Single-Machine Group Scheduling Problems with General Deterioration and Linear Learning Effects

Ning, Lei; Sun, Li

doi:https://doi.org/10.1155/2023/1455274

Mathematical Problems in Engineering

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Research Article | Open Access

Volume 2023 | Article ID 1455274 | https://doi.org/10.1155/2023/1455274

Single-Machine Group Scheduling Problems with General Deterioration and Linear Learning Effects

Lei Ning¹and Li Sun²

Academic Editor: Bogdan Smolka

Received29 Jun 2022

Revised16 Dec 2022

Accepted21 Dec 2022

Published17 Jan 2023

Abstract

In this paper, we introduce a group scheduling model with deterioration and a general linear learning effect. The group setup time is a linear deterioration function of its starting time. The general learning effect is time-dependent and position-dependent. The objectives are to minimize the makespan, total completion time, and total weighted completion time, respectively. We show the problems remain solvable in polynomial time under the proposed model.

1. Introduction

Machine scheduling problems with deterioration or learning effect have been paid more attention in recent years. Scheduling with deterioration was first independently introduced by Gupta and Gupta [1] and Browne and Yechiali [2]. Since then, related scheduling models with deterioration have been extensively studied from a variety of perspectives. Gao et al. [3] study a two-agent scheduling on a parallel-batch machine to minimize makespan. Kim and Kim [4] consider single scheduling problems incorporating a rate-modifying activity (RMA) and processing time deterioration rates. For more recent literature involving deterioration, readers can refer to Sánchez–Herrera et al. [5]; Wang et al. [6]; Ding et al. [7]; Zhang et al. [8]; Li et al. [9]; Soleimani et al. [10]; Cheng et al. [11]. Wright [12] and Biskup [13] are among the pioneers that brought the topic of learning into the field of scheduling. Since then, the learning effect has been widely employed in management science. Jemmali and Hidri [14] investigate a parallel machine scheduling model with DeJong’s learning effect. Bai et al. [15] address a flow shop scheduling problem with a learning effect to optimize maximum lateness. Cheng et al. [11] introduce the optimization scheduling problems of batch operations with batch-position-dependent learning effect and aging effect. For more recent scheduling papers with learning effect, the readers are suggested to refer to the studies by Yang and Lu [16]; Jiang et al. [17]; Rudek [18]; Jiang et al. [19]; Wu et al. [20]; Bai et al. [15]; Li et al. [9]; and Li et al. [21].

In many manufacturing processes, the production efficiency can be increased by grouping various parts and products with similar designs or production processes. This phenomenon is known as group technology in the literature. Group technology layout is widely used in modern enterprise production layout, which is to use group technology to improve production efficiency. Many advantages have been claimed through the wide applications of group technology in industrial production, such as the studies by Yin et al. [22]; Zhang et al. [23]; Sun et al. [24]; Costa et al. [25]; Bai et al. [26], consider single-machine group scheduling problems with learning effect and deterioration at the same time. They show the makespan minimization problem can be solved polynomially. Under a certain assumption, they show the total completion time minimization problem can be solved polynomially too. In this paper, we study another group scheduling model with deterioration and general linear learning effects. In our model, the actual group setup time is followed as that in Bai et al. [26]; and the actual processing time of job scheduled in the rth position is linear function of the general learning effect. We will give polynomial time algorithm for the makespan minimization problem. And, we will prove the total completion time minimization problem is also polynomially solvable under certain conditions.

The remaining part is organized as follows: In Section 2, a precise formulation of the problem is given. We consider the makespan minimization problem in Section 3. The total completion time minimization problem is discussed in Section 4. We study the total weighted completion time minimization problem in Section 5. Section 6 contains some conclusion.

2. Problem Formulation

There are jobs which are classified into groups and will be processed on a single-machine. All jobs are available at time , and jobs preemption is not allowed. A group setup time is required if the machine switches to process from one group to another group. Jobs in the same group are processed consecutively and need no setup time. The machine can handle one job at a time. We assume that the actual setup time of group is , where is the basic (normal) setup time, is the setup deterioration rate, and is the starting setup time of group . If job in group is scheduled in the th position of group , then the actual job processing time iswhere is the basic (normal) processing time, , , and . is a differentiable nonincreasing function and . is a nonincreasing function with . It is the Wang [27] model if , the Wang et al. [28] model if . The weight of is . The objectives are to find the optimal job sequence in each group and the optimal group sequence so that the makespan, the total completion time, and the total weighted completion time are minimized, respectively. denotes the general linear learning effect. Using the three-field notation for scheduling problems, we denote our problems as

Before proving the problems, a lemma is introduced as follows.

