An EOQ inventory model for items with Weibull distribution deterioration,ramp type demand rate and partial backlogging

In this paper, an EOQ inventory model is presented depleted not only by demand but also by Weibull distribution deterioration, in which the demand rate is assumed that with a ramp type function of time. In the model, shortages are allowed partial backlogging and the backlogging rate is variable and is dependent on waiting time for the next replenishment. The method is illustrated by three numerical examples, and sensitivity analysis of the optimal solution with respect to parameters of the system is carried out.     

Application of the knapsack model for budgeting

The budget problem of selecting projects (or activities) with known values (or payoffs) and associated costs, subject to a prescribed maximum budget, is akin to the knapsack problem, which is well documented in the literature. The optimal solution to maximise the total value of selected projects for a given budget constraint can readily be obtained. In practice, budgets are often somewhat flexible, or subject to possible changes, so that an optimal solution for a given budget value may not remain optimal when the budget is modified. It is, therefore, sensible in many situations to consider a budget range, instead of a single budget value. In addition to their original objective of maximising the total value of selected projects, decision makers are often concerned to get ‘value for money’, indicated by the ratio of payoff to cost. This paper examines how these questions can be tackled through the introduction of a stability index, to guide project selection within a defined budget range, and the use of a portfolio diagram, to help in the ranking of projects with respect to the stated twin objectives.     

A comparison of several approximations to the lead time demand distribution

Several distributions have been used for approximating the lead time demand distribution in inventory systems. We compare five distributions, the normal, the logistic, the lognormal, the gamma and the Weibull for obtaining the expected number of back orders, the reorder levels to have a given protection and the optimal order quantity, reorder levels in continuous review models of (Q, r) type. The normal and the logistic distributions are inadequate to represent the situations where the coefficient of variation (the ratio of the standard deviation to the mean) of the lead time demand distribution is large. The lognormal, the gamma and the Weibull distributions are versatile and adequate; however the lognormal seems to be a viable candidate because of its computational simplicity.     

Note on inventory models with Weibull distribution deterioration

This article explores the inventory model with a general demand rate function in which both the Weibull distributed deterioration and partial backlogging are considered. The inventory model discussed here is based on the important finding by Wu [Wu, K.S., 2001. An EOQ inventory model for items with Weibull distribution deterioration, ramp type demand rate and partial backlogging. Production Planning and Control, 12, 787–793]. There are four parts in our research. First, we derive the analytical framework of the inventory model for a general demand rate function1 1.?Based on detailed suggestions of Professor Wafik H. Iskander, Email: whiskander@mail.wvu.edu . Second, for the ramp type demand, we improve Wu's model to find the criterion to guarantee the existence and uniqueness of the optimal solution. Third, we develop a new model to compensate for the missing case in Wu's article. Fourth, we combine our results to show that our findings are applicable to the ramp type demand inventory model, so that the optimal solution is independent of the demand function. Finally, some numerical examples and graphs are provided to illustrate our discovery and demonstrate the application of our analytical framework.     

KBSS: a knowledge-based job-shop scheduling system*

Abstract. A knowledge based job-shop scheduling system (KBSS), which employs artificial intelligence techniques to solve general job-shop problems, is presented. The system consists of two main components: a knowledge base (KBSS model) and a control mechanism. The KBSS model has a three-level architecture which captures, in frames, knowledge about the job-shop domain and the characteristics of individual job-shop problems. The control mechanism consists of two modules: the primary scheduler (PS) and the heuristic scheduler (HS). PS is a rule-based scheduler built around a time-transcending heuristic which attempts to minimize the makespan and meet the due dates. If PS fails to meet the due dates then HS applies a combination of A* and beam search techniques in the space of alternative solutions and returns the best alternative it finds. The emphasis on due dates is a distinctive feature of KBSS. Experience with evaluating the presented scheduling system is reported and discussed.     

