New Optimization Approach for Construction Site Layout

Transcription

1 1 st International Construction Specialty Conference 1ère Conférence internationale spécialisée sur le génie de la construction Calgary, Alberta, Canada May 23-26, 2006 / mai 2006 New Optimization Approach for Construction Site Layout S. Easa 1, F. Sadeghpour 2 and A. Hossain 1 1 Department of Civil Engineering, Ryerson University, Toronto, Ontario, Canada 2 Department of Architectural Science, Ryerson University, Toronto, Ontario, Canada Abstract: In planning construction projects, layout of temporary construction facilities (objects) is an important activity that can affect productivity and safety. The construction area may consist of unavailable area that include existing facilities (sites) and available area in which the objects can be located. The layout of objects is accomplished by optimizing a specific objective function and satisfying a variety of constraints. Thus, construction site layout is a complex problem whose solution requires the use of analytical methods. Existing methods, that include genetic algorithms and graphical methods, are based on dividing the construction area into a grid in x-y coordinate system. This paper presents a new approach, based on nonlinear integer optimization that considers a continuous x-y space in locating a single object. The available construction area is divided into regions that are formulated using binary variables. The objective function of the model minimizes the total weighted distance between the objects and the sites. The proposed approach offers many capabilities, including accommodating non-convex regions, non-rectangular objects, and object-region constraints. Application of the model is illustrated using an example. The results show that the proposed model is more efficient and more flexible than existing methods, and as such should be of interest to construction engineers and practitioners. 1. Introduction The importance of site layout and its effect on productivity of construction sites, as well as safety and security has been repeatedly reported in the literature (Tommelein et. al 1992, Hegazy and Elbetagi 1999, Sadeghpour et al. 2006). Due to its complexity, from the early studies in 1960s, a computerized approach deemed essential to implement the process of site layout. Since then, the development of tools to aid planners in the task of site layout has been a topic of research, and various site layout models have been developed using different approaches and assumptions. Regardless of the utilized techniques, two methodologies have been used in large to develop layout models: improvement methods and construction methods (Moore 1980, Li and Love 1998, Sirinaovakul and Thajchayapong 1996). Improvement methods start from a complete initial site design, which is then altered by moving objects around the site to create new layouts. Each layout is then evaluated against an objective function and the best solution is identified accordingly. This methodology has been used in an earlier model by Moore (1971) and was later implemented in models using genetic algorithms (Li and Love 1998, Elbeltagi and Hegazy 1999). Improvement methods provide near-optimum solutions, but there is no guarantee that they reach the optimum solution. Construction methods locate the objects one at a time. In each step, one object is located on the site, considering the updated status of the site before adding each object. A median placement method is partial improvement, which has the characteristics of both abovementioned algorithms (Li and Love 1998). Here, a facility is located in all the possible positions and a set of possible partial layouts are generated and evaluated. In this way, the most suitable alternative location CT-036-1

