Evaluating the robustness of lead time demand models

This paper examines the robustness of lead time demand models for the continuous review (r, Q) inventory policy. A number of classic distributions, (e.g. normal, lognormal, gamma, Poisson and negative binomial) as well as distribution selection rules are examined under a wide variety of demand conditions. First, the models are compared to each other by assuming a known demand process and evaluating the errors associated with using a different model. Then, the models are examined using a large sample of simulated demand conditions. Approximation results of inventory performance measures—ready rate, expected number of backorders and on-hand inventory levels are reported. Results indicate that distribution selection rules have great potential for modeling the lead time demand.     

Variance amplification and the golden ratio in production and inventory control

A discrete linear control theory model of a generic model of a replenishment rule is presented. The replenishment rule, which we term a “Deziel Eilon—automatic pipeline, inventory and order-based production control system”, is guaranteed to be stable. From a z-transform model of the policy, an analytical expression for bullwhip is derived that is directly equivalent to the common statistical measure often used in simulation, statistical and empirical studies to quantify the bullwhip effect. This analytical expression clearly shows that we can reduce bullwhip by taking a fraction of the error between the target and actual inventory and pipeline (or work in progress (WIP) or “orders placed but not yet received”) positions. This is in contrast to the common situation where ordering policies account for all of the error every time an order is placed. Furthermore, increasing the average age of the forecast reduces bullwhip, as does reducing the production/distribution lead-time. We then derive an analytical expression for inventory variance using the same procedure to identify the closed form bullwhip expression.We assume that a suitable objective function is linearly related to the bullwhip and inventory variance amplification ratios and then optimise the PIC system for different weightings of order rate and inventory level variance. We highlight two forms of the objective function, one where “the golden ratio” can be used to determine the optimal gain in the inventory and WIP feedback loop and another that allows the complete range of possible solutions to be visualised. It is interesting that the golden ratio, which commonly describes the optimum behaviour in the natural world, also describes the optimal feedback gain in a production and inventory control system.     

An inventory model with non-resuming randomly interruptible lead time

We assume that an unreliable supplier in a single-item stochastic inventory system alternates randomly between two possible states (i.e., available and unavailable), following a two-state continuous-time homogeneous Markov chain. For a compound Poisson stream of demands and Erlang lead times, our model considers the scenario where the processing of the outstanding order (if any) is interrupted at every supplier's transition epoch from the available to the unavailable state, and is restarted from the outset upon the supplier's regaining its available state. We derive the stationary distribution of the on-hand inventory under a continuous-review policy and provide some numerical results.     

Near-optimal (r,Q) policies for a two-stage serial inventory system with Poisson demand

We consider a two-stage serial inventory system whose cost structure exhibits economies of scale in both stages. In the system, stage 1 faces Poisson demand and replenishes its inventory from stage 2, and the latter stage in turn orders from an outside supplier with unlimited stock. Each shipment, either to stage 2 or to stage 1, incurs a fixed setup cost. We derive important properties for a given echelon-stock (r, Q) policy for an approximation of the problem where all states are continuous. Based on these properties, we design a simple heuristic algorithm that can be used to find a near-optimal (r, Q) policy for the original problem. Numerical examples are given to demonstrate the effectiveness of the algorithm.     

Impact of frequency of alignment of physical and information system inventories on out of stocks: A simulation study

Inaccuracy in the information system inventory as compared to the physical inventory may lead to out of stocks. Inaccuracy may occur for many reasons, a principal one being random losses such as theft. One way to reduce this inaccuracy is to adjust the inventory information in the systems at some regular frequency. Such alignments are quite expensive in practice. Thus how often to align the two inventories is the focus of this research. A simulation model is employed to investigate the effect of such loss defined by the stock loss parameter (λ) and the frequent alignment of physical and information system inventories on the stockout (S_out) and average inventory (I). A term to be called the effective value of stock loss parameter is introduced to signify the effect of frequency of alignment (f) on S_out. The results derived in this study provide a powerful tool in the hands of an inventory manager. It has been noted that, so far as stockout is concerned, by selecting a moderate value of alignment frequency (f), the effective value of stock loss parameter (λ_e) can be reduced to∼ λ/f. The accuracy of S_out and I values across a number of runs in the simulation studies, sensitivity of S_out and I on various parameters and the nature of stochastic demand distribution, and application of these results with or without deployment of RFID to reduce the loss due to stockout are also discussed. The results, verified under various scenarios, indicate that there is a significant reduction in stockout loss when the alignment is done monthly vs. annually, but it does not add much value beyond a monthly check.     

