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Question

In this paper we introduce a robust optimization approach to solve the Vehicle Routing Problem (VRP) with demand uncertainty.This approach yields routes that minimize transportation costs while satisfying all demands in a given bounded uncertainty set. We show that for the Miller–Tucker–Zemlin formulation of the VRP and specific uncertainty sets, solving for the robust solution is no more difficult than solving a single deterministic VRP.Our computational results on benchmark instances and on families of clustered instances show that the robust solution can protect from unmet demand while incurring a small additional cost over deterministic optimal routes.This is most pronounced for clustered instances under moderate uncertainty,where remaining vehicle capacity is used to protect against variations within each cluster at a small additional cost. We compare the robust optimization model with classic stochastic VRP models for this problem to illustrate the differences and similarities between them. We also observe that the robust solution amounts to a clever management of the remaining vehicle capacity compared to uniformly and non-uniformly distributing this slack over the vehicles.

Keywords:Robust optimization,vehicle routing,demand uncertainty

1.Introduction
Many industrial applications deal with the problem of routing a fleet of vehicles from a depot to service a set of customers that are geographically dispersed.This type of problem is faced daily by courier services (e.g.,Federal Express,United Parcel Service and the Overnight United States Postal Service), local trucking companies and demand responsive transportation services,just to name a few.The setypes of services have experienced tremendous growth inrecent years.For example, both United Parcel Service and the Federal Express have annual revenue of well over $10 billion,and the dial-a-ride service for the disabled and handicapped is today a $ 1.2 billionin dustry (Palmer etal.,2004).However, congestion and variability in demand and travel times affects these industries on three major service dimensions: (i) travel time; (ii) reliability; and (iii) cost (Meyer,1996). Therefore, there is a need to develop routing and scheduling tools that directly account for the uncertainty.In this paper,we focuson the uncertainty in demand.

Answer 1

在本文我们介绍一种健壮优化方法解决车发送问题(VRP)以需求不确定性。这种方法产生使运输费用减到最小，当满足在一个特定一定不确定性集合时的所有要求的路线。 我们表示，为VRP和具体不确定性集合的Miller–Tucker–Zemlin公式化，解决健壮解答的比解决唯一确定VRP.Our计算结果在基准事例和在成群的事例家庭表示困难，健壮解答可能保护免受为满足的需求，当招致在确定优选的路线时的一个小追加成本。这为成群的事例是最显著的在适度不确定性之下，剩余的车能力用于防止受到在每群之内的变异在一个小追加成本。 我们健壮最优化模型与这个问题的经典随机VRP模型比较能说明在他们之间的区别和相似性。 我们也观察健壮解答共计剩余的车容量的聪明的管理与一致地比较和non-uniformly分布在车的这松驰。

Keywords ：健壮优化，车发送，需求不确定性

1.Introduction
Many工业应用应付寻址车队的问题从集中处的为地理上被分散的一套顾客服务。此种问题由递送急件服务(即，联邦快递公司、United Parcel Service和隔夜美国邮政局)每日面对，地方载重汽车运输公司和要求敏感运输服务，命名一些。服务setypes体验了巨大成长inrecent岁月。例如， United Parcel Service和联邦快递公司有年收入远远超出$10十亿，并且残疾的定时接送的公交和有残障今天是dustry $ 1.2的billionin (等Palmer， 2004)。然而，受欢迎的壅塞和的可变性和旅行时间影响在三个主要服务维度的这些产业： (i)旅行时间; (ii)可靠性; 并且(iii)费用(迈尔1996)。 所以，有直接地占不确定性的需要开发发送和预定的工具。在本文，我们focuson受欢迎的不确定性。