A Credibility Approach on Fuzzy Slacks Based Measure (SBM) DEA Model
IIranian Journal of Fuzzy Systems
Volume *, Number *, (****), pp. **
A Credibility Approach on Fuzzy Slacks Based Measure (SBM) DEAModel
D. Mahla and S. Agarwal Department of Mathematics, Birla Institute of Technology and Science, Pilani, India [email protected], [email protected]
Abstract
Data Envelopment Analysis (DEA) is a multi-criteria technique based on linear programming to deal with many real-lifeproblems, mostly in nonprofit organizations. The slacks-based measure (SBM) model is one of the DEA model used toassess the relative efficiencies of decision-making units (DMUs). The SBM DEA model directly used input slacks andoutput slacks to determine the relative efficiency of DMUs. In order to deal with qualitative or uncertain data, a fuzzySBM DEA model is used to assess the performance of DMUs in this study. The credibility measure approach, transformthe fuzzy SBM DEA model into a crisp linear programming model at different credibility levels is used. The resultscame from the fuzzy DEA model are more rational to the real-world situation than the conventional DEA model. Inthe end, the data of Indian oil refineries is collected, and the efficiency behavior of the companies obtained by applyingthe proposed model for its numerical illustration.
Keywords:
Data envelopment analysis, Decision-making units, Fuzzy slacks-based measure, Relative efficiency. : 90C70.
Data Envelopment Analysis (DEA) is a robust non-parametric linear programming based technique to asses the per-formance of non-profit organizations. Many researchers used this technique in education, supply chain management,banking, transport, etc. The ranking of decision-making units (DMUs) using multiple conflicting criteria is one of theuses of DEA [17]. There are different types of DEA models in literature, some of them are, Charnes, Cooper, Rhodes(CCR) [8], Banker, Charnes, Cooper (BCC) [5], slacks-based measure (SBM) [32], new slack model (NSM) [1], etc. Thepresent study used the SBM DEA model, which was proposed by Tone [32], to evaluate the relative efficiencies of thedecision-making units(DMUs). The SBM DEA model directly deals with the input excess and output shortfalls.DEA, as the name suggests, used frontier analysis to assess the relative efficiencies of DMUs where the minutechange in data can significantly change the frontier. One of the challenges in real-world situations is that the availabledata might be present in an uncertain or qualitative form, or sometimes some data might be missing. Therefore, theconventional DEA models are absurd to use with these types of data [11]. The fuzzy set theory, which was developedby Zadeh [37] in 1965, integrates with DEA [9] to deal with this types of situations by creating more rational formsof DEA models. Many researchers applied DEA models to evaluate the efficiency of DMUs under fuzzy environments[15, 4]. Over the three-decade of research, the four primary approaches namely, the tolerance approach [36], the α -level-based approach [25], the fuzzy ranking approach [10], the possibility approach [18] used by researcher to solvethe fuzzy DEA models. Wen et al. [34] extended the traditional DEA models to a fuzzy environment proposed afuzzy DEA model based on credibility measure. Chen et al. [7] concluded that the use of the fuzzy SBM DEA modelfor estimating efficiency values not only represents the characteristic of the uncertainty of the efficiency values, it alsopresents the potential effect of risk volatility on efficiency values. Haiso et al. [14] also concluded that linguistic termscould not entirely fit with the conventional DEA models. Puri et al. [30] used fuzzy SBM DEA models to handle the Corresponding Author: D. MahlaReceived: **; Revised: **; Accepted: **. a r X i v : . [ m a t h . O C ] F e b D. Mahla, S. Agarwal imprecise data, and calculate the mix-efficiencies of State Bank of Patiala in the Punjab state of India. Wanke et al.[33] presented an analysis of the efficiency of Angolan banks using fuzzy DEA and stochastic DEA models based on the α − level approach and different tail dependence structure, respectively. Recently, Bakhtavar et al. [6] used a specialrisk prioritization algorithm by failure mode and effects analysis by SBM DEA model under fuzzy conditions. As perthe Zadeh examination of the fuzzy theory, he concluded the possibility distribution of the fuzzy variable replicates theprobability distribution of the random variable in probability theory. Fuzzy LP models can be considered the evolutionof conventional LP models, in which fuzzy variables play the role of fuzzy coefficients, and fuzzy events construct theconstraints controlling the model. The possibility theory determines the possibility of fuzzy events. Lertworasirikul etal. [18, 19] studied the fuzzy DEA models built by Guo et al. [13], which took the possibility criterion and the necessitycriterion as a measure and solved the ranking problem with two distinct approaches namely, the possibility approachand the credibility approach. Recently, Agarwal [1] applied possibility measure to solve the fuzzy SBM DEA model.The Possibility measure is used extensively, but it has no self-dual property, which is undoubtedly needed for practice.Liu and Liu [23] proposed credibility measure in 2002, which shows the self-dual character. The credibility theory whichmanaged personal conviction degree numerically given by Liu [21] and refined it in his next researches [25, 22]. In thepresent study, the SBM DEA model is extended with a fuzzy environment for evaluating the efficiency of DMUs andsolved by credibility measure.The rest of the paper organized as follows, section 2 recalls the basic SBM and fuzzy SBM DEA models. In section3, the credibility measure is used to solve the fuzzy SBM DEA model. In section 4, the relative efficiency of Indian oilcompanies is calculated and compared with existing methods. In the end, the conclusion is given. This section discusses the SBM DEA model, the fuzzy numbers, and the fuzzy SBM DEA model in detail. Somedefinitions which will be used in this study are also discussed in this section.
