Capacitated Lot-sizing Problem Research Articles

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.

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

AbstractUsing 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 heuristic approach for the integrated production–transportation problem with process flexibility

We study an integrated multi-item production and distribution problem considering a network of multiple plants and clients, who are geographically dispersed, with direct shipments from the plants to the clients. In addition to the decisions on production and distribution, a decision needs to be taken on the level of process flexibility in the network, i.e., which items can be produced in which plants. There is a clear trade-off between these decisions. On the one hand, a network with total flexibility where each plant can produce all types of items allows for lower transportation costs, but requires large investments in flexibility and frequent setups. On the other hand, a network with a limited amount of flexibility where each plant produces only a few items, will increase the transportation costs, but requires a lower investment in flexibility. We model this problem as an extension of the capacitated lot-sizing problem. We limit the investment in flexibility by a budget constraint and minimize the operational costs. Varying this budget allows us to analyze different levels of flexibility. In this paper, we propose mathematical models and a hybrid solution method that combines a mixed integer programming-based approach and a kernel search heuristic. Our computational results using data sets from the literature show that the proposed hybrid method produces on average better solutions with significantly lower computational times when compared with the results produced by a state-of-the-art optimization software. Additional computational results are presented by varying key parameters and analyzing their impact on the value of flexibility. These computational experiments indicate that some of the main managerial insights which were derived in the literature for the case without transportation costs are no longer valid when we consider transportation costs.

A matheuristic approach for the multi-level capacitated lot-sizing problem with substitution and backorder

The lot-sizing problem aims at determining the products to be produced and their quantities for each time period, which is a difficult problem in production planning. This problem becomes even more complicated when practical aspects such as limited production capacity, bill of materials, and item substitution are considered. In this paper, we study a new variant of the lot-sizing problem, called the multi-level capacitated lot-sizing problem with substitution and backorder. Unlike previous studies, this variant considers substitutions at both the product and component levels, which is based on the real needs of manufacturers to increase planning flexibility. Backorders are allowed, but should be delivered within a certain time limitation. We formulate this problem using a mathematical programming model. A matheuristic approach is proposed to solve the problem. This first generates an initial feasible solution using a relax-and-fix algorithm, and then improves it using a hybrid fix-and-optimise algorithm. The proposed algorithm is calibrated with a full factorial design of experiments, and its efficiency is well validated. Finally, through extensive numerical experiments, we analyse the properties of this new lot-sizing problem, such as the effect of substitution options, and the influence of backorder time limitation, and provide several useful managerial insights for manufacturing companies to save costs in production planning.

Improvement to an existing multi-level capacitated lot sizing problem considering setup carryover, backlogging, and emission control

This paper presents a multi-level, multi-item, multi-period capacitated lot-sizing problem. The lot-sizing problem studies can obtain production quantities, setup decisions and inventory levels in each period fulfilling the demand requirements with limited capacity resources, considering the Bill of Material (BOM) structure while simultaneously minimizing the production, inventory, and machine setup costs. The paper proposes an exact solution to Chowdhury et al. (2018)'s[1] developed model, which considers the backlogging cost, setup carryover & greenhouse gas emission control to its model complexity. The problem contemplates the Dantzig-Wolfe (D.W.) decomposition to decompose the multi-level capacitated problem into a single-item uncapacitated lot-sizing sub-problem. To avoid the infeasibilities of the weighted problem (WP), an artificial variable is introduced, and the Big-M method is employed in the D.W. decomposition to produce an always feasible master problem. In addition, Wagner & Whitin's[2] forward recursion algorithm is also incorporated in the solution approach for both end and component items to provide the minimum cost production plan. Introducing artificial variables in the D.W. decomposition method is a novel approach to solving the MLCLSP model. A better performance was achieved regarding reduced computational time (reduced by 50%) and optimality gap (reduced by 97.3%) in comparison to Chowdhury et al. (2018)'s[1] developed model.© 2023 Society of Manufacturing Engineers (SME). Published by Elsevier Ltd. All rights reserved.This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/)Peer-review under responsibility of the Scientific Committee of the NAMRI/SME.

