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کد مقاله : 202412291194889 بازدید : 18 صفحه: -

نوع مقاله: پژوهشی

Ranking of decision-making units with fuzzy inputs and outputs using cross-efficiency model and fuzzy ranking function

محورهای موضوعی : International Journal of Data Envelopment Analysis

ٍEskandar Abdolahi 1

1 - گروه ریاضی، دانشگاه آزاد اسلامی واحد کرمان، کرمان، ایران

تاریخ دریافت : 1403/10/09 تاریخ پذیرش : 1403/11/15 تاریخ انتشار : 1403/11/20

کلید واژه: Data Envelopment Analysis, Cross-Sectional Efficiency, Ranking, Ranking Function,

چکیده مقاله :

Many definitions and concepts are uncertain. Here it is necessary to compare uncertain data, in this comparison the decision maker faces a kind of uncertainty that is related to the lack of precise and firm demarcation of concepts. These concepts cannot be argued, inferred and decided with Aristotelian logic, which requires precise and quantitative data. Fuzzy set theory, by using specific models, is able to give mathematical formulation to many concepts, variables and systems that are imprecise and vague and provides the basis for inference and decision-making under conditions of uncertainty. In this article, we will use our efforts in this direction. We will evaluate the decision-making units while the inputs and outputs are fuzzy, and in this evaluation, we will rank the decision-making units with the help of the cross-efficiency model and the use of the fuzzy ranking function.

چکیده انگلیسی:

Many definitions and concepts are uncertain. Here it is necessary to compare uncertain data, in this comparison the decision maker faces a kind of uncertainty that is related to the lack of precise and firm demarcation of concepts. These concepts cannot be argued, inferred and decided with Aristotelian logic, which requires precise and quantitative data. Fuzzy set theory, by using specific models, is able to give mathematical formulation to many concepts, variables and systems that are imprecise and vague and provides the basis for inference and decision-making under conditions of uncertainty. In this article, we will use our efforts in this direction. We will evaluate the decision-making units while the inputs and outputs are fuzzy, and in this evaluation, we will rank the decision-making units with the help of the cross-efficiency model and the use of the fuzzy ranking function.

منابع و مأخذ:

