Modeling and Solving the Deadline Satisfaction Problem in Line-of-Balance Scheduling Downloaded from ascelibrary.org by University Of Florida on 10/28/17. Copyright ASCE. For personal use only; all rights reserved. Xin Zou1; Qian Zhang2; and Lihui Zhang3 Abstract: This study investigates the deadline satisfaction problem in line-of-balance (LOB) scheduling, where each activity can be done concurrently in several units by hiring additional crews, and all employed crews are allocated to implement an activity in sequential units smoothly with no interruption. Furthermore, the project must be completed within a given deadline. In previous studies, the objective was to minimize the total number of crews under the assumption that the cost of hiring a crew is the same for all activities. In fact, this cost usually differs among activities because of the differences in type of work, crew size, and technical level of workers. This paper presents a mixed-integer linear programming model with an extended objective of minimizing the crew employment cost of all activities. The model is equipped with an exact procedure to validate the feasibility of a prespeciﬁed deadline. A highway project was used to show the application of the proposed method, and then extensive computational experiments were conducted to investigate the performance of the proposed method by comparing the obtained results with those obtained by the existing methods. The results highlight the signiﬁcance of integrating the time-cost trade-off analysis, and indicate that the proposed method can ﬁnd optimal solutions for large-scale projects within a reasonable amount of time. DOI: 10.1061/(ASCE)ME.1943-5479.0000565. © 2017 American Society of Civil Engineers. Author Keywords: Repetitive projects; Line-of-balance; Time-cost trade-off; Deadline constraint; Mixed-integer linear programming. Introduction Repetitive projects are those that consist of 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. In real life, most multistage projects (e.g., high-rise buildings and multiple similar houses) and infrastructure projects (e.g., highways, railways, and pipelines) fall into this category. Repetitive projects represent a large portion of the construction industry. According to China’s 13th Five-Year Plan, regional road networks will be expanded to 5,000,000 km, with an increase of 420,000 km; approximately 30,000 km of railways will be built and start operation by 2020. Thus, efﬁcient planning of this type of project is an important determinant of the success of construction companies. In repetitive projects, each crew usually needs to perform the same or similar activities in different units by moving from one unit to the next. Therefore, the needless waste of resources owing to waiting for preceding activities to ﬁnish should be eliminated by maintaining work continuity of crews (Long and Ohsato 2009). Moreover, maintaining work continuity is beneﬁcial for the crews, as it allows them to develop the so-called learning effect that will 1 Lecturer, Dept. of Economic Management, North China Electric Power Univ., Hebei 071003, People’s Republic of China. E-mail: zoux788@126.com 2 Ph.D. Degree Candidate, School of Economics and Management, North China Electric Power Univ., Beijing 102206, People’s Republic of China (corresponding author). E-mail: zhangqian881111@163.com 3 Professor, School of Economics and Management, North China Electric Power Univ., Beijing 102206, People’s Republic of China. E-mail: zlh6699@126.com Note. This manuscript was submitted on February 13, 2017; approved on July 3, 2017; published online on October 25, 2017. Discussion period open until March 25, 2018; separate discussions must be submitted for individual papers. This paper is part of the Journal of Management in Engineering, © ASCE, ISSN 0742-597X. © ASCE likely reduce the time and money required for completing later units (Arditi et al. 2001a). The critical-path method (CPM) was widely implemented for construction project planning in the late 50s (Mohammadipour and Sadjadi 2016). When used to schedule repetitive projects, however, CPM becomes less practical because it is not oriented toward providing work continuity for crews. Since the 1960s, many techniques with the capability of eliminating CPM deﬁciencies in scheduling repetitive projects have been developed (Arditi and Albulak 1986; Al Sarraj 1990; Reda 1990; Harmelink and Rowings 1998; Harris and Ioannou 1998; Lucko 2009). As shown in Table 1, although these techniques differ somewhat in their focus, they all plan a schedule in a space-time, two-dimensional coordinate system, and have the ability to maintain work continuity of crews. In the literature, several studies have been conducted to improve existing methods at different levels of detail. For instance, Tang et al. (2014) developed a two-stage scheduling model for resource leveling based on the linear-scheduling model (LSM), where optimal or near-optimal schedules are obtained using constraint programming techniques. Hyari and El-Raye (2006) presented a multiobjective optimization model to minimize the project duration and maximize crew work continuity. Hsie et al. (2009) constructed a LSMbased resource-constrained scheduling model with the objective of minimizing the project duration by searching for the optimal set of production rates for activities in different time periods. Recently, Huang et al. (2016) designed a genetic algorithm to solve the multimode discrete time-cost trade-off problem considering soft logic. The objective was to ﬁnd a set of activity modes, start times, and work sequences between units such that the total cost was minimized while meeting a given deadline. A notable assumption adopted by the aforementioned studies is that only one crew could be used for each activity. Fan et al. (2012) highlighted that, if the units in an activity are technically independent (i.e., there is no precedence relation between them), in theory, these units can be performed at the same time by hiring additional crews. Considering multiple crews can 04017044-1 J. Manage. Eng., 2018, 34(1): 04017044 J. Manage. Eng. Table 1. Review of