Dr Mehdi Toloo

You et al. (2013) indicated two errors in Amin and Toloo (2007). The first error was the infeasibility of Amin and Toloo's (2007) model and the second drawback was the lack of a suitable value for the non-Archimedean epsilon in the proposed approach of Amin and Toloo (2007). This paper deals with the raised issues and proves that the model of Amin and Toloo (2007) is always feasible. In addition, we also formulate a new model for finding a suitable value for the epsilon.

Data envelopment analysis seeks a frontier to envelop all data with data acting in a critical role in the process and in such a way measures the relative efficiency of each decision making unit in comparison with other units. There is a statistical and empirical rule that if the number of performance measures is high in comparison with the number of units, then a large percentage of the units will be determined as efficient, which is obviously a questionable result. It also implies that the selection of performance measures is very crucial for successful applications. In this paper, we extend both multiplier and envelopment forms of data envelopment analysis models and propose two alternative approaches for selecting performance measures under variable returns to scale. The multiplier form of selecting model leads to the maximum efficiency scores and the maximum discrimination between efficient units is achieved by applying the envelopment form. Also individual unit and aggregate models are formulated separately to develop the idea of selective measures. Finally, in order to illustrate the potential of the proposed approaches a case study using a data from a banking industry in the Czech Republic is utilized. (C) 2014 Elsevier Ltd. All rights reserved.

Data envelopment analysis-discriminant analysis (DEA-DA) has been used for predicting cluster membership of decision-making units (DMUs). One of the possible applications of DEA-DA is in the marketing research area. This paper uses cluster analysis to cluster customers into two clusters: Gold and Lead. Then, to predict cluster membership of new customers, DEA-DA is applied. In DEA-DA, an arbitrary parameter imposing a small gap between two clusters (η) is incorporated. It is shown that different η leads to different prediction accuracy levels since an unsuitable value for η leads to an incorrect classification of DMUs. We show that even the data set with no overlap between two clusters can be misclassified. This paper proposes a new DEA-DA model to tackle this issue. The aim of this paper is to illustrate some computational difficulties in previous DEA-DA approaches and then to propose a new DEA-DA model to overcome the difficulties. A case study demonstrates the efficacy of the proposed model.

Data envelopment analysis (DEA) is a data based mathematical approach, which handles large numbers of variables, constraints, and data. Hence, data play an important and critical role in DEA. Given a set of decision making units (DMUs) and identified inputs and outputs (performance measures), DEA evaluates each DMU in comparison with all DMUs. According to some statistical and empirical rules, a balance between the number of DMUs and the number of performance measures should exist. However, in some situations the number of performance measures is relatively large in comparison with the number of DMUs. These cases lead us to choose some inputs and outputs in a way that produces acceptable results. We refer to these selected inputs and outputs as selective measures. This paper presents an approach toward a large number of inputs and outputs. Individual DMU and aggregate models are recommended and expanded separately for developing the idea of selective measures. The practical aspect of the new approach is illustrated by two real data set applications.

Data envelopment analysis (DEA) deals with the evaluation of efficiency score of peer decision making units (DMUs) and divides them in two mutually exclusive sets: efficient and inefficient. There are various ranking methods to get more information about the efficient units. Nevertheless, finding the most efficient unit is a scientific challenge and hence has been the subject of numerous studies. Here, the main contribution is an integrated model that is able to determine the most efficient unit under a common condition is developed. The current research formulates a new minimax mixed integer linear programming (MILP) model for fining the most efficient DMU. Three different case studies from different contexts are taken as numerical examples to compare the proposed model with other methods. These numerical examples also illustrate the various potential applications of the suggested model.

