A note on “The EPQ with partial backordering and phase-dependent backordering rate”

Pentico et al. [The EPQ with partial backordering and phase-dependent backordering rate. Omega 2011;39(5):574-7] recently proposed an EPQ model with partial backordering. They tried to determine the minimum average total cost per year from the three cases: to lose all sales, to meet all demand, and to allow stockouts with partial backordering. In this paper, we solve the same model without differential calculus and present a different decision procedure. Without the comparison of those costs, a numerical example is illustrated to determine the optimal production policy by our criteria.     

A note on the deterministic EPQ with partial backordering

In a recent paper published in Omega [Pentico DW, Drake MJ, Toews C. The deterministic EPQ with partial backordering: a new approach. Omega 2009; 37(3):624–36], the authors proposed an EPQ model with partial backordering. In the EPQ model, a critical value of backordering rate was developed, below which the optimal production policy is either to lose all sales or meet all demand, above which the optimal policy is to allow stockouts with partial backordering and meet fractional demand. However, there exists a flaw in the critical value of backordering rate, which will be amended in this note.     

The deterministic EPQ with partial backordering: A new approach

Several authors have developed models for the EOQ when only a percentage of stockouts will be backordered. Most of these models are complicated, with equations unlike those for the EOQ with full backordering. In this paper we extend work by Pentico and Drake [The deterministic EOQ with partial backordering: a new approach. European Journal of Operational Research 2008; in press] that developed equations for the EOQ with partial backordering that are more like those for the EOQ with full backordering to develop a comparable model for the EPQ with partial backordering.     

On the optimality of the optimal policies for the deterministic EPQ with partial backordering

In a note published in Omega [Zhang RQ. A note on the deterministic EPQ with partial backordering. Omega 2009;37(5):1036–8], the amended decision procedure for the Pentico et al.'s EPQ with partial backordering (EPQ-PBO) is proposed, by developing another critical value of the backordering rate. However, there is a case when a decision made with this amended procedure is not optimal, which will be shown in this paper. A new decision procedure will be proposed based on the derived necessary and sufficient conditions for considering the policy of losing all sales or the policy to meet all demand as possible optimal decisions. The proposed decision procedure is adapted for one of the extensions of the EPQ-PBO.     

A partial backordering approach to inventory control

We study the effects of using a partial backordering approach to controlling inventories under deterministic and stochastic demands, respectively. With a deterministic demand, our model is built with the objective of minimizing the total cost of ordering, holding, backordering and lost sales. The conditions for the partial backordering policy to be feasible are identified and a pair-wise comparison among the no-backordering, complete backordering, and partial backordering doctrines is conducted. In the stochastic case, we focus on a make-to-stock system with a Poisson demand and exponential production time, which allows us to establish a queuing model to examine the cost-effectiveness of using partial backorders. The conditions under which the partial backordering policy outperforms the complete backordering policy are identified.     

Optimal inventory model for items with imperfect quality and shortage backordering

In this study, we develop an optimal inventory model for items with imperfect quality and shortage backordering. Such implicit assumption is reasonable in view of the fact that poor-quality items do exist during production. They are usually picked up during the screening process and are withdrawn from stock instantaneously. It is assumed that all customers are willing to wait for new supply when there is a shortage. The analysis shows that our model is a generalization of the models in current literatures. An algorithm and numerical analysis developed show that our model always results in better performance.     

Analysis of an EOQ inventory model with partial backordering and non-linear unit holding cost

In this paper, an economic order quantity inventory model is analyzed, considering that the unit cumulative holding cost has two significant components: a fixed cost which represents the cost of accommodating the item in the warehouse and a variable cost given by a potential function of the length of time over which the item is held in stock. Shortages are allowed and, during the stockout period, only a fraction of demand is partially backordered. The backordering cost includes a fixed cost and a cost linearly dependent on the length of time for which backorder exists. A solution procedure is developed for determining the optimal inventory policy. Moreover, to illustrate the effects of some parameters on the optimal policy and the minimum total inventory cost, a numerical study is developed.     

A supplement to the EPQ with partial backordering and phase-dependent backordering rate

In this note, we extend Pentico et al. (2011)'s [1] EPQ model where the authors considered two different backordering rates during shortage period; they assumed the backordering rate increases only from the start of production phase. We propose two models that allow the backordering rate to increase anytime during the production phase, and thus provide more flexibility for decision-makers to vary the backordering rate.     

Exact closed-form solutions for “optimal inventory model for items with imperfect quality and shortage backordering”

Wee et al. [Optimal inventory model for items with imperfect quality and shortage backordering. Omega 2007;35(1):7–11] recently contributed an optimal inventory model for items with imperfect quality and shortage backordering. This article revisits their study and applies the well-known renewal-reward theorem to obtain a new expected net profit per unit time. We derive the exact closed-form solutions to determine the optimal lot size, backordering quantity and maximum expected net profit per unit time, specifically without differential calculus. We also solve the same model algebraically from another direction, which has been mentioned, but the process has not been finished yet. The problem parameter effects upon the optimal solutions are examined analytically and numerically.     