Techniques for Scheduling Repetitive Projects Method LOB LSM Repetitive scheduling method Productivity scheduling method Repetitive project modeling Downloaded from ascelibrary.org by University Of Florida on 10/28/17. Copyright ASCE. For personal use only; all rights reserved. a Work Typical activitya interruption Author(s) Multiple crew assignment Deadline constraint satisfaction Graphic-based method Arditi and Albulak (1986); Al Sarraj (1990) Harmelink and Rowings (1998) Harris and Ioannou (1998) Yes No No No No allowed Yes No No No No No Yes Yes Yes Lucko (2009) No No No No No Reda (1990) Yes No No Yes Yes A typical activity means that the duration of the activity is constant along all units (locations). Unit Buffer time 6 Crew 1 5 4 Crew 2 Crew 2 Crew 1 Crew 1 Crew 1 Crew 1 2 1 Crew 1 Activity A Crew 1 3 Crew 3 Crew 2 Crew 1 Activity C Crew 3 Crew 2 Crew 2 Activity B Crew 1 Crew 1 Crew 1 Time Buffer time Fig. 1. Sample LOB schedule provide more ﬂexibility in scheduling repetitive projects, this allows the project to be rescheduled by adjusting the number of crews assigned to each activity. As a result, a more competitive schedule in terms of time, cost, and/or resource distribution may be obtained. In the literature, the concept of multiple crews has been integrated into many project scheduling problems [e.g., cost optimization (Hegazy and Wassef 2001), time-cost trade-off (Hegazy et al. 2004), deadline satisfaction (Tokdemir et al. 2006), proﬁt maximization considering cash ﬂow (Ali and Elazouni 2009), resource leveling (Damci et al. 2013), and schedule compression (Bakry et al. 2014)]. It is worth noting that most studies involving multiple crews prefer to construct their optimal schedules in a lineof-balance (LOB) diagram, because LOB permits more than one crew to be assigned to an activity concurrently compared with other scheduling techniques. This paper focuses on the classic deadline satisfaction problem in LOB scheduling. The objective is to construct a LOB schedule by determining the optimal number of crews for each activity, such that each crew has a continuous workﬂow and the project is completed within a given deadline. LOB is typically represented by a two-dimensional diagram similar to the one illustrated in Fig. 1, in which each activity is represented by a sloped bar and crews are allocated to implement this activity in sequential units smoothly with no interruption. To the best of the authors’ knowledge, the integrated CPM and LOB method (CPM/LOB) proposed by Suhail and Neale (1994) may be the ﬁrst approach to put forward a formulation for determining crews needed to meet a given deadline. This method integrates the capabilities of CPM, as an analytical method, into the LOB technique to schedule repetitive projects in a nongraphical way. An improved version of CPM/LOB can be found in the work by © ASCE Ammar (2013). Other alternative methods include the repetitive unit scheduling system (RUSS) proposed by Arditi et al. (2001b), and the advanced linear scheduling system (ALISS) developed by Tokdemir et al. (2006). Both RUSS and ALISS start with an initial LOB schedule in which each activity is performed by only one crew without interruption. Then, the schedule is updated iteratively by increasing the number of crews by one for the selected activity. The process continues either until the given deadline is met or until the project cannot be compressed by accelerating any one activity. Dolabi et al. (2014) pointed out that these two methods cannot guarantee optimal/near-optimal crew distribution, because the selection of consecutive activities with identical progress rates is prohibited. Therefore, the authors presented a heuristic iterative algorithm called search-based heuristic line of balance (SHLOB). In each iteration, a blocking process is executed such that parallel and consecutive activities are grouped together as a block. Then, SHLOB identiﬁes a decision domain including blocks that, when increased, their progress rates may decrease the project completion time. Next, the better selection within the decision domain is determined based on several heuristic rules, and the number of crews for each individual activity in the selected block is increased by one. Although several methods have been developed for solving the deadline satisfaction problem in LOB scheduling, they suffer from the following limitations. First, they may fail to obtain a LOB schedule that satisﬁes the deadline constraint. Although CPM/LOB allocates crews in light of the stated deadline, the ﬁnal schedule may still be infeasible if the calculated number of crews is not an integer. ALISS and SHLOB also violate the deadline constraint if no activity is selected by their heuristic rules in an iteration. Second, SHLOB is valid only for serial projects in which the precedence relations between activities are ﬁnish to start and each activity has 04017044-2 J. Manage. Eng., 2018, 34(1): 04017044 J. Manage. Eng. Downloaded from ascelibrary.org by University Of Florida on 10/28/17. Copyright ASCE. For personal use only; all rights reserved. just one successor. Third, as heuristic procedures, CPM/LOB, ALISS, and SHLOB provide a way to obtain good solutions but do not guarantee optimality. Therefore, extensive experiments should be conducted to validate the effectiveness of these methods by comparing their results with some benchmark results. Unfortunately, no such experiments are found in existing research works. Fourth, their objective is to minimize the total number of crews under the assumption that the cost of hiring a crew is the same for all activities. In fact, this cost is usually different between activities owing to the differences in type of work, crew size, and technical level of workers. Therefore, an extended objective of minimizing the crew employment cost of all activities should be considered. Finally, when an infeasible schedule is generated, existing methods are unable to provide any useful information for planners to make decisions. Any infeasible solution of the problem under study is caused by either the problem itself, which includes an infeasible deadline constraint, or