In 2001, Tone [32] proposed the SBM DEA model, which measured the efficiency of DMUs by minimizing the ratioof the reduction rate input slacks to the expansion rate of output slacks. This model is units invariant in nature andmonotone decreasing for input excess and output shortfall. Consider there are m -inputs, n -outputs, r -number of DMUs, x iz = amount of i th input used by z th DMU, y jz = amount of j th output used by z th DMU, S − iz = slack in the i th input ofthe z th DMU, S + jz = slack in the j th output of the z th DMU, and λ oz are intensity variables. Then, the SBM [16] DEAmodel for DMU z is given by, min ρ z = t − m (cid:80) mi =1 S − iz x iz t + n (cid:80) nj =1 S + rz y jz subject to r (cid:88) o =1 λ oz x io + S − iz = tx iz ∀ i = 1 , · · · , m r (cid:88) o =1 λ oz y jo − S + jz = ty jz ∀ j = 1 , · · · , nλ oz ≥ , S − iz ≥ , S + jz ≥ , t > ∀ o = 1 , · · · , r. (1)The crisp SBM DEA model requires the accurate value of attributes. But, sometimes, in real-life situations, the availabledata is vague due to human error, qualitative in nature, or non-obtainable. The above problem is solved by using fuzzyset theory. The definition of the fuzzy number and fuzzy SBM DEA model is discussed in the next subsections. The fuzzy set theory was introduced by Zadeh [37] to deal with the vagueness of the data. A fuzzy set on universal set M is defined by ˜ M = ( x, µ ˜ M ( x )) | x ∈ M ; µ ˜ M ( x ) ∈ [0 ,
1] in which µ ˜ M ( x ) is called the membership function of the fuzzyset. The fuzzy numbers are special kind of fuzzy sets defined on real numbers R which satisfy the following properties:1. Fuzzy numbers are normal (i.e. ∃ x ∈ R : µ ˜ M ( x ) = 1).2. Fuzzy numbers are convex (i.e. µ ˜ M ( x ) ≥ min { µ ˜ M ( b ) , µ ˜ M ( a ) } ∀ a ≤ x ≤ b ). Credibility Approach on Fuzzy Slacks Based Measure (SBM) DEA Model
33. The membership function of fuzzy numbers is an upper semi-continuous function.There are several types of fuzzy numbers; However, a triangular fuzzy number is used in this study. A triangular fuzzynumber ˜ M is a special kind of fuzzy number. It is fuzzy variable determined by triplet ( r, s, u ) such that ( r < s < u ),with membership function as, µ ( ˜ M ) = x − rs − r , if r ≤ x ≤ s = u − xu − s , if s ≤ x ≤ u = 0 , otherwise . (2) SBM DEA model is changed into a fuzzy SBM DEA model by considering the inputs and outputs are fuzzy numbers.The i th input of the z th DMU is indicated by ˜ x iz , and the j th output of the z th DMU is indicated by ˜ y jz , are fuzzyinput and output for DMU z , respectively.min ρ z = t − m (cid:80) mi =1 S − iz / ˜ x iz t + n (cid:80) nj =1 S + rz / ˜ y jz subject to r (cid:88) o =1 λ oz ˜ x io + S − iz = t ˜ x iz ∀ i = 1 , · · · , m r (cid:88) o =1 λ oz ˜ y jo − S + jz = t ˜ y jz ∀ j = 1 , · · · , nλ oz ≥ , S − iz ≥ , S + jz ≥ , t > ∀ o = 1 , · · · , r. (3)The model (3) is a fuzzy SBM DEA model, which can not be solved directly. The credibility measure based approachis used to convert the fuzzy SBM DEA model into a crisp linear programming problem (lpp). The credibility measureis discussed in the next section. In this study, the fuzzy SBM DEA model is approached by credibility measure, and the credibility measure is definedas,
Credibility Measure:
