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In this paper, we consider the case that sharing many secrets among a set ofparticipants using the threshold schemes. All secrets are assumed to bestatistically independent and the weak secure condition is focused on. Undersuch circumstances we investigate the infimum of the (average) informationratio and the (average) randomness ratio for any structure pair which consistsof the number of the participants and the threshold values of all secrets.  For two structure pairs such that the two numbers of the participants are thesame and the two arrays of threshold values have the subset relationship, twoleading corollaries are proved following two directions. More specifically, thebound related to the lengths of shares, secrets and randomness for the complexstructure pair can be truncated for the simple one; and the linear schemes forthe simple structure pair can be combined independently to be a multiplethreshold scheme for the complex one. The former corollary is useful for theconverse part and the latter one is helpful for the achievability part.  Three new bounds special for the case that the number of secretscorresponding to the same threshold value $ t $ is lager than $ t $ and twonovel linear schemes modified from the Vandermonde matrix for two similar casesare presented. Then come the optimal results for the average information ratio,the average randomness ratio and the randomness ratio. We introduce a tinyexample to show that there exists another type of bound that may be crucial forthe information ratio, to which we only give optimal results in three cases.

Multiple Threshold Schemes Under The Weak Secure Condition