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Procedia Computer Science 109C (2017) 148–155
The 8th International Conference on Ambient Systems, Networks and Technologies The 8th International Conference on(ANT Ambient Systems, Networks and Technologies 2017) (ANT 2017)
Triangular intuitionistic fuzzy number theory for driver-pedestrians Triangular intuitionistic fuzzy number theory for driver-pedestrians interactions and risk exposure modeling interactions and risk exposure modeling Meriem MANDARaa, Lamia KARIMbb, Azedine BOULMAKOULbb*, Ahmed LBATHcc Meriem MANDAR , Lamia KARIM , Azedine BOULMAKOUL *, Ahmed LBATH a National School of Applied Sciences Bd Béni Amir, BP 77 Khouribga, Morocco a National School of Applied Sciences Bd Béni Amir,ofBPcomputer 77 Khouribga, Morocco Faculty of Sciences and Technologies of Mohammedia, Department sciences, LIM/IOS Lab, Morocco b c Faculty of Sciences and Technologies of Grenoble Mohammedia, of computer sciences, LIM/IOS Lab, Morocco University Alpes,Department CNRS, LIG/MRIM, France. c University Grenoble Alpes, CNRS, LIG/MRIM, France. b
Abstract Abstract We travel every day as pedestrians on the road network, if only for a short time, in order to reach another means of transport. But We every pedestrians on the be. roadThese network, if only for a short in order reachsystem anothercombining means of the transport. But this travel activity is notday as as risk-free as it should accidents represent the time, product of an to overall individual this activityofis pedestrians not as risk-free it shouldtransport be. Theseactivity, accidents represent the product ofsystem, an overall system combining individual behaviors andasdrivers, and the environmental under conditions thatthemake these behaviors of both pedestrians andand drivers, transport environmental under that makeinthese malfunctions predictable avoidable. This activity, predictiveand skillthe is an essential step system, in ensuring the conditions safety of pedestrians road malfunctions predictable and avoidable. This predictive is an to essential stepexposure in ensuring the safety oftopedestrians in risks road traffic. In thisboth paper, we use intuitionistic fuzzy number theoryskill in order model the of pedestrians the accidents traffic. In this paper, we use intuitionistic fuzzy number theory in order to model the exposure of pedestrians to the accidents risks by new indicators. This approach aligns the behavioral psychology of pedestrians and drivers with intuitive methods based on the by newofindicators. This approachsolution aligns the behavioral for psychology of pedestrians intuitive methods based on the theory fuzzy sets. A software is developed this instance by reusingand the drivers models with of pedestrian simulation developed theory of fuzzy sets. A software solution is article developed for thisa novelty instancein bypartial reusing the models pedestrian simulation developed in our previous works. The attempt in this considers aspect of riskof modeling. The fuzzy intuitionistic in our previous works. The attempt in hesitation this articleand considers a novelty partial aspect ofand risk modeling. The fuzzymust intuitionistic theory has the flexibility to investigate indecision. Someinother behavioral environmental factors be taken theory has the flexibility to investigate hesitation and indecision. Some other behavioral and environmental factors must be taken into account in order to explore deeply the issue. into account in order to explore deeply the issue. © 2017 The©Authors. Published Elsevier by B.V. 1877-0509 2017 The Authors.by Published Elsevier B.V. © 2017 The under Authors. Published by Elsevier B.V. Peer-review responsibility of the Conference Peer-review under responsibility of the Conference Program Program Chairs. Chairs. Peer-review under responsibility of the Conference Program Chairs. Keywords:Intuitionistic fuzy number, virtual pedestrian model; intelligent agent; simulation; accident risk; transportation theory Keywords:Intuitionistic fuzy number, virtual pedestrian model; intelligent agent; simulation; accident risk; transportation theory
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1877-0509 © 2017 The Authors. Published by Elsevier B.V. Peer-review under responsibility of the Conference Program Chairs. 10.1016/j.procs.2017.05.309
