are developed. All models are proposed in both radial and non-radial situations. In radial models, inputs and outputs are changing simultaneously but in non-radial models, one of the inputs or outputs would change per request and the changes are separate. In order to making a better sense for readers, Models are proposed in two main classes known as the WND models, and weak managerial disposability models. In each classes several models are discussed. On the other hand, in each classes radial and non-radial models are proposed, and for each case a model is proposed to calculate for efficiency evaluation, and a procedure is proposed to determine the scale efficiency and RTS situation.

Table 1 presents indices, parameters, decision variables, and sets which are used in this section.

Table 1

To calculate Unified Efficiency (UE) under the WNDNR² approach, Models (23)-(31) are proposed.

(20)

(21)

(22)

(23)

(24)

Again, the unified Efficiency (θ*) under WNDNR model can be calculated same as the WND Model.

The duality of the WNDNR Model (23)-(31) can be written as multiplier form Model (33)-(38).

(25)

(26)

(27)

(28)

(29)

(30)

It is notable that due to the duality theorem in linear programming, the optimum objective values of both Models (23)-(31) and (33)-(38) are equal.

The range adjusted measure (RAM) of efficiency model which was first proposed by Cooper, Park and Pastor (2000), is used here in order to adjust the range of inputs and outputs.

Scale Efficiency (SE) score in both radial and non-radial states are the same, but the RTS sign would be calculated separately based on the following algorithm.

Now, in order to define economic scale of inefficient units, first an image of inefficiency was extracted and the rate of inefficiency obtained using (43).

(31)

In order to calculate RTS for radial models, first the two maximization and minimization models (44)-(59) should be solved.

(32)

(33)

(34)

(35)

(36)

(37)

(38)

(39)

(40)

(41)

(42)

(43)

Using the calculated upper and lower limit of μ* per each DMU, the RTS of DMUk is determined as follows:if there is increasing return to scale, if there is constant return to scale, and if there is decreasing return to scale.

In order to calculate RTS for non-radial models, first the two maximization and minimization models (60)-(75) should be solved.

Min (Max)

(45)

(46)

(47)

(48)

(49)

(50)

(51)

(52)

(53)

(54)

(55)

Using the calculated upper and lower limit of π* per each DMU, the RTS of DMUk is determined as follows: if there is increasing return to scale, if there is constant return to scale, and if there is decreasing return to scale.

The Scale Efficiency (SE) score in both radial and non-radial states are the same, but RTS sign would be calculated separately based on the following algorithm. If DMUk is efficient under the WMD assumption, the efficiency score would be calculated using (42). Now, in order to define economic scale of inefficient units, first an image of inefficiency was extracted and the rate of inefficiency obtained using (43). In order to calculate RTS for radial models, first the two maximization and minimization models (109)-(124) should be solved.

(80)

(81)

(82)

(83)

(84)

(85)

(86)

(87)

(88)

(89)

(90)

(91)

Similar to the natural disposability conditions, using the calculated upper and lower limit of π* per each DMU, the RTS of DMUk is determined as follows: if there is increasing return to scale, if there is constant return to scale, and ifthere is decreasing return to scale.

In order to calculate RTS for non-radial models, first the two maximization and minimization models (125)-(140) should be solved.

(92)

(93)

(94)

(95)

(96)

(97)

(98)

(99)

(100)

(101)

(102)

(103)

(104)

(105)

(106)

(107)

Again, using the calculated upper and lower limit of π* per each DMU, the RTS of

DMUk is determined as follows: if there is increasing return to scale, if there is constant return to scale, and if there is decreasing return to scale.

The proposed models are applied in a real case study involving 33 public health organization in Tehran, Iran. A hospital is assumed as a DMU which uses inputs in order to produce outputs. Both WND and WMD situations are assumed for DMUs and all models are run and the results are discussed in this section. On the other hand, the efficiency score, scale efficiency, and RTS are calculated for both WND and WMD situations, using both radial and non-radial models. The schematic vies of a DMU is shown in Figure 1.

Figure 1

It is notable that due to anonymity of results of this research the name of public health organization are not reported.

RESULTS OF WND MODELS

In this part, the results of WND models in radial and non-radial cases are presented. Table 2 presents the result of Radial WND models in radial case. The results include efficiency scores, RTS of each DMU, slack variables of each input, and slack variable of each output.

Table 2

Based on Table 2, 19 public health organization are efficient, and 14 public health organization are inefficient. Seventeen out of 19 efficient public health organization have constant RTS while 2 of efficient public health organization have decreasing RTS. The slack variables for all 19 efficient public health organization are equal to zero, and this means that these public health organization are strong efficient. It is notable that slack variables can help an inefficient hospital to find its projection towards an efficient frontier. On the other hand, the slack variables represent extra input used, and shortage of production of outputs. For instance, consider DMU2 which is inefficient among all 33 DMUs. The efficiency score of DMU2 is equal to

0.72. DMU2 can improve its efficiency score using projection. If DMU2 reduces 1.94 of its doctors, 71.30 of its active beds, 7.28 of its hospital area, and 0.15 of its budget the inputs will be set. Moreover, DMU2 should improve the resident time 0.98, the death rate 12.57 in order to set its outputs. Under such changes DMU2 will be an efficient DMU among the other DMUs. The same analysis can be done based on Table 2 for all other inefficient DMUs.

In a similar way, Table 3 presents the result of WND models in non-radial case. Again, the results include efficiency scores, RTS of each DMU, slack variables of each input, and slack variable of each output.

Table 3

Similar analysis can be made on contents of Table 3. Twenty five public health organization are efficient, and 8 public health organization are inefficient. Eighteen out of 25 efficient public health organization have constant RTS, 2 of them have increasing RTS and 5 of efficient public health organization have decreasing RTS. Again the slack variables of all 25 efficient public health organization are equal to zero, and this means that public health organization are strong efficient. Slack variables can help an inefficient hospital to find its projection towards efficient frontier.

As it is clear from the contents of Table 2 and Table 3, the number of efficient DMUs in non-radial models are more than radial models. In order to make a better sense of the results under WND conditions, the efficiency scores of all DMUs for both radial and non-radial models are plotted in Figure 2.

Figure 2

It could be concluded that radial models measure efficiency in a more precise way than non-radial models and the discrimination power of DEA approaches is decreased when using non-radial models as DMU can change its inputs and outputs independently, so it has more opportunity to close towards efficient frontier.

In this paper, several models were proposed to measure the efficiency scores of systems in the presence of undesirable outputs. In this regards, two new assumptions on production possibility sets (PPS) were proposed. In the first assumption, the WND was proposed. Under the WND situation, a DMU’s interest is to reduce its inputs in order to increase its good outputs while producing a constant value of bad outputs. In the second assumption, the WMD was proposed. Under the WMD situation, a DMU’s interest is to increase its inputs in order to increase its good outputs while producing a constant value of bad outputs. On the other hand, the WND perspective seeks to meet the environmental issues while improving efficiency and the WMD perspective seeks to develop the production rate in order to improve the efficiency. Based on these two PPSs and using radial and non-radial models, several models based on DEA were developed in order to measure the efficiency scores of DMUs, to determine the return to scale, and to determine the DTS. The proposed models and procedures were applied in 33 public health organization in Iran and the results were discussed for the case study. The main contribution of this study were as follows:

Introducing two new PPSs as WND and WMD;

As there are several qualitative criteria in performance assessment, development of the procedure of this paper in presence of qualitative and uncertain criteria can be interesting. Other applications such as baking, energy sector, production and services can be handled using proposed models of this study.