An integrated algorithm for solving multi-customer joint replenishment problem with districting consideration

https://doi.org/10.1016/j.tre.2020.101896Get rights and content

Highlights

  • Districting consideration is successfully integrated with the multi-customer JRP.

  • This study is vital to a company outsourcing logistics to a 3PL service provider.

  • Our numerical experiments are based on a real-life example in a bank system.

  • An effective GA-based algorithm is proposed for solving the problem.

  • Managerial insights provide baseline information before contract negotiation.

Abstract

This paper studies a multi-customer joint replenishment problem with districting consideration (MJRPDC) which is of particular importance to a company that outsources its transportation and delivery operations to a third-party logistics (3PL) service provider. To solve the problem, we first propose an innovative search algorithm for solving the traditional multi-customer joint replenishment problem in a given zone. Then we design a GA-based framework to handle the corresponding districting problem based on the performance of each district evaluated by using the proposed search algorithm. The proposed methodologies are demonstrated by using an example of solving MJRPDC for a bank.

Introduction

It is well-known that supply chain management is critical for suppliers, manufacturers, and consumers. Companies rely on strategic plans with advanced logistic services and inventory control to enhance their core competencies for long-term profits (see Simchi-Levi et al. (2013) and Chopra and Meindl (2015)). This study investigates a logistics plan that integrates inventory control and districting problems for supply chain management.

Joint replenishment is a common practice in operations management to reduce the unit setup cost of order picking, packaging, preparation, and dispatch through economies of scale. The joint replenishment problem (JRP) aims to determine the lot size and the delivery schedule of n items of goods sourced from a single supplier for one customer such that the average total cost of setup and inventory holding is minimized over an infinite (and continuous) planning horizon. Typically, there are two types of setup costs: (i) a major setup cost, A0, for collectively processing a subset of items and (ii) a minor setup cost, ai, for handling each item i. A major setup often requires a significant amount of time and cost. JRP is commonly adopted to coordinate the replenishment schedule of each item i such that the major setup cost is economically shared to balance the inventory holding cost of each item.

Shu (1971) presented an early work on JRP. More recently, Cohen-Hillel and Yedidsion (2018) showed that the periodic JRP is strongly NP-hard. A majority of the solution methods for JRP in the literature enumerate through all combinations of the basic replenishment cycle time B and its multiples ki of item i to determine a global optimal solution (Fung and Ma, 2001, Goyal, 1974, Van Eijs, 1993, Viswanathan, 2002). Different from this approach, Wildeman et al. (1997) proposed solving JRP using Lipschitz optimization. Lee and Yao (2003) explored the characteristics of the objective function for JRP under the power-of-two (PoT) policy that requires multipliers ki=2p for some pN. They showed that the corresponding objective function is piece-wise convex and proposed an efficient search algorithm to determine the global optimum of JRP under the PoT policy. Their theoretical analysis inspires us to investigate the general integer (GI) policy that only mandates every ki to be a positive integer. For a thorough review on JRP, see Khouja and Goyal (2008) and Bastos et al. (2017).

If many customers order the same group of items from a single supplier, such as a chain of retail stores or several branches in the same business group, we can take advantage of the joint replenishment and extend JRP to multiple customers. Such an extension of JRP is called a multi-customer joint replenishment problem (MJRP), which aims to determine the lot size and the delivery schedule of n items of goods from one supplier for multiple customers such that the average total cost of setup and inventory holding for all customers will be minimized over an infinite horizon. In the case of numerous customers scattered across a wide area, it is necessary to take certain practical considerations such as geography, transportation, or fleet size to divide the customers into several mutually exclusive zones of a planning region. Then, MJRP is employed to coordinate the replenishment for customers in the same zone. Real-world examples have motivated researchers to study optimal replenishment and delivery coordination policies in distribution and supply networks that share decision-making scenarios with MJRP. Liu et al. (2018), for example, studied the case of China’s Zhongbai Holdings Group Co., Ltd., which transports products from a huge warehouse to numerous supermarkets and medium markets. Blumenfeld et al. (1987) indicated that General Motors manages a supply network with a similar structure. Wu and Mao (2017) discussed a drug pooling system with one drug supplier and multiple hospitals in China. Liu et al. (2017) investigated more complicated cases of large companies owning multiple warehouses in China, for example, JD.com, China’s largest online direct sales company (www.jd.com/); Alibaba Group Holding Ltd. (www.alibaba.com/); and HaiXun Logistics Co., Ltd.

In general, customers in the planning region can be partitioned on the basis of zip codes or an administrative definition of districts. Chan et al. (2003), for example, solve MJRP in each zone corresponding to eight predetermined groups (Fig. 1.1). Different from their study, we further consider categorizing customers into a fixed number of groups instead of having given partitions. We term this integrated problem a multi-customer JRP with districting consideration (MJRPDC), which determines an optimal districting setting such that the average total cost of MJRP for all zones is minimized.

MJRPDC is of particular importance to a company that outsources its transportation and delivery operations to a third-party logistics (3PL) service provider. Ogorelc (2007) indicated that by contracting out to a 3PL provider with the right capabilities and resources, companies could focus on their core competencies and leverage logistics to improve their service levels. As of 2014, 80% of all Fortune 500 companies and 96 Fortune 100 companies use some form of 3PL services (Menner (2015)). Prior to signing an outsourcing contract with a 3PL service provider, a company must provide a zoning plan with a replenishment schedule to cover all customers for negotiating the freight rate in the contract.

To design a zoning plan, we consider a districting problem that groups individual units into clusters (or districts) according to certain relevant planning criteria (Kalcsics, 2015). Ricca et al. (2013) listed some commonly used principles:

  • (i)

    Contiguity: A zone is said to be contiguous if it is possible to travel between any two territorial units without leaving the zone.