An EOQ model with partial delayed payment and partial backordering

In many transactions concerning selling and buying, a specified delay of payment is offered or accepted by the seller. This can be regarded as a kind of discount and has potential consequences for the order size. These kinds of effects are not explicitly incorporated in the classical formulas for economic order quantities (EOQ). In this research we consider an EOQ problem under partial delayed payment. A fraction of the purchasing cost must be paid at the beginning of the period and the remaining amount can be paid later. Shortages are permitted and occur as a combination of backorders and lost sales. The aim of this paper is to determine the order and shortage quantities.     

提前期、构建成本和短缺量拖后率均可控的EOQ模型

传统库存模型通常将提前期和构建成本视为不可控制。事实上可以通过追加投资缩短提前期和降低构建成本。缺货期间,为减少订单丢失量和补偿顾客的损失,供应商会给予一定的价格折扣。现实库存系统中,容易得到需求的期望值和标准差,但较难得到其分布规律。基于此,考虑短缺量拖后率与价格折扣和缺货期间库存水平相关,提出了一种需求为任意分布且提前期和构建成本均可控的EOQ模型,证明了模型存在唯一最优解,给出了一种寻优算法。数值仿真分析表明,一般情况下,压缩提前期和降低构建成本能降低订购批量和安全库存,降低库存总成本;短缺量拖后系数和缺货概率对库存总成本影响较大,企业应尽量降低缺货概率,尤其在短缺量拖后系数较小时。     

Multi-product lot scheduling with backordering and shelf-life constraints

In this paper, we revisit the economic lot scheduling problem (ELSP), where a family of products is produced on a single machine, or facility, on a continual basis. Our focus is on the determination of a feasible production schedule, including the manufacturing batch size of each item. We assume that total backordering is permissible and that each of the products has a limited post-production shelf life. Several studies examining this problem have suggested a rotational common cycle approach, where each item is produced exactly once every cycle. To ensure schedule feasibility, we resort to the technique of reducing individual production rates and allow the flexibility of producing any item more than once in every cycle, in conjunction with appropriate timing adjustments. In order to solve this more generalized model, which is NP hard, we suggest a two-stage heuristic algorithm. A numerical example demonstrates our solution approach.     

Algorithms with limited number of preemptions for scheduling on parallel machines

In previous study on comparing the makespan of the schedule allowed to be preempted at most i times and that of the optimal schedule with unlimited number of preemptions, the worst case ratio was usually obtained by analyzing the structures of the optimal schedules. For m identical machines case, the worst case ratio was shown to be 2m/(m+i+1) for any 0≤i≤m?1 (Braun and Schmidt in SIAM J. Comput. 32(3):671–680, 2003), and they showed that LPT algorithm is an exact algorithm which can guarantee the worst case ratio for i=0. In this paper, we propose a simpler method which is based on the design and analysis of the algorithm and finding an instance in the worst case. It can not only obtain the worst case ratio but also give a linear algorithm which can guarantee this ratio for any 0≤i≤m?1, and thus we generalize the previous results. We also make a discussion on the trade-off between the objective value and the number of preemptions. In addition, we consider the i-preemptive scheduling on two uniform machines. For both i=0 and i=1, we give two linear algorithms and present the worst-case ratios with respect to s, i.e., the ratio of the speeds of two machines.     

风险质检行为下的模糊生产-库存决策

本文研究在风险质检行为下需求与生产率具有模糊属性的由一个制造商与一个零售商构成的生产-库存问题。在假设制造商生产的产品存在质量缺陷,零售商在质检过程中存在质检风险的基础上,分别建立了需求模糊下的生产-库存总成本模型,需求与生产率模糊下的总成本模型,运用符号距离法进行了逆模糊化处理,将模糊化的生产-库存总成本转化为确定性成本。证明了两类情形下的总成本均是关于最优订购量与最优缺货量的联合凸函数。数值分析结果表明：随着期望缺陷率的增加,最优订购量增加,最优缺货量减少,最优成本增加,且最优成本增加的速度越来越快。一类质检风险增大引起总成本增加,二类质检风险增大引起总成本减少。最优订购策略对质检一类风险敏感,对质检二类风险不敏感。     

Efficiency of well-diversified portfolios: Evidence from data envelopment analysis

In this work, we evaluate eight exchange traded funds (ETFs) and their benchmark index (the KOSPI 200 Index), based on the Sharpe ratio and the Treynor ratio and find that the performance of these well-diversified portfolios are quite poor relative to individual stocks. Investors׳ preference to avoid the well-diversified portfolios would be related to this poor performance. However, we empirically show that ETFs and the KOSPI 200 Index are the most efficient investment instruments with respect to the new performance measure designed on the basis of the data envelopment analysis (DEA) methodology. Examining the panel data over the period between 2003 and 2014 indicates that well-diversified portfolios improve the efficiency by adjusting the input variables (σ and β). Furthermore, they do so more effectively as they mature.     