[۱] جهانشاهلو غلامرضا، حسین زاده لطفی فرهاد ، نیک مرام هاشم، تحلیل پوشش داده ها و کاربردهای آن، دانشگاه آزاد اسلامی واحد علوم و تحقیقات.
[2] Allaliviranloo, T. Sancifard, R., Hosseinzadeh, F., & Mikaelvand. N. Euclidean Ranking DMUs with fuzzy data in DEA. Applied Mathematical Sciences, 60, (2007) 2989-2998
[3] Alka Arya Shiv Prasad Yadav Development of FDEA Models to Measure the Performance Efficiencies of DMUSTaiwan Fuzzy Systems Association and Springer Verlag Berlin Heidelberg 2017
[4] Bagherzadeh Valami, H. and R. Racinojehdehi, Ranking units in Data Envelopment Analysis with fuzzy data. Journal of Intelligent and Fuzzy Systems,
30(5): 2016 p. 2505-2516
[5] Çakir, S., Efficiency measurement in caykur factories with fuzzy data envelopment analysis Journal ofthe Faculty of Engineering and Architecture of Gazi University. 31(2): 2016 p. 369-381.
[6] Dotoli, M., Epicoco, N., Falagario, M., Sciancalepore, F. A cross-efficiency fuzzy data envelopment analysis technique for performance evaluation of decision making units under uncertainty. Comput. Ind Eng 79, 2015.103-114
[7] Hatami-Marbim, A. Emrouznejad, A., & Tavana, M. A taxonomy and review of the fuzzy data envelopment analysis literature; two decades in the making. European Journal of Operational Research, 214(3), 2011. 457-472
[8] Ignatius, J., M. Ghasemi, F. Zhang, A. Emrooznejad and A. Hatami-Marbini, Carbon Efficiency Evaluation: An Analytical Framework Using Fuzzy DEA. European Journal of
[9] Operational Research. 253(2): 2016. p. 428-440 [9] Jin, M., S. Ang. F. Yang and G. Bi. An improved cross-ranking method in data envelopment analysis, INFOR, 54(1): 2016. p. 19-32
[10] Jahanshahloo, G. R., Soleimani-Damanch, M., & Nasrabadi. E Measure of efficiency in DEA with fuzzy input-output levels a methodology for assessing, ranking and imposing of weights restrictions. Applied Mathematics and Computation, 156(1), (2004). P: 175-187.
[11] Khodabakhshi, M. Aryavash, K: Ranking units with fuzzy data in DEA. Data Envel, Anal. Decis Sci, 2014, 1-10 (2014)
[12] Lo Storto, C. Ecological efficiency based ranking of cities: A combined dea crossefficiency and Shannon's entropy method. Sustainability (Switzerland), 2016, 8(2).
[13] Sauti, S. Hatami-Marbini, A., & Makui, A. An Application of Fuzzy Numbers Ranking in Performance Analysis. Journal of Applied Sciences, 9(9), (2009).
[14] Wen, M.. You, C. & Kang, R. A new ranking method to fuzzy data cvelopment analysis Computers & Mathematics with Applications 5911), (2010) P 3398-3404.
[15] Won, M. Li, H. Fuzzy data envelopment analysis (DEA): model and ranking method J Comput Appl. Math. 223, (2009), P:872-878
[16] Zerafat Angiz L, M, Mustafa, A., & Emrouznejad, A Ranking efficient decisionmaking units in data envelopment analysis using fuzzy concept. Computers & Industrial.
[17] Engineering, 59(4), (2010a). P: 712-719 [17] Zerafat Angaz L, M., Emrouznejad, A., Mustafa, A., & Al-Eraqi, A. S. Aggregating preference ranking with fuzzy data envelopment analysis Knowledge-Based Systems, 23(6), (2010b).P-512-519
[18] Ziari, S., An alternative transformation in ranking using II -norm in data evelopment analysis. Journal of Industrial Engineering International, 12(3) 2016 P:401-40
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Available online at http://ijdea.srbiau.ac.ir

Int. J. Data Envelopment Analysis (ISSN 2345-458X)

Vol. 12, No. 4, Year 2024 Article ID IJDEA-00422, Pages 8-20

Description: Description: C:\Users\Dell\Desktop\GGGG.jpgDescription: Description: C:\Users\Dell\Desktop\MMMM.jpgResearch Article

 

 

International Journal of Data Envelopment Analysis                                                                            Science and Research Branch (IAU)

 

 

 

 

Ranking decision-making units with fuzzy inputs and outputs using the cross-efficiency model and fuzzy ranking function

 

E. Abdollahi 1

Department of Mathematics, Kerman Branch, Islamic Azad University of Kerman, Iran

 

Received 12 March 2024, Accepted 29 July 2024 

 

Abstract

Data envelopment analysis is a mathematical technique for examining the performance of decision-making units with multiple inputs and multiple outputs. In data envelopment analysis, one of the methods that evaluate decision-making units is the intersection efficiency method. In this paper, this method is used to evaluate decision-making units with fuzzy inputs and outputs, and we use the ranking function for ranking.

 

Keywords: Data envelopment analysis, intersection efficiency, ranking, ranking function

[1]  Corresponding author: Email: abodlahi.eskandar@gmail.com

 

 

 

 

 


 

1.       Introduction

Many definitions and concepts are characterized by uncertainty. It is necessary that uncertain data be compared. In this comparison, the decision-maker is faced with a type of uncertainty that is related to the lack of precise and firm boundaries of concepts. These concepts cannot be reasoned, inferred, or decided upon using Aristotelian logic, which requires precise and quantitative data. The fuzzy set theory, by employing specific models, is able to mathematically formulate many concepts, variables, and systems that are imprecise and ambiguous, thereby providing a foundation for inference and decision-making under conditions of uncertainty. In this paper, efforts are made in this regard to evaluate decision-making units while their inputs and outputs are fuzzy. In this evaluation, decision-making units are ranked with the aid of the cross-efficiency model and the fuzzy ranking function.

