1 Introduction

According to the World Health Organization (WHO), “Transplantation is the transfer of human cells, tissues or organs from a donor to a recipient with the aim of restoring function(s) in the body” [1]. The term “organ transplantation” refers to the transplantation of solid transplantable organs including heart, lung, kidney, liver, pancreas and intestine. There are two types of donated organs: (1) cadaveric organ (i.e., organ procured from deceased donor) and (2) living donor organ (i.e., organ procured from living donor). Many patients fail to find a living donor organ so they register for a cadaveric organ transplant [2]. According to Organ Procurement and Transplantation Network (OPTN) national database a total number of 744,534 organ transplants have been performed as of 2017, out of which 80.27% were from deceased donors [3]. Among all transplantable organs, kidney is the most demanded one. For instance, in USA as of April 2017 and in Iran as of December 2011, 82.82% and 89.72% of waitlisted candidates were kidney transplant candidates respectively [4, 5].

Transplantation has become a popular and successful cure for many fatal diseases (e.g., end stage liver diseases and end stage renal diseases) in the past 50 years. The number of successful transplantations has increased worldwide. In Iran the number of successful cadaveric organ transplants including kidney, liver, heart and lung transplants has increased from 47 in 2000 to 1180 in 2011. However, the total number of registered patients waiting for at least one of the aforementioned organs was 19,761 in 2011, out of which 86% remained on the waitlist or rarely received organ from living donors [5]. This implies the unmet demand of organs and crucial role of organ transplantation network administers in managing organs as rare public resources.

Several international and national organizations are established in different countries to administer the transplantation activities (e.g., matchmaking and finding the best recipients in candidate pools). Eurotransplant in Europe, United Network for Organ Sharing (UNOS) in the United States of America and Iranian Network for Organ Procurement and Transplantation (IRNOPT) in Iran are examples of international and national organizations. The allocation of organs in countries with national organizations is usually done based on a hierarchical conduction, i.e., the organs are first allocated locally then regionally and eventually nationally [6]. The whole country is categorized into regions, and further into local donor service areas. Each of these local areas is under the jurisdiction of one Organ Procurement Organization (OPO) [7]. Once an organ is available, the local OPO’s waitlist is searched for a suitable recipient and this is called the matchmaking process. During this process the incompatible patients are screened out, and others are ranked according to some organ-specific criteria. If no suitable patient is found in the local area, the search takes place regionally, and thereafter nationally [8]. There is another allocation method named centralized method, in which the search takes place centrally in the national-level waitlist. Eurotransplant is a good instance of organizations following the latter allocation conduction [9].

The management of organ transplantation process is a sophisticated task due to many restrictions such as time constraint. Deceased donors are kept artificially alive and this status cannot last for a long time. Therefore, organ allocation and distribution activities such as matchmaking and transportation of organs/recipients should be accomplished as fast as possible. In addition, once an organ is harvested from donor body, it should be implanted into the recipient’s body within the organ’s maximum allowable Cold Ischemia Time (CIT) (four-six hours for heart and lung, eight-twelve hours for liver, twenty four-thirty six hours for kidney) [10]; otherwise it will be wasted. CIT is the time during which the organ can survive in the absence of blood perfusion. It starts after organ harvesting operation and ends up by implantation of organ to the recipient body. It is desirable to minimize the CIT of organ in order to obtain better post-transplantation outcomes.

An important issue in the optimization of organ transplantation network that arises from the ever-growing gap between supply of and demand for organs is the design and evaluation of fair and efficient distribution and allocation policies [9]. The concept of distributive justice –how to fairly divide resources– arises around organ transplantation because of the aforementioned huge gap. Distributive justice theory states that there is not one universally accepted right way to distribute organs, but rather many ways a person could justify giving an organ to one particular individual over someone else [11]. According to University of Washington School of Medicine website some possible distributive justice criteria include: to each person an equal share; to each person according to need, effort, contribution, merit, free-market exchanges [12].

One set of distributive justice criteria concern equal access (equity). Organs allocated according to equity criteria are distributed to patients based on objective factors aimed to limit bias and unfair distribution. An example of equity criteria is length of time waiting on the waitlist (i.e., first come, first served).

Equity supporters believe that organ transplantation is a valuable medical procedure and worth offering to those who need it. They also argue that because the procedure is worthy, everyone should be able to access it equally [13]. To encourage equality in organ transplantation, the equity theory encourages a distribution process for transplantable organs that is free of biases based on race, sex, income level and geographic distance from the organ [14].

Another type of distributive justice criteria is maximum benefit (medical efficiency). The goal for maximum benefit criteria is to maximize the number of successful transplants. Examples of maximum benefit criteria include: Medical need (i.e., the sickest people are given the first opportunity for a transplantable organ) and probable success of a transplant (i.e., giving organs to the person who will be most likely to live the longest). People who support the maximum benefit philosophy believe organ transplants are medically valuable procedures and wish to avoid the wasting of organs because they are very scarce [15]. To avoid waste, they support ranking transplant candidates by taking into account how sick the patient is and how likely it is that the patient will live after he or she receives a transplant.

Successful transplants are measured by the number of life years gained (e.g., Life Years From Transplant (LYFT)). Life years are the number of years that a person will live with a successful organ transplant that they would not have lived otherwise. This philosophy allows organ procurement organizations to take into account several things when distributing organs that the equity philosophy does not – like giving a second organ transplant to someone who has already had one or factoring in the probability of a successful medical outcome [16, 17].

An allocation system tends to prioritize waitlisted patients according to some organ-specific factors (e.g., blood type, waiting time, illness severity and other medical criteria) for receiving a procured organ. Some of these factors fall in the category of equity criteria, while others serve the medical efficiency of organ allocation and distribution so the allocation policy should balance these two categories of factors. An effective integration of such factors in designing a medically efficient and equitable allocation system, which determines to whom the gift of life goes, is essential and should be studied precisely. Different organ allocation methods have been developed, evaluated and implemented by organ transplantation networks around the world so far. The ever-changing behavior of organ allocation systems admits that there is no guarantee that an allocation system currently in use is the best in satisfying both equity and medical efficiency goals of organ allocation. Therefore policy makers are always engaged in the problem of evaluating allocation rules and finding the best allocation systems.

Among all the organ allocation systems, kidney allocation system has undergone many reforms and has been receiving a great deal of attention from researchers because kidney is the most demanded organ. This issue urged us to put the focus of our work on designing a kidney allocation system rather than other organ allocation systems. It is to be noted that the presented allocation approach could be applied to other organs by some modifications such as altering the set of allocation criteria which is unique for each organ-type.

The focus of this paper is specifically on the most popular and convenient cadaveric kidney allocation method which is based on a point system and is currently in use by many organizations such as UNOS and IRNOPT.

A point system prioritizes compatible candidates for receiving a cadaveric kidney based on a scoring rule. A scoring rule which is composed of a set of elements and their weights, assigns a ranking score to each candidate by calculating the weighted sum of the aforementioned elements. Kidney Allocation Score (KAS) employed by US Kidney Transplantation Committee (KTC) is an example of such rules. The following notation for KAS adopted from paper by Bertsimas et al. [18] facilitates the notion of a scoring rule:

$$ KAS\left(c,k\right)={\sum}_j{w}_j{f}_{j,\left(c,k\right)} $$
(1)

where c is a given candidate who is compatible for receiving the procured kidney k, f j , (c, k)is the value of jth element of the scoring rule belonging to the pair (c, k) (e.g., time spent on dialysis, time spent on registration, life expectancy of the patient in case he remained on dialysis and the life expectancy in case he received the procured organ), w j is the weight of jth score element and KAS(c, k) is the calculated score of pair (c, k).

A challenge faced by policy makers in designing allocation rules is to select scoring rule elements from a set of allocation factors and determine their associated weights so as to get the best medical efficiency and equity outcomes. Related studies in the organ allocation indicate the high sensitivity of transplantation outcomes to very small changes of the aforementioned weights [18].

