Online Dial-A-Ride Problem with Time Windows: an exact algorithm using status vectors

Transcription

1 Online Dial-A-Ride Problem with Time Windows: an exact algorithm using status vectors A. Fabri 1 and P. Recht 2 1 Universität Dortmund a.fabri@wiso.uni-dortmund.de 2 Universität Dortmund p.recht@wiso.uni-dortmund.de Abstract. The Dial-A-Ride Problem (DARP) has often been used to organize transport of elderly and handicapped people, assuming that these people can book their transport in advance. But the DARP can also be used to organize usual passenger or goods transportation in real online scenarios with time window constraints. This paper presents an efficient exact algorithm with significantly reduced calculation times. 1 Introduction The problem consists in routing a vehicle that serves a set N of demands growing in time. The road network is represented by a weighted graph G = (V,E,d) whose vertices v V correspond to the possible pickup and delivery points and whose arcs e E represent shortest paths between these points. The cost of each arc is given by a function d: E R +, indicating the length of the shortest path. The planning horizon is denoted by the interval [0,T]. Demands for transportation pop up dynamically, without any information in advance about time of appearance, location or quantity of incoming demands. Each demand i N consists of a pickup point u i V with a pickup time window [a i,b i ] [0,T], a delivery point v i V with a delivery time window [c i,e i ] [0,T] and the quantity of goods to be transported. The vehicle is characterized by its starting point s where it is located at time 0, its end point z where it should be at time T and its capacity k. When at any time a new demand pops up, it must be decided immediately whether the demand is accepted or rejected. If it is accepted, a valid tour has to be found that serves all accepted demands meeting all time windows exactly. The vehicle may wait for service. The objective consists of two parts: first, as many demands as possible shall be served. Second, the traveled distance needed to serve all accepted demands shall be minimized. This paper presents a special graph structure to handle the DARP and

2 2 A. Fabri and P. Recht sketches the solution algorithm. The results show that calculation efforts and therefore calculation times are reduced significantly. 2 The Status Vector Graph The basic status vector graph G n = (V n,e n,d ) forms the basis for a dynamic programming solution sequencing the pickup and delivery points of all n accepted demands. Each accepted demand i N is characterized by its status j(i) {0,1,2}. Here, j(i) = 0 means demand i is still waiting for service, j(i) = 1 denotes demand i has been picked up, but not yet delivered and j(i) = 2 says demand i has been served completely. A status vector is a vector (j(i 1 ),j(i 2 ),...j(i n )) that represents the status of each demand i N with {i 1,i 2,...i n } = N. The set of all possible status vectors of all demands i N forms the set of vertices V n. Two vertices v,v V n are connected by an edge (v,v ) E n whenever vector v can be obtained from vector v by adding 1 to exactly one vector element of v. This change corresponds to a pickup or delivery of one demand, implying a vehicle movement between two nodes in G. The arc (v,v ) is therefore weighted with the corresponding costs (see [1]). Note that the single status vector does not contain any information about the vehicle s location. The source vertex is the status vector representing the actual status when the calculation is started, the sink vertices are (2,...,2). Figure 1 shows G 2 for both demands still waiting for pickup. Fig. 1. Basic Status Vector Graph for two Demands Calculating a tour now consists in finding a shortest path from the source to one of the sink vertices subject to time window and capacity constraints. Attention is attracted by the fact that many status vectors are identical. Calculation times could be significantly reduced, if these multiple status vectors could be merged to obtain a graph of reduced size, say red G n =

3 Online Dial-A-Ride Problem with Time Windows 3 ( red V n, red E n, red d ). Due to the cost function depending on the vehicles location, two identical vectors can only be merged if also the vehicle location is identical. Hence, one more piece of information has to be stored. See figure 2 Fig. 2. Graph red G 2 for both demands still waiting for pickup for an example of red G 2 with pickup places A,B and delivery places C,D for demand 1,2, both still waiting for service. Regarding the number of nodes of red G n, one must state that it is worth storing the location additionally: Lemma 1. For a set of n demands not yet served the following equations hold: ( ) n V n n! j (j + t)! = 1 + (n j)! 2 t (j t)!t! red V n = 1+ k/2 n k=1 l=0 j=0 t=1 n! (k l) 2n l! (k 2l)! (n k + l)! + k/2 k=n+1 l=k n n! (k l) l! (k 2l)! (n k + l)! The proof is technical and can be found in [3]. It essentially uses the special structure of red G n which allows a recursive calculation of the number of nodes. Table 1 shows the number of nodes of both graphs for n = 1..6 demands. It is obvious that the reduced graph may lead to significantly reduced calculation times considering the savings in the number of nodes. The following section treats the algorithm highlighting the implementation to realize this reduction. 3 The Algorithm To find a shortest path from the source to one of the sink vertices in red G n satisfying the time window and capacity constraints, the A*-algorithm is used.

