Optimality of Base-Stock and (s, S) Policies for Inventory Problems with Information Delays

Information delays exist when the most recent inventory information available to the inventory manager (IM) is dated; namely, the IM observes only the inventory level of an earlier period. We introduce information delays into the standard multiperiod stochastic inventory problem with backorders. We consider two types of information delays: (i) a constant delay and (ii) a random delay. We define an appropriate reference inventory position, which is a sufficient statistics for finding the optimal order quantity. We show that the optimal ordering policy is of base-stock type with respect to the reference inventory position and is of (s, S) type if there is also a fixed cost of ordering.All authors were supported in part by NSF Grant DMS-0509278.     

Age-dependent production planning and maintenance strategies in unreliable manufacturing systems with lost sale

In this paper, we formulate an analytical model for the joint determination of an optimal age-dependent buffer inventory and preventive maintenance policy in a production environment that is subject to random machine breakdowns. Traditional preventive maintenance policies, such as age and periodic replacements, are usually studied based on simplified and non-realistic assumptions, as well as on the expected costs criterion. Finished goods inventories and the age-dependent likelihood of machine breakdowns are usually not considered. As a result, these policies could significantly extend beyond the anticipated financial incomes of the system, and lead to crises. In order to solve this problem, a more realistic analysis model is proposed in this paper to consider the effects of both preventive maintenance policies and machine age on optimal safety stock levels. Hence, a unified framework is developed, allowing production and preventive maintenance to be jointly considered. We use an age-dependent optimization model based on the minimization of an overall cost function, including inventory holdings, lost sales, preventive and corrective maintenance costs. We provide optimality conditions for the manufacturing systems considered, and use numerical methods to obtain an optimal preventive maintenance policy and the relevant age-dependent threshold level production policy. In this work, this policy is called the multiple threshold levels hedging point policy. We include numerical examples and sensitivity analyses to illustrate the importance and the effectiveness of the proposed methodology. Compared with other available optimal production and maintenance policies, the numerical solution obtained shows that the proposed age-dependent optimal production and maintenance policies significantly reduce the overall cost incurred.     

Optimal control of a facility with periodic interrupted demand

The system investigated consists of a stochastic periodic stream of raw material, a continuous processing operation with controllable deterministic service rates, and a storage facility. The arrival stream is periodically interrupted and divided into alternating on-off intervals of fixed length. The processing facility is allowed to operate during the off-interval. Superimposed on this system is a cost structure composed of processing and holding costs. Such operations may be found in manufacturing as well as service systems (for example, dry cleaners, machine shops, repair and maintenance shops, printers, information processing centers, etc). A service rate control rule that minimizes the infinite-horizon discounted expected total cost is found. Existence and uniqueness of long-term optimal cost and policy functions is shown. Since the optimal policy cannot be expressed explicitly, an approximate solution was obtained. An error bound on the optimal cost associated with this solution is exhibited. The approximate solution is characterized by a service rate control rule that is a linear function of the level of inventory at the start of each on-interval and a piecewise linear function of inventory at the start of each off-interval. The optimal discounted expected total cost is quadratic in the inventory level at the start of each interval. Computational results indicate relative cost errors in the order of 2–3 percent.This research was performed at the Sanitary Engineering Research Laboratory and Operations Research Center of the University of California, Berkeley. It was made possible by US Public Health Research Grant UI-00547 from the Environmental Control Administration-Bureau of Solid Waste Management and by National Science Foundation Grant GK-1684.The author thanks Professor C. R. Glassey for not only suggesting this research, but for his constant encouragement and suggestions throughout its duration. He also thanks Professors W. S. Jewell and P. H. McGauhey whose comments on the draft were very helpful.     

