Optimal repair-cost limit for a consumer following expiry of a warranty

This paper develops a methodology for obtaining the optimalrepair-cost limit for the repair-replace decision that a consumerfaces on the expiry of a product warranty, where the producthas been minimally repaired on failure during the warranty.Three cases are considered; (i) an exponentially distributedtime to failure with an arbitrary repair-cost distribution,(ii) an exponentially distributed time to failure with a uniformrepair-cost distribution, and (iii) a Weibull-distributed timeto failure with a beta-distributed repair cost. Numerical optimizationsare performed for the third case, and conclusions drawn as tothe sensitivity of the optimal repair-cost limit on the parametervalues of the distributions involved.     

The optimal repair-time limit replacement policy with imperfect repair: Lorenz transform approach

In this paper, we consider a simple repair-time limit replacement problem with imperfect repair, and focus on the problem of determining the optimal repair-time limit which minimizes the expected cost per unit time in the steady-state. Applying the Lorenz transform, we develop a nonparametric method to estimate the optimal repair-time limit from the empirical repair-time data. Numerical examples are considered to calculate the optimal policy and to examine the asymptotic properties of the estimator.     

A comparison of adaptive and stationary repair-cost limit criteria for a producer's warranty

A producer sells a product together with a warranty valid fora specified duration.Whenever the product fails during the warrantyperiod, the producer is responsible for making the product functional,either by replacing or repairing the equipment,according towhether the anticipated repair cost exceeds or does not exceeda specified repair-cost limit. It is assumed that repairs areminimal repairs, and that the distribution of repair costs isknown, as is the time-to-failure distribution of the product.In this paper, we compare two models in discrete time. The firstone involves an adaptive repair-cost limit which is set dynamicallyaccording to the age of the product, and the length of warrantyremaining. The second model assumes a constant or stationaryrepair-cost limit throughout the warranty period. Empiricalresults are summarized.     

The average optimality of a repair-limit replacement policy

We consider a minimal-repair and replacement problem of a reliability system whose state at a failure is described by a pair of two attributes, i.e., the total number of its past failures and the current failure level. It is assumed that the system is bothered by more frequent and more costly failures as time passes. Our problem is to find and/or characterize a minimal-repair and replacement policy of minimizing the long-run average expected maintenance cost per unit time over the infinite time horizon. Formulating the problem as a semi-Markov decision process, we show that a repairlimit replacement policy is average optimal. That is, for each total number of past system failures, there exists a threshold, called a repair limit, such that it is optimal to repair minimally if the current failure level is lower than the repair limit, and to replace otherwise. Furthermore, the repair limit is decreasing in the total number of past system failures.     

Optimal replacement model with age-dependent failure type based on a cumulative repair-cost limit policy

This paper presents a replacement model with age-dependent failure type based on a cumulative repair-cost limit policy, whose concept uses the information of all repair costs to decide whether the system is repaired or replaced. As failures occur, the system experiences one of the two types of failures: a type-I failure (minor), rectified by a minimal repair; or a type-II failure (catastrophic) that calls for a replacement. A critical type-I failure means a minor failure at which the accumulated repair cost exceeds the pre-determined limit for the first time. The system is replaced at the nth type-I failure, or at a critical type-I failure, or at first type-II failure, whichever occurs first. The optimal number of minimal repairs before replacement which minimizes the mean cost rate is derived and studied in terms of its existence and uniqueness. Several classical models in maintenance literature are special cases of our model.     

An optimal control problem in cancer chemotherapy

We consider a mathematical model for the control of the growth of tumor cells which is formulated as a problem of optimal control theory. It is concerned with chemotherapeutic treatment of cancer and aims at the minimization of the size of the tumor at the end of a certain time interval of treatment with a limited amount of drugs. The treatment is controlled by the dosis of drugs that is administered per time unit for which also a limit is prescribed. It is shown that optimal controls are of bang-bang type and can be chosen at the upper limit, if the total amount of drugs is large enough.     

A fast heuristic for minimising total average cycle stock subject to practical constraints

We address a problem of setting reorder intervals (time supplies) of a population of items, subject to a restricted set of possible intervals as well as a limit on the total number of replenishments per unit time, both important practical constraints. We provide a dynamic programming formulation for obtaining the optimal solution. In addition, a simple and efficient heuristic algorithm has been developed. Computational experiments show that the performance of the heuristic is excellent based on a set of realistic examples.     

Optimal age-replacement time with minimal repair based on cumulative repair-cost limit for a system subject to shocks

An operating system is subject to random shocks that arrive according to a non-homogeneous Poisson process and cause the system failed. System failures experience to be divided into two categories: a type-I failure (minor), rectified by a minimal repair; or a type-II failure (catastrophic) that calls for a replacement. An age-replacement model is studied by considering both a cumulative repair-cost limit and a system’s entire repair-cost history. Under such a policy, the system is replaced at age T, or at the k-th type-I failure at which the accumulated repair cost exceeds the pre-determined limit, or at any type-II failure, whichever occurs first. The object of this article is to study analytically the minimum-cost replacement policy for showing its existence, uniqueness, and the structural properties. The proposed model provides a general framework for analyzing the maintenance policies, and presents several numerical examples for illustration purposes.     