Nowadays, algorithms and computer programs, which are going to speed up, short time to run and less memory to occupy have special importance. Toward these ends, researchers have always regarded suitable strategies and algorithms with the least computations. Since linear programming (LP) has been introduced, interest in it spreads rapidly among scientists. To solve an LP, the simplex method has been developed and since then many researchers have contributed to the extension and progression of LP and obviously simplex method. A vast literature has been grown out of this original method in mathematical theory, new algorithms, and applied nature. Solving an LP via simplex method needs an initial basic feasible solution (IBFS), but in many situations such a solution is not readily available so artificial variables will be resorted. These artificial variables must be dropped to zero, if possible. There are two main methods that can be used to eliminate the artificial variables: two-phase method and Big-M method. Data envelopment analysis (DEA) applies individual LP for evaluating performance of decision making units, consequently, to solve these LPs an IBFS must be on hand. The main contribution of this paper is to introduce a closed form of IBFS for conventional DEA models, which helps us not to deal with artificial variables directly. We apply the proposed form to a real-data set to illustrate the applicability of the new approach. The results of this study indicate that using the closed form of IBFS can reduce at least 50 % of the whole computations.

Our peripheral environment is changing rapidly and globalization of organizations has made them more complex. Therefore, organizations should codify their strategic plans and executive methods more accurately. However, some executive methods are not properly fulfilling the organization's strategic priorities. This paper proposes a comprehensive framework in order to evaluate and prioritize strategies and rank executive methods. To do this, firstly, the strategic plans are developed with SWOT (Strength, Weakness, Opportunity, Threat) analysis and then plans are weighted and diminished by using FQSPM-Gap (Fuzzy Quantitative Strategic Planning Matrix) model. Finally, the executive methods of the company are prioritized by QFD (Quality Function Deployment) matrix to accomplish its strategic plans. The model is implemented in a textile and clothing Company.

Finding and classifying all efficient assignments for a Multi-Criteria Assignment Problem (MCAP) is one of the controversial issues in Multi-Criteria Decision Making (MCDM) problems. The main aim of this study is to utilize Data Envelopment Analysis (DEA) methodology to tackle this issue. Toward this end, we first state and prove some theorems to clarify the relationships between DEA and MCAP and then design a new two-phase approach to find and classify a set of efficient assignments. In Phase I, we formulate a new Mixed Integer Linear Programming (MILP) model, based on the Additive Free Disposal Hull (FDH) model, to gain an efficient assignment and then extend it to determine a Minimal Complete Set (MCS) of efficient assignments. In Phase II, we use the BCC model to classify all efficient solutions obtained from Phase I as supported and non-supported. A 4 x 4 assignment problem, containing two cost-type and single profit-type of objective functions, is solved using the presented approach. (C) 2014 Elsevier Ltd. All rights reserved.

Finding and classifying all efficient solutions for a Bi-Objective Integer Linear Programming (BOILP) problem is one of the controversial issues in Multi-Criteria Decision Making problems. The main aim of this study is to utilize the well-known Data Envelopment Analysis (DEA) methodology to tackle this issue. Toward this end, we first state some propositions to clarify the relationships between the efficient solutions of a BOILP and efficient Decision Making Units (DMUs) in DEA and next design a new two-stage approach to find and classify a set of efficient solutions. Stage I formulates a two-phase Mixed Integer Linear Programming (MILP) model, based on the Free Disposal Hull (FDH) model in DEA, to gain a Minimal Complete Set of efficient solutions. Stage II uses a variable returns to scale DEA model to classify the obtained efficient solutions from Stage I as supported and non-supported. A BOILP model containing 6 integer variables and 4 constraints is solved as an example to illustrate the applicability of the proposed approach.

The determination of a single efficient decision making unit (DMU) as the most efficient unit has been attracted by decision makers in some situations. Some integrated mixed integer linear programming (MILP) and mixed integer nonlinear programming (MINLP) data envelopment analysis (DEA) models have been proposed to find a single efficient unit by the optimal common set of weights. In conventional DEA models, the non-Archimedean infinitesimal epsilon, which forestalls weights from being zero, is useless if one utilizes the well-known two-phase method. Nevertheless, this approach is inapplicable to integrated DEA models. Unfortunately, in some proposed integrated DEA models, the epsilon is neither considered nor determined. More importantly, based on this lack some approaches have been developed which will raise this drawback. In this paper, first of all some drawbacks of these models are discussed. Indeed, it is shown that, if the non-Archimedean epsilon is ignored, then these models can neither find the most efficient unit nor rank the extreme efficient units. Next, we formulate some new models to capture these drawbacks and hence attain assurance regions. Finally, a real data set of 53 professional tennis players is applied to illustrate the applicability of the suggested models.

