Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 452602, 9 pages

http://dx.doi.org/10.1155/2015/452602

## Note on a Single-Machine Scheduling Problem with Sum of Processing Times Based Learning and Ready Times

^{1}Department of Business Administration, Fu Jen Catholic University, New Taipei City 24205, Taiwan^{2}Department of Statistics, Feng Chia University, Taichung 40724, Taiwan

Received 5 September 2014; Accepted 20 January 2015

Academic Editor: Chuangxia Huang

Copyright © 2015 Shang-Chia Liu et al. 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.

#### Abstract

In the recent 20 years, scheduling with learning effect has received considerable attention. However, considering the learning effect along with release time is limited. In light of these observations, in this paper, we investigate a single-machine problem with sum of processing times based learning and ready times where the objective is to minimize the makespan. For solving this problem, we build a branch-and-bound algorithm and a heuristic algorithm for the optimal solution and near-optimal solution, respectively. The computational experiments indicate that the branch-and-bound algorithm can perform well the problem instances up to 24 jobs in terms of CPU time and node numbers, and the average error percentage of the proposed heuristic algorithm is less than 0.5%.

#### 1. Introduction

In the classical scheduling models, most researchers consider that the job processing times are all constant numbers. In fact, it can be seen in many real situations that a production time can be shortened if it is operated later due to the fact that the efficiency of the production facility (e.g., a machine or a worker) continuously improves with time. This situation is named as the “*learning effect*” in the research community [1, 2]. Furthermore, Biskup [3] provided a survey paper to discuss different learning models in the scheduling research.

More recently, the* learning effect *has continued to receive a lot of effort. For more recent problems with time-dependent processing times on single-machine settings, we refer readers to Cheng et al. [4], Eren and Güner [5, 6], Eren [7], Janaik and Rudek [8], Toksarı et al. [9], Toksari and Güner [10], Wang and Liu [11], Wang et al. [12], Yin et al. [13], Yin et al. [14–17], Wu et al. [18–20], Wang et al. [21–23], Bai et al. [24], Vahedi-Nouri et al. [25], Lu et al. [26], and so forth.

Meanwhile, the concept of the learning process with ready times is relatively limited. For example, Bachman and Janiak [27] showed that the makespan single-machine job position-based learning problem is NP-hard in the strong sense. Following the same model by Bachman and Janiak [27], Lee et al. [28] built exact and heuristic algorithms to solve the optimal and near-optimal solutions, respectively. Eren [29] formulated a nonlinear mathematical programming model for the single-machine learning scheduling problem with different ready times. Lee et al. [30] explore a single-machine position-based learning scheduling problem with ready times to minimize the sum of makespan and total completion time. Wu and Liu [31] dealt with a single-machine problem with the learning effect and release times where the objective is to minimize the makespan. They proposed a branch-and-bound algorithm and three two-stage heuristic algorithms for the problem. Wu et al. [32] considered a single-machine problem with the sum of processing times based learning effect and release times to minimize the total completion times. They developed a branch-and-bound algorithm for the optimal solution. Then they proposed a simulated-annealing heuristic algorithm for a near-optimal solution. Wu et al. [33] considered a single-machine problem with learning effect and ready times where the objective is to minimize the total weighted completion time. They developed a branch-and-bound algorithm and a simulated annealing algorithm for the problem. Wu et al. [34] considered a single-machine problem with the sum of processing time based learning effect and release times to minimize the total weighted completion times. They developed a branch-and-bound algorithm incorporating with several dominance properties and two lower bounds for the optimal solution and then used a genetic heuristic-based algorithm for a near-optimal solution. J.-B. Wang and J.-J. Wang [35] investigated a single-machine scheduling problem with time-dependent processing times and group technology assumption to minimize the makespan with ready times of the jobs. They showed that the problem can be solved in polynomial time when starting time-dependent processing times and group technology simultaneously.

Besides, Dessouky [36] pointed out the importance of ready time in semiconductor manufacturing where it is common to find newer, more modern machines running side by side with older, less efficient machines which are kept in operation because of high replacement cost. Moreover, all of the above works deal with job position-based learning. Following Kuo and Yang [37] model, in this paper, we explored the jobs with different release times into the sum of processing time based learning mode.

