A note on “Single-machine scheduling problems with both deteriorating jobs and learning effects”

The main results in a recent paper [1] (J.-B. Wang, D. Wang, G.-D. Zhang, Single-machine scheduling problems with both deteriorating jobs and learning effects, Appl. Math. Model. 34 (2010) 2831–2839) are incorrect because job processing times are variable due to both deteriorating jobs and learning effects, which is not taken into account by the authors. In this note, we show by counter-examples that some published results are incorrect.     

A note on “Two-warehouse inventory model with deterioration under FIFO dispatch policy”

A recently published paper by Lee [C.C. Lee, Two-warehouse inventory model with deterioration under FIFO dispatching policy, European Journal of Operational Research 174 (2006) 861–873] considers different dispatching models for the two-warehouse inventory system with deteriorating items, in which Pakkala and Achary’s LIFO (last-in–first-out) model [T.P.M. Pakkala, K.K. Achary, A deterministic inventory model for deteriorating items with two warehouses and finite replenishment rate, European Journal of Operational Research 57 (1992) 71–76] is first modified, and then the author concludes that the modified LIFO model always has a lower cost than Pakkala and Achary’s LIFO model under a particular condition specified by him. The present note points out that this conclusion is incorrect and misleading. Alternatively, we provide a new sufficient condition such that the modified LIFO model always has a lower cost than Pakkala and Achary’s model. Besides, we also compare Pakkala and Achary’s original LIFO model with Lee’s FIFO (first-in–first-out) model for the special case where the two warehouses have the same deteriorating rates. Finally, numerical examples are provided to investigate and examine the impact of corresponding parameters on policy choice. The results in this note give a much clearer picture than those at Lee’s paper about the relationships between the different dispatching policies for the two-warehouse inventory system with deterioration items.     

A note on single-machine scheduling with nonlinear deterioration

Some results in a recent paper (Wang and Wang in Optim Lett doi:) are incorrect because job processing times are variable due to nonlinear deterioration. In this note, we show by a counter-example that the results are incorrect.     

A note on a single-vendor and multiple-buyers production-inventory policy for a deteriorating item

In this note we examine the total cost function of a single-vendor multiple-buyers production-inventory policy for a deteriorating item by Yang and Wee [P.C. Yang and H.M. Wee, A single-vendor multiple-buyers production-inventory policy for a deteriorating item, European Journal of Operational Research 143 (2002) 570–581]. Two possible flaws in the cost function of Wee and Yang’s model are pointed out. A proposal to eradicate the flaws is given.     

A note on the inventory models for deteriorating items with ramp type demand rate

In this research we study the inventory models for deteriorating items with ramp type demand rate. We first clearly point out some questionable results that appeared in (Mandal, B., Pal, A.K., 1998. Order level inventory system with ramp type demand rate for deteriorating items. Journal of Interdisciplinary Mathematics 1, 49–66 and Wu, K.S., Ouyang, L.Y., 2000. A replenishment policy for deteriorating items with ramp type demand rate (Short Communication). Proceedings of National Science Council ROC (A) 24, 279–286). And then resolve the similar problem by offering a rigorous and efficient method to derive the optimal solution. In addition, we also propose an extended inventory model with ramp type demand rate and its optimal feasible solution to amend the incompleteness in the previous work. Moreover, we also proposed a very good inventory replenishment policy for this kind of inventory model. We believe that our work will provide a solid foundation for the further study of this sort of important inventory models with ramp type demand rate.     

A note on the article “Fuzzy alpha-sets and alpha-continuous maps”

In this note we show that some results which appeared in the article by M.K. Singal and N. Rajvanshi [Fuzzy Sets and Systems 48 (1992) 383–390] are incorrect.     

On the E-Epigraph of an E-Convex Function

In Ref 1, Yang shows that some of the results obtained in Ref. 2 on E-convex programming are incorrect, but does not prove that the results which make the connection between an E-convex function and its E-epigraph are incorrect. In this note, we show that the results obtained in Ref. 2 concerning the characterization of an E-convex function f in terms of its E-epigraph are incorrect. Afterward, some characterizations of E-convex functions using a different notion of epigraph are given.     