Another problem in designing the organ allocation algorithms is the uncertainty in estimated and measured values of medical factors and parameters. Patients cannot describe exactly what has happened to them or how they feel, doctors and nurses cannot tell exactly what they observe, laboratories report results only with some degree of error, physiologists do not understand precisely how the human body works, medical researchers cannot precisely characterize how diseases alter the normal functioning of the body, pharmacologists do not fully understand the mechanisms accounting for the effectiveness of drugs, and no one can precisely determine one’s prognosis [19]. Uncertainties due to laboratory measurements and parameter estimation exist in organ transplantation realm as other parts of medicine. Thus, the allocation algorithm should be able to tackle the inherent uncertainty of input values.

In this paper, a Credibility-based Fuzzy Common Weight Data Envelopment Analysis (CFCWDEA) method is proposed in order to provide solutions for the mentioned challenges and enhance the outcomes of kidney matchmaking process. The kidney allocation method is designed based on a set of four kidney allocation criteria out of which three are commonly used in point systems employed by OPTN and IRNOPT and the other one is related to geographical distance. The presented approach can help decision makers to design and evaluate various organ allocation methods by altering different allocation criteria and opt for the best allocation method while it can handle imprecise input data. As the results of our study will demonstrate the proposed method is practically helpful in the following ways:

  1. 1.

    Employing Data Envelopment Analysis (DEA): Charnes et al. [20] introduced DEA to assess the relative efficiency of a homogeneous group of n operating Decision Making Units (DMUs), such as authority departments, schools, hospitals, or sales outlets. The DMUs usually use a set of I resources, referred to as input indices, and transform them into a set of R outcomes, referred to as output indices. In other words, DEA converts multiple inputs and outputs (associated with each DMU) into a scalar measure of efficiency and accordingly divides DMUs into two categories: efficient DMUs (with identical efficiency scores of 1) and inefficient DMUs. The relative efficiency of a DMU is defined as the ratio of its total weighted output to its total weighted input. In mathematical programming terms, this ratio, which is to be maximized, forms the objective function for the particular DMU being evaluated. A set of normalizing constraints is required to reflect the condition that the output to input ratio of every DMU be less than or equal to unity [21]. Our method treats the candidate-organ pairs (c, k) and their aforementioned elements f j , (c, k) as the DMUs and outputs (inputs) of the DEA method, respectively. It calculates a measure of efficiency for each pair and ranks them accordingly. It helps decision makers to rank the pairs based on the value of their efficiency scores without the need for weights (w j ) or in other words without knowing the coefficients in scoring rule. Therefore, using this approach for candidate prioritization eliminates the mentioned challenge of finding the best weights for scoring rules. The presented approach enables decision makers to build and evaluate various allocation systems based on various sets of scoring rule elements without the difficulty of finding score weights.

  2. 2.

    Using common weights: In conventional DEA methods each DMU is likely to be evaluated with different importance weights for inputs and outputs, which may not be a desired outcome for a decision maker in organ allocation field who expects all DMUs (candidate-organ pairs) to be evaluated with common attribute weights for fair prioritization purposes. DEA allows each DMU to specify its own weights so as to obtain a maximum efficiency score for itself. The flexibility of a DMU to choose its input and output weights in DEA can produce an efficient DMU in two extremes: weighting a spread of inputs and outputs to achieve efficiency; or weighting a single input and/or output by placing inconsiderable weights on other inputs and/or outputs to appear efficient. The latter extreme, which is unrealistic, may result in false efficient DMUs, and aggravate the discriminating power. Hence, using Common Weight DEA (CWDEA) with an improved discriminating power is required to avoid relatively efficient DMUs with an unrealistic weighting structure [21].

  3. 3.

    Using fuzzy sets theory and its principles: Medical errors exist in organ allocation process due to the involvement of human beings and machines. Matchmaking as an important and sensitive part of organ transplantation process needs precise numerical data since it determines who survives and who dies. In practice, available, extracted or estimated data (e.g., Estimated Post Transplant Survival Score (EPTS), LYFT, patient’s Calculated Panel Reactive Antibody (CPRA) level and transportation time) is not precise to accomplish this process up to a desired degree of accuracy due to various practical and economic reasons. Thus, collected data may have some sort of uncertainties and the proposed fuzzy CWDEA approach can be utilized to cope with this important challenge.

  4. 4.

    Incorporating transportation time of kidneys into the allocation criteria set: By the use of the proposed approach, any potential criterion could be added to the set of inputs (outputs) as an allocation criterion and its impact on equity and efficiency of outcomes could be further evaluated through simulation examinations. This paper involves the estimated time of transportation between organ and recipient in an allocation method (based on a point system) as a direct nonmedical allocation factor for the first time. Transportation time is an important criterion in all developed allocation policies. Therefore, organ transplantation organizations, especially those allocating organs on a hierarchical conduction may try to minimize total transportation time of organs through regional and physical configuration design of organ transplantation networks (e.g., location allocation of transplant network facilities). Iranian kidney allocation system allocates kidneys to the candidates in the same Organ Procurement Unit (OPU) service area/region. In this way it prioritizes nearer candidates (in the same OPU service area) over more distant candidates (in other OPU service areas), thus indirectly takes into account the transportation time in kidney allocation process. But the results of our study show that using this criterion directly in calculating an allocation score for further prioritization of candidates within the same OPU service area brings in an improvement in overall outcomes of the allocation policy. To the best of our knowledge no prior work has included transportation time as a direct allocation criterion in the set of allocation criteria in designing and evaluating organ allocation systems. Longer transportation of procured organs leads to organ wastage, damage and increased CIT which is a predictor of delayed graft function in kidney transplantation [22]. Thus choosing nearest candidates decreases CIT of organs and total number of wasted organs; increases total number of transplantations and results in better post-transplant outcomes (e.g., LYFT). As previously mentioned LYFT is viewed as a medical efficiency measure and since involving transportation time brings in an improvement in post-transplant outcomes (e.g., LYFT) and consequently medical efficiency, in the present paper we conceptualize transportation time as a measure of efficiency.

Medical “efficiency” of a transplant must be distinguished from “efficiency” scores that are determined for each DMU by the DEA method. The term “efficiency” in transplantation context refers to the medical benefit of a transplant which cannot be calculated straightforwardly as a single value, although it can be measured by means of some allocation criteria that represent or relate to the medical success of the transplant. On the other hand “efficiency” in the DEA context denotes a score between 0 and 1 which is calculated for each DMU by the DEA model. In our case the efficiency score of each pair describes the priority of its candidate over other candidates for receiving a procured kidney.

Rest of the paper is organized as follows. A brief literature review is presented in Section 2. In Section 3, the proposed CFCWDEA approach is explained and then is followed by its application in Iran’s kidney allocation system in Section 4. Finally, concluding remarks and future research directions are presented in Section 5.

2 Literature review

The existing organ transplantation network planning problems can be categorized with respect to different planning horizons ranging from long-term horizon to mid-term and short-term horizons.

According to Fig. 1, organ transplantation activities comprise organ donation, organ allocation, organ/recipient/medical staff transportation, surgery and recovery/rehabilitation. The long-term decisions including location-allocation of facilities (e.g., OPOs, Transplant Centers (TCs) and shipping agents) [9, 24,25,26], regional configuration design [27,28,29] and contracting problems are related to the structure of organ transplantation network. Organ allocation, transportation of organs/recipients/medical staff and personnel scheduling are examples of short-term decisions among which allocation and distribution of organs have received much attention from researchers.

Fig. 1
figure 1

Organ transplantation network hierarchical planning matrix (Adopted from [23])

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Organ allocation systems and related policies always undergo rapid changes because of the controversial and questionable nature of the optimal allocation problem and the existence of different alternatives [30]. So far various mathematical programming models including analytical and computational models have been developed and implemented to study organ allocation problem from different points of views. A number of mathematical programming models consider the problem from individual patient’s (i.e., who may wish to accept or reject an organ offer) perspective [31,32,33,34] while others concern the society’s (e.g., UNOS policy makers) perspective [18, 33, 35]. From the patient’s perspective the objective is to maximize some measures of that patient’s benefit, typically LYFT. The goal from society’s point of view is to design an organ allocation system so as to maximize some given criteria. Some examples of these criteria include total medical benefit (e.g., total number of transplants or total number of LYFTs) and some measure of equity [30]. In another classification a category of works tackles the development of an allocation policy whereas the other one deals with performance evaluation of allocation systems [36].

