Multicriteria Approach to Sustainable Transport Evaluation under Incomplete Knowledge: Electric Bikes Case Study

: The problem of sustainable city transport is a growing ﬁeld of study, and will be addressed in this paper. With the rising signiﬁcance of present transportation systems’ negative externalities on the environment, such as the unavoidable increase of air pollution levels, cities seek sustainable means of transport and reduction of combustion cars’ utilization. Moreover, improvements in the area of renewable energy sources have led to rising trends in sustainability, driving the usage and production of electric vehicles. Currently, there is an increasing tendency of looking for more sustainable transport solutions, especially in highly congested urban areas. It seems that in that case, electric bicycles can be a good option, as they yield more beneﬁts in comparison to cars, especially combustion cars. In this paper, we identify an assessment model for the selection of the best electric bicycle for sustainable city transport by using incomplete knowledge. For this purpose, the Characteristic Objects METhod (COMET) is used. The COMET method, proven effective in the assessment of sustainable challenges, is a modern approach, utterly free of the rank reversal phenomenon. The evaluated model considers investigated multiple criteria and is independent of chosen alternatives in the criteria domain. Hence, it can be easily modiﬁed and extended for diverse sets of decisional variants. Moreover, the presented approach allows assessing alternatives under conditions of incomplete knowledge, where some data are presented as possible interval numbers.


Introduction
More than 50% of the world's population lives in urban areas, and it is estimated that by 2045 the number of people living in cities will increase up to 1.5 times-which is around 6 billion people [1]. There are more than 1 billion motorized vehicles worldwide, and with rising income levels, one should expect continuous expansion [2,3]. As a result of high flows of people into cities, transport systems are on the verge of transformation. With rising numbers of people dwelling in urban areas, many challenging needs and problems arise regarding sustainable development, where the critical element is the sustainable transport. The importance of this issue is not only proven by the critical postulates of the Sustainable Development Goals [4], but also by negative aspects current urban transport poses, such as elevated levels of air pollution or more frequent premature deaths from human exposure to harmful pollutants [5]. Diseases caused by pollution were responsible for an estimated 9 million modern cities are limiting the usage of combustion vehicles and some are planning to completely withdraw existing ones or ban future sales [53]. With the encouragement of sustainable means of transport, one can expect an increasing trend towards electric vehicles in the foreseeable future [54]. The dynamically developing technology causes that the set of available sustainable transport variants (e.g., ebikes, electric powered or hybrid cars, emotors) grows rapidly. In such context, it becomes necessary to develop methodological foundations for the evaluation of sustainable transport options. The specifics of this problem, implicating the natural lack of selected data (e.g., malfunction reports data, loss of value of newly-introduced variants over a few-year time span), result in the necessity to develop the methodological guidelines in the environment of incomplete knowledge in the model.
In this paper, the ongoing challenge of sustainable transport will be addressed, which is the selection of the most rational electric bicycle. The paper shows step by step how to identify multi-criteria decision-making assessment model. For this purpose, the Characteristic Objects METhod (COMET) is used, which was proven successful in other sustainable development problems [55]. The COMET method, as one of the multi-criteria decision-analysis technique based on a rule set, is completely free of the rank reversal paradox, which is a key in a proper analysis [56,57]. The obtained model is used to rank the considered bicycles and choose the most rational from the set of ten proposed products. Methodically, work presents a successful attempt of building integrated, inference based model, that aggregates full data with incomplete expert knowledge. In a practical sense, the authors' contribution is an attempt to build a reference ranking of ebikes for a carefully selected extensive set of ebikes, which can provide useful knowledge both for particular decision makers as well as for local communities and policy makers, where the evaluation and selection of sustainable transport variants is at stake.
The rest of the paper is organized as follows: in the next section, the Fuzzy Set Theory preliminaries are outlined. In Section 3, the COMET method is described as a useful tool to identify multi-criteria decision models. Subsequently, in Section 4, an experiment to build a decision model for an assessment of electric bicycles is described step by step, and the results are presented. The conclusions and the possible future directions are presented in Section 5.

Fuzzy Set Theory: Preliminaries
The fuzzy set theory was developed by Lofti Zadeh, who introduced the idea of fuzzy sets in [58]. The growing importance of the Fuzzy Set Theory in model creation in numerous scientific fields has proven to be an effective way to approach and solve multi-criteria decision problems [59][60][61]. The necessary concepts of the Fuzzy Set Theory are described as follows [62][63][64]: Definition 1. The fuzzy set and the membership function-the characteristic function µ A of a crisp set A ⊆ X assigns a value of either 0 or 1 to each member of X, and the crisp sets only allow a full membership (µ A (x) = 1) or no membership at all (µ A (x) = 0). This function can be generalized to a function µÃ so that the value assigned to the element of the universal set X falls within a specified range, i.e., µÃ : X → [0, 1]. The assigned value indicates the degree of membership of the element in the set A. The function µÃ is called a membership function and the setÃ = (x, µÃ(x)), where x ∈ X, defined by µÃ(x) for each x ∈ X is called a fuzzy set [65,66].

