International Construction Specialty Conference of the Canadian Society for Civil Engineering (ICSC) (5th : 2015)

Optimizing linear schedule : congestion-minimization approach Esfahan, Nazila Roofigari; Razavi, Saiedeh Jun 30, 2015

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5th International/11th Construction Specialty Conference 5e International/11e Conférence spécialisée sur la construction    Vancouver, British Columbia June 8 to June 10, 2015 / 8 juin au 10 juin 2015   OPTIMIZING LINEAR SCHEDULES: CONGESTION-MINIMIZATION APPROACH Nazila Roofigari Esfahan1,3, Saiedeh Razavi2  1 PhD Candidate, Dept. Of Civil Engineering, McMaster University, 1280, Main West, Hamilton, Canada 2 Assistant Professor, Dept. Of Civil Engineering, McMaster University, 1280, Main West, Hamilton, Canada 3 corresponding_author_roofign@mcmaster.ca Abstract: Space is a strictly limited resource on a construction site. For linear type of construction projects, the importance of effectively managing space is crucial as their schedules is generated with due consideration to both time and space. As a result, spatio-temporal congestions between activities of these projects could substantially hinder the performance of interfering activities and cause deviations from planned schedules. The existence of such congestions decreases work productivity, and causes accidents to occur. The current literature focuses on minimizing the workspace conflicts in order to perform efficient work and increase productivity. However, the other side of this problem, i.e. the changes in productivities which give rise to such spatio-temporal congestions is overlooked. To tackle this limitation, this paper proposes a constraint satisfaction approach to quantify and minimize potential space-time congestions in the schedules of linear projects using space-time floats. The method is able to detect not only potential conflicts between each activity with its immediate successors, but also any possible conflict between resources of any activity with all other project activities’ throughout the life cycle of the project. In order to optimize the potentially congested schedules in the planning stage, either the range of productivities available to activities are narrowed down, or the overlapped activities are rescheduled to minimize the conflict. A numerical example is analyzed to demonstrate the added benefits of the proposed method.  1 INTRODUCTION Workspaces associated with construction activities and materials continuously go through variations in space and time throughout the ifecycle of a project. Selecting construction methods, scheduling activities, and planning the use of site space over time have been known to be the key to constructing a construction project efficiently (Zouein and Tommelein, 2001). Construction planning, scheduling and control need to be studied and employed not only on temporally but also spatially.  Integrating spatial information of project resources and activities into project scheduling and control has long been studied. However, compared to the available methods for sequencing tasks, provided by critical path analysis and its derivatives, still relatively lower attention is paid to the allocation of tasks to spaces (Winch and North, 2006).  The space planning problem in construction has been studied in two different domains: the space scheduling problem, which focuses on planning of task execution spaces (e.g. (Thabet and Beliveau, 1997, Dawood and Mallasi, 2006, Koo et al., 2013, Choi et al., 2014)) and the site layout problem which 125-1 focuses on the location of temporary facilities of various kinds (e.g. (Zhou et al., 2009, Andayesh and Sadeghpour, 2014, Pradhananga and Teizer, 2014, Karan and Ardeshir, 2008)).  Acknowledging  activity work  space  requirements  and  integrating  this  requirement  as  a  constraint  in  the  scheduling  process  provides  a  number  of  benefits  such  as:  improved  safety,  decreased  conflicts  among  workers,  reduced  crew  waiting  and  idling,  improved  efficiency,  increased productivity,  better  quality,  and  reduced  project  delays (Thabet and Beliveau, 1994, Su and Cai, 2013). Among all, the main purpose of such integration of spatial information into schedules is to prevent available and/or potential space-time conflicts and congestions in the construction site. Existence of congestions in construction site not only decreases resource productivity, but also impacts safety, and may lengthen project durations. Linear projects such as highways, bridges, pipelines and railways not only aren’t exception to this rule, but also need more careful considerations in terms of spaces allocated to the movement of their resources. This is due to the fact that, this class of project is characterized by a series of repetitive activities; where construction crews are often required to repeat the same work in various locations and therefore, move from one location to another. This frequent movement of resources makes it important to continuously integrate the location and movement of resources into the schedules of these projects (Roofigari-Esfahan et al., 2015). As a result, the progress of these projects highly depends on the productivities achieved from their resources. In other words, activity production rates derive the development and accuracy of linear schedules (Duffy et al., 2011). The productivities achieved in these projects are dependent upon the space paved by resources allocated to activities over time. Congestion on construction