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A fuzzy Full Consistency Method-Dombi-Bonferroni model for prioritizing transportation demand management measures
Journal Pre-proof A fuzzy Full Consistency Method-Dombi-Bonferroni model for priorititizing transportation demand management measures Dragan Pamucar, Muhammet Deveci, Fatih Canıtez, Darko Bozanic
PII: DOI: Reference:
S1568-4946(19)30733-1 https://doi.org/10.1016/j.asoc.2019.105952 ASOC 105952
To appear in:
Applied Soft Computing Journal
Received date : 25 June 2019 Revised date : 12 November 2019 Accepted date : 18 November 2019 Please cite this article as: D. Pamucar, M. Deveci, F. Canıtez et al., A fuzzy Full Consistency Method-Dombi-Bonferroni model for priorititizing transportation demand management measures, Applied Soft Computing Journal (2019), doi: https://doi.org/10.1016/j.asoc.2019.105952. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Elsevier B.V. All rights reserved.
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A Fuzzy Full Consistency Method-Dombi-Bonferroni Model for Priorititizing Transportation Demand Management Measures Dragan Pamucar a,*, Muhammet Devecib,c, Fatih Canıtezd, Darko Bozanice a
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Department of Logistics, Military academy, University of Defence, Belgrade 11000, Serbia, +381668700363, fax: +381113603832, email: [email protected]; [email protected] b Department of Industrial Engineering, Naval Academy, National Defense University, 34940, Turkey c ASAP Research Group, School of Computer Science, University of Nottingham, NG8 1BB, Nottingham, UK, [email protected], [email protected] d Department of Management Engineering, Faculty of Management, Istanbul Technical University, 34367 Maçka, Istanbul, Turkey, [email protected] e University of Defence, Belgrade 11000, Serbia, email: [email protected]
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Abstract:The selection and prioritization of appropriate Transportation Demand Management (TDM) measures is a common problem faced by transport planners and decision makers. The problem involves many uncertainties due to changing economic conditions, uncertainty in project success, changes in mobility and population characteristics etc. In this study, the multi-criteria decision making (MCDM) based fuzzy Full Consistency Method-Dombi-Bonferroni (fuzzy FUCOM-D'Bonferroni) model is proposed for a case study in Istanbul's urban mobility system. Istanbul’s historical peninsula is considered to be a pilot area for the implementation of TDM projects by the local government. The proposed model is compared with other well-known four MCDM methods in order to show its validity and consistency. The results show that public transport capacity improvements is the best alternative among the other TDM measures. Keywords: Transport demand management, sustainable mobility, pull-push measures, urban transport, fuzzy FUCOM, multi-criteria decision making. 1. Introduction
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Promoting sustainable urban transport measures is one of the key challenges that cities face and grapple with [1]. Public transport, cycling and walking are the urban mobility options [2-4] which can bring about change towards sustainable urban transport [3]. Achieving a modal shift from private cars to public transport options requires a coordinated action by cities encouraging use of public transport and discouraging private car use [5]. The problem is especially acute in developing megacities such as Istanbul, Sao Paulo, New Delhi, etc. where rapid population growth and rising transport demand are key urban problems due to inefficient and insufficient urban transport. The results of these urban problems cause chronic traffic jams, air and noise pollution, crowded and unreliable public transport [6-9]. Insufficient public transport supply, heavy reliance on private cars and the lack of effective and consistent transport policies add to these urban problems, making the cities unsustainable in terms of economic, social, and environmental outcomes.
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Transportation demand management (TDM) is a set of transport policies which seek to change people’s travel behaviour towards more sustainable modes of transport and reduce the travel demand by providing alternative transport options [10-12]. By relocating and shifting the transport demand, more effective use of existing road space is aimed. Similar to balancing supply and demand in economics, transportation supply is sought to be balanced with transportation demand. The set of measures under TDM is commonly grouped into two categories [13-15]:
Push (hard) measures which aim to discourage people from the use of road space, often involve penalising or charging for use of road space. Congestion charging, parking pricing, road pricing are some examples of push measures.
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Pull (soft) measures aim at encouraging and incentivizing the use of sustainable transport modes such as public transport, cycling, and walking. Changing workplace travel plans (teleworking), providing active transport modes and associated infrastructure such as pedestrian amenities, cycling lanes, etc., car-sharing and ride-sharing activities, and enhancement of public transport services are some examples for the implementation of pull measures.
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The city of Istanbul, having a population around 15 million inhabitants [16], faces many urban mobility problems ranging from traffic congestion to insufficient and often crowded public transport services, particularly during peak times. Istanbul Metropolitan Municipality (IMM), the local government in Istanbul which is also responsible for regulating and managing urban transport, set out a new transport policy and plan named Integrated Urban Transport Master Plan for Istanbul. The plan aims to prioritize the use of public transport and discourage the use of private cars. To promote and prioritize public transport, IMM has dedicated significant financial resources to increase the modal share of rail-based modes including metro, tram, light rail transit (LRT) and funicular modes in Istanbul’s public transport. Rail-based public transport increased from 45.1 km in 2014 to 233 km in 2019. Surface transport modes such as bus and Bus Rapid Transit (BRT) services are promoted and their capacities increased with fleet investments. BRT connects European and Asian sides across Bosporus through a totally segregated right-of-way. Besides public transport investment, road infrastructure projects have increased the road capacity of Istanbul, which further triggered car use, hence contradicting the public transport-oriented policies. Transport supply, therefore, has improved the supply side of urban transport in Istanbul [15]. TDM measures are proposed by IMM to also improve the demand side of urban transport in Istanbul. Both push measures and pull measures are proposed as a set of policies to address urban mobility problems. The following push measures are proposed by IMM:
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Congestion charging, Increasing parking prices, Road pricing.
On the other hand, the following pull measures are put forward by IMM:
Increasing the capacity of public transport, Synchronized travel plans, Ride sharing.
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The measures are combined into a master TDM project by the urban planners, transport experts and transport managers in IMM. However, due to budget and time constraints, the measures (sub-projects) should be prioritized before implementation based on the costs and outcomes or impacts of the measures. Capital and operating costs as well as economic, social and environmental impacts are considered for each measure for prioritization. The prioritization should also include the uncertainty, hence leading the experts to employ a fuzzy multi-criteria decision making (MCDM) method. In TDM literature, however, a prioritization of those measures to facilitate implementation has not been examined. There are multiple TDM projects and measures which can be implemented in cities, yet these measures need to be prioritized and selected based on their importance and contribution. This topic is important in TDM projects due to a lack of prioritization methodology addressing the uncertainty in those projects.
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Real-life applications of multi-criteria decision making approaches require the processing of imprecise, uncertain, qualitative or vague data. An efficient way to model uncertainty and imprecision is the use of fuzzy sets theory [16]. Fuzzy sets provide the flexibility required to represent and handle the uncertainty and imprecision resulting from a lack of knowledge or ill-defined information [17]. Due to the availability and uncertainty of information as well as the vagueness of human feeling and recognition, it is relatively difficult to provide exact numerical values for the criteria and to make an exact evaluation and to convey the feeling and recognition of objects for decision-makers [18]. In the last few years, numerous studies attempting to handle this uncertainty, imprecision and subjectiveness have been carried out basically by means of fuzzy set theory, as fuzzy set theory might provide the flexibility needed to represent the imprecision or vague information resulting from a lack of knowledge
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The research questions of this paper are as follows:
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or information. Therefore, the application of fuzzy set with multi-criteria evaluation methods for evaluation of a TDM measures has proved to be an effective approach. Most of the TDM measures cannot be given precisely and the evaluation data of the alternatives’ suitability for various subjective criteria and the weights of the criteria are usually expressed in linguistic terms by the decision-makers. The transition from vagueness provided by linguistic values like ‘‘equally importance/very poor, ‘‘weakly important/poor’’, ‘‘fairly important/medium’’, ‘‘very important/high’’, ‘‘absolutely important/very high’’, etc. to quantification is performed by applying the fuzzy set theory. Therefore, this study extends the previous studies by providing a fuzzy MCDM methodology for the implementation of TDM measures. Can TrFN Dombi Bonferroni (TrFN D'Bonferroni) mean operator for processing fuzzy information be used in MCDM problems? How can we adapt a new fuzzy FUCOM to process fuzzy information described by TrFNs? How does this new fuzzy operator work in a real-life case?
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This study examines the proposal and implementation of the TrFN Dombi Bonferroni (TrFN D'Bonferroni) mean operator in Istanbul’s TDM project for Historical Peninsula area. The remainder of this paper is organized as follows. In Section 2, a background to the techniques are presented. An overview of Istanbul’s urban transport system is introduced in Section 3. Section 4 gives preliminaries of used approaches. The problem description and proposed method are also described in Section 5. Section 6 presents experimental results and scenario analysis. Results are discussed in Section 7. Section 8 concludes the study. 2. Background
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A wide variety of TDM measures have been implemented in the world to reduce car use and increase public transport modal share. 2.1. Overview of Transport Demand Management
Table 1
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The effectiveness and preferability of TDM policies depend on many factors including political priorities, sustainability awareness, perceived benefits and costs, and potental public reaction [14]. The literature invastigates the examination of a variety of TDM measures in the world [15,28-31]. Table 1. shows some application of these TDM measures among the cities. TDM measure application examples in the world. Measures
City
Bianco [20] Seik [21] Goh [22] Santos and Shaffer [23] Beevers and Carslaw [24] Eliasson [25] Rotaris et al. [26] Chu [27]
Parking Pricing Electronic road pricing Electronic road pricing Congestion charging Congestion charging Congestion charging Electronic road pricing Car restraint policies and mileage
Portland Singapore Singapore London London Stockholm Milan Singapore
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Author(s)
Batur and Koc [14] examine the potential of TDM measures for Istanbul by carrying out a survey for the residents to understand their perception of the proposed TDM measures. They found that TDM measures provide significant potential for reducing the traffic congestion in Istanbul. Recently,
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2.2. Literature Review of FUCOM Method
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information-based TDM measures such as providing environmental impacts and health benefits for the transport mode being used has gained popularity. E Silva et al. [31] analyze the influence of these information-based TDM measures on commuting mode choice in Lisbon Metropolitan Area. Hammadou and Mahieux [32] explore the potential for TDM for ensuring low carbon mobility by estimating a mode choice model with a nested logit specification. Habibian and Kermanshah [33] examine the role of TDM on commuters’ mode choice in the city of Tehran by analysing the impact of five TDM measures: increasing parking cost, increasing fuel cost, cordon pricing, transit time reduction, and transit access improvement. Their model shows that the single policies main impact and interactions among the multiple policies are significant in influencing the commuters’ mode choice.
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One of the recent models which is the in the same way as the Analytic Hierarchy Processes (AHP) method [34] and the Best Worst Method (BWM) [35], based on the principles of comparisons in pairs of criteria and the validation of results through deviation from the maximum consistency, is the Full Consistency Method (FUCOM) [36]. The benefits specific to the application of the FUCOM are: (1) small number of comparisons in pairs of criteria (only n-1 comparison), (2) possibility of results validation by defining the deviation from the maximum consistency (OMC) of the comparison, (3) taking into consideration transitivity in the comparison of pairs of criteria; and (4) eliminating the issue of redundancy of comparisons in pairs of criteria, which is present in some subjective models for determining weights of criteria [37]. Although this is a new model, there are certain studies in which the benefits of the FUCOM are employed. Badi and Abdulshahed [38] showed the application of the FUCOM in the evaluation of lines in air traffic. Noureddine and Ristic [39] used hybrid FUCOMMABAC model for evaluating routes in the transport of dangerous goods by road traffic. Pamucar et al. [37] showed the application of the FUCOM-MAIRCA multi-criteria model in the evaluation of level crossings during the installation of security equipment. In addition to the above studies, the FUCOM was also applied in the field of logistics: the selection of equipment for storage systems [40], sustainable supplier selection [41] and supply chain management [42-43]. According to the author's knowledge, the application of the FUCOM in fuzzy environment has not been explored so far. Therefore, one of the motives for the development of this paper is the expansion of the FUCOM in fuzzy environment. 2.3. Motivation of Dombi and Bonferroni Aggregators
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Generally, aggregation operators are important tools for fusing information in MCDM problems. The most widely used operators in fuzzy theory are the min and the max operator. Hereby, we would like to emphasize their main advantages: (1) They are easy to calculate and (2) They can be extended into a lattice structure. However, in the case of min-max operators, the main disadvantage is that the result is determined only by one variable and the other has no influence. Also, the min-max operators are not analytic, their second derivative is not continuous [44]. These disadvantages of traditional minmax operators in fuzzy environment are successfully eliminated by generalized Dombi operator class. Also, Dombi T-norms (TN) and T-conorms (TCN) have general parameters of general TN and TCN, and this can make the aggregation process more flexible.
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However, one of the Dombi TN and TCN limitations is the manipulation with the numbers in the interval [0,1]. Therefore, so far Dombi TN and TCN have only been used to transform uncertain numbers meeting this requirement. For example, Dombi TN and TCN are used in the aggregation of information in group decision-making in intuitionistic fuzzy environment [45], interval-valued intuitionistic fuzzy environment [46], single-valued neutrosophic environment [47], hesitant fuzzy environment [48] etc. In all of the above examples, uncertain numbers meet the requirement that the values are in the interval [0,1]. In real decision making systems, the attributes are often represented by values that are not within the interval [0,1], w especially when it is necessary to present the distance (in kilometers) between two cities. Transformation of such attributes into MCDM models by using traditional Dombi TN and TCN is not possible. In order to eliminate this limitation, in this paper Dombi TN and TCN are modified with the purpose of the aggregation of fuzzy numbers regardless of the values they are presented with. Making decisions in real systems requires rational understanding of the relationship between attributes and elimination of the impact of awkward data. For this purpose, Bonferroni [49] introduced the Bonferroni mean (BM) operator, allowing the presentation of interconnections between elements
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and their fusion into unique score function. Zhu et al. [50] presented the geometric Bonferroni mean through the implementation of geometric mean operator in the BM operator. He et al. [51] and He & He [52] introduced the expansion of the BM operator in intuitionistic fuzzy environment. In order to consider the benefits of hybrid aggregators, He et al. [53-55] demonstrated the possibility of the power average (PA) and BM operator fusion. In recent years, the advantages of the Bonferroni aggregator have been implemented through MCDM models in numerous fuzzy uncertainty theories: intuitionistic fuzzy sets [49], interval-valued intuitionistic fuzzy sets [56], rough sets [54] etc. One of the demands of the decision making process in real systems is flexible decision making and taking into consideration the interaction between the attributes of a decision. Obviously, Bonferroni and Dombi aggregators can successfully achieve this goal. According to the author's knowledge, there has been no research up to date considering the fusion of the Dombi and Bonferroni aggregators in fuzzy environment represented by triangular fuzzy numbers (TrFNs). Therefore, logical goal and motivation for this study is to show hybrid Dombi-Bonferroni aggregator for the transformation by TrFN. However, BM cannot be processed by Dombi operations. In this study, TrFN D'Bonferroni aggregators for TrFN aggregation are proposed and the multi-criteria model (fuzzy FUCOM-D'Bonferroni model) is proposed for group decision making. In the first part of the multi-criteria model, the FUCOM was expanded using TrFNs, while in the second part was presented the application of the TrFN D'Bonferroni operator for decision aggregation in the multi-criteria model.
3. Preliminaries
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The main advantages of the proposed fuzzy FUCOM-D'Bonferroni model are: (1) The fuzzy FUCOMD'Bonferroni model provides flexible decision making and takes into consideration the interaction between the attributes of a decision; (2) The model considers interconnections between attributes and eliminates the impact of awkward data. Thus, the main contribution and novelty of this study are threefold: (1) One of the contributions developed in this paper is the introduction of the new fuzzy based FUCOM-D'Bonferroni model that provides more objective expert evaluation of criteria in a subjective environment; (2) The improved MCDM methodology suggested provides purchasing managers with a powerful tool for prioritizing transportation demand management measures; (3) The presented methodology enables the evaluation of alternatives despite dilemmas in the decision making process and lack of quantitative information. In this section, the fundamental elements of the Dombi TNs/TCNs and BM operators are briefly represented in the field. 3.1. Bonferroni Mean Operator
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Definition 1 [52]: Let (a1, a2 ,..., an ) be a set of non-negative numbers and p, q 0 . If 1
BM
then
p,q
pq n 1 (a1 , a2 ,..., an ) aip a qj n(n 1) i , j 1 i j
BM p , q is
(1)
called a Bonferroni mean (BM) operator.
Definition 2 [57]: Let p, q 0 , (a1, a2 ,..., an ) be a set of non-negative numbers, wi (a1, a2 ,..., an ) the relative weight of ai , wi 0,1 and in1 wi 1 . If 1
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n wi w j p q p q NWBM p , q (a1 , a2 ,..., an ) ai a j i , j 1 1 wi
then NWBM p,q is called a normalized weighted BM (NWBM) operator.
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(2)
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3.2. Dombi operations of TrFN Definition 3. Let p and q be any two real numbers. Then, the Dombi T-norm and T-conorm between p and q are defined as follows [58]: 1 1/
1 p 1 q 1 p q
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OD ( p, q)
and ODc ( p, q) 1
1 1/
p q 1 1 p 1 q
where 0 and ( p, q) 0,1 .
(3)
(4)
According to the Dombi T-norm and T-conorm, we define the Dombi operations of TrFNs.
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Definition 4. Suppose 1 1l ,1m ,1u and 2 2l ,2m ,2u are two TrFNs, , 0 and let it be l m u f i f (il ), f (im ), f (iu ) n i l , n i m , n i u fuzzy function, i i i i 1 i 1 i 1
then some operational laws of fuzzy
numbers [18] based on the Dombi T-norm and T-conorm can be defined as follows: (1) Addition “+”
2 l 2 j 1 lj j 1 mj m j 1 j l l 1/ , j 1 j m m 1/ f f f 1 2 1 f 2 1 1 l l m m 1 f 1 1 f 2 1 f 1 1 f 2 1 2 2 u 2 u j 1 j j 1 j u u 1/ f 1 f 2 1 u u 1 f 1 1 f 2 2
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2
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(2) Multiplication “x” 2 j 1 lj 1 2 1/ l 1 f l 1 f 1 l 1 l 2 f 1 f 2
j 1 mj
m 1 f 1 1 m f 1
1 f 2m f 2m
j 1 uj
2
,
,
2
1/
,
u 1 f 1 1 u f 1
1 f 2u f 2u
1/
(5)
(6)
(3) Scalar multiplication, where 0.
1 1l
1/
l 1 1 l 1 1
,
1m
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1l
(4) Power, where 0
6
1m
1/
m 1 1 m 1 1
,
1u
1u
1/
u 1 1 u 1 1
(7)
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1l
1
1 11l 1l
1/
,
1m
1/
1 m 1 m1 1
,
1u
1/
1 u 1 u1 1
(8)
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3.3. Dombi Bonferroni Mean Operator with TrFNs
Based on the TrFNs operators (3)-(8) we propose the TrFN Dombi Bonferroni mean (TrFNDBM) operator. Theorem 1. Let it be j lj , mj , uj ; j 1, 2,..., n , collection of TrFNs in 𝑅, then TrFNDBM operator is defined as follows
1
n il i 1 1/ 1 n ( n 1) 1 p q n 1 i , j 1 l l i j p 1 f i q 1 f j l l f j f i
n
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pq n p q 1 ( 1 , 2 ,..., n ) i j n( n 1) i , j 1 i j
n
im ,
i 1
iu
1/
n ( n 1) 1 1 n 1 pq i , j 1 1 f m 1 f mj i i j q p m f mj f i
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TrFNDBM
p,q,
where f i f (il ), f (im ), f (iu )
il
n il i 1
,
im
i 1im n
,
iu
,
n ( n 1) 1 1 n 1 pq i , j 1 1 f u 1 f uj i i j p q u f uj f i
i 1iu n
i 1
1/
(9)
represents fuzzy function.
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About the proof of this Theorem 1, please see Appendix 1. Example (1). Let 1 (2,3, 4) , 2 (3, 4,5) , 3 (3, 4,5) and 4 (2,3,4) be four TrFNa and p=q=ρ=1, then we can show the following calculations: (1) (2)
f 1l 2 10 0.2 ; f 2l 3 10 0.3 ;...; f 4m 3 14 0.214 ;...; f 3u 5 18 0.278 ; f 4u 4 18 0.222 .
1 f 1l f 1l
5.75
;
1 f 2l f 2l
3.50
;...;
1 f 3m f 3m
4.17
;...;
1 f 4u f 4u
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(3) TrFNDBM 1,1,1 (2,3,4);(3,4,5);(3,4,5);(2,3,4)
7
7.20
.
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2 3 3 2 1/1 , 4(4 1) 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ... 1 1 1 1 1 1 5.75 3.50 5.75 3.50 5.75 5.75 5.75 2.86 3.50 5.75 2.86 3.50 2.86 5.75 3 4 4 3 1/1 , 1 1 4(4 1) 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ... 1 1 1 1 1 1 4.17 2.88 4.17 2.88 4.17 4.17 4.17 9.33 2.88 4.17 9.33 2.88 9.33 4.17 455 4 1/1 1 4(4 1) 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4.13 3.10 4.13 3.10 4.13 4.13 4.13 7.20 3.10 4.13 ... 7.20 3.10 7.20 4.13 2.63, 2.95,3.99
In the following part, we shall explore some desirable properties of TrFNDBM operator.
Theorem 2 (Idempotency): Set j lj , mj , uj ; j 1, 2,..., n , collection of TrFNs in R, if i , then TrFNDBM p , q , ( 1 , 2 ,.., n ) TrFNDBM p , q , ( , ,.., ) . About the proof of this Theorem 2, please see Appendix
2.
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Theorem 3 (Boundedness): Set j lj , mj , uj ; j 1, 2,..., n , collection of TrFNs in R, let min il , min im , min iu and max il , max im , max iu then
TrFNDBM p , q , (1, 2 ,..., n ) . About the proof of this Theorem 3, please see Appendix 3.
Theorem 4 (Commutativity): Let the grey set (1, 2 ,..., n ) be any permutation of ( 1 , 2 ,.., n ) . '
'
'
'
'
TrFNDBM p , q , (1 , 2 ,.., n ) TrFNDBM p , q , (1 , 2 ,.., n ) .
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Then
'
3.4. Normalized Dombi Bonferroni Mean Operator with TrFNs Based on the TrFN operators (3)-(8) we propose the TrFN Normalized Weighted Dombi Bonferroni mean (TrFNDNGBM) operator. Theorem 5. Let it be j lj , mj , uj ; j 1, 2,..., n , collection of TrFNs in R, then TrFNDNGBM operator
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is defined as follows
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TrFNDNGBM p , q , ( 1 , 2 ,..., n )
n 1 p i q j p q i , j 1
wi w j 1 wi
n n il im m i 1 i 1 l 1/ , i 1/ i i 1 i 1 1 wi 1 wi 1 1 1 1 ( p q ) wi w j ( p q ) wi w j n n 1 1 l l m m i , j 1 i , j 1 f i f f i f i j i j q j q j p p 1 f lj 1 f mj 1 f il 1 f im n iu n i 1 iu 1/ i 1 1 wi 1 1 ( p q ) wi w j n 1 u u i , j 1 f i f i j q j p 1 f uj 1 f iu n
n
where f i f (il ), f (im ), f (iu )
il
i 1 n
l i
,
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pro of
,
im
i 1 n
this Theorem 5, please see Appendix 4.
m i
,
iu
i 1iu n
(10)
represents fuzzy function. About the proof of
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Example 2. 1 (2,3, 4) , 2 (3, 4,5) , 3 (3, 4,5) and 4 (2,3,4) be four TrFNa and p=q=ρ=1 and w j 0.18,0.32,0.33,0.17 then we can show the following calculations an w j 0.18,0.32,0.33,0.17 , then we can show the following calculations: TrFNDNGBM w1 ,1,1 (2,3,4);(3,4,5);(3,4,5);(2,3,4)
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2 3 3 4 10 1/1 , 1 1 1 1 0.180.33 1 0.18 0.17 1 0.32 0.18 1 0.32 0.33 1 0.17 0.33 1 11 0.180.32 ... 1 1 1 1 1 1 1 1 1 1 1 1 1 0.18 1 0.18 1 0.18 1 0.32 1 0.32 1 0.17 0.25 0.43 0.25 0.43 0.25 0.25 0.43 0.25 0.43 0.43 0.25 0.43 3 4 4 3 14 1/1 , 1 1 1 1 0.180.33 1 0.18 0.17 1 0.32 0.18 1 0.32 0.33 1 0.17 0.33 1 11 0.180.32 ... 1 0.18 0.271 0.401 1 0.18 0.271 0.401 1 0.18 0.271 0.271 1 0.32 0.401 0.271 1 0.32 0.401 0.401 1 0.17 0.271 0.401 4 5 5 4 18 1/1 1 1 111 0.180.32 1 0.180.33 1 0.18 0.17 1 0.32 0.18 1 0.32 0.33 1 0.17 0.33 1 ... 1 0.18 0.221 0.2781 1 0.18 0.221 0.2781 1 0.18 0.221 0.221 1 0.32 0.2781 0.221 1 0.32 0.2781 0.2781 1 0.17 0.221 0.2781 2.61,3.61, 4.61
In the following part, we shall explore some desirable properties of TrFNDNGBM operator.
