Extend (r, Q) Inventory Model Under Lead Time and Ordering Cost Reductions When the Receiving Quantity is Different from the Ordered Quantity

This paper investigates the continuous review inventory model involving variable lead time with partial backorders, where the amount received is uncertain. The options of investing in ordering cost reduction is included, and lead time can be shortened at an extra crashing cost. The objective of this article is to simultaneously optimize the order quantity, reorder point, ordering cost and lead time. We first assume that the lead time demand follows a normal distribution and develop an algorithm to find the optimal solution. Then, we relax the assumption of normality to consider a distribution free case where only the mean and standard deviation of lead time demand are known. We apply the minimax distribution free procedure to solve this problem. For both cases, we also show that the objective cost function to be minimized is jointly convex in the decision variables. Furthermore, two numerical examples are given to illustrate the results.     

A general method for addressing forecasting uncertainty in inventory models

In practice, inventory decisions depend heavily on demand forecasts, but the literature typically assumes that demand distributions are known. This means that estimates are substituted directly for the unknown parameters, leading to insufficient safety stocks, stock-outs, low service, and high costs. We propose a framework for addressing this estimation uncertainty that is applicable to any inventory model, demand distribution, and parameter estimator. The estimation errors are modeled and a predictive lead time demand distribution obtained, which is then substituted into the inventory model. We illustrate this framework for several different demand models. When the estimates are based on ten observations, the relative savings are typically between 10% and 30% for mean-stationary demand. However, the savings are larger when the estimates are based on fewer observations, when backorders are costlier, or when the lead time is longer. In the presence of a trend, the savings are between 50% and 80% for several scenarios.     

A Note on Minimax Mixture of Distributions Free Procedure for Inventory Model with Variable Lead Time

In recent papers, Moon and Choi (1998) and Hariga and Ben-Daya (1999)considered a continuous review inventory model with a mixture of backordersand lost sales in which the lead time, the order quantity, and the reorder pointare decision variables was studied. Moreover, they also develop a minimaxdistribution free procedure for the problem. While the demands of differentcustomers are not identical in the lead time, then we can't only use a singledistribution (such as Moon & Choi (1998) and Hariga & Ben-Daya (1999))to describe the demand of the lead time. Hence, we correct and extend the modelof Moon and Choi (1998) and Hariga and Ben-Daya (1999) by considering thelead time demand with the mixture of distributions. In addition, we also applythe minimax mixture of distributions free approach to the model by simultaneouslyoptimizing the order quantity, the reorder point, and the lead time to devise a practical procedure which can be used without specific information on demand distribution.     

The interchangeability of safety stocks and safety lead time

Due to the uncertainty in estimating both the demand for end products and the supply of components from lower levels, buffering techniques should be included before the loading of a material requirement planning (MRP) system. Safety stocks and safety lead time are two techniques of providing buffering for loading. There have been many studies made concerning the determination of the amount of safety stocks and safety lead time. Some guidelines for choosing between safety stocks and safety lead time for dealing with uncertainty in both demand and supply also have been established. Although these two different methods have been used successfully, it has not been documented that using these two methods in a given situation will yield essentially the same results; that is, the interchangeability of these two buffering techniques has not been explored quantitatively.Since the net influence of safety stocks and safety lead time and their quantitative interchangeability are of major interest, an analytical model is proposed for this study. The lead-time offset procedure for components loading are represented by a matrix model that is based on a lot-for-lot lot-sizing technique. This lead-time offset matrix model is the product of the precedence matrix and the fixed-duration matrix. The precedence matrix is formed according to the total requirement factor matrix and the duration matrix is formed by each component process time. Thus, the lead-time offset matrix will generate the starting period of each component.When the lead-time offset procedure is modeled, the net influence of buffering quantity can be analyzed. The planned safety stock that is normally used to accommodate unexpected demand, shortage in supply, and defects from the operation at each process can be combined with demand to form the master production schedule. The revised lead time due to the integration of the safety stocks can be calculated through the lead-time offset model. The safety lead time may extend the component process time as well as overall production lead time if the designated safety lead time is longer than the available slack time in a fixed lead-time loading system.When the proposed lead-time offset model is further examined, it is found that planned safety stocks at the higher level can buffer the fluctuations of lower level components quantity as well as the fluctuations of same level components quantity. Safety stocks can also buffer shortages that are caused by the delay of raw material and manufacturing processes. Thus, safety stocks can be used to buffer unexpected delay time up to certain limits. A planned safety lead time at higher level component process can buffer the fluctuations of lower level components process time, as well as the same level component process time. The safety lead time can be used to produce additional products to meet unexpected excessive demand up to certain limits under the following conditions: 1. The excessive demand is known before the actual processing of the components in the lowest level. 2. The raw material at the lowest level is available.Although safety stocks and safety lead time are interchangeable in terms of the ability to buffer variations in quantity, the conditions for safety lead time are seldom met in actual practices. Thus, the slack time in a fixed lead-time loading system cannot be considered as an effective measure to substitute safety stocks. However, all or part of the delay in manufacturing processes or the supply from the lower level components can be buffered by the safety stock and the MPS will still be met. From this study, it is obvious that the slack time can be reduced when safety stocks are planned for an MRP system. The reduction of fixed lead-time duration will be beneficial to the overall planning and scheduling in MRP systems.     

