Schedule control model for linear projects based on linear scheduling method and constraint programming

Automation in Construction 37 (2014) 22–37

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Automation in Construction journal homepage: www.elsevier.com/locate/autcon

Schedule control model for linear projects based on linear scheduling method and constraint programming Yuanjie Tang a, Rengkui Liu a,⁎, Quanxin Sun b a b

State Key Laboratory of Rail Traffic Control and Safety, Beijing Jiaotong University, No. 3 of Shangyuan Residence, Haidian District, Beijing, 100044, PR China MOE Key Laboratory for Urban Transportation Complex Systems Theory, Beijing Jiaotong University, No. 3 of Shangyuan Residence, Haidian District, Beijing, 100044, PR China

a r t i c l e

i n f o

Article history: Accepted 18 September 2013 Available online 25 October 2013 Keywords: China railway construction project Linear scheduling method Resource leveling Schedule control Constraint programming

a b s t r a c t China Railway is undertaking massive construction and development projects. A reasonable and resource-leveled schedule that allows for adjustments for unforeseen circumstances during construction is critical for managing railway construction projects. Currently, most construction projects use traditional network planning methods or the Gantt schedule for project management. However, these methods have limited applicability to railway construction projects, which are typically linear. This study uses the linear scheduling method and constraint programming techniques for solving schedule control problems faced during railroad construction. The proposal comprises a schedule control model, scheduling model, and schedule control system; the scheduling model is central to the schedule control model. Characteristics such as high flexibility and practicality facilitate multi-objective optimization during scheduling and modification of the linear schedule. The proposed model and algorithm were validated by comparing results with actual data from a highway construction project and the Urumqi–Dzungaria railway construction project. © 2013 Elsevier B.V. All rights reserved.

1. Introduction China is currently undertaking the largest railway construction operation in the world. According to China's Twelfth Five-Year Development Plan, the total mileage of the country's operating railway will increase from 91,000 km to 120,000 km by the end of 2015. Railway construction projects may be considered typical “linear projects,” which means that they are characterized by a series of repetitive activities. In addition to railways, other linear projects include construction of highways, pipelines, tunnels, etc. In the field of construction in China, scheduling often relies on network planning methods such as the critical path method (CPM) [1]. Such traditional network planning methods, though, do not apply to the kind of linear projects mentioned above. Among others, the linear scheduling method (LSM) has been suggested as a new method for scheduling such linear projects, and has been shown to offer unique advantages [2–6]. Scheduling is one of the most fundamental functions of construction project management [7]. Moreover, the schedule is regarded as the core part of a railway construction organization plan, and even of railway construction project management [1]. Changes are quite common in construction projects, so schedules often need to be updated or rescheduled [8–11]. This is especially true for railway construction projects. The construction periods of railway projects often take years, and during such a long time, environments and situations routinely experience change. To accommodate these changes, appropriate adjustments ⁎ Corresponding author. Tel./fax: +86 10 51687137. E-mail address: [email protected] (R. Liu). 0926-5805/$ – see front matter © 2013 Elsevier B.V. All rights reserved. http://dx.doi.org/10.1016/j.autcon.2013.09.008

must be made to schedules and to the implementation of schedules in response to these changes. Thus, schedules undergo a continuous process of revision to ensure their ongoing validity, so that the project can be implemented in a defined, organized manner [12]. Construction resources usually consist of manpower, machines, materials, money, information, and management decisions [13]. To guarantee that construction is completed within a predetermined duration and budget, resources need to be well managed. In this regard, it has aptly been said, “in some degree, construction management is nothing but resource management” [14]. Resource management typically includes resource allocation and resource leveling. The major challenge is for management to achieve leveling of resources in construction projects with a fixed duration [7]. Resource leveling makes the resource demand curve as smooth as possible, avoiding short-term peaks and troughs, to reduce resource costs and management costs and avoid unnecessary losses. In recent years, to enhance the standardized management of railway construction, the Chinese government has strengthened the management of resources for railway construction and the requirements of resource leveling during scheduling [1,15]. This paper mainly studies the use of the LSM in the scheduling of China's railway construction projects under the constraint of a fixed duration. In doing so, we thus establish a schedule control model and a schedule control system for China's railway construction projects. The core part of the schedule control model is a multi-objective optimization scheduling model, which is combined with constraint programming technique. Lastly, the proposed model and algorithm were validated through a comparison of the model results with actual data

Y. Tang et al. / Automation in Construction 37 (2014) 22–37

from a highway construction project [7,20] and the Urumqi–Dzungaria railway construction project undertaken by the Urumqi Railway Bureau. 2. Literature review 2.1. Resource leveling through the use of LSM The LSM is directly related to the LOB technique, a scheduling method developed by the U.S. Navy in the early 1950s. However, the exact origin of the LSM is not quite clear [2,7]. Based on the characteristics of the construction of linear projects, the LSM uses a Cartesian coordinate system to describe the construction schedule of a linear project. The horizontal axis usually represents the spatial location of the project, while the vertical axis represents the time progress of the project. This makes it possible for any activity to be expressed on the basis of its construction time and spatial location in the two-dimensional coordinate system, using a certain icon. The two-dimensional coordinate system used to describe the schedule for the projects and the elements within it, is called an LSM diagram. The LSM includes activities, the rate at which activities take place, and the buffer between activities, among other key elements. The LSM divides construction project activities into three types: linear, block, and bar types, and the linear-type activity can be further sub-divided into continuous linear activity and intermittent linear activity [16]. The concept of rate is used to describe a linear type activity, which reflects the progress of a linear activity in a given location per unit time. This is the most important characteristic of linear type activities, and it also represents a major difference between the LSM and CPM. In an LSM diagram, the rate is expressed as the slope of a linear type of activity. At the same time, the rate of a linear type activity reflects the resource usage of the activity and varies according to the increase or decrease in resource usage; in an LSM diagram, the slope of activities becomes smaller or larger with an increase or decrease in resource usage. In an LSM diagram, the distance between two activities in the horizontal direction is called the distance buffer, and the distance between two activities in the vertical direction is called the time buffer. A buffer represents the requirements of a given activity's technological, managerial, or other external constraints. The minimum time and minimum distance between two activities that cannot be exceeded are called the minimum time buffer and the minimum distance buffer, while the maximum time and maximum distance between two activities that cannot be exceeded are called the maximum time buffer and maximum distance buffer. The critical path of a schedule developed by the network planning method can be calculated. Similarly, there is a critical path in a schedule developed by the LSM, which is called the controlling activity path (CAP) [16]. A number of scholars have studied the role of the critical path in the LSM [3,16–19]. Alexandros Kallantzis and Sergios Lambropoulos [17] discussed the problem of scheduling using the LSM, but did not take into account the fixed duration constraint. Furthermore, the resource usage of activities and the buffer between activities are fixed, so a schedule was developed on the basis of merely satisfying the buffer constraints, and further optimization of the schedule was limited. The model proposed by Maged E. Georgy [7], which has a scheduling function, considered the satisfaction of the fixed duration constraint and resource variability, but the initial solutions that satisfied the constraints had to be obtained in a random manner, so their efficiency is relatively low. In addition, although the author introduced the concept of a variable buffer, a maximum buffer constraint between activities had to be added artificially when there is no maximum buffer between activities. This then affects both the flexibility of the model and the quality of the solutions it can offer. In addition, a number of scholars have conducted research on resource-leveling problems based on the LSM. Kris G. Mattila and