Consider ξ be a nonempty set with P { ξ } be the power set of ξ . Liu and Liu [23] defined thecredibility set function Cr { . } as credibility measure if it holds the following conditions:1. Cr { ξ } = 1,2. Cr { Y } ≤ Cr { Z } whenever Y ⊂ Z ∈ ξ ,3. Cr { Y } +Cr { Y } C = 1 for any event Y ∈ ξ ,4. Cr {∪ i Y i } =Sup i Cr { Y i } for any events Y i with Sup i Cr { Y i } < . ξ, P ( ξ ) , Cr ) are called the credibility space [20]. The theory of credibility of fuzzy events and chance-constrained programming (CCP) is used in this study to solve the fuzzy SBM DEA model. Wen et al. [35] has giventhe following results which are used in solving process of our fuzzy model. Theorem 1.
Consider ψ and ψ are two fuzzy variables defined on credibility space ( ξ, P ( ξ ) , Cr ). If Cr { ψ = y } andCr { ψ = y } are quasi concave, then1. Cr { ψ + ψ ≤ d } ≥ α iff ( ψ ) U − α ) + ( ψ ) U − α ) ≤ d ,2. Cr { ψ + ψ ≤ d } ≤ α iff ( ψ ) U − α ) + ( ψ ) U − α ) ≥ d . Here, 0 . ≤ α ≤ Theorem 2.
Consider ( ψ ) Lα and ( ψ ) Uα are the lower and upper bounds of α -cut of ψ , respectively. Then, D. Mahla, S. Agarwal
1. if k ≥
0, then ( kψ ) Uα = k ( ψ ) Uα and ( kψ ) Lα = k ( ψ ) Lα ,2. if k ≤
0, then ( kψ ) Uα = k ( ψ ) Lα and ( kψ ) Lα = k ( ψ ) Uα .The credibility distribution of triangular fuzzy number ˜ M (2) is defined as,Cr( ˜ M ≤ b ) = 0 , if r ≥ b = b − r s − r ) , if r ≤ b ≤ s = b − s + u u − s ) , if s ≤ b ≤ u = 1 , r ≤ b. (4)Cr( ˜ M ≥ b ) = 1 , if r ≥ b = 2 s − r − b s − r ) , if r ≤ b ≤ s = u − b u − s ) , if s ≤ b ≤ u = 0 , r ≤ b. (5)According to credibility measure, converting fuzzy-chance constraints into their equivalent crisp ones [28] in oneparticular confidence level α ≥ . M ≤ b ) ≥ α ⇐⇒ (2 − α ) s + (2 α − u Cr( ˜ M ≥ b ) ≥ α ⇐⇒ (2 − α ) s + (2 α − r (6) In the procedure to transform the model in the credibility programming SBM DEA model, each fuzzy coefficient isconsidered as a fuzzy variable, and each constraint is defined as a fuzzy event. The fuzzy SBM DEA model (3) becomesthe following credibility SBM DEA model: min f z subject to :Cr (cid:26) t − m (cid:80) mi =1 S − iz / ˜ x iz t + n (cid:80) nj =1 S + rz / ˜ y jz ≤ f z (cid:27) ≥ α Cr (cid:26) (cid:80) ro =1 λ oz ˜ x io + S − iz − t ˜ x iz = 0 (cid:27) ≥ α ∀ i = 1 , · · · , m Cr (cid:26) (cid:80) ro =1 λ oz ˜ y jo − S + jz − t ˜ y jz = 0 (cid:27) ≥ α ∀ j = 1 , · · · , nλ oz ≥ , S − iz ≥ , S + jz ≥ , t > ∀ o = 1 , · · · , r. (7)In order to solve the credibility SBM DEA model, the equality sign of constraints are converted into the inequality.Thus, model (7) becomes, Credibility Approach on Fuzzy Slacks Based Measure (SBM) DEA Model min f z subject to :Cr (cid:26) t − m (cid:80) mi =1 S − iz / ˜ x iz t + n (cid:80) nj =1 S + rz / ˜ y jz ≤ f z (cid:27) ≥ α Cr (cid:26) (cid:80) ro =1 λ oz ˜ x io + S − iz − t ˜ x iz ≤ (cid:27) ≥ α ∀ i = 1 , · · · , m Cr (cid:26) (cid:80) ro =1 λ oz ˜ x io + S − iz − t ˜ x iz ≥ (cid:27) ≥ α ∀ i = 1 , · · · , m Cr (cid:26) (cid:80) ro =1 λ oz ˜ y jo − S + jz − t ˜ y jz ≤ (cid:27) ≥ α ∀ j = 1 , · · · , n Cr (cid:26) (cid:80) ro =1 λ oz ˜ y jo − S + jz − t ˜ y jz ≥ (cid:27) ≥ α ∀ j = 1 , · · · , nλ oz ≥ , S − iz ≥ , S + jz ≥ , t > ∀ o = 1 , · · · , r. (8)If the membership functions of the fuzzy variable are normal and convex and α ≥ .
