Construction rescheduling based on a manufacturing rescheduling ...

Automation in Construction 18 (2009) 715–723

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Automation in Construction j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / a u t c o n

Construction rescheduling based on a manufacturing rescheduling framework Shu-Shun Liu a,⁎, Kuo-Chuan Shih b,1 a b

Department of Construction Engineering, National Yunlin University of Science and Technology, No. 123, Section 3, University Road, Touliu, 640 Taiwan, ROC Graduate School of Engineering Science and Technology, National Yunlin University of Science and Technology, No. 123, Section 3, University Road, Touliu, 640 Taiwan, ROC

a r t i c l e

i n f o

Article history: Accepted 17 February 2009 Keywords: Construction project scheduling Resource-constrained project scheduling Rescheduling Constraint Programming

a b s t r a c t Changes during project execution frequently require schedule updating and rescheduling. However, few studies have discussed rescheduling issues or implemented rescheduling solutions for construction projects. This study investigates resource-constrained construction rescheduling issues using concepts associated with manufacturing rescheduling. Based on an initial schedule and actual progress, a novel rescheduling optimization model using Constraint Programming (CP) techniques is developed to reschedule projects. Two rescheduling methods: (1) complete regeneration (CR); and, (2) partial rescheduling (PR) while minimizing overall project variation are implemented in the proposed model to demonstrate the model capability and applications. PR requiring additional treatments to decrease overall project variation is performed using a novel constraint-release mechanism. Finally, using a case study, optimization results obtained using two rescheduling methods are analyzed and discussed. © 2009 Elsevier B.V. All rights reserved.

1. Introduction

2. Literature review

Rescheduling is common in project management, especially in the manufacturing industry. Generally, unexpected events adversely affect projects when necessary treatments are not adopted. Therefore, the dominant issues in rescheduling are how to adapt to a changing environment and reschedule incomplete work and resources. A rescheduling problem consists of general scheduling problems that develop after a schedule is updated. Project information modifications and schedule updating may generate additional constraints due to the altered environment. Based on schedule updating results, rescheduling must rearrange incomplete work and resources while generating a practical schedule that meets the project goal. Compared to the manufacturing industry, construction projects have more unpredictable factors, such as environmental and productivity issues, that make maintaining schedules difficult. Although construction schedules are regularly updated and controlled during construction, few studies have investigated the effects of rescheduling issues on the rescheduling process. Therefore, applying manufacturing rescheduling concepts to the construction field is worthy of investigation. This study presents an overview of construction rescheduling issues, including characteristics of construction rescheduling and appropriate rescheduling methods, and proposes a novel rescheduling mechanism for solving issues that cater to management needs.

Rescheduling has been widely discussed in the manufacturing industry. Vieira et al. [1] defined rescheduling as the modification of an existing production schedule in response to disruptions or other changes. Additionally, Vieira et al. [1] proposed a framework for manufacturing rescheduling and defined terms in the rescheduling problem. Wu and Li [2] proposed a similar framework to Vieira et al. [1]. Joh et al. [3] identified characteristics of scheduling and rescheduling problems and developed a model for examining scheduling and rescheduling processes. Haldun et al. [4] analyzed four aspects of risk: cause, context, impact and inclusion to identify rescheduling factors. Yang [5], who demonstrated that new jobs always influence schedules, attempted to minimize the effects of negative disruptions through total cost optimization. According to Vieira et al. [1], scheduling problems can be formulated as combinatorial optimization problems. Herroelen et al. [6] and Brucker et al. [7] collected, classified, and solved various scheduling optimization problems using mathematical models. According to Herroelen et al. [6], a resource-constrained project rescheduling problem can be preliminarily identified as a discrete time-resource trade-off problem. Additionally, Herroelen et al. [6] recommended using the branch-and-bound method to optimize the resource-constrained project rescheduling problem. Kelleher and Cavichiollo [8] demonstrated that a constraintbased approach is superior for generating schedules when combined with dependency analysis techniques based on reason maintenance systems and partial order backtracking. To reduce rescheduling frequency, ElMekkawy and ElMaraghy [9] developed a deadlock-free rescheduling algorithm that used a heuristic routine to reschedule some jobs rather than all jobs. Yu et al. [10] applied

