Optimizing a multi-vendor multi-retailer vendor managed inventory problem: Two tuned meta-heuristic algorithms Javad Sadeghi a , Seyed Mohsen Mousavi b , Seyed Taghi Akhavan Niaki c,⇑ , Saeid Sadeghi d a Young Researchers and Elite Club, Islamic Azad University, Qazvin Branch, Qazvin, Iran b Faculty of Industrial and Mechanical Engineering, Islamic Azad University, Qazvin Branch, Qazvin, Iran c Department of Industrial Engineering, Sharif University of Technology, Tehran, Iran d Faculty of Industrial Engineering, Islamic Azad University, South Tehran Branch, Tehran, Iran article info Article history: Received 27 August 2012 Received in revised form 5 June 2013 Accepted 13 June 2013 Available online 25 June 2013 Keywords: Multi-vendor Multi-retailer Vendor managed inventory model Meta-heuristic Taguchi method abstract The vendor-managed inventory (VMI) is a common policy in supply chain management (SCM) to reduce bullwhip effects. Although different applications of VMI have been proposed in the literature, the multi- vendor multi-retailer single-warehouse (MV-MR-SW) case has not been investigated yet. This paper develops a constrained MV-MR-SW supply chain, in which both the space and the annual number of orders of the central warehouse are limited. The goal is to find the order quantities along with the num- ber of shipments received by retailers and vendors such that the total inventory cost of the chain is min- imized. Since the problem is formulated into an integer nonlinear programming model, the meta- heuristic algorithm of particle swarm optimization (PSO) is presented to find an approximate optimum solution of the problem. In the proposed PSO algorithm, a genetic algorithm (GA) with an improved oper- ator, namely the boundary operator, is employed as a local searcher to turn it to a hybrid PSO. In addition, since no benchmark is available in the literature, the GA with the boundary operator is proposed as well to solve the problem and to verify the solution. After employing the Taguchi method to calibrate the parameters of both algorithms, their performances in solving some test problems are compared in terms of the solution quality. Ó 2013 Elsevier B.V. All rights reserved. 1. Introduction In today’s global markets, supply chain management (SCM) plays an important role to integrate different parts of organizations and companies in order to achieve better performances. SCM is a set of approaches used to efficiently integrate suppliers, manufac- tures, warehouses, and stores, so that merchandise is produced and distributed at the right quantities, to the right locations, and the right time, in order to minimize system-wide costs while satisfying service level requirements [44]. Inventory is one of the key factors in SCM. In a supply chain, undesirable or fluctuating inventory, referred to the bullwhip effect, may even make companies to go near bankruptcy [9,44]. Although there are several strategies in retailer–supplier partnerships such as quick response, continuous partnerships, and advanced continuous replenishments, the vendor-managed inventory (VMI) policy has shown to have the best performances in many companies [8,44]. VMI is a strategy based on which suppliers (or vendors) manage buyers’ (or retailers’) inventory level such that the chain total inventory cost is minimized. Information technology is required in VMI to share retailers’ sales and inventory information with vendors. The obtained infor- mation is used to schedule deliveries, program production, and determine the amount of the orders and inventory levels such that total inventory cost is minimized. The possible benefits of the VMI models include reduction of inventory costs for the supplier and the retailer and improvement of customers’ service levels [1]. Wal-Mart, Kmart, and JC-Penny are just a few examples of success- ful companies that took advantage of the VMI policy [43]. Although the classical economic order quantity (EOQ) and eco- nomic production quantity (EPQ) [2,48] due to their simple and easy to understand natures have been the most employed inven- tory model in VMI implementation, some researchers extended them in order to bring the models closer to reality based on real-world constraints. While there are several constraints in an inventory system, in this research, the number of orders and the warehouse capacity in a multi-vendor multi-retailer VMI problem are assumed limited. In other words, a supply chain problem consisting of several vendors and several retailers with a single constrained-capacity central warehouse is investigated in which the number of orders is limited. The objective is to determine the order quantity and number of shipments received by retailers 0950-7051/$ - see front matter Ó 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.knosys.2013.06.006 ⇑ Corresponding author. Tel.: +98 21 66165740; fax: +98 21 66022792. E-mail addresses: sadeqi@qiau.ac.ir (J. Sadeghi), mousavi.mohsen8@gmail.com (S.M. Mousavi), niaki@Sharif.edu (S.T.A. Niaki), saeed.sadeghi.85@gmail.com (S. Sadeghi). Knowledge-Based Systems 50 (2013) 159–170 Contents lists available at SciVerse ScienceDirect Knowledge-Based Systems journal homepage: www.elsevier.com/locate/knosys