Lot-sizing Problem Research Articles

Spatial-temporal deep learning method for solving data-driven multi-echelon stochastic lot sizing problems with intermediate demands

ABSTRACT For the production processes and inventory decisions of a multichannel supply chain, a multi-echelon stochastic lot sizing problem with intermediate demands is extended to a constrained problem that considers the production (inventory) capacity constraint and to multi-item problems that consider feature and item correlations. Three spatial-temporal deep learning methods combining convolutional, recurrent and graph convolutional neural networks are proposed to transform the extended problems as spatial-temporal sequential prediction problems and learn data-driven decisions. Real-world data are used in the computational experiments, and the results show that considering both temporal and spatial features and/or feature correlations can provide improved prediction performance.

A genetic algorithm to solve the production lot-sizing problem with capacity adjustment

When dynamic production capacity is considered in production lot-sizing plans, it becomes very challenging to efficiently determine the optimal production plan. Many papers apply “at most one fractional production period” to develop efficient algorithms, but those algorithms are still time consuming. In this paper, in a special situation where costs are non-speculative, we provide a novel proposal based on “at least one balance period”, in which the products made in this period not only satisfy the demands in this period but also backlogged demands and some demands after this period, to obtain an efficient algorithm. This algorithm complexity is one level lower than the algorithm without non-speculative cost assumptions in the literature regarding the number of periods in their time complexity function. Then, in a general situation, we propose a combination of two complementary algorithms as an efficient heuristic method. Moreover, in the literature, the estimation of computation time complexity in searching for the optimal production plan considers only the number of capacity levels and production periods on theoretical view. However, with numerical experiments, we observe that demand variation could also have significant effects on the computation time in practice.

A Modeling and Hybridized Decomposition Approach for the Multi-level Capacitated Lot-Sizing Problem with Setup Carryover, Backlogging, and Emission Control

A Modeling and Hybridized Decomposition Approach for the Multi-level Capacitated Lot-Sizing Problem with Setup Carryover, Backlogging, and Emission Control

An Effective Particle Swarm Optimization for Lot-Sizing Problem with Particulate Matter Emission Constraint

An Effective Particle Swarm Optimization for Lot-Sizing Problem with Particulate Matter Emission Constraint

Forecast Horizon of Dynamic Lot Sizing Problem with Perishable Inventory and Multiple Chain Stores: Shipping and Stockout Cost

Perishable products are very common, but managing inventory of perishable products can be very challenging for firms, especially in distribution systems, including multiple chain stores. In this environment, we consider a dynamic lot-sizing problem faced by a distribution center that dis-patches a single perishable product to multiple chain stores. Demand cannot be backlogged, but it does not have to be satisfied; unsatisfied demand means stockout (lost sale). The first step is to transform the total profit function into a special total cost function. Our next step is to explore the properties of the optimal solution and use them to formulate a dynamic programming algorithm to solve the problem. Furthermore, we establish forecast and decision horizon results, which help the operation manager to decide the precise forecast horizon in a rolling decision-making process. Based on the model setting and the methods of dynamic programming, we obtained two interesting findings: (1) the maximized profit objective function is equivalent to the minimized cost objective function, and (2) the famous zero inventory property conditionally holds in the inventory management of perishable products. On an extensive test bed, useful insights were obtained on the impact of the lifetime of the product and cost parameters on the total cost and length of the forecast horizon. Thus, the contributions of this study are as follows: (1) we explore two structure policies in an optimal solution to devise efficient algorithms to reduce computational complexity; (2) we provide a sufficient condition for forecasting and decision horizons; and (3) we determine that, for a given fixed cost, the median forecast horizon first increases with the lifetime of the product and stockout cost and then remains invariable when it reaches a certain level.