Definition 2.
The triangular fuzzy number (TFN)-a fuzzy setÃ, defined on the universal set of real numbers , is told to be a triangular fuzzy numberÃ(a, m, b) if its membership function has the following form [65,66] (1): and the following characteristics, Equations (2) and (3): Definition 3. The support of a TFN-the support of a TFNÃ is defined as a crisp subset of theÃ set in which all elements have a non-zero membership value in theÃ set [65,66], Equation (4): Definition 4. The core of a TFN-the core of a TFNÃ is a singleton (one-element fuzzy set) with the membership value equal to 1 [65,66], Equation (5): Definition 5. The fuzzy rule-the single fuzzy rule can be based on the Modus Ponens tautology [65,66]. The reasoning process uses the IF-THEN, OR and AND logical connectives.

Definition 6.
The rule base-the rule base consists of logical rules determining the causal relationships existing in the system between the input and output fuzzy sets [66,67].

Definition 7.
The T-norm operator (intersection)-the T-norm operator is a T function modelling the AND intersection operation of two or more fuzzy numbers, e.g.,Ã andB. Basic requirements for a function T is described by four properties: boundary Equation (6), monotonicity Equation (7), commutativity Equation (8), and associativity Equation (9) (for any a, b, c, d ∈ [0, 1]).
T(a, b) < T(c, d) ⇔ i f a < c and b < d T(a, b) = T(b, a) T(a, T(b, c)) = T(T(a, b), c) In this paper, the product is used as the T-norm operator [65][66][67], Equation (10): Definition 8. The S-norm operator (union), or T-conorm-the S-norm operator is an S function modelling the OR union operation of two or more fuzzy numbers, e.g.,Ã andB. Basic requirements for a function S is described by four properties: boundary Equation (11), monotonicity Equation (12), commutativity Equation (13), and associativity Equation (14) (for any a, b, c, d ∈ [0, 1]).

The Characteristic Objects Method
Many MCDM methods exhibit the rank reversal phenomenon; however, the Characteristic Objects Method (COMET) is completely free of the this problem. In previous works, the accuracy of the COMET method was verified [68]. The formal notation of the COMET method should be briefly recalled [62][63][64]. The whole decision-making process by using the COMET method is presented in Figure 1.  Step 1. Definition of the space of the problem-the expert determines the dimensionality of the problem by selecting r criteria, C 1 , C 2 , · · · , C r . Then, a set of fuzzy numbers is selected for each criterion C i , e.g., {C i1 ,C i2 , · · · ,C ic i } (16): where C 1 , C 2 , · · · , C r are the ordinals of the fuzzy numbers for all criteria.
Step 2. Generation of the characteristic objects-the characteristic objects (CO) are obtained with the usage of the Cartesian product of the fuzzy numbers' cores of all the criteria: As a result, an ordered set of all CO is obtained: where t is the count of COs and is equal to: Step 3. Evaluation of the characteristic objects-the expert determines the Matrix of Expert Judgment (MEJ) by comparing the COs pairwise. The matrix is presented below: where α ij is the result of comparing CO i and CO j by the expert. The function f exp denotes the mental judgement function of the expert. It depends solely on the knowledge of the expert. The expert's preferences can be presented as: After the MEJ matrix is prepared, a vertical vector of the Summed Judgments (SJ) is obtained as follows: Eventually, the values of preference are approximated for each characteristic object. As a result, a vertical vector P is obtained, where the ith row contains the approximate value of preference for CO i .
Step 4. The rule base-each characteristic object and its value of preference is converted to a fuzzy rule as: In this way, a complete fuzzy rule base is obtained.
Step 5. Inference and the final ranking-each alternative is presented as a set of crisp numbers, e.g., A i = {α i1 , α 2i , α ri . This set corresponds to the criteria C 1 , C 2 , · · · , C r . Mamdani's fuzzy inference method is used to compute the preference of the i − th alternative. The rule base guarantees that the obtained results are unequivocal. The bijection makes the COMET completely rank reversal free.

