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.     

Coordinating a one-warehouse N-retailer distribution system under retailer-reporting

We consider a distribution system consisting of a System Administrator (SA) and N retailers managed under dynamic allocation and static routing policies. We study the system under two scenarios: (a) the up-to-date information scenario, under which the SA has access to the inventory levels at the retailers in real time and makes the replenishment and allocation decisions based on these information, aiming to minimize total supply chain cost; (b) the retailer-reporting scenario, under which the SA makes the replenishment and allocation decisions based on the information reported by the retailers. Under the retailer-reporting scenario, the retailers compete against each other by reporting their inventory levels falsely to get the desired allocations, which minimize the retailers’ expected cost. For N=2 retailers, we show that, in general, reporting truthfully does not minimize either retailer's expected cost/cycle over their respective replenishment/allocation cycles, and that the sum of the retailers' expected cost/cycle is strictly greater than in the up-to-date information scenario. We then show that by including a particular transfer-payment between the two retailers, truth-telling is a Bayesian equilibrium, and that the same system expected cost/period as in up-to-date information scenario can be achieved.     

A queueing-inventory system with two classes of customers

We consider a queueing-inventory system with two classes of customers. Customers arrive at a service facility according to Poisson processes. Service times follow exponential distributions. Each service uses one item in the attached inventory supplied by an outside supplier with exponentially distributed lead time. We find a priority service rule to minimize the long-run expected waiting cost by dynamic programming method and obtain the necessary and sufficient condition for the priority queueing-inventory system being stable. Formulating the model as a level-dependent quasi-birth-and-death (QBD) process, we can compute the steady state probability distribution by Bright–Taylor algorithm. Useful analytical properties for the cost function are identified and extensive computations are conducted to examine the impact of different parameters to the system performance measures.     

Near cost-optimal inventory control policies for divergent networks under fill rate constraints

We deal with the optimisation of stock levels in general divergent networks under a periodic review, order-up-to ($\text{R,}\text{S}$) policy. The goal is to attain target fill rates, while the total holding costs in the entire network are minimised. To this end, we first present a method for the fast calculation of the control parameters, given central and intermediate stock levels. Next we develop an approximate procedure to determine stock levels sequentially. Extensive numerical experimentation shows that this procedure yields satisfactory results. It also shows that significant stocks at intermediate stockpoints are only useful if unit holding costs in these stockpoints are considerably less than in the end stockpoints that deliver directly to the final customers.     

Logistics scheduling to minimize inventory and transport costs

We study a logistics scheduling problem where a manufacturer receives raw materials from a supplier, manufactures products in a factory, and delivers the finished products to a customer. The supplier, factory and customer are located at three different sites. The objective is to minimize the sum of work-in-process inventory cost and transport cost, which includes both supply and delivery costs. For the special case of the problem where all the jobs have identical processing times, we show that the inventory cost function can be unified into a common expression for various batching schemes. Based on this characteristic and other optimal properties, we develop an O(n) algorithm to solve this case. For the general problem, we examine several special cases, identify their optimal properties, and develop polynomial-time algorithms to solve them optimally.     

Coordination in retailer-carrier channels for long term planning

We study a retailer-carrier channel for the purpose of long term planning and coordination. Here, the term channel represents the business interaction between the retailer and the carrier. The retailer sells a particular item with price-dependent demand, whereas the carrier is responsible for transporting the item to the retailer's site. We characterize the profit functions of each channel member as well as the total channel profit. We consider two specific channel structures: (i) the centralized channel and (ii) the decentralized channel. Under the first channel structure, the goal is to set the retail price so as to maximize the total channel profit. Under the latter, the carrier and the retailer choose their own policy parameters, i.e., the freight rate for the carrier and the retail price for the retailer, so as to maximize their individual profits. We model the decentralized channel as a Stackelberg Game and propose a coordination mechanism between the retailer and the carrier in which the retailer signals a price multiplier to the carrier. We illustrate that this mechanism could provide win-win solutions for both parties and present analytical and numerical results on the efficiency of channel coordination. We demonstrate that coordination in retailer-carrier channels can be as promising as supplier-retailer channels. We also discuss the effects of retailer-carrier coordination on inventory levels.     

