16 Sep 04Transport Workshop at Queen's1 A numerical comparison of three heuristic methods for path reassignment for dynamic user equilibrium Ying-en Ge and Malachy Carey 16 September 2004 School of Management & Economics Queen’s University Belfast BT7 1NN

16 Sep 04Transport Workshop at Queen's2 Introduction Dynamic traffic assignment (DTA) 1.Network loading, with inflows/ assignment to spatial paths taken as given compute new path travel times 2.Spatial path reassignment (based on travel-times from 1) Three methods for path reassignment –Pair-wise swapping method –Wu et al. (1998) method –Lo & Szeto (2002) method

16 Sep 04Transport Workshop at Queen's3 Pair-wise swapping method Step 1 At iteration n, for each time interval i, note the path with current highest cost (travel time) and path with lowest cost [ or variants of this, e.g. choose the same paths for several time intervals, etc.] Step 2 For each time interval i, switch proportion s i n of inflow from higher cost to lower cost path s i n =  n where  n is a chosen parameter (1 >  n > 0)

16 Sep 04Transport Workshop at Queen's4 Wu et al. (1998) method VI formulation The solution of the VI formulation is obtained by solving a series of quadratic programs below (1) where  is a positive constant.

16 Sep 04Transport Workshop at Queen's5 Lo & Szeto (2002) method Step 1. Compute g ip n and v i n for all i and p by: g ip n = max{0, f ip n –  [  ip n – ( u ip n –  (  p f ip n – d i ))]} v i n = u i n -  (  p g ip n – d i ) Step 2. Compute f ip n and u i n for all i and p f ip n+1 = f ip n - t n  n (f ip n - g ip n ) u i n+1 = u i n - t n  n ( u i n - v i n ) where t n =  n (  -1 ),  n  (0,2) such that t n  (0,1) and  n = r 1 / r 2 with r 1 =  ip ( f ip n - g ip n ) 2 +  2  i ( g ip n – g i ) 2 and r 2 = r 1 +  i (  p f ip n – g ip n ) 2

16 Sep 04Transport Workshop at Queen's6 Numerical experiments Scenario Settings –2-link network –Network loading –Travel demand Convergence measure –Maximum absolute difference Numerical experiments –Effects of parameters in three methods –Convergence measure values over iterations, and –Accuracy of numerical solutions

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16 Sep 04Transport Workshop at Queen's9 Minimum values of convergence measure Convergence measure Maximum absolute difference Pair-wise swapping method (  n = 1) Wu et al. method (  = 2) Lo & Szeto method (  = 0.5 and t n =1.0)

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16 Sep 04Transport Workshop at Queen's11 Stopping iterations when given tolerances for maximum absolute difference are satisfied tolerance for maximum absolute difference (5%) (1%) (0.5%) Pair-wise swapping method (  n = 1)  Wu et al. method (  = 2) 290  Lo & Szeto method (  = 0.50 and t n = 1.00) Note: The percentages given in the round brackets after each tolerance represent the proportion of a tolerance to the free-flow travel time of the shorter of the two paths [1.25 minutes].

16 Sep 04Transport Workshop at Queen's12 Summary Preferred parameter values Not able to set an arbitrarily small tolerance Performance of three methods