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

Construction space float definition, quantification, and analysis Said, Hisham M. M.; Lucko, Gunnar 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  CONSTRUCTION SPACE FLOAT DEFINITION, QUANTIFICATION, AND ANALYSIS Hisham M. M. Said1, 2, 4 and Gunnar Lucko3 1 Department of Civil Engineering, Santa Clara University, USA 2 Structural Engineering Department, Cairo University, Egypt 3 Department of Civil Engineering, Catholic University of America, USA 4 hsaid@scu.edu Abstract: Schedule float is a fundamental concept in construction planning and control that refers to the flexibility of delaying project activities. However, traditional schedule floats ofer limited help in congested construction sites and cannot answer a common field question of ‘how much time is available to use this space to  stage  material?’  This  paper therefore  presents the  development  of  new theory  and  metrics  of space float for  construction  activities  based  on  a  previously  developed  spatial  scheduling  model. It is structured into three  main  sections.  First, it  presents  a review  of  a  previously  developed  spatial scheduling model that utilizes singularity functions to represent and schedule activities as interdependent and  overlapping  workspaces.  Second, it  presents the  diferent  possible  activity float types  and the detailed  calculation  of  activities  shift float.  Third, it  describes the  new  space float  algorithm to  generate dynamic position float contours that change over the project time. The calculation of proposed activity and space floats are ilustrated with an example of a smal construction jobsite. The proposed concept has the potential to  strengthen the relation  between  construction  scheduling  and  other  management functions, such as e.g. lean operations and material layout planning. 1 INTRODUCTION Al  construction  activities  occur  within their integrated  environment  of temporal  and  spatial  constraints. While the time aspect has been thoroughly explored in construction scheduling, the linkage to the spatial dimensions  has received less  atention,  potentialy  due to  a lack  of  an integrated  model that is  able to jointly  express these  aspects  mathematicaly,  while  adequately  considering their  very  diferent  nature: Time is a single dimension that can only move forward, inevitably passes whether progress is made on the jobsite or not, and is germane to al activities without being influenced by them. Space, on the other hand,  has  additional  dimensions  of the  workspace  wherein  each  productive  activity  occurs,  which  can move into  either  direction  or remain  stationary,  and is typicaly  occupied  exclusively  by  a  single  activity and thus represents an important resource of limited availability. Previous research (Lucko et al. 2014a) has introduced  how to  mathematicaly  describe  both time  and  space  with  singularity functions – spatial scheduling, as is explained in the folowing section – but has not yet examined the float that is generated in such schedules, which wil provide valuable information to the project manager about the criticality and flexibility of activities at diferent locations. The model assumes that spatial height can be simplified based on a safety rule that no two activities work underneath one another. Scheduling with explicit consideration of a spatial aspect thus has potential to improve eficiency while fulfiling constraints and ensuring safety. 200-1 Relevant previous studies can be clustered into four main areas, spatial float analysis, modeling activity workspaces, spatial scheduling, and space criticality. First, studies of spatial float were performed for one-dimensional linear schedules. The application of total float and rate float concepts in linear schedule were investigated by Awwad and Ioannou (2007), Ammar (2003), and Harmelink (2001). In addition, Kalantzis and Lambropoulos (2004) developed a methodology to determine the critical path and segments within a linear  schedule.  Lucko  and  Peña  Orozco (2009) formulated  diverse float types (total, free, interfering, independent,  and  safety) in linear  schedules  using  singularity functions.  Second,  studies  developed formulation and conflict analysis methodologies of construction activity workspaces using 4D topological metrics (Chua  et  al.  2010;  Su  and  Cai  2014a/b),  GIS (Bansal  2011),  CAD (Gu  2002),  and taxonomies (Akinci et al. 2002). Third, spatial scheduling systems were developed to generate and track construction schedules considering their spatial needs and behaviors. Thabet and Beliveau (1994) created a system that formulates and quantifies the workspaces of construction activity to be considered in sequencing of the work. Malasi (2009) developed a 4D optimization model in search of an optimum execution strategy to  minimize the  conflicts  between  activity  workspaces.  Esfahan et  al. (2013)  described  spatio-temporal schedule tracking  and  updating  of resource  movements through  an  enveloping  prism  of  maximum  and minimum  expected  progress rates. Fourth,  Winch  and  North (2006)  were  explicit in  acknowledging the jobsite space as a resource and developed a system that reports its overloading by activity workspaces. Despite the contributions of these diverse previous studies, no methodologies or metrics were developed to quantify float of activities in two-dimensional (2D) spatial schedules (Lucko et al. 2014b). Absence of float in spatial schedules prevents project managers and schedulers from being able to properly evaluate the impact  of  potential  delays  and interruptions  within its  actual  physical  environment. Furthermore, the newly envisioned space float metric can be used as a link between the managerial tasks of construction scheduling  and jobsite layout  planning.  