the method itself, which fails to provide a feasible solution. It is important to tell planners the real reason, because if the deadline constraint is infeasible, they have the right to ask for a looser deadline; otherwise, using other methods is recommended. Thus, this study aimed to eliminate the limitations of existing methods by presenting a novel method with time-cost trade-off consideration. The proposed method consists of two modules: feasibility test and scheduling optimization. In the feasibility test module, an exact procedure is developed to evaluate the shortest project duration (SPD), and then, the rationality of a prespeciﬁed deadline can be checked by comparing it with the SPD. Once the deadline passes the feasibility test, the scheduling optimization module is implemented to minimize the crew employment cost of all activities without exceeding a given deadline. Repetitive projects usually have a smaller number of activities, which makes exact optimization techniques a feasible option (Bakry et al. 2014). Based on this, the approach of mixed-integer linear programming was adopted to guarantee optimal crew distribution, where the integer variables relate to the crew assignment, whereas the continuous variables involve the start times of activities in the ﬁrst and last units. Finally, a highway project was used to show the application of the proposed method, and extensive computational experiments were conducted to investigate the performance of the proposed method, with the obtained results compared against the existing methods. The rest of the paper is organized as follows. The next section provides a capsule description for CPM/LOB, ALISS, and SHLOB. The third section describes how the proposed method is developed. Computational experiments are reported in the fourth section, and concluding remarks are given in the last section. the prespeciﬁed deadline and completion time of the unit network, respectively; Rj is the desired progress rate of activity j; and TFj is the corresponding total ﬂoat calculated from the previous step Rj ¼ m1 Td þ TFj T1 (1) 3. Calculate the number of crews (Cj) required to maintain the desired progress rate of activity j using Eq. (2). If Cj is not an integer, it should be rounded using Eq. (3) to determine the actual number of crews (Caj). Eq. (3) also requires that Caj be j ) of activless than or equal to the maximal crew availability (C ity j C j ¼ Rj d j (2) jg Caj ¼ minfdCj e; C (3) 4. Draw the ﬁnal LOB schedule by further considering precedence constraints between activities. ALISS This method ﬁrst initializes a LOB schedule by assigning only one crew for each activity. If the deadline constraint is satisﬁed, then the procedure stops and an optimal solution is found; otherwise, the following procedure is executed to improve the initial LOB schedule: 1. List the activities that have more than one crew available, and sort them in descending order of durations along all units. If no such activity exists, the procedure stops and it is reported that no feasible solution is found; otherwise, the ﬁrst activity in the list is selected. 2. Run the LOB analysis to see if the project duration will be reduced by assigning an additional crew to the selected activity. If such a project compression happens, update the schedule by increasing the number of crews by one for the selected activity and go to the next step; otherwise, select the next activity in the list and repeat this step. If the deadline constraint is still not satisﬁed after all listed activities have been investigated, the procedure fails to ﬁnd a feasible schedule. 3. If the deadline constraint is met, the procedure stops and a local optimum is found; otherwise, return to Step 1 in the ALISS method. SHLOB Existing Solution Methods As mentioned earlier, the available methods for solving the deadline satisfaction problem in LOB scheduling include, but are not limited to, CPM/LOB, ALISS, and SHLOB. These methods are designed to minimize the total number of crews without exceeding a predetermined deadline. CPM/LOB The integrated CPM/LOB method proposed by Ammar (2013) was considered in this study. Generally, this method can be decomposed into the following four steps: 1. Perform the CPM calculation to obtain the completion time of a unit network and the ﬂoat times of activities. 2. Evaluate the desired progress rates of all activities using Eq. (1), where m denotes the number of units; Td and T1 represent © ASCE This method is designed for repetitive projects with serial activities. It ﬁrst constructs an initial schedule using CPM/LOB. If the deadline constraint is met, the procedure terminates and an optimal solution is found; otherwise, the following procedure is implemented: 1. Classify parallel and consecutive activities into the same block. 2. Determine the set of permitted blocks that may lead to a reduced project duration when their progress rates increase. If the set of permitted blocks is empty, the procedure stops and provides feedback that no feasible solution is found. 3. Calculate the potential reduction time (RTj) for each permitted block j, and determine the corresponding reduction time index using RTIj = RTj/nj, where nj is the number of activities existing in block j. For more detailed information regarding the calculation of RTj, please refer to Dolabi et al. (2014). 4. Generate a random number (RN 2 ½0; 1) and compare it with the prespeciﬁed constant C 2 ½0; 1. If RN is greater than C, the 04017044-3 J. Manage. Eng., 2018, 34(1): 04017044 J. Manage. Eng. Downloaded from ascelibrary.org by University Of Florida on 10/28/17. Copyright ASCE. For personal use only; all rights reserved. block with the highest reduction time index is selected, and the number of crews of each activity in this block is increased by one; otherwise, a block from the set of permitted blocks is selected randomly for schedule modiﬁcation. 