Batch scheduling of deteriorating reworkables

The problem of scheduling the production of new and recoverable defective items of the same product manufactured on the same facility is studied. Items are processed in batches. Each batch comprises two sub-batches processed consecutively. In the first sub-batch, all the items are newly manufactured. Some of them are of the required good quality and some are defective. The defective items are remanufactured in the second sub-batch. They deteriorate while waiting for rework. This results in increased time and cost for their remanufacturing. All the items in the same sub-batch complete at the same time, which is the completion time of the last item in the sub-batch. Each remanufactured defective item is of the required good quality. It is assumed that the percentage of defective items in each batch is the same. A setup time is required to start batch processing and to switch from manufacturing to remanufacturing. The demands for good quality items over time are given. The objective is to find batch sizes such that the total setup and inventory holding cost is minimized and all the demands are satisfied. Dynamic programming algorithms are presented for the general problem and some important special cases.     

Integrated production-inventory models with imperfect production processes and a limited capacity for raw materials

This paper deals with inventory models that unify the decisions for raw materials and the finished product for a single product manufacturing system. The product is manufactured in batches and raw materials are jointly replenished from outside suppliers. The system is assumed to deteriorate during the production process. As a result, some proportion of nonconforming items is produced. The objective is to minimize the total variable cost for the system. A solution procedure is developed to find a near optimal solution for the basic model. The analysis for the basic model is extended to cases where the proportion of defective items is not constant or the defective rate is a function of production setup cost.     

Determining the optimal run time for EPQ model with scrap,rework, and stochastic breakdowns

This paper is concerned with determination of optimal run time for an economic production quantity (EPQ) model with scrap, rework, and stochastic machine breakdowns. In real life manufacturing systems, generation of defective items and random breakdown of production equipment are inevitable. In this study, a portion of the defective items is considered to be scrap, while the other is assumed to be repairable. Total production-inventory cost functions are derived respectively for both EPQ models with breakdown (no-resumption policy is adopted) and without breakdown taking place. These cost functions are integrated and the renewal reward theorem is used to cope with the variable cycle length. Theorems on conditional convexity of the integrated overall costs and bounds of the production run time are proposed and proved. We conclude that the optimal run time falls within the range of bounds and it can be pinpointed by the use of the bisection method based on the intermediate value theorem. Numerical example is provided to demonstrate its practical usages.     

Fuzzy scheduling of job orders in a two-stage flowshop with batch-processing machines

In this paper, we present a mixed-integer fuzzy programming model and a genetic algorithm (GA) based solution approach to a scheduling problem of customer orders in a mass customizing furniture industry. Independent job orders are grouped into multiple classes based on similarity in style so that the required number of setups is minimized. The family of jobs can be partitioned into batches, where each batch consists of a set of consecutively processed jobs from the same class. If a batch is assigned to one of available parallel machines, a setup is required at the beginning of the first job in that batch. A schedule defines the way how the batches are created from the independent jobs and specifies the processing order of the batches and that of the jobs within the batches. A machine can only process one job at a time, and cannot perform any processing while undergoing a setup. The proposed formulation minimizes the total weighted flowtime while fulfilling due date requirements. The imprecision associated with estimation of setup and processing times are represented by fuzzy sets.     

分类数很大时的成组排序问题

成组排序具有深刻的实际应用背景,是近年来国外研究得较多的一个热点.已有的某些动态规划算法的复杂性随分类数的增长呈指数型增长趋势,本文用“归并”和解不超过四个新的子问题的方法把分类数较大时的问题转化为分类数较小时的相应问题,简化了问题的求解.     

Multi-product lot-sizing and sequencing on a single imperfect machine

A problem of lot-sizing and sequencing several products on a single machine is studied. The machine is imperfect in two senses: it can produce defective items and it can breakdown. The number of defective items for each product is given as an integer valued non-decreasing function of the manufactured quantity. The total machine breakdown time is given as a real valued non-decreasing function of the manufactured quantities of all the products. A sequence-dependent setup time is required to switch the machine from manufacturing one product to another. Two problem settings are considered. In the first, the objective is to minimize the completion time of the last item, provided that all the product demands for the good quality items are satisfied. In the second, the goal is to minimize the total cost of demand dissatisfaction, subject to an assumption that the completion time of the last item does not exceed a given upper bound. Computational complexity and algorithmic results are presented, including an FPTAS for a special case of the cost minimization problem, and computer experiments with the FPTAS.     