23

Duley M. Abraham [20] employed an integer programming method to model and solve resource-leveling problems based on the LSM. In this approach, however, the process relies upon an existing schedule and only considers the adjustment of resources for segments of non-critical activities, which does not provide enough flexibility [7]. Moreover, the adjustment of the segments of non-critical activities may disrupt the continuity of resources, which is a natural advantage of the LSM. Maged E. Georgy [7] considered the adjustment of resources for the whole activity, but a model built upon a genetic algorithm cannot guarantee the quality of the solution. Moreover, the maximum buffer constraints that are artificially added to the model would also affect the optimization results. In summary, the existing models suffer from the following shortcomings: ■ Scheduling: Automatic preparation of the scheduling plan cannot be achieved. Even if it is achieved, the efficiency is low. ■ Buffer handling: Some models set the buffer limit as a constant, which is ineffective. Although other models adopted a variable buffer limit, the constraint is artificially increased, thereby lowering the flexibility of the model and quality of the solution. ■ Optimization of schedule based on resource leveling: Optimization cannot be achieved by some models, while others cannot guarantee the quality of the solution due to their inherent limitations. Some models are not sufficiently flexible owing to critical path limitations or dependence on the initial schedule, which affects the solution quality. In addition to the aforementioned shortcomings, the proposed scheduling models based on the LSM mentioned above [7,17,20] are limited to providing schedules before the beginning of a project. They are incapable of revising the schedule or performing rescheduling according to changes in environments and situations that take place during construction. In addition, construction site conditions are considered incompletely, since a single activity involving multiple crews, and intermittent linear-type activities were not considered in the model's development. The problem of construction project resource leveling is a category of combinatorial optimization problem; in addition to mathematical programming methods and heuristic methods, the constraint programming (CP) technique has been employed as a new approach to handling this kind of problem [21–24]. Shu-Shun Liu applied the CP technique to the problem of resource allocation optimization on a bridge project, which is a linear project [25]. This paper adopts the LSM to construct a constraint satisfaction problem (CSP)-based multi-objective optimization scheduling model under the constraint of a fixed duration, to realize railway construction project schedule control. The model adopts both resource-leveling and minimum schedule changes as objectives. Compared with the findings of existing studies, the proposed model is innovative in the following ways: ■ The proposed model can automatically prepare and optimize the schedule of a linear project within the constraints of a fixed duration without depending on any existing schedule. In addition, it is more efficient and can generate the optimized solution in shorter time owing to the use of the highly efficient CP technology. ■ This model makes use of constraint programming (CP) technology and benefits from the efficiency of the latter. The model does not require additional constraints to arrive at a solution—that is, no artificial limitations are imposed on the data range of the buffer. At the same time, model flexibility is enhanced, and solution quality is assured. ■ This model provides a comprehensive understanding of the on-site situation. It can prepare and optimize the schedule even when multiple construction teams are at work and when intermittent linear activities are involved.

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■ This model can describe various scenarios of time constraints faced during construction, giving it greater flexibility. Examples of time constraints include: (i) those that exist between different types of activities; (ii) those imposed by activities that are being carried out at different rates; (iii) those that concurrently exist between a particular activity and multiple predecessor activities; (iv) when multiple construction teams are working simultaneously; and (v) those that exist at any arbitrary mileage point between activities. In addition, the model can simultaneously consider the two objectives of resource leveling and minimization of schedule revisions, allowing schedule revisions to be made based on multiple optimized objectives. These various features allow for strong process control. Lastly, the effectiveness of the proposed model and algorithm were validated through a comparison of its results with actual data from a highway construction project [7,20] and the Urumqi–Dzungaria railway construction project undertaken by the Urumqi Railway Bureau. The optimized solution was then obtained using CP technology.

2.2. Current state of schedule control and management of China's railroad construction projects In China, railway construction project scheduling mainly adopts the Gantt chart method, visual progress method, and network planning method, as railway construction-related management specifications have explicit representations [1,15]. Table 1 summarizes a number of different scheduling methods and their scopes of application. Table 1 shows that in the scheduling of China's railway construction projects, different methods are adopted according to different aspects of the schedule. Whether guidance (overall) or operative scheduling is used, it generally adopts some of the scheduling methods presented in Table 1, combined with a method of arranging a project (activities) from back to front, artificially. Any necessary adjustments to the schedule are then made in response to actual situations [12]. Thus, the quality of the schedule is largely dependent on the personal experience of the scheduler, and the schedule's rationality cannot be guaranteed. Moreover, the unsuitability of traditional methods such as the network planning method for a railway construction project, which is a linear project, has already been discussed [2–6]. In China, the specifications [1,15] for railway construction project management require that during the scheduling process, the resource allocation plan should be developed according to the schedule and other conditions of the project, so as to fulfill the aim of effective management and resource leveling. However, the scheduling of China's railway construction projects and the allocation of resources do not proceed simultaneously. Resource allocation planning is subordinate to the schedule, and a level allocation of resources can only be realized through local adjustments to the schedule after its completion. This approach to scheduling is not conducive to a level allocation of resources, and may even lead to problems

Table 1 China railway construction project scheduling methods and their scopes of application. Scheduling method

Advantage

Scope of application

Gantt chart

Direct, simple, and easy to understand Arranges time according to location, expresses the progress of project more intuitively than the Gantt chart method Clearly reflects the logical relationship between activities Reflects the scheduled dates of specific milestones

Every stage of the construction process Mainly applied in the design phase, and the guidance (overall) schedule of construction stage Mainly applied to the operating schedule Guidance schedule

Visual progress graph

Network planning method Milestone plan

such as an insufficient supply of resources, which of course affects the project's progress. Dynamic management is employed in China's railway construction project schedule control, which means that revision of the schedule is unavoidable. The specifications [1,15] require that when there is a large deviation between the schedule and the actual progress being made, adjustments should be made to the schedule, which of course disrupts the original plan and the management of manpower, materials, machines, money, etc. In particular, when there are multiple crews working on construction parallel to one another, this typically has a highly disruptive impact on the coordination of the project [26]. However, adjustments to China's railway construction project scheduling mainly rely on the experience of the scheduler, and due to time pressures, there may be an even larger deviation between the new schedule and the original schedule, which exacerbates the negative effects on the construction that follows. 3. Schedule control model To solve the problems involved in schedule control on China's railway construction projects, combined with the management status of these projects, we present a circulating schedule control model that consists of scheduling and adjustments, implementation, and checking, to realize schedule control of a railway project, as shown in the dashed box in Fig. 1. Due to the advantage offered by the LSM in schedule management of linear projects, the schedule control model is developed based upon the LSM. The proposed schedule control model is facilitated by a schedule control system (which is briefly introduced in Section 6). The core part of the schedule control system is a multi-objective optimization scheduling module with objectives of resource leveling and minimum schedule changes, under the constraint of a fixed duration. Combined with a schedule display module, and a comparison and analysis module, the proposed schedule control model can provide managers with an optimal schedule, as well as a comparison of the schedule with the actual progress being made. By circulating on the basis of this information, the project manager is assisted in realizing schedule control of a railway construction project. The database module acts as the basis for the control model. It mainly records and stores, and executes progress control based on three types of data: (i) outstanding work, (ii) from the original schedule, and (iii) current progress. The first two types of data are used by the scheduling module for preparing and adjusting the schedule. The second and third types of data are used by the comparative analysis module for comparing the actual progress and against the original plan. Outstanding work (activities) arises from adjustments made to the schedule. The following data on the attributes of each outstanding activity are recorded: ■ Name of activity; ■ ID of activity; ■ Type of activity, which includes linear and block (bar-type activity is processed as a special block-type activity); ■ Mileage at start of activity; ■ Mileage at end of activity; ■ Minimum amount of resources for a type of activity; ■ Maximum amount of resources for a type of activity; ■ Resource productivity of an activity; ■ predecessor activity; and ■ Buffer between activities, which includes buffer type (minimum and maximum time constraints) and buffer value. Data from the original schedule refer to that generated by the proposed model and comprise the following: ■ Serial number of activity; ■ Resources allocated to activity;