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TrFNDNGBM operator also contains the following properties: (1) Idempotency: Set TrFNDNGBM
p,q,
j lj , mj , uj ;
( 1 , 2 ,.., n ) TrFNDNGBM
j 1, 2,..., n , collection of TrFNs in R, if i , then
p,q,
( , ,.., ) .
(2) Boundedness: Set j lj , mj , uj ; j 1, 2,..., n , collection of TrFNs in R, let min il , min im , min iu and max il , max im , max iu then TrFNDNGBM p, q, (1, 2 ,..., n ) .
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'
TrFNDNGBM
p,q,
(1 , 2 ,.., n ) TrFNDNGBM
'
be any permutation of ( 1 , 2 ,.., n ) . Then
'
(1 , 2 ,..., n )
(3) Commutativity: Let the grey set
'
'
'
(1 , 2 ,.., n ) .
p,q,
The proof of these properties is the same as that for TrFNDNGBM operator and because of that it is omitted here.
pro of
In the following part, some special cases of the TrFNDBM and TrFNDNGBM operator will be discussed. (1) If q=0, then:
a) Eq. (7) reduces to the TrFN Dombi generalized mean (TrFNDGM) operator as below. 1
1
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n p p 1 1 n p p TrFNDGM p ,0, ( 1 , 2 ,..., n ) i i n i 1 n(n 1) i 1 n n n il im iu i 1 i 1 i 1 1/ , 1/ , 1/ 1 n n n 1 1 1 n 1 n 1 p n p p 1 1 1 i 1 l i 1 1 f im i 1 1 f iu p 1 f i p p m u f il f i f i
b) Eq. (8) reduces to the TrFN Dombi generalized weighted geometric (TrFNGWG) operator. wi
n
n
,
im
im i 1
1 1 p w i
1/
n
1 m f i
p 1 f i 1
n
iu
n
i 1
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n n il i 1 il 1/ i 1 1 1 1 p n f l i w p i l 1 f i 1 i
1 n p i p i , j 1
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TrFNGWG p ,0, ( 1 , 2 ,..., n )
m i
,,
iu i 1
i 1
1 1 p w i
n
1/
1 u f i
p 1 f i 1
u i
,
(2) If p=1 and q=0, then:
a) Eq. (7) reduces to the TrFN Dombi arithmetic average (TrFNDAA) operator. TrFNDAA1,0, ( 1 , 2 ,..., n )
n
n
,
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n n il i 1 il 1/ n f il 1 i 1 1 l n i 1 1 f i
1 n i n i 1
i 1
m i
im i 1
1 1 n
n
f im
1 f i 1
m i
1/
,
n
n
i 1
u i
iu i 1
1 1 n
n
f iu
1 f i 1
u i
1/
b) Eq. (8) reduces to the TrFN Dombi weighted geometric (TrFNDWG) operator.
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i 1
n il i 1 1/ n 1 f l 1 wi l i i 1 f i
wi
n
n
im ,
i 1
1 wi
1 f im f im i 1 n
iu ,
1/
i 1
1 wi
1 f iu f iu i 1 n
1/
(3) If p→0 and q=0, then
pro of
n
TrFNWG1,0, ( 1 , 2 ,..., n ) i
a) Eq. (7) reduces to the TrFN Dombi geometric average (TrFNDGA) operator 1/ n
n lim TrFNDGAwp ,0, ( 1 , 2 ,..., n ) i p 0 i 1 n il i 1 1/ n 1 f l 1 1 l i n i 1 f i
n
n
im ,
i 1
1 1 n
iu ,
1/
1 f im m f i 1 i
n
i 1
1 1 n
1/
1 f iu u f i 1 i
n
b) Eq. (8) reduces to the TrFN Dombi weighted geometric average (TrFNDWGA) operator p 0
wi
i 1
n il i 1 1/ n 1 f l 1 wi l i i 1 f i
n
n
im ,
i 1
1 wi
iu 1/
1 f im m f i 1 i n
,
i 1
1 wi
1/
1 f iu u f i 1 i n
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n
lim TrFNDWGAwp ,0, ( 1 , 2 ,..., n ) i
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4. Fuzzy FUCOM-D’Bonferroni multicriteria model
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Fuzzy FUCOM-D'Bonferroni multi-criteria model is implemented in two phases as shown in Fig. 3. In the first phase, weight coefficients are calculated using fuzzy FUCOM. The weight coefficients obtained in the first phase of the model are further used in the D'Bonferroni model for the evaluation of alternatives. A detailed presentation of the steps of the FUCOM-D'Bonferroni multi-criteria model is shown in the next section.
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Expert comparison of criteria
Expert evaluation of alternatives
Defining the limitations of fuzzy model
Aggregation of matrices – TrFNDBM operator
Forming FUCOM nonlinear model Aggregation of weight coefficients TrFNDBM
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Identificationof alternatives
Phase 1: TrF FUCOM
Criteria identification and their ranking
Aggregated initial decision matrix (N) Evaluation of alternatives TrFNDNGBM aggregator
Optimal values of criteria weights
Phase 2: D’Bonferroni model
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Initial alternative ranking
Influence of dynamic matrices to ranks
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Phase 3: Sensitivity analysis and discussion
Changes of criteria weights
Comparisons with other MCDM models
Influence of parameters p, q and on the rankings results
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Final alternative ranking
Fig 3. FUCOM-D’Bonferroni multicriteria model. Phase I: Fuzzy FUCOM
Table 3
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Assuming that there are n evaluation criteria in the multicriteria model that are designated as 𝑤𝑗 , 𝑗 = 1,2, . . . , 𝑛, and their weight coefficients need to be determined. Subjective models for determining weights based on comparison in pairs of criteria require from the decision maker to determine the degree of influence of the criterion 𝑖 on the criterion 𝑗. This degree of influence of the criterion 𝑖 on the criterion j is presented as the value of the comparison (𝑎𝑖𝑗 ). Since the comparison values 𝑎𝑖𝑗 are not based on accurate measurements, but on subjective estimates, it is expected that existing uncertainties will be presented with fuzzy numbers. In the application of fuzzy numbers in the models of multi-criteria decision making, linguistic scales are most often used. Thus, in this paper for the presentation of expert preferences in fuzzy FUCOM, fuzzy linguistic scale [59] is used, which is represented by triangular fuzzy numbers, as shown in Table 3. Fuzzy linguistic scale for criteria evaluation.
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Linguistic terms Equally importance (EI) Weakly important (WI) Fairly Important (FI) Very important (VI) Absolutely important (AI)
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Membership function (1,1,1) (2/3,1,3/2) (3/2,2,5/2) (5/2,3,7/2) (7/2,4,9/2)
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In accordance with the defined settings, in the next section is presented the fuzzy FUCOM algorithm in five steps. Step 1. Determining a set of evaluation criteria and their ranking. The initial step in multicriteria models is defining a set of evaluation criteria. If we assume that there is n (𝑗 = 1,2, . . . , 𝑛) evaluation criteria, then we can present them with a set C C1 , C2 ,..., Cn .
pro of
Let's assume that a group of experts E1 , E2 ,..., Et is involved in the research. After defining the set of criteria, the experts determine their rank in accordance with their preferences. The criteria rank is determined according to the significance, respectively, the first rank is assigned to the criterion that is expected to have the highest weight coefficient. The last place occupies the criterion for which we expect to have the lowest value of the weight coefficient. Thus, we obtain the criteria ranked according to their expected influence on decision making in the MCDM model. C j (1) C j (2) ... C j ( k )
(11)
where 𝑘 represents the rank of the observed criterion. If two or more criteria have the same rank, the equality sign is placed between the criteria instead of ">".
j(k )
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Step 2. Comparisons of the criteria. Mutual comparison of the criteria is done using fuzzy linguistic expressions from the defined scale (see Table 3). Mutual comparisons are performed by each expert individually Ee ( 1 e t ) according to his preferences. The comparison is made with respect to the first-ranked (most significant) criterion. Thus, for every expert are obtained fuzzy significances of the e criteria C for all the criteria ranked in the step 2. Based on the defined significances of the criteria, e
using the expression (12) are determined fuzzy comparative significances k / ( k 1) of expert comparisons. C
j ( k 1)
e C j(k )
( Celj ( k 1) , Cemj ( k 1) , Ceuj ( k 1) ) ( Celj ( k ) , Cemj ( k ) , Ceuj ( k ) )
(12)
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e
e
k / ( k 1)
Thus are obtained fuzzy vectors of comparative significances of the criteria for every expert, according to the expression (13)
e
e
e
e
1/ 2 , 2/3 ,..., k / ( k 1)
e
(13)
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Where k / ( k 1) present the significance which according to the expert e the criterion with the rank C j ( k ) has in comparison with the criterion with the rank C j ( k 1) . Step 3. Defining the limitations of the fuzzy model. Values of weight coefficients should meet the condition where the relation of the weight coefficients of criteria ( C j ( k ) and C j ( k 1) ) is the same as their e
comparative significance k / ( k 1) , respectively, e
wk w
e k 1
e
k / ( k 1)
(14)
In addition to the condition (14), final values of weight coefficients should meet the condition of transitivity, respectively, k /(k 1) (k 1)/( k 2) k /( k 2) , that is
wk wk 1 wk wk 1 wk 2 wk 2
. Thus is obtained the
second condition which final values of weight coefficients should meet e
e
e
k / ( k 1) ( k 1) / ( k 2)
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wk
w
e k 1
(15)
Step 4. Forming fuzzy model for the calculation of optimal values of the weights of criteria. In the fourth
e
e
e T
step is calculated final values of fuzzy weight coefficients of the criteria for every expert w1 , w2 ,..., wn .
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The conditions defined in the previous step (step 3) should be met with minimal deviation from the e
wk
maximal consistency (OMC). In other words, the conditions should be met
w
e
e k 1
k / ( k 1) 0
and
e
w
e k 2
e
e
k / ( k 1) ( k 1) / ( k 2) 0 .
Meeting these conditions, the OMC amounts to 0 . From the previously
e
presented relations follows that the values of the weight coefficients of criteria w1 , w2 ,..., wn meet the condition where
w
e k
e
w k 1
e
k / ( k 1)
e k
w
and
e
wk 2
e
e
k / ( k 1) ( k 1) / ( k 2)
values.
e T
e
pro of
wk
should
, with the minimization of the
Based on the defined settings, we can set nonlinear model for determining optimal fuzzy values of the
e
e
e T
weight coefficients of the evaluation criteria w1 , w2 ,..., wn . min
e
w j ( wlj , wmj , wuj )
e
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Where
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s.t. wek e e k / ( k 1) , j w k 1 wek e e e k / ( k 1) ( k 1) / ( k 2) , j wk 2 n e w j 1, j , j 1 l m u w j w j w j , l w j 0, j j 1, 2,..., n
(16)
and k / ( k 1) (kl / ( k 1) ,km/ ( k 1) ,ku/ ( k 1) ) .
e
Considering that the maximum consistency requires meeting the condition where e
w
e k 2
e
e
k / ( k 1) ( k 1) / ( k 2) 0 ,
w
e k 1
e
k / ( k 1) 0
and
the model (16) can be transformed into fuzzy linear model (17) and by
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wk
wk
e
e
e T
solving it are obtained optimal fuzzy values of the weight coefficients w1 , w2 ,..., wn . min s.t.
(17)
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wek wek 1 e k / ( k 1) , j e e e e wk wk 2 k / ( k 1) ( k 1) / ( k 2) , j n e w j 1, j , j 1 wl wm wu , j j j wlj 0, j j 1, 2,..., n
Where
e
w j ( wlj , wmj , wuj )
e
and k / ( k 1) (kl / ( k 1) ,km/ ( k 1) ,ku/ ( k 1) ) .
Step 5. Aggregation of the weight coefficients of criteria. Solving the model (16) or (17) are obtained weight coefficients of criteria by experts. By applying TrFNDBM operator (9), it is performed the
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e
e
e T
aggregation of the weight coefficients w1 , w2 ,..., wn
and final optimal values of the weight coefficients
of criteria are obtained w1, w2 ,..., wn . T
Phase II: D’Bonferroni model for the evaluation of alternatives
pro of
Step 1. Forming initial decision making matrix. In this research participated five experts which evaluated the alternatives. For every expert Ee ( 1 e t ) is obtained correspondent matrix X e xij e
mn
( 1 e t ).
By the application of TrFNDBM operator (9) it is performed the aggregation of experts’ matrices and it is formed the initial decision making matrix X xij m n . Step 2. Normalization of the initial decision making matrix. Generally, in MCDM models there are two types of criteria, benefit criteria and cost criteria. Therefore, it is necessary to normalize initial decision making matrices, forming that way normalized matrix N nij m n . The elements of the normalized matrix are obtained by applying the expression (18). jB j C
(18)
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x ij / n x ij if i 1 nij n 1 x ij / i 1 x ij if
Step 3. Determining score function S (ni ) . By applying the TrFNDNGBM aggregator (10) the values of the score function S (ni ) TrFNDNGBM p , q , r n1, n2 ,..., nm are obtained, presenting final values of preferences by alternatives. The TrFNDNGBM aggregator implies the application of the values of criteria which meet the condition nj w j 1 . In the phase I of the application of the FUCOM-D'Bonferroni model are
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obtained fuzzy values of the weights which meet the condition nj wlj 1 , nj wmj 1 and nj wmj 1 . So as to meet the previously defined condition, by applying the expression wj wlj 4wmj wuj 61 it is performed the defuzzification. After the defuzzification, by applying additive normalization, defuzzified values are normalized so that nj w j 1 . Step 4. Ranking alternatives. Ranking alternatives A1 , A1 ,..., Am and the selection of the best alternative from the considered set. Alternatives are ranked based on the values of the score function S (ni ) , wherein the highest possible value of the alternative S (ni ) is more favorable.
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5. Case Study
The overview of Istanbul’s urban transport system is provided by the following steps: 1- Showing the increasing transport demand in Istanbul 2- Considering TDM in Historical Peninsula to address the increasing transport demand a. Considering push measures: congestion charging, parking pricing, road pricing. b. Considering pull measures: synchronized travel plans, public transport capacity improvement, ride sharing.
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The motorization rate in Istanbul has rapidly increased with increasing car ownership and car use rates. Parallel to this development, the use of public transport modes such as buses, metro, tram, funicular, bus rapid transit (BRT), light rail transit (LRT), cable car, paratransit modes such as minibuses, dolmus, company and school services and taxis also increased in terms of daily ridership figures [15]. Fig. 1 indicates the motorization rate along with population growth between 1980 and 2014. Both values are indexed to 100 for year 1980 to illustrate the wide divergence in the population and vehicle growth rates over the years.
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Fig. 1. Population and motorization growth in Istanbul between 1980 – 2014 [14].
The fact that motorization growth far outweighed the population growth over the years is an indication of transport supply saturation, which needs to be balanced with demand-side interventions.
Table 2
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Number of daily trips for public transport modes which include cars, public transport, walking and taxis is nearly 32 million. Table 2 shows the number of average daily trips for each transport mode. Fig. 2 shows the modal share of Istanbul’s public transport [60]. Number of daily transport trips in Istanbul’s public transport.
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Transport Mode Rail Transport Metro Tram Cable Car, Nostalgic tram and Funicular Marmaray Surface Transport IETT Buses Private Bus Operator-1 Private Bus Operator-2 Minibus Taxi Company and School Services Waterborne Transport Sea buses (IDO) Ferries Boats Total
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Daily Ridership 2,709,914 1,728,555 640,351 54,168 286,840 11,717,979 1,940,750 1,571,393 835,422 3,098,963 1,403,949 2,867,502 565,472 163,434 231,444 170,594 14,993,365
Modal share (%) 18.07 11.53 4.27 0.36 1.91 78.15 12.94 10.48 5.57 20.67 9.36 19.13 3.77 1.09 1.54 1.14 100
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4%
78%
Rail Transport
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18%
Surface Transport
Waterborne Transport
Fig. 2. The modal share of public transport.
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Besides public transport, the total road kilometre in Istanbul increased up to 36,000 km with the road infrastructure investments. IMM forecasts for future daily trip numbers (36 million by 2023) expect a pressure on the current road and public transport infrastructure. This leads IMM to pursue policies to meet increasing mobility demands by increasing transport supply or rebalancing transport demand, particularly away from peak hours and locations with high mobility demand.
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TDM measures have been considered by IMM recently to achieve a balanced transport supply and demand. Congestion charging is suggested around the Historical Peninsula to limit the traffic flow entering the area and improving the air quality as well as reducing traffic congestion levels. A feasibility study by IMM is undertaken to analyse the impacts of such a measure. The results suggest a 20% reduction in traffic congestion levels. Supporting this measure, IMM also looks to implementing parking pricing in the area that would decrease the car use. Road tolls are in place for the bridges across the Bosporus and Eurasia Tunnel which reduces the demand for car use in Bosporus crossings, which is the bottleneck for Istanbul’s traffic during peak times.
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On the other hand, the push measures listed above are planned to be supported by pull measures as well. Travel plans for employees getting to work around the same peak hours are planned to mitigate the concentration of travel demand within an hour at peak times (08:00 – 09:00). Flexible work schedules, especially for public agencies and schools, can redistribute the travel around the peak times. Travel information tools, such as “IBB Navi”, travel awareness campaigns making people shift their travel preferences towards public transport, cycling and walking, teleworking, car sharing and ride sharing, home shopping, road space rationing, are considered by IMM for pull measures. IMM decided to focus on increasing the public transport capacity, synchronized travel plans and ride sharing as the options to implement through the TDM project. 5.1. Problem Description
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IMM identified the set of pull and push measures under a TDM project package. Six sub-projects (measures) comprising of the TDM project are grouped as shown in the following Fig. 4. This figure shows the list of measures as a complete TDM policy package recommended for implementation by Istanbul’s local authority.
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Push Measures
Pull Measures
- Increasing the capacity of public transport
- Congestion charging
pro of
- Increasing parking prices
- Synchronized travel plans
- Road pricing
- Ride sharing
Transport Demand Management
Fig. 4. Transport demand management policy package.
Table 4 Experts (decision makers) of TDM project.
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The costs and outcomes of these measures are discussed with following five experts (see Table 4) working in IMM and IETT, Istanbul’s public bus operator to determine the priority scores of each measure:
Department
Agency
Director Manager Transport planner Transport planner Urban planner
Public Transport Services Strategy Planning Transport Directorate Transport Planning Directorate Traffic Coordination Centre
IMM IETT IMM IETT IMM
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Position
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Each measure is evaluated based on the project costs, economic, social and environmental impact. Fig. 5 shows these evaluation criteria and how they are related with each TDM measure.
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MCDM Approach Priorititizing transportation demand management measures
Economic impact
Capital costs Operating costs
A1 (Congestion charging)
Travel time Public transport trip revenues
A2 (Parking prices)
A3 (Road pricing)
Environmental impact
Social impact
pro of
Costs
Social inclusion Vulnerable users Public opposition
A4 (Public transport capacity)
Push measures
Decreasing carbon emissions Fuel saving
A5 (Synchronized travel plans)
A6 (Ride sharing)
Pull measures
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Fig. 5. The evaluation indicator system for prioritizing TDM. Alternatives:
Alternatives in the form of push and pull measures are proposed as part of Transportation Demand Management (TDM) strategy. An effective and comprehensive TDM strategy need both positive incentives in the form of pull measures and negative incentives in the form of push measures. In Istanbul, the following pull and push measures are proposed:
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Push measures: Push measures make private vehicles less attractive to use. - Congestion charging (A1): A congestion charging scheme is proposed for Historical Peninsula, which includes the touristic and historical downtown of the city. It is proposed to encourage mode shift, hence reduce air pollution. Only electric buses and other zero-emission vehicles are allowed to enter the congestion area without any charge. As part of the initiative, bus fleet used in the area is planned to be electrified. However, the details of the scheme are not yet announced.
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- Parking prices (A2): Increasing number of private cars in Istanbul has driven up the number of parking areas, which caused induced traffic phenomenon whereby new parking facilities spurred the demand for transport. By increasing the parking prices and dynamically setting the prices to curb the demand, the local government aims to make private vehicles less attractive in central areas. - Road pricing (A3): Variable road pricing is proposed for 3 bridges across the Bosporus, which creates most serious traffic congestion problem, especially during the rush hours. Adjusting bridge tolls based on the demand aims to curb the increasing demand for the bridge crossings. Pull Measures: Pull measures make other modes of transport more attractive. -Public transport capacity improvements (A4): Public transport capacity improvements in the form of new metro lines, bus services, and ferry services are put forward to complement the push measures by increasing the attractiveness of public transport lines.
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- Synchronized travel plans (A5): The high traffic congestion during rush hours is mainly driven by commuting to workplaces, schools and hospitals opening nearly the same hours (08:30 – 09:00). Therefore, the local government seeks to implement a synchronized travel plan so that the overlapped commuting transport demand is curbed. - Ride sharing (A6): The increasing popularity of ride-sharing applications has the potential to decrease the use of road space by cars. The average number of people in each car can be reduced by increasing ride-sharing. However, there is a potential risk that ride-sharing applications can reduce the demand for mass transit options such as metro and bus services, and lead to an increase in the number of cars. Being 19
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a somewhat controversial measure, the local government seeks to spread ride-sharing through increasing partnerships with ride-sharing provider companies. 5.2. Criteria Definition
pro of
Four main and nine sub-critera are determined by experts for priorititizing transport demand managmenet measures. The decision hierarchy of TDM measures is shown in Fig. 5. Costs: Allocating budget for TDM measures is a key part of project implementation. While some measures such as congestion charging and public transport capacity improvements require high upfront capital costs, many measures such as synchronized travel plans and ride sharing require less costs. Cities facing financial difficulties tend to favor less costly projects.
Capital costs (C1): Capital costs are investment costs for the large-scale TDM measures. Congestion charging, for example, involve high initial capital costs such as technological installements, automatic number plate recognition (ANPR) and charging systems. Operating costs (C2): Operating costs play an important role in the selection of TDM measures. While TDM mesaures involving infrastructural investments need higher operational costs, other measures such as sycnhronized travel plans require less operational costs.
Economic Impact:
Travel time (C3): One of the objectives of TDM measures is to reduce travel time, hence increase accessibility in the city. By encouraging modal shift, thereby reducing the traffic congestion, TDM measures help reduce the travel time for passengers. The measures involving greater degree of travel time reduction impact increase the selection probability by decision makers. Public transport trip revenues (C4): Some measures involve generation of revenue such as congestion charging which increases the likelihood of selection by decision makers. While push measures are more likely to generate revenue due to penalising road use, pull measures are less likely to generate revenue.
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Social inclusion (C5): Although TDM measures do not directly have an impact on social inclusion, the increased accessibility as a result of TDM lead to more social inclusion. Some measures such as ride-sharing can lead to more social interaction. Similarly public transport capacity improvements lead to increased accessiblity of citizens to the urban opportunities. Vulnerable users (C6): Vulnerable users such as disabled people, senior people, and children have more difficulty in accessing the urban amenities. TDM measures, by promoting a modal shift to sustainable transport options such as public transport, walking and cycling, facilitate the urban transport experience for the vulnerable users. Public opposition (C7): TDM measures, especially push measures involving penalising private car usage can trigger reaction from the public. Therefore, policy makers can tend to choose less controversial TDM measures. This is one of the reasons congestion charging is controversially implemented in many cities, leading to reaction form business inside the congestion charging area.
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Social impact: TDM measures are implemented to have a positive social impact in the form of social inclusion and participation in urban opportunities.
Environmental Impact: Reducing the greenhouse gas emissions and improving the air quality is one of the key objectives of TDM measures. By promoting modal shift from private car to sustainabile transport modes, TDM measures help reduce the emissions and hence enhancing the air quality. Decreasing Carbon Emissions (C8): Cleaner transport modes such as walking, cycling and public transport decrease the carbon emissions. The impact on environment of each measure depends on the degree to which it contributes to modal shift. Congestion charging, for example, can improve air quality by substantially reducing the car use within central city areas. Fuel Saving (C9): Car usage leads to increased levels of fossil fuel consumption, with negative impacts on natural environment. TDM measures can bring about higher fuel saving, hence reducing the negative impacts on the environment. Modal shift from private car to public
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transport tremendously reduce the fuel consumption per person. TSM measures leading to more fuel saving, therefore, are more likely to be selected by decision makers. 5.3. Proposed Method
pro of
The phase I of the application of the FUCOM-D'Bonferroni implies the application of fuzzy FUCOM model and the determination of the weight coefficients of criteria. The criteria for the evaluation of alternatives are grouped into four sets of criteria: Cost (C1), Economic impact (C2), Social impact (C3) and Environmental impact (C4) criteria. The C1-C4 criteria constitute the first hierarchical level, while the second hierarchical level consists of sub-criteria classified within the four sets of criteria in Table 5. Table 5 Criteria/sub-criteria for evaluation.
