Integrated lot-sizing and scheduling: Mitigation of uncertainty in demand and processing time by machine learning

Highlights

• A rescheduling approach is proposed for a dynamic lot-sizing and scheduling problem.

• A new formulation is presented under satisfiability modulo theories (SMT).

• Two safety parameters are employed to deal with the uncertainties.

• A data-driven algorithm is developed to estimate the safety parameters.

Abstract

Production rescheduling is one of the most challenging problems in production management, in which some parameters, such as customer demand and job processing time, are subject to uncertainty during the planning horizon. This paper develops a scheduling and rescheduling method for the simultaneous lot-sizing and job shop scheduling problem, considering sequence-dependent setup times and capacity constraints. The objective is to find a trade-off between safety levels and production costs focusing on schedulability and optimality as the two most important performance indicators in the face of uncertainty. Therefore, a new adjustable formulation based on satisfiability modulo theories has been developed to tackle the demand and processing time uncertainty. Then, a combination of neural networks and a K-means based heuristic was applied to calibrate the model by determining the profitable value of the safety levels as a strategy to increase the robustness of the schedule. We also developed a Monte Carlo simulation to assess the performance of the proposed algorithm and compare it with other approaches addressed in the literature. The computational results show that the proposed algorithm is efficient and promising in protecting the schedulability of the model, taking the optimality criterion into account.

Introduction

The integrated lot-sizing and scheduling problem (ILSP) is one of the most frequent problems in production planning that belongs to medium or long-term planning. In this problem, two types of decision variables are usually determined in a finite planning horizon, including master planning variables (the lot size, that is, the quantity of product to be produced in a given period) and the variables related to the detailed schedule (sequence of jobs within periods). The capacitated lot-sizing and scheduling problem (CLSP) with the resource capacity constraint is known to be NP-hard (Bitran and Yanasse, 1982). In addition, finding a feasible solution for CLSP is also NP-complete (Maes et al., 1991). This complexity intensifies in a dynamic environment so that meeting the client’s delivery time (without any shortage) is often challenging where initial plans are affected by unpredictable noise such as demand, processing time or lead time fluctuations, machine breakdowns, due date changes, order cancellations, etc. These disruptions generally trigger rescheduling and usually result in updating decision variables. Predictive, completely reactive, and mixed predictive-reactive strategies are three standard scheduling and rescheduling strategies for dealing with uncertainty. In predictive scheduling, all decision variables are made in advance, before realizing uncertain events, and no rescheduling takes place (Ouelhadj and Petrovic, 2009). Predictive scheduling requires a sufficient amount of information for prediction. However, the majority of real-world applications are indeed dynamic due to hard-to-predict events (Baykasoğlu and Ozsoydan, 2018). Completely reactive rescheduling is proposed for the problem, where all decision variables are taken in response to real-time events (Ouelhadj and Petrovic, 2009). The reactive rescheduling can be done either regularly after a specific time (periodic) or immediately after events are observed. The predictive-reactive strategy involves initial and predictive scheduling while all or some decisions regarding control variables are made in advance before uncertainty is observed. Then, in the reactive phase, all or some parts of the control variables are made or revised in response to real-time events (Gupta et al., 2016). Strengthening the schedule in the predictive phase reduces the cost of changes in the reactive phase. Improving forecasting and estimation capabilities in the predictive phase is crucial for this type of rescheduling. Hereupon, in recent years, the application of machine learning techniques in the form of data-driven optimization has been taken into attention. Data-driven optimization has been introduced as a new paradigm in optimization problems where evaluating the probability function or uncertainty is not straightforward. This approach emerged due to the explosion of data availability during the last decade and uses historical data derived from discrete event simulation or real production.

This paper proposes a predictive and periodic reactive strategy for rescheduling a dynamic CLSP in a job shop environment, while a data-driven approach using machine learning is proposed to tackle the demand and processing time uncertainty. We take a rolling horizon algorithm where scheduling is divided into periods, which means customer demand is updated periodically (weekly or monthly). This setup implies that even though we have information about upcoming periods, the more distant they are, the less certain we are about the uncertain parameters. Here we address this challenge by introducing safety reserves into the schedule. We consider two types of safety reserves: safety stock (a reserve in stock) and safety slack (leeway in the schedule). The question arises, how to set the values of these reserves? The values depend on external factors—for instance, seasons typically influence customer behavior. For this purpose, we employ use machine learning (ML) to predict values of the safety reserves for the next periods based on historical data.

Fig. 1 gives an overview of our approach, which is going through periods $t=1,2,\dots$. The process might terminate early if the current demand or processing time does not enable a feasible schedule—this is undesirable, and that is what we are trying to prevent by setting appropriate reserves for the schedule. In each period, we first query the newly revealed data of the system (e.g., new customers’ demand) and then apply ML to predict the values of reserves. The existing information of the system and the values of reserves are then used to calculate the future schedule. For this purpose, we have incorporated a new model based on satisfiability modulo theories (SMT) in such a way that the model is calibrated using historical data and a custom-designed ML during the scheduling horizon. We use the solver Z3 as an exact optimization solver for solving the proposed SMT model. If the solver comes back with a feasible schedule, the current period $t$ is put into production, and the process moves on to the next one. If there is no feasible schedule, the whole process terminates. The algorithm’s performance is evaluated in a Monte Carlo simulation regarding the schedulability and optimality criteria as two critical measures in the face of uncertainty.

The rest of this article is structured as follows. Section 2 reviews the most relevant papers in the literature and the contribution of this paper. The description of the problem and the assumptions are defined in Section 3. Section 4 presents a novel SMT formulation of the problem. Section 5 illustrates the proposed solution method. Section 6 presents the computational results and Monte Carlo simulation to evaluate the proposed algorithm. Ultimately, Section 7 provides the conclusions and directions for future studies.