Y. Tang et al. / Automation in Construction 37 (2014) 22–37

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Fig. 1. Railway construction project schedule control model.

■ Commencement date of activity; and ■ Completion date of activity. Data on current progress include all data from work (activities) that have been completed since the previous adjustment to the schedule. These are ■ Serial number of activity; ■ Actual commencement date of activity; and ■ Actual completion date of activity. Next, the three steps of scheduling and adjustment, implementation, and checking shown in Fig. 1 are briefly reviewed. Scheduling and adjustment: Schedule with the objective of resource leveling for an initial schedule. When deviations appear between the schedule and the actual progress being made, rescheduling takes place with multiple objectives of resource leveling and minimum changes in the schedule, and guide for construction based on the schedule. This is the core part of the schedule control model, which is facilitated by the scheduling module, whose core part is a CSPbased scheduling model (which is introduced in Section 5). Implementation: Organize construction according to the schedule, while the project manager periodically collects data on the project's progress. This step is facilitated by the display module. Checking: Compare and analyze the deviations based on the schedule versus data on the actual progress being made, to determine whether adjustments must be made to the schedule to meet the constraint of completion on a fixed duration. If the schedule does not need to be adjusted, organize construction based on the original schedule; otherwise, reschedule according to the actual degree of progress. This step is facilitated by the comparison and analysis module.

4. Constraint programming Constraint programming (CP) is a programming paradigm that integrates mathematics, artificial intelligence, and operations research techniques with the aim of solving CSPs and combinatorial problems [24,27]. The basic CP formulation consists of problem specification, consistency techniques, and systematic search strategies for problem solving. A constraint satisfaction problem consists of three components: (1) an n-tuple number of variables X = {x1, …, xn}; (2) for each variable xi, there exists a finite domain Di of possible values.; (3) an m-tuple number of constraints C = {c1, …, cm} restricting the values that the variables can simultaneously take, and that can then be recognized [28–30]. To improve the computational efficiency of solving problems, CP provides users with different consistency techniques, such as node consistency, arc consistency, and path consistency, for variable domain reduction and different search strategies, including generate and test (GT), backtracking (BT), forward checking (FC), etc. [28,31,32]. Compared to GT, BT is more efficient. Compared to FC, BT requires less constraint propagation, making the use of BT for the processing at each node less expensive [27,28]. In addition, appropriate variable selection and value selection under some heuristics also reduces the computational effort required and promotes the search ability [25,29,32]. When solving an optimization problem, the objective function in the problem is treated as a constraint, and this additional constraint forces the new feasible solution to have a better objective value than the current one, and the upper or lower bounds of the constraint are replaced as soon as a better objective function value is found. While recording the current best solution, the propagation mechanism narrows the domains of the decision variables to reduce the size of the search space. When no feasible solution is found, the search terminates, and the last feasible solution found is the optimal solution [21,28].

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Several approaches can be employed to handle a resource-leveling problem that belongs to a constraint satisfaction problem, such as mathematical methods, heuristic methods that include genetic algorithms, ant colony optimization, etc. [7,33–35]. Mathematical methods identify specific solutions, but the problem-solving stage usually costs much time and labor. Heuristic methods can obtain solutions in a short time, but the quality of the solutions is not assured, and the effect of the algorithm is affected by the experience of the users. Compared with heuristic methods and mathematical methods, CP can search for solutions more simply, depending upon the algorithm chosen by users. In addition, it is not restricted by any particular model formulation, such as linear equations [27,30], and the quality of the solution could also be ensured. For scheduling problems based on the LSM, CP has another advantage, namely that the prioritization of activities in linear scheduling problems becomes clear due to the logical and sequential constraints in CP [25]. As regards the proposed model, in the problem specification stage, the objective and the variables are determined. In this study, resource leveling and minimum changes in the schedule are regarded as the objectives, and decision variables include the start date and resource usage of activity. To narrow the search space and find feasible solutions, a consistency checking technique is used for constraint propagation. Considering the fact that BT search is used for the majority of CSP problems, as well as the unique characteristics of BT as stated in literature [27,28], we have adopted BT as the search strategy. In this study, IBM ILOG CPLEX Optimization Studio is used, and the ILOG OPL language [36] is adopted as the model formulation language. 5. Scheduling model The proposed scheduling model is a multi-objective optimization scheduling model based on CSP and LSM. It is able to realize the aim of scheduling and rescheduling of linear projects automatically based on one or more objectives. In addition, the optimal solution of the model can be obtained in combination with the CP technique, in which the model is the core part of the schedule control model. Both resource leveling (minimum deviation of resource usage) and minimum changes in the schedule are treated as objectives in the realization of resource leveling, while reducing as much as possible the negative effects on the construction that follows. The fixed duration and logical relationships between activities, such as sequence and buffer, are treated as constraints. To strengthen the practicability and flexibility of the model, both the rate and start date of an activity are treated as decision variables, and their domains are determined according to the following methods. Determination of the rate domain: First, construct the linear relationship between the resource usage and rate of activity according to the quota [1,37], reflect the rate of an activity in the resource usage of the activity, and combine resource allocation and scheduling together, to realize better resource leveling. The domain of resource usage is determined by the actual situations at the construction site. The proposed scheduling model involves two concepts of rate: ■ Resource production rate: the amount of work that can be accomplished by a unit of resource in a unit time period, and ■ Production rate: the amount of work that was accomplished during a unit of time. Determination of the start date domain: An activity's start date is constrained by the start date of its predecessors, which have a constraint relationship with it, and by the buffer between them. In the case of identification of the start date of an activity's predecessor, once the buffer between them is identified, the start date of the activity will then be determined. Therefore, the flexibility of the start date is reflected by the flexibility of the buffer. The proposed model treats the buffer as a decision variable that can vary between the minimum time buffer and the maximum time buffer, to improve the model's flexibility and the quality of the solution.