Kidney is the most demanded transplantable organ therefore it has obtained much attention rather than other organs in mathematical modeling papers. It is also the focus of the present paper and for that reason we review kidney allocation models in Section 2.1.

2.1 Kidney allocation models

David and Yechiali [37] provided a mathematical model in order to help patients decide whether to accept a living donor organ. They applied their model for the case of kidney. In another work by David and Yechiali [38] the allocation of multiple organs to multiple recipients is formulated as a sequential stochastic assignment model. The objective of this problem is to find an allocation policy which maximizes various optimality criteria including average-reward optimality criterion. Furthermore, David and Yechiali [39] considered a sequential matching problem for the assignment of multiple live donors to multiple waiting recipients. They innovated a reward structure that reflects the tissue matching reality of organ transplants.

Ahn and Hornberger [40] designed a decision making model to take into consideration the patients’ preferences for acceptance of an offered organ. They developed a Markov decision process model to maximize patients’ total expected quality adjusted life years employing expected one-year graft survival rate as an index for kidney acceptability measurement.

Zenios [35] developed a queuing model with reneging to represent the kidney transplant waiting list. He classified patients and organs according to their immunological and demographic characteristics and developed closed-form expressions to identify the main factors underlying the performance of the transplant waiting list. Zenios et al. [41] proposed a dynamic resource allocation model to find an optimal allocation policy. The concerned objective was to maximize the quality adjusted life expectancy and minimize two measures of inequity. A dynamic index policy was obtained through approximate analysis of the optimal control problem and compared with the UNOS policies using a large-scale simulation model. Su and Zenios [32] offered a sequential stochastic model to study the effect of patients’ preferences on the allocation of kidneys. Their approach aimed at finding an optimal allocation policy that maximizes total expected reward for patients. In another paper by Su and Zenios [33] the kidney allocation problem was framed as a sequential stochastic problem which considers both patients’ and society’s perspective. The goal was to determine an allocation policy which maximizes patients’ total expected reward and society’s equity.

Last but not least, Bertsimas et al. [18] proposed a mechanism for estimating the optimal weights of scoring rules involved in the point systems for kidney allocation. Their method generally designs a point system, which is based on the selected scoring components and maximizes medical efficiency (e.g., LYFT), while simultaneously enforcing selected fairness constraints. This mechanism enables decision makers to dynamically change the scoring rules and evaluate the implementation results by means of a simulation model which is provided and used by OPTN. They input historical data, alternative score components and fairness constraints into a Mixed Integer Linear Programming (MINLP) model and obtain optimal scoring rules by determining score elements weights using a linear regression.

In many of the reviewed models, the authors restricted their definition of equity or efficiency to one or two measures and neglected other applicable components of equity/efficiency. Neglecting other components of equity and efficiency and models entailing multiple equity/efficiency components as well as the absence of mathematical and clearer definitions of equity and efficiency can be regarded as gaps in the body of organ allocation literature. All the reviewed allocation models involve medical aspects of kidney transplantation. However, the central decision makers may be willing to consider the economic and logistics aspects of the allocation as well as medical efficiency and equity in designing an allocation system, since they face time and budget limitations in real environment. This issue arouses the need for combining the logistics planning perspective with common organ allocation problems, which is a novel area for research. The present study proposes the first allocation model that takes transportation time (which is a logistics and geographical related parameter) as an undesirable allocation criterion.

Uncertainty due to estimation of parameters, lab measurement errors and unavailability of data is an inherent property of the organ allocation problem, notably kidney allocation problem. However, no prior research paper in the body of literature has incorporated uncertainty into the point system-based allocation of organs. One contribution of our method is that it tackles the uncertainty tainted kidney allocation problem by means of fuzzy mathematical programming.

Another challenge in designing point system based allocation rules after selecting desired score elements, is to find their suitable weights. A number of researchers address the problem of finding best weights for a given set of score components in an allocation rule. The trend of changes in score weights indicates that there are consistently ways that an allocation rule can be improved [10]. The present paper proposes a methodology which works on the same basis as point systems while excludes the need for score elements weights.

3 Methodology

The process of finding the best recipient for a procured kidney is called the matchmaking process, which should last no longer than the maximum CIT of the kidney. In Iran non-profit OPUs are responsible for accomplishment of this process as one of the most important activities in organ transplantation network. There are many quantifiable and unquantifiable factors (e.g., weight, blood type and tissue type of donor and recipient) that determine with whom the procured kidney is compatible. As previously mentioned, in an allocation system in the form of a point system once the compatible patients are identified they are ranked in a descending order, according to summation of points that they are awarded for some specific criteria. These criteria could be:

  • Dialysis Time (DT): The time patients have spent on dialysis or reached end-stage kidney failure. In IRNOPT’s allocation system Waiting Time (WT) of patients is considered the main allocation criterion. But if a patient has been on dialysis before they are registered for kidney transplantation the WT will be calculated from the point dialysis time has started.

  • Number of shared Human Leukocyte Antigens (HLA)Footnote 1 matches between donor and recipient: After determining HLA, the “cross-match” test is performed to indicate if there is specific immune reactivity between the donor and recipient [42].

  • Donor and recipient age.

  • LYFT: LYFT is the estimated incremental years of life that are expected to be achieved with a transplant from a specific available deceased donor, computed as the difference in expected median lifespan with that transplant compared with remaining on dialysis. Expected median lifespan is the expected median survival (lifetime) for a candidate. Expected lifetimes with and without a kidney transplant are calculated based on the medical and demographic characteristics of each candidate. Survival with a kidney transplant incorporates characteristics of the donor kidney as well [43]. The formula used to calculate LYFT is shown in Fig. 2.

  • Kidney Donor Profile Index (KDPI): This is a percentage score that ranges from zero to 100%. The score is associated with how long the kidney is likely to function when compared to other kidneys. A KDPI score of 20% means that the kidney is likely to function longer than 80% of other available kidneys [10].

Fig. 2
figure 2

Formula used to calculate LYFT (Adapted from Scientific Registry of Transplant Recipients (SRTR), 2007 [44])

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  • CPRA: In order to determine whether or not a patient already has any specific HLA antibodies, a lab specialist will test a patient’s blood (serum) against lymphocytes (white blood cells) obtained from a panel of about 100 blood donors. These 100 donors represent the potential HLA makeup for a donor from that area. Percent PRA (%PRA) is the number of reactions within that panel. If a candidate’s serum does not react with any of the donor samples (antigens), the candidate is not sensitized and has a PRA of 0. If a candidate’s serum reacts in 80 out of 100 samples, the patient has a PRA of 80%. Theoretically, that means that if a donor becomes available from that donor pool, the recipient would experience acute rejection 8 out of 10 times. That patient might have to wait a very long time until a compatible donor becomes available. This is why the kidney allocation algorithm gives patients with a PRA of 80% or higher, some additional points. By knowing the number of times various antigens appear in the donor pool, PRA of a specific candidate can be calculated as the likelihood of incompatibility of recipient with the donor. This value which is calculated by computers would be known as the CPRA [45].

In Iran prioritization of eligible HLA matched candidates for kidney allocation is mainly based on the time patients spend on waiting lists. The current kidney allocation system does not take any criterion to measure post-transplant outcomes such as LYFT. For this reason, some kidney recipients have not received a kidney that works as long as they may need and consequently rejoined the list. Another issue with the current system is that some patients must wait much longer than others for a kidney due to their high sensitization level or CPRA. There exist other problems arising from the logistics mismanagement of kidney allocation system which exacerbates the lack of kidneys. This kind of problem arises when a chosen candidate can not be reached within the allowable CIT of procured kidney because of long geographical distance or transportation time. Many of such cases caused the valuable kidneys to be discarded. This paper suggests IRNOPT policy makers calculating and adding LYFT, CPRA and transportation time to the aforementioned current set of criterion (i.e., waiting time of patients) in order to choose pairs that lead to a better equity level (i.e., giving some priority to highly sensitized patients) and better post-transplantation outcomes. Based on previous justifications, WT and CPRA are considered as equity measures and LYFT and transportation time are regarded as measures of medical efficiency in the present study. Since the values of these criteria are tainted with high degree of uncertainty due to recording errors, imprecise medical measurements and in some cases unavailability of exact data, a fuzzy decision making approach is recommended rather than a point system like those being used by KTC in OPTN (OPTNKTC). Another strength of the proposed method over point systems is that it does not need determining score component weights based on experts opinions and this reduces the effort of simulating and evaluating performance of different scoring rules under various weights [18].

