In this paper, we are interested in minimizing the sum of block sizes in a pairwise balanced design, where there are some constraints on the size of one block or the size of the largest block. For every positive integers n, m, where m≤ n, let S (n, m) be the smallest integer s for which there exists a PBD on n points whose largest block has size m and the sum of its block sizes is equal to s. Also, let S′(n, m) be the smallest integer s for which there exists a PBD on n points which has a block of size m and the sum of it block sizes is equal to s. We prove some lower bounds for S (n, m) and S′(n, m). Moreover, we apply these bounds to determine the asymptotic behaviour of the sigma clique partition number of the graph K n− K m, the Cocktail party graphs and complement of paths and cycles.