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Modelling parking choice behaviour using Possibility Theory a
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Michele Ottomanelli , Mauro Dell'Orco & Domenico Sassanelli
b
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Department of Environmental Engineering and Sustainable Development, Technical University of Bari, Viale Del Turismo, 8-74100, Taranto, Italy b
Department of Highways and Transportation, Technical University of Bari, Via Orabona, 4-70125, Bari, Italy Available online: 22 Aug 2011
To cite this article: Michele Ottomanelli, Mauro Dell'Orco & Domenico Sassanelli (2011): Modelling parking choice behaviour using Possibility Theory, Transportation Planning and Technology, 34:7, 647-667 To link to this article: http://dx.doi.org/10.1080/03081060.2011.602846
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Transportation Planning and Technology Vol. 34, No. 7, October 2011, 647667
Modelling parking choice behaviour using Possibility Theory Michele Ottomanellia*, Mauro Dell’Orcob and Domenico Sassanellib a
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Department of Environmental Engineering and Sustainable Development, Technical University of Bari, Viale Del Turismo, 8-74100 Taranto, Italy; bDepartment of Highways and Transportation, Technical University of Bari, Via Orabona, 4-70125 Bari, Italy (Received 6 September 2010; accepted 16 April 2011) This article presents a discrete choice model for evaluating parking users’ behaviour. In order to explicitly take into account imprecision and uncertainty underlying a user’s choice process, the proposed model has been developed within the framework of Possibility Theory. This approach is an alternative way to represent imperfect knowledge (uncertainty) of users about both parking and transportation system status, as well as the approximate reasoning of the human decision maker (imprecision). The resulting model is a quantitative soft computing tool that could support traffic analysts in planning parking policies and Advanced Traveller Information Systems. In fact, effects of information on user choice can be incorporated into the model itself. Thus, we consider the parking user be a decision maker who assumes a certain choice set (set of perceived parking alternatives); the user has some information about the parking supply system and he/she associates each parking alternative with an approximate perceived cost/utility that is represented by a possibility distribution; and, finally, the user chooses the alternative which minimises/maximises his/her perceived parking cost/utility. The results show how the model is able to represent the effect of various parking policies on users’ behaviour and how the single component of parking policy affects the decision process. Keywords: parking policy; user behaviour; uncertainty; Possibility Theory; fuzzy logic; approximate reasoning
Introduction The supply of car parking plays a crucial role in the transportation system. From a theoretical and practical standpoint, the literature shows how research in the field of parking behaviour modelling leads to useful tools for the planning and design of parking policies and facilities. In fact, the models proposed can be specified to face the design problem at different planning levels: long term (strategic), middle term (tactical) and short term (real-time). Thus, parking models can be classified with respect to the particular problem planners and managers have to cope with (Young 2000). Whatever the hierarchy of the planning problem, parking choice has to be analysed and represented through appropriate simulation models. Thus, in the literature, parking choice simulation is studied as a ‘stand-alone’ model as well as a
*Corresponding author. Email:
[email protected] ISSN 0308-1060 print/ISSN 1029-0354 online # 2011 Taylor & Francis DOI: 10.1080/03081060.2011.602846 http://www.informaworld.com
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module of wider transportation system models (e.g. travel demand models, location models, design models, decision support systems). In this article, we deal with parking choice behaviour modelling, since we are interested in the explicit simulation of how parking systems management policies can affect drivers’ choices. Consequent results are useful for strategic and tactical level planning of parking facilities supply system. Parking policy in urban areas can be considered as one of the most powerful travel demand management (TDM) tools that consist of different kinds of measures on the parking supply system such as number and location of parking facilities, parking lot supply, pricing policies, illegal parking control strategies, and so on. The effectiveness and power of parking policy is shown practically by the wide and strong effects they may cause on the urban system: here, it is worthwhile mentioning the effects on travel demand distribution and/or modal choice, on urban functions accessibility, on the real estate market, as well as on traffic congestion and consequent air and noise pollution levels. In order to reduce, and possibly avoid, such unwanted consequences, as well as reduce traffic congestion, it is necessary to develop and use mathematical models and methods to forecast the impacts of different parking supply management strategies (parking policies) on drivers’ behaviour when they face the parking choice problem. Thus, it is important to forecast how drivers react to possible parking management policies, so that it is possible to understand how the design variable is perceived by parking users. In order to contribute to other studies in this field, this article proposes an alternative way to model drivers’ parking choices by trying to take into account explicitly the uncertainty of users in perceiving available parking alternatives. A consistent approach for modelling uncertainty is probability theory that is commonly used for modelling the randomness underlying the transportation problem. Actually, different sources of uncertainty are involved in human decision-making processes, such as in choice behaviour of transportation system users. In this article, we deal with the problem of parking choice behaviour and the embedded uncertainty relevant to vagueness of information, imprecise and/or incomplete data or knowledge about the parking system. Possibility Theory seems to be an alternative theoretical framework for modelling user parking choice and incorporating such uncertainty sources and in particular for incorporating various issues relevant to users when a choice has to be made (Kikuchi and Chakroborty 2006):
decision maker attitude (i.e. optimistic and conservative approach); anxiety of the decision maker relevant to the uncertainty level; and residual unknown data and the impact of additional information.