5, then by using theorem 1 and 2,the credibility programming SBM model (8) is transformed into the model (9): min f z subject to : (cid:26) t − m (cid:80) mi =1 S − iz / ˜ x iz t + n (cid:80) nj =1 S + rz / ˜ y jz (cid:27) L − α ) ≤ f z (cid:26) (cid:80) ro =1 λ oz ˜ x io + S − iz − t ˜ x iz (cid:27) U − α ) ≤ ∀ i = 1 , · · · , m (cid:26) (cid:80) ro =1 λ oz ˜ x io + S − iz − t ˜ x iz (cid:27) L − α ) ≥ ∀ i = 1 , · · · , m (cid:26) (cid:80) ro =1 λ oz ˜ y jo − S + jz − t ˜ y jz (cid:27) U − α ) ≤ ∀ j = 1 , · · · , n (cid:26) (cid:80) ro =1 λ oz ˜ y jo − S + jz − t ˜ y jz (cid:27) L − α ) ≥ ∀ j = 1 , · · · , nλ oz ≥ , S − iz ≥ , S + jz ≥ , t > ∀ o = 1 , · · · , r. (9)The lower α -cut of triangular fuzzy number ˜ a ˜ b , given as, { ˜ a ˜ b } Lα = ˜ a Lα ∗ / (˜ b Uα ). Therefore, the model (9) is proportionate D. Mahla, S. Agarwal to the model (10) as follows: min (cid:26) t − m (cid:80) mi =1 S − iz / ˜ x iz (cid:27) L − α ) (cid:26) t + n (cid:80) nj =1 S + rz / ˜ y jz (cid:27) U − α ) subject to : (cid:26) (cid:80) ro =1 λ oz ˜ x io + S − iz − t ˜ x iz (cid:27) U − α ) ≤ ∀ i = 1 , · · · , m (cid:26) (cid:80) ro =1 λ oz ˜ x io + S − iz − t ˜ x iz (cid:27) L − α ) ≥ ∀ i = 1 , · · · , m (cid:26) (cid:80) ro =1 λ oz ˜ y jo − S + jz − t ˜ y jz (cid:27) U − α ) ≤ ∀ j = 1 , · · · , n (cid:26) (cid:80) ro =1 λ oz ˜ y jo − S + jz − t ˜ y jz (cid:27) L − α ) ≥ ∀ j = 1 , · · · , nλ oz ≥ , S − iz ≥ , S + jz ≥ , t > ∀ o = 1 , · · · , r. (10)The fractional model (10) can be changed into a crisp linear programming model by normalization given as follows: min (cid:26) t − m (cid:80) mi =1 S − iz / ˜ x iz (cid:27) L − α ) subject to : (cid:26) t + n (cid:80) nj =1 S + rz / ˜ y jz (cid:27) U − α ) = 1 (cid:26) (cid:80) ro =1 λ oz ˜ x io + S − iz − t ˜ x iz (cid:27) U − α ) ≤ ∀ i = 1 , · · · , m (cid:26) (cid:80) ro =1 λ oz ˜ x io + S − iz − t ˜ x iz (cid:27) L − α ) ≥ ∀ i = 1 , · · · , m (cid:26) (cid:80) ro =1 λ oz ˜ y jo − S + jz − t ˜ y jz (cid:27) U − α ) ≤ ∀ j = 1 , · · · , n (cid:26) (cid:80) ro =1 λ oz ˜ y jo − S + jz − t ˜ y jz (cid:27) L − α ) ≥ ∀ j = 1 , · · · , nλ oz ≥ , S − iz ≥ , S + jz ≥ , t > ∀ o = 1 , · · · , r. (11)The fuzzy SBM DEA model is transformed into a crisp lpp model (11) using equation (6) and can be solved using anysoftware programs like MATLAB, PYTHON, LINGO, etc. A numerical example is presented to illustrate the proposed methodology. Here, the data of Indian oil refineries iscollected for the financial year 2017-18, and the relative efficiencies are calculated by using the proposed methodology.The Indian oil sector is one of the largest and core areas which influences the other core sectors effectively. India is oneof the biggest importers of oil, and its imports are increasing year by year. Compare to 4.56 million barrels of oil perday consumption by India in 2016 which is increased to 4.69 million barrels of oil per day in 2017 [26].The total of 249.4 Million Metric Tons (MMT) oil installed in their refineries made it the second-largest refiner inAsia. In India, there are 10 (6 PSUs+ 2 Joint Venture+ 2 private limited) oil companies with 23 refineries and 249.4
Credibility Approach on Fuzzy Slacks Based Measure (SBM) DEA Model • Capital expenditure (CE):
Capital expenditure is funds utilized by an organization to require all the physicalresources needed for the company. The CE makes new projects or new investments. This financial outlay is madeby organizations to keep up or increment the extent of their tasks. CE can be used for creating new buildings,repairing the equipment, or constructing the room. In our model, CE is used as crisp input. • Oil Throughput Utilization:
Crude throughput is the aggregate sum of unrefined that goes into a refinerybefore it comes out processed. Oil throughput utilization is given in the company’s annual reports; therefore, itshould be taken as a crisp input. • Revenue:
The amount of money generated by a company from the selling of the product within a stipulatedperiod is revenue. Every company announces its annual revenue in its yearly reports precisely; therefore, it shouldbe taken as a crisp output. • Nelson Complexity Index (NCI):
In India, it has been observed due to the proper utilization of capacity levelsand high gross refining margins (GRMs), the NCI is improving from the last few years. The NCI compares thecost of upgrading units to the price of the crude distillation unit. The precise calculation of NCI requires repeatedconstruction of the process units and consistent, standardized public reporting of the cost of such installations.In reality, the creation of crude distillation units are made of different sizes with the various technologies and thedata regarding the value of such construction is reported could be of poor quality and not homogeneous. Due topreexisting limitations and heterogeneous data, one must be cautious while processing the data and should expecta high level of uncertainty. Therefore, to deal with this uncertainty, NCI is converted into a fuzzy number in thisstudy.In our particular data, we divide two different types of companies based on scale size. The large companies arethose which having employees more than 10000, and the small companies are which having employees number less than10000. The companies BPCL, ONGC, HPCL, IOCL, and RIL, are large companies and CPCL, NRL, BORL, and NELare small companies. The relative efficiency of small companies calculated separately as the comparison of the relativeefficiency of small companies with large companies is not reasonable. Still, large companies can be compared to smallcompanies. The data is given in table 1 as,Table 1: Data of Indian Oil Companies
Company CapitalExpendi-ture (Rs.in Cr) Crude oilthrough-put(MMT) NCI Revenue(Rs.in Cr)
BPCL 8161 28.2 5.8 279312.70ONGC+MRPL 73664 16.27 9.5 928877HPCL 6722.45 28.7 12 243226.66CPCL 963 10.8 8.2 44227.2404IOCL 20345 69 9.5 506428NRL 387 2.8 10.5 15923.19BORL 3000 6.7 9.1 88454.4808RIL 40000 70.5 12.7 529120.00NEL 961.66 20.7 11.8 86636.66The values of NCI have some uncertainty; therefore, in the proposed work, the NCI is converted into triangular fuzzynumber by fuzzification process. The Saaty’s scale is used to fuzzify NCI and fuzzified triangular fuzzy numbers areshown in table 2. Further, the efficiencies of all companies are calculated by the proposed approach on the fuzzySBM DEA model, and calculations are done using MATLAB. The relative efficiencies of the companies using α − cut D. Mahla, S. Agarwal
Table 2: Fuzzy data of Indian Oil Companies
CompanySize Company CapitalExpen-diture Crude oilthrough-put NCI Revenue