Code C1 C11 C12 C2 C21 C22 C3 C31 C32 C33 C4 C41 C42
Min/max min min min max max max min max max
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Criteria/Sub-criteria Cost Capital costs Operating costs Economic impact Travel time Public transport trip revenues Social impact Social inclusion Vulnerable users Public opposition Environmental Impact Decreasing Carbon Emissions Fuel Saving
Using fuzzy FUCOM are calculated the values of local weights of the sub-criteria. After defining the local weights of the sub-criteria, the weight coefficients of the criteria are multiplied by the group of weight coefficients of the sub-criteria. Thus are obtained global values that are further used to evaluate alternatives in the D'Bonferroni model. In order to solve this problem, five fuzzy FUCOM models are defined:
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1) Model 1 - Calculation of the values of the weight coefficients of the criteria C1, C2, C3 and C4; 2) Model 2 - Calculation of local values of the weight coefficients of the sub-criteria C11 and C12; 3) Model 3 - Calculation of local values of the weight coefficients of the sub-criteria C21 and C22; 4) Model 4 - Calculation of local values of the weight coefficients of the sub-criteria C31, C32 and C33 and 5) Model 5 - Calculation of local values of the weight coefficients of the sub-criteria C41 and C42. Detailed overview of the models 1 - 5 is presented in the next section of the paper.
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Step 1 and 2. After defining the criteria and the sub-criteria (see Table 5), their ranking is performed. Based on the defined ranks of the criteria/sub-criteria and the preferences of the experts (E1-E5), linguistic values of the comparative significances of the criteria/sub-criteria are determined as given in Table 6. To determine the comparative significances of the criteria/sub-criteria, linguistic values from the Table 3 are used. Table 6
Linguistic evaluations of the criteria/sub-criteria. Criteria/sub-criteria Criteria R
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E1 C1>C2>C3>C4
E2 C2>C1>C4>C3
E3 C4>C3>C2>C1
E4 C1>C4>C2>C3
E5 C1>C2>C4>C3
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EI,FI,VI,VI C11>C12 EI,FI C22>C21 EI,AI C33>C31>C32 EI,AI,AI C42>C41 EI,VI
EI,EI,AI,AI C11>C12 EI,FI C21>C22 EI,FI C33>C31>C32 EI,WI,VI C41>C42 EI,WI
EI,FI,VI,VI C11>C12 EI,VI C22>C21 EI,FI C32>C31>C33 EI,VI,AI C41>C42 EI,WI
EI,WI,AI,AI C11>C12 EI,FI C21>C22 EI,VI C31>C33>C32 EI,FI,AI C41>C42 EI,FI
pro of
C1-C4 C EI,WI,AI,AI Sub-criteria R C11>C12 C11-C12 C EI,VI Sub-criteria R C21>C22 C21-C22 C EI,AI Sub-criteria R C32>C31>C33 C31-C33 C EI,VI,VI Sub-criteria R C42>C41 C41-C42 C EI,VI R – Rank; C- Comparisons
Based on the comparative significances of the criteria/sub-criteria (see Table 6), using the expressions (2) and (3), the vectors of comparative significances are defined by levels of the criteria as given in Table 7. Table 7
Vectors of comparative significances of the criteria/sub-criteria. Experts
Criteria/sub-criteria C11-C12
(0.67,1,1.5);(2.33,4,6.72);(0.78,1,1.29)
(2.5,3,3.5)
E2
(1.5,2,2.5);(1,1.5,2.33);(0.71,1,1.4)
E3
(1,1,1);(3.5,4,4.5);(0.78,1,1.29)
E4
(1.5,2,2.5);(1,1.5,2.33);(0.71,1,1.4)
E5
(0.67,1,1.5);(2.33,4,6.72);(0.78,1,1.29)
C21-C22
C31-C33
C41-C42
(3.5,4,4.5)
(2.5,3,3.5);(0.71,1,1.4)
(2.50,3,3.5)
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C1-C4
E1
(1.5,2,2.5)
(3.5,4,4.5)
(3.5,4,4.5);(0.78,1,1.29)
(2.50,3,3.5)
(1.5,2,2.5)
(1.5,2,2.5)
(0.67,1,1.5);(1.67,3,5.22)
(0.67,1,1.5)
(2.5,3,3.5)
(1.5,2,2.5)
(2.5,3,3.5);(1,1.33,1.8)
(0.67,1,1.5)
(1.5,2,2.5)
(2.5,3,3.5)
(1.5,2,2.5);(1.4,2,3)
(1.50,2,2.5)
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Comparative significances for the expert E1 and the criteria C1-C4 are obtained by applying the expression (2): C1/ C 2 C1 C 2 (2/3,1,3/2) (1,1,1)= 2/3,1,3/2 ; C 2/ C 3 C 2 C 3 (7/2,3,9/2) (2/3,1,3/2) =(2.33,4.00,6.72) ;
C 3/ C 4 C 2 C 3 (7/2,3,9/2) (7/2,3,9/2) =(0.78,1.00,1.29) .
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Thus is obtained the vector of comparative significances 0.67,1,1.5 ; 2.33, 4,6.72 ; 0.78,1,1.29 . The remaining values of comparative significances from the Table 7 are obtained in the same way. 1
In the following section (Step 3), based on the vector of comparative significance are defined the limitations of the model (16). By applying the expression (14), it is defined the first group of limitations for the expert E1 and the criteria C1-C4: wC1 / wC 2 2/3,1,3/2 , wC 2 / wC 3 2.33,4,6.72 and wC 3 / wC 4 0.78,1.00,1.29 .
condition
of
By applying the expression (15) are defined two limitations arising from the
transitivity
of
relations:
wC 2 / wC 4 2.33,4,6.72 0.78,1,1.29 1.81,4.00, 8.64 .
wC1 / wC 3 0.67,1.00,1.50 2.33,4,6.72 (1.56,4,10.07) and
The limitations for the remaining models are defined
in the same way.
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On the basis of the defined limitations, the models (16) were formed for determining optimal values of the weight coefficients of the criteria/sub-criteria. In the following section, nonlinear models are presented for determining optimal fuzzy values of the weight coefficients of the criteria. The models for determining the weight coefficients of the sub-criteria are presented in the Appendix 5.
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Expert 1 (C1-C4) min
Expert 5 (C1-C4) min s.t.
w1l w1m w1u u 0.67 ; m 1.00 ; l 1.5 ; w w w2 2 2 l m u w w 2 2.33 ; 2 4.00 ; w2 6.72 ; w3u w3m w3l w3l w3m w3u u 0.78 ; m 1.00 ; l 1.29 ; w w w4 4 4 l w1m w1u w1 wu 1.56 ; wm 4.00 ; wl 10.07 ; 3 3 3 m u wl w w 2u 1.81 ; 2m 4.00 ; 2l 8.64 ; w4 w4 w4 l m u l m u ( w1 4 w1 w1 ) / 6 ( w2 4 w2 w2 ) / 6 ( wl 4 wm wu ) / 6 1; 3 3 3 wlj wmj wuj ; wlj , wmj , wuj 0, j 1, 2,..., n
w1l w1m w1u u 0.67 ; m 1.00 ; l 1.5 ; w w w2 2 2 l m u w w 2 2.33 ; 2 4.00 ; w2 6.72 ; w4u w4m w4l w4l w4m w4u u 0.78 ; m 1.00 ; l 1.29 ; w w w3 3 3 l w1m w1u w1 wu 1.56 ; wm 4.00 ; wl 10.07 ; 4 4 4 m u wl w w 2u 1.81 ; 2m 4.00 ; 2l 8.64 ; w3 w3 w3 l m u l m u ( w1 4 w1 w1 ) / 6 ( w2 4 w2 w2 ) / 6 ( wl 4 wm wu ) / 6 1; 3 3 3 wlj wmj wuj ; wlj , wmj , wuj 0, j 1, 2,..., n
pro of
s.t.
Table 8
re-
By solving the models presented, optimal local values of weight coefficients by experts are obtained. By multiplying local values of the sub-criteria with the weight coefficients of the criteria, global values of the sub-criteria for every expert are obtained, as in the Table 8. The Lingo 17.0 software is used to solve the model. By solving fuzzy nonlinear models, the average value χ ≈ 0.0 is obtained, which shows high consistency of the obtained values of the weights of criteria. Global values of fuzzy weight coefficients of the sub-criteria. E1 (0.166,0.289,0.557) (0.057,0.096,0.186) (0.254,0.292,0.452) (0.065,0.073,0.112) (0.038,0.062,0.086) (0.015,0.021,0.025) (0.011,0.021,0.035) (0.044,0.066,0.117) (0.015,0.022,0.039)
E2 (0.108,0.17,0.18) (0.06,0.085,0.087) (0.051,0.081,0.135) (0.226,0.322,0.478) (0.014,0.036,0.039) (0.011,0.036,0.05) (0.05,0.145,0.175) (0.024,0.038,0.045) (0.08,0.114,0.116)
E3 (0.038,0.069,0.1) (0.021,0.034,0.048) (0.049,0.071,0.074) (0.027,0.035,0.036) (0.156,0.168,0.176) (0.03,0.056,0.105) (0.105,0.168,0.264) (0.131,0.209,0.233) (0.141,0.209,0.215)
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Criteria C11 C12 C21 C22 C31 C32 C33 C41 C42
E4 (0.183,0.31,0.46) (0.063,0.103,0.153) (0.03,0.053,0.053) (0.057,0.106,0.106) (0.018,0.046,0.049) (0.045,0.137,0.172) (0.01,0.034,0.049) (0.073,0.129,0.142) (0.079,0.129,0.131)
E5 (0.133,0.259,0.518) (0.074,0.129,0.249) (0.228,0.276,0.424) (0.078,0.092,0.141) (0.029,0.049,0.108) (0.006,0.012,0.031) (0.019,0.025,0.043) (0.039,0.071,0.089) (0.022,0.036,0.043)
Optimal values (0.114,0.199,0.311) (0.051,0.083,0.126) (0.088,0.121,0.167) (0.069,0.093,0.12) (0.031,0.059,0.079) (0.017,0.037,0.059) (0.026,0.055,0.082) (0.052,0.086,0.11) (0.052,0.08,0.09)
urn a
By solving fuzzy nonlinear models, the average value of the OMC 0.07 is obtained, which shows high consistency of the obtained values of the weights of the criteria. Using TrFNDBM operator (9), it is performed the aggregation of the weight coefficients and final optimal values of the weight coefficients of the criteria are obtained w1 , w2 ,..., wn , as shown in the Table 8. The final global fuzzy T
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values of the weights of the criteria are presented in the Fig. 6.
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0.3
Weights
0.2
0.15
0.1
0.05
C11
C12
C21
C22
C31
pro of
0.25
C32
C33
C41
C42
re-
Fig. 6. Final values of the weight coefficients of the criteria.
Final values of the weight coefficients are used to evaluate and select optimal alternative in the D'Bonferroni multi-criteria model. Evaluation of the alternatives is done using fuzzy linguistic scale, as shown in the Table 9. Table 9
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Fuzzy linguistic scale for the evaluation of the alternatives. Linguistic terms Very poor (VP) Poor (P) Medium (M) High (H) Very high (VH)
Membership function (1,1,3) (1,3,5) (3,5,7) (5,7,9) (7,9,9)
Table 10
urn a
The first step of the D'Bonferroni model involves the formation of the aggregated initial decision making matrix. The aggregated decision making matrix is obtained on the basis of expert correspondence matrices (see Table 10) in which the experts evaluated the alternatives by criteria. Expert correspondent matrices. A1 VH,H,H,VI, VH VH,H,H,AI, VH VH,H,H,AI, VH VH,M,VH,A I,P VP,P,VP,EI, VP VP,P,P,EI,V P VH,H,H,VI, VH H,VH,M,AI, VH M,H,M,FI,H
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Crit .C1 C2 C3 C4 C5 C6 C7 C8 C9
24
A2 M,M,M,FI, H H,VH,H,FI, H VH,H,P,VI, M VH,VH,H,V I,H VP,VP,P,WI ,P VP,P,VP,WI ,M VH,VH,H,F I,M H,H,P,VI,H P,P,M,WI,P
A3 H,VH,H,FI,H H,H,VH,WI, M VH,VH,H,AI, H VH,M,H,VI, M VP,M,VP,EI, VP VP,VP,M,EI, P VH,VH,VH,V I,H H,H,M,VI,H M,M,H,FI,M
A4 H,M,P,VI,P VH,M,M,VI,V H VH,VH,VH,FI, VH VH,H,VH,AI, VH H,VH,VH,AI, VH VH,H,H,VI,V H VP,P,VP,WI,V P VH,H,VH,AI, VH VH,H,VH,VI, VH
A5 VP,VP,P,WI, VP VP,P,VP,WI, VP P,VP,P,FI,V P P,VP,P,WI,V P H,VH,H,VI, M M,P,P,FI,M H,M,P,FI,VP M,P,M,EI,M M,P,P,WI,V P
A6 VP,P,VP,EI, VP VP,M,P,EI, P P,M,M,WI,P P,P,M,FI,P VH,H,VH,F I,H P,M,H,FI,H VP,P,P,FI,P P,VP,H,FI,P VH,H,H,VI, M
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Aggregation of expert matrices is performed using the TrFNDBM operator (9). The aggregated initial decision making matrix is shown in the Table 11. Table 11 Initial decision making matrix. A1 (3.06,3.65,4.15) (3.65,4.15,3.06) (4.15,3.06,3.65) (3.06,3.65,4.15) (3.65,4.15,3.06) (4.15,3.06,3.65) (3.06,3.65,4.15) (3.65,4.15,2.21) (4.15,2.21,2.77)
A2 (1.95,2.6,3.21) (2.6,3.21,2.86) (3.21,2.86,3.47) (2.86,3.47,4.07) (3.47,4.07,2.04) (4.07,2.04,2.6) (2.04,2.6,3.14) (2.6,3.14,3.06) (3.14,3.06,3.65)
A3 (2.86,3.47,4.07) (3.47,4.07,2.59) (4.07,2.59,3.21) (2.59,3.21,3.81) (3.21,3.81,3.06) (3.81,3.06,3.65) (3.06,3.65,4.15) (3.65,4.15,2.33) (4.15,2.33,2.93)
A4 (1.44,1.96,2.58) (1.96,2.58,2.51) (2.58,2.51,3.11) (2.51,3.11,3.64) (3.11,3.64,3.5) (3.64,3.5,4.09) (3.5,4.09,4.6) (4.09,4.6,3.27) (4.6,3.27,3.87)
A5 (1.14,1.25,1.53) (1.25,1.53,1.14) (1.53,1.14,1.25) (1.14,1.25,1.53) (1.25,1.53,1.14) (1.53,1.14,1.38) (1.14,1.38,1.64) (1.38,1.64,1.14) (1.64,1.14,1.38)
A6 (1.14,1.25,1.53) (1.25,1.53,1.15) (1.53,1.15,1.48) (1.15,1.48,2.04) (1.48,2.04,1.37) (2.04,1.37,1.87) (1.37,1.87,2.41) (1.87,2.41,1.15) (2.41,1.15,1.61)
pro of
Crit. C1 C2 C3 C4 C5 C6 C7 C8 C9
Table 12 Normalized matrix. A1
A2
A3
(0.64,0.74,0.82) (0.69,0.77,0.84) (0.71,0.79,0.85) (0.12,0.17,0.25) (0.07,0.09,0.12) (0.07,0.10,0.17) (0.67,0.76,0.83) (0.14,0.20,0.30) (0.12,0.18,0.28)
(0.72,0.82,0.89) (0.69,0.79,0.85) (0.78,0.85,0.90) (0.16,0.23,0.32) (0.06,0.09,0.13) (0.08,0.12,0.19) (0.69,0.78,0.85) (0.11,0.16,0.24) (0.06,0.10,0.18)
(0.65,0.76,0.83) (0.71,0.80,0.87) (0.71,0.79,0.85) (0.12,0.18,0.26) (0.08,0.10,0.14) (0.08,0.12,0.19) (0.66,0.75,0.82) (0.12,0.18,0.27) (0.10,0.17,0.26)
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Crit. C1 C2 C3 C4 C5 C6 C7 C8 C9
re-
In the next step, the initial decision making matrix is modified using the expression (18), and the normalized matrix is formed, as in the Table 12.
A4
A5
A6
(0.78,0.86,0.92) (0.73,0.81,0.87) (0.68,0.76,0.83) (0.17,0.24,0.33) (0.19,0.26,0.36) (0.20,0.30,0.42) (0.88,0.92,0.94) (0.16,0.23,0.32) (0.17,0.25,0.36)
(0.87,0.91,0.93) (0.88,0.92,0.94) (0.88,0.92,0.94) (0.06,0.09,0.12) (0.15,0.22,0.31) (0.08,0.15,0.25) (0.81,0.87,0.91) (0.07,0.12,0.2) (0.06,0.10,0.17)
(0.87,0.91,0.93) (0.85,0.91,0.94) (0.83,0.89,0.93) (0.06,0.10,0.16) (0.18,0.25,0.33) (0.13,0.21,0.31) (0.86,0.92,0.95) (0.07,0.11,0.16) (0.14,0.20,0.30)
urn a
For the calculation of the score functions of the alternatives, the elements of the normalized matrix (Table 12) and defuzzified values of the weight coefficient of the criteria are used
0.203, 0.085, 0.123, 0.094, 0.058, 0.037, 0.055, 0.084, 0.077
T
.
Using
the
TrFNDNGBM
aggregator (10), final values of the score function are obtained S ( ni ) . Based on the value S ( ni ) the alternatives are ranked and the optimal alternative is selected from the set of considered alternatives. Score functions and ranking of alternatives are shown in the Table 13. Table 13
Rank of alternatives.
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Alternative Congestion charging (A1) Parking prices (A2) Road pricing (A3) Public transport capacity improvements (A4) Synchronized travel plans (A5) Ride sharing (A6)
25
S (ni ) (0.190,0.263,0.354) (0.198,0.273,0.367) (0.195,0.266,0.361) (0.255,0.335,0.420) (0.220,0.279,0.355) (0.227,0.298,0.385)
Rank 6 4 5 1 3 2
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The alternatives are ranked based on the value of the score function S ( ni ) , wherein it is more preferable for the alternative to have the highest possible value of S ( ni ) . Thus, on the basis of the obtained values of S ( ni ) the first rank is assigned to the alternative A4.
pro of
6. Stability of the Obtained Results Analysis of the stability of the obtained results is carried out through four parts. In the first part, the sensitivity analysis of fuzzy FUCOM-D'Bonferroni model is performed by changing the weight coefficients of the criteria. The analysis of the influence of the change in the weight coefficients of the criteria is carried out through 15 scenarios. In the second part, the analysis of the influence of dynamic matrices of decision making to the change of the rank of the alternatives is performed. In the third part, the comparison of the obtained results with other MCDM models is made: TrFNDWG operator (proposed), Dombi weighted arithmetic averaging (TrFNDWAA) operator (proposed), fuzzy MABAC [61] and fuzzy VIKOR [62-63] model. In the fourth part, the analysis of the dependence of the obtained results on the change of ρ, p and q parameters is demonstrated. More detailed overview of the sections of the discussion on the results is shown in the next part of the paper. 6.1. Changing the Weights of the Criteria
re-
After determining the weight coefficients of the criteria using fuzzy FUCOM, the "most effective criterion" is identified for the purpose of the sensitivity analysis. The objective of the sensitivity analysis is to evaluate the influence of the most effective criterion on ranking performance of the proposed model. Based on the recommendations of Kirkvood [64] and Kahraman [65], the proportion of the weights of the criteria in the sensitivity analysis and the elasticity coefficient [66] are defined. The elasticity coefficient is used to express relative compensation of the values of other weight coefficients in relation to changes in the weight of the most important criterion.
Table 14
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In this research, C11 criterion is identified as the most effective one since it has the highest value of the weight coefficient w11 0.203 . In the next step, the coefficient of weight elasticity s of the most significant criterion (see Table 14) is determined and the limits for change of the weight coefficient of the most significant criterion are defined. Coefficient of elasticity for changing weights. C2 0.139
C3 0.201
C4 0.153
urn a
Criteria C1 s 1.00
C5 0.094
C6 0.061
C7 0.090
C8 0.138
C9 0.125
The limit values of the C11 criterion are obtained, which are in the range of -0.2032≤Δx≤0.61299. On the basis of the defined limit values, the scenarios for sensitivity analysis are determined. The interval 0.2032≤ x ≤0.61299 is divided into 15 sequences based on which a total of 15 scenarios are established. For every scenario, new values of the weight coefficients are set, so 15 new groups of weight coefficients are obtained, as shown in the Table 15. Table 15
New criteria weights. C1 0.000 0.050 0.100 0.150 0.200 0.250 0.300
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Scenario S1 S2 S3 S4 S5 S6 S7
26
C2 0.139 0.132 0.125 0.118 0.111 0.104 0.097
C3 0.201 0.191 0.181 0.171 0.161 0.151 0.141
C4 0.153 0.145 0.137 0.130 0.122 0.115 0.107
C5 0.094 0.089 0.085 0.080 0.075 0.071 0.066
C6 0.061 0.058 0.055 0.052 0.049 0.045 0.042
C7 0.090 0.085 0.081 0.076 0.072 0.067 0.063
C8 0.138 0.131 0.124 0.117 0.110 0.103 0.096
C9 0.125 0.119 0.113 0.107 0.100 0.094 0.088
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0.350 0.400 0.450 0.500 0.550 0.600 0.650 0.700
0.090 0.083 0.077 0.070 0.063 0.056 0.049 0.042
0.131 0.121 0.111 0.101 0.090 0.080 0.070 0.060
0.099 0.092 0.084 0.076 0.069 0.061 0.053 0.046
0.061 0.056 0.052 0.047 0.042 0.038 0.033 0.028
0.039 0.036 0.033 0.030 0.027 0.024 0.021 0.018
0.058 0.054 0.049 0.045 0.040 0.036 0.031 0.027
0.089 0.083 0.076 0.069 0.062 0.055 0.048 0.041
0.081 0.075 0.069 0.063 0.056 0.050 0.044 0.038
pro of
S8 S9 S10 S11 S12 S13 S14 S15
The influence of the new values of the weight coefficients from the Table 15 to the change of the ranks of alternatives is presented in the Fig. 7. S1 6
S15
S2
5 S14
S3
4 3 2
S13
S4
re-
1
S12
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S11
S10
A1
S9 A2
A3
A4
S5
S6
S7 S8 A5
A6
Fig. 7. Analysis of sensitivity of the ranks of alternatives through 15 scenarios.
urn a
The results (Fig. 7) show that assigning different weights to the criteria through scenarios leads to the change in the ranks of individual alternatives, which confirms that the model is sensitive to changes in weight coefficients. By comparing the first two alternatives in the rank (A4 and A6) through the scenarios, we can conclude that both first-ranked alternatives (A4 and A6) retain their ranks through all 15 scenarios. The remaining alternatives (A1, A2, A3 and A5) retain their ranks in 12 out of total of 15 scenarios. Just in the first three scenarios, the alternatives A1, A2, A3 and A5 change their rank. In the scenarios S1 and S2, the ranks A4> A6> A2> A3> A1> A5 are obtained, while in the S3 scenario, the alternatives are ranked as follows A4> A6> A2> A5> A3> A1. From the above results, we can conclude that the rank of the alternative A4 is credible and there is sufficient advantage of the mentioned alternative compared to the second-ranked (A6) and other alternatives. The results are also confirmed by the correlation of ranks through the scenarios.
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Correlation of ranks is determined using Spearman's correlation coefficient. Spearman's coefficient (SCC) is used to determine statistical significance of the difference between the ranks obtained through the scenarios [36]. By analyzing the obtained correlation values, we note that there is high correlation of ranks, since in 13 out of 15 scenarios, the SCC exceeds 0.940. In the remaining two scenarios, the SCC values are 0.657. The mean value of the SCC across all scenarios is 0.950, which shows high correlation of ranks, respectively, confirms the results shown in the Table 14. 6.2. Influence of dynamic matrices on changing the rank of alternatives Internal changes in the decision making matrix, such as the introduction of new ones or the elimination of the existing alternatives from the set of considered alternatives, can cause changes in final
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pro of
preferences. Accordingly, in this paper, the performance of the proposed model is analyzed in the conditions of dynamic initial decision making matrix. Five scenarios are formed. For every scenario, a change in the number of alternatives is made and the ranks obtained are analyzed. The scenarios are formed by removing one inferior (the worst) alternative in every scenario from further considerations. At the same time, within the scenarios, the remaining alternatives are ranked according to the newlyobtained initial decision making matrix. The initial solution using fuzzy FUCOM-D'Bonferroni model is generated as A4> A6> A5> A2> A3> A1. It is clear that the alternative A1 is the worst option, and in the first scenario the alternative A1 is eliminated from the set and new decision making matrix is obtained with a total of five alternatives. A new solution for the decision-making matrix is generated and the rank A4> A6> A5> A2> A3 is obtained. The ranking in the first scenario shows that A4 is still the best alternative, while A3 is the worst alternative. Further implementation of the described procedure results in the following ranks through the remaining three scenarios: S2: A4> A6> A5> A2; S3: A4> A6> A5 and S4: A4> A6. Through the modification of the initial matrix, which is done by the elimination of the worst alternative, we notice that there is no rank reversal among the alternatives in the FUCOM-D'Bonferroni model. The alternative A4 remained the best ranked through all scenarios, which confirmed the robustness and accuracy of the obtained ranks of alternatives in dynamic environment. 6.3. Comparison of the FUCOM-D'Bonferroni model ranks with other models
Table 16
re-
In this section, we compared the results of the FUCOM-D'Bonferroni model with the TrFNDWG operator (proposed), Dombi weighted arithmetic averaging (TrFNDWAA) operator, fuzzy MABAC [61] and fuzzy VIKOR [62-63] models. The comparative overview of ranks by various MCDM techniques are presented as given in Table 16. The ranking of alternatives in terms of various MCDM techniques.
