Limited resources in Data Envelopment Analysis with imprecise and integer data

In this paper a new model which is in the framework of centralized models in data envelopment analysis (DEA) is proposed. The existing models which have been presented for the human resource rightsizing organized in two stage and the models were suggested especially for integer data. In this paper we improve the existing models in one stage only and consider both imprecise data and integer data in our suggested model. With the numerical results, the limited resources are reallocated among different α-level sets along with the consideration of integer data in the proposed model.


Introduction
One of the well known methods for evaluating the relative efficiency of decision making units is data envelopment analysis (DEA) which has been proposed by Charnes et al. [1].DEA model originally is a non-linear fractional mathematical programming model, known as the Charnes, Cooper and Rhodes (CCR) model.The objective function in this model is considered to reach the best set of weights for the single ratio of the weighted outputs to the weighted inputs for a particular decision making unit (DMU) denoted by DMUo.Evaluating the relative efficiency is not the only usage of DEA; the another attitudes of DEA are target setting and resource allocation.For instance, there are some situations in which all the units belong to the same organization and there is a centralized decision maker (DM) who supervises these units and he/she also desire to set a target for future planning of his/her organization.Therefore, according to the relative efficiency of each decision making units the DM will reallocate the resources for inputs or outputs.Many models and methods have been proposed for target setting and resource allocation which include the decision maker's preference in target setting process, for example, Athanassopoulos ( [2], [3], [4]), Golany [5], Korhonen and Syrjänen [6], Lozano and Villa [7], Thanassoulis and Dyson [8] and Malekmohammadi et al. ([9], [10]).In the applications of DEA (such as bank branches, hospitals, university departments, secondary schools, police stations, etc.), there is a centralized decision maker who supervises and oversees these units.Lozano et al. [11] considered a situation in which the centralized DM decides to decrease total input consumption or increase total output production simultaneously.In their models instead of running n mathematical models just one model is needed to be run.That was the main differences of their approach to the previously introduced models in DEA.More details can be found in (Malekmohammadi et al.,[10]).Lozano and Gutiérrez [12] considered models for integer data with the wide application in allocating a number of glass containers among the different municipalities within a region.M.-M.Yu et al. [13], contributed more comprehensive approach in centralized DEA by applying their models to airport human resource reallocation under different scenario.Their attitude contains in two fold, firstly, provides a systematic and centralized perspective (i.e., reallocation'perspective) of resource downsizing in Taiwanese airports.Secondly, by using the three strategies that consider obstacles under different scenarios in human resource adjustment which shows the flexibility of their model over the previous models in centralized DEA.Considering the importance of imprecise data such as fuzzy, ordinal and interval data in organizations the DEA models with exact data have been extended to imprecise data.Cooper et al. [14] introduced applications of DEA whose data was imprecise.In imprecise data envelopment analysis (IDEA), the data can be ordinal, interval or fuzzy, which results in a non-linear DEA model.Despotis and Smirlis [15] also studied the problem of IDEA and developed an alternative approach to deal with imprecise data in DEA.They converted a nonlinear DEA model to linear programming equivalent by applying transformations only to the variables, resulting in efficiency scores which were intervals.According to their approach, Wang et al. [16] developed a new pair of interval DEA models that resulted in the best lower bound efficiency and the best upper bound efficiency of each DMU.They used a fixed and uniformed production frontier to determine the efficiencies of decision-making units (DMUs) with interval input and output data.M.-M.Yu et al. [13] proposed their models in two stage and the contributed models was suitable especially for exact data which is mentioned mostly in a form of integer data.In this paper a new model is presented for limited resources in total input consumption and total output production especially for integer data which is inspired from the conventional centralized DEA models.Considering the importance of imprecise data in organizations, we defined our model for imprecise data types such as interval and fuzzy data along with dealing integer data.This paper proceeds as follows.In section 2, centralized data envelopment analysis model is presented.In section 3, we introduce the new model with imprecise and integer data.A numerical example and conclusions are provided in sections 4 and 5, respectively.

