Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 456719, 9 pages

http://dx.doi.org/10.1155/2015/456719

## No-Wait Flexible Flow Shop Scheduling with Due Windows

Department of Business Administration, Fu Jen Catholic University, New Taipei City 24205, Taiwan

Received 12 January 2015; Accepted 19 February 2015

Academic Editor: Yunqiang Yin

Copyright © 2015 Rong-Hwa Huang 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

To improve capacity and reduce processing time, the flow shop with multiprocessors (FSMP) system is commonly used in glass, steel, and semiconductor production. No-wait FSMP is a modern production system that responds to periods when zero work is required in process production. The production process must be continuous and uninterrupted. Setup time must also be considered. Just-in-time (JIT) production is very popular in industry, and timely delivery is important to customer satisfaction. Therefore, it is essential to consider the time window constraint, which is also very complex. This study focuses on a no-wait FSMP problem with time window constraint. An improved ant colony optimization (ACO), known as ant colony optimization with flexible update (ACOFU), is developed to solve the problem. The results demonstrate that ACOFU is more effective and robust than ACO when applied to small-scale problems. ACOFU has superior solution capacity and robustness when applied to large-scale problems. Therefore, this study concludes that the proposed algorithm ACOFU performs excellently when applied to the scheduling problem discussed in this study.

#### 1. Introduction

Just-in-time (JIT) manufacturing is important in industry. JIT scheduling can decrease inventory, resource consumption, and storage needs and optimize delivery time. Managers prefer that product completion time be close to the due date, but achieving this creates a tight time window. The time interval is called the* due window* of a job, and the left and right ends of the window are called its starting and finishing times, respectively. If a job is finished before its due window, it must be stored as inventory, which creates an earliness penalty. On the other hand, if a job is finished after its due window, it incurs whatever tardiness penalty is stipulated in the contract [1]. For both firms and customers, these two costs (earliness penalty and tardiness penalty) are disadvantageous. Good scheduling should ensure the product is finished and delivered on time to preserve mutual benefit. This study lists some of the research on due window. Yeung et al. [2] developed a branch and bound algorithm and heuristic to solve the nonpreemptive two-stage flow shop scheduling problem to minimize earliness and tardiness under conditions where a common due window exists. According to the computational result, a strong lower bound is derived for the branch and bound algorithm that can efficiently solve a 15-job problem in approximately 5 minutes. The heuristic is shown to be efficient and effective and can provide a near-optimal solution to a 150-job problem in about 20 seconds. Huang et al. [3] developed two pheromone ant colony optimizations (2PH-ACO) to solve the flexible job shop with due window scheduling problem. This problem aims to minimize the sum of the earliness and tardiness costs. Computational results indicate that 2PH-ACO outperforms ant colony optimization (ACO). Chen and Lee [4] proposed a branch and bound algorithm to solve a parallel machine scheduling problem. All the jobs have a given common due window, and the aim is to optimize the schedule so as to minimize total earliness-tardiness penalty. The computational results show that the proposed algorithm can solve problems involving up to 40 jobs and any number of machines within a reasonable computational time.

No-wait flow shop is a popular production system. Because this system relies on continuous production, it is necessary to deploy a continuous production environment. Many industries suffer this constraint, such as the metal melting industry. To prevent deterioration of heated metal, a series of operations must be completed before it cools. Similarly, plastic, silver, and glass molding also require a series of operations before cooling. However, the related literature is scarce, and so further discussion of this topic is necessary. Besides continuity, setup time is another important consideration in such facilities. These considerations significantly increase the problem complexity.