2 for the facility at hand is identified. The process is then repeated until all facilities are located. The limitation of the latter methods is that they do not provide a global optimum layout. When locating an object at every stage of layout, the object is located in the optimum location considering the existing situation of the layout. As new objects are entered to the layout, that location may not necessarily be the optimum for that object anymore. As a result, these methods do not result in a final layout that is totally optimal. Mathematical optimization has also been considered for construction site layout. Indeed developing layout models originally started in Operations Research in the area of facility layout in industrial engineering. This trend, however, did not last long and mathematical optimization lost the focus of attention to heuristic models. The main reason for this tendency, apart from the excitement of exercising heuristics and artificial intelligence, was that only a few of these mathematical models seemed to be successful and even then, the models worked only for the specific cases introduced in the problem (Tommelein 1992). As well, these approaches were rendered computationally infeasible for large problems due to the limitations of computer technology at the time (Hamiani 1987). They also demanded a large amount of data to run and required great skill from the user to fit a given problem into the model (Tommelein et al. 1992). Despite the research efforts in developing mathematical models for construction site layout, the use of these models was rather limited. The models were difficult to learn and use, and their computer implementations were complex, and hence received resistance toward being commonly used (Hamiani and Popescu 1988). As well, these models have considerably simplified the real world, resulting in loss of practical information (Tam and Tong 2003). For example, site facilities were generally dealt with as points where their dimensions were ignored, and thus the layout problem was limited to location allocation problem (Tommelein et al. 1992). Furthermore, most models have focused on distance as the objective. In practice, however, the quality of a proposed construction site layout is judged by a multitude of criteria. This paper presents a new approach for optimization of construction site layout that overcomes the limitations of existing methods. Compared to earlier methods, with recent computational advances both in software and hardware, the computation time and effort is not an issue anymore. The proposed approach allows for consideration of objects with different sizes and configurations, and construction sites with different shapes. Several types of constraints can also be modeled. The concept of the proposed approach, which is similar to that used for resource leveling and cash-flow scheduling ((Easa 1989, Easa 1992), is based on nonlinear-integer optimization. For purposes of this paper, the proposed approach is presented in the context of modeling the location of a single object. The following section presents details regarding model development, including formulations for regions, non-rectangular objects, object-site proximity, object-region relationship, non-rectangular regions, and objective function. Example application of the model is then presented, followed by the conclusions. 2. Model Development Consider a construction area that is rectangular with width A and breadth B (Fig. 1). A coordinate system (X, Y) is established such that the origin lies at one of the corners of the construction area. The construction area has two types of areas: available areas and unavailable areas. The available area is an area in which construction objects (temporary facilities) are required to be located. The unavailable area includes construction sites (permanent facilities) whose locations are fixed and other sites that are not available for locating construction objects. The key concept behind the proposed optimization model is to divide the available area into rectangular regions as shown by the dashed lines in Fig. 1. The way in which these rectangular regions are established is arbitrary. However, the smaller the number of these rectangular regions is, the faster the solution of the model would be. Let the number of regions be M. The boundaries of the region i along the x-axis are x1 i and x2 i and along the y-axis are y1 i and y2 i, where i = 1, 2,, M. CT-036-2

3 Y Available Area Unavailable Area Construction Boundary Site B Site j Region i 0 A X Figure 1. Illustration of Regions (Available Area) and Sites (Unavailable Area) The proposed approach is formulated here for a single construction object to be located in the construction area. The object assumed to be rectangular and may have different sizes (Fig. 2a). The object has a width b and breadth h. The coordinates of the centroid of the construction object are XO and YO, which are the decision variables. There are N construction sites. The coordinates of the centroid of construction site k are XS j and YS j, where j = 1, 2,, N. 2.1 Basic Region Constraints In order for the optimization model to explore all the regions that are available for locating the objects, a binary variable is created. Let, [1] λ i = 1 if region i is selected for locating construction object j 0, Otherwise, i = 1, 2,, M Then, the constraints associated with region i are given by [2] XO - b/2 λ i x1 i [3] XO + b/2 λ i x2 i + (1 λ i ) Q [4] YO - h/2 λ i y1 i [5] YO + h/2 λ i y2 i + (1 λ i ) Q where Q = large number greater than the larger dimension of the construction area and i = 1, 2,, M. For λ i = 1, Eqs. 2 and 3 ensure that the object boundaries along the y-axis lie within the region (i.e., XO - b/2 x1 i and XO + b/2 x2 i. Similarly, Eqs. 4 and 5 ensure that the object boundaries along the x-axis lie within the region. For For λ i = 0, Eqs. 2 and 4 will have no effect on the solution since they are true for any region. Eqs. 3 and 5 yield XO + b/2 Q and YO + h/2 Q which will not be binding. Since only one region should be selected, then CT-036-3