sites often leads to lowered efficiency and productivity of resources (Watkins et al., 2009). Consequently, a reasonable and resource-leveled schedule that allows for adjustments for unforeseen circumstances and minimizes possible congestions during construction is critical for managing linear construction projects. The spaces considered by some methods are the spaces associated with construction objects and not the ones required for the movement of resources. Also, the current linear scheduling methods still do not consider the whole range of possible production rates that are available to each activity at each point in time and space. Also, As a result, they disregard all other possible production rates that can be achieved without delaying activities. This is a limitation that can be tackled by using Space-Time float, as explained in authors’ previous work (Roofigari-Esfahan et al., 2015).  As a continuation of the previous work, this paper proposes a constraint satisfaction approach to quantify and minimize potential space-time congestions in the schedules of linear projects using space-time floats. The space-time congestion in this research specifically means the workspace interference when two activities have both spatial and temporal overlaps. The method is able to detect not only conflicts between each activity with its immediate successors, but also any possible conflict between resources of any activity with all other project activities, throughout the life cycle of the project.  2 RELEVANT LITERATURE Early research studies in construction site planning have recognized the importance of space as a construction resource and have subsequently incorporated it as an integral part of planning constraints (Winch and North, 2006, Hildum and Smith, 2007). Watkins et al. (Watkins et al., 2009) proposed an agent-based modeling method to represent the construction site as a system of complex interactions and explore whether labor efficiency can be treated as an emergent property resulting from individual and crew interactions in space with positive results. Zouein and Tommelein (Zouein and Tommelein, 2001) noted that construction sequences were often constrained by the sequential occupation of workspaces. The utilization of space associated with these sequences is then analyzed from a comparison of space supply and demand. Winch and North (Winch and North, 2006) further refined this idea by defining and analyzing the criticality of space in a manner analogous to critical-path method. Akinci et al. (Akinci et al., 2002) introduced a taxonomy of space conflicts which correctly defines conflict as a high-level knowledge construct, encompassing various forms 125-2 including congestion, unavailability of access, safety hazards, damage of finished products, and design conflict.  Guo (Guo, 2002) analyzed spatial conflict and temporal conflict separately, introducing two independent interference indicators called the interference space percentage and the interference duration percentage. Additionally, the spatial requirements of movement paths (space for moving workers, equipment, and materials on-site) have not been adequately modeled. Other research used graphical methods to explain potential congestions in collided areas and detection of interferences among trades (e.g. (Chua et al., 2010, Koo et al., 2013). Riley and Sanvido (1995) argued that abstracting workspaces in “solid” CAD models was not truly representative of on-site construction. Four-dimensional 4D computer-aided design CAD overcomes this difficulty by incorporating the temporal element in three-dimensional 3D models. Many construction practitioners and researchers have developed four-dimensional (4D) models by linking the three-dimensional (3D) components of buildings with the network activities of a project schedule (Wang et al., 2014, Mallasi, 2006, Moon et al., 2014). However, these methods mainly model the space required for objects in the construction site. They do not consider the space that is occupied throughout execution of the project only as a result of movement of its resources. This way, the space floats that are available to each resource at each point during activity execution is not taken into account while dealing with unforeseen events. As mentioned in the introduction section, due consideration to space requirements becomes more important when it comes to scheduling linear type of construction projects. This is because the dynamic resources on sites of linear projects are more likely to interact with each other in a complex spatial-temporal manner. Currently  available  network  techniques  and  linear  scheduling  methods  mainly  consider  technological  constraints  and  resource  requirements in  the  generation  of schedules. Such techniques and methods overlook the   requirements  of  activities  for  the requisite work  space   for  material  storage  and movement  of manpower  and equipment. There are methods that attempted to visualize linear schedules, using 4D CAD (Staub-French et al., 2008) and MS-Excel (Lluch, 2009) as well as methods that consider variable productivity rates for linear activities; e.g. (Lucko, 2008, Duffy et al., 2011). Yamin (2001) developed an approach to analyze the cumulative effect of productivity rate variability (CEPRV) on linear activities in highway projects. The focus of the research was to advance the risk analysis capabilities of linear scheduling to allow managers to forecast the probability of project delay. Duffy, et al.’s method (2011) divided linear schedules to areas of time and location for which unique production variables, called working windows can be assigned. Working windows display when and where production variables may change along the linear project. Similarly, (Lucko et al., 2014) models linear schedules using singularity functions to model changes in productivity along the execution time of each activity.  