The branch-and-bound method is a general algorithm for finding optimal solutions of various optimization problems. However, the execution time required is impractical when the number of activities increases. Therefore, a branch-and-bound algorithm usually incorporates with dominance properties and lower bounds to derive the optimal solution (Lee et al. [28, 30]). Thus, we also developed a branch-and-bound algorithm incorporating with several dominances and two lower bounds to derive the optimal solution.

This paper is organized as follows. In Section 2, we define the problem formulation. In Section 3, we derive some dominance properties and two lower bounds used in the branch-and-bound method and state the descriptions of the proposed heuristic. In Section 4, we report the results of a computational experiment. Conclusions could be found in the last section.

#### 2. Problem Statement and Notation Definition

Below are stated some notations used throughout the paper. denotes the size number of given jobs. , denote the sequences of jobs. denotes the basic processing time of job . denotes the ready time of job . is defined as the completion time of job scheduled in a sequence . denotes the learning effect with . denotes the basic processing time for the job scheduled in the th position in . denotes the ready time of a job scheduled in the th position in . denotes the completion time of a job scheduled in the th position in . denotes a nondecreasing order of processing times . denotes a nondecreasing order of ready times .

At first, we assume that there are given jobs to be operated on a single machine. Let each job have its processing time and a ready time . Due to the learning effect, we assume that the actual processing time of job is if it is scheduled in the th position where is a learning ratio common for all jobs. The objective of this paper is to minimize the makespan of given jobs in a sequence.

#### 3. Solution Methods

Due to the fact that the proposed problem is a difficult one, we attempt to use a branch-and-bound method and a heuristic algorithm for this problem. In order to speed up the searching process, we derive several dominance propositions and two lower bounds used in the branch-and-bound method.

Next, we derive five properties based on a pairwise interchange of two adjacent jobs. The proofs of Propositions 2 to 5 are omitted since they are similar to that of Proposition 1.

Proposition 1. *If and , then there is an optimal schedule in which job is scheduled before job .*

*Proof. *Consider two sequences and , where and denote partial sequences. To show that dominates , it suffices to show that . In addition, let be the completion time of the last job in the subsequence . Since , we have After taking the difference of (1) and (2), we haveOn substituting , , and into (3) and simplifying, we obtain Equation (4) can be easily obtained by taking the first and the second derivatives to for , , and , and we have . Therefore, dominates .

Proposition 2. *If and , then dominates .*

Proposition 3. *If and , then there is an optimal schedule in which job is scheduled before job .*

Proposition 4. *If , , and , then there is an optimal schedule in which job is scheduled before job .*

Proposition 5. *If and , then there is an optimal schedule in which job is scheduled before job .*

In what follows, two more properties to determine the ordering of the remaining unscheduled jobs are developed. Let and be two sequences of jobs in which denotes the scheduled part containing jobs, denotes the unscheduled part, and the jobs in are scheduled in the SPT rule; that is, .

Proposition 6. *If there exists job such that , then is a dominated sequence.*

Proposition 7. *If , then dominates sequences of the type for any unscheduled sequence .*

Following that, we will propose two simple lower bounds to curtail the branching tree. According to the definition, the completion time for the th job is given byThe makespan for is easily obtained as follows:According to (6), we have where . In a similar way, we havewhere denote the processing times of the unscheduled jobs arranged in a nondecreasing order.

In order to make the lower bound tighter, we choose the maximum value from (7) and (8) as the lower bound. That is,

A typical approach to a NP-hard problem is to provide a heuristic algorithm. In what follows, a hybrid SPT heuristic algorithm is proposed. Chen et al. [38] used the concept hybrid to move from local optimal solutions to near-optimal solutions. Liu and MacCarthy [39] used to arrange jobs in a nondecreasing order of weight-factors of processing time and their corresponding ready time for jobs for the flow time criterion. Thus, we attempt to adopt a dynamic weight which is dependent on the job size due to learning effect. The steps of the proposed algorithm are described as follows.

The details of the first heuristic algorithm (HA) are provided as shown in Algorithm 1.