Inventory management with advance supply information

We consider a single stage inventory system with stochastic capacity. The manager receives forecasts from the upstream source about future capacity availability within an information horizon, referred to as “advance supply information”. We study two main questions: (i) How can advance supply information be utilized when making replenishment decisions? (ii) What is the value of such information sharing, compared to a fixed base-stock policy? We show that state-dependent base-stock policies are optimal. We develop easily computable and implementable heuristic policies. We numerically test the accuracy of the approximations and analyze the value of collaboration under various business scenarios.     

Base-stock policies for the lost sales inventory system with Poisson demand and Erlangian lead times

The base-stock policies for the studied inventory system can easily be evaluated through Erlang's loss formula when the lead times are mutually independent. This is often the case only if the base-stock $S$ is one. If $S$ is larger than one, the Erlangian lead times become stochastically dependent under the realistic assumption that the replenishment orders do not cross in time. We make this assumption and show for any positive $S$ that the number of replenishment orders outstanding has an equilibrium distribution which is a slightly modified truncated version of a negative binomial distribution. It turns out to be easy to compute the stock-out frequency recursively for $S=1,2,\dots .$ For each $S$, the average stock can be specified in terms of this frequency. We prove that the frequency is convex in $S$. It is therefore straightforward to compute the base-stock for which the average cost is minimized and to compute the minimum average cost. Our numerical study illustrates that the minimum average cost is very sensitive to the shape parameter describing the Erlangian lead times, which is in sharp contrast to the complete insensitivity when lead times are independent.     

Economic lot-sizing and dynamic quantity competition

We study a problem of dynamic quantity competition in continuous time with two competing retailers facing different replenishment cost structures. Retailer 1 faces fixed ordering costs and variable procurement costs and all inventory kept in stock is subject to holding costs. Retailer 2 only faces variable procurement costs. Both retailers are allowed to change their sales quantities dynamically over time. Following the structure of the economic order quantity (EOQ) model, retailer 1 places replenishment orders in batches and retailer 2 follows a just-in-time (JIT) policy. The objective of both retailers is to maximize their individual average profit anticipating the competitor's replenishment and output decisions. The problem is solved by a two-stage hierarchical optimization approach using backwards induction. The second-stage model is a differential game in output quantities between the two retailers for a given cycle length. At the first stage, the replenishment policy is determined. We prove the existence of a unique optimal solution and derive an open-loop Nash equilibrium. We show that both retailers follow contrary output strategies over the order cycle. The EOQ retailer, driven by inventory holding costs, decreases his market share whereas the output of the JIT retailer increases. Moreover, depending on the cost structure, the EOQ retailer might partially be a monopolist. At the first stage, the EOQ retailer determines the cycle length, anticipating the optimal output trajectories at the second stage.     

A Continuous-Time Examination of End-of-Life Parts Acquisition With Limited Customer Information

Manufacturers often encounter difficulties in supplying an adequate number of spare parts for a product that is in its post-production phase. Current solution methods for optimal spare part order amounts use assumptions that require knowledge of the number of products in operation at any time t. In the business situation presented here, the manufacturer only has knowledge of the part and product failure rates and a probability distribution on the number of products still in operation; unless the manufacturer and customer have a close working relationship, it is likely that this information is all that the manufacturer will have available in a typical business environment. We model time-discounted revenues and costs as a function of the spare parts order amount. We consider two different versions of the problem—incremental replenishment and no replenishment—that operate by determining the rate of change of the total profit with respect to the order amount q.     

Replenish-up-to inventory control policy with random replenishment intervals

The multi period inventory problems have been studied under two main assumptions. Continuous review assumption where an order can be made at any time depending on inventory position and periodic review assumption where an order can be initiated only at discrete time epochs. In this study, we analyze a multi period inventory problem that falls under neither of these two assumptions. In the case we consider, there are periodic replenishments but the replenishment intervals are taken to be i.i.d. random variables. This setting represents the real life cases where a supplier visits a retailer with random inter arrival times and the retailer replenishes his inventories based on a replenish-up-to-level inventory control policy. We also assume that only a certain fraction of unmet demand is backordered and the rest of it is lost.In this setting under general distribution between replenishment epochs, we show the concavity of the expected profit function and give the condition that must hold for the optimal replenish-up-to-level. We also present the specific solutions and analysis under two different distributions, namely, uniform and exponential distributions, together with some numerical examples.     

