Selection of possible scenarios for improving the quality of public transport services through the use of hybrid Fuzzy-MCDM models

A unified calculating approach is needed for public passenger transportation. All public transport companies and other stakeholders would have additional opportunities to create a transport offer if the unified methodology was made available to them and if calculations and calculation criteria were harmonized. Thus, the main goal – improving citizen mobility – would be accomplished. For this reason, in the study, we suggested the hybrid fuzzy methods for evaluating and improving the quality of public transport service. Unreliable responses of survey participants often distort group decision-making regarding the problem of public services, negatively affecting the end of the calculation procedure. Fuzzy multicriteria decision-making approach has been used. The suggested technique has the advantage of taking into account the degree of fuzzification of respondents' judgments about the choice scenario, while also using two MCDM models to eliminate bias in the responses.


Summary.
A unified calculating approach is needed for public passenger transportation. All public transport companies and other stakeholders would have additional opportunities to create a transport offer if the unified methodology was made available to them and if calculations and calculation criteria were harmonized. Thus, the main goalimproving citizen mobilitywould be accomplished. For this reason, in the study, we suggested the hybrid fuzzy methods for evaluating and improving the quality of public transport service. Unreliable responses of survey participants often distort group decision-making regarding the problem of public services, negatively affecting the end of the calculation procedure. Fuzzy multicriteria decision-making approach has been used. The suggested technique has the advantage of taking into account the degree of fuzzification of respondents' judgments about the choice scenario, while also using two MCDM models to eliminate bias in the responses.

INTRODUCTION
Sustainable development requires the implementation of activities in accordance with the existing needs [1], while being aware of the limited possibilities of natural resources [2]. This means the necessity to search for and develop new technologies that will enable the reduction of energy consumption. And the improvement should not include travel limitation [3] It should be remembered that in relation to transport, many guidelines indicate the implementation of shared journeys (public collective transport) as a solution that positively affects the transport system, but also the environment, through reducing the load on the transport network or reducing the emission of harmful substances [4], [5], [6], [7]. The increase in the number of public transport passengers requires a constant improvement in the quality of services in order to maintain the current mode of travel, as well as to convince other people. The quality of collective public transport is determined using a set of different parameters. Starowicz [8] distinguished such parameters as: frequency, punctuality, reliability, comfort, information, travel cost, contact with the customer, time availability and travel time. In the handbook [9] reliability and travel time are called the key parameters related to the perception of the public transport system by users. In other publications, attention was also paid to the safety aspect (among others [10], [11]). Redman et al. [12] also identified vehicle condition among the quality parameters. The quality of the fleet is also indicated in the reports of the collective transport assessment by BEST [13].
One of the decision-making problems is the assessment of the quality of public collective transport, and thus the possibility of indicating places that require improvement. The organization and implementation of efficiently functioning public transport should mainly result from the cooperation of three stakeholders -cities, carriers and public transport managers in a selected area (depending on the adopted management pattern). Based on the literature review, the article divides selected quality parameters of public collective transport, creating a hierarchical system from them, and then identifies a number of activities (scenarios) aimed at improving the functioning of this type of transport. In addition to the scenarios indicated below, the considerations also included, among others, construction of transfer stations and implementation of dynamic passenger information. However, due to the fact that the city of Katowice was selected for the case study, where both solutions have already been implemented, the list of scenarios has been reduced. Possible actions (scenarios) ultimately included: on the city side -separation of bus lanes and bus priority at intersections (to a greater extent than at present), and on the carrier's side -fleet replacement. Two scenarios were identified for the manager of public transport -changing (improving) timetables and introducing new lines and bus stops. Finally, for the selected set of parameters (criteria) and scenarios, a fuzzy multicriteria decision-making approach (MCDM) was used. It is just an option for decision makers to support them with decision-making process.

