Periodic-review Model Research Articles

Coordinating pricing and inventory control in a fluctuating environment

This paper addresses the simultaneous determination of pricing and inventory replenishment strategies under a fluctuating environment. Specifically, we analyze the single item, periodic review model. The demand consists of two parts: the deterministic component, which is influenced by the price, and the stochastic component (perturbation). The distribution of the stochastic component is determined by the current state of an exogenous Markov chain. The price that is charged in any given period can be specified dynamically. A replenishment order may be placed at the beginning of some or all of the periods, and stockouts are fully backlogged. Ordering costs that are lower semicontinuous, and inventory/backlog (or surplus) costs that are continuous with polynomial growth. Finite-horizon and infinite-horizon problems are addressed. Existence of optimal policies is established. Furthermore, optimality of (s,S,p)-type policies is proved when the ordering cost consists of fixed and proportional cost components and the surplus cost (these costs are all state-dependent) is convex.

The Value of Component Commonality in a Dynamic Inventory System with Lead Times

Component commonality has been widely recognized as a key factor in achieving product variety at low cost. Yet the theory on the value of component commonality is rather limited in the inventory literature. The existing results were built primarily on single-period models or periodic-review models with zero lead times. In this paper, we consider a continuous-review system with positive lead times. We find that although component commonality is in general beneficial, its value depends strongly on component costs, lead times, and dynamic allocation rules. Under certain conditions, several previous findings based on static models do not hold. In particular, component commonality does not always generate inventory benefits under certain commonly used allocation rules. We provide insight on when component commonality generates inventory benefits and when it may not. We further establish some asymptotic properties that connect component lead times and costs to the impact of component commonality. Through numerical studies, we demonstrate the value of commonality and its sensitivity to various system parameters in between the asymptotic limits. In addition, we show how to evaluate the system under a new allocation rule, a modified version of the standard first-in-first-out rule.

Joint optimization on pricing, promotion and inventory control with stochastic demand

Firms often utilize and coordinate dynamic adjustment of price, non-price promotions such as advertisement and recruitment of excellent salespeople and proper inventory replenishment policies to satisfy customer demands and achieve maximal profit. This paper provides an analytical model for obtaining optimal decisions jointly on pricing, promotion and inventory control. Specifically, we study a single item, finite horizon, periodic review model in which the demand is influenced by price and promotion, and the objective is to maximize the total profit. We characterize the optimal policy and provide solution for coordinating promotion, pricing and inventory replenishment. We show that the optimal promotion policy is a threshold policy under some reasonable assumptions and once the promotion is determined the optimal inventory/pricing policy is a base-stock-list-price policy. Numerical study is also provided to demonstrate the contribution of our model.

Periodic review inventory models with stochastic supplier's visit intervals

Periodic review inventory models are widely used in practice, especially for inventory systems in which many different items are purchased from the same supplier. However, all periodic review models have assumed a fixed length of the review periods. In practice, it is possible that the review periods are of a variable length. Such periodic systems result mainly from supply uncertainties. For example, the supplier visits the downstream retailers and replenishes inventories for them, but does not always come in constant intervals. This may be because retailers are geographically dispersed in the supply chain, the supplier is in a relatively more powerful position, or the supplier simply does not have a reliable visit schedule. In such situations, the replenishment cycle length is random in nature. In this paper, we use dynamic programming to model such institutional contexts. We assume that the supplier's visit intervals are independently and identically distributed. With a suitable transformation, the backlogged periodic review model derived becomes a standard discrete-time model. The computation shows that ignoring the variability of the supplier's visit intervals can incur extremely large losses, especially if shortage is costly, demand variability is low, and/or the replenishment lead-time is short.

A periodic review (R, T) model with investment to reduce lost sales rate

In 1973 Montgomery et al. [1] proposed a (R, T) type of periodic review inventory model in which the lost sales rate is given. The purpose of this paper is to investigate the impact in this heuristic periodic review inventory model with increasing investment to reduce the lost sales rate, where the review period and lost sales rate are decision variables. In this paper, the protection interval demand is assumed to be normal distributed, and two commonly used investment cost functionalforms, logarithmic and power, are employed for lost sales rate reduction. A procedure of finding the optimal investment decision is developed, and three numerical examples are given to illustrate the results.