The proposed CFCWDEA approach is intended to work on the same concept as popular point systems. In point systems the aim is ranking patients according to weighted sum of points they acquire for a set of criteria. An example of such systems is the one discussed by Bertsimas et al. [18] which was the dominant proposal of OPTNKTC up to that point. The KAS under this proposal entails DT, KDPI, CPRA and LYFT as score components and is as follows:

$$ {\displaystyle \begin{array}{l} KAS\left(p,o\right)=0.8 LYFT\left(p,o\right)\times \left(1- KDPI(o)\right)+\\ {}0.8 DT(p)\times KDPI(O)+0.2 DT(p)+0.04 CPRA(p)\end{array}} $$
(2)

According to this system higher LYFT, DT and CPRA result in a higher KAS and therefore a higher priority. As it is obvious from Eq. (2), the KAS aims for an equitable allocation by giving some priorities to patients who have waited the longest on waiting lists and have a higher sensitization level while it seeks an efficient allocation by giving some points to patients with higher LYFT. In contrast with other score components that are pre-known physiological or non-physiological characteristics of the recipient or kidney, LYFT is an estimated outcome of transplantation that we wish to maximize. In other words it is an estimated value which measures the transplantation success.

One issue faced by decision makers in designing an allocation score is selecting the score elements from a set of allocation factors. In the next step the challenge is determining weights associated with these elements. Our approach eliminates the problem of determining these weights by using DEA method. In the proposed approach compatible patient-kidney pairs (according to compatibility criteria, e.g., blood type compatibility, tissue matches, age and weight of candidates and kidney donor) are considered as DMUs of DEA model. Among selected score components, score elements measuring desirable (undesirable) post-transplant outcomes are considered as desirable (undesirable) DEA outputs. Among other score elements, those with positive coefficients in scoring rules that bring in priority points for patient-kidney pairs are considered as undesirable DEA inputs which we wish to maximize.

According to this concept, the selected scoring rule elements in this problem are classified and treated in the following way. LYFT is considered as the desirable output, WT and CPRA are undesirable inputs (to be maximized) and transportation time is a desirable input (to be minimized) of the DEA method. The aim is to choose the best suitable recipient who gains the longest LYFT while has spent the longest time on the waiting list and has highest CPRA and shortest transportation time to the relevant OPU. Finally, the proposed DEA approach ranks all the eligible patient-kidney pairs according to efficiencies they acquire. The efficiency score of each DMU in present approach could be viewed as the counterpart of an allocation score of each pair in a point system. Decision makers can exclude or add any scoring rule elements to the system and assess the overall outcomes of the new system through simple simulation experiments. The functionality of the proposed approach is schematically shown in Fig. 3.

Fig. 3
figure 3

Schematic view of the proposed kidney allocation approach

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In conventional DEA models each DMU tends to acquire a favorable set of input and output weights, which maximizes its efficiency score. Ranking pairs according to efficiencies which are calculated by conventional DEA methods results in some sort of inequity and unfair evaluation of pairs. As one key factor in assessment of a good allocation system in addition to its efficiency level is the equity level, a DEA approach which uses a common set of weights for all DMUs (candidate-organ pairs) is applied in the proposed method. There are many research papers in the literature that have proposed methodologies to determine one common set of weights for the performance indices of only DEA efficient DMUs [46,46,47,48,49,50,52] over which we opted for the CWDEA approach proposed by Karsak, Ahiska [21] since it better complies with the requirements of the problem at hand. The proposed framework enables further ranking of DEA-efficient DMUs with a notable saving in the number of mathematical programming models solved. As previously mentioned in conventional DEA methods should be solved to determine the efficiency of DMUs, while the method proposed by Karsak, Ahiska [21] reduces this number to one and since the present case involves hundreds of eligible pairs or in other words DMUs, their method is chosen to overcome this complexity.

As previously stated a part of required data related to organs and patients is imprecise due to errors in estimation or measurement of parameters. Our methodology by employing fuzzy sets theory and its principles enables decision makers to cope with the epistemic uncertainty (i.e., lack of knowledge about the precise values of input data) in organ allocation problem.

3.1 Common-weights DEA

Since the seminal work on DEA of Charnes et al. [53], it has been repeatedly applied to performance evaluation of many different kinds of entities engaged in many different activities, in many different contexts including health care, as can be concluded from the review of Hollingsworth [54] who considers 317 efficiency studies, among which 99 are DEA based studies on hospital efficiency [55].

To the best of our knowledge, the paper by Ozcan et al. [56] was the first and only paper which applied DEA method to organ transplantation realm so far. Here we apply DEA method to assess the efficiency of candidate-organ pairs in order to prioritize candidates and allocate organs accordingly.

In this section the method proposed by Karsak, Ahiska [21] is briefly described. Since interested readers may see the original paper for more details the same notation as the one used in the original paper is used here for the sake of integrity and readers convenience. The formulation of the aforementioned CWDEA method for a problem of determining the efficiency of DMU j with outputs r and inputs i is as follows

$$ \mathit{\min}\ M $$
(3)

Subject to

$$ M-{d}_j\ge 0,\forall j, $$
(4)
$$ \sum \limits_r{u}_r{y}_{rj}-\sum \limits_i{v}_i{x}_{ij}+{d}_j=0,\forall j, $$
(5)
$$ \sum \limits_r{u}_r+\sum \limits_i{v}_i=1, $$
(6)
$$ {u}_r,{v}_i,{d}_j\ge 0,\forall r,i,j $$
(7)

where M is the maximum deviation from efficiency, d j is the deviation of jth DMU’s efficiency score from its ideal value of 1, x and y are the values of inputs and outputs and variables v and u represent the weights of inputs and outputs.

The above-mentioned formulation originated from the conventional DEA method proposed by Charnes et al. [20]. It has undergone following changes and replacements:

  • The objective function of maximizing the efficiency of jth DMU (\( {E}_{j_0} \)) is replaced by an equivalent one defined as minimization of jth DMU’s deviation (d j ) from its ideal efficiency score of 1 (\( {d}_{j_0} \)).

  • To avoid unrealistic weight distribution and improve the discriminating power of DEA, the minimax efficiency measure is used. In classical DEA model the objective is to maximize \( {E}_{j_0} \) (or to minimize \( {d}_{j_0} \)). When applying minimax efficiency measure the objective function \( {d}_{j_0} \) is replaced with M. The feasible region for variables v and u in minimax model is the same as that in classical DEA model. It is obvious that jth DMU is efficient (in the classical sense) if and only if the value of \( {d}_{j_0} \) corresponding to the solution that optimizes the objective function of classical DEA model (\( {d}_{j_0} \)) is zero and it is minimax efficient if and only if the value of \( {d}_{j_0} \) corresponding to the solution that minimizes the objective function of minimax model (M) is zero. The minimax criterion does not give favorable consideration to the DMU under evaluation, as the classical DEA criterion does. Therefore, efficiencies acquired by minimax criterion are more restrictive than those obtained under classical DEA. More precisely, if jth DMU is minimax efficient, it must also be DEA efficient, because, by definition, minimax efficiency requires\( {d}_{j_0}=0 \). However, if jth DMU is DEA efficient, it may or may not be minimax efficient, because \( {d}_{j_0}=0 \) does not necessarily mean that M is minimized. Accordingly, we can deduce that the minimax criterion generally yields fewer efficient DMUs. The discriminating power of DEA method can be improved by introducing this new criterion into it. On the other hand, since M is a function of all deviation variables (d j ) and each of them is related to a constraint, minimizing M is, equivalent to imposing tighter constraints on weight variables. In this way, weight flexibility is effectively restricted [57].