⁎ Corresponding author. Tel.: +886 5 5342601x4724; fax: +886 5 5312049. E-mail addresses: [email protected] (S.-S. Liu), [email protected] (K.-C. Shih). 1 Tel.: +886 5 5342601x4701; fax: +886 5 5312049. 0926-5805/$ – see front matter © 2009 Elsevier B.V. All rights reserved. doi:10.1016/j.autcon.2009.02.002

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S.-S. Liu, K.-C. Shih / Automation in Construction 18 (2009) 715–723

an immune algorithm, which is a heuristic optimization algorithm, to solve a flexible dynamic scheduling problem. The theory of constraints proposed by Goldratt [11] employed buffers to generate robust schedules that accommodated minor risks. Chua et al. [12] proposed a constraint-based project planning method with activity buffers. Hegazy and Petzold [13] proposed a genetic algorithm-based scheduling model and used buffers to determine the appropriate time to implement corrective actions. 3. Construction rescheduling Table 1 classifies rescheduling problems and can be used as a reference for construction rescheduling problems. For classification details, refer to Vieira et al. [1]. The terms used in this study are quoted from Vieira et al. [1] as follows: • Rescheduling is the process of updating an existing production schedule in response to disruptions or other changes. This includes arrival of new jobs, machine failures, and machine repairs. • Rescheduling environment identifies the set of jobs that the schedule should include. • A Rescheduling strategy describes whether or not production schedules are generated. • A Rescheduling policy specifies when and how rescheduling is done. The policy specifies the events that trigger rescheduling. • Rescheduling methods generate and update production schedules. • Complete regeneration reschedules the entire set of operations (jobs) not processed before the rescheduling point, including those not affected by the disruption. • Partial rescheduling reschedules only the operations affected directly or indirectly by the disruption. 3.1. Characteristics of construction rescheduling Compared with manufacturing, construction environments have more uncertainties such as relatively long project durations and issues related to subcontracting, outsourcing, and weather. To identify construction rescheduling characteristics, rescheduling factors that alter a project environment must first be recognized. Some factors that distinguish construction rescheduling issues from those in the manufacturing industry are as follows: (1) Productivity variation Manufacturing projects depend on linear operations, and products are manufactured using standardized methods. There-

Table 1 Classification of construction rescheduling (modified from Vieira et al. [1]).

fore, productivity has minimal variation. However, in the construction industry, supply sources typically vary, and outsourcing options are generally available. Moreover, resource and work methods impact productivity and activity duration. Assessing productivity is critical in construction project scheduling during the planning stage. During construction, maintaining a productivity level that adheres to the initial schedule is extremely important. (2) Operational environment Compared with manufacturing operations in factories, construction operations are typically performed outdoors and influenced significantly by numerous external factors such as weather and temperature. These uncertainties may alter an environment making productivity difficult to maintain. (3) Demand–supply relationship Most rescheduling factors in the manufacturing industry are related to uncertain customer demands; conversely, the principal rescheduling factors in construction projects are due to production processes. In the construction industry, planners must manage projects in response to uncertainties occurring during construction. The primary goal of construction planners is to complete a project before its due date and to execute most activities according to contracts. When uncertainties occur, planners must execute effective reactions and adjustments based on actual progress. Project tasks must be monitored, controlled, updated, and even rescheduled during construction. Therefore, factors impacting project schedules in manufacturing and construction industries differ fundamentally. 3.2. Definition of rescheduling in construction This work classifies and discusses construction rescheduling characteristics as the following (Table 1): Deterministic environment. Based on the predictability of the tasks required, the rescheduling environment for construction projects is deterministic because a schedule consists of certain activities awarded to contractors on a fixed basis. A completed project design identifies all tasks and itemizes contract content. As a situation in which work content rarely changes, construction projects are typically scheduled and executed in a deterministic environment. Predictive–reactive strategy. Deterministic environments frequently employ a predictive–reactive strategy to generate and update an initial schedule for most rescheduling problems (Vieira et al. [1]). A


