A cutting plane algorithm for the site layout planning problem with travel barriers

Computers and Operations Research 82 (2017) 36–51

Contents lists available at ScienceDirect

Computers and Operations Research journal homepage: www.elsevier.com/locate/cor

A cutting plane algorithm for the site layout planning problem with travel barriers Ahmed W.A. Hammad∗, David Rey, Ali Akbarnezhad School of Civil and Environmental Engineering, University of New South Wales, Australia

a r t i c l e

i n f o

Article history: Received 23 January 2016 Revised 29 September 2016 Accepted 9 January 2017 Available online 16 January 2017 Keywords: Site layout planning Mixed integer programming Cutting plane algorithms Decomposition methods Global optimisation d-visibility

a b s t r a c t Site layout planning is an imperative procedure that may significantly impact the productivity and the efficiency of logistical operations undertaken on a construction site. This paper considers the site layout planning problem (SLPP) which entails the allocation of temporary facilities on a construction site in the presence of travel barriers such that the total transportation cost between facilities is minimised. In order to account for travel barriers, the SLPP is typically solved under the assumption that the available region for facility layout can be discretised. In this paper, we propose a general Mixed Integer Programming (MIP) model to represent the SLPP, accounting for the presence of barriers, and we show how space-discretised formulations can be derived from this model. In particular, we propose a novel MIP model, which permits facilities to cover multiple locations. This is then benchmarked against a commonly adopted MIP model in the literature. We also highlight a systematic procedure to convert the continuous feasible space in SLPP to a set of discretised locations based on the concept of d-visibility, enabling us to approximate the barrier distance function embedded in the objective function. In particular, we focus on presenting a simple space discretisation approach for converting the continuous SLP into a discrete problem for which the discrete SLP models would be applicable. Space-discretised MIP formulations are highly combinatorial and we introduce a cutting plane algorithm to improve their tractability. Specifically, we propose a novel exact location-decomposition algorithm which works from a relaxed MIP formulation and iteratively generates feasibility cuts to converge to an optimal solution. Both space-discretised MIP models and the decomposition algorithm are tested on a large group of instances to analyse their effectiveness in solving the SLPP. Computational results indicate that the proposed location-decomposition algorithm improves on the pure MIP approach and provides a competitive framework to solve realistic SLPP instances. © 2017 Elsevier Ltd. All rights reserved.

1. Introduction In engineering construction and management, the site layout planning problem (SLPP) is a well-studied layout problem which requires finding an appropriate physical arrangement of temporary facilities operating on construction sites [15,21,26,47,51]. The physical layout of the facilities should respect design constraints, i.e. site boundaries and non-overlapping restrictions, and a layout is said to be feasible if all design constraints are satisfied. The total transportation cost, measured in terms of distance-weighted travel frequencies among facilities, is commonly used as a criterion for assessing the suitability of the construction site layout [50]. This is because the construction process of various types of projects, including infrastructure, buildings and heavy civil works, is one ∗

Corresponding author. E-mail addresses: [email protected] (A.W.A. Hammad), d.rey@unsw. edu.au (D. Rey), [email protected] (A. Akbarnezhad). http://dx.doi.org/10.1016/j.cor.2017.01.005 0305-0548/© 2017 Elsevier Ltd. All rights reserved.

that entails many logistical and building activities, requiring the envelopment of different types of facilities throughout the construction period. These activities often involve the transportation of construction material from one facility to another. Hence the layout of the construction site may have a considerable impact on the efficiency of such integral operations. The SLPP can be then stated as follows: given a construction site and a set of permanent and temporary facilities with a priori known dimensions, find a feasible arrangement of the temporary facilities on the available space of the construction site such that the total transport cost among all facilities is minimised. The decision process for locating facilities in the SLPP problem is commonly formalised using the Euclidean plane, adopting either a continuous or a discrete space approach [27]. The solution to the SLPP is a block layout showing the dimensions along with the relative positions of the facilities within a given area [7]. In relation to similar problems presented in the operational research field, the SLPP falls in the general category of layout-location problems,