Simple expressions for finding recovery system inventory control parameter values

In recent years considerable effort has been devoted to the development of inventory control models for joint manufacturing and remanufacturing. Optimality of control policies is analyzed and algorithms for the determination of parameter values have been developed. However, there is still a lack of formulae or algorithms that allow for an easy computation of optimal or near optimal policy parameter values. This paper addresses the problem of computing the produce-up-to level S and the remanufacture-up-to level M in a periodic review inventory control model. We provide simple formulae for the policy parameter values, which can easily be implemented within spreadsheet applications. The approach is to derive news-vendor-type formulae that are based on underage and overage cost considerations. We propose different formulae depending on whether lead times for production and remanufacturing are identical or not. A numerical study shows that the obtained solutions provide relatively small cost deviations compared to the optimal solution within the investigated class of inventory control policies.     

On the use of phase‐type distributions for inventory management with supply disruptions

Maintaining the continuity of operations becomes increasingly important for systems that are subject to disruptions due to various reasons. In this paper, we study an inventory system operating under a (q, r) policy, where the supply can become inaccessible for random durations. The availability of the supply is modeled by assuming a single supplier that goes through ON and OFF periods of stochastic duration, both of which are modeled by phase‐type distributions (PTD). We provide two alternative representations of the state transition probabilities of the system, one with integral and the other employing Kolmogorov differential equations. We then use an efficient formulation for the analytical model that gives the optimal policy parameters and the long‐run average cost. An extensive numerical study is conducted, which shows that OFF time characteristics have a bigger impact on optimal policy parameters. The ON time characteristics are also important for critical goods if disasters can happen. Copyright © 2011 John Wiley & Sons, Ltd.     

Inventory control with fixed cost and price optimization in continuous time

We continue to study the problem of inventory control, with simultaneous pricing optimization in continuous time. In our previous paper [8], we considered the case without set up cost, and established the optimality of the base stock-list price (BSLP) policy. In this paper we consider the situation of fixed price. We prove that the discrete time optimal strategy (see [11]), i.e., the (s, S, p) policy can be extended to the continuous time case using the framework of quasi-variational inequalities (QVIs) involving the value function. In the process we show that an associated second order, nonlinear two-point boundary value problem for the value function has a unique solution yielding the triplet (s, S, p). For application purposes the explicit knowledge of this solution is needed to specify the optimal inventory and pricing strategy. Se- lecting a particular demand function we are able to formulate and implement a numerical algorithm to obtain good approximations for the optimal strategy.     

Optimal Solutions in the Multi-location Inventory System with Transshipments

We consider a single-period multi-location inventory system where inventory choices at each location are centrally coordinated. Transshipments are allowed as recourse actions in order to reduce the cost of shortage or surplus inventory after demands are realized. This problem has not been solved to optimality before for more than two locations with general cost parameters. In this paper we present a simple and intuitive model that enables us to characterize optimal inventory and transshipment policies for three and four locations as well. The insight gained from these analytical results leads us to examine the optimality conditions of a greedy transshipment policy. We show that this policy will be optimal for two and three locations. For the n location model we characterize the necessary and sufficient conditions on the cost structure for which the greedy transshipment policy will be optimal.      

Analysis of deteriorating inventory/production systems using a linear quadratic regulator

We consider an inventory-production system where items deteriorate at a constant rate. The objective is to develop an optimal production policy that minimizes the cost associated with inventory and production rate. The inventory problem is first modeled as a linear optimal control problem. Then linear quadratic regulator (LQR) technique is applied to the control problem in order to determine the optimal production policy. Examples are solved for three different demand functions. Sensitivity analysis is then conducted to study the effect of changing the cost parameters on the objective function.     

Analysis of a two-sided production policy with inventory-level-dependent production rates

In this paper we analyse a stochastic production/inventory problem with compound Poisson demand and state (i.e. inventory level) dependent production rates. Customers arrive according to a Poisson process where the amount demanded by each customer is assumed to have a general distribution. When the inventory W(t) falls below a critical level m, production is started at a rate of r[W(t)], i.e. production rate dynamically changes as a function of the inventory level. Production continues until a level M (œ w m) is reached. Excess demand is assumed to be lost. We identify a dam content process X that is a dual for the inventory level W and develop the stationary distribution for the X process. To achieve this we use tools from renewal and level crossing theories. The two-sided (m, M) policy is optimized using the expected cost obtained from the stationary density of W and a conditional (on w) expected cost function for this process. For a special case, we obtain explicit results for all the relevant expressions. Numerical examples are provided for several test problems. © 1996 John Wiley & Sons, Ltd.     