Independence of Capacity Ordering and Financial Subsidies to Risky Suppliers

The risk of supply disruptions because of suppliers' financial problems plays a prominent role in manufacturers' risk portfolios. Even large suppliers (e.g., Delphi) could file for bankruptcy, and manufacturers' actions, such as financial subsidies to suppliers, profoundly affect suppliers' financial health. Using a dynamic, stochastic, periodic-review model of the manufacturer's joint capacity reservation and financial subsidy decisions and a general firm-value model of the supplier's financial state, this paper addresses the following questions: What is the optimal joint capacity ordering and financial subsidy policy for the manufacturer? Must subsidy and capacity ordering decisions be made jointly? How good are the recommendations from the traditional procurement models, which ignore the benefits of controlling the supplier's financial state through subsidies? The paper presents general assumptions that allow the manufacturer to make ordering decisions independent of subsidy decisions and investigates interactions between ordering and subsidy decisions when these assumptions are violated. Conditions are presented for the optimal subsidy policy to have a “subsidize-up-to” structure and for the optimal ordering decisions to be newsvendor fractiles.

Joint Inventory Replenishment and Pricing Control for Systems with Uncertain Yield and Demand

We study the joint inventory replenishment and pricing problem for production systems with random demand and yield. More specifically, we analyze the following single-item, periodic-review model. Demands in consecutive periods are independent random variables and their distributions are price sensitive. The production yield is uncertain so that the quantity received from a replenishment is a random variable whose distribution depends on the production quantity. Stockouts are fully backlogged. Our problem is to characterize the optimal dynamic policy that simultaneously determines the production quantity and the price for each period to maximize the total discounted profit. We show that the optimal replenishment policy is of a threshold type, i.e., it is optimal to produce if and only if the starting inventory in a period is below a threshold value, and that both the optimal production quantity and the optimal price in each period are decreasing in the starting inventory. We further study the operational effects of uncertain yield. We prove that, in the single-period case, the threshold of replenishment is independent of the yield variability, and, in the multiperiod case, it is higher in a system with uncertain yield than in one with certain yield. In addition, the system with uncertain yield always charges a higher price.

Information Sharing to Improve Retail Product Freshness of Perishables

We explore the value of information (VOI) in the context of a retailer that provides a perishable product to consumers and receives replenishment from a single supplier. We assume a periodic review model with stochastic demand, lost sales, and order quantity restrictions. The product lifetime is fixed and deterministic once received by the retailer, although the age of replenished items provided by the supplier varies stochastically over time. Since the product is perishable, any unsold inventory remaining after the lifetime elapses must be discarded (outdated). Without the supplier explicitly informing the retailer of the product age, the age remains unknown until receipt. With information sharing, the retailer is informed of the product age prior to placing an order and hence, can utilize this information in its decision‐making. We formulate the retailer's replenishment policies, with and without knowing the age of the product upon receipt, and measure the VOI as the marginal improvement in profit that the retailer achieves with information sharing, relative to the case when no information is shared. We establish the importance of information sharing and identify the conditions under which substantial benefits can be realized.

Inventory models with inventory-level-dependent demand: A comprehensive review and unifying theory

Marketing researchers and practitioners have long recognized the demand of many retail items is proportional to the amount of inventory displayed. Recently, two distinct types of inventory control models reflecting this relationship have appeared in the literature, models in which the demand rate of an item is a function of the initial inventory level and those in which it is dependent on the instantaneous inventory level. We present a comprehensive overview of this literature and demonstrate the equivalence of the two types of models through the use of a simple, periodic-review model. An alternative approach to sensitivity analysis for inventory models with inventory-level-dependent demand is also presented.