Accordingly, this  paper  presents new  methodologies to  quantify activity and space floats by  using singularity functions. The folowing sections  wil therefore sequentialy explain the application of singularity functions in spatial scheduling, the new metrics of activity floats (shift, rate, and combined), space float quantification and contours for individual positions, and the ilustration of the proposed methodology by calculating the new activity and space floats of an application example. 2 SPATIAL SCHEDULING USING SINGULARITY FUNCTIONS Singularity functions are mathematicaly defined so that they can specificaly express singularities, which are locations  of  discontinuities in the  value  or  behaviour  of  a function.  They  employ  an  operator that is denoted with pointed brackets 〈  〉 and a shape exponent n to indicate the type of behaviour – constant, linear, quadratic, etc. – that is being modeled. Within the brackets, the activation cutof a is compared to the input x to determine whether the entire function t(x) remains zero or yields a non-zero value. Finaly, the strength factor s determines the intensity of whatever behaviour n provides. Equation 1 provides the operator for a single term. As each term only captures exactly one behaviour (from its activation onward), multiple terms are added to compose any more complex singularity function. Multiple singularity functions can be added; multiple terms can be simplified if they have an identical activation a and exponent n. [1] ( ) ( )≥<−⋅=−⋅= xxaxififaxsaxsxt nn 0  To express one activity within a linear schedule, which measures quantity of work and durations across time, the former is  selected  as the independent  variable x and the later  as the  dependent  variable t(x) (Lucko et al. 2014b) as Figure 1 shows. The reason is that time is typicaly supposed to be minimized on construction  projects,  while work is a given input. For an activity  with start  position axS (often  zero) and finish position axF, a start time of tS and a productivity of Δx work units that are produced in Δt time units, the singularity function is t(x) = tS·〈x – axS〉0 + Δt/Δx·[〈x – axS〉1 – 〈x – axF〉1] – Δt·〈x – axF〉0, where tS is an upstep  on the time  axis (intercept), Δt/Δx is  a  slope (inverse  of  productivity,  due to  charting time  over work). The singularity function models the activity: 1) Its first term adds the activity start tS to the schedule only between the its start and finish positions, 2) the second term adds its rate of progress and removes it beyond its finish position axF, else this activity would  continue to  produce forever;  and  3) the last term 200-2 removes the activity duration Δt beyond its finish position axF, which is accumulated by the second term. By evaluating such equation for any work quantity, the time when it wil be completed can be determined.  xaxaxtSaxS axFtFΔxΔta) Nonlinear Singularity Functionb) Linear Singularity Functionc) Linear Activity as a Singularity Function  Figure 1: Singularity Functions and Their Use to Represent Linear Activities 2.1 Activity Progress Representation Previous work by the authors has broadened the definition of Equation 1 beyond its confines of a single independent variable (Lucko et al. 2014a). Since activities are located within the site, a two-dimensional plot of land, reflecting it in the planning and scheduling efort adds realism and value, because it enables that spatial interaction can be modeled and analyzed. Thus each activity can be located by its coordinates on two length axes x and y. The third dimension – height – is beyond the scope of this research and wil be addressed in future work. Activities are assumed to progress paralel to one or both axes as folows: • Activity may be stationary and completely occupies an x-y-area for a specific period of time; • Activity can only progress into positive, negative, or both positive and negative x-direction’ • Activity can only progress into positive, negative, or both positive and negative y-direction; • Activity can progress into a combination of positive or negative x-direction and y-direction. Equation 1 can only express progress in terms of the plane of time t over a single length axis. Viewing a projection of the  progress  along both x and y provides a solution to the chalenge of how to  extend the model into the third  dimension:  Combining  projections in the x-t-plane  and y-t-plane  via their  common variable t. Each projection is a regular singularity function per Equation 1 of order n = 0 for a stationary activity or n = 1 for a directional one. Equation 2 models an activity with three dimensions, two of space and one of time. Its ranges are {axS to axF} on the x-axis and {ayS to ayF} on the y-axis, respectively. In this example, it grows into the positive x-direction as a multiplicative combination of linear behaviour in the x-t-plane and constant behaviour in the y-t-plane. Al other such directions can be modeled analogously. [2] ( ) [ ][ ]0000, yFySxFxSS ayayaxaxtyxt −−−⋅−−−⋅=         [ ] ( )[ ]01100 xFxSxFxFxSyFyS axaaaxaxayayxz −⋅−−−−−⋅−−−⋅∆∆+  The last terms in Equation 2 subtract slope and height at axF to ensure that the activity does not continue to occupy space beyond its boundaries. If it is only evaluated for values within valid x- and y-ranges, it is not strictly necessary to subtract the duration (height on the t-axis) that the activity has consumed beyond axF to yield correct results. Indeed, even the negative terms in the rectangular brackets could formaly be omited, and are merely used to convey the information about the upper boundary of the interval itself.  