Compared with the results found in the available research, the proposed scheduling model has stronger practicability and flexibility for four reasons: ■ Various scenarios of time constraint that can occur between the activities during the construction progress can be analyzed comprehensively. This includes the following time constraints: (i) between different types of activities (linear and linear, and linear and block); (ii) arising from construction activities that progress at different speeds; (iii) existing between multiple activities (e.g. between a particular activity and its various predecessor activities, or arising from multiple construction teams working simultaneously); and (iv) existing at an arbitrary mileage point between activities. The various time constraints between activities are classified into two categories: minimum and maximum. The model has great flexibility and can simultaneously take into consideration the two objectives of resource leveling and minimization of schedule revisions. Changes to the scheduling plan are made on the basis of multiple optimized objectives, thereby realizing process control. ■ Ref. [1] requires that a railway construction project may adopt a streamlined operation mode in the same section, while adopting a parallel operation mode between different sections. The possibility of a parallel operating mode between different sections increases the flexibility of resource allocation, and thus has a positive effect upon resource leveling. The existing research that has been conducted on scheduling based on the LSM did not account for the case in which more than one crew is working on one activity at the same time, while the model proposed in this paper does well at solving this problem. ■ Previous studies of scheduling using the LSM become inadequate when dealing with intermittent linear activities, which are defined as activities that are paced by some other activity and are typically in progress intermittently so as to stay within a reasonable distance of the pacing activity [16]. The model proposed here describes the constraint between an intermittent linear activity and a pacing activity as a minimum time constraint and maximum time constraint that exist at the same time, and is thus able to schedule them using the LSM. ■ The proposed model was combined with the CP technique, whose efficiency makes it unnecessary to add additional constraints while problem solving. The buffer is treated as a variable, there do not need to be any additional constraints for the buffer, which makes the model more flexible, while ensuring the quality of the solution. The following variables, constraints, and objectives were adopted in the construction of the CSP-based model. 5.1. Constants The values of constants will not change during problem solving of the CSP; these constants include the following: fa la ui SDi EDi qi

First activity of the project; Last activity of the project; Resource production rate of activity i; Start location of activity i; End location of activity i; Total mileage of activity i qi ¼ EDi −SDi ;

minri maxri minbi,j maxbi,j Ci D

Minimum resource usage of activity i; Maximum resource usage of activity i; Minimum time buffer between activity i and activity j; Maximum time buffer between activity i and activity j; Type of activity i: Ci ∈ {block,line} Total duration of the project.

ð1Þ

The values of the decision variables are determined during the solution search process. The decision variables include the following:

F

Resource usage of activity i: ri ri ∈ [minri, maxri] (fixed for block type activity); STi Start date of activity i: STi ∈ [0,D].

minbD , F

E C

5.3. Decision expressions

pui ¼ r i ui ; di

ð3Þ

ð4Þ

Time needed for activity i to progress from location SDi to location x: T i;x ¼ ðx−SDi Þ=pui ;

Wi, j

ð2Þ

End date of activity i: ET i ¼ ST i þ di ;

Ti,x

minbC , E

Duration of activity i: di ¼ qi =pui ðfixed for block‐type activityÞ;

ETi

D

Production rate of activity i:




0; ST j ≥i or ET j b i 1; ST j b i and ET j ≥ i;

minbB ,C

B minbA, B minbA,G

A

G

Locationa

i

j

Minimum time buffer between activity i and activity j

Fig. 2. Conditions of minimum time constraints that may exist between activities.

ð5Þ

Boolean variable that identifies whether activity j is being executed on day i or not: W i; j ¼

minbA,C minbC , D

pui

minbE , F

5.2. Decision variables

27

Time

Y. Tang et al. / Automation in Construction 37 (2014) 22–37

ð6Þ

(1) The minimum time constraints between activity i and its predecessor activity h are described as follows: When Ci = linear and Ch = linear: ST i ≥ minbi;h þ ST h þ T h;SDi ; if pui ≤ puh and SDi ≥ SDh : ð8Þ Corresponds to case 1 in Fig. 3;

Ri

Total resource usage of the project on day i: Ri ¼

n X

r j W i; j :

ST i ≥ minbi;h þ ST h −T i;SDh ; if pui ≤ puh and SDi b SDh : ð7Þ

j¼1

5.4. Constraints Previous models can only describe the constraint between an activity and its immediate predecessor activity (such as the constraint between activity G and activity A in Fig. 2). They are unable to deal with constraints between more than two activities, such as the case in which an activity has more than one predecessor activity that has constraint with it (such as the constraint between activities A, B and C in Fig. 2), or the case in which one activity has more than one crew working on it (such as the constraint between activities C, D and E, or the constraint between activities D, E and F in Fig. 2). If we take the minimum time constraint, for example, Fig. 2 shows the conditions in which there may be minimum time constraints between activities, in which activity D and activity E are of the same type, yet are worked upon by two crews. In Fig. 2, there are constraints between activities A and B, activities B and C, activities A and C, activities D and C, activities E and C, activities D and F, activities E and F, and activities A and G. Due to the complexity of a construction site, adjustments made to the schedule during the construction process will involve a variety of situations. Unlike the case in which constraints between activities usually occur at the beginning or at the end of activities, there may be constraints at any point in the activity, such as a constraint that arises due to a cessation of work due to inclement weather [11]. For this reason, the proposed model needs to be able to deal with a variety of conditions in its realization of schedule control.

ð9Þ

Corresponds to case 2 in Fig. 3; ST i ≥ minbi;h þ ST h þ T h;EDi −di ; if pui N puh and EDi ≤ EDh : ð10Þ Corresponds to case 3 in Fig. 3; ST i ≥ minbi;h þ ST h −T i;EDh þ dh ; if pui Npuh and EDi NEDh :ð11Þ Corresponds to case 4 in Fig. 3; When Ci = block and Ch = linear: ST i ≥ minbi;h þ ST h þ T h;EDi ;

ð12Þ

Corresponds to case 5 in Fig. 3; When Ci = linear and Ch = block: ST i ≥ minbi;h þ ST h þ dh −T i;SDh :

ð13Þ

Corresponds to case 6 in Fig. 3; (2) The maximum time constraints between activity i and its predecessor activity h are described as follows: When Ci = linear and Ch = linear: ST i ≤ maxbi;h þ ST h þ T h;EDi −di ; if pui ≤ puh and EDi ≤ EDh : ð14Þ

Y. Tang et al. / Automation in Construction 37 (2014) 22–37

Time

Time

28

B

B

x2

minbA, B

x1

minbA, B

x2

A

TA, SDB x0 Location minbA, B CB = line and C A = line Case 1: puB ≤ pu A and SDB ≥ SDA

TB , SDA

x1

A

x0

Location

Time

Time

CB = line and C A = line Case 2: pu ≤ pu and SD < SD B A B A

x3

dB

x2

TA, EDB

x1

A

TB , EDA

x3 minbA, B

minbA, B

B

x2 x1

A

dA

x0 Location

x0 Location

CB = line and C A = line puB > pu A and EDB ≤ EDA

CB = line and C A = line puB > pu A and EDB > EDA

Time

Case 4:

Time

Case 3:

B

B x2

minbA, B

TA, EDB

x1

TB , SDA

x2

A

minbA, B

B

x1 x0

x0

A

Location

Location

Case 5: CB = block and C A = line

Case 6: CB = line and C A = block

Fig. 3. Method of calculating the minimum time constraint between activities.

