Integrated Lot Sizing Research Articles

Matheuristic for the lot-sizing and scheduling problem in integrated pulp and paper production

Vertical pulp and paper production is challenging from a process point of view. Managers must deal with floating bottlenecks, intermediate storage levels, and by-product production to control the whole process while reducing unexpected downtimes. Thus, this paper aims to address the integrated lot sizing and scheduling problem considering continuous digester production, multiple paper machines, and a chemical recovery line to treat by-products. The aim is to minimize the total production cost to meet customer demands, considering all productive resources and encouraging steam production (which can be used in power generation). Production planning should define the sizes of production lots, the sequence of paper types produced in each machine, and the digester working speed throughout the planning horizon. Furthermore, it should indicate the rate of by-product treatment at each stage of the recovery line and ensure the minimum and maximum storage limits. Due to the difficulty of exactly solving the mixed integer programming model representing this problem for real-world instances, mainly with planning horizons of over two weeks, constructive and improvement heuristics are proposed in this work. Different heuristic combinations are tested on hundreds of instances generated from data collected from the industry. Comparisons are made with a commercial Mixed-Integer and Linear Programming solver and a hybrid metaheuristic. The results show that combining the greedy constructive heuristic with the new variation of a fix-and-optimize improvement method delivers the best performance in both solution quality and computational time and effectively solves realistic size problems in practice. The proposed method achieved 69.41% of the best solutions for the generated set and 55.40% and 64.00% for the literature set for 1 and 2 machines, respectively, compared with the best solution method from the literature and a commercial solver.

Optimizing integrated lot sizing and production scheduling in flexible flow line systems with energy scheme: A two level approach based on reinforcement learning

Many production environments are faced with the need to simultaneously determine the planning of lot sizing and the scheduling of production sequences while ensuring cost minimization. This issue becomes even more complex when integrating multiple energy sources with the goal of a low-carbon economy. To address this challenge, this paper proposes an integrated lot sizing and flexible flow line production scheduling model under a time-of-use pricing scheme. In addition, the model takes into account conventional grid power, on-site renewable energy sources, and an energy storage system. The associated objective function is solved adopting a two-level approach to optimize energy costs while maintaining production throughput and meeting customer demand. The implementation relies on reinforcement learning capabilities to tackle complexity issues. The proposed approach is evaluated on a benchmark case and its results are compared with those obtained with First-In-First-Out heuristic, genetic algorithm and CPLEX. These results highlight the promising aspect of the proposed approach in terms of performance.

A flexible planning approach for integrated lot sizing and rework planning with random proportion of defective products

We consider a stochastic capacitated lot sizing problem with in-line rework. Both defective and defect-free products can be the result of an imperfect production process in real-world production systems. However, the proportion of defective items in a production lot is subject to uncertainty. For economic and environmental reasons, the possibility to rework defective products appears reasonable since these products usually have considerable value. In this paper, a nonlinear model formulation for integrated lot sizing and rework planning is proposed. We use a sample average approach to approximate the generic nonlinear model formulation. To cope with the randomness of the proportion of defective products, we apply a flexible planning approach that allows the adjustment of production and rework quantities. We conduct extensive numerical investigations to evaluate the performance of the proposed planning approach.

Relax-and-Fix Heuristics Applied to a Real-World Lot Sizing and Scheduling Problem in the Personal Care Consumer Goods Industry

This paper addresses an integrated lot sizing and scheduling problem in the industry of consumer goods for personal care, a very competitive market in which good customer service and cost management are crucial in the competition for clients. In this research, a complex operational environment composed of unrelated parallel machines with limited production capacity and sequence-dependent setup times and costs is studied. There is also a limitation in the total storage capacity for finished goods, a characteristic not found in the literature. Backordering is allowed, but it is extremely undesirable. The problem is described through a mixed integer linear programming formulation. Since the problem is NP-hard, relax-and-fix heuristics with hybrid partitioning strategies are investigated. Computational experiments with randomly generated and real-world instances are presented. The results show the efficacy and efficiency of the proposed approaches. Compared to the current solutions used by the company, the best proposed strategies yield results with substantially lower costs, primarily from the reduction in inventory levels and better allocation of production batches on the machines.

