On the on-line number of snacks problem

In the number of snacks problem (NSP), 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 inf(sigma) C-A (sigma) /C-OPT (sigma), where sigma denotes a sequence of numbers of customers, C-OPT (sigma) is the cost of satisfying by an optimal offline algorithm, and C-A (sigma) is the cost of satisfying sigma by an on-line algorithm. In this paper we give a competitive algorithm for on-line number of snacks problem P1, the Extreme Numbers Harmonic Algorithm (ENHA), with competitive ratio 1 + p . ( M - m) / (M + m),where M and are two extreme numbers of customers over the total period of the game, and p 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 NSP and give some results for it. Finally, we propose a conjecture for the on-line NSP.