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1. Introduction and backgroud How does the code road define a pedestrian? If the most common use of the pedestrian term is to characterize the "person who walks on foot (in town, on a road)", the road code rakes wider. Pedestrians are also persons who drive a child, sick or infirm car or any other small vehicle without a motor. Moreover pedestrians are also considered as persons who drive a cycle or a moped by hand, the infirm who move in a wheelchair driven by themselves or moving at the walking pace. According to the World Health Organization, every year nearly 1.25 million people die in a road accident and 20 to 50 million are injured, sometimes even handicapped. The majority of these victims are pedestrians hit by motor vehicles in various situations. Road traffic injuries result in considerable economic losses for those affected, their families and countries as a whole. There are few global estimates of the cost of road accidents, but studies conducted in 2010 show that they account for about 3% of national gross product. This number even reaches 5% in some low- and middle-income countries. Pedestrians are at risk of accidents either by standing or walking on the sidewalk or crossing the road. The number of accidents depends on the local transport policy and the resources devoted to pedestrian safety. It should be noted that there is a close relationship between pedestrian safety and their environment. The design of streets rests on the specific characteristics of cars, namely their speeds, dimensions, the spaces required for rotating movements, and so on. Without taking into account those of pedestrians, which increases the risk of pedestrian accidents. This is because walking is not yet considered as an essential and necessary part of the overall transport system. Although, in recent years, streets and areas dedicated to pedestrians, "zones 30" and restricted traffic have been developed, but this approach is still limited, due to divergence between the design of the planner And that of the user, which leads to dysfunctions observed in the system failures in the form of accidents. These dysfunctions are both predictable and avoidable as they represent the product of an overall system combining the individual behavior of pedestrians and drivers, transport activity and the environmental system. The reduction or elimination of risk to pedestrians is an important and achievable goal. This risk is greater for children, the elderly and pedestrians who are intoxicated because of their inability to adapt to both the reflexes of road traffic and the rules governing interaction with the road environment. From an early age, men transgress these rules and are more likely than women to be involved in a road accident. Almost three-quarters (73%) of the victims killed on the roads are men. In young drivers, young men under the age of 25 are nearly three times more likely to be killed in a car accident than young women. This can be explained by the fact that women are more susceptible to potential losses, while men are compared to earnings. This behavioral difference results from the dissimilarity of the sex roles of the stereotypes to which they belong. These roles are defined culturally and strongly influence the belonging of the individuals to these stereotypes, without of course neglecting the possibility of adherence to the stereotypes of the opposite sex. On the other hand, it should be noted that the factor of adherence to sexual stereotypes constitutes the best predictor of risk taking by referring to both the age and sex of the individual concerned, whether pedestrian or driver. In our study, we assume that pedestrians are only at risk when crossing a road, as accidents outside the crossing area are generally rare. This crossing can take several forms of routes involving different types of crossing depending on the characteristics of the road traffic, including volume and traffic speed. Thus crossing pedestrians can be perceived as one between the characteristics of road traffic, adherence to sexual stereotypes and the desire to carry out this activity with maximum possible comfort. Why fuzzy intuitionistic risk? The study of the exposure of pedestrians to the risks of accidents is based mainly on relevant kinematic parameters of the vehicle