Data Envelopment Analysis (DEA) is a well-established non-parametric technique for evaluating the relative efficiency of decision-making units (DMUs) that utilize multiple inputs to produce multiple outputs. Introduced by Charnes et al. (1978), [1]. DEA has been widely applied in various sectors, including healthcare, banking, education, and manufacturing, to assess performance and identify best practices [2,3,4]. By constructing an efficient frontier based on observed data, DEA compares each DMU against the most efficient units, providing insights into operational effectiveness and areas for improvement [5].

DEA models can generally be categorized into input-oriented and output-oriented approaches, depending on whether the goal is to minimize inputs for a given level of outputs or maximize outputs for a given level of inputs [6]. Over the years, numerous extensions of DEA have been proposed to address challenges such as the presence of undesirable outputs, negative data, and uncertain information [7]. Among these extensions, cross-efficiency DEA has gained significant attention for ranking DMUs by incorporating both self-evaluation and peer-evaluation mechanisms, leading to more comprehensive performance assessments [8].

In practical applications, DEA has demonstrated significant advantages, such as its ability to handle multiple inputs and outputs without requiring an explicit functional form. However, it also has limitations, including sensitivity to data quality and the challenge of distinguishing between efficient DMUs when multiple units achieve the highest efficiency score. Recent advancements, such as the integration of fuzzy logic and artificial intelligence with DEA, aim to enhance its applicability and robustness in complex decision-making environments [9,10].

Puri and Yadav (2014) developed a fuzzy DEA model that incorporates undesirable fuzzy outputs, addressing the challenge of imprecise input/output data in real-world scenarios [11]. They applied their model to evaluate the efficiency of Indian public sector banks from 2009 to 2011, demonstrating how undesirable outputs and data uncertainty impact efficiency assessments. Dotoli et al. (2015) developed a cross-efficiency fuzzy DEA method to evaluate the performance of decision-making units (DMUs) under uncertainty using triangular fuzzy numbers. They applied their approach to assess and rank healthcare systems in Southern Italy, demonstrating its effectiveness in handling uncertainty and supporting policy reforms [12]. Mashayekhi et al. (2016) introduced a multi-objective portfolio selection model that integrates DEA cross-efficiency with the Markowitz mean-variance model, incorporating trapezoidal fuzzy numbers to handle uncertainty in asset returns. Their model, tested on 52 firms from the Iranian stock exchange, demonstrated better performance compared to traditional Markowitz and DEA models by considering return, risk, and efficiency simultaneously [13]. Meng and Xiong (2021) introduced a logical efficiency decomposition approach for general two-stage systems by incorporating cross-efficiency evaluation, addressing the limitations of traditional "black-box" DEA models. They proposed a leader-follower method to decompose system efficiency and applied multiplicative hesitant fuzzy elements (MHFEs) to represent cross-efficiency relationships between DMUs. Their approach enhances evaluation accuracy by ensuring consistent preference relations and was successfully applied to assess the efficiency of nine top universities in China [14]. Liu et al. (2021) proposed a novel fuzzy cross-efficiency evaluation method in DEA that simultaneously considers all possible weight combinations for DMUs, eliminating the need for weight selection. They employed the α-level-based approach to develop a pair of linear programs that calculate the lower and upper bounds of fuzzy efficiency scores, demonstrating enhanced discrimination power in ranking DMUs under fuzzy conditions [15]. Sharafi et al. (2022) proposed a novel fuzzy DEA model for green supplier selection, incorporating expert votes to enhance decision-making in green supply chain management. They introduced an improved cross-efficiency method using a secondary goal model based on the fuzzy CODAS approach, which was applied to an automotive group, achieving a complete ranking of green suppliers [16]. Soltanifar et al. (2022) introduced a modified DEA cross-efficiency method that addresses the challenges of negative data and the limitations of traditional cross-efficiency ranking methods. They proposed a new non-radial model to handle negative data and developed a secondary goal model to resolve the issue of multiple optimal solutions. Additionally, they integrated a hybrid MADM-DEA approach using the fuzzy VIKOR method to improve result aggregation. The proposed models were applied to a real-world supplier selection problem, demonstrating their effectiveness in ranking suppliers under complex conditions [17]. Song et al. (2023) proposed a novel group decision-making (GDM) method that integrates the DEA cross-efficiency approach with regret theory to handle multi-granular hesitant fuzzy linguistic information (MGDM-RCE). Their approach accounts for decision-makers' non-rational behavior and varying granularity scales by developing cross-efficiency models based on regret-rejoice utility values, providing cross-efficiency intervals for DMUs. An extended stochastic cross-efficiency technique is introduced to finalize rankings, with the method demonstrating superior stability and robustness compared to traditional techniques like VIKOR and TOPSIS through sensitivity and comparative analyses [18]. Zhang et al. (2024) introduced a stochastic cross-efficiency DEA approach based on prospect theory to enhance fairness and transparency in public procurement tenders. Their method includes cross-efficiency DEA models that consider experts' risk behaviors to maximize gains and minimize losses, and employs a stochastic Benefit-of-the-Doubt (BoD) model with Monte Carlo simulation to aggregate diverse evaluations without pre-defined weights. Additionally, hesitant fuzzy linguistic term sets are used to handle uncertainty in qualitative assessments, ensuring more robust and fair bidder rankings [19].

