Reducing lost-sales rate in (T, R, L) inventory model with controllable lead time

In order to establish a good image and to enhance customer’s loyalty, many efforts such as upgrading the servicing facilities, maintaining a high quality of products and increasing expenditure on advertisement could be made by a selling shop. Naturally, an extra-added cost must be spent for these efforts and it is expected to have a result to reduce the shortage cost of lost-sales and the total expected annual cost. This paper explores a probabilistic inventory model with optimal lost-sales caused by investment due to two different types of cost functions. We consider that the lead time can be shortened at an extra crashing cost, which depends on the length of the lead time. Moreover, we assume that the lost-sales rate can also be reduced by capital investment. The purpose of this paper is to establish a (T, R, L) inventory model with controllable lead time and to analyze the effects of increasing two different types of investments to reduce the lost-sales rate, in which the review period, lead time and lost-sales rate are treated as decision variables. We first formulate the basic periodic review model mathematically with the capital investment to reduce lost-sales rate. Then two models are discussed, one with normally distributed protection interval demand and another with distribution-free case. For each model, two investment cost functional forms, logarithmic and power, are employed for lost-sales rate reduction. Two computational algorithms with the help of the software Matlab are furnished to determine the optimal solution. In addition, six numerical examples and sensitivity analysis are presented to illustrate the theoretical results and obtain some managerial insights. Finally, the effect of lost-sales rate reduction is investigated. By framing this new model, we observe that a significant amount of savings can be easily achieved to increase the competitive edge in business. The results in the numerical examples indicate that the savings of expected annual total cost are realized through lost-sales reduction.     

Analysis of optimal opportunistic replenishment policies for inventory systems by using a (s,S) model with a maximum issue quantity restriction

The analysis of optimal inventory replenishment policies for items having lumpy demand patterns is difficult, and has not been studied extensively although these items constitute an appreciable portion of inventory populations in parts and supplies types of stockholdings. This paper studies the control of an inventory item when the demand is lumpy. A continuous review (s,S) policy with a maximum issue quantity restriction and with the possibility of opportunistic replenishment is proposed to avoid the stock of these items being depleted unduly when all the customer orders are satisfied from the available inventory and to reduce ordering cost by coordinating inventory replenishments. The nature of the customer demands is approximated by a compound Poisson distribution. When a customer order arrives, if the order size is greater than the maximum issue quantity w, the order is satisfied by placing a special replenishment order rather than from the available stock directly. In addition, if the current inventory position is equal to or below a critical level A when such an order arrives, an opportunistic replenishment order which combines the special replenishment order and the regular replenishment order will be placed, in order to satisfy the customer's demand and to bring the inventory position to S. In this paper, the properties of the cost function of such an inventory system with respect to the control parameters s, S and A are analysed in detail. An algorithm is developed to determine the global optimal values of the control parameters. Indeed, the incorporation of the maximum issue quantity and opportunistic replenishment into the (s,S) policy reduces the total operating cost of the inventory system.     

Rationing policies for some inventory systems

This paper considers inventory systems which maintain stocks to meet various demand classes with different priorities. We use the concept of a support level control policy. That is rationing is accomplished by maintaining a support level, say K, such that when on hand stock reaches K, all low priority demands are backordered. We develop four analytical and simulation models to improve the existing models. Firstly, multiple support levels are used instead of using a single support level. Secondly, a simulation model with a more realistic assumption on the demand process has been provided. Thirdly, a single period deterministic cost minimisation model has been developed analytically. Finally, we address a continuous review (Q, r) model with a compound Poisson process.     

Modeling and analysis for determining optimal suppliers under stochastic lead times

The policy of simultaneously splitting replenishment orders among several suppliers has received considerable attention in the last few years and continues to attract the attention of researchers. In this paper, we develop a mathematical model which considers multiple-supplier single-item inventory systems. The item acquisition lead times of suppliers are random variables. Backorder is allowed and shortage cost is charged based on not only per unit in shortage but also per time unit. Continuous review (s,Q)$(s\text{,}Q)$ policy has been assumed. When the inventory level depletes to a reorder level, the total order is split among n suppliers. Since the suppliers have different characteristics, the quantity ordered to different suppliers may be different. The problem is to determine the reorder level and quantity ordered to 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 with the models in the literature. According to our extensive experiments, the model developed in this paper is the best model in the literature which considers order splitting for n-supplier inventory systems since it is the nearest model to the real inventory system.     