5. Update the schedule according to the modiﬁed crew distribution. If the deadline constraint is satisﬁed, then the procedure stops and a local optimum is found; otherwise, return to Step 1. Because SHLOB introduces a random search operator, the authors decided to equip this algorithm with an iterative construct to gain better results. In each iteration, an improvement on the initial LOB schedule is made, which may be different from previous solutions owing to the randomness of parameter RN. When the maximum number of iterations is reached, the feasible solution with the minimal number of crews is treated as the ﬁnal LOB schedule. Proposed Method This section presents a mathematical programming-based optimization method for the deadline satisfaction problem in LOB scheduling. As mentioned earlier, the proposed method consists of two modules: feasibility test and scheduling optimization. For a given problem, the feasibility test module is designed to calculate the SPD and then to check the feasibility of the predetermined deadline by comparing it with the SPD. If it fails to pass the test, the procedure reports a message that the current problem is infeasible due to a tight deadline and shows the corresponding SPD to planners; otherwise, the scheduling optimization module is used to search for a feasible solution with the lowest project cost. To facilitate the understanding of the proposed method, the authors ﬁrst introduce the scheduling optimization module. min k j XX © ASCE (4) where binary variables xjk need to meet the following constraints to guarantee that the number of crews assigned to each activity is deﬁnite: k j X xjk ¼ 1; j 2 A (5) k¼1 In LOB scheduling, the precedence constraints between each pair of activities [ð p; jÞ 2 E] are satisﬁed in all units only if they are preserved in the ﬁrst and last units. More precisely, if the progress rate of Activity j is less than that of its predecessor (p; see Activities A and B in Fig. 1), the precedence constraints between these two activities can be met by preserving that in the ﬁrst unit; otherwise, the last unit has to be considered (see Activities B and C in Fig. 1). Based on the assumption that the buffer time is equal to zero for each activity pair in E, the precedence constraints read as follows: sp1 þ dp sj1 ; ð p; jÞ 2 E (6) spm þ dp sjm ; ð p; jÞ 2 E (7) The start times of each Activity j in the ﬁrst and last units are linked, with reference to Fig. 2, by sj1 þ ðm 1Þ=Rj ¼ sjm . In contrast, the progress rate (Rj) of Activity j must equal to the ratio of the actual number of crews employed by this activity and its unit P duration to maintain work continuity of crews (i.e., k j kxjk =dj ). By combining these two equations, the folRj ¼ k¼1 lowing linear constraints are obtained: Scheduling Optimization Module In this module, an exact mixed-integer linear programming (MILP) model is developed for solving the problem. This idea is inspired by the conclusion that constructing a model with linear constraints and a linear objective function is a key factor for obtaining exact solutions to combinatorial optimization problems in a reasonable time (Rieck et al. 2012). The parameters used for model development are described. Let set A ¼ f0; 1; :::; n þ 1g consist of all activities and set U ¼ f1; 2; :::; mg consist of all units. Without loss of generality, the authors assume that Activity 0 is the single dummy start activity, and Activity n þ 1 is the single dummy end activity. For each activity i = 1, ..., n, denote by dj the unit duration, k j , the maximal number of available crews, and l jk the cost for employing k =1, ..., k j , crews. For dummy Activities 0 and n þ 1, set d0 ¼ dnþ1 ¼ 0; λ01 ¼ λnþ1;1 ¼ 0, and k 0 ¼ k nþ1 ¼ 1. Pre-cedence constraints between activities are represented by a set E A A in which each pair of activities [ð p; jÞ 2 E] indicates that Activity p is a predecessor of Activity j. Let parameter d be the given deadline. The present objective is to determine the optimal number of crews for each activity and then construct a feasible LOB schedule such that the project cost is minimized. Based on this, the authors deﬁne two sets of decision variables: (1) sj1 ; sjm 2 Rþ , which represents the start times of Activity j in the ﬁrst and last units, respectively; and (2) xjk, which takes a value of 1 if k crews are employed for Activity j and zero otherwise. Then, the objective function can be mathematically described as follows: λjk xjk j2A k¼1 sj1 þ ðm 1Þdj k j X xjk k¼1 k ¼ sjm ; j 2 A (8) The deadline constraint forces the project to be completed before the given deadline (d ). Because Activity n þ 1 is the single dummy end activity, the deadline constraint can be mathematically described by snþ1;m d (9) Unit Activity j 5 unit duration (d j ) = 3 number of crews (C j ) = 2 4 progress rate ( R j ) = 2 3 3 2 1 Crew 2 Cj Crew 1 Rj di Time Fig. 2. Synchronization and work continuity of crews 04017044-4 J. Manage. Eng., 2018, 34(1): 04017044 J. Manage. Eng. Finally, the exact MILP model for solving the deadline satisfaction problem in LOB scheduling is stated as follows: k j XX min λjk xjk ESm jkj can be iteratively calculated for each j 2 A and kj ¼ 1; :::; k j , then the value of the SPD is obtained by Eq. (11), because Activity n þ 1 is the single dummy end activity with only one available crew SPD ¼ ESm nþ1;1 (11) j2A k¼1 k j X xjk ¼ 1; j 2 A k¼1 sp1 þ dp sj1 ; ð p; jÞ 2 E Downloaded from ascelibrary.org by University Of Florida on 10/28/17. Copyright ASCE. For personal use only; all rights reserved. spm þ dp sjm ; ð p; jÞ 2 E sj1 þ ðm 1Þdj k j X xjk k¼1 k ¼ sjm ; j 2 A snþ1; m d sj1 ; sjm 0; xjk 2 f0; 1g For the aforementioned model, if l jk = k is assumed for each j and k, the reduced model, with the objective to minimize the total number of crews, represents the exact version of existing LOB scheduling methods, such as CPM/LOB, ALISS, and SHLOB. In fact, the value of l jk is the sum of all costs incurred in performing Activity j with k crews. Generally, it can be divided into material cost (MC), labor cost (LC), and equipment cost (EC). In addition, the costs that cannot be associated with a certain activity but may exist in the contract can also be integrated in the objective function. These cost elements may include project indirect cost (IC), early completion incentives (ECI), and liquidated damage cost (LDC). IC is a function of the time needed to