 This study aims to address these challenges by proposing a cross-efficiency DEA model with fuzzy inputs and outputs, incorporating a suitable fuzzy ranking function to achieve a reliable and interpretable ranking of DMUs. A practical application in the banking sector is provided to validate the effectiveness of the proposed approach.

 

2.       The fuzzy Sexton model (fuzzy cross-efficiency) is utilized with the fuzzy ranking function when all inputs and outputs are fuzzy.

In this paper, the process of forming the cross-efficiency table (Sexton model) for fuzzy inputs and outputs is followed. Triangular fuzzy numbers are considered, and decision-making units are ranked using the fuzzy ranking function. To achieve this goal, the fuzzy inputs and outputs are considered as follows.

img0 

This means that all inputs and outputs are considered as triangular fuzzy numbers.

Now, the following fuzzy CCR model is considered.

img0 

The above fuzzy model is expanded as follows:

img0 

img0 

Model (1) is the dual form of the CCR envelopment model. Considering the relationships between the dual model and the original model, and taking into account that in the envelopment form img0, the corresponding constraint img0 can be less than or equal to one. This means that:

img0 

Therefore, model (2) is considered as follows:

img0 

img0 

In model (3), img0 is a positive parameter representing the importance of the objective functions such thatimg0and the model is solved and img0is assumed to be the optimal solution of the model. In this case, the efficiency is calculated as follows:

img0 

Therefore, it is obtained that:

The img0 element of the fuzzy cross-efficiency table is img0.

Considering the above calculations, the fuzzy cross-efficiency table is presented as follows:

Therefore, it is obtained that:

 

Table (1): Fuzzy Cross-Efficiency Table

 

average

img0 

⋯

img0 

 

img0 

img0 

⋯

img0 

img0 

img0 

img0 

 

img0 

img0 

⋮

⋮

⋮

⋮

 

img0 

img0 

 

 

img0 

The above table, in which all elements are triangular fuzzy numbers, represents the efficiency of decision-making units in the fuzzy state. As observed, the average is also a triangular fuzzy number. These triangular fuzzy averages must be compared with each other, and for this comparison, the fuzzy ranking function is used. For this purpose, one of the fuzzy number ranking methods that better aligns with the problem's conditions is applied. Various methods have been proposed for comparing and ordering fuzzy numbers, which is a very important process in decision-making. Each method has its advantages and disadvantages depending on its practical application.

The following method is used to compare the averages.

Let img0 and img0be assumed as triangular fuzzy numbers. They are defined as follows:

  img0

img0 

img0 

 

3.       Practical Example

Ten branches of a commercial bank are studied, and the required information from these ten bank branches is obtained as shown in Tables (2), (3), (4), (5), and (6). Model (3) is solved, and the cross-efficiency tables are obtained as presented in Tables (8), (9), and (10). The efficiency results of the decision-making units are provided in Tables (11), (12), and (13). The fuzzy averages are calculated according to Table (14), and the ranking of the decision-making units is performed in Table (15).