Modelling and computing (Rn, Sn) policies for inventory systems with non-stationary stochastic demand

This paper addresses the single-item, non-stationary stochastic demand inventory control problem under the non-stationary (R, S) policy. In non-stationary (R, S) policies two sets of control parameters—the review intervals, which are not necessarily equal, and the order-up-to-levels for replenishment periods—are fixed at the beginning of the planning horizon to minimize the expected total cost. It is assumed that the total cost is comprised of fixed ordering costs and proportional direct item, inventory holding and shortage costs. With the common assumption that the actual demand per period is a normally distributed random variable about some forecast value, a certainty equivalent mixed integer linear programming model is developed for computing policy parameters. The model is obtained by means of a piecewise linear approximation to the non-linear terms in the cost function. Numerical examples are provided.     

A stocking policy for spare part provisioning under age based preventive replacement

A new policy, called stocking policy for ease of reference, has been advanced for joint optimization of age replacement and spare provisioning. It combines age replacement policy with continuous review (s, S) type inventory policy, where s is the stock reorder level and S is the maximum stock level. The policy is generally applicable to any operating situation having either a single item or a number of identical items. A simulation model has been developed to determine the optimal values of the decision variables by minimizing the total cost of replacement and inventory. The behaviour of the stocking policy has been studied for a number of case problems specifically constructed by 5-factor second order rotatory design and the effects of different cost elements and item failure characteristics have been highlighted. For all case problems, optimal (s, S) policies to-support the Barlow-Proschan age policy have also been determined. Simulation results clearly indicate that the optimal stocking policy is, in general, more cost-effective than the Barlow-Proschan policy.     

Optimality of (s, S) policies for jump inventory models

This paper is concerned with the optimality of (s, S) policies for a single-item inventory control problem which minimizes the total expected cost over an infinite planning horizon and where the demand is driven by a piecewise deterministic process. Our approach is based on the theory of quasi-variational inequality.     

Inventory control in multi-channel retail

In recent years multi-channel retail systems have received increasing interest. Partly due to growing online business that serves as a second sales channel for many firms, offering channel specific prices has become a common form of revenue management. We analyze conditions for known inventory control policies to be optimal in presence of two different sales channels. We propose a single item lost sales model with a lead time of zero, periodic review and nonlinear non-stationary cost components without rationing to realistically represent a typical web-based retail scenario. We analyze three variants of the model with different arrival processes: demand not following any particular distribution, Poisson distributed demand and a batch arrival process where demand follows a Pòlya frequency type distribution. We show that without further assumptions on the arrival process, relatively strict conditions must be imposed on the penalty cost in order to achieve optimality of the base stock policy. We also show that for a Poisson arrival process with fixed ordering costs the model with two sales channels can be transformed into the well known model with a single channel where mild conditions yield optimality of an (s, S) policy. Conditions for optimality of the base stock and (s, S) policy for the batch arrival process with and without fixed ordering costs, respectively, are presented together with a proof that the batch arrival process provides valid upper and lower bounds for the optimal value function.     

Periodic review (<Emphasis Type="Italic">s</Emphasis>, <Emphasis Type="Italic">S</Emphasis>) inventory model with permissible delay in payments

This paper investigates the effect of permissible delay in payments on ordering policies in a periodic review (s, S) inventory model with stochastic demand. A new mathematical model is developed, which is an extension to that of Veinott and Wagner (Mngt Sci 1965; 11: 525) who applied renewal theory and stationary probabilistic analysis to determine the equivalent average cost per review period. The performance of the model is validated using a custom-built simulation programme. In addition, two distribution-free heuristic methods of reasonable accuracy develop approximate optimal policies for practical purposes based only on the mean and the standard deviation of the demand. Numerical examples are presented with results discussed.     

Controlling inventories with stochastic item returns: A basic model

Environmental legislation and customer expectations increasingly force manufacturers to take back their products after use. Returned products may enter the production process again as input resources. Material management has to be modified accordingly.One of the areas concerned is inventory management. The present paper provides a step towards a systematic analysis of inventory control in the context of reuse. A basic inventory model is presented comprising Poisson demand and returns. For this model, an optimal control policy is derived and optimal control parameters are computed. Moreover, a numerical analysis is provided of the impact of the return-flow on the inventory system. Comparison with traditional (s,Q)-inventory models is central throughout the analysis.     