Lemma 1 (see [29]). The sum of products is minimized if sequence is ordered in a nondecreasing manner and sequence is ordered in a nonincreasing manner or vice versa, and it is maximized if the sequences are ordered in the same way.

3. Makespan Minimization

In this section, we consider the group scheduling problem to minimize the makespan under the new model.

Theorem 1. For the problem, the optimal schedule is obtained by sequencing the jobs in each group in nondecreasing order of , i.e., (the SPT rule).

Proof. Without loss of generality, we assume the starting time of group in a sequence is . Hence,Here, the first part is a constant. Therefore, to minimize the makespan is to minimize . can be viewed as the scalar product of two vectors, and is already sorted in nonincreasing order. From Lemma 1, the should be sorted in nondecreasing order. Based on the above analysis, the optimal job sequence in each group can be obtained by the shortest normal processing time first (the SPT rule).

Theorem 2. For the problem, the groups are arranged in nondecreasing order of

Proof. Let and be two job schedules where the difference between and is a pairwise interchange of two adjacent groups and . That is, , , where and are partial sequences and and may be empty. Furthermore, we assume that t denote the completion time of the last job in . To show dominates , it is sufficient to show and for any job in . By definition, the job’s completion times of group and group in are given by the following equation:By substitutingHence,The job’s completion times of group and group in are given by the following equation:Taking the difference between equations (6) and (7), the following is obtained:To have , if and only ifWithout loss of generality, suppose is the first job in .
Hence,If , it is easy to obtain . Therefore, we have shown that the first job in , which starts earlier in , completes earlier in . Similarly, we have for any job in . This completes the proof.
From Theorems 1 and 2, the problem can be solved by Algorithm 1.

Algorithm 1. Step 1. Jobs in each group are scheduled in nondecreasing order of the normal processing time , i.e., (the SPT rule) Step 2. Calculate Step 3. Groups are scheduled in nondecreasing order of , i.e. Obviously, it is easy to show that the total time complexity for Algorithm 1 is .

Example 1. We consider six jobs are divided into three groups and , . The basic job and group processing times, the group deterioration rates, and job’s learning effect rates are given, respectively, as follows:For the problem, we suppose the start time is .

Solution 1. According to Algorithm 1, we solve the example as follows: Step 1: in group , the optimal job sequence is . In group , the optimal job sequence is . In group , the optimal job sequence is . Step 2: calculate , , . It is easy to say . Therefore, the optimal group sequence is and the optimal schedule is . The completion times of the job are , , , , , and , and the makespan is .

4. Total Completion Time Minimization

In this section, we consider the group schedule to minimize total completion time. In our model, we will consider this problem with the agreeable condition.

Theorem 3. For the problem, the optimal schedule is obtained by sequencing the jobs of each group in nondecreasing order of , i.e., (the SPT rule).

Proof. Without loss of generality, we assume that the starting time of group in a sequence is . Hence,Here, is a constant. Therefore, to minimize is equivalent to minimize . can be viewed as the scalar product of two vectors. The is already sorted in nonincreasing order. From Lemma 1, the should be sorted in nondecreasing order. Based on the above analysis, the optimal job sequence of each group can be obtained by the shortest normal processing time first (the SPT rule).

Theorem 4. For the problem, the optimal group sequence are arranged in nondecreasing order of , if and are in agreeable condition, i.e., implies , , and .

Proof. Here, we still use the same notations as that in the proof of Theorem 2. To show dominates , it suffices to show and for any job in .
From Theorem 2, the total completion times of group and group in are obtained as follows:and the total completion times of group and group in are obtained as follows:Hence, we have the following equation:To obtain when .
If and only if,From the case of equation (16), we have , i.e., .
And, from the result of equation (17), we can obtain , i.e., . From Theorem 2, it is easy to have for any job in . Therefore, if and are under agreeable condition, the optimal sequence between groups are arranged in nondecreasing order of . This completes the proof.
From Theorems 3 and 4, if and are under agreeable condition, the problem can be solved by Algorithm 2.

Algorithm 2. Step 1. Jobs in each group are scheduled in nondecreasing order of the normal processing time , i.e., (the SPT rule) Step 2. Calculate Step 3. Groups are scheduled in nondecreasing order of , i.e., Obviously, it is easy to show that the total time complexity for Algorithm 2 is . We conjecture the general version of minimizing total completion time is NP-hard. Therefore, the two following special cases are shown to be polynomially solvable.