Study Case
This paper presents the assessment model of decision-making concerning sustainable city transport. Selecting an optimal electric bike, especially considering multiple criteria, is a complex challenge. Such compound problems require the expert's knowledge to establish the requirements for the model and therefore, based on the expert's opinion, eight criteria were selected. Hence, the space of the investigated problem is equal to 8. Based on [69][70][71][72][73][74][75][76][77][78][79][80], the following criteria are specified:  [74,79].
In this study case, the considered problem is simplified to a simple structure which is presented in Figure 2. In that way, we have to identify four related models, where each one requires a lot smaller number of queries. The decision model can be presented as the following models: • battery effectiveness assessment model with two inputs (9 characteristic objects and 36 pairwise comparisons are needed); • engine assessment model with two inputs (9 characteristic objects and 36 pairwise comparisons are needed);  Figure 2. The hierarchical structure of the electric bicycle assessment problem.
In Table 1, criteria are presented along with their respective linguistic values. Table 2 presents 10 exemplary ebikes along with the explanation of the model foundations, whereas the complete model for 64 bikes is presented in Section 4.7.  In order to identify the final model for the electric bicycle assessment, we first have to determine the following assessment models, i.e., battery effectiveness, engine, drive system and comfort of the usage.

Battery Effectiveness Assessment Model
The expert identified two important criteria for the battery effectiveness assessment model, battery capacity, expressed in Ampere-hours [Ah] and charging time expressed in hours [H]. The former is a profit type criterion; hence the value increase implies the preference increase, whereas the latter is a cost-type criterion; hence the value increase implies the preference decrease. It should be noted that the relationship is rarely linear in such complex problems. The triangular fuzzy numbers of criteria C 1 and C 2 are depicted in Figure 3. The characteristic objects CO 1 -CO 9 are created using the Cartesian product of the fuzzy numbers' cores of criteria C 1 and C 2 and are presented in Table 3. The space of the problem, including characteristic objects and alternatives, is presented in Figure 4.    The expert performed 36 pairwise comparisons of the characteristic objects. As a result, the Matrix of Expert Judgment (MEJ) was determined, where each α ij value was calculated using the Equation (21). The MEJ matrices are presented in Figures 5 and 6, where the α ij values of either 0, 0.5 or 1 are represented by white, black and gray boxes, respectively. Consequently, the vector of the Summed Judgements (SJ) was calculated using Equation (22), and hence, it was to used to calculate the values of preference (P), presented in Table 3.

Engine Assessment Model
The expert identified two criteria for the engine assessment model, namely engine power, expressed in Watts [W] and maximum speed reached by electric bicycle in electric mode expressed in kilometers per hour [km/h]. Both are profit type criteria, hence as stated before, with the increase in values, preference increases. The triangular fuzzy numbers of criteria C 4 and C 5 are depicted in Figure 3. The characteristic objects CO 1 -CO 9 are created using the Cartesian product of the fuzzy numbers' cores of criteria C 4 and C 5 and are presented in Table 4. The space of the problem, including characteristic objects and alternatives, is presented in Figure 7.  Figure 5. The matrix of expert judgement for P 1 (a), P 2 (b), and P 3 (c).

Comfort of Usage Assessment Model
Two important criteria were selected for the engine assessment model, range, expressed in kilometers [km] and weight of the bicycle in kilograms [kg]. The first one is a profit type criterion and the second one is a cost-type criterion. The triangular fuzzy numbers of criteria C 6 and C 7 are shown in Figure 3. The characteristic objects CO 1 -CO 9 are created using the Cartesian product of the fuzzy numbers' cores of criteria C 6 and C 7 and are presented in Table 5. The space of the problem, including characteristic objects and alternatives, is presented in Figure 8. Table 5. The results for criteria C 6 (range) and C 7 (weight) and their characteristic objects' values of preference P for CO i .

Drive System Assessment Model
For the model of drive system assessment, the variables from previous models are used, namely battery effectiveness assessment model and the engine assessment model, P 1 and P 2 respectively. Additionally, the third criterion is included-the number of gears available in the bicycle derailleur system C 3 . The aggregated variables P 1 , P 2 and criterion C 3 are all profit type. The triangular fuzzy numbers of criterion C 3 is presented in Figure 3. The characteristic objects CO 1 -CO 12 are presented in Table 6.

Electric Bicycle Assessment Model
The final model for the electric bicycle assessment has three input variables. Two aggregated variables were used, the output variable from the drive system assessment and the output variable from the comfort of usage assessment. Moreover, we include the third input-the criterion price C 7 . The aggregated variables P 1 , P 2 are both profit type, whereas the criterion C 7 is cost type. The triangular fuzzy numbers of criterion C 7 is shown in Figure 3. The characteristic objects CO 1 -CO 12 are presented in Table 7.

