Inventory management for dual sales channels with inventory-level-dependent demand

This paper studies the inventory management problem of dual channels operated by one vendor. Demands of dual channels are inventory-level-dependent. We propose a multi-period stochastic dynamic programming model which shows that under mild conditions, the myopic inventory policy is optimal for the infinite horizon problem. To investigate the importance of capturing demand dependency on inventory levels, we consider a heuristic where the vendor ignores demand dependency on inventory levels, and compare the optimal inventory levels with those recommended by the heuristic. Through numerical examples, we show that the vendor may order less for dual channels than those recommended by the heuristic, and the difference between the inventory levels in the two cases can be so large that the demand dependency on inventory levels cannot be neglected. In the end, we numerically examine the impact of different ways to treat unmet demand and obtain some managerial insights.     

Bayesian demand updating in the lost sales newsvendor problem: A two-moment approximation

We consider Bayesian updating of demand in a lost sales newsvendor model with censored observations. In a lost sales environment, where the arrival process is not recorded, the exact demand is not observed if it exceeds the beginning stock level, resulting in censored observations. Adopting a Bayesian approach for updating the demand distribution, we develop expressions for the exact posteriors starting with conjugate priors, for negative binomial, gamma, Poisson and normal distributions. Having shown that non-informative priors result in degenerate predictive densities except for negative binomial demand, we propose an approximation within the conjugate family by matching the first two moments of the posterior distribution. The conjugacy property of the priors also ensure analytical tractability and ease of computation in successive updates. In our numerical study, we show that the posteriors and the predictive demand distributions obtained exactly and with the approximation are very close to each other, and that the approximation works very well from both probabilistic and operational perspectives in a sequential updating setting as well.     

Estimating need and demand for prehospital care

Models estimating demand and need for emergency transportation services are developed. These models can provide reliable estimates which can be used for planning purposes, by complementing and/or substituting for historical data. The model estimating demand requires only four independent variables: population in the area, employment in the area, and two indicators of socioeconomic status which can be obtained from census data. The model can be used to estimate demand according to 4 operational categories and 11 clinical categories. The parameters of the model are calibrated with 1979 data from 82 ambulance services covering over 200 minor civil divisions in Southwestern Pennsylvania. This model was tested with data from another 55 minor civil divisions, also in Southwestern Pennsylvania, and it provided good estimates to total demand. The model to estimate need evolves from the demand model. It enables planners to estimate unmet need occurring in the region. The effect of emergency transportation service (ETS) provider characteristics on demand was also investigated. Statistical tests show that, for purposes of forecasting demand, when the sociodemographic factors are taken into account, provider characteristics are not significant.     

On two-echelon inventory systems with Poisson demand and lost sales

We consider a two-echelon, continuous review inventory system under Poisson demand and a one-for-one replenishment policy. Demand is lost if no items are available at the local warehouse, the central depot, or in the pipeline in between. We give a simple, fast and accurate approach to approximate the service levels in this system. In contrast to other methods, we do not need an iterative analysis scheme. Our method works very well for a broad set of cases, with deviations to simulation below 0.1% on average and below 0.36% for 95% of all test instances.     

Multi-period empty container repositioning with stochastic demand and lost sales

This paper considers repositioning empty containers between multi-ports over multi-periods with stochastic demand and lost sales. The objective is to minimize the total operating cost including container-holding cost, stockout cost, importing cost and exporting cost. First, we formulate the single-port case as an inventory problem over a finite horizon with stochastic import and export of empty containers. The optimal policy for period n is characterized by a pair of critical points (A _n , S _n ), that is, importing empty containers up to A _n when the number of empty containers in the port is fewer than A _n ; exporting empty containers down to S _n when the number of empty containers in the port is more than S _n ; and doing nothing, otherwise. A polynomial-time algorithm is developed to determine the two thresholds, that is, A _n and S _n , for each period. Next, we formulate the multi-port problem and determine a tight lower bound on the cost function. On the basis of the two-threshold optimal policy for a single port, a polynomial-time algorithm is developed to find an approximate repositioning policy for multi-ports. Simulation results show that the proposed approximate repositioning algorithm performs very effectively and efficiently.     

Estimating market shares in each market segment using the information entropy concept

Sales data of a certain product for the various competitors are usually available at the aggregate level. However these data give no clue to the heterogeneities in the sales pattern across different market segments. Heterogeneities are caused by different purchasing behavior in each market segment; as a purchaser in a segment will be attracted to the attributes of the product most important to that segment. This concept can be formalized via a simple attraction model that utilizes an elasticity measure for each quality or price attribute [G.S. Carpenter, L.G. Cooper, D.M. Hanssens, D.F. Midgley, Modeling asymmetric competition, Marketing Science 7 (4) (1998) 393–412]. Assessment of these elasticities is not difficult since customer response – in each market segment – to perception of quality and price is tracked by most firms [J. Ross, D. Georgoff, A survey of productive and quality issues in manufacturing. The state of the industry, Industrial Management 3 (5) (1991) 22–25]. This paper attempts to formulate a generic framework based on the information entropy concept that utilizes such an attraction model to estimate competitors’ sales in each market segment.     

