Techniques of Finding Lower Bounds in Multi Objective Functions

In this paper, the problm of sequencing n jobs on one machine is considered with a multi objective function.Two problems have been studied, sum of completion times added with the maximum tardiness (  + N i i T c max ) and sum of completion times with the maximum tardiness ( N i i T and c max ), the first one has optimal solution solved by Branch and bound technique, the second has efficient solutions founded by Van Wassenhove algorithm.A theorem is presented to show a relation between the number of efficient solutions, lower bound (LB) and optimal solution.This theorem 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.


ABSTRACT
In this paper, the problm of sequencing n jobs on one machine is considered with a multi objective function.Two problems have been studied, sum of completion times added with the maximum tardiness Although there are a lot of published results on single machine problems with tardiness ( i T ), there are only some papers dealing with multi objective function [Lauff and Werner , 2004]. The problem class considered is as follows : n jobs 1,2,3,…,n have to be processed on a single machine (m=1) and become available at time zero , require a positive processing time Pi [Potts,1991].For each job i ,a processing time Pi ,a due date di , are specified .Given a schedule,we can compute for each job i the completion time ci , the tardiness

2.Notations and Definitions :
N=the set {1,2,3,…,n}. Pi =processing time for job i . di =Due date for job i. ci =Completion time for job i. Li =Lateness of job i. Ti =Tardiness of job i. EDD-rule: (Early due date) meaning the jobs are sequenced in nondecreasing order of di SPT-rule: (Short processing time) meaning the jobs are sequenced in nondecreasing order of pi . LB: ( Lower bound ) is a value of objective function, which is less than or equal to optimal value. UB: ( Upper bound ) is a value of objective function, which is greater than or equal to optimal value. Example: For this schedule ( 1,2,3 ) we find ci and Tmax as follows : c1 = p1 , c2 =c1+ p2 , c3 = c2 + p3 and Ti =max{ci-di , 0}. The Algorithm : Step(0) : Put ∆ =  N i i p Step (1) : Let Di =di +∆ for all i.
Step (2) : Solve using modified smith rule , if a solution exists then it is efficient.Else,go to step (4).

4.Relation Between Optimal and Efficient Solutions :
We know that a lower bound is less than the optimal solution.The question is: " What is the difference between lower bound and the optimal solution ?"of course , this depends on the lower bound and the objective Hence M  N2 + 1.We will prove N1-1  M by induction on N1. If N1 = 1,that is there is only one efficient solution which is SPT as well as EDD then M=0ptimal value -LB = That is M ε [N1-1,N2+1], and so the theorem is true for N1 = 1. If N1 = 2, i.e, the number of efficient solutions is two which are SPT and σ , say. N1=2 implies that N1-1=1, if SPT is optimal then M = And now if σ is optimal then M = , and so M ε [N1-1 , N2+1] and hence the theorem is true for N1 = 2.
Thus the theorem is true for N1 = 3.
Suppose the theorem is true for N1 = k , i.e., the theorem is true for the k efficient solutions SPT, σ , σ1 , …, σk-2 ,that is for these k efficient solutions N1-1  M  N2+1.

5.Conclutions and Suggestions
At the end of this paper , we conclude that the lower bound of a problem is one of the important factors to understand the nature of objective function and the method which is used to solve the problem .Also the efficient solutions used to find optimal solution ,but in our objective function , the relation between them will lead to a new area of study , that is the difference between optimal value and lower bound with the help of efficient solutions . This study opens algebraic operations and concepts to solve any problem of this type.
Lastly , using the new lower bound of this objective function certainly leads to other results.