Performances Management When Modelling Internal Structure of a Production Process

Performance management is a central point for both public and private organizations. In the data envelopment analysis (DEA) method, performance management takes the form of measuring relative efficiency. Furthermore, considering each organization and or production process as a black box, inputs are transformed into outputs. In reality, production organizations or processes are composed of different parts that carry out different related activities. For this reason, modeling the internal structure of a production process like a system of interconnected parts makes it possible to measure its performance at the sub-process level. In this paper, we hypothesized a production process, made up of three interconnected parts. It is a new strategy to acquire relative efficiency consisting of building a block inside the system with at least two sub-processes. This step refers to a basic model of relational Network Data Envelopment Analysis (NDEA). Also, we used the additive decomposition formula to measure the efficiency of the whole process. We highlighted the differences in the measurement, between the direct application of the relational NDEA model and the measurement with the block approach model.We compared the cumulative empirical distribution functions of the efficiency scores of a sub-process with the decomposition formula multiplicative and our approach. In conclusion, the paper proposes, a new strategy to measure the relative performances of a production process model as a network system of three subprocesses, which combines the NDEA and the DEA. This allows us to reevaluate, the indications of policy at the individual sub-process level (block). Moreover, it is a versatile approach which allows aggregation of the sub-processes in blocks, according to the particular policy requirements, legislative technological constraints, etc.


Introduction
The measurement as the response at real problems solution is the core of the Operation Research (OR) among others (Hiller, et al., 2001). The Data Envelopment Analysis (DEA)  within the management science (MS) and operations research (OR) tradition, occupy an important place as a method for shaping production process and measuring different concepts of efficiency (Note 1)(i.e. technical efficiency, scale efficiency, scope efficiency and so on). The two aspects so far outlined, modelling and measurement,are two fundamental steps in the organizational performances management. Economists and/or operation researchers used mainly two different approaches to economically modeling the production process and measure its efficiency: 1) the econometric approach and 2) the mathematical approach. Inside the first approach, the preferences of the economists fall on the Stochastic Frontier Approach (SFA) (Aigner, et al., 1977); (Meeusen, et al., 1977), Correctly Ordinary Least Square (COLS), Modified Ordinary Least Square (MOLS) (Robinson, 2008) and Maximum Likelihood Estimation (MLE) (Greene, 1980). Inside the operations researcher group the most used approaches are the DEA  and the Free Disposable Hull (FDH) (Deprins, et al., 1984). The main difference between the two groups of scholars is that for those that uses DEA and FDH it is more easy to carry out multi-output production process, meanwhile in the case of the econometric approach it is possible consider error terms (although in the DEA approach much is doing done) (Olesen, et al., 2016). Inside the DEA approach, some author noted that the DEA approach have some weakness that can misleading the efficiency measurement[among others (Daraio, et al., 2008). In particular some of they noted that the DEA do not allow to see inside the production process, modeling it as a "black box" that transform inputs in the outputs (Note 2). Different authors, following this observation developed different approaches/models to modeling the internal structure of the DMU inside the DEA approach as well as proposed several formulas/approach to measure its efficiency in the case of basic model with two stages as well as in the case of more than two stages among others (Fa¨re, et al., 1995); (Fa¨re, et al., 1996a); (Fa¨re, et al., 1996b); (Kao, et al., 2008); (Liang, et al., 2006); (Liang, et al., 2008) with different extensions among others (Chen, et al., 2010 b); (Premachandra, et al., 2012), (Castelli, et al., 2004). Inside this literature the internal structure in the case of health care services it has been also considered [among others (Chilingerian, et al., 2004) as well as in others sectors i.e. (Kao, et al., 2008) apply NDEA in non-life insurance companies, (Chen, et al., 2004) in the bank branch, (Sexton, et al., 2003) at major league baseball and so on. The objective of this paper is to modeling the internal structure of a production process with three interdependent subprocesses. At this end, we take as a cue the relational Network Data Envelopment Analysis (NDEA) model proposed in (Pinto, 2016) where the author modeling and measure the relative efficiency of the hospital acute care production process. In particular, here, differently to (Pinto, 2016) are proposing a relational model with tree stages corresponding three different activity taken in the acute care in the hospital setting. In particular we are assuming that the hospital acute care provides the medical activity, the rehabilitative activity, and the assistance activity. The consideration of a third activity (rehabilitative) relatively at the two (medical and assistance) considered in (Pinto, 2016) need to reconsider the relationship among these three activities, as well as will produce a different functioning of the process of the hospital care of the acute. This allow us to build a more general NDEA model to apply in others sectors/activities. Once obtained the relational NDEA model we propose a novel strategy consisting in building a block of two subprocess to estimate the efficiency of the its parties, The paper is structured as follow: in the section 2 we modeling graphically the internal structure of a production process with three stages (three subprocess), and describe its functioning (subsection 2.1). In the section 3 we formulate our relational NDEA model (subsection 3.1) and the way to calculate the relative efficiency of its subprocess (subsection 3.2), in the section 4 we apply the relational NDEA model expanding the model in (Pinto, 2016). Finally in the section 5 we show discussion and conclusions.