and the pedestrian and certain geometric criteria of the road. And this without taking into consideration the behavioral factors related to the perception of space and the decision-making of the two road users. To overcome this problem, the intuitionist approach makes it possible to relate the two realities perceived by the two users, taking into consideration both the objective and subjective information of the driver and the pedestrian. On the one hand, we have integrated the two antagonistic perceptions into intuitionistic fuzzy numbers and, on the other hand, developed a relative ranking method to derive the indicators of risk exposure. The remainder of this paper is organized as follows: section 2 recalls our fuzzy ant pedestrians’ model and situates it in relation with Pedestrians’ risk accident exposure. The section 3 introduces intuitionistic fuzzy theory and gives first concepts for risk exposure modeling. Finally, section 4 concludes.
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2. Fuzzy ant pedestrians’ model Either inside public building, open areas or networks’ roads, prediction of infrastructure influences planning on people behavior, comfort and mobility, become very important presses. In this purpose, we developed a model of pedestrians’ motion based on the behaviors of natural swarms, notably ants. We used the metaheuristic ACO of swarm intelligence in a cellular space, assuming that virtual pedestrians communicate through a virtual trace that evaporates over time3. We have modified the term visibility of ants in the ACO metaheuristic to adapt it to the visibility or desirability of virtual pedestrians, using the basic mechanisms of artificial potential fields. Virtual pedestrians begin their walking process by choosing a given destination. The latter applies a field of attractive potential, while the obstacles apply a repellent. The superposition of the forces applied by the static components of the environment guides the virtual pedestrians towards their objectives while pushing them away from obstacles. The environment can then be perceived as a hill sloping towards a point of destination, which is at the lowest point in space, in terms of distances, allowing virtual pedestrians to "roll" to their destinations while being repelled from obstacles. In this model, the fuzzyfication of pedestrians' utility concerns only spatial perception (obstacles, preferred direction, amount of pheromone for dynamic floor, etc.). While we chose not to fuzzy the cell occupation parameter, because it does not contain a degree of uncertainty. This is because each cell can only be occupied by one pedestrian at a time. The possibility of movement is given through a fuzzy general utility of movement, expressed by: 𝑃𝑃̃𝑖𝑖𝑖𝑖 (𝑡𝑡) = ∑ While
𝛼𝛼
̃𝑖𝑖𝑖𝑖(𝑡𝑡)] [𝜏𝜏̃𝑖𝑖𝑖𝑖(𝑡𝑡)] [𝜂𝜂
𝛽𝛽
̃𝑖𝑖𝑖𝑖 (𝑡𝑡)]𝛽𝛽 ̃ 𝑖𝑖𝑖𝑖 (𝑡𝑡)]𝛼𝛼 [𝜂𝜂 𝑇𝑇 [𝜏𝜏 𝑙𝑙∈𝐽𝐽 𝑘𝑘 𝑖𝑖
(1)
× 𝐼𝐼𝑉𝑉 𝑖𝑖 (𝑗𝑗) 8
𝜏𝜏̃𝑖𝑖𝑖𝑖 (𝑡𝑡)is the pheromone quantity in the cell (𝑗𝑗) describing a parameter of dynamic floor field; 𝛼𝛼 is an influence control parameter of 𝜏𝜏̃𝑖𝑖𝑖𝑖 ; 𝛽𝛽 is an influence control parameter of 𝜂𝜂̃𝑖𝑖𝑖𝑖 ; and 𝐼𝐼𝑉𝑉 𝑖𝑖 is the set of eight cells 8
neighboring cell (𝑖𝑖). 𝜂𝜂 ̃𝑖𝑖𝑖𝑖 (𝑡𝑡)is
Where
the pedestrian desirability or visibility and is given by:
𝜂𝜂̃𝑖𝑖𝑖𝑖 (𝑡𝑡) = 𝑒𝑒𝑒𝑒𝑒𝑒 [− (𝑈𝑈𝑎𝑎𝑎𝑎𝑎𝑎 (𝑐𝑐𝑗𝑗 ) + 𝑈𝑈𝑟𝑟𝑟𝑟𝑟𝑟 (𝑐𝑐𝑗𝑗 ))] × (1 − 𝑂𝑂𝑖𝑖𝑖𝑖 (𝑡𝑡))
(2)
Attractive potential field is expressed by: 1
𝑈𝑈𝑎𝑎𝑎𝑎𝑎𝑎 (𝑐𝑐𝑗𝑗 ) = ∗ 𝑘𝑘𝑎𝑎𝑎𝑎𝑎𝑎 ∗ 𝑑𝑑𝑔𝑔2
(3)
2
And repulsive potential field is expressed by: 𝑈𝑈𝑟𝑟𝑟𝑟𝑟𝑟 (𝑐𝑐𝑗𝑗 ) = {
0
1 2
∗ 𝑘𝑘𝑎𝑎𝑎𝑎𝑎𝑎 ∗ [
1
𝑑𝑑0
−
1
𝜌𝜌0
] ∗ 𝑑𝑑𝑔𝑔2
𝑖𝑖𝑖𝑖 𝑑𝑑0 ≤ 𝜌𝜌0
𝑖𝑖𝑖𝑖 𝑑𝑑0 > 𝜌𝜌0
(4)
While 𝜌𝜌0 is a specific parameter for each obstacle in the simulation grid, 𝑑𝑑0 and 𝑑𝑑𝑔𝑔 are the distances between pedestrian position and nearest obstacle and the goal respectively. We choose to apply the Brush Fire algorithm in calculating distance between the next target cell and an obstacle, and Wave Front Algorithm for the one between the next target cell and the goal. The virtual pheromone evaporating through time is provided by the formula: 𝜏𝜏̃𝑖𝑖𝑖𝑖 (𝑡𝑡) = 𝜌𝜌𝜏𝜏̃𝑖𝑖𝑖𝑖 (𝑡𝑡 − 1) + ∆𝜏𝜏̃𝑖𝑖𝑖𝑖
(5)
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𝑘𝑘 Where 𝜌𝜌 is pheromone vaporization rate, and ∆𝜏𝜏̃𝑖𝑖𝑖𝑖 (𝑡𝑡) = ∑𝑚𝑚 𝑘𝑘=1 ∆𝜏𝜏̃ 𝑖𝑖𝑖𝑖 (𝑡𝑡) is the sum of pheromone laid down by all pedestrians at time step t. In a time step only one pedestrian occupy a cell, so∆𝜏𝜏̃𝑖𝑖𝑖𝑖 (𝑡𝑡) = 𝑂𝑂𝑖𝑖𝑖𝑖 (𝑡𝑡) .