Ranking A4>A6>A5>A2>A3>A1 A4>A6>A5>A2>A1>A3 A4>A6>A5>A2>A3>A1 A4>A6>A5>A2>A3>A1 A4>A6>A5>A2>A3>A1
urn a
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MCDM techniques Fuzzy FUCOM-D’Bonferroni TrFNDWG TrFNDWAA Fuzzy MABAC Fuzzy VIKOR
The rank of alternatives according to the presented methods shows that the alternatives A4 and A6 are the best-ranked alternatives through all models. Fuzzy MABAC and TrFNDWAA models have fully confirmed the ranking of the FUCOM-D'Bonferroni model. In the TrFNDWG model, only the last ranked alternatives (A3 and A1) changed, while the remaining alternatives kept their ranks. In fuzzy VIKOR model, the third-ranked and fourth-ranked alternatives (A5 and A2) replaced the positions, while the ranks of the remaining alternatives were confirmed.
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We can conclude that A4 and A6 are the best alternatives by all models, while the A1 alternative in four models (FUCOM-D'Bonferroni, TrFNDWAA, MABAC and VIKOR) is the worst one. The results also show that there is high correlation between the ranks. All the correlation values are significantly higher than 0.90, so we can conclude that there is high correlation between the proposed approach and other tested MCDM models. 6.4. Influence of parameters ρ, p and q on the ranking results In the above steps, the values of the parameters p, q and ρ were initially assumed to be 1, however, it can easily be observed the effects of the changing value of p, q and in the proposed D'Bonferroni model.
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Sceanrio 1 A1 ρ=50
A2
A3 ρ=1 6
Sceanrio 2
A4
A5
A6
A1
ρ=2
A2
A3 p=1
6
p=50
5
A4
A5
A6
p=2
5
4
ρ=30
pro of
In order to examine the influence of the parameters p, q and ρ on the obtained results, three scenarios are formed. In the first scenario, the values of the parameter ρ are changed in the range from 1 to 100, while for the parameters p and q the values are p=q=1. In the second scenario (S2), the values of the parameter p are changed in the range from 1 to 100, while for the parameters ρ and q the values are p=q=1. In the third scenario (S3), the values of the parameter q are changed in the range from 1 to 100, while for the parameters ρ and 𝑝 the values are ρ= p=1. The Fig. 8 shows the influence of the parameters p, q and ρ on the ranks of the alternatives.
ρ=3
4
p=30
3
p=3
3
2
2
ρ=20
ρ=4 p=20
1
p=4
1
0 ρ=10
ρ=5
ρ=10
p=10
ρ=6
ρ=7
p=10
re-
ρ=9
p=5
p=9
ρ=8
Sceanrio 3
A1
p=6
A2
q=50
A3 q=1 6
A4
A5
p=7 p=8
A6
q=2
5
4
q=3
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q=30
3
2
q=20
q=4
1
q=10
q=5
urn a
q=10
q=9
q=6
q=7 q=8
Fig. 8. Ranking orders for varying values of parameters 𝑝, 𝑞 and .
Jo
Generally, the higher the values of the parameters p, q and ρ, the more complex the calculation becomes, and the more the interrelations between the attributes are emphasized. DMs usually choose the parameters p, q and ρ according to their preferences [37]. In real decision making, we generally recommend that the parameter values be 1 from a practical point of view, which is not only intuitionistic and simple, but also able to consider the inner connections between attributes. From the Fig. 8, it can be clearly pointed out that when the parameters p, q and ρ have different values, the ranking orders of the considered alternatives remain almost the same. Minor changes occur when the parameter ρ is changed, while changing the parameters p and q there is no change in the ranking. In the scenario S1, for the values ρ=5 and ρ=9, the ranks A4> A6> A2> A5> A3> A1 are obtained, while the rank A4> A6> A5> A2> A1> A3 is obtained when ρ=20. The changes in ranks that occurred when the values of ρ=5, ρ=9 and ρ=20 are minimal and did not impact the rank of alternatives A4 and A6 that were identified as the best alternatives. From the above analysis we can conclude that the parameters p and q have no effect on final preferences. The parameter ρ has minor influence on the ranking of certain alternatives, but it does
29
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not impact final preferences when choosing the most effective alternatives, respectively, the alternatives A4 and A6. 7. Results and Discussion
pro of
It is noteworthy that results suggest the prioritization of pull measures over push measures. This is largely because pull measures cause less public reaction and involve less costs compared to push measures which are often based on penalizing the use of cars, hence triggering public reaction. Push measures also require higher investment costs, which make them less attractive in the eyes of decision makers. Results also indicate that public transport capacity improvements should be prioritized over other measures. Public transport capacity improvements include the following actions:
Increasing the number of vehicles in the bus fleet, Increasing the frequency of bus services, Increasing the length of total metro network, Adding new rolling stock to the metro
re-
Capacity improvements in public transport can require high capital and operational costs compared to other measures of the TDM project. However, it has the highest impact on environmental, economic and social outcomes. Increasing the capacity of public transport would enhance the service quality levels of public transport services. For example, frequent bus services increase the attractiveness of public transport by reducing the crowding level in buses which discourage people from using public buses.
lP
Ride sharing comes next in implementation ranking. There is no cost involved by IMM to implement ride sharing. Third party private network companies provide ride sharing services. IMM can support the use of those services by including ride sharing option in its IBB Navi application, which is a journey planning application suggesting travel routes to the desired destinations. Incorporation of ride sharing as one of the travel options would increase the use of ride-sharing by passengers. By helping reduce the number of cars used by people, ride-sharing can contribute a great deal to the improvement of urban mobility. Economic, social and environmental benefits of ride-sharing are considerably high when one considers its impact on reducing car use.
urn a
The third pull measure, synchronized travel plans, particularly helps ameliorate the peak hour predicament in Istanbul. The economic, social and environmental impacts of this measure are quite significant. Most workplace including public schools and agencies start working between 8:00-08:30, which makes millions of passengers rushing towards cars, public buses, and metros at the same time. By shifting the start times of public agencies and schools, IMM can achieve a more balanced distribution of trips around the peak hours. This would relieve the load on public buses, and make them more comfortable during those hours. This measure requires coordination with other public bodies such as Governorate of Istanbul, National Education Agency, etc. After pull measures, IMM should also implement push measures to discourage the use of cars. Increasing the parking prices is the less costly method among the push measures. By charging drivers for occupying public space which could also be used by passengers, cyclists, public transport users, parking prices can bring about a more public-transport friendly mobility in Istanbul. In terms of economic impacts, IMM can generate significant resources through charging for parking lots. It can use the resources obtained in reinvesting in public transport services (e.g. capacity improvements, etc.). IMM can both increase parking prices and increase the number of parking lots and areas to be charged around the city, hence preventing on-the-street parking without any charge, which makes the urban roads quite busy and passenger-unfriendly.
Jo
In addition to parking prices, road pricing can also be considered to discourage car use. However, road pricing is a political decision which might not be feasible to implement during the times when economy is in decline or stagnation. It can bring about a public reaction by the drivers as well as private transport operators such as taxi and minibus services. Its economic and social impacts are not as straightforward as pull measures, although it can lead to a potential improvement in air quality by reducing the number of cars on the streets and roads.
30
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Finally, congestion charging can be considered by IMM as the last option to implement. High initial upfront investment costs and operational costs make congestion charging as an option not prioritized by IMM given the budget constraints and economic difficulties. It can also create a public reaction similar to road pricing. Therefore, congestion charging had better stay as a last option to implement after implementing pull measures. Push measures can complement the prioritized pull measures.
pro of
The practical application of this model allows obtaining credible results when deciding under uncertainty conditions and when the data on which it is based is partially known and imprecise. This model practically helps managers to deal with their own subjectivity in prioritizing criteria and attributes. The use of the fuzzy FUCOM-D’Bonferroni approach reduces imprecision when assessing the impacts of TDM measures. The research has shown that, in addition to predictable indicators, the assessing of the TDM measures is influenced by numerous unknown and partially known indicators. The results of the research indicate the justification of the selected MCDM model. 8. Conclusion
re-
IMM’s problem of prioritizing the TDM measures is solved in this study by applying a fuzzy-based method. Without such a prioritization, IMM could not achieve the implementation of the entire TDM policy set at the same time. Pull measures being prioritized is quite expected given the incentive-based approach in those measures. They involve making peoples’ travel decisions shift towards public transport modes without any penalty for car use. Public transport capacity improvements stand out as the most highly suggested TDM measure, as followed by ride sharing and synchronized travel options, all of which are pull measures. On the other hand, push measures being more penalty-based which discourage car use make them less prioritized in the eyes of IMM. The push measures, such as increasing the parking prices, road and congestion charging measures come after the pull measures. Decision makers may avoid generating public reaction by implementing penalty-based push measures. The study demonstrated that implementing pull measures should be prioritized over push measures given the budget constraints and impacts of TDM measures.
Jo
urn a
lP
The developed fuzzy FUCOM-D’Bonferroni model has shown a great degree of flexibility and it can be used in other branches of transport and for other real problems. It should then be adapted to the needs of managers from those branches and the issues considered. The developed model is based on expert knowledge and on the basis of expert assessments it can rank certain alternatives. However, it is possible to use real data for the assessment of alternatives since the fuzzy FUCOM-D’Bonferroni model is not limited to certain scales. It can help the DMs to use this method in any situation when it is necessary to rank certain alternatives. The main points of this paper are as follows: - The development of a new method of multi-criteria analysis based on hierarchical and methodological procedures for solving real-world business problems, and in this study applied for evaluation of alternatives in the form of push and pull measures as part of TDM strategy. - The application of a decision-making model based on FUCOM and Dombi-Bonferroni aggregators was tested by qualitative analysis and it helped to classify push and pull measures from the best to the worst by applying four main and nine sub-critera. - The model is very flexible and simple so it can also be applied to other problems of multi-criteria analysis (in transport, logistics and other sectors of the economy). - Fuzzy FUCOM-D’Bonferroni model provides the ranking of alternatives, and helps managers and all other decision-makers to address the problem easily and solve it by applying the ranking of alternatives. - The applied fuzzy FUCOM-D’Bonferroni model has shown good performance with a classifying push and pull measures. - Future studies can explore the use of fuzzy-based methods (fuzzy FUCOM and DombiBonferroni operators) on selecting other transport measures being considered by decision makers and planners. In contrast to previous studies in TDM measures, we applied original hybrid Dombi-Bonferroni aggregation operators to address fuzzy values. This allow us to make the information aggregation process more flexible by a parameter. The second advantage of the proposed MCDM model is the
31
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pro of
development of fuzzy FUCOM for determining criteria weights. Moreover, pairwise comparisons for criteria were conducted using linguistic variables instead of crisp values in the decision-making process. Fuzzy FUCOM provides credible weight coefficients with only n-1 pairwise comparisons, which contribute to reasonable evaluation in decision-making. As a result, the fuzzy FUCOM is an effective decision-tool that aids decision-makers in dealing with their own subjectivity while prioritizing criteria. Therefore fuzzy approach helped us solve this complex and multi-dimensional problem in a convenient way. In the future, we plan to implement for working on topics as; implementing the proposed model in other transport problems measures being considered by decision makers and planners and the supply chain problems such as logistic and facility location. In addition, improving the model using rough set approach or neutrosophic fuzzy values might be an option for further research work. References
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pro of
[63] Nikolic, V., Milovancevic, M., Petkovic, D., Jocic, D., Savic, M. (2018). Parameters forecasting of laser welding by the artificial intelligence techniques. Facta Universitatis, Series: Mechanical Engineering, 16(2), 193-201. [64] Kirkwood, C. W. (1997). Strategic decision making: multiobjective decision analysis with spreadsheets (Vol. 59). Belmont, CA: Duxbury Press. [65] Kahraman, Y. R. (2002). Robust sensitivity analysis for multi-attribute deterministic hierarchical value models (No. AFIT/GOR/ENS/02-10). AIR FORCE INST OF TECH WRIGHTPATTERSONAFB OH. Appendix 1
Theorem 1. Let it be j , , l j
m j
u j
; j 1, 2,..., n , collection of TrFNs in R, then TrFNDBM
operator is defined as follows 1
n
im
,
i 1
n ( n 1) 1 1 n 1 pq m 1 f 1 f m i , j 1 i j i j q p f m m f i j
lP
n il i 1 1/ 1 1 n ( n 1) p q n 1 i l l 1 f , j 1 1 f i j i j q p f l l f i j
re-
pq n p q 1 TrFNDBM p , q , ( 1 , 2 ,..., n ) i j n(n 1) i , j 1 i j
1/
,
urn a
Proof.
i 1
n ( n 1) 1 1 n 1 pq u 1 f 1 f u i , j 1 i j i j q p f u u f i j
l m u where f i f (il ), f (im ), f (iu ) n i l , n i m , n i u i i i i 1 i 1 i 1
n
iu
1/
represents fuzzy function.
We need to prove the Eq. (7) is kept. According to the operational laws of TrFNs, we get il i 1/ 1 p 1l il i p
,
im
m 1 1 p mi i
1/
il lj i j 1/ 1 p 1il q 1 lj il lj p
q
Jo
further we get
35
,
,
iu
u 1 1 p u i i
1/
q l , j j 1/ 1 q 1 lj lj
im mj
m 1 1 p mi i
1 mj q m j
1/
,
,
mj 1/
u 1 1 p u i i
1 uj q u j
1/
and , 1/ u 1 j 1 q u j uj
m 1 j 1 q m j
iu uj
(7)
Journal Pre-proof
n
im i 1
n 1 i , j 1 1 f m i i j p f m i
n p q 1 i j n(n 1) i , j 1 i j
im
n
i 1
Thereafter,
,
1/
1
1 f mj q f m j
im i 1
i 1
n 1 i , j 1 1 f m i i j p f m i
im i 1
n
i 1
1/
n 1 1 n ( n 1) i , j 1 1 f m i i j p f im
Therefore,
1
1 f mj q f mj
,
im i 1
pq n p q 1 ( 1 , 2 ,..., n ) i j n(n 1) i , j 1 i j
n
Jo
urn a
n il i 1 1/ 1 1 n ( n 1) pq n 1 i , j 1 1 f il 1 f lj i j q p f l l f i j
lP
TrFNDBM
36
,
iu
i 1
n ( n 1) 1 1 n 1 pq 1 f m i , j 1 1 f m i j i j q p f m m f i j
n
im
1/
,
i 1
n ( n 1) 1 1 n 1 pq 1 f u i , j 1 1 f u i j i j q p f u u f i j
1/
1 f mj q f m j
n
im i 1
n 1 1 n ( n 1) i , j 1 1 f m i i j p f im
1
p,q,
1/
1
n
im
n
,
re-
n n il i 1 il 1/ i 1 n 1 1 1 n ( n 1) i , j 1 1 f l 1 f lj i i j q p f il f lj
n
im
n
,
pro of
n n n l il p q i 1 1/ i j i i , j 1 i 1 i j n 1 1 i , j 1 1 f l 1 f lj i i j p q f l f l i j
1/
1
1 f mj q f mj
Journal Pre-proof
Appendix 2 Theorem 2 (Idempotency): Set j lj , mj , uj ; p, q,
(1 , 2 ,.., n ) TrFNDBM
j 1, 2,..., n , collection of TrFNs in R, if
(,,.., ) .
Proof: Since i , i.e. , , u then l i
m i
l
u i
m
TrFNDBM p 1, q 1, 1 ( 1 , 2 ,..., n ) ( , ,..., )
,
n ( n 1) 1 1 n 1 pq 1 f mj i , j 1 1 f m i i j q p f m m f i j
2im 1 1 n 11 i 1
1/
2
1
1i m i
m
1im m i
urn a
2il 1/ 1 1 n 1 l i1 (11) 1l i i
,
,
2im
1 n i 1
1/
1 1 m (11) 1i m i
,
1 n i 1
Jo
The proof of Theorem 2 is completed.
37
n
iu
i 1
,
1/
2iu
1 1 n 11 i 1
,
i 1
n ( n 1) 1 1 n 1 pq 1 f uj i , j 1 1 f u i i j q p f u u f i j
2iu
1/
1 1 u (11) 1i u i
1/
2 1 u u 1i 1i u u i i
lP
2il 1/ 2 1 111 n 1 l l i 1 1i 1i l l i i
n
im
re-
n il i 1 1/ 1 1 n ( n 1) pq n 1 i , j 1 1 f il 1 f lj i j q p f l l f i j
pro of
then TrFNDBM
p, q,
l , m , u
1/
i ,
Journal Pre-proof
Appendix 3 Theorem 3 (Boundedness): Set j , , l j
min , min , min
l i
m i
u i
m j
u j
; j 1, 2,..., n ,
and max , max
l i
m i
, max
u i
collection of TrFNs in R, let
then
pro of
TrFNDBM p, q, (1, 2 ,..., n ) .
Proof: Let min(1 , 2 ,..., n ) min il , min im , min iu and
max(1 , 2 ,..., n ) max il , max im , max iu .
Then,
it
can
be
stated
that
l min(il ) , i
m min(im ) , u min(iu ) , l max(il ) , m max(im ) and u max(iu ) . Based on that, the i
i
i
i
following inequalities can be formulated:
i ; min(il ) il max(il ); i
i
min(im ) im max(im ); i
i
min(iu ) iu max(iu ); i
re-
i
i
According to the inequalities shown above, it can be concluded that TrFNDBM p,q, (1, 2 ,..., n ) holds. Theorem 5. Let it be j lj , mj , uj ;
lP
n 1 p i q j p q i , j 1
wi w j
1 wi
n
n
urn a
n n il i 1 il 1/ i 1 1 wi 1 1 ( p q ) w w i j n 1 l l f f i , j 1 i i j q j p 1 f l 1 f il j
,
i 1
m i
1 1 ( p q ) wi w j
n
im
n
i 1
1/
1 wi n 1 f m i , j 1 f im j i j q p 1 f m m 1 f i j
,
i 1
u i
l m u where f i f (il ), f (im ), f (iu ) n i l , n i m , n i u i i i i 1 i 1 i 1
Jo
Appendix 4
j 1, 2,..., n , collection of TrFNs in R, then TrFNDNGBM
operator is defined as follows TrFNDNGBM p , q , ( 1 , 2 ,..., n )
iu i 1
1 1 ( p q ) wi w j
1/
1 wi n 1 f u i , j 1 f iu j i j q p 1 f u u 1 f i j
(8) represents fuzzy function.
Proof: We need to prove the Eq. (8) is kept. According to the operational laws of TrFNs, we get
38
Journal Pre-proof
pro of
l il im iu m u p i i 1/ , i 1/ , i 1/ , l m u 1 p i l 1 p i m 1 p i u 1i 1i 1i lj mj uj m u and q j lj , j , j 1/ 1/ 1/ l m u j j j 1 q 1 q 1 q l m u 1 j 1 j 1 j
il lj im mj iu uj m u u p i q j il lj mj 1/ , i 1/ , i j 1/ l l m m u u j j j 1 p i l q 1 p i m q 1 p i u q l m u 1 1 1 1 1 1 i j i j i j
further we get
i
q j
wi w j 1 wi
,
im mj 1 1 1 wi 1 m p mi i
Thereafter,
i
q j
1 mj q m j
,
,
iu uj
1 1 1 wi 1 u p ui i
1/
wi w j 1 uj q u j
n
n
im
iu
i 1
wi w j n 1 1 wi i , j 1 i j
urn a
i , j 1 i j
wi w j 1 wi
n il i 1 1/ wi w j n 1 1 1 wi l l ii, jj1 p 1 f i q 1 f j l l f j f i
wi w j
lP
p n
1/
re-
p
il lj 1/ 1 wi w j 1 1 wi l l p 1l i q 1l j i j
1/
1
1 f im p f m i
1 f mj q f m j
,
i 1
wi w j n 1 1 wi i , j 1 i j
1/
1
1 f iu p f u i
1 f uj q f u j
Therefore,
TrFNDNGBM p , q , ( 1 , 2 ,..., n )
n 1 p i q j p q i , j 1
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n n il i 1 il 1/ i 1 1 wi 1 1 ( p q ) w w i j n 1 f l i , j 1 f il i j q j p 1 f l 1 f il j
39
n
,
i 1
m i
wi w j
1 wi
1 1 ( p q ) wi w j
n
n
im
n
i 1
1/
1 wi n i , j 1 i j
1
f m i p 1 f m i
f m j q 1 f m j
,
i 1
u i
iu i 1
1 1 ( p q ) wi w j
1/
1 wi n i , j 1 i j
1
f u i p 1 f u i
f u j q 1 f u j
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Appendix 5 Expert 1 (C11-C12) min
Expert 5 (C11-C12) min s.t.
w11l w11m w11u u 2.50 ; m 3.00 ; l 3.5 ; w12 w12 w12 l m u l m u ( w 4 w w ) / 6 ( w 4 w 11 11 11 12 12 w12 ) / 6 1; wl wm wu , j 1, 2 j j j wl , wm , wu 0, j 1, 2 j j j
w11l w11m w11u u 1.50 ; m 2.00 ; l 2.5 ; w12 w12 w12 l m u l m u ( w 4 w w ) / 6 ( w 4 w 11 11 11 12 12 w12 ) / 6 1; wl wm wu , j 1, 2 j j j wl , wm , wu 0, j 1, 2 j j j
Expert 1 (C21-C22) min
Expert 5 (C21-C22) min
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s.t.
s.t.
l m u w21 w21 w21 u 3.50 ; m 4.00 ; l 4.5 ; w w w 22 22 22 l m u l m u ( w21 4 w21 w21 ) / 6 ( w22 4 w22 w22 ) / 6 1; wl wm wu , j 1, 2 j j j wl , wm , wu 0, j 1, 2 j j j
l m u w21 w21 w21 u 2.50 ; m 3.00 ; l 3.5 ; w w w 22 22 22 l m u l m u ( w21 4 w21 w21 ) / 6 ( w22 4 w22 w22 ) / 6 1; wl wm wu , j 1, 2 j j j wl , wm , wu 0, j 1, 2 j j j
Expert 1 (C31-C33) min
Expert 5 (C31-C33) min
s.t.
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s.t.
s.t.
w w w 2.50 ; 3.00 ; 3.5 ; w w w l m u w31 0.71 ; w31 1.00 ; w31 1.4 ; u m l w33 w33 w33 l m u w w w32 32 32 wu 1.79 ; wm 3.00 ; wl 4.90 ; 33 33 33 ( wl 4 wm wu ) / 6 ( wl 4 wm wu ) / 6 31 31 31 32 32 32 l m u ( w33 4 w33 w33 ) / 6 1; l m u l m u w j w j w j , w j , w j , w j 0, j 1, 2,3
l m u w31 w31 w31 u 1.50 ; m 2.00 ; l 2.5 ; w33 w33 w33 l m u w33 1.40 ; w33 2.00 ; w313 3.00 ; u m l w32 w32 w32 l m u w31 w31 w31 wu 2.10 ; wm 4.00 ; wl 7.50 ; 32 32 32 ( wl 4 wm wu ) / 6 ( wl 4 wm wu ) / 6 31 31 31 32 32 32 l m u ( w33 4 w33 w33 ) / 6 1; l m u l m u w j w j w j , w j , w j , w j 0, j 1, 2,3
Expert 1 (C41-C42) min
Expert 5 (C41-C42) min
m 32 m 31
u 32 l 31
s.t.
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l 32 u 31
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l m u w42 w42 w42 u 2.50 ; m 3.00 ; l 3.5 ; w41 w41 w41 l m u l m u ( w 4 w w ) / 6 ( w 4 w 41 41 41 42 42 w42 ) / 6 1; wl wm wu , j 1, 2 j j j wl , wm , wu 0, j 1, 2 j j j
40
s.t. l m u w41 w41 w41 u 1.50 ; m 2.00 ; l 2.5 ; w42 w42 w42 l m u l m u ( w 4 w w ) / 6 ( w 4 w 41 41 41 42 42 w42 ) / 6 1; wl wm wu , j 1, 2 j j j wl , wm , wu 0, j 1, 2 j j j
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*Highlights (for review)
1) Pull and push measures under a TDM project are identified. 2) Transport demand management measures are assessed. 3) A new fuzzy FUCOM-D'Bonferroni method for solving MCDM problem is proposed.
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4) Economic, cost, social and environmental criteria are considered.
5) The proposed method can be adapted for other transport problems.
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6) Different scenarios are employed for stability of results.
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*Declaration of Interest Statement
Declaration of interests
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☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☒The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
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To the author’s knowledge, no conflict of interest, financial or other, exists. Each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content.
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*Author Contributions Section
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Author contribution statements
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Dragan Pamucar and Muhammet Deveciand conceived of the presented idea, writing–review and editing. Muhammet Deveci developed the theory and Dragan Pamucar performed the computations. Fatih Canitez and Darko Bozanic verified the analytical methods. All authors discussed the results and contributed to the final manuscript.
