Using aggregate fill rate for dynamic scheduling of multi-class systems

For dynamic scheduling of multi-class systems where backorder cost is incurred per unit backordered regardless of the time needed to satisfy backordered demand, the following models are considered: the cost model to minimize the sum of expected average inventory holding and backorder costs and the service model to minimize expected average inventory holding cost under an aggregate fill rate constraint. Use of aggregate fill rate constraint in the service model instead of an individual fill rate constraint for each class is justified by deriving equivalence relations between the considered cost and service models. Based on the numerical investigation that the optimal policy for the cost model is a base-stock policy with switching curves and fixed base-stock levels, an alternative service model is considered over the class of base-stock controlled dynamic scheduling policies to minimize the total inventory (base-stock) investment under an aggregate fill rate constraint. The policy that solves this alternative model is proposed as an approximation of the optimal policy of the original cost and the equivalent service models. Very accurate heuristics are devised to approximate the proposed policy for given base-stock levels. Comparison with base-stock controlled First Come First Served (FCFS) and Longest Queue (LQ) policies and an extension of LQ policy (Δ policy) shows that the proposed policy performs much better to solve the service models under consideration, especially when the traffic intensity is high.     

Optimal control of a production-inventory system with customer impatience

We consider the control of a production-inventory system with impatient customers. We show that the optimal policy can be described using two thresholds: a production base-stock level that determines when production takes place and an admission threshold that determines when orders should be accepted. We describe an algorithm for computing the performance of the system for any choice of base-stock level and admission threshold. In a numerical study, we compare the performance of the optimal policy against several other policies.     

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.     

On Optimality of a Class of Dynamic Myopic Policies for Continuous-Time Replenishment with Periodic Updates

This paper is motivated by inventory problems arising in supply chains characterized by continuous replenishment programs based on information exchanged (reviewed) only intermittently between a manufacturing system (supplier) and a customer (retailer). When the replenishment is once-per-period, rather than at any point of time, a well-known result is the optimality of the so-called myopic base-stock policy. We generalize the notion of the base-stock policy and study the optimality of the corresponding class of dynamic myopic policies. We identify a myopic policy and prove that although the replenishment rule is dynamic, this policy is optimal when the demands are stationary and the number of review periods tends to infinity.     

A comparison of inventory policies for perishable goods

Past research on inventory management of perishables introduced models in which demand is sensitive to the age of the product. For such models, we prove that a fixed-order quantity policy is optimal under certain conditions and show that its expected cost is closer to optimal than that of the base-stock level policy when there is demand for units of all ages. We also show numerically when substituting older products to fulfill the demand for new (or vice versa) is beneficial.     

Stochastic Multiproduct Inventory Models with Limited Storage

This paper studies multiproduct inventory models with stochastic demands and a warehousing constraint. Finite horizon as well as stationary and nonstationary discounted-cost infinite-horizon problems are addressed. Existence of optimal feedback policies is established under fairly general assumptions. Furthermore, the structure of the optimal policies is analyzed when the ordering cost is linear and the inventory/backlog cost is convex. The optimal policies generalize the base-stock policies in the single-product case. Finally, in the stationary infinite-horizon case, a myopic policy is proved to be optimal if the product demands are independent and the cost functions are separable.     

Optimal ordering policies for periodic-review systems with replenishment cycles

In this paper, we consider inventory models for periodic-review systems with replenishment cycles, which consist of a number of periods. By replenishment cycles, we mean that an order is always placed at the beginning of a cycle. We use dynamic programming to formulate both the backorder and lost-sales models, and propose to charge the holding and shortage costs based on the ending inventory of periods (rather than only on the ending inventory of cycles). Since periods can be made any time units to suit the needs of an application, this approach in fact computes the holding cost based on the average inventory of a cycle and the shortage cost in proportion to the duration of shortage (for the backorder model), and remedies the shortcomings of the heuristic or approximate treatment of such systems (Hadley and Whitin, Analysis of Inventory Systems, Prentice-Hall, Englewood Cliffs, NJ, 1963). We show that a base-stock policy is optimal for the backorder model, while the optimal order quantity is a function of the on-hand inventory for the lost-sales model. Moreover, for the backorder model, we develop a simple expression for computing the optimal base-stock level; for the lost-sales model, we derive convergence conditions for obtaining the optimal operational parameters.     

The influence of demand variability on the performance of a make-to-stock queue

Variability, in general, has a deteriorating effect on the performance of stochastic inventory systems. In particular, previous results indicate that demand variability causes a performance degradation in terms of inventory related costs when production capacity is unlimited. In order to investigate the effects of demand variability in capacitated production settings, we analyze a make-to-stock queue with general demand arrival times operated according to a base-stock policy. We show that when demand inter-arrival distributions are ordered in a stochastic sense, increased arrival time variability indeed leads to an augmentation of optimal base-stock levels and to a corresponding increase in optimal inventory related costs. We quantify these effects through several numerical examples.     

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.     

