非线性折旧下设备在线租赁策略及其竞争分析

The strategy and its competitive analysis of online leasing problem that depreciates based on the nonlinear function

经典的Karp租雪橇模型假定投资者购买的设备在使用结束后再无剩余价值,这与实际情况不符,因此本文从租赁决策的实际出发,提出了设备随使用时间以默比乌斯变换折旧的在线租赁问题。本文首先建立了该问题在线分析的数学模型,然后给出了最优离线策略以及一个在线策略。最后,利用解决在线问题常用的在线竞争分析方法,证明了该在线策略是该问题唯一最优策略,而且该策略具有比Karp模型更小的竞争比,竞争分析结果表明考虑设备可折旧因素能够改进在线策略的竞争比从而提高在线决策效率。

Leasing,as an important financing method that integrates the functions of financing capitals and financing goods,is increasingly favored and paid attention by the enterprise decision-makers.However,the investment environment in the market is complex and changeable,the decision maker should usually make a decision immediately in case they miss the investment opportunity,this situation makes the decision of leasing financing often have strong online characteristics.Because online algorithm theory can give a global optimal strategy in the case of an uncertain future information,this theory has been widely applied in the field of economic management and decision-making.Based on this,this article explores the issue of online leasing.The research on the online leasing problem originated from the famous online ski-rental problem proposed by Karp,that is,how to choose to rent or buy a pair of skis in the case of unknown skiing time in the future.In the ski-rental model of Karp,it is assumed that the skis have no surplus value left afterwards.But in the actual rental decision,there is residual value after the end of the use of the equipment.Although the existing literatures consider the depreciation factor,they only analyze the online competitive strategy under depreciation of fixed amount in each period in order to simplify the model.However,in practice,the depreciation price of a device is often a monotonous decreasing function of the time after the lessee purchases the equipment.Considering the good characteristics of the M?bius transform,this paper defines the M?bius transform depreciation function of the purchasing price of the equipment,the depreciation factor and the usage time after the lessee purchases the equipment.Therefore,this paper makes use of the online algorithm and competitive analysis method,and studies the online leasing problem based on the M?bius transform depreciation which is a more reasonable and more specific depreciation function.In the first part,this paper establishes a mathematical model of online leasing problem based on the M?bius transform depreciation function,and analyzes the relationship between the two cost functions,renting equipment to the end and purchasing equipment from the beginning.In the second part,because the optimal offline decision-makers know the total duration of the equipment in advance,this paper first gives the optimal offline strategy by analyzing the total cost of offline decision-makers in three different situations:"always renting equipment","purchasing equipment from the beginning",and"renting equipment for the first t periods continuously and then purchasing in the t+1-th period".Next,this paper presents an online strategy for this problem,that is,the lessee rents the equipment in each period before the rent-purchase critical time point of the optimal offline strategy,and purchases the equipment at the critical time point of the optimal offline strategy.By using the competitive analysis method commonly used to solve online problems,the competitive ratio is obtained.Also,the online strategy this paper presented is proved to be the unique optimal strategy for this online rental problem,and its competitive ratio is the optimal competitive ratio.Finally,this paper analyzes the competitive performance of the optimal online strategy.It is found that this online strategy possesses a much smaller competitive ratio than that of Karp’s model.Particularly,when the depreciation factor is 0,that is,when the depreciation is not considered,the optimal online strategy given in this paper and its competitive ratio are the same as the optimal online strategy and competitive ratio of Karp’s model.Besides,the competitive analysis results indicate that the larger the depreciation factor is the smaller the competitive ratio and the better the competitive performance of the optimal online strategy.In addition,considering both the purchasing price and the depreciation factor of the equipment,it can be found that the competitive ratio of the optimal online strategy decreases rapidly as the purchasing price decreases and depreciation factor increases,but the increase of the depreciation factor plays a leading role in reducing the competitive ratio.In general,taking the consideration of equipment depreciation factor can improve the competitive ratio of online strategy,thereby helping the lessee improve the efficiency of online decision-making,and making the optimal online strategy more realistic and economical.