Static Problems

DARP problem is an NP-hard problem and difficult to solve because of its paired constraints and precedence constraints. Psarafits (1983) proposed a dynamic

i

i k

MRT =

LPTi LDT_i

Time DRTi

DRTi

) (DDT_i

EPTi EDT_i

programming approach that could find an exact solution, but the problem size was limited to less than and equal to 9 customers with user-specified time windows on departure and arrival time. In practice, the customer sizes must be larger, it would be hundreds or even thousands requests combined with PDP pairs and also time window constraints, therefore, this problem cannot be solved for an exact solution in polynomial time (Mitrovic-Minic 1998). However, some papers used heuristic methods to find nearly exact solution in short time.

Heuristics for solving the Dial-a-ride problem have been studied for a number of decades. The examples of the heuristics method are cheapest insertion heuristic (Jaw et al. 1986; Madsen et al. 1995), new regret heuristic (Diana and Dessouky 2004) and tabu heuristic (Cordeau and Laporte 2003a; Mitrovic-Minic et al. 2004)

One of the mostly cited references in this area is due to Jaw et al. (1986). They proposed the Advanced Dial-A-Ride with Time Windows (ADARTW) algorithm for the Dial-A-Ride Problem (DARP) of advance-request, with multiple vehicles and service quality constrains. In their algorithm, the time windows for each customer i are determined with the customers specified desired pick-up time (DPT_i) or desired delivery time (DDT_i), by calculating the values of earliest pick-up time (EPT_i), latest pick-up time (LPT_i), earliest delivery time (EDT_i), latest delivery time (LDT_i). In their assumptions, the vehicle capacity was limited, and loading and unloading time were specified for picking-up and delivering customers and the vehicles were not allowed to idle when carrying customers. In the algorithm, for a number of customers, the sequence of insertion was labeled with the order of earliest pick-up times EPTi

(i=1,…,N). Starting with a single vehicle fleet size, each customer was inserted to the position which incurred the smallest cost among all feasible insertion positions, concerned the parameters of disutility to the system’s customers and operator costs. If a customer was infeasible to be assigned to any of the vehicle fleet, an additional vehicle would be introduced. The algorithm finally terminated when all customers were inserted to the vehicles.

Diana and Dessouky (2004) adopted and modified the time settings by Jaw et al.

(1986), and formulated their static DARP with time windows without capacity constraints. They developed a route initialization procedure which exclusively keeps

into account the spatial and temporal effects of the demand, and a parallel regret insertion heuristic to improve some degree of flexibility for further insertions. Instead of ranking the requests with a certain criteria, for example, earliest pickup time or latest delivery time as in classic insertion heuristics, the regret insertion builds up an incremental cost matrix for each of the unassigned requests assigning to each of the existing vehicle routes. A regret cost, which is a measure of the potential difficulty if a request is not immediately assigned, is calculated for each request, and the algorithm seeks for the one with the largest regret cost, and inserts it into the existing schedules.

The whole procedure is repeated until all requests are inserted.

Cordeau and Laporte (2003a) proposed a tabu search heuristic for the dial-a-ride problem. Customers specified their requests for origins and destinations, also their time window on the arrival time and departure time of their outbound trip and inbound trip respectively. The model aimed to design a set of vehicle routes, with a supplied fleet size, to satisfy all requests with least operating cost. The algorithm started from an initial feasible solution s₀, and when the best solution was found in a neighborhood N(s_t) at iteration t, the new solution was changed to s_t. Besides, the recent visited solutions were declared forbidden for a number of iterations so as to avoid cycling, unless they contributed a new incumbent. When the solution was being searched, the time window and vehicle capacity constraints were allowed to violate. If the current solution was feasible with the constraints, the cost function was re-calculated by dividing the cost parameters, otherwise by multiplying them. Finally, the best feasible solution can be reached by repeating the several iterations.

In the problem of not considering time window constraints, Tseng (1992) proposed three different insertion criteria and eight dispatching headways types, for a case study of Science Park Administration in Hsin-Chu, Taiwan. The three different insertions, namely minimum incremental time, minimum incremental distance and minimum incremental cost, were compared, and the minimum incremental time would be a better heuristic algorithm taking account of operating cost and level of service. In the division of 43 zones in the case study, eight dispatch types were presented, including dispatch a vehicle with certain time intervals, with certain accumulated requests and combined with both the decisions. The Simulation Language for Alternative Modeling (SLAM) was used to evaluate the result. Finally, the combined certain time intervals

and accumulated requests with minimum incremental time algorithm led a better service type in the simulation.

Cordeau and Laporte (2003b) surveyed over 30 publications and showed a review on the features and variants of the Dial-a-Ride problem. They summarized the important algorithms which have been published over the last thirty years in static and dynamic problems with single- or multi-vehicle type. They concluded that excellent heuristics were existed to solve static problems, but dynamic problems are rarely studied. Since DARP is focused on carrying people, the level of service is an important index for operation. Combined with the intelligent technologies in order to respond new requests in real-time, the operating service level will be enhanced.

Therefore, solving DDARP will be an important issue in practice.