Coordinating Inventory Control and Pricing Strategies with Random Demand and Fixed Ordering Cost: The Finite Horizon Case

We analyze a finite horizon, single product, periodic review model in which pricing and production/inventory decisions are made simultaneously. Demands in different periods are random variables that are independent of each other and their distributions depend on the product price. Pricing and ordering decisions are made at the beginning of each period and all shortages are backlogged. Ordering cost includes both a fixed cost and a variable cost proportional to the amount ordered. The objective is to find an inventory policy and a pricing strategy maximizing expected profit over the finite horizon. We show that when the demand model is additive, the profit-to-go functions are k-concave and hence an (s, S, p) policy is optimal. In such a policy, the period inventory is managed based on the classical (s, S) policy and price is determined based on the inventory position at the beginning of each period. For more general demand functions, i.e., multiplicative plus additive functions, we demonstrate that the profit-to-go function is not necessarily k-concave and an (s, S, p) policy is not necessarily optimal. We introduce a new concept, the symmetric k-concave functions, and apply it to provide a characterization of the optimal policy.

Coordinating Inventory Control and Pricing Strategies with Random Demand and Fixed Ordering Cost: The Infinite Horizon Case

We analyze an infinite horizon, single-product, periodic review model in which pricing and production/inventory decisions are made simultaneously. Demands in different periods are identically distributed random variables that are independent of each other, and their distributions depend on the product price. Pricing and ordering decisions are made at the beginning of each period, and all shortages are backlogged. Ordering cost includes both a fixed cost and a variable cost proportional to the amount ordered. The objective is to maximize expected discounted, or expected average, profit over the infinite planning horizon. We show that a stationary (s, S, p) policy is optimal for both the discounted and average profit models with general demand functions. In such a policy, the period inventory is managed based on the classical (s, S) policy, and price is determined based on the inventory position at the beginning of each period.

Optimal Policies for Production/Inventory Systems with Finite Capacity and Markov-Modulated Demand and Supply Processes

In many production/inventory systems, not only is the production/inventory capacity finite, but the systems are also subject to random production yields that are influenced by factors such as breakdowns, repairs, maintenance, learning, and the introduction of new technologies. In this paper, we consider a single-item, single-location, periodic-review model with finite capacity and Markov modulated demand and supply processes. When demand and supply processes are driven by two independent, discrete-time, finite-state, time-homogeneous Markov chains, we show that a modified, state-dependent, inflated base-stock policy is optimal for both the finite and infinite horizon planning problems. We also show that the finite-horizon solution converges to the infinite-horizon solution.

A periodic-review model with serially-correlated, inventory-level-dependent demand

It has been established that the demand rates of many consumer products are serially correlated and are dependent on the amount of inventory displayed to the customer. A periodic-review model is developed that considers both of these effects. An adaptive, base-stock policy based on the critical fractile is presented, and a solution methodology is developed. The effect of this demand pattern on the safety stock requirement is investigated.

Application of an Adaptive Neuro-Fuzzy Inference System in Inventory Control

A model based on adaptive neuro-fuzzy inference systems is proposed for inventory control. The periodic review model of inventory control with variable order quantity is considered. The model takes into account the dynamics of production-inventory system in a control theoretic approach. The control module combines fuzzy logic and the conventional proportional-integral (PI) algorithm. The fuzzy logic membership functions are adapted to fit the training data set of production-inventory. It simulates the decision support system to maintain the inventory, of the finished product, at the desired level, in spite of variations in demand. The effectiveness of the proposed control model is illustrated using the real data of a typical packaging organization operating in the Sultanate of Oman.

A note on periodic review inventory model with controllable setup cost and lead time

In this study, we investigate the periodic review inventory models with a mixture of backorders and lost sales by controlling lead time and setup cost simultaneously to reduce the inventory operating cost. It is assumed that the probability distribution of the protection interval, i.e., review period plus lead time, demand is unknown but its first two moments are given, we apply the minimax distribution free procedure to solve this problem. An algorithm to find the optimal solutions is developed. Specifically, from the results of numerical examples, it can be shown that, the significant savings can be achieved through the reductions of lead time and setup cost. Scope and purpose In most of the literature dealing with periodic review inventory problems, both lead time and setup cost are treated as constants. Recently, Ouyang and Chuang (J. Inf. Manage. Sci. 9 (1998) 25) presented a minimax distribution free procedure for the periodic review inventory model which involves a controllable lead time. We note that the paper is focusing on the benefits from lead time reduction in which setup cost is viewed as a fixed constant. From the Japanese experience of Just-In-Time (JIT) production, it has been observed in many manufacturing settings including job shops, batch shops and flow shops, whose setup cost can be reduced by investing capital. For this reason, we attempt to extend Ouyang and Chuang's model by formulating a modified periodic review model to accommodate more practical features of the real inventory systems.