Integration and efficiency convergence in EU banking markets

Evidence of financial integration and convergence are considered of importance in assessing the outcome of EU deregulation policies aimed at improving the efficiency and performance of banking sectors. This paper evaluates the recent dynamics of bank cost efficiency by means of data envelopment analysis (DEA). Borrowing from the growth literature, we apply dynamic panel data models (GMM) to the concepts of β-convergence and σ-convergence to assess the speed at which banking markets are integrating. We also employ a partial adjustment model to evaluate convergence towards best practice. Results seem to provide supporting evidence of convergence of efficiency levels towards an EU average. Nevertheless, there is no evidence of an overall improvement of efficiency levels towards best practice.     

Structural empowerment,psychological empowerment,and work engagement: A cross-country study

In this cross-country study we drew on job demands-resources theory to investigate whether psychological empowerment mediates the positive association between structural empowerment and work engagement and, consequently, task performance and intention to quit. A total of 1033 employees working in the service sector in Spain (N = 515) and the United Kingdom (N = 518) participated in the study. Multi-group structural equation modeling analyses revealed that psychological empowerment partially mediated the positive relationship between structural empowerment and work engagement, and that work engagement associated positively with task performance and negatively with intention to quit. Invariance analyses suggested that the positive link between psychological empowerment and work engagement was stronger for employees working in the UK than in Spain, providing support for partial structural invariance of the hypothesized model. These findings suggest that psychological empowerment is an underlying mechanism that may explain why structural empowerment relates positively to work engagement with implications for theory (i.e., extend the nomological network of the investigated constructs) and management practice (e.g., emphasize the role of structural empowerment for work design).     

Adjusting financial ratios: a Bayesian analysis of the Spanish manufacturing sector

In this paper, we propose a Bayesian hierarchical model based on the partial adjustment model described by Wu and Ho (Rev. Quant. Finance Acc. 9 (1997) 71). The proposed model allows us to estimate the average adjustment coefficients associated with the error correction component and with the sensitivity of the firm to exogenous factors that have an industry-wide effect. Using the proposed model, we analyse the financial ratios calculated by The Bank of Spain’s Central Balance Sheet Office (CBSO) corresponding to the Spanish manufacturing sector during the period 1986–1997. In almost all the ratios analysed, we find that the error correction component exerts a greater influence, with the Interest Expense to Liabilities ratio demonstrating a greater sensitivity to this effect; by contrast, factors endogenous to the firm have more influence over the Indebtedness ratio. When considered by sectors, we find that it is the Transport sector which enjoys the greatest capacity for manoeuvre in the Profitability and Indebtedness ratios.     

Appointment scheduling in tandem-type service systems

Appointment-based service systems arise in a broad variety of healthcare settings (for example an outpatient clinic or a dentist). Where most existing algorithms specifically consider the situation of the patient undergoing a single service, in many practical situations multiple services have to be sequentially performed. Modeling the service system as a tandem queue, the main objective of this paper is to generate schedules that soundly balance the interests of patients (i.e., low waiting times) and staff (i.e., low idle times). Importantly, following up on prior work for the single-node queue, we advocate a phase-type based technique that can deal with any service-time distribution (which may, in addition, vary across patients). Relying on a novel recursive scheme to evaluate the sojourn-time distribution of clients in such tandem systems, we show how optimal schedules can be computed. Our technique is illustrated by extensive numerical experimentation, also leading to practical guidelines that apply to a broad range of parameter settings.     

Computing globally optimal (s,S,T) inventory policies

We consider a single-echelon inventory installation under the (s,S,T) periodic review ordering policy. Demand is stationary random and, when unsatisfied, is backordered. Under a standard cost structure, we seek to minimize total average cost in all three policy variables; namely, the reorder level s, the order-up-to level S and the review interval T. Considering time to be continuous, we first model average total cost per unit time in terms of the decision variables. We then show that the problem can be decomposed into two simpler sub-problems; namely, the determination of locally optimal solutions in s and S (for any T) and the determination of the optimal T. We establish simple bounds and properties that allow solving both these sub-problems and propose a procedure that guarantees global optimum determination in all policy variables via finite search. Computational results reveal that the usual practice of not treating the review interval as a decision variable may carry severe cost penalties. Moreover, cost differences between (s,S,T) and other standard periodic review policies, including the simple base stock policy, are rather marginal (or even zero), when all policies are globally optimized. We provide a physical interpretation of this behavior and discuss its practical implications.