Analysis of a two-echelon inventory system with two supply modes

In this paper, we consider a serial two-echelon periodic review inventory system with two supply modes at the most upstream stock point. As control policy for this system, we propose a natural extension of the dual-index policy, which has three base-stock levels. We consider the minimization of long run average inventory holding, backlogging, and both per unit and fixed emergency ordering costs. We provide nested newsboy characterizations for two of the three base-stock levels involved and show a separability result for the difference with the remaining base-stock level. We extend results for the single-echelon system to efficiently approximate the distributions of random variables involved in the newsboy equations and find an asymptotically correct approximation for both the per unit and fixed emergency ordering costs. Based on these results, we provide an algorithm for setting base-stock levels in a computationally efficient manner. In a numerical study, we investigate the value of dual-sourcing in supply chains and illustrate that dual-sourcing can lead to significant cost savings in cases with high demand uncertainty, high backlogging cost or long lead times.     

Computing a Stationary Base-Stock Policy for a Finite Horizon Stochastic Inventory Problem with Non-linear Shortage Costs

Abstract

We consider a periodic-review stochastic inventory problem in which demands for a single product in each of a finite number of periods are independent and identically distributed random variables. We analyze the case where shortages (stockouts) are penalized via fixed and proportional costs simultaneously. For this problem, due to the finiteness of the planning horizon and non-linearity of the shortage costs, computing the optimal inventory policy requires a substantial effort as noted in the previous literature. Hence, our paper is aimed at reducing this computational burden. As a resolution, we propose to compute “the best stationary policy.” To this end, we restrict our attention to the class of stationary base-stock policies, and show that the multi-period, stochastic, dynamic problem at hand can be reduced to a deterministic, static equivalent. Using this important result, we introduce a model for computing an optimal stationary base-stock policy for the finite horizon problem under consideration. Fundamental analytic conclusions, some numerical examples, and related research findings are also discussed.     

Integrating inventory control and a price change in the presence of reference price effects: a two-period model

Demand and procurement planning for consumer electronics products must cope with short life cycles, limited replenishment opportunities and a willingness to pay that is influenced by past prices and decreases over time. We therefore propose the use of an integrated pricing and inventory control model with a two-period linear demand model, in which demand also depends on the difference between a price-history-based reference price and the current price. For this model we prove that the optimal joint pricing/inventory policy for the replenishment opportunity after the first period is a base-stock list-price policy. That is, stock is either replenished up to a base-stock level and a list-price is charged, or it is not replenished and a discount is given that increases with the stock-level. Furthermore, we use real-world cell phone data to study the differences between an integrated policy and traditional sequential optimization, where prices are initially optimized based on the expected demand and ordering cost, and the resulting demand distribution is used to determine an optimal inventory policy. Finally, we discuss possible extensions of the model.     

Lost-sales inventory systems with a service level criterion

Competitive retail environments are characterized by service levels and lost sales in case of excess demand. We contribute to research on lost-sales models with a service level criterion in multiple ways. First, we study the optimal replenishment policy for this type of inventory system as well as base-stock policies and (R, s, S) policies. Furthermore, we derive lower and upper bounds on the order-up-to level, and we propose efficient approximation procedures to determine the order-up-to level. The procedures find values of the inventory control variables that are close to the best (R, s, S) policy and comply to the service level restriction for most of the instances, with an average cost increase of 2.3% and 1.2% for the case without and with fixed order costs, respectively.     

Scheduling policies for a repair shop problem

In this paper, we analyze a repair shop serving several fleets of machines that fail from time to time. To reduce downtime costs, a continuous-review spare machine inventory is kept for each fleet. A spare machine, if available on stock, is installed instantaneously in place of a broken machine. When a repaired machine is returned from the repair shop, it is placed in inventory for future use if the fleet has the required number of machines operating. Since the repair shop is shared by different fleets, choosing which type of broken machine to repair is crucial to minimize downtime and holding costs. The optimal policy of this problem is difficult to characterize, and, therefore, is only formulated as a Markov Decision Process to numerically compute the optimal cost and base-stock level for each spare machine inventory. As an alternative, we propose the dynamic Myopic(R) policy, which is easy to implement, yielding costs very close to the optimal. Most of the time it outperforms the static first-come-first-served, and preemptive-resume priority policies. Additionally, via our numerical study, we demonstrate that repair shop pooling is better than reserving a repair shop for each fleet.     

Production-inventory control policy under warm/cold state-dependent fixed costs and stochastic demand: partial characterization and heuristics

The motivation for our study comes from some production and inventory systems in which ordering/producing quantities that exceed certain thresholds in a given period might eliminate some setup activities in the next period. Many examples of such systems have been discussed in prior research but the analysis has been limited to production settings under deterministic demand. In this paper, we consider a periodic-review production-inventory model under stochastic demand and incorporate the following fixed-cost structure into our analysis. When the order quantity in a given period exceeds a specified threshold value, the system is assumed to be in a “warm” state and no fixed cost is incurred in the next period regardless of the order quantity; otherwise the system state is considered “cold” and a positive fixed cost is required to place an order. Assuming that the unsatisfied demand is lost, we develop a dynamic programming formulation of the problem and utilize the concepts of quasi-K-convexity and non-K-decreasing to show some structural results on the optimal cost-to-go functions. This analysis enables us to derive a partial characterization of the optimal policy under the assumption that the demands follow a Pólya or uniform distribution. The optimal policy is defined over multiple decision regions for each system state. We develop heuristic policies that are aimed to address the partially characterized decisions, simplify the ordering policy, and save computational efforts in implementation. The numerical experiments conducted on a large set of test instances including uniform, normal and Poisson demand distributions show that a heuristic policy that is inspired by the optimal policy is able to find the optimal solution in almost all instances, and that a so-called generalized base-stock policy provides quite satisfactory results under reasonable computational efforts. We use our numerical examples to generate insights on the impact of problem parameters. Finally, we extend our analysis into the infinite horizon setting and show that the structure of the optimal policy remains similar.     