Corresponds to case 1 in Fig. 4;

Corresponds to case 2 in Fig. 4;

ST i ≤ maxbi;h þ ST h −T i;EDh þ dh ; if pui ≤ puh and EDi N EDh : ð15Þ

ST i ≤ maxbi;h þ ST h −T i;SDh ; if pui N puh and SDi ≤ SDh : ð16Þ

Y. Tang et al. / Automation in Construction 37 (2014) 22–37

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Fig. 4. Method of calculating the maximum time constraint between activities.

Corresponds to case 4 in Fig. 4; When Ci = block and Ch = linear:

Corresponds to case 3 in Fig. 4; ST i ≤ maxbi;h þ ST h þ T h;SDi ; if pui N puh and SDi N SDh : ð17Þ

ST i ≤ maxbi;h þ ST h þ T h;EDi :

ð18Þ

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consumption on a day-to-day basis. Resource consumption deviation is expressed by the absolute difference in resource consumption of two consecutive days:

Table 2 Weight allocation scheme. Project schedule

α weight

β weight

0 0–10% 10–20% 20–30% 30–40% 40–50% 50–60% 60–70% 70–80% 80–90% 90–100%

1 0–0.1 0.1–0.2 0.2–0.3 0.3–0.4 0.4–0.5 0.5–0.6 0.6–0.7 0.7–0.8 0.8–0.9 0.9–1

0 0.9–1 0.8–0.9 0.7–0.8 0.6–0.7 0.5–0.6 0.4–0.5 0.3–0.4 0.2–0.3 0.1–0.2 0–0.1

Min

Here, Ri represents the total consumption of resources for the entire project on day i of the revised schedule, while x represents the xth day, the point at which the initial schedule had to be revised. 5.5.2. Objective function to minimize schedule changes This function is used to minimize the differences between the revised and initial schedules, so as to reduce the negative effects on the remaining construction. It is expressed as the absolute value of the difference in resource consumption by the two schedules on the same day.

ð19Þ

Min

ð20Þ

ET i ≤ D:

ð21Þ

Here, Ri represents the total consumption of resources for the entire project on day i of the revised schedule, R′i represents the total consumption of resources for the entire project on day i of the initial schedule, while x represents the xth day, the point at which the initial schedule had to be revised. Finally, to derive the optimal Pareto solution, a weighted summation is utilized to convert the multi-objective optimization problem into a single-objective problem. The optimized solution for the singleobjective problem is then obtained [38,39]. Upon conversion, the optimal objective function is as follows:

The first activity starts on day 0, and the last activity ends on day D: ET la ¼ D:

ð22Þ

ST fa ¼ 0:

ð23Þ

D X jRi −R′i j: i¼x

Corresponds to case 6 in Fig. 4; (3) The fixed duration constraints are described as follows: All the activity starts on or after day 0, and ends on or before day D: ST i ≥ 0:

jRiþ1 −Ri j:

i¼x

Corresponds to case 5 in Fig. 4; When Ci = linear and Ch = block: ST i ≤ maxbi;h þ ST h þ dh −T i;SDh :

D−1 X

Min α

(4) Constraint between an intermittent linear activity and its pacing activity. A realistic buffer needs to be maintained between an intermittent linear activity and its pacing activity, which thus leads to intermittent resource usage. To maintain continuity of resources, the constraint is described as a minimum time constraint and a maximum time constraint that work simultaneously, and the process is realized by the adjustment of the rates of the relative activities. The constraints are described by Eqs. (2) and (3).

D −1 X

jRiþ1 −Ri j þ β

i¼x

D X jRi −R′i j: i¼x

Here, α and β are the weights for the two targets, which are set by the administrators based on their relative degree of importance, in accordance with the equation α + β = 1. In particular, when α = 0 or β = 0 (when the schedule was first prepared, without any constraints imposed by the minimum changes in schedule, then β = 0.), the multiobjective optimization model becomes a single-objective optimization model with minimal schedule changes or a resource-leveling optimization model.

5.5. Objectives 5.6. Weight allocation The proposed model realizes multi-objective optimization of resource leveling and minimum changes in schedules.

In this study, weights are allocated according to the relative importance of each objective function toward achieving the objective. In addition, the allocated weights are adjusted as the construction project progresses, based on changes to the relative importance of each objective function.

5.5.1. Objective function of resource leveling In this study, the objective function of resource leveling from literature [7] is used for achieving minimum deviation in total resource

Table 3 Data on highway construction project. Activity

Type

Buffer

Start location

End location

Minimum resource

Ditch excavation Culvert installation Concrete pavement Removal Peat excavation and swamp backfill Embankment Utility work Sub-base Gravel Paving

Line Block Line

2 0 2

0 42 0

50 42 50

1 1 2

Maximum resource 3 1 7

Prod/res/day 10/3 3 day (duration) 0.34

Block Line Line Line Line Line

2 2 2 2 2 2

8 0 30 0 0 0

12 50 50 50 50 50

8 2 2 4 4 4

8 7 4 10 10 10

3 day (duration) 1.25 5 25/39 1.25 25/12

Y. Tang et al. / Automation in Construction 37 (2014) 22–37 Table 4 Optimization results using proposed model. Activity

Resource

Start date

End date

Duration (day)

Ditch excavation Culvert installation Concrete pavement removal Peat excavation and swamp backfill Embankment Utility work Sub-base Gravel Paving

2 1 3 8 3 3 10 10 10

0 0 3 8 11 25 23 31 35

8 3 23 11 25 27 31 35 38

8 3 20 3 14 2 8 4 3

During preparation of the initial schedule prior to commencement of construction, no adjustments are made to the schedule. Thus, at that point, schedule preparation and optimization constitute a resourceleveling problem. The selected target weights are α = 1 and β = 0. After commencement of construction, schedule adjustments have to be made to achieve progress control. These adjustments disrupt the original plan and affect the management of various resources, including manpower, materials, machinery, and expenditure of the project budget. Problems with overall coordination are exacerbated when multiple teams are involved in the construction work [26]. For reducing the differences between the revised and original schedules and the effects of adjustments on the outstanding construction activities, the objective of minimizing adjustments to the schedule must be introduced. At this stage, the schedule adjustment and optimization problem become a multi-objective optimization problem that involves resource leveling and minimum changes in schedule.

31

Table 5 Comparison of resource allocation by different studies to activities for highway construction project. Activity

Non-optimized case

Study by Mattila & Abraham [20]

Study by George [7]

Current study

Ditch excavation Culvert installation Concrete pavement removal Peat excavation and swamp backfill Embankment Utility work Sub-base Gravel Paving

3 1 4

3/2 1 4

2 1 7

2 1 3

8

8

8

8

5 2 6 8 8

5 2 6 4/8 8

7 3 7 7 9

3 3 10 10 10

Adjustments made to the schedule in the preliminary phase have broader effects on the subsequent construction process; this makes the objective to minimize changes to the schedule more important. As the project progresses, the effect of adjustments to the schedule on subsequent construction is reduced because there is less time remaining on the project schedule. As such, the objective to minimize adjustments to the schedule becomes less important. In contrast, resource leveling becomes increasingly important as the project progresses. Based on the above arguments, we would like to suggest a possible weight allocation scheme, which is summarized in Table 2. 6. Model validation Using the Schedule Control Model explained in Section 3 as a basis, Visual Studio 2010 IDE and C# language were used for the programming of the system. The system comprises four modules: data input, scheduling, plan display, and plan comparison. Among the four, the scheduling module was the core of the system. The proposed scheduling model described in Section 5 was used as a basis. Integrated with the IBM ILOG CPLEX Optimization Studio, the ILOG OPL modeling language was used to program the model. The graphic interface for the plan display and comparison modules was created using MapXtreme. The system uses Microsoft SQL Server 2008 as a database. Fig. 1 shows the relationship between the various modules of the system and the interaction between data flows. Based on the information shown, the system's auxiliary project manager would be able to carry out closed-loop schedule control management, including planning and adjustment, implementation, and checking. Using the progress control system, the proposed model was validated using actual construction data from a highway construction project [7,20] and the DK53+000–DK60+000 subgrade project of the Urumqi–Dzungaria railway undertaken by the Urumqi Railway Bureau. 6.1. Case study 1 In the literature [7,20], a highway construction project was used for verifying the resource leveling, scheduling, and optimization capabilities of the proposed model. We used the same construction project to verify the capabilities of our proposed model with regard to resource leveling, Table 6 Comparison of key project parameters for highway construction project by different studies.