Lead time reduction strategies in a single-vendor-single-buyer integrated inventory model with lot size-dependent lead times and stochastic demand

This paper studies alternative methods for reducing lead time and their impact on the safety stock and the expected total costs of a (Q,s) continuous review inventory control system. We focus on a single-vendor-single-buyer integrated inventory model with stochastic demand and variable, lot size-dependent lead time and assume that lead time consists of production and setup and transportation time. As a consequence, lead time may be reduced by crashing setup and transportation time, by increasing the production rate, or by reducing the lot size. We illustrate the benefits of reducing lead time in numerical examples and show that lead time reduction is especially beneficial in case of high demand uncertainty. Further, our studies indicate that a mixture of setup time and production time reduction is appropriate to lower expected total costs.     

Retailer policy, uncertainty reduction, and supply chain performance

We investigate how the retailer’s inventory policy affects the total cost of a serial supply chain. When the retailer uses the locally optimal (s,S) policy, there is randomness in order time and order quantity to the supplier whereas the supplier sees randomness only in order quantity for the suboptimal (R,T) policy and only in order time for another suboptimal (Q,r) policy. Using an extensive computational study, we find that the suboptimal policies perform better from the total supply chain perspective. The benefit of policy changes is magnified when the retailer costs are low, when the supplier costs are high, and when there is information sharing.     

Non-cooperative strategies for production and shipment lot sizing in one vendor-multi-buyer system

The supply chain structure examined in this paper consists of a single vendor (or manufacturer) with multiple heterogeneous buyers (or retailers). A continuous deterministic model is presented. To satisfy buyers demands, the vendor will deliver the product in JIT shipments to each buyer. The production rate is constant and sufficient to meet the buyers’ demands. The product is shipped in discrete batches from the vendor's stock to buyers’ stocks and all shipments are realized instantaneously. Special production-replenishment policies of the vendor and the buyers are analyzed. That is, the production batch is transferred to each buyer in several sub-batches in each production distribution cycle (PDC).This paper offers game model without prices, where agents minimize individual costs. It is a non-cooperative (1+N)-person game model with agents (a single vendor and N-buyers) choosing numbers and sizes of transferred batches. The model describes inventory patterns and cost structure of PDC. It is proved that there exist Nash equilibria in several types of sub-games of the considered game.     

Single item inventory control under periodic review and a minimum order quantity

In this paper we study a periodic review single item single stage inventory system with stochastic demand. In each time period the system must order none or at least as much as a minimum order quantity Q_min. Since the optimal structure of an ordering policy with a minimum order quantity is complicated, we propose an easy-to-use policy, which we call (R, S, Q_min) policy. Assuming linear holding and backorder costs we determine the optimal numerical value of the level S using a Markov Chain approach. In addition, we derive simple news-vendor-type inequalities for near-optimal policy parameters, which can easily be implemented within spreadsheet applications. In a numerical study we compare our policy with others and test the performance of the approximation for three different demand distributions: Poisson, negative binomial, and a discretized version of the gamma distribution. Given the simplicity of the policy and its cost performance as well as the excellent performance of the approximation we advocate the application of the (R, S, Q_min) policy in practice.     

Inventory control with a modified Croston procedure and Erlang distribution

This paper considers an inventory control system, primarily for a finished goods inventory. The purpose is to create a procedure that can handle both fast-moving items with regular demand and slow-moving items. The suggested procedure should be easy to implement in a modern computerized ERP-system. Essentially, the system is a periodic review system built around a Croston forecasting procedure. An Erlang distribution is fitted to the observed data using the mean and variance of the forecasted demand rate. According to probabilities for stock shortages, derived from the probability distribution, the system decides if it is time to place a new order or not. The Croston forecasting method is theoretically more accurate than ordinary exponential smoothing for slow-moving items. However, it is not evident that a Croston forecasting procedure (with assumed Erlang distribution) outperforms ordinary exponential smoothing (with assumed normal distribution) applied in a “practical” inventory control system with varying demand, automatically generated replenishment, etc. Our simulation study shows that the system in focus will present fewer shortages at lower inventory levels than a system based on exponential smoothing and the normal distribution.     

Evaluation of postponement in the soluble coffee supply chain: A case study

This paper, through a case study, evaluates postponement as an option to improve supply chain performances in a soluble coffee manufacturer. The main focus of the paper is on the development of a numerical data-driven methodology for quantifying the benefits of postponement obtained from the total inventory reduction. Our study shows that significant cost savings are achievable by delaying the labelling and packaging processes until actual orders from retailers are known. These savings include the reduction of the cycle stock, safety stock, and obsolete stock unsold from promotional events. We also discuss the cost implications of the proposed configuration that may offset the estimated potential benefits. The methodology presented in this paper can also be used in many other supply chains, particularly in the food industry with similar environments. This paper contributes to the literature in extending the very short list of published work presenting a systematic methodology of characterising the postponement benefits in real world settings and driven by numerical data.     

