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 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.     

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 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.     

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 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.     

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.     

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.     

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．     

A mathematical model for order splitting in a multiple supplier single-item inventory system

In this paper we develop a mathematical model which considers multiple-supplier single-item inventory systems. The lead times of the suppliers and demand arrival rate are random variables. All shortages are backordered. Continuous review (s, Q) policy has been assumed. When the inventory level hits the reorder level, the total order is split among n suppliers. The problem is to determine the reorder level and order quantity for each supplier so that the expected total cost per time unit, including ordering cost, procurement cost, inventory holding cost and shortage cost is minimized. We also conduct extensive numerical experiments to show the advantages of our model compared to the relevant models in the literature. In addition, some managerial insights are observed.     

Reactive scheduling in a job shop where jobs arrive over time

The paper considers the dynamic job shop scheduling problem (DJSSP) with job release dates which arises widely in practical production systems. The principle characteristic of DJSSP considered in the paper is that the jobs arrive continuously in time and the attributes of the jobs, such as the release dates, routings and processing times are not known in advance, whereas in the classical job shop scheduling problem (CJSSP), it is assumed that all jobs to be processed are available at the beginning of the scheduling process. Reactive scheduling approach is one of the effective approaches for DJSSP. In the paper, a heuristic is proposed to implement the reactive scheduling of the jobs in the dynamic production environment. The proposed heuristic decomposes the original scheduling problem into a number of sub problems. Each sub problem, in fact, is a dynamic single machine scheduling problem with job release dates. The scheduling technique applied in theproposed heuristic is priority scheduling, which determines the next state of the system based on priority values of certain system elements. The system elements are prioritized with the help of scheduling rules (SRs). An approach based on gene expression programming (GEP) is also proposed in the paper to construct efficient SRs for DJSSP. The rules constructed by GEP are evaluated in the comparison of the rules constructed by GP and several prominent human made rules selected from literatures on extensive problem sets with respect to various measures of performance.     

Analyzing survival curves at a fixed point in time for paired and clustered right-censored data

In clinical trials, information about certain time points may be of interest in making decisions about treatment effectiveness. Therefore, rather than comparing entire survival curves, researchers may wish to focus the comparison on fixed time points with potential clinical utility. For two independent samples of right-censored data, Klein et al. (2007) compared survival probabilities at a fixed time point by studying a number of tests based on transformations of the Kaplan-Meier estimators of the survival function. To compare the survival probabilities at a fixed time point for paired right-censored data or clustered right-censored data, however, their approach requires modification. In this paper, we extend the statistics to accommodate possible within-pair and within-cluster correlation. We use simulation studies to present comparative results. Finally, we illustrate the implementation of these methods using two real data sets.     

A multiple-vendor single-buyer integrated inventory model with a variable number of vendors

Supply chain management is concerned with the coordination of material and information flows in multi-stage production systems. A closer look at the literature reveals that previous research on the coordination of multi-stage production systems has predominantly focused on the sales side of the supply chain, whereas problems that arise on the supply side have often been neglected. This article closes this gap by studying the coordination of a supplier network in an integrated inventory model. Specifically, we consider a buyer sourcing a product from heterogeneous suppliers and tackle both the supplier selection and lot size decision with the objective to minimise total system costs. First, we provide mathematical formulations for the problem under study, and then suggest a two-stage solution procedure to derive a solution. Numerical studies indicate that our solution procedure reduces the total number of supplier combinations that have to be tested for optimality, and that it may support initiatives which aim on increasing the efficiency of the supply chain as a heuristic planning tool.     

A multi-class closed queueing maintenance network model with a parts inventory system

The system we address is a maintenance network of repairable items where a set of bases is supported by a centrally located repair depot and a consumable replacement parts inventory system. If an item fails, a replacement part must be obtained at the parts inventory system before the failed item enters the repair depot. The ordering policy for the parts is the (S,Q) inventory policy. An approximation method for this system is developed to obtain performance measures such as steady-state probabilities of the number of items at each site and the expected backorders at the parts inventory system. The proposed system is modelled as a multi-class closed queueing network with a synchronization station and analyzed using a product-form approximation method. Particularly, the product-form approximation method is adapted so that the computational effort on estimating the parameters of the equivalent multi-class network is minimized. In analyzing a sub-network, a recursive method is used to solve balance equations by exploiting the special structure of the Markov chain. Numerical tests show that the approximation method provides fairly good estimation of the performance measures of interests.     

A new pricing constrained single-product inventory-production model in perishable food for maximizing the total profit

In this paper, a new constrained single-product pricing and inventory model is presented. The goal of the proposed model is to decide on the prices as well as on the inventory and production decisions in order to maximize the total profit. The developed model is a nonlinear programming model which is solved by using hybrid genetic algorithm (HGA) and simulated annealing. Comparison of results obtained from the two algorithms shows that HGA is better. To enhance the performance of our algorithms, we apply the Taguchi experimental design method to tune their parameters. Finally, some recommendations for future developments are presented.     

A note on a random walk model for texture analysis

Wechsler and Citron recently published a random walk model with applications in texture analysis. It is shown in the current note that their chi-square test of homogeneity based on this model is misleading.     

Optimal policy for set-up time reduction in a multistage production inventory system

A batch production-inventory system consisting of multiple stages with an optimal policy of set-up time reduction and a fixed increment cost are discussed. The ratio of set-up time reduction as a decision variable under various cases of demand in the batch production-inventory model is considered. The ratio of set-up time reduction and lot size are solved simultaneously to obtain an optimal value of the total annual cost. A numerical example is presented to demonstrate the accuracy of the proposed method.