Some remarks on an optimal order quantity and reorder point when supply and demand are uncertain

In the reports in the literature on inventory control, the effects of the random capacity on an order quantity and reorder point inventory control model have been integrated with lead time demand following general distribution. An iterative solution procedure has been proposed for obtaining the optimal solution. However, the resulting solution may not exist or it may not guarantee to give a minimum to the objective cost function, the expected cost per unit time. The aim of this study was to introduce a complete solution of the order quantity/reorder point problem, optimality, properties and bounds on the optimal order quantity and reorder point. The two most appealing distributions of lead time demand, normal and uniform distributions, in conjunction with an exponentially distributed capacity, are used to illustrate our findings in determining the optimal order quantity and reorder point.     

Lot sizing under (Q, R) policy in a capcity constrained manufacturing facility

Traditional approaches to lot sizing and inventory control consider uncertainties in either the demand or manufacturing process but not both. In these models it is usually assumed that the lead times are independent and identically distributed, but this is not realistic in many practical instances. In this paper we consider the lot sizing problem for items with stochastic demands and manufacturing lead times. It is assumed that the inventory of the finished product is controlled by continuous review policy of Q,R type—order quantity, order point system—and the problem is to determine optimal Q and R. We examine the decision parameters under a variety of conditions using exact and approximate methods.     

Optimal reorder decision utilizing centralized stock information in a two-echelon distribution system

The objective of this paper is to develop an optimal reorder policy for a two-echelon distribution system with one central warehouse and multiple retailers. We assume the warehouse has centralized stock information and each facility uses continuous-review batch ordering policy. Since echelon stock policies may show poor performance for distribution systems, we propose a new type of policy that utilizes the centralized stock information more effectively. We define the order risk policy, which decides reorder time based on the order risk which represents the relative cost increase due to immediate order compared to delayed order. We formulate the order risk and prove the optimality of the order risk policy under the system assumption that the warehouse guarantees delivery within the fixed lead time. The order risk is derived from the marginal analysis. Since exact calculation of the order risk is complex, an approximation method is provided. Computational experiment that compares our policy with existing policies shows that a significant cost savings is obtained. The concept of the order risk can be extended to the other models.Scope and purposeDue to the improvement of modern information technologies, many companies start tracking the real-time stock information of the supply chain members. Thus, the reorder policy based on the real-time centralized stock information becomes very important. In this paper, we consider the reorder policy for a continuous-review batch-ordering two-echelon distribution system, utilizing the centralized stock information. Existing reorder policies are classified into installation stock policies and echelon stock policies. Since installation stock policies consider only local stock information, echelon stock policies have been used when the centralized stock information is available. For serial and assembly systems, it has been known that the echelon stock policies are superior to the installation stock policies. However, for distribution systems, both policies may outperform each other in different situations. The purpose of this study is to develop the optimal reorder policy for a distribution system with one-warehouse and multiple retailers, where the real-time stock information is centralized at the warehouse. We define a new type of reorder policy of which the reorder decision is based on the ‘order risk’. The order risk is defined as the relative cost increase due to immediate order compared to delayed order. We formulate the order risk and prove the optimality. For computational simplicity, we provide an approximation method to calculate the order risk. Computational experiment shows that a significant cost savings is obtained.     

A bi-objective continuous review inventory control model: Pareto-based meta-heuristic algorithms

In this paper, a bi-objective multi-product (r,Q) inventory model in which the inventory level is reviewed continuously is proposed. The aim of this work is to find the optimal value for both order quantity and reorder point through minimizing the total cost and maximizing the service level of the proposed model simultaneously. It is assumed that shortage could occur and unsatisfied demand could be backordered, too. There is a budget limitation and storage space constraint in the model. With regard to complexity of the proposed model, several Pareto-based meta-heuristic approaches such as multi-objective vibration damping optimization (MOVDO), multi-objective imperialist competitive algorithm (MOICA), multi-objective particle swarm optimization (MOPSO), non-dominated ranked genetic algorithm (NRGA), and non-dominated sorting genetic algorithm (NSGA-II) are applied to solve the model. In order to compare the results, several numerical examples are generated and then the algorithms were analyzed statistically and graphically.     