200-3 2. The earliest execution time (EET) is defined as the minimum value of any successor activities that are scheduled  at  position  P,  which  here is  calculated  as the  minimum  of t(6,  10)A and t(6,  10)B as  9 hours; 3. The position space float PF(x, y, t) is calculated as the diference between EET and the current time t that must be known. Accordingly, at position P = {6, 10} at time t = 0, here its PF(6, 10, 0)P = 9 hours; 4. If no successor activities are executed at the position that is analyzed, its PF is simply the diference between project finish and current time. Thus at Q = {28, 25} at t = 0, the PF(28, 25, 0)Q = 45 hours; 5. The previous steps can be applied to al discrete positions within the jobsite to generate position float contours for any time of interest. Figure 5 shows the position float contours for this example at t = 0. Position float contours wil dynamicaly vary across the jobsite as time passes until the project finish.  y xPoint P (6,10)tA(6,10)tB(6,10)Point Q (28,25) Point P (6,10)Space Float   PF(x,y)Point Q (28,25)t Figure 5: Example of Position Float Calculations 5 APPLICATION EXAMPLE Calculating the new activity and position float metrics are ilustrated in an application example that folows the approach that Lucko et al. (2014a) had originaly analyzed. It comprises nine construction activities of a slab-on-grade (SOG) and underground utilities (electrical and drainage) as Figure 6 shows. Its activities are  one-directional (B,  C,  D,  E,  G,  and  H); two-directional (A  and I),  or  stationary (F).  Resource-driven relations are considered between those of the same trade, e.g. electrical (B, C, D, E, and F) or drainage activities (G, and H). SOG concrete pouring activity (I) has a finish-to-start dependency with al activities, while rebar placement (A) has a concurrent relation with its predecessors (F, G, and H). The mentioned spatial  scheduling  model (Lucko et  al. 2014a) generated the  space  schedule. Its optimized total  project duration is 53 hours. The final scheduled direction, start, and finish of each activity are listed in Table 2.  Table 2: Spatial Scheduling Results and Activity Floats of the Application Example Number Activity Name Duration [h] Direction S [h] F [h] SF [h] RF [h/m] CF [h.m2] 1 A 16 Pos. x 06 26 0 0.0 0750.0 2 B 24 Pos. x 25 28 0 0.0 0375.0 3 C 14 Pos. y 31 33 0 0.0 0150.0 4 D 6 Pos. x 28 31 0 0.0 0375.0 5 E 14 Pos. y 33 35 0 0.0 0450.0 6 F 6 Stationary 00 08 12 0.0 1400.0 7 G 16 Pos. x 00 08 0 0.0 0100.0 8 H 20 Pos. x 08 16 8 0.8 0437.5 9 I 10 Neg. y 37 53 0 0.0 0000.0  200-7 Space Float S PF(x,y) Figure 8: Position Float Contours of the Application Example (t = 33) 6 CONCLUSIONS AND RECOMMENDATIONS • This  paper  has  provided  an  overview  of  a  newly  developed  spatial  scheduling  model  wherein singularity functions  express the  geometry  and  progress  of  each  activity.  However,  previous research had not yet provided any metrics of space float (including criticality of activities without such flexibility). Therefore, shift, rate, and combined float have been discussed, and position float contours have been introduced. Together, they provide a detailed view of the spatio-temporal constraints and opportunities that  exist  within  a  project  schedule from  a  spatial  perspective. In  summary, it is envisioned that  spatial  scheduling  and floats  can  greatly  support  professionals in the  construction industry in integrating space-aware project schedules, designing eficient site layouts, and facilitating lean execution of their construction activities. • Future research should mathematicaly extend the new model with its two dimensions of space and one dimension of time to more dimensions, including height or even other aspects, such as e.g. cost, by  generalizing the  multiplicative  approach (Lucko  and  Su  2014),  even if it  cannot  be  visualized completely. References Akinci, B., Fischer, M. A., Levit, R. E., Carlson, R. C. 2002. Formalization and automation of time-space conflict analysis. Journal of Computing in Civil Engineering 16(2): 124-134. Ammar,  M.A.  2003.  Float  analysis  of  non-serial repetitive  activities. Construction  Management  and Economics 21(5): 535-542. Awwad,  R.  E.,  P.  G. Ioannou.  2007.  Floats in  RSM:  Repetitive  scheduling  method. Proceedings  of the Construction  Research  Congress,  Grand  Bahama Island,  Commonwealth  of the  Bahamas,  May  6-8, 2007, American Society of Civil Engineers, Reston, VA: 8 pp. Bansal, V. K. 2011. Use of GIS and topology in the identification and resolution of space conflicts. Journal of Computing in Civil Engineering 25(2): 159-171. Chua, D. K. H., Yeoh, Y., Song, Y. 2010. Quantification of spatial temporal congestion in four-dimensional computer-aided design. Journal of Construction Engineering and Management 136(6): 641-649. Esfahan, N. R., Páez A., Razavi, S. 2013. Spatio-temporal progress estimation for highway construction. Proceedings  of the  ASCE International  Workshop  on  Computing in  Civil  Engineering,  Los  Angeles, CA, June 23-25, 2013, American Society of Civil Engineers, Reston, VA: 541-548. Guo, S.