However, none of these methods considers the space available to the activities’ resources throughout their execution. As such, the potential space-time conflicts that can happen as a result of variation in planned productivity rates cannot be detected. This is a limitation that can be tackled by using Space-Time float, as introduced in authors’ previous work (Roofigari-Esfahan et al., 2015). As a continuation of the previous work, this paper proposes a constraint satisfaction approach to quantify and minimize potential space-time congestions in the schedules of linear projects using space-time floats. The space-time congestion in this research specifically means the workspace interference when two activities have both spatial and temporal overlaps. The method is able to detect not only conflicts between each activity with its immediate successors, but also any possible conflict between resources of any activity with all other project activities, throughout the life cycle of the project. The details of the method are presented in next section. 3 RESEARCH METHOD The optimization method presented in this paper aims at generating minimized-congestion schedules for linear schedules. The method utilizes the output of the duration optimization method presented by the authors (Roofigari-Esfahan and razavi 2015) to identify and quantify the potential congestions and and re-125-3 maximum achievable resource productivity rates (polygon boundaries) and activity’s optimum productivity rate.  As it can be seen in Figure 2, the intersection between polygons, i.e. potential congestion between activities, are automatically detected. Subsequently, a matrix is generated in which the exact areas of the congestions are calculated. In that matrix, if the element aij is filled with a number, it means there would exist potential congestion between activities i and j. Essentially, the element aij will be equal to zero if no potential congestion exist between the two activities i and j. It is important to note that in the generation of congestion matrix, these congestions are considered for any two activities within project network. This helps detect not only potential congestions between each activity with its immediate successors, but also will detect any potential congestion in the construction job site. The schedule is then revised in the next phase to minimize the detected congestions. Further, as shown in that figure, the time and space that these congestions would potentially happen are also identified.  Figure 2: a) Generated Linear Schedule b) Realization of Potential Space-Time Congestions with Due Consideration to Space-Time Float prisms 3.2  Congestion Minimization The optimization framework proposed here is a Constraint Satisfaction Problems CSP-based optimization model. Constraint programming (CP) is a programming paradigm being used for solving Constraint Satisfaction Problems (CSPs) through using a combination of mathematics, artificial intelligence, and operations research techniques (Chan and Hu 2002; Liu and Wang 2012 (Tang et al., 2014b). It has been successfully used to solve complex combinatorial problems in a wide variety of domains. To improve the computational efficiency of solving problems, CP provides users with different consistency techniques such as node, arc, and path consistency for variable domain reduction. Constraint programming provides different search strategies such as generate and test (GT), backtracking (BT), and forward checking (Liu and Wang 2008; Marriott and Stucky 1998). Its Selection of appropriate variables and values through heuristics reduces the computational effort required and improves the search ability (Liu and Wang 2007; Russell and Norvig 2009). Apart from being applicable to solve a variety of problems, it has particular advantages in solving scheduling problems due to: (1) its efficient solution search mechanism, (2) flexibility to consider a variety of constraint types, and (3) convenience of model formulation (Menesi et al., 2013). In other words, the highly constrained problems associated with project scheduling can be best modeled and optimized using  CP because of its characteristics (constraints are naturally incorporated into the problem description); (Chan and Hu 2002), as well as its flexibility in description of constraints, and its capability in processing complex and special constraints (Tang et al., 2014a). When solving an optimization problem, the objective function in the problem is treated as a constraint and this additional constraint forces the new feasible schedule to have a better objective value than the current schedule. The upper or lower bounds of the constraint are replaced as soon as a better objective function value is found. The propagation 125-5 mechanism narrows the domains of decision variables to reduce the size of the search space while recording the current best schedule. The search terminates when no feasible schedule is found and the last feasible schedule is the optimal schedule (Liu and Wang 2008). CP is suitable for modeling and solution finding of the project scheduling optimization of linear projects and accordingly applied to schedule optimization phase based on LSM. This is because for LSM-based scheduling problems, prioritization of activities in linear scheduling problems becomes clear owing to the logical and sequential constraints in CP (Liu and Wang 2007). Also, the procedure for the solution of a problem does not require complex mathematical models and formula derivation, eliminating unavoidable simplification and ignorance, truthfully reflecting the original appearance of the problems and ensuring the quality of the solution.  