Optimizing replenishment polices using Genetic Algorithm for single-warehouse multi-retailer system
Introduction
Supply chain management (SCM) ensures that the firm’s supply chain is running efficiently, cost-effectively and helps them to improve their competitive position in a highly turbulent marketplace. A well managed supply chain can lead to better resource utilization, profitability and also help firms to achieve sustainable competitive advantage. In contrast to a traditional supply chain where companies work independently, SCM involves close coordination and synchronization of information and material flows. Aided by the shared information on the demand, inventory, and supply between the parties in a supply chain, companies are increasingly using coordinated replenishment to manage their distribution systems, with the aim to reduce transportation and warehousing costs.
One of the well-studied inventory models is known as single-warehouse multi-retailer (SWMR) problem (Chan et al., 2002, Levi et al., 2005, Muckstadt and Roundy, 1987). In the general SWMR problem, retailers face known external demand of products over a finite planning horizon. Goods are shipped from the suppliers to the warehouse and distributed from the warehouse to the retailers. The goal of this SWMR problem is to find an optimal replenishment policy to minimize the total transportation time and inventory costs in the system. However, the SWMR problem becomes more complicated when third-party logistics providers transport goods from suppliers through warehouses to retailers. In this paper, we study SWMR problem with a delivery capacity and transportation cost structures that models all-unit discount cost functions. The transportation cost structure representing quantity discounts, volume-based price incentives, and other forms of economies of scale, has a major impact on the replenishment strategy. The objective is to find an optimal distribution strategy to minimize the total transportation, inventory and shortage costs over the finite horizon.
The SWMR model plays a fundamental role in broad planning issues and has been studied extensively in literature. Federgruen and Zipkin (1984) considered a single period, one warehouse, multi retailer problem where the demands was uncertain. Their objective was to obtain vehicle routes and replenishment quantities for the retailers to minimize the expected one period cost consisting of transportation, inventory holding and shortage costs. Chan and Kumar, 2009a, Chan and Kumar, 2009b also studied a complex warehouse-scheduling problem in a complex manufacturing environment. A RFID case-based logistics resource management system for managing order-picking operations in warehouses was proposed by Poon et al. (2009). Roundy (1985) analyzed the problem that permits no shortages or backlogging, and shows how to find a policy with 98% effectiveness in O(N log N) time. Mitchell (1987) extended the Roundy’s results to situations with backlogging and that resulted in a 98%-effective policy for the backlogging problem in O(N log N) time. Hwang and Cho (2006) proposed a performance evaluation model for the order picking facility for warehouse design in a supply center (SC) by reducing the travel distance of transporters. Levi, Roundy, and Shmoys (2005) proposed constant approximation algorithms for the dynamic one-warehouse multi-retailer problem with fixed-charge ordering costs. The outcome of their research showed better results for the joint replenishment problem (JRP). A network flow formulation for a two-stage supply chain using a Lagrangian decomposition procedure to provide solutions for this integrated production and transportation planning problem was studied by Eksioğlu, Eksioğlu, and Romeijn (2007).
A number of research papers consider piece-wise linear transportation costs. Anily and Tzur (2005) developed a Lagrangian relaxation based procedure to solve the problem for multi-echelon distribution systems with general piece-wise linear ordering and transportation cost functions. Yoo, Kim, and Rhee (1997) also proposed a DRP method to schedule the multi-echelon distribution network. They aimed to apply DRP method to meet just in time concept and minimize the out-of-stock probability. The effectiveness of ZIO policies for a SWMR model with a modified all-unit discount freight cost structure was analyzed by Chan et al. (2002). Such an ordering cost function represents transportation costs charged by many carries. They also show that the cost associated with the optimal ZIO policy is no more than 4/3rd of the optimal policy for their SWMR problem. Shen, Shu, Levi, Teo, and Zhang (2009) extended Chan’s model to a more general SWMR system and showed that a solution that is within €-optimality can be obtained by solving a related piece-wise linear concave cost multi-commodity network flow problem. Seo (2006) extended the order risk policy to general multi-echelon systems utilizing the real-time shared stock information. The outcome of the findings showed superior performance of the order risk policy, compared to the existing reorder policies. Hill and Galbreth (2008) presented a new heuristic procedure to provide good solutions to problems involving all-unit discount cost functions while significantly reducing solution times. These research findings show that although researchers have explored the single-warehouse multi-retailer (SWMR) problem, replenishment issue has not been addressed extensively.
In this study, we extend the model of Chan et al. (2002) and focus on obtaining an optimal replenishment policy for the SWMR system with transportation capacity and backlogging. The remainder of this paper is organized in the following way: in Section 2, the mathematical formulation of the model is presented and the basic notations are defined. In Section 3 we propose Genetic Algorithm to find an optimal solution to the problem. Computational results are presented in Section 4 and summary and conclusions are discussed in Section 5.
Section snippets
Problem formulation
In this paper, a SWMR system is considered where ‘M’ suppliers provide different products to ‘N’ retailers. Each retailer provides suppliers with a forecast demand for the next ‘T’ time periods. Products are moved from the suppliers to the retailers, using the warehouse as a cross-docking point (see Fig. 1). Cross-docking is considered to be the most efficient means of facilitating replenishment coordination among the several methods used to handle the flow of goods across the supply chain such
Implementation of Genetic Algorithm
The general SWMR problem described here can also be used to model the Joint Replenishment Problem (JRP), which is a special case of SWMR by setting the cargo capacity to a sufficiently large quantity and making the holding costs at the warehouse identical to those at the retailer (Joneja, 1990). Since the Joint Replenishment problem is a NP-hard problem (Arkin, Joneja, & Roundy, 1989), the single-warehouse multi-retailer problem is also considered to be a NP-hard even if all the transportation
Experimentation and results
In this section the performance of the Genetic Algorithm used to optimize the replenishment policies in terms of computational time has been tested. The algorithm was coded in C++ programming language and were then compiled and executed on a PC with Pentium IV CPU 2.0 GHz processor with 2 GB RAM. In our experimentation, the parameter values were tuned after extensive experiments. These parameter values are as follows: population size = 80, maximum number of generations = 1000, crossover rate = 0.5 and
Summary and conclusions
One of the challenges faced by many producers and distributors of consumer goods is the task of distributing their products to many small retailers located in different cities. To manage a local presence at different geographical locations, most of the companies maintain at least one warehouse in each city. Therefore, a single-warehouse, multiple-retailer (SWMR) system is very common for the manufacturing firms. In this paper, we proposed a multi-retailer, multi-product and multi-period problem
Acknowledgements
The work described in this paper was supported by a grant from the Hong Kong Polytechnic University (Project No. G-YJ02). The authors would like to thank Hong Kong Polytechnic University Research Committee for the financial support.
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