4 M [6] λ i = 1 i = Non-rectangular Objects The basic region constraints of Eqs. 2-5 assume that the object is rectangular with width b and height h. If the object is non-rectangular, it can always be converted to a rectangular object by enveloping the nonrectangular object with rectangular sides. For example, the trapezoidal object of Fig. 2b can be converted to a rectangular object by adding the dashed lines. This enveloping rectangular object, with width b and height h, is then used in the analysis. In the case of non-rectangular objects, the location of the centroid (which lies at distances b/2 and h/2 from the sides of the enveloping rectangle) would be approximate. 2.3 Object-Site Proximity Constraint The object may be required to be positioned within a minimum distance from construction site j. For example, if it is required to have a minimum distance between the object and site j, d j, then [7] d j = [(XO XS k ) 2 + (XO XS k ) 2 ] 0.5 [8] d j Dmin j where Dmin j = minimum distance allowed between the object and site j. 2.4 Object-Region Constraint It may be required that the object should not be located in certain regions. For example, if the object should not be located in region i, the following constraint will ensure this condition [9] λ i = 0, for all applicable regions i 2.5 Non-rectangular Regions In the previous formulation (Eqs. 2-5), a rectangular region has been assumed. If the region is not rectangular, the proposed approach requires that the region be convex, as shown in Fig. 3a and 3b. A convex region can simply be defined as a region in which any line connecting two points on the region boundaries will lie inside the region. Therefore, the region in Fig 3c is not convex because a line between two points on the right-side boundaries will lie outside the region. For the convex region of Fig. 4, the constraints of Eqs. 2, 4, and 5 remain the same, and the constraint of Eq. 3 becomes [10] XO + b/2 λ i [(x2 i + h/2 y1 i ) S i ] + (1 λ i ) Q where S i = slope of the right boundary of region i and b = x2 i x1 i (Fig. 4). Note that a non-convex region such as that in Fig. 5d can be converted into two convex regions and modeled as described above. (a) h (b) h b b Figure 2. Geometry of Construction Object: (a) Rectangular Object and (b) Non-rectangular Object CT-036-4

5 2.6 Objective Function The objective function for the site layout optimization problem minimizes the total weighted distance between the object and construction sites. That is, N [11] Minimize z = WSO j d j j = 1 where d j = distance between the object and site j and WSO j = unit resource requirements between site j and the object. The optimization model consists of Eqs. 1-11, and can be solved using any optimization software. LINGO version 8 was used for the application of this optimization model (Schrage 2003). LINGO is very useful optimization software that performs iterations to determine the best decision variables, subject to given constraints and objective function. It takes a few seconds to perform thousands of iterations and provide the optimum solution. (a) (b) (c) (d) Figure 3. Illustration of convex and non-convex regions: (a) convex region (rectangular), (b) convex region (non-rectangular), (c) non-convex region, (d) converting a non-convex region into two convex regions Y B Construction Boundary Site y2 i i S i y1 i 0 x1 i x2 i A X Figure 4. Geometry of Non-rectangular Region CT-036-5

6 3. Example Application It is required to determine the best location for a water fountain on the corridors of a manufacturing facility, so as to minimize the overall travel distance of employees between their offices and the water fountain (Zouein 1996). Fig. 5 shows the layout of the facility. Although the facility has 20 departments, only seven departments that house employees (shaded in Fig. 5) are considered in the optimization process. The number of employees in each department is shown in parentheses. The shaded area (corridor space) represents the area available for locating the water fountain. The construction site was divided into five regions, and the boundaries of the regions are shown in Table 1. The characteristics of the seven construction sites (departments) are shown in Table 2. In this example, the departments represent construction sites, while the water fountain represents the single construction object. The objective function minimizes the sum of the weighted distances. The weights are the number of employees in the associated department (Table 2). It is assumed that the employees are uniformly distributed in each department. Hence, distances are measured from the geometric centroid of each department. In addition, the dimensions of the fountain are ignored. These assumptions are identical to those used by Zouein (1996). The input data and model formulation for LINGO are given below. Note that the regions and sites are first defined as vectors and their numerical values are then specified. The data include region boundaries, site locations, weights, and other data. In other data, the variable KK = 1000 (an arbitrarily large number) represents the variable Q used in Eqs. 3 and 5. The binary variables λ i are represented by the vector OPEN which is defined as a vector of binary variables at the end of the formulation. Expansion Space Dept (25) PCB Manufacturing (34) Exp. Space 2 Dept (30) A Expansion Space 3 Test / QC Rest. Burn-in Rooms Dept (44) Dept (60) Finished Goods Storage Shipping Exp. 4 Offices (90) Small Parts Storage (12) PCB Manufacturing Lobby Cafeteria Figure 5. Layout of Construction Sites and Available Regions for Application Example CT-036-6