Mean-variance analysis and the single-period inventory problem

The classic single-period inventory problem, colloquially referred lo in the literature as the newsboy problem, is examined assuming the objective of maximizing a mean-variance utility function. This objective function is intended to reflect different attitudes of decision makers towards risk. The optimal order quantity is derived and it is shown that under risk aversion the amount that is ordered is less than the amount ordered under risk neutrality. On the other hand, it is shown that the optimal order quantity when the decision maker is a risk lover is greater than it would be were he risk neutral. The problem is also considered when the ordering cost includes a fixed cost component in addition to the purely variable quantity purchase cost. For this case, an optimal (s, Q) policy is derived (where s is the reorder point and Q is the order quantity).     

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.     

Optimal production lot size and reorder point of a two-stage supply chain while random demand is sensitive with sales teams' initiatives

The paper develops a production-inventory model of a two-stage supply chain consisting of one manufacturer and one retailer to study production lot size/order quantity, reorder point sales teams’ initiatives where demand of the end customers is dependent on random variable and sales teams’ initiatives simultaneously. The manufacturer produces the order quantity of the retailer at one lot in which the procurement cost per unit quantity follows a realistic convex function of production lot size. In the chain, the cost of sales team's initiatives/promotion efforts and wholesale price of the manufacturer are negotiated at the points such that their optimum profits reached nearer to their target profits. This study suggests to the management of firms to determine the optimal order quantity/production quantity, reorder point and sales teams’ initiatives/promotional effort in order to achieve their maximum profits. An analytical method is applied to determine the optimal values of the decision variables. Finally, numerical examples with its graphical presentation and sensitivity analysis of the key parameters are presented to illustrate more insights of the model.     

On the distribution free continuous-review inventory model with a service level constraint

We consider a continuous-review (Q, r) inventory model with a fill rate service constraint and relax the assumption that the distribution of lead time demand is known. We adopt a distribution free approach: We assume that only the first two moments of the lead time demand distribution are known, and then, optimize the policy parameters against the worst possible distribution. We are able to derive closed-form expressions for the optimal order quantity and reorder point.     

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.     

Simultaneous coordination of order quantity and reorder point in a two-stage supply chain

In this paper, we investigate the simultaneous coordination of order quantity and reorder point in a two-stage supply chain (SC). While coordination of order quantity has received much attention in the supply chain management literature, coordination of the reorder point has been less-studied. The retailer's reorder point has a direct impact on product availability and customer service level (CSL) and therefore has a great impact on SC profitability. Our proposed model adopts a two-stage SC with stochastic demand and lead times over multiple periods. The proposed coordination model assures global optimization of order quantity–reorder point decisions. Using a pricing scheme with a discount factor, we extract conditions in which both downstream and upstream members have sufficient motivation to participate in the coordination scheme. Numerical experiments demonstrate that the proposed model can achieve channel coordination. Results of the modeling and analyses show that coordination of both reorder point and order quantity can lead to increased SC profitability as well as CSL improvement.     

Controlling general multi-echelon distribution supply chains with improved reorder decision policy utilizing real-time shared stock information

The objective of this paper is to introduce an improved reorder decision policy for controlling general multi-echelon distribution systems utilizing shared stock information. Since traditional reorder policies sometimes show poor performances when applied to distribution systems, the order risk policy, which utilizes shared stock information more accurately, was developed for the two-echelon distribution system in the previous research. In this paper, the order risk policy is extended to general multi-echelon systems. Since the calculation of the exact order risk for general multi-echelon systems is complex, a practical approximation method is developed. Through computational experiments, the superior performance of the order risk policy, compared to the existing reorder policies, is shown and the value of shared stock information varying with the characteristics of the supply chain is analyzed.     

Analyzing an inventory system with multiple reorder points and periodic replenishment

This paper investigates multiple reorder point, periodic replenishment systems similar to those utilized on board some U.S. naval vessels for Selected Item Management (SIM). This inventory system is of interest since it involves both regular and priority replenishment, three reorder points, and cyclical restocking of regular orders. A simulation model is developed to study the impact of demand distribution, cycle time, priority order leadtime, and the quantity of units ordered on the performance of the system. It is found that the frequency of service and days without shortages performance measures can be maximized by setting the reorder points at equal intervals between the zero level and the high level. Also, a power approximation model is presented that estimates the high level that will lead to a desired percent days without shortage percentage.     

Secure interoperation in a multidomain environment employing RBAC policies

Multidomain application environments where distributed multiple organizations interoperate with each other are becoming a reality as witnessed by emerging Internet-based enterprise applications. Composition of a global coherent security policy that governs information and resource accesses in such environments is a challenging problem. In this paper, we propose a policy integration framework for merging heterogeneous role-based access control (RBAC) policies of multiple domains into a global access control policy. A key challenge in composition of this policy is the resolution of conflicts that may arise among the RBAC policies of individual domains. We propose an integer programming (IP)-based approach for optimal resolution of such conflicts. The optimality criterion is to maximize interdomain role accesses without exceeding the autonomy losses beyond the acceptable limit.     