Material Studied
To consider the internal structure of a production process/organization inside DEA context is relatively recent. The classification in (Castelli, et al., 2010) define the main research in this areas as shared flow, multilevel and network models, depending on the assumptions made. For shared flow models the subunits of the DMU have shared inputs and shared outputs. In the multilevel models DMU inputs (outputs) are not necessarily inputs (outputs) of its subunits. In network models the subunits have at least one output which is an input of another different subunits.  from measurement point of view classify nine types of models used to measure efficiency of network production process: 1) independent models, 2) system distance measure model, 3) process distance measure model, 4) factor distance measure model 5) slacks-based measure model,6) ratio-form system efficiency model, 7) ratio-form process efficiency model, 8) game theoretic model, 9) value-based model. The independent model treat each part process as independent DMUs and measure their efficiencies separately. The system distance measure model specify a model to find either the minimum input distance measure or maximum output distance measure for the system efficiency. In the system distance measure model every process is required to have the same distance parameter when measuring the system efficiency. An extension of this model is to allow each process to be associated with a different parameter, , which represents the efficiency of each k process, and the system efficiency is a weighted average of these individual process efficiencies. The factor distance measure allow every factor to have different parameters, and the objective function is to minimize the weighted average of these. Slacks-based measure model measure the system and process efficiencies of a network system. Ratio-form system efficiency model is those proposed by (Kao, 2009(a)) where for general network systems, which requires the same factor to have the same multiplier in the aggregation, no matter which process the factor corresponds to. In contrast to the system efficiency model, where system efficiency is the primary concern in searching for the most favorable multipliers, and a relationship between the system and process efficiencies is then derived, the primary concern of a process efficiency model is process efficiencies, with a driving force of their aggregation. The value-based model is concerned with maximizing either the aggregate output, or the profit, of a network system within the constraints. By structure point of view  classify the network system by: 1) basic two-stage structure, 2) general two-stage structure, 3) series structure, 4) parallel structure, 5) mixed structure. 6) dynamic structure, 7) hierarchical structure. In this paper we propose a network system of three subprocesses differently connected, so following  our model can fall in to mixed structure and by measurement point of view in to ratio form process efficiency. Following (Castelli, et al., 2010) our network process can be classified as a shared variables models. The distinctive point of the work is that to measure the relative efficiency of a mixed structure we do not apply directly one of the measurement method classify above in  but we suggest as first thing to individuated blocks inside it in a way to reduce our model at more simple series or parallel structure and then adopt the some type of measurement, i.e. Description of modeling a pro e Figure 1) an vided services te 3).