Final Ranking
The final preference and the ranking for the ten considered alternatives of electric bicycles are presented in Table 8. Concerning the case with the full knowledge, when we always assume the best value of the interval, then the best variant is bicycle Emu Crossbar (preference value 0.6119), slightly exceeding the Rattan bicycle (preference value 0.6056). Undoubtedly, the worst alternative is the ANCHEER Plus model (preference value 0.3752). The second worst, however, not that close to the least optimal one, is the Xiaomi QiCycle model (preference value 0.4063). The Emu Crossbar scored such high results due to the fact, that has the most extended range (up to 100 km) and the highest battery capacity (14.5 Ah), which outweighs its disadvantages, such as high price. However, when considering Rattan model (second best), it is worth stating that, although it has lower battery capacity than Emu Crossbar, it has also lower charging time (which is cost type), hence the overall preference for battery assessment is greater in Rattan model. Compared to Emu, Rattan is also two times cheaper and has a higher preference regarding engine assessment. However, its rather average range contributed to only second place. However, these were the rankings under assuming full knowledge as the best value from the uncertainty intervals. When we do take them into account, the final ranking has to be interpreted differently-taking into example the Emu Crossbar, which the final preference interval is equal to [0.4219, 0.6119]. While considering the highest values from intervals made Emu clearly the the first, it is worth noticing how broad the interval is, mainly due to the fact that his range C 6 and his preference value for comfort of usage assessment P 4 have extensive intervals, 55-100 (km) and [0.3781, 0.8000], respectively. Thus, under some degree of uncertainty, the final ranking varies greatly. Hence, the Emu still might be first, but one has to be aware of the fact that considering the lowest values from the interval, Emu belongs closer to the end of the ranking.
On the contrary, the Rattan bicycle (second place), which also exhibits the problem of uncertainty, has a much lower range of the interval [0.5800, 0.6056]. Even with the lowest value, Rattan still has a high preference value (P = 0.5800), whereas Emu had (P = 0.4219). Therefore, interval numbers can be a useful indicator of the stability of the final ranking, under conditions of incomplete knowledge, which can easily change the definitive ranking of alternatives.

Practical Exploitation of the Identified Model
The proposed methodological aspects of the identification of the assessment model in the conditions of incomplete knowledge are presented on a set of 10 arbitrarily selected decision variants. In practice, the number of possible decision options is often much higher. Therefore, as a result of the analysis of literature and websites [81][82][83][84][85][86][87], the set of evaluated variants was extended by another 54 decision alternatives. The extended set was assessed using the identified model and decision-making system available on wwww.comet.edu.pl. Table 9 presents the analyzed set together with the determined assessments of preferences (P). The worst option was the California Bicycle Factory Retro S bike (with 0.3669 point). It had a low battery capacity (it was 8 ah), which was not compensated by charging time (it was 4 hours), low number of gears, low engine power, high weight, high price ($2499) and average maximum speed. The next variant was only improved by 0.0229 of point. On the other hand, the best option was the Blix Sol bike, which was rated 0.687 of point. The parameters were mostly medium; however, the low price and big range were crucial to the victory of this option. Table 9. The results of the model for a wider domain of e-bikes according to [81][82][83][84][85][86][87]. It can often happen that some parameters describing variants are uncertain, e.g., the maximum distance may depend on the weight of the passenger, orography of the city and the climate (cold climate shorter distance). The given catalog values are very often averaged experimental results, therefore it seems justified to use interval numbers. It is difficult to assume that the vehicle will be moved in the same conditions as in laboratory tests.

Conclusions
Sustainable development is becoming a more complex problem, one that requires a modern approach with the use of various methods. In this paper, the sustainable city transport challenge was addressed in the context of selecting the electric bicycle. Methodically, the paper presents a successful attempt to adapt an MCDA method called COMET to the needs of evaluation of sustainable transport.
The fuzzy set theory with the COMET method was used to investigate and create the decision model under the condition of full knowledge and with some degree of uncertainty, using the interval numbers. As a result of the preformed research, also a practical contribution in the sustainable transport domain was obtained. The presented model for ebikes evaluation based on an extensive set of available options contains important domain knowledge supporting the decision makers in the process of evaluation and selection of proper variant of sustainable transport. Eight criteria were taken into consideration (battery capacity, charging time, number of gears, engine power, maximum speed, range, weight, and price); however, to reduce the amount of pairwise comparison, the final assessment model was restructured into sub-models, significantly reducing the complexity of the problem and restructuring the criteria into groups. Ten alternatives to electric bicycles were examined in details (from an extensive set of bikes presented in Table 9. The practical implications in the domain of sustainable transport provide a reference model for ebikes. It is worth adding that the proposed approach has considerable practical usability. The freely available software on the comet.edu.pl website results in the model being easy to validate even for non-experienced users.
For further study, it would be valuable to extend this model with additional criteria and sub-models, creating an even more detailed selection of electric bicycles. Additionally, further works concerning the incomplete knowledge case could be performed.