Optimal control of an inventory-production system with state-dependent random yield

In this paper we study a single stage, periodic-review inventory problem for a single item with stochastic demand. The inventory manager determines order sizes according to an order-up-to logic and observes a random yield due to quality problems in the production. We distinguish between two different states of the production process combined with different probabilities to produce a defective unit. In order to improve the production process, periodic inspections are conducted and in case of a failure the machine is repaired. Approximations are developed to evaluate the average cost for a given order-up-to level and a given inspection interval and we illustrate the existence of optimal policy parameters. The approximations are tested in a simulation study and reveal an excellent performance as they lead to near optimal policy parameters. Moreover, we decompose the problem and test different methods to compute the policy parameters either sequentially or separately. Our results show that a joint optimization of the inventory and maintenance policy leads to a better system performance and reduced costs.     

Optimal inventory threshold for a dynamic service make‐to‐stock system with strategic customers

We consider a make‐to‐stock production system with one product type, dynamic service policy, and delay‐sensitive customers. To balance the waiting cost of customers and holding cost of products, a dynamic production policy is adopted. If there is no customer waiting in the system, instead of shutting down, the system operates at a low production rate until a certain threshold of inventory is reached. If the inventory is empty and a new customer emerges, the system switches to a high production rate where the switching time is assumed to be exponentially distributed. Potential customers arrive according to the Poisson process. They are strategic in the sense that they make decisions on whether to stay for product or leave without purchase on the basis of on their utility value and the system information on whether the number of products is observable to customers or not. The strategic behavior is explored, and a Stackelberg game between production manager and customers is formulated where the former is the game leader. We find that the optimal inventory threshold minimizing the cost function can be obtained by a search algorithm. Numerical results demonstrate that the expected cost function in an observable case is not greater than that in an unobservable case. If a customer's delay sensitivity is relatively small, these two cases are entirely identical. With increasing of delay sensitivity, the optimal inventory threshold might be positive or zero, and hence, a demarcation line is depicted to determine when a make‐to‐stock policy is advantageous to the manager.     

Cost-effective inventory control in a value-added manufacturing system

This paper considers the cost-effective inventory control of work-in-process (WIP) and finished products in a two-stage distributed manufacturing system. The first stage produces a common WIP, and the second stage consists of several production sites that produce differentiated products with different capacity and service level requirements. The unit inventory holding cost is higher at the second stage. This paper first uses a network of inventory-queue model to evaluate the inventory cost and service level achievable for given inventory control policy, and then derives a very simple algorithm to find the optimal inventory control policy that minimizes the overall inventory holding cost and satisfies the given service level requirements. Some managerial insights are obtained through numerical examples.     

Optimal control of a finite-capacity inventory system with setup cost and lost sales

One of the most fundamental results in inventory theoryis the optimality of (s, S) policy for inventory systems withsetup cost. This result is established based on a key assumptionof infinite production/ordering capacity. Several studies haveshown that, when there is a finite production/ordering capacity,the optimal policy for the inventory system is very complicatedand indeed, only partial characterization for the optimal policyis possible. In this paper, we consider a continuous reviewinventory system with finite production/ordering capacity andsetup cost, and show that the optimal control policy for thissystem has a very simple structure. We also develop efficientalgorithms to compute the optimal control parameters.     

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.     

Monotonicity properties of a class of stochastic inventory systems

We consider inventory systems which are governed by an (r,q) or (r,nq) policy. We derive general conditions for monotonicity of the three optimal policy parameters, i.e., the optimal reorder level, order quantity and order-up-to level, as well as the optimal cost value, as a function of the various model primitives, be it cost parameters or complete cost rate functions or characteristics of the demand and leadtime processes. These results are obtained as corollaries from a few general theorems, with separate treatment given to the case where the policy parameters are continuous variables and that where they need to be restricted to integer values. The results are applied both to standard inventory models and to those with general shelf age and delay dependent inventory costs.     

