Terminal inventory level constraints for online production scheduling

Highlights

• Online production scheduling is studied for high quality of implemented schedules.

• A framework for the analysis of general classes of terminal constraints is proposed.

• New inventory terminal constraints for multiple production environments are proposed.

• Proposed constraints lead to better implemented solutions.

Abstract

We study online production scheduling, that is, the iterative solution of scheduling optimization problems taking into account feedback, to ensure high quality of implemented, as opposed to predicted, production schedules. While the addition of terminal constraints on inventory levels can be used to obtain high quality implemented schedules, traditional approaches based on safety stocks may be ineffective. Accordingly, we first propose a framework for the analysis of general classes of terminal constraints, and then propose new classes of linear terminal constraints for specific production environments. We provide proofs on the validity of these constraints as well as extensions to more general environments. Finally, through a computational study, we show that the implementation of the proposed constraints leads to better implemented solutions.

Introduction

Production scheduling arises in many industrial sectors and has been widely studied (Barker & McMahon, 1985; Blazewicz, Dror & Weglarz, 1991; Drexl & Kimms, 1997; Graves, 1981; Hall & Potts, 2003; Harjunkoski et al., 2014; Kreipl & Pinedo, 2004; Maravelias, 2012; Proth, 2007; Verderame, Elia, Li & Floudas, 2010). As advances in computational hardware and optimization software enable the solution of medium and large size problems in a timely fashion, researchers have started to look into the design of algorithms for rescheduling or, more generally, online scheduling to react to disturbances (e.g., production delays), uncertainty (e.g., in process yield), and arrival of new information (e.g., arrival of “rush” orders). After observing uncertainty or simply a “trigger” event, the existing schedule needs to be modified (Ouelhadj & Petrovic, 2009; Vieira, Herrmann & Lin, 2003). Another approach is to recompute a schedule periodically in a moving horizon fashion (Ovacik & Uzsoy, 1995; Sand, Engell, Märkert, Schultz & Schulz, 2000). Interestingly, online scheduling, carried out periodically, leads to better solutions even in the absence of uncertainty (Gupta & Maravelias, 2016, 2020; Gupta, Maravelias & Wassick, 2016; McAllister, Rawlings & Maravelias, 2020).

Consider a setting where we carry out online scheduling once a day and we study a 30-day period. At the first iteration, we obtain a solution which is essentially a prediction of how the schedule will look over the next 30 days. We refer to this solution as the open-loop solution. After 30 iterations and the implementations of the first day in each of the 30-day open-loop solutions, we record the solution that was implemented, after all uncertainties were observed and feedback was used to take recourse on the production floor. We will refer to this solution as the closed-loop solution. We will also refer to the optimization problem we solve at each iteration as the open-loop problem, and the problem of optimizing the implemented solution (which is solved via multiple online optimizations) as the closed-loop problem. It has been shown that, often counterintuitive, modifications to the open-loop problem can lead to higher quality closed-loop solutions (Gupta & Maravelias, 2016).

Before we discuss such modifications, we make an important remark regarding the treatment of uncertainty. There are different approaches for scheduling under uncertainty, including robust optimization (Ben-Tal, Goryashko, Guslitzer & Nemirovski, 2004; Bertsimas, Brown & Caramanis, 2011; Lappas & Gounaris, 2016; Li & Ierapetritou, 2008), stochastic programming (Bonfill, Bagajewicz, Espuña & Puigjaner, 2004; Dupačová, Gröwe-Kuska & Römisch, 2003), and fuzzy programming (Balasubramanian & Grossmann, 2003). However, all these methods seek to find a better open-loop schedule, under the assumption that this open-loop schedule is more likely to lead to a better closed-loop schedule. However, even when the open-loop problem accounts for uncertainty, rescheduling is still necessary.

In terms of modifications to the open-loop problem, terminal constraints can be used, in general, to ensure that the open-loop problem does not yield myopic solutions. For example, inventory constraints prevent inventory depletion, which would otherwise occur due to inventory cost minimization (Lima, Grossmann & Jiao, 2011), which would in turn lead to infeasibilities or high cost in the closed-loop solution. The inclusion of such terminal constraints is similar, in principle, to the approach followed in model predictive control (MPC), where state variables are constrained to lie in a terminal region (Mayne, Rawlings, Rao & Scokaert, 2000). However, generating good terminal constraints for scheduling remains an open challenge (Stadtler, 2000).