Estimating the demand pattern for C category items

This paper presents a methodology for estimating the demand pattern for the slowest-moving C category inventory items. The methodology uses an aggregation-by-items scheme and a forecasting procedure based on conditional demand analysis whereby aggregate demand is assumed to be an arbitrarily mixed, heterogeneous Poisson distribution. Practical aspects of demand heterogeneity, parameter estimation and model implementation are illustrated using a case study in retail inventory planning and control.     

Estimating vaccine coverage by using computer algebra

Email: altmannd{at}rki.deEmail: altmann{at}mathematik.hu-berlin.de The approach of N Gay for estimating the coverage of a multivalentvaccine from antibody prevalence data in certain age cohortsis complemented by using computer aided elimination theory ofvariables. Hereby, Gay's usage of numerical approximation canbe replaced by exact formulae which are surprisingly nice, too.     

A two-step method for forecasting spare parts demand using information on component repairs

Forecasting spare parts demand is notoriously difficult, as demand is typically intermittent and lumpy. Specialized methods such as that by Croston are available, but these are not based on the repair operations that cause the intermittency and lumpiness of demand. In this paper, we do propose a method that, in addition to the demand for spare parts, considers the type of component repaired. This two-step forecasting method separately updates the average number of parts needed per repair and the number of repairs for each type of component. The method is tested in an empirical, comparative study for a service provider in the aviation industry. Our results show that the two-step method is one of the most accurate methods, and that it performs considerably better than Croston’s method. Moreover, contrary to other methods, the two-step method can use information on planned maintenance and repair operations to reduce forecasts errors by up to 20%. We derive further analytical and simulation results that help explain the empirical findings.     

Estimating the solutions of linear stochastic equations by the information criterion

The information criterion is applied to estimate the state of linear dynamic systems in the presence of random disturbances on the input and the output. The problem of finding the best information estimator is reduced to an optimal control problem. An expression for the maximum mean information is derived. Recursive Kalman-Bucy filters are constructed.Translated from Vychislitel'naya i Prikladnaya Matematika, No. 60, pp. 100–107, 1986.     

Estimating the lead-time demand distribution when the daily demand is non-normal and autocorrelated Hon-Shiang LAU

This paper develops convenient formulas for estimating the probability distribution of lead time demand when the inventory item's daily demand follows a probability distribution of any arbitrary shape, and when simple univariate models can be found for the autocorrelated daily demand series. Numerical examples are presented to illustrate the use of these formulas. The numerical results also indicate that significant error may be incurred when inventory decisions are made without proper consideration of the autocorrelations or the arbitrary distribution shapes of daily demands.     

Estimating the demand distributions of single-period items having frequent stockouts

Very often, the service level of a single-period newsboy-type product is set at such a low level that: (i) stockouts occur in the majority of the periods, and (ii) a large right-hand side of the empirical demand distribution is never observable. This paper reports a practical approach for estimating the periodic-demand distribution of such a product. The approach has three components: (i) using the non-parametric ‘product limit’ method to estimate the fractiles of the observable left-hand side of the empirical distribution; (ii) using a subjective approach and an ‘extrapolation of hourly sales’ approach to ‘fill in’ the missing right-hand side of the empirical distribution; (iii) fitting the estimates obtained in the preceding two components to a Tocher curve — which can handle the diversity of shapes of a realistic demand distribution and is also computationally very convenient for subsequent calculations for production/inventory decisions. The entire approach is shown to be simpler but more powerful than existing alternatives for the problem.     

Optimal contracts under endogenous demand information acquisition

We consider a supplier selling to a retailer who decides whether or not to exert a fixed cost to acquire private demand information. We show that quantity discounts, established by the extant literature to be the optimal response for exogenous information acquisition, are also optimal under the setting with endogenous information acquisition.     

Using imperfect advance demand information in forecasting

Email: t.tan{at}tue.nl Received on 4 January 2007. Accepted on 11 January 2008. In this paper, we consider the demand-forecasting problem ofa make-to-stock system operating in a business-to-business environmentwhere some customers provide information on their future orders,which are subject to changes in time, hence constituting imperfectadvance demand information (ADI). The demand is highly volatileand non-stationary not only because it is subject to seasonalityand changing trends but also because some individual clientdemands have significant influence on the total demand. In suchan environment, traditional forecasting methods may result inhighly inaccurate forecasts, since they are mostly developedfor the total demand based only on the demand history, not makinguse of demand information and ignoring the effects of individualorder patterns of the customers. We propose a forecasting methodologythat makes use of individual ordering pattern histories of theproduct–customer combinations and the current build upof orders. Moreover, we propose making use of limited judgementalupdates on the statistical forecasts prior to the use of ADI.     

