A periodic review inventory model with loss-averse retailer,random supply capacity and demand

This paper studies a periodic review inventory model with random supply capacity and demand, where the retailer is loss-averse. For the single-period problem, it is shown that the retailer will not order unless the initial inventory level is less than a critical value, and the order-up-to level is generally not a constant. Moreover, the critical value and optimal order quantity are only dependent on the random demand and independent on the random capacity. We also investigate the impacts of loss aversion, price and cost on the optimal order quantity and maximum expected utility. Then a dynamic programming approach is used to analyse the retailer’s ordering policy in the multi-period case. In each period, if the initial inventory level is above a given value, an order will not be placed. Otherwise, the upper bound on the optimal order quantity is given. The numerical experiments are conducted to illustrate the optimal ordering policies.     

Measure of the bullwhip effect considering the market competition between two retailers

In order to consider the market competition, a new supply chain with one supplier and two retailers is established in this paper. Two retailers employ different AR(1) demand processes, respectively, and an order-up-to inventory policy characterises the inventory decisions. The bullwhip effect in this supply chain is measured under the moving average forecasting technique. We investigate the effects of the lead time, the span of forecast, market competition and the consistency of demand volatility on the bullwhip effect using the algebraic analysis and numerical simulation. Conclusions indicate that different factors lead to the bullwhip effect following different patterns in the supply chain. Moreover, some suggestions are present to help managers to control parameters that yield the lower bullwhip effect.     

An efficient procedure for non-stationary inventory control

We present an efficient solution method - Direct Derivative Estimation (DDE) - for computing optimal order-up-to levels for a discrete time non-stationary inventory control model. We generalize a number of non-stationary inventory control models (which underlie larger models of supply chains) that include forecast updates, seasonality, information sharing, and exchange-rate fluctuations. This procedure is different from the existing ones in that it computes, in a recursive manner, the derivative of the cost function and not the cost function itself. It can also handle a much wider variety of fluctuations in the problem parameters. In our computational testing it was found to be considerably faster than Dynamic Programming and Infinitesimal Perturbation Analysis.     

Can the desired service level be achieved when the demand and lost sales are unobserved?

In this paper we consider an inventory management problem in the retail industry with unobserved lost sales. The retailer does not know the demand for a particular product but she has access to Point-Of-Sale (POS) data. The retailer uses a fixed review period, order-up-to level system to control the inventory. The objective of the retailer is to achieve a pre-specified service level. We define the service level as the fraction of demand satisfied from inventory. However, due to the unobserved lost sales nature of the problem, the retailer cannot exactly measure the current service level. We propose a POS data-based mechanism for periodic updating of the order-up-to level. We show that the periodic updating approach yields the desired service level without using any inventory information. Once the periodic updating scheme converges, it also gives the actual demand distribution.     

Stockout propagation and amplification in supply chain inventory systems

We investigate the existence and magnitude of stockout propagation and stockout amplification in the context of supply chain inventory systems. Stockout amplification is a stage-to-stage increase in overall stockout rates. Stockout propagation is the tendency for stockout at one node to instigate a stockout at a neighbouring node and is conceptually related to the idea of cascading failures in physical systems, such as electrical power grids. We study these concepts in both upstream (‘supply side’) and downstream (‘demand side’) directions in the context of normal operating conditions for an adaptive R, S (periodic, order-up-to) inventory policy. We build a simulation model of a 5-stage serial supply chain that experiences normally distributed customer demands and gamma distributed lead times. We find that stockout propagation exists, but contrary to conventional wisdom, it occurs in the upstream direction. There is little indication that stockout propagation is occurring to any significant degree in the downstream direction. We also find stockout amplification occurring in the upstream direction in scenarios where more aggressively adaptive inventory parameter updating is performed. We discuss implications of this work in the areas of supply chain inventory modelling, ordering decisions, safety stock determination, and the use of adaptive inventory policies.     

Revisiting rescheduling: MRP nervousness and the bullwhip effect

We study the material requirements planning (MRP) system nervousness problem from a dynamic, stochastic and economic perspective in a two-echelon supply chain under first-order auto-regressive demand. MRP nervousness is an effect where the future order forecasts, given to suppliers so that they may plan production and organise their affairs, exhibits extreme period-to-period variability. We develop a measure of nervousness that weights future forecast errors geometrically over time. Near-term forecast errors are weighted higher than distant forecast errors. Focusing on replenishment policies for high volume items, we investigate two methods of generating order call-offs and two methods of creating order forecasts. For order call-offs, we consider the traditional order-up-to (OUT) policy and the proportional OUT policy (POUT). For order forecasts, we study both minimum mean square error (MMSE) forecasts of the demand process and MMSE forecasts coupled with a procedure that accounts for the known future influence of the POUT policy. We show that when retailers use the POUT policy and account for its predictable future behaviour, they can reduce the bullwhip effect, supply chain inventory costs and the manufacturer’s MRP nervousness.     

