Pareto and scalar bicriterion optimization in scheduling deteriorating jobs

In the paper two problems of a single machine bicriterion scheduling of a set of deteriorating jobs are considered. The jobs are independent, nonpreemptable and are ready for processing at time 0. The processing time pj of each job is a linear function of the starting time Sj of the job, pj=1+αjSj, where Sj 0 and αj>0 for j=0,1,...,n. The first problem is to find a schedule which is Pareto optimal with respect to ΣCj and Cmax criteria. The second problem is to find an optimal schedule subject to the minimization of the criterion in the form of λΣCj+ (1-λ)Cmax, where λ [0,1]. There are given necessary and sufficient conditions for a schedule to be Pareto optimal for the first problem. It is proved that there exists 0<λ0<1 such that for all λ [0,λ0] the second problem is solvable in time. It is also proved that an optimal schedule for the second problem has a V-shape for all λ [λ1,1], where λ0<λ1<1.