The literature on the no-wait flow shop production scheduling problem mostly deals with a single condition, and there have been few integrated discussions dealing with multiple conditions. For the no-wait production scheduling problem, Aldowaisan [5] developed a new heuristic and dominance relations for the two-machine no-wait separate setup flow shop problem, where performance is assessed in terms of total flow time. Schuster and Framinan [6] proposed two local search algorithms to solve the no-wait job shop problem for minimum makespan. Shyu et al. [7] proposed an application of the ACO to a two-machine no-wait flow shop scheduling problem. The problem aims to devise a schedule that minimizes total completion time. Numerical results show that the ACO algorithm performs well and has a small error ratio. Wang and Cheng [8] proposed a heuristic to solve the two-machine flow shop scheduling problem in a no-wait processing environment to minimize maximum lateness. The computational results show that the heuristic can rapidly obtain near-optimal solutions for problems with a realistic scale. Su and Lee [9] studied the scheduling problem where a set of jobs are available for processing in a no-wait and separate setup two-machine flow shop system with a single server. Both the heuristic and the branch and bound algorithms are developed to solve the problem. Computational experiments indicate that the heuristic and branch and bound algorithm outperform existing algorithms in solution quality and number of branching nodes. In parallel machine scheduling research, Pereira Lopes and de Carvalho [10] developed a new branch-and-price optimization algorithm to solve the problem of scheduling independent jobs on unrelated parallel machines with sequence-dependent setup times. The computational results show that this approach quickly obtains optimal solutions to large-scale problems. Biskup et al. [11] proposed a new heuristic approach to solve parallel machine scheduling problems and developed heuristic algorithms to minimize total tardiness on identical parallel machines. The computational results show that the proposed approach effectively obtains optimal or near optimal schedules to minimize total tardiness on parallel machines. de Paula et al. [12] proposed a nondelayed relax-and-cut algorithm, based on a Lagrangian relaxation of a time indexed formulation to schedule a set of jobs, on any one of a set of unrelated parallel machines, without preemption. The objective function considered is to minimize total weighted tardiness. Mor and Mosheiov [13] proposed heuristics that comprise an allocation of jobs to the machines (based on LPT for parallel identical machines, and modified LPT for uniform machines) to study due date and due window assignment problems based on flow-allowance with a min-max objective function. Numerical tests indicate that the heuristics produce near optimal schedules, and the average optimality gaps remain minimal. Paul et al. [14] proposed a new scheduling heuristic for multi-item hoist production lines with flexible processing times in the context of an existing industrial application problem. Computational results show that the new heuristic can compete with previous approaches that rely on permutation schedules and works best on problems with uniformly distributed processing times. Owing to the highly efficient way of schedule building, productivity can be increased by at least 20%. Kumar et al. [15] proposed a methodology takes care of all the parameters of the ant colony optimization (ACO) algorithms and also incorporates preventive measures to overcome the difficulties in using the ACO algorithms.

Recent research has devoted great effort to no-wait, setup time, and time window problem of flow shop production. However, discussion of multiprocessor flow shop is limited. This study takes the literature as a basis to examine the no-wait due window flow shop with multiprocessors (FSMP) problem in a two-stage flow shop production environment. For schedule optimization, an improved ant colony optimization (ACO) named ant colony optimization with flexible update (ACOFU) is developed and applied. ACO was extensively applied to solve various scheduling problems. To summarize, the problem discussed in this study is very complex and considers various production constraints and objectives. The problem complexity resembles that of a real world production problem. The ACOFU proposed in this study solves the scheduling problem more effectively and helps improve production efficiency.

#### 2. Problem Definition

##### 2.1. Problems

Consider the following:

This study considered jobs with processing time and two stages. represents a two-stage flow shop production environment, with machines in the first stage and machines in the second stage; denotes no-wait production; represents the setup time of job ; denotes the multiple criteria objective, namely, to minimize costs; and represent the weights of the earliness and tardiness, respectively; and and denote the earliness and tardiness of the due window constraint.

##### 2.2. Notations

: total job number; : total stage number (in this study ); : set of unscheduled jobs; : number of multiprocessors in stage ; : job of number ; : stage of number ; : machine of number in stage of number ; : processing time of job on machine ; : setup time of job on machine ; : time of job starting processing on machine ; : time of job completing processing on machine ; : beginning of setup time of job on machine ; : completion of setup time of job on machine ; : starting time of processing of job with sequence on machine ; : completion time of processing of job with sequence on machine ; : beginning of setup time of job when sequence is being processed on machine ; : completion of setup time of job when sequence is being processed on machine ; : completion time of job ; : earliness of job ; : tardiness of job ; : weight factor of ; : weight factor of ; : a binary variable deciding whether job that is processed on machine of sequence ( or 1; denotes that job is processed on machine of sequence , otherwise ).