Measurement of performance is an important activity in identifying weaknesses in managerial efficiency and devising goals for improvement. Data envelopment analysis (DEA) is a mathematical quantitative approach for measuring the performance of a set of similar units. Toloo (2013) extended a DEA approach for finding the most efficient unit considering a data set without explicit inputs. The aim of this paper is to develop DEA models without explicit outputs, henceforth called DEA-WEO, to find the most efficient unit when outputs are not directly considered. The suggested models directly utilize the data without the need of adding a virtual output, whose value is equal to for all units. A real data set involving 139 different alternatives for long-term asset financing provided by Czech banks and leasing companies is taken to illustrate the potential application of the proposed approach.

Data envelopment analysis (DEA), considering the best condition for each decision making unit (DMU), assesses the relative efficiency and partitions DMUs into two sets: efficient and inefficient. Practically, in traditional DEA models more than one efficient DMU are recognized and these models cannot rank efficient DMUs. Some studies have been carried out aiming at ranking efficient DMUs, although in some cases only discrimination of the most efficient unit is desirable. Furthermore, several investigations have been done for finding the most CCR-efficient DMU. The basic idea of the majority of them is to introduce an integrated model which achieves an optimal common set of weights (CSW). These weights help us identify the most efficient unit in an identical condition. Recently, Toloo (2012) [13] proposed a new mixed integer programming (MIP) model to find the most BCC-efficient unit. Based on this study, we propose a new basic integrated linear programming (LP) model to identify candidate DMUs for being the most efficient unit; next a new MIP integrated DEA model is introduced for determining the most efficient DMU. Moreover, these models exclude the non-Archimedean epsilon and consequently the optimal solution of these models can be obtained, straightforwardly. We claim that the most efficient unit, which could be obtained from all other integrated models, has to be one of the achieved candidates from the basic integrated LP model. Two numerical examples are illustrated to show the variant use of these models in different important cases. (C) 2013 Elsevier Inc. All rights reserved.

Cook and Zhu (2007) introduced an innovative method to deal with flexible measures. Toloo (2009) found a computational problem in their approach and tackled this issue. Amirteimoori and Emrouznejad (2012) claimed that both Cook and Zhu (2007) and Toloo (2009) models overestimate the efficiency. In this response, we prove that their claim is incorrect and there is no overestimate in these approaches.

Vendor’s performance evaluation is an important subject which has strategic implications for managing an efficient company. However, there are many important criteria for prospering company. These criteria may contradict together. In other words, while a criterion is improved, the other may worsen. Indeed, similar to manufacturing manager in global market, purchasing manager who has significant practical implications deals with this issue. The vendor selection problem (VSP) is obviously affected by the complexity and uncertainty due to the lack of information associated with related business environment of countries in a global market. On the other hand, in the automotive industry which plays an important role in the worldwide market, these decisions will be exacerbated by increasing the outsourcing and opportunities. There are varieties of techniques, from simple weighted scoring methods to complex mathematical programming, for handling VSP. In this study, we propose a new cost efficiency data envelopment analysis (CE–DEA) approach with price uncertainty for finding the most cost efficient unit. Potential uses are then illustrated with an application to automotive industry involving 73 vendors in Turkey.

Supplier selection, a multi-criteria decision making (MCDM) problem, is one of the most important strategic issues in supply chain management (SCM). A good solution to this problem significantly contributes to the overall supply chain performance. This paper proposes a new integrated mixed integer programming ‐ data envelopment analysis (MIP‐DEA) model for finding the most efficient suppliers in the presence of imprecise data. Using this model, a new method for full ranking of units is introduced. This method tackles some drawbacks of the previous methods and is computationally more efficient. The applicability of the proposed model is illustrated, and the results and performance are compared with the previous studies.