The effect of lead-time variability: The case of independent demand

We show in this study, through analysis and examples, the impact on stockouts and stockout risk if the variability of lead time in independent demand systems is ignored. In calculating safety stocks, we recommend that the compound distribution of demand during lead time, or a good approximation to it, be used. We motivate the article by a case study on lead-time variability at the U.S. Air Force and show the impact of lead-time variability by means of numerical examples and by marginal analysis. Having established that it is essential to consider lead-time variability, we take advantage of theoretical developments and show how to calculate reorder points and safety stocks in some common situations.It is important to use the proper form for the compound distribution of demand during lead time. A normal approximation to it will often yield significant errors. This is because the true distribution is usually very much skewed to the right.     

一种新的超市门店需求预测方法与库存管理模型

探讨了VMI模式下如何确定连锁超市门店的订货点的问题。利用专家判定法改进了基于历史数据的零售商需求预测模型，并在此需求预测模型基础上提出一种具有时间和数量柔性的自动补货策略模型。最后通过实倒分析表明，研究结果有助于帮助企业在快速响应市场需求的同时。降低企业的成本，提高企业的竞争力。此模型将在某连锁百货的订货系统中得到应用。     

Optimal designs for Cox regression

Optimal designs under a survival analysis framework have been rarely considered in the literature. In this paper, an optimal design theory is developed for the typical Cox regression problem. Failure time is modeled according to a probability distribution depending on some explanatory variables through a linear model. At the end of the study, some units will not have failed and thus their time records will be censored. In order to deal with this problem from an experimental design point of view it will be necessary to assume a probability distribution for the time an experimental unit enters the study. Then an optimal conditional design will be computed at the beginning of the study for any possible given time. Thus, every time a new unit enters the study, there is an experimental design to be determined. A particular and simple case is used throughout the paper in order to illustrate the procedure.     

A Note on EOQ Model for Items with Mixtures of Exponential Distribution Deterioration,Shortages and Time-varying Demand

An inventory model is considered in which inventory is depleted not only by demand, but also by deterioration. In this paper, we derive the EOQ model for inventory of items that deteriorates at a mixtures of exponential distributed rate, assuming the demand rate with a continuous function of time. Moreover, the proposed model cannot be solved directly in a closed form, thus we used the computer software IMSL MATH/LIBRARY (1989) to find the optimal reorder time Further, we also find that the optimal procedure is independent of the form of the demand rate. Finally, we also assume that the holding cost is a continuous, nonnegative and non-decreasing function of time in order to generalize EOQ model. Moreover, four numerical examples and sensitivity analyses are provided to assess the solution procedure.     

