Controlling lead times and minor ordering costs in the joint replenishment problem with stochastic demands under the class of cyclic policies

• Published date: 01 January 2021
• Author: Davide Castellano,Marcello Braglia,Liberatina Santillo,Dongping Song
• DOI: http://doi.org/10.1111/itor.12571
• Date: 01 January 2021

Intl. Trans. in Op. Res. 28 (2021) 376–400

DOI: 10.1111/itor.12571

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Controlling lead times and minor ordering costs in the joint

replenishment problem with stochastic demands under the class

of cyclic policies

Marcello Bragliaa, Davide Castellanob,∗, Liberatina Santilloband Dongping Songc

aDipartimento di Ingegneria Civile e Industriale, Universit`

adiPisa,Pisa,Italy

bDipartimento di Ingegneria Chimica, dei Materiali e della Produzione Industriale, Universitadegli Studi di Napoli

“Federico II”, Napoli, Italy

cUniversity of Liverpool Management School,Liverpool, UK

E-mail: m.braglia@ing.unipi.it[Braglia]; davide.castellano@unina.it [Castellano]; santillo@unina.it [Santillo];

Dongping.Song@liverpool.ac.uk[Song]

Received 14 February2017; received in revised form 28 March 2018; accepted 9 June 2018

Abstract

In this paper, we consider the periodic review joint replenishment problem under the class of cyclic policies.

For each item, the demand in the protection interval is assumed stochastic. Moreover, a fraction of shortage

is lost, while the other quota is backordered. We suppose that lead times and minor ordering costs are

controllable. The problem concerns determining the cyclic replenishment policy, the lead times,and the minor

ordering costs in order to minimize the long-run expected total cost per time unit. We established several

properties of the cost function, which permit us to derive a heuristic algorithm. A lower bound on the

minimum cost is obtained, which helps us to evaluatethe effectiveness of the proposed heuristic. The heuristic

is also compared with a hybrid genetic algorithm that is specifically developed for benchmarking purposes.

Numerical experiments have been carried out to investigate the performance of the heuristic.

Keywords:inventory;joint replenishment problem; stochastic; optimization; heuristics; lower bound; investments;stockout

1. Introduction

In practical contexts, the coordination of replenishments among several items often leads to eco-

nomic benefits. Examples may include the case of different products that are ordered from the

same supplier or that are processed on the same piece of equipment (Nilsson and Silver, 2008). The

problem related to the optimization of coordinated inventory replenishmentpolicies among several

items is typically referred to as the joint replenishment problem (JRP).

∗Corresponding author.

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2018 The Authors.

International Transactionsin Operational Research C

2018 International Federation ofOperational Research Societies

Published by John Wiley & Sons Ltd, 9600 Garsington Road, Oxford OX4 2DQ, UK and 350 Main St, Malden, MA02148,

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M. Braglia et al. / Intl. Trans. in Op. Res.28 (2021) 376–400 377

Because of the cost structure characterizing the JRP that includes two types of ordering cost (i.e.,

a minor cost and a major cost), economies of scale may be exploited (Kiesm¨

uller, 2010). In fact,

group replenishments may lead to substantial cost savings, which potentially grow as the major

ordering cost increases (Khouja and Goyal, 2008).

The JRP has been widely studied in the literature. The reader is referred to Khouja and Goyal

(2008) for a review of papers published in the period 1989–2005. Since then more papers have been

published. These works can be classified into two main groups, depending on whether the demand

is supposed to be stochastic or not: deterministic JRP and stochastic JRP.

Although the assumption of deterministic demand is little practical, the first group or researches

has attracted much attention over the years. The papers by Zhang et al. (2012, 2018), Amaya et al.

(2013), Tsaoand Teng (2013), Wanget al. (2013), Konur and Schaefer (2016), and Mohammaditabar

and Ghodsypour (2016) are examples of contributions in the first group.

As stochastic demand is more representative of reality, our paper is focused on this context.

In the stochastic JRP group, we can include the following works. Tsai et al. (2009) presented an

approach to cluster items according to the correlation between their demands. Items are then

managed using the can-order policy. Kiesm¨

uller (2010) compared two different continuous review

policies assuming that each item demand follows a compound renewal process, and that the total

amount of products to be ordered is constrained. Narayanan and Robinson (2010b) carried out

a study to evaluate the performance of nine lot-sizing heuristics in a dynamic rolling horizon

system, where demands are Gaussian. Mustafa Tanrikulu et al. (2010) developed a continuous

review policy that takes into account a limited transportation capacity, assuming that each item

demand follows a Poisson process. Feng et al. (2015) analyzed a discounted cost model in which

demands are correlated. Qu et al. (2015) approached the location-inventory problem under two

different strategies, that is, coordinated and independent replenishments, under the assumption

that shortages are fully backordered. Finally, Braglia et al. (2016a, 2016c, 2017) studied various

extensions of the JRP to take into account different aspects, such as backorders–lost sales mixtures,

controllable lead time, investments to reduce the major ordering cost, or adopting distribution-free

approach.

In many practical circumstances, lead time can be shortened at the expense of an additional

cost. In other words, lead time can be controllable. The just-in-time (JIT) philosophy suggests that,

if lead time is reduced, several benefits can be achieved, such as lower investment in inventory,

better product quality, less scrap, reduced storage space requirements, higher flexibility, increased

productivity, and improved competitive position of the company (Hariga and Ben-Daya, 1999;

Hariga, 2000; Glock, 2012). The concept of controllable lead time has been widely endorsed in the

inventory management literature (Glock, 2012; Braglia et al., 2016b; Castellano et al., 2017; Sarkar

and Mahapatra, 2017).

Further actions, other than shortening lead time, can be tackled to reach JIT goals. One of these

initiatives is concerned with the setup/ordering cost reduction, which can be achieved in practice

by means of various activities, such as procedural changes, specialized equipments acquisition,

and workers training (Leschke, 1996). As observed in literature, decreasing the setup/ordering cost

permits to improve quality and flexibility, lower investment in inventory, and increase effective

capacity (Leschke and Weiss, 1997). Setup cost control has been a topic of interest for many

researchers in the field of production/inventory management (Castellano et al., 2017; Priyan and

Uthayakumar, 2017).

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2018 The Authors.

International Transactionsin Operational Research C

2018 International Federation of OperationalResearch Societies