Multilocation Combined Pricing and Inventory Control

We consider the problem of managing inventories and dynamically adjusting retailer prices in distribution systems with geographically dispersed retailers. More specifically, we analyze the following single item, periodic review model. The distribution of demand in each period, at a given retailer, depends on the item's price according to a stochastic demand function. These stochastic demand functions may vary by retailer and by period. The replenishment process consists of two phases: In some or all periods, a distribution center may place an order with an outside supplier. This order arrives at the distribution center after an “order leadtime” and is then, in the second phase, allocated to the retailers. Allocations arrive after a second “allocation leadtime.” We develop an approximate model that is tractable and in which an optimal policy of simple structure exists. The approximate model thus provides analytically computable approximations for systemwide profits and other performance measures. Moreover, the approximate model allows us to prove how various components of the optimal strategy (i.e., prices and order-up-to levels) respond to shifts in the model parameters, e.g., to shifts in the retailers' demand functions. In addition, we develop combined pricing, ordering, and allocation strategies and show that the system's performance under these strategies is well gauged by the above approximations. We use this model to assess the impact of different types of geographic dispersion on systems with dynamically varying prices and how different system parameters (e.g., leadtimes, coefficients of variation of individual retailers' demand, price elasticities) contribute to this impact. Similarly, we use the model to gauge the benefits of coordinated replenishments under dynamic pricing, and how these benefits increase as the allocation decisions of the systemwide orders to individual retailers are postponed to a later point in the overall replenishment leadtime. We report on a comprehensive numerical study based on data obtained from a nationwide department store chain.

Integrated optimization of inventory-distribution systems by random local search and a genetic algorithm

A new model and its solution procedure for the commodity distribution system consisting of distribution centers and consumer points are discussed. Demand is assumed to be a random variable that obeys a known, stationary probability distribution. An integrated optimization model is built where both the order-up-to- R policy, which is one of the typical inventory policies for periodic review models, and the transportation problem are considered simultaneously. The assignment of consumer points to distribution centers is not fixed. The problem is to determine the target inventory and the transportation quantity in order to minimize the expectation of the sum of inventory related costs and transportation costs. Simulation and linear programming are used to calculate the expected costs, and a random local search method is developed in order to determine the optimum target inventory. A genetic algorithm is also tested and compared with the proposed random local search method. The model and effectiveness of the proposed solution procedure are clarified by computational experiments.

An inventory control model using fuzzy logic

A model based on fuzzy logic is proposed for inventory control. The periodic review model of inventory control with variable order quantity is considered. The model takes into account the dynamics of production–inventory system in a control theoretic approach. The control module combines fuzzy logic with proportional-integral-derivative (PID) control algorithm. It simulates the decision support system to maintain the inventory of the finished product at the desired level in spite of variations in demand. The effectiveness of the proposed control model is illustrated using the actual data of a typical packaging organization operating in the Sultanate of Oman.

Product recovery in stochastic remanufacturing systems with multiple reuse options

In many product recovery situations, returned products can be reused in multiple ways. Under these circumstances, the problem arises in which quantities reusable items should be allocated to the different remanufacturing options, especially in case of insufficient stock of returns. For this problem a periodic review model is formulated which also includes a disposal option and incorporates uncertainties in returns and demands for the different serviceable options. The structure of the optimal policy is analyzed, and it is shown that under specific allocation rules a near-optimal policy with a simple structure exists. An efficient computational procedure for determination of the optimal policy parameters is presented. This procedure is applied in a numerical investigation that gives interesting managerial insights into important product recovery issues.

Analysis of an inventory system under backorder correlated deterministic demand and geometric supply process

In this article we propose a single item, periodic review model that investigates the effects of changes in the demand process that occur after stockout realizations. We investigate a system where the demands in successive periods are deterministic but affected by the backorder realizations. In order to capture the effects of changes in the demand process we use a geometric type supply availability. We analytically derive the necessary components for obtaining profit related performance measures and provide computational analysis.