Periodic Review Inventory Policy Research Articles

Single item periodic review inventory control with sales dependent stochastic return flows

Retailers have to deal with increasing levels of product returns as the shares of e-commerce sales soars. With this increase, it is no longer feasible to dispatch returned products to outlets or landfills, hence retailers must re-evaluate them both to maximize profit and to minimize their environmental impact. Our objective is to study a retailer’s optimal inventory control policy under product returns to maximize expected profit which is the sales revenue minus the procurement, backorder, holding, and salvage costs incurred in a finite horizon. We model a period’s returns to be stochastically dependent on the previous period’s sales quantity. Using dynamic programming formulation, we solve for the optimal periodic review inventory policy and provide structural results on the optimal policy of the final period. Through numerical studies, we show that incorporating detailed sales-dependent returns could increase a retailer’s expected profit by 23%. Ignoring this dependency in determining the optimal inventory policy results with increased order frequency, higher levels of backorders and more leftovers which could eventually end up in a landfill, but above all could lead to a significant overestimation of the resulting profit.

A hybrid differential evolution algorithm for a stochastic location-inventory-delivery problem with joint replenishment

A practical stochastic location-inventory-delivery problem with multi-item joint replenishment is studied. Unlike the conventional location-inventory model with a continuous-review (r, Q) inventory policy, the periodic-review inventory policy is adopted with multi-item joint replenishment under stochastic demand, and the coordinated delivery cost is considered. The proposed model considers the integrated optimization of strategic, tactical, and operational decisions by simultaneously determining (a) the number and location of distribution centers (DCs) to be opened, (b) the assignment of retailers to DCs, (c) the frequency and cycle interval of replenishment and delivery, and (d) the safety stock level for each item. An intelligent algorithm based on particle swarm optimization (PSO) and adaptive differential evolution (ADE) is proposed to address this complex problem. Numerical experiments verified the effectiveness of the proposed two-stage PSO-ADE algorithm. A sensitivity analysis is presented to reveal interesting insights that can guide managers in making reasonable decisions.

Fast evaluation of a periodic review inventory policy

We consider a single-echelon inventory system facing stochastic demand controlled by the standard (r,nQ,T) batch ordering policy. While an exact algorithm for optimizing all three policy variables exists, excessive computational requirements hinder industrial applications, particularly in multi-product environments. Aiming at reducing computation time, we revisit the exact total average cost function. After establishing a new analytical property, we show that the exact ordering cost can be very accurately approximated by two convex functions. Using these, the original problem is decomposed in two separate sub-problems easier to solve. For each sub-problem we establish properties linking each solution with the classical EOQ, which lead to a fast heuristic search algorithm for the policy variables evaluation. Numerical comparisons with both the exact algorithm and two popular meta-heuristics demonstrate the excellent performance of the heuristic in terms of solution quality and run-time.

A procedure for the approximation of the waiting time distribution in a discrete-time (r, S) inventory system

We study a single-item periodic review inventory policy. Demands arrive on a discrete (e.g. daily) time axis. This is the time model underlying the Advanced Planning Software systems used for supply chain management in industrial practice. We present a procedure for the calculation of the probability distribution of the customer waiting time. We discuss optimisation problems which can be solved on the basis of the probability distribution of the waiting time.

An optimal joint maintenance and spare parts inventory model

Optimised maintenance of plant machineries is one of the important tasks to reduce plant operating cost. To carry out these maintenance tasks smoothly, development of spare parts inventory policy is the most prioritised issue. This research aims at the derivation of a joint maintenance and spare parts inventory model for a multi-component system. The decision variables are the order quantity and the ratio of ordering interval to preventive maintenance interval. We used block-based maintenance policy and periodic review inventory policy in this study. At first, a cost function is developed assuming that failure occurrences follow Poisson process with a constant failure rate and machine failures are non-repairable. Second, the model is further expanded for the case when failure occurrences follow non-homogeneous Poisson process and machine failures are repairable. The applicability of the developed models is demonstrated by numerical examples. Sensitivity analysis is also conducted to examine the effects of input parameters.

Periodic review inventory policy for non-instantaneous deteriorating items with time dependent deterioration rate

The paper studies a periodic review inventory model with no shortages and different demand rates during pre- and post- deterioration periods . Deterioration of units start after a fixed time interval, and the deterioration rate is time dependent. The model determines the optimal reorder interval and the optimal order quantity so as to minimize the total cost per unit length of an inventory cycle. An extension of the model to include price discount has been further considered. Numerical examples are presented to illustrate the model and a sensitivity analysis is also reported.