Note on solving inventory models from operational research view

In this article, we will show that an optimal solution can be derived without complicated mathematical derivation, an improvement on the paper [Yang, G.K., et al., 2011. Note on inventory models with Weibull distribution deterioration. Production Planning & Control, 22 (4), 437–444] that was recently published in Production Planning & Control. We extend their inventory model with a generalised deterioration and backlog rates. Our findings will allow researchers to consider inventory models directly from the view of operational research instead of complex analysis.     

考虑广告主投资竞争的关键词拍卖研究

关键词拍卖是搜索引擎盈利手段之一，同时给广告主带来高额回报。在搜索引擎注重质量权重的拍卖规则的推动下和广告主自身利益的驱使下，广告主通过投资来提高自身表现水平，赢得更好排位增加点击量。引入广告主投资，并用连续可变的努力水平来表征参与竞价的广告主投资过程中所付出的各种要素和资源投入，考虑投资和竞价两阶段模型，分析高低两类广告主的努力水平决策及均衡竞价策略。研究表明当满足初始投入最低努力水平时的边际成本大于边际收益的条件时，决定低类型广告主类型转换的估价阈值存在且唯一，并且在情形一中，潜力广告主的最优努力水平总是大于高类型广告主，在情形二中，随着估价的增大，两类广告主的最优努力水平趋于一致；同时，搜索引擎拍卖规则会影响广告主投资过程中最优努力水平的决策：搜索引擎给低类型广告主的质量权重越大，转换类型的估价阈值就越高，且潜力广告主取内点解时的最优努力水平和高类型广告主的最优努力水平均随之减少。最后，通过数值算例分析了两类广告主最优努力水平以及拍卖规则对广告主努力水平的影响。     

Burn-in Time and Estimation of Change-Point with Weibull-Exponential Mixture Distribution*

Many industrial products have three phases in their product lives: infant-mortality, normal, and wear-out phases. In the infant-mortality phase, the failure rate is high, but decreasing; in the normal phase, the failure rate remains constant; and in the wear-out phase, the failure rate is increasing. A burn-in procedure may be used to reduce early failures before shipping a product to consumers. A cost model is formulated to find the optimal burn-in time, which minimizes the expected sum of manufacturing cost, burn-in cost, and warranty cost incurred by failed items found during the warranty period. A mixture of Weibull hyperexponential distribution with shape parameter less than one and exponential distribution is used to describe the infant-mortality and the normal phases of the product life. The product under consideration can be either repairable or non-repairable. When the change-point of the product life distribution is unknown, it is estimated by using the maximum-likelihood estimation method. The effects of sample size on estimation error and the performance of the model are studied, and a sensitivity analysis is performed to study the effects of several parameters of the W-E distribution and costs on the optimal burn-in time.     

Asymptotically optimal policy for stochastic job shop scheduling problem to minimize makespan

This paper studies the large-scale stochastic job shop scheduling problem with general number of similar jobs, where the processing times of the same step are independently drawn from a known probability distribution, and the objective is to minimize the makespan. For the stochastic problem, we introduce the fluid relaxation of its deterministic counterpart, and define a fluid schedule for the fluid relaxation. By tracking the fluid schedule, a policy is proposed for the stochastic job shop scheduling problem. The expected value of the gap between the solution produced by the policy and the optimal solution is proved to be O(1), which indicates the policy is asymptotically optimal in expectation.     

同时集散货物的开放式车辆路径问题研究

本文研究了车辆工作时间限制下同时集散货物的多配送中心开放式车辆路径问题,以车辆数和运输里程最小为目标,建立了多目标规划模型,提出了基于拉格朗日松弛技术和禁忌搜索算法的混合求解算法。 该算法首先求出最优解的最大下界,然后采用客户点的分配和调整策略实现解的可行化,其中禁忌搜索引入了4种领域搜索方法,采用了随机变领域搜索方法和重起策略。算例分析表明,该算法能有效地找到满意解,且采用开放式安排路线比闭合式安排路线更加经济合理。     