hybrid rescheduling policy that combines periodic and eventdriven policies can be a choice in response to productivity variation and uncertainties during construction. Robust schedules. To reduce potential impact of risks, planners generally employ robust schedules that accommodate minor risks to allow for uncertainties. Buffer management in the theory of constraints can be used as a solution allowing project duration extension as a project buffer. By adopting the theory of constraints, project duration in this study equals contract duration minus buffers thereby avoiding unnecessary duration extensions for activities. For further buffer issues, refer to Goldratt [11]. Although Vieira et al. [1] concluded three rescheduling methods, this study does not discuss right-shift rescheduling, which postpones all activities not yet executed until rescheduling reasons have been solved. No schedule rearrangement is needed except to postpone activity times and then consume project buffers while following the network structure of initial schedule. Complete regeneration (CR). The principal goal of large construction companies is typically total cost minimization. Consequently, CR produces a new schedule irrespective of whether the initial schedule is feasible for some construction projects. Partial rescheduling (PR). Construction projects exhibit a deterministic environment with contractual constraints including budget and duration. Although productivity variation and the operational environment disturb schedules, planners try to create reliable initial schedules during the design stage. Notably, most risks can be identified, predicted, and even avoided prior to scheduling. Maintaining the initial schedule is a typical project goal. Furthermore, subcontracting and outsourcing are common practices during construction. Contractors are typically responsible for contract execution and dealing with suppliers. Therefore, PR, while minimizing overall project variation compared with the initial schedule, is suitable and practical for construction rescheduling problems. Executing all activities on time in compliance with client and subcontractor contracts eliminates contract conflicts. Choosing PR or CR depends on the potential for contract conflicts. As possible contract conflicts increase, CR becomes increasingly unrealistic. Furthermore, construction projects having long project durations often have adequate reaction time to deal with accidents and perform rescheduling. Methods for rescheduling can generate a new practical schedule compared to the initial schedule to identify rescheduled activity impact. Further analyses of schedule changes are

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beneficial to select the most appropriate rescheduling method before adopting a new schedule. When implementing rescheduling methods, planners should technically treat CR as a new scheduling problem. By comparison, PR requires treatments that decrease overall project variation. Therefore, this study illustrates PR as a rescheduling method for avoiding contract conflicts.

4. Schedule updating versus rescheduling Compared with rescheduling, schedule updating, which is not a new practice in the construction field, is typically performed to monitor and control project progress. Furthermore, project information used in schedule updating yields an index useful for measuring project performance. The following tasks are essential in schedule updating: (1) compare the initial schedule with project progress; (2) identify all delayed activities; (3) identify who or what is responsible for delays; and, (4) forecast and modify projected work progress based on actual progress. Schedule updating involves all scheduled inspections and may include rescheduling. The relationship between schedule updating and rescheduling can be further identified as follows: (1) Schedule updating recognizes actual progress. If any unavoidable factor conflicts with the initial schedule, project information must be consistent with environmental changes. The primary tasks in schedule updating are determining an applicable rescheduling policy, identifying rescheduling factors, and evaluating the effects of rescheduling factors on the initial schedule. (2) The initial schedule is modified by rearranging activities and resources during rescheduling. The primary tasks in rescheduling are determining which rescheduling method to use and producing new schedules. Fig. 1 presents the difference between schedule updating and rescheduling to clearly define the scope of this study.

5. Schedule optimization and schedule updating This study requires a schedule in a resource-constrained environment, and applies the following model to generate schedules, including scheduling or rescheduling. When an initial schedule has been generated, start times, durations, resource requirements, and direct costs, are elements that represent the initial schedule activities. Some can be parameters while rescheduling.

Fig. 1. Difference between schedule updating and rescheduling.