To facilitate the use of CP algorithms in scheduling problems, a powerful optimization package, termed ILOG CPLEX Optimization Studio (Beck et al. 2011), was developed incorporating a CP optimizer engine that offers features specially adapted to solving scheduling problems. ILOG CPLEX Optimization Studio was used and the ILOG OPL language was adopted as the model formulation language. For the proposed model, the objective and variables were determined in the problem specification stage. In the research reported in this paper, the objective was considered as minimizing the total congestion area between prisms of different activities, and the decision variables include the start date, minimum and maximum productivity rates of linear activities. For this purpose, the optimum productivity rates of activities and their respective time interval (including start and times) are used as known information. In this optimization phase, different decision variables are taken into account in the optimization process. These variables include prism boundaries, i.e. min and max productivity rates of activities, as well as start time of activities. In other words, in order to minimize the congestion (intersection) areas between activities, the float prism for the either of the intersecting activities (or both) becomes narrower on the congested side; or the start time of the activities in moved to reduce the intersection area.  Changing the boundaries of the float prism on the congested side only, provides the advantage of not reducing the floats available to activities where it is not necessary (Figure 4). In other words, the floats are only compensated where their existence leads to occurrence of congestion between activities. This way, the generated schedule gives the planners a better sense of spaces and times that are practically available to activities. This capability enables the planners to make more practical decisions in using floats of activities when delays happen during execution of activities. It should be noted that, the deadline constraint and precedence relationships between activities are considered as strict constraints which cannot be changed. The optimization data are listed in detail below: Constants: SLi:  Start location of activity i; ELi:  End location of activity i; OptProi:   Optimum resource production rate of activity i; Pmini:   Minimum productivity rate of activity i; Pmaxi:   Maximum productivity rate of activity i; D:   Project deadline Decision variables: P1mini:   Minimum productivity rate of activity i which minimizes congestions; P1maxi:   Maximum productivity rate of activity i which minimizes congestions;   STi    Start time of activity i;  Decision expressions: ETi:   End time of activity i; 125-6 [1]   ; [2]    [3]    Where Acong i,j is the area of the intersection between space-time prisms of activities i and j respectively, Acong is the total congestion area available in the construction site, x1 is the x coordinate of vertex 1 and yn is the y coordinates of the nth vertex. For minimizing the potential congestion areas between construction activities, first, such areas need to be calculated taking into account the decision variables. To formulate this area, a mathematical formula for finding the area of polygons with known coordinates of vertices is used as presented in the Equation 1. This area represents the possible interactions between successive prisms.  Figure 3: Different intersection areas between prisms In order to use this equation, the coordinates of the intersection polygons between each two activities need to be formulated in terms of the decision variable; i.e. start times of the two activities and their prism boundaries. The intersection points are identified by getting the information from the second phase which includes the intersecting lines of the two polygons and the coordinate of the intersection points. Having identified the intersected lines, the system of line equations is set to find the intersection points in terms of decision variables.  [4]                      After solving the parametric system the intersection points are found in terms of start time (ST) and minimum and maximum productivity rates (P1min and P1max) of the intersecting polygons. The coordinates of the intersection points are feed into the area equation. Subsequently, the objective function of the optimization process is formed by summing up all the intersection areas in the project network schedule. The optimization process then optimizes the schedule by minimizing the total congestion in the project network. This is done through narrowing down the space –time float prism of one or both of the intersected prisms by changing their boundary slopes and/or changing start time of succeeding activities. 