Discrete Dynamics in Nature and Society

Volume 2018, Article ID 4657368, 13 pages

https://doi.org/10.1155/2018/4657368

## Metaheuristics for Order Scheduling Problem with Unequal Ready Times

^{1}Department of Business Administration, Cheng Shiu University, Kaohsiung 1037, Taiwan^{2}Business School, Sichuan University, Chengdu 610064, China^{3}College of Information Science and Engineering, Northeastern University, Shenyang 110819, China^{4}Department of Statistics, Feng Chia University, Taichung 40724, Taiwan^{5}Shandong Yingcai University, Shandong, Jinan 250104, China

Correspondence should be addressed to Win-Chin Lin; wt.ude.ucf@cwnil

Received 9 June 2018; Revised 14 August 2018; Accepted 4 September 2018; Published 18 September 2018

Academic Editor: Filippo Cacace

Copyright © 2018 Jan-Yee Kung 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 recent years, various customer order scheduling (OS) models can be found in numerous manufacturing and service systems in which several designers, who have developed modules independently for several different products, convene as a product development team, and that team completes a product design only after all the modules have been designed. In real-life situations, a customer order can have some requirements including due dates, weights of jobs, and unequal ready times. Once encountering different ready times, waiting for future order or job arrivals to raise the completeness of a batch is an efficient policy. Meanwhile, the literature releases that few studies have taken unequal ready times into consideration for order scheduling problem. Motivated by this limitation, this study addresses an OS scheduling model with unequal order ready times. The objective function is to find a schedule to optimize the total completion time criterion. To solve this problem for exact solutions, two lower bounds and some properties are first derived to raise the searching power of a branch-and-bound method. For approximate solution, four simulated annealing approaches and four heuristic genetic algorithms are then proposed. At last, several experimental tests and their corresponding statistical outcomes are also reported to examine the performance of all the proposed methods.

#### 1. Introduction

In recent years, various researchers have described customer order scheduling (OS) models (Framinan et al. and [1]). These models can be found in numerous manufacturing and service systems in which several designers, who have developed modules independently for several different products, convene as a product development team, and that team completes a product design only after all the modules have been designed. A practical example is that a manufacturer produces semifinished lenses and competes globally provided by Ahmadi et al. [2].

Regarding the literature on order scheduling with order ready times, only Leung et al. [3] discussed and analyzed the order ready times in a customer order scheduling model with regard to minimizing the total weighted completion time. Garg et al. [1] analyzed the complexities of order schedules to optimize the weighted flow time or offline weighted completion time. Therefore, in this paper we investigate a multiple-machine OS model with different order ready times in which the goal is to find a schedule to minimize the total completion time of orders. Regarding the importance of ready times in wafer fabrication, readers may refer to Monch et al. [4]. Moreover, the jobs or orders to be processed may have different priorities such as the due dates or weights or ready times. In a model with different ready times, waiting for future order or job arrivals to raise the completeness of a batch is an efficient policy.