Disaggregation and consolidation of imperfect quality shipments in an extended EPQ model

We consider a standard economic production quantity (EPQ) model. Due to manufacturing variability, a fraction P of the produced inventory will have imperfect quality, where P is a random variable with a known distribution. We consider a 100% inspection policy and further assume that the inspection rate is larger than that of production. Thus, all imperfect quality items will be detected by the end of the production cycle. For such an augmented EPQ model, we first derive the new optimal production quantity assuming that the imperfect quality items are salvaged once at the end of every production cycle. Then, we extend this base model to allow for disaggregating the shipments of imperfect quality items during a single production run. Finally, we consider aggregating (or consolidating) the shipments of imperfect items over multiple production runs. Under both scenarios we derive closed-form expressions for both the economic production quantity and the batching policy, and show that our desegregation/consolidation schemes can lead to significant cost savings over the base model.     

Integer-ratio policies for distribution/inventory systems

We study a multistage distribution/inventory system with a central warehouse and N retailers. Customer demand arrives at each retailer at a constant rate. The retailers replenish their inventories from the warehouse, which in turn orders from an outside supplier. It is assumed that shortages are not allowed and lead times are negligible. The goal is to determine policies which minimize the overall cost in the system, that is, the sum of the costs at each facility consisting of a fixed charge per order and a holding unit cost. We propose a heuristic procedure to compute near-optimal policies. Computational results on several randomly generated problems are reported.     

An investigation of setup instability in non-stationary stochastic inventory systems

In stochastic inventory systems unfolding uncertainties in demand lead to the revision of earlier replenishment plans which in turn results in an instability or so-called system nervousness. In this paper, we provide the grounds for measuring system nervousness in non-stationary demand environments, and gauge the stability and the cost performances of (R,S) and (s,S) inventory policies. Our results reveal that, both the stability and the cost performance of inventory policies are affected by the demand pattern as well as the cost parameters, and the (R,S) policy has the potential to replace the cost-optimal (s,S) policy for systems with limited flexibility.     

Modified critical fractile approach for a class of partial postponement problems

This paper studies a single period partial postponement problem, which is motivated by an inventory planning problem encountered in Reebok NFL replica jersey supply chain. We consider a group of regular products each facing a random demand. There are two stocking options. One is to procure the regular products. The other is to stock a common component which can be customized later to any one of the regular products, after demand realization.We obtain an insightful interpretation of the optimality conditions for this class of problems, and use it to obtain rules of thumb that practitioners can incorporate into their inventory models to determine the stocking levels to minimize the supply chain cost. Instead of proposing a numerical procedure to obtain the optimal solution, we propose an adaptation of the classical critical fractile approach for this class of partial postponement problem. The closed-form formula obtained are surprisingly effective. Our numerical results suggest that this simple approach to inventory planning often comes close to the performance of the optimal solution obtained from numerical method.     

考虑质押物损耗的存货质押融资决策模型

质押物的完整性是关系存货质押融资业务能否顺利开展的重要问题,质押物损耗长期困扰各参与方,阻碍了存货质押融资业务的顺利开展。考虑质押物损耗和第三方物流企业对质押物监管所带来的损耗节约的基础上,研究了存货质押融资最优决策。研究表明:当零售商的初始现金余额很低时,零售商的最优订购数量随着初始现金余额的增加而减少,之后零售商最优订货量保持不变,直到初始现金余额大于某一水平时,零售商的最优订货量开始增加;在零售商贷款且不存在破产风险时,银行的收益随着零售商初始现金余额增加而减少;当风险估值是需求的增函数时,零售商的最优订货量随利率减少。     

Eliminating drift in inventory and order based production control systems

An Inventory and Order Based Production Control System lies at the heart of many commercial and bespoke ordering systems based on periodic review of stock and production targets. This simple and elegant control system works well, even when dealing with scenarios in which there are many competing value streams. However, such “interferences” inevitably cause some uncertainty in pipeline delivery times. We show via linear z-transform analysis that the consequences may include the possibility of inventory drift and instability. In this paper we establish the stability boundaries for such systems, and demonstrate an innovative method of eliminating inventory drift due to lead-time effect. This new principle is confirmed by simulation results.     

An integrated product planning model for pricing and bundle selection using Markov decision processes and data envelope analysis

When bundling products during the product planning stage, there are a number of possible combinations that can be offered to customers. Consider a firm that offers N distinctive products, then there are 2^N−(N+1) possible bundling combinations. Now, if we wish to make pricing and bundle selection decisions, keeping in my mind limited inventory and a finite time horizon, then the size of the state space could be very large and finding an optimal solution could be impossible. To tackle this issue, we formulate an integrated model that utilizes a Markov decision process and data envelope analysis. Bundle selections are made using data envelope analysis in each decision epoch. Once the efficient bundles are selected they are priced by solving a Markov decision process using dynamic programming. Numerical examples are solved to exhibit the model's potential in solving real-world problems.