Moreover, these issues are always present when uncertainty exists, but particularly when the information to handle is approximate. This article presents a discrete choice model in which parking user is supposed to be a rational decision maker that:
considers a certain choice set (set of his/her perceived parking facilities alternatives);
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has some (vague, incomplete and/or imprecise) information about the parking supply system; assigns to each parking alternative an approximate perceived cost/utility that is represented by a fuzzy set (possibility distribution); and chooses the alternative which minimises/maximises his/her perceived (fuzzy) parking cost/utility.
Even if researcher attention on this topic is intermittent, the literature proposes different parking choice simulation models that have been founded on different modelling and theoretical approaches, depending on the particular design problem to face. Traditionally, parking choice models can be divided into at least two main groups according to the modelling approach: explicit discrete choice modelling and network-based choice modelling. In the first approach, parking choice is explicitly simulated, assuming that a parking user chooses his/her parking alternative among a discrete number of parking alternatives that constitute his/her choice set. Parking choice can also be assumed to be the subsequent choice step after mode choice. The choice set is defined as a function of the practical needs and analysis aims and can be specified for different classes of users. Generally, parking choice is represented by an objective (cost or utility) function to be optimised (minimised or maximised) according to the assumed theoretical framework. The specification of the objective function depends both on the aim of the model and on the basic theoretical assumptions (i.e. deterministic, probabilistic). The earliest of the models proposed addressed the evaluation of the effects on mode choice due to different parking pricing policies. Then, more detailed formulations were suggested with more disaggregate modelling, with respect to both users’ class and supply system whose attributes are to be included in the objective function. The most used parking choice models are based on random utility theory (Ben Akiva and Lerman 1985) that assume a user to be a rational decision maker who associates with each known alternative a utility function that is represented by means of a set of attributes that are relevant to the socio-economic system under study and to the characteristics of the alternatives, and by a random residual. Such theory assumes that a user chooses the alternative having the highest utility. Different assumptions can be made about the distribution of the random residual, so that different models can be defined. The simplest and most used are the Multinomial Logit models based on the assumption that random residuals are distributed as a Weibull-Gumbel random variable. Such Logit-based parking models have been presented by Lambe (1996) and the CLAMP model (Bates and Bradley 1986, Polak et al. 1990). More recent developments in this field are the Mixed Multinomial Logit specification (Hensher and King 2001, Hess and Polak 2004). Network-based models aim to simulate implicitly the parking choice process by extending the simulation capability of traffic assignment models where the road network is modelled using graph theory, and users’ choice is represented by means of a route choice model (deterministic or probabilistic). From this view, to model the parking supply system, the usual graph representation of the road network is modified through a sequence of additional links that replace the connector links (i.e. fictitious arcs that connect road links to centroid nodes) (Cascetta 2001), so that
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parking is represented together with route choice. Each additional link represents an activity relevant to the parking choice process and the relevant costs. For example, we could represent, for a given parking facility, a parking access link, a parking link (per type), pedestrian links, and so on (Bifulco 1993, Cascetta 2001). In this approach, as shown by Bifulco (1993), it is possible to consider also a parking searching time link, with the relevant cost, as a function of the parking occupancy level (i.e. parking demand), so that a traffic equilibrium problem is introduced. In this framework, parking choice modelling can be viewed as a minimum cost path searching problem where the choice set is made up of routes connecting drivers’ trip origins and destinations and passing through the available parking facilities. The consequent traffic assignment model and the solution algorithms depend on the assumptions on the link cost functions as well as on the route choice model. Nour Eldin et al. (1981) solve the network and parking loading problem by means of an incremental assignment approach. Bifulco (1993) proposes a congested network to model both the road network and parking supply, and the resulting traffic assignment model is of the Probit-User Equilibrium type. Lam et al. (2002) propose a network-based model to determine the optimal price for charged parking. A network-based approach is also used in