        

 

 

Table (2): Lower Bounds of Inputs for Decision-Making Units

Paid Interest img0

Generated Claims img0

Personnel Score img0

Bank Branch

347912609

166965005

5/42

DMU1

321087157

1364254263

6/5

DMU2

439622053

1021540167

5/13

DMU3

247470622

1023094065

7/58

DMU4

28332000

244442242

7/58

DMU5

175107405

150114017

3/89

DMU6

55843067

41603512

4/44

DMU7

5079356

1025368685

2/69

DMU8

321956067

1259611949

2/26

DMU9

58700000

1720212885

2/77

DMU10

 

Table (3): Midpoint of Inputs for Decision-Making Units

Paid Interest img0

Generated Claims img0

Personnel Score img0

Bank Branch

6267251735

3204527225

11/505

DMU1

13974801379

5420093131

17/77571429

DMU2

2430817731

4062997330

14/77714286

DMU3

3264590279

5811247992

14/939228571

DMU4

8935005782

3728154627

14/79285714

DMU5

5148882350

4825940830

11/85928571

DMU6

7333046577

4557896111

10/82642857

DMU7

4346972893

3925577955

10/1457142

DMU8

7062857806

5262472780

15/90214286

DMU9

8880600175

4451249734

13/45428571

DMU10

 

Table (4): Upper Bounds of Inputs for Decision-Making Units

Paid Interest img0

Generated Claims img0

Personnel Score img0

Bank Branch

30033076818

5826283949

21/61

DMU1

94870216509

15713640424

56/23

DMU2

9958384916

11969476089

38/59

DMU3

17958774018

30435770419

31/43

DMU4

82231846069

11450174945

26/41

DMU5

29728030018

15105382313

19/31

DMU6

36434239083

11461622508

25/01

DMU7

19571582641

12424040548

22/8

DMU8

57849361275

18113748298

31/9

DMU9

91314625872

12448647772

33/24

DMU10

 

Table (5): Lower Bounds of Outputs for Decision-Making Units

Received Fee 

05

Received Profit 

04

Facilities

 03

Other Deposits 

02

Total Deposits

01

Branch

5162000

38368691

848671179

28948043462

7098487595

DMU1

4405000

42883893

566162650

23631050649

7949322656

DMU2

7300000

36411899

864134766

38631972139

14995970790

DMU3

18275000

77239155

895838606

26432773959

18514914833

DMU4

3925000

1431607

18843288

19275628277

17332785899

DMU5

11749913

56474

4024246

21779585799

20385936597

DMU6

1106670

391780

569445

9327588934

7452604318

DMU7

650000

25128335

221934579

28111168328

6146639414

DMU8

17733000

25582456

1367871203

20439226024

27850207872

DMU9

3500000

27773398

5179767795

2845411692

10096103316

DMU10

 

Table (6): Midpoint of Outputs for Decision-Making Units

Received Fee 

05

Received Profit 

04

Facilities

 03

Other Deposits 

02

Total Deposits

01

Branch

5162000

38368691

848671179

5572985238

52171726468

DMU1

4405000

42883893

566162650

91501177806

92582493774

DMU2

7300000

36411899

864134766

7200492720

80053680037

DMU3

18275000

77239155

895838606

93783934491

80184379532

DMU4

3925000

1431607

18843288

84640654618

89211750661

DMU5

11749913

56474

4024246

72319356101

80255164183

DMU6

1106670

391780

569445

46608470345

69565403708

DMU7

650000

25128335

221934579

59838074926

50638122921

DMU8

17733000

25582456

1367871203

88236906513

81830300111

DMU9

3500000

27773398

5179767795

85908459121

86094189632

DMU10

 

Table (7): Upper Bounds of Outputs for Decision-Making Units

Received Fee 

05

Received Profit 

04

Facilities

 03

Other Deposits 

02

Total Deposits

01

Branch

747392532

305467932

6914738665

121751849411

122467919047

DMU1

15347945440

2306896375

22330988268

329970969669

378785621113

DMU2

5720495923

1057234112

14534600915

167941180111

144962760424

DMU3

17734125040

1683454030

24239290627

353170012187

320467175205

DMU4

1939809308

457407119

19805543528

390987804421

26782666394

DMU5

11142908153

2688388239

35304269319

211466703060

311973279912

DMU6

3956028971

792206863

15460389283

124678248906

266434914435

DMU7

6055939136

570261939

8069534218

140587061901

188565753715

DMU8

18411137172

517953877

20717137949

259004083181

232634001693

DMU9

30782513082

2316300360

216266923578

355691625890

443720575872

DMU10

 