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RUCDi;

5.1. Initial schedule optimization Eqs. (1)–(4) comprise the model used to generate schedules in scheduling or rescheduling problems. Eqs. (5)–(11) are optional when performing schedule updating and rescheduling. 5.1.1. Objective function " # r n X X Minimize RUCDi ;j ×CDi Þ×rcj + ½Se + CDe ×idc j=1

ð1Þ

i=1

or

5.1.2. Basic constraints Precedence relationships: i = 1; 2; N ; e;

all ði; sÞaP;

8CDi aDi

ð3Þ

Resource limitation: ASk X rlj z RUCDi ;j ;

8iaSk ;

k = 1; 2; N ; ½Se + CDe ;

ð4Þ

i=1

5.1.3. Optional constraints for rescheduling Constraints of in-progress activities: Si = rt; 8iaL

ð5Þ

Si zrt; 8iaL

ð6Þ

Constraints of being-scheduled activities for partial scheduling: Si = osi ;

ð7Þ

8iaM

ð8Þ

8iaM

Di = odi ;

Productivity modifications: Di = Ceil½odi ×ð1 − pai Þ×ðadi odi Þ pai Di = odi − adi

ð9Þ ð10Þ

Budget limitation: cbz

r X j=1

"

n X

# RUCDi ;j ×CDi ×rcj + ½Se + CDe ×idc

i=1

where parameter i is the index of an activity; j e r n rcj rlj idc osi odi rt pai adi cb

where set P is a set of pairs of activities with precedence relationships; Di Sk M L Ceil

is the set of duration options for activity i; is the in-progress activities on day k; is the set of being-scheduled activities; is the set of in-progress activities; is a function that extracts the minimum integer that is greater than an input value.

ð2Þ

Minimize ½Se + CDe

Ss − Si zCDi

is the amount of resource type j utilized for activity i according to CDi; Si and Ss are start time of activity i and its successor, respectively. j

is the index of a resource; is the index of the last activity in a project network; is the total number of resource types; is the total number of activities; is the unit cost per day of resource type j; is the daily limit of resource type j per day; is the daily indirect cost; is the start time for activity m in the initial schedule; is the duration of activity m in the initial schedule; is the time at which rescheduling is conducted; is the completion percentage for activity i; is the actual number of work days for activity i; is project budget.

where variable CDi is the duration of activity i;

ð11Þ

Eq. (1) summarizes total project cost. Total project cost includes resource usage costs (the quantity of resources used multiplied by the unit cost of a resource) and indirect costs (the sum of daily indirect project costs). Eq. (2) determines minimal project duration. Minimal total project cost and minimal project duration are typical objective functions used for construction projects. Furthermore, planner assessment can determine which objective function depends on project goals. Eq. (3) represents the relationships among activities using activity pairs. The finish-to-start (FS) relationship represents activity relationships. Additionally, in the proposed model, activity duration (CDi) derives from a set of durations (Di) and is assigned to an activity. Moreover, Eq. (4) restrains daily resource usage for all activities. Eqs. (1)–(4) basically form a resource-constrained project rescheduling problem including scheduling and rescheduling. For additional operations of schedule update and rescheduling, optional constraints in Eqs. (5)–(11) are explained as follows: 5.2. Schedule updating Identifying project changes due to actual progress is the first task in schedule updating. Environmental changes may require information modifications, which are represented as parameter revisions. Additional constraints may be required. A schedule can be updated for the following four activity types: (1) Finished activity Scheduling updating removes finished activities from the rescheduling activity list and retains information regarding actual progress and expenses to determine the impact of project changes on the initial schedule. The information for finished activities must be corrected. If any inconsistency is discovered, the causes, which may be due to environmental factors or an incorrect productivity assessment, must be investigated. (2) In-progress activity In-progress activity may be the primary reason for requiring rescheduling. Generally, in-progress activities can be classified as splitting and non-splitting activities. An in-progress activity is split into two activities, the finished one and the incomplete one. When an in-progress activity cannot be split, Eq. (5) is applied to the activity that must be continually executed after rescheduling. Conversely, Eq. (6) presents a condition when in-progress activities can be split. Thus, planners may postpone some incomplete activities to release resources for urgent in-progress activities such as a delayed activity on the critical path. (3) Being-scheduled activity To reschedule a project, the most important task is rearranging incomplete activities and being-scheduled activities. However, such a rearrangement generates a new schedule that may significantly influence all remaining activities and project participants. As mentioned, PR is an applicable rescheduling method for construction projects requiring additional treatments. Minimizing overall project variation, as compared with