Within the relevant literature on taking the total weighted completion times into consideration, it has been shown as an NP-hard by Sung and Yoon [5]. Some effective heuristics were developed by Wang and Cheng [6] and Leung et al. [3, 7, 8] for approximate solutions, and a branch-and-bound algorithm was proposed by Yoon and Sung [9] for the exact solution. For the total completion time criterion minimization, Ahmadi et al. [2] showed that it is NP-hard and built some heuristics accordingly. More recently, Framinan and Perez-Gonzalez [10] studied an OS model with completion time objective but zero ready times. They proposed a constructive heuristic with the same complexity compared to the ECT. The key element of this heuristic lies in its ability to estimate the expected contribution of the unscheduled orders, therefore mitigating the greedy behavior of the ECT. In addition, they proposed two improvement mechanisms that are further embedded into a greedy search procedure (GSA) for finding high-quality solutions. For the OS models with the due dates, readers may refer to Leung et al. [11] for minimizing the maximum lateness and to Lee [12] for minimizing the total tardiness. Extending the problem by Lee [12], Xu et al. [13] assumed that order processing times can be shortened by a position-based learning factor. Framinan and Perez-Gonzalez [14] studied the same problem as Lee [12] did. As extremely fast (negligible time) solutions are required, Framinan and Perez-Gonzalez [14] derived a new constructive heuristic that incorporates a look-ahead mechanism to estimate the objective function at the time that the solution is being built. For the scenarios where longer decision intervals are allowed, they provided a novel matheuristic strategy to provide extremely good solutions. Wu et al. [15] addressed an OS problem with sum of-processing-time-based learning effect to minimize total tardiness of the orders. For other researches on OS models, readers may refer to Yang and Posner [16], Leung et al. [3, 17], and Leung, Li et al. [18] on identical but parallel machine settings. Lin et al. [19] utilized three variants of a particle swarm optimization (PSO) algorithm including a basic PSO, an opposite-based PSO, and a PSO with a declined inertia weight for a OS model with two agents and ready times. The criterion is to minimize the total completion time for the orders from agent A while the total completion time for the orders from agent B has a given upper bound. Wu et al. [20] considered an OS model to minimize the linear sum of the total flowtime and the maximum tardiness. In particular, they proposed a branch-and-bound (B&B) algorithm along with several dominance relations and a lower bound, three modified heuristics, and a hybrid iterated greedy algorithm to solve the problem. For some substantial works on approximation algorithms for , readers may refer to Chekuri et al. [21], Hochbaum and Shmoys [22], Sevastianov and Woeginger [23], Alon et al. [24], and Schuurnman and Woeginger [25]. Readers may refer to Chen et al. [26] for a general overview of approximation techniques. The main contributions of this work are described as follows: an order scheduling problem of introducing ready times for each orders and minimizing the total completion time criterion is studied. A B&B method with four propositions and two lower bounds is derived for finding the exact solution. Four versions of simulated annealing algorithm (SA) and four versions of genetic algorithm (GA) are provided for searching the good-quality approximate solutions. The instance simulation experiments and some statistical methods are utilized to determine the performances for all proposed algorithms.

The rest of this study is organized as follows: Section 2 introduces the notation and formulates the problem. Section 3 provides some dominance propositions and two lower bounds used in a branch-and-bound (B&B) algorithm. Section 4 proposes four variants of a simulated annealing (SA) approach and four heuristic genetic algorithms (GA) for finding approximate solutions. Section 5 presents computational results to evaluate the performance of the eight proposed algorithms. The conclusion and suggestions for future study are provided in Section 6.

#### 2. Problem Statement

This section first introduces some notation that is applied in this study. *n*: total order number *m*: total machine number : machine* k*, and : schedules of orders where and denote partial sequences of the orders : processing time for order on machine* k*, ; : ready time for order* i*, : completion time for the last order in on machine* k*,* k* = 1, 2, …,* m* : completion time of order in* S*, i.e., is the order which is arranged in the* l*th position in a given schedule.

The problem is formulated as follows. Suppose that orders come in from different clients at different times (ready times), and each order comprises components. Let a facility with different machines be operated in parallel form. A machine can produce its own item; that is, an item can be done on a dedicated machine. If each order has its own ready time, then unforced idleness is also allowed. Preemptions are not forbidden. Let denote the processing time for component of order to be processed on machine* k* and let denote order to be placed in the* r*th position in a given sequence. The objective is to seek a sequence to let the total completion time criterion be minimized. To find that the optimal sequence is equivalent to finding a schedule that optimizes the objective function , an illustrative example follows.

*Example 1. *Given an instance problem with* n* = 2,* m* = 2. The processing times of each component of order ,* i* = 1, 2, on machine , and the ready times of order ,* i* = 1, 2, are provided in Table 1.

Given an order schedule* S* = (O_{1}, O_{2}), according to the definition of , one have the following computation, and Figure 1 shows the Gantt diagram.