Integrated shelf-life rules for multi-level pharmaceutical tablets manufacturing processes

This paper discusses the multi-level capacitated lot-sizing problem with linked lot sizes and backorders (MLCLSP-L-B) considering deterministic product shelf-life applied to pharmaceutical tablets manufacturing processes. This paper's motivation, essential concepts, and conclusions for the pharmaceutical industry were already presented and discussed in the International Workshop on Lot-Sizing (IWLS), see Simonis and Nickel (2023a. International Workshop on Lot-Sizing-IWLS'2023, Vol. 13, 30–33). Material's shelf-life depends on the remaining shelf-life of issued ingredients in tablets manufacturing processes. This particular shelf-life behaviour is named integrated shelf-life rules. The MLCLSP-L-B is extended by integrated shelf-life rules (MLCLSP-L-B-SL). Moreover, an exact mathematical problem formulation is provided. First In-First Out (FIFO) and First Expire-First Out (FEFO) heuristics are developed for the MLCLSP-L-B. The MLCLSP-L-B-SL, FIFO, and FEFO heuristics are evaluated based on anonymised real-world data of five multi-level tablets manufacturing problem instances. Additionally, proposed solutions regarding manufacturing costs and shelf-life conflicts are compared. Finally, planning rules and managerial insights are derived for tablets manufacturing processes.

Simultaneous Pickup-and-Delivery Production-Routing Problem in closed-loop supply chain with remanufacturing and disassembly consideration

In a context of Closed-Loop Supply Chain, Production-Routing Problem with Pickup and Delivery is presented. In its basic form, Production-Routing Problem (PRP) attempts to solve jointly Dynamical Lot-Sizing and Inventory-Routing Problems. In this paper, a novel PRP model seeking to minimize the total cost of manufacturing, remanufacturing, disassembly, inventory, and routing by taking into account the remanufacturing and disassembly processes of End-of-Life returned products is provided. Novel hybrid heuristics based on Two-Phase Iterative and Relax-and-Fix heuristics were developed and outperform Branch-and-Cut algorithm for large size instances with a small vehicle capacity. Main insights derived from sensitivity analyses are (i) managers can use the model to trade off the cost saving resulting from the integration of forward and reverse flows, on the one hand, and any additional cost incurred by the organizational changes related to the integration, on the other hand, (ii) a high level of remanufacturing rate for returned products is not necessarily profitable, and managers should develop a return policy that leads to the appropriate level of remanufacturing rate, and (iii) investing in expanding the remanufacturing or the disassembly capacities can significantly reduce the total cost until a threshold level, beyond which any further investment is not beneficial.

An integrated optimization model for procurement and production lot sizing and scheduling problems

Abstract Lot sizing is a prevalent issue within manufacturing companies, where determining the optimal procurement and production lot sizes is crucial for maximizing profits. This problem has become more complex, given that numerous suppliers can provide the same raw materials with different prices and quantity discount schemes. A company should also determine optimal carriers to deliver materials to the company’s warehouse. In a manufacturing process, the company should determine the optimal production lot size and its schedules. In this paper, a model was developed to solve simultaneously procurement and production lot sizing, as well as production scheduling problems. The model encompasses multiple suppliers offering quantity discounts, aiming to maximize company profit by accounting for various costs, including procurement, production, inventory, and quality costs. A case study is taken from a company producing noodles and its related derivative products to illustrate the application of the model. Based on the optimization results, the company obtained a total profit of IDR. 14,656,550,000 or $950,921.30 (the exchange rate of $1 at IDR. 15,413). The sensitivity analysis results show that the objective function is sensitive to changes in the purchase cost, sale revenue, and discount rate parameters. The decision variables for accepted product demand, product quantity, and the starting and completion time of product family are only sensitive to changes in certain parameters. Meanwhile, the decision variables for product inventory, product backlog, raw material inventory, and purchased raw material quantity are sensitive to the changes in all the analyzed parameters.

Complexity analysis of integrated dynamic lot sizing and maintenance planning problems

We consider a problem where the planning decisions of producing on and maintaining a single machine are integrated. The age of the machine, and thus when maintenance should be performed, is influenced by the production decisions. Decisions are taken on a planning horizon discretized in periods. The computational complexity of several variants of the single-item problem are studied depending on whether (i) Maintenance can be performed on the machine at any point in time in a period or must be performed at the end of a period, (ii) There is a fixed aging component when starting production, and (iii) There is a minimum age before maintenance can be performed on the machine. The multi-item case is proved to be NP-hard.