A.W.A. Hammad et al. / Computers and Operations Research 82 (2017) 36–51

having a close association with the facility layout problem where the aim is to allocate departments, workshops, facilities etc. to locations [39]. Adopting an efficient on-site layout of temporary facilities serves the purpose of improving the overall work organization aspect on a construction site in several ways. In construction, especially for expansive construction sites where large equipment is utilised for transporting the material between facilities, it is expected that a site layout configuration which reduces frequencyweighted travel distances would have a significant impact on the operating costs, measured in terms of material movement between facilities [16,34]. Productivity is improved when an adequate construction site plan is put into use [36], as it is expected that a more coordinated material flow will result between the temporary facilities, leading to a reduction in the overall costs expended on material handling equipment, and a better use of such systems. In the area of manufacturing, facility layout and its effect on the total material costs inflicted by its underlying structure has been assessed to reach levels of up to 50% of the total operating costs [44]. Improvement to the layout of facilities can also lead to reductions of 10% - 30% in the total material handling costs expended [37]. In terms of safety, it has been acknowledged that a well planned construction site helps in reducing the number of accidents and injuries occurring to workers [12,32]. Other aspects that are also affected include the safety of people in the vicinity of the construction site, along with the environmental and social influences caused by the adopted site layout [2]. Consensus on the hardness of layout problems is that they are NP- hard [10,23,51]; this explains the great number of studies in the literature in which non-exact solution techniques have been proposed. Many of the studies reviewed on facility layout problems therefore tend to propose heuristic or meta-heuristic approaches for solving the problem [19, 29, 28, 40]. In the SLPP, considerable attention has been directed at the applications of various solution techniques. This includes the deployment of: 1) exact methods such as Linear Programming [52], Mixed Integer Programming (MIP) [11,46] and Mixed Integer Non-Linear Programming (MINLP) [16]; 2) Heuristics, mostly developed originally for facility layout planning and which have been suggested as appropriate for the SLPP, such as entire layout algorithms [17], solution improvement algorithms [31], partial improvement algorithms [41] and priority facility selection [43]; 3). Meta-heuristics such as genetic algorithms [24], ant colony optimisation [33], particle swarm optimisation [49] and bee colony algorithm [48]. In this paper, we focus on the application of exact approaches for the SLPP. Currently, exact methods adopted in the SLPP literature are able to solve for cases involving only 4 facilities [11]. The maximum number of facilities for which the SLPP has been solved is 16 facilities, though this is achieved through use of metaheuristics [1]. To the best of our knowledge, no exact approach capable of solving large instances of the SLPP has yet been proposed. A distinctive element of construction sites that plays an important role in the determination of the distance measure adopted between facilities is the presence of work area zones, known as building footprints, where most construction activities take place. These regions are barriers that impose an obstacle to travel, as mobility across these regions by on-land material handling equipment is normally prohibited. Studies on the SLPP tend to neglect the impacts that the barriers can have on the distance measure adopted in the formulations. The construction site layout planning problem considered in this paper therefore concerns the arrangement of temporary facilities around such barriers, labelled as permanent facilities, such that the distance metric adopted between the temporary facilities accounts for the presence of the barriers. In this paper, we propose a general MIP formulation for the SLPP accounting for travel barriers, and we show how integer-