A note on Fukuyama (2000)

This brief note discusses Fukuyama’s paper [Fukuyama, H., 2000. Returns to scale and scale elasticity in data envelopment analysis. European Journal of Operational Research 125, 93–112]. We prove that the Proof of Lemma 1 by Fukuyama (2000) is incorrect due to incorrect substitution. In addition, we exemplify our conclusion by a counterexample taken from Fukuyama (2000). Thus all conclusions, based on Lemma 1, of Fukuyama (2000) cannot be reached and are left for further research.     

Remarks on oscillation and nonoscillation for second-order linear difference equations

In this remark, we shall show the main results of the earlier work [W.T. Li, S.S. Cheng, Remarks on two recent oscillation theorems for second-order linear difference equations, Appl. Math. Lett. 16 (2003) 161–163] are incorrect.     

Two-agent scheduling with position-based deteriorating jobs and learning effects

Scheduling with deteriorating jobs and learning effects has been widely studied. However, multi-agent scheduling with simultaneous considerations of deteriorating jobs and learning effects has hardly been considered until now. In view of this, we consider a two-agent single-machine scheduling problem involving deteriorating jobs and learning effects simultaneously. In the proposed model, given a schedule, we assume that the actual processing time of a job of the first agent is a function of position-based learning while the actual processing time of a job of the second agent is a function of position-based deterioration. The objective is to minimize the total weighted completion time of the jobs of the first agent with the restriction that no tardy job is allowed for the second agent. We develop a branch-and-bound and several simulated annealing algorithms to solve the problem. Computational results show that the proposed algorithms are efficient in producing near-optimal solutions.     

多货栈及变质情形下两种可替代物品的经济订货批量模型

研究了多货栈及变质情形下两种可替代物品的经济订货批量问题.在计划期内,若某一种易变质物品发生缺货,则可以被另一种易变质物品以一定的替代率代替补充,不同物品有不同的变质率,且要决定租用货栈的数量.以库存系统的总费用最小为目标函数,分别对货栈容量无限与有限的情形建立模型,证明了最优策略存在的唯一性,并分别给出了求解最优订购策略的算法,最后通过一个算例验证了算法的最优性.     

Optimal Ordering Policy for Deteriorating Items with Partial Backlogging under Permissible Delay in Payments

In 1985, Goyal developed an Economic order quantity (EOQ) model under conditions of permissible delay in payments. Jamal et al. then generalized Goyal’s model for deteriorating items with completely backlogging. However, they only ran several simulations to indicate that the total relevant cost may be convex. Recently, Teng amended Goyal’s model by considering the difference between unit price and unit cost, and provided an alternative conclusion that it makes economic sense for some retailers to order less quantity and take the benefits of the permissible delay more frequently. However, he did not consider deteriorating items and partial backlogging. In this paper, we establish a general EOQ model for deteriorating items when the supplier offers a permissible delay in payments. For generality, our model allows not only the partial backlogging rate to be related to the waiting time but also the unit selling price to be larger than the unit purchase cost. Consequently, the proposed model includes numerous previous models as special cases. In addition, we mathematically prove that the total relevant cost is strictly pseudo-convex so that the optimal solution exists and is unique. Finally, our computational results reveal six managerial phenomena.     

A note on due-date assignment and single-machine scheduling with deteriorating jobs

Journal of the Operational Research Society - In this note, we study a single-machine scheduling problem with deteriorating jobs whose processing times are an increasing function of their start...     

An optimal production-inventory model for deteriorating items with multiple-market demand

The global markets of today offer more selling opportunities to the deteriorating items’ manufacturers, but also pose new challenges in production and inventory planning. From a production management standpoint, opportunities to exploit the difference in the timing of the selling season between geographically dispersed markets for deteriorating items are important to improving a firm’s profitability. In this paper, we examined the above issue with an insightful production-inventory model of a deteriorating items manufacturer selling goods to multiple-markets with different selling seasons. We also provided a solution procedure to find the optimal replenishment schedule for raw materials and the optimal production plan for finished products. A numerical example was then used to illustrate the model and the solution procedure. Finally, sensitivity analysis of the optimal solution with respect to major parameters was carried out.     