Fig. 5. Corresponding LSM diagram of Pc.

Parameters

Non-optimized case

Study by Mattila and Abraham [20]

Study by George [7]

Current study

Total duration Average resource usage Sum of day-to-day fluctuations

38 8 52

38 8 32

38 8 30

38 8 23

32

Y. Tang et al. / Automation in Construction 37 (2014) 22–37

Table 7 Overview of DK53+000–DK60+000 subgrade construction activities. Activity

Type

Ground shaping Linear Treatment of weak foundation Block General embankment filling Linear Embankment filling for base of foundation (team 1) Embankment filling for base of foundation (team 2) Embankment filling for foundation surface Trim of subgrade Ancillary works for subgrade Trim and ending

Partial-span linear Partial-span linear Linear

Predecessor

Start point

End point

Resource domain (people) Resource production rate (m/5 people/day)

Preparation for construction Ground shaping Ground shaping Treatment of weak foundation General embankment filling

DK53+000 DK60+000 [5,10] DK53+350 DK53+450 [10] DK53+000 DK60+000 [25,30,35]

334 6 days (duration) 20

DK53+000 DK56+500 [30,35,40]

14

General embankment filling

DK56+500 DK60+000 [30,35,40]

14

Embankment filling for base of foundation

DK53+000 DK60+000 [15,20,25,30]

31.25

Intermittent linear Embankment filling for foundation surface DK53+000 DK60+000 [15,20] Linear Trim of subgrade DK53+000 DK60+000 [15,20,25] Linear Ancillary works for subgrade DK53+000 DK60+000 [10,15]

Table 8 Constraints on the construction activities for the DK53+000 to DK60+000 subgrade project. Buffer between activities

Buffer type

Ground shaping & treatment of weak foundation Treatment of weak foundation & general embankment filling Ground shaping & general embankment filling General embankment filling & embankment filling for base of foundation (teams 1 & 2) Embankment filling for base of foundation (teams 1 & 2) & Embankment filling for foundation surface Embankment filling for foundation surface & trim of subgrade Trim of subgrade & ancillary works for subgrade Ancillary works for subgrade & trim and ending

Minimum time buffer

Value (day) 2

Minimum time buffer

1

Minimum time buffer Minimum time buffer

5 10

Minimum time buffer

10

Minimum time buffer Maximum time buffer Minimum time buffer

5 20 20

Minimum time buffer

5

automatic scheduling, and optimization of the scheduling model and its algorithm. Through the comparison, we hope to demonstrate the proposed model's strengths. The project comprised 50 stations and nine activities with the total work duration being limited to 38 d. Project details are summarized in Table 3. This case involved only one-time schedule planning for the purpose of resource leveling, and no adjustments were made to the schedule during construction. There was only one objective, i.e., resource leveling, so α = 1 and β = 0. The data in Table 3 were applied to the proposed scheduling model for preparing a linear schedule for the project. The schedule is denoted by Pc, and its related data are listed in Table 4. The optimized

Table 9 Initial construction schedule.

39 27.8 350

schedule was able to satisfy the 38-d constraint and the time buffer constraints between activities. The objective function value was calculated to be 23. Fig. 5 shows the corresponding LSM diagram for Pc that was generated using the proposed schedule control system. Tables 5 and 6 show the results of a comparison between the initial schedule for the highway construction project and the schedule after various optimization processes (including those by [7,20] and the proposed model). It can be inferred from Table 5 that the resource allocation scheme of the proposed model significantly different from that of the initial construction schedule as well as that in the research by Mattila and Abraham [20], and George [7]. Table 6 lists the different construction-related parameters. The initial construction schedule with the various optimization processes satisfied the 38-d time constraint with an average resource consumption of 8 units. The objective function value of the schedule obtained by the proposed model was superior to the initial plan as well as that by Mattila and Abraham [20], and George [7]. Therefore, the proposed model in this paper has certain advantages compared over models proposed in previous studies. 6.2. Case study 2 Actual data from the DK53+000–DK60+000 subgrade project for the Xiaohuangahan–Wucaiwan portion of the Urumqi–Dzungaria railway construction project were used for verifying the following capabilities of the proposed model: ■ Ability to handle more comprehensive constraint types (those that exist between more than two activities and those at an arbitrary mileage point between activities); ■ Ability to schedule and optimize as multiple construction teams work concurrently; Table 10 Optimized initial schedule.

Activity

Resource

Start date

End date

Ground shaping Treatment of weak foundation General embankment filling Embankment filling for base of foundation (team 1) Embankment filling for base of foundation (team 2) Embankment filling for foundation surface Trim of subgrade Ancillary works for subgrade Trim and ending

Duration (day)

Activity

Resource

5 10 35 35

0 8 13 27

35 20 15 20 10

Start date

End date

21 14 63 63

21 6 50 36

52

88

36

50 60 82 145

106 120 145 155

56 60 63 10

Ground shaping Treatment of weak foundation General embankment filling Embankment filling for base of foundation (team 1) Embankment filling for base of foundation (team 2) Embankment filling for foundation surface Trim of subgrade Ancillary works for subgrade Trim and ending

Duration (day)

10 10 35 30

0 5 11 21

11 11 61 63

11 6 50 42

35

61

97

36

25 15 25 15

63 68 97 148

108 128 148 155

45 60 51 7

Y. Tang et al. / Automation in Construction 37 (2014) 22–37

Fig. 6. Comparison of pre- and post-optimized initial schedules.

33

34

B) Resource consumption based on the optimized initial schedule Fig. 7. Difference between resource consumption in pre- and post-optimized initial schedules.

