A single-period inventory model with fuzzy demand

The single-period inventory models have wide applications in the real world in assisting the decision maker to determine the optimal quantity to order. Due to lack of historical data, the demand has to be subjectively determined in many cases. In this paper, a single-period inventory model for cases of fuzzy demand is constructed. The costs considered include the procurement cost, shortage cost, and holding cost. For different fuzzy total cost resulted from different order quantity, a method for ranking fuzzy numbers is adopted to find the optimal order quantity in terms of the cost. When the profit gained from selling one item is less (greater) than the loss incurred due to one unsold item, the optimal order quantity lies in the range defined for the left-shape (right-shape) function of the fuzzy demand. If the unit profit is equal to the unit loss, then all quantities with a membership grade 1 are optimal to be ordered. The methodology of this paper can be applied to construct other inventory models with fuzzy demand.     

A single-period inventory and payment model with partial trade credit

In this paper, we consider the ordering and payment issues for a retailer facing stochastic demand. We assume that the retailer can enjoy the partial trade credit from his supplier and borrow money from bank as well if needed, and he can also earn return by investing his superfluous on-hand cash (if any). The retailer’s objective is to maximize the expected cash level at the end of the selling period. We formulate the model of this problem by taking initial inventory and capital levels as the two-dimensional state. First, given the exogenous fraction of immediate payment, we show that unlike the critical fractile solution the retailer’s optimal ordering strategy is a two-threshold policy, which is independent of the retailer’s initial inventory level and capital level. Second, we consider an extensive model where the fraction of immediate payment is decided by the retailer. We employ the sequential optimization procedure to solve the extensive problem, and present the structure of the retailer’s optimal policies under different partial-trade-credit penalty rates. Numerical experiments show that if the fraction of immediate payment is exogenous, both partial trade credit and loan opportunity are detrimental to the capital-constrained retailer in many cases, although they can stimulate the retailer to order more.     

Decision-making in a single-period inventory environment with fuzzy demand

This paper first defines the profitability to be the probability of achieving a target profit under the optimal ordering policy, and introduces a new index (achievable capacity index; I_A) which can briefly analyze the profitability for newsboy-type product with normally distributed demand. Note that since the level of profitability depends on the demand mean μ and the demand standard deviation σ if the related costs, selling price, and target profit are given, the index I_A is a function of μ and σ. Then, we assess level performance which examines if the profitability meets designated requirement. The results can determine whether the product is still desirable to order/manufacture. However, μ and σ are always unknown, and the demand quantity is common to be imprecise, especially for new product. To tackle these problems, a constructive approach combining the vector of fuzzy numbers is introduced to establish the membership function of the fuzzy estimator of I_A. Furthermore, a three-decision testing rule and step-by-step procedure are developed to assess level performance based on fuzzy critical values and fuzzy p-values.     

A mixed inventory model with variable lead time and random back-order rate

In this paper, we study the determination of the optimal lead time, reorder point and order quantity considering that the back-order probability of a demand made during a stock-out period depends on the interval from the moment in which the order is placed until the next replenishment. Two models are analysed for the specification of the back-order probability: exponential functions and piecewise constant functions. The distribution of the lead time demand is assumed to be Poisson. An algorithm for the determination of the optimal order quantity, reorder point and lead time is given. A numerical example is presented to illustrate the results.     

A finite time horizon inventory model with deterioration and time-value of money under the conditions of permissible delay in payments

A varying deterioration rate, time-value of money and the condition of permissible delay in payments used in conjunction with the EOQ model are the focus of discussion. The replenishment number and fraction of each cycle in which there is no shortage are both determined to minimize the present value of inventory cost over a finite planning horizon. Two special cases and numerical examples are presented to illustrate the model.     

Lot size reorder point inventory model with controllable lead time and set-up cost

This paper deals with the lead time and set-up cost reductions problem on the modified lot size reorder point inventory model in which the production process is imperfect. We consider that the lead time can be shortened at an extra crashing cost, which depends on the length of lead time to be reduced and the ordering lot size. The option of investing in reducing set-up cost is also included. Two commonly used investment cost functional forms, logarithmic and power, are employed for set-up cost reduction. We assume that the stochastic demand during lead time follows a Normal distribution. The objective is simultaneously to optimize the lot size, reorder point, set-up cost and lead time. An algorithm of finding the optimal solution is developed, and two numerical examples are given to illustrate the results.     

A single period inventory model with a truncated normally distributed fuzzy random variable demand

In this article, a single period inventory model has been considered in the mixed fuzzy random environment by assuming the annual customer demand to be a fuzzy random variable. Since assuming demand to be normally distributed implies that some amount of demand information is being automatically taken to be negative, the model has been developed for two cases, using the non-truncated and the truncated normal distributions. The problem has been developed to represent scenarios where the aim of the decision-maker is to determine the optimal order quantity such that the expected profit is greater than or equal to a predetermined target. This ‘greater than or equal to’ inequality has been modelled as a fuzzy inequality and a methodology has been developed to this effect. This methodology has been illustrated through a numerical example.     