On approximation of the fixed charge transportation problem

In this paper we present a new approximation for computing lower bound for the fixed charge transportation problem (FCTP). The lower bounds thus generated delivered 87% optimal solutions for 56 randomly generated small, up to 6×10 in size, problems in an experimental design. For somewhat larger, 10×10 and 10×15 size problems, the lower bounds delivered an average error of 5%, approximately, using a fraction of CPU times as compared to CPLEX to solve these problems. The proposed lower bound may be used as a superior initial solution with any other existing branch-and-bound method or tabu search heuristic procedure to enhance convergence to the optimal solution for large size problems which cannot be solved by CPLEX due to time constraints.     

具有遗憾值约束的鲁棒供应链网络设计模型研究

考虑不确定性环境,研究战略层次的供应链网络鲁棒设计问题,目标是设计参数发生摄动时,供应链性能能够保持稳健性。基于鲁棒解的定义,建立从上游供应商选择到下游设施选址-需求分配的供应链网络设计鲁棒优化模型;提出确定遗憾值限定系数上限和下限的方法,允许决策者调节鲁棒水平,选择多种供应链网络结构;通过模型分解与协调,设计了供应链节点配置的禁忌搜索算法。算例的计算结果表明了禁忌搜索算法具有良好的收敛特性,以及在处理大规模问题上的优越性;同时也反映了利用鲁棒优化模型进行供应链网络设计,可以有效规避投资风险。     

静态市场下考虑重复购买的价格和质保期联合决策

本文研究静态市场下的以质保期和价格为决策变量的最优控制决策问题。考虑了初次购买和重复购买两个过程以及重复购买过程是否受价格影响两种情况。模型中的生产成本考虑了产品生产过程中的学习效应, 质保成本使用了免费维修质保策略并假设产品两次故障之间的时间服从韦伯分布。在完整建模的基础上, 应用最大化原理对基本模型进行了求解, 之后本文给出了一种基本模型的具体应用场景, 并运用Lingo11软件进行了数值试验。最后对主要研究结论进行了应用讨论。     

On statistical bounds of heuristic solutions to location problems

Combinatorial optimization problems such as locating facilities frequently rely on heuristics to minimize the objective function. The optimum is often sought iteratively; a criterion is therefore necessary to be able to decide when the procedure attains such an optimum. Pre-setting the number of iterations is dominant in OR applications, however, the fact that the quality of the solution cannot be ascertained by pre-setting the number of iterations makes it less preferable. A small and, almost dormant, branch of the literature suggests usage of statistical principles to estimate the minimum and its bounds as a tool to decide upon the stopping criteria and also to evaluate the quality of the solution. In the current work we have examined the functioning of statistical bounds obtained from four different estimators using simulated annealing. P-median test problems taken from Beasley’s OR-library were used for the sake of testing. Our findings show that the Weibull estimator and 2nd order Jackknife estimators are preferable and the requirement of sample size to be about 10. It should be noted that reliable statistical bounds are found to depend critically on a sample of heuristic solutions of high quality; we have therefore provided a simple statistic for checking the quality. The work finally concludes with an illustration of applying statistical bounds to the problem of locating 70 post distribution centers in a region in Sweden.     

Penalty guided genetic search for redundancy optimization in multi-state series-parallel power system

This paper presents a genetic algorithm (GA) for parallel redundancy optimization in series-parallel power systems exhibiting multi-state behavior, optimizing the reliability subject to constraints. The components are binary and chosen from a list of products available in the market, and are being characterized by their feeding capacity, reliability, cost and weight. System reliability is defined as the ability to satisfy consumer demand and is presented as a piecewise cumulative load curve. In GA, to handle infeasible solutions penalty strategies are used. Penalty technique keep a certain amount of infeasible solutions in each generation so as to enforce genetic search towards an optimal solution from sides of, both, feasible and infeasible regions. We here present a dynamic adaptive penalty function which helps the algorithm to search efficiently for optimal/near optimal solution. To evaluate system reliability, a fast procedure, based on universal generating function, is used. An example considering a multi-state series-parallel power system is solved considering both homogeneous and heterogeneous types of redundancy. Also an example considering price discounts is solved. The effectiveness of the penalty function and the proposed algorithm is studied and shown graphically.     