37

linear space-discretised formulations can be derived from this model. In particular, two space-discretised SLP models are benchmarked. The first is commonly adopted in the literature and requires facilities to cover at most a single discretised location. In this model, the underlying space discretisation structure implicitly prevents overlap with barriers. The second is a novel model which permits facilities to cover multiple discretised locations and which explicitly accounts for the presence of barriers through non-overlap constraints. We then introduce an exact cutting plane algorithm for the space-discretised SLPP. The proposed solution method works from a relaxed MIP formulation and iteratively adds cutting planes to achieve feasibility. We show that this decomposition algorithm improves on a pure MIP approach and is able to solve the problem on realistic sized instances in reasonable time. Further, we quantify the trade-off between computational performance and model accuracy induced by the space-discretisation procedure. The importance of our work is thus in the formation of a method to optimally solve reasonable-sized site layout planning problems within a realistic time frame, all the while incorporating the additional restrictions imposed by the presence of travel barriers. The contributions of this paper can be summarised as follows: 1) we propose a general MIP model to represent the SLPP while accounting for travel barriers in the design and in the objective function; 2) we introduce a systematic space-discretisation approach for the SLPP and we propose a method to approximate travel distance in the presence of travel barriers using the concept of d-visibility; 3) we present a novel multi-cover discretised SLP model which is benchmarked against the common MIP formulation used in the literature to solve the SLPP; 4) we improve on the existing exact approaches in the literature by presenting an exact cutting plane algorithm which is able to solve large scale instances of the SLPP to optimality, hence enhancing tractability; and 5) we conduct numerical experiments to quantify the impact of the proposed space discretisation scheme on the solution quality of the space-discretised MIPs. To describe the contributions of this work, the structure of this paper is organised as follows: In Section 2, we formally define the SLPP and introduce a general MIP model to represent it. In Section 3, we present a space-discretisation approach, based on the concept of d-visibility, which provides a network representation of the SLPP. In Section 4, two MIP models for solving the discrete SLPP are presented. In Section 5, we introduce a cutting plane algorithm to solve the space-discretised SLPP. In Section 6, we perform numerical experiments to contrast the proposed SLP models and quantify the impacts of the space discretisation scheme on the solution quality. Section 7 summarises the findings of the paper and discusses possible extensions of this work. 2. Problem definition and general formulation for slp In this section we provide a description of the feasible region available for locating temporary facilities within a construction site in the presence of barriers, along with a formal definition of the SLPP and a general MIP representation. First the notation is given as follows: let F denote the set of temporary facilities to be positioned on the construction site. We do not consider positioning permanent facilities in the SLPP since they are presumed to have a priori defined locations. We also assume that all facilities can be represented as rectangles. Other shapes, possibly non-convex, can be handled by fitting a minimum bounding rectangle. For each temporary facility i ∈ F, let Wi and Hi denote the input width and length of the facility in the SLP model, respectively. Adopting rectangular shapes for facilities enables the definition of constraints to control requirements such as non-overlap between facilities and facility positioning within

38

A.W.A. Hammad et al. / Computers and Operations Research 82 (2017) 36–51

site boundaries, based on the centroid coordinates of the facilities and their dimensions. Let WS and HS represent the width and the length, respectively, of the area of a rectangular construction site S ⊂ R2 , whose origin is assumed to be positioned at (0, 0) in the Euclidean plane R2 . Let B be the set of rectangular barriers. Each barrier b ∈ B is assumed to have a pre-defined position on the plane and is assumed to be disjoint from other barriers in B. Barriers represent areas through which travel is forbidden and in which facilities cannot be located. For instance, on a typical construction site, these are regions occupied by the permanent facilities, such as the building footprint. We assume rectangular-shaped barriers since it is common on a construction site for a rectangular shape to represent the building footprint. This also comprises the excavation area at the onset of a project’s construction stage, which we model as an obstacle region in our formulations. Throughout this paper the terms obstacle, barrier, building and permanent facility are used interchangeably. Let Tij be the frequency of travel between two facilities i and j, where i, j ∈ F: i = j. The feasible space of a given construction site area is defined as the space within which temporary facilities can be positioned without overlapping with the present barriers. The position of temporary facilities on the plane is denoted by the continuous decision variables (xi , yi ) ∈ R2 ∀i ∈ F, where xi represents the Cartesian x coordinate of the centroid of facility i and yi is the Cartesian y coordinate of the centroid of the facility. We assume that 90° rotations of the rectangular facilities are permitted. Hence, we define the binary rotation variable ri , which equals 1 if facility i is rotated 90°, and zero otherwise, and the continuous variables, W i and H i to represent the final resulting dimensions of the facilities, once the binary rotation variables are decided in the model. As a result, the dimensions of facilities are incorporated as decision variables in the model [5]. To specify the non-overlap condition among temporary facilities we define the y following binary variables: let μxi j (resp. μi j ) equal to 1 if facilities i and j are separated in the x (resp. y) direction, and 0 otherwise. To ensure that the facilities do not overlap with the present x barriers, we define the following binary variables: let β ib and y