Data envelopment analysis (DEA) has been a very popular method for measuring and benchmarking relative efficiency of each decision making units (DMUs) with multiple inputs and multiple outputs. DEA and Discriminant Analysis (DA) are similar in classifying units to exhibit either good or poor performance. On the other hand, selecting the most efficient unit between several efficient ones is one of the main issues in multi-criteria decision making (MCDM). Some proponents have suggested some approaches and claimed their methodologies involve discriminating power to determine the most efficient DMU without explicit input. This paper focuses on the weakness of a recent methodology of these approaches and to avoid this drawback presents a mixed integer programming (MIP) approach. To illustrate this drawback and compare discriminating power of the recent methodology to our new approach, a real data set containing 40 professional tennis players is utilized.

This paper proposes a new integrated mixed integer programing – data envelopment analysis (MIP–DEA) model to improve the integrated DEA model which was introduced by Toloo & Nalchigar [M. Toloo, S. Nalchigar. A new integrated DEA model for finding most BCC–efficient DMU. Appl. Math. Model. 33 (2009) 597–60]. In this study some problems of applying Toloo & Nalchigar’s model are addressed. A new integrated MIP–DEA model is then introduced to determine the most BCC-efficient decision making unit (DMU). Moreover, it is mathematically proved that the new model identifies only a single BCC-efficient DMU by a common set of optimal weights. To show applicability of proposed models, a numerical example is used which contains a real data set of nineteen facility layout designs (FLDs).

In conventional data envelopment analysis (DEA) models, a performance measure whether as an input or output usually has to be known. Nevertheless, in some cases, the type of a performance measure is not clear and some models are introduced to accommodate such flexible measures. In this paper, it is shown that alternative optimal solutions of these models has to be considered to deal with the flexible measures, otherwise incorrect results might occur. Practically, the efficiency scores of a DMU could be equal when the flexible measure is considered either as input or output. These cases are introduced and referred as share cases in this study specifically. It is duplicated that share cases must not be taken into account for classifying inputs and outputs. A new mixed integer linear programming (MILP) model is proposed to overcome the problem of not considering the alternative optimal solutions of classifier models. Finally, the applicability of the proposed model is illustrated by a real data set.

The success of a supply chain is highly dependent on selection of best suppliers. These decisions are an important component of production and logistics management for many firms. Little attention is given in the literature to the simultaneous consideration of cardinal and ordinal data in supplier selection process. This paper proposes a new integrated data envelopment analysis (DEA) model which is able to identify most efficient supplier in presence of both cardinal and ordinal data. Then, utilizing this model, an innovative method for prioritizing suppliers by considering multiple criteria is proposed. As an advantage, our method identifies best supplier by solving only one mixed integer linear programming (MILP). Applicability of proposed method is indicated by using data set includes specifications of 18 suppliers.

This paper suggests new data envelopment analysis (DEA) models for input and output scaling in advanced manufacturing technology (AMT). For a given group of AMT observations using the traditional DEA models, it is not possible to evaluate the units when a specified input (or specified output) is required to be scaled for all units. The paper provides theoretical results for obtaining the relationship between the original AMT observations and the corresponding scaled data. Also, the paper uses numerical illustrations to show the usefulness of the suggested contribution.

Data mining techniques, extracting patterns from large databases have become widespread in business. Using these techniques, various rules may be obtained and only a small number of these rules may be selected for implementation due, at least in part. to limitations of budget and resources. Evaluating and ranking the interestingness or usefulness of association rules is important in data mining. This paper proposes a new integrated data envelopment analysis (DEA) model which is able to find most efficient association rule by solving only one mixed integer linear programming (MILP). Then, utilizing this model, a new method for prioritizing association rules by considering Multiple criteria is proposed. As an advantage, the proposed method is computationally more efficient than previous works. Using an example of market basket analysis, applicability of our DEA based method for measuring the efficiency of association rules with multiple criteria is illustrated. (C) 2008 Elsevier Ltd. All rights reserved.

Cook and Zhu [Cook, W.D., Zhu, J., 2007. Classifying inputs and outputs in data envelopment analysis. European Journal of Operational Research 180, 692–699] introduced a new method to determine whether a measure is an input or an output. In practice, however, their method may produce incorrect efficiency scores due to a computational problem as result of introducing a large positive number to the model. This note introduces a revised model that does not need such a large positive number.