Capacitated lot sizing problem with periodic carbon emission constraints and multiple resources

We study the single item capacitated lot sizing problem with multiple resources and periodic carbon emission constraints that impose an upper bound for the average emission per product produced in any period. Although the uncapacitated version of this problem can be solved in polynomial time, generalisation of the problem including the resource capacities is NP-Hard, in general. We present important structural properties for the optimal solutions of the problem. We consider the special cases with two resources and under non-speculative costs, construct the piecewise linear total production cost function when the resource capacities, and the emission and cost parameters are time-invariant, and develop a polynomial time dynamic programming algorithm (DP) to solve them. Then, we generalise the procedure to construct the total production cost function and the DP for the general setting with fixed number of capacitated resources. We test our algorithm for different problem instances, and compare it with a commercial solver and a DP available in the literature for solving the lot sizing problem with piecewise concave production cost functions. The results reveal that our DP outperforms the other one, and it performs better than the commercial solver when the number of breakpoints of the total production cost function is small.

Learning Optimal Solutions via an LSTM-Optimization Framework

In this study, we present a deep learning-optimization framework to tackle dynamic mixed-integer programs. Specifically, we develop a bidirectional Long Short Term Memory (LSTM) framework that can process information forward and backward in time to learn optimal solutions to sequential decision-making problems. We demonstrate our approach in predicting the optimal decisions for the single-item capacitated lot-sizing problem (CLSP), where a binary variable denotes whether to produce in a period or not. Due to the dynamic nature of the production and inventory levels that must meet the periodic demand, the CLSP can be treated as a sequence labeling task where a recurrent neural network can capture the problem’s temporal dynamics. Computational results show that our LSTM-Optimization (LSTM-Opt) framework significantly reduces the solution time of benchmark CLSP problems without much loss in feasibility and optimality. For example, the predictions at the 85% level reduce the CPLEX solution time by a factor of 9 on average for over 240000 test instances with an optimality gap of less than 0.05% and 0.4% infeasibility in the test set. Also, models trained using shorter planning horizons can successfully predict the optimal solution of the instances with longer planning horizons. For the hardest data set, the LSTM predictions at the 25% level reduce the solution time of 70 CPU hours to less than 2 CPU minutes with an optimality gap of 0.8% and without any infeasibility. The LSTM-Opt framework outperforms classical ML algorithms, such as the logistic regression and random forest, in terms of the solution quality, and exact approaches, such as the ( $$\ell$$ , S) and dynamic programming-based inequalities, with respect to the solution time improvement. Our machine learning approach could be beneficial in tackling sequential decision-making problems similar to CLSP, which need to be solved repetitively, frequently, and in a fast manner.

New construction heuristic for capacitated lot sizing problems

We consider the classical single-level multi-item capacitated lot-sizing problem (CLSP) which is the core model for production planning. It is NP-hard and if setup operations consume capacity the feasibility problem itself is NP-complete. Several construction heuristics have been proposed in the research literature, but none of them achieves a sufficient solution quality and generality at the same time - meaning that they can be applied to different variations of the problem easily. We propose a general two-step construction heuristic (2-SCH) which sorts the customer orders in the first step and iteratively adds them to a preliminary production plan in the second step. Hence, various problem classes can be solved easily and fast. Computational experiments on the CLSP without setup times show that the 2-SCH outperforms the well-known Dixon–Silver (Dixon & Silver, 1981) and the ABC heuristics Maes and Van Wassenhove (1986) and provides better results than the genetic programming approach proposed recently by Hein et al. (2018). We also apply it to the CLSP with setup times where it outperforms the construction heuristic proposed by Trigeiro (1989). Finally, we show the flexibility of the 2-SCH by applying it to the CLSP with backorders and the multi-level CLSP as well as the single-level CLSP in an online environment.