Analysis of a multi-component periodic review inventory system in an assembly environment

Inventory systems within production or distribution chains typically interact with preceding supply and successive demand processes. We observe that the more clearly the performance characteristics of an inventory system are described and communicated, the better the planning and configuration of the surrounding systems can be. This paper is thus dedicated to the analysis of a component inventory model that is typically found within multi-level supply chain structures. Operating on a discrete time axis, we assume that lead times are discretely distributed, and may also depend on preceding occurrences. Customer demand for the components emerges from a single end product that requires all of the components, and whose demand per period is continuously i.i.d. The components are replenished according to a periodic review order-up-to (r, S ^[i]) policy, where r is common to all components and S ^[i] may be selected individually. Particularly, we discuss the modeling of the lead time process and give a detailed analysis of the delay time process versus the successive level of the supply chain. The analysis is exact and has not so far been conducted on the delay time of a comparable system. We have given special emphasis to elaborating the relevant formulae and computational approaches to a level at which they are ready to be computed.     

An Inventory System with Periodic Regular Review and Flexible Emergency Review

We analyze a discrete time inventory model with two supply modes. Regular orders are placed periodically and arrive after a deterministic lead time. Emergency orders, characterized by a shorter deterministic lead time but higher purchase costs, can be placed in case of imminent stock-outs. We propose and analyze flexible order-up-to policies with emergency orders that arrive one, two,⃛, up to some fixed number of time units before a regular order arrives. Based on an approximate cost model, we develop two sets of approximate explicit optimality conditions. The first set of conditions are rather complicated, but simulation shows that they do lead to near-optimal policies in most cases. The second set of conditions are very simple and seem more practical, but they only lead to a near-optimal policy if there are enough emergency supply opportunities in a review period to prevent most backorders.     

Exploring nonlinear supply chains: the dynamics of capacity constraints

While most supply chain models assume linearity, real production and distribution systems often operate in constrained contexts. This article aims to analyse the consequences of capacity limits in the order-up-to replenishment policy with minimum mean squared error forecasting under independently and identically distributed random demand. Our study shows that the impact of this nonlinearity is often significant and should not be ignored. In this regard, we introduce the concept of a settling capacity, which informs when our knowledge from a linear analysis is a reasonable approximation in a nonlinear context. If the available capacity is less than the settling capacity, the nonlinear effects can have a significant impact. We compare the Bullwhip Effect and Fill Rate in constrained contexts to well-established results for linear supply chains. We reveal the capacity limit acts as a production smoothing mechanism, at the expense of increasing inventory variability. We proceed to analyse the economic consequences of the capacity constraint and show that it can actually reduce costs. We provide an approximate solution for determining the optimal capacity depending on the demand, the unit costs and the lead time.     

The multilocation transshipment problem

We consider a supply chain, which consists of several retailers and one supplier. The retailers, who possibly differ in their cost and demand parameters, may be coordinated through replenishment strategies and transshipments, that is, movement of a product among the locations at the same echelon level. We prove that in order to minimize the expected long-run average cost for this system, an optimal replenishment policy is for each retailer to follow an order-up-to S policy. Furthermore, we demonstrate how the values of the order-up-to quantities can be calculated using a sample-path-based optimization procedure. Given an order-up-to S policy, we show how to determine an optimal transshipment policy, using a linear programming/network flow framework. Such a combined numerical approach allows us to study complex and large systems.     

Economic production quantity model for deteriorating inventory with random machine unavailability and shortage

Both random machine unavailability and shortage are common occurrences in manufacturing industries. In this paper, machine unavailability is considered in deriving an economic production quantity (EPQ) model for deteriorating items. We develop four EPQ models for deteriorating inventory with random machine unavailability and shortage. The study considers lost sales, backorder and two kinds of machine unavailability distributions. Numerical examples are provided to illustrate the theory. Key parameter changes that affect costs are shown in the sensitivity analysis. From the results of the sensitivity analysis, it is shown that the random machine unavailability parameter and holding cost have significant effects on the optimal total cost and production down-time.     

An optimised target-level inventory replenishment policy for vendor-managed inventory systems

In vendor-managed inventory (VMI) systems the supplier is responsible for replenishing customers and for deciding when and how much to deliver. One of two inventory policies is typically employed by the supplier. The first one, called the maximum level (ML) policy, gives full freedom to the supplier to deliver any quantity as long as it respects customer inventory capacities. The alternative, which is more constrained, is called the order-up-to (OU) policy. It states that the supplier has to bring the customer inventory up to its maximum capacity level upon delivery. We propose a new tactical policy in the context of VMI systems, called optimised target-level (OTL), under which when the supplier visits a customer, the quantity delivered is such that the final inventory will always be at the same customer-dependent OTL. We perform a computational evaluation of this new policy against both traditional strategies on benchmark instances. We show that it yields lower costs and inventory levels than the OU policy, and is only marginally more expensive than the ML policy, while being easier to implement.     