2.1. Pedestrians’ risk accident exposure
The interaction with road traffic at crossing exposes us to the possibility of being hited by a vehicle. This level of contact with potentially harmful traffic defines our exposure to the risks of accidents. Several measures of exposure to risks to pedestrians have been established but none has been specified as being the best. Nevertheless, some are qualified better adapted than others according to specific needs and objectives. In addition, the identification of accident risk factors for pedestrians can significantly improve road safety. These factors are related, on the one hand, to the environmental characteristics of the cross-section belonging to the road network and, on the other, to the behavioral characteristics of pedestrians and drivers. In this work, we assume that pedestrians are at risk when crossing the road. This is due to the low rate of accidents outside the crossing zone. This crossing activity can take several forms of routes, the choice of which is often a compromise between the perception of the risk by the pedestrian and his ability to cross as comfortably as possible. However, these routes take a rectilinear form perpendicular to the road in crossing zone. While they take the form of oblique lines once pedestrians move away from the crossing areas. On the other hand, it should be noted that in the face of a crossing situation, pedestrians continually tend to adjust their speed and their routes. This implies that the process of crossing integrates several cognitive processes more than that of motor activity, while relying heavily on the theory of maximization of utility. And this by selecting the infrastructure and the most satisfactory locations in terms of comfort and walking. On the other hand, it should be noted that drivers cannot stop immediately after the perception of pedestrians because of their speeds. They need a braking distance, also known as a safety distance, to be able to stop without the risk of collision with pedestrians or other obstacles on their roads. The security distance can be expressed by the following formula Cohen4: 𝑆𝑆 = 𝐿𝐿 + 𝑇𝑇𝑟𝑟 𝜈𝜈 + 𝜈𝜈 ⁄2𝛾𝛾
(6)
𝑇𝑇 = 𝑆𝑆⁄𝜈𝜈 = 𝐿𝐿⁄𝜈𝜈 + 𝑇𝑇 + 1⁄2𝛾𝛾
(7)
Where: 𝑇𝑇𝑟𝑟 is the reaction time of the driver; 𝐿𝐿 is the vehicle length; 𝜈𝜈 is the vehicle speed; 𝛾𝛾 is the vehicle acceleration. From the equation (6) we can get what we call the safety stopping time for a vehicle, which is given by: Statistical studies in France defer the security distance for a vehicle to the following formula: 𝑆𝑆 = 8 + 0.2𝜈𝜈 + 0.03𝜈𝜈 2 ≅ 0.2𝜈𝜈 + 0.03𝜈𝜈 2 And the safety stop time for the vehicle becomes 𝑇𝑇𝑆𝑆 = 0.2 + 0.03𝜈𝜈. Moreover, the vehicle flow can be expressed in terms of their densities and speeds, which they reach their maximum value if the density is null, and vanish in the case of maximum density, according to the following equation𝜈𝜈 = 𝜈𝜈𝑚𝑚𝑚𝑚𝑚𝑚 (1 − 𝜌𝜌⁄𝜌𝜌𝑚𝑚𝑚𝑚𝑚𝑚 ). Where: 𝜈𝜈 is the vehicle speed; 𝜈𝜈𝑚𝑚𝑚𝑚𝑚𝑚 is the maximum vehicle speed; 𝜌𝜌 is the vehicles density; 𝜌𝜌𝑚𝑚𝑚𝑚𝑚𝑚 is the maximum vehicles density; therefore the safety stopping time for a vehicle becomes: 𝑇𝑇𝑠𝑠 = 𝑇𝑇𝑟𝑟 +
𝜈𝜈𝑚𝑚𝑚𝑚𝑚𝑚 2𝛾𝛾
(1 − 𝜌𝜌⁄𝜌𝜌𝑚𝑚𝑚𝑚𝑚𝑚 )
(8)