A simulation–optimization approach for a cyclic production scheme in a tablets packaging process

This paper discusses an iterative simulation–optimization approach to estimate high-quality solutions for the capacitated lot-sizing problem with linked lot sizes and backorders (CLSP-L-B) based on probabilistic demands. An uncertainty framework for incorporating the impact of simulated demand scenarios is embedded into the CLSP-L-B. The framework is generalized by a variable neighborhood search (VNS) approach, which accelerates the search for stable and cyclic production schemes. Moreover, an exact mathematical problem formulation is introduced for the generalized model framework. Anonymized real-world data of four tablets packaging robots is used for evaluation. The experimental design covers nine classes of demand uncertainty per packaging robot. Additionally, proposed solutions are evaluated against an established benchmark approach from literature for each uncertainty class in terms of manufacturing costs and customer service levels. Finally, planning rules and managerial insights are given for the packaging robots.

A deep learning approach for integrated production planning and predictive maintenance

This paper considers a multi-period multi-product capacitated lot-sizing problem. It develops an integrated predictive maintenance and production planning framework using deep learning and mathematical programming. The objective is to minimise the sum of maintenance, setup, holding, backorder, and production costs, while satisfying the demand for all products over the horizon under consideration. Based on a rolling horizon approach, the model dynamically integrates data-driven predictive maintenance and production planning. The used maintenance policy includes replacements and minimal repairs that are considered as preventive and corrective maintenance, respectively. To select preventive maintenance actions, a long short-term memory model is employed to accurately predict the health condition of the machine. Each rolling horizon consists of ordinary and forecast stages, and by collecting new sensor data, the maintenance and production decisions are simultaneously updated. The resulting integrated framework is validated using a benchmarking data set. The results are compared for different approaches to highlight the advantages of the proposed framework.

A capacitated lot-sizing problem in the industrial fashion sector under uncertainty: a conditional value-at-risk framework

In this study, we present a multi-product, multi-period inventory control problem under uncertainty in product demands that emerges in the fashion industry. A two-stage stochastic model is proposed to design a planning strategy where the total cost incurred by purchase orders, inventory and shortage is minimised. We incorporate the Conditional Value at Risk (CVaR) within the formulation to address exogenous uncertainty. An industrial case study involving a Mexican fashion retail company was considered to assess the performance of the two-stage stochastic model. Scenarios were considered using historical data provided by the company. A sensitivity analysis was also conducted on risk-aversion parameters to assess how the values of these parameters affect the behaviour of the proposed formulation. The results show that the proposed two-stage stochastic formulation is an efficient and practical approach to handle exogenous uncertainty in industrial-scale capacitated lot-sizing problems.

Exact algorithms for multi-module capacitated lot-sizing problem, and its generalizations with two-echelons and piecewise concave production costs

We study new generalizations of the classic capacitated lot-sizing problem with concave production (or transportation), holding, and subcontracting cost functions in which the total production (or transportation) capacity in each time period is the summation of capacities of a subset of n available modules (machines or vehicles) of different capacities. We refer to this problem as Multi-module Capacitated Lot-Sizing Problem without or with Subcontracting, and denote it by MCLS or MCLS-S, respectively. These are NP-hard problems if n is a part of the input and polynomially solvable for n = 1. In this article we address an open question: Does there exist a polynomial time exact algorithm for solving the MCLS or MCLS-S with fixed ? We present exact fixed-parameter tractable (polynomial) algorithms that solve MCLS and MCLS-S in time for a given It generalizes algorithm of Atamtürk and Hochbaum [Management Science 47(8):1081–1100, 2001] for MCLS-S with n = 1. We also present exact algorithms for two-generalizations of the MCLS and MCLS-S: (a) a lot-sizing problem with piecewise concave production cost functions (denoted by LS-PC-S) that takes time, where m is the number of breakpoints in these functions, and (b) two-echelon MCLS that takes time. The former reduces run time of algorithm of Koca et al. [INFORMS J. on Computing 26(4):767–779, 2014] for LS-PC-S by 93.6%, and the latter generalizes algorithm of van Hoesel et al. [Management Science 51(11):1706–1719, 2005] for two-echelon MCLS with n = 1. We perform computational experiments to evaluate the efficiency of our algorithms for MCLS and LS-PC-S and their parallel computing implementation, in comparison to Gurobi 9.1. The results of these experiments show that our algorithms are computationally efficient and stable. Our algorithm for MCLS-S addresses another open question related to the existence of a polynomial time algorithm for optimizing a linear function over n-mixing set (a generalization of the well-known 1-mixing set).