Lam et al. (2006) where parking location and parking duration in road networks is handled assuming multiple user classes and multiple parking facilities. (For a comprehensive state-of-the-art review about classical approaches to parking choice modelling, the reader is referred to the work by Polak and Vythoulkas (1993) and Young (2000)). Even if the main interest of researchers in this field has been mostly addressed to traffic assignment problems, more recently, thanks to the development of Intelligent Transport Systems (ITSs) and Advanced Traveller Information Systems (ATISs), more attention has been paid to parking choice analysis assuming information provision. ATISs provide drivers with information about the road network and/or parking system status (i.e. congestion level, accidents occurring, working zones, occupancy, pricing) in order to help the user in his/her travel/parking choice and traffic managers in reducing the negative impacts of traffic, such as congestion and/ or air and noise pollution. Mathematical models are both useful and necessary to simulate (and forecast) the effects of information on user behaviour and then to design an ATIS effectively, such as parking guidance systems, by explicitly taking into account uncertainty underlying users’ perception. Most of this research work has been developed using the framework of probability theory in order to model the uncertainty underlying the variation in cost attributes and relevant user perceptions (Thompson and Richardson 1998, Wong et al. 2000, Lam et al. 2006, Teodorovic´ and Dell’Orco 2005). This approach is able to represent the random facets (stochastic and dynamic) of transportation system evolution over the considered time period (day-to-day or within-day dynamics) and may not fully represent other kinds of uncertainties that may arise during the processing of information by users such as imprecision that are ‘closed’ to human perception (Klir and Folger 1988). Indeed, when users have to choose among different alternatives, two kinds of uncertainty may occur (Dubois and Prade 1997, Henn 2005, Henn and Ottomanelli 2006):
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the lack of precision in the variables’ values, or to some error in those values; and/or random effects (accident occurring, roadside works, etc).
Thus, in order to model transportation user perception and consequent choice behaviour the different uncertainty sources should be considered (Henn and Ottomanelli 2006, Kikuchi and Chakroborty 2006):
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imprecision-uncertainty: linked to the natural approximate reasoning of humans; and/or random-uncertainty in cost-attribute perception: related to unpredictable events that may occur over the network as well as to the different way of perceiving a considered cost attribute among users (say, subjective/individual evaluation of attributes).
In this framework, the uncertainty underlying in parking (and/or travel) choice can be properly modelled by means of fuzzy sets and handled mathematically through Possibility Theory (Zadeh 1978, Dubois and Prade 1988). This implies that users, during the choice process, are assumed to handle the costs relevant to parking as fuzzy (imprecise)-costs, that is by following the approximate reasoning of human decision makers and the random-uncertainty about transport system conditions. Thus, in this article, we present a parking choice model based on Possibility Theory in which parking users are assumed to be rational decision makers whose aim is to minimise his/her perceived generalised parking costs. The generalised parking costs are represented by a combination of different cost attributes that are modelled by fuzzy cost functions. In the first part, we examine related research in the field of transportation choice modelling by Fuzzy Set Theory and briefly report the background to Possibility Theory with respect to the decision-making problem and utility theory (Klir and Folger 1988, Dubois et al. 1983). Then, the general assumptions of the proposed model and the specification of the fuzzy cost functions are described. The parking facility choice procedure to determine the users’ choice possibilities is shown. In addition, parking choice probabilities are determined. In the final part of this article, a numerical application of the model is presented. Related research Recently, great attention has been given to new paradigms and theories in order to model uncertainty underlying the transportation problem and, in particular, in users’ choice behaviour. Such an interest is due to the concept of uncertainty that is closely related to the traveller information issue as well as to the vagueness, subjectivity and imprecision of many variables and data relevant to traffic and transportation problems. In this environment, Fuzzy Set Theory and related techniques seem to be a consistent paradigm in order to model traffic and transportation problems characterised by ambiguity, subjectivity and uncertainty. The literature reviews (Teodorovic´ and Vukadinovic´ 1998, Teodorovic´ 1999, Avineri 2005) show how many issues of transportation science have been covered by fuzzy set-based applications starting from the pioneer work by Pappis and Mamdani (1977) relevant to traffic light control.