Optimal price,warranty length and production rate for free replacement policy in the static demand market

In the present paper, a decision model is developed for producers in the static demand market to determine the optimal price, warranty length and production rate of a product to maximize profit based on the pre-determined life cycle. The free renewal warranty policy is considered under which failed products are renewed before the end of warranty length at no cost to consumers. The expected number of renewals based on warranty length is derived for Weibull life distributed products. The objective function includes both demand and cost functions, where production cost, warranty cost and inventory cost are involved. A solution approach using the maximum principle is described, and is applied to two specific cases of markets. The first case of market considers positive discount rate, and the second case of market considers zero discount rate. The economic sensitivity analysis is conducted to evaluate the effect of model parameters on the optimal solution. Some conclusions are drawn based on the sensitivity analysis.     

The falling tide algorithm: A new multi-objective approach for complex workforce scheduling

We present a hybrid approach of goal programming and meta-heuristic search to find compromise solutions for a difficult employee scheduling problem, i.e. nurse rostering with many hard and soft constraints. By employing a goal programming model with different parameter settings in its objective function, we can easily obtain a coarse solution where only the system constraints (i.e. hard constraints) are satisfied and an ideal objective-value vector where each single goal (i.e. each soft constraint) reaches its optimal value. The coarse solution is generally unusable in practise, but it can act as an initial point for the subsequent meta-heuristic search to speed up the convergence. Also, the ideal objective-value vector is, of course, usually unachievable, but it can help a multi-criteria search method (i.e. compromise programming) to evaluate the fitness of obtained solutions more efficiently. By incorporating three distance metrics with changing weight vectors, we propose a new time-predefined meta-heuristic approach, which we call the falling tide algorithm, and apply it under a multi-objective framework to find various compromise solutions. By this approach, not only can we achieve a trade off between the computational time and the solution quality, but also we can achieve a trade off between the conflicting objectives to enable better decision-making.     

JOB-SHOP VS. HYBRID FLOW-SHOP ROUTING IN A DUAL RESOURCE CONSTRAINED SYSTEM

This paper studies the impact of job routing pattern on the performance of dual resource constrained (DRC) systems. Three markedly different job routing patterns are simulated using a DRC system model that incorporates labor transfer times. The problem is viewed from two perspectives: one portion of the analysis focuses on the relative performances of five different dispatching rules as the routing pattern varies; the other examines the effect of the routing pattern on each rule's individual performance. One generalized conclusion of this study is that routing pattern has no significant impact on the relative effectiveness of various dispatching rules. Consequently, findings from pure job-shop research studies may be applied to situations where a mixture of job-shop and flow-shop routings is present.     

突发疫情环境下考虑订单释放时间的零售物流配送路径优化研究

“新冠肺炎”疫情突发严重危及了人们的生活安全，而物流保障在疫情防控、保民生活、稳定社会方面却起到了重要支撑作用。以保证民生的零售物流配送为背景，本文研究考虑订单释放时间的车辆路径优化问题，以配送完工时间与运输成本的加权之和最小为目标，构建了线性规划模型,并设计了改进的迭代局部搜索求解算法。该算法采用基于多种邻域结构和改进的US算法的局部搜索过程以增强算法的寻优能力，通过引入大路径和最优分割过程获得高质量的初始解和打破机制以增强算法的搜索效率。同时，提出一些性质用于拉格朗日松弛算法，为本文研究问题提供高质量的下界。最后，数值实验部分通过求解标杆算例和文中算例验证了模型和算法的有效性，并对订单释放时间和模型参数进行灵敏度分析，结果表明本文模型和算法能够为零售物流配送的效率提升和成本控制提供有效的决策支持。