制造商为小本经营情况下零售商购货机制的合同设计

考虑零售商采用逆向拍卖方式到个小本经营的制造商处购货,研究供应链中如何设计合同使得零售商单位时间的支出费用最低,采用最优控制的方法和委托代理理论中的显示原理,得出零售商的最优购货机制,同时给出两个简单的购货机制——打分拍卖机制和固定生产时间下的价格拍卖。结果表明由制造商承担的制造费用越多,零售商的进货价格越高;在简单机制下,制造商承担制造费用的大小并不影响最优生产率的制定。     

Heavy traffic analysis of controlled multiplexing systems

The paper develops the mathematics of the heavy traffic approach to the control and optimal control problem for multiplexing systems, where there are many mutually independent sources which feed into a single channel via a multiplexer (or of networks composed of such subsystems). Due to the widely varying bit rates over all sources, control over admission, bandwidth, etc., is needed to assure good performance. Optimal control and heavy traffic analysis has been shown to yield systems with greatly improved performance. Indeed, the heavy traffic approach covers many cases of great current interest, and provides a useful and practical approach to problems of analysis and control arising in modern high speed telecommunications. Past works on the heavy traffic approach to the multiplexing problem concentrated on the uncontrolled system or on the use of the heavy traffic limit control problem for applications, and did not provide details of the proofs. This is done in the current paper. The basic control problem for the physical system is hard, and the heavy traffic approach provides much simplification. Owing to the presence of the control, as well as to the fact that the cost function of main interest is “ergodic”, the problem cannot be fully treated with “classical” methods of heavy traffic analysis for queueing networks. A basic result is that the optimal average costs per unit time for the physical problem converge to the optimal cost per unit time for the limit stationary process as the number of sources and the time interval goes to infinity. This convergence is both in the mean and pathwise senses. Furthermore, a “nice” nearly optimal control for the limit system provides nearly optimal values for the physical system, under heavy traffic, in both a mean and pathwise sense. This revised version was published online in June 2006 with corrections to the Cover Date.     

Maximizing the profit per unit of time for the TSP with convex resource-dependent travelling times

This paper introduces a new problem that is an extension of the travelling salesman problem (TSP) in which the travelling times are resource dependent and the objective is to maximize the profit per unit of time. We present an optimal solution approach comprised of three main steps: (1) calculating the optimal amount of total resource required (regardless of the selected tour); (2) constructing the tour; and (3) assigning the optimal resource to each connection between vertices using the equivalent load method. This solution approach finds the optimal solution with the same computational complexity for solving the classic TSP.     

Control Parametrization and Finite Element Method for Controlling Multi-species Reactive Transport in a Rectangular Diffuser Unit

The paper describes a problem involving optimal effluent release in a rectangular diffuser unit. The total amount of effluent released is to be maximized while observing given concentration bounds at the release end of the unit. The effluent degrades while passing through the unit. A computational scheme using combined control parametrization and finite element method is developed for solving the problem. Numerical examples have been solved to illustrate the efficiency of our method.     

考虑产品与单元匹配的单元装配系统构建模型

针对日本式单元生产过程中流水装配线向单元装配系统转换的决策优化问题,本文以最小化总完工时间和工人总工时为目标,同时考虑产品与单元的匹配关系,构建了单元装配系统多目标优化模型,基于NSGA-II算法设计了适用于该问题的三段染色体的编码方式。通过算例分析了产品与单元匹配关系变化对总完工时间和工人总工时的影响,结果表明,在工人数不变的情况下,产品与单元匹配关系变化会导致总完工时间产生较大差异,随着工人数增多,这种差异逐渐变小。     

基于重新排序时间错位限制下的最小化总完工时间的平行分批排序（英文）

In the rescheduling on a single machine, a set of the original jobs has already been scheduled, in order to make a given objective function is optimal. The decision maker needs to insert the new jobs into the existing schedule without excessively disrupting it. A batching machine is a machine that can handle up to some jobs simultaneously. In this paper,we consider the total completion time under a limit on the sequence disruptions for parallel batching based on rescheduling. For the parallel batching problem based on rescheduling, we research the properties of feasible schedules and optimal schedules on the total completion time under a limit on the maximum time disruptions or total time disruptions, in which the jobs are sequenced in SPT order, and give out the pseudo-polynomial time algorithms on the number of jobs and the processing time of jobs by applying the dynamic programming method.     

A hybrid algorithm for solving the economic lot and delivery scheduling problem in the common cycle case

The ELDSP problem is a combined lot sizing and sequencing problem. A supplier produces and delivers components of different types to a consumer in batches. The task is to determine the cycle time, i.e., the time between deliveries, which minimizes the total cost per time unit. This includes the determination of the production sequence of the component types within each cycle.We investigate the computational behavior of two published algorithms, a heuristic and an optimal algorithm. With large number of component types, the optimal algorithm has long running times. We devise a hybrid algorithm, which is both optimal and efficient.     