7 TITLE Example Application; SETS: REGIONS / R1, R2, R3, R4, R5/: X1, X2, Y1, Y2, OPEN; SITES / S1, S2, S3, S4, S5, S6, S7/ : XS, YS, WSO; ENDSETS DATA:! Boundaries of each region; X1 = 0, 84, 178, 84, 114; X2 = 280, 94, 188, 280, 134; Y1 = 84, 94, 94, 177, 28; Y2 = 94, 177, 177, 187, 84;! Centroids of each site; XS = 49.54, , 34.96, , , 62.00, ; YS = 43.68, 62.00, , , , , ;! Weights for site-object combinations; WSO = 90, 12, 34, 44, 60, 25, 30;! Other data; KK = 1000; ENDDATA! The objective; [TTL_WEIGHTED_DISTANCE] MIN SITES ( J): WSO( J)*((XO - XS( J))^2 + (YO - YS( J))^2)^0.5);! Region REGIONS( I): [LOWERX] XO - b/2 >= X1( I)*OPEN( I); [UPPERX] XO + b/2 <= X2( I)*OPEN( I) + (1 - OPEN( I))*KK; [LOWERY] YO - h/2 >= Y1( I)*OPEN( I); [UPPERY] YO + h/2 <= Y2( I)*OPEN( I) + (1 - OPEN( I))*KK);! Only one region to be REGIONS ( I): OPEN( I)) = 1;! Make OPEN REGIONS OPEN)); END CT-036-7

8 Table 1. Boundaries of the Regions of Available Areas for Example Application Region No x 1 (m) x 2 (m) y 1 (m) y 2 (m) Table 2. Characteristics of Construction Sites for Example Application (Zouein 1996) Site Number a, j Department Name Number of Employees Centroid Coordinates (m) XS j YS j 1 Offices Small part storage PCB Manufacturing Dept Dept Dept Dept a Note that departments that do not house employees are not considered in the optimization. Table 3. Results of Proposed and Existing Optimization Methods for Example Application Method Optimum Location of Object (m) XO YO Objective Value (m) Solution Time (sec) Proposed Zouein (1996) ,762 31, a Sadeghpour et al. (2006) , a Solution time was not provided. The optimum solution was found as XO = 94 m and YO = m (Point A in Fig. 5), with an objective function of 30,762. Table 3 shows a comparison between the solutions obtained by the proposed model and those obtained by Zouein (1996) and Sadeghpour et al. (2006). Note that the optimum location of the CT-036-8