While conventional data envelopment analysis (DEA) models set targets separately for each decision making unit (DMU), Lozano and Villa (2004) introduced the concept of "centralized" DEA models, which aim at optimizing the combined resource consumption by all units in an organization rather than considering the consumption by each unit, separately. In these models, there is a centralized decision maker (DM) who supervises all DMUs. The main aim is optimizing total input consumption and output production. In this paper, firstly we present centralized output product model. Then we introduce parametric centralized additive model, which during one phase minimizes total consumption inputs and maximizes total output production simultaneously, in the direction of optimization vector. Some numerical examples of the proposed models and their results are presented.

Cost efficiency (CE) assesses the ability to produce current output at minimal cost. There are some models which are introduced to measure cost efficiency with certain and uncertain input prices. Normally, by using data envelopment analysis (DEA) models, more than one cost efficient decision making units (DMUs) are recognized. The main contribution of this paper consists of development of a model which was proposed by Amin and Toloo (2007) to some models for finding the most cost efficient DMU in various situations of input prices. These models find the most cost efficient DMU by solving only one mixed integer linear programming (MILP) in each case.

In many applications of widely recognized technique, DEA, finding the most efficient DMU is desirable for decision maker. Using basic DEA models, decision maker is not able to identify most efficient DMU. Amin and Toloo [Gholam R. Amin, M. Toloo, Finding the most efficient DMUs in DEA: an improved integrated model. Comput. Ind. Eng. 52 (2007) 71–77] introduced an integrated DEA model for finding most CCR-efficient DMU. In this paper, we propose a new integrated model for determining most BCC-efficient DMU by solving only one linear programming (LP). This model is useful for situations in which return to scale is variable, so has wider range of application than other models which find most CCR-efficient DMU. The applicability of the proposed integrated model is illustrated, using a real data set of a case study, which consists of 19 facility layout alternatives.

This paper presents a framework where data envelopment analysis (DEA) is used to measure overall profit efficiency with interval data. Specifically, it is shown that as the inputs, outputs and price vectors each vary in intervals, the DMUs cannot be easily evaluated. Thus, presenting a new method for computing the efficiency of DMUs with interval data, an interval will be defined for the efficiency score of each unit. As well as, all the DMUs are divided into three groups which are defined according to the interval obtained for the efficiency value of DMUs.

This paper presents an Improved MCDM Data Envelopment Analysis (DEA) model in order to evaluate the best efficient DMUs in Advanced Manufacturing Technology (AMT). This model is capable of ranking the next most efficient DMUs after removing the previous best one.

This paper presents a new algorithm for computing the non-Archimedean ε in DEA models. It is shown that this algorithm is polynomial-time of O(n), where n is the number of decision making units (DMUs). Also it is proved that using only inputs and outputs of DMUs, the non-Archimedean ε can be found such that, the optimal values of all CCR models, which are corresponding to all DMUs, are bounded and an assurance value is obtained.

Countries need robust long-term plans to keep up with the global pace of transitioning from pollutant fossil fuels towards clean, renewable energies. Renewable energy generation expansion plans can be either centralized, decentralized, or a combination of these two. This paper presents a novel approach to obtain an optimal multi-period plan for generating each type of renewable energy (solar, wind, hydro, geothermal, and biomass) via multi-objective mathematical modeling. The proposed model has integrated with Autoregressive Integrated Moving Average (ARIMA) econometric method to forecast the country’s demand during the planning horizon. The optimal energy mix based on several socio-economic aspects of renewable sources was obtained using the Passive and Active Compensability Multicriteria ANalysis (PACMAN) multi-attribute decision-making method. The model has been solved by a Non-dominated Sorting Genetic Algorithm (NSGA-II) metaheuristic algorithm. Each solution in the Pareto front contains a plan for each electricity generation region under a certain combination of centralization and decentralization strategies.

