Separable programming/duality approach to solving the multi-product Newsboy/Gardener Problem with linear constraints

The Gardener Problem is an extension of the multi-product Newsboy Problem with constraints. It deals with situations where not only the demand is random but also the yield (the supply). Separable programming and duality approaches are utilized to solve the constrained Newsboy/Gardener Problem. The solution methodologies are developed for the common probability distribution functions for the demand, and uniform distribution for the supply, rendering exact and approximate solutions to the problem. Numerical examples are given and when applicable, the performance of the developed approach is compared to those of existing works in this arena. The results reveal that the developed solution methods efficiently converge to the optimal or near optimum solutions. Also, a salient feature of the proposed methodologies is that they can utilize readily available commercial software to solve the considered problems. This feature facilitates the portability of the developed models to the classroom environment.     

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.     

The single-item newsboy problem with dual performance measures and quantity discounts

This paper considers models for the single-item newsboy problem with quantity discount and the following dual performance measure: “maximize the expected profit subject to a constraint that the probability of achieving a target profit level is no less than a predetermined risk level.” We also consider two types of quantity discount: all-unit and incremental. For our models with zero shortage cost, a closed-form solution for determining the optimal order quantity is derived. However, models with positive shortage cost can only be solved numerically.     

带有二次订购和二次销售的报童问题

本文提出二次进货二次销售的报童模型 ,并分析了新模型与经典报童问题和带有反馈生产模型的最优订购量及收益关系 ,分析了模型的灵敏度 ,服务水平等 .     

The distribution-free newsboy problem under the worst-case and best-case scenarios

New theoretical foundations for analyzing the newsboy problem under incomplete information about the probability distribution of random demand are presented. Firstly, we reveal that the distribution-free newsboy problem under the worst-case and best-case demand scenarios actually reduces to the standard newsboy problem with demand distributions that bound the allowable distributions in the sense of increasing concave order. Secondly, we provide a theoretical tool for seeking the best-case and worst-case order quantities when merely the support and the first k moments of the demand are known. Using this tool we derive closed form formulas for such quantities in the case of known support, mean and variance, i.e. k = 2. Consequently, we generalize all results presented so far in literature for the worst-case and best-case scenarios, and present some new ones. Extensions of our findings to the cases of the known mode of a unimodal demand distribution, the known median, and to other stochastic inventory problems are indicated.     

Unreliable newsboy problem with a forecast update

In this paper, we derive an optimal ordering policy for an unreliable newsboy who can place two sequential orders before the start of a single selling season by using a demand forecast update. Supply yield is modeled using a uniform distribution considering both the minimum order guarantee and the maximum yield. Our results indicate that a firm should focus on increasing the minimum order guarantee from a first stage supplier to reduce its total supply chain cost.     

基于委托—代理下的模糊报童问题

在委托-代理关系中.代理人的私人信息对委托人的决策会产生重要影响.因此,为了获取代理人的真实信息.设计一个有效的激励机制是委托人的核心问题.本文基于需求依赖于代理人的努力水平和模糊市场条件的假设,利用委托一代理相关理论,探讨了模糊报童问题的激励机制问题.首先讨论了委托人最优产量的确定问题.然后分别给出了可观测和不可观测两种努力水平下的最优激励机制,最后分析了模糊市场条件对委托一代理双方利益的影响.     

The Newsboy problem when customer demand is a compound renewal process

The Newsboy (Newsvendor) problem is probably the simplest of all stochastic inventory problems, involving a one-time purchase decision and a stochastic sales outcome. As an investment, it can be interpreted as the simplest stochastic version of the point-in, point-out investment problem of Jevons [Jevons, W.S., Theory of Political Economy, Macmillan, London 1871].     

The multi-product newsboy problem with supplier quantity discounts and a budget constraint

This paper considers the multi-product newsboy problem with both supplier quantity discounts and a budget constraint, while each feature has been addressed separately in the literature. Different from most previous nonlinear optimization models on the topic, the problem is formulated as a mixed integer nonlinear programming model due to price discounts. A Lagrangian relaxation approach is presented to solve the problem. Computational results on both small and large-scale test instances indicate that the proposed algorithm is extremely effective for the problem. An extension to multiple constraints and preliminary computational results are also reported.     

Incorporating one-way substitution policy into the newsboy problem with imprecise customer demand

This paper presents an approach for solving an inventory model for single-period products with maximizing its expected profit in a fuzzy environment, in which the retailer has the opportunity for substitution. Though various structures of substitution arise in real life, in this study we consider the fuzzy model for two-item with one-way substitution policy. This one-way substitutability is reasonable when the products can be stored according to certain attribute levels such as quality, brand or package size. Again, to describe uncertainty usually probability density functions are being used. However, there are many situations in real world that utilize knowledge-based information to describe the uncertainty. The objective of this study is to provide an analysis of single-period inventory model in a fuzzy environment that enables us to compute the expected resultant profit under substitution. An efficient numerical search procedure is provided to identify the optimal order quantities, in which the utilization of imprecise demand and the use of one-way substitution policy increase the average expected profit. The benefit of product substitution is illustrated through numerical example.