-J. 2002. Identification  and resolution of  work space conflicts in building  construction. Journal of Construction Engineering and Management 128(4): 287-295. Harmelink,  D.  J.  2001.  Linear  scheduling  model:  Float  characteristics. Journal  of  Construction Engineering and Management 127(4): 255-260. 200-9 Kalantzis,  A.,  Lambropoulos,  S.  2004.  Critical  path  determination  by incorporating  minimum  and maximum time  and  distance  constraints into linear  scheduling. Engineering,  Construction  and Architectural Management 11(3): 211-222. Lucko,  G.,  Peña  Orozco,  A.  A.  2009.  Float types in linear  schedule  analysis  with  singularity functions. Journal of Construction Engineering and Management 135(5): 368-377. Lucko, G., Said, H. M. M., Bouferguene, A. 2014a. Spatialy-constrained scheduling with multi-directional singularity functions. Proceedings  of the  Construction  Research  Congress,  Atlanta,  GA,  May  19-21, 2014, American Society of Civil Engineers, Reston, VA: 1448-1457. Lucko,  G.,  Said,  H.  M.  M.,  Bouferguene,  A.  2014b.  Construction  spatial  modeling  and  scheduling  with three-dimensional singularity functions. Automation in Construction 43(July): 132-143. Lucko, G., Su, Y. 2014. Singularity functions as new tool for integrated project management. Proceedings of the  Creative  Construction  Conference,  eds.  Hajdu,  M.,  Skibniewski,  M.,  Prague,  Czech  Republic, June 21-24, 2014, Szent István University, Budapest, Hungary: 414-420. Malasi,  Z.  2009.  Towards  minimizing  space-time  conflicts  between  site  activities  using  simple  generic algorithm – the  best  execution  strategy. Journal  of Information  Technology in  Construction 14:  154-179. Roofigari,  E.  N.,  Razavi,  S.  2013.  GIS-based resource integrated  progress tracking for  construction projects using spatio-temporal data. Proceedings of the 4th Construction Specialty Conference, May 29 to June 1, 2013, Canadian Society of Civil Engineers, Montréal, Québec: CON-35-1 – CON-35-10. Su,  X.,  Cai,  H.  2014a.  Enabling  construction  4D topological  analysis for  efective  construction  planning. Journal of Computing in Civil Engineering: 04014123(10). Su, X., Cai, H. 2014b. Life cycle approach to construction workspace modeling and planning. Journal of Construction Engineering and Management 140(7): 04014019(12). Thabet,  W.Y.,  Beliveau,  Y.J.  1994. Modeling  workspace to  schedule repetitive floors in  multistory buildings. Journal of Construction Engineering and Management 120(1): 96-116. Winch,  G.  M.,  North,  S.  2006.  Critical  space  analysis. Journal  of  Construction  Engineering  and Management 132(5): 473-481.  200-10  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  CONSTRUCTION SPACE FLOAT DEFINITION, QUANTIFICATION, AND ANALYSIS Hisham M. M. Said1, 2, 4 and Gunnar Lucko3 1 Department of Civil Engineering, Santa Clara University, USA 2 Structural Engineering Department, Cairo University, Egypt 3 Department of Civil Engineering, Catholic University of America, USA 4 hsaid@scu.edu Abstract: Schedule float is a fundamental concept in construction planning and control that refers to the flexibility of delaying project activities. However, traditional schedule floats ofer limited help in congested construction sites and cannot answer a common field question of ‘how much time is available to use this space to  stage  material?’  This  paper therefore  presents the  development  of  new theory  and  metrics  of space float for  construction  activities  based  on  a  previously  developed  spatial  scheduling  model. It is structured into three  main  sections.  First, it  presents  a review  of  a  previously  developed  spatial scheduling model that utilizes singularity functions to represent and schedule activities as interdependent and  overlapping  workspaces.  Second, it  presents the  diferent  possible  activity float types  and the detailed  calculation  of  activities  shift float.  Third, it  describes the  new  space float  algorithm to  generate dynamic position float contours that change over the project time. The calculation of proposed activity and space floats are ilustrated with an example of a smal construction jobsite. The proposed concept has the potential to  strengthen the relation  between  construction  scheduling  and  other  management functions, such as e.g. lean operations and material layout planning. 1 INTRODUCTION Al  construction  activities  occur  within their integrated  environment  of temporal  and  spatial  constraints. While the time aspect has been thoroughly explored in construction scheduling, the linkage to the spatial dimensions  has received less  atention,  potentialy  due to  a lack  of  an integrated  model that is  able to jointly  express these  aspects  mathematicaly,  while  adequately  considering their  very  diferent  nature: Time is a single dimension that can only move forward, inevitably passes whether progress is made on the jobsite or not, and is germane to al activities without being influenced by them. Space, on the other hand,  has  additional  dimensions  of the  workspace  wherein  each  productive  activity  occurs,  which  can move into  either  direction  or remain  stationary,  and is typicaly  occupied  exclusively  by  a  single  activity and thus represents an important resource of limited availability. Previous research (Lucko et al. 2014a) has introduced  how to  mathematicaly  describe  both time  and  space  with  singularity functions – spatial scheduling, as is explained in the folowing section – but has not yet examined the float that is generated in such schedules, which wil provide valuable information to the project manager about the criticality and flexibility of activities at diferent locations. The model assumes that spatial height can be simplified based on a safety rule that no two activities work underneath one another. Scheduling with explicit consideration of a spatial aspect thus has potential to improve eficiency while fulfiling constraints and ensuring safety. 