Single machine batch scheduling with resource dependent setup and processing times

Jobs are processed by a single machine in batches. A batch is a set of jobs processed contiguously and completed together when the processing of all jobs in the batch is finished. Processing of a batch requires a machine setup time common for all batches. Both the job processing times and the setup time can be compressed through allocation of a continuously divisible resource. Each job uses the same amount of the resource. Each setup also uses the same amount of the resource, which may be different from that for the jobs. Polynomial time algorithms are presented to find an optimal batch sequence and resource values such that either the total weighted resource consumption is minimized, subject to meeting job deadlines, or the maximum job lateness is minimized, subject to an upper bound on the total weighted resource consumption. The algorithms are based on linear programming formulations of the corresponding problems.     

Serial-batching scheduling with time-dependent setup time and effects of deterioration and learning on a single-machine

This paper deals with serial-batching scheduling problems with the effects of deterioration and learning, where time-dependent setup time is also considered. In the proposed scheduling models, all jobs are first partitioned into serial batches, and then all batches are processed on a single serial-batching machine. The actual job processing time is a function of its starting time and position. In addition, a setup time is required when a new batch is processed, and the setup time of the batches is time-dependent, i.e., it is a linear function of its starting time. Structural properties are derived for the problems of minimizing the makespan, the number of tardy jobs, and the maximum earliness. Then, three optimization algorithms are developed to solve them, respectively.     

An optimization problem of manufacturing systems with stochastic machine breakdown and rework process

This paper is concerned with optimization of production run time that takes stochastic breakdown and the reworking of defective items into consideration. In a real‐life manufacturing process, production of imperfect quality items as well as random breakdowns of production equipment is inevitable. All defective items produced are assumed to be repairable through a rework process right after the regular production stops in each cycle. This research starts with derivations of the cost functions for production systems with breakdown (no‐resumption policy is considered) and without breakdown taking place, respectively. Then cost functions of both cases are integrated. Theorems on conditional convexity of the overall cost function and bounds for optimal production run time are proposed and proved. This study concludes that although the optimal run time cannot be expressed in a closed form, it falls within the range of bounds. Hence, it can be pinpointed by the use of the bisection method based on the intermediate value theorem. A numerical example is provided to demonstrate its practical usages. Copyright © 2007 John Wiley & Sons, Ltd.     

Economic production quantity with the presence of imperfect quality and random machine breakdown and repair based on the artificial bee colony heuristic

This article considers a production-inventory system consisting of a single imperfect unreliable machine. The items manufactured by the system are either perfect items or imperfect items, which require a rework to be restored to perfect quality. The rework rate is permitted to be different from the production rate if the rework process is different from the main manufacturing process. The fraction of the number of imperfect items is random following a general distribution function. The time to failure of the machine is random, following a general distribution function. If the machine fails before the lot is completed, the production is interrupted and the machine repair is started immediately. A random machine repair time is assumed, with a general distribution function. Unlike a common assumption in the literature, after the repair of the machine is completed, the production resumes. During the machine repair, a shortage can occur. A single-variable expected average cost function is derived to find the optimal lot size. Because of the complexity in the model, the ABC heuristic is proposed and implemented to find a near optimal value for the lot size. The article also provides a sensitivity analysis of the model's key parameters. It has been observed that the lot interruption-resumption policy leads to smaller lot sizes.     

多台机器流水作业中存在调整时间的Lot—streaming问题

多台机器流水作业的Lot-streaming问题（简称LS）,以往的研究都不考虑调整时间,固定分批数,寻找最优分批大小;本文对机器引入调整时间,研究同时决定最优分批数及分批大小,并给出了相应最优算法。     

Scheduling deteriorating jobs with a learning effect on unrelated parallel machines

In this study we consider unrelated parallel machines scheduling problems with learning effect and deteriorating jobs, in which the actual processing time of a job is a function of joint time-dependent deterioration and position-dependent learning. The objective is to determine the jobs assigned to corresponding each machine and the corresponding optimal schedule to minimize a cost function containing total completion (waiting) time, total absolute differences in completion (waiting) times and total machine load. If the number of machines is a given constant, we show that the problems can be solved in polynomial time under the time-dependent deterioration and position-dependent learning model.     

The Coordination of Scheduling and Batch Deliveries

This paper considers several scheduling problems where deliveries are made in batches with each batch delivered to the customer in a single shipment. Various scheduling costs, which are based on the delivery times of the jobs, are considered. The objective is to minimize the scheduling cost plus the delivery cost, and both single and parallel machine environments are considered. For many combinations of these, we either provide efficient algorithms that minimize total cost or show that the problem is intractable. Our work has implications for the coordination of scheduling with batch delivery decisions to improve customer service.     

