روش اکتشافی برای مشکل قیمت گذاری و مسیریابی موجودی در زنجیره تامین

The inventory routing problem (IRP) in a supply chain (SC) is to determine delivery routes from suppliers to some geographically dispersed retailers and inventory policy for retailers. In the past, the pricing and demand decisions seem ignored and assumed known in most IRP researches. Since the pricing decision affects the demand decision and then both inventory and routing decisions, it should be considered in the IRP simultaneously to achieve the objective of maximal profit in the supply chain. In this paper, a mathematical model for the inventory routing and pricing problem (IRPP) is proposed. Since the solution for this model is an NP (non-polynomial) problem, a heuristic method, tabu search adopting different neighborhood search approaches, is used to obtain the optimal solution. The proposed heuristic method was compared with two other methods considering the IRPP separately. The experimental results indicate that the proposed method is better than the two other methods in terms of average profit.

The inventory routing problem (IRP) in a supply chain (SC) is to determine delivery routes from suppliers to some geographically dispersed retailers and inventory policy for retailers. It is consisted of two sub-problems: inventory problem for retailers and vehicle routing problem (VRP) for suppliers. The IRP considering inventory and routing simultaneously has gained attentions since the coordination of the inventory and routing decisions between the supplier and retailers leads to a better overall performance (Vidal & Goetschalckx, 1997). According to the literature (Raa and Aghezzaf, 2009 and Zhao et al., 2007), the pricing and demand decisions seem ignored and assumed known in most IRP researches. Since the pricing decision affects the demand decision and then both inventory and routing decisions, it should be made in the IRP simultaneously to achieve the objective of maximal profit in the supply chain. For example, higher pricing causes lower demand and then lower order quantity and lower inventory. In contrast, lower pricing causes higher demand and then higher order quantity and higher inventory. Since the pricing decision is interrelated to inventory routing decisions, the profit may decrease when they are made separately. Hence, how to determine inventory, routing and price simultaneously becomes an important issue in supply chain management. Because the inventory routing and pricing problem (IRPP) is a NP-hard problem (Since inventory routing decisions is a NP-hard problem (Lenstra & Rinnooy, 1981), the IRPP is more complex than the IRP.), a heuristic method is adopted to resolve this problem. Until now, there are few researches about IRPP. Hence, this paper presented a survey for two related areas: inventory routing problem and pricing problem, in the following. Bell, Dalberto, and Fisher (1983) adopted an optimization method to resolve the IRP. After that, some other optimization methods were developed to resolve the IRP (Anily and Federgruen, 2004, Dror and Ball, 1987, Gallego and Simchi-Levi, 1990, Kleywegt et al., 2002, Qu et al., 1999 and Yu et al., 2008). Since the IRP is an NP-hard problem, heuristic methods are needed. Federgruen and Zipkin (1984) developed a nonlinear integer programming model and adopted an exchange method to resolve the IRP. Golden, Assad, and Dahl (1984) adopted an insertion method to resolve the IRP. Viswanathan and Mathur (1997) adopted a stationary nested joint replenishment policy heuristic (SNJRP) to resolve the IRP. The results show the method simultaneously making inventory and routing decisions is better than that making inventory and routing decisions separately. Campbell and Savelsbergh (2004) adopted a two-phase method to resolve the IRP. The first phase adopted an integer programming method to obtain the initial solution. The second phase adopted an insertion method to improve the initial solution. Gaur and Fisher (2004) adopted a randomized sequential matching algorithm (RSMA) to resolve the IRP. An insertion method was adopted to obtain the initial solution. Then a cross-over method was adopted to improve the initial solution. Sindhuchao, Romeijn, Akcali, and Boondiskulchok (2005) adopted a two-phase method for the IRP. The first phase adopted a column generation method to obtain the initial solution. The second phase adopted a very large-scale neighborhood search (VLSN) to improve the initial solution. Lee, Jung, and Lee (2006) adopted a tabu search method to resolve the IRP. Raa and Aghezzaf (2008) adopted a heuristic method to resolve the IRP. A column generation method was adopted to find the initial solution. Then a saving heuristic method was adopted to improve the initial solution. Zhao et al. (2007) adopted a heuristic method to resolve the IRP. The initial solution was generated randomly. Then a tabu search method adopting the GENI neighborhood search was used to improve the initial solution. Zhao, Chen, and Zang (2008) adopted a variable large neighborhood search (VLNS) method to resolve the three-echelon (suppliers, distributors, retailers) IRP. The results show the proposed method is better than the tabu search method. In summary, tabu search (TS) adopting the GENI neighborhood search approach and VLNS have been adopted to find the optimal solution for the inventory routing problem effectively and efficiently (Gaur and Fisher, 2004, Lee et al., 2006, Zhao et al., 2007 and Zhao et al., 2008). Hence, they will be adopted to resolve the IRP sub-problem in IRPP in this paper. As for the pricing problem, some researchers (Jung and Klein, 2006, Kotler, 1971, Lau and Lau, 2003 and Ray et al., 2005) determined the prices and demands using calculus according to the known demand function based on the maximal profit criterion. Nachiappan and Jawahar (2007) adopted a genetic algorithm (GA) method to find the prices and demands based on the maximal profit criterion in a supply chain. The pricing problem is a nonlinear integer programming (NIP) problem. Searching for the optimal solution is an NP problem. According to the literature (Costa and Oliveria, 2001, Exler et al., 2008, Schlüter et al., 2009 and Yin and Wang, 2008), genetic algorithm (GA), particle swarm optimization (PSO), ant colony optimization (ACO) and tabu search (TS) have been adopted to resolve the NIP problem. Since tabu search is adopted to resolve the IRP sub-problem in IRPP mentioned above, if GA, PSO or ACO is adopted to resolve the pricing sub-problem in IRPP, the IRPP would be resolved separately by different methods. Hence, tabu search is adopted to resolve the IRPP simultaneously in this paper.

In this paper, we have developed an effective heuristic method for the inventory routing and pricing problem. The proposed heuristic method making the inventory, routing and pricing decisions simultaneously is better than two other heuristic methods making the inventory, routing and pricing decisions separately based on the maximal average profit criterion. In addition, when the slope decreases, the vehicle capacity increases, or the supplier capacity increases, the average profits for all heuristic methods increase. When the slope, vehicle capacity or supplier capacity increases, the average CPU time of the proposed heuristic method decreases. Due to the limitations of this paper, some factors such as multiple products, different vehicle fleet, etc. are not considered. So considering these factors would help the inventory routing and pricing decisions made more realistically.