200-1 Relevant previous studies can be clustered into four main areas, spatial float analysis, modeling activity workspaces, spatial scheduling, and space criticality. First, studies of spatial float were performed for one-dimensional linear schedules. The application of total float and rate float concepts in linear schedule were investigated by Awwad and Ioannou (2007), Ammar (2003), and Harmelink (2001). In addition, Kalantzis and Lambropoulos (2004) developed a methodology to determine the critical path and segments within a linear  schedule.  Lucko  and  Peña  Orozco (2009) formulated  diverse float types (total, free, interfering, independent,  and  safety) in linear  schedules  using  singularity functions.  Second,  studies  developed formulation and conflict analysis methodologies of construction activity workspaces using 4D topological metrics (Chua  et  al.  2010;  Su  and  Cai  2014a/b),  GIS (Bansal  2011),  CAD (Gu  2002),  and taxonomies (Akinci et al. 2002). Third, spatial scheduling systems were developed to generate and track construction schedules considering their spatial needs and behaviors. Thabet and Beliveau (1994) created a system that formulates and quantifies the workspaces of construction activity to be considered in sequencing of the work. Malasi (2009) developed a 4D optimization model in search of an optimum execution strategy to  minimize the  conflicts  between  activity  workspaces.  Esfahan et  al. (2013)  described  spatio-temporal schedule tracking  and  updating  of resource  movements through  an  enveloping  prism  of  maximum  and minimum  expected  progress rates. Fourth,  Winch  and  North (2006)  were  explicit in  acknowledging the jobsite space as a resource and developed a system that reports its overloading by activity workspaces. Despite the contributions of these diverse previous studies, no methodologies or metrics were developed to quantify float of activities in two-dimensional (2D) spatial schedules (Lucko et al. 2014b). Absence of float in spatial schedules prevents project managers and schedulers from being able to properly evaluate the impact  of  potential  delays  and interruptions  within its  actual  physical  environment. Furthermore, the newly envisioned space float metric can be used as a link between the managerial tasks of construction scheduling  and jobsite layout  planning.  Accordingly, this  paper  presents new  methodologies to  quantify activity and space floats by  using singularity functions. The folowing sections  wil therefore sequentialy explain the application of singularity functions in spatial scheduling, the new metrics of activity floats (shift, rate, and combined), space float quantification and contours for individual positions, and the ilustration of the proposed methodology by calculating the new activity and space floats of an application example. 2 SPATIAL SCHEDULING USING SINGULARITY FUNCTIONS Singularity functions are mathematicaly defined so that they can specificaly express singularities, which are locations  of  discontinuities in the  value  or  behaviour  of  a function.  They  employ  an  operator that is denoted with pointed brackets 〈  〉 and a shape exponent n to indicate the type of behaviour – constant, linear, quadratic, etc. – that is being modeled. Within the brackets, the activation cutof a is compared to the input x to determine whether the entire function t(x) remains zero or yields a non-zero value. Finaly, the strength factor s determines the intensity of whatever behaviour n provides. Equation 1 provides the operator for a single term. As each term only captures exactly one behaviour (from its activation onward), multiple terms are added to compose any more complex singularity function. Multiple singularity functions can be added; multiple terms can be simplified if they have an identical activation a and exponent n. [1] ( ) ( )≥<−⋅=−⋅= xxaxififaxsaxsxt nn 0  To express one activity within a linear schedule, which measures quantity of work and durations across time, the former is  selected  as the independent  variable x and the later  as the  dependent  variable t(x) (Lucko et al. 2014b) as Figure 1 shows. The reason is that time is typicaly supposed to be minimized on construction  projects,  while work is a given input. For an activity  with start  position axS (often  zero) and finish position axF, a start time of tS and a productivity of Δx work units that are produced in Δt time units, the singularity function is t(x) = tS·〈x – axS〉0 + Δt/Δx·[〈x – axS〉1 – 〈x – axF〉1] – Δt·〈x – axF〉0, where tS is an upstep  on the time  axis (intercept), Δt/Δx is  a  slope (inverse  of  productivity,  due to  charting time  over work). The singularity function models the activity: 1) Its first term adds the activity start tS to the schedule only between the its start and finish positions, 2) the second term adds its rate of progress and removes it beyond its finish position axF, else this activity would  continue to  produce forever;  and  3) the last term 200-2 removes the activity duration Δt beyond its finish position axF, which is accumulated by the second term. By evaluating such equation for any work quantity, the time when it wil be completed can be determined.  