  • The weight restriction constraint, i.e., Eq. (6), is added to overcome the difficulty of solving one separate linear programming model for each DMU in order to evaluate a set of n DMUs. This is motivated from the typical Multi Criteria Decision Making (MCDM) approaches in which attribute weights are normalized in a way that they add up to 1. In the optimal solution of formulation (3)–(7), the DMUs with d j  = 0 are declared as minimax efficient, since their efficiency scores, i.e., the ratio of weighted output to weighted input, equal 1. If a ranking among inefficient DMUs is required, the efficiency scores of inefficient DMUs (i.e., the ones with d j  > 0) can be calculated by\( {\sum}_r{u}_r^{\ast }{y}_{rj}/{\sum}_i{v}_i^{\ast }{x}_{ij} \) where\( {u}_r^{\ast } \) and\( {v}_i^{\ast } \) denote the optimal weights for output r and input i, respectively. The use of formulation (3)–(7) enables the computation of efficiency scores for all DMUs through a single formulation. This one-step efficiency computation allows for the evaluation of the relative efficiency of all DMUs based on a common set of weights in contrast with conventional DEA methods where each DMU is evaluated by different weights and n linear programming models need to be solved.

Equations (3)–(7) may still result in more than one efficient DMU. The following CWDEA method can be used in this case:

$$ \mathit{\min}\kern0.5em M-k\sum \limits_{j\epsilon EF}{d}_j $$
(8)

Subject to

$$ M-{d}_j\ge 0,\forall j, $$
(9)
$$ M-\sum \limits_{j\epsilon EF}{d}_j\ge 0, $$
(10)
$$ \sum \limits_r{u}_r{y}_{rj}-\sum \limits_i{v}_i{x}_{ij}+{d}_j=0,\forall j, $$
(11)
$$ \sum \limits_r{u}_r+\sum \limits_i{v}_i=1, $$
(12)
$$ {u}_r,{v}_i,{d}_j\ge 0,\forall r,i,j. $$
(13)

where EF is the minimax efficient DMUs that are determined using Eqs. (3)–(7). Eqs. (8) and (10) are introduced to the model such that the maximum deviation from efficiency could be minimized while the sum of deviations from efficiency of minimax efficient DMUs are maximized. As the new model evaluates aggressively the minimax efficient DMUs, i.e., in a way to decrease their efficiencies, the number of efficient DMUs will decrease. The discriminating parameter kϵ(0, 1] is assigned values ranging from 0 to 1 with a predetermined step size until the model outputs a single efficient DMU [58].

3.2 Credibility-based fuzzy common weights DEA

One advantage of the present allocation approach is that it can tackle uncertain medical and nonmedical parameters involved in organ transplantation. The uncertain parameters in the problem at hand include LYFT, CPRA and transportation time. The value of LYFT is tainted with uncertainties since it is estimated based on historical data from past recipients. CPRA value is calculated based on the results of a test performed on patient’s serum against lymphocytes (white blood cells) obtained from a panel of about 100 blood donors [2]. Thus, CPRA is an uncertainty tainted parameter due to errors in lab measurements and approximation of number of various antigens in a pool of donors. Transportation time of organs/patients is another uncertain parameter of the problem at hand which fluctuates because of changes in climate, unknown transportation modes and routes and traffic conditions. The significance of transplantation outcomes and their high sensitivity to small differences in allocation criteria values urged us to take uncertainties (in values of allocation criteria) into account in kidney allocation for the first time.

One way to manipulate the uncertain data in DEA is via probability distributions. However, probability distributions require either a priori predictable regularity or a posteriori frequency determinations which are difficult to construct in most of real-life problems [59]. Since the uncertainty we encounter in the concerned DEA problem is epistemic, fuzzy set theory and related possibility distributions are applied to model the imprecise parameters. In other words the imprecise parameters involved in our problem are treated as fuzzy variables, which are associated with possibility distributions first introduced by Zadeh [60].

The CWDEA model with fuzzy coefficients, i.e., fuzzy CWDEA is shown as Eqs. (14)–(18).

$$ \mathit{\min}\kern0.5em M $$
(14)

Subject to

$$ M-{d}_j\ge 0,\forall j, $$
(15)
$$ \sum \limits_r{u}_r{\overset{\sim }{y}}_{rj}-\sum \limits_i{v}_i{\overset{\sim }{x}}_{ij}+{d}_j=0,\forall j, $$
(16)
$$ \sum \limits_r{u}_r+\sum \limits_i{v}_i=1, $$
(17)
$$ {u}_r,{v}_i,{d}_j\ge 0,\forall r,i,j. $$
(18)

Parameters with a tilde (~) on represent the uncertain coefficients.

Fuzzy DEA models cannot be solved by standard Linear Programming (LP) solvers because of the fuzzy coefficients. Many approaches have been developed to solve fuzzy DEA models, but each has shortcomings in the way it treats uncertain data as mentioned in [61]. Fuzzy DEA models take the form of fuzzy linear programming models, which are not well defined since ranking of fuzzy sets, such as \( {\sum}_r{u}_r{\overset{\sim }{y}}_{rj}-{\sum}_i{v}_i{\overset{\sim }{x}}_{ij}+{d}_j=0\kern0.5em \)is not clear. Hence, there is no universally accepted approach for solving fuzzy DEA models. According to [62], the proposed approaches for solving fuzzy DEA models can be categorized into four distinct approaches: tolerance approach, defuzzification approach, α-level based approach, fuzzy ranking approach. Interested readers are referred to the paper by Lertworasirikul et al. [62] for more details on advantages and disadvantages of these approaches. Moreover, possibilistic chance constrained programming approaches, which are based on the concept of α-level and employ fuzzy measures (i.e., possibility and credibility measures) to rank fuzzy numbers, have been utilized by Lertworasirikul et al. [61, 62] to solve fuzzy DEA models. This approach does not specifically belong to any of the four mentioned approaches. However, it is related to both α-level based and fuzzy ranking approaches.

In this paper credibility-based possibilistic programming method [63, 64], which is one of the approaches of possibilistic chance constrained programming, is applied to solve the fuzzy CWDEA. Possibilistic chance constrained programming and flexible programming are two main branches of fuzzy mathematical programming. The first one is used to cope with the epistemic uncertainty in values of the parameters. The latter is applied to handle flexible target value of goals and soft constraints fuzziness [65].

Possibility theory was first proposed by Zadeh in the context of fuzzy set theory as a mathematical framework for modeling problems involving epistemic uncertainty [61]. For more details on possibility theory consult Dubois and Prade [66].

Credibility-based possibilistic programming is one of the confident methods of possibilistic chance constrained programming which has the following advantages over other methods:

  • It is based on valid mathematical concepts such as credibility measure of fuzzy numbers. Credibility measure, which is derived from necessity and possibility measures, was first introduced by Liu [63];

  • It can support all different types of fuzzy numbers namely triangular and trapezoidal fuzzy numbers;

  • And it enables decision makers to adjust the levels of confidence under which the constraints hold.

The most significant advantage of this method is that it uses credibility measure, which is a self-dual measure rather than possibility and necessity measures [63]. In other words once the credibility measure equals one it is obvious that the fuzzy event certainly takes place and when the credibility measure equals zero the fuzzy event certainly does not take place theoretically. But when the possibility measure of a fuzzy event equals one there is a chance that the event does not take place. Similarly a necessity measure of zero does not assure that the event won’t take place [65].

The credibility-based possibilistic programming approach treats fuzzy constraints as fuzzy events and converts fuzzy CWDEA model into a CFCWDEA model by using credibility measures of fuzzy constraints. It is shown that for trapezoidal and triangular fuzzy membership functions the CFCWDEA becomes a linear programming model.

