Component replenishment planning for a single-level assembly system under random lead times: A chance constrained programming approach

Abstract

In practice, production planning and inventory control, both managed in the assembly systems framework, are often subject to various sources of exogenous uncertainty. In this regard, the present paper focuses on a single-level multi-component inventory control problem for the assembly systems replenishment under stochastic component procurement lead times. In order to be closer to the common assumption of MRP software tools, the case of discrete distributions of component lead times is considered, these latter being thus expressed as a number of periods. Since the finished product is assembled by using several types of component at the same time, the assembly process is stopped even if only a type of component is delayed. The assembly stopping forced by a components delay or stock-out is penalised by backlogging costs. Hence, the problem objective aims to minimize the total cost composed of holding and backlog ones. To address this problem, a joint chance constrained model is proposed and solved via an equivalent linear reformulation, the special structure of which is also deeply discussed. Apart from the effectiveness of the provided equivalent linear reformulation, the practical advantage of the proposed approach resides in its release from backlogging costs, which are often difficult to be quantified in real-life industrial applications.

Introduction

Assembly systems presuppose the coordination of a chain of serially disposed plants or workshops, where the upstream facilities supply components and/or sub-assemblies to the downstream ones. Parts move down the chain until the final assembly facility caters the demand needed for the finished products accomplishment. Forming part of large amounts of standardized goods, sub-assembly components represent the basis of the production in a great number of industries. Often, material requirements planning (MRP) software techniques are used for planning and controlling component replenishments in such systems.

Nonetheless, the premise of deterministic environments limits the potential of the MRP approach to suitably meet industrial applications. In the real life set of circumstances, production planning and inventory control, both managed in the assembly systems framework, are subject to miscellaneous sources of exogenous uncertainty. According to Tajbakhsh et al. (2007), upstream variability refers preponderantly to three types of uncertainty in supply chains: uncertainty in the supply timing, uncertainty in the supply quantity (or quality), and uncertainty in the purchase price. More specifically, this paper considers the case of single-level assembly systems under uncertainty in the supply timing. Usually, the supply timing is denominated in the literature as a component procurement lead time or simply called lead times.

By handling assembly system planning problems jointly with stochastic inventory control methods, it is possible to gain significant deep insights on the optimal components ordering to satisfy the customer finished products demand. Moreover, these insights can also be exploited to assess and improve the risk management strategies in order to support the MRP parametrization for assembly systems under lead time uncertainty.

For those purposes and drawn on problem features, this paper applies the joint (also called, integrated) chance constrained programming. To the best of our knowledge, this paper represents one of the first contributions that provides optimal solutions jointly with a certain reliability level against random lead times in the context of assembly systems.

Earlier, Louly et al. (2008) studied the problem of component replenishment planning for single-level assembly systems by replacing backlogging costs with a so-called service level in the framework of a non-linear optimization problem. This study was conducted in order to better meet the industrial realities, where backlogging costs are often difficult to be quantified.

In the continuity of the work of Louly et al. (2008), the contribution of the present paper has both practical and theoretical importance: (i) on the one side, it proves that the service (reliability) level absolves the backlog exigence, a fact that exempts managers to fix backlogging costs which are difficult to be quantified in industrial practice (e.g. damage to the business reputation and customer inconveniences), (ii) on the other side, an equivalent linear reformulation, for which pseudo-polynomial solution approaches exist, is also provided.

Pioneered by Charnes et al. (1958), separable (or individual) chance constrained programming seeks to satisfy each probabilistic constraint with enough high probability taken separately. On the other hand, Miller and Wagner (1965) initiated the joint chance constrained programming, an approach which allows us to trade the solution conservatism versus its performance by imposing a certain confidence level. In the framework of this latter approach, all soft constraints (i.e. those affected by uncertainty) are satisfied at least with a prescribed confidence level (threshold probability). In fact, based on existing theoretical contributions on the joint chance constrained programming approach (Kall and Wallace, 1994, Shapiro et al., 2009, Dentcheva et al., 2002), we tackle the single-level multi-component inventory control problem under random lead times, by providing a deterministic equivalent reformulation jointly with optimal solution characterization (its value and the associated confidence level).

The remainder of this paper is structured as follows. In the upcoming section, a brief review of the state of the art related to the supply planning under uncertainty in MRP environments is provided by zooming specifically on studies devoted to handling random lead times. Thereafter, Section 3 delineates the problem statement and its formal stochastic formulation. A joint chance constrained model is presented in Section 4 and solved subsequently in Section 5 via an equivalent linear counterpart program. In Section 6, an illustrative example is conducted in order to highlight the industrial interest and the managerial potential of the proposed joint chance constrained approach in terms of decision support under uncertainty. Finally, Section 7 outlines some concluding remarks and suggests future research avenues of this paper.