The Integrated Production-Inventory-Routing Problem with Reverse Logistics and Remanufacturing: A Two-Phase Decomposition Heuristic

Sustainable supply chains depend on three critical decisions: production, inventory management, and distribution with reverse flows. To achieve an effective level of operational performance, policymakers must consider all these decisions, especially in Closed-Loop Supply Chains (CLSCs) with remanufacturing option. In this research paper, we address the Integrated Production-Inventory-Routing Problem with Remanufacturing (IPIRP-R) of returned End-Of-Life (EOL) products. The aim behind solving this optimization problem is to minimize conjointly the total manufacturing, remanufacturing, setup, inventory, and routing costs over the planning horizon. A two-phase decomposition heuristic is developed to solve the model iteratively. Our study finds its originality in the fact of jointly optimizing the Capacitated Lot-Sizing Problem with Remanufacturing (CLSP-R) option and the Vehicle Routing Problem with Simultaneous Pick-up and Delivery (VRPSPD) in a single framework. Numerical results showed that our solution approach provides good solutions regarding small and medium-scale size instances under acceptable computational time, especially for problems occurring with significant manufacturing and remanufacturing costs under relatively low pickup requests.

A dynamic programming approach for the two-product capacitated lot-sizing problem with concave costs

In this paper, we analyze a two-product multi-period dynamic lot-sizing problem with a fixed capacity constraint in each period. Each product has a known demand in each period that must be satisfied over a finite planning horizon. The aim of this problem is to minimize the overall cost of placing orders and carrying inventory across all periods. The structure of an optimal solution is analyzed with respect to a type of period called regeneration period, which is a period where the inventory of one or both products reach zero. We show that there is an optimal arrangement of placing orders between consecutive regeneration periods. We propose a pseudo-polynomial algorithm to solve the two-product problem. First, we show how the optimal ordering pattern between two consecutive regeneration periods can be solved using a shortest path problem. Then, we explain how the optimal locations for regeneration periods can be found by solving a shortest path problem on a different network, where each arc corresponds to the shortest path in a subproblem network. We then show how this approach can be scaled up to a three-product problem and generalize this technique to any number of products, as long as it is small.

The robust multi-plant capacitated lot-sizing problem

The robust multi-plant capacitated lot-sizing problem

Self-adaptive randomized constructive heuristics for the multi-item capacitated lot sizing problem

The Capacitated Lot-Sizing Problem (CLSP) and its variants are important and challenging optimization problems. Constructive heuristics are known to be the most intuitive and fastest methods for finding good feasible solutions for the CLSPs and therefore are often used as a subroutine in building more sophisticated exact or metaheuristic approaches. Classical constructive heuristics, such as period-by-period heuristics and lot elimination heuristics, are widely used by researchers. This paper introduces four perturbation strategies to the period-by-period and lot elimination heuristics to further improve the solution quality. We propose a new procedure to automatically adjust the parameters of the randomized period-by-period (RPP) heuristics. The procedure is proved to offer better solutions with reduced computation times by improving time-consuming parameter tuning phase. Combinations of the self-adaptive RPP heuristics with Tabu search and lot elimination heuristics are tested to be effective. Computational experiments provided high-quality solutions with a 0.88% average optimality gap on benchmark instances of 12 periods and 12 items, and an optimality gap within 1.2% for the instances with 24 periods and 24 items.