Suppose that \( \overset{\sim }{\xi } \) is a trapezoidal fuzzy number \( \overset{\sim }{\xi }=\left({\xi}^1,{\xi}^2,{\xi}^3,{\xi}^4\right) \) and r is a real number, according to Liu [64] and Zhu and Zhang [67] we have the following equations for credibility level of α ≥ 0.5:

$$ cr\left\{\overset{\sim }{\xi}\le r\right\}\ge \alpha \leftrightarrow r\ge \left(2-2\alpha \right){\xi}^3+\left(2\alpha -1\right){\xi}^4 $$
(19)
$$ cr\left\{\overset{\sim }{\xi}\ge r\right\}\ge \alpha \leftrightarrow r\le \left(2\alpha -1\right){\xi}^1+\left(2-2\alpha \right){\xi}^2 $$
(20)

The readers are referred to the original papers for further details and proofs. Equations (19) and (20) can directly be used for transforming fuzzy chance constraints to their deterministic counterparts. To convert fuzzy chance constraint \( cr\left\{\overset{\sim }{\xi}\approx r\right\} \) to its deterministic counterpart the approach proposed by Rouhani [68] is applied. In this approach the credibility measure of equality chance constraints for α ≥ 0.5 could be stated as follows:

$$ cr\left\{\overset{\sim }{\xi}\approx r\right\}\ge \alpha \leftrightarrow \left\{\begin{array}{c}\alpha \leftrightarrow r\ge \left(\frac{\alpha }{4}\right)\left({\xi}^3+{\xi}^4\right)+0.5\left({\xi}^1+{\xi}^2\right)\left(1-\left(\frac{\alpha }{2}\right)\right)\\ {}\alpha \leftrightarrow r\le \left(\frac{\alpha }{4}\right)\left({\xi}^1+{\xi}^2\right)+0.5\left({\xi}^3+{\xi}^4\right)\left(1-\left(\frac{\alpha }{2}\right)\right)\end{array}\right. $$
(21)

According to concepts of chance constrained programming, that deals with uncertainty by specifying the desired levels of confidence with which the constraints hold [69] and credibility of fuzzy events the suggested CFCWDEA model is reformulated as Eqs. (22)–(27).

$$ \mathit{\min}\kern1.25em M $$
(22)

Subject to

$$ M-{d}_j\ge 0,\kern1.25em \forall j, $$
(23)
$$ {\displaystyle \begin{array}{l}- dj\ge \left(\frac{\alpha }{4}\right)\left(\begin{array}{c}\hfill \left(\sum \limits_r{u}_r{y^3}_{rj}-\sum \limits_i{v}_i{x^2}_{ij}\right)+\hfill \\ {}\hfill \left(\sum \limits_r{u}_r{y^4}_{rj}-\sum \limits_i{v}_i{x^1}_{ij}\right)\hfill \end{array}\right)\\ {}+0.5\left(\begin{array}{c}\hfill \left(\sum \limits_r{u}_r{y^1}_{rj}-\sum \limits_i{v}_i{x^4}_{ij}\right)+\hfill \\ {}\hfill \left(\sum \limits_r{u}_r{y^2}_{rj}-\sum \limits_i{v}_i{x^3}_{ij}\right)\hfill \end{array}\right)\left(1-\left(\frac{\alpha }{2}\right)\right),\forall j,\end{array}} $$
(24)
$$ {\displaystyle \begin{array}{l}-{d}_j\le \left(\frac{\alpha }{4}\right)\left(\begin{array}{c}\hfill \left(\sum \limits_r{u}_r{y^1}_{rj}-\sum \limits_i{v}_i{x^3}_{ij}\right)+\hfill \\ {}\hfill \left(\sum \limits_r{u}_r{y^2}_{rj}-\sum \limits_i{v}_i{x^3}_{ij}\right)\hfill \end{array}\right)\\ {}+0.5\left(\begin{array}{c}\hfill \left(\sum \limits_r{u}_r{y^3}_{rj}-\sum \limits_i{v}_i{x^2}_{ij}\right)+\hfill \\ {}\hfill \left(\sum \limits_r{u}_r{y^4}_{rj}-\sum \limits_i{v}_i{x^1}_{ij}\right)\hfill \end{array}\right)\times \left(1-\left(\frac{\alpha }{2}\right)\right),\forall j,\end{array}} $$
(25)
$$ \sum \limits_r{u}_r+\sum \limits_i{v}_i=1, $$
(26)
$$ {u}_r,{v}_i,{d}_j\ge 0,\forall r,i,j. $$
(27)

It should be noted, CFCWDEA model at credibility level of 1 is equivalent to CWDEA model when trapezoidal fuzzy numbers are symmetrical. Since the presented problem entails a mixture of desirable and undesirable DEA inputs as the set of allocation criteria, the method proposed by Zhu [70] is used to deal with undesirable inputs. According to this method undesirable and desirable inputs are denoted as \( {x}_{ij}^U \) and\( {x}_{ij}^D \), respectively. The parameter \( {x}_{ij}^U \) is first multiplied by “-1” and further added to a proper c i to let\( {\overset{-}{x}}_{ij}^U=-{x}_{ij}^U+{c}_i>0 \). Therefore, formulation (22)–(27) for triangular fuzzy coefficients changes to formulation (28)–(35).

$$ \mathit{\min}\kern0.75em M $$
(28)

Subject to

$$ M-{d}_j\ge 0,\forall j, $$
(29)
$$ {\displaystyle \begin{array}{l}- dj\ge \left(\frac{\alpha }{4}\right)\left(\begin{array}{c}\hfill \left(\sum \limits_r{u}_r{y^2}_{rj}-\sum \limits_i{v}_i{{\overline{x}}^{U2}}_{ij}\right)+\hfill \\ {}\hfill \left(\sum \limits_r{u}_r{y^3}_{rj}-\sum \limits_i{v}_i{{\overline{x}}^{U1}}_{ij}\right)\hfill \end{array}\right)\\ {}+0.5\left(\begin{array}{c}\hfill \left(\sum \limits_r{u}_r{y^1}_{rj}-\sum \limits_i{v}_i{{\overline{x}}^{U3}}_{ij}\right)+\hfill \\ {}\hfill \left(\sum \limits_r{u}_r{y^2}_{rj}-\sum \limits_i{v}_i{{\overline{x}}^{U2}}_{ij}\right)\hfill \end{array}\right)\times \left(1-\left(\frac{\alpha }{2}\right)\right),\forall j,\end{array}} $$
(30)
$$ {\displaystyle \begin{array}{l}- dj\le \left(\frac{\alpha }{4}\right)\left(\begin{array}{c}\left({\sum}_r{u}_r{y^1}_{rj}-{\sum}_i{v}_i{{\overline{x}}^{U3}}_{ij}\right)+\\ {}\left({\sum}_r{u}_r{y^2}_{rj}-{\sum}_i{v}_i{{\overline{x}}^{U2}}_{ij}\right)\end{array}\right)\\ {}+0.5\left(\begin{array}{c}\left({\sum}_r{u}_r{y^2}_{rj}-{\sum}_i{v}_i{{\overline{x}}^{U2}}_{ij}\right)+\\ {}\left({\sum}_r{u}_r{y^3}_{rj}-{\sum}_i{v}_i{{\overline{x}}^{U1}}_{ij}\right)\end{array}\right)\left(1-\left(\frac{\alpha }{2}\right)\right),\forall j,\end{array}} $$
(31)
$$ {\displaystyle \begin{array}{l}- dj\ge \left(\frac{\alpha }{4}\right)\left(\begin{array}{c}\hfill \left({\sum \limits}_r{u}_r{y^2}_{rj}-{\sum \limits}_i{v}_i{x^{D2}}_{ij}\right)+\hfill \\ {}\hfill \left({\sum \limits}_r{u}_r{y^3}_{rj}-{\sum \limits}_i{v}_i{x^1}_{ij}\right)\hfill \end{array}\right)\\ {}+0.5\left(\begin{array}{c}\hfill \left({\sum \limits}_r{u}_r{y^1}_{rj}-{\sum \limits}_i{v}_i{x^{D3}}_{ij}\right)+\hfill \\ {}\hfill \left({\sum \limits}_r{u}_r{y^2}_{rj}-{\sum \limits}_i{v}_i{x^{D2}}_{ij}\right)\hfill \end{array}\right)\left(1-\left(\frac{\alpha }{2}\right)\right),\forall j,\end{array}} $$
(32)
$$ {\displaystyle \begin{array}{l}- dj\le \left(\frac{\alpha }{4}\right)\left(\begin{array}{c}\kern1.00em \left({\sum}_r{u}_r{y^1}_{rj}-{\sum}_i{v}_i{x^{D3}}_{ij}\right)+\kern1.00em \\ {}\kern1.00em \left({\sum}_r{u}_r{y^2}_{rj}-{\sum}_i{v}_i{x^{D2}}_{ij}\right)\kern1.00em \end{array}\right)\\ {}+0.5\left(\begin{array}{c}\kern1.00em \left({\sum}_r{u}_r{y^2}_{rj}-{\sum}_i{v}_i{x^{D2}}_{ij}\right)+\kern1.00em \\ {}\kern1.00em \left({\sum}_r{u}_r{y^3}_{rj}-{\sum}_i{v}_i{x^{D1}}_{ij}\right)\kern1.00em \end{array}\right)\left(1-\left(\frac{\alpha }{2}\right)\right),\forall j,\end{array}} $$
(33)
$$ \sum \limits_r{u}_r+\sum \limits_i{v}_i=1, $$
(34)
$$ {u}_r,{v}_i,{d}_j\ge 0,\forall r,i,j. $$
(35)