Data Envelopment Analysis and Resource Allocation
In this section, the CCR envelopment model (the first DEA model) proposed by Charnes et al. in 1978 is introduced.Suppose we have a set of j = 1, ..., n decision making units (DMUs) and each unit uses input X ∈ R m + quantities to produce output quantities Y ∈ R s + .We consider the index sets of inputs,I = 1, ..., m and outputs, O = 1, ...s.Also o(o ∈ 1, ..., n) is the DMU under assessment (usually denoted by DMU o ).For evaluating the relative efficiency of DMU o , we need to solve the following linear mathematical programming problem: DMU o is CCR efficient if θ * = 1 and the slacks s + * k , s − * i in every optimal solution are zero.The target input and output for DMU o can be obtained as follows: k , ∀k ∈ O Data envelopment analysis can also be used for target setting and resource allocation.Thereafter along with evaluating the efficiency, the acceptance of the projected DMU into the efficient frontier is also considerable for researchers.

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Various type of models and methods for target setting and resource allocation can be found in Athanassopoulos ([2], [3], [4]), Golany [5], Korhonen and Syrjänen [6], Lozano and Villa [7], Thanassoulis and Dyson [8]and Malekmohammadi et al. [9], [10] .Lozano and Villa [7] established model (2.2) for resource allocation.Their main goals were firstly, reducing total inputs consumption and total outputs production were guaranteed not to decrease.Secondly, instead of projecting each DMU separately, all of them will usually project to their most productive scale size (MPSS) position simultaneously.The proposed model (2.2) for centralized resource allocation consist of two phases.In the first phase, the decrease along all input dimensions is considered while, in the second phase, they focus to the reduction of any input and/or expansion of any output.Let j, r = 1, 2, ..., n, be indexes for DMUs; i = 1, 2; ..., m, be index for inputs; k = 1, 2, ..., p, be index for outputs; x i j , amount of input i consumed by DMU j ; y k j , quantity of output k produced by DMU j , θ , radial contraction of total input vector; s i , slack along the input dimension i; t k , additional increase along the output dimension k; (λ 1r , λ 2r ,...,λ nr ) vector is used for projecting DMUr.The phase I model (2.2) is: Once the phase II model is solved, each DMU r uses the corresponding vector (λ * 1r , λ * 2r ,...,λ * nr ) for the operating point at which it shall be gained.The inputs and outputs of each such point can be computed as, More comprehensive attitudes about centralized resource allocation can be found in Malekmohammadi et al., [10].
International Scientific Publications and Consulting Services 3 The New Centralized Model with Fuzzy and Integer Data M.-M.Yu et al. [13] contributed new model in two stage which was suitable for human resource rightsizing with the wide application in Taiwan's Airports.In this section we introduce new model which can be defined mostly for integer and fuzzy data.In this new model we also considered a situation when the decision maker encounters limited resources in total input consumption or total output production as follows: Let j, r = 1, ..., n be the indices for decision making units (DMUs) while each unit uses input quantities X ∈ R m + to deliver output quantities Y ∈ R s + .We can also consider the indices sets of integer inputs, h = 1, ..., m h , non-integer inputs, i = 1, ..., p i and outputs ,k = 1, ..., s.The vector (λ 1r , λ 2r , ..., λ nr ) such that ∑ λ jr = 1, is imposed for convex combination between inputs or outputs for n DMUs.In order to define Model (3.3) for fuzzy data, we find that the method which is introduced by Wang et al. [16] is more considerable.These types of data (fuzzy data) should be transformed into interval data by using the α-level sets (Zimmermann [17]).Therefore by transforming fuzzy data into interval data, the new model will be a kind of interval DEA models (more information about interval DEA models can be found in [16]).Let the inputs xi j and outputs ỹk j ( also for ỹkr ) be fuzzy data with membership functions µ xi j and µ ỹk j , respectively, and S( xi j ), S( ỹk j ) be the support of xi j and ỹk j , respectively.Then the α-level sets of and xi j and ỹk j , can be defined as, Where 0 ≤ α ≤ 1 .By setting different levels of confidence, namely 1 − α, fuzzy data are accordingly transformed into different α-level sets , which are all intervals.The widest input and output intervals will be( ] and x L i j , x U i j , y L k j , y U k j are the lower and upper bounds of fuzzy data xi j and outputs ỹk j , respectively.The production frontier will obviously be determined by interval data ] and ] (i = 1, ..., p i ; j = 1, ..., n; k = 1, ..., s).Any α-level sets input and output data ] should be measured using the identical production frontier.So, the interval DEA models for fuzzy, interval, exact and integer input and output data will be as follows: International Scientific Publications and Consulting Services Where G i , G ′ k indicate the bounds for total consumption of non-integer input x L i j and total production of output y U kr , respectively.(θ NU i ) α and (Z U k ) α are, respectively, the best contraction rate of non-integer input i and the best expansion rate of output k under given α-level sets.Where the interval form of the contraction rate of non-integer input i and the expansion rate of output k under different alpha level set is denoted by . Note that we also use one production frontier for every α-level rather than different production frontiers for different α-levels.Note that x L h j and G I h indicate the lower bound of integer input and the bounds for total consumption of integer input.Also, θ IU h is considered as the best contraction rate of integer input h.h + k , v − i are the user-specified constants reflecting the decision makers' preferences over the improvement of input/output components.
Proposition 3.1.For any DMU j and DMU r under given α-level set the points

