Capacitated Lot-sizing Problem Research Articles

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.

Discrete multi-module capacitated lot-sizing problems with multiple items

We study single-item discrete multi-module capacitated lot-sizing problems without and with backlogging where the amount produced in each time period is equal to the summation of binary multiples of the capacities of n available different modules (or machines). For fixed n≥2, we develop fixed-parameter tractable (polynomial) exact algorithms that generalize the algorithms of van Vyve (2007) for n=1. We utilize these algorithms within a Lagrangian decomposition framework to solve their multi-item versions. Our algorithms are computationally efficient and stable, in comparison to Gurobi 9.0.

Variable neighborhood search for the capacitated lot sizing problem with remanufacturing and overtime

This study proposes a mixed integer linear programming (MILP) model for the single-item multi-period capacitated lot-sizing problem with remanufacturing. The tackled problem is capacitated in terms of production and storage. The production capacity is overcome by allowing for overtime in some periods. Moreover, the customer demand is assumed to be dynamic and deterministic. Demand is satisfied by either producing new items or remanufacturing returned items. The objective is to obtain an optimal plan over a finite planning horizon. The structure of the problem results in strongly NP-hard problem. Consequently, we develop a variable neighborhood search (VNS) metaheuristic algorithm for solving this problem. We demonstrate the robustness of the proposed VNS approach by examining different problem instances with different sizes. The obtained results are compared with the output of CPLEX solver in GAMS. The results obtained from the proposed metaheuristic outperform the solver output in terms of solutions quality and computational time.

Parallelized Decomposition Approaches to Solve the Capacitated Lot-Sizing Problem with Setup Times and Constraints on Ending Inventories

Parallelized Decomposition Approaches to Solve the Capacitated Lot-Sizing Problem with Setup Times and Constraints on Ending Inventories

Parallelized Decomposition Approaches to Solve the Capacitated Lot-Sizing Problem with Setup Times and Constraints on Ending Inventories

Parallelized Decomposition Approaches to Solve the Capacitated Lot-Sizing Problem with Setup Times and Constraints on Ending Inventories

Motivations and analysis of the capacitated lot-sizing problem with setup times and minimum and maximum ending inventories

This paper first analyzes the negative impact of the end-of-horizon effect when solving the capacitated multi-item lot-sizing problem with setup costs and times on a rolling horizon. Maximum ending inventories for items and a global minimum ending inventory are considered to define a new optimization problem whose optimal solutions are much less impacted by the end-of-horizon effect. Then, a generation scheme is proposed to create new instances with initial inventories and ending inventories. This scheme relies on the analysis of the cyclical production planning problem to derive relevant parameters. Computational experiments are carried out to compare the solutions obtained for original instances of the literature and for the new instances, and to analyze the relevance of the new instances on a rolling horizon.

Clustering-based solution approach for a capacitated lot-sizing problem on parallel machines with sequence-dependent setups*

This paper studies an industrial lot-sizing and scheduling problem coming from the food-industry that extends the multi-item capacitated lot-sizing and includes lost sales, overtimes, safety stock and non-uniform sequence-dependent setups on parallel machines. We introduce two different formulations and adapt the well-known Relax-and-Fix and Fix-and-Optimise heuristics in order to quickly obtain feasible solutions on large industrial instances. The complexity of our problem prevents the procedure to obtain good solutions within the time allocated by practitioners on real-life cases, hence we propose to use a clustering approach to approximate the sequence-dependent setup times. The resulting problem is significantly smaller to solve and experimental results suggest that this transformation effectively improves the solutions found on industrial instances. In particular, the combination of this clustering method and Relax-and-Fix and Fix-and-Optimise procedure turns out to be a promising approach to obtain good solutions in the given time-limit.

A risk-averse two-stage stochastic programming model for a joint multi-item capacitated line balancing and lot-sizing problem

In this paper, a comprehensive production planning problem under uncertain demand is investigated. The problem intertwines two NP-hard optimization problems: an assembly line balancing problem and a capacitated lot-sizing problem. The problem is modelled as a two-stage stochastic program assuming a risk-averse decision maker. Efficient solution procedures are proposed for tackling the problem. A case study related to mask production is presented. Several insights are provided stemming from the COVID-19 pandemic. Finally, the results of a series of computational tests are reported.

Design of efficient multiobjective binary PSO algorithms for solving multi‐item capacitated lot‐sizing problem

In this paper, a multi-item capacitated lot-sizing problem with setup times and backlogging (MICLSP_SB) is addressed with respect to two conflicting objective functions. This nondeterministic polynomial time (NP)-hard problem consists of finding optimal production plans while minimizing, simultaneously, the total cost and the total inventory level. To effectively solve the considered problem, two new versions of multiobjective binary particle swarm optimization are designed. In the first proposed version “standard multiobjective particle swarm optimization” (S-MOBPSO) algorithm, two main contributions are introduced. First, a new particle encoding and decoding method is developed to treat the MICLSP_SB decision variables. Second, the overall violation minimization method and the constraint handling method are combined to effectively handle the problem constraints. To further enhance the overall performance, an improvement procedure that minimizes the capacity constraints violations is developed and embedded into the S-MOBPSO algorithm. This second version is named improved MOBPSO (I-MOBPSO). An illustrative example is presented to explain the application of the proposed algorithms. Five performance metrics are considered to measure and evaluate the effectiveness of the proposed algorithms. Experimental results demonstrate the efficiency of the S-MOBPSO and I-MOBPSO algorithms compared to each other as well as the basic MOBPSO and nondominated sorting genetic algorithm II.

