MIP formulation and heuristics for multi-stage capacitated lot-sizing and scheduling problem with availability constraints

In this paper, the problem of lot-sizing and scheduling of multiple product types in a capacitated flow shop with availability constraints for multi-period planning horizon is considered. In many real production systems, machines may be unavailable due to breakdowns or preventive maintenance activities, thus integrating lot-sizing and scheduling with maintenance planning is necessary to model real manufacturing conditions. Two variants are considered to deal with the maintenance activities. In the first, the starting times of maintenance tasks are fixed, whereas in the second one, maintenance must be carried out in a given time window. A new mixed-integer programming (MIP) model is proposed to formulate the problem with sequence-dependent setups and availability constraints. The objective is to find a production and preventive maintenance schedule that minimizes production, holding and setup costs. Three MIP-based heuristics with rolling horizon framework are developed to generate the integrated plan. Computational experiments are performed on randomly generated instances to show the efficiency of the heuristics. To evaluate the validity of the solution methods, problems with different scales have been studied and the results are compared with the lower bound. Computational experiments demonstrate that the performed methods have good-quality results for the test problems.     

Collaborative production planning with production time windows and order splitting in make-to-order manufacturing

In this paper, we study a generalized production planning problem, that simultaneously investigates the two decisions that play critical roles in most firms, namely, production planning and order splitting and assignment. The problem takes into consideration the production time windows and capacities. We formulate the integrated problem as a linear mixed-integer program with a minimized total cost. A particle swarm optimization-based approach is developed to address the problem. Extensive computational experiments show that the proposed approach outperforms a commercial optimization package. Some managerial insights are also explored and reported. Finally, concluding remarks and future research directions are provided.     

Quasi-optimal integrated production, inventory and maintenance policies for a single-vendor single-buyer system with imperfect production process

In this paper we deal with the integrated supply chain management problem in the context of a single vendor-single buyer system for which the production unit is assumed to randomly shift from an in-control to an out-of-control state. At the end of each production cycle, a corrective or preventive maintenance action is performed, depending on the state of the production unit, and a new setup is carried out. Two different integrated production, shipment and maintenance strategies are proposed to satisfy the buyer’s demand at minimum total cost. The first one suggests that the buyer orders batches of size nQ and the vendor produces nQ and makes equal shipments of size Q. The second policy proposes that to satisfy the same ordered quantity, the vendor produces separately smaller batches of size Q, n times. The total integrated average cost per time unit corresponding to each strategy is considered as the performance criterion allowing choosing the best policy for any given situation.     

Optimal lot-sizing problem with imperfect maintenance and imperfect production

In this paper, we develop an integrated model for the joint determination of both economic production quantity and level of preventive maintenance (PM) for an imperfect production process. This process has a general deterioration distribution with increasing hazard rate. The effect of PM activities on the deterioration pattern of the process is modelled using the imperfect maintenance concept. In this concept, it is assumed that after performing PM, the ageing of the system is reduced in proportion to the PM level. After a period of time in production, the process may shift to out-of-control states, either type I or type II. A minimal repair will remove the type I out-of-control state. If a type II out-of-control state occurs, the production process has to stop, and then restoration work is carried out. Examples of Weibull shock models are given to show that the use of PM reduces costs.     

Use of (max, +) algebra for scheduling and optimization of HVLV systems subject to preventive maintenance

High-Variety, Low-Volume (HVLV) manufacturing systems are built to produce parts of several types in small quantities and under multiple production objectives. They relate to job-shop systems well known by researchers. One of the most studied assumptions of HVLV systems scheduling is considering that machines may be periodically unavailable during the production scheduling. This article deals with an analytical integrating method using (max, +) algebra to model HVLV scheduling problems subject to preventive maintenance (PM) while considering machines availability constraints. Each machine is subject to PM while maintaining flexibility for the start time of the maintenance activities during the planning period. The proposed model controls the placement of maintenance activities along the production operations. Indeed, the sequencing of maintenance activities on the machines depends on the criteria to minimize and may be different for each criteria value. For preventive maintenance, the proposed model aims to generate the best sequencing between activities while respecting the planning program that satisfy the optimal criteria values. In order to illustrate the performance of the proposed methodology, a simulation example is given.     

Optimal control of system with continuous and discrete state variables

An optimization problem for a system composed of continuous and discrete subsystems is considered. The discrete subsystem is introduced to express a combinatorial constraint in conventional control problems. The objective, the state equation and the constraint are assumed linear. The problem is formulated as a mixed-integer linear program with a staircase structure. A feasible decomposition method is developed for obtaining a suboptimal solution to the problem. In applications to an optimal energy control and planning problem for a large-scale production plant, our method uses less computing time than the non-decomposition method.     

