Single-Machine Group Scheduling Problems with Deterioration to Minimize the Sum of Completion Times

We consider two single-machine group scheduling problems with deteriorating group setup and job processing times. That is, the job processing times and group setup times are linearly increasing or decreasing functions of their starting times. Jobs in each group have the same deteriorating rate. The objective of scheduling problems is to minimize the sum of completion times. We show that the sum of completion times minimization problems remains polynomially solvable under the agreeable conditions.


Introduction
In classical scheduling problems, the scheduling models routinely assume that job processing times are known and fixed throughout the period of job processing.However, this assumption may be unrealistic in many situations that the processing times of jobs may be prolonged due to deterioration or shortened due to learning over time.Scheduling with deteriorating jobs was first independently introduced by J. N. D. Gupta and S. K. Gupta 1 and Browne and Yechiali 2 .Since then, related models of scheduling with deteriorating jobs have been extensively studied from a variety of perspectives.Cheng et al. 3 give a detailed review of scheduling problems with deteriorating jobs.More recent papers which have considered scheduling jobs with deteriorating jobs include Wang 4  On the other hand, the production efficiency can be increased by grouping various parts and products with similar designs and/or production processes.This phenomenon is known as group technology in the literature.To the best of our knowledge, only a few results concerning scheduling problems with deteriorating effect under group technology are given.Wei and Wang 17 consider single-machine scheduling problems with group technology GT and deteriorating jobs.Yang 18 investigate group scheduling problems with simultaneous considerations of learning and deterioration effects on a single-machine setting.The learning phenomenon is implemented to model the setup time of groups.Three models of deteriorating for the job processing time within a group are examined.Bai et al. 19 consider single-machine group scheduling problems with effects of learning and deterioration at the same time.Lee and Lu 20 discuss a single-machine scheduling problem of minimizing the total weighted number of late jobs with deteriorating jobs and setup times.A branch-and-bound with several dominance properties and a lower bound is developed to solve the problem optimally.S. J. Yang and D. L. Yang 21 study two models of learning for the job processing time, they provide polynomial time solutions for the makespan minimization problems.Huang et al. 22 present a single-machine group scheduling problems with both learning effects and deteriorating jobs.Wu and Lee 23 consider a situation where both setup times and job-processing times are lengthened as jobs wait to be processed.Two single-machine group-scheduling problems are investigated where the group setup times and the job-processing times are both increasing functions of their starting times.
In this paper, we study the group scheduling model proposed by Lee and Wu 24 , who have considered the makespan problems.In our study, we investigate the sum of completion times minimization problem under the same model as Lee and Wu 24 .In addition, we suppose jobs in each group have the same deteriorating rate.Hence, the problems considered will be noted by 1/GT p ij a ij ± b i t, s i θ i ± δ i t/ C ij .We show that the problems remain polynomially solvable under the agreeable conditions.
The remainder of this paper is organized in four sections.The solution procedure for the linear increasing function problem is given in the next section, and the solution procedure for the linear decreasing function is described in Section 3.Then, some examples are offered in Section 4. The conclusion is given in the last section.

The 1/GT, p ij a ij b i t, s i θ i δ i t/ C ij Problem
In this section, we consider the single-machine group scheduling problem with deterioration to minimize the sum of completion times of all jobs.In our model, we follow the notation and terminology used by Lee and Wu 24 .In addition, a i j is the basic normal processing time of job J i j scheduled in the jth position of group G i , and C i j is the completion time scheduled in the jth position of group G i .We suppose the jobs in each group have the same deterioration rate noted by b i .Theorem 2.1.For the 1/GT , p ij a ij b i t, s i θ i δ i t/ C ij problem, the optimal schedule is obtained by sequencing the jobs in each group in nondecreasing order of a i j , that is,

m, (the SPT rule).
Proof.In the same group, the result can be easily obtained by using interchanging technology and is omitted.
Proof.Let π 1 and π 2 be two job schedules where the difference between π 1 and π 2 is a pairwise interchange of two adjacent groups G i and G j .That is, where S 1 and S 2 are partial sequences and S 1 and S 2 may be empty.Furthermore, we assume that t denotes the completion time of the last job in π 1 .To show π 1 dominates π 2 , it suffices to show that 2.1 By definition, the completion times of jobs in π 1 are given by