The term “exposure” originates from the field of epidemiology and is described as the situation when an agent, located at specified, is subject to potentially hazardous situation or substance. In our context, pedestrians are exposed to accident risk in a road segment involved by a flow of vehicle during crossing time. Thus a first formulation of pedestrians' exposure to accidents risk is defined as9,10, see also8 : 𝑇𝑇
𝐸𝐸(𝑡𝑡) = ∫0 𝑃𝑃 𝑞𝑞𝜈𝜈 𝑑𝑑𝑑𝑑 = 𝑞𝑞𝜈𝜈 ∙ 𝑇𝑇𝑃𝑃
(9)
Where 𝑞𝑞𝜈𝜈 and 𝑇𝑇𝑃𝑃 are respectively the vehicles flow and pedestrians crossing time. We assume that the pedestrian cross a road having a given width d in a rectilinear line, with a given speed 𝜈𝜈𝑝𝑝 as follow:𝑇𝑇𝑃𝑃 𝑑𝑑 ⁄𝜈𝜈𝑝𝑝 . Knowing that flow
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of an organism is the product of its density and speed, the pedestrians' exposure to accidents risk becomes 𝐸𝐸 = 𝑞𝑞𝜈𝜈 ∙ 𝑇𝑇𝑃𝑃 . Moreover, the vehicle flow can be expressed in terms of their densities and speeds, which they reach their maximum value if the density is null, and vanish in the case of maximum density. Therefore the pedestrians' exposure to accidents risk becomes 𝐸𝐸(𝑡𝑡) = 𝑇𝑇𝑃𝑃 ∙ 𝜌𝜌 ∙ 𝜈𝜈𝑚𝑚𝑚𝑚𝑚𝑚 (1 − 𝜌𝜌⁄𝜌𝜌𝑚𝑚𝑚𝑚𝑚𝑚 ) = 𝑇𝑇𝑃𝑃 ∙ 𝜌𝜌 ∙ 𝜈𝜈𝑚𝑚𝑚𝑚𝑚𝑚 − 𝑇𝑇𝑃𝑃 ∙ 𝜌𝜌2 𝜈𝜈𝑚𝑚𝑚𝑚𝑚𝑚 ⁄𝜌𝜌𝑚𝑚𝑚𝑚𝑚𝑚
(10)
̃′ = (𝑇𝑇̃𝑆𝑆 − 𝑇𝑇̃𝑃𝑃 ) ∙ 𝜌𝜌 ∙ 𝜈𝜈 = 𝑇𝑇̃𝑆𝑆 ∙ 𝑞𝑞 − 𝐸𝐸̃ 𝐸𝐸
(11)
𝜈𝜈 ̃ 𝐸𝐸 ′ = (𝑇𝑇𝑟𝑟 + 𝑚𝑚𝑚𝑚𝑚𝑚 (1 − 𝜌𝜌⁄𝜌𝜌𝑚𝑚𝑚𝑚𝑚𝑚 )) ∙ 𝑞𝑞 − 𝑇𝑇̃𝑃𝑃 ∙ 𝜌𝜌 ∙ 𝜈𝜈 = (𝛼𝛼̃ − 𝛽𝛽𝛽𝛽) ∙ 𝑞𝑞
(12)
In this paper, we modify the formulation of the pedestrians' exposure to accidents risk, given by the equation 6, and we consider that both safety stopping time for a vehicle and pedestrian crossing time are triangular fuzzy numbers because their knowledge are not deterministic and remain imprecise as to pedestrians than vehicles. When crossing the road, the pedestrian carries a time cross equal to𝑇𝑇𝑃𝑃 . The knowledge of both safety stopping time for a vehicle and pedestrian crossing time is not deterministic and remain imprecise as to pedestrians than vehicles. That’s why in this paper we choose to represent them by the triangular fuzzy numbers 𝑇𝑇̃𝑃𝑃 = 𝑡𝑡𝑡𝑡𝑡𝑡(𝑇𝑇𝑃𝑃 , 𝛼𝛼𝑙𝑙 , 𝛼𝛼𝑟𝑟 )and 𝑇𝑇̃𝑆𝑆 = 𝑡𝑡𝑡𝑡𝑡𝑡(𝑇𝑇𝑆𝑆 , 𝛽𝛽𝑙𝑙 , 𝛽𝛽𝑟𝑟 ) respectively as shown in the figure 1. Therefore the pedestrians' exposure to accidents risk becomes:
Moreover, by referring the safety stopping time for a vehicle given by equation 6 in the new formulation of pedestrians' exposure to accidents risk, we obtain: 2𝛾𝛾
Where: 𝛼𝛼̃ = 𝑇𝑇𝑟𝑟 − 𝑇𝑇̃𝑃𝑃 and 𝛽𝛽 = 1⁄2𝛾𝛾
The new formulation of pedestrians' exposure to accidents risk can be expressed by the following formula: ̃ ̃2 ∙ 𝜌𝜌2 + 𝑎𝑎 ̃1 ∙ 𝜌𝜌 𝐸𝐸 ′ = 𝑎𝑎3 ∙ 𝜌𝜌3 + 𝑎𝑎
(13)
2 2 ; 𝑎𝑎̃2 = 𝜈𝜈𝑚𝑚𝑚𝑚𝑚𝑚 (2𝛽𝛽𝜈𝜈𝑚𝑚𝑚𝑚𝑚𝑚 − 𝛼𝛼̃)⁄𝜌𝜌𝑚𝑚𝑚𝑚𝑚𝑚 ; and 𝑎𝑎 𝛽𝛽 ⁄𝜌𝜌𝑎𝑎𝑎𝑎𝑎𝑎 ̃1 = 𝜈𝜈𝑚𝑚𝑚𝑚𝑚𝑚 (𝛼𝛼̃ + 𝛽𝛽𝜈𝜈𝑚𝑚𝑚𝑚𝑚𝑚 ) Where:𝑎𝑎3 = − 𝜈𝜈𝑚𝑚𝑚𝑚𝑚𝑚 Consequently, the new and old accident risk indicators are related by to the following formula: 𝐸𝐸̃ ′ + 𝐸𝐸̃ = 𝑇𝑇𝑟𝑟 𝜈𝜈𝑃𝑃 − 𝛽𝛽𝜈𝜈 2 𝜌𝜌 At a crossing situation two cases can be discussed. The first is when 𝑇𝑇̃𝑃𝑃 > 𝑇𝑇̃𝑆𝑆 , which means that pedestrian has enough time to cross the road before that the vehicle stops. And there for the accident risk tends to zero. Thereby pedestrian cross the road if and only if 𝑃𝑃(𝑇𝑇̃𝑆𝑆 < 𝑇𝑇̃𝑃𝑃 ) ≫ 𝑃𝑃(𝑇𝑇̃𝑃𝑃 < 𝑇𝑇̃𝑆𝑆 ). The second is when 𝑇𝑇̃𝑃𝑃 < 𝑇𝑇̃𝑆𝑆 , and then pedestrian could not cross the road safely because he will be collided by the vehicle as the stopping time of this later is greater. Then the accident risk increase in this case. Consequently the vehicle moves if and only if 𝑃𝑃(𝑇𝑇̃𝑆𝑆 < 𝑇𝑇̃𝑃𝑃 ) ≪ 𝑃𝑃(𝑇𝑇̃𝑃𝑃 < 𝑇𝑇̃𝑆𝑆 ).