The maximum recorded value of WT is the c i parameter for WT and the maximum recorded value of CPRA is the c i parameter for CPRA.

4 Case study and numerical analysis

In this section, the proposed CFCWDEA approach is implemented for the studied case, i.e., IRNOPT, and the corresponding numerical results are evaluated and analyzed. The information, data and statistics on the allocation process provided in this section is collected from literature and interviews conducted with kidney transplantation specialists and experts from Department of Transplantation and Special Diseases, Ministry of Health and Medical Education (MOHME) of Iran.

In Iran like other countries kidney is the most demanded and transplanted organ. The sluggish growth of donation rate from 0.25 per million population (pmp) in year 2000 to 9.36 pmp in 2015 and the ever growing gap between supply (i.e., 2273 cadaveric and living kidney transplants in 2011) and demand (i.e., 17,910 waitlisted patients for kidney in 2011) of kidney require new and more serious interventions [5]. One required action by IRNOPT could be improving public awareness and enhancing positive public attitude towards donation and another helpful intervention is making the best use of available organs by distributing them in efficient ways.

IRNOPT is in charge of central management and allocation of organs. It manages the transplantation activities through local OPUs set up in each university of medical science. These nonprofit units are in charge of identification and evaluation of brain death cases, obtainment of donor consent and procurement of organs. Each unit has a chief transplant coordinator who is responsible for making connection with IRNOPT. The allocation of organs in IRNOPT is done based on a hierarchical conduction, i.e., the organs are first allocated locally then regionally and eventually nationally [6]. The whole country is categorized into 31 regions, and further into 53 local donor service areas. Each of these local areas is under the jurisdiction of one OPU [7]. Once an organ is available, the local OPU’s waitlist is searched for a suitable recipient. This is called the matchmaking process. During this process the incompatible patients are screened out, and others are ranked according to some organ-specific criteria. As mentioned in Section 3 the only key factor in ranking compatible patient-kidney pairs in Iranian kidney allocation system is WT of patients. If no eligible patient is found in the local area, the search takes place regionally, and thereafter nationally.

In order to implement and evaluate our method we used 2011’s second three months of kidney transplantation data provided by Department of Transplantation and Special Diseases. It is notable that the estimated values of LYFT parameter were not included in the recorded database, but rather they are estimated by transplant specialists according to historical data and collected through a series of interviews.

4.1 Numerical analysis

Department of Transplantation and Special Diseases offered us a dataset of the second three months of 2011 which includes a sample of 1000 registered patients and 97 kidney offers from deceased donors. We ran the CWDEA model 97 times for each kidney offer on a COREi3 PC with 4 GB of RAM using GAMS 23.9.5 software. The proposed deterministic method contains (R + I + 3n + 1) constraints and (n + R + I) variables. R, I and n denote the number of outputs, inputs and DMUs of the DEA method respectively. Since in the proposed method DMUs are eligible candidate-organ pairs, the number of DMUs differs in each run and is averagely 150. Thus, the average numbers of constraints and variables involved in our problem are 455 and 154 respectively.

As the number of candidate-kidney pairs in our case is almost close to the number of actual compatible candidate-kidney pairs in OPU waitlist search (100–300), and the proposed method produces high quality results in a reasonable computational time, we claim that this method can be used in practice. The developed allocation method is validated in two stages. Firstly, the deterministic CWDEA model is implemented by the use of available nominal data which was used by IRNOPT at year 2011 and its results are compared with those of current IRNOPT’s allocation system. The corresponding results are recorded and tabulated in Table 1.

Table 1 Comparing results of CWDEA and IRNOPT’s system over a dataset covering a period of three months in 2011
Full size table

Secondly, the performance of proposed CFCWDEA approach is compared to the performance of deterministic CWDEA model in order to demonstrate the applicability of the proposed CFCWDEA method under uncertainty. To do so, a set of 10 realizations for all uncertain parameters including CPRA, LYFT and transportation time is simulated for 30 randomly chosen runs (kidney offers). Thereafter, the CWDEA model is run over 10 realizations of data for each of 30 kidney offers which are chosen randomly out of all 97 kidney offers and results are recorded. The CFCWDEA and CWDEA models are also implemented over nominal fuzzy and crisp data respectively for the same 30 runs. The concluding observations are shown in Table 2.

Table 2 Comparing similarities of simulation results with results obtained by CWDEA and CFCWDEA models under nominal crisp and fuzzy data over 10 data realizations for 30 randomly chosen runs
Full size table

According to Table 1, the proposed method is capable of increasing the average LYFT by 21.99%, average CPRA of chosen recipients by 75.88% and reducing the transportation time of chosen recipients by 32.89% for 97 under study kidney offers during a three months period. But as it is expected, our method decreases the average spent WT of chosen recipients by 32.86% since WT was the only determinative factor in IRNOPT’s allocation system. This loss in average WT as an equity measure in organ allocation is compensable since our method makes substantial improvements in other measures of equity (e.g., total CPRA of recipients) and all discussed measures of medical efficiency (i.e., LYFT, transportation time) which were never taken into account as ranking criteria in IRNOPT’s allocation system so far. The higher average of LYFT implies less number of future rejoining recipients in the waiting lists and less total WT of waitlisted patients consequently. As a matter of fact the rate of rejoining is substantial due to the importance of candidates’ lives and lack of organ’s supply. Unfortunately there is not an exact recorded value for this rate in Iran, but according to OPTN data base 11.4% of transplants in USA are repeat transplants and this obviously represents the considerable rate of rejoining and the vital need for reducing this amount [3].

It also results in better use of organs and less organ wastage. The resulted higher average CPRA of chosen recipients in the proposed method indicates that patients with higher sensitivity level have higher chances of getting a kidney offer in comparison to IRNOPT’s allocation system. The substantial decrease in transportation time due to choosing nearest candidates results in less organ loss or more number of transplantations, higher quality of kidneys at the time of transplantation, less delayed kidney functionality and better post-transplant outcomes. That is why transportation time of recipients/organs was treated as a measure of inefficiency in our approach. It is obvious that the proposed method not only made an important and logical tradeoff between equity and efficiency measures but also had taken into consideration other equity measures rather than only WT in allocation process. To assure functionality of the proposed CFCWDEA model under uncertain conditions an eight-step performance evaluation process is designed. It prognosticates which patients would be chosen if the real values of uncertain data were revealed and which of two models (i.e., CFCWDEA model and CWDEA model) provide closest responses to the simulation results. The corresponding steps are as follows:

  • Step 1: Choose 30 kidney offers randomly among all 97 available kidney offers.

  • Step 2: Simulate a set of 10 data realizations for each kidney offer by generating uniformly distributed realized values for uncertain parameters including LYFT, CPRA and transportation time.

  • Step 3: Run the deterministic CWDEA model with each of 10 real datasets for 30 kidney offers and record chosen recipients as the results of each run.

  • Step 4: Run the deterministic CWDEA model with nominal crisp data for each of 30 kidney offers and record the obtained results for each run.

