This paper presents a new approach to solve the joint replenishment problem under deterministic demand and resource constraints (C-JRP). To solve the problem, a heuristic framework based on linear programming is presented. The proposed method can be extended to solve different versions of the problem and can be extended with linear programming modeling. This method is analyzed and some tests are performed, the results of which show that the proposed algorithm outperforms other well-known algorithms in total cost.
Coordinated procurement policies have shown great potential to save companies money by reducing costs in logistical operations. In environments in which a set of products are obtained from one vendor, it is advantageous to exploit the economies of scale that result from coordinating transportation, production runs or item replenishment. However, warehouse capacities, budget limitations and other factors often create resource constraints. The problem of obtaining a common replenishment period under resource constraints is known as the joint replenishment problem with resource constraint (C-JRP).
Many efforts have been made to solve the unconstrained joint replenishment problem (JRP), which involves obtaining coordinated replenishment policies without regard for resources constraints. However, given the complexity of the problem, most authors work on the development of heuristic methods to obtain approximated solutions instead of the optimal one. An overview of approaches used to solve this problem can be found in Viswanathan (1996), Silver (1976), Kaspi and Rosenblatt (1991), Viswanathan (2002), and Li (2004), who have developed different methods and models that allow for obtaining near-optimal solutions.
Silver (1976) presents a simple non-iterative method to solve the C-JRP based on solving the differential equation system. The results show that the solutions are near optimal and with low penalties associated with use. Nilsson and Silver (2008) have updated the Silver method with an improved routine that allows for significant outperformance of the previous results. On the other hand, the RAND heuristic (Kaspi and Rosenblatt, 1991) and the subsequent changes made by Viswanathan (1996) and Viswanathan (2002) have proven to be successful at finding near-optimal solutions in less computational time than the counterparts. Porras and Dekker (2008) improve upon the results obtained by Viswanathan (2002) through the inclusion of a correction factor at the objective function. However, the algorithms presented above have been applied to the constrained version of the problem and may not be useful when applied to systems with limited resources different from the original constraint. (Hoque, 2006, Porras and Dekker, 2006, Khouja and Goyal, 2008, Moon and Cha, 2006 and Goyal and Giri, 2003).
Several authors propose solution methods to deal with the constrained version of the joint replenishment problem (C-JRP). Most of the research into C-JRP is centered on the inclusion of constraints like transportation, budget and physical space. Proposed methods are usually only useful for solving specific versions of the problem and cannot be extended to other variations. The new proposals in the literature have included different constraints on the JRP, such as minimum-order constraints, storage constraints, purchase-budget constraints and transportation constraints. This shows that the solution methods must still be improved, so that coordinated replenishment policies may be obtained in real environments. The JRP has recently been used as a basis to propose a model for supply-chain distribution problems, including those involving freight consolidation, full truckloads and joint delivery orders (Goyal, 1975, Kiesmüller, 2009, Cha and Moon, 2008 and Moon et al., 2011).
Goyal (1975) proposed a heuristic based on the Lagrangian Multipliers method to solve the C-JRP. Moon and Cha (2006) use Goyal's algorithm as a basis to modify the RAND heuristic and proposed the C-RAND heuristic to solve the C-JRP. His experimental results showed that their new method outperforms the results obtained by using Goyal's heuristic. Furthermore, the C-RAND was compared with an evolutionary algorithm (GA) proposed to solve the C-JRP (Khouja et al., 2000) showing that the results are favorable to the former.
The aim of this paper is to propose a solution method based on linear programming that can be used to solve a general version of the constrained joint replenishment problem. The proposed method can be extended to solve the joint replenishment problem under different constraints, depending on the context in which it is used.
This paper is organized as follows. Section 2 presents the nomenclature used in the paper and a non-linear version of the mathematical model for the JRP and C-JRP. Section 3 presents the structure of the algorithm. Section 4 describes the way in which the instances were designed and the parameters used as an input this process. 5 and 6 evaluate the algorithm and compare them with the obtained by using the C-RAND heuristic to present the advantages and disadvantages of the proposed approach.
The proposed algorithm is a good alternative to solve the C-JRP. The method outperforms previous results. The heuristic obtains better objective function values than the C-RAND heuristic. The results show that in 93% of cases the heuristic gives better objective function values than the C-RAND, although taking an average of 12 s more.
Unlike the known models in the literature, the presented method can be extended easily to solve the problem of coordinated supply under many kinds of constraints, such as transportation, storage capabilities, etc.
Future work aims to use the unimodular property of the problem, implementing a Lagrangian relaxation method as a strategy to solve the integer-programming model faster. This method can deal with multiple and different kinds of constraints in more realistic contexts.