Risk-averse two-stage stochastic programming for assembly line reconfiguration with dynamic lot sizes

In this paper, a comprehensive optimization problem is developed for a composite of an assembly line reconfiguration problem with multiple lines and a capacitated lot-sizing problem. Multiple products are considered, whose demand is uncertain and is dynamically forecasted. The production planner is assumed to be risk-averse, and decisions are made contingent upon the risk preference. To model the problem, a stochastic program with two stages is utilized. A solution approach is devised using a divide-and-conquer algorithm, which incorporates a set of valid inequalities. The effectiveness and efficiency of the proposed solution approach are assessed through a series of computational tests. Finally, a case study focusing on an engine production process is presented, leading to the derivation of several valuable insights.

Training Demand Prediction Models by Decision Error for Two-Stage Lot-Sizing Problems

Demand prediction to support appropriate production decisions is being actively studied. Many prediction models are designed to minimize the prediction error, which is measured by determining the difference between the predicted and ground-truth demand. However, these models ignore the effect of the prediction error on downstream production decisions. This prompted our study, which focuses on demand prediction models for two-stage uncapacitated lot-sizing problems. In this paper, we present a prediction model that minimizes the decision error, which is measured by the optimization objective of lot-sizing problems. Our model mitigates the impact of prediction errors by leveraging the structure of the lot-sizing problems. To enhance the ability to accommodate imperfect data, such as data based on inaccurate information, we subsequently extend the prediction models to distributionally robust versions. We consider the worst-case formulation in the feature space to enhance the robustness of the model to data imperfection. Numerical experiments demonstrate that the proposed prediction models are significantly superior to the traditional prediction methods when the model being trained is misspecified. In addition, the robust extension enables the models to train well on imperfect datasets while requiring less training data. <italic xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">Note to Practitioners</i> —This study is motivated by the two-stage lot-sizing problem in manufacturing systems which involves the determination of the time at which (the setup decisions) to produce before the demand is revealed, and the production quantity decisions are adjusted during production processing. Predicting the demand properly while making the set-up decisions helps to decrease the production cost. This paper proposes a novel method that aims to minimize the total production cost to train demand prediction models. Practitioners can deploy the proposed method to train popular prediction models including linear prediction, decision tree, and neural network models. It is shown that the production cost associated with the decisions based on our models is lower than that associated with the traditional prediction models. Furthermore, our method enhances the ability of the model to handle model misspecification and imperfect data. Practitioners can apply our method to overcome the problems caused by imperfect data such as those containing contaminated and inaccurate measurements.

A Mixed-integer Linear Programming Model for Solving Multi-item Uncapacitated Lot-sizing Problems with Storage Capacities Limitations: A Case Study of an Apparel Manufacturing Company

This study aimed to use a mixed-integer linear programming model to solve the multi-item uncapacitated lot size problem with storage capacity limitations for apparel manufacturing. In real-world operations, sometimes the storage capacity of cloth rolls can be insufficient at any given time. This would be because the sum of the remaining cloth rolls from the previous period and the receiving cloth rolls from the existing period would be a vast number. Furthermore, the extra cloth on the rolls would have to be carried and kept in the production line. Consequently, this caused the late issue of the distribution of the material, the loss of the cloth rolls, and the failure to count the cloth rolls. As a result, we conducted a multi-item cloth roll replenishment plan to meet the production line's demand and keep all the inventory and receiving cloth rolls. Using mixed-integer linear programming, the multi-item lot size of each period and the sum of the cloth, ordering, and holding costs was calculated, including the number of occupied cloth racks that did not exceed the number of available cloth racks in the cloth storage. Then, this result was compared with the calculation of a mixed-integer linear programming model based on the procurement policies of a case study company. This study used OpenSolver and GAMS/CPLEX software to calculate the solutions. The results found that in using the procurement policy, we could provide the maximum number of occupied cloth racks, about 368 racks, or 2.78 times the racks on the available shelves of the cloth storage. The result of the proposed model indicated that the number of occupied cloth racks to be installed did not exceed the number of available racks on the shelves. The total cost of the model of the procurement policy was less than the proposed model by about 0.26 percent because ordering occurred only about 25 times.