A model for line capacity design for PWB assembly systems

In a multi-product, flexible manufacturing environment, line capacity of printed wiring board (PWB) assembly systems may need to be designed at the beginning of each aggregate planning period because of demand fluctuation over multiple periods. A model of line capacity design problem and production planning at the aggregate level is developed, in which production and subcontracting are assumed to be two options for a firm to meet market demand. The model presented is a large-scale integer programming problem, it cannot be solved by using standard- or mixed-integer programming codes. Under the assumption that each machine line is dedicated to produce one product family, the model can be decomposed as a relatively small subproblem, and each subproblem has good properties by which the subproblems can be further simplified and decomposed over multiple planning periods. As the result, the original large-scale two-stage integer programming problem can be approximately solved by solving a series of small-scale mixed-integer programming, which can be implemented on a workstation or a PC. Computational studies show that the solution method is developed which gives near-optimal solutions with much less computational effort.     

A discounted integrated inspection-maintenance model for a single deteriorating production facility

In this paper, we address the problem of determining optimum inspection schedules for a single deteriorating production system with a predetermined replacement cycle. It is assumed that, at different discrete points in time over the fixed planning horizon, the facility is inspected to detect its operating state and then it goes over an imperfect preventive maintenance routine to enhance its operating performance. Moreover, the facility undergoes minimal repair once detected in an “out‐of‐control” state. We also adopt the concept of discounted cash flow analysis to account properly for the effect of time value of money on the inspection policies. Under these settings, we formulate the discounted integrated inspection‐maintenance problem as a dynamic programming model with general time to failure distribution. After illustrating the model with a numerical example, we perform sensitivity analysis to investigate the effects of some input parameters on the expected present worth and the number of inspections.     

基于二阶段时间延迟的多产品生产系统生产与维修联合优化研究

针对多产品生产部件串联系统的生产和维修问题进行了研究,提出了基于二阶段时间延迟的联合优化模型。首先,基于生产周期分段理论,将整个周期等分成若干单位时间段,生产与维修共用每段时间,且若干时间段后采取一次预防维修。其次,考虑生产系统的实际生产时间、可用生产时间和维修耗费时间,建立了生产计划与维修计划总成本模型。其中,维修计划考虑缺陷和故障维修费用、维修检查费用,以及非正常状态下设备运行可能产生的不合格产品损失费用;生产计划考虑生产成本、库存成本、延期未交货成本和维修停机后恢复生产的设备启动成本。最后,通过算例分析,计算最优预防维修周期和各单位时间段各产品产量,验证了模型的有效性。     

Combined production-maintenance decisions in situations with process deterioration

This article addresses an imperfect production system with process deterioration and how best to maintain the system. In the system there is a constant percentage to produce defective products in the ‘in-control’ state and a higher percentage to produce defective products in the ‘out-of-control’ state. The production process may shift randomly from an in-control state to an out-of-control state during a production run; i.e. system process deterioration. This article considers both preventive maintenance and corrective maintenance. The objective is to determine the optimal production run time and preventive maintenance frequency while still minimising the total annual cost. We provide two efficient algorithms for solving the problems when considering the preventive maintenance frequency as discrete and continuous, respectively. Finally, we apply a numerical analysis to study the effects of changing parameter values on the optimal solutions and to point out some management implications.     

Buffer allocation and preventive maintenance optimization in unreliable production lines

In this paper, we consider a serial production line consisting of \(n\) unreliable machines with \(n-1\) buffers. The objective is to determine the optimal preventive maintenance policy and the optimal buffer allocation that will minimize the total system cost subject to a given system throughput level. We assume that the mean time between failure of all machines will be increased after performing periodic preventive maintenance. An analytical decomposition-type approximation is used to estimate the production line throughput. The optimal design problem is formulated as a combinatorial optimization one where the decision variables are buffer levels and times between preventive maintenance. To solve this problem, the extended great deluge algorithm is proposed. Illustrative numerical examples are presented to illustrate the model.     

A methodology for joint optimization for maintenance planning, process quality and production scheduling

Performance of a manufacturing system depends significantly on the shop floor performance. Traditionally, shop floor operational policies concerning maintenance scheduling, quality control and production scheduling have been considered and optimized independently. However, these three aspects of operations planning do have an interaction effect on each other and hence need to be considered jointly for improving the system performance. In this paper, a model is developed for joint optimization of these three aspects in a manufacturing system. First, a model has been developed for integrating maintenance scheduling and process quality control policy decisions. It provided an optimal preventive maintenance interval and control chart parameters that minimize expected cost per unit time. Subsequently, the optimal preventive maintenance interval is integrated with the production schedule in order to determine the optimal batch sequence that will minimize penalty-cost incurred due to schedule delay. An example is presented to illustrate the proposed model. It also compares the system performance employing the proposed integrated approach with that obtained by considering maintenance, quality and production scheduling independently. Substantial economic benefits are seen in the joint optimization.     

Maintenance scheduling incorporating dynamics of production system and real-time information from workstations

In this paper, a new approach to maintenance scheduling for a multi-component production system which takes into account the real-time information from workstations including remaining reliability of equipments as well as work-in-process inventories in each workstation is proposed. To model dynamics of the system, other information like production line configuration, cycle times, buffers’ capacity and mean time to repair of machines are also considered. Using factorial experiment design the problem is formulated to comprehensively monitor the effects of each possible schedule on throughput of the production system. The optimal maintenance schedule is searched by genetic algorithm-based optimization engine implemented in a simulation optimization platform. The proposed approach exploits all of makespans of planning horizon to find the best opportunity to perform maintenance actions on degrading machines in a way that maximizes the system throughput and mitigates the production losses caused by imperfect traditional maintenance strategies. Finally the proposed method is tested in a real production line to magnify the accuracy of proposed scheduling method. The experimental results indicate that the proposed approach guarantees the operational productivity and scheduling efficiency as well.     