Methods for Determining the Re-order Point of an (<Emphasis Type="Italic">s</Emphasis>, <Emphasis Type="Italic">S</Emphasis>) Ordering Policy when a Service Level is Specified

This paper deals with a periodic review inventory system. Methods are discussed for determining the re-order point s of an (s, S) order policy, when a certain service level is required. The results differ from those presented for a (Q, s) model which is usually considered in literature and implemented in practice. Methods are discussed for determining the re-order point of an (s, S) policy when demand is normal or gamma distributed. A numerical investigation demonstrates the applicability of the described methods. In particular, it is shown that these methods are superior to a formula that is implemented in many inventory control systems.     

The capital cost of holding inventory with stochastically mean-reverting purchase price

Most models of inventory control assume that the per unit purchase price is constant. The capital cost of holding inventory can then be taken into account by adding a fixed interest rate, r, times the purchase price, C, to the out-of pocket holding cost. However, it is not uncommon that the purchase price varies over time. How the capital cost then should be calculated is the focus of the present paper. The paper studies the common single-item inventory model with a fixed set-up cost and assumes that the stochastic purchase price follows the mean-reverting Ornstein–Uhlenbeck process. Methods for computing an adjusted interest rate, r, are suggested along with modifications of well-known heuristics and formulas for lot-sizing. Simulation tests, where the optimal policy has been compared to policies obtained using modified versions of the Silver–Meal method, the Part Period algorithm and the EOQ formula, suggest that r should be estimated as the sum of the unadjusted interest rate and the average expected purchase price decrease, measured over a period between 1/3 and 2/3 of the length of the order cycle.     

Inventory control with indivisible units of stock transfer

We consider an infinite horizon, single item inventory model with backorders and a fixed lead time. Demand is stationary stochastic and review is periodic. Inventory may only be replenished in multiples of a fixed package size q but demands may be of any size. Ordering costs are linear and combined holding and shortage costs can be expressed as a convex function of the inventory position. The control policy is defined as (s, S, q), where an order is placed if the inventory position falls to or below s and the order size is the largest multiple of q which results in the inventory position not exceeding S. The parameters s and S are restricted to be multiples of q. The objective is to find the control policy that minimizes the long run average cost per unit time. The optimal solution procedure requires renewal theory and a structured search. Fortunately, a heuristic based on the ‘quantized ordering’ approach of Zheng and Chen provides solutions that are near optimal over a broad range of parameter values.     

Optimal selection of process mean for a stochastic inventory model

It is very common to assume deterministic demand in the literature of integrated targeting – inventory models. However, if variability in demand is high, there may be significant disruptions from using the deterministic solution in probabilistic environment. Thus, the model would not be applicable to real world situations and adjustment must be made. The purpose of this paper is to develop a model for integrated targeting – inventory problem when the demand is a random variable. In particular, the proposed model jointly determines the optimal process mean, lot size and reorder point in (Q, R) continuous review model. In order to investigate the effect of uncertainty in demand, the proposed model is compared with three baseline cases. The first of which considers a hierarchical model where the producer determines the process mean and lot-sizing decisions separately. This hierarchical model is used to show the effect of integrating the process targeting with production/inventory decisions. Another baseline case is the deterministic demand case which is used to show the effect of variation in demand on the optimal solution. The last baseline case is for the situation where the variation in the filling amount is negligible. This case demonstrates the sensitivity of the total cost with respect to the variation in the process output. Also, a procedure is developed to determine the optimal solution for the proposed models. Empirical results show that ignoring randomness in the demand pattern leads to underestimating the expected total cost. Moreover, the results indicate that performance of a process can be improved significantly by reducing its variation.     