Setup cost reduction in (<Emphasis Type="Italic">Q,r</Emphasis>) policy with lot size,setup time and lead-time interactions

It is often assumed in most deterministic and stochastic inventory models that lead-time is a given parameter and the optimal operating policy is determined on the basis of this unrealistic assumption. However, the manufacturing lead-time is made up of several components (moving time, waiting time, setup time, lot size, and rework time) most of which should be treated as controllable variables. In this paper the effect of setup cost reduction is addressed in a stochastic continuous review inventory system with lead-time depending on lot size and setup time. An efficient iterative procedure is developed to determine the near optimal lot size, reorder point and setup time. Furthermore, a sensitivity analysis is carried out to assess the cost savings that can be realised by investing in setup.     

Economic production quantity model with repair failure and limited capacity

We develop an economic production quantity (EPQ) model with random defective items and failure in repair. The existence of only one machine results with limited production capacity and shortages. The aim of this research is to derive the optimal cycle length, the optimal production quantity and the optimal back ordered quantity for each product so as to minimize the total expected cost (holding, shortage, production, setup, defective items and repair costs). The convexity of the model is derived and the objective function is proved convex. Two numerical examples illustrate the practical usage of the proposed method.     

Joint determination of production run length and number of standbys in a deteriorating production process

Two economic manufacturing quantity models with unrepairable and repairable standby key modules are proposed in this study that determine the economic production run length and the economic number of standbys in a deteriorating production process, where the key module of the production unit deteriorates over time and incurs some portion of defective items. For the model with unrepairable standbys, the active key module, once deteriorating, is replaced by a standby and the module itself is disposed. For the model with repairable standbys, the deteriorating key module is replaced by a standby and the module is then sent to the service center for maintenance. When completing the maintenance, it then joins the standbys for later production use. By minimizing the annual cost, which takes into account setup cost, holding cost, costs due to standbys and defective items, the economic production run length and the economic number of standbys are obtained for each of the proposed models.     

Multiproduct single-machine production system with stochastic scrapped production rate, partial backordering and service level constraint

In this paper, a multiproduct single-machine production system under economic production quantity (EPQ) model is studied in which the existence of only one machine causes a limited production capacity for the common cycle length of all products, the production defective rates are random variables, shortages are allowed and take a combination of backorder and lost sale, and there is a service rate constraint for the company. The aim of this research is to determine the optimal production quantity, the allowable shortage level, and the period length of each product such that the expected total cost, including holding, shortage, production, setup and defective items costs, is minimized. The mathematical model of the problem is derived for which the objective function is proved to be convex. Then, a derivative approach is utilized to obtain the optimal solution. Finally, two numerical examples in each of which a sensitivity analysis is performed on the model parameters, are provided to illustrate the practical usage of the proposed methodology.     

A bi-objective coordination setup problem in a two-stage production system

This paper addresses a problem arising in the coordination between two consecutive departments of a production system, where parts are processed in batches, and each batch is characterized by two distinct attributes. Due to the lack of interstage buffering between the two stages, these departments have to follow the same batch sequence. In the first department, a setup occurs every time the first attribute of a new batch is different from the one of the previous batch. In the downstream department, there is a setup when the second attribute changes in two consecutive batches. The problem consists in finding a batch sequence optimizing the number of setups paid by each department. This case results in a particular bi-objective combinatorial optimization problem. We present a geometrical characterization for the feasible solution set of the problem, and we propose three effective heuristics, as shown by an extensive experimental campaign. The proposed approach can be also used to solve a class of single-objective problems, in which setup costs in the two departments are general increasing functions of the number of setups.     

Heuristic procedures for a stochastic batch service problem

This paper considers a multi-class batch service problem that involves a class-dependent waiting cost and a service cost in determining customer batch sizes. Unlike a fixed service cost used widely in standard models, the service cost considered in this work is incurred only if the total service time is over the capacity. We formulate this problem as an infinite horizon Markov decision process, and exploit its structural properties to establish theoretical results, including bounds on the optimal action space. We use the results to improve the value iteration procedure. Furthermore, we design heuristic algorithms for large problems. The numerical experiments demonstrate that the class-dependent waiting cost has a considerable influence on the optimal customer batch size. Finally, we evaluate the efficiency of the proposed value iteration procedure and the quality of the heuristic solutions.     

Multiobjective evolutionary algorithms to identify highly autocorrelated areas: the case of spatial distribution in financially compromised farms

In this paper, we address the optimal batch sizing and just-in-time scheduling problem where upper and lower bounds on the size of the batches are imposed. The objective is to find a feasible schedule that minimizes the sum of the weighted earliness and tardiness penalties as well as the setup costs, which involves the cost of creating a new batch. We present some structural properties of the optimal schedules and describe solving algorithms for the single machine problem.