PII: DOI: Reference:
S1568-4946(19)30733-1 https://doi.org/10.1016/j.asoc.2019.105952 ASOC 105952
To appear in:
Applied Soft Computing Journal
Received date : 25 June 2019 Revised date : 12 November 2019 Accepted date : 18 November 2019 Please cite this article as: D. Pamucar, M. Deveci, F. Canıtez et al., A fuzzy Full Consistency Method-Dombi-Bonferroni model for priorititizing transportation demand management measures, Applied Soft Computing Journal (2019), doi: https://doi.org/10.1016/j.asoc.2019.105952. This is a PDF file of an article that has undergone enhancements after acceptance, such as the addition of a cover page and metadata, and formatting for readability, but it is not yet the definitive version of record. This version will undergo additional copyediting, typesetting and review before it is published in its final form, but we are providing this version to give early visibility of the article. Please note that, during the production process, errors may be discovered which could affect the content, and all legal disclaimers that apply to the journal pertain.
© 2019 Elsevier B.V. All rights reserved.
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A Fuzzy Full Consistency Method-Dombi-Bonferroni Model for Priorititizing Transportation Demand Management Measures Dragan Pamucar a,*, Muhammet Devecib,c, Fatih Canıtezd, Darko Bozanice a
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Department of Logistics, Military academy, University of Defence, Belgrade 11000, Serbia, +381668700363, fax: +381113603832, email: [email protected]; [email protected] b Department of Industrial Engineering, Naval Academy, National Defense University, 34940, Turkey c ASAP Research Group, School of Computer Science, University of Nottingham, NG8 1BB, Nottingham, UK, [email protected], [email protected] d Department of Management Engineering, Faculty of Management, Istanbul Technical University, 34367 Maçka, Istanbul, Turkey, [email protected] e University of Defence, Belgrade 11000, Serbia, email: [email protected]
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Abstract:The selection and prioritization of appropriate Transportation Demand Management (TDM) measures is a common problem faced by transport planners and decision makers. The problem involves many uncertainties due to changing economic conditions, uncertainty in project success, changes in mobility and population characteristics etc. In this study, the multi-criteria decision making (MCDM) based fuzzy Full Consistency Method-Dombi-Bonferroni (fuzzy FUCOM-D'Bonferroni) model is proposed for a case study in Istanbul's urban mobility system. Istanbul’s historical peninsula is considered to be a pilot area for the implementation of TDM projects by the local government. The proposed model is compared with other well-known four MCDM methods in order to show its validity and consistency. The results show that public transport capacity improvements is the best alternative among the other TDM measures. Keywords: Transport demand management, sustainable mobility, pull-push measures, urban transport, fuzzy FUCOM, multi-criteria decision making. 1. Introduction
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Promoting sustainable urban transport measures is one of the key challenges that cities face and grapple with [1]. Public transport, cycling and walking are the urban mobility options [2-4] which can bring about change towards sustainable urban transport [3]. Achieving a modal shift from private cars to public transport options requires a coordinated action by cities encouraging use of public transport and discouraging private car use [5]. The problem is especially acute in developing megacities such as Istanbul, Sao Paulo, New Delhi, etc. where rapid population growth and rising transport demand are key urban problems due to inefficient and insufficient urban transport. The results of these urban problems cause chronic traffic jams, air and noise pollution, crowded and unreliable public transport [6-9]. Insufficient public transport supply, heavy reliance on private cars and the lack of effective and consistent transport policies add to these urban problems, making the cities unsustainable in terms of economic, social, and environmental outcomes.
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Transportation demand management (TDM) is a set of transport policies which seek to change people’s travel behaviour towards more sustainable modes of transport and reduce the travel demand by providing alternative transport options [10-12]. By relocating and shifting the transport demand, more effective use of existing road space is aimed. Similar to balancing supply and demand in economics, transportation supply is sought to be balanced with transportation demand. The set of measures under TDM is commonly grouped into two categories [13-15]:
Push (hard) measures which aim to discourage people from the use of road space, often involve penalising or charging for use of road space. Congestion charging, parking pricing, road pricing are some examples of push measures.
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Pull (soft) measures aim at encouraging and incentivizing the use of sustainable transport modes such as public transport, cycling, and walking. Changing workplace travel plans (teleworking), providing active transport modes and associated infrastructure such as pedestrian amenities, cycling lanes, etc., car-sharing and ride-sharing activities, and enhancement of public transport services are some examples for the implementation of pull measures.
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The city of Istanbul, having a population around 15 million inhabitants [16], faces many urban mobility problems ranging from traffic congestion to insufficient and often crowded public transport services, particularly during peak times. Istanbul Metropolitan Municipality (IMM), the local government in Istanbul which is also responsible for regulating and managing urban transport, set out a new transport policy and plan named Integrated Urban Transport Master Plan for Istanbul. The plan aims to prioritize the use of public transport and discourage the use of private cars. To promote and prioritize public transport, IMM has dedicated significant financial resources to increase the modal share of rail-based modes including metro, tram, light rail transit (LRT) and funicular modes in Istanbul’s public transport. Rail-based public transport increased from 45.1 km in 2014 to 233 km in 2019. Surface transport modes such as bus and Bus Rapid Transit (BRT) services are promoted and their capacities increased with fleet investments. BRT connects European and Asian sides across Bosporus through a totally segregated right-of-way. Besides public transport investment, road infrastructure projects have increased the road capacity of Istanbul, which further triggered car use, hence contradicting the public transport-oriented policies. Transport supply, therefore, has improved the supply side of urban transport in Istanbul [15]. TDM measures are proposed by IMM to also improve the demand side of urban transport in Istanbul. Both push measures and pull measures are proposed as a set of policies to address urban mobility problems. The following push measures are proposed by IMM:
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Congestion charging, Increasing parking prices, Road pricing.
On the other hand, the following pull measures are put forward by IMM:
Increasing the capacity of public transport, Synchronized travel plans, Ride sharing.
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The measures are combined into a master TDM project by the urban planners, transport experts and transport managers in IMM. However, due to budget and time constraints, the measures (sub-projects) should be prioritized before implementation based on the costs and outcomes or impacts of the measures. Capital and operating costs as well as economic, social and environmental impacts are considered for each measure for prioritization. The prioritization should also include the uncertainty, hence leading the experts to employ a fuzzy multi-criteria decision making (MCDM) method. In TDM literature, however, a prioritization of those measures to facilitate implementation has not been examined. There are multiple TDM projects and measures which can be implemented in cities, yet these measures need to be prioritized and selected based on their importance and contribution. This topic is important in TDM projects due to a lack of prioritization methodology addressing the uncertainty in those projects.
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Real-life applications of multi-criteria decision making approaches require the processing of imprecise, uncertain, qualitative or vague data. An efficient way to model uncertainty and imprecision is the use of fuzzy sets theory [16]. Fuzzy sets provide the flexibility required to represent and handle the uncertainty and imprecision resulting from a lack of knowledge or ill-defined information [17]. Due to the availability and uncertainty of information as well as the vagueness of human feeling and recognition, it is relatively difficult to provide exact numerical values for the criteria and to make an exact evaluation and to convey the feeling and recognition of objects for decision-makers [18]. In the last few years, numerous studies attempting to handle this uncertainty, imprecision and subjectiveness have been carried out basically by means of fuzzy set theory, as fuzzy set theory might provide the flexibility needed to represent the imprecision or vague information resulting from a lack of knowledge
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The research questions of this paper are as follows:
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or information. Therefore, the application of fuzzy set with multi-criteria evaluation methods for evaluation of a TDM measures has proved to be an effective approach. Most of the TDM measures cannot be given precisely and the evaluation data of the alternatives’ suitability for various subjective criteria and the weights of the criteria are usually expressed in linguistic terms by the decision-makers. The transition from vagueness provided by linguistic values like ‘‘equally importance/very poor, ‘‘weakly important/poor’’, ‘‘fairly important/medium’’, ‘‘very important/high’’, ‘‘absolutely important/very high’’, etc. to quantification is performed by applying the fuzzy set theory. Therefore, this study extends the previous studies by providing a fuzzy MCDM methodology for the implementation of TDM measures. Can TrFN Dombi Bonferroni (TrFN D'Bonferroni) mean operator for processing fuzzy information be used in MCDM problems? How can we adapt a new fuzzy FUCOM to process fuzzy information described by TrFNs? How does this new fuzzy operator work in a real-life case?
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This study examines the proposal and implementation of the TrFN Dombi Bonferroni (TrFN D'Bonferroni) mean operator in Istanbul’s TDM project for Historical Peninsula area. The remainder of this paper is organized as follows. In Section 2, a background to the techniques are presented. An overview of Istanbul’s urban transport system is introduced in Section 3. Section 4 gives preliminaries of used approaches. The problem description and proposed method are also described in Section 5. Section 6 presents experimental results and scenario analysis. Results are discussed in Section 7. Section 8 concludes the study. 2. Background
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A wide variety of TDM measures have been implemented in the world to reduce car use and increase public transport modal share. 2.1. Overview of Transport Demand Management
Table 1
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The effectiveness and preferability of TDM policies depend on many factors including political priorities, sustainability awareness, perceived benefits and costs, and potental public reaction [14]. The literature invastigates the examination of a variety of TDM measures in the world [15,28-31]. Table 1. shows some application of these TDM measures among the cities. TDM measure application examples in the world. Measures
City
Bianco [20] Seik [21] Goh [22] Santos and Shaffer [23] Beevers and Carslaw [24] Eliasson [25] Rotaris et al. [26] Chu [27]
Parking Pricing Electronic road pricing Electronic road pricing Congestion charging Congestion charging Congestion charging Electronic road pricing Car restraint policies and mileage
Portland Singapore Singapore London London Stockholm Milan Singapore
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Author(s)
Batur and Koc [14] examine the potential of TDM measures for Istanbul by carrying out a survey for the residents to understand their perception of the proposed TDM measures. They found that TDM measures provide significant potential for reducing the traffic congestion in Istanbul. Recently,
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2.2. Literature Review of FUCOM Method
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information-based TDM measures such as providing environmental impacts and health benefits for the transport mode being used has gained popularity. E Silva et al. [31] analyze the influence of these information-based TDM measures on commuting mode choice in Lisbon Metropolitan Area. Hammadou and Mahieux [32] explore the potential for TDM for ensuring low carbon mobility by estimating a mode choice model with a nested logit specification. Habibian and Kermanshah [33] examine the role of TDM on commuters’ mode choice in the city of Tehran by analysing the impact of five TDM measures: increasing parking cost, increasing fuel cost, cordon pricing, transit time reduction, and transit access improvement. Their model shows that the single policies main impact and interactions among the multiple policies are significant in influencing the commuters’ mode choice.
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One of the recent models which is the in the same way as the Analytic Hierarchy Processes (AHP) method [34] and the Best Worst Method (BWM) [35], based on the principles of comparisons in pairs of criteria and the validation of results through deviation from the maximum consistency, is the Full Consistency Method (FUCOM) [36]. The benefits specific to the application of the FUCOM are: (1) small number of comparisons in pairs of criteria (only n-1 comparison), (2) possibility of results validation by defining the deviation from the maximum consistency (OMC) of the comparison, (3) taking into consideration transitivity in the comparison of pairs of criteria; and (4) eliminating the issue of redundancy of comparisons in pairs of criteria, which is present in some subjective models for determining weights of criteria [37]. Although this is a new model, there are certain studies in which the benefits of the FUCOM are employed. Badi and Abdulshahed [38] showed the application of the FUCOM in the evaluation of lines in air traffic. Noureddine and Ristic [39] used hybrid FUCOMMABAC model for evaluating routes in the transport of dangerous goods by road traffic. Pamucar et al. [37] showed the application of the FUCOM-MAIRCA multi-criteria model in the evaluation of level crossings during the installation of security equipment. In addition to the above studies, the FUCOM was also applied in the field of logistics: the selection of equipment for storage systems [40], sustainable supplier selection [41] and supply chain management [42-43]. According to the author's knowledge, the application of the FUCOM in fuzzy environment has not been explored so far. Therefore, one of the motives for the development of this paper is the expansion of the FUCOM in fuzzy environment. 2.3. Motivation of Dombi and Bonferroni Aggregators
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Generally, aggregation operators are important tools for fusing information in MCDM problems. The most widely used operators in fuzzy theory are the min and the max operator. Hereby, we would like to emphasize their main advantages: (1) They are easy to calculate and (2) They can be extended into a lattice structure. However, in the case of min-max operators, the main disadvantage is that the result is determined only by one variable and the other has no influence. Also, the min-max operators are not analytic, their second derivative is not continuous [44]. These disadvantages of traditional minmax operators in fuzzy environment are successfully eliminated by generalized Dombi operator class. Also, Dombi T-norms (TN) and T-conorms (TCN) have general parameters of general TN and TCN, and this can make the aggregation process more flexible.
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However, one of the Dombi TN and TCN limitations is the manipulation with the numbers in the interval [0,1]. Therefore, so far Dombi TN and TCN have only been used to transform uncertain numbers meeting this requirement. For example, Dombi TN and TCN are used in the aggregation of information in group decision-making in intuitionistic fuzzy environment [45], interval-valued intuitionistic fuzzy environment [46], single-valued neutrosophic environment [47], hesitant fuzzy environment [48] etc. In all of the above examples, uncertain numbers meet the requirement that the values are in the interval [0,1]. In real decision making systems, the attributes are often represented by values that are not within the interval [0,1], w especially when it is necessary to present the distance (in kilometers) between two cities. Transformation of such attributes into MCDM models by using traditional Dombi TN and TCN is not possible. In order to eliminate this limitation, in this paper Dombi TN and TCN are modified with the purpose of the aggregation of fuzzy numbers regardless of the values they are presented with. Making decisions in real systems requires rational understanding of the relationship between attributes and elimination of the impact of awkward data. For this purpose, Bonferroni [49] introduced the Bonferroni mean (BM) operator, allowing the presentation of interconnections between elements
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and their fusion into unique score function. Zhu et al. [50] presented the geometric Bonferroni mean through the implementation of geometric mean operator in the BM operator. He et al. [51] and He & He [52] introduced the expansion of the BM operator in intuitionistic fuzzy environment. In order to consider the benefits of hybrid aggregators, He et al. [53-55] demonstrated the possibility of the power average (PA) and BM operator fusion. In recent years, the advantages of the Bonferroni aggregator have been implemented through MCDM models in numerous fuzzy uncertainty theories: intuitionistic fuzzy sets [49], interval-valued intuitionistic fuzzy sets [56], rough sets [54] etc. One of the demands of the decision making process in real systems is flexible decision making and taking into consideration the interaction between the attributes of a decision. Obviously, Bonferroni and Dombi aggregators can successfully achieve this goal. According to the author's knowledge, there has been no research up to date considering the fusion of the Dombi and Bonferroni aggregators in fuzzy environment represented by triangular fuzzy numbers (TrFNs). Therefore, logical goal and motivation for this study is to show hybrid Dombi-Bonferroni aggregator for the transformation by TrFN. However, BM cannot be processed by Dombi operations. In this study, TrFN D'Bonferroni aggregators for TrFN aggregation are proposed and the multi-criteria model (fuzzy FUCOM-D'Bonferroni model) is proposed for group decision making. In the first part of the multi-criteria model, the FUCOM was expanded using TrFNs, while in the second part was presented the application of the TrFN D'Bonferroni operator for decision aggregation in the multi-criteria model.
3. Preliminaries
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The main advantages of the proposed fuzzy FUCOM-D'Bonferroni model are: (1) The fuzzy FUCOMD'Bonferroni model provides flexible decision making and takes into consideration the interaction between the attributes of a decision; (2) The model considers interconnections between attributes and eliminates the impact of awkward data. Thus, the main contribution and novelty of this study are threefold: (1) One of the contributions developed in this paper is the introduction of the new fuzzy based FUCOM-D'Bonferroni model that provides more objective expert evaluation of criteria in a subjective environment; (2) The improved MCDM methodology suggested provides purchasing managers with a powerful tool for prioritizing transportation demand management measures; (3) The presented methodology enables the evaluation of alternatives despite dilemmas in the decision making process and lack of quantitative information. In this section, the fundamental elements of the Dombi TNs/TCNs and BM operators are briefly represented in the field. 3.1. Bonferroni Mean Operator
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Definition 1 [52]: Let (a1, a2 ,..., an ) be a set of non-negative numbers and p, q 0 . If 1
BM
then
p,q
pq n 1 (a1 , a2 ,..., an ) aip a qj n(n 1) i , j 1 i j
BM p , q is
(1)
called a Bonferroni mean (BM) operator.
Definition 2 [57]: Let p, q 0 , (a1, a2 ,..., an ) be a set of non-negative numbers, wi (a1, a2 ,..., an ) the relative weight of ai , wi 0,1 and in1 wi 1 . If 1
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n wi w j p q p q NWBM p , q (a1 , a2 ,..., an ) ai a j i , j 1 1 wi
then NWBM p,q is called a normalized weighted BM (NWBM) operator.
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(2)
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3.2. Dombi operations of TrFN Definition 3. Let p and q be any two real numbers. Then, the Dombi T-norm and T-conorm between p and q are defined as follows [58]: 1 1/
1 p 1 q 1 p q
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OD ( p, q)
and ODc ( p, q) 1
1 1/
p q 1 1 p 1 q
where 0 and ( p, q) 0,1 .
(3)
(4)
According to the Dombi T-norm and T-conorm, we define the Dombi operations of TrFNs.
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Definition 4. Suppose 1 1l ,1m ,1u and 2 2l ,2m ,2u are two TrFNs, , 0 and let it be l m u f i f (il ), f (im ), f (iu ) n i l , n i m , n i u fuzzy function, i i i i 1 i 1 i 1
then some operational laws of fuzzy
numbers [18] based on the Dombi T-norm and T-conorm can be defined as follows: (1) Addition “+”
2 l 2 j 1 lj j 1 mj m j 1 j l l 1/ , j 1 j m m 1/ f f f 1 2 1 f 2 1 1 l l m m 1 f 1 1 f 2 1 f 1 1 f 2 1 2 2 u 2 u j 1 j j 1 j u u 1/ f 1 f 2 1 u u 1 f 1 1 f 2 2
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2
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(2) Multiplication “x” 2 j 1 lj 1 2 1/ l 1 f l 1 f 1 l 1 l 2 f 1 f 2
j 1 mj
m 1 f 1 1 m f 1
1 f 2m f 2m
j 1 uj
2
,
,
2
1/
,
u 1 f 1 1 u f 1
1 f 2u f 2u
1/
(5)
(6)
(3) Scalar multiplication, where 0.
1 1l
1/
l 1 1 l 1 1
,
1m
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1l
(4) Power, where 0
6
1m
1/
m 1 1 m 1 1
,
1u
1u
1/
u 1 1 u 1 1
(7)
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1l
1
1 11l 1l
1/
,
1m
1/
1 m 1 m1 1
,
1u
1/
1 u 1 u1 1
(8)
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3.3. Dombi Bonferroni Mean Operator with TrFNs
Based on the TrFNs operators (3)-(8) we propose the TrFN Dombi Bonferroni mean (TrFNDBM) operator. Theorem 1. Let it be j lj , mj , uj ; j 1, 2,..., n , collection of TrFNs in 𝑅, then TrFNDBM operator is defined as follows
1
n il i 1 1/ 1 n ( n 1) 1 p q n 1 i , j 1 l l i j p 1 f i q 1 f j l l f j f i
n
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pq n p q 1 ( 1 , 2 ,..., n ) i j n( n 1) i , j 1 i j
n
im ,
i 1
iu
1/
n ( n 1) 1 1 n 1 pq i , j 1 1 f m 1 f mj i i j q p m f mj f i
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TrFNDBM
p,q,
where f i f (il ), f (im ), f (iu )
il
n il i 1
,
im
i 1im n
,
iu
,
n ( n 1) 1 1 n 1 pq i , j 1 1 f u 1 f uj i i j p q u f uj f i
i 1iu n
i 1
1/
(9)
represents fuzzy function.
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About the proof of this Theorem 1, please see Appendix 1. Example (1). Let 1 (2,3, 4) , 2 (3, 4,5) , 3 (3, 4,5) and 4 (2,3,4) be four TrFNa and p=q=ρ=1, then we can show the following calculations: (1) (2)
f 1l 2 10 0.2 ; f 2l 3 10 0.3 ;...; f 4m 3 14 0.214 ;...; f 3u 5 18 0.278 ; f 4u 4 18 0.222 .
1 f 1l f 1l
5.75
;
1 f 2l f 2l
3.50
;...;
1 f 3m f 3m
4.17
;...;
1 f 4u f 4u
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(3) TrFNDBM 1,1,1 (2,3,4);(3,4,5);(3,4,5);(2,3,4)
7
7.20
.
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2 3 3 2 1/1 , 4(4 1) 1 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ... 1 1 1 1 1 1 5.75 3.50 5.75 3.50 5.75 5.75 5.75 2.86 3.50 5.75 2.86 3.50 2.86 5.75 3 4 4 3 1/1 , 1 1 4(4 1) 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ... 1 1 1 1 1 1 4.17 2.88 4.17 2.88 4.17 4.17 4.17 9.33 2.88 4.17 9.33 2.88 9.33 4.17 455 4 1/1 1 4(4 1) 1 11 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 4.13 3.10 4.13 3.10 4.13 4.13 4.13 7.20 3.10 4.13 ... 7.20 3.10 7.20 4.13 2.63, 2.95,3.99
In the following part, we shall explore some desirable properties of TrFNDBM operator.
Theorem 2 (Idempotency): Set j lj , mj , uj ; j 1, 2,..., n , collection of TrFNs in R, if i , then TrFNDBM p , q , ( 1 , 2 ,.., n ) TrFNDBM p , q , ( , ,.., ) . About the proof of this Theorem 2, please see Appendix
2.
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Theorem 3 (Boundedness): Set j lj , mj , uj ; j 1, 2,..., n , collection of TrFNs in R, let min il , min im , min iu and max il , max im , max iu then
TrFNDBM p , q , (1, 2 ,..., n ) . About the proof of this Theorem 3, please see Appendix 3.
Theorem 4 (Commutativity): Let the grey set (1, 2 ,..., n ) be any permutation of ( 1 , 2 ,.., n ) . '
'
'
'
'
TrFNDBM p , q , (1 , 2 ,.., n ) TrFNDBM p , q , (1 , 2 ,.., n ) .
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Then
'
3.4. Normalized Dombi Bonferroni Mean Operator with TrFNs Based on the TrFN operators (3)-(8) we propose the TrFN Normalized Weighted Dombi Bonferroni mean (TrFNDNGBM) operator. Theorem 5. Let it be j lj , mj , uj ; j 1, 2,..., n , collection of TrFNs in R, then TrFNDNGBM operator
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is defined as follows
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TrFNDNGBM p , q , ( 1 , 2 ,..., n )
n 1 p i q j p q i , j 1
wi w j 1 wi
n n il im m i 1 i 1 l 1/ , i 1/ i i 1 i 1 1 wi 1 wi 1 1 1 1 ( p q ) wi w j ( p q ) wi w j n n 1 1 l l m m i , j 1 i , j 1 f i f f i f i j i j q j q j p p 1 f lj 1 f mj 1 f il 1 f im n iu n i 1 iu 1/ i 1 1 wi 1 1 ( p q ) wi w j n 1 u u i , j 1 f i f i j q j p 1 f uj 1 f iu n
n
where f i f (il ), f (im ), f (iu )
il
i 1 n
l i
,
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pro of
,
im
i 1 n
this Theorem 5, please see Appendix 4.
m i
,
iu
i 1iu n
(10)
represents fuzzy function. About the proof of
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Example 2. 1 (2,3, 4) , 2 (3, 4,5) , 3 (3, 4,5) and 4 (2,3,4) be four TrFNa and p=q=ρ=1 and w j 0.18,0.32,0.33,0.17 then we can show the following calculations an w j 0.18,0.32,0.33,0.17 , then we can show the following calculations: TrFNDNGBM w1 ,1,1 (2,3,4);(3,4,5);(3,4,5);(2,3,4)
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2 3 3 4 10 1/1 , 1 1 1 1 0.180.33 1 0.18 0.17 1 0.32 0.18 1 0.32 0.33 1 0.17 0.33 1 11 0.180.32 ... 1 1 1 1 1 1 1 1 1 1 1 1 1 0.18 1 0.18 1 0.18 1 0.32 1 0.32 1 0.17 0.25 0.43 0.25 0.43 0.25 0.25 0.43 0.25 0.43 0.43 0.25 0.43 3 4 4 3 14 1/1 , 1 1 1 1 0.180.33 1 0.18 0.17 1 0.32 0.18 1 0.32 0.33 1 0.17 0.33 1 11 0.180.32 ... 1 0.18 0.271 0.401 1 0.18 0.271 0.401 1 0.18 0.271 0.271 1 0.32 0.401 0.271 1 0.32 0.401 0.401 1 0.17 0.271 0.401 4 5 5 4 18 1/1 1 1 111 0.180.32 1 0.180.33 1 0.18 0.17 1 0.32 0.18 1 0.32 0.33 1 0.17 0.33 1 ... 1 0.18 0.221 0.2781 1 0.18 0.221 0.2781 1 0.18 0.221 0.221 1 0.32 0.2781 0.221 1 0.32 0.2781 0.2781 1 0.17 0.221 0.2781 2.61,3.61, 4.61
In the following part, we shall explore some desirable properties of TrFNDNGBM operator.