Optimization of the cutting process integrated to the lot sizing in multi-plant paper production industries

Many industrial production and cutting processes have costs associated with setup, production of objects, waste of material in the cutting process, inventory, and transport which can vary according to each plant’s location (factory). The production lot sizing, the planning of cuts, and the transport between plants of a company are interconnected problems that, if optimized in an integrated way, can lead to reduction in total costs. In the literature, such problems are widely addressed independently. Recently, the integrated lot sizing and cutting stock problem has been the subject of several studies. However, none have addressed the integrated problem, which also considers multiple plants. This paper presents an integer linear programming formulation for the integrated lot sizing and cutting stock problem with multiple plants and a solution methodology. The formulation is inspired by a practical situation in the paper-making industry. The developed solution methodology consists of a column generation method and a relax-and-fix with a feasibility heuristic. Computational tests were performed to evaluate the performance of the proposed approach and the solution methodology. The proposed approach presented lower total costs, demonstrating the potential for reducing all considered costs. The applied solution methodology showed satisfactory performance for all the studied instances.

Impact analysis of flexibility on the integrated lot sizing and supplier selection problem

Impact analysis of flexibility on the integrated lot sizing and supplier selection problem

A matheuristic approach for solving a simultaneous lot sizing and scheduling problem with client prioritization in tire industry

This paper introduces an integrated lot sizing and scheduling problem inspired from a real-world application in off-the-road tire industry. This problem considers the assignment of different items on parallel machines with complex eligibility constraints within a finite planning horizon. It also considers a large panel of specific constraints such as: backordering, a limited number of setups, upstream resources saturation and customers prioritization. A novel mixed integer formulation is proposed with the objective of optimizing different normalized criteria related to the inventory and service level performance. Based on this mathematical formulation, a problem-based matheuristic method that solves the lot sizing and assignment problems separately is proposed to solve the industrial case. A computational study and sensitivity analysis are carried out based on real-world data with up to 170 products, 70 unrelated parallel machines and 42 periods. The obtained results show the effectiveness of the proposed approach on improving the company’s solution. Indeed, the two most important KPIs for the management have been optimized of respectively 32% for the backorders and 13% for the overstock. Moreover, the computational time have been reduced significantly.

Hybrid matheuristics to solve the integrated lot sizing and scheduling problem on parallel machines with sequence-dependent and non-triangular setup

This paper approaches the integrated lot sizing and scheduling problem (ILSSP), in which non-identical machines work in parallel with non-triangular sequence-dependent setup costs and times, setup carry-over and capacity limitation. The aim of the studied ILSSP, here called ILSSP-NT on parallel machines, is to determine a production planning and tasks sequencing that meet period demands without delay and in such a way that the total costs of production, machine setup and inventory are minimized. The dearth of literature on the ILSSP-NT, despite the increasing amount of applications in the industrial sector, mainly in the food processing industry, motivated us to conduct this study. In this paper, we propose efficient methods to solve the ILSSP-NT on parallel machines. The methods virtually consist in the hybridization of the relax-and-fix and fix-and-optimize methods with the path-relinking and kernel search heuristics. To assess how well the heuristics solve the ILSSP-NT on parallel machines, we compared their results with those of the CPLEX solver with a fixed CPU time limit. The proposed matheuristics significantly outperformed CPLEX in most of the tested instances.

Integrated lot sizing and blending problems

The standard blending problem consists of combining components to produce a final product with a given demand, while satisfying specific criteria with respect to the global blend and minimizing the total cost. The Bill-Of-Material (BOM) (or recipe) indicates which components are used and in which proportion. Typically, there is some flexibility in the planning process with respect to the proportion used for each of the components, where it may vary between a minimum and a maximum level instead of being fixed. This problem has been widely studied in a single period setting. However, the problem becomes more complex when we take into account a longer time frame. In such a case, demand for the final product occurs in several time periods, and both the final product and the components can be held in stock. In the integrated lot sizing and blending problem, the decisions relate to the production of the final product via the blending process, and the production (or procurement) of the components over an extended time horizon. We propose three mathematical formulations for this integrated problem and present a theoretical analysis of their Linear Programming relaxation bounds. In a computational experiment, we analyse the impact of important parameters such as the level of flexibility in the BOM, the variance in the procurement cost among the components, and the variance of the proportion of the components in the total mix. Furthermore, we analyse the value of integration by comparing the solution of the integrated models to the solutions of approaches that do not fully capture this integration such as a lot-for-lot approach, just-in-time models without inventory for the final product or components, and a hierarchical approach.