 

Table (8): Cross-Efficiency (Lower Bound) for Decision-Making Units

0092/0

0032/0

0075/0

0092/0

0073/0

0115/0

0035/0

0078/0

0089/0

0063/0

0124/0

0040/0

0095/0

0102/0

0054/0

0091/0

0030/0

113/0

0057/0

0071/0

0128/0

0041/0

0096/0

0103/0

0058/0

0094/0

0031/0

0115/0

0060/0

0075/0

0129/0

0042/0

0096/0

0103/0

0059/0

0095/0

0032/0

0116/0

0060/0

0076/0

0079/0

0028/0

0071/0

0085/0

0083/0

0131/0

0039/0

0070/0

0097/0

0058/0

0092/0

0032/0

0075/0

0092/0

0073/0

0115/0

0035/0

0078/0

0089/0

0063/0

0091/0

0032/0

0076/0

0093/0

0077/0

0121/0

0037/0

0079/0

0091/0

0063/0

0094/0

0033/0

0077/0

0095/0

0076/0

0119/0

0036/0

0081/0

0089/0

0064/0

0039/0

0013/0

0033/0

0045/0

0029/0

0049/0

0014/0

0019/0

0065/0

0023/0

0133/0

0044/0

0096/0

0102/0

0064/0

0097/0

0033/0

0117/0

0063/0

0082/0

 

Table (9): Cross-Efficiency (Midpoint) for Decision-Making Units

0589/0

0694/0

0680/0

0809/0

0707/0

0871/0

0780/0

0693/0

0660/0

0837/0

0537/0

0656/0

0626/0

0874/0

0590/0

0802/0

0671/0

0696/0

0605/0

0797/0

0544/0

0668/0

0628/0

0877/0

0598/0

0810/0

0681/0

0703/0

0611/0

0809/0

0545/0

0671/0

0628/0

0879/0

0601/0

0812/0

0684/0

0705/0

0613/0

0812/0

0563/0

0631/0

0637/0

0688/0

0721/0

0797/0

0763/0

0642/0

0641/0

0777/0

0589/0

0694/0

0680/0

0809/0

0707/0

0871/0

0780/0

0693/0

0660/0

0837/0

0584/0

0690/0

0676/0

0799/0

0712/0

0861/0

0780/0

0692/0

0658/0

0834/0

0591/0

0703/0

0684/0

0824/0

0711/0

0875/0

0785/0

0704/0

0663/0

0841/0

0431/0

0453/0

0491/0

0495/0

0496/0

0633/0

0454/0

0424/0

0467/0

0541/0

0550/0

0682/0

0627/0

0874/0

0608/0

0813/0

0690/0

0706/0

0615/0

0824/0

 

Table (10): Cross-Efficiency (Upper Bound) for Decision-Making Units

4765/0

7833/0

4427/0

6014/0

1,0000

1,0000

1,0000

9400/0

7954/0

1,0000

2504/0

7960/0

4875/0

6400/0

1,0000

1,0000

6943/0

8106/0

7337/0

1,0000

2506/0

7965/0

4882/0

6403/0

1,0000

1,0000

6942/0

8105/0

7349/0

1,0000

2507/0

7966/0

4884/0

6404/0

1,0000

1,0000

6941/0

8104/0

7353/0

1,0000

2742/0

6783/0

3609/0

5307/0

1,0000

7665/0

1,0000

9321/0

7773/0

9008/0

2765/0

7833/0

4427/0

6014/0

1,0000

1,0000

1,0000

9400/0

7954/0

1,0000

2776/0

7771/0

4300/0

5932/0

1,0000

9571/0

1,0000

9502/0

7842/0

1,0000

2791/0

7941/0

4438/0

5987/0

1,0000

1,0000

1,0000

9453/0

7865/0

1,0000

1927/0

5024/0

3173/0

4108/0

6880/0

8522/0

8559/0

6312/0

6463/0

6308/0

2511/0

7949/0

4874/0

6423/0

1,0000

1,0000

6953/0

8031/0

7301/0

1,0000

 