., m h from model (3.3) indicate the best lowest total input consumption for non-integer input i and integer input h and the highest total output production for output k, respectively.
Proof.Let us assume that the proposition is false and we will always arrive at a contradiction.If ((x * 11 ) α , (x * 22 ) α , ..., (x * p i n ) α ) is not the smallest total input consumption then, there exist xL i j and ( θU i ) α (i = 1, ..., p i ) and ) α and at least for one input i ′ the inequality is strict.Let us assume that it is for input i ′ for which contradiction, because by model (3.3) we will get another minimum feasible solution forθ U i , i = 1, ..., m.The same proof can be done for (y * kr ) α and (x * h j ).

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It should be mentioned that the exact data can be considered as interval data which the lower and upper bound of them are equal.For more emphasis, model (3.3) is also suitable for interval data.The same model can be done for outputs.

Numerical Results
The application of model (3.3) will be illustrated in this section.Table 1 presents the data set which can be found in (Wang et al. [16]), but the data have been changed a little to be more suitable for models.There are eight dependent manufacturing enterprises (DMUs) with two inputs and one output.Each manufacturing enterprise manufactures the same type of product.All of the manufacturing enterprises are under the control of a centralized decision maker (DM) who supervises them.The DM prefers to reduce total input consumption and increase total output production.The output, gross output values (GOV) have been shown with triangular fuzzy numbers.The inputs include capital (C) and the number of employees (NOE), whose data can be considered as exact and integer data respectively(the interval and ordinal data have been used in Malekmohammadi et al., [10]).It in mentionable that in our previous approach (Malekmohammadi et al., [10]) we did not consider fuzzy and integer data in the numerical result.The data are presented in Table 1.We have to mention that we have changed the purchase cost (PC) to capital (C) in Wang et al.'s [16] data set.This was done since it is not logical to reallocate these two data item (PC) among the units.Since the GOV index for DMU2, DMU4 and DMU6 is given in the form of triangular fuzzy number, i.e.GOV j = (GOV L j , GOV M j , GOV U j )( j = 2, 4, 6), their membership functions can be expressed as, where GOV L j , GOV M j and GOV U j are the lower bound, most likely and upper bound values of GOV j , respectively.For a given α-level, the corresponding α-level sets are given by, For considering exact data, they can be viewed as a special case of interval data with the lower and upper bounds being equal.We have chosen h + k = v − i = 1 for i = 1, ..., p i and k = 1, ..., s and also M = 1.Table 2 shows the results obtained from model (3.3) (solved by LINGO, a powerful software package).We have chosen this model for reaching the targets, since with this model we can define the optimum target (Proposition 1).In Table 2, we have shown the optimum total input consumption and output production for the related α-level set and also we have compared them with the existing total inputs and outputs.Each column of Table 2 shows the target of the input and output for which the DMUs should aim.According to the results, the decision maker along with his/her priority is able to decrease total inputs consumption such as capital and number of employees and increase total output production at the same time.The main difference between model (3.3) and conventional DEA models is the consideration of integer data (for example, number of employees)in the model.

Conclusion
In this paper, we suggested a new centralized model which is more suitable for limited resources with fuzzy and integer data.The main difference between our new model and the previously introduced models in DEA is the human resource allocation in every decision making units.By the numerical result the proposed model has been applied to manufacturing enterprises and limited resources have been considered among different alpha level sets.

Table 1 :
The input-output data for the eight DMUs

Table 2 :
The total input-output data for the eight DMUs by given α-level set