A web-based interactive decision support system for a multi-objective lot-sizing and production scheduling model

In the course of globalization and the corresponding steadily growing market dynamics, companies are faced with an increasing challenge to remain competitive. Therefore, the efficient use of the available resources, while maintaining a high level of service and customer satisfaction, is a key instrument for ensuring success. Within this framework, production scheduling can be seen as one of the most decisive operational activities of a manufacturing company, with the production orders being sequenced in a way that allows the achievement of an efficient allocation of the available resources. Furthermore, the correlated lot-sizing represents a crucial influencing factor in production scheduling. In this article, a novel methodology to solve multi-objective lot-sizing and production scheduling problems is proposed. Essentially, it provides an interactive three-phase framework that supports a procedure for simultaneous decision-making regarding lot-sizing and production scheduling. The main goal is not only to find an appropriate production schedule that best meets the decision-maker’s preferences but also to gain a deeper insight into the relationship among criteria and a better understanding of possible trade-offs between competing objectives. To guide the user through the whole decision-making process, a web-based interactive application was developed. The proposed methodology was applied to a case study regarding an automotive supplier company, revealing itself as a viable and practical technique able to solve real-world multi-objective lot-sizing and production scheduling problems in the manufacturing industry.

Matheuristics vs. metaheuristics for joint lot-sizing and dynamic pricing problem with nonlinear demands

This paper considers a joint dynamic pricing and production planning decisions problem for a profit-maximizing firm that produces and sells multiple products. The objective is to develop a coordinated decision approach for multi-product pricing and lot sizing decisions for a manufacturer considering a limited production capacity. The demand for each product is assumed to be iso-elastic and integrates the complementarity and the substitution effects between the products. First, the problem is formulated as a non-convex mixed integer nonlinear programming model (MINLP) incorporating capacity constraints, setup costs, and nonlinear demand functions. Then, since the model is nonlinear and non-convex, a set of approximate approaches based on the Genetic algorithm, Late Acceptance Hill Climbing and Simulated Annealing methods are designed to solve this problem. Based on this study, the performances of two variants of approximate methods: matheuristics and metaheuristics are discussed and analyzed. The extensive experimental study, performed on real-world inspired instances, shows that matheuristic methods with setup-variables encoding scheme outperform the rest of the methods. The research outcomes show that coordinating a decision-making process by optimizing both prices and production plans simultaneously can result in significant profit for a company. However, one must consider the joint effect of the parameters of the demand function as well as the impact of the production capacity. This comprehensive understanding enables the company to avoid excessive investments in less lucrative products with lower sales potential, thereby ensuring resource allocation aligns with profitability and market demand.

Exact Method for Production Hub Location

This study proposes a production hub location (PHL) problem that integrates the classical multiplant lot-sizing and hub location problems. The PHL problem is to determine the location of production facilities, lot-sizing, inventory, hub location, and the distribution of multiple commodities from plants to customers, with an objective to minimize the total production, setup, inventory, hub operating, and transportation costs. The PHL problem applies to manufacturing companies that either have built a hub-and-spoke distribution network or are accessible to such a network through collaborations with other third-party logistics companies. Because the PHL problem is [Formula: see text]-hard, we propose an exact method that integrates dynamic programming and Benders decomposition (DPBD) for solving the problem. The DPBD method is enhanced by exploring several problem properties, such as a multicut reformulation, the generation of Pareto-optimal cuts, a two-stage hub elimination and restoration procedure, and the inclusion of a novel heuristic procedure. We compare the PHL model with several related models theoretically and computationally with newly created benchmark instances. The computational results indicate that the proposed model can reduce the total costs and facilitate better network designs and system decisions, highlighting the value of an integrated approach. We also provide managerial insights into the benefits of integration and show the efficiency of the DPBD method through an extensive number of computational tests. History: Accepted by Andrea Lodi, Area Editor for Design and Analysis of Algorithms—Discrete. Supplemental Material: The online supplement is available at https://doi.org/10.1287/ijoc.2023.0339 .