xaxaxtSaxS axFtFΔxΔta) Nonlinear Singularity Functionb) Linear Singularity Functionc) Linear Activity as a Singularity Function  Figure 1: Singularity Functions and Their Use to Represent Linear Activities 2.1 Activity Progress Representation Previous work by the authors has broadened the definition of Equation 1 beyond its confines of a single independent variable (Lucko et al. 2014a). Since activities are located within the site, a two-dimensional plot of land, reflecting it in the planning and scheduling efort adds realism and value, because it enables that spatial interaction can be modeled and analyzed. Thus each activity can be located by its coordinates on two length axes x and y. The third dimension – height – is beyond the scope of this research and wil be addressed in future work. Activities are assumed to progress paralel to one or both axes as folows: • Activity may be stationary and completely occupies an x-y-area for a specific period of time; • Activity can only progress into positive, negative, or both positive and negative x-direction’ • Activity can only progress into positive, negative, or both positive and negative y-direction; • Activity can progress into a combination of positive or negative x-direction and y-direction. Equation 1 can only express progress in terms of the plane of time t over a single length axis. Viewing a projection of the  progress  along both x and y provides a solution to the chalenge of how to  extend the model into the third  dimension:  Combining  projections in the x-t-plane  and y-t-plane  via their  common variable t. Each projection is a regular singularity function per Equation 1 of order n = 0 for a stationary activity or n = 1 for a directional one. Equation 2 models an activity with three dimensions, two of space and one of time. Its ranges are {axS to axF} on the x-axis and {ayS to ayF} on the y-axis, respectively. In this example, it grows into the positive x-direction as a multiplicative combination of linear behaviour in the x-t-plane and constant behaviour in the y-t-plane. Al other such directions can be modeled analogously. [2] ( ) [ ][ ]0000, yFySxFxSS ayayaxaxtyxt −−−⋅−−−⋅=         [ ] ( )[ ]01100 xFxSxFxFxSyFyS axaaaxaxayayxz −⋅−−−−−⋅−−−⋅∆∆+  The last terms in Equation 2 subtract slope and height at axF to ensure that the activity does not continue to occupy space beyond its boundaries. If it is only evaluated for values within valid x- and y-ranges, it is not strictly necessary to subtract the duration (height on the t-axis) that the activity has consumed beyond axF to yield correct results. Indeed, even the negative terms in the rectangular brackets could formaly be omited, and are merely used to convey the information about the upper boundary of the interval itself.  200-3 2. The earliest execution time (EET) is defined as the minimum value of any successor activities that are scheduled  at  position  P,  which  here is  calculated  as the  minimum  of t(6,  10)A and t(6,  10)B as  9 hours; 3. The position space float PF(x, y, t) is calculated as the diference between EET and the current time t that must be known. Accordingly, at position P = {6, 10} at time t = 0, here its PF(6, 10, 0)P = 9 hours; 4. If no successor activities are executed at the position that is analyzed, its PF is simply the diference between project finish and current time. Thus at Q = {28, 25} at t = 0, the PF(28, 25, 0)Q = 45 hours; 5. The previous steps can be applied to al discrete positions within the jobsite to generate position float contours for any time of interest. Figure 5 shows the position float contours for this example at t = 0. Position float contours wil dynamicaly vary across the jobsite as time passes until the project finish.  y xPoint P (6,10)tA(6,10)tB(6,10)Point Q (28,25) Point P (6,10)Space Float   PF(x,y)Point Q (28,25)t Figure 5: Example of Position Float Calculations 5 APPLICATION EXAMPLE Calculating the new activity and position float metrics are ilustrated in an application example that folows the approach that Lucko et al. (2014a) had originaly analyzed. It comprises nine construction activities of a slab-on-grade (SOG) and underground utilities (electrical and drainage) as Figure 6 shows. Its activities are  one-directional (B,  C,  D,  E,  G,  and  H); two-directional (A  and I),  or  stationary (F).  Resource-driven relations are considered between those of the same trade, e.g. electrical (B, C, D, E, and F) or drainage activities (G, and H). SOG concrete pouring activity (I) has a finish-to-start dependency with al activities, while rebar placement (A) has a concurrent relation with its predecessors (F, G, and H). The mentioned spatial  scheduling  model (Lucko et  al. 2014a) generated the  space  schedule. Its optimized total  project duration is 53 hours. The final scheduled direction, start, and finish of each activity are listed in Table 2.  