Mathematical Problems in Engineering

Volume 2018, Article ID 6534021, 12 pages

https://doi.org/10.1155/2018/6534021

## A Biobjective Optimization Model for Deadline Satisfaction in Line-of-Balance Scheduling with Work Interruptions Consideration

^{1}Department of Economic Management, North China Electric Power University, Hebei 071003, China^{2}School of Economics and Management, North China Electric Power University, Beijing 102206, China^{3}State Grid Energy Research Institute, Beijing 102209, China

Correspondence should be addressed to Xin Zou; moc.621@887xuoz

Received 11 December 2017; Accepted 25 March 2018; Published 3 May 2018

Academic Editor: Anna Vila

Copyright © 2018 Xin Zou 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

The line-of-balance (LOB) technique has demonstrated many advantages in scheduling repetitive projects, one of which is that it allows more than one crew to be hired by an activity concurrently. The deadline satisfaction problem in LOB scheduling (DSPLOB) aims to find an LOB schedule such that the project is completed within a given deadline and the total number of crews is minimized. Previous studies required a strict application of crew work continuity, which may lead to a decline in the competitiveness of solutions. This paper introduces work interruptions into the DSPLOB and presents a biobjective optimization model that can balance the two conflicting objectives of minimizing the total number of crews and maximizing work continuity. An efficient version of the -constraint method is customized to find all feasible tradeoff solutions. Then, these solutions are further improved by an automated procedure to reduce the number of interruptions for each activity without deteriorating the performance in both the objectives. The effectiveness and practicability of the proposed model are verified using a considerable number of instances. The results show that introducing work interruptions provides more flexibility in reducing the total number of crews under the LOB framework, especially for serial projects with a tight deadline constraint.

#### 1. Introduction

Repetitive projects are characterized by a number of units to be finished and a set of activities that need to be repeated for each unit in the length of the job [1]. Such projects are frequently encountered in multistage projects (e.g., high-rise buildings and multiple similar houses) and infrastructure networks (e.g., highways, railways, and pipelines). Repetitive projects account for a substantial part of the construction industry [2]. In China, the government has long been concerned with improving the nation’s transport system; the “13th Five-Year Plan (2016–2020)” predicted that regional road networks will be increased by 420,000 kilometers and approximately 30,000 kilometers of railways will be built and start operation by 2020. Therefore, efficient planning and scheduling of this type of project is crucial.

In repetitive projects, construction resources (crews) are often required to perform the same or similar work in various units by moving from one to another. Due to this frequent crew movement, an effective schedule is important to maintain work continuity of crews by eliminating their needless interruption days from their movement between units [3]. However, a strict compliance with work continuity by preventing all crews from having interruptions may lead to a longer overall project duration [4]. In recent decades, numerous researches began to focus on repetitive projects scheduling with work interruptions [5–9].

The critical path method (CPM) is the most common and extensively used scheduling technique in construction projects [10]. However, its application in repetitive projects is limited as CPM is incapable of minimizing work interruptions [8]. As such, several scheduling methods applicable to repetitive projects have been developed since the 1960s [11–13], among which one of the most representative is the line-of-balance (LOB) technique.

Dolabi et al. [14] enumerated several advantages of LOB over the other techniques, one of which is that it allows more than one crew to be employed by an activity concurrently. In fact, if the units of an activity are technically independent (i.e., the precedence relations between them are not mandatory), these units in theory can be performed at the same time by hiring additional crews [15]. Considering multiple crews allows a project to be rescheduled by adjusting the crew distribution of activities, which enhances the practicability of the output schedules. In literature, multiple crew usage has attracted considerable attention and has been considered by many scheduling optimization problems, for example, cost optimization [16], time-cost tradeoff [17], resource leveling [18], deadline satisfaction [19], and profit maximization with cash flow [20]. In this paper, the deadline satisfaction problem under the LOB framework (DSPLOB) will be investigated.

The DSPLOB involves a typical project in which the duration of each activity is assumed to be uniform along all units. The objective is to find an LOB schedule such that the project is completed within a given deadline and the total number of crews is minimized. Available methods for solving this problem can be classified into heuristic procedures and mathematical optimization. Heuristic procedures provide a way to obtain good solutions but do not guarantee optimality, including the integrated CPM and LOB method (CPM/LOB) [21], repetitive unit scheduling system (RUSS) [22], advanced linear scheduling system (ALISS) [23], and search-based heuristic line-of-balance (SHLOB) [14]. Mathematical optimization methods convert the problem into mathematical programming models and then use commercial software to seek the optimal solution. Such works include the mixed-integer linear model developed by Zou et al. [19] and the mixed-integer nonlinear model presented by Dolabi et al. [14]. Since all the above studies required a strict application of work continuity, the optimal schedule constructed by each of them is similar to the one shown in Figure 1, where each activity is represented by a sloped bar and the slope of the bar stands for the progress rate of the activity. A brief review of existing solution methods is as follows:(i)CPM/LOB starts with a CPM calculation to determine the completion time of the standard unit and float time of each activity . Then, the theoretical number of crews employed for activity is determined by , where is the given deadline, is the unit duration of activity , and is the number of units. However, this method may fail to obtain a feasible LOB schedule that satisfies the deadline constraint if there exists an activity such that is not an integer.(ii)ALISS (or RUSS) is a heuristic iterative algorithm that initializes an LOB schedule by assigning only one crew to each activity. Then, the schedule is updated iteratively by increasing the number of crews by one for the selected activity. The procedure ends when the given deadline is satisfied or the project duration cannot be shortened by accelerating any one activity, and in the latter case, an infeasible LOB schedule will be obtained.(iii)SHLOB is also a heuristic iterative algorithm that uses the CPM/LOB solution as the initial LOB schedule. In each iteration, parallel and consecutive activities are considered as a block, and the set of permitted blocks that may lead to a reduced project duration by increasing their progress rates is identified. If this set is null, the procedure fails to obtain a feasible solution; otherwise, one block within the set is selected on the basis of several heuristic rules and the number of crews for each activity in the selected block is increased by one. Then, the procedure goes to the next iteration until the deadline constraint is met. SHLOB is valid only for serial projects in which precedence relations between activities are finish to start and each activity has just one successor.(iv)The performance of the mixed-integer linear model presented by Zou et al. [19] was investigated using 2700 instances with up to 120 activities. The results revealed that their model can find optimal schedules for all instances within 20 seconds. In comparison, although SHLOB provided each instance with a feasible schedule, only 11.93% were solved to optimality. Of the 2700 stances, 57.59% can be solved by ALISS, of which only 1.7% were found to be optimal. Even worse, CPM/LOB failed to solve any instance. The exact model developed by Dolabi et al. [14] is less practical because it took approximately 31 hours to handle a serial project with 24 activities.