Joint optimization of preventive maintenance and inventory policies for multi-unit systems subject to deteriorating spare part inventory

The interconnection of maintenance and spare part inventory management often puzzles managers and researchers. The deterioration of the inventory affects decision-making and increases losses. Block replacement and periodic review inventory policies were here used to evaluate a joint optimization problem for multi-unit systems in the presence of inventory deterioration. The deterministic deteriorating inventory (DDI) model was used to describe deteriorating inventory when deteriorating inventory data were available and the stochastic deteriorating inventory (SDI) model was used when they were not. Analytical joint optimization models were established, and the preventive replacement interval and the maximum inventory level served as decision variables to minimize the expected system total cost rate. This work proved the existence of the optimal maximum inventory level and gave the uniqueness condition under the DDI model. Numerical experiments based on the electric locomotives in Slovenian Railways were performed to confirm the effectiveness of the proposed models. Results showed the total cost rate to be sensitive to the maximum inventory level, which indicates that the research of this work is necessary. Further, the optimal preventive replacement interval was reduced and the optimal maximum inventory level was increased to balance the influence of deteriorating inventory. Monte Carlo experiments were used to show that the proposed policy is better than policies that do not take deteriorating inventory into account.

Supply Chain Performance Evaluation Using Spreadsheet Simulation

The objective of this paper is to develop a spreadsheet-based simulation methodology to evaluate the performance of a serial supply chain under different inventory position based ((r, Q) and (r, S)) and non-inventory position based (Fixed Order Quantity (FOQ) and Demand Flow Strategy (DFS)) periodic review inventory policies. The performance measures comprises of Fill Rate, Risk of Shortage, Bullwhip Effect and Total Cost of Supply Chain. An experimental study with four scenarios are carried out for two customer demand distributions under information sharing and no information sharing scenarios to determine performance measures in lost sales environment. The best performing inventory control policy for each scenario is found by conducting Grey Relational Analysis and the analysis show that one of non-inventory position based policy, FOQ is the better policy compared to other three policies.

Determining the fill rate for a periodic review inventory policy with capacitated replenishments, lost sales and zero lead time

In this paper we consider a periodic review order-up-to inventory system with capacitated replenishments, lost sales and zero lead time. We consider discrete demand. It is shown that the initial stock levels of the different review periods form a Markov chain and we determine the transition matrix. Furthermore we study for what probability mass functions of the review period demand the Markov chain has a unique stationary distribution. Finally, we present a method to determine the fill rate.

Analysis and Optimization of a Single-item Production System with Lead Time under General Periodic Review Inventory Policy

In this paper, a single-product inventory system with a general periodic review policy is considered on a discrete time. The number of products in inventory is reviewed periodically with a positive integer time interval, and new products are ordered under base stock policy. The amounts of demand are mutually independent and identically distributed with a general distribution. There is an integer production lead time, which may not be a multiple of the order interval. Zhang and Zhang (2007) derived an expression for the fill-rate of the demand. In this paper, we derive an expression for the average inventory and backlogs theoretically. We also consider an optimization problem for determining the optimal amount of base stock to minimize the long-time average inventory holding and backlog costs. An algorithm for deriving the optimal value is developed by using reverse Laplace transforms. The number of base stock for minimizing the average cost is also compared with the one satisfying the required fill rate.

On the exact calculation of the fill rate in a periodic review inventory policy under discrete demand patterns

The primary goal of this paper is the development of a generalized method to compute the fill rate for any discrete demand distribution in a periodic review policy. The fill rate is defined as the fraction of demand that is satisfied directly from shelf. In the majority of related work, this service metric is computed by using what is known as the traditional approximation, which calculates the fill rate as the complement of the quotient between the expected unfulfilled demand and the expected demand per replenishment cycle, instead of focusing on the expected fraction of fulfilled demand. This paper shows the systematic underestimation of the fill rate when the traditional approximation is used, and revises both the foundations of the traditional approach and the definition of fill rate itself. As a result, this paper presents the following main contributions: (i) a new exact procedure to compute the traditional approximation for any discrete demand distribution; (ii) a more suitable definition of the fill rate in order to ignore those cycles without demand; and (iii) a new standard procedure to compute the fill rate that outperforms previous approaches, especially when the probability of zero demand is substantial. This paper focuses on the traditional periodic review, order up to level system under any uncorrelated, discrete and stationary demand pattern for the lost sales scenario.