9 object for the proposed method and Sadeghpour et al. s method is based on the Euclidean coordinate system. However, Zouein s optimal solution is based on rectilinear measurements (x + y), not Euclidean. To allow comparison among the three methods, the objective function for the Zouein s optimal solution was calculated based on Euclidean distances. As noted, the proposed optimization model is slightly better (smaller objective value) than existing methods since it considers continuous decision variables. It is also noted that the proposed model is much faster since it is based on algorithmic rather than exhaustive search-based technique. To examine the sensitivity of the computation time to the number of regions modeled, the available area was divided into ten regions instead of the five regions used previously. The region boundaries were as follows: X1 = 0, 84, 114, 178, 84, 178, 84, 178, 114, 114; X2 = 84, 114, 178, 280, 94, 188, 178, 280, 134, 134; Y1 = 84, 84, 84, 84, 94, 94, 177, 177, 28, 50; Y2 = 94, 94, 94, 94, 177, 177, 187, 187, 50, 84; The model was used to solve the same example but with ten regions, and the computation time was 7 seconds, which is slightly twice the time needed for five regions. Clearly, as the number of regions increases, the computation time will increase at a greater rate. 4. Conclusions This paper has presented a new approach for the optimization of construction site layout. Unlike existing methods of construction site layout, which are mostly grid-based, the proposed approach is based on nonlinear integer optimization that considers a continuous x-y space in locating the object. Such a representation not only provides more precise results, but is also more efficient. The preliminary results showed that the optimization model is almost ten times more efficient than the graphical-based method, and is expected to be more efficient than other grid-based methods. The objective function of the model minimizes the total weighted distance between the object and the construction sites. However, other objectives can be easily accommodated. The proposed approach has many capabilities, including accommodating non-convex regions, non-rectangular objects, and objectregion constraints. The regions of the construction site can be defined in any arbitrary manner. However, the user should make a special effort to use as fewer regions as possible. As for future extensions, the proposed approach offers flexibility to accommodate other features of construction site layout, such as multiple objects, object-object relationships, line of sight analysis, and multiple-criteria objective function. In addition, the proposed approach can be integrated with a graphical site layout model that provides flexible setup of the project (Sadeghpour et al. 2006), and the achieved optimum layout can be represented visually. The proposed optimization approach opens the way for additional innovative features that can be addressed in construction site layout, and as such should be of interest to construction engineers and practitioners. 5. References Easa, S Resource Levelling in Construction by Optimization. J. Cons. Engrg. and Mgt., 115(2): Easa, S Optimum Cash-flow Scheduling of Construction Projects. J. Civ. Engrg. Sys., 9: Hamiani, A CONSITE: A Knowledge-Based Expert System Framework for Construction Site Layout. Ph.D. Dissertation, University of Texas, Austin, TX, USA. Hamiani, A. and Popescu, C Consite: A Knowledge-Based Expert System for Site Layout. Proc., 5 th Conference on Comp. in Civil Engrg., ASCE, Alexandria, Virginia, USA, CT-036-9

10 Hegazy, T.M. and Elbeltagi, E EvoSite: Evolution-Based Model for Site Layout Planning. J. Comp. in Civ. Engrg., 13(3): Li, H. and Love, P.E.D Site-level Facilities Layout Using Genetic Algorithms. J. Comp. in Civ. Engrg., 12(4): Moore, J Computer Methods in Facility Layout. Industrial Engineering, 19: Sadeghpour, F., Moselhi, O., and Alkass, S Computer-Aided Site Layout Planning. J. Cons. Engrg. and Mgt., 132(2): Schrage, L Optimization Modeling with LINGO. LINDO Systems, Palo Alto, California, USA. Sirinaovakul, B. and Thajchayapong, P An Analysis of Computer-Aided Facility Layout Techniques. J. Computer-Aided Manufacturing, 9(4): Tam, C.M. and Tong, T.K.L GA-ANN Model for Optimizing the Locations of Tower Crane and Supply Points for High-rise Public Housing Construction. Construction Management and Economics, Spon Press, 21(3): Tommelein, I.D Construction Site Layouts Using Blackboard Reasoning with Layered Knowledge. Expert Systems for Civil Engineers: Knowledge Representation, ASCE, Tommelein, I.D., Levitt, R.E., and Hayes-Roth, B How can Artificial Intelligence Help? J. Cons. Engrg. and Mgt., 118(3): Zouein, P.P MoveSchedule: A Planning Tool for Scheduling Space Use on Construction Sites. Ph.D. Dissertation, Univ. of Michigan, Anna Arbor, MI, USA. CT