Table 2: Spatial Scheduling Results and Activity Floats of the Application Example Number Activity Name Duration [h] Direction S [h] F [h] SF [h] RF [h/m] CF [h.m2] 1 A 16 Pos. x 06 26 0 0.0 0750.0 2 B 24 Pos. x 25 28 0 0.0 0375.0 3 C 14 Pos. y 31 33 0 0.0 0150.0 4 D 6 Pos. x 28 31 0 0.0 0375.0 5 E 14 Pos. y 33 35 0 0.0 0450.0 6 F 6 Stationary 00 08 12 0.0 1400.0 7 G 16 Pos. x 00 08 0 0.0 0100.0 8 H 20 Pos. x 08 16 8 0.8 0437.5 9 I 10 Neg. y 37 53 0 0.0 0000.0  200-7 Space Float S PF(x,y) Figure 8: Position Float Contours of the Application Example (t = 33) 6 CONCLUSIONS AND RECOMMENDATIONS • This  paper  has  provided  an  overview  of  a  newly  developed  spatial  scheduling  model  wherein singularity functions  express the  geometry  and  progress  of  each  activity.  However,  previous research had not yet provided any metrics of space float (including criticality of activities without such flexibility). Therefore, shift, rate, and combined float have been discussed, and position float contours have been introduced. Together, they provide a detailed view of the spatio-temporal constraints and opportunities that  exist  within  a  project  schedule from  a  spatial  perspective. In  summary, it is envisioned that  spatial  scheduling  and floats  can  greatly  support  professionals in the  construction industry in integrating space-aware project schedules, designing eficient site layouts, and facilitating lean execution of their construction activities. • Future research should mathematicaly extend the new model with its two dimensions of space and one dimension of time to more dimensions, including height or even other aspects, such as e.g. cost, by  generalizing the  multiplicative  approach (Lucko  and  Su  2014),  even if it  cannot  be  visualized completely. References Akinci, B., Fischer, M. A., Levit, R. E., Carlson, R. C. 2002. Formalization and automation of time-space conflict analysis. Journal of Computing in Civil Engineering 16(2): 124-134. Ammar,  M.A.  2003.  Float  analysis  of  non-serial repetitive  activities. Construction  Management  and Economics 21(5): 535-542. Awwad,  R.  E.,  P.  G. Ioannou.  2007.  Floats in  RSM:  Repetitive  scheduling  method. Proceedings  of the Construction  Research  Congress,  Grand  Bahama Island,  Commonwealth  of the  Bahamas,  May  6-8, 2007, American Society of Civil Engineers, Reston, VA: 8 pp. Bansal, V. K. 2011. Use of GIS and topology in the identification and resolution of space conflicts. Journal of Computing in Civil Engineering 25(2): 159-171. Chua, D. K. H., Yeoh, Y., Song, Y. 2010. Quantification of spatial temporal congestion in four-dimensional computer-aided design. Journal of Construction Engineering and Management 136(6): 641-649. Esfahan, N. R., Páez A., Razavi, S. 2013. Spatio-temporal progress estimation for highway construction. Proceedings  of the  ASCE International  Workshop  on  Computing in  Civil  Engineering,  Los  Angeles, CA, June 23-25, 2013, American Society of Civil Engineers, Reston, VA: 541-548. Guo, S.-J. 2002. Identification  and resolution of  work space conflicts in building  construction. Journal of Construction Engineering and Management 128(4): 287-295. Harmelink,  D.  J.  2001.  Linear  scheduling  model:  Float  characteristics. Journal  of  Construction Engineering and Management 127(4): 255-260. 200-9 Kalantzis,  A.,  Lambropoulos,  S.  2004.  Critical  path  determination  by incorporating  minimum  and maximum time  and  distance  constraints into linear  scheduling. Engineering,  Construction  and Architectural Management 11(3): 211-222. Lucko,  G.,  Peña  Orozco,  A.  A.  2009.  Float types in linear  schedule  analysis  with  singularity functions. Journal of Construction Engineering and Management 135(5): 368-377. Lucko, G., Said, H. M. M., Bouferguene, A. 2014a. Spatialy-constrained scheduling with multi-directional singularity functions. Proceedings  of the  Construction  Research  Congress,  Atlanta,  GA,  May  19-21, 2014, American Society of Civil Engineers, Reston, VA: 1448-1457. Lucko,  G.,  Said,  H.  M.  M.,  Bouferguene,  A.  2014b.  Construction  spatial  modeling  and  scheduling  with three-dimensional singularity functions. Automation in Construction 43(July): 132-143. Lucko, G., Su, Y. 2014. Singularity functions as new tool for integrated project management. Proceedings of the  Creative  Construction  Conference,  eds.  Hajdu,  M.,  Skibniewski,  M.,  Prague,  Czech  Republic, June 21-24, 2014, Szent István University, Budapest, Hungary: 414-420. Malasi,  Z.  2009.  Towards  minimizing  space-time  conflicts  between  site  activities  using  simple  generic algorithm – the  best  execution  strategy. Journal  of Information  Technology in  Construction 14:  154-179. Roofigari,  E.  N.,  Razavi,  S.  2013.  GIS-based resource integrated  progress tracking for  construction projects using spatio-temporal data. Proceedings of the 4th Construction Specialty Conference, May 29 to June 1, 2013, Canadian Society of Civil Engineers, Montréal, Québec: CON-35-1 – CON-35-10. Su,  X.,  Cai,  H.  2014a.  Enabling  construction  4D topological  analysis for  efective  construction  planning. Journal of Computing in Civil Engineering: 04014123(10). Su, X., Cai, H. 2014b. Life cycle approach to construction workspace modeling and planning. Journal of Construction Engineering and Management 140(7): 04014019(12). Thabet,  W.Y.,  Beliveau,  Y.J.  1994. Modeling  workspace to  schedule repetitive floors in  multistory buildings. Journal of Construction Engineering and Management 120(1): 96-116. Winch,  G.  M.,  North,  S.  2006.  Critical  space  analysis. Journal  of  Construction  Engineering  and Management 132(5): 473-481.  