Multi-period descriptive sampling for scenario generation applied to the stochastic capacitated lot-sizing problem

Using scenarios to model a stochastic system’s behavior poses a dilemma. While a large(r) set of scenarios usually improves the model’s accuracy, it also causes drastic increases in the model’s size and the computational effort required. Multi-period descriptive sampling (MPDS) is a new way to generate a small(er) set of scenarios that yield a good fit both to the periods’ probability distributions and to the convoluted probability distributions of stochastic variables (e.g., period demands) over time. MPDS uses descriptive sampling to draw a sample of S representative random numbers from a period’s known (demand) distribution. Now, to create a set of S representative scenarios, MPDS heuristically combines these random numbers (period demands) period by period so that a good fit is achieved to the convoluted (demand) distributions up to any period in the planning interval. A further contribution of this paper is an (accuracy) improvement heuristic, called fine-tuning, executed once the fix-and-optimize (FO) heuristic to solve a scenario-based mixed integer programming model has been completed. Fine-tuning uses linear programming (LP) with fixed binary variables (e.g., setup decisions) generated by FO and iteratively adapts production quantities so that compliance with given expected service level constraints is reached. The LP is solved with relatively little computational effort, even for large(r) sets of scenarios. We show the advancements possible with MPDS and fine-tuning by solving numerous test instances of the stochastic capacitated lot-sizing problem under a static uncertainty approach.

A cooperative coevolutionary hyper-heuristic approach to solve lot-sizing and job shop scheduling problems using genetic programming

Lot-sizing and scheduling in a job shop environment is a fundamental problem that appears in many industrial settings. The problem is very complex, and solutions are often needed fast. Although many solution methods have been proposed, with increasingly better results, their computational times are not suitable for decision-makers who want solutions instantly. Therefore, we propose a novel greedy heuristic to efficiently generate production plans and schedules of good quality. The main innovation of our approach represents the incorporation of a simulation-based technique, which directly generates schedules while simultaneously determining lot sizes. By utilising priority rules, this unique feature enables us to address the complexity of job shop scheduling environments and ensures the feasibility of the resulting schedules. Using a selection of well-known rules from the literature, experiments on a variety of shop configurations and complexities showed that the proposed heuristic is able to obtain solutions with an average gap to Cplex of 4.12%. To further improve the proposed heuristic, a cooperative coevolutionary genetic programming-based hyper-heuristic has been developed. The average gap to Cplex was reduced up to 1.92%. These solutions are generated in a small fraction of a second, regardless of the size of the instance.

Reinforcement learning approaches for the stochastic discrete lot-sizing problem on parallel machines

This paper addresses the stochastic discrete lot-sizing problem on parallel machines, which is a computationally challenging problem also for relatively small instances. We propose two heuristics to deal with it by leveraging reinforcement learning. In particular, we propose a technique based on approximate value iteration around post-decision state variables and one based on multi-agent reinforcement learning. We compare these two approaches with other reinforcement learning methods and more classical solution techniques, showing their effectiveness in addressing realistic size instances.

A modified variable neighborhood search algorithm for dynamic lot-sizing with supplier selection under varying delivery time quotation

This paper investigates a dynamic lot-sizing problem with supplier selection, where the manufacturer develops an aggregate plan to meet product delivery demand with the objective of minimizing the cost. Specifically, a novel supplier selection scenario with varying delivery time quotation is considered, i.e., the manufacturer places raw material replenishment orders with varying delivery time quotation and dynamically selects suppliers, and the suppliers guarantee delivery within the quoted delivery lead time. In the dynamic lot-sizing problem with supplier selection, a joint decision on the production plan, replenishment order, and supplier selection needs to be made. To tackle this complex problem, a modified Variable Neighborhood Search algorithm with nested two parallel neighborhoods (PN-VNS) is first developed for the decision on the production plan and replenishment order. Then, the decision on supplier selection needs to be made for the given replenishment order, and a pre-pruning optimization rule is proposed to obtain the optimal supplier scheme. Computational experiments are conducted to evaluate the performance of the proposed algorithm, and the experimental results demonstrate that our algorithm performs better both in terms of solution quality and running time.

Efficient approximate algorithms for a class of dynamic lot size problems under product substitution

Efficient approximate algorithms for a class of dynamic lot size problems under product substitution