Machine-scheduling Problems With The Learning Effect And Deteriorated Jobs

Scheduling is an important combinatorial optimization problem. In the classical schedulingit is assumed that the processing times of jobs are constant values. However, there are manysituations where the processing times of the job depends on its starting time, its resource allocation or its position in the sequence. So some morden scheduling problems emerged: Machinescheduling problems with the effects of learning and deterioration. For example, industrial workers repeat processing of the similar tasks continuously, efficiency is able to become more and morehigher. This phenomenon is known as the "learning effect" in the literature. However, operationof machines become slower then jobs prcocessing times also become longer. This phenomenon isknown as the "deterioration" in the literature (Pinedo, 2002 [31]).In this paper, we consider the machine scheduling problem with the effects of learning anddeterioration. The main conclutions can be summarized as follows:1. Single-machine scheduling problems with the effect of learning and deterioration(a) For the problem 1|P_j[r]=p_jα(t)r^a|C_max can be solved optimally by sequencing jobs innon-decreasing processing times (SPT rule).(b) For the problem (?), if p_j≤p_k implies w_j≤w_k for all jobsJ_j,J_k, then optimal sequence can be attained by seqeucingthe jobs non-decreasingorder of(?).(c) For the problem 1|P_j[r]=p_jα(t)r^a|L_max, if p_j≤p_k implies d_j≤d_k for all jobs J_j, J_k,then optimal sequence can be attained by seqeucingthe jobs non-decreasing order ofd_j.Where deteriorating general convex non-decreasing function a(t)≥0, r denote the jobposition and a denote the learning effect.2. Resource constrained problems with the effects of learning and deteriorationConsidering two problem 1|p_j[r]=(p_j+βt)r^a,r_j=f(u_j),C_max≤C|(?)and 1|1|P_j[r]=(p_j+βt)r^a,r_j=f(u_j),(?)≤U|C_max give optimal algorithms for above twoproblems, respectively. Where u_j is the amount of resource allocated to job J_j, f is a strictly decreasing function, U is the global amount of continuously divisible non-renewableresource. C, satisfying with (?), is a constant.3. Scheduling problem with the learning effect(a) For the problem(?), we give the optimal algorithm;however, for the problem (?), we give the optimalalgorithm under certain agreeable restriction.(b) For scheduling problems: (?) and F2|p_j[r] =(?), we can attain the optimal sequence by the SPTrule. Where The term ord in the problem definitions as follows: a_j≤b_j for each jobsj and with b_j≤b_k whenever a_j≤a_k for any two jobs j, k. a_j,b_j denote the lenghof the first and second operation respectively of job j. The term prp in the problemdefinitions bj = ca_j (c≥1 is a constant factor).