用二叉树解决配送线路问题

物流配送是物流中的核心环节,配送成本在整个物流过程费用中占有很大的比重,因此为了减少配送成本有必要对其配送路线进行合理优化。运用二叉树遍历的知识并结合节约算法的思想,将货物需求点作为叶子结点并适当增加一些需求量为零的叶子结点构造一种有特殊意义的二又树,提出了一种运用这种特殊二叉树在满足车辆额定载货量的前提下寻求最优配送路线的方法．并通过实例证明了其正确性。     

随机需求和提前时间存储系统的建模与仿真

讨论了需求量、需求发生的间隔以及提前时间都为随机变量的一般随机存储系统。以平均库存总费用和缺货率为系统性能指标建立仿真模型，将二者加权平均得到策略的综合系数。在各种费用参数给定后，利用计算机仿真技术来选定最优存储策略。     

The part-period balancing algorithm and its look ahead-look back feature: A theoretical and experimental analysis of a single stage lot-sizing procedure

This paper investigates the discrete Part-Period Balancing (PPB) lot-sizing algorithm and its optional feature, the Look Ahead-Look Back tests. PPB is the most commonly used dynamic lot-sizing procedure in practice and it has also been tested extensively in simulation experiments. Although its overall cost performance, relative to other heuristics, have been fairly good, a fundamental flaw with the model has been noted in the literature. This deficiency leads to poor performance under certain conditions.In this paper a simple adjustment to the main algorithm is analytically derived under the assumptions of a constant demand rate and an infinite planning horizon. The adjustment leads to an optimal behavior for the PPB heuristic under the stated conditions. Subsequent experimental analysis through simulation of lot-sizing performance in environments with time-varying, discrete demand shows that the proposed adjustment leads to significant cost reductions.This paper also analyzes the Look Ahead-Look Back tests which is the distinguishing feature between the PPB procedure and the Least Total Cost algorithm. The tests were devised to improve the cost performance of the PPB heuristic by marginally adjusting each tentative lot-size. The effect of the Look Ahead-Look Back tests have, however, never been verified in the literature. The tests have undergone some changes over time, when they have been included in commercial software packages for inventory management. We suggest yet another modified version in this paper.In the last portion of the paper, the cost effectiveness of the Look Ahead-Look Back tests is confirmed through simulation. That is, when used together with the original PPB procedure, they lead to an improved cost performance. It is also shown that a combination of these tests and the adjustment to the PPB procedure mentioned earlier leads to an even lower average total cost. All cost improvements are statistically significant. It is finally noted that the Look Ahead-Look Back tests perform poorly in certain constant demand situations. Additional analytic and experimental analysis shows that these results stem from a dominance of the Look Back test over the Look Ahead test, leading to the former test being performed more often. This can easily be corrected, however, by checking for sufficient variability in the data before the Look Back test is employed.     

具有模糊—随机性的易腐货品最优补货策略

结合易腐物品的变质特点,考虑不确定环境下配送中心以固定周期为连锁门店送货、连锁商单位时间需求为模糊变量的分销系统,通过利用三参数Weibull函数来描述易腐货品的变质特性,运用可信性理论的逆模糊化和随机理论,建立了模糊-随机下的每周期易腐货品的最优补货策略模型。借助MATLAB得到了求解最优补货策略的方法,并通过仿真模拟验证了方法的合理性。     

Long-term forecasting of fuel demand at theater entry points

Managing the distribution of fuel in theater requires Army fuel planners to forecast demand at the strategic level to ensure that fuel will be in the right place, at the right time, and in the amounts needed. This work presents a simulation approach to forecasting that accounts for the structure of the supply chain network when aggregating the demand of war fighters across the theater over the forecasting horizon. The resulting empirical distribution of demand at the theater entry point enables planners to identify forecast characteristics that impact their planning process, including the amplitudes and temporal positions of peaks in demand, and the estimated lead time to the point of use. Experimentation indicates that the forecasts are sensitive to the pattern of war fighter demand, the precise structure of the in-theater supply chain network, and the constraints and uncertainty present in the network, all of which are critical planning considerations.     

Determining the optimal scheme of zoned effluent charges for the control of air pollution

A minimum costly zoned effluent charge program for the control of air pollution is considered paying attention to the general situation where polluters' cost functions of treating pollutant are unknown to the policy authority, and an iterative procedure by which the authority can attain a set of optimal charges is presented. The algorithm consists of steps of estimating the unknown treatment cost functions by observing polluter behavior and steps of revising charges based on the estimated cost functions. In the latter steps, a newly developed solution procedure for the zoned charge programming problem is involved. Simulated use of the algorithm indicates that the iterative charge revision procedure proposed in this paper can effectively provide the optimal scheme of zoned effluent charges.     