4 NUMERICAL EXAMPLE The method presented here was applied to a case study to demonstrated and validated. The minimum and maximum productivities are assumed here, while in the real-world examples, project management teams are requested to enter general information about the project and respective activities. The project 125-7 includes construction of a 4-kilometer altering access road. Four activities (A, C, D and F) are assumed to be linear activities while the other two (B and E) were considered to be non-repetitive for this specific example. Accordingly, maximum, minimum were estimated (the inputs to the optimization module) for repetitive activities. The project was initially estimated to finish in 17 days. The description of each activity as well as inputs and outputs of the optimization process are included in Table 1 and Table 2, respectively.  As it is shown in Table 1, the input data of the congestion minimization process include Activity IDs, their start and end location, successors, their minimum and maximum achievable productivity rates,as well as their planned (optimum) productivity rates and its associated duration. The developed optimization engine uses this information to search for the optimum duration for the project for which the least potential congestion in the project is achieved, considering all logical and spatio-temporal (productivity) constraints to each activity. As shown in Table 1, the project deadline (17 days in this case) is another constraint which is inputted to the optimization process. The output of this process includes optimum minimum and maximum productivity rates for all repetitive activities (that also is representative of floats available to the resources at each point in time and space) as well as start and end times of all the activities, considering precedence relationships. The optimum duration for each activity and its associated duration are kept the same. The optimum duration achieved in this process might be longer than the previously duration for the project, but still satisfies the deadline constraint.   Figure 4: Initial Schedule versus the Optimum Congestion-less Schedule Table 1: Input of the Optimization Process NbTasks 6       Deadline 17       Initial Schedule Task SL EL Pmin Pmax SuccsId Duration productivity A 0 4 0.3 0.9 {2,3} 8 0.5 B 0 4 Non repetitive Non repetitive {4} Non repetitive Non repetitive C 0 4 0.3 0.8 {4,5} 8 0.5 D 0 4 0.2 0.7 {6} 10 0.4 E 0 4 Non repetitive Non repetitive {6} Non repetitive Non repetitive F 0 4 0.5 1 - 6 0.75 125-8 Figure 4 shows the initially generated schedule for this project along with the optimum schedule. As it can be seen in that figure, in the generated schedule the intersection between space-time float polygons for succeeding activities are detected and marked. These potential congestions are totally eliminated in the optimum schedule. The potential congestion area between intersecting activities as well as the total congestion inherent to the project schedule is calculated. As it can be seen in the Figure 4a, potential congestion in location/time exists between activities C and D, as well as between activities D and F. The project schedule’s total potential congestion is calculated to be 4.418 (km.day). Table 2 shows the results of the schedule after the proposed optimization process. It can be inferred from the table 2 that the scheme of the schedule generated using proposed model is significantly different from that of the initial schedule in which congestion was not considered. The schedule finishes at 16 days which still satisfies the 17 day deadline constraint. Although the achieved duration is 1 days longer than the original minimum duration, the potential congestions are reduced to zero in the optimized schedule. For this purpose, as can be seen in Table 2 and Figure 4b, activities D and F are moved by one day, i.e. from day 5 to 6 and 9 to 10, respectively. Also, the available productivity interval for activity D (its minimum and maximum productivity rates) is reduced from [0.2, 0.7] to [0.3, 0.5], which accordingly means less float will be available to the resources of this activity. The result attained through this example ensures the applicability of the method. This way, the approach presented here helps in providing realistic plans for linear projects both in terms of time and space, while satisfying all there logical constraints. Table 2: Optimization Output Optimum Schedule Activity Pmin(opt) Pmax(opt) ST ET Duration A 0.3 0.9 0 8 8 C 0.3 0.8 4 12 8 D 0.3 0.5 6 16 10 F 0.5 1 10 16 6 5 CONCLUDING REMARKS In order to avoid or eliminate space-time conflicts in construction job sites, the subcontractors may frequently  change  their  schedules.  This occasionally cause a change in the materials delivery dates and/or the operation productivity or production rates Such changes can be eliminated through detecting the potential space-time congestions in the planning phase and minimizing them before they happen. In this paper a method of detection, quantification and minimization of potential space time congestions of linear projects is presented. The method minimizes the potential space time congestions in the schedules of linear projects with due consideration to all their logical and precedence constraints. This is done by first, identifying the space time floats available to the resources of each activity, and then using this flexibility in predicting and accordingly minimizing the potential congestions. The method for time-space scheduling was subsequently validated through a numerical example. The results obtained promise applicability of the proposed method in managing linear construction projects. 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