Y. Tang et al. / Automation in Construction 37 (2014) 22–37

A) Resource consumption based on the initial schedule (pre-optimized)

Y. Tang et al. / Automation in Construction 37 (2014) 22–37

35

Table 11 Activities not yet completed when the initial construction schedule was revised. Activity

Type

Predecessor

Start point

End point

Resource domain (people) Resource production rate (m/5 people/day)

General embankment filling

Linear

DK58+300 DK60+000 [25,30,35]

20

Embankment filling for base of foundation (team 1) Embankment filling for base of foundation (team 2) Embankment filling for foundation surface Trim of subgrade Ancillary works for subgrade Trim and ending

Partial-span linear

Ground shaping Treatment of weak foundation General embankment filling

DK56+000 DK56+500 [30,35,40]

14

Partial-span linear

General embankment filling

DK56+950 DK60+000 [30,35,40]

14

Linear

Embankment filling for base of foundation DK53+000 DK60+000 [15,20,25,30]

31.25

Intermittent linear Embankment filling for foundation surface DK53+000 DK60+000 [15,20] Linear Trim of subgrade DK53+000 DK60+000 [15,20,25] Linear Ancillary works for subgrade DK53+000 DK60+000 [10,15]

■ Ability to schedule and optimize in the presence of intermittent linear-type activities; and ■ Ability to adjust a schedule with multi-objective optimization. The model's schedule control ability was fully validated based on the above. The Xiaohuangshan–Wucaiwan section of the Urumqi–Dzungaria railway construction project is in the Changji prefecture within the territory of the Xinjiang Uyghur Autonomous Region. The line departs from the Xiaohuangshan dedicated rail station, travels north along the Tuwuda (Turpan–Urumqi–Huangshan) Expressway, eastwards through Tudunzi, and down south via Quanchun, Zini Quanzi, and Wutong Zaozi before terminating at Wucaiwan station. The duration of the project lasted 2 years, from November 15, 2007 to November 18, 2009 (operational date). The length of the line is 95.01 km, and the origin– destination (OD) mileage is DK1+000–DK96+000. The length of the DK53+000 to DK60+000 subgrade is 7 km. The project duration was limited to 155 days. The breakdown of the project and related data are shown in Table 7. At the DK56+500 mileage point, the construction of the underlying foundation was divided into two segments, each undertaken by a separate team. Trim of the subgrade was defined as an intermittent linear activity. There was both a minimum and a maximum time constraint between this activity and the one prior to it. The constraints on the various construction activities are listed in Table 8. During the actual construction process, the schedule was prepared twice. The first schedule was created prior to the start of the project as an initial plan to guide construction, while the second was created two months (60 days) later, due to unforeseen circumstances that had affected the project. The proposed scheduling model was used to optimize both the initial and revised schedules, so that a comparison could be made, and for purposes of model validation. 6.2.1. Comparison of pre- and post-optimized initial schedules As stated in Section 5.5, since the initial schedule did not involve any adjustments, it was purely a resource-leveling optimization problem. The parameters selected for the objective function were α = 1 and β = 0. Table 12 Revised construction schedule.

39 27.8 350

Table 9 is the initial schedule (denoted as Pinitial), which was prepared based on the experience of the field engineers. This schedule complied with the constraints of the minimum and maximum time buffer stated in Table 8. It also fulfilled the criteria set for the total number of work days, which was limited to 155. The corresponding objective function value was calculated to be 335. The data in Table 7 were optimized using the proposed scheduling model. The optimized initial schedule (denoted as Pinitial_opt) is shown in Table 10. Like Table 9, this optimized schedule satisfied the various constraints and criteria that were set. The corresponding objective function value was calculated to be 135. The system developed by this research was used to generate the LSM diagram of the two schedules for purposes of comparison (Fig. 6). “A” represents the LSM diagram for Pinitial, while “B” is the LSM diagram for Pinitial_opt (post-optimization). Fig. 7 presents a histogram showing the difference between the resource consumption in the pre- and post-optimized initial schedules, which are indicated as “A” and “B,” respectively. After optimization, the objective function value of 335 for Pinitial was reduced to 135 for Pinitial_opt. The peak value for resource consumption was reduced from 140 to 75, while the average daily consumption of resources remained unchanged, at 50. These data show that optimization would result in obvious advantages. 6.2.2. Comparison of pre- and post-optimized revised schedules In response to unforeseen circumstances, the field engineers had to revise the construction schedule on the 60th day. The activities that had not yet been completed and related information are shown in Table 11. These activities still had to meet the buffer constraints stipulated in Table 8. At this point, out of the total limit of workdays, there were only 95 workdays left. Table 12 shows the revised schedule (denoted as Prevised), which was created in accordance with the actual on-site progress and was based on the experience of the field engineers. This schedule complied with the constraints stated in Table 8. It also fulfilled the criteria set for a total of 95 work days. Based on the weight allocation scheme proposed in Section 5.6, for α = 0.4, β = 0.6, the corresponding objective function

Table 13 Optimized revised schedule.

Activity

Resource

Start date

End date

General embankment filling Embankment filling for base of foundation (team 1) Embankment filling for base of foundation (team 2) Embankment filling for foundation surface Trim of subgrade Ancillary works for subgrade Trim and ending

Duration (day)

Activity

Resource

25 35

60 66

40 25 15 25 15

Start date

End date

77 72

17 6

66

94

28

65 70 99 148

110 130 150 155

45 60 51 7

General embankment filling Embankment filling for base of foundation (team 1) Embankment filling for base of foundation (team 2) Embankment filling for foundation surface Trim of subgrade Ancillary works for subgrade Trim and ending

Duration (day)

35 40

60 60

73 65

13 5

40

60

88

28

25 15 20 15

60 65 85 148

105 125 148 155

45 60 63 7

36 Y. Tang et al. / Automation in Construction 37 (2014) 22–37

A) pre-optimized revised schedule

B) post-optimized revised schedule Fig. 8. Comparison of pre- and post-optimized revised schedules.

Y. Tang et al. / Automation in Construction 37 (2014) 22–37

value was calculated to be 1,288. Specifically, when α = 1 and β = 0, the corresponding objective function value was 320. When α = 0 and β = 1, the corresponding objective function value was 1,970. The actual data in Table 11 were optimized using the proposed scheduling model. The optimized revised schedule (denoted as Prevised_opt) is shown in Table 13. Like Table 12, this optimized schedule satisfied the various constraints and criteria that were set. Based on the proposed weight allocation scheme, the values α = 0.4, β = 0.6 were selected; the corresponding objective function value was calculated to be 603. Specifically, when α = 1 and β = 0, the corresponding objective function value was 165. When α = 0 and β = 1, the corresponding objective function value was 895. Fig. 8 shows the LSM diagrams of the two schedules for purposes of comparison. Here, “A” represents the LSM diagram for Prevised, while “B” is the LSM diagram for Prevised_opt (post-optimization). Again, as was the case with the initial construction schedule, the data also show that the optimization of the revised schedule resulted in obvious advantages. The two case studies presented earlier verified the advantages of the proposed model with regard to scheduling and optimization over the methods proposed in previous studies as well as its ability to perform schedule control.

