A Neutrosophic-Based Approach in Data Envelopment Analysis with Undesirable Outputs

Data Envelopment Analysis is one of the paramount mathematical methods to compute the general performance of organizations, which utilizes similar sources to produce similar outputs. Original DEA schemes involve crisp information of inputs and outputs that may not always be accessible in real-world applications. Nevertheless, in some cases, the values of the data are information with indeterminacy, impreciseness, vagueness, inconsistent, and incompleteness. Furthermore, the conventional DEA models have been originally formulated solely for desirable outputs. However, undesirable outputs may additionally be present in the manufacturing system, which wishes to be minimized. To tackle the mentioned issues and in order to obtain a reliable measurement that keeps original advantage of DEA and considers the influence of undesirable factors under the indeterminate environments, this paper presents a neutrosophic DEAmodel with undesirable outputs. (e recommended technique is based on the aggregation operator and has a simple construction. Finally, an example is given to illustrate the new model and ranking approach in details.


Introduction
All organizations, whether governmental or private, require an effective performance assessment for development, growth, and sustainability in the competitive world of today so that within this growth it can appraise the efficiency and effectiveness of its organizational programs and its human resources processes. In other words, senior executive managers have always been seeking for a solution to ensure that their strategies are executed and, hence, have selected performance assessment methods as tools to implement their strategies. We should be mindful that novel measurement systems have been developed to target the existing strategies, and in this system, the indexes could be viewed as a crucial factor, in the present and future successes of the company. If these factors are alleviated or improved, then the company has launched its strategy.
e Data Envelopment Analysis (DEA) is a nonparametric method to analyse and assess the performance of decision-making units (DMUs), which converts several inputs into several outputs and considers the qualitative and quantitative criterions. Based on the work of [1], Charnes et al. [2] proposed this methodology that is called the CCR model. After that, Banker et al. [3] extended this model and established a new model to measure efficiency that is the socalled BBC model. With DEA, directors can achieve the relative efficiency of a set of DMUs. Note that the production function does not need to be known in this technique. In the recent years, an extensive application of DEA in numerous fields, such as banking institutions [4], insurance industry [5], financial services [6], education [7], sustainability [8], energy [9], and health-care services [10], has been observed. Furthermore, in the past four decades, numerous reports and articles have been published in esteemed global journals verifying that this method is operational, see [11][12][13][14][15][16].
However, in some cases, the values of the data are often information with indeterminacy, impreciseness, vagueness, inconsistent, and incompleteness. Inaccurate assessments are mainly the outcome of information that is unquantifiable, incomplete, and unavailable. By fuzzy sets [17], it is possible to model the uncertainty in information. In addition, there are several models of DEA with the fuzzy set, see [17][18][19] and references therein. However, fuzzy sets (FSs) deliberate only the membership function and cannot arrange other parameters of vagueness. erefore, intuitionistic fuzzy sets (IFSs) have been introduced in [20], see also the Pythagorean fuzzy sets [21][22][23].
In the DEA literature, there are also some models of DEA with IFSs, see [52][53][54][55][56][57][58][59][60]. However, in actual activities, there are some indeterminate information. If the indeterminacy factors are ignored, the relative effectiveness of DMUs will still be evaluated using the DEA model established on the basis of determinate values and biased or even wrong information will be obtained, which will bring some errors to management decisions. erefore, it seems convenient and necessary to consider the neutrosophic DEA model.
As far as we know, there are few studies concerning DEA with neutrosophic information. e utilization of neutrosophic set in DEA can be traced to Edalatpanah [61] and additional investigations have been accessible in [62][63][64][65][66][67]. However, these neutrosophic DEA methods are formulated solely for desirable outputs and cannot eliminate the influence of undesirable factors on the efficiency evaluation. We know that the undesirable outputs may additionally be present in the manufacturing system that needs to be minimized. For example, in banking industry nonperforming loans/assets and in manufacturing systems, the production of a variety of emissions, pollutions, and industrial waste gas are undesirable outputs [68][69][70][71][72][73]. erefore, there is still a need from the neutrosophic DEA method to develop a new model that keeps original advantage and considers the influence of undesirable factors.
Consequently, in this study, we establish a novel method of DEA with undesirable outputs in which all data are singlevalued neutrosophic sets (SVNSs). Furthermore, a competent algorithm for solving the new DEA model has been presented.
e main contributions of this paper are four folds: (1) we model the indeterminacies in the input and output data using SVNSs; (2) the proposed approach considers the impact of undesirable outputs on the performance of DMUs; (3) we provide a theorem regarding the feasibility and boundedness of the solution of new model; (4) we determine the efficiency scores of the DMUs as crisp values. e paper unfolds as follows. Some basic knowledge and concepts on DEA and its models are deliberated in Section 2. In Section 3, some knowledge and arithmetic operations on neutrosophic sets are discussed. In Section 4, we propose a new model of neutrosophic DEA with undesirable outputs and establish an algorithm to solve it. In Section 5, the proposed model and the related algorithm are illustrated with a numerical example to ensure their validity and usefulness over the existing models. Finally, conclusions and future direction are offered in Section 6.