200-10  Construction Space Float Definition, Quantification and Analysis5th International/11th Construction Specialty ConferenceJune 8, 2015Gunnar LuckoCatholic University of AmericaHisham SaidSanta Clara UniversityOutlineBackgroundResearch Needs and ObjectiveSpatial SchedulingActivity FloatsSpace FloatApplication ExampleSpace is a central and complex element of construction management (Bernold 2002).BackgroundConstruction SchedulingSite SpaceSequencing & Productivity ImpactSpace Needs & Patterns Relevant CEM Studies:1)  Modeling Activity Workspaces – formulation and conflict analysis.4D topological metrics (Chua et al. 2010; Su and Cai 2014a/b), GIS (Bansal 2011), CAD (Gu 2002), and taxonomies (Akinci et al. 2002)2) Spatial Float Analysis – one-dimensional spatial schedulesAwwad and Ioannou (2007), Ammar (2003), and Harmelink (2001), Kallantzis and Lambropoulos (2004), Lucko and Peña Orozco (2009)3) Spatial Scheduling – space-aware scheduling.Thabet and Beliveau (1994), Mallasi (2009), Esfahan et al. (2013), Lucko et al. (2014)4) Space Criticality – assigning space as a resource.Winch and North (2006)BackgroundRelevant CEM Studies:1)  Modeling Activity Workspaces2) Spatial Float Analysis3) Spatial Scheduling4) Space CriticalityResearch Need & ObjectiveNo methodologies or metrics were developed to quantify the criticality of two-dimensional (2D) spatial schedules Develop Float metrics and algorithms for construction activities and space in 2D space setting Spatial SchedulingSpatial Scheduling Singularity FunctionsxaxaxtSaxS axFtFΔxΔta) Nonlinear Singularity Functionb) Linear Singularity Functionc) Linear Activity as a Singularity FunctionSpatial SchedulingSpatial Scheduling Singularity Functions2 space dimensions + timextSaxS axFtFΔxΔt            yxtFtSaxS axFaySayFΔtΔx( ) [ ] [ ]0000, yFySxFxSS ayayaxaxtyxt −−−⋅−−−⋅=[ ] [ ]00011 yFySxFxFxS ayayaxxaxaxxt−−−⋅−⋅∆−−−−⋅∆∆+Lucko et al. (2014)Activity Spatial RepresentationActivity RelationsTime Time Timea) SequentialB depends on Ab-1) ConcurrentB depends on Ab-2) ConcurrentIndependentaxS_B axF_A axF_B axS_B axF_A axF_B axS_B axF_A axF_BtS_BtF_BtF_AbuffABtF_AtF_BtS_AtS_BtF_BtF_Bx x xAByayF_BayF_AayS_BayS_AaxS_A axS_B axF_A axF_BxActivity WorkspacesSequentialConcurrentActivity Relation DependentIndependentSpatial Scheduling Algorithm1. Sort Activities using selected heuristic rule2. Loop over all activities2.2. Loop over all predecessors2.2.2. Select the scheduling direction with the earliest start time2.3. Output Activity Singularity Function and Scheduling Times2.1. Generate Initial Singularity Function2.2.1 Loop over all possible scheduling directions2.2.1.1 “Stack” the activity at its start timeActivity FloatsAByayF_BayF_AayS_BayS_AaxS_AaxS_B axF_A axF_BxActivity WorkspacesAreaoverlapTimebuffABaxS_AaxS_B axF_A axF_BxTimebuffABxTimebuffABxSFABRFABCFABa) Shift Float (SFAB) b) Rate Float (RFAB) c) Combined Float (CFAB)axS_AaxS_B axF_A axF_B axS_AaxS_B axF_A axF_BtF_AtF_BtS_AtS_BtF_AtF_BtS_AtS_BtF_AtF_BtS_AtS_BShift Float (SF) Acceptable delay of the activity start timeRate Float (RF) Acceptable rate drop of the activity productivityCombined Float (CF) Overall criticality metricActivity FloatsExample – Shift FloatyxActivity AActivity B13 282520DurA = 14 hDurB = 15 hyxSFABbuffAB= 6 hrRxARyBtS_B = 30 hrtF_A = 20 hrOverlapping zonetF_B = 45 hrtS_A = 6 ht1 t2t4t3  SFAB = min [t1,t2,t3,t4] AB – buffABSpace FloatPosition space float PF(x, y, t)It is the amount of time that is available to use the said location (x,y) at time t before it will be occupied by the earliest successor activity whose workspace includes this point also.yxPoint P (6,10)tA(6,10)tB(6,10)Point Q (28,25)tyxActivity AActivity B13 282520DurA = 14 hDurB = 15 h Point QPoint PPF(P,t) = min [tA(6,10) , tB(6,10)]  – t PF(Q,t) = Project Finish Time   – t Space Float ContoursyxPoint P (6,10)tA(6,10)tB(6,10)Point Q (28,25)tPoint P (6,10)Space Float   PF(x,y)Point Q (28,25)Space Float Contours @ t = 0Repeat PF calculations over a grid of site locationsApplication ExampleLucko et al. (2014)Application ExampleElect. Ducts (1)Elect. Ducts (4)Elect. Ducts (2)Elect. Ducts (3)Two-directional Activities for the whole area: (A)SOG Rebar,and (I) Concrete Pouring.Drainage Pipes (1)XYBFGDEElectrical Room InletsCDrainage Pipes (2)HActivity Layout Activity SequencingLucko et al. (2014)Application ExampleLucko et al. (2014)Elect. Ducts (1)Elect. Ducts (4)Elect. Ducts (2)Elect. Ducts (3)Two-directional Activities for the whole area: (A)SOG Rebar,and (I) Concrete Pouring.Drainage Pipes (1)XYBFGDEElectrical Room InletsCDrainage Pipes (2)HXYZF53 hrsApplication ExampleXYZF53 hrsApplication ExampleXYZF53 hrsSpace Float   PF(x,y)Space Float Contours @ t = 0Application ExampleXYZF53 hrsSpace Float Contours @ t = 33Space Float S PF(x,y)Summary and Future ResearchNew 2D activity floats (shift, rate, combined)New Space Float Metric and Representation (contours)These metrics can help in integrating:o Space-aware scheduleso Schedule-driven site layoutso Efficient Lean ExecutionFuture Research:o Extend Model Dimensionality (3rd space dimension, cost, ..)o Site Layout using Space ContoursThank you!Questions and Feedback are welcome!Gunnar Lucko, Ph.D.Director, CE&M ProgramDepartment of Civil EngineeringCatholic University of America202-319-4381lucko@cua.eduHisham Said, Ph.D.Department of Civil EngineeringSanta Clara University408-551-7156hsaid@scu.eduConstruction Space Float Definition, Quantification and Analysis

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