Optimal structure of joint inventory-pricing management with dual suppliers and different lead times

We consider a joint inventory-pricing control problem in a single-product, periodic-review, dual-supplier inventory system. The two suppliers have different lead times. One expedited supplier offers instantaneous replenishment, and one regular supplier requires an L-period lead time for delivery. The supply quantity is stochastic and the demand is price-dependent. For the expedited inventory replenishment, we characterize the optimal policy as a state-dependent almost-threshold policy by extending the stochastically linear in mid-point to a multidimensional setting. To investigate the optimal regular inventory replenishment and pricing policy, we propose the notions of partially stochastic translation (PST) and increasing partially stochastic translation (IPST), which help in obtaining the antimultimodularity preservation in dynamic programming problems. We provide properties, sufficient conditions, and examples for PST and IPST functions. By applying PST and IPST, we obtain the antimultimodularity of the profit functions. The antimultimodular profit functions ensure that the optimal regular ordering quantity and the optimal price are monotone in the current inventory level and outstanding order quantities. Moreover, we reveal that as the time interval increases, the effects of previous outstanding orders on the optimal regular ordering and pricing decisions are decreasing and increasing, respectively. PST and IPST also enable us to further characterize the optimal expedited ordering quantity as decreasing in the inventory level. However, the optimal expedited ordering quantity can be non-monotone with respect to the outstanding order quantities, as shown in the example.     

Fixed Effects and Random Effects Estimation of Higher-order Spatial Autoregressive Models with Spatial Autoregressive and Heteroscedastic Disturbances

Abstract

This paper develops a unified framework for fixed effects (FE) and random effects (RE) estimation of higher-order spatial autoregressive panel data models with spatial autoregressive disturbances and heteroscedasticity of unknown form in the idiosyncratic error component. We derive the moment conditions and optimal weighting matrix without distributional assumptions for a generalized moments (GM) estimation procedure of the spatial autoregressive parameters of the disturbance process and define both an RE and an FE spatial generalized two-stage least squares estimator for the regression parameters of the model. We prove consistency of the proposed estimators and derive their joint asymptotic distribution, which is robust to heteroscedasticity of unknown form in the idiosyncratic error component. Finally, we derive a robust Hausman test of the spatial random against the spatial FE model.     

不确定条件下缺货不补货的库存模型分析

对以前研究得很少的一种情况——连续检查库存、提前期固定、需求随机且缺货不补进行了分析,详细推导了其最佳订货点和订货批量,并探讨了简化计算的合理性,然后用EXCEL进行了实例计算,最后进一步推广到提前期也随机的情况。     

Price asymptotics

This paper examines the pricing decisions of a seller facing an unknown demand function. It is assumed that partial information, in the form of an independent random sample of values, is available. The optimal price for the inferred demand satisfies a consistency property—as the size of the sample increases, the maximum profit and price approach the values for the case where demand is known. The main results deduced here are asymptotics for prices. Prices converge at a rate of O _p (n ^−1/3) with a limit that can be expressed as a functional of a Gaussian process. Implications for the comparison of mechanisms are discussed.      

Valuing lead time

When do short lead times warrant a cost premium? Decision makers generally agree that short lead times enhance competitiveness, but have struggled to quantify their benefits. Blackburn (2012) argued that the marginal value of time is low when demand is predictable and salvage values are high. de Treville et al. (2014) used real-options theory to quantify the relationship between mismatch cost and demand volatility, demonstrating that the marginal value of time increases with demand volatility, and with the volatility of demand volatility. We use the de Treville et al. model to explore the marginal value of time in three industrial supply chains facing relatively low demand volatility, extending the model to incorporate factors such as tender-loss risk, demand clustering in an order-up-to model, and use of a target fill rate that exceeded the newsvendor profit-maximizing order quantity. Each of these factors substantially increases the marginal value of time. In all of the companies under study, managers had underestimated the mismatch costs arising from lead time, so had underinvested in cutting lead times.     

Kostenoptimale Prüfpläne für ein quantitatives Merkmal

Summary A sample inspection plan is said to be optimal in the sense of the minimax regret principle, if it minimizes the difference between the expected total costs and the unavoidable costs. The results of this article can be used to calculate such sample inspection plans for a quantitative quality control with one-sided tolerance limits and known or unknown variance of the test variate. As an example of practical importance the case of a normal variate with unknown variance is considered. Formulae are given to estimate the error that arises if the assumed distribution of the test variate differs from the actual distribution.