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TrFNDNGBM operator also contains the following properties: (1) Idempotency: Set TrFNDNGBM
p,q,
j lj , mj , uj ;
( 1 , 2 ,.., n ) TrFNDNGBM
j 1, 2,..., n , collection of TrFNs in R, if i , then
p,q,
( , ,.., ) .
(2) Boundedness: Set j lj , mj , uj ; j 1, 2,..., n , collection of TrFNs in R, let min il , min im , min iu and max il , max im , max iu then TrFNDNGBM p, q, (1, 2 ,..., n ) .
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'
TrFNDNGBM
p,q,
(1 , 2 ,.., n ) TrFNDNGBM
'
be any permutation of ( 1 , 2 ,.., n ) . Then
'
(1 , 2 ,..., n )
(3) Commutativity: Let the grey set
'
'
'
(1 , 2 ,.., n ) .
p,q,
The proof of these properties is the same as that for TrFNDNGBM operator and because of that it is omitted here.
pro of
In the following part, some special cases of the TrFNDBM and TrFNDNGBM operator will be discussed. (1) If q=0, then:
a) Eq. (7) reduces to the TrFN Dombi generalized mean (TrFNDGM) operator as below. 1
1
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n p p 1 1 n p p TrFNDGM p ,0, ( 1 , 2 ,..., n ) i i n i 1 n(n 1) i 1 n n n il im iu i 1 i 1 i 1 1/ , 1/ , 1/ 1 n n n 1 1 1 n 1 n 1 p n p p 1 1 1 i 1 l i 1 1 f im i 1 1 f iu p 1 f i p p m u f il f i f i
b) Eq. (8) reduces to the TrFN Dombi generalized weighted geometric (TrFNGWG) operator. wi
n
n
,
im
im i 1
1 1 p w i
1/
n
1 m f i
p 1 f i 1
n
iu
n
i 1
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n n il i 1 il 1/ i 1 1 1 1 p n f l i w p i l 1 f i 1 i
1 n p i p i , j 1
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TrFNGWG p ,0, ( 1 , 2 ,..., n )
m i
,,
iu i 1
i 1
1 1 p w i
n
1/
1 u f i
p 1 f i 1
u i
,
(2) If p=1 and q=0, then:
a) Eq. (7) reduces to the TrFN Dombi arithmetic average (TrFNDAA) operator. TrFNDAA1,0, ( 1 , 2 ,..., n )
n
n
,
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n n il i 1 il 1/ n f il 1 i 1 1 l n i 1 1 f i
1 n i n i 1
i 1
m i
im i 1
1 1 n
n
f im
1 f i 1
m i
1/
,
n
n
i 1
u i
iu i 1
1 1 n
n
f iu
1 f i 1
u i
1/
b) Eq. (8) reduces to the TrFN Dombi weighted geometric (TrFNDWG) operator.
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i 1
n il i 1 1/ n 1 f l 1 wi l i i 1 f i
wi
n
n
im ,
i 1
1 wi
1 f im f im i 1 n
iu ,
1/
i 1
1 wi
1 f iu f iu i 1 n
1/
(3) If p→0 and q=0, then
pro of
n
TrFNWG1,0, ( 1 , 2 ,..., n ) i
a) Eq. (7) reduces to the TrFN Dombi geometric average (TrFNDGA) operator 1/ n
n lim TrFNDGAwp ,0, ( 1 , 2 ,..., n ) i p 0 i 1 n il i 1 1/ n 1 f l 1 1 l i n i 1 f i
n
n
im ,
i 1
1 1 n
iu ,
1/
1 f im m f i 1 i
n
i 1
1 1 n
1/
1 f iu u f i 1 i
n
b) Eq. (8) reduces to the TrFN Dombi weighted geometric average (TrFNDWGA) operator p 0
wi
i 1
n il i 1 1/ n 1 f l 1 wi l i i 1 f i
n
n
im ,
i 1
1 wi
iu 1/
1 f im m f i 1 i n
,
i 1
1 wi
1/
1 f iu u f i 1 i n
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n
lim TrFNDWGAwp ,0, ( 1 , 2 ,..., n ) i
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4. Fuzzy FUCOM-D’Bonferroni multicriteria model
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Fuzzy FUCOM-D'Bonferroni multi-criteria model is implemented in two phases as shown in Fig. 3. In the first phase, weight coefficients are calculated using fuzzy FUCOM. The weight coefficients obtained in the first phase of the model are further used in the D'Bonferroni model for the evaluation of alternatives. A detailed presentation of the steps of the FUCOM-D'Bonferroni multi-criteria model is shown in the next section.
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Expert comparison of criteria
Expert evaluation of alternatives
Defining the limitations of fuzzy model
Aggregation of matrices – TrFNDBM operator
Forming FUCOM nonlinear model Aggregation of weight coefficients TrFNDBM
pro of
Identificationof alternatives
Phase 1: TrF FUCOM
Criteria identification and their ranking
Aggregated initial decision matrix (N) Evaluation of alternatives TrFNDNGBM aggregator
Optimal values of criteria weights
Phase 2: D’Bonferroni model
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Initial alternative ranking
Influence of dynamic matrices to ranks
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Phase 3: Sensitivity analysis and discussion
Changes of criteria weights
Comparisons with other MCDM models
Influence of parameters p, q and on the rankings results
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Final alternative ranking
Fig 3. FUCOM-D’Bonferroni multicriteria model. Phase I: Fuzzy FUCOM
Table 3
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Assuming that there are n evaluation criteria in the multicriteria model that are designated as 𝑤𝑗 , 𝑗 = 1,2, . . . , 𝑛, and their weight coefficients need to be determined. Subjective models for determining weights based on comparison in pairs of criteria require from the decision maker to determine the degree of influence of the criterion 𝑖 on the criterion 𝑗. This degree of influence of the criterion 𝑖 on the criterion j is presented as the value of the comparison (𝑎𝑖𝑗 ). Since the comparison values 𝑎𝑖𝑗 are not based on accurate measurements, but on subjective estimates, it is expected that existing uncertainties will be presented with fuzzy numbers. In the application of fuzzy numbers in the models of multi-criteria decision making, linguistic scales are most often used. Thus, in this paper for the presentation of expert preferences in fuzzy FUCOM, fuzzy linguistic scale [59] is used, which is represented by triangular fuzzy numbers, as shown in Table 3. Fuzzy linguistic scale for criteria evaluation.
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Linguistic terms Equally importance (EI) Weakly important (WI) Fairly Important (FI) Very important (VI) Absolutely important (AI)
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Membership function (1,1,1) (2/3,1,3/2) (3/2,2,5/2) (5/2,3,7/2) (7/2,4,9/2)
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In accordance with the defined settings, in the next section is presented the fuzzy FUCOM algorithm in five steps. Step 1. Determining a set of evaluation criteria and their ranking. The initial step in multicriteria models is defining a set of evaluation criteria. If we assume that there is n (𝑗 = 1,2, . . . , 𝑛) evaluation criteria, then we can present them with a set C C1 , C2 ,..., Cn .
pro of
Let's assume that a group of experts E1 , E2 ,..., Et is involved in the research. After defining the set of criteria, the experts determine their rank in accordance with their preferences. The criteria rank is determined according to the significance, respectively, the first rank is assigned to the criterion that is expected to have the highest weight coefficient. The last place occupies the criterion for which we expect to have the lowest value of the weight coefficient. Thus, we obtain the criteria ranked according to their expected influence on decision making in the MCDM model. C j (1) C j (2) ... C j ( k )
(11)
where 𝑘 represents the rank of the observed criterion. If two or more criteria have the same rank, the equality sign is placed between the criteria instead of ">".
j(k )
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Step 2. Comparisons of the criteria. Mutual comparison of the criteria is done using fuzzy linguistic expressions from the defined scale (see Table 3). Mutual comparisons are performed by each expert individually Ee ( 1 e t ) according to his preferences. The comparison is made with respect to the first-ranked (most significant) criterion. Thus, for every expert are obtained fuzzy significances of the e criteria C for all the criteria ranked in the step 2. Based on the defined significances of the criteria, e
using the expression (12) are determined fuzzy comparative significances k / ( k 1) of expert comparisons. C
j ( k 1)
e C j(k )
( Celj ( k 1) , Cemj ( k 1) , Ceuj ( k 1) ) ( Celj ( k ) , Cemj ( k ) , Ceuj ( k ) )
(12)
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e
e
k / ( k 1)
Thus are obtained fuzzy vectors of comparative significances of the criteria for every expert, according to the expression (13)
e
e
e
e
1/ 2 , 2/3 ,..., k / ( k 1)
e
(13)
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Where k / ( k 1) present the significance which according to the expert e the criterion with the rank C j ( k ) has in comparison with the criterion with the rank C j ( k 1) . Step 3. Defining the limitations of the fuzzy model. Values of weight coefficients should meet the condition where the relation of the weight coefficients of criteria ( C j ( k ) and C j ( k 1) ) is the same as their e
comparative significance k / ( k 1) , respectively, e
wk w
e k 1
e
k / ( k 1)
(14)
In addition to the condition (14), final values of weight coefficients should meet the condition of transitivity, respectively, k /(k 1) (k 1)/( k 2) k /( k 2) , that is
wk wk 1 wk wk 1 wk 2 wk 2
. Thus is obtained the
second condition which final values of weight coefficients should meet e
e
e
k / ( k 1) ( k 1) / ( k 2)
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wk
w
e k 1
(15)
Step 4. Forming fuzzy model for the calculation of optimal values of the weights of criteria. In the fourth
e
e
e T
step is calculated final values of fuzzy weight coefficients of the criteria for every expert w1 , w2 ,..., wn .
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The conditions defined in the previous step (step 3) should be met with minimal deviation from the e
wk
maximal consistency (OMC). In other words, the conditions should be met
w
e
e k 1
k / ( k 1) 0
and
e
w
e k 2
e
e
k / ( k 1) ( k 1) / ( k 2) 0 .
Meeting these conditions, the OMC amounts to 0 . From the previously
e
presented relations follows that the values of the weight coefficients of criteria w1 , w2 ,..., wn meet the condition where
w
e k
e
w k 1
e
k / ( k 1)
e k
w
and
e
wk 2
e
e
k / ( k 1) ( k 1) / ( k 2)
values.
e T
e
pro of
wk
should
, with the minimization of the
Based on the defined settings, we can set nonlinear model for determining optimal fuzzy values of the
e
e
e T
weight coefficients of the evaluation criteria w1 , w2 ,..., wn . min
e
w j ( wlj , wmj , wuj )
e
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Where
re-
s.t. wek e e k / ( k 1) , j w k 1 wek e e e k / ( k 1) ( k 1) / ( k 2) , j wk 2 n e w j 1, j , j 1 l m u w j w j w j , l w j 0, j j 1, 2,..., n
(16)
and k / ( k 1) (kl / ( k 1) ,km/ ( k 1) ,ku/ ( k 1) ) .
e
Considering that the maximum consistency requires meeting the condition where e
w
e k 2
e
e
k / ( k 1) ( k 1) / ( k 2) 0 ,
w
e k 1
e
k / ( k 1) 0
and
the model (16) can be transformed into fuzzy linear model (17) and by
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wk
wk
e
e
e T
solving it are obtained optimal fuzzy values of the weight coefficients w1 , w2 ,..., wn . min s.t.
(17)
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wek wek 1 e k / ( k 1) , j e e e e wk wk 2 k / ( k 1) ( k 1) / ( k 2) , j n e w j 1, j , j 1 wl wm wu , j j j wlj 0, j j 1, 2,..., n
Where
e
w j ( wlj , wmj , wuj )
e
and k / ( k 1) (kl / ( k 1) ,km/ ( k 1) ,ku/ ( k 1) ) .
Step 5. Aggregation of the weight coefficients of criteria. Solving the model (16) or (17) are obtained weight coefficients of criteria by experts. By applying TrFNDBM operator (9), it is performed the
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e
e
e T
aggregation of the weight coefficients w1 , w2 ,..., wn
and final optimal values of the weight coefficients
of criteria are obtained w1, w2 ,..., wn . T
Phase II: D’Bonferroni model for the evaluation of alternatives
pro of
Step 1. Forming initial decision making matrix. In this research participated five experts which evaluated the alternatives. For every expert Ee ( 1 e t ) is obtained correspondent matrix X e xij e
mn
( 1 e t ).
By the application of TrFNDBM operator (9) it is performed the aggregation of experts’ matrices and it is formed the initial decision making matrix X xij m n . Step 2. Normalization of the initial decision making matrix. Generally, in MCDM models there are two types of criteria, benefit criteria and cost criteria. Therefore, it is necessary to normalize initial decision making matrices, forming that way normalized matrix N nij m n . The elements of the normalized matrix are obtained by applying the expression (18). jB j C
(18)
re-
x ij / n x ij if i 1 nij n 1 x ij / i 1 x ij if
Step 3. Determining score function S (ni ) . By applying the TrFNDNGBM aggregator (10) the values of the score function S (ni ) TrFNDNGBM p , q , r n1, n2 ,..., nm are obtained, presenting final values of preferences by alternatives. The TrFNDNGBM aggregator implies the application of the values of criteria which meet the condition nj w j 1 . In the phase I of the application of the FUCOM-D'Bonferroni model are
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obtained fuzzy values of the weights which meet the condition nj wlj 1 , nj wmj 1 and nj wmj 1 . So as to meet the previously defined condition, by applying the expression wj wlj 4wmj wuj 61 it is performed the defuzzification. After the defuzzification, by applying additive normalization, defuzzified values are normalized so that nj w j 1 . Step 4. Ranking alternatives. Ranking alternatives A1 , A1 ,..., Am and the selection of the best alternative from the considered set. Alternatives are ranked based on the values of the score function S (ni ) , wherein the highest possible value of the alternative S (ni ) is more favorable.
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5. Case Study
The overview of Istanbul’s urban transport system is provided by the following steps: 1- Showing the increasing transport demand in Istanbul 2- Considering TDM in Historical Peninsula to address the increasing transport demand a. Considering push measures: congestion charging, parking pricing, road pricing. b. Considering pull measures: synchronized travel plans, public transport capacity improvement, ride sharing.
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The motorization rate in Istanbul has rapidly increased with increasing car ownership and car use rates. Parallel to this development, the use of public transport modes such as buses, metro, tram, funicular, bus rapid transit (BRT), light rail transit (LRT), cable car, paratransit modes such as minibuses, dolmus, company and school services and taxis also increased in terms of daily ridership figures [15]. Fig. 1 indicates the motorization rate along with population growth between 1980 and 2014. Both values are indexed to 100 for year 1980 to illustrate the wide divergence in the population and vehicle growth rates over the years.
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Fig. 1. Population and motorization growth in Istanbul between 1980 – 2014 [14].
The fact that motorization growth far outweighed the population growth over the years is an indication of transport supply saturation, which needs to be balanced with demand-side interventions.
Table 2
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Number of daily trips for public transport modes which include cars, public transport, walking and taxis is nearly 32 million. Table 2 shows the number of average daily trips for each transport mode. Fig. 2 shows the modal share of Istanbul’s public transport [60]. Number of daily transport trips in Istanbul’s public transport.
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Transport Mode Rail Transport Metro Tram Cable Car, Nostalgic tram and Funicular Marmaray Surface Transport IETT Buses Private Bus Operator-1 Private Bus Operator-2 Minibus Taxi Company and School Services Waterborne Transport Sea buses (IDO) Ferries Boats Total
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Daily Ridership 2,709,914 1,728,555 640,351 54,168 286,840 11,717,979 1,940,750 1,571,393 835,422 3,098,963 1,403,949 2,867,502 565,472 163,434 231,444 170,594 14,993,365
Modal share (%) 18.07 11.53 4.27 0.36 1.91 78.15 12.94 10.48 5.57 20.67 9.36 19.13 3.77 1.09 1.54 1.14 100
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4%
78%
Rail Transport
pro of
18%
Surface Transport
Waterborne Transport
Fig. 2. The modal share of public transport.
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Besides public transport, the total road kilometre in Istanbul increased up to 36,000 km with the road infrastructure investments. IMM forecasts for future daily trip numbers (36 million by 2023) expect a pressure on the current road and public transport infrastructure. This leads IMM to pursue policies to meet increasing mobility demands by increasing transport supply or rebalancing transport demand, particularly away from peak hours and locations with high mobility demand.
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TDM measures have been considered by IMM recently to achieve a balanced transport supply and demand. Congestion charging is suggested around the Historical Peninsula to limit the traffic flow entering the area and improving the air quality as well as reducing traffic congestion levels. A feasibility study by IMM is undertaken to analyse the impacts of such a measure. The results suggest a 20% reduction in traffic congestion levels. Supporting this measure, IMM also looks to implementing parking pricing in the area that would decrease the car use. Road tolls are in place for the bridges across the Bosporus and Eurasia Tunnel which reduces the demand for car use in Bosporus crossings, which is the bottleneck for Istanbul’s traffic during peak times.
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On the other hand, the push measures listed above are planned to be supported by pull measures as well. Travel plans for employees getting to work around the same peak hours are planned to mitigate the concentration of travel demand within an hour at peak times (08:00 – 09:00). Flexible work schedules, especially for public agencies and schools, can redistribute the travel around the peak times. Travel information tools, such as “IBB Navi”, travel awareness campaigns making people shift their travel preferences towards public transport, cycling and walking, teleworking, car sharing and ride sharing, home shopping, road space rationing, are considered by IMM for pull measures. IMM decided to focus on increasing the public transport capacity, synchronized travel plans and ride sharing as the options to implement through the TDM project. 5.1. Problem Description
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IMM identified the set of pull and push measures under a TDM project package. Six sub-projects (measures) comprising of the TDM project are grouped as shown in the following Fig. 4. This figure shows the list of measures as a complete TDM policy package recommended for implementation by Istanbul’s local authority.
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Push Measures
Pull Measures
- Increasing the capacity of public transport
- Congestion charging
pro of
- Increasing parking prices
- Synchronized travel plans
- Road pricing
- Ride sharing
Transport Demand Management
Fig. 4. Transport demand management policy package.
Table 4 Experts (decision makers) of TDM project.
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The costs and outcomes of these measures are discussed with following five experts (see Table 4) working in IMM and IETT, Istanbul’s public bus operator to determine the priority scores of each measure:
Department
Agency
Director Manager Transport planner Transport planner Urban planner
Public Transport Services Strategy Planning Transport Directorate Transport Planning Directorate Traffic Coordination Centre
IMM IETT IMM IETT IMM
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Position
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Each measure is evaluated based on the project costs, economic, social and environmental impact. Fig. 5 shows these evaluation criteria and how they are related with each TDM measure.
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MCDM Approach Priorititizing transportation demand management measures
Economic impact
Capital costs Operating costs
A1 (Congestion charging)
Travel time Public transport trip revenues
A2 (Parking prices)
A3 (Road pricing)
Environmental impact
Social impact
pro of
Costs
Social inclusion Vulnerable users Public opposition
A4 (Public transport capacity)
Push measures
Decreasing carbon emissions Fuel saving
A5 (Synchronized travel plans)
A6 (Ride sharing)
Pull measures
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Fig. 5. The evaluation indicator system for prioritizing TDM. Alternatives:
Alternatives in the form of push and pull measures are proposed as part of Transportation Demand Management (TDM) strategy. An effective and comprehensive TDM strategy need both positive incentives in the form of pull measures and negative incentives in the form of push measures. In Istanbul, the following pull and push measures are proposed:
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Push measures: Push measures make private vehicles less attractive to use. - Congestion charging (A1): A congestion charging scheme is proposed for Historical Peninsula, which includes the touristic and historical downtown of the city. It is proposed to encourage mode shift, hence reduce air pollution. Only electric buses and other zero-emission vehicles are allowed to enter the congestion area without any charge. As part of the initiative, bus fleet used in the area is planned to be electrified. However, the details of the scheme are not yet announced.
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- Parking prices (A2): Increasing number of private cars in Istanbul has driven up the number of parking areas, which caused induced traffic phenomenon whereby new parking facilities spurred the demand for transport. By increasing the parking prices and dynamically setting the prices to curb the demand, the local government aims to make private vehicles less attractive in central areas. - Road pricing (A3): Variable road pricing is proposed for 3 bridges across the Bosporus, which creates most serious traffic congestion problem, especially during the rush hours. Adjusting bridge tolls based on the demand aims to curb the increasing demand for the bridge crossings. Pull Measures: Pull measures make other modes of transport more attractive. -Public transport capacity improvements (A4): Public transport capacity improvements in the form of new metro lines, bus services, and ferry services are put forward to complement the push measures by increasing the attractiveness of public transport lines.
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- Synchronized travel plans (A5): The high traffic congestion during rush hours is mainly driven by commuting to workplaces, schools and hospitals opening nearly the same hours (08:30 – 09:00). Therefore, the local government seeks to implement a synchronized travel plan so that the overlapped commuting transport demand is curbed. - Ride sharing (A6): The increasing popularity of ride-sharing applications has the potential to decrease the use of road space by cars. The average number of people in each car can be reduced by increasing ride-sharing. However, there is a potential risk that ride-sharing applications can reduce the demand for mass transit options such as metro and bus services, and lead to an increase in the number of cars. Being 19
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a somewhat controversial measure, the local government seeks to spread ride-sharing through increasing partnerships with ride-sharing provider companies. 5.2. Criteria Definition
pro of
Four main and nine sub-critera are determined by experts for priorititizing transport demand managmenet measures. The decision hierarchy of TDM measures is shown in Fig. 5. Costs: Allocating budget for TDM measures is a key part of project implementation. While some measures such as congestion charging and public transport capacity improvements require high upfront capital costs, many measures such as synchronized travel plans and ride sharing require less costs. Cities facing financial difficulties tend to favor less costly projects.
Capital costs (C1): Capital costs are investment costs for the large-scale TDM measures. Congestion charging, for example, involve high initial capital costs such as technological installements, automatic number plate recognition (ANPR) and charging systems. Operating costs (C2): Operating costs play an important role in the selection of TDM measures. While TDM mesaures involving infrastructural investments need higher operational costs, other measures such as sycnhronized travel plans require less operational costs.
Economic Impact:
Travel time (C3): One of the objectives of TDM measures is to reduce travel time, hence increase accessibility in the city. By encouraging modal shift, thereby reducing the traffic congestion, TDM measures help reduce the travel time for passengers. The measures involving greater degree of travel time reduction impact increase the selection probability by decision makers. Public transport trip revenues (C4): Some measures involve generation of revenue such as congestion charging which increases the likelihood of selection by decision makers. While push measures are more likely to generate revenue due to penalising road use, pull measures are less likely to generate revenue.
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Social inclusion (C5): Although TDM measures do not directly have an impact on social inclusion, the increased accessibility as a result of TDM lead to more social inclusion. Some measures such as ride-sharing can lead to more social interaction. Similarly public transport capacity improvements lead to increased accessiblity of citizens to the urban opportunities. Vulnerable users (C6): Vulnerable users such as disabled people, senior people, and children have more difficulty in accessing the urban amenities. TDM measures, by promoting a modal shift to sustainable transport options such as public transport, walking and cycling, facilitate the urban transport experience for the vulnerable users. Public opposition (C7): TDM measures, especially push measures involving penalising private car usage can trigger reaction from the public. Therefore, policy makers can tend to choose less controversial TDM measures. This is one of the reasons congestion charging is controversially implemented in many cities, leading to reaction form business inside the congestion charging area.
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Social impact: TDM measures are implemented to have a positive social impact in the form of social inclusion and participation in urban opportunities.
Environmental Impact: Reducing the greenhouse gas emissions and improving the air quality is one of the key objectives of TDM measures. By promoting modal shift from private car to sustainabile transport modes, TDM measures help reduce the emissions and hence enhancing the air quality. Decreasing Carbon Emissions (C8): Cleaner transport modes such as walking, cycling and public transport decrease the carbon emissions. The impact on environment of each measure depends on the degree to which it contributes to modal shift. Congestion charging, for example, can improve air quality by substantially reducing the car use within central city areas. Fuel Saving (C9): Car usage leads to increased levels of fossil fuel consumption, with negative impacts on natural environment. TDM measures can bring about higher fuel saving, hence reducing the negative impacts on the environment. Modal shift from private car to public
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transport tremendously reduce the fuel consumption per person. TSM measures leading to more fuel saving, therefore, are more likely to be selected by decision makers. 5.3. Proposed Method
pro of
The phase I of the application of the FUCOM-D'Bonferroni implies the application of fuzzy FUCOM model and the determination of the weight coefficients of criteria. The criteria for the evaluation of alternatives are grouped into four sets of criteria: Cost (C1), Economic impact (C2), Social impact (C3) and Environmental impact (C4) criteria. The C1-C4 criteria constitute the first hierarchical level, while the second hierarchical level consists of sub-criteria classified within the four sets of criteria in Table 5. Table 5 Criteria/sub-criteria for evaluation.