Inventory models with minimal service level constraints

This paper investigates inventory models in which the stockout cost is replaced by a minimal service level constraint (SLC) that requires a certain level of service to be met in every period. The minimal service level approach has the virtue of simplifying the computation of an optimal ordering policy, because the optimal reorder level is solely determined by the minimal SLC and demand distributions. It is found that above a certain “critical” service level, the optimal (s,S) policy “collapses” to a simple base-stock or order-up-to level policy, which is independent on the cost parameters. This shows the minimal SLC models to be qualitatively different from their shortage cost counterparts. We also demonstrate that the “imputed shortage cost” transforming a minimal SLC model to a shortage cost model does not generally exist. The minimal SLC approach is extended to models with negligible set-up costs. The optimality of myopic base-stock policies is established under mild conditions.     

Single-vendor multi-buyer integrated production-inventory model with controllable lead time and service level constraints

This paper presents an integrated production-inventory model where a vendor produces an item in a batch production environment and supplies it to a set of buyers. The buyer level demand is assumed to be independent normally distributed and lead time of every buyer can be reduced at an added crash cost. The buyers review their inventory using continuous review policy, and the unsatisfied demand at the buyers is completely backordered. A model is formulated to minimize the joint total expected cost of the vendor–buyers system to determine the optimal production-inventory policy. Since it is often difficult to estimate the stock-out cost in inventory systems, and so instead of having stock-out cost component in the objective function, a service level constraint (SLC) corresponding to each buyer is included in the model. A Lagrangian multiplier technique based algorithmic approach is proposed, which evaluates a very limited number of combinations of lead time of the buyers to find simultaneously the optimal lead time, order quantity and safety factor of the buyers and the number of shipments between the vendor and the buyers in a production cycle. Finally, a numerical example and effects of the key parameters are included to illustrate the results of the proposed model.     

An ordering policy with allowable shortage and permissible delay in payments

It is common business practice to purchase inventory on an open account. Purchased inventory can be considered to be financed in whole or in part with permissible delay in payments. This paper develops a model to determine an optimal ordering policy under conditions of allowable shortage and permissible delay in payment and shows that the total annual variable cost function possesses some kinds of convexities. With those convexities, a theorem is presented to determine the optimal order quantity. Numerical examples are given to illustrate the theorem.     

考虑顾客退货和固定成本的联合补货与定价策略

为了吸引更多顾客,许多电子商务零售商允许顾客在一定时间内退货,导致其利润明显减少。同时,在补货时不仅产生依赖补货量的变动成本,而且会产生与补货量无关的固定成本。基于此,以最大化电子商务零售商的利润为目标,建立考虑顾客退货和固定成本的联合补货与定价模型,其中顾客的退货量与满足的需求呈正比。在一般需求情形下,部分刻画多期问题的最优策略;在特殊需求情形下,证明(s,S,p)策略对单期问题最优,并对多期问题的最优策略进行严格刻画。根据已有刻画为多期问题构造启发式策略。数值结果表明启发式策略近似最优;当初始库存水平足够高/低时,最优补货水平和定价随退货率与固定成本单调变化。关键词:联合补货与定价模型;顾客退货;固定成本;随机动态规划;最优策略     

Optimal policy for brownian inventory models with general convex inventory cost

We study an inventory system in which products are ordered from outside to meet demands, and the cumulative demand is governed by a Brownian motion. Excessive demand is backlogged. We suppose that the shortage and holding costs associated with the inventory are given by a general convex function. The product ordering from outside incurs a linear ordering cost and a setup fee. There is a constant leadtime when placing an order. The optimal policy is established so as to minimize the discounted cost including the inventory cost and ordering cost.