β xib (resp. β ib and β yib ) equal 1 if facilities are positioned outside

the upper and lower limits, respectively delineating the boundary of the barrier in the x (resp. y) direction, and 0 otherwise. In order to determine the travel distance, dij , between two temporary facilities i and j, we need to account for the presence of travel obstacles b ∈ B. Due to the restriction imposed by travel barriers, and since movement is only possible around the boundaries of such barriers, the presence of an obstruction in the direct line of travel between two points on the planar region causes an increase in the overall distance to be covered [3]. Let fd (xi , xj , yi , yj , B) be a function, which, depending on the underlying distance metric, d, returns the shortest travel distance between facilities i and j, accounting for the presence of barriers. The function, referred to as the barrier distance function, can be thought of as an oracle, defined relative to the centroids of the facilities and the locations of the barriers, which returns a distance measure in nonconvex space. The MIP model defined by Eqs. (1)–(20), named G-SLP, can be used to represent the SLPP in the presence of travel barriers.

minimise



Ti j di j

(1)

i, j∈F i= j

subject to

di j = f d ( xi , x j , yi , y j , B )

∀i, j ∈ F : i < j

(2)

W i = ri Hi + (1 − ri )Wi

∀i ∈ F

(3)

H i = (1 − ri )Hi + riWi

∀i ∈ F

(4)

  xi − x j  ≥ 0.5(W i + W j )μxi j ∀i, j ∈ F : i < j

(5)

  yi − y j  ≥ 0.5(H i + H j )μy ∀i, j ∈ F : i < j ij

(6)

μxi j + μyi j ≥ 1 ∀i, j ∈ F : i < j

(7)

















yi ≥ yb + 0.5H i yi ≤ yb − 0.5H i

xi ≥ xb + 0.5W i xi ≤ xb − 0.5W i x

y

β ib ∀i ∈ F , ∀b ∈ B

(8)

  β yib + 1 − β yib HS ∀i ∈ F , ∀b ∈ B

(9)

x

β ib ∀i ∈ F , ∀b ∈ B

(10)

  β xib + 1 − β xib WS ∀i ∈ F , ∀b ∈ B

(11)

y

β ib + β xib + β ib + β yib ≥ 1 ∀i ∈ F , ∀b ∈ B

(12)

∀i ∈ F

(13)

xi − 0.5W i ≥ 0 xi + 0.5W i ≤ WS

∀i ∈ F

(14)

yi + 0.5W i ≤ HS

∀i ∈ F

(15)

yi − 0.5H i ≥ 0 ri ∈ {0, 1}

∀i ∈ F

(16)

∀i ∈ F

(17)

μxi j , μyi j ∈ {0, 1} ∀i, j ∈ F : i < j y

x

β ib , β ib , β xib , β yib ∈ {0, 1} W i , H i , xi , yi ≥ 0

∀i ∈ F

∀i ∈ F , ∀b ∈ B

(18) (19) (20)