An exact method for scheduling a yard crane

This paper studies an operational problem arising at a container terminal, consisting of scheduling a yard crane to carry out a set of container storage and retrieval requests in a single container block. The objective is to minimize the total travel time of the crane to carry out all requests. The block has multiple input and output (I/O) points located at both the seaside and the landside. The crane must move retrieval containers from the block to the I/O points, and must move storage containers from the I/O points to the block. The problem is modeled as a continuous time integer programming model and the complexity is proven. We use intrinsic properties of the problem to propose a two-phase solution method to optimally solve the problem. In the first phase, we develop a merging algorithm which tries to patch subtours of an optimal solution of an assignment problem relaxation of the problem and obtain a complete crane tour without adding extra travel time to the optimal objective value of the relaxed problem. The algorithm requires common I/O points to patch subtours. This is efficient and often results in obtaining an optimal solution of the problem. If an optimal solution has not been obtained, the solution of the first phase is embedded in the second phase where a branch-and-bound algorithm is used to find an optimal solution. The numerical results show that the proposed method can quickly obtain an optimal solution of the problem. Compared to the random and Nearest Neighbor heuristics, the total travel time is on average reduced by more than 30% and 14%, respectively. We also validate the solution method at a terminal.     

Optimal replacement policies for two-unit machines with increasing running costs 1

A machine consists of two stochastically failing units. Failure of either of the units causes a failure of the machine and the failed unit has to be replaced immediately. Associated with the units are running costs which increase with the age of the unit because of increasing maintenance costs, decreasing output, etc.A preventive replacement policy is proposed under which, at failure points, we also replace the second unit if its age exceeds a predetermined control limit. It is proved that, for two identical units with exponential life-time distributions and linear running costs, this policy is optimal and the optimal control limit is calculated. In an additional model we take into consideration the length of time it takes to replace one unit or both units.The method of solution is a variation of dynamic semi-Markov programming. Analytical results are obtained and the influence of the various parameters on them is investigated. Finally, we study the saving due to our policy in comparison with a policy in which only failed units are replaced.     

Branch-and-price-and-cut for the multiple traveling repairman problem with distance constraints

In this paper, we extend the multiple traveling repairman problem by considering a limitation on the total distance that a vehicle can travel; the resulting problem is called the multiple traveling repairmen problem with distance constraints (MTRPD). In the MTRPD, a fleet of identical vehicles is dispatched to serve a set of customers. Each vehicle that starts from and ends at the depot is not allowed to travel a distance longer than a predetermined limit and each customer must be visited exactly once. The objective is to minimize the total waiting time of all customers after the vehicles leave the depot. To optimally solve the MTRPD, we propose a new exact branch-and-price-and-cut algorithm, where the column generation pricing subproblem is a resource-constrained elementary shortest-path problem with cumulative costs. An ad hoc label-setting algorithm armed with bidirectional search strategy is developed to solve the pricing subproblem. Computational results show the effectiveness of the proposed method. The optimal solutions to 179 out of 180 test instances are reported in this paper. Our computational results serve as benchmarks for future researchers on the problem.     

An Optimal Mixed Batch Shipment Policy for Multiple Items in a Single-Supplier Multiple-Retailer Integrated System

This study addresses mixed batch shipment policy with common replenishment cycle for multiple items in a single-supplier multiple-retailer integrated system. The supplier produces multiple items on a single facility under a common replenishment cycle and delivers products to retailer utilizing a mixed batch shipment policy. The objective is to determine the optimal replenishment cycle, the number of shipments, and the structure of mixed shipment, all of which minimize the integrated total cost per unit time. The single-item isolation model is constructed first, and the single-item isolation model is then integrated into the single-item integration model. Moreover, the single-item integration model is integrated into the multi-item integration model. The minimum total cost model is transformed into a maximum replenishment cycle model to optimize the structure of the mixed batch shipment. The replenishment cycle division method is then developed to obtain the optimal solutions to the subject problem. Examples are presented to illustrate the procedures involved in the replenishment cycle division method.     

Comparative analysis of the machine repair Problem with imperfect coverage and service pressure condition

In this paper we analyze the warm-standby M/M/R machine repair problem with multiple imperfect coverage which involving the service pressure condition. When an operating machine (or warm standby) fails, it may be immediately detected, located, and replaced with a coverage probability c by a standby if one is available. We use a recursive method to develop the steady-state analytic solutions which are used to calculate various system performance measures. The total expected profit function per unit time is derived to determine the joint optimal values at the maximum profit. We first utilize the direct search method to measure the various characteristics of the profit function followed by Quasi-Newton method to search the optimal solutions. Furthermore, the particle swarm optimization (PSO) algorithm is implemented to find the optimal combinations of parameters in the pursuit of maximum profit. Finally, a comparative analysis of the Quasi-Newton method with the PSO algorithm has demonstrated that the PSO algorithm provides a powerful tool to perform the optimization problem.