Interval and fuzzy optimization. Applications to data envelopment analysis

Enhancing concern in the efficiency assessment of a set of peer entities termed Decision Making Units (DMUs) in many fields from industry to healthcare has led to the development of efficiency assessment models and tools. Data Envelopment Analysis (DEA) is one of the most important methodologies to measure efficiency assessment through the comparison of a group of DMUs. It permits the use of multiple inputs/outputs without any functional form. It is vastly applied to production theory in Economics and benchmarking in Operations Research. In conventional DEA models, the observed inputs and outputs possess precise and realvalued data. However, in the real world, some problems consider imprecise and integer data. For example, the number of defect-free lamps, the fleet size, the number of hospital beds or the number of staff can be represented in some cases as imprecise and integer data. This thesis considers several novel approaches for measuring the efficiency assessment of DMUs where the inputs and outputs are interval and fuzzy data. First, an axiomatic derivation of the fuzzy production possibility set is presented and a fuzzy enhanced Russell graph measure is formulated using a fuzzy arithmetic approach. The proposed approach uses polygonal fuzzy sets and LU-fuzzy partial orders and provides crisp efficiency measures (and associated efficiency ranking) as well as fuzzy efficient targets. The second approach is a new integer interval DEA, with the extension of the corresponding arithmetic and LU-partial orders to integer intervals. Also, a new fuzzy integer DEA approach for efficiency assessment is presented. The proposed approach considers a hybrid scenario involving trapezoidal fuzzy integer numbers and trapezoidal fuzzy numbers. Fuzzy integer arithmetic and partial orders are introduced. Then, using appropriate axioms, a fuzzy integer DEA technology can be derived. Finally, an inverse DEA based on the non-radial slacks-based model in the presence of uncertainty, employing both integer and continuous interval data is presented

Measuring Technical Efficiency of Dairy Farms with Imprecise Data: A Fuzzy Data Envelopment Analysis Approach

This article integrates fuzzy set theory in Data Envelopment Analysis (DEA) framework to compute technical efficiency scores when input and output data are imprecise. The underlying assumption in convectional DEA is that inputs and outputs data are measured with precision. However, production agriculture takes place in an uncertain environment and, in some situations, input and output data may be imprecise. We present an approach of measuring efficiency when data is known to lie within specified intervals and empirically illustrate this approach using a group of 34 dairy producers in Pennsylvania. Compared to the convectional DEA scores that are point estimates, the computed fuzzy efficiency scores allow the decision maker to trace the performance of a decision-making unit at different possibility levels.fuzzy set theory, Data Envelopment Analysis, membership function, α-cut level, technical efficiency, Farm Management, Production Economics, Productivity Analysis, Research Methods/ Statistical Methods, Risk and Uncertainty, D24, Q12, C02, C44, C61,

A fuzzy DEA slacks-based approach

This paper deals with the problem of efficiency assessment using Data Envelopment Analysis (DEA) when the input and output data are given as fuzzy sets. In particular, a fuzzy extension of the measure of inefficiency proportions, a well-known slacks-based additive inefficiency measure, is considered. The proposed approach also provides fuzzy input and output targets. Computational experiences and comparison with other fuzzy DEA approaches are reported.The first author was partially supported by the research project MTM2017-89577-P (MINECO, Spain). The second author was partially supported by the Spanish Ministry of Economy and Competitiveness, grant AYA2016-75931-C2-1-P and from the Consejería de Educación y Ciencia, Spain (Junta de Andalucía, reference TIC-101). The third author acknowledges the financial support of the Spanish Ministry of Science, Innovation and Universities, grant PGC2018-095786-B-I00

Performance Evaluation of Airport Construction Energy-saving based on DEA Network Design and Performance Analysis

AbstractFuzzy Data Envelopment Analysis (DEA) evaluation model in Airport Construction Energy-saving have improved incomplete weight deficiency of information processing. Firstly, indices values were converted to trapezoid fuzzy numbers, then with incomplete information on indices weights as constraints, a fuzzy DEA model with outputs only and preference was established, and then by applying the α-cut approach, the model was transformed to a family of crisp DEA models and was solved. Experiments demonstrated the feasibility and applicability of the method

