  Tiedemann, et al. [Proc. of WALCOM, LNCS 8973, 2015, pp.210-221] defined
multi-objective online problems (as an online version of multi-objective
optimization problems) and the competitive analysis for multi-objective online
problems and showed that (1) with respect to the worst component competitive
analysis, the online algorithm RPP-HIGH is best possible for the
multi-objective time series search~problem; (2) with respect to the arithmetic
mean component competitive analysis, the online algorithm RPP-MULT is best
possible for the bi-objective time series search problem; (3) with respect to
the geometric mean component competitive analysis, the online algorithm
RPP-MULT is best possible for the bi-objective time series search problem. In
this paper, we first point out that the definitions and frameworks of the
competitive analysis due to Tiedemann, et al. do not necessarily capture the
efficiency of online algorithms for multi-objective online problems and provide
modified definitions of the competitive analysis for multi-objective online
problems. Then under the modified framework, we present a simple online
algorithm Balanced Price Policy BPP_{k} for the multi-objective (k-objective)
time series search problem, and show that the algorithm BPP_{k} is best
possible with respect to any measure of the competitive analysis (defined by a
monotone continuous function f). Under the modified framework, we derive exact
values of the competitive ratio for the multi-objective time series search
problem with respect to the worst component competitive analysis, the
arithmetic mean component competitive analysis, and the geometric mean
component competitive analysis.