in the function reported in ( t for acute pat ation activity tation services of hospitaliza Note 5). All 3 e process (gre e 1) or the me he assistance es ( to the exampl the producti ivity (develop eloped with su yellow path, Y ing other med f hospital bed lved. In fact, -health care s ires technicalhabilitation pe r the assistanc y staff becom between the 3 t included in th whole proces e days of hosp other two sub tment process e and the reha modeling the p pe of Decision 07); (Cook, et is not consid (Castelli, et al fication in (Ca (see Figure 1) pose a novel s . 15, No. 7; ges of a productio on will be show o, 2018). l rently connect (Castelli, et (i.e In the fo efficiency displayed

The N
In this se efficiency of this, ac correspon above[ (K Where:     (2) , Y 4 (2) , Y 5 (2) = outputs II sub-process X 5 = resources III sub-process X 2 (3) , X 3 (3) ,= shared resources between I and III sub-process Y 1 (13) , Y 2 (13) = shared outputs resources between I and III sub-process ( ) With this formulation the same inputs and outputs will receive the same weights [ (Kao, 2009 (a))]. The operation of each process is described with the constraints in the model 1. For example the constraints sub 1 write down the operations of the first subprocesses. The constraint sub 2 consider the operations of the second subprocesses, and so on. The relational nature of the model 1 is that the outputs of the first subprocesses ( , ) receive the same weights ( , ) of the relational variables that connect it with the second subprocesses ( ( ) , ( ) ) and third subprocess ( ( ) , ( ) ). So, as stated in others parts in the paper the same variables (in this case ) receive the same weights. The proportion assigned to the variables of each subprocesses is differently defined (Kao, 2017). Here we assigned a fixed proportion (i.e. , for the variables , ) without any specific intention. This latter step (the assignment of the proportions) is of crucial interest for the purposes of policy indications.

The Efficiency Measurement: Two Approaches
According to the relational approach (Kao, 2009a) once solved the model 1 above the efficiency of the system (E sys ) is given by : = * * * * * * * * = * + * + * (formula 1) While the relative efficiency of the subprocesses will be produced with its constraints as follows:: Variables with an asterisk present the optimal value of the weights, once model 1 has been solved. As evident from formulas 2, 3, 4 the weights are the same for the same variable. Studies have proposed that, in the case of unstructured systems with multiple stages, remodeling the system as a structure in parallel with a series of subsystems using a dummy process is required (Kao, 2009(a)). In our opinion, the use of dummy processes remains an excellent solution for such systems. However, to estimate the relative efficiency of such systems and its sub-processes, we propose to build blocks by combining at least two sub-processes together without any dummy process. So, for the process represented in Figure 1, we build a block (Block 1) using subprocess 1 and 2 and we get a new system (see Figure 3). The initial system will then be transformed as follows: Block 2 − ( + + + + ) ≤ 0 To the model 3 we will obtain the following optimal values: * , * , * , * , * , * . So, the efficiency of the third subprocess will be: The efficiency of the whole subprocess can be calculate using the weighted additive efficiency decomposition (Chen, et al., 2009b) as follow: Where ≥ 0, ≥ 0 with + = 1, represent the importance of sub-process 3 and block 1 in measuring efficiency for the entire system. With this strategy, we treat the variables of subprocess 3 and those of the block as non-relational variables. In fact, as presented above, we will have different weights for each subprocess input variable. Thus, the variables X2 (of block 1) and X6 (of subprocess 3) represent the same resource (ie read in this example). The variable associated with it is assigned to block 1 and subprocess 3 as independent variables each with an own weight and in a given proportion. All data will be covered in the section dedicated to the application. In our opinion, this measurement strategy is very useful in technological, organizational and / or legislative constraints.

The Application to the Hospitals
The efficiency measurement in the health care/hospitals setting when we consider its internal structure is relatively recent.[i.e. (Chilingerian, et al., 2004); (Kawaguchi, et al., 2014); (Pinto, 2016)]. Chilingerian et al. 2004 consider a two stage process in measuring the physicians care and apply two separate DEA. The first stage has as inputs registered nurses, medical supplies, and capital and fixed costs. These inputs generate the outputs as patient days, quality of treatment, drugs dispensed, among others. These first stage outputs are the inputs of the second stage to generate as outputs research grants, quality patients, and quantity of individual trained, by speciality. Kawaguchi et al 2014 test the policy effects of the health reform in Japan on the hospital efficiency considering this latter as organizations with two internal heterogenous organizations. In particular the authors apply the dynamic-network data envelopment analysis. Pinto 2016 consider a two stage process in the hospitals acute care applying the network DEA approach to estimates the relative efficiency of it. In Pinto,2016 the second stage has an exogenous inputs conferring the non linearity to the model.In this paper we proposed in the subsection 3.2,according our opinion, a new approach in the case of a three stages process. In this section we apply it to the hospital acute care services adding a third process at the process in (Pinto, 2016). The variables used here are the same in (Pinto, 2016) (see Table 1) The role of the some variables inside the relational model will depend to how the production process will be modeled. Here, the variables of the relational NDEA model in the case of hospitals acute care production process with three stages as in the Figure 1 above will be: X1, X2, X3, X4, X5= system resources: physicians, nurses, beds, rehabilitative staff, medical-technical-staff.
