Production lot size problem with failure in repair and backlogging derived without derivatives

Conventional approaches for solving the production lot size problems are by using the differential calculus on the long-run average production-inventory cost function with the need to prove optimality first. This note presents a simple algebraic method to replace the use of calculus for determining the optimal lot size. This study refers to the approach used by Grubbström and Erdem [Grubbström, R.W., Erdem, A., 1999. The EOQ with backlogging derived without derivatives, International Journal of Production Economics 59, 529–530] and extends it to the model examined by Chiu and Chiu [Chiu, S.W., Chiu, Y.-S.P., 2006. Mathematical modelling for production system with backlogging and failure in repair. Journal of Scientific and Industrial Research 65(6), 499–506]. This paper demonstrates that the lot size solution and the optimal production-inventory cost of an imperfect EMQ model can be derived without derivatives. As a result, the practitioners or students with little or no knowledge of calculus may be able to manage or understand with ease the realistic production systems.     

Inspection scheduling for imperfect production processes under free repair warranty contract

The paper considers scheduling of inspections for imperfect production processes where the process shift time from an ‘in-control’ state to an ‘out-of-control’ state is assumed to follow an arbitrary probability distribution with an increasing failure (hazard) rate and the products are sold with a free repair warranty (FRW) contract. During each production run, the process is monitored through inspections to assess its state. If at any inspection the process is found in ‘out-of-control’ state, then restoration is performed. The model is formulated under two different inspection policies: (i) no action is taken during a production run unless the system is discovered in an ‘out-of-control’ state by inspection and (ii) preventive repair action is undertaken once the ‘in-control’ state of the process is detected by inspection. The expected sum of pre-sale and post-sale costs per unit item is taken as a criterion of optimality. We propose a computational algorithm to determine the optimal inspection policy numerically, as it is quite hard to derive analytically. To ease the computational difficulties, we further employ an approximate method which determines a suboptimal inspection policy. A comparison between the optimal and suboptimal inspection policies is made and the impact of FRW on the optimal inspection policy is investigated in a numerical example.     

Utilizing an approximative solution to obtain the real optimal solution for a production and inspection model

Previous studies have proposed an integrated production and inspection model for a deteriorating production system whose process is characterized by an exponential distribution. Since the simultaneous determination of the optimal scheduled inspection times and the optimal production run length is difficult, an approximative production and inspection solution is obtained under the condition that the optimal inspection policy is equally-spaced. That is, obtaining an approximative production run length and number of inspections. This study further investigates this approximative solution and demonstrates that how to utilize it to obtain the real optimal solution more efficiently.     

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本文考虑了一个其产品保修期内免费小修的退化 生产系统的定期检修策略. 系统的退化过程包括三个状态: 可控制状态, 不可控制状态, 故障状态. 过程呆在可控制状态和不可控制状态的时间假设都服从指数分布. 生产系统在固定的时刻t或发生故障时进行检修, 两者以先发生为准. 本文讨论了使单位产品每周期期望成本最小的最优定期检修时间本文考虑了一个其产品保修期内免费小修的退化生产系统的定期检修策略.系统的退化过程包括三个状态:可控制状态,不可控制状态,故障状态.过程呆在可控制状态和不可控制状态的时间假设都服从指数分布.生产系统在固定的时刻t﹡或发生故障时进行检修,两者以先发生为准.本文讨论了使单位产品每周期期望成本最小的最优定期检修时间t﹡,三种特殊情况显示了最优值t的性质.此外,灵敏性分析和数字实例说明了模型中的参数对最优定期检修策略的影响.     

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.     

Cost-effective production policy for a stochastic unreliable manufacturing system

^** Email: dohi{at}rel.hiroshima-u.ac.jp The paper deals with an economic manufacturing quantity (EMQ)problem for an unreliable manufacturing system in both continuous-and discrete-time settings. The time to machine failure andcorrective and preventive repair times of the production facilityare assumed to follow arbitrary probability distributions. Thetraditional method of determining the EMQ policy for a failure-pronemanufacturing system is based on the minimization of the long-runaverage cost in the steady state. In this paper, an alternativecriterion of optimality called cost effectiveness is introduced.The criteria for the existence and uniqueness of the optimalproduction time maximizing the cost effectiveness are derivedanalytically under general failure and specific repair (correctiveand preventive) time distributions. The optimal cost-effectiveand average cost production policies are numerically calculatedand compared in terms of their performances.