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In particular, the interesting issue of representing the decision process of human being in the role of transportation user has also been faced. Namely, fuzzy logic has been used to model the approximate reasoning of users when they have to choose among transport alternatives, such as departure time, trip destination, mode of transport, route, parking, and so on. Most effort, however, has been devoted to the problem of route choice modelling by means of rule-based Fuzzy Logic Systems (FLSs) constituted by a set of IF . . . THEN rules in which input and output variables are fuzzy sets. This approach allows the handling of both approximate reasoning and managing/control users’ attitudes by means of linguistic attributes such as: IF travel time on route 1 is very short AND travel time on route 2 is long, THEN preference for route 1 is strong where ‘very short’, ‘long’ and ‘strong’ are fuzzy sets representative of user perceptions. The first attempt to apply the fuzzy logic inference to route choice modelling was presented by Teodorovic´ and Kikuchi (1990). A fuzzy inference-based model of route choice was also presented by Akiyama and Kawahara (1997). The simulation of route choice drivers’ behaviour with information provision has been proposed by Lotan (1992) and Lotan and Koutsopoulos (1993) also using a Neuro-Fuzzy approach (Koutsopoulos et al. 1994). (For a wider discussion of the use of fuzzy logic in transportation engineering and relevant problems, the reader should refer to Teodorovic´ (1999)). More recently, research in transportation science has been showing how approximate reasoning of transportation users and relevant uncertainty can be conveniently represented using Fuzzy Set Theory within the framework of Possibility Theory (Dubois and Prade 1988, Zadeh 1978). In fact, recent studies in the field of psychology and cognitive science are demonstrating that the decision process of humans can be assumed, in a more realistic way, to evolve within a possibility rather than a probabilistic structure. In particular, Raufaste et al. (2003) report some experimental results, suggesting that Possibility Theory might be a more suitable framework for human uncertainty than probability theory, and that ‘subjective probabilities’ could advantageously be renamed ‘subjective possibilities’. Such conclusions are logically related to the way human beings process information to be used in decision-making that is based on subjective, approximate (sometimes qualitative) judgements. Also in this field of study, attention has been addressed mainly to route choice modelling and traffic assignment. It is worthwhile mentioning the early work in this direction proposed by Perincherry (1994) where discussion of uncertainty in the transportation problem is given as well as different applications of Possibility Theory presented. The use of Possibility Theory for modelling route choice with a theoretical comparison with other approaches is presented in Henn and Ottomanelli (2006). This article differs from other apparently similar ones since it is based on recent developments in the field of decision theory by considering separately randomuncertainty and imprecision (Dubois et al. 1996). A more general and updated discussion about uncertainty and the role of Possibility Theory in transportation problem is presented in Kikuchi and Chakroborty (2006). The traffic assignment problem assuming fuzzy travel cost is discussed within a static approach by Henn (2000), while a dynamic formulation is proposed by Liu et al. (2003).
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Very few works deal with the parking planning and design problem in the framework of soft-computing or artificial intelligence. With respect to Fuzzy Logic, Teodorovic´ and Dell’Orco (2005) proposed a multi-agent system for parking facilities management where the parking choice process is based on the negotiation of two agents: driver-agent and parking-agents. The core of the system is the driver-agent choice simulator that is a Fuzzy Rule-based inference engine consisting of a set of 12 rules. The rules assume as fuzzy variables the cost attributes of parking (parking fee, parking distance, searching time, enforcement) and the parking-agents (parking facilities manager and city government) are modelled by two fuzzy logic-based inference engines that update/adjust, respectively, the parking fee and enforcement on the basis of driver choice feedback (i.e. on the basis of the parking demand and illegal parking). Teodorovic´ and Lucic (2006) proposed an intelligent system for parking space inventory control based on a real-time parking guidance system that manages (i.e. reject or accept) the pre-trip requests for free parking lots. The system is based on the combination of simulation, optimisation techniques (integer programming) and fuzzy logic that is the real-time decision module. These approaches are useful tools for the real-time/dynamic management of the existing parking supply system or single parking infrastructures. Our paper aims to address the tactical and/ or strategic planning issue of the parking supply system that requires different modelling approaches, such as Possibility Theory, that has not until now been used for representing parking choice behaviour and in the field of route choice, as will be shown, seems to be an appreciable and consistent alternative modelling approach.
Modelling uncertainty by Possibility Theory Use of stochastic models is well established in representing uncertainty in transportation systems analysis. Random utility-based models assume that the user associates each alternative with a perceived utility, made up of the sum of a systematic term and a random term that takes into account both user’s perception error and modeller’s error (Ben Akiva and Lerman 1985, Cascetta 2001). Then, he/ she chooses the alternative with the maximum perceived utility. These models allow calculating the choice probability for each alternative belonging to the ith user’s choice set. Errors due to the modeller are relevant to the approximation in systems representation that depends on his/her subjectivity/approach. For example, attributes representing a transport supply system’s level of service, which are used within the systematic utility, are often estimated by a model of a real supply system. The values used in the traffic model (or utility functions) are also fixed values that do not actually take into account the distribution of the perceived (different perception) values among the users, but an average one. In the last 30 years, however, new theories have been developed to model imprecision and uncertainty: among them, Possibility Theory sounds very promising for representing the uncertainty of human perception, and then useful in the simulation and understanding of effects of information/knowledge on users’ choice behaviour. It constitutes a mathematical tool preserving uncertainty when comparing different values of perceived cost (or utility), expressed through imprecise, vague numbers or by intervals, qualitative judgements, even if partially overlapping.