A periodic review inventory model involving variable lead time with a service level constraint

In this study, we consider a mixture periodic review inventory model in which both the lead time and the review period are considered as decision variables. Instead of having a stock-out term in the objective function, a service level constraint is added to the model. In our paper, we first assume that the protection interval (i.e. the review period plus the lead time) demand follows a normal distribution, and then we relax this assumption and only assume that the first two moments of the protection interval demand are given. For each case, we develop an algorithm to find the optimal review period and optimal lead time. Furthermore, a sensitivity analysis is also performed.     

Change point detection in time series data with random forests

A large class of monitoring problems can be cast as the detection of a change in the parameters of a static or dynamic system, based on the effects of these changes on one or more observed variables. In this paper, the use of random forest models to detect change points in dynamic systems is considered. The approach is based on the embedding of multivariate time series data associated with normal process conditions, followed by the extraction of features from the resulting lagged trajectory matrix. The features are extracted by recasting the data into a binary classification problem, which can be solved with a random forest model. A proximity matrix can be calculated from the model and from this matrix features can be extracted that represent the trajectory of the system in phase space. The results of the study suggest that the random forest approach may afford distinct advantages over a previously proposed linear equivalent, particularly when complex nonlinear systems need to be monitored.     

A discrete-in-time deteriorating inventory model with time-varying demand,variable deterioration rate and waiting-time-dependent partial backlogging

A new inventory system is considered for deteriorating items in which unsatisfied demands are partially backlogged depending on the waiting time until the next replenishment and deterioration rates are constant, but different from period to period. Time is considered as a discrete variable, which better represents some real-life scenarios. An optimal solution can be easily derived. A sensitivity study is provided through substantial numerical experiments to illustrate the use and robustness of the proposed model. The sensitivity analysis reveals some useful and significant insights.     

随机系数自回归模型变均值点在线监测与应用

对随机系数自回归模型的变均值点进行在线监测时, 如果变均值点的位置远离开始监测点, 则平均地说, 需要较长的运行时间方能检测到该变均值点. 为此, 笔者引进一个窗宽参数, 提出了一种改进的在线监测方法. 给出了监测统计量在原假设下的极限分布, 并证明了此方法的一致性. 模拟结果显示新方法明显优于已有的方法. 最后将该方法应用于两组股票价格均值点的监测问题中, 说明了方法的有效性.     

A production inventory model with fuzzy random demand and with flexibility and reliability considerations

The classical inventory control models assume that items are produced by perfectly reliable production process with a fixed set-up cost. While the reliability of the production process cannot be increased without a price, its set-up cost can be reduced with investment in flexibility improvement. In this paper, a production inventory model with flexibility and reliability (of production process) consideration is developed in an imprecise and uncertain mixed environment. The aim of this paper is to introduce demand as a fuzzy random variable in an imperfect production process. Here, the set-up cost and the reliability of the production process along with the production period are the decision variables. Due to fuzzy-randomness of the demand, expected average profit of the model is a fuzzy quantity and its graded mean integration value (GMIV) is optimized using unconstraint signomial geometric programming to determine optimal decision for the decision maker (DM). A numerical example has been considered to illustrate the model.     

基于时间点过程的时间序列预测模型

对城市中发生的事件进行有效预测,可以为政府避免、控制或减轻相关的社会风险提供决策支撑.首先,提出基于积分求导法的条件强度函数式,提高序列预测精度;其次,构建基于递归神经网络和累积危险函数的时间点过程模型,通过递归神经网络捕获历史事件的非线性依赖关系,利用全连接网络获得累积危险函数;最后,选择具有代表性的合成数据集和真实...     

A note on a production‐inventory model under stock‐dependent demand,Weibull distribution deterioration,and shortage

Recently, Roy and Chaudhuri presented a production‐inventory model for deteriorating items when the demand rate depends on the instantaneous inventory level and the production rate depends on both stock level and demand. The models without shortages and with shortages were discussed. These models impose continuity constraints on the inventory levels, and consequently the optimization problems so obtained are constrained ones. However, the authors treated them as unconstrained ones.     

A random fuzzy minimum spanning tree problem through a possibility-based value at risk model

This paper considers a minimum spanning tree problem under the situation where costs for constructing edges in a network include both fuzziness and randomness. In particular, this article focuses on the case that the edge costs are expressed by random fuzzy variables. A new decision making model based on a possibility measure and a value at risk measure is proposed in order to find a solution which fully reflects random and fuzzy information. It is shown that an optimal solution of the proposed model is obtained by a polynomial-time algorithm.     