Code C1 C11 C12 C2 C21 C22 C3 C31 C32 C33 C4 C41 C42
Min/max min min min max max max min max max
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Criteria/Sub-criteria Cost Capital costs Operating costs Economic impact Travel time Public transport trip revenues Social impact Social inclusion Vulnerable users Public opposition Environmental Impact Decreasing Carbon Emissions Fuel Saving
Using fuzzy FUCOM are calculated the values of local weights of the sub-criteria. After defining the local weights of the sub-criteria, the weight coefficients of the criteria are multiplied by the group of weight coefficients of the sub-criteria. Thus are obtained global values that are further used to evaluate alternatives in the D'Bonferroni model. In order to solve this problem, five fuzzy FUCOM models are defined:
urn a
1) Model 1 - Calculation of the values of the weight coefficients of the criteria C1, C2, C3 and C4; 2) Model 2 - Calculation of local values of the weight coefficients of the sub-criteria C11 and C12; 3) Model 3 - Calculation of local values of the weight coefficients of the sub-criteria C21 and C22; 4) Model 4 - Calculation of local values of the weight coefficients of the sub-criteria C31, C32 and C33 and 5) Model 5 - Calculation of local values of the weight coefficients of the sub-criteria C41 and C42. Detailed overview of the models 1 - 5 is presented in the next section of the paper.
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Step 1 and 2. After defining the criteria and the sub-criteria (see Table 5), their ranking is performed. Based on the defined ranks of the criteria/sub-criteria and the preferences of the experts (E1-E5), linguistic values of the comparative significances of the criteria/sub-criteria are determined as given in Table 6. To determine the comparative significances of the criteria/sub-criteria, linguistic values from the Table 3 are used. Table 6
Linguistic evaluations of the criteria/sub-criteria. Criteria/sub-criteria Criteria R
21
E1 C1>C2>C3>C4
E2 C2>C1>C4>C3
E3 C4>C3>C2>C1
E4 C1>C4>C2>C3
E5 C1>C2>C4>C3
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EI,FI,VI,VI C11>C12 EI,FI C22>C21 EI,AI C33>C31>C32 EI,AI,AI C42>C41 EI,VI
EI,EI,AI,AI C11>C12 EI,FI C21>C22 EI,FI C33>C31>C32 EI,WI,VI C41>C42 EI,WI
EI,FI,VI,VI C11>C12 EI,VI C22>C21 EI,FI C32>C31>C33 EI,VI,AI C41>C42 EI,WI
EI,WI,AI,AI C11>C12 EI,FI C21>C22 EI,VI C31>C33>C32 EI,FI,AI C41>C42 EI,FI
pro of
C1-C4 C EI,WI,AI,AI Sub-criteria R C11>C12 C11-C12 C EI,VI Sub-criteria R C21>C22 C21-C22 C EI,AI Sub-criteria R C32>C31>C33 C31-C33 C EI,VI,VI Sub-criteria R C42>C41 C41-C42 C EI,VI R – Rank; C- Comparisons
Based on the comparative significances of the criteria/sub-criteria (see Table 6), using the expressions (2) and (3), the vectors of comparative significances are defined by levels of the criteria as given in Table 7. Table 7
Vectors of comparative significances of the criteria/sub-criteria. Experts
Criteria/sub-criteria C11-C12
(0.67,1,1.5);(2.33,4,6.72);(0.78,1,1.29)
(2.5,3,3.5)
E2
(1.5,2,2.5);(1,1.5,2.33);(0.71,1,1.4)
E3
(1,1,1);(3.5,4,4.5);(0.78,1,1.29)
E4
(1.5,2,2.5);(1,1.5,2.33);(0.71,1,1.4)
E5
(0.67,1,1.5);(2.33,4,6.72);(0.78,1,1.29)
C21-C22
C31-C33
C41-C42
(3.5,4,4.5)
(2.5,3,3.5);(0.71,1,1.4)
(2.50,3,3.5)
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C1-C4
E1
(1.5,2,2.5)
(3.5,4,4.5)
(3.5,4,4.5);(0.78,1,1.29)
(2.50,3,3.5)
(1.5,2,2.5)
(1.5,2,2.5)
(0.67,1,1.5);(1.67,3,5.22)
(0.67,1,1.5)
(2.5,3,3.5)
(1.5,2,2.5)
(2.5,3,3.5);(1,1.33,1.8)
(0.67,1,1.5)
(1.5,2,2.5)
(2.5,3,3.5)
(1.5,2,2.5);(1.4,2,3)
(1.50,2,2.5)
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Comparative significances for the expert E1 and the criteria C1-C4 are obtained by applying the expression (2): C1/ C 2 C1 C 2 (2/3,1,3/2) (1,1,1)= 2/3,1,3/2 ; C 2/ C 3 C 2 C 3 (7/2,3,9/2) (2/3,1,3/2) =(2.33,4.00,6.72) ;
C 3/ C 4 C 2 C 3 (7/2,3,9/2) (7/2,3,9/2) =(0.78,1.00,1.29) .
urn a
Thus is obtained the vector of comparative significances 0.67,1,1.5 ; 2.33, 4,6.72 ; 0.78,1,1.29 . The remaining values of comparative significances from the Table 7 are obtained in the same way. 1
In the following section (Step 3), based on the vector of comparative significance are defined the limitations of the model (16). By applying the expression (14), it is defined the first group of limitations for the expert E1 and the criteria C1-C4: wC1 / wC 2 2/3,1,3/2 , wC 2 / wC 3 2.33,4,6.72 and wC 3 / wC 4 0.78,1.00,1.29 .
condition
of
By applying the expression (15) are defined two limitations arising from the
transitivity
of
relations:
wC 2 / wC 4 2.33,4,6.72 0.78,1,1.29 1.81,4.00, 8.64 .
wC1 / wC 3 0.67,1.00,1.50 2.33,4,6.72 (1.56,4,10.07) and
The limitations for the remaining models are defined
in the same way.
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On the basis of the defined limitations, the models (16) were formed for determining optimal values of the weight coefficients of the criteria/sub-criteria. In the following section, nonlinear models are presented for determining optimal fuzzy values of the weight coefficients of the criteria. The models for determining the weight coefficients of the sub-criteria are presented in the Appendix 5.
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Expert 1 (C1-C4) min
Expert 5 (C1-C4) min s.t.
w1l w1m w1u u 0.67 ; m 1.00 ; l 1.5 ; w w w2 2 2 l m u w w 2 2.33 ; 2 4.00 ; w2 6.72 ; w3u w3m w3l w3l w3m w3u u 0.78 ; m 1.00 ; l 1.29 ; w w w4 4 4 l w1m w1u w1 wu 1.56 ; wm 4.00 ; wl 10.07 ; 3 3 3 m u wl w w 2u 1.81 ; 2m 4.00 ; 2l 8.64 ; w4 w4 w4 l m u l m u ( w1 4 w1 w1 ) / 6 ( w2 4 w2 w2 ) / 6 ( wl 4 wm wu ) / 6 1; 3 3 3 wlj wmj wuj ; wlj , wmj , wuj 0, j 1, 2,..., n
w1l w1m w1u u 0.67 ; m 1.00 ; l 1.5 ; w w w2 2 2 l m u w w 2 2.33 ; 2 4.00 ; w2 6.72 ; w4u w4m w4l w4l w4m w4u u 0.78 ; m 1.00 ; l 1.29 ; w w w3 3 3 l w1m w1u w1 wu 1.56 ; wm 4.00 ; wl 10.07 ; 4 4 4 m u wl w w 2u 1.81 ; 2m 4.00 ; 2l 8.64 ; w3 w3 w3 l m u l m u ( w1 4 w1 w1 ) / 6 ( w2 4 w2 w2 ) / 6 ( wl 4 wm wu ) / 6 1; 3 3 3 wlj wmj wuj ; wlj , wmj , wuj 0, j 1, 2,..., n
pro of
s.t.
Table 8
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By solving the models presented, optimal local values of weight coefficients by experts are obtained. By multiplying local values of the sub-criteria with the weight coefficients of the criteria, global values of the sub-criteria for every expert are obtained, as in the Table 8. The Lingo 17.0 software is used to solve the model. By solving fuzzy nonlinear models, the average value χ ≈ 0.0 is obtained, which shows high consistency of the obtained values of the weights of criteria. Global values of fuzzy weight coefficients of the sub-criteria. E1 (0.166,0.289,0.557) (0.057,0.096,0.186) (0.254,0.292,0.452) (0.065,0.073,0.112) (0.038,0.062,0.086) (0.015,0.021,0.025) (0.011,0.021,0.035) (0.044,0.066,0.117) (0.015,0.022,0.039)
E2 (0.108,0.17,0.18) (0.06,0.085,0.087) (0.051,0.081,0.135) (0.226,0.322,0.478) (0.014,0.036,0.039) (0.011,0.036,0.05) (0.05,0.145,0.175) (0.024,0.038,0.045) (0.08,0.114,0.116)
E3 (0.038,0.069,0.1) (0.021,0.034,0.048) (0.049,0.071,0.074) (0.027,0.035,0.036) (0.156,0.168,0.176) (0.03,0.056,0.105) (0.105,0.168,0.264) (0.131,0.209,0.233) (0.141,0.209,0.215)
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Criteria C11 C12 C21 C22 C31 C32 C33 C41 C42
E4 (0.183,0.31,0.46) (0.063,0.103,0.153) (0.03,0.053,0.053) (0.057,0.106,0.106) (0.018,0.046,0.049) (0.045,0.137,0.172) (0.01,0.034,0.049) (0.073,0.129,0.142) (0.079,0.129,0.131)
E5 (0.133,0.259,0.518) (0.074,0.129,0.249) (0.228,0.276,0.424) (0.078,0.092,0.141) (0.029,0.049,0.108) (0.006,0.012,0.031) (0.019,0.025,0.043) (0.039,0.071,0.089) (0.022,0.036,0.043)
Optimal values (0.114,0.199,0.311) (0.051,0.083,0.126) (0.088,0.121,0.167) (0.069,0.093,0.12) (0.031,0.059,0.079) (0.017,0.037,0.059) (0.026,0.055,0.082) (0.052,0.086,0.11) (0.052,0.08,0.09)
urn a
By solving fuzzy nonlinear models, the average value of the OMC 0.07 is obtained, which shows high consistency of the obtained values of the weights of the criteria. Using TrFNDBM operator (9), it is performed the aggregation of the weight coefficients and final optimal values of the weight coefficients of the criteria are obtained w1 , w2 ,..., wn , as shown in the Table 8. The final global fuzzy T
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values of the weights of the criteria are presented in the Fig. 6.
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0.3
Weights
0.2
0.15
0.1
0.05
C11
C12
C21
C22
C31
pro of
0.25
C32
C33
C41
C42
re-
Fig. 6. Final values of the weight coefficients of the criteria.
Final values of the weight coefficients are used to evaluate and select optimal alternative in the D'Bonferroni multi-criteria model. Evaluation of the alternatives is done using fuzzy linguistic scale, as shown in the Table 9. Table 9
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Fuzzy linguistic scale for the evaluation of the alternatives. Linguistic terms Very poor (VP) Poor (P) Medium (M) High (H) Very high (VH)
Membership function (1,1,3) (1,3,5) (3,5,7) (5,7,9) (7,9,9)
Table 10
urn a
The first step of the D'Bonferroni model involves the formation of the aggregated initial decision making matrix. The aggregated decision making matrix is obtained on the basis of expert correspondence matrices (see Table 10) in which the experts evaluated the alternatives by criteria. Expert correspondent matrices. A1 VH,H,H,VI, VH VH,H,H,AI, VH VH,H,H,AI, VH VH,M,VH,A I,P VP,P,VP,EI, VP VP,P,P,EI,V P VH,H,H,VI, VH H,VH,M,AI, VH M,H,M,FI,H
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Crit .C1 C2 C3 C4 C5 C6 C7 C8 C9
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A2 M,M,M,FI, H H,VH,H,FI, H VH,H,P,VI, M VH,VH,H,V I,H VP,VP,P,WI ,P VP,P,VP,WI ,M VH,VH,H,F I,M H,H,P,VI,H P,P,M,WI,P
A3 H,VH,H,FI,H H,H,VH,WI, M VH,VH,H,AI, H VH,M,H,VI, M VP,M,VP,EI, VP VP,VP,M,EI, P VH,VH,VH,V I,H H,H,M,VI,H M,M,H,FI,M
A4 H,M,P,VI,P VH,M,M,VI,V H VH,VH,VH,FI, VH VH,H,VH,AI, VH H,VH,VH,AI, VH VH,H,H,VI,V H VP,P,VP,WI,V P VH,H,VH,AI, VH VH,H,VH,VI, VH
A5 VP,VP,P,WI, VP VP,P,VP,WI, VP P,VP,P,FI,V P P,VP,P,WI,V P H,VH,H,VI, M M,P,P,FI,M H,M,P,FI,VP M,P,M,EI,M M,P,P,WI,V P
A6 VP,P,VP,EI, VP VP,M,P,EI, P P,M,M,WI,P P,P,M,FI,P VH,H,VH,F I,H P,M,H,FI,H VP,P,P,FI,P P,VP,H,FI,P VH,H,H,VI, M
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Aggregation of expert matrices is performed using the TrFNDBM operator (9). The aggregated initial decision making matrix is shown in the Table 11. Table 11 Initial decision making matrix. A1 (3.06,3.65,4.15) (3.65,4.15,3.06) (4.15,3.06,3.65) (3.06,3.65,4.15) (3.65,4.15,3.06) (4.15,3.06,3.65) (3.06,3.65,4.15) (3.65,4.15,2.21) (4.15,2.21,2.77)
A2 (1.95,2.6,3.21) (2.6,3.21,2.86) (3.21,2.86,3.47) (2.86,3.47,4.07) (3.47,4.07,2.04) (4.07,2.04,2.6) (2.04,2.6,3.14) (2.6,3.14,3.06) (3.14,3.06,3.65)
A3 (2.86,3.47,4.07) (3.47,4.07,2.59) (4.07,2.59,3.21) (2.59,3.21,3.81) (3.21,3.81,3.06) (3.81,3.06,3.65) (3.06,3.65,4.15) (3.65,4.15,2.33) (4.15,2.33,2.93)
A4 (1.44,1.96,2.58) (1.96,2.58,2.51) (2.58,2.51,3.11) (2.51,3.11,3.64) (3.11,3.64,3.5) (3.64,3.5,4.09) (3.5,4.09,4.6) (4.09,4.6,3.27) (4.6,3.27,3.87)
A5 (1.14,1.25,1.53) (1.25,1.53,1.14) (1.53,1.14,1.25) (1.14,1.25,1.53) (1.25,1.53,1.14) (1.53,1.14,1.38) (1.14,1.38,1.64) (1.38,1.64,1.14) (1.64,1.14,1.38)
A6 (1.14,1.25,1.53) (1.25,1.53,1.15) (1.53,1.15,1.48) (1.15,1.48,2.04) (1.48,2.04,1.37) (2.04,1.37,1.87) (1.37,1.87,2.41) (1.87,2.41,1.15) (2.41,1.15,1.61)
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Crit. C1 C2 C3 C4 C5 C6 C7 C8 C9
Table 12 Normalized matrix. A1
A2
A3
(0.64,0.74,0.82) (0.69,0.77,0.84) (0.71,0.79,0.85) (0.12,0.17,0.25) (0.07,0.09,0.12) (0.07,0.10,0.17) (0.67,0.76,0.83) (0.14,0.20,0.30) (0.12,0.18,0.28)
(0.72,0.82,0.89) (0.69,0.79,0.85) (0.78,0.85,0.90) (0.16,0.23,0.32) (0.06,0.09,0.13) (0.08,0.12,0.19) (0.69,0.78,0.85) (0.11,0.16,0.24) (0.06,0.10,0.18)
(0.65,0.76,0.83) (0.71,0.80,0.87) (0.71,0.79,0.85) (0.12,0.18,0.26) (0.08,0.10,0.14) (0.08,0.12,0.19) (0.66,0.75,0.82) (0.12,0.18,0.27) (0.10,0.17,0.26)
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Crit. C1 C2 C3 C4 C5 C6 C7 C8 C9
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In the next step, the initial decision making matrix is modified using the expression (18), and the normalized matrix is formed, as in the Table 12.
A4
A5
A6
(0.78,0.86,0.92) (0.73,0.81,0.87) (0.68,0.76,0.83) (0.17,0.24,0.33) (0.19,0.26,0.36) (0.20,0.30,0.42) (0.88,0.92,0.94) (0.16,0.23,0.32) (0.17,0.25,0.36)
(0.87,0.91,0.93) (0.88,0.92,0.94) (0.88,0.92,0.94) (0.06,0.09,0.12) (0.15,0.22,0.31) (0.08,0.15,0.25) (0.81,0.87,0.91) (0.07,0.12,0.2) (0.06,0.10,0.17)
(0.87,0.91,0.93) (0.85,0.91,0.94) (0.83,0.89,0.93) (0.06,0.10,0.16) (0.18,0.25,0.33) (0.13,0.21,0.31) (0.86,0.92,0.95) (0.07,0.11,0.16) (0.14,0.20,0.30)
urn a
For the calculation of the score functions of the alternatives, the elements of the normalized matrix (Table 12) and defuzzified values of the weight coefficient of the criteria are used
0.203, 0.085, 0.123, 0.094, 0.058, 0.037, 0.055, 0.084, 0.077
T
.
Using
the
TrFNDNGBM
aggregator (10), final values of the score function are obtained S ( ni ) . Based on the value S ( ni ) the alternatives are ranked and the optimal alternative is selected from the set of considered alternatives. Score functions and ranking of alternatives are shown in the Table 13. Table 13
Rank of alternatives.
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Alternative Congestion charging (A1) Parking prices (A2) Road pricing (A3) Public transport capacity improvements (A4) Synchronized travel plans (A5) Ride sharing (A6)
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S (ni ) (0.190,0.263,0.354) (0.198,0.273,0.367) (0.195,0.266,0.361) (0.255,0.335,0.420) (0.220,0.279,0.355) (0.227,0.298,0.385)
Rank 6 4 5 1 3 2
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The alternatives are ranked based on the value of the score function S ( ni ) , wherein it is more preferable for the alternative to have the highest possible value of S ( ni ) . Thus, on the basis of the obtained values of S ( ni ) the first rank is assigned to the alternative A4.
pro of
6. Stability of the Obtained Results Analysis of the stability of the obtained results is carried out through four parts. In the first part, the sensitivity analysis of fuzzy FUCOM-D'Bonferroni model is performed by changing the weight coefficients of the criteria. The analysis of the influence of the change in the weight coefficients of the criteria is carried out through 15 scenarios. In the second part, the analysis of the influence of dynamic matrices of decision making to the change of the rank of the alternatives is performed. In the third part, the comparison of the obtained results with other MCDM models is made: TrFNDWG operator (proposed), Dombi weighted arithmetic averaging (TrFNDWAA) operator (proposed), fuzzy MABAC [61] and fuzzy VIKOR [62-63] model. In the fourth part, the analysis of the dependence of the obtained results on the change of ρ, p and q parameters is demonstrated. More detailed overview of the sections of the discussion on the results is shown in the next part of the paper. 6.1. Changing the Weights of the Criteria
re-
After determining the weight coefficients of the criteria using fuzzy FUCOM, the "most effective criterion" is identified for the purpose of the sensitivity analysis. The objective of the sensitivity analysis is to evaluate the influence of the most effective criterion on ranking performance of the proposed model. Based on the recommendations of Kirkvood [64] and Kahraman [65], the proportion of the weights of the criteria in the sensitivity analysis and the elasticity coefficient [66] are defined. The elasticity coefficient is used to express relative compensation of the values of other weight coefficients in relation to changes in the weight of the most important criterion.
Table 14
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In this research, C11 criterion is identified as the most effective one since it has the highest value of the weight coefficient w11 0.203 . In the next step, the coefficient of weight elasticity s of the most significant criterion (see Table 14) is determined and the limits for change of the weight coefficient of the most significant criterion are defined. Coefficient of elasticity for changing weights. C2 0.139
C3 0.201
C4 0.153
urn a
Criteria C1 s 1.00
C5 0.094
C6 0.061
C7 0.090
C8 0.138
C9 0.125
The limit values of the C11 criterion are obtained, which are in the range of -0.2032≤Δx≤0.61299. On the basis of the defined limit values, the scenarios for sensitivity analysis are determined. The interval 0.2032≤ x ≤0.61299 is divided into 15 sequences based on which a total of 15 scenarios are established. For every scenario, new values of the weight coefficients are set, so 15 new groups of weight coefficients are obtained, as shown in the Table 15. Table 15
New criteria weights. C1 0.000 0.050 0.100 0.150 0.200 0.250 0.300
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Scenario S1 S2 S3 S4 S5 S6 S7
26
C2 0.139 0.132 0.125 0.118 0.111 0.104 0.097
C3 0.201 0.191 0.181 0.171 0.161 0.151 0.141
C4 0.153 0.145 0.137 0.130 0.122 0.115 0.107
C5 0.094 0.089 0.085 0.080 0.075 0.071 0.066
C6 0.061 0.058 0.055 0.052 0.049 0.045 0.042
C7 0.090 0.085 0.081 0.076 0.072 0.067 0.063
C8 0.138 0.131 0.124 0.117 0.110 0.103 0.096
C9 0.125 0.119 0.113 0.107 0.100 0.094 0.088
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0.350 0.400 0.450 0.500 0.550 0.600 0.650 0.700
0.090 0.083 0.077 0.070 0.063 0.056 0.049 0.042
0.131 0.121 0.111 0.101 0.090 0.080 0.070 0.060
0.099 0.092 0.084 0.076 0.069 0.061 0.053 0.046
0.061 0.056 0.052 0.047 0.042 0.038 0.033 0.028
0.039 0.036 0.033 0.030 0.027 0.024 0.021 0.018
0.058 0.054 0.049 0.045 0.040 0.036 0.031 0.027
0.089 0.083 0.076 0.069 0.062 0.055 0.048 0.041
0.081 0.075 0.069 0.063 0.056 0.050 0.044 0.038
pro of
S8 S9 S10 S11 S12 S13 S14 S15
The influence of the new values of the weight coefficients from the Table 15 to the change of the ranks of alternatives is presented in the Fig. 7. S1 6
S15
S2
5 S14
S3
4 3 2
S13
S4
re-
1
S12
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S11
S10
A1
S9 A2
A3
A4
S5
S6
S7 S8 A5
A6
Fig. 7. Analysis of sensitivity of the ranks of alternatives through 15 scenarios.
urn a
The results (Fig. 7) show that assigning different weights to the criteria through scenarios leads to the change in the ranks of individual alternatives, which confirms that the model is sensitive to changes in weight coefficients. By comparing the first two alternatives in the rank (A4 and A6) through the scenarios, we can conclude that both first-ranked alternatives (A4 and A6) retain their ranks through all 15 scenarios. The remaining alternatives (A1, A2, A3 and A5) retain their ranks in 12 out of total of 15 scenarios. Just in the first three scenarios, the alternatives A1, A2, A3 and A5 change their rank. In the scenarios S1 and S2, the ranks A4> A6> A2> A3> A1> A5 are obtained, while in the S3 scenario, the alternatives are ranked as follows A4> A6> A2> A5> A3> A1. From the above results, we can conclude that the rank of the alternative A4 is credible and there is sufficient advantage of the mentioned alternative compared to the second-ranked (A6) and other alternatives. The results are also confirmed by the correlation of ranks through the scenarios.
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Correlation of ranks is determined using Spearman's correlation coefficient. Spearman's coefficient (SCC) is used to determine statistical significance of the difference between the ranks obtained through the scenarios [36]. By analyzing the obtained correlation values, we note that there is high correlation of ranks, since in 13 out of 15 scenarios, the SCC exceeds 0.940. In the remaining two scenarios, the SCC values are 0.657. The mean value of the SCC across all scenarios is 0.950, which shows high correlation of ranks, respectively, confirms the results shown in the Table 14. 6.2. Influence of dynamic matrices on changing the rank of alternatives Internal changes in the decision making matrix, such as the introduction of new ones or the elimination of the existing alternatives from the set of considered alternatives, can cause changes in final
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pro of
preferences. Accordingly, in this paper, the performance of the proposed model is analyzed in the conditions of dynamic initial decision making matrix. Five scenarios are formed. For every scenario, a change in the number of alternatives is made and the ranks obtained are analyzed. The scenarios are formed by removing one inferior (the worst) alternative in every scenario from further considerations. At the same time, within the scenarios, the remaining alternatives are ranked according to the newlyobtained initial decision making matrix. The initial solution using fuzzy FUCOM-D'Bonferroni model is generated as A4> A6> A5> A2> A3> A1. It is clear that the alternative A1 is the worst option, and in the first scenario the alternative A1 is eliminated from the set and new decision making matrix is obtained with a total of five alternatives. A new solution for the decision-making matrix is generated and the rank A4> A6> A5> A2> A3 is obtained. The ranking in the first scenario shows that A4 is still the best alternative, while A3 is the worst alternative. Further implementation of the described procedure results in the following ranks through the remaining three scenarios: S2: A4> A6> A5> A2; S3: A4> A6> A5 and S4: A4> A6. Through the modification of the initial matrix, which is done by the elimination of the worst alternative, we notice that there is no rank reversal among the alternatives in the FUCOM-D'Bonferroni model. The alternative A4 remained the best ranked through all scenarios, which confirmed the robustness and accuracy of the obtained ranks of alternatives in dynamic environment. 6.3. Comparison of the FUCOM-D'Bonferroni model ranks with other models
Table 16
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In this section, we compared the results of the FUCOM-D'Bonferroni model with the TrFNDWG operator (proposed), Dombi weighted arithmetic averaging (TrFNDWAA) operator, fuzzy MABAC [61] and fuzzy VIKOR [62-63] models. The comparative overview of ranks by various MCDM techniques are presented as given in Table 16. The ranking of alternatives in terms of various MCDM techniques.