Y 1 , Y 2 = outputs of the I sub-process :surgical interventions, days on hospitals.
Y 1 (12) ,Y 2 (12) = relational resources between the I and the II sub-process:surgical interventions, days on hospitals.
Y 1 (13) , Y 2 (13) = relational resources between the I and the III sub-process:surgical interventions, days on hospitals.
X4= exogenous resources of the II sub-process:nurses.
As noted, differently to (Pinto, 2016) here we added a third subprocesses to modeling the rehabilitative activity using as dedicated variable the rehabilitative staff. (see Table 1). This latter variable characterizing the third subprocesses in the model (lacking in (Pinto, 2016)). Solving the model 1 and applying the formulas 1,2,3,4 using data on these variables we will have the following optimal weights (additional file1 .xlsx ) and the following relative efficiencies (see Table 2 below for its descriptive statistics).  Table 5 net.org igure 5. Empir curve is the ndependent st urve is the cum 3 but calculate 4). The two c e variable in t receives the ss 3 becomes r when it is an oaches, relatio (Smith, et al.,    y inputs, whil case we posed cretionary ass Table 5 abov n the Figure 6 s and histogra ing the produc ces. Within an cy similar to tion process ). In these tw veral connecte nnected parts this direction tructure and th measure its re hree interconn 15, No. 7; le the column d the variable sumption, in ve and in the 6 below.  Figure 1. For demonstration purposes, we applied this model to the case of the hospital acute care production process by exploiting the conceptualization (Pinto, 2016). The process in (Pinto, 2016) was enriched by connecting the rehabilitation activity along with medical care and assistance with activities / subprocesses, in a treatment process for hospital acute care. We later developed a relational NDEA model to measure its relative efficiency(model 1) and calculate its subprocess efficiency using multiplicative decomposition formula (Kao, 2009(a)). As innovative way to measure its relative efficiency we proposed as an alternative solution for construction of an internal block of two subprocesses. In this latter case we not apply the relational NDEA approach directly to the whole process to estimates its relative efficiency. Once the relative efficiency of the block 1 in Figure 3 was calculated with a relational NDEA model, we applied the multiplicative decomposition formula for the block to calculate the relative efficiency of its subprocesses. To calculate the efficiency of the entire process, we have adopted the additive decomposition formula with the efficiency of the block and the efficiency of the remaining sub-process (Chen, et al., 2009b). In other words, our solution, one obtained an internal block, combines relational NDEA with DEA to measure the relative efficiency of network systems with more than two subprocesses. However, once the block has been constructed, a specific weight can be assigned to it, reflecting its importance within the measurement. Our approach, according our opinion, is useful for specific policy indications. Obviously, blocking can be differently constructed based on the requirements of the policy objectives, and under the constraint of a network model of two sub-processes resulting from the basic ones in series and in parallel (Kao, 2009(a)). The relational NDEA 1 proposed model, is characterized by intermediate flows, shared variables and exogenous variables (Castelli, et al., 2010) and the same variable receive the same weight-A variant not considered here, is that which uses exogenous variables to give the model a multi-stage nature (Kao, 2017). The difference between the direct application to the relational NDEA model (model 1) and our strategy, is that the former requires the use of the same weights for the same variables regardless of whether they belong to one or the other subprocesses (Kao, 2009a). In our approach to the same variable they can be assigned to different weight depending on whether it belongs to the residual block or sub-process, and avoid to insert a dummy process (Kao, 2009a). In this way, generating different optimal values in the weights and therefore different policy considerations (Smith, et al., 2005). We would like to conclude, that the study represents an innovative work for introducing the block strategy in the measurement of relative efficiency of systems with more than two subprocesses. Moreover, it is new in view of the measurement of relative hospital efficiency with NDEA models. In addition, it can be applied for different technological, legislative, etc, constraints where two subprocesses in a network system are joint.