Another related problem is the widely studied inventory management (control) problem. Most research in this area has focused on finding re-ordering policies given uncertainty in the supply chain (SC), including uncertainty in supply (Schwartz, Wang & Rivera, 2006), demand (Song & Zipkin, 1993; van der Laan, Salomon, Dekker & van Wassenhove, 1999), lead time (Song, 2009), and price (Wu & Chen, 2010). Material requirement planning techniques have been developed to address such problems (Dolgui & Prodhon, 2007). Inventory procurement review policies can be classified as continuous (Feng & Xiao, 2002; Muckstadt & Isaac, 1981) or periodic (Ignaciuk & Bartoszewicz, 2010; Paschalidis & Liu, 2003). The problem has also been approached as an optimal control one, adopting a wide range of modeling approaches, including Markov chains (Song & Zipkin, 1993), dynamic programming (Flynn & Garstka, 1990), and model predictive control (Braun, Rivera, Flores, Carlyle & Kempf, 2003; Yi & Reklaitis, 2014). Simultaneous determination of pricing and inventory management has also been studied (Chen, Pang & Pan, 2014; Federgruen & Heching, 1999). Finally, different production environments, including single- and multi-product, single- and multi-stage environments, have been studied (Braun et al., 2003; Clark & Scarf, 1960; Kapuściński & Tayur, 1998; Qiu & Loulou, 1995). Nevertheless, despite the extended research on inventory management, to the best of our knowledge, there are no systematic ways to generate terminal constraints for scheduling problems, and the methods that can be adopted have some important limitations, as discussed next, especially when applied to systems with fluid (gas and liquid) materials and batch size restrictions.

One inventory-management-inspired solution is to enforce the terminal inventory level to be greater than a lead-time-based inventory threshold, which includes a buffer term named “safety stock” (Eppen & Martin, 1988; Kreipl & Pinedo, 2004; Sana & Goyal, 2015; You & Grossmann, 2008). Another production-scheduling-inspired approach is to enforce the terminal inventory levels to be equal to their initial values, at the start of the horizon (Baker, 1981; Shah, Pantelides & Sargent, 1993), or enforce periodicity (Subramanian, Maravelias & Rawlings, 2012). There is also an approach to add a quadratic valuation term of the terminal inventory level to the objective function (Fisher, Ramdas & Zheng, 2001). However, all the aforementioned approaches do not account for the relationship of inventory levels among different materials. For instance, in a single-stage two-product problem, if the inventory level of one product is high, then a low inventory level of the other product can be adequate, because more resources can be allocated towards the production of the latter without leading to a stockout of the former. In this way, we can reduce the total inventory levels, and therefore have better closed-loop solutions without affecting demand satisfaction levels.

Accordingly, the goal of this paper is to first provide a framework for the derivation of terminal constraints for short-term scheduling problems, and then propose new constraints explicitly accounting for the relationship among the materials produced in the same facility using shared resources. The proposed terminal constraints, used for online scheduling, are based on (1) the structure of the production environment and (2) an expected value of demand rate. The resulting solutions are better, compared to the ones obtained by traditional approaches, in terms of avoidance of stockouts and inventory holding cost savings. The proposed constraints can be easily incorporated in any mixed integer programming (MIP) scheduling model. Theoretically, we prove that, for deterministic problems, if the terminal inventory levels satisfy the proposed constraints, then the open-loop problem remains feasible, a property often referred to as recursive feasibility (Löfberg, 2012; Risbeck, Maravelias & Rawlings, 2019).

We considered different production environments that are industrially important, including multi-stage and multi-product environments. One industrial example, of multi-stage single-product environment, is the production of Terephthalic acid (PTA), a commodity chemical (Tomas, Bordada & Gomes, 2013). PTA is produced in two stages. In the first stage, mixed xylene is converted to paraxylene (PX); while PX is converted in the second stage to PTA. Interestingly, PX is also a product by itself, as it is often shipped to other facilities to be converted to PTA. Therefore, one must keep track and constrain the inventories of both PX and PTA. Moreover, PTA can be converted to multiple resins, and therefore, the production of resins from xylene happens in a multi-stage multi-product environment. Another industrial example of a multi-product environment is pharmaceutical production, where the same resources are used in campaign mode to produce multiple products (Sundaramoorthy & Karimi, 2004).

The paper is structured as follows. In Section 2, we present motivating examples and the problem statement. The proposed framework is given in Section 3. In Sections 4 through 6, we present the terminal constraints for different production environments with one machine (or unit) per stage, including multi-stage single-product (Section 4); single-stage multi-product (Section 5); and multi-stage multi-product (Section 6). In Section 7, we generalize the proposed constraints to problems with multiple machines in each stage. In Section 8, we present computational results, using instances with and without uncertainty. The proofs of the theoretical results are given in the Appendix. The supplementary material includes proofs that were not included in the main text due to space limitations, as well as additional examples, parameters and statistics of all the studied instances.