Comparing the value of information sharing under different inventory policies in construction supply chain

Measuring the value of information sharing (VIS) in different inventory policies can help general contractor to improve materials and equipments management in construction supply chain. The models for measuring the VIS under (s, S) and Periodic Review (PR) inventory policies are established. A method to compare the VIS is given in a two-level construction supply chain, which aims to measure the impacts of supply quantity information of materials on general contractor's service level and total inventory cost under different inventory policies. A numerical example is proposed to explore the gap of VIS between (s, S) and PR inventory policies in a rebar supply chain. The results show that general contractor's service level in (s, S) inventory is higher than PR's and total inventory cost in (s, S) inventory is lower than PR's in the case of no information sharing. In the case of information sharing, general contractor's service level in PR inventory policy is higher than (s, S) policy; however, it is implemented by the increase of total inventory cost. This research provides valuable information to help general contractor to make correct decision on inventory management in supply chain at firm-level.

Exact and approximate calculation of the cycle service level in periodic review inventory policies

The parameters of stock policies are usually determined to minimize costs while satisfying a target service level. In a periodic review policy the time between reviews can be selected to minimize costs while the order-up-to-level is based on the fulfilment of a target service level. Generally, the calculation of this service measurement is obtained using approximations based on an additional hypothesis related to the demand pattern. Previous research has shown that there is a substantial difference between exact and approximate calculations in some general circumstances, so in these cases the service level is not accomplished or the stock level is overestimated. Although an exact calculation of CSL was developed in previous work, the computational effort required to apply it in practical environments leads to the proposal of two approximate methods ( PI and PII) that, with the classic approximation, are analysed and evaluated in this paper. This analysis points out the risks of using the classic approximation and leads one to suggest PII as the most suitable and accurate enough procedure to compute the CSL straightforwardly in practice. Additionally, a heuristic approach based on PII is proposed to accept or reject an inventory policy in terms of fulfilling a given target CSL. This paper focuses on uncorrelated, discrete and stationary demand with a known distribution pattern and without backlog.

Sharing shipment quantity information in the supply chain

This paper evaluates the benefit of a strategy of sharing shipment information, where one stage in a supply chain shares shipment quantity information with its immediate downstream customers—a practice also known as advanced shipping notice. Under a periodic review inventory policy, one supply-chain member places an order on its supplier every period. However, due to supplier's imperfect service, the supplier cannot always exactly satisfy what the customer orders on time. In particular, shipment quantities arriving at the customer, after a given lead-time, may be less (possibly more) than what the customer expects—we define this phenomenon as shipment quantity uncertainty. Where shipment quantity information is not shared with customers, the only way to respond is through safety stock. However, if the supplier shares such information, i.e. customers are informed every period of the shipment quantity dispatched, the customer may have enough time to adapt and resolve this uncertainty by adjusting its future order decisions. Our results indicate that in most circumstances this strategy, enabled by information technologies, helps supply-chain members resolve shipment quantity uncertainty well. This study provides an approach to quantify the value of shared shipment information and to help supply-chain members evaluate the cost-benefit trade-off during information system construction. Numerical examples are provided to indicate the impact of demand/shipment parameters on strategy implementation. While previous studies mainly focus on the information receiver's perspective, we evaluate a more general three-tier linear supply chain model via simulation, studying how this strategy affects the whole supply chain: the information sender, the information receiver and the subsequent downstream tier.

An Inventory Model Embedded in Designing a Supply Contract

To reduce lead-time and its variability, modern supply and transportation contracts often specify the frequency of, and volume available for, future deliveries in advance even when final demand is somewhat uncertain (Yano and Gerchak [Yano, C. A., Y. Gerchak. 1989. Transportation contracts and safety stocks for just-in-time deliveries. Manufacturing and Oper. Management 2 314–330.]). We explore the joint optimization of contract parameters and inventory control policy in such environments. We first model and derive the optimal periodic review inventory policy corresponding to a given supply contract, which generates piecewise-linear convex ordering costs. The optimal policy has two critical levels, and there is a range of stock levels for which the quantity ordered equals the contract volume. To numerically compute the critical levels, we model consecutive inventory levels as a Markov Chain, whose steady-state distribution is used to compute the holding, shortage and transportation costs. We then use the resulting total costs to derive the optimal contract volume. Various examples are provided. The optimal contracted delivery frequency can also be computed.

Lot sizing for products with finite demand horizon and periodic review inventory policy

This paper analyzes and develops an efficient algorithm for solving a simple generalization of the classical economic order quantity (EOQ) model where the demand horizon is finite and a periodic review inventory policy is followed. Such problems could arise in production shops where the product under consideration is going to be discontinued and/or replaced by some other product after a certain known number of periods. Such problems could also arise as subproblems in large prodution-inventory problems were demand rate is constant in some periods and varying in others.