A fuzzy expected value approach under generalized data envelopment analysis

Fuzzy data envelopment analysis (DEA) models emerge as another class of DEA models to account for imprecise inputs and outputs for decision making units (DMUs). Although several approaches for solving fuzzy DEA models have been developed, there are some drawbacks, ranging from the inability to provide satisfactory discrimination power to simplistic numerical examples that handles only triangular fuzzy numbers or symmetrical fuzzy numbers. To address these drawbacks, this paper proposes using the concept of expected value in generalized DEA (GDEA) model. This allows the unification of three models - fuzzy expected CCR, fuzzy expected BCC, and fuzzy expected FDH models - and the ability of these models to handle both symmetrical and asymmetrical fuzzy numbers. We also explored the role of fuzzy GDEA model as a ranking method and compared it to existing super-efficiency evaluation models. Our proposed model is always feasible, while infeasibility problems remain in certain cases under existing super-efficiency models. In order to illustrate the performance of the proposed method, it is first tested using two established numerical examples and compared with the results obtained from alternative methods. A third example on energy dependency among 23 European Union (EU) member countries is further used to validate and describe the efficacy of our approach under asymmetric fuzzy numbers

A novel hybrid fuzzy DEA-Fuzzy MADM method for airlines safety evaluation

Safety is a critical element in the air transport industry. Although fatal air accidents are rare compared to other transport industries, the rapid growth in air travel demands has resulted in a growing aviation risk exposure and new challenges in the aviation sector. Although the issue of airline safety is of serious public concern, notably few studies have investigated the safety efficiency of airlines. This paper aims to propose a novel hybrid method using fuzzy data envelopment analysis (DEA) and fuzzy multi-attribute decision making (F-MADM) for ranking the airlines’ safety. In this study, fuzzy DEA is utilized to calculate criteria weights, in contrast to the conventional approach of using DEA for measuring the efficiency of alternatives. A ranking of each airline (DMU) on the basis of obtained weights is then assessed using MADM methods. Six MADM methods including Fuzzy SAW, Fuzzy TOPSIS, Fuzzy VIKOR, ARAS-F, COPRAS-F and Fuzzy MULTIMOORA are implemented to rank the alternatives, and finally, the results are compounded with the utility interval technique. This new hybrid method can efficiently overcome the pitfalls of traditional hybrid DEA-MADM models. The method proposed in this study is used to evaluate the safety levels of seven Iranian airlines and to select the safest one. © 2018 Elsevier Lt

The role of multiplier bounds in fuzzy data envelopment analysis

The file attached to this record is the author's final peer reviewed version. The Publisher's final version can be found by following the DOI link.The non-Archimedean epsilon ε is commonly considered as a lower bound for the dual input weights and output weights in multiplier data envelopment analysis (DEA) models. The amount of ε can be effectively used to differentiate between strongly and weakly efficient decision making units (DMUs). The problem of weak dominance particularly occurs when the reference set is fully or partially defined in terms of fuzzy numbers. In this paper, we propose a new four-step fuzzy DEA method to re-shape weakly efficient frontiers along with revisiting the efficiency score of DMUs in terms of perturbing the weakly efficient frontier. This approach eliminates the non-zero slacks in fuzzy DEA while keeping the strongly efficient frontiers unaltered. In comparing our proposed algorithm to an existing method in the recent literature we show three important flaws in their approach that our method addresses. Finally, we present a numerical example in banking with a combination of crisp and fuzzy data to illustrate the efficacy and advantages of the proposed approach

A geometrical approach for fuzzy DEA frontiers.

Interval DEA frontiers are here used in situation where one input or output is subject to uncertainty in its measurement and is presented as an interval data. We built an efficient frontier without any assumption about the probability distribution function of the imprecise variable. We take into account only the minimum and the maximum values of each imprecise variable. Two frontiers are constructed: the optimistic and the pessimistic ones. We use fuzszy relationships to introduce a new efficiency index based on a set of some Fuzzy T Norms. We will explore only the case where only on single variable presents a certain degree of uncertainty

An Integrated Fuzzy Clustering Cooperative Game Data Envelopment Analysis Model with application in Hospital Efficiency

Hospitals are the main sub-section of health care systems and evaluation of hospitals is one of the most important issue for health policy makers. Data Envelopment Analysis (DEA) is a nonparametric method that has recently been used for measuring efficiency and productivity of Decision Making Units (DMUs) and commonly applied for comparison of hospitals. However, one of the important assumption in DEA is that DMUs must be homogenous. The crucial issue in hospital efficiency is that hospitals are providing different services and so may not be comparable. In this paper, we propose an integrated fuzzy clustering cooperative game DEA approach. In fact, due to the lack of homogeneity among DMUs, we first propose to use a fuzzy C-means technique to cluster the DMUs. Then we apply DEA combined with the game theory where each DMU is considered as a player, using Core and Shapley value approaches within each cluster. The procedure has successfully been applied for performances measurement of 288 hospitals in 31 provinces of Iran. Finally, since the classical DEA model is not capable to distinguish between efficient DMUs, efficient hospitals within each cluster, are ranked using combined DEA model and cooperative game approach. The results show that the Core and Shapley values are suitable for fully ranking of efficient hospitals in the healthcare systems