Range of lower bounds

Each of n jobs is to be processed without interruption on a single machine. Each job becomes available for processing at time zero. The objective is to find a processing order of the jobs which minimizes the sum of weighted completion times added with maximum weighted tardiness. In this paper we give a general case of the theorem that given in [6]. This theorem shows a relation between the number of efficient solutions, lower bound LB and optimal solution. It restricts the range of the lower bound, which is the main factor to find the optimal solution. Also, the theorem opens algebraic operations and concepts to find new lower bounds.