需求随机波动下多设备批量生产系统的视情维护策略研究

针对需求随机波动情况下多设备批量生产系统的设备维护问题,提出了一种基于滚动生产计划和设备退化状况的视情维护策略。首先,通过滚动时域规划方法预测不同产品的随机需求并在此基础上以总生产成本最小确定滚动生产计划。其次,在每一滚动生产周期开始前检测系统中各设备的退化水平,利用Gamma过程描述退化增量,以最小维护成本率确定当前退化状态下各设备的最佳维护时间,同时为避免生产过程中断利用提前延后维护策略对预防维护进行动态调整。在系统层,利用生产转换时机对需要维护的组件进行组合维护。再次,引入时间约束和服务水平约束,建立批量生产与视情维护的联合优化模型,以总成本最小为目标,确定实际生产计划和维护计划。最后,通过算例以整个生产计划期内的总成本和故障次数为度量验证了所提出的多设备批量生产系统视情维护策略的有效性。     

汽车同步装配线生产计划与调度集成优化汽车同步装配线生产计划与调度集成优化

为克服传统的"自顶向下"方式下生产计划与调度不协调的缺陷,针对汽车同步装配线,构造了生产计划与调度集成优化混合整数规划模型,并采用拉格朗日松弛法将其分解为批量计划及调度等子问题.将调度子问题转化为与时间相关的旅行商问题,并采用dynasearch算法求解.对于拉格朗日对偶问题,采用均衡方向策略法求解.仿真实验结果验证了模型及算法的有效性.     

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.     

Maintenance scheduling and production control of multiple-machine manufacturing systems

This paper deals with the production and preventive maintenance control problem for a multiple-machine manufacturing system. The objective of such a problem is to find the production and preventive maintenance rates for the machines so as to minimize the total cost of inventory/backlog, repair and preventive maintenance. A two-level hierarchical control model is presented, and the structure of the control policy for both identical and non-identical manufacturing systems is described using parameters, referred to here as input factors. By combining analytical formalism with simulation-based statistical tools such as experimental design and response surface methodology, an approximation of the optimal control policies and values of input factors are determined. The results obtained extend those available in existing literature to cover non-identical machine manufacturing systems. A numerical example and a sensitivity analysis are presented in order to illustrate the robustness of the proposed approach. The extension of the proposed production and preventive maintenance policies to cover large systems (multiple machines, multiple products) is discussed.     

A polynomial algorithm for a lot-sizing problem with backlogging,outsourcing and limited inventory

This paper addresses a real-life production planning problem arising in a manufacturer of luxury goods. This problem can be modeled as a single item dynamic lot-sizing model with backlogging, outsourcing and inventory capacity. Setup cost is included in the production cost function, and the production level at each period is unbounded. The holding, backlogging and outsourcing cost functions are assumed to be linear. The backlogging level at each period is also limited. The goal is to satisfy all demands in the planning horizon at minimal total cost. We show that this problem can be solved in O(T⁴ log T) time where T is the number of periods in the planning horizon.     

Two-stage Maintenance of a Production System with Exponentially Distributed On- and Off-periods

We consider an on–off production system which is subject to failure during on-periods. In case of a failure, the production system is maintained correctively. In addition, preventive maintenance is carried out to prevent failures. The costs of both preventive and corrective maintenance are modelled in terms of the net down-time of the production system, that is, the time that the system is not available when needed for production. Since preventive maintenance can be planned during off-periods, there is a perspective of significant savings if some freedom is built in concerning the starting time of preventive maintenance. Therefore, a two-stage maintenance policy is considered, which—in a first stage—provides the maintenance manager with a finite interval during which preventive maintenance must be carried out, and—in a second stage—determines the optimal starting time for preventive maintenance within this interval. Computational results offer useful insights, and indicate that significant savings can be achieved in comparison with a classical age maintenance policy.     

Scheduling a single mobile robot for part-feeding tasks of production lines

This study deals with the problem of sequencing feeding tasks of a single mobile robot which is able to provide parts for feeders of machines on production lines. The mobile robot has to be scheduled in order to stoppage from lack of parts in the production line. A method based on the characteristics of feeders and inspired by the ( \(s,Q\) ) inventory system, is thus applied to define time windows for the feeding tasks of the robot. The capacity of the robot is also taken into consideration. The performance criterion is to minimize total traveling time of the robot for a given planning horizon. A genetic algorithm-based heuristics is presented which results in a significant increase in the speed of finding near-optimal solutions. To evaluate the performance of the genetic algorithm-based heuristic, a mixed-integer programming model has been developed for the problem. A case study is implemented at an impeller production line in a real factory and computational experiments are also conducted to demonstrate the effectiveness of the proposed approach.