BPCL 1.89 4.20 (4,5,6) 3.41ONGC+MRPL 9.00 2.85 (6,7,8) 9.00Large HPCL 1.73 4.26 (7,8,9) 3.09IOCL 3.21 8.83 (6,7,8) 5.36RIL 5.34 9.00 (8,9,9) 5.56CPCL 3.57 5.17 (6,7,8) 5.00Small NRL 2.03 2.08 (7,8,9) 2.44BORL 9.00 3.59 (6,7,8) 9.00NEL 3.56 9.00 (8,9,9) 8.84and possibility measure approach on SBM DEA are also computed. Table 3 shows the efficiencies of small and largecompanies using credibility measure as well as the ranking of the companies based on relative efficiency. The results oftable 3 can be interpreted as, companies NRL, BORL, and NEL are efficient companies with relative efficiency 1. CPCLis inefficient among small companies at every credibility level from 0.5 to 1, with given four criteria. It is noticeablefrom the table that the value of relative efficiencies is increasing with the increase of credibility level of small companies.It is also evident from Table 3 that ONGC and HPCL are efficient among large companies at all credibility level. TheTable 3: Relative efficiency of companies using credibility measure approachCredibility Level 0.5 0.6 0.7 0.8 0.9 1Large Company RankingBPCL 0.8571 0.8484 0.8387 0.8275 0.8148 0.8 3ONGC+MRPL 1 1 1 1 1 1 1HPCL 1 1 1 1 1 1 1IOCL 0.7595 1 1 1 1 1 2RIL 0.6655 0.6937 0.701 0.7014 0.7025 0.7029 4Small CompanyCPCL 0.5748 0.6836 0.6854 0.6872 0.6391 0.6309 2NRL 1 1 1 1 1 1 1BORL 1 1 1 1 1 1 1NEL 1 1 1 1 1 1 1BPCL and RIL are inefficient while BPCL is relatively more efficient than RIL at every credibility level with givenfour criteria. The rank of IOCL is third among large companies followed by ONGC, HPCL (ranked 1), and BPCL atcredibility level 0.5. The overall rank of IOCL is second, as it is efficient at every credibility level except 0.5. Further,the efficiency of these companies are also computed using the ” α − cut approach” and ”possibility approach” on fuzzySBM DEA model. Tables 4 and 5 show the relative efficiency using a α − cut approach for small and large companies.Table 4: Relative efficiency of small companies using α − cut approach Credibility level CPCL NRL BORL NEL
Rank
Credibility Approach on Fuzzy Slacks Based Measure (SBM) DEA Model α − cut approach Credibility level BPCL ONGC HPCL IOCL RIL
Rank α is moving from 0.5 to 1.0. This present study has established a credibility approach to solve the fuzzy SBM DEA model. This approach convertedthe fuzzy model into a credibility SBM DEA model. The credibility measure, which is the mean of necessity andpossibility measure and self-dual in nature used to find the desired level of confidence. The credibility approachdeals with the fuzzy environment in the fuzzy SBM DEA model significantly. The method is more reasonable andconceivable as compares to the tolerance approach, α − cut approach, fuzzy ranking approach, and possibility approach.In the numerical example, the relative efficiency and ranking of Indian oil and refineries are computed, and results arecompared with existing methods to solve the fuzzy SBM DEA model. Future Work
The present work proposed a new approach to solving the fuzzy SBM DEA model, and the numerical method demon-strated the compatibility of the proposed method. From the mathematical problem, we can conclude that there aretwo limitations. First, we can not rank the efficient DMUs directly from this method, and the other is, the targets cannot be computed because the data is rationalized. In the future, we will extend this model to rank the efficient DMUs,and to find out the input and output targets.
Acknowledgement
The authors thank DST for providing partial support under FIST grant SR/FST/MSI-090/2013(C) and CSIR forproviding financial assistance under JRF programme (Award No.: 09/719/(0079)/2017-EMR-I. The authors wish toexpress their appreciation for several excellent suggestions for improvements in this paper made by the referees.0
D. Mahla, S. Agarwal
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