3. Intuitionistic fuzzy theory and risk exposure
Intuitionistic fuzzy set is very useful in providing a flexible model to elaborate uncertainty and vagueness involved in decision making. In fuzzy set theory introduced by Zadeh 13,14, the membership of an element to a fuzzy set is a single value in the interval [0,1]. Nevertheless, it may not always be true that the degree of non-membership of an element in a fuzzy set is equal to 1 minus the membership degree because there may be some indecision degree. Hence, a generalization of fuzzy sets was proposed by Atanassov1,2 .as intuitionistic fuzzy sets which incorporate the degree of uncertainty or hesitation called indecision margin.
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Definition An intuitionistic fuzzy set (IFS) A in E is defined as an object of the following form: 𝐴𝐴 = {〈𝑥𝑥, 𝜇𝜇𝐴𝐴 (𝑥𝑥), 𝜗𝜗𝐴𝐴 (𝑥𝑥)〉|𝑥𝑥 ∈ 𝐸𝐸}
(14)
Where the functions: 𝛍𝛍𝐀𝐀 : 𝐄𝐄 → [𝟎𝟎, 𝟏𝟏] And 𝛝𝛝𝐀𝐀 : 𝐄𝐄 → [𝟎𝟎, 𝟏𝟏]
Define the degree of membership and the degree of non-membership of the element𝒙𝒙 ∈ 𝑬𝑬, respectively, and for every𝒙𝒙 ∈ 𝑬𝑬: 𝟎𝟎 ≤ 𝛍𝛍𝐀𝐀 (𝐱𝐱) + 𝛝𝛝𝐀𝐀 (𝐱𝐱) ≤ 𝟏𝟏 Obviously, each ordinary fuzzy set may be written as: {〈𝐱𝐱, 𝛍𝛍𝐀𝐀 (𝐱𝐱), 𝟏𝟏 − 𝛍𝛍𝐀𝐀 (𝐱𝐱)〉|𝐱𝐱 ∈ 𝐄𝐄}
Definition The value of 𝛑𝛑(𝐀𝐀) = 𝟏𝟏 − 𝛍𝛍𝐀𝐀 (𝐱𝐱) − 𝛝𝛝𝐀𝐀 (𝐱𝐱)is called the degree of non-determinacy (or uncertainty) of the element 𝒙𝒙 ∈ 𝑬𝑬 to the intuitionistic fuzzy set A.
In this work, we will need intuitionistic fuzzy numbers. The concept of the Triangular Intuitionistic Fuzzy Number (TIFN) is a generalization of that of the triangular fuzzy number. Definition A TIFN 𝑢𝑢̃ = (𝑢𝑢, 𝑢𝑢, 𝑢𝑢, 𝛼𝛼𝑢𝑢̃ , 𝛽𝛽𝑢𝑢̃ ) is a special IF set on a real number set R, whose membership function and non-membership function are given in the following: (𝑢𝑢 − 𝑥𝑥) + (𝑥𝑥 − 𝑢𝑢) × 𝛽𝛽𝑢𝑢̃ (𝑥𝑥 − 𝑢𝑢) × 𝛼𝛼𝑢𝑢̃ 𝑖𝑖𝑖𝑖𝑢𝑢 ≤ 𝑥𝑥 < 𝑢𝑢 𝑖𝑖𝑖𝑖𝑢𝑢 ≤ 𝑥𝑥 < 𝑢𝑢 (𝑢𝑢 − 𝑢𝑢) (𝑢𝑢 − 𝑢𝑢) 𝛼𝛼𝑢𝑢̃ 𝑖𝑖𝑖𝑖 𝑥𝑥 = 𝑢𝑢 𝛽𝛽𝑢𝑢̃ 𝑖𝑖𝑖𝑖 𝑥𝑥 = 𝑢𝑢 𝜗𝜗𝑢𝑢̃ (𝑥𝑥) = 𝜇𝜇𝑢𝑢̃ (𝑥𝑥) = (𝑢𝑢 − 𝑥𝑥) × 𝛼𝛼𝑢𝑢̃ (𝑥𝑥 − 𝑢𝑢) + (𝑢𝑢 − 𝑥𝑥) × 𝛽𝛽𝑢𝑢̃ 𝑖𝑖𝑖𝑖𝑖𝑖 < 𝑥𝑥 ≤ 𝑢𝑢 𝑖𝑖𝑖𝑖𝑖𝑖 < 𝑥𝑥 ≤ 𝑢𝑢 (𝑢𝑢 − 𝑢𝑢) (𝑢𝑢 − 𝑢𝑢) 0 𝑖𝑖𝑖𝑖 𝑢𝑢 < 𝑥𝑥 𝑜𝑜𝑜𝑜 𝑥𝑥 < 𝑢𝑢 1 𝑖𝑖𝑖𝑖 𝑢𝑢 < 𝑥𝑥 𝑜𝑜𝑜𝑜 𝑥𝑥 < 𝑢𝑢 { { The values 𝛼𝛼𝑢𝑢̃ and 𝛽𝛽𝑢𝑢̃ represent the maximum degree of membership and the minimum degree of non-membership, respectively, such that they satisfy the conditions 0 ≤ 𝛼𝛼𝑢𝑢̃ ≤ 1 , 0 ≤ 𝛽𝛽𝑢𝑢̃ ≤ 1 and 0 ≤ 𝛼𝛼𝑢𝑢̃ ≤ 1 0 ≤ (𝛼𝛼𝑢𝑢̃ + 𝛽𝛽𝑢𝑢̃ ) ≤ 1. TIFN (𝑢𝑢, 𝑢𝑢, 𝑢𝑢, 𝛼𝛼𝑢𝑢̃ , 𝛽𝛽𝑢𝑢̃ ) may express an expression quantity like “approximate u'', which is approximately equal to u (see Fig. 2). The additional parameters 𝛼𝛼𝑢𝑢̃ and 𝛽𝛽𝑢𝑢̃ are introduced to reflect the confidence level and non-confidence level of the TIFN. 𝑢𝑢̃ = (𝑢𝑢, 𝑢𝑢, 𝑢𝑢, 𝛼𝛼𝑢𝑢̃ , 𝛽𝛽𝑢𝑢̃ ) respectively. The TIFNs may express more Fig. 1. A TIFN (𝑢𝑢, 𝑢𝑢, 𝑢𝑢, 𝛼𝛼𝑢𝑢̃ , 𝛽𝛽𝑢𝑢̃ ) uncertainty than the triangular fuzzy numbers (see figure 1). Let ũi = (ui , ui , ui , αũi , βũi ) (I = 1.., N) be TIFNs. A ratio ranking process for ranking the ITFNs ũi (I = 1.., N) is described below. After simplification of the ranking process given by Deng e al.6. A ratio of the value index to the ambiguity index for a TIFN is defined as follows: [ui + 4 × ui + ui ] [(λ − 1) × αũi + λ × (1 − βũi )] R(ũi ) = 2 × [3 + (ui − ui ) × [(1 − λ) × (1 − βũi ) + λ × αũi ]]
