The mutually consistent calculations of signal tim ings and equilibrium flows for the bi-level problem (6.0) can be perform ed in the follow ing steps.
Step 0 . Set k = 0 for given signal timings , find the corresponding equilibrium
flows by means of the convex com bination m ethod given in Steps 6.0- 6.3.
Step 1. Run TRANS YT program m e to obtain the TRA N S YT- optimal signal timings
^(^+1) for the flows •
Steji2_. Update the travel time function (6.1a) of all links to obtain
c(9 ,
Step 3 . Calculate the corresponding equilibrium flows by means of the
convex com bination method given in Steps 6.0- 6.3.
Step 4 . Run TRANS YT program m e again to obtain the optimal signal timings
given by the equilibrium flows ^*(\}/*^^''''^) .
Step 5 . Compare the values of and , if there is no change between
and then go to Step 6\ otherw ise, increase k by 1 and go to Step 2.
Step 6 . S to p :. and q are the m utually consistent signal tim ings and
equilibrium flows.
6.3 Mixed Search Procedure
The proposed solution m ethod for the bi-level problem in (6.0) has been given in Chapter 5. For any one given solution, the proposed solution method can be described as a m ixed search procedure in terms of the following three type of steps:
Type 1. Search for the optimal signal setting variables with respect to Ç , 0 , (j) over the w hole feasible region such that the value o f the perform ance index is the m inim al one and the num ber of the search dim ension is the total num ber of signal setting variables. The descent direction at each iteration is determ ined by means of the gradient projection method, along which a good step length is decided by the one dim ensional bisection method, both of which have been discussed in problem (5.25).
Type I I . Similarly as in type I but search for the optim al signal settings variables with
respect to 0 , (|) for given com m on cycle tim e by specifying Ç as , so that the
num ber of search dim ensions is the num ber o f signal setting variables 0 and (j) . The descent direction at each iteration is determ ined by means of the gradient projection m ethod, along which a good step length is decided by the one dim ensional bisection m ethod, both of which have been discussed in problem (5.26).
Type III. Search for the optimal signal setting variables with respect to 0 only for given common cycle time and the durations of green times for all signal groups by
specifying ^ , ({) as , (1)° respectively, and changing the starts of greens for all
signal groups at each junction by the same am ount specific to that junction, so that the number of search dim ensions is reduced to the num ber of signal-controlled junctions but the step length in the chosen direction is unconstrained. The search direction for the unconstrained optimisation problem (5.27) at each iteration is determ ined by the steepest descent direction, along which a good step length is decided by the global search throughout the whole length lying within the feasible region, both of which have been discussed in problem (5.27).
The purpose of the m ixed search procedure for the bi-level problem given in (6.0) is to find a good local optimum by adopting the TRA N SY T-like hill-clim bing technique. A step of Type I allows each elem ent of the signal setting variables to vary within the feasible constraint set, therefore a good local solution can be obtained. A step of Type II specifies the com mon cycle time as a fixed value while still allowing other signal setting variables to vary within the restricted feasible region by which less com putation effort will be incurred but a good local solution for that cycle time can be located. Furtherm ore, a step of Type III specifies the com m on cycle time and the durations of green times for all signal groups at each junction as fixed values, and makes equal and simultaneous changes in the starts of green times for all signal groups at any one junction which corresponds to making changes in the offset variables; therefore not only will less com putation effort be incurred but also the search in this restricted set of directions will be extended over the whole feasible region. Adopting an equal and sim ultaneous change in the starts of green tim es for all signal groups at any one junction can help us to reach a more favourable point of another part of the feasible region along the restricted set of directions because no practical constraint applies with respect to the offset variables. In sum m ary, the rationale for this mixed search procedure is to alternately apply the three type of search steps to finding the near-optim al in m any parts of the feasible region and locating them within the corresponding neighbourhoods.
Tw o variants of the mixed search procedure obtained by applying various sequences of the three types of step are stated as follows. Firstly m ethod a starts from any initial
solution with negative values of the total derivatives for the perform ance index with respect to the move size a and follows the sequence of search steps of type I-II, III, and I-II and so on until a good local optim um for the bi-level form ulation is found and located. Secondly, m ethod b adopts the optim al search technique used in TRA N SY T program m e which is carried out by means of the m ixed com bination of the small and large steps for the starts of green times. The m ethod b starts with the search step of type III by simultaneous and equal changes of the starts of greens at any one junction and therefore in searching many parts of the feasible region a good initial solution can be found and used as a new starting point for the subsequent search process carried out by making steps of type I and II of the m ixed search procedure; the sequence of search steps is type III, I-II, and type III, I-II and so on. Stopping criterion for perform ing the mixed search will be m et if a predeterm ined threshold is satisfied between successive iterations.