The capacitated lot-sizing and energy efficient single machine scheduling problem with sequence dependent setup times and costs in a closed-loop supply chain network

The capacitated lot-sizing and energy efficient single machine scheduling problem with sequence dependent setup times and costs in a closed-loop supply chain network

Adaptive Genetic Algorithm Based on Fuzzy Reasoning for the Multilevel Capacitated Lot-Sizing Problem with Energy Consumption in Synchronizer Production

The multilevel capacitated lot-sizing problem (MLCLSP) is a vital theoretical problem of production planning in discrete manufacturing. An improved algorithm based on the genetic algorithm (GA) is proposed to solve the MLCLSP. Based on the solution results, the distribution of energy consumption in a synchronous production case is analyzed. In the related literature, the GA has become a much-discussed topic in solving these kinds of problems. Although the standard GA can make up for the defects of the traditional algorithm, it will lead to the problems of unstable solution results and easy local convergence. For these reasons, this research presents an adaptive genetic algorithm based on fuzzy theory (fuzzy-GA) to solve the MLCLSP. Firstly, the solving process of the MLCLSP with the fuzzy-GA is described in detail, where algorithms for key technologies such as the capacity constraint algorithm and the algorithm of solving fitness value are developed. Secondly, the auto-encoding of decision variables for MLCLSPs is studied; within this, the decision variables of whether to produce or not are encoded into a hierarchical structure based on the bill of material; combined with external demand, the decision variables of lot-sizing are constructed. Thirdly, the adaptive optimization process of parameters of the GA for the MLCLSP based on fuzzy theory is expounded, in which membership function, fuzzy rule, and defuzzification of the MLCLSP is mainly presented. Experimental studies using the processed dataset collected from a synchronizer manufacturer have demonstrated the merits of the proposed approach, in which the energy consumption distribution of the optimized production plan is given. The optimal lot-sizing is closer to the average value of the optimal value compared with the standard GA, which indicates that the proposed fuzzy-GA approach has better convergence and stability.

Using the proximal policy optimisation algorithm for solving the stochastic capacitated lot sizing problem

This paper studies the multi-item stochastic capacitated lot-sizing problem with stationary demand to minimise set-up, holding, and backorder costs. This is a common problem in the industry, concerning both inventory management and production planning. We study the applicability of the Proximal Policy Optimisation (PPO) algorithm in this problem, which is a type of Deep Reinforcement Learning (DRL). The problem is modelled as a Markov Decision Process (MDP), which can be solved to optimality in small problem instances by using Dynamic Programming. In these settings, we show that the performance of PPO approaches the optimal solution. For larger problem instances with an increasing number of products, solving to optimality is intractable, and we demonstrate that the PPO solution outperforms the benchmark solution. Several adjustments to the standard PPO algorithm are implemented to make it more scalable to larger problem instances. We show the linear growth in computation time for the algorithm, and present a method for explaining the outcomes of the algorithm. We suggest future research directions that could improve the scalability and explainability of the PPO algorithm.

Solving a parallel-line capacitated lot-sizing and scheduling problem with sequence-dependent setup time/cost and preventive maintenance by a rolling horizon method

This paper presents a novel Mixed-Integer Nonlinear Programming (MINLP) model for a parallel-line Capacitated Lot-Sizing Problem (CLSP) with the sequence-dependent setup time/cost, due date, and preventive maintenance planning. The target of this integrated model is to specify optimal lot sizes, inventory, and shortage levels along with the optimal preventive maintenance plan and the best sequence, starting time, and completion time of the lots. The production and scheduling aspect of the problem is modeled as the CLSP with the sequence-dependent setup time/cost and due date, while maintenance sub-problem is modeled by using the time-based maintenance technique and age-reduction modeling concept with the actual run-time and Exponential lifetime distribution. For the complexity of the model, the MIP-based Rolling Horizon (RH) heuristic algorithms are developed to solve the linearized model in a reasonable truncated time. The computational results obtained from generated small- to large-sized instances show that the proposed RH1 and RH2 heuristic methods report the final solutions with better quality (lower relative gap and total cost) compared with the solutions reported by the truncated simple CPLEX method. Especially, the RH2 method provides a reasonable production plan, better-adjusted schedules with respect to the due dates, and a proper PM plan in comparison with the other solution methods. Also, the featured model develops an optimal production, scheduling, and maintenance plan with higher feasibility, more available production time, and lower total cost in comparison with the classic CLSP with the sequence-dependent setups that neglects the effects of sudden failures and corrective maintenance actions.