Designing an uncertain bi-objective green leagile capacitated lot sizing problem considering FM/M/C queue system

PurposeThis paper aims to present a novel bi-objective mathematical model for a production-inventory system under uncertainty.Design/methodology/approachThis paper addresses agile and lean manufacturing concepts alongside with green production methods to design an integrated capacitated lot sizing problem (CLSP). From a methodological perspective, the problem is solved in three phases. In the first step, an FM/M/C queuing system is used to minimize the number of customers waited to receive their orders. In the second step, an effective approach is applied to deal with the fuzzy bi-objective model and finally, a hybrid metaheuristic algorithm is used to solve the problem.FindingsSome numerical test problems and sensitivity analyzes are conducted to measure the efficiency of the proposed model and the solution method. The results validate the model and the performance of the solution method compared to Gams results in small size test problems and prove the superiority of the hybrid algorithm in comparison with the other well-known metaheuristic algorithms in large size test problems.Originality/valueThis paper presents a novel bi-objective mathematical model for a CLSP under uncertainty. The proposed model is conducted on a practical case and several sensitivity analysis are conducted to assess the behavior of the model. Using a queue system, this problem aims to reduce the items waited in the queue to receive service. Two objective functions are considered to maximize the profit and minimize the negative environmental effects. In this regard, the second objective function aims to reduce the amount of emitted carbon.

A capacitated lot-sizing model with sequence-dependent setups, parallel machines and bi-part injection moulding

This paper aims to propose a model for solving a capacitated lot-sizing problem with sequence-dependent setups (CLSD) and parallel machines in a bi-part injection moulding (BPIM) context that consists of injecting two different parts or products into the same mould in separate injection cavities. These two parts require the same sequence order and available capacity at the same instant, but generate different part inventories. The addressed problem should be considered a multi-machine CLSD-BPIM one. We provide a mixed integer linear programming (MILP) model for it. The proposed model is based on a real-world case study from a second-tier supplier of the automotive sector. In sequence setup, inventory and resource setup costs terms, the benefits of the proposed multi-machine CLSD-BPIM model are validated by comparing it with a single-machine CLSD-BPIM model that replicates the current planning scenario carried out in the company under study. Additionally, random instances based on the real problem have been generated and tested. The computational results exhibit optimal solutions for the small instances with an average of 0.2-second solution time, the medium instances present average gaps of 0.47% with 2.6 h and large instances present average gaps of 4% with 6 h solution time.

Data-driven branching and selection for lot-sizing and scheduling problems with sequence-dependent setups and setup carryover

The capacitated lot-sizing and machine scheduling problem with sequence-dependent setup time and setup carryover is a challenging problem with a wide application in industries. For the problem, two mixed integer programming models are proposed in order to explore their relative efficiencies in obtaining optimal solutions and linear programming relaxation lower bounds. Furthermore, due to the fact that the complicating constraints involve pairs of items (the sequence-dependent setups) and pairs of consecutive periods (the setup carryovers), making it difficult to decompose the problem per item or per period, we instead present a Dantzig-Wolfe decomposition reformulation per machine to improve lower bounds. We propose a branching and selection method to solve the problem, in which a collection of variables rather than individual variables are put into a data-driven process to generate useful information which is then adopted in the branching and selection process. Extensive numerical experiments show that the proposed algorithm can obtain numerically near-optimal solutions for small-scale problems and outperforms CPLEX and other heuristics in terms of both solution quality and runtime when the scale of instances increases. More experiments have been conducted to extract some insightful features related to the model and algorithm.

A multistage stochastic lot-sizing problem with cancellation and postponement under uncertain demands

A multistage stochastic capacitated discrete procurement problem with lead times, cancellation and postponement is addressed. The problem determines the procurement of a product under uncertain demand at minimal expected cost during a time horizon. The supply of the product is made through the purchase of optional distinguishable orders of fixed size with delivery time. Due to the unveiling of uncertainty over time it is possible to make cancellation and postponement corrective decisions on order procurement. These decisions involve costs and times of implementation. A model of the problem is formulated as an extension of a discrete capacitated lot-sizing problem under uncertain demand and lead times through a multi-stage stochastic mixed-integer linear optimization approach. Valid inequalities are generated, based on a conventional inequalities approach, to tighten the model formulation. Experiments are performed for several problem instances with different uncertainty information structure. Their results allow to conclude that the incorporation of a subset of the generated inequalities favor the model solution.

