Production, Manufacturing and Logistics The traveling purchaser problem with stochastic prices: Exact and approximate algorithms Seungmo Kang a , Yanfeng Ouyang b,⇑ a School of Civil, Environmental and Architectural Engineering, Korea University, Seoul 136-701, Republic of Korea b Department of Civil and Environmental Engineering, University of Illinois at Urbana-Champaign, Urbana, IL 61801, USA article info Article history: Received 13 February 2009 Accepted 8 September 2010 Available online 17 September 2010 Keywords: Traveling purchaser problem Stochastic price Dynamic programming Approximation Heuristic abstract The paper formulates an extension of the traveling purchaser problem where multiple types of commod- ities are sold at spatially distributed locations with stochastic prices (each following a known probability distribution). A purchaser’s goal is to find the optimal routing and purchasing strategies that minimize the expected total travel and purchasing costs needed to purchase one unit of each commodity. The pur- chaser reveals the actual commodity price at a seller upon arrival, and then either purchases the com- modity at the offered price, or rejects the price and visits a next seller. In this paper, we propose an exact solution algorithm based on dynamic programming, an iterative approximate algorithm that yields bounds for the minimum total expected cost, and a greedy heuristic for fast solutions to large-scale appli- cations. We analyze the characteristics of the problem and test the computational performance of the proposed algorithms. The numerical results show that the approximate and heuristic algorithms yield near-optimum strategies and very good estimates of the minimum total cost. Ó 2010 Elsevier B.V. All rights reserved. 1. Introduction A new challenge faced by the location-based service (LBS) industry is to provide a traveling purchaser with an open tour (from an origin to a destination) visiting a subset of points-of- interest (POIs), or commodity sellers, so as to minimize the sum of travel and commodity purchasing costs. Such problems fre- quently arise in our daily life as we try to consolidate multiple cost-incurring social-economic activities into a trip chain. For example, one commuter may ask the following question: ‘‘On my way home today, where shall I stop for gas, pick up groceries, and shop for some furniture, respectively?” We define a market to be the subset of sellers who offer a cer- tain type of commodity. If the purchaser wants to purchase multi- ple types of commodities, the tour should visit at least one seller from each market. Kang et al. (2006) suggest a heuristic method using pre-selection scheme with the Euclidean distance, and Kang and Kim (2009) suggested a dynamic programming (DP) approach with a market permutation scheme to solve this problem. When the commodity prices are fixed and known, the LBS prob- lem can be formulated as a traveling purchaser problem (TPP). TPP includes a set of sellers and a set of commodities. Each seller pro- vides one or more commodities and offers a certain price for each. Traveling between sellers involves costs, which is often computed along the shortest paths in the real transportation network. The problem is to construct a tour through a subset of the sellers so that all commodities are purchased while the sum of total travel and purchase costs is minimized. We consider a case that the commodity prices are not determin- istically known but follow known probability distributions. 1 The probability distributions are independent but not necessarily identi- cal, and can often be estimated from historical data. The purchaser needs to determine the optimal routing strategy based on these probability distributions. After arriving at a seller, the purchaser will be offered a price (a realization from the known distribution). The purchaser has an option whether to buy the commodity at the of- fered price, or reject the offer and visit another seller. We assume that the purchaser is not allowed to revisit any seller that has already been visited before. Without losing generality, the origin and desti- nation locations are assumed to be different. This paper proposes the traveling purchaser problem with sto- chastic prices (TPPSP) in the LBS application context, and suggests effective exact and heuristic solution algorithms. We propose (i) a dynamic programming algorithm that finds exact optimal solutions; (ii) an efficient iterative approximation algorithm that provides tight cost bounds; and (iii) a greedy heuristic that provides quick solutions to large-scale problems. We study the characteristics of the TPPSP with numerical examples and extract 0377-2217/$ - see front matter Ó 2010 Elsevier B.V. All rights reserved. doi:10.1016/j.ejor.2010.09.012 ⇑ Corresponding author. Tel.: +1 217 333 9858; fax: +1 217 333 1924. E-mail address: yfouyang@illinois.edu (Y. Ouyang). 1 This case is realistic when real-time price information for each commodity at each seller is difficult to obtain (or simply not available). European Journal of Operational Research 209 (2011) 265–272 Contents lists available at ScienceDirect European Journal of Operational Research journal homepage: www.elsevier.com/locate/ejor