Machine repairing models for production systems

In this paper we consider manufacturing systems of m identical unreliable machines producing one type of product. The operating time of each machine is exponentially distributed. The repairing process of a machine requires more than one phase. In each phase, the repair time is exponentially distributed and more than one operator may be required for fixing a broken machine. Here we consider two models of manufacturing systems. In the first model, there are r operators assigned in one server to repair a broken machine. The repairing rate in each phase depends on the number of operators there. This is a generalized model discussed in Buzacott and Shanthikumar [11]. We then consider a two-phase repairing model. Two groups of operators are assigned in the two phases. Each operator can handle one broken machine in each phase individually. This model is a generalization of Eben–Chaime's model [8]. Average profits are derived for both models and can be optimized by suitable allocation of the number of machines and operators in the systems.     

Integrated inventory models with controllable lead time and backorder discount considerations

Many companies use time as a means of differentiating themselves in the marketplace. In many literatures, the controllable lead time is regarded as a decision variable and decomposed into several components, each having a crashing cost function for the respective reduced lead time. When an item is out of stock, the loyal, patient and captive customers will wait until the outstanding orders arrive and are served from them. To compensate for the inconvenience of backordering and to secure orders, the supplier may offer a price discount on the stockout item. In this paper, an integrated inventory system in which shortage is allowed and both lead time and backordering are negotiable is investigated. The lead time crashing cost is represented as a function of reduced lead time and the quantities in the orders. There are two inventory models proposed in the paper, one with normally distributed demand, and another with generally distributed demand.     

A perishable inventory model with Markovian renewal demands

In the inventory model, people usually assume that the inter-demand time is independently identical distributed which may not be true in reality. Here we study an (s,S) continuous review model for items with an exponential random lifetime and a general Markovian renewal demand process. By constructing Markovian renewal equations, we derive the mean and the variance of the reorder cycle time and lead to a simple expression for the total expected long run cost rate. The numerical results illustrate the system behavior and lead to managerial insights into controlling such inventory systems.     

A local search algorithm for the optimization of the stochastic economic lot scheduling problem

This paper describes a model for the stochastic economic lot scheduling problem (SELSP) and a Local Search heuristic to find close to optimal solutions to this model. The SELSP considers multiple products, which have to be scheduled on a single facility with limited capacity and significant setup times and costs. The demand is modeled as a stationary compound renewal process. The objective is to find a schedule that minimizes the long-run average costs for setups and inventories while satisfying a given fill rate. We use a cyclic scheduling approach in which the individual cycle time of each product is a multiple of some basic period (fundamental cycle).For the deterministic version of the SELSP, efficient heuristics have been developed which guarantee the feasibility of the solution by adding an additional constraint to the problem. In our case this is not sufficient, because for the calculation of the average inventory levels and fill rates we need to develop a schedule with detailed timing of the lots. We present an efficient heuristic for this scheduling problem, which can also be used to check the feasibility of the solution. Thereby, the most time-consuming step (the calculation of average inventory levels and fill rates) is only performed for a limited set of candidates.The algorithm was tested on deterministic benchmark problems from literature and on a large set of stochastic instances. We report on the performance of the heuristic in both cases and try to identify the main factors influencing the objective.     

Joint pricing and inventory control for non-instantaneous deteriorating items with partial backlogging and time and price dependent demand

In this paper, a joint pricing and inventory control for non-instantaneous deteriorating items is developed. We adopt a price and time dependent demand function. Shortages is allowed and partially backlogged. The major objective is to determine the optimal selling price, the optimal replenishment schedule and the optimal order quantity simultaneously such that, the total profit is maximized. We first show that for any given selling price, optimal replenishment schedule exists and unique. Then, we show that the total profit is a concave function of price. Next, we present a simple algorithm to find the optimal solution. Finally, we solve a numerical example to illustrate the solution procedure and the algorithm.     

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.     

Production planning and inventory allocation of a single-product assemble-to-order system with failure-prone machines

We consider the optimal production and inventory allocation of a single-product assemble-to-order system with multiple demand classes and lost sales. Each component is replenished by a dedicated machine that is subjected to unpredictable breakdowns. We find that the machine state not only influences the production and allocation decisions on its own component but also influences the decisions on the other components. Specifically, the optimal component production policy is a base-stock policy with the base-stock level non-decreasing in the inventory levels of the other components and the states of the other machines. The optimal component allocation policy is a rationing policy with the rationing level non-increasing in the inventory levels of the other components, the states of the other machines, and its own machine state. We use an exponential distribution to approximate the distribution of the total processing times and propose two heuristic policies to address the production and allocation decisions. The importance of taking machine failures into consideration is revealed through computational experiments.