The distribution free continuous review inventory system with a service level constraint

The stochastic inventory models require the information on the lead time demand. However, the distributional information of the lead time demand is often limited in practice. We relax the assumption that the cumulative distribution function, say F, of the lead time demand is completely known and merely assume that the first two moments of F are known and finite. The distribution free approach for the inventory model consists of finding the most unfavorable distribution for each decision variable and then minimizing over the decision variable. We apply the distribution free approach to the continuous review inventory system with a service level constraint. We develop an iterative procedure to find the optimal order quantity and reorder level.     

Single period stochastic inventory problems with ordering or returns policies

In recent years, there has been an increasing adoption of returns policies in the coordination of the supply chain, where market demand is always assumed to be satisfied by manufacturing or by ordering from suppliers. However, many industries face the important decision of how to balance their inventory level. This problem has long been studied in financial institutions such as banks. This study presents an optimal inventory policy under a given stochastic demand such as a uniformly distributed demand, single-item, and single period review inventory system. The optimal inventory control policy obtained in this study is called a four-point policy: that is, when the entity’s inventory level is below a reorder point, the entity must increase his stock level by ordering and order up-to a fixed level (second point); when the entity’s inventory level is over a return point (third point); the stock level must be decreased by returns and decreased to a fixed level (fourth point); otherwise, nothing should be done. We also analyze the (K, S)-convex properties of the inventory cost function.     

Integrated vendor--buyer inventory system with sublot sampling inspection policy and controllable lead time

This article investigates the impact of inspection policy and lead time reduction on an integrated vendor--buyer inventory system. We assume that an arriving order contains some defective items. The buyer adopts a sublot sampled inspection policy to inspect selected items. The number of defective items in the sublot sampling is a random variable. The buyer's lead time is assumed reducible by adding crash cost. Two integrated inventory models with backorders and lost sales are derived. We first assume that the lead time demand follows a normal distribution, and then relax the assumption about the lead time demand distribution function and apply the minimax distribution-free procedure to solve the problem. Consequently, the order quantity, reorder point, lead time and the number of shipments per lot from the vendor to the buyer are decision variables. Iterative procedures are developed to obtain the optimal strategy.     

Sole versus dual sourcing under order dependent lead times and prices

In this paper, we consider a dual-sourcing model with constant demand and stochastic lead times. Two suppliers may be different in terms of purchasing prices and lead-time parameters. The ordering takes place when the inventory level depletes to a reorder level, and the order is split among two suppliers. Unlike previous works in the order splitting literature, the supply lead time between vendor and buyer as well as unit purchasing prices is considered to be order quantity dependent. The proposed model finds out the optimal reorder point, order quantity and splitting proportion, using a solution procedure. Numerical results show that neglecting the relationship between ordering batch size and lead times is a shortcoming that hides one of order splitting advantages. Moreover, connecting unit prices to order quantity can decrease the percentage saving from dual sourcing compared to sole sourcing. Furthermore, sensitivity analysis shows some managerial insights.     

Lyapunov exponents of the logistic map with periodic forcing

The iterative map x_n+1 = r_nx_n (1 - x_n) is investigated with r_n changing periodically between two values A and B. Different periodicities are assumed, e.g., {r_n} = {BABA …} or {r_n} = {BBABA BBABA …}. The Lyapunov exponent (a measure of average stability) is displayed with high resolution on the A-B-plane. The resulting images have aesthetically appealing self-similar structures. Furthermore, these images allow with one glimpse the identification of a number of system properties: coexistence of attractors, superstable curves, order by alternation of chaotic processes, and chaos by periodic resetting from a stable into an unstable fixed point.     

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．     

An optimal algorithm for Gaussian elimination of band matrices on an MIMD computer

This paper describes a parallel algorithm for the LU decomposition of band matrices using Gaussian elimination. The matrix dimension is n × n with 2r−1 diagonals. In the case when 1 r 2 p an optimal number of the processors, , is determined according to the equation . When 2 p r n a number of processors, p, statged by Veldhorst is adopted (see [7]). For band matrix with 2r-1 diagonals (1 r 2p) the task scheduling procedure with the aim to obtain maximal parallelism in system operation, i.e. good load balancing, is defined. The architecture of the system is of MIMD type. The connection between the processors is realised via a common bus. Communication and synchronization is performed by message passing technique.