the initial schedule, is another goal of PR, and additional constraints (Eqs. (7) and (8)) on all being-scheduled activities are applied to ensure these activities comply with the initial schedule at the beginning of PR rescheduling process. (4) New activity Changed orders and other risks can add new activities to a construction project. Although such additions sometimes significantly influence the initial schedule, these new activities reflect real situations and resource requirements. Parameters also define information about these new activities and Eq. (3) determines the relationships between established activities and new activities. 5.3. Productivity modification Incomplete activities and being-scheduled activities require productivity modifications when an incorrect productivity assessment in the initial schedule is identified in a finished activity, which shares identical resources with those activities. Furthermore, productivity modifications can be represented as an activity duration adjustment. In case of an incomplete activity, this work adopts Eq. (9), proposed by Hegazy and Petzold [13]. Eq. (10) is adopted when productivity modifications are unnecessary for incomplete activities. In case of being-scheduled activities, productivity modifications can be made directly by adjusting activity duration. Moreover, a project budget limitation can be a constraint if necessary as shown in Eq. (11). 5.4. Constraint Programming The proposed model in this study adopts Constraint Programming (CP) as an optimization technique. CP is a standard approach in the field of artificial intelligence for solving scheduling problems (Kelleher and Cavichiollo [8]). For details of CP, refer to [14–16]. During the CP search procedure, a variable is considered as a branch origin and its variable domain determines how many branches are included in the branch origin. The search algorithm powered by CP consists of the following three search techniques which improve search function and efficiency: Forward checking. Confirms feasible branches of the subsequent branch origin for the current branch. If no branch of the subsequent branch origin is feasible, then this branch is cut. Backtracking. Returns to the previous branch origin to search for feasible solutions when no feasible branch for the current branch origin exists. Consistency check. Refines constraints and variable domains using feedback from the branching procedure.

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Whenever a branch extends, it means that a value is assigned to a variable. Therefore, a consistency check action confirms that all constraints are obeyed and adds a constraint to assign the corresponding value. Simultaneously, a forward checking action deletes infeasible branches of subsequent branch origin due to added constraints. If the current branch origin has no feasible branch after forward checking, a backtracking action is activated to avoid unnecessary search effort. A solution in which all branch origins have a feasible branch is kept temporarily as a constraint to enhance subsequent forward checking to compare objective function value with other solutions. The best objective function value is retained as a new solution constraint. When the search process ends, the surviving objective function value comprises the optimal solution. According to Herroelen et al. [6] and Brucker et al. [7], the variable sequence users employ significantly influences solution-seeking efficiency during a search procedure. Fig. 2 presents the variable sequence this study adopts. First, project environment variables such as number of activities, number of resources, resource limitations, and project due date can be predefined. Activity duration and corresponding resource usage are then defined as pair variables. Once an activity duration option is selected as a branch, the corresponding resource usage is determined. Finally, an activity start time is assigned to each activity as a variable and activity relationships lead to different branching priorities. To create schedule solutions, this study executes forward and backward calculations by logical activity relationships derived using the Critical Path Method (CPM). For a PR problem, constraints generated from Eqs. (7) and (8) for being-scheduled activities may lead to a situation in which no solution exists. Therefore, a backtracking action returns to the first activity duration variable during the search procedure. Several constraints generated from Eqs. (7) and (8) are removed according to the constraint-release mechanism. This procedure continues until finding a feasible optimized solution. 5.5. Constraint-release mechanism for PR After updating a schedule, all in-progress activities, beingscheduled activities, and new activities are in the rescheduling activity list. Activities in the list are first grouped into levels (Fig. 3). In-process activities and being-scheduled activities without predecessors are deemed first-level activities. Subsequent activities are then second-level activities. This grouping process continues until all activities are grouped. This study determines the scheduled activity group according to relationships between activities and the degree of activity freedom in the constraints generated from Eqs. (7) and (8).

Fig. 2. Search strategy.

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Fig. 3. Constraint-release mechanism through recursive optimization operations.

Fig. 4. Schedule updating and rescheduling optimization procedure.


project characteristics. The optimization model acquires the initial schedule using Eqs. (1), (3) and (4). Table 3 presents optimal solution output in which optimized total project cost is $5,640,000 and project duration is 630 days. The contract stipulates project cost and duration.

