On the On-line Number of Snacks Problem

  • Authors:

  • Weimin Ma; Jane You; Yinfeng Xu; James Liu; Kanliang Wang

  • Affiliations:

  • School of Management, Xi'an Jiaotong University, Xi'an, Shaanxi, P.R. China 710049/Department of Computing, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong (e-mail: );Department of Computing, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong (e-mail: );School of Management, Xi'an Jiaotong University, Xi'an, Shaanxi, P.R. China 710049/;Department of Computing, The Hong Kong Polytechnic University, Hung Hom, Kowloon, Hong Kong (e-mail: );School of Management, Xi'an Jiaotong University, Xi'an, Shaanxi, P.R. China 710049/

  • Venue:
  • Journal of Global Optimization
  • Year:
  • 2002

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Abstract

In the inumber of snacks problem (iNSP), which was originally proposed by our team, an on-line player is given the task of deciding how many shares of snacks his noshery should prepare each day. The on-line player must make his decision and then finish the preparation before the customers come to his noshery for the snacks; in other words, he must make decision in an on-line fashion. His goal is to minimize the competitive ratio, defined as ∈σ: CA(σ)/COPT(σ), where σ denotes a sequence of numbers of customers, iCOPT(σ) is the cost of satisfying σ by an optimal off-line algorithm, and iCA(σ) is the cost of satisfying σ by an on-line algorithm. In this paper we give a competitive algorithm for on-line number of snacks problem iP1, the iExtreme Numbers Harmonic Algorithm (iENHA), with competitive ratio 1+pċ(M-m)/(M+m), where iM and im are two extreme numbers of customers over the total period of the game, and ip is a ratio concerning the cost of the two types of situations, and then prove that this competitive ratio is the best one if an on-line player chooses a fixed number of shares of snacks for any sequence of numbers of customers. We also discuss several variants of the iNSP and give some results for it. Finally, we propose a conjecture for the on-line iNSP.