۱- Introduction

۲- Model and heuristic algorithm

۳- Exact algorithm

۴- Numerical computations

۵- Conclusion

References

Abstract

In a paper published in the International Journal of Production Economics (IJPE) [Zhang, R., Kaku, I., Xiao, Y., 2012. Model and heuristic algorithm of the joint replenishment problem with complete backordering and correlated demand. International Journal of Production Economics 139 (1), 33–۴۱], the authors proposed a joint replenishment problem (JRP) model with complete backordering and correlated demand caused by cross-selling. The model was transformed into minimizing a function with respect to multiples of a major item’s order cycle, and a heuristic algorithm was developed for near-optimal solutions. In this paper, we reinvestigate the problem and analyze the mathematical property of the model to develop an exact algorithm. The algorithm can obtain global optima and exhibits polynomial complexity.

Introduction

The multi-item inventory management problem has been studied for several decades, for which the joint replenishment problem (JRP) is the most representative topic (Goyal, 1974; Khouja and Goyal, 2008). The classic JRP supposes that the demand for items is deterministic and inventory replenishments are related to one another due to sharing of the common/major ordering cost. In recent years, the JRP model has been further extended to deal with transportation costs (Venkatachalam and Narayanan, 2016), stochastic demand (Braglia et al., 2016a; Lee and Lee, 2018), and perishable or deteriorating items (Kouki et al., 2016; Ai et al., 2017), among others. Certain extensions simultaneously consider multiple factors, including stochastic demand, the backorders-lost sales mixture, controllable lead time, and changeable ordering costs (Braglia et al., 2016b, 2017). Cunha et al. (2017) proposed a model for the multi-item economic lot-sizing problem, which extends JRP to remanufacturing contexts. It is noticeable that the above work related to the multi-item inventory/production problem holds an implicit assumption: the item has no externalities, which means that the demand for a given item does not affect that for any other items. However, item demands are frequently interrelated in numerous economic systems due to the externalities of products and consumption (Turnovsky and Monteiro, 2007; Hashimoto and Matsubayashi, 2014).