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To this end, two measures, Possibility and Necessity, are used to represent, respectively, the optimistic and pessimistic point of views in comparison. In this article, we deal with the Possibility measure. From a theoretical standpoint, an imprecise (or expressed by an interval) number is represented by a function h defined in the interval [0, 1]:
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h : X ! ½0; 1 8x 2 X
(1)
where X is the domain set. The function h associates for each x X its membership degree to the set defining the imprecise (fuzzy) number and is therefore called the membership function. The shape and values of a membership function can be defined and calibrated through different mathematical tools, such as neural networks or genetic algorithms (Medasani et al. 1998). For the sake of simplicity, in the following discussion the triangular shape, defined by its centre value and left and right limits, will be used. For example, in Figure 1 the fuzzy set of number ‘close to 5’ is reported and this set can be expressed, numerically, by A (2, 5, 8) where 5 is the centre value, 2 and 8 are the left and right limits, respectively. To the centre value corresponds the maximum value of the membership degree [i.e. h(5) 1], whilst at the left and right limits the value of function is equal to zero. A fuzzy set A induces a Possibility distribution that is numerically equivalent to its membership function (Zadeh 1978), that is: hðxÞ ¼ PossðxÞ
(2)
where Poss(x) is the Possibility distribution induced by h(x) and represents how much x belongs to the set A. In other words, by saying that for the number 3 the membership degree to the set of numbers ‘close to 5’ is 0.4 means that, from a numerical point of view, the possibility that the number 3 belongs to the set A is 0. Possibility theory allows comparisons between two (or more) fuzzy numbers by defining the Possibility that a fuzzy number hj(x) belongs to the fuzzy set greater or equal to w, or that the fuzzy number hw(x) belongs to the fuzzy set less or equal to j. 1
h 0,5 0,4
0 0
1
2
3
Figure 1. Example of membership function.
4
5
6
7
8
9
x
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Let hw(x) be the membership function of the set greater or equal to w; the Possibility that hj(x) belongs to this set is given by (Zadeh 1978, Dubois and Prade 1988, Zimmermann 1996):
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Possðhj ðxÞ hw ðxÞÞ ¼ Max Min ðhj ðxÞ; h > wðxÞÞ 8x
(3)
Figure 2, in which the sets h 5w(x) and h ]w(x) are pointed out, explains the meaning of Eq. (3). In order to better explain the computing of the Possibility measure, let us consider two fuzzy sets hj(2, 4, 6) and hw(5, 7, 9), as reported in Figure 3, and suppose we are interested in computing the possibility that hj(x) ] hw(x). In Figure 3, the dotted line represents the set ‘greater or equal to w’ while the bold line represents the set J of the points Min[(hj(x), h]w(x)] whose maximum value is equal to a and represents the possibility that the alternative j is perceived as better than w. The proposed parking choice model In the proposed model, uncertainty due to users’ imperfect knowledge of the system, and therefore inherent in their decision-making, is modelled explicitly. In fact, without precise and complete information, a user chooses on the basis of a priori expected costs he/she associates with each parking alternative. Uncertainty and imprecision related to this kind of cost estimate (perception) can be represented through fuzzy sets. We assume the kth user (or class of users) as a rational decision maker that has a choice set Ik made up of n parking alternatives. The user associates each parking alternative with a perceived cost that is modelled by a set of fuzzy attributes. The choice of parking is given by a comparison among the alternatives that is carried out through applying Possibility Theory (Dubois and Prade 1983). The assumption of a rational user is similar to other discrete choice models, but, as already stated in previous sections, within Possibility Theory we are able to handle imprecise, incomplete data/information and consequently the uncertainty underlying decision makers’ system perceptions and to model it coherently by an approximate approach. In addition, modelling system attributes by fuzzy sets implicitly allows the strong hypothesis of perfect knowledge of choice set and alternatives by users to be partially relaxed, as assumed in the random utility theory approach. hj(x), hw(x)
h≤w(x)
h≥w(x)
1
Poss[hj(x)≤hw(x)] Poss[hj(x)≥hw(x)] hj(x)
0
Figure 2. Comparison between two fuzzy numbers.
hw(x)
X
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Figure 3. Example of calculation of possibility measure.