METHOD
It is always problematic, which multi-criteria decision-making approach should be chosen. Sometimes a very general approach can be used like in [14] to support finding the best option to make travel by using all sharing modes and public transport services. Broniewicz and Ogrodnik [15] analysed a lot of scientific papers about the transport sector published between 2000 and 2021 and proofed that AHP is the most popular MCDM method, and the next one is TOPSIS. In multi-criteria approach, the relevant parameters or criteria often have different dimensions, which may result in difficulties in assessment. To prevent this problem, the Fuzzy approach is required [16]. Authors already used fuzzy MCDM to choose scenarios for the management of the railway transportation company [17]. In the current paper Fuzzy AHP for calculating weights is used, and it is mixed with Fuzzy Topsis for determining priority in scenarios.
The steps of the calculation procedure are described below: Step 1: The Decision Maker evaluates the criteria or scenarios by comparing them using the linguistic concepts in Table 1. It includes comparison of the Saaty scale and fuzzy triangular scale (FTS). The FTS is used to represent the statement "Criterion 1 (C1) is Weakly Important than Criterion 2 (C2)" if the decision maker states "Criterion 1 (C1) is Weakly Important than Criterion 2 (C2)" (2,3,4). On the other side, the pairwise contribution matrix of the criteria will be compared using the FTS between C2 and C1.  (9,9,9) The intermittent values between two adjacent scales 2 (1,2,3) 4 (3,4,5) 6 (5,6,7) 8 (7,8,9) The pairwise contribution matrices are shown in Eq. 1, where ̃i j k denotes the preference of the k th decision maker for the i th criterion over j th criterion. The triangular number demonstration in this instance is denoted by "tilde," and the preference of the first decision maker for the first criterion over the second criterion is denoted by ̃1 2 1 , which equals ̃1 2 1 = (2, 3, 4).
Step 2: When there are several decision-makers, the average of each person's preferences (̃ij k ) is used to calculate (̃ij) , which is then computed according to Eq. 2.
Step 4: As stated in Eq. 4, according to Buckley [20], the geometric mean of fuzzy comparison values for each criterion is produced. Represents triangular values in this situation.
Step 5: Using Eq. 5, get the fuzzy weights of each criterion by integrating the next three substages.
Step 5a: Calculate the vector sum of each ĩ.
Step 5c: Multiply each ĩ with this reverse vector to get the fuzzy weight of criteria i (̃i).
To find the weights criterion, follow these five steps.
Step 6: The decision-maker is advised to quickly examine the ratings of scenarios on a number of subjective factors using the linguistic variables (given in Table.2). These linguistic variables could be described by the triangle-shaped fuzzy number x ij = (aij, bij, cij). . .
x mn ] where x ij ,i,j is the fuzzy rating of scenario Ai (i = 1,2,…, m) with respect to criterion Cj and w j (j = 1, 2,…, n) is the weight of criterion Cj.
Step 7: The transformation of the several criterion scales into a single scale using the linear scale method ensures consistency between language evaluations of subjective criteria and objective criteria evaluation. The normalized fuzzy decision matrix R may be represented as follows: where B and C are the set of benefit criteria and cost criteria, respectively.
The normalizing procedure discussed above preserves the condition that the ranges of normalized fuzzy numbers correspond to [0,1].
Step 8: Calculate the weighted normalized fuzzy decision matrix: Step Step 10: Compute the distance from each scenario to the FPIS and FNIS: Step 11: Compute the closeness coefficient CCi for each scenario. For each scenario Ai, we calculate the closeness coefficient CCi as follows: Step 12: Rank the scenarios. The scenario with the highest closeness coefficient represents the best scenario.

Determining weights of criteria
As a case study, Katowice city was selected. In the case, exactly three stakeholders existurban authorities (responsible for transport infrastructure), carriers (responsible for bus fleet) and public transport managers (responsible for the organization of public transport in the area). Figure 1 contains selected criteria with a hierarchical structure. Possible scenarios are also shown. The list of criteria and scenario is based on literature review (described in the Introduction section). To determine the criteria and evaluate the scenarios, a meeting was arranged with a group who specializes in urban transportation. In order to determine the mean pair-wise comparison of the criterion and sub-criteria based on his preferences, questionnaires were employed. After the first three stages of the procedure have been finished, Eq. 4 is used to obtain the geometric mean of the fuzzy comparison values for each criterion and sub-criterion. For the "Quality of realization of passenger transport" criteria, for instance, Eq. 14 produces thegeometric mean of fuzzy comparison values.
For all criteria and sub-criteria, The fuzzy weight of the sub-criteria in the step 5 is determined using Eq. 5. In Table 7

Determining scenarios
The decision-maker evaluates the rating of scenarios in regard to each criterion by using the linguistic rating variables (given in Table 2).
Tab. 8 Linguistic variables for the ratings The Fuzzy Positive Ideal Solution (FPIS) and the Fuzzy Negative Ideal Solution (FNIS) are computed in accordance with step 9 after steps 6, 7, and 8 have been completed to produce the fuzzy weighted matrix. As a consequence,

CONCLUSIONS
The constant increase in the quality of public collective transport is one of the foundations of the city's transport system. In this approach, the presented analysis may support decision makers. Of course, it is only an example of how the fuzzy-MCDM approach can be used in relation to collective public transport problems.
The use of the above method allowed for the evaluation of the public transport system. In the case, four scenarios (A1-A4) were taken into account. Changing in timetables (A3) is the best choice, according to table 10, which is the outcome of picking scenarios. Introducing new lines and bus stops (A4), fleet replacement (A1), and separation of bus lanes (A2) are rated lower.
The approach presented in the article relates only to one of the pillars of the transport system elements organized by the city. The aim of further research will be to build a model for assessing the functioning of the city's transport system in a broader sense (taking into account other municipal services and individual transport as a whole).