A note on the can-order policy for the two-item stochastic joint-replenishment problem

In this paper, a two-item continuous-review inventory system is studied. Demands for item 1 and item 2 occur at epochs generated by independent Poisson processes. In addition to the standard cost structure, there is economy of scale in joint replenishment. For the continuous joint replenishment problem, the literature proposes the can-order policy. Under this policy, an order is triggered by item 1 at its demand epoch, when its inventory position falls to its reorder level. In this situation, if the inventory position of item 2 is at or below its “can-order” level, item 2 is also included in this order and a discounted fixed ordering cost is charged for it. As a result, the inventory positions of both items are raised to their respective order-up-to levels. Reciprocally, the same procedure is valid at the demand epoch of item 2. In this study, this two-item inventory system is modeled as a semi-Markov decision process and a simple enumeration algorithm is proposed for its solution. We show that previous formulations of the problem do not necessarily converge to the best can-order policy by providing numerical examples.     

A Note on Numerical Evaluation of Compound Poisson Distributions

The discrete compound Poisson distribution can be used to model a number of practical phenomena. In this note we draw attention to a recursion formula which may be used to numerically evaluate the compound Poisson probabilities. Computation of each probability point requires the evaluation of only one simple convolution, and can be carried out in most cases by hand or with the aid of a small calculator. The procedure is illustrated by two simple inventory system examples.     

The impact of consumer price forecasting behaviour on the bullwhip effect

As prices fluctuate over time, a strategic consumer may buy more in advance to reduce his or her future needs in anticipation of higher prices in the future, or may choose to postpone a purchase in anticipation of lower prices in the future. We investigate the bullwhip effect from a consumer price forecasting behavioural perspective in the context of a simple two-level supply chain composed of a supplier and a retailer. We consider two different forms for the demand function – linear and iso-elastic demand functions, both depending on the prices in multiple periods. Assuming that the retailer employs an order-up-to inventory policy with exponential smoothing forecasting technology, we derive analytical expressions for the bullwhip effect under the two demand functions, and extend the results to the multiple-retailer case. We find that consumer forecasting behaviour can reduce the bullwhip effect, most significantly when the consumer sensitivity to price changes is medium (approximately 0.5) for both the demand forms. In addition, for iso-elastic demand, the mitigation of the bullwhip effect induced by consumer price forecasting behaviour becomes more significant as the product price sensitivity coefficient and standard deviation of the price decrease. These findings are applicable to the development of managerial strategies by supply chain members that are conducive to bullwhip effect reduction through customer behaviour.     

Analysis of supply contracts with total minimum commitment

In this paper we analyze a supply contract for a single product that specifies that the cumulative orders placed by a buyer, over a finite horizon, be at least as large as a (contracted upon) given quantity. We assume that the demand for the product is uncertain, and the buyer places orders periodically. We derive the optimal purchase policy for the buyer for a given total minimum quantity commitment and a discounted price. We show that the policy is characterized by the order-up-to levels of the corresponding finite horizon and a single-period standard newsboy problem with no commitment but with discounted price. We show that this policy can be computed easily. We can use this to evaluate any discount schedule characterized by a set of (price discount, minimum commitment) pairs.     

The routed inventory pooling problem with multiple lateral transshipments

We consider a single item, single-period inventory problem with two identical retailers who can pool stocks at multiple predetermined points in time. Since routing decisions are often taken at the multi-item level in practice, we use a predetermined route for redistribution at the single-item level. The objective is to determine order-up-to levels and transshipment decisions that minimise the sum of inventory holding, backorder and fixed as well as variable transshipment costs. We present a Dynamic Programming formulation to solve the problem and we compare the performance of our routed pooling policy to situations without pooling and with complete pooling in an extensive numerical study. The results reveal when routed pooling performs well and when multiple pooling moments provide significant benefits.     

Optimization and optimality of a joint pricing and inventory control policy in periodic-review systems with lost sales

This paper studies a periodic review dynamic pricing and inventory control problem with fixed ordering cost included. Demand is uncertain and price-sensitive in a general form. At the end of each period, all unmet demand is lost. We focus on the optimization and optimality of an (s, S, p)-type policy, which operates as follows: whenever the inventory level is less than or equal to s, an order is placed to order-up-to level S; when the inventory level is larger than s, no order is issued; the price p is specified by the inventory level. We further numerically investigate the optimal solutions and their sensitivity to cost parameters and demand uncertainty.     

The Generalized Newsboy Problem, Contract Negotiations and Secondary Vendors

The single period inventory model, known as the newsboy problem, has been extended by considering the supply of inventory as a random variable. A rule for acceptance of inventory from the primary vendor (supplier) is given. Analyses for the newsboy's use in contract negotiations with his supplier are presented. The analyses ensure that the newsboy suffers no decrease in expected revenue by determining values for fixed and proportional penalties when shortages exist. In addition, the unit cost that the newsboy can afford to pay to purchase the excess supply is determined so that there is no decrease in expected revenue. The optimal use of secondary vendors is derived when the specific amount of shortage is known and when the specific amount (if any) of shortage is not known. Illustrative distributions for supply and demand and specific numerical cases are given.     

考虑残次品和碳排放的供应链多级库存的EOQ模型

针对供应链多级库存系统,探析残次品与碳排放对该系统的影响作用与供应链库存持有成本、订购成本、运输成本和检查成本的构成函数,建立多级库存的EOQ模型,提供考虑残次品和碳排放的供应链订货经济批量和库存控制策略,并用算例验证了模型的有效性。