Multi-Objective Combinatorial Optimization Problems and Solution Methods discusses the results of a recent multi-objective combinatorial optimization achievement that considered metaheuristic, mathematical programming, heuristic, hyper heuristic and hybrid approaches. In other words, the book presents various multi-objective combinatorial optimization issues that may benefit from different methods in theory and practice. Combinatorial optimization problems appear in a wide range of applications in operations research, engineering, biological sciences and computer science, hence many optimization approaches have been developed that link the discrete universe to the continuous universe through geometric, analytic and algebraic techniques. This book covers this important topic as computational optimization has become increasingly popular as design optimization and its applications in engineering and industry have become ever more important due to more stringent design requirements in modern engineering practice.

In this paper, a metaheuristic-based design approach is developed in which the structural design optimization of large-scale steel frame structures is concerned. Although academics have introduced form-dominant methods, yet using artificial intelligence in structural design is one of the most critical challenges in recent years. However, the Charged System Search (CSS) is utilized as the primary optimization approach, which is improved by using the main principles of quantum mechanics and fuzzy logic systems. In the proposed Fuzzy Adaptive Quantum Inspired CSS algorithm, the position updating procedure of the standard algorithm is developed by implementing the center of potential energy presented in quantum mechanics into the general formulation of CSS to enhance the convergence capability of the algorithm. Simultaneously, a fuzzy logic-based parameter tuning process is also conducted to enhance the exploitation and exploration rates of the standard optimization algorithm. Two 10 and 60 story steel frame structures with 1026 and 8272 structural members, respectively, are utilized as design examples to determine the performance of the developed algorithm in dealing with complex optimization problems. The overall capability of the presented approach is compared with the Charged System Search and other metaheuristic optimization algorithms. The proposed enhanced algorithm can prepare better results than the other metaheuristics by considering the achieved results.

•Infeasibility under the super-efficiency problem aggravates under nonconvexity.•New super-efficiency cost frontier is feasible under constant returns to scale•Super-efficiency cost frontier may be infeasible under variable returns to scale.•The super-efficiency decomposition is new in the literature.•New cost super-efficiency model under incomplete price data is proposed. This contribution extends the literature on super-efficiency by focusing on ranking cost-efficient observations. To the best of our knowledge, the focus has always been on technical super-efficiency and this focus on ranking cost-efficient observations may well open up a new topic. Furthermore, since the convexity axiom has both an impact on technical and cost efficiency, we pay a particular attention to the effect of nonconvexity on both super-efficiency notions. Apart from a numerical example, we use a secondary data set guaranteeing replication to illustrate these efficiency and super-efficiency concepts. Two empirical conclusions emerge. First, the cost super-efficiency notion ranks differently from the technical super-efficiency concept. Second, both cost and technical super-efficiency notions rank differently under convex and nonconvex technologies.

•We develop robust equivalents for fractional DEA models.•The proposed models give a proper interpretation of robust efficiency.•The superiorities of our approach models over the existing ones have been investigated.•Duality relation in robust DEA is established according to the “primal worst equal dual best” theorem in robust optimization.•We show an equivalent relation between robust input-and output-oriented models.•We illustrate our proposed models with a study from the largest airports in Europe. Robust Data Envelopment Analysis (RDEA) is a DEA-based conservative approach used for modeling uncertainties in the input and output data of Decision-Making Units (DMUs) to guarantee stable and reliable performance evaluation. The RDEA models proposed in the literature apply robust optimization techniques to the linear and conventional DEA models which lead to the difficulty of obtaining a robust efficient DMU. To overcome this difficulty, this paper tackles uncertainty in DMUs from the original fractional DEA model. We propose a robust fractional DEA (RFDEA) model in both input and output orientation which enables us to overcome the deficiency of existing RDEA models. The linearized models of the fractional DEA are further used to establish duality relations from a pessimistic and optimistic view of the data. We show that the primal worst of the multiplier model is equivalent to the dual best of the envelopment model. Furthermore, we show that the robust efficiency in the input- and output-oriented DEA models are still equivalent in the new approach which is not the case in conventional RDEA models. We finally present a study of the largest airports in Europe to illustrate the efficacy of the proposed models. The proposed RDEA is found to provide an effective management evaluation strategy under uncertain environments.