Time consistency and risk averse dynamic decision models: Definition,interpretation and practical consequences

This paper aims at resolving a major obstacle to practical usage of time-consistent risk-averse decision models. The recursive objective function, generally used to ensure time consistency, is complex and has no clear/direct interpretation. Practitioners rather choose a simpler and more intuitive formulation, even though it may lead to a time inconsistent policy. Based on rigorous mathematical foundations, we impel practical usage of time consistent models as we provide practitioners with an intuitive economic interpretation for the referred recursive objective function. We also discourage time-inconsistent models by arguing that the associated policies are sub-optimal. We developed a new methodology to compute the sub-optimality gap associated with a time-inconsistent policy, providing practitioners with an objective method to quantify practical consequences of time inconsistency. Our results hold for a quite general class of problems and we choose, without loss of generality, a CVaR-based portfolio selection application to illustrate the developed concepts.     

Myopic control of stochastic inventories with intermittent updates: continuous <Emphasis Type="Italic">versus</Emphasis> periodic replenishment

A manufacturer who is responsible for supplying a retailer with a single product is considered. The retailer sells the product in response to stochastic demand and provides the manufacturer with periodic updates about his inventories. Replenishing the retailer's inventory under two myopic base-stock policies is addressed. These policies, referred to as vendor managed inventory, represent a relatively new approach to allocating responsibility in the replenishment process. Specifically, the manufacturer, who is responsible for the retailer's inventories, can replenish them either continuously at any point in time or periodically, at one point in time for each period. The myopic replenishment policies that are considered are of a base-stock type. It is shown that the selected policies become optimal as the number of review periods tends to infinity. Furthermore, the two replenishment alternatives are compared in terms of both base-stock levels and expected costs, including those for inventory holding/shortage and transportation costs. Although continuous rather than periodic replenishment is evidently more expensive in terms of transportation costs, it is shown that even when the transportation cost constitutes more than 55% of the total average cost, it may still be preferable to replenish continuously rather than periodically.     

Optimal control of a dual service rate M/M/1 production-inventory model

We analyse a dual-source, production-inventory model in which the processing times at a primary manufacturing resource and a second, contingent resource are exponentially distributed. We interpret the contingent source to be a subcontractor, although it could also be overtime production. We treat the inventory and contingent sourcing policies as decision variables in an analytical study and, additionally, allow the primary manufacturing capacity to be a decision variable in a subsequent numerical study. Our goal is to gain insight into the use of subcontracting as a contingent source of goods and whether it can fulfill real-world managers' expectations for improved performance. We prove that a stationary, non-randomised inventory and subcontracting policy is optimal for our M/M/1 dual-source model and, moreover, that a dual base-stock policy is optimal. We then derive an exact closed-form expression for one of the optimal base stocks, which to our knowledge is the first closed-form solution for a dual-source model. We use that closed-form result to advantage in a numerical study from which we gain insight into how optimal capacity, subcontracting, and inventory policies are set, and how effectively a contingent source can reduce total cost, capacity cost, and inventory cost. We find that (i) the contingent source can reduce total cost effectively even when contingent sourcing is expensive and (ii) contingent sourcing reduces capacity cost more effectively than it does inventory cost.     

An optimal ordering policy for a spare unit with lead time

We consider a generalised ordering policy in which a spare unit for replacement can be delivered only by order after a constant lead time. Introducing the costs for ordering, shortage and holding, we derive the expected cost per unit time in the steady-state. We discuss the optimal ordering policy which minimises the expected cost, and show in a main theorem that the optimal policy is reduced to either one of two typical ordering policies depending on some conditions. We further discuss a similar ordering policy with varying lead times.     

基于设备状态监控的单级备件库存系统补货策略研究

本文研究一个周期性订货的多设备同备件库存系统,将备件库存策略与设备状态监控相结合,讨论了存在设备状态监控情形下的备件库存策略。针对设备状态自然腐蚀过程和人 为修复过程的复合过程,运用一个新的马尔科夫概率转移矩阵对设备需求概率进行刻画,并在此基础上给出静态订货模型和状态监控下的动态订货模型的最优订货策略。通过对比以上两种订货策略优缺点,本文提出一种新的启发式订货策略: 基于关键状态的订货策略模型。该策略可以有效降低对全部设备实行动态监控的信息成本,且成本节省优于静态订货策略,对于企业的现实问题有着较好的指导意义。