##### 2.3. Mathematical Model

Objective is as follows: Constraints are as follows: Equation (2) minimizes total earliness and tardiness. Equation (3) represents early completion of job . Equation (4) represents late completion of job . Equation (5) denotes job completion time. Equation (6) denotes that job completion time equals job completion time during the last stage. Equation (7) transforms the completion time on machine , stage , and sequence into the completion time of job in stage . Equation (8) transforms the complete setup time on machine , stage , and sequence into the complete setup time of job in stage . Equation (9) transforms the start time on machine , stage , and sequence into that of job in stage . Equation (10) represents the complete setup time on machine , stage , and sequence equals the initial setup time of the job plus the setup time. Equation (11) suggests that setup time can be separated from processing time. Equation (12) transforms the start of the setup time on machine , stage , and sequence into the start time of setup of job in stage . Equation (13) transforms the start time of processing on machine , stage , and sequence into the start time of processing of job in stage . Equations (14) and (15) satisfy the no-wait constraint. Equation (16) limits jobs to being processed only once by a given machine in every stage. Equation (17) limits processing to a single job on machine , stage , and sequence . Equation (18) decides whether job is processed on machine at sequence ( or 1; suggests job is processed on machine at sequence , otherwise ).

#### 3. The ACOFU Procedure

For the scheduling problem, this study proposes a more effective method than conventional ACO, namely, ant colony optimization with flexible update (ACOFU). The following illustrates the development of the proposed algorithm and the steps involved.

##### 3.1. Algorithm Principle

ACOFU uses the concept of the state transition rule in ACO and removes the original state transition rule of ACO. To solve the due window problem discussed in this study, the performance criterion equals . The principle is to prioritize jobs with a narrower due window to minimize both earliness and tardiness.

In ACOFU, artificial ants begin from node and select the next node using where .

determines exploitation or exploration behaviors of artificial ants to select the next job; represents the unvisited job set when ant arrives at job ; is a random variable based on the probability distribution of (20). Every time an ant is at job and has to choose the next job, a random number is selected. If , a job with the peak value of in the set is chosen, and the ant moves to this job. This process is called exploitation. Otherwise, exploration behavior is used; namely, the next job is randomly selected using the probability distribution stated in where represents the probability of ant selecting in from job .

##### 3.2. Dynamic Update Rules

In each iteration, the ants search are based on the due window information . At the end of each iteration, the routes searched by each ant are compared. The best route identified in this iteration is enhanced using a probability enhancing factor , where the probability of the route being selected in the next iteration is increased; additionally, the inferior routes will be reduced by a probability reducing factor , as represented in

From (21), the routes searched in each iteration are updated. Ants thus are inclined to select better routes in the next iteration search, while worse routes are less likely to be repeatedly searched. The dynamic update rule of ACOFU differs from the pheromone update in conventional ACO. In conventional ACO, the lower bound of pheromone deposit is the starting amount of pheromone. However, the dynamic pheromone update involves a larger total update and thus increases the effectiveness of the update.

##### 3.3. End of Algorithm

An iteration is defined as the period from all ants commencing their search to the dynamic update. The algorithm ends on completion of a predetermined number of iterations.

#### 4. Data Test and Analysis

The software Lingo 9.0 is used to optimize the solutions to the scheduling problem in this paper. This study then compiled ACO and ACOFU with Microsoft Visual C++ 6.0 to solve the problem and performed effectiveness and robustness data tests and comparisons.

##### 4.1. Data Generation and Test Environment

This first stage of data generation comprised a small-scale data test to verify algorithm effectiveness and robustness. The second stage involved a large-scale data test. Because Lingo 9.0 required an extensive calculation time, only ACO and ACOFU were implemented to verify effectiveness and robustness. Both parts of the test were executed on a AMD Athlon(tm) 64 X2 Dual Core Processor 5200+ 2.71 GHz with 2.00 GB RAM desktop PC. Comparison and analysis were carried out after program execution.

The notations used in this section include number of jobs (), number of stages (), number of multiprocessors per stage (), and average program execution time (CPU time). To simulate data generation, the processing time of jobs at each stage () was randomly generated from [], and the setup time () was generated from []. The due window was generated based on the design of Zheng et al. [16] as , . Table 1 lists the parameters used in due window generation.