The integrated lot sizing and cutting stock problem in an automotive spring factory

In this paper, a manufacturer of automotive springs is studied in order to reduce inventory costs and losses in the steel bar cutting process. For that, a mathematical model is proposed considering parallel machines and operational constraints, besides the demand, inventory costs and limits for items and final products. This one-dimensional integrated lot sizing and cutting stock problem is solved by a price-and-branch approach, using column generation and a method for obtaining integer solutions. Results using real data show that the proposed method achieved, in reasonable computational time, considerably better solutions than current practice at the company. Furthermore, focused on managerial insights, tests with random data were performed to analyze the influence that different parameters have in the solution, as well as the overall performance of the method.

An optimal integrated lot sizing and maintenance strategy for multi-machines system with energy consumption

This paper proposes an integrated model for multi-machines dynamic lot sizing aiming to produce a single item, considering the energy consumption during the production horizon. The objective is to find, firstly, the optimal lot size as well as the number of machines that satisfy a random demand under given service level and secondly, maintenance plan depended to production planning to minimise the total production, energy and maintenance costs. In fact, the problem of energy consumption is one of the most evoked topics especially with the decision of many governments to reduce theirs (For example France is willing to reduce the total consumption by 20% by 2020). The keys of this study are to consider, firstly, the correlation between the forecasting of demand, the variation of the working machines as well as their production rates under energy constraint and secondly the correlation between the production cadences and the maintenance strategy of all machines.

Heuristic approaches to solve a two-stage lot sizing and scheduling problem

This paper addresses the two-stage lot sizing and scheduling problem considering the production environment with parallel machines. The aim of this problem is to ensure the synchrony between the stages through of the integrated lot sizing and scheduling decisions. In addition, we consider the presence of a buffer between the production stages with minimum and maximum time lags of the items. Reprocessing, discarding and acceleration costs are considered when the permanence times of the items in the buffer are not respected. Two mathematical models and several heuristics strategies based on decomposition of the problem are presented to solve real data adapted instances. The results showed that our heuristic strategies can achieve competitive solutions in short computational time, outperforming the solutions provided by the optimization solver.

A holistic framework for lot sizing problem in fast-moving perishable products

One of the major challenges being faced by manufacturing companies today is the issue of addressing various aspects of perishable products in a supply chain environment. To address this issue, the integrated lot size problem for a perishable product has been investigated in the present work. The problem is modeled as a single vendor multiple buyer system. A variant of the truckload discount scheme is applied and the proposed model is formulated as a mixed integer program (MIP). The traditional warehouses are replaced by ‘cross-docks’ and situations in which, cross-docking would be more beneficial are highlighted. The problem of fleet selection is also addressed and various strategies to minimize the vendor cost are also highlighted for centralized and decentralized supply chains. Sensitivity analysis is then carried out on various input parameters such as setup cost at plant, variable transportation cost, fixed transportation cost, setup cost per order, holding cost and, lost cost, which underscores the significant impact of economies of scale in transportation on the total supply chain cost. Analysis of lead time-cost trade-off reveals that alternate modes of transportation could be explored, which significantly reduces the lead time of transportation, thereby minimizing the total supply chain cost.

An optimal integrated lot sizing policy of inventory in a bi-objective multi-level supply chain with stochastic constraints and imperfect products

This paper provides a developed mathematical model to derive the optimal integrated lot sizing in a multi-level supply chain with imperfect quality products. The developed mathematical model has a bi-objective function, with conflicting goals, minimizing the chain inventory costs and maximizing the chain total profit aided to find optimum policy for integrated lot sizing. We further actualize the problem by assuming multiple stochastic constraints. The mathematical formulation of the problem is stochastic, nonlinear, and large. In this regard, the interior point algorithm that is developed as more effective algorithm with less iteration is used for solving the recent convex nonlinear model. Numerical example shows that the developed interior point method is an efficient method with excellent performance in terms of the quality of the solution. Also, Sensitivity analysis shows that the developed interior point method is an efficient method with excellent performance in terms of the quality of the solution.

Integrated lot sizing and safety stock placement in a network of production facilities

In a network of production facilities, safety stock placement is determined at the network level to hedge against uncertainty in demand, while considering the production cycle time (PCT) of each stage as an exogenous parameter. The lot sizing policy at each stage affects setup time, capacity, and is a major determinant of queuing behavior and PCT. This paper integrates strategic safety stock placement and tactical lot sizing, which are typically optimized independently. We model the integrated problem to minimize system-wide production and inventory costs subject to meeting service-level requirements of external customers. We establish bounds for optimal lot sizes and extend existing safety stock placement dynamic programming algorithm to solve the integrated problem. Based on numerical experiments, we assess the value of integrating the two decisions and find that integration leads to smaller lot sizes, shorter lead times, and favors pooling of safety stocks. The setup cost, setup time, and the added value at a production stage are the most significant factors in determining the amount of savings generated through integration. In addition, simulation studies show that integration reduces PCT variability and decreases the fraction of delayed items.