Where λ ∈ [0,1]
3.1. Intuitionistic risk exposure model Hereinafter we give specifications of an intuitionistic fuzzy set theory to capture the processing of pedestrians' risk exposure (figure 2).
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Fig. 2. Intuitionistic decision and indecision.
The fundamental idea is to define an actor's decision-making by two schemas. For a given actor, the first scheme assumes that his antagonist actor makes the right decision that remains consistent with his knowledge to take the right or wrong decision. In the second scheme, one of the actors assumes that his antagonist actor takes the wrong decision that remains consistent with his knowledge to take the right or wrong decision. Precisely this hypothetical indecision can be modeled by the theory of fuzzy intuitive numbers. Let (𝜎𝜎𝑠𝑠 , 𝜎𝜎𝑝𝑝 ) two parameters corresponding to standard deviation of 𝑇𝑇𝑠𝑠 the safety stopping time for a vehicle, and 𝑇𝑇𝑝𝑝 pedestrians crossing time, respectively. We define the following TIFN.
Let ℛ d = (μdS (x), ϑdS (x)) = (Tp − σd , Tp , Tp + σd , αd , βd ) be a triangular intuitionistic fuzzy number that denote an incorrect assessment of the risk from the driver point of view. Modeling the appreciation of the crossing time of the pedestrian from the driver's point of view. p p Let ℛ p = (μS (x), ϑS (x)) = (Ts − σp , Ts , Ts + σp , αp , βp ) be a triangular intuitionistic fuzzy number with an incorrect assessment of the risk from the pedestrian point of view. Modeling the assessment of the safety stop time from the pedestrian point of view. μdS (x) Denotes the degree of indecision available to the vehicle driver to appreciate properly the pedestrian crossing time. ϑdS (x) Denotes the degree of uncertainty that the driver has to not properly appreciate the crossing time of pedestrian. p μS (x) Denotes the degree of indecision that the pedestrian has to properly estimate the time of vehicle safety shutdown. p ϑS (x) Denotes the degree of uncertainty that the pedestrian has to not properly appreciate the safety stopping time of vehicle. TAB. 1 – Hazard’s premises according to intuitionistic fuzzy set theory.
Pedestrian decision 𝑝𝑝 𝜇𝜇𝑆𝑆 (𝑥𝑥) 𝑝𝑝 𝜗𝜗𝑆𝑆 (𝑥𝑥)
Driver decision 𝜇𝜇𝑆𝑆𝑑𝑑 (𝑥𝑥) 𝜗𝜗𝑆𝑆𝑑𝑑 (𝑥𝑥) + 𝜇𝜇 𝜇𝜇𝜇𝜇 +++ 𝜇𝜇𝜇𝜇 ++ 𝜈𝜈 ++++
With these considerations, we obtain the risk scenarios given in Table 1. One of the requirements highlighted in the table above endorse accident occurrence clauses or dangerous pedestrian-vehicle conflicts. Other situations are plausible and will be considered in our future work. The risk levels of risk are ranked as follows: 𝜇𝜇 + < 𝜇𝜇𝜇𝜇 ++ < 𝜇𝜇𝜇𝜇 +++ < 𝜈𝜈 ++++ .