4 4 A. Fabri and P. Recht n V n red V n reduction 0% 31,58% 79,70% 97,05% 99,75% 99,99% Table 1. Number of nodes in the basic / reduced status vector graph for up to 6 demands The A*-algorithm bases on the Dijkstra algorithm but expands the cost function by adding an estimated value for the remaining distance to the sink. The optimal solution is obtained whenever the distances satisfy the triangle inequality and the estimated distance to the sink does not overestimate the real value (see [5]). With a given estimate that calculates the maximum distance of all left demands assuming that only one of them was left to be served (see [2]) and the distances between two vertices consisting of the shortest paths, these assumptions are met. Due to the special structure of red G n, the graph does not have to be given in advance but is constructed iteratively and only as far as necessary during the algorithm. As shown in [2], this algorithm performs well on G n, especially with the presented additional feasibility checks. To adapt the algorithm to red G n, it is necessary to check quickly whether the status vector recently generated is really new or just a copy of another one. Therefore, an efficient search structure is crucial. 3.1 Efficient Search Structure During the A*-algorithm, the generated nodes of red G n have to be searched for two criteria: the A*-algorithm needs to find the node with minimal (partly estimated) cost that can be expanded next, while the graph structure itself leads to searching for a node copy. Therefore, all nodes are concatenated twice: in the order of their costs and in the order of their status vectors. To give a unique numerical order to the status vectors, each of them is assigned its status number, interpreting the vector elements as numerals in the numeral system modulo 3. For a quick search in both node chains a double skip list is constructed, as skip lists are nearly as efficient in search as a balanced tree but significantly better when inserting and deleting elements (see [6]). A skip list is an ordered linked list with randomly added additional links. Every list element gets a random number of link levels with a minimum of one and a maximum of 6 levels and growing probability for a lower number of levels. Every element is linked on each of its levels to the next element having that level. The source vertex has the maximum number of level. To search this ordered list for a special value, one starts at the source and follows the highest level links until the value was exceeded with the next step. Then the search is continued one level below and so on. To be able to search for both cost and status number,

5 Online Dial-A-Ride Problem with Time Windows 5 Fig. 3. Example of a status skip list for a graph under construction with 2 demands each level has two links: one to the node with the next best cost having the same level and one to the node with the next higher status number having the same level. Figure 3 shows a partly generated red G 2 with status numbers and locations and its representation as a skip list. For a clearer intuition, only the status links are shown. 4 Results and Conclusion To test the algorithm, the further modified Li and Lim test instances for dynamic pickup and delivery vehicle routing with time windows with about 100 places to visit are used (see [4]). These test instances are changed in the way that only one vehicle is used. The results are given as an average value of the lc, lr and lrc test cases representing concentrated, random or a mixture of concentrated and random located places to visit. Due to the capacity constraints, only the very first demands can be served. The algorithm running on the basic status vector graph is stopped as soon as 10 million nodes have been generated. Table 2 shows the average number of orders treated completely until then, the maximal and average calculation times to accept or reject one order and the number of nodes generated to treat these orders, divided in generated nodes over all, unfeasible nodes, multiple nodes (reduced to one) and expanded nodes. The algorithm running on the reduced status vector graph is stopped at the same number of orders to compare number of nodes and calculation times in seconds.

6 6 A. Fabri and P. Recht nr of graph order time (sec) nr of nodes orders average max. over all unfeasible multiple expanded lc 26,82 basic 18, , , , ,76 reduced 0,06 7, , , , ,53 lr 31,13 basic 39, , , , ,30 reduced 0,01 0, , , , ,83 lrc 33,50 basic 29, , , , ,19 reduced 0,01 0, , , , ,31 Table 2. Nr of nodes and calculation times for the test cases 100 Table 2 shows that the number of nodes and the calculation times could be significantly reduced. This significant reduction is due to the fact that many nodes have been recognized as multiple nodes and could be merged into only one node implying that only this one node is further expanded. To conclude, we can summarize that it is really worth relying on the more complicated data structure of the reduced graph due to the impressing reduction of calculation times. References 1. Caramia, M., Italiano, G. F., Oriolo, G., Pacifici, A. and Perugia, A. (2001) Routing a fleet of vehicles for dynamic combined pickup and delivery services. Proceedings of the Symposium on Operation Research 2001, pp 3-8. Berlin/Heidelberg: Springer-Verlag 2. Fabri, A., Recht, P. (2006) On dynamic pickup and delivery vehicle routing with several time windows and waiting times. Transportation Research Part B, 40, pp Elsevier 3. Fabri, A. (2006) Die Größe der Statusgraphen des DARP unter Ladebedingungen. Working Paper. Universität Dortmund, Fachgebiet Operations Research 4. Fabri, A. (2005) Benchmarkprobleme, P. E. Hart, N. J. Nilsson, B. Raphael: A Formal Basis for the Heuristic Determination of Minimum Cost Paths; IEEE transactions of systems science and cybernetics, Vol. 4, No.2, Pugh, William (1990) Skip lists: a probabilistic alternative to balanced trees. Commun. ACM, vol. 33 nr. 6ACM Press,pp New York: ACM Press