New Best Solutions for the Close-Open Mixed Vehicle Routing Problem by using an Improved Harmony Search Algorithm Tantikorn Pichpibul Ruengsak Kawtummachai Faculty of Business Administration Panyapiwat Institute of Management, Thailand {pichpibul@gmail.com, ruengsak@yahoo.com} Abstract This paper addresses the close-open mixed ve- hicle routing problem (COMVRP) that both close and open variants of the vehicle routing problem are simultaneously considered. The objective is to find the solution which minimizes the total cost of delivery. In this paper, we propose an improved harmony search algorithm (IHSA) composed of two main approaches. First, the probabilistic Clarke-Wright savings algorithm is applied to improve an initial COMVRP solution. Second, that solution is used to produce the best COMVRP solution by the music improvisation process. The computational results on well-known benchmark problems show that the IHSA is superior to the best existing algorithm. Moreover, new best solutions for 10 benchmark instances are also found. Keywords: Close-open mixed vehicle routing problem, Clarke-Wright savings algorithm, Harmony search algorithm 1 Introduction The close-open mixed vehicle routing problem (COMVRP) [1] is a new variant of the vehicle routing problems which is the combination be- tween the capacitated vehicle routing problem (CVRP) [2] and the open vehicle routing problem (OVRP) [3]. Both CVRP and OVRP are known as NP-hard [4] [5]. Therefore, COMVRP is also a NP-hard. Liu and Jiang [1] mentioned that COMVRP is more complex and even harder to be tackled than CVRP and OVRP. The COMVRP arises in the real-world applications. For example, the companies which have their own trucks de- liver their products to many customers with fluctuant demand. But in the situation that the total demand of customers is greater than the total capacity of own trucks, the decision makers may consider to deal with third party carriers. In COMVRP, their own trucks must return to their warehouse after completing the delivery which is one characteristic of CVRP, but the outsourcing trucks do not return to the warehouse and stop the delivery at the last customer which is one cha- racteristic of OVRP. The other characteristics of COMVRP can be described as the problem of determining a set of vehicle routes to serve a set of customers with known demand. Each route represents a sequence of customers that must be visited by a vehicle. The COMVRP involves a single warehouse, a homogeneous fleet of ve- hicles and a set of customers. The objective is to find the solution which minimizes the total cost of delivery while serving all customers’ demand. The example routes for COMVRP are shown in Figure 1. 2 Literature Review The vehicle routing problems have attracted many researchers since Dantzing and Ramser [2] proposed the problem in 1959. Many efficient heuristics were presented to solve CVRP and OVRP, such as a tabu search [6] [5], an ant co- lony system [7] [8], a particle swarm optimiza- tion [9] [10], and a genetic algorithm [11] [12]. Although CVRP and OVRP were tackled by many researchers, Liu and Jiang [1] is only one who tackled COMVRP by using a memetic al- gorithm. In the literature, we have found that a harmony search algorithm (HSA) proposed by Geem et al. [13] is one of recent algorithms which had successful applications to various optimization problems [14] [15]. However, there is no previously published research that presents HSA to solve the vehicle routing problems. This is our contribution to develop a new HSA which can efficiently solve the COMVRP in terms of solution quality.