Integrated lot sizing and energy-efficient job shop scheduling problem in manufacturing/remanufacturing systems

In this paper, a system designed to produce multi-class single-level products through both manufacturing of raw materials and remanufacturing of return products is taken into consideration, with the aim of defining and solving an integrated lot sizing and energy-efficient job shop scheduling problem. A mixed-integer programming formulation is proposed for the problem. This model minimizes not only the manufacturing and remanufacturing costs, the setup cost and the inventory holding and backlogging costs over the planning horizon, but also the energy costs paid for the utilization of machines and the compression of processing times. Since the model is NP-hard, a relax-and-fix heuristic is proposed to solve the problem. The proposed algorithm is illustrated with a numerical example, and its performance is tested on a set of randomly generated experimental problems. The results show the efficiency of the algorithm. Besides, the performance of the proposed energy-efficient model has been compared with classical models (that consider only the minimization of manufacturing/remanufacturing, holding and setup costs); the results indicate that the proposed model not only diminishes the energy consumption and the machines idle times, but it actually reduces the overall cost of the system.

A genetic algorithm for integrated lot sizing and supplier selection with defective items and storage and supplier capacity constraints

The single product, multi-period inventory lot-sizing problem is one of the most common and basic problems in the production and inventory management literature. In this paper, we consider an environment with multiple suppliers and multiple periods with supplier capacity and storage capacity constraints. Moreover, considering defective items, we move one-step toward a real environment of inventory problems. In this paper, we present the nonlinear programming of the problem. Since complexity of lot sizing problems belongs to a class of NP-hard problems, we propose a genetic algorithm to solve the problem. We develop a unique encoding-decoding procedure, which creates feasible solutions. Using the Taguchi experimental design method, the optimum parameters of the proposed genetic algorithm are selected. The result comparison between proposed GA and GAMS software as an exact solution for small and medium size problems shows that we can trust the proposed GA as a solution methodology for larger problems.

Food production in batch manufacturing systems with multiple shared-common resources: a scheduling model and its application in the yoghurt industry

The industry-specific characteristics of food-processing make production scheduling a hard and complex issue. Food processing has involved relatively minor concerns in several scheduling researches. In this paper, the authors address the integrated lot sizing and scheduling problem in batch production systems and propose a new mixed integer linear programming (MILP) formulation with multiple objective functions. The considered production system contains semi-continuous flow lines with series-parallel machines fed by parallel shared common resources. The model has been numerically and empirically validated using six weekly demand plan of a dairy company. The results demonstrate that both the operational cost and total completion time has been optimised.

A genetic algorithm for integrated lot sizing and supplier selection with defective items and storage and supplier capacity constraints

The single product, multi-period inventory lot-sizing problem is one of the most common and basic problems in the production and inventory management literature. In this paper, we consider an environment with multiple suppliers and multiple periods with supplier capacity and storage capacity constraints. Moreover, considering defective items, we move one-step toward a real environment of inventory problems. In this paper, we present the nonlinear programming of the problem. Since complexity of lot sizing problems belongs to a class of NP-hard problems, we propose a genetic algorithm to solve the problem. We develop a unique encoding-decoding procedure, which creates feasible solutions. Using the Taguchi experimental design method, the optimum parameters of the proposed genetic algorithm are selected. The result comparison between proposed GA and GAMS software as an exact solution for small and medium size problems shows that we can trust the proposed GA as a solution methodology for larger problems.

Optimization of an Integrated Lot Sizing and Cutting Stock Problem in the Paper Industry

Two important optimization problems occur in the planning and production scheduling in paper industries: the lot sizing problem and the cutting stock problem. The lot sizing problem must determine the quantity of jumbos of different types of paper to be produced in each machine over a finite planning horizon.These jumbos are then cut in order to meet the demand of items for each period. In this paper, we deal with the integration of these two problems, aiming to minimize costs of production and in- ventory of jumbos, as well as the trim loss of paper generated during the cutting process. Two mathematical models for the integrated problem are considered, and these models are solved both heuristically and using an optimization package. Attempting to get lower bounds for the problem, relaxed versions of the models also have been solved. Finally, computational experiments are presented and discussed.