Delayed demand information and dampened bullwhip effect

This paper considers an inventory setting in which the historical data used for demand forecasting is delayed. When the replenishment is controlled via an order-up-to policy, we show that such delays reduce the variability of the order history and dampen the bullwhip effect. We discuss the intuition behind this result.     

Exploiting timely demand information to reduce inventories

Electronic Data Interchange (EDI) and related technologies are making it less expensive to frequently transmit demand information up the supply chain from the point of sale. However, little is known about how the various participants in the supply chain might exploit more timely demand information. This research is intended to be an opening contribution to the analytical study of timely demand information. We study a simple two-level system composed of a supplier (or component plant) and a single customer (or final assembly plant). The two factories use a standard periodic base-stock policy for one particular item, but the equal-length production cycles of the two factories do not necessarily coincide. Both participants hold inventories to buffer the effects of uncertain orders and uncertain deliveries. The supplier is in turn supplied by a source with infinite capacity. Final demand occurs only at the final assembly plant, which communicates demand over its order cycle to the component plant. If the component plant begins a production cycle between orders from the final assembly plant, it is uninformed about some number of days' demand which has been observed at the customer but not yet communicated. Any technology, such as EDI, that makes it cheaper for the customer to communicate demand as it occurs allows the supplier to base its decisions on more accurate information. With more accurate demand information the supplier could reduce inventories, or improve the reliability of its deliveries to its customer, or both. The customer, in turn, could reduce its inventories if its supplier were more reliable. We examine the changes brought about by the exchange of timely demand information in inventories and service levels at both the supplier and customer. Our results show that inventory-related benefits are particularly sensitive to demand variability, the service level provided by the supplier, and the degree to which the order and production cycles are out of phase.     

Salespeople's management of customer information: Impact on optimal territory and sales force sizes

Collecting and processing territory and account information are major aspects of a salesperson's task. To a large extent, salespeople's effectiveness depends on the amount and quality of the market information available to them (such as their customers' needs and potential, the likelihood of getting an order after some contact time, etc.). Although they are not always easy to disentangle, these information gathering and processing activities on one hand, and the effective contact time devoted to selling to clients and prospects on the other, vie for the limited time resources available to a salesperson. This paper provides a simple statistical procedure for estimating the costs of information gathering and processing by a salesperson. The model can be used by management for estimating the most profitable territory size to be assigned to each salesperson, and consequently for estimating the optimal sales force size.     

Ordering and pricing of service products in an advance sales system with price-dependent demand

This investigation addresses a service inventory control problem in which a firm orders and sells a service which will be used or consumed by customers on a specific future date. The firm sells the product through an advance booking system, aiming to optimize product price to maximize the total expected profit. Considering situations in which product demand is price-dependent and customers with reservations may cancel advance orders, this work develops a continuous-time model to simultaneously determine the order quantity and selling prices. The analytical results reveal that the optimal ordering quantity and prices are derived via closed-form solutions. In addition, sensitivity analysis of the optimal prices with respect to the system parameters is also conducted to illustrate optimal decision characteristics.     

Contracting with asymmetric demand information in supply chains

We solve a buyback contract design problem for a supplier who is working with a retailer who possesses private information about the demand distribution. We model the retailer’s private information as a space of either discrete or continuous demand states so that only the retailer knows its demand state and the demand for the product is stochastically increasing in the state. We focus on contracts that are viable in practice, where the buyback price being strictly less than the wholesale price, which is itself strictly less than the retail price. We derive the optimal (for the supplier) buyback contract that allows for arbitrary allocation of profits to the retailer (subject to the retailer’s reservation profit requirements) and show that in the limit this contract leads to the first-best solution with the supplier keeping the entire channel’s profit (after the retailer’s reservation profit).     

The value of information for price dependent demand

Predicting demand and determining optimal pricing are essential components of operations management. It is often useful to think in terms of the price elasticity of demand when reasoning about the demand curve. Firms wishing to invest in demand prediction and information gathering should reason about the relationship between the expected value of perfect information (EVPI) on demand and demand elasticity. Should firms pay more/less for information on demand if elasticity is high/low? Furthermore, when considering different product prices, correlation may exist between demand at different prices. Should firms pay more/less for information if the correlation between demand at different prices is high or low? This paper derives analytic and numeric results to answer these questions. We start with the assumption that demand is uncertain and follows a uniformly distributed band around a deterministic demand curve where the upper and lower bounds of the demand distribution vary with price. This formulation enables a closed form expression for EVPI that provides a useful benchmark. We find nuanced behavior of EVPI that depends on both the elasticity and the initial price preference. The EVPI approaches zero as elasticity increases (decreases) for a firm that initially prefers the low (high) price. Numerical results using the truncated normal and beta distributions relax assumptions about the uniform distribution and show EVPI is similar when the distribution variances are similar. Finally, we relax the assumption of perfect information and show the expected value of imperfect information (EVOI) follows similar patterns as EVPI with respect to demand elasticity.