  • Step 5: Run the CFCWDEA model with nominal fuzzy data under desired credibility levels for each of 30 kidney offers and record the results.f each run.

  • Step 6: For each of 30 kidney offers, record the number of times (out of 10 runs) in which the CWDEA model under real data and CFCWDEA model with nominal fuzzy data resulted in choosing the same recipients.

  • Step 7: For each kidney offer, record the number of times (out of 10 runs) in which CWDEA model under real data and the deterministic CWDEA model with nominal crisp data resulted in the same recipient choices.

  • Step 8: For each kidney offer compare the recorded numbers in Steps 6 and 7. The model with the highest associated recorded number is the most reliable model capable of providing more realistic solutions under uncertain data.

The results for 30 × 10 runs and four various credibility levels (0.5, 0.75, 0.9 and 1) are provided in Table 2. In Table 2 each row belongs to each of 30 kidney offers and columns belongs to different models solved under nominal data. Each cell represents total number of times (out of ten realizations) that solving the model in related column under nominal data has the same results as deterministic CWDEA model under realized data. The results can be interpreted in the following way. CFCWDEA approach at credibility levels of 0.5 and 0.9 covered 94% and 82.33% of simulation results respectively while CWDEA approach was able to cover 70.33% of simulation results. In other words as it is shown in Fig.4, CWDEA model under nominal crisp data failed to choose right recipients in 29.67% of simulation runs according to data realizations. This measure of failure was less for CFCWDEA approach at credibility levels lower than 1. Up to this point the functionality of CFCWDEA approach and its superiority over CWDEA approach is illustrated.

Fig. 4
figure 4

Comparing failure rates of CWDEA and CFCWDEA models in choosing right recipients according to simulation results (CFCWDEA model at credibility level of 1 is equivalent to the deterministic CWDEA model)

Full size image

From Fig.4 one may interpret that using CFCWDEA at credibility level of 0.5 is the best way of choosing suitable recipients under uncertainty. However, the fact is that CFCWDEA like other fuzzy DEA methods (e.g., the method proposed by Lertworasirikul et al. [61]) chooses more than one efficient DMUs. The number of introduced efficient DMUs decreases when credibility level increases. In our case the number of efficient DMUs introduced by CFCWDEA varied from 1 to 4 at credibility level of 0.5 and from 1 to 2 at credibility level of 0.9.

In almost all 30 studied kidney offers the set of efficient DMUs at a higher credibility level is a subset of those at lower credibility levels. That is the reason why the sum of all cells in each row exceeds ten. Another issue that is obvious from both Table 2 and Fig.4 is that CFCWDEA at credibility level of 1 and CWDEA are equivalent since our uncertain data in form of triangular fuzzy numbers are symmetrical. To find out which credibility level except 1 is superior for solving our problem, the Spearman’s rank correlation method is adopted like in the paper by Babazadeh et al. [71]. For this purpose the correlation between two sets of ranks obtained by CWDEA model under real data (simulations) and CFCWDEA model is calculated at different credibility levels using measure ρ computed as Eq. (36) for each of 30 × 10 runs.

$$ \rho =1-\frac{6\times \sum \limits_{i=1}^n{d}_i^2}{n\times \left({n}^2-1\right)} $$
(36)

where d i is the difference between ranks of two methods for ith DMU and n is the number of ranked DMUs in each run. The average calculated correlation of ranks obtained by simulation and CFCWDEA is demonstrated in Fig.5.

Fig. 5
figure 5

Average correlation between ranks obtained by CFCWDEA (at various α levels) and ranks obtained by simulations

Full size image

According to Fig.5, simulation results illustrate that the higher the credibility level the higher the correlation between ranks provided by CFCWDEA and ranks obtained by simulations. It can be concluded that selecting CFCWDEA at higher credibility levels (e.g., 0.9) results in the most realistic ranking and most credible efficient recipient choices.

The observed results of above performance evaluation process are illustrated in Fig.6. As it is shown in this figure, the CFCWDEA area indicates the number of times in which running CFCWDEA model (at credibility level of 0.9 under uncertainty) has resulted in the same recipient choices as CWDEA model (under real data). The CWDEA area illustrates the number of times in running CWDEA model (under nominal crisp data) and CWDEA model (under real data) has resulted in the same recipient choices. It is obvious that the CFCWDEA area is 1.17 times bigger than the CWDEA area. This demonstrates the applicability and validation of the proposed CFCWDEA approach.

Fig. 6
figure 6

Comparing results of CWDEA and CFCWDEA at credibility level of 0.9 based on a common set of weights

Full size image

5 Conclusions

Transplantation is the only life-saving treatment for many end-stage diseases (e.g., end-stage liver and heart failure) and the most cost effective treatment for end-stage kidney failure. Despite all the advances in medications and increasing number of implantations in most of the countries, there still exist thousands of patients waiting for organs and many of them die before they are able to receive an organ transplant. Among all organs kidney is the most demanded organ for which the demand exceeds the supply. The huge and ever-growing gap between demand for and supply of kidneys results in the need for fair and efficient allocation and distribution of kidneys.

The allocation systems undergo non-stop changes in search of the most medically efficient and equitable ones. This paper introduces a new point system based allocation method which has many advantages over common point systems. In the proposed method candidate-organ pairs (DMUs) are ranked according to efficiency scores they achieve through a Common Weights DEA (CWDEA) approach. Since fairness is an important objective of allocation systems that policy makers seek to maximize, a CWDEA approach has been applied to rank patients based on a common set of weights.

The kidney allocation criteria (scoring rule elements) are taken as desirable and undesirable inputs (outputs) of the CWDEA method. In this study according to definitions of equity and efficiency the use of LYFT and transportation time as efficiency measures and use of CPRA and WT as measures of equity are suggested. The aim is to make improvements and logical tradeoffs between these measures compared to the current IRNOPT’s allocation method.

One advantage of the proposed DEA approach over conventional scoring rules in ranking pairs is that it eliminates the complexity of finding best weights for scoring rule elements by using a common set of weights. The proposed method has taken the simple criterion of organ transportation time as a logistics criterion in allocation process, which has resulted in a substantial decrease in transportation time of organs and consequently a major potential improvement in post-transplant outcomes and total number of transplants by reducing CIT and organ wastage. The other advantage of this method is its ability of coping with uncertainty in estimated and measured criteria using fuzzy credibility measure which is not found in any other allocation systems. At the end, some simulation experiments are conducted to validate and verify the functionality of CWDEA and Credibility-based Fuzzy CWDEA models when data are certain and uncertain, respectively. According to simulation results, policy makers are strongly recommended to apply the proposed method by taking any desirable set of allocation criteria. Researchers can utilize different sets of criteria and evaluate the outcomes to find the best set of criteria for kidney allocation that brings in improvements in both equity and efficiency measures in future. In addition, the proposed method can be used for other transplantable organs as a future research direction with some little modifications.

Modifying existing models by considering other components of equity and efficiency, developing new models entailing multiple components and proposing a mathematical and clearer definition of equity and efficiency in organ allocation can be studied in future.

As previously mentioned organ allocation problem can be viewed from two perspectives; i.e., patient’s and society’s perspectives. From patient’s perspective the goal is to maximize the patient’s benefit; however from society’s perspective the aim is to maximize the overall benefit of the society. No research work has taken a combination of two perspectives of decision making in organ allocation system design up to this point. One can combine the two perspectives and assess the resulting changes in outcomes in order to find out whether the overall performance of the allocation system improves, if patients’ preferences are taken into account. Accordingly, the method proposed here could further be extended to a bi-perspective approach.

Our method produces reasonable results for a problem containing average numbers of 455 and 154 constrains and variables respectively. The size of this problem is almost near to that of actual allocation (matchmaking) problems in OPU level list. Thus, this method can be used in practice. However, for matchmaking in an upper level list (e.g., regional or national level list) or in a more populated country with larger OPO service areas which contains a greater number of compatible pairs (maybe thousands of pairs), the problem becomes more complex. Thus developing and utilizing suitable algorithms to handle larger problems, tainted with more computational complexity by researchers in future is helpful.