Table 2 Example project information. Activity Cost

Resource 1 500/unit

Resource 2 400/unit

Resource 3 300/unit

Indirect cost Succeeding 2000/day activities

10 units

10 units

10 units

–

Duration Demand

Demand

Demand

Direct cost

50 60 90 120 130 150 120 130 140 160 170 180 190 130 140 70 80 90

4 4 5 6 6 2 5 5 5 4 3 2 1 3 2 4 3 5

5 5 2 6 5 4 6 4 2 4 3 2 1 6 5 3 2 5

280,000 276,000 414,000 744,000 702,000 675,000 516,000 481,000 434,000 928,000 782,000 612,000 418,000 585,000 532,000 385,000 384,000 540,000

Daily limit A B C D E

F

G H I

5 3 4 4 3 5 1 1 1 6 5 4 3 3 3 6 6 5

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6.2. Scheduling updating B, D, F

Assume the project proceeds to day 120 and total cost-to-date is $700,000. Activity A is completed. Activity B is in-progress, 90% complete and ahead of schedule; consequently, no productivity modification is required. According to Eq. (10), the remaining duration of activity B is nine days. Activity D is in-progress, 42% complete and behind schedule; thus, activity D requires productivity modification. According to Eq. (9), the remaining duration of activity D increases from eighty days to ninety-seven days. No other activity starts. A rescheduling action is triggered by activity D, which is behind schedule.

C H C, E H, I

E, G

I

6.3. Case illustration

I

This investigation discusses and analyzes three cases. Table 4 and Fig. 5 present a detailed comparison of these cases. Notably, Case 2 adopts Eq. (2) as the objective function, and adds a constraint of project budget limitation, Eq. (11), (confined to $5,640,000).

–

The search procedure first constrains all being-scheduled activities using Eqs. (7) and (8). If the process identifies an optimal solution, the process ends and generates a schedule. The optimal solution demonstrates that the initial schedule is retained by only rearranging in-progress activities. Along with the strong likelihood that a current PR solution is infeasible, the constraint-release mechanism, which includes backtracking actions, executes recursively in a level-by-level manner to search for a solution and obtain a feasible optimized solution (Fig. 3). Once the constraints of last-level activities, generated from Eqs. (7) and (8), are released, the PR method guarantees an optimal solution as it fails and becomes a CR solution. The constraintrelease mechanism attains the multi-objective goal of minimizing total project cost and overall project variations. Fig. 4 presents schedule updating and rescheduling optimization procedures in the proposed model.

(1) Case 1: CR with total cost minimization Table 3 presents the optimized schedule with a total cost of $5,482,900 and project duration of 647 days. Compared with the initial schedule, the new schedule changes entirely. The CR rearranges all activities in the initial schedule regardless of the activity start time. All activity start times differ from those in the initial schedule (Fig. 5). This completely regenerated schedule with the objective of total cost minimization does not consider schedule variations and thereby typically results in initial schedule disturbances. All scheduled activities retain their original duration which is the most economical. (2) Case 2: CR with project duration minimization This case has a limited project budget of $5,640,000. The minimized project duration is 627 days and total cost is $5,531,900. The contractor can complete the project before the project due duration (630 days) within the project budget ($5,640,000). This experimental result observes disturbances to the initial schedule resulting in schedule variations (Fig. 5). To shorten project duration, the duration of activities C and E decreases from 130 days to 120 days, and 140 days to 130 days, respectively.

6. Rescheduling practice 6.1. Initial schedule This study adopts and modifies the example project used by Leu and Yang [17] to illustrate the rescheduling process. Table 2 shows Table 3 Optimal case results.