The objective function, Eq. (1), minimises the total pairwise frequency-weighted travel distance among facilities. Eq. (2) defines the barrier distance function used to weigh the contribution of each pair of facilities in the objective function, which is dependent on the centroids of the temporary facilities, along with the positions of the barriers. Eqs. (3) and (4) specify the resulting facility dimensions, considering any associated rotations implemented. Eqs. (5)–(7) impose that every pair of facilities must be separated in space in at least one dimension. To ensure that temporary facilities do not overlap with the barriers, Eqs. (8)–(12) are introduced. Eqs. (13)–(16) ensure that the facilities are contained entirely within the boundaries of the construction site. Eqs. (17)–(20) define the domain of the variables employed in the model. Model G-SLP provides a general formulation to represent the SLPP. This model requires a representation of the barrier distance function, fd (xi , xj , yi , yj , B). Since the continuous available space for locating facilities on construction sites is non-convex, establishing a general distance function may be difficult. To tackle this issue, a discretisation scheme needs to be adopted to partition the space into convex regions connected through a transport network. In the next section, we present a systematic approach for discretising space and approximating travel distance among facilities.

A.W.A. Hammad et al. / Computers and Operations Research 82 (2017) 36–51

39

Fig. 1. Space discretisation procedure.

3. Space discretisation approach Solving Model G-SLP presents many challenges, since it is complex to model the distance oracle relative to facilities that are yet to be positioned. Most studies in the literature adopt a spacediscretised formulation where the distance oracle is approximated without accounting for travel barriers. We note that the space discretisation approaches presented in the literature are arbitrary in nature [11,46]; a clear set of rules defining a systematic space discretisation process is currently lacking and it is our intention to highlight one approach in this section. We assume that the input design of any SLPP follows known practices in the construction industry, in terms of keeping a safe distance between the building working zones, considered as the barriers in this problem. As a result, a design which brings barriers tightly close to one another is prohibited as an input. The space discretisation approach relies on superimposing a grid consisting of horizontal and vertical lines onto the region, such that the continuous available space in G-SLP is converted into discrete regions, henceforth referred to as cells, which form the set of discrete locations L. An illustration depicting the process is shown in Fig. 1. As can be noticed from Fig. 1, some cells overlap with the barriers. Hence, the barriers are extended to accommodate those locations whose interior is partly overlaid by the barrier. Once a cell is merged with a barrier, the extreme points of the barriers in the discrete models are redefined. This merging process is replicated in reality on a construction site in cases where the region surrounding the building footprint i.e. barrier, is barricaded to allow for the safe operations of material lifting operations [18]. The underlying principle of the discretisation scheme is that all resulting discrete locations are convex, with no intrusion of any of the barriers into the space occupied by the discrete locations. Distance among the generated discrete locations can be determined using a d-visibility network [22] and finding shortest paths between locations’ centroids. Let d be a metric in the Euclidean space. A point is said to be d-visible from another if the length of the shortest path, accounting for travel barriers, joining these two points, fd , is equal to their distance as measured by the adopted distance metricd. Formally, if we let A be the feasible region available for positioning the temporary facilities, the set of points p ∈ A which are d-visible from q ∈ A, such that p = q, is defined as Vd (q): ={p ∈ A: fD (p, q, B) = d(p, q)}. The distance metric that we adopt is the Manhattan (L1) metric, as it was shown in various works in the literature to be a representative measure that accounts for hindered travel path [6]. The nodes of the visibility network are the available grid centroids for assigning facilities to, and the extreme barrier points. The edges of the L1-visibility network represent the unobstructed L1 distance connecting the grid centroids to one another and to

extreme points of the barriers [4]. This is shown in Fig. 2. Once the transportation network is constructed, an all-to-all shortest path algorithm such as Floyd-Warshall’s algorithm [13,45] is used to get the shortest path distance matrix [Dmn ]m, n ∈ L for all pairs of cells’ centroids. The distance matrix is then adopted as a proxy for the distance between the temporary facilities, depending on the respective assignment of facilities to locations by the optimisation models. Even though this may be an approximation, the difference in distance calculations is expected to decrease as space discretisation becomes finer. We use the grids’ centroids to approximate the pairwise distance function, fd , between any two facilities i, j ∈ F which are assigned to discrete locations. The need to approximate the distance measure between facilities in the SLPP arises due