When 𝑇𝑇̃𝑃𝑃 < 𝑇𝑇̃𝑆𝑆 then pedestrians could not cross the road safely because they will be collided by the vehicle. Therefore,
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we have to compute Ϝ =
Meriem MANDAR et al. Computer / Procedia Science Computer 109C (2017) 148–155 Mandar et al./ Procedia 00 Science (2017) 000–000 R(𝑇𝑇̃𝑃𝑃 ) . R(𝑇𝑇̃𝑆𝑆 )
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When Ϝ ≪ 1 the risk of accidents increases considerably, and consequently it is the
fuzzy intuitionistic indicator of the risk that we will adopt. Certainly, the idea is interesting in that it can model uncertain hypothetical decision making for drivers and pedestrians. Obviously, other psychological factors must be brought into our approach. In all cases, the approach is strong by its simplicity and by its ability to integrate both abductive reasoning and uncertainty, thanks to intuitionistic fuzzy sets theory. 4. Conclusion
In this work we have aligned the theory of intuitionistic fuzzy on pedestrian risk modeling goals. We have proposed new indicators to model pedestrian exposure to accidents. The proposed approach allows addressing behavioral psychology of pedestrians with intuitive methods based on fuzzy set theory. A software system is currently being developed for this purpose and reuses simulation models of pedestrian developed in our previous work. Future developments of this work will consider the multi-attribute preference problem and intuitionistic fuzzy numbers' ranking process5,7,11,12. The use of intuitionistic fuzzy theory for modeling risk for pedestrians is completely new. The first work referring to this theory and giving some foundation and first development was proposed by Boulmakoul et al10. In this work we have made extensions by the use of the ranking of triangular intuitionistic fuzzy numbers. The proposed approach does not pretend to cover the risk problem in all its dimensions. However, intuitionist theory can largely manage uncertainty and hesitation for drivers and pedestrians sharing conflictual space-time. Evaluations of this model are planned and will be carried out first by the simulation and subsequently be validated in the field. References 1. Atanassov, K., T. (1986) Intuitionistic fuzzy sets, Fuzzy Sets and Systems 20 (1986) 87-96. 2. Atanassov, K., T. (1999) On Intuitionistic fuzzy sets, Theory and Applications, ISBN 978-3-7908-2463-6, (Studies in fuzziness and soft computing; Vol. 35), Springer-Verlag Berlin Heidelberg GmbH 1999. 3. Boulmakoul, A., Mandar, M. (2011) Fuzzy ant colony paradigm for virtual pedestrian simulation. The Open Operational Research Journal, DOI:10.2174/1874243201105010019. ISSN: 18742432, 2011, 19-29. 4. Cohen S.,(1993) Ingénierie du trafic routier, 01/02/1993, 246 pages, Presses des ponts, EAN13 : 9782859781545 5. Cuiping Wei, Na Zhao, Xijin Tang. (2014) Operators and Comparisons of Hesitant Fuzzy Linguistic Term Sets. IEEE Transactions on Fuzzy Systems 22:3, 575-585. 6. Deng-Feng Li (2010) A ratio ranking method of triangular intuitionistic fuzzy numbers and its application to MADM problems, Computers and Mathematics with Applications 60(2010)1557-1570, Elsevier. 7. Jian Wu. (2016) Consistency in MCGDM Problems with Intuitionistic Fuzzy Preference Relations Based on an Exponential Score Function. Group Decision and Negotiation 25:2, 399-420. Online publication date: 1-Mar-2016. 8. Karim L., Boulmakoul A. Mandar M. (2016) Un système intelligent temps réel basé sur les nombres flous gaussiens pour la génération des messages d’alerte concernant la dangerosité de la conduite sur route et pour l’analyse du comportement des conducteurs vis-à-vis de la signalisation routière. Brevet OMPIC 2016. 9. Mandar, M., Boulmakoul, A. (2014) Virtual pedestrians’ risk modelling, International Journal of Civil Engineering and Technology. 10/2014; 5(10):32-42. 10. Mandar, M., Boulmakoul, A. (2016) Pedestrians' fuzzy intuitionistic risk exposure model: foundation and first development, in Workshop International sur l'Innovation et Nouvelles Tendances dans les Systèmes d'Information INTIS 2016, 25-26 Novembre 2016 - Fès, Maroc. ISBN 978-9954-34-378-4, ISSN 2351-9215, At Fes, Morocco. 11. Xia Liang, Cuiping Wei. (2014) An Atanassov’s intuitionistic fuzzy multi-attribute group decision making method based on entropy and similarity measure. International Journal of Machine Learning and Cybernetics 5:3, 435-444. 12. Yujia Liu, Jian Wu, Changyong Liang. (2015) Attitudinal ranking and correlated aggregating methods for multiple attribute group decision making with triangular intuitionistic fuzzy Choquet integral. Kybernetes 44:10, 1437-1454. 13. Zadeh, L. (1975) Information Sciences 8:199-249, 1975. 14. Zadeh, L. A. (1964) Fuzzy Sets, Memo. ERL, No. 64–44. Univ. of California, Berkeley. 1964.