In our model, we assume that parking demand is known and defined by the dwell time as well as by trip origin o and destination d. The parking supply model Let S1, S2, . . ., Sn be n parking alternatives. Each one may be characterised by a vector of attributes A1, A2, . . . Am. The cost (or utility) of the ith alternative can be expressed as a linear combination of A(i) attributes, that is: ðiÞ
ðiÞ
Ci ¼ c1 A1 þ c2 A2 þ cm AðiÞ m
(4)
Uncertainty, or the imprecise knowledge of the attributes, is expressed in Eq. (4) ðiÞ ðiÞ through approximated values of attributes A1 ; A2 ; . . . ; AðiÞ m . Usually, these attributes are divided into monetary (e.g. cost of gasoline, toll) and non-monetary (i.e. lotsearching time). The units of corresponding coefficients cj are such that the resulting quantities (i.e. cj AðiÞ j) are uniform. The costs associated with all alternatives belonging to the choice set are arranged in a preference order. Let Rp be the preference order of parking alternatives S, defined as: Sj Rp Sw ! fSj is better than Sw g:
(5)
To verify this condition, we compare the attributes of Sj with those of Sw, that is: ðiÞ
ðwÞ
Ak Rp Ak ;
k ¼ 1; . . . :m:
If attributes were expressed through precise values, the preference order is held when all attributes of Sj are better than those of Sw. Since in our case the attributes are expressed through approximated values, the comparison is carried out within the framework of Possibility Theory and expressed in the range [0, 1] how much the attributes of Sj are better than those of Sw, that is: ðjÞ
ðwÞ
PossðSj Rp Sw Þ ¼ MinfPossðAk Rp Ak Þg 8k; j 6¼ w
(6)
The Min sign means that the Possibility is based on the attribute having the lowest preference value.
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Moreover, let Poss(Sj) be the possibility that Sj is the best alternative, then: PossðSj Þ ¼ Min ðPossðSj Rp Sw Þg 8w 6¼ j
(7)
In this model, the alternatives are defined for each pair of trip origin o and destination d, for parking typology t and capacity q. The generalised parking cost Cj, associated with the parking alternative Sj(o, d, t, q) is specified through an attribute vector A(Sj), whose components represent:
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the the the the
pricing structure for charged parking; control and enforcement policy of illegal parking; distance between parking facility and final destination; and congestion level of the parking facility.
These elements are known with respect to a given case study and depend on the system management strategy (parking policy) and the technical characteristics of the parking supply systems (i.e. capacity of parking facilities, location). In particular, the generalised cost Cj, associated with parking alternative Sj(o,d, t, q), has been specified by the following cost attributes: TVo,j TRj TARj FCi MU CTj,d a
travel cost for reaching parking Sj from origin o of the trip (hour); parking lot searching time (hour); hourly fee for charged parking (t EURO); hourly frequency of controls for illegal parking (controls/hour); fine due to illegal parking (t EURO); travel cost from parking Sj to final destination d of the trip (hour); value of time (VoT) (t EURO/hour)
Therefore, the generalised cost for parking facility j is defined as: Cj ¼ cj þ mj ð1=aÞ ðj ¼ 1; 2; . . . ; nÞ
(8)
where: cj ¼ TVo;j þ TRj þ CTj;d 8 0 > < MU ½minð1; FCj DSi Þ mj ¼ > : TAR ½minimum integer DS j i
for free parking for illegal parking for charged parking
DS ¼ parking staying time ðhourÞ: The values of the specified attributes can be expressed both by intervals (or approximated numbers) and by precise numbers. Of course, using approximated numbers, imprecision and uncertainty of perception can be explicitly introduced in the mathematical expression of perceived cost.
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From choice possibility to choice probability Each value for Poss(Si) represents the possibility of choosing parking alternative Si, that is the degree to which the ith parking facility belongs to the ‘best alternative’ set. In practical cases, probability is used in its frequency meaning to find out for each alternative the users’ share choosing that alternative. In contrast, the possibility values cannot be seen as frequency values, and thus cannot be used to calculate the users’ share. Therefore, possibility values must be transformed into probability values. This transformation can be solved using the Uncertainty Invariance Principle (Klir 1990). This principle asserts that, in a given situation, Uncertainty must be the same, whatever the mathematical framework used to describe that situation. In stochastic choice models, Uncertainty is measured through the Shannon entropy (Shannon and Weaver 1948), given by H ¼ Rj ðpj log2 pj Þ
(9)
where H is the measure of entropy and pj is the choice probability of alternative j. In the context of Possibility Theory, as suggested in Klir and Folger (1988) and Klir (1990), the Uncertainty measure is given by U ¼ Rj ½ðPossðSj Þ PossðSjþ1 ÞÞ log2 ðjÞ
(10)
where Poss(Sj) and Poss(Sj1) are the possibility values of alternatives j and j 1, ordered in a decreasing way. Applying the Uncertainty Invariance Principle, both the Uncertainties measured by Eqs. (8) and (9) must be the same. That is: Rj ðpj log2 pj Þ ¼ Rj ½ðPossðSj Þ PossðSjþ1 ÞÞ log2 ðjÞ
(11)
Additionally, Possibility and Probability distributions are subject to the normalisation requirement, claiming that: Rj pj ¼ 1 maxðPossðSj ÞÞ ¼ 1
ðprobabilistic normalizationÞ ðpossibilistic normalizationÞ
In our case, a suitable transformation from possibility to probability is the loginterval scale transformation that has the following form (Geer and Klir 1992): c
PossðSj Þ ¼ bðpj Þ
(12)
where g and b are positive constants. From Eq. (16), we obtain pj (Poss(Sj)/b)1/g and, applying the probabilistic normalisation requirement: 1=c
1 ¼ Rj ðPossðSj ÞÞ b1=c ) b1=c ¼ Rj ðPossðSj ÞÞ
1=c
:
Hence: Pj ¼ ðPossðSj ÞÞ
1=c
1=c
=Ri ðPossðSj ÞÞ
(13)
Replacing Eq. (13) in Eq. (11) and solving numerically the resulting equation, the constant g can be calculated; then, the probability value can be found by Eq. (13).