Subcontracting and lot-sizing with constant capacities

In this paper, we consider the subcontracting and single-item lot-sizing problem with constant capacities, which is uncapacitated in subcontracting but capacitated in production. For a holistic understanding of the problem, an infinite-period model is proposed. Such a model provides a unified view of a capacitated lot-sizing problem. The usefulness of the infinite-period model is shown by the principle that the firm’s production schedule drives the subcontractor’s supply schedule. For efficient construction of the production schedule, an innovative technique–coined using the concept of the shadow period–is presented to designate the optimal marginal supply cost. The success of the infinite-period model is finally proved with the concept of the effective period, indicating the supply coverage over periods. This fills the gap between the uncapacitated techniques and the capacitated techniques, making it possible to apply the best techniques from the lot-sizing literature. With this novel approach, we improve existing algorithms and provide new ones for concave production and supply cost structures involving setups.

Product line selection of fast-moving consumer goods

The fast-moving consumer goods sector relies on economies of scale. However, its assortments have been overextended as a means of market share appropriation and top-line growth. This paper studies the selection of the optimal set of products for fast-moving consumer goods producers to offer, as there is no previous model for product line selection that satisfies the requirements of the sector. Our mixed-integer programming model combines a multi-category attraction model with a capacitated lot-sizing problem, shared setups and safety stock. The multi-category attraction model predicts how the demand for each product responds to changes within the assortment. The capacitated lot-sizing problem allows us to account for the indirect production costs associated with different assortments. As seasonality is prevalent in consumer goods sales, the production plan optimally weights the trade-off between stocking finished goods from a long run with performing shorter runs with additional setups. Finally, the safety stock extension addresses the effect of the demand uncertainty associated with each assortment. With the computational experiments, we assess the value of our approach using data based on a real case. Our findings suggest that the benefits of a tailored approach are at their highest in scenarios typical fast-moving consumer goods industry: when capacity is tight, demand exhibits seasonal patterns and high service levels are required. This also occurs when the firm has a strong competitive position and consumer price-sensitivity is low. By testing the approach in two real-world instances, we show that this decision should not be made based on the current myopic industry practices. Lastly, our approach obtains profits of up to 9.4% higher than the current state-of-the-art models for product line selection.

A fix-and-optimize heuristic for the capacitated multi-item stochastic lot-sizing problem

This study addresses the stochastic multi-item capacitated lot-sizing problem. Here, it is assumed that all items are produced on a single production resource and unmet demands are backlogged. The literature shows that the deterministic version of this problem is NP-Hard. We consider the case where period demands are time-varying random variables. The objective is to determine the minimum expected cost production plan so as to meet stochastic period demands over the planning horizon. We extend the mixed integer programming formulation introduced in the literature to capture the problem under consideration. Further, we propose a fix-and-optimize heuristic building on an item-period oriented decomposition scheme. We then conduct a numerical study to evaluate the performance of the proposed heuristic as compared to the heuristic introduced by Tempelmeier and Hilger [16]. The results clearly show that the proposed fix-and-optimize heuristic arises as both cost-efficient and time-efficient solution approach as compared to the benchmark heuristic.

Capacitated Lot-Sizing Problem with Sequence-Dependent Setup, Setup Carryover and Setup Crossover

Since setup operations have significant impacts on production environments, the capacitated lot-sizing problem considering arbitrary length of setup times helps to develop flexible and efficient production plans. This study discusses a capacitated lot-sizing problem with sequence-dependent setup, setup carryover and setup crossover. A new mixed integer programming formulation is proposed. The formulation is based on three building blocks: the facility location extended formulation; the setup variables with indices for the starting and the completion time periods; and exponential number of generalized subtour elimination constraints (GSECs). A separation routine is adopted to generate the violated GSECs. Computational experiments show that the proposed formulation outperforms models from the literature.

New integration of preventive maintenance and production planning with cell formation and group scheduling for dynamic cellular manufacturing systems

This paper proposes a new integration method for cell formation, group scheduling, production, and preventive maintenance (PM) planning problems in a dynamic cellular manufacturing system (CMS). The cell formation sub-problem aims to form part families and machine groups, which minimizes the inter-cell material handling, under-utilization, and relocation costs. The production planning aspect is a multi-item capacitated lot-sizing problem accompanied by sub-contracting decisions, while the group scheduling problem deals with the decisions on the sequential order of the parts and their corresponding completion times. The purpose of the maintenance sub-problem is to determine the availability of the system and the time when the noncyclical perfect PM must be implemented to reduce the number of corrective actions. Numerical examples are generated and solved by Bender’s decomposition pack in GAMS to evaluate the interactions of the proposed model. Statistical analysis, based on a nonparametric method, is also used to study the behavior of the model’s cost components in two different situations. It is shown that by adding the PM planning decisions to the tactical decisions of the dynamic CMS, the optimal configuration and production plans of the system are heavily affected. The results indicate that omitting the PM actions increases the number of sudden failures, which leads to a higher total cost. Finally, it is concluded that the boost in the total availability of the dynamic CMS is one of the main advantages of the proposed integrated method.