Managing an integrated production inventory system with information on the production and demand status and multiple non-unitary demand classes

In this paper, we study a system consisting of a manufacturer or supplier serving several retailers or clients. The manufacturer produces a standard product in a make-to-stock fashion in anticipation of orders emanating from n retailers with different contractual agreements hence ranked/prioritized according to their importance. Orders from the retailers are non-unitary and have sizes that follow a discrete distribution. The total production time is assumed to follow a k₀-Erlang distribution. Order inter-arrival time for class l demand is assumed to follow a k_l-Erlang distribution. Work-in-process as well as the finished product incur a, per unit per unit of time, carrying cost. Unsatisfied units from an order from a particular demand class are assumed lost and incur a class specific lost sale cost. The objective is to determine the optimal production and inventory allocation policies so as to minimize the expected total (discounted or average) cost. We formulate the problem as a Markov decision process and show that the optimal production policy is of the base-stock type with base-stock levels non-decreasing in the demand stages. We also show that the optimal inventory allocation policy is a rationing policy with rationing levels non-decreasing in the demand stages. We also study several important special cases and provide, through numerical experiments, managerial insights including the effect of the different sources of variability on the operating cost and the benefits of such contracts as Vendor Managed Inventory or Collaborative Planning, Forecasting, and Replenishment. Also, we show that a heuristic that ignores the dependence of the base-stock and rationing levels on the demands stages can perform very poorly compared to the optimal policy.     

An inventory model with returns and lateral transshipments

In this paper, we propose a single-item inventory model with returns. The model allows lateral transshipment of returns from one inventory system to another. Each inventory system is under continuous review and an (r, Q) policy is employed as the inventory control. An approximated closed-form solution of the system steady-state probability distribution is derived when Q is large. The approximated inventory cost and replenishment cost can be written in terms of this distribution. We show that the rejection rate of returns is reduced significantly when transshipment of returns is allowed between the inventory systems.     

Updating incomplete factorization preconditioners for shifted linear systems arising in a wind model

The efficiency of a finite element mass consistent model for wind field adjustment depends on the stability parameter α which allows adjustment from a strictly horizontal wind to a pure vertical one. Each simulation with the wind model leads to the resolution of a linear system of equations, the matrix of which depends on a function ε(α), i.e., (M+εN)x_ε=b_ε, where M and N are constant, symmetric and positive definite matrices with the same sparsity pattern for a given level of discretization. The estimation of this parameter may be carried out by using genetic algorithms. This procedure requires the evaluation of a fitness function for each individual of the population defined in the searching space of α, that is, the resolution of one linear system of equations for each value of α. Preconditioned Conjugate Gradient algorithm (PCG) is usually applied for the resolution of these types of linear systems due to its good convergence results. In order to solve this set of linear systems, we could either construct a different preconditioner for each of them or use a single preconditioner constructed from the first value of ε to solve all the systems. In this paper, an intermediate approach is proposed. An incomplete Cholesky factorization of matrix A_ε is constructed for the first linear system and it is updated for each ε at a low computational cost. Numerical experiments related to realistic wind field are presented in order to show the performance of the proposed preconditioning strategy.     

Dynamic inventory and pricing policy in a capacitated stochastic inventory system with fixed ordering cost

We study a dynamic inventory and pricing optimization problem in a periodic review inventory system with setup cost and finite ordering capacity in each period. We show that the optimal inventory control is characterized by an (s,s^′,p) policy in four regions of the starting inventory level.     

Inventory rationing in an (<Emphasis Type="Italic">s,Q</Emphasis>) inventory model with lost sales and two demand classes

Whenever demand for a single item can be categorised into classes of different priority, an inventory rationing policy should be considered. In this paper we analyse a continuous review (s, Q) model with lost sales and two demand classes. A so-called critical level policy is applied to ration the inventory among the two demand classes. With this policy, low-priority demand is rejected in anticipation of future high-priority demand whenever the inventory level is at or below a prespecified critical level. For Poisson demand and deterministic lead times, we present an exact formulation of the average inventory cost. A simple optimisation procedure is presented, and in a numerical study we compare the optimal rationing policy with a policy where no distinction between the demand classes is made. The benefit of the rationing policy is investigated for various cases and the results show that significant cost reductions can be obtained.     

一种改进企业在框架协议下库存管理的方法

企业为了货源的稳定而与供应商签订一定期限内的订购框架协议,但如何实施订购策略以保障协议量的完成,保证满足需求的同时最小化企业的库存成本成为企业必须考虑的一个问题.传统的企业库存管理方法如(r,Q)、(s,S)等订购策略在一定程度上只考虑了如何满足需求,避免缺货,而没有考虑供应商的供货能力,在协议期内的总订购量只考虑了需求的影响,因而不能保证框架协议的完成.本文在(s,S)策略的基础上,考虑了框架协议的约束,提出了一种既能够保障框架协议的完成,又能够最大限度满足需求,同时降低企业库存成本的方法一“双表作业法”,并验证了方法的有效性.