complete the project (i.e., IC ¼ snþ1;m ICR, where ICR is the indirect cost rate per day of the project, and snþi,m is used to evaluate the project duration). ECI is the monetary incentive provided by the owner to the contractor if the project is completed before the given deadline. This cost is calculated by ECI ¼ I p , where I is the daily ECI paid to the contractor, and p – is the time when project completion is early. LDC is the ﬁnancial penalty that needs to be paid by the contractor to the owner if the project is not completed by the desired deadline. This cost is given by LDC ¼ DLD p þ , where DLD is the daily liquidated damage paid by the contractor, and p þ is the time when project completion is late. Therefore, if necessary, anyone can easily replace the objective function [Eq. (4)] by a reﬁned one [see Eq. (10)]. In doing so, additional constraints should be further considered to determine the values of p – and p þ min k j XX Eqs. (12)–(17) demonstrate formulas for calculating parameters ES1jkj and ESm jkj . In these equations, Pj is the set of predecessors of Activity j. Rjkj is the progress rate of Activity j when kj crews are hired and therefore can be calculated by Rjkj ¼ kj =dj . In particular, Rjkj is assumed to be inﬁnite for Activities 0 and n þ 1, as their unit duration is zero. On the condition that kj crews are assigned to Activity j and kp crews are employed by Activity p, parameters 1 m and tj;k denote the earliest start times of Activity j in tj;k j ð p;kp Þ j ð p;kp Þ the ﬁrst and last units, respectively, such that the precedence constraint between Activity j and its predecessor p is satisﬁed. Because the dummy Activity 0 has no predecessor, parameters ES101 and ESm 01 are calculated as zero [see Eq. (12)]. For each j 2 A f0g and 1 kj ¼ 1; :::; k j ; ES1jkj is determined by the minimum of those tj;k j ð p;kp Þ that can preserve precedence constraints between Activity j and all its predecessors [see Eq. (13)]. When the value of ES1jkj is determined, ESm jkj can be calculated using Eq. (16) ES101 ¼ ESm 01 ¼ 0 ES1jkj ¼ max p2Pj n 1 min ftj;k g j ð p;kp Þ (12) o (13) 1kp k p ðMCjk þ LCjk þ ECjk Þxjk þ IC ECI þ LDC j2A k¼1 (10) Feasibility Test Module This module aims to calculate the SPD of the project and then to analyze the feasibility of the given deadline (d ) by comparing it with the SPD. Obviously, the deadline (d ) is feasible if and only if the inequality d ≥ SPD is true. Note that, owing to the presence of work continuity, ﬁnishing a project at the SPD does not mean that the number of crews employed by each activity can reach its maximal value—that is, it is impossible to determine the value of the SPD directly by assigning k j crews to each Activity j. Let ES1jkj and ESm jkj represent the earliest possible start times of Activity j in the ﬁrst and last units, respectively, when kj crews are employed by this activity. It is not difﬁcult to see that if ES1jkj and © ASCE Fig. 3. Computational procedure of proposed method 04017044-5 J. Manage. Eng., 2018, 34(1): 04017044 J. Manage. Eng. 1 tj;k j ð p;kp Þ 8 1 > > < ESpkp þ dp ; ¼ Rpkp Rjkj ; dj m > > : tj;kj ð p;kp Þ ðm 1Þ k ; otherwise: (14) j m tj;k j ð p;kp Þ 8 > dj > 1 < tj;k þ ðm 1Þ ; Rpkp Rjkj ; j ð p;kp Þ kj ¼ > > : ESm otherwise: pkp þ dp ; Downloaded from ascelibrary.org by University Of Florida on 10/28/17. Copyright ASCE. For personal use only; all rights reserved. 1 ESm jkj ¼ ESjkj þ Rjkj m1 Rjkj (15) (16) 8 > < kj ; j 2 A; f0; n þ 1g; ¼ dj > : þ1; j ¼ 0 or n þ 1 (17) Fig. 3 introduces the computational procedure of the proposed method. Because MILP problems are nondeterministic polynomial (NP)-hard, the CPLEX MIP Solver may not terminate in a reasonable amount of time. Hence, a maximum running time (e.g., 1 min, 1 h, or 1 day) is enforced. If the procedure stops before the maximum running time, an optimal solution will be found; otherwise, a nearoptimal solution is reported. Illustrative Example To show an application of the proposed method, a highway project that was ﬁrst presented by Dolabi et al. (2014) is considered. From the practitioners’ point of view, the project is assumed to consist of 24 serial activities with no buffers and is divided into 10 units. Activity descriptions, staff-hour estimates to ﬁnish a unit, crew sizes, daily working hours, and unit durations are given in Table 2. The unit duration of an activity (Column 6) is calculated by dividing the required staff-hours to ﬁnish a unit (Column 3) by the product of the number of workers in a crew (Column 4) and the daily working hours (Column 5). Considering that the cost of an activity for employing a crew is usually proportional to the crew size, the authors adopt a simple but realistic setting of λjk ¼ kNWj , in which NWj represents the number of workers in a crew of Activity j (Column 4). For the illustrative example, the maximal number of available crews is 10 for each activity, and the deadline is set to 238 days. The proposed method is used to schedule the project considering two cases. In Case 1, the authors do not consider the timecost trade-off analysis and adopt the objective of minimizing the totalPnumber Pk j of crews as existing methods have done (i.e., kxjk ). In Case 2, the proposed objective function min j2A k¼1 [Eq. (4)] is considered. The optimal crew distribution for each case is presented in the last two columns in Table 2. Then, the project cost for each case can be calculated by the sum of the crew employment costs of all activities, where the crew employment cost of an activity is the product of the crew size (Column 4) and the calculated number of crews (Column 7 or 8). The results show that, in total, 66 crews should be employed for project completion in Case 1 with a project cost of 373. In comparison, several activities in Case 2 have to be rescheduled by adjusting their numbers of crews under pressure to reduce the project cost. Speciﬁcally, the numbers of crews are reduced by one for Activities 3, 22, and 23; Activities 5 and 6 get an additional crew. The aforementioned adjustment results in a total number of crews of 65, and Table 2. Project Information and Computational Results of Different