2.2
And the completion times of jobs in π 2 are given by . . .

Mathematical Problems in Engineering
Therefore, we have

2.4
By substituting we can obtain

2.6
To have when t ≥ 0, if and only if From the result of 2.8 , we have And from the case of 2.9 , we can obtain B i /A i ≤ B j /A j .Therefore, if ω G i and B i /A i have agreeable conditions, the optimal sequence between groups are arranged in nondecreasing order of ω G i , This completes the proof.
From Theorems 2.1 and 2.2, if ω G i and B i /A i have agreeable condition, the problem 1/GT, p ij a ij b i t, s i θ i δ i t/ C ij can be solved by the following algorithm.Algorithm 2.3.Step 1. Jobs in each group are scheduled in nondecreasing order of the basic processing time a i j , that is, Step 3. Groups are scheduled in nondecreasing order of ω G i , that is, Obviously, It is easy to show that the total time for Algorithm 2.3 is O n log n .

The 1/GT, p ij a ij − b i t, s i θ i − δ i t/ C ij Problem
In order to discuss the problem 1/GT, problem, the optimal schedule is obtained by sequencing the jobs in each group in nondecreasing order of basic processing time a i j , that is,

m, (the SPT rule).
Proof.In the same group, the result can be easily obtained by using interchanging technology and is omitted.
Proof.Here, we still use the same notations as that in Theorem 2.2.To show π 1 dominates π 2 , it suffices to show that

3.1
By definition, the completion times of jobs in π 1 are given by . . .

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And the completion times of jobs in π 2 are given by . . .

3.3
Therefore, we have

3.4
By substituting we can obtain

3.6
To have Mathematical Problems in Engineering 7 when t ≥ 0 if and only if From the result of 3.8 , we have And from the case of 3.9 , we can obtain B i /A i ≤ B j /A j .Therefore, if ω G i and B i /A i have agreeable conditions, the optimal sequence between groups are arranged in nondecreasing order of ω G i , This completes the proof.
From Theorems 3.1 and 3.2, if ω G i and B i /A i have agreeable condition, the problem 1/GT, p ij a ij − b i t, s i θ i − δ i t/ C ij can be solved by the following algorithm.Algorithm 3.3.Step 1. Jobs in each group are scheduled in nondecreasing order of the basic processing time a i j , that is, Step 3. Groups are scheduled in nondecreasing order of ω G i , that is, Obviously, It is easy to show that the total time for Algorithm 3.3 is O n log n .

Examples
Example 4.1.For the 1/GT, p ij a ij b i t, s i θ i δ i t/ C ij problem, we consider six jobs are divided into three groups.The basic job and group processing times, the job and group deterioration rates for each group are given, respectively, as follows: Step 1.In group G 1 , the optimal job sequence is J 11 → J 12 .
In group G 2 , the optimal job sequence is J 21 → J 23 → J 22 .
In group G 3 , the optimal job sequence is J 31 . Step

4.2
Solution 2. According to Algorithm 2.3, we solve the example as follows.
Step 1.In group G 1 , the optimal job sequence is J 13 → J 12 → J 11 .
In group G 2 , the optimal job sequence is J 21 .
In group G 3 , the optimal job sequence is J 32 → J 31 . Step

Conclusions
In this paper, we investigate the single-machine group scheduling problems with deterioration which is studied by Lee and Wu 24 .We consider the sum of completion times minimization problem, and we show the problem to minimize total completion time of all jobs is polynomially solvable under agreeable conditions in our study.