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Numerical application The proposed model has been applied to a choice set made up of three parking alternatives Sj (j 1, 2, 3):
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j 1 (free legal parking FP); j 2 (illegal parking IP); and j 3 (charged parking CP).
Parking demand is defined through linguistic propositions referring to dwell time DS, where: DS ¼ fvery short; short; average; longg: and modelled by the fuzzy sets represented in Figure 4. As depicted in Figure 5, the VoT has been represented by a triangular fuzzy set whose values are (2, 4, 6) [t/ hour]. Attributes representing travel cost TA(o, Sj), from origin o to parking facility Sj, the parking searching time cost TR(Sj), the transfer time cost (in this case, walking time) CT(Sj, d) from parking Sj to final destination d, have also been specified by triangular fuzzy sets and reproduced in Figure 6. Three different scenarios have been considered by varying the dwell time and the controls frequency (enforcement) for illegal parking, as shown in Table 1. Through Eq. (6), starting with each cost attribute, the total costs for each scenario and each parking choice have been calculated. The parking supply models for three different scenarios are reproduced in Figure 7ac where the linguistic propositions used for the cost attributes can be recognised. In order to model the supply system, we define a network-based model where each link is associated with the fuzzy cost relevant to the considered attribute of the user’s perceived parking cost. Thus, in general, a parking alternative is modelled by a sequence of links that are representative of a path that starts from the origin of the trip o and ends at the final destination passing through parking facility Pj. Each link corresponds to an activity and the relevant cost (Figure 8).
Figure 4. Membership functions of fuzzy sets for ‘Dwell Time’ (DS).
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Figure 5. Membership functions of fuzzy sets for ‘Value of Time’.
The perceived cost associated with the links is represented by a membership function according to the fuzzy attributes defined in Figure 6. Thus, in Figure 7 the following parking alternatives in the scenarios given in Table 1 have been represented: (1) free parking; (2) illegal parking; (3) charged parking. In order to estimate the effects of parking policies in terms of illegal parking control, the free-parking alternative has been purposely penalised, assuming high values for its cost attributes (i.e. long distances from the final destination and from the origin of the trip). In the three scenarios, the values of the free-parking attributes have been considered unchanged, whilst those of illegal or charged parking vary according to the scheme (Table 1). In each scenario, the total cost Cj of the parking alternatives has been compared to each other using Eq. (6), and the choice possibilities of each alternative have been evaluated (Figure 9). The comparisons have been carried out using a software programme written in MatlabTM. Once the choice possibilities for each alternative and each scenario have been defined, the probability values for each alternative can also be calculated, by applying the Uncertainty Invariance Principle described in the above section. Table 2 summarises the outcomes from the proposed model, namely the values of the parking choice possibility for each scenario (in brackets) and the values of the choice probabilities. With reference to the illegal parking alternative, the model is able to evaluate effectively the user’s risk perception to be fined. In fact, as the histogram in Figure 10 shows, the stronger the illegal parking controls, the higher the perceived risk of being fined. This effect is more evident when control frequency increases from average to strong. In fact, assuming the same dwell time, the illegal parking choice probability decreases while free and charged parking choices increase, even if free parking was assumed to have high costs. In addition, by increasing the dwell time (from short to
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average), the illegal parking choice probability decreases, while the charged parking and free parking choice probability increase. These results are quite intuitive and can also be obtained with other parking choice models proposed in literature, but the model proposed here allows uncertain and/incomplete data to be handled as well as to consider explicitly the approximate reasoning of the human decision maker in processing and perception of real systems. Such considerations are important if we are also interested in affecting/modifying user’s choice behaviour by using information systems to help travellers perceive what is useful and to achieve traffic managers aims.
