Quadratic Approximation of the Newsvendor Problem with Imperfect Quality

The paper presents a newsvendor problem in a fuzzy environment by introducing product quality as a fuzzy variable, and product demand as a probability distribution in an economic and supply chain management environment. In order to determine the optimal order quantity, a methodology is developed where the solution is achieved using a fuzzy ranking method combined with a quadratic programming problem approximation. Numerical examples are provided and compared in both situations, namely fuzzy and crisp. The results of these numerical examples show that the decision maker has to order a higher quantity when product quality is a fuzzy variable. The model can be useful for real world problems when historical data are not available

A Novel Method for Optimal Solution of Fuzzy Chance Constraint Single-Period Inventory Model

A method is proposed for solving single-period inventory fuzzy probabilistic model (SPIFPM) with fuzzy demand and fuzzy storage space under a chance constraint. Our objective is to maximize the total profit for both overstock and understock situations, where the demand D~j for each product j in the objective function is considered as a fuzzy random variable (FRV) and with the available storage space area W~, which is also a FRV under normal distribution and exponential distribution. Initially we used the weighted sum method to consider both overstock and understock situations. Then the fuzziness of the model is removed by ranking function method and the randomness of the model is removed by chance constrained programming problem, which is a deterministic nonlinear programming problem (NLPP) model. Finally this NLPP is solved by using LINGO software. To validate and to demonstrate the results of the proposed model, numerical examples are given

Evaluating alternative estimators for optimal order quantities in the newsvendor model with skewed demand

This paper considers the classical Newsvendor model, also known as the Newsboy problem, with the demand to be fully observed and to follow in successive inventory cycles one of the Exponential, Rayleigh, and Log-Normal distributions. For each distribution, appropriate estimators for the optimal order quantity are considered, and their sampling distributions are derived. Then, through Monte-Carlo simulations, we evaluate the performance of corresponding exact and asymptotic confidence intervals for the true optimal order quantity. The case where normality for demand is erroneously assumed is also investigated. Asymptotic confidence intervals produce higher precision, but to attain equality between their actual and nominal confidence level, samples of at least a certain size should be available. This size depends upon the coefficients of variation, skewness and kurtosis. The paper concludes that having available data on the skewed demand for enough inventory cycles enables (i) to trace non-normality, and (ii) to use the right asymptotic confidence intervals in order the estimates for the optimal order quantity to be valid and precise.Inventory Control; Newsboy Problem; Skewed Demand; Exact and Asymptotic Confidence Intervals; Monte-Carlo Simulations

The distribution-free newsboy problem with resalable returns

We study the case of a catalogue/internet mail order retailer selling seasonal productsand receiving large numbers of commercial returns. Returned products arriving beforethe end of the selling season can be resold if there is sufficient demand. A single orderis placed before the season starts. Excess inventory at the end of the season is salvagedand all demands not met directly are lost. Since little historical information is available,it is impossible to determine the shape of the distribution of demand. Therefore, weanalyze the distribution-free newsboy problem with returns, in which only the mean andvariance of demand are assumed to be known. We derive a simple closed-form expressionfor the distribution-free order quantity, which we compare to the optimal order quantities whengross demand is assumed to be normal, lognormal or uniform. We find that the distribution-freeorder rule performs well in most realistic cases.inventory;product returns;distribution-free newsboy problem

The Newsboy Problem with Resalable Returns

We analyze a newsboy problem with resalable returns. A single order isplaced before the selling season starts. Purchased products may bereturned by the customer for a full refund within a certain timeinterval. Returned products are resalable, provided they arrive backbefore the end of the season and are undamaged. Products remaining atthe end of the season are salvaged. All demands not met directly arelost. We derive a simple closed-form equation that determines theoptimal order quantity given the demand distribution, the probabilitythat a sold product is returned, and all relevant revenues and costs.We illustrate its use with real data from a large catalogue/internetmail order retailer.inventory;reverse logistics;product returns;mail order retailer;newsboy problem

Aggregate constrained inventory systems with independent multi-product demand: control practices and theoretical limitations

In practice, inventory managers are often confronted with a need to consider one or more aggregate constraints. These aggregate constraints result from available workspace, workforce, maximum investment or target service level. We consider independent multi-item inventory problems with aggregate constraints and one of the following characteristics: deterministic leadtime demand, newsvendor, basestock policy, rQ policy and sS policy. We analyze some recent relevant references and investigate the considered versions of the problem, the proposed model formulations and the algorithmic approaches. Finally we highlight the limitations from a practical viewpoint for these models and point out some possible direction for future improvements