The Basic Concepts of DEA and its Models
DEA is a mathematical programming methodology that allows performance measurement of homogeneous DMUs that have several inputs and outputs. In DEA approach, there is no need to determine the specific form of the production function, and linear programming is needed to construct a piecewise linear surface (frontier) to cover all data, and then the efficiency of each of the DMUs is calculated from this frontier. e frontier obtained is the efficiency boundary, where the points on it are efficient, and other DMUs that are inside the cover surface are inefficient.
Let DMUO be under consideration; then, the production possibility set of the BCC model proposed by Banker et al. [3] is as follows: erefore, the BCC model is as follows: For the BCC model with undesirable outputs, Guo and Wu [68] proposed In Model (3), we assume that there are n DMUs that use inputs x pq (p � 1, . . . , m), to achieve desirable outputs (good) y g sq (s � 1, . . . , k) and undesirable outputs (bad)

Neutrosophic Sets
Smarandache has given a number of actual examples for potential applications of NS; nevertheless, usage of NSs in real applied problems is difficult. Hence, NSs reduced into a type of SVNSs that will preserve the processes of the NSs. Here, we will discuss some basic definitions related to neutrosophic sets and SVNSs, respectively [30][31][32][33].

Definition 2.
Suppose that C and D be two SVNSs, then C ⊆ D if and only if Definition 3. LetΜandΝbe two SVNNs. en, equations (5) to (8) are true: en, the related weighted arithmetic average operator is and the aggregation operator is defined as follows: where W � (ω 1 , ω 2 , . . . , ω n ) is the weight vector of Μ q ,ω q ∈ [0, 1] and n q�1 ω q � 1.

Mathematical Problems in Engineering
We called this model as single-valued neutrosophic BCC with undesirable outputs (SVNBCC-UO). Now, we present an algorithm to solve Model (11).
In the following, we show that our model is feasible and bounded.

Theorem 1.
e model of SVNBCC-UO is feasible and bounded. Furthermore, its optimal objective function is 1.
Proof. From Algorithm 1, we can transform the model SVNBCC-UO into Step 4 in Algorithm 1. So, with the solution α q � 1, q � o, 0, else, and η o � 1, it is easy to see that the Step 4 in Algorithm 1 is always feasible. us, regardless of the values of inputs and outputs, there is always at least one feasible solution for Step 4 in Algorithm 1. Because the above solution is feasible along with the objective function of Step 4 Step 1. Construct the problem based on Model (10).
Step 5. Using Step 4, get the optimal efficiency of each DMU. ALGORITHM 1: Solution of the SVNBCC-UO model. 4 Mathematical Problems in Engineering in Algorithm 1 is minimization, the best value regarding the objective function is certainly lower than or equal to 1. □

Numerical Experiment
In this section, the proposed SVNBCC-UO and the related algorithm are illustrated with a numerical example to ensure their validity and usefulness over the existing models.

Example 1.
Consider an efficiency problem with 15 DMUs, three inputs, two desirable outputs, and one undesirable output that all information are presented as SVNNs (see Tables 1 and 2). Here, we used Algorithm 1 for solving the SVNBCC-UO model. Suppose that DMU 1 is under consideration, so based on Algorithm 1 we have the following.
First, we construct the model using Steps 1-3. en, by means of Step 4, we construct the Min η 1 with the constraints of Step 4 in Algorithm 1. For example, by Step 4 in Algorithm 1 for Input 1, we have the following relation: So, after calculations with Matlab, we get η * 1 � 0.9068 for DMU 1 . Correspondingly, the technical efficiencies of each DMU i are measured using the proposed algorithm and are presented in Table 3.
To validate the presented efficiencies, these efficiencies were compared with the efficiencies obtained by the existing neutrosophic DEA methods and are given in Figure 1. In this figure, we can see the impact of undesirable outputs on the performance of DMUs. From Figure 1, we can see that the efficiencies of DMUs are found to be smaller for SVNBCC-UO compared to the existing neutrosophic DEA methods. It is interesting that DMUs 1-4, 6-7, and 9 are efficient in the existing neutrosophic DEA methods; however, they are actually inefficient with the efficiency scores of 0.8329, 0.7504, 0.9451, 0.8197, 0.5822, 0.7430, and 0.9821 using   erefore, SVNBCC-UO is more realistic than the existing neutrosophic DEA methods. erefore, if we solve the mentioned problem with the existing methods, we cannot get to a reliable evaluation.

Conclusions and Future Work
is paper, established a new strategy to solve a neutrosophic data envelopment analysis model with undesirable outputs. In comparison with the existing neutrosophic DEA methods, the significant characteristic of the new DEA method is that it can handle the undesirable outputs simply and effectively. A numerical experiment and comparison results with the existing models have been demonstrated to display the competence of the presented method. e proposed methodology has created hopeful consequences from computing and performance facets. It is worth mentioning that the uncertainty, ambiguity, and indeterminacy in this paper are limited to single-valued neutrosophic numbers. Nevertheless, the other types of numbers such as bipolar NSs and interval-valued neutrosophic numbers, Pythagorean fuzzy set, and q-rung orthopair fuzzy set can also be used to indicate variables characterizing the core in world-wide problems. As for future research, we intend to extend the proposed approach to these kinds of tools.

Data Availability
e data used to support the findings of this study are included within the article.

Conflicts of Interest
e authors declare no conflicts of interest.