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This paper studies a single machine scheduling problem with time-dependent learning and setup times. Time-dependent learning means that the actual processing time of a job is a function of the sum of the normal processing times of the jobs already scheduled. The setup time of a job is proportional to the length of the already processed jobs, that is, past-sequence-dependent (psd) setup time. We show that the addressed problem remains polynomially solvable for the objectives, i.e., minimization of the total completion time and minimization of the total weighted completion time. We also show that the smallest processing time (SPT) rule provides the optimum sequence for the addressed problem.

In classical scheduling problems, it is reasonable and necessary to consider scheduling problems with setup times. In many realistic situations, the setup times are considered either sequence independent or sequence dependent. In the first case, the setup times are usually added to the job processing times while in the second case, the setup time for the job currently being scheduled depends on the previous one or ones already scheduled. Koulamas and Kyparisis [

Recently, there is a growing interest in the literature to study scheduling problems with a learning effect [

In this section, addressing single machine scheduling problems, the actual processing time of a job is assumed to be a function of the sum of the normal processing times of the jobs already scheduled and the setup time of a job is proportional to the length of the jobs already processed. Let

where

where

Before you begin to format your paper, first write and save the content as a separate text file. Keep your text and graphic files separate until after the text has been formatted and styled. Do not use hard tabs, and limit use of hard returns to only one return at the end of a paragraph. Do not add any kind of pagination anywhere in the paper. Do not number text headsâ€•the template will do that for you. In this section, we consider a single machine scheduling problem with the objective of minimizing the total completion time. We show that the problem

presented as follows.

Lemma 1.

Proof. Let

Hence,

Lemma 2.

Proof. Let

Since

Theorem 1. For the minimization of total completion time on a single machine scheduling problem

order of

Proof. For two adjacent jobs

tion time of the job

the problem is minimized by sequencing the jobs in a SPT order, sufficient to show that (a)

First, the proof of part (a) is given as follows.

we have

By substituting

From Lemma 2, we have

Note the proof of part (a) also shows that the makespan is minimized by the SPT rule. Furthermore, the proof of part (b) is given as follows.

and

we have

Since

the fourth terms are non-negative as well. Therefore, this implies that

This completes the proof of (b) and thus of the theorem.

Hence, the optimal job-sequence of the single machine scheduling problem

be solved in polynomial time.

For the problem to minimize the total weighted completion time, we show that an optimal solution if the processing times and the weights are agreeable, i.e.,

Lemma 3.

Proof. Let

Hence,

Since

Lemma 4.

Proof. Let

Theorem 2. For minimization of the total weighted completion time on a single machine scheduling problem

and

Proof. Since

By substituting

From

Hence,

Thus, the proof is completed.

Hence, the optimal job-sequence of the scheduling problem

solved in polynomial time.

In this study, we analyzed a single machine scheduling problem with time-dependent learning and setup times. Time-dependent learning means that the actual processing time of a job is a function of the sum of the normal processing times of the jobs already scheduled. The setup time of a job is proportional to the length of the already processed jobs, that is, past-sequence-dependent (psd) setup time. The problem addressed with the two objectives, i.e., minimization of the total completion time and total weighted completion time, was studied in depth. We proved that the SPT rule can provide the optimal schedule for both the total completion time and total weighted completion time objectives. We also show that both the total completion time problem remains polynomially solvable, as does the total weighted completion time problem, under certain agreeable conditions.

Yuling Yeh,Chinyao Low,Wen-Yi Lin, (2015) Single Machine Scheduling with Time-Dependent Learning Effect and Non-Linear Past-Sequence-Dependent Setup Times. Journal of Applied Mathematics and Physics,03,10-15. doi: 10.4236/jamp.2015.31002