•Propose a framework for measuring a maturity level of performance-based budgeting.•Develop a parallel network data envelopment analysis model.•Consider the hierarchical configuration of performance indicators.•Use fuzzy sets theory to deal with vagueness and ambiguity.•Present a case study to demonstrate the applicability of the developed framework. Performance-based budgeting (PBB) aims to formulate and manage public budgetary resources to improve managerial decisions based on actual performance measures of agencies. Although the PBB system has been overwhelmingly applied by various agencies, the progress and maturity of its implementation process are not satisfactory at large. Therefore, it warrants to find, evaluate and improve the performance of organisations in relation to implementing a PBB system. To do so, the composite indicators (CIs) have been proposed to aggregate multiple indicators associated with the PBB system, but their employment is contentious as they often lean on ad-hoc and troublesome assumptions. Data envelopment analysis (DEA) methods as a powerful and established tool help to contend with key limitations of CIs. Although the original DEA method ignores an internal production process, the knowledge of the internal structure of the PBB systems and indicators is of importance to provide further insights when assessing the performance of PBB systems. In this paper, we present a budget assessment framework by breaking a PBB system into two parallel stages including operations performance (OP) and financial performance enhancement (FPE) to open up the black-box structure of the system and consider the indicator hierarchy configuration of each stage. In situations of the hierarchical configuration of indicators, we develop a multilayer parallel network DEA-based CIs model to measure the PBB maturity levels of the system and its stages. It is shown that the discrimination power of the proposed multilayer model is better than the existing models with one layer and in situations of relatively small number of DMUs the model developed in this paper can be a good solution to the dimension reduction of indicators. Moreover, this research leverages fuzzy logic to surmount the subjective information that is often available in collecting indicators of the PBB systems. The major contribution of this research is to examine a case study of a PBB maturity award in Iran, as a developing country with a myriad of financial challenges, to adopt a PBB maturity model as well as point towards the efficacy and applicability of the proposed framework in practice.

•Our approach addresses the sharing risk problem between government and investors.•It includes constraints that limit the pollution effects on population centers.•It considers social responsibility, economics factors, and benefits of waste recycling. The public–private partnership (PPP) is a practical and standard model that has been at the center of attention over the past two decades. Sharing risk between government and investors has been a challenging issue over the last year. This study formulates a model that aims to define the investors’ longing and allocate risks to the government in a logical range. Besides, in some real-world conditions, foreign investors with lower cost, higher quality, and better technology than domestic investors partner with the government. Under this condition, it is essential to consider the disruption risks because of sanctions and currency price fluctuations. Furthermore, the limited budget of the government for investing in infrastructure projects is intended. In this paper, the government's disruption risks and limited budget are added to the risk-sharing ratio model for the first time in literature. Moreover, the Pythagorean fuzzy sets (PFSs) are applied to cope with the uncertainty of real-world conditions. The PFSs are more potent than classical and intuitionistic fuzzy sets (IFSs) in dealing with uncertainty. The PFSs provide the membership, non-membership, and hesitancy degree for experts to better address the derived uncertainty of real-world conditions. Also, compared with the IFSs, PFSs prepare more space, consequently providing more freedom to address the uncertainty. Finally, a case study is presented to illustrate the applicability and susceptibility of the suggested model. As disruption risks increase, general utility degree, government utility, and investor’s effort decrease, and the guarantee risk ratio by government increases. Note that, investor’s effort decreases because the government is forced to give the unfinished project to the domestic investor; consequently, exclusive terms arise for the domestic investor.

Fractional programming (FP) refers to a family of optimization problems whose objective function is a ratio of two functions. FP has been studied extensively in economics, management science, information theory, optic and graph theory, communication, and computer science, etc. This paper presents a bibliometric review of the FP-related publications over the past five decades in order to track research outputs and scholarly trends in the field. The reviews are conducted through the Science Citation Index Expanded (SCI-EXPANDED) database of the Web of Science Core Collection (Clarivate Analytics). Based on the bibliometric analysis of 1811 documents, various theme-related research indicators were described, such as the most prominent authors, the most commonly cited papers, journals, institutions, and countries. Three research directions emerged, including Electrical and Electronic Engineering, Telecommunications, and Applied Mathematics.