Table (11): Lower Bound Average Efficiency of Decision-Making Units

Efficiency

DMUs

0100/0

DMU1

0034/0

DMU2

0079/0

DMU3

0091/0

DMU4

0065/0

DMU5

0103/0

DMU6

0032/0

DMU7

0087/0

DMU8

0076/0

DMU9

0064/0

DMU10

Table (12): Midpoint Efficiency of Decision-Making Units

Efficiency

DMUs

0552/0

DMU1

0654/0

DMU2

0636/0

DMU3

0793/0

DMU4

0645/0

DMU5

0814/0

DMU6

0716/0

DMU7

0666/0

DMU8

0619/0

DMU9

0791/0

DMU10

 

Table (13): Upper Bound Average Efficiency of Decision-Making Units

Efficiency

DMUs

2579/0

DMU1

7503/0

DMU2

4389/0

DMU3

5899/0

DMU4

9688/0

DMU5

9576/0

DMU6

8634/0

DMU7

8574/0

DMU8

7519/0

DMU9

6532/0

DMU10

 

Table (14): Based on the calculations, the following table is obtained

Fuzzy Average Efficiency

DMUs

img0 

DMU1

img0 

DMU2

img0 

DMU3

img0 

DMU4

img0 

DMU5

img0 

DMU6

img0 

DMU7

img0 

DMU8

img0 

DMU9

img0 

DMU10

 

 

 

Table (15): Ranking of Decision-Making Units

10

9

8

7

6

5

4

3

2

1

img0 

9

2

7

8

1

10

5

4

6

3

Rank

 

 

Here, the fuzzy averages related to the DMUs must be compared. In this comparison, any DMU with a better average is considered more efficient.

The following method is used to compare the fuzzy numbers:

Let img0and img0be triangular fuzzy numbers.

We define: img0

Therefore, we have:

img0 

Based on the above definition and Table (14), it is obtained that:

img0 

By observing the above calculations, it is observed that:

img0 

The ranking of the decision-making units is presented as follows.

Table (15) shows the ranking of decision-making units (DMUs), where each DMU represents a branch of a commercial bank.

DMU (6) has been assigned the 1st rank. By examining the input and output Tables (3), (4), (5), (6), (7), and (8), it is observed that this DMU, on average, has the lowest input and the highest output compared to other DMUs. Therefore, assigning the 1st rank to this unit is justifiable.

DMU (5) has been assigned the 10th rank, meaning it is the weakest decision-making unit among the ten DMUs. By reviewing the input and output Tables (3), (4), (5), (6), (7), and (8), it is evident that this DMU, on average, has the highest input and the lowest output compared to other DMUs. Thus, assigning the 10th rank to this unit is also reasonable. Similarly, the rankings of other DMUs can be interpreted through their comparisons.

 

4. Conclusion

In this study, the cross-efficiency DEA model was employed to evaluate decision-making units (DMUs) with fuzzy inputs and outputs. The proposed approach incorporated a fuzzy ranking function to achieve a comprehensive ranking of DMUs, ensuring a more accurate and robust assessment in uncertain environments. A practical case study involving ten branches of a commercial bank was conducted to demonstrate the applicability of the method. The obtained rankings were analyzed and validated against real-world data, confirming the effectiveness of the model in handling fuzzy data and providing meaningful insights for decision-makers.

The results of this study highlight the importance of using fuzzy cross-efficiency models in environments characterized by uncertainty and imprecision, as they offer a more flexible and reliable alternative to traditional crisp DEA models. The application of triangular fuzzy numbers and the fuzzy ranking function allowed for better differentiation among DMUs, addressing potential limitations in conventional efficiency evaluation techniques.

Future research can further extend the proposed model by incorporating advanced fuzzy methods, such as intuitionistic fuzzy sets or interval-valued fuzzy sets, to enhance the model's ability to handle more complex uncertainties. Additionally, integrating artificial intelligence techniques, such as machine learning, could improve the adaptability and scalability of DEA models in large-scale applications across various industries.


 

References

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