Methods Act 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 Description Mobilization Surveying Access road construction Clearing grubbing Earthmoving Ditch excavation Culvert installation Peat excavation and swamp backﬁll Embankment Utility work Fine grading Granular base (bulk) Barrier walls Granular base (ﬁne) Signage (foundations) Paving (base) Signage (installation) High mast lighting Paving (wearing course) Shoulder granular Lane marking Landscaping Checkout and acceptance Demobilization Number of crews Required staff-hours to finish a unit Number of workers in a crew Daily working hours Unit duration (days) Case1 Case 2 192 64 576 576 1,040 400 80 192 4 2 6 8 10 5 5 6 8 8 8 8 8 8 8 8 6 4 12 9 13 10 2 4 3 2 5 3 4 3 1 2 3 2 4 3 5 4 1 2 432 160 256 336 576 384 1,600 336 288 224 288 320 16 72 80 160 6 4 4 6 8 6 10 6 4 4 6 5 2 3 2 5 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 9 5 8 7 9 8 20 7 9 7 6 8 1 3 5 4 4 2 3 2 2 2 5 2 3 2 2 3 1 3 4 3 4 2 3 2 2 2 5 2 3 2 2 3 1 2 3 3 Note: The boldface means that the number of crews assigned to the activity is different in the two cases. © ASCE 04017044-6 J. Manage. Eng., 2018, 34(1): 04017044 J. Manage. Eng. Computational Experiments This section reports the results of computational experiments that were used to analyze the performance of the proposed method in solving large-size problems, with the obtained results compared against three existing LOB scheduling methods, including CPM/ LOB, ALISS, and SHLOB. All of these methods are coded with MATLAB (R2015b) and run on a PC with an i7-4700MQ CPU, an 8-GB RAM memory, and Windows 7 (64 bit) operating system. The CPLEX MIP Solver (Version 12.6) is used for solving the MILP model presented in this paper. Test Problems All test problems are generated randomly based on the following four parameters: number of real activities (n), maximal number of available crews (k j ), coefﬁcient of network complexity (CNC), and deadline tightness (u ). Table 3 presents the settings for these parameters. Furthermore, the number of units (m) is ﬁxed at 30. Because Computational Results and Discussions An intuitive report on the results of the computational experiments is shown in Fig. 4, where the symbol %svd is the percentage of instances that can be solved, and %opt is the percentage of instances solved to optimality. Fig. 4 shows that the proposed method is able to obtain optimal solutions for all 2,700 instances. In comparison, SHLOB provides each instance with a feasible solution, but only 11.93% (161 out of 1,350) are found to be optimal. Of the 2,700 instances, 57.59% can be solved by ALISS, of which only 1.7% (46 out of 2,700) are solved to optimality. CPM/LOB fails to solve any instance (i.e., %svd = 0). The results highlight the good performance of the proposed method in deadline constraint satisfaction and crew number minimization. Table 3. Parameters of Test Problems Parameter Number of real activities (n) Maximal crew available number (k j ) CNC Deadline tightness (u ) Values 60, 90, or 120 Randomly chosen from the interval [2,5], [6,10], or [11,20] 1 or 1.5 0.05, 0.2, 0.35, 0.5, and 0.65 %svd Percentage (%) Downloaded from ascelibrary.org by University Of Florida on 10/28/17. Copyright ASCE. For personal use only; all rights reserved. existing LOB scheduling methods are designed to minimize the total number of crews and do not consider the time-cost trade-off, a fair test environment should be considered by assuming l jk = k for each j and k. When the values of parameters n, k j , CNC, and u are determined, a random instance is obtained using the following four steps: 1. Randomly construct an activity-on-node (AON) network (A, E) using the problem generator (ProGen) developed by Kolisch et al. (1995) based on parameters n and CNC. The parameter CNC represents the average number of nonredundant arcs per node in an AON network in which each node represents an activity and each arc indicates the precedence constraint between the corresponding activity pair. If the value of CNC is set to 1, then a serial network will be obtained in which each activity has only one successor. 2. Randomly determine the unit duration (dj) for each Activity j from a uniform distribution over the set of f1; 2; :::; 50g. 3. Calculate the SPD and longest project duration (LPD) of the random instance. The SPD is determined according to the feasibility test module, whereas the LPD is equal to the makespan of the LOB schedule in which each activity is assigned to only one crew. 4. Set the deadline d ¼ SPD þ ðLPD SPDÞu . For each parameter level combination, the authors generate 30 random instances, which gives a total of 3 3 2 5 30 = 2,700 instances. Each instance is solved separately by using CPM/LOB, ALISS, and the proposed method. For SHLOB, only the instances with CNC = 1 are considered. Besides that, a maximum running time of 1 min is set for the proposed method. reduces the project cost to 363 with a savings of 2.75%. Despite the limited scope of the analysis, the results support the thesis that choosing whether or not to implement the time-cost trade-off analysis may have a signiﬁcant impact on crew distribution and project cost. The authors discussed this study with some researchers and practitioners, and they all agree that the deadline satisfaction problem in LOB scheduling should be extended to integrate the timecost trade-off analysis. Generally, the chief target of construction companies is to maximize proﬁts; however, this target may not be achieved if the project cost cannot be evaluated accurately. If the planners schedule the project according to the objective of minimizing the total number of crews rather than minimizing the project cost, the ﬁnal crew distribution may be far from optimal and bring construction companies a loss. 100 90 80 70 60 50 40 30 20 10 0 100 %opt 100 57.59 11.93 1.7 Proposed method SHLOB ALISS 0 CPM/LOB Fig. 4. Comparison of computational results for different methods © ASCE 04017044-7 J. Manage. Eng., 2018, 34(1): 04017044 J. Manage. Eng. Table 4. Computational Results for Instances with Serial Activities (CNC = 1) CNC = 1 ALISS Proposed method k j n u Nfea Nopt Avg dev (%) Max dev (%) Nfea Nopt Nfea Nopt Avg dev (%) Max dev (%) Avg time (s) Max time (s) [2,5] 60 0.05 0.2 0.35 0.5 0.65 0.05 0.2 0.35 0.5 0.65 0.05 0.2 0.35 0.5 0.65 0.05 0.2 0.35 0.5 0.65 0.05 0.2 0.35 0.5 0.65 0.05 0.2 0.35 0.5 0.65 0.05 0.2 0.35 0.5 0.65 0.05 0.2 0.35 0.5 0.65 0.05 0.2 0.35 0.5 0.65 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 30 1 1 9 18 26 0 1 0 3 16 0 0 0 1 5 0 0 0 4 19 0 0 0 0 15 0 0 0 0 5 0 0 1 4 18 0 0 0 1 8 0 0 0 1 4 3.23 2.79 1.34 0.54 0.19 4.77 3.82 2.77 1.28 0.49 5.63 3.78 2.55 1.47 0.77 7.31 5.04 3.15 2.06 0.52 8.75 6.05 3.85 2.51 0.63 10.27 7.02 4.65 2.69 1.03 8.44 4.70 3.96 1.63 0.66 10.18 6.41 4.17 2.39 0.85 11.07 7.71 5.18 2.76 1.23 7.02 6.45 3.70 1.43 1.49 10.34 7.97 5.13 2.75 1.96 9.25 5.91 4.88 3.97 1.50 12.28 9.02 6.98 5.19 1.43 15.50 10.49 6.43 6.78 2.83 15.93 12.24 8.42 6.25 2.16 20.00 10.00 9.89 4.82 2.78 19.23 10.23 8.97 4.24 2.75 19.95 12.02 8.52 4.94 3.57 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 14 30 30 30 0 14 28 30 30 0 20 30 30 30 0 5 25 29 30 0 2 25 30 30 0 3 25 29 30 0 2 20 30 30 0 1 17 30 30 0 2 19 30 30 0 0 0 2 3 0 0 0 0 0 0 0 1 0 0 0 0 1 1 2 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0 0 0 — 4.39 2.80 2.58 2.71 — 5.30 3.65 3.30 3.75 — 5.24 3.22 3.31 3.48 — 7.00 6.64 3.52 3.39 — 7.35 7.45 2.91 2.88 — 12.88 7.24 3.44 3.18 — 7.44 6.38 4.02 3.47 — 9.82 5.68 3.50 3.62 — 6.81 8.87 3.60 3.28 — 7.95 9.76 5.71 5.88 — 9.29 8.47 6.42 6.73 — 9.34 6.92 6.16 5.88 — 13.51 23.66 8.75 7.04 — 9.70 24.09 7.50 4.76 — 22.37 17.88 8.44 7.19 — 7.48 20.45 13.75 8.45 — 9.82 9.63 7.32 7.62 — 7.02 24.32 12.43 5.00 0.09 0.05 0.03 0.03 0.03 0.18 0.09 0.06 0.05 0.04 0.29 0.12 0.08 0.06 0.06 1.05 0.14 0.07 0.05 0.04 2.42 0.27 0.11 0.07 0.06 4.44 0.50 0.17 0.10 0.08 2.08 0.21 0.09 0.07 0.06 4.85 0.40 0.14 0.10 0.08 9.12 0.75 0.22 0.13 0.11 0.21 0.10 0.06 0.04 0.04 0.37 0.16 0.11 0.07 0.06 0.56 0.18 0.14 0.11 0.09 2.50 0.31 0.19 0.14 0.06 4.01 0.70 0.23 0.11 0.11 12.72 2.14 0.50 0.20 0.12 3.52 0.49 0.20 0.10 0.08 11.51 0.99 0.26 0.13 0.13 18.80 1.65 0.40 0.24 0.16 90 Downloaded from ascelibrary.org by University Of Florida on 10/28/17. Copyright ASCE. For personal use only; all rights reserved. CPM/ LOB SHLOB 120 [6,10] 60 90 120 [11,20] 60 90 120 Tables 4 and 5 detail the computational results of all tested methods. In Tables 4 and 5, the symbol Nfea (or Nopt) denotes the number of instances for which a feasible (or an exact) solution is found. The column labeled Avg dev (or Max dev) displays the average (or maximal) percentage of deviation from optimality (only for the instances of column Nfea). The column labeled Avg time (or Max time) indicates the average (or maximal) running time in seconds needed to solve the instances. Considering that no feasible solution is found by CPM/LOB for all instances, the corresponding columns Avg dev and Max dev are ignored. Similarly, the columns Nfea, Nopt, Avg dev, and Max dev are ignored for the proposed method because an optimal solution can be given for each instance. © ASCE As shown in Tables 4 and 5, all instances with up to 120 activities can be solved by the proposed method within 20 s. Dolabi et al. (2014) also developed an exact model in the form of mixed-integer nonlinear programming, but their model is less practical because it took approximately 31 h to deal with a project with 24 activities. This means that the proposed method outperforms the existing exact model and is capable of providing exact solutions for large problems in a reasonable amount of time. By comparison, the maximal running times of SHLOB, ALISS, and CPM/LOB are less than 1 s (the speciﬁc data are not listed). The result is foregone, because these methods belong to heuristic algorithms that trade optimality, accuracy, or precision for speed. 04017044-8 J. Manage. Eng., 2018, 34(1): 04017044 J. Manage. Eng. Table 5. Computational Results for Instances with Nonserial Activities (CNC = 1.5) CNC = 1.5 ALISS Proposed method k j n u Nfea Nopt Nfea Nopt Avg dev (%) Max dev (%) Avg time (s) Max time (s) [2,5] 60 0.05 0.2 0.35 0.5 0.65 0.05 0.2 0.35 0.5 0.65 0.05 0.2 0.35 0.5 0.65 0.05 0.2 0.35 0.5 0.65 0.05 0.2 0.35 0.5 0.65 0.05 0.2 0.35 0.5 0.65 0.05 0.2 0.35 0.5 0.65 0.05 0.2 0.35 0.5 0.65 0.05 0.2 0.35 0.5 0.65 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 3 14 21 27 29 2 14 27 28 30 5 15 23 29 29 0 7 20 27 29 0 4 14 24 27 0 6 17 24 29 0 4 15 27 29 0 2 12 26 30 0 1 13 25 27 0 1 2 1 6 0 0 0 1 4 0 0 0 3 9 0 0 0 1 1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 2 0 0 0 0 1 0 0 0 0 0 2.71 4.15 4.80 4.77 3.04 3.61 7.45 5.83 4.44 3.34 4.70 6.41 4.89 2.71 2.02 — 10.88 11.34 9.50 6.73 — 10.85 13.26 7.98 4.86 — 12.92 8.93 6.76 3.96 — 6.98 10.56 7.69 4.43 — 14.21 10.37 9.44 5.52 — 20.00 9.95 8.04 4.31 3.53 7.69 12.50 8.96 7.94 4.17 15.38 19.81 13.00 11.46 11.54 14.56 13.10 10.00 7.26 — 18.37 19.51 20.29 20.00 — 18.52 29.17 20.00 16.33 — 19.05 12.84 15.91 8.73 — 12.24 35.53 14.29 16.42 — 17.56 20.97 21.10 15.46 — 20.00 23.68 20.45 9.85 0.04 0.05 0.04 0.03 0.03 0.10 0.11 0.07 0.05 0.03 0.15 0.15 0.11 0.06 0.04 0.35 0.20 0.14 0.07 0.04 1.20 0.64 0.35 0.12 0.07 2.41 1.64 0.51 0.17 0.08 0.59 0.29 0.16 0.08 0.06 3.37 1.22 0.51 0.20 0.12 5.00 1.93 0.76 0.21 0.12 0.09 0.09 0.11 0.07 0.05 0.23 0.21 0.13 0.12 0.06 0.22 0.33 0.22 0.14 0.07 1.56 0.34 0.29 0.17 0.07 3.49 2.61 1.33 0.28 0.13 4.51 3.95 2.02 0.72 0.19 1.48 0.70 0.30 0.14 0.15 7.22 3.81 1.08 0.50 0.32 16.08 3.79 3.05 0.49 0.27 90 Downloaded from ascelibrary.org by University Of Florida on 10/28/17. Copyright ASCE. For personal use only; all rights reserved. CPM/LOB 120 [6,10] 60 90 120 [11,20] 60 90 120 Now, this paper further investigates the performance of SHLOB and ALISS using the data listed in Table 4. It is not necessary to discuss CPM/LOB in more detail because the corresponding %svd and %opt are equal to zero. The computational results of SHLOB and ALISS for different n, k j , and u are shown in Figs. 5–7, respectively. The results conclude that SHLOB is superior to ALISS owing to the following reasons: ﬁrst, although their average deviations (Avg dev) are more or less the same, SHLOB can provide feasible solutions for more instances and keeps the