基于重新排序的退化工件最小化总延误时间问题

考虑了错位限制下的含有退化工件的重新排序问题,即工件的实际加工时间看作是工件开工时间的线性函数．重新排序就是在原始工件已经按照某种规则使目标函数达到最优时有一新工件集到达,新工件的安排使得原始工件重新排序进而产生错位．研究了最大序列错位和总序列错位限制下的退化工件最小化总延误时间问题,其最优排序的结构性质是使得原始工件集和新工件集中的工件是按加工率αj非减的序列排列,基于此通过分阶段排序和动态规划方法给出了两个问题的多项式时间的最优算法．     

A note on the complexity of flow-shop scheduling with deteriorating jobs

This paper is a note on “Complexity analysis of job-shop scheduling with deteriorating jobs” [G. Mosheiov, Complexity analysis of job-shop scheduling with deteriorating jobs, Discrete Applied Mathematics 117 (2002) 195-209]. A proportional deterioration rate is assumed and the objective is the minimization of the makespan. Mosheiov presents NP-hardness results for flow-shops and open-shops with three or more machines and for job-shops with two or more machines. The proof of NP-hardness for the flow-shop case is however not correct. This paper provides a correct proof.     

Parallel machines scheduling to minimize job tardiness and machine deteriorating cost with deteriorating jobs

This paper studies the parallel machines bi-criteria scheduling problem (PMBSP) in a deteriorating system. Sequencing and scheduling problems (SSP) have seldom considered the two phenomena concurrently. This paper discusses the parallel machines scheduling problem with the effects of machine and job deterioration. By the machine deterioration effect, we mean that each machine deteriorates at a different rate. This deterioration is considered in terms of cost which depends on the production rate, the machine’s operating characteristics and the kind of work done by each machine. Moreover, job processing times are increasing functions of their starting times and follow a simple linear deterioration. The objective functions are minimizing total tardiness and machine deteriorating cost. The problem of total tardiness on identical parallel machines is NP-hard, thus the problem with machine deteriorating cost as an additional term is also NP-hard. We propose the LP-metric method to show the importance of our proposed multi-objective problem. A metaheuristic algorithm is developed to locate optimal or near optimal solutions based on a Tabu search mechanism. Numerical examples are presented to show the efficiency of this model.     

Remarks on oscillation of second-order linear difference equation

In this remark, we shall show that the Lemma 1 and parts of the proof of Theorem 1 in the paper “Oscillation criteria for second-order linear difference equations” (Applied Mathematics and Computation 145 (2003) 591-601) are incorrect.     

机器具有优势关系下的工件加工时间可控的流水作业排序问题

本文主要研究机器具有优势关系下的工件加工时间可控的流水作业排序问题.我们主要对以下两种情形进行了讨论:工件加工时间为线性恶化和线性学习.对于每一种加工模型,我们分别研究了几类不同的优势机器,并且对每种情况均给出了多项式时间算法.     

考虑保质期的非即时易腐品最优订购与定价决策

针对有保质期约束的非即时易腐品,零售商可以在产品非腐败阶段和腐败阶段进行差异化定价来调整市场需求。即零售商可以调整产品非腐败阶段的定价策略,使得非即时易腐品在进入腐败阶段之前全部售出;或者在产品腐败阶段降低零售价格以刺激市场需求。因此,在需求依赖价格的假设下,本文研究了零售商关于有保质期约束的非即时易腐品的最优订购与定价决策。通过分析零售商单位时间利润函数的理论性质,得到了零售商的最优订购与定价策略。基于线性和指数型两种需求函数形式进行数值算例,可以发现针对保质期较长或者非腐败阶段较长的易腐品,零售商会延长订货周期来增加销售收入。特别是在指数型需求函数情境下,当市场需求与零售价格高度相关时,零售商更有意愿降价来刺激销量,从而使得易腐品在非腐败阶段内销售完毕。