Discussion The application of the proposed model has shown how the Possibility Theory approach is a valid framework for modelling parking users’ choice behaviour by Table 1. Characteristics of considered scenarios. Scenario 1 2 3
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Figure 7. Evaluated parking supply models.
simulating their approximate reasoning based on uncertain, imprecise and vague data. Since illegal parking is a real problem in many Italian towns, in our numerical tests we have highlighted the effects of enforcement policy. In fact, as we have experienced in real cases, illegal parking is perceived as ‘free parking’ when charged parking is not supported by strong enforcement. As shown in Table 2, if staying time is short and enforcement is weak and walking distance is null then illegal parking is preferred (55%) to charged parking (39%) even if travel time and walking distance are both short. Thus, costs relevant to the parking fee are perceived to be higher than the risk to be fined. If the dwell time is average, by increasing the enforcement from
Transportation Planning and Technology Free parking lot searching
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Figure 8. Representation of the parking supply system.
average to strong, then illegal parking choice decreases from 28% to 10%. In the same way, the choice of free parking also increases even if walking distance is long. The results show that the model is consistent and realistic with respect to observed behaviour in urban areas, and this is very promising since the theoretical background is actually consistent with human approximate reasoning. The proposed model is very responsive to the cost attribute variations considered and could be easily used to define parking policies in terms of pricing, enforcement, facilities location and types. For example, by varying the frequency of control as well as the amount of fine, the optimal human resource for parking enforcement can be determined assuming a given parking pricing policy. At the same time, by varying
Figure 9. Comparison among alternatives for different scenarios.
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Table 2. Choice probability and possibility for each alternative for different supply scenarios. Scenario Parking facility 1 free parking 2 illegal parking 3 charged parking
1
2
3
0.06 (0.56) 0.55 (1.00) 0.39 (0.92)
0.19 (0.89) 0.28 (0.93) 0.53 (1.00)
0.32 (0.89) 0.10 (0.74) 0.58 (1.00)
Conclusions This article has presented a model to evaluate users’ parking choice behaviour with respect to different parking system management policies. The modelling approach is based on the Possibility Theory that constitutes a suitable mathematical framework to deal with uncertainty in user’s perception and also to handle numerically quantities expressed and processed through verbal propositions and/or judgements such as imprecise numbers (or intervals). Possibility Theory seems to fit in a good way the human decision-making process; in fact, human approximate reasoning could be considered more possibilistic than probabilistic. For example, we do not process and handle enforcement by some crisp frequency or random distributions, but we perceive it to be an imprecise, approximate value, depending on our experience and degree of information. Thus, the possibilistic model is based on linguistic attributes of costs, or judgements, as to how the decision maker logically perceives and processes the real system; such quantities are similar to the ones that can be provided by ATIS and whose effects on user’s choice behaviour can be evaluated. From this point of view, this approach could be useful to analyse and define in a better and more coherent way the information to provide users within order to deliver the less uncertain ones. From the quantitative standpoint, the results given in the numerical application are interesting and show the capability of the model to represent how users respond free-parking
illegal parking
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travel cost between parking places and final destination, a park-and-ride system solution could be considered in the model and then evaluated.
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Figure 10. Parking choice probabilities by parking type and scenario.
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to different parking management policies. The application has considered only three types of parking, but the generalised parking cost function can be specified to take into account other types of parking facilities, such as metered parking and time limited parking. The model also assumes that the VoT can be specified as a fuzzy number to take into account the imprecision of such an element for the ith user (or class of users) that may not be perceived in a crisp (or same) way among users of the same class. Another issue to highlight is the implementation of such an approach for real cases that, and also assuming other studies in the same field, is not different from the traditional parking choice model since a parking supply system inventory is necessary and a supply model has to be defined. In practice, no commercial software is currently available for possible computation applications other than to meet our research aim. Indeed, updated parking and traffic management policies are mainly based on information systems and then uncertainty problems in information to be delivered have to be faced, so that Possibility Theory could be a valid alternative in order to model the field of drivers’ behaviour. This article has shown that a parking choice model is an important step towards the development of parking demand assignment models in an ATIS environment as a support tool to help design Traveller Information Systems and to evaluate parking policies for the reduction of traffic network congestion. Since the model is quite flexible it could also be used to extend the capability of a Possibility Theory-based route choice model, in order to jointly evaluate route and parking place choice. Acknowledgements The authors owe an anonymous referee a debt of gratitude for the comments and suggestions that helped us improve the earlier version of this article.
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