maximal deviation (Max dev) at a lower level for each parameter level combination; and second, ALISS is effective only for the instances with a loose deadline constraint (i.e., the value of u is larger). In particular, if the © ASCE value of u is set to 0.2, only 14% (63 out of 450) of the instances can be solved by ALISS. This ratio will be reduced to zero if a smaller deadline tightness (u ) of 0.05 is considered, as shown in Fig. 7. For nonserial projects (i.e., the instances with CNC = 1.5), SHLOB is invalid owing to its own ﬂaws, and the computational results listed in Table 5 suggest that ALISS outperforms CPM/ LOB. Examining Figs. 5 and 6 again, the authors observe that the average deviation of SHLOB (ALISS) does not increase signiﬁcantly as the project size (measured in terms of n and k j ) grows. To be speciﬁc, when the number of activities (n) increases from 60 to 120, the average deviation of SHLOB (ALISS) slightly increases from 2.36 04017044-9 J. Manage. Eng., 2018, 34(1): 04017044 J. Manage. Eng. 30 Avg. Dev. (SHLOB) Avg. Dev. (ALISS) Max. Dev. (SHLOB) Max. Dev. (ALISS) Percentage (%) 25 20 15 10 0 60 90 %svd SHLOB ALISS n =60 100 61.11 120 n =90 100 59.33 n =120 100 61.78 Fig. 5. Average and maximal deviations of SHLOB and ALISS for different n 30 Avg. Dev. (SHLOB) Avg. Dev. (ALISS) Max. Dev. (SHLOB) Max. Dev. (ALISS) 25 Percentage (%) Downloaded from ascelibrary.org by University Of Florida on 10/28/17. Copyright ASCE. For personal use only; all rights reserved. 5 20 15 10 5 0 U[2,5] U[6,10] %svd SHLOB ALISS U[2,5] 100 70.22 U[6,10] 100 58.44 U[11,20] U[11,20] 100 53.56 Fig. 6. Average and maximal deviations of SHLOB and ALISS for different k j (3.47) to 4.76 (4.46%). Moreover, when the maximal number of available crews (k j ) increases from U[2,5] to U[11,20], the average deviation of SHLOB (ALISS) slightly increases from 3.04 (3.91) to 4.52 (4.35%). This phenomenon may indicate that SHLOB and ALISS are robust to the activities and maximal crew available numbers. In contrast, Fig. 7 reveals that as the deadline tightness (u ) decreases, the average deviation of SHLOB (ALISS) rapidly increases from 0.71 (3.72) to 7.74 (5.92%). Meanwhile, the maximal deviation of SHLOB (ALISS) rapidly increases from 3.57 (8.45) to 20 (24.32%). The reason lies in their limited ability in solving exact or high-quality solutions for the instances with a tight deadline constraint. The aforementioned experiment fully explains the limitations of existing methods, and the computational results validate the © ASCE effectiveness of the method presented in this paper. However, the proposed method also has some limitations. First, coding the objective function as well as the required constraints in machine language may be time-consuming and prone to errors. Second, the proposed method is based on MILP, and it is difﬁcult for practitioners to obtain the optimal LOB schedule if no commercial software is available. Concluding Remarks This paper presents an exact MILP-based method for solving the deadline satisfaction problem in LOB scheduling. The method may be the ﬁrst LOB scheduling method that considers the time-cost 04017044-10 J. Manage. Eng., 2018, 34(1): 04017044 J. Manage. Eng. 30 Avg. Dev. (SHLOB) Avg. Dev. (ALISS) Max. Dev. (SHLOB) Max. Dev. (ALISS) Percentage (%) 25 20 15 10 Downloaded from ascelibrary.org by University Of Florida on 10/28/17. Copyright ASCE. For personal use only; all rights reserved. 5 0 0.05 %svd SHLOB ALISS 0.2 0.05 100 0 0.35 0.2 100 14 0.5 0.35 100 48.67 0.5 100 59.56 0.65 0.65 100 60 Fig. 7. Average and maximal deviations of SHLOB and ALISS for different u trade-off and deadline constraint satisfaction and is capable of providing optimal solutions for large projects in a reasonable amount of time. More importantly, the method is easy for practitioners to use and does not need them to master any optimization theory. The ﬁnal schedule constructed by the method is more competitive than ever. As discussed in the previous studies, the proposed method can be applied to repetitive projects with typical activities (e.g., highways, high-rise buildings, and housing projects with identical units). The main contributions of this study include the following three aspects. First, the proposed method integrates the time-cost tradeoff analysis; it is thus capable of determining a crew distribution that minimizes the project cost without exceeding the given deadline. Moreover, the proposed method is also valid for instances with a tight deadline constraint. Second, this paper developed the ﬁrst procedure (i.e., the feasibility test module) for calculating the shortest duration of a project and judging the feasibility of the given deadline, which enhances the value of this work for practical applications. Third, this paper conducted extensive computational experiments, and the results will help practitioners fully understand the performance of the proposed method and existing procedures. Similar experiments have never been mentioned in the literature. Moreover, the computational results presented in this paper may be used as benchmarks to evaluate the performance of other future heuristic methods. In the future, a potential research direction lies in integrating LOB scheduling with other realistic settings. These settings may result from the needs of planners or the characteristics of the project itself. A meaningful extension could be the deadline satisfaction problem in LOB scheduling with nontypical projects (i.e., the activity duration may be variable for different units). Nontypical projects are as common as typical projects in real life; however, the scheduling decision for nontypical projects is very different from that for typical projects. Acknowledgments The authors acknowledgment the Natural Science Foundation of China (Grants 71271081 and 71701069) and Fundamental © ASCE Research Funds for the Central Universities (Grant 2017MS175). The authors are grateful to the anonymous reviewer and editor for a careful scrutiny of details and for comments that helped improved this paper. References Ali, M. M., and Elazouni, A. 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