Validity and precision of estimates in the classical newsvendor model with exponential and rayleigh demand

In this paper we consider the classical newsvendor model with profit maximization. When demand is fully observed in each period and follows either the Rayleigh or the exponential distribution, appropriate estimators for the optimal order quantity and the maximum expected profit are established and their distributions are derived. Measuring validity and precision of the corresponding generated confidence intervals by respectively the actual confidence level and the expected half-length divided by the true quantity (optimal order quantity or maximum expected profit), we prove that the intervals are characterized by a very important and useful property. Either referring to confidence intervals for the optimal order quantity or the maximum expected profit, measurements for validity and precision take on exactly the same values. Furthermore, validity and precision do not depend upon the values assigned to the revenue and cost parameters of the model. To offer, therefore, a-priori knowledge for levels of precision and validity, values for the two statistical criteria, that is, the actual confidence level and the relative expected half-length are provided for different combinations of sample size and nominal confidence levels 90%, 95% and 99%. The values for the two criteria have been estimated by developing appropriate Monte-Carlo simulations. For the relative-expected half-length, values are computed also analytically.Inventory Control; Classical newsvendor model; Exponential and Rayleigh Distributions; Confidence Intervals; Monte-Carlo Simulations

Constructive solution methodologies to the capacitated newsvendor problem and surrogate extension

The newsvendor problem is a single-period stochastic model used to determine the order quantity of perishable product that maximizes/minimizes the profit/cost of the vendor under uncertain demand. The goal is to fmd an initial order quantity that can offset the impact of backlog or shortage caused by mismatch between the procurement amount and uncertain demand. If there are multiple products and substitution between them is feasible, overstocking and understocking can be further reduced and hence, the vendor\u27s overall profit is improved compared to the standard problem. When there are one or more resource constraints, such as budget, volume or weight, it becomes a constrained newsvendor problem. In the past few decades, many researchers have proposed solution methods to solve the newsvendor problem. The literature is first reviewed where the performance of each of existing model is examined and its contribution is reported. To add to these works, it is complemented through developing constructive solution methods and extending the existing published works by introducing the product substitution models which so far has not received sufficient attention despite its importance to supply chain management decisions. To illustrate this dissertation provides an easy-to-use approach that utilizes the known network flow problem or knapsack problem. Then, a polynomial in fashion algorithm is developed to solve it. Extensive numerical experiments are conducted to compare the performance of the proposed method and some existing ones. Results show that the proposed approach though approximates, yet, it simplifies the solution steps without sacrificing accuracy. Further, this dissertation addresses the important arena of product substitute models. These models deal with two perishable products, a primary product and a surrogate one. The primary product yields higher profit than the surrogate. If the demand of the primary exceeds the available quantity and there is excess amount of the surrogate, this excess quantity can be utilized to fulfill the shortage. The objective is to find the optimal lot sizes of both products, that minimize the total cost (alternatively, maximize the profit). Simulation is utilized to validate the developed model. Since the analytical solutions are difficult to obtain, Mathematical software is employed to find the optimal results. Numerical experiments are also conducted to analyze the behavior of the optimal results versus the governing parameters. The results show the contribution of surrogate approach to the overall performance of the policy. From a practical perspective, this dissertation introduces the applications of the proposed models and methods in different industries such as inventory management, grocery retailing, fashion sector and hotel reservation

Однопериодная модель управления запасами с непрерывным нечетким случайным спросом

The paper describes an algorithm for searching the optimal enterprise inventory capacity, where the demand for this resource is a fuzzy random variable. In particular, the case of continuously distributed demand with the expected value which is a triangular fuzzy number has been discussed. A numerical example is given to illustrate the model.У статті описано алгоритм пошуку оптимального обсягу запасу ресурсу підприємства, в якому попит на цей ресурс є нечіткою випадковою величиною. Зокрема, розглянуто випадок, коли закон розподілу випадкового попиту відомий або може бути оцінений на основі статистичних даних і математичне сподівання якого – нечітке трикутне число. Теоретичний матеріал проілюстровано числовим прикладом.В статье описан алгоритм поиска оптимального размера запаса ресурса предприятия, спрос на который является нечеткой случайной величиной. В частности, рассмотрен случай, когда закон распределения спроса известен или может быть оценен на основании статистических данных и математическое ожидание которого – нечеткое треугольное число. Теоретический материал проиллюстрирован числовым примером