Special case 1. Suppose the numbers of jobs in each group and the setup deterioration rates of groups are the same, respectively, that is, , .
The model reduces to . From Theorems 3 and 4, the problem can be solved by Algorithm 3.

Algorithm 3. Step 1. Jobs in each group are scheduled in nondecreasing order of the normal processing time , i.e., (the SPT rule) Step 2. Calculate Step 3. Groups are scheduled in nondecreasing order of , i.e., Obviously, it is easy to show that the total time complexity for Algorithm 3 is .

Special case 2. Suppose the setup deterioration rates of groups are zero. The model reduces to . From Theorems 3 and 4, the problem can be solved by Algorithm 4.

Algorithm 4. Step 1. Jobs in each group are scheduled in nondecreasing order of the normal processing time , i.e., (the SPT rule) Step 2. Calculate Step 3. Groups are scheduled in nondecreasing order of , i.e., Obviously, it is easy to show that the total time complexity for Algorithm 4 is .

Example 2. We still consider the above example. For the problem, we suppose the start time is .

Solution 2. According to Algorithm 2, we solve the example as follows: Step 1: in group , the optimal job sequence is . In group , the optimal job sequence is . In group , the optimal job sequence is . Step 2: calculate: , , and and , , and . Hence, and . Therefore, the optimal group sequence is and the optimal schedule is . The completion times of jobs are , , , , , and , and the sum of completion times is .

5. Total Weighted Completion Time Minimization

In this section, we consider the group scheduling problem to minimize the total weighted completion time. In our model, we will consider this problem with the agreeable condition.

Theorem 5. For the problem, if the jobs in each group have disagreeable weights, i.e., implies for the jobs and , the optimal schedule is obtained by sequencing the jobs of each group in nondecreasing order of , i.e., , (the WSPT rule).

Proof. The problem of job sequence in each group reduces to a single machine scheduling problem with a learning effect. By carrying out a pairwise interchange of adjacent jobs, the result can be easy to obtain. Therefore, we omit it.

Theorem 6. For the problem, if the jobs in each group have disagreeable weights, i.e., implies for the jobs and , and if and are in agreeable condition for group , i.e., implies , , , the optimal group schedule is obtained by sequencing the groups in non-decreasing order of .

Proof. Here, we still use the same notations as in the proof of Theorem 2. To show dominates , it suffices to show that and for any job in .
From Theorem 2, we have the following equation:To obtain when .
If and only if,From the case of equation (19), we have the following equation:and from equation (20), we can obtain the following equation:From Theorem 2, it is easy to have for any job in . Therefore, if and are under agreeable condition, the optimal sequence between groups are arranged in nondecreasing order of . This completes the proof.
From Theorems 5 and 6, if the jobs in each group have disagreeable weights and and are under agreeable condition for group , the problem can be solved by Algorithm 5.

Algorithm 5. Step 1. Jobs in each group are scheduled in nondecreasing order of the normal processing time , i.e., (the WSPT rule) Step 2. Calculate Step 3. Groups are scheduled in nondecreasing order of , i.e., Obviously, it is easy to show that the total time complexity for Algorithm 2 is .

Example 3. We still consider the above-given example. Six jobs are divided into three groups and , . The basic job and group processing times, the group deterioration rates, and job’s learning effect rates and weight are given, respectively, as follows:For the problem, we suppose the start time is . Obviously, the jobs in each group have disagreeable weights.

Solution 3. According to Algorithm 5, we solve the example as follows: Step 1: in group , the optimal job sequence is . In group , the optimal job sequence is . In group , the optimal job sequence is . Step 2: calculate , , and , , and . Hence, and . Therefore, the optimal group sequence is and the optimal schedule is . The completion times of jobs are , , , , , and , and the sum of completion times is .

6. Conclusions

In this paper, we study single-machine group scheduling problems with deterioration and general linear learning effect simultaneously. The objectives are to minimize the makespan, the total completion time, and the total weighted completion time. We first show that the makespan minimization problem is polynomially solvable for our model. Meanwhile, we prove the total completion time minimization and the total weighted completion time minimization remain polynomially solvable under agreeable conditions. Future research may investigate more general deteriorating and learning models under other environments or consider other objective functions.

Data Availability

All data generated or analysed during this study are included in this article.

Conflicts of Interest

The authors declare that they have no conflicts of interest.

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Copyright © 2023 Lei Ning and Li Sun. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

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