Multiobjective heuristic search in road maps q E. Machuca ⇑ , L. Mandow ⇑⇑ Dpto. Lenguajes y Ciencias de la Computación, Universidad de Málaga, Boulevar Louis Pasteur, 35, Campus de Teatinos, 29071 Málaga, Spain article info Keywords: Multiobjective shortest path problem Best-first search Heuristic search Artificial intelligence Road networks abstract This article considers the application of exact multiobjective techniques to search in large size realistic road maps. In particular, the NAMOA / algorithm is successfully applied to several road networks from the DIMACS shortest path implementation challenge with two objectives. An efficient heuristic function previously proposed by Tung and Chew is evaluated. Heuristic values are precalculated with search. The precalculation effort is shown to pay off during the multiobjective search stage. An improvement to the calculation procedure is also proposed, resulting in added improved time performance in many problem instances. Ó 2011 Elsevier Ltd. All rights reserved. 1. Introduction Real decision problems frequently involve the consideration of multiple conflicting objectives. Solution to these problems nor- mally results in a set of so-called Pareto optimal or nondominated solutions. These define the optimal tradeoffs between the objec- tives under consideration. The multiobjective shortest path problem is an extension of the shortest path problem with several objectives. Multiobjective prob- lems are more complex computationally than their scalar counter- parts. In fact, a number of objections question the application of exact multiobjective heuristic search techniques to large size realis- tic problems. In the first place, the number of solutions to these prob- lems is known to grow exponentially with goal depth in the worst case, even for the two-objective case. The problem is therefore formally intractable (Hansen, 1979). In the second place, multiobjec- tive heuristic search has been regarded for some time as lacking the formal properties of its scalar counterparts. In particular, the work of Stewart and White presented MOA / , a multiobjective extension of the A / algorithm (Hart, Nilsson, & Raphael, 1968), but could not establish a significant relation between the accuracy of heuristic information and search efficiency in MOA / (Stewart & White, 1991). Finally, good heuristic functions are frequently hard to find even for a single objective. Particularly, precalculating heuristic estimates with search was formally shown to be more inefficient than blind search in certain cases (Hansson, Mayer, & Valtorta, 1992; Valtorta, 1984). However, several recent results promise to reduce the impact of the above objections in practical applications. In the first place, a new multiobjective heuristic search algorithm (NAMOA / ) has been proposed. This algorithm has been found to be optimal over the class of admissible multiobjective algorithms, and to strictly dom- inate MOA / (Mandow & Pérez de la Cruz, 2010). Furthermore, the efficiency of NAMOA / has been formally linked to heuristic accu- racy in the sense that more informed heuristics can never decrease efficiency (Mandow & Pérez de la Cruz, 2010). Rather, efficiency normally increases with better informed heuristics, just as in the case of single-objective A / . In the second place, several authors have pointed out that par- ticular classes of multiobjective search problems do not exhibit exponential worst-case behavior (Mandow & Pérez de la Cruz, 2009; Müller-Hannemann & Weihe, 2006). In particular, in polyno- mial state spaces with bounded integer costs and a fixed number of objectives, the number of nondominated solutions can be shown to grow only polynomially with goal depth in the worst case (Man- dow & Pérez de la Cruz, 2009). Finally, pattern database heuristics (Culberson & Schaeffer, 1998) and hierarchical graph search (Holte, Perez, Zimmer, & Mac- Donald, 1996; Holte, Grajkowski, & Tanner, 2005) have overcome previous formal limitations, and shown that under certain circum- stances heuristics precalculated with search can be used to boost search efficiency. Regarding multiobjective search, the work of Tung and Chew proposed a precalculated multiobjective heuristic (TC) that calls for solving one single-objective problem for each objective under consideration (Tung & Chew, 1992). However, this multiobjective heuristic has not received much attention and its impact in many potential practical applications still lacks adequate evaluation. Applications of multiobjective search in road maps range from off-line planning of routes for hazardous material (hazmat) trans- portation (Caramia, Giordani, & Iovanella, 2010; Dell’Olmo, Gentili, 0957-4174/$ - see front matter Ó 2011 Elsevier Ltd. All rights reserved. doi:10.1016/j.eswa.2011.12.022 q This work is partially funded by Este trabajo está parcialmente financiado por: Consejería de Innovación, Ciencia y Empresa. Junta de Andalucía (España), P07-TIC- 03018. ⇑ Main corresponding author. Tel.: +34 (9) 52 132863; fax: +34 (9) 52 131397. ⇑⇑ Corresponding author. E-mail addresses: machuca@lcc.uma.es (E. Machuca), lawrence@lcc.uma.es (L. Mandow). Expert Systems with Applications 39 (2012) 6435–6445 Contents lists available at SciVerse ScienceDirect Expert Systems with Applications journal homepage: www.elsevier.com/locate/eswa