  • (ii)

    Population equity: A partition is said to be in population equity if the number of desired units such as eligible voters and sales potential in each zone is roughly the same.

  • (iii)

    Compactness: A zone is said to be compact if it is geographically round shaped.

Our study on MJRPDC explicitly considers the population equity and compactness criteria because population size determines demand and compactness affects transportation distance and cost.

The districting problem has been widely studied in the context of electoral districting (Ricca et al., 2013), school districting (Ferland and Guénette, 1990), and sales territories (Zoltners and Sinha, 2005). In each context, districting serves a different purpose and can significantly impact the performance of operations. When flexibility and complexity are considered, various heuristics such as simulated annealing (D’Amico et al., 2002), tabu search (Bozkaya2003), and genetic algorithms (GA) (Cowgill et al., 1999, Maulik and Bandyopadhyay, 2000) are adopted to solve the districting problem. Xu and Wunsch (2005) conducted a survey that highlights GA is more favorable. In this study, we design a GA-based solution method to determine the optimal districting setting to solve MJRPDC.

The integration of the districting problem with MJRP is a rather novel idea, although some studies have integrated JRP with delivery decisions (e.g., Cha et al., 2008, Coelho and Laporte, 2014, Chen et al., 2016, Pasandideh et al., 2018, Liu et al., 2017, Liu et al., 2018, Qu et al., 2013, Wang et al., 2018, Zeng et al., 2016). This study aims to determine the optimal districting setting in which the average total cost of MJRP in all zones is minimized. MJRPDC is an extension of MJRP studied in Chan et al. (2003), although our study significantly differs from other research on JRP with location considerations.

To solve the problem, we first propose an innovative search algorithm for solving the traditional multi-customer joint replenishment problem (MJRP) in a given zone by adopting the concept of “junction point” analysis (Lee and Yao (2003)) for the power-of-two (PoT) policy and extending the analysis for the general integer (GI) policy. We then design a GA-based framework to solve the districting problem based on the performance of each district evaluated by using the proposed search algorithm. The analysis employs an example of MJRPDC for a bank to demonstrate the efficiency and effectiveness of the proposed solution method. We also conduct a sensitivity analysis on the parameters corresponding to the demand equity and compactness constraints for better decision making.

This study makes at least three contributions. First, it presents an efficient search algorithm to solve MJRP under the GI policy. Second, by incorporating the new search algorithm for MJRP, we propose a GA-based solution method to solve MJRPDC. Finally, we use the example of a bank to provide managerial insights and strategic implications that can guide decision makers when signing a contract with a 3PL service provider.

The remainder of this paper is organized as follows. Section 2 introduces a decision-making scenario and a mathematical model for MJRPDC. Section 3 proposes an effective search algorithm to solve MJRP by examining the optimal properties of a junction point analysis. Section 4 designs a GA-based framework incorporating the proposed search algorithm to solve MJRPDC. Section 5 presents computational experiments to demonstrate the effectiveness of the proposed GA-based solution method. In addition, we conducted a sensitivity analysis to discuss the impact of the “population equity” and “compactness” criteria. Section 6 concludes the paper with some managerial insights and strategic implications.

Section snippets

Mathematical model

This section introduces a decision-making scenario for MJRPDC, presents notations and assumptions applied in the study, and then provides a mathematical model for MJRPDC.

An effective solution method for solving MJRP

This section provides a theoretical analysis of the MJRP model and design a search algorithm for solving it. Following the MJRPDC model in Eqs. (2.1)–(2.8), the MJRP in a given zone z is formulated as follows:(MJRP)minTCz(Kz,Bz)=1BzA0+jNzi=1qaijkij+jNzi=1qHij2kijBzsubject to kijN,i,jBz>0Nz{j:rjz=1,j}Kz{kij:rjz=1,i,j}The objective function in Eq. (3.1) is to minimize the average total cost that include the major setup cost, minor setup cost and inventory holding cost per unit of

The integrated GA-based solution approach

This section proposes a modified GA that incorporates the search algorithm proposed in Section 3.2.4 to solve MJRPDC. Fig. 4.1 presents the framework for the GA-based solution approach. The latter half of this section introduces the chromosome representation and the initialization procedure. We then discuss the evaluation of chromosomes and certain genetic operators such as selection, crossover, mutation, and termination conditions.

Computational experiments

This section presents two categories of numerical experiments for MJRPDC. First, we study a case of 80 customers (corresponding to 80 branches of a bank) to derive deeper managerial insights on the decision-making scenario. Then, we conduct random experiments to test the effectiveness of the proposed solution approach. Our integrated GA-based solution approach is coded in MATLAB R2017a and we perform our experiments on a Dell personal computer with an Intel Core i7 3.4 GHz CPU and 8.0 GB RAM.

We

Conclusions

In this paper, we study a multi-customer joint replenishment problem with districting consideration, which is of particular importance to a company that outsources its transportation and delivery operations to a 3PL service provider. To solve MJRPDC, we first design an innovative search algorithm for solving the traditional MJRP in a given zone. Then a GA-based framework is proposed to handle the corresponding districting problem using the proposed search algorithm to evaluate the performance

CRediT authorship contribution statement

Ming-Jong Yao: Conceptualization, Methodology, Writing - review & editing. Jen-Yen Lin: Software, Investigation, Visualization. Yu-Liang Lin: Software, Formal analysis, Writing - original draft. Shu-Cherng Fang: Supervision, Writing - review & editing.

Acknowledgments

This work is supported by MOST, Taiwan (NSC 101-2221-E-009 -064 -MY3) and the U.S. Army Research Office (Grant W911NF- 15-1-0223).

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