Ranking A4>A6>A5>A2>A3>A1 A4>A6>A5>A2>A1>A3 A4>A6>A5>A2>A3>A1 A4>A6>A5>A2>A3>A1 A4>A6>A5>A2>A3>A1
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MCDM techniques Fuzzy FUCOM-D’Bonferroni TrFNDWG TrFNDWAA Fuzzy MABAC Fuzzy VIKOR
The rank of alternatives according to the presented methods shows that the alternatives A4 and A6 are the best-ranked alternatives through all models. Fuzzy MABAC and TrFNDWAA models have fully confirmed the ranking of the FUCOM-D'Bonferroni model. In the TrFNDWG model, only the last ranked alternatives (A3 and A1) changed, while the remaining alternatives kept their ranks. In fuzzy VIKOR model, the third-ranked and fourth-ranked alternatives (A5 and A2) replaced the positions, while the ranks of the remaining alternatives were confirmed.
Jo
We can conclude that A4 and A6 are the best alternatives by all models, while the A1 alternative in four models (FUCOM-D'Bonferroni, TrFNDWAA, MABAC and VIKOR) is the worst one. The results also show that there is high correlation between the ranks. All the correlation values are significantly higher than 0.90, so we can conclude that there is high correlation between the proposed approach and other tested MCDM models. 6.4. Influence of parameters ρ, p and q on the ranking results In the above steps, the values of the parameters p, q and ρ were initially assumed to be 1, however, it can easily be observed the effects of the changing value of p, q and in the proposed D'Bonferroni model.
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Sceanrio 1 A1 ρ=50
A2
A3 ρ=1 6
Sceanrio 2
A4
A5
A6
A1
ρ=2
A2
A3 p=1
6
p=50
5
A4
A5
A6
p=2
5
4
ρ=30
pro of
In order to examine the influence of the parameters p, q and ρ on the obtained results, three scenarios are formed. In the first scenario, the values of the parameter ρ are changed in the range from 1 to 100, while for the parameters p and q the values are p=q=1. In the second scenario (S2), the values of the parameter p are changed in the range from 1 to 100, while for the parameters ρ and q the values are p=q=1. In the third scenario (S3), the values of the parameter q are changed in the range from 1 to 100, while for the parameters ρ and 𝑝 the values are ρ= p=1. The Fig. 8 shows the influence of the parameters p, q and ρ on the ranks of the alternatives.
ρ=3
4
p=30
3
p=3
3
2
2
ρ=20
ρ=4 p=20
1
p=4
1
0 ρ=10
ρ=5
ρ=10
p=10
ρ=6
ρ=7
p=10
re-
ρ=9
p=5
p=9
ρ=8
Sceanrio 3
A1
p=6
A2
q=50
A3 q=1 6
A4
A5
p=7 p=8
A6
q=2
5
4
q=3
lP
q=30
3
2
q=20
q=4
1
q=10
q=5
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q=10
q=9
q=6
q=7 q=8
Fig. 8. Ranking orders for varying values of parameters 𝑝, 𝑞 and .
Jo
Generally, the higher the values of the parameters p, q and ρ, the more complex the calculation becomes, and the more the interrelations between the attributes are emphasized. DMs usually choose the parameters p, q and ρ according to their preferences [37]. In real decision making, we generally recommend that the parameter values be 1 from a practical point of view, which is not only intuitionistic and simple, but also able to consider the inner connections between attributes. From the Fig. 8, it can be clearly pointed out that when the parameters p, q and ρ have different values, the ranking orders of the considered alternatives remain almost the same. Minor changes occur when the parameter ρ is changed, while changing the parameters p and q there is no change in the ranking. In the scenario S1, for the values ρ=5 and ρ=9, the ranks A4> A6> A2> A5> A3> A1 are obtained, while the rank A4> A6> A5> A2> A1> A3 is obtained when ρ=20. The changes in ranks that occurred when the values of ρ=5, ρ=9 and ρ=20 are minimal and did not impact the rank of alternatives A4 and A6 that were identified as the best alternatives. From the above analysis we can conclude that the parameters p and q have no effect on final preferences. The parameter ρ has minor influence on the ranking of certain alternatives, but it does
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not impact final preferences when choosing the most effective alternatives, respectively, the alternatives A4 and A6. 7. Results and Discussion
pro of
It is noteworthy that results suggest the prioritization of pull measures over push measures. This is largely because pull measures cause less public reaction and involve less costs compared to push measures which are often based on penalizing the use of cars, hence triggering public reaction. Push measures also require higher investment costs, which make them less attractive in the eyes of decision makers. Results also indicate that public transport capacity improvements should be prioritized over other measures. Public transport capacity improvements include the following actions:
Increasing the number of vehicles in the bus fleet, Increasing the frequency of bus services, Increasing the length of total metro network, Adding new rolling stock to the metro
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Capacity improvements in public transport can require high capital and operational costs compared to other measures of the TDM project. However, it has the highest impact on environmental, economic and social outcomes. Increasing the capacity of public transport would enhance the service quality levels of public transport services. For example, frequent bus services increase the attractiveness of public transport by reducing the crowding level in buses which discourage people from using public buses.
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Ride sharing comes next in implementation ranking. There is no cost involved by IMM to implement ride sharing. Third party private network companies provide ride sharing services. IMM can support the use of those services by including ride sharing option in its IBB Navi application, which is a journey planning application suggesting travel routes to the desired destinations. Incorporation of ride sharing as one of the travel options would increase the use of ride-sharing by passengers. By helping reduce the number of cars used by people, ride-sharing can contribute a great deal to the improvement of urban mobility. Economic, social and environmental benefits of ride-sharing are considerably high when one considers its impact on reducing car use.
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The third pull measure, synchronized travel plans, particularly helps ameliorate the peak hour predicament in Istanbul. The economic, social and environmental impacts of this measure are quite significant. Most workplace including public schools and agencies start working between 8:00-08:30, which makes millions of passengers rushing towards cars, public buses, and metros at the same time. By shifting the start times of public agencies and schools, IMM can achieve a more balanced distribution of trips around the peak hours. This would relieve the load on public buses, and make them more comfortable during those hours. This measure requires coordination with other public bodies such as Governorate of Istanbul, National Education Agency, etc. After pull measures, IMM should also implement push measures to discourage the use of cars. Increasing the parking prices is the less costly method among the push measures. By charging drivers for occupying public space which could also be used by passengers, cyclists, public transport users, parking prices can bring about a more public-transport friendly mobility in Istanbul. In terms of economic impacts, IMM can generate significant resources through charging for parking lots. It can use the resources obtained in reinvesting in public transport services (e.g. capacity improvements, etc.). IMM can both increase parking prices and increase the number of parking lots and areas to be charged around the city, hence preventing on-the-street parking without any charge, which makes the urban roads quite busy and passenger-unfriendly.
Jo
In addition to parking prices, road pricing can also be considered to discourage car use. However, road pricing is a political decision which might not be feasible to implement during the times when economy is in decline or stagnation. It can bring about a public reaction by the drivers as well as private transport operators such as taxi and minibus services. Its economic and social impacts are not as straightforward as pull measures, although it can lead to a potential improvement in air quality by reducing the number of cars on the streets and roads.
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Finally, congestion charging can be considered by IMM as the last option to implement. High initial upfront investment costs and operational costs make congestion charging as an option not prioritized by IMM given the budget constraints and economic difficulties. It can also create a public reaction similar to road pricing. Therefore, congestion charging had better stay as a last option to implement after implementing pull measures. Push measures can complement the prioritized pull measures.
pro of
The practical application of this model allows obtaining credible results when deciding under uncertainty conditions and when the data on which it is based is partially known and imprecise. This model practically helps managers to deal with their own subjectivity in prioritizing criteria and attributes. The use of the fuzzy FUCOM-D’Bonferroni approach reduces imprecision when assessing the impacts of TDM measures. The research has shown that, in addition to predictable indicators, the assessing of the TDM measures is influenced by numerous unknown and partially known indicators. The results of the research indicate the justification of the selected MCDM model. 8. Conclusion
re-
IMM’s problem of prioritizing the TDM measures is solved in this study by applying a fuzzy-based method. Without such a prioritization, IMM could not achieve the implementation of the entire TDM policy set at the same time. Pull measures being prioritized is quite expected given the incentive-based approach in those measures. They involve making peoples’ travel decisions shift towards public transport modes without any penalty for car use. Public transport capacity improvements stand out as the most highly suggested TDM measure, as followed by ride sharing and synchronized travel options, all of which are pull measures. On the other hand, push measures being more penalty-based which discourage car use make them less prioritized in the eyes of IMM. The push measures, such as increasing the parking prices, road and congestion charging measures come after the pull measures. Decision makers may avoid generating public reaction by implementing penalty-based push measures. The study demonstrated that implementing pull measures should be prioritized over push measures given the budget constraints and impacts of TDM measures.
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The developed fuzzy FUCOM-D’Bonferroni model has shown a great degree of flexibility and it can be used in other branches of transport and for other real problems. It should then be adapted to the needs of managers from those branches and the issues considered. The developed model is based on expert knowledge and on the basis of expert assessments it can rank certain alternatives. However, it is possible to use real data for the assessment of alternatives since the fuzzy FUCOM-D’Bonferroni model is not limited to certain scales. It can help the DMs to use this method in any situation when it is necessary to rank certain alternatives. The main points of this paper are as follows: - The development of a new method of multi-criteria analysis based on hierarchical and methodological procedures for solving real-world business problems, and in this study applied for evaluation of alternatives in the form of push and pull measures as part of TDM strategy. - The application of a decision-making model based on FUCOM and Dombi-Bonferroni aggregators was tested by qualitative analysis and it helped to classify push and pull measures from the best to the worst by applying four main and nine sub-critera. - The model is very flexible and simple so it can also be applied to other problems of multi-criteria analysis (in transport, logistics and other sectors of the economy). - Fuzzy FUCOM-D’Bonferroni model provides the ranking of alternatives, and helps managers and all other decision-makers to address the problem easily and solve it by applying the ranking of alternatives. - The applied fuzzy FUCOM-D’Bonferroni model has shown good performance with a classifying push and pull measures. - Future studies can explore the use of fuzzy-based methods (fuzzy FUCOM and DombiBonferroni operators) on selecting other transport measures being considered by decision makers and planners. In contrast to previous studies in TDM measures, we applied original hybrid Dombi-Bonferroni aggregation operators to address fuzzy values. This allow us to make the information aggregation process more flexible by a parameter. The second advantage of the proposed MCDM model is the
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pro of
development of fuzzy FUCOM for determining criteria weights. Moreover, pairwise comparisons for criteria were conducted using linguistic variables instead of crisp values in the decision-making process. Fuzzy FUCOM provides credible weight coefficients with only n-1 pairwise comparisons, which contribute to reasonable evaluation in decision-making. As a result, the fuzzy FUCOM is an effective decision-tool that aids decision-makers in dealing with their own subjectivity while prioritizing criteria. Therefore fuzzy approach helped us solve this complex and multi-dimensional problem in a convenient way. In the future, we plan to implement for working on topics as; implementing the proposed model in other transport problems measures being considered by decision makers and planners and the supply chain problems such as logistic and facility location. In addition, improving the model using rough set approach or neutrosophic fuzzy values might be an option for further research work. References
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[63] Nikolic, V., Milovancevic, M., Petkovic, D., Jocic, D., Savic, M. (2018). Parameters forecasting of laser welding by the artificial intelligence techniques. Facta Universitatis, Series: Mechanical Engineering, 16(2), 193-201. [64] Kirkwood, C. W. (1997). Strategic decision making: multiobjective decision analysis with spreadsheets (Vol. 59). Belmont, CA: Duxbury Press. [65] Kahraman, Y. R. (2002). Robust sensitivity analysis for multi-attribute deterministic hierarchical value models (No. AFIT/GOR/ENS/02-10). AIR FORCE INST OF TECH WRIGHTPATTERSONAFB OH. Appendix 1
Theorem 1. Let it be j , , l j
m j
u j
; j 1, 2,..., n , collection of TrFNs in R, then TrFNDBM
operator is defined as follows 1
n
im
,
i 1
n ( n 1) 1 1 n 1 pq m 1 f 1 f m i , j 1 i j i j q p f m m f i j
lP
n il i 1 1/ 1 1 n ( n 1) p q n 1 i l l 1 f , j 1 1 f i j i j q p f l l f i j
re-
pq n p q 1 TrFNDBM p , q , ( 1 , 2 ,..., n ) i j n(n 1) i , j 1 i j
1/
,
urn a
Proof.
i 1
n ( n 1) 1 1 n 1 pq u 1 f 1 f u i , j 1 i j i j q p f u u f i j
l m u where f i f (il ), f (im ), f (iu ) n i l , n i m , n i u i i i i 1 i 1 i 1
n
iu
1/
represents fuzzy function.
We need to prove the Eq. (7) is kept. According to the operational laws of TrFNs, we get il i 1/ 1 p 1l il i p
,
im
m 1 1 p mi i
1/
il lj i j 1/ 1 p 1il q 1 lj il lj p
q
Jo
further we get
35
,
,
iu
u 1 1 p u i i
1/
q l , j j 1/ 1 q 1 lj lj
im mj
m 1 1 p mi i
1 mj q m j
1/
,
,
mj 1/
u 1 1 p u i i
1 uj q u j
1/
and , 1/ u 1 j 1 q u j uj
m 1 j 1 q m j
iu uj
(7)
Journal Pre-proof
n
im i 1
n 1 i , j 1 1 f m i i j p f m i
n p q 1 i j n(n 1) i , j 1 i j
im
n
i 1
Thereafter,
,
1/
1
1 f mj q f m j
im i 1
i 1
n 1 i , j 1 1 f m i i j p f m i
im i 1
n
i 1
1/
n 1 1 n ( n 1) i , j 1 1 f m i i j p f im
Therefore,
1
1 f mj q f mj
,
im i 1
pq n p q 1 ( 1 , 2 ,..., n ) i j n(n 1) i , j 1 i j
n
Jo
urn a
n il i 1 1/ 1 1 n ( n 1) pq n 1 i , j 1 1 f il 1 f lj i j q p f l l f i j
lP
TrFNDBM
36
,
iu
i 1
n ( n 1) 1 1 n 1 pq 1 f m i , j 1 1 f m i j i j q p f m m f i j
n
im
1/
,
i 1
n ( n 1) 1 1 n 1 pq 1 f u i , j 1 1 f u i j i j q p f u u f i j
1/
1 f mj q f m j
n
im i 1
n 1 1 n ( n 1) i , j 1 1 f m i i j p f im
1
p,q,
1/
1
n
im
n
,
re-
n n il i 1 il 1/ i 1 n 1 1 1 n ( n 1) i , j 1 1 f l 1 f lj i i j q p f il f lj
n
im
n
,
pro of
n n n l il p q i 1 1/ i j i i , j 1 i 1 i j n 1 1 i , j 1 1 f l 1 f lj i i j p q f l f l i j
1/
1
1 f mj q f mj
Journal Pre-proof
Appendix 2 Theorem 2 (Idempotency): Set j lj , mj , uj ; p, q,
(1 , 2 ,.., n ) TrFNDBM
j 1, 2,..., n , collection of TrFNs in R, if
(,,.., ) .
Proof: Since i , i.e. , , u then l i
m i
l
u i
m
TrFNDBM p 1, q 1, 1 ( 1 , 2 ,..., n ) ( , ,..., )
,
n ( n 1) 1 1 n 1 pq 1 f mj i , j 1 1 f m i i j q p f m m f i j
2im 1 1 n 11 i 1
1/
2
1
1i m i
m
1im m i
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2il 1/ 1 1 n 1 l i1 (11) 1l i i
,
,
2im
1 n i 1
1/
1 1 m (11) 1i m i
,
1 n i 1
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The proof of Theorem 2 is completed.
37
n
iu
i 1
,
1/
2iu
1 1 n 11 i 1
,
i 1
n ( n 1) 1 1 n 1 pq 1 f uj i , j 1 1 f u i i j q p f u u f i j
2iu
1/
1 1 u (11) 1i u i
1/
2 1 u u 1i 1i u u i i
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2il 1/ 2 1 111 n 1 l l i 1 1i 1i l l i i
n
im
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n il i 1 1/ 1 1 n ( n 1) pq n 1 i , j 1 1 f il 1 f lj i j q p f l l f i j
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then TrFNDBM
p, q,
l , m , u
1/
i ,
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Appendix 3 Theorem 3 (Boundedness): Set j , , l j
min , min , min
l i
m i
u i
m j
u j
; j 1, 2,..., n ,
and max , max
l i
m i
, max
u i
collection of TrFNs in R, let
then
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TrFNDBM p, q, (1, 2 ,..., n ) .
Proof: Let min(1 , 2 ,..., n ) min il , min im , min iu and
max(1 , 2 ,..., n ) max il , max im , max iu .
Then,
it
can
be
stated
that
l min(il ) , i
m min(im ) , u min(iu ) , l max(il ) , m max(im ) and u max(iu ) . Based on that, the i
i
i
i
following inequalities can be formulated:
i ; min(il ) il max(il ); i
i
min(im ) im max(im ); i
i
min(iu ) iu max(iu ); i
re-
i
i
According to the inequalities shown above, it can be concluded that TrFNDBM p,q, (1, 2 ,..., n ) holds. Theorem 5. Let it be j lj , mj , uj ;
lP
n 1 p i q j p q i , j 1
wi w j
1 wi
n
n
urn a
n n il i 1 il 1/ i 1 1 wi 1 1 ( p q ) w w i j n 1 l l f f i , j 1 i i j q j p 1 f l 1 f il j
,
i 1
m i
1 1 ( p q ) wi w j
n
im
n
i 1
1/
1 wi n 1 f m i , j 1 f im j i j q p 1 f m m 1 f i j
,
i 1
u i
l m u where f i f (il ), f (im ), f (iu ) n i l , n i m , n i u i i i i 1 i 1 i 1
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Appendix 4
j 1, 2,..., n , collection of TrFNs in R, then TrFNDNGBM
operator is defined as follows TrFNDNGBM p , q , ( 1 , 2 ,..., n )
iu i 1
1 1 ( p q ) wi w j
1/
1 wi n 1 f u i , j 1 f iu j i j q p 1 f u u 1 f i j
(8) represents fuzzy function.
Proof: We need to prove the Eq. (8) is kept. According to the operational laws of TrFNs, we get
38
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pro of
l il im iu m u p i i 1/ , i 1/ , i 1/ , l m u 1 p i l 1 p i m 1 p i u 1i 1i 1i lj mj uj m u and q j lj , j , j 1/ 1/ 1/ l m u j j j 1 q 1 q 1 q l m u 1 j 1 j 1 j
il lj im mj iu uj m u u p i q j il lj mj 1/ , i 1/ , i j 1/ l l m m u u j j j 1 p i l q 1 p i m q 1 p i u q l m u 1 1 1 1 1 1 i j i j i j
further we get
i
q j
wi w j 1 wi
,
im mj 1 1 1 wi 1 m p mi i
Thereafter,
i
q j
1 mj q m j
,
,
iu uj
1 1 1 wi 1 u p ui i
1/
wi w j 1 uj q u j
n
n
im
iu
i 1
wi w j n 1 1 wi i , j 1 i j
urn a
i , j 1 i j
wi w j 1 wi
n il i 1 1/ wi w j n 1 1 1 wi l l ii, jj1 p 1 f i q 1 f j l l f j f i
wi w j
lP
p n
1/
re-
p
il lj 1/ 1 wi w j 1 1 wi l l p 1l i q 1l j i j
1/
1
1 f im p f m i
1 f mj q f m j
,
i 1
wi w j n 1 1 wi i , j 1 i j
1/
1
1 f iu p f u i
1 f uj q f u j
Therefore,
TrFNDNGBM p , q , ( 1 , 2 ,..., n )
n 1 p i q j p q i , j 1
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n n il i 1 il 1/ i 1 1 wi 1 1 ( p q ) w w i j n 1 f l i , j 1 f il i j q j p 1 f l 1 f il j
39
n
,
i 1
m i
wi w j
1 wi
1 1 ( p q ) wi w j
n
n
im
n
i 1
1/
1 wi n i , j 1 i j
1
f m i p 1 f m i
f m j q 1 f m j
,
i 1
u i
iu i 1
1 1 ( p q ) wi w j
1/
1 wi n i , j 1 i j
1
f u i p 1 f u i
f u j q 1 f u j
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Appendix 5 Expert 1 (C11-C12) min
Expert 5 (C11-C12) min s.t.
w11l w11m w11u u 2.50 ; m 3.00 ; l 3.5 ; w12 w12 w12 l m u l m u ( w 4 w w ) / 6 ( w 4 w 11 11 11 12 12 w12 ) / 6 1; wl wm wu , j 1, 2 j j j wl , wm , wu 0, j 1, 2 j j j
w11l w11m w11u u 1.50 ; m 2.00 ; l 2.5 ; w12 w12 w12 l m u l m u ( w 4 w w ) / 6 ( w 4 w 11 11 11 12 12 w12 ) / 6 1; wl wm wu , j 1, 2 j j j wl , wm , wu 0, j 1, 2 j j j
Expert 1 (C21-C22) min
Expert 5 (C21-C22) min
pro of
s.t.
s.t.
l m u w21 w21 w21 u 3.50 ; m 4.00 ; l 4.5 ; w w w 22 22 22 l m u l m u ( w21 4 w21 w21 ) / 6 ( w22 4 w22 w22 ) / 6 1; wl wm wu , j 1, 2 j j j wl , wm , wu 0, j 1, 2 j j j
l m u w21 w21 w21 u 2.50 ; m 3.00 ; l 3.5 ; w w w 22 22 22 l m u l m u ( w21 4 w21 w21 ) / 6 ( w22 4 w22 w22 ) / 6 1; wl wm wu , j 1, 2 j j j wl , wm , wu 0, j 1, 2 j j j
Expert 1 (C31-C33) min
Expert 5 (C31-C33) min
s.t.
re-
s.t.
s.t.
w w w 2.50 ; 3.00 ; 3.5 ; w w w l m u w31 0.71 ; w31 1.00 ; w31 1.4 ; u m l w33 w33 w33 l m u w w w32 32 32 wu 1.79 ; wm 3.00 ; wl 4.90 ; 33 33 33 ( wl 4 wm wu ) / 6 ( wl 4 wm wu ) / 6 31 31 31 32 32 32 l m u ( w33 4 w33 w33 ) / 6 1; l m u l m u w j w j w j , w j , w j , w j 0, j 1, 2,3
l m u w31 w31 w31 u 1.50 ; m 2.00 ; l 2.5 ; w33 w33 w33 l m u w33 1.40 ; w33 2.00 ; w313 3.00 ; u m l w32 w32 w32 l m u w31 w31 w31 wu 2.10 ; wm 4.00 ; wl 7.50 ; 32 32 32 ( wl 4 wm wu ) / 6 ( wl 4 wm wu ) / 6 31 31 31 32 32 32 l m u ( w33 4 w33 w33 ) / 6 1; l m u l m u w j w j w j , w j , w j , w j 0, j 1, 2,3
Expert 1 (C41-C42) min
Expert 5 (C41-C42) min
m 32 m 31
u 32 l 31
s.t.
urn a
lP
l 32 u 31
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l m u w42 w42 w42 u 2.50 ; m 3.00 ; l 3.5 ; w41 w41 w41 l m u l m u ( w 4 w w ) / 6 ( w 4 w 41 41 41 42 42 w42 ) / 6 1; wl wm wu , j 1, 2 j j j wl , wm , wu 0, j 1, 2 j j j
40
s.t. l m u w41 w41 w41 u 1.50 ; m 2.00 ; l 2.5 ; w42 w42 w42 l m u l m u ( w 4 w w ) / 6 ( w 4 w 41 41 41 42 42 w42 ) / 6 1; wl wm wu , j 1, 2 j j j wl , wm , wu 0, j 1, 2 j j j
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*Highlights (for review)
1) Pull and push measures under a TDM project are identified. 2) Transport demand management measures are assessed. 3) A new fuzzy FUCOM-D'Bonferroni method for solving MCDM problem is proposed.
pro of
4) Economic, cost, social and environmental criteria are considered.
5) The proposed method can be adapted for other transport problems.
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6) Different scenarios are employed for stability of results.
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*Declaration of Interest Statement
Declaration of interests
pro of
☒ The authors declare that they have no known competing financial interests or personal relationships that could have appeared to influence the work reported in this paper. ☒The authors declare the following financial interests/personal relationships which may be considered as potential competing interests:
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To the author’s knowledge, no conflict of interest, financial or other, exists. Each author has participated and contributed sufficiently to take public responsibility for appropriate portions of the content.
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*Author Contributions Section
pro of
Author contribution statements
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Dragan Pamucar and Muhammet Deveciand conceived of the presented idea, writing–review and editing. Muhammet Deveci developed the theory and Dragan Pamucar performed the computations. Fatih Canitez and Darko Bozanic verified the analytical methods. All authors discussed the results and contributed to the final manuscript.