Pedestrian route-choice and activity scheduling theory and models

More documents

Recommendations

Info

178 S.P. Hoogendoorn, P.H.L. Bovy / Transportation Research Part B 38 (2004) 169–190 We can argue that, depending on trip-purpose, age, gender, etc. pedestrians may be inclined to walk in areas where there are at least some pedestrians. In this case, small density values may have an attracting effect on pedestrians route choice behavior, yielding that fðkÞ < 0 for some k < k cr . 5.4.5. Stimulation of the environment Finally, we consider the stimulating effects of the environment. This can be described relatively easy by considering the benefit cðt; xÞ 6 0 of walking at a certain location x at instant t. Note that the benefit has a negative sign (negative cost). We have L 5 ðt; x; vÞ ¼cðt; xÞ In substituting the contributions (9)–(15) into the running cost (8), we get X Lðt; x; vÞ ¼c 1 þ c 2 a m e dðx;O mÞ=b m þ 1 2 c 3v 0 v þ c 4 fðt; xÞþc 5 cðt; xÞ m ð15Þ ð16Þ Note that generally the weights c k as well as the weights a m cannot be determined uniquely from data, since only the ratio between the weights are determinant for pedestrian behavior. Without loss of generality, we can set c 1 ¼ 1. The approach is easily adapted when considering a heterogeneous population with taste-variation and differences in the physical abilities. In fact, these differences are the main reasons for including the different cost factors: empirical research has indicated large differences in how different types of pedestrians value route attributes (Bovy and Stern, 1990; Hill, 1982; Senevarante and Morall, 1986; Guy, 1987). The pedestrian population is stratified into homogeneous groups, and route-choice and activity scheduling is solved for each group. The characteristics of each homogeneous group is characterized by the utilities U ij gained when performing activity i at A ij , the weights c k , and the maximum walking speed v 0 ðt; xÞ. For instance, when travel time is the dominant factor (e.g. for commuting pedestrians), c 1 will be relatively large compared to c 2 , c 4 and c 5 ; when shopping, stimulation of the environment will be a relatively important attribute, which is expressed by c 5 . Furthermore, the weights c k also depend on the situation: in case of an emergency, travel time (or distance) will be the dominantly important attribute. In the latter case however, the applicability of utility optimization approaches may be limited, depending on the evacuation situation. 5.5. Route choice and activity area choice in continuous space Let us now formulate the combined route-choice and activity area (destination) choice problem. Eq. (6) shows that the expected (predicted) subjective cost C is a function of initial instant t, location xðtÞ ¼^x, and velocity path v ½t;Ti Þ. 5.5.1. Modeling principle and problem formulation The subjective utility optimization paradigm implies that the pedestrian will choose the velocities v ½t;Ti Þ yielding the predicted route and used activity area that minimize the subjective expected cost (6) subject to the internal prediction model (4). The subjective utility optimization paradigm thus that the pedestrian determines the optimal velocity v ½t;T i Þ satisfying
S.P. Hoogendoorn, P.H.L. Bovy / Transportation Research Part B 38 (2004) 169–190 179 Z Ti v ½t;T i Þ ¼ arg minC i t; ^x; v ½t;Ti Þ; fA ij g ¼ arg minE Lðs; xðsÞ; vðsÞÞds þ /ðT i ; xðT i ÞÞ ð17Þ where L and / are given by Eqs. (8) and (7) respectively. To solve the path choice problem, let us define the so-called expected minimum perceived disutility function W ðt; ^xÞ (often referred to as the value function in optimal control theory) by the expected value of the costs upon applying the optimal velocity v ½t;T i Þ Z Ti W ðt; ^xÞ :¼ E Lðs; x ðsÞ; v ðsÞÞ þ /ðT i ; x ðT i ÞÞ ð18Þ subject to t dx ¼ v dt þ rðx ; v Þdw subject to x ðtÞ ¼^x To derive the dynamic programming equation, consider the period [t; t þ h). According to BellmanÕs optimization principle (Bellman, 1957), we have Z tþh W ðt; ^xÞ ¼E Lðs; x ðsÞ; v ðsÞÞ þ W ðt þ h; x ðt þ hÞÞ ð20Þ t Eq. (20) describes that the expected minimal cost of walking from ðt; ^xÞ to A ij equals the minimal expected cost of both walking from ðt; ^xÞ to ðt þ h; x ðt þ hÞÞ and walking from ðt þ h; x ðt þ hÞÞ to A ij . For small h, the following approximation is valid Z tþh E Lðs; xðsÞ; vðsÞÞ ¼ Lðt; xðtÞ; vðtÞÞh þ Oðh 2 Þ ð21Þ t The random variate xðt þ hÞ describing the predicted location at instant t þ h subject to Eq. (4) can be expanded using a Taylor series p xðt þ hÞ ¼^x þ hvðtÞþr ffiffiffi h w þ Oðh 3=2 Þ ð22Þ where rh 1=2 w is a Nð0; hrr 0 Þ distributed random variate. We can rewrite the expected value of the second term of the right-hand-side of Eq. (20) E½W ðt þ h; xðt þ hÞÞŠ ¼ W ðt þ h; ^x þ hvÞþ h X H ij ðx; vÞ o2 W ðt; ^xÞ þ Oðh 3=2 Þ ð23Þ 2 ox i ox j where Hðx; vÞ :¼ rðx; vÞr 0 ðx; vÞ. Substitution of Eqs. (21) and (23) into Eq. (20), using the appropriate Taylor series expansions, and taking the limit h ! 0 yields the so-called Hamilton– Jacobi–Bellman (HJB) or dynamic programming equation for decision making in continuous time and space under uncertainty o W ðt; xÞ ¼Hðt; x; rW ; DW Þ ot ð24Þ with terminal conditions W ðt 1 ; xÞ ¼/ i ð25Þ ij t ð19Þ
Page 1 and 2: Transportation Research Part B 38 (
Page 3 and 4: S.P. Hoogendoorn, P.H.L. Bovy / Tra
Page 9: Substitution of running cost (9) yi
Page 19 and 20: 6.2. Activity order choice S.P. Hoo

Pedestrian route-choice and activity scheduling theory and models

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?