A numerical solution for a two-stage production and inventory system with random demand arrivals

This paper is about the study of a production lot sizing problem consisting of customers, one retailer, and one manufacturer. Demand from customers arrives randomly at a retailer one unit at a time. The retailer replenishes inventory from the manufacturer upon receiving a customer's order after its inventory depleted to zero. The manufacturer's production rate is assumed to be a finite constant. A production cycle starts when the manufacturer's inventory falls to or below zero and stops when its on-hand inventory reaches its optimal level. During the uptime in a production cycle, inventory is being built while randomly arriving orders from retailer are being fulfilled. The order arrival times from customers are independently and identically distributed, hence the inventory processes at both the manufacturer and the retailer become a renewal process that is difficult to solve analytically for a general distribution of order arrival time. Therefore, a numerical approach is used in developing a search procedure to obtain the optimal solution to the problem. Employing such a numerical approach, we also investigate how optimal solutions in different cases will change over the spectrum of some key parameters of the problem.     

The production size and inventory policy for a manufacturer in a two-echelon inventory model

This paper proposes a two-echelon inventory model for a periodical commodity, in which the market and manufacturing channels are combined. This model can be used to solve the production policy, the order policies of the raw materials for the manufacturer, and order size for the retailer. By assuming that the retailers’ demand obeys normal distribution and that the retailer makes orders according to the Newsboy Rule, we derive the necessary and sufficient conditions for the optimal solution of production size, wholesale price, and replenishment cycle of raw materials for the manufacturer. Also, the necessary condition is explored in order to gain managerial insights and economic implications based on numerical examples and sensitivity analysis.     

A minimax distribution free procedure for mixed inventory model involving variable lead time with fuzzy demand

In a recent paper, Ouyang and Wu applied the minimax decision approach to solve a continuous review mixed inventory model in which the lead time demand distribution information is unknown but the annual demand is fixed and given. However, in the practical situation, the annual demand probably incurs disturbance due to various uncertainties. In this article, we attempt to modify Ouyang and Wu's model by considering two fuzziness of annual demand (i.e., fuzzy number of annual demand and statistic-fuzzy number of annual demand) and to investigate a computing schema for the continuous review inventory model in the fuzzy sense. We give an algorithm procedure to obtain the optimal ordering strategy for each case.Scope and purposeIn most of the early literature dealing with inventory problems, either using deterministic or probabilistic models, lead time is viewed as a prescribed constant or a stochastic variable. Recently, some researchers (e.g., Liao and Shyu, Ben-Daya and Raouf, and Ouyang and Wu) incorporated the crashing lead time idea to continuous review inventory models, in which the annual demand is given and fixed. However, in the real situation, the annual demand will probably have a little disturbance due to various uncertainties. The purpose of this article is to modify the Ouyang and Wu's model to accommodate this reality, specifically, we apply the fuzzy set concepts to deal with the uncertain annual demand. We first consider a case where the annual demand is treated as the triangular fuzzy number. Then, we employ the statistical method to construct a confidence interval for the annual demand, and through it to establish the corresponding fuzzy number (namely, the statistic-fuzzy number). For each fuzzy case, we investigate a computing schema for the new model and develop an algorithm to find the optimal ordering strategy.     

模糊随机决策环境下的核电站不常用条件(Q,r)库存模型

针对固定提前期内的需求为三角模糊变量,且用户总需求为梯形模糊随机变量的情形下,构建了不常用备件连续盘点模式下的（Q,r）模型,并推导出模糊成本最小化函数,进而利用基于模糊数期望值理论的去模糊化方法,求出最优订货点及订货量．最后,通过一个实例验证了模型的科学性和实用性．     

Deterministic inventory model with two levels of storage,a linear trend in demand and a fixed time horizon

A deterministic inventory model is developed for a single item having two separate storage facilities (owned and rented warehouses) due to limited capacity of the existing storage (owned warehouse) with linearly time-dependent demand (increasing) over a fixed finite time horizon. The model is formulated by assuming that the rate of replenishment is infinite and the successive replenishment cycle lengths are in arithmetic progression. Shortages are allowed and fully backlogged. As a particular case, the results for the model without shortages are derived. Results are illustrated with two numerical examples.Scope and purposeThroughout the world, the production of food grains is periodical. Normally, in countries where state control is less, the demand of essential food grains is lowest at the time of harvest and goes up to the highest level just before the next harvest. This phenomenon is very common in developing third world countries where most of the people are landless or marginal farmers. At the time of harvest, they share some grain/product with landowners and as soon as the small inventory is exhausted, they are forced to buy food grains from the open market. As a result, demand for food grains increases with time in a period along with the number of the people whose initial stock of food grains gets exhausted.In this paper, a two-storage inventory model with time-dependent demand and fixed time horizon is developed and solved by a mathematical programme based on gradient method. This methodology of model development and its solution are quite general and it can be applied to inventory models of any product whose production is periodical and demand increases linearly with time.