7. Conclusion This study used LSM as the basis to examine the issue of schedule control for linear projects (such as railway construction) which are constrained by fixed duration. A progress control model, a CSP-based scheduling model, and a schedule control system were proposed. The proposed CSP-based scheduling model is central to the schedule control model. It uses the fixed duration as the constraint and resource leveling and minimum changes to the schedule as the optimization objectives. Automatic preparation of a linear schedule and optimization with resource leveling and minimum changes to the schedule as objectives are achieved based on LSM. The CSP-based scheduling model sets the resources usage to the activity and buffer between activities as variables and can handle constraints between multiple (more than two) activities, constraints at any arbitrary mileage points between activities, and other more complex and broad constraints. The model has great flexibility and practicality. It can schedule and optimize under special circumstances such as scenarios in which multiple construction teams work simultaneously and in the presence of intermittent linear activities. The model is solved using CP technology. The planner can obtain the optimal solution in a relatively short period of time owing to the high efficiency of CP technology. The schedule control system was developed based on the schedule control model and the scheduling model. It can automatically prepare and adjust the schedule with resource leveling and minimum changes to the schedule as its optimization objectives. With the graphical interface from MapXtreme, the LSM diagram from the scheduling module can be displayed. The actual construction progress can then be compared to the schedule on this interface for assisting the supervisor with progress control. The effectiveness of the proposed model was validated using actual data from a highway construction project [7,20] and the Xiaohuangshan–Wucaiwan section of the Urumqi–Dzungaria railway construction project undertaken by the Urumqi Railway Bureau.

Acknowledgments This research was funded by State Key Laboratory of Rail Traffic Control and Safety of Beijing Jiaotong University of China under Grant RCS2009ZT007 and National Key Technology R&D Program under Grant 2009BAG12A10.

37

References [1] The Ministry of Railways of the People's Republic of China, Guide for Construction Organization Plan of Railway Construction, China Railway Publishing House, Beijing, 2010. (in Chinese). [2] D.W. Johnston, Linear scheduling method for highway construction, J. Constr. Div. 107 (CO2) (1981) 247–261. [3] R.B. Harris, P.G. Ioannou, Scheduling projects with repeating activities, J. Constr. Eng. Manag. 124 (4) (1998) 269–278. [4] R.A. Yamín, D.J. Harmelink, Comparison of linear scheduling model (LSM) and critical path method (CPM), J. Constr. Eng. Manag. 127 (5) (2001) 374–381. [5] R.M. Reda, RPM: repetitive project modeling, J. Constr. Eng. Manag. 116 (2) (1990) 316–330. [6] D.J. Harmelink, Linear scheduling model: float characteristics, J. Constr. Eng. Manag. 127 (4) (2001) 255–260. [7] M.E. Georgy, Evolutionary resource scheduler for linear projects, Autom. Constr. 17 (2008) 573–583. [8] B.G. Hwang, L.K. Low, Construction project change management in Singapore: status, importance and impact, Int. J. Proj. Manag. (2011), http://dx.doi.org/10.1016/ j.ijproman.2011.11.001. [9] I.A. Motawa, C.J. Anumba, S. Lee, F. Peña-Mora, An integrated system for change management in construction, Autom. Constr. 16 (2007) 368–377. [10] J. Bröchner, U. Badenfelt, Changes and change management in construction and IT projects, Autom. Constr. 20 (2011) 767–775. [11] T.Y. Hsieh, S.T. Lu, C.H. Wu, Statistical analysis of causes for change orders in metropolitan public works, Int. J. Proj. Manag. 22 (8) (2004) 679–686. [12] S.S. Zhao, L.X. Zhang, Railway Construction Project Management, China Railway Publishing House, Beijing, 2004. (in Chinese). [13] D.W. Halpin, R.W. Woodhead, Construction Management, 2nd ed. John Wiley & Sons, Inc., New York, 1998. [14] M. Park, Model-based dynamic resource management for construction projects, Autom. Constr. 14 (2005) 585–598. [15] Ministry of Housing and Urban–Rural Development of the People's Republic of China, General Administration of Quality Supervision, Inspection and Quarantine of the People's Republic of China, The Code of Construction Project Management, China Architecture & Building Press, Beijing, 2006. [16] D.J. Harmelink, J.E. Rowings, Linear scheduling model: development of controlling activity path, J. Constr. Eng. Manag. 124 (4) (1998) 263–268. [17] A. Kallantzis, S. Lambropoulos, Critical path determination by incorporating minimum and maximum time and distance constraints into linear scheduling, Eng. Constr. Archit. Manag. 11 (3) (2004) 211–222. [18] A. Kallantzis, J. Soldatos, S. Lambropoulos, Linear versus network scheduling: a critical path comparison, J. Constr. Eng. Manag. 133 (7) (2007) 483–491. [19] M.A. Ammar, E. Elbeltagi, Algorithm for determining controlling path considering resource continuity, J. Comput. Civ. Eng. 15 (4) (2001) 292–298. [20] K.G. Mattila, D.M. Abraham, Resource leveling of linear schedules using integer linear programming, J. Constr. Eng. Manag. 124 (3) (1998) 263–268. [21] M.L. Pinedo, Scheduling—Theory, Algorithms, and Systems, Third ed. SpringerVerlag, Inc., New York, 2008. [22] S.C. Brailsford, C.N. Potts, B.M. Smith, Constraint satisfaction problem: algorithms and applications, Eur. J. Oper. Res. 119 (3) (1999) 557–581. [23] V. Jain, I.E. Grossmann, Algorithms for hybrid MILP/CP models for a class of optimization problems, INFORMS J. Comput. 13 (4) (2001) 258–276. [24] W.T. Chan, H. Hu, Constraint programming approach to precast production scheduling, J. Constr. Eng. Manag. 128 (6) (2002) 513–521. [25] S.S. Liu, C.J. Wang, Optimization model for resource assignment problems of linear construction projects, Autom. Constr. 16 (2007) 460–473. [26] D.F. Lu, Railway Construction Project Management, China Railway Publishing House, Beijing, 2004. (in Chinese). [27] S.S. Liu, C.J. Wang, Optimizing linear project scheduling with multi-skilled crews, Autom. Constr. 24 (2012) 16–23. [28] S.S. Liu, C.J. Wang, Resource-constrained construction project scheduling model for profit maximization considering cash flow, Autom. Constr. 17 (2008) 966–974. [29] S.J. Russell, P. Norvig, Artificial Intelligence: A Modern Approach, Third ed. Prentice Hall, New Jersey, 2009. [30] S. Heipcke, Comparing constraint programming and mathematical programming approaches to discrete optimization—the change problem, J. Oper. Res. Soc. 50 (1999) 581–595. [31] K. Marriott, P.J. Stucky, Programming with Constraints: An Introduction, MIT Press, Cambridge, MA, 1998. [32] K.R. Apt, Principles of Constraint Programming, First ed. Cambridge University Press, Cambridge, 2003. [33] T. Hegazy, Optimization of resource allocation and leveling using genetic algorithms, J. Constr. Eng. Manag. 125 (3) (1999) 167–175. [34] S. Christodoulou, Ant Colony Optimization in Construction Scheduling, Proc. 2005 ASCE International Conference on Computing in Civil Engineering, Cancun, Mexico, 2005. [35] R. Kolisch, S. Hartmann, Experimental investigation of heuristics for resourceconstrained project scheduling: an update, Eur. J. Oper. Res. 174 (2006) 23–37. [36] International Business Machines Corporation, IBM ILOG OPL Language Reference Manual V6.3, 2009. [37] Railway Engineering Quota Institute, Budget Quota of Railway Engineering, China Zhijian Publishing House, Beijing, 2005. (in Chinese). [38] R.T. Marler, J.S. Arora, Survey of multi-objective optimization methods for engineering, Struct. Multidiscip. Optim. 26 (2004) 369–395. [39] M. Ehrgott, Multicriteria Optimization, Second ed. Springer-Verlag, Berlin, 2005.