Initial plan Case 1 Case 2 Case 3

Project duration

Total cost

630 647 627 630

5,640,000 5,482,900 5,531,900 5,537,900

A

B

C

D

E

F

G

H

I

Finished

Inprogress

Beingscheduled

Inprogress

Beingscheduled

Beingscheduled

Beingscheduled

Beingscheduled

Beingscheduled

D

ST

D

ST

D

ST

D

ST

D

ST

D

ST

D

ST

D

ST

D

ST

50 – – –

0 – – –

90 9 9 9

50 120 120 120

130 130 120 120

200 217 217 217

150 97 97 97

50 120 120 120

140 140 130 130

330 347 337 337

190 190 190 190

140 129 129 129

140 140 140 140

330 319 337 337

70 70 70 70

470 487 467 470

90 90 90 90

540 557 537 540

D = duration; ST = start time. Table 4 Rescheduling methods of cases.

Characteristic Objective Special constraint

Case 1 (CR)

Case 2 (CR)

Case 3 (PR)

Total cost control Minimize total project cost None

Total time control Minimize project duration Eq. (11)

Overall project variation control Minimize total cost Eqs. (7) and (8); constraint-release mechanism

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Fig. 5. Network of case results.

(3) Case 3: PR Partial rescheduling (PR) produces an optimized schedule based on the proposed constraint-release mechanism. Total project cost is $5,537,900 and project duration is 630 days. The optimization process is as follows: activities can be grouped into three levels (Fig. 3). The first-level contains activities B, D, and F; the second-level contains activities C, E, and G; and the third-level contains activities H and I. The optimal solution is found by the constraint-release mechanism after second-level grouped activities are released, indicating that constraints generated from Eqs. (7) and (8) for activities H and I are retained, respectively. Therefore, overall project duration of the new schedule and the schedules for activities H and I are retained as in the initial schedule. 6.4. Result analysis Based on rescheduling results, project duration with minimized total cost in Case 1 is longest, and project duration is shortest with limited project budget (Case 2). Furthermore, total cost for Case 1 is lowest, and PR (Case 3) generates the highest total cost. These cases have three basic rescheduling goals: minimize total project cost, minimize project duration, and minimize overall project variation. In Case 1, project cost is optimized. However, the long project duration generates new risks such as activity delays, contract conflicts, and problems when reassigning resources. If these risks are acceptable, producing a new schedule for the contractor is worthwhile. Conversely, the output of Case 2 demonstrates that the project can be completed within the contract budget and on time. PR in Case 3 eliminates effects caused by the delay resulting from activity D, by rearranging activities C, E, F, and G. The schedules for activities H and I are retained as in the initial schedule. Compared with Cases 1 and 2, Case 3 indicates that PR increases total cost and does not generate the shortest project duration. However, PR maintains the stability of subsequent activities and avoids possible contractual disputes, which are common in construction projects. PR can be beneficial in such situations. In PR, the constraintrelease mechanism is a procedure that releases resources by adjusting being-scheduled activities to resolve the negative influence of rescheduling causes. For example, activities C, E, F, and G in Case 3 are adjusted to eliminate the adverse impact of delayed activity D. The proposed optimization model successfully reschedules these three cases, common in construction projects. This study practices construction rescheduling to distinguish between CR and PR. Any specific rescheduling problem can be treated as an extension of these three cases.

7. Conclusions This study applies a rescheduling framework commonly adopted in manufacturing to construction projects. Construction rescheduling problems are located within the manufacturing framework, based on construction project characteristics. The proposed model implements two rescheduling methods, complete regeneration (CR) and partial rescheduling (PR). To reduce overall project variation for PR, this work also integrates a constraint-release mechanism into PR method. Since each construction project possesses unique characteristics, the advantages of the proposed model lie in choosing between different rescheduling options and thus adhering to project goals. The proposed model not only provides planners the opportunity to “fine-tune” ongoing schedules, but offers possible rescheduling options prior to making decisions to adjust project tasks and resources. Depending on the various management goals of rescheduling methods, generating new schedules utilizing the proposed rescheduling optimization mechanism requires a trade-off between cost, time and the influence of activity delay. Such a trade-off can be exchanged for a “steady” schedule. Additional research warrants examining this trade-off from the perspective of dynamic project control. Acknowledgements The authors would like to thank the National Science Council of the Republic of China, Taiwan, for financially supporting this research under Contract No. NSC93-2211-E-224-025. Appendix A. Notation The following symbols are used in this paper: Parameter i j e r n rcj rlj idc osi odi rt

is the index of an activity; is the index of a resource; is the index of the last activity in a project network; is the total number of resource types; is the total number of activities; is the unit cost per day of resource type j; is the daily limit of resource type j per day; is the daily indirect cost; is the start time for activity m in the initial schedule; is the duration of activity m in the initial schedule; is the time at which rescheduling is conducted;


pai adi cb

is the completion percentage for activity i; is the actual number of work days for activity i; is project budget. Variable

CDi RUCDi;

is the duration of activity i; is the amount of resource type j utilized for activity i according to CDi; Si and Ss are start time of activity i and its successor, respectively. j

Set P Di Sk M L Ceil

is a set of pairs of activities with precedence relationships; is the set of duration options for activity i; is the in-progress activities on day k; is the set of being-scheduled activities; is the set of in-progress activities; is a function that extracts the minimum integer that is greater than an input value.

References [1] G.E. Vieira, J.W. Herrmann, E. Lin, Rescheduling manufacturing system: a framework of strategies, policies, and methods, Journal of Scheduling, vol. 6, Kluwer Academic Publishers, 2003, pp. 39–62. [2] H.H. Wu, R.K. Li, Lecture review and analysis for rescheduling decision problem in production schedule, Journal of the Chinese Institute of Industrial Engineers 14 (2) (1997) 147–158.

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[3] C.H. Joh, Theo A. Arentze, Harry J.P. Timmermans, Understanding activity scheduling and rescheduling behaviour: theory and numerical illustration, GeoJournal, vol. 53, Kluwer Academic Publishers, 2001, pp. 359–371. [4] A. Haldun, A.L. Mark, M. Kenneth, M. Shantha, U. Reha, Executing production schedules in the face of uncertainties: a review and some future directions, European Journal of Operational Research, vol. 161, Elsevier, 2005, pp. 86–110. [5] B. Yang, Single machine rescheduling with new jobs arrivals and processing time compression, International Journal of Advanced Manufacturing Technology, vol. 34 (3–4), Springer, 2007 N.A. [6] W. Herroelen, B.D. Reyck, E. Demeulemeester, Resource-constrained project scheduling: a survey of recent developments, Computers and Operations Research, vol. 25(4), Elsevier, 1998, pp. 279–302. [7] P. Brucker, A. Drexl, R. Mohring, K. Neumann, E. Pesch, Resource-constrained project scheduling: notation, classification, models, and methods, European Journal of Operational Research, vol. 112, Elsevier, 1999, pp. 3–41. [8] G. Kelleher, P. Cavichiollo, Supporting rescheduling using CSP, RMS, and POB — an example application, Journal of Intelligent Manufacturing, vol. 12 (4), Springer, 2001, pp. 343–357. [9] T.Y. ElMekkawy, H.A. ElMarraghy, Real-time scheduling with deadlock avoidance in flexible manufacturing systems, International Journal of Advanced Manufacturing Technology, vol. 22(3–4), Springer, 2003, pp. 259–270. [10] J. Yu, S. Sun, J. Hau, Flexible Dynamic Scheduling Based on Immune Algorithm, PROLAMAT, Shanghai, 2006, pp. 887–895. [11] E.M. Goldratt, What is this Thing Called Theory of Constraints and How Should It be Implemented? North River Press, New York, 1990. [12] D.K.H. Chua, L.J. Shen, S.H. Bok, Constraint-based planning with integrated production scheduler over Internet, Journal of Construction Engineering and Management, ASCE 129 (3) (2003) 293–301. [13] T. Hegazy, K. Petzold, Genetic optimization for dynamic project control, Journal of Construction Engineering and Management, ASCE 129 (4) (2003) 396–404. [14] ILOG, ILOG Solver 5.1 User's Manual, France, 2001. [15] ILOG, ILOG Scheduler 5.1 User's Manual, France, 2001. [16] ILOG, ILOG OPL Studio3.6.1 Language Manual, France, 2001. [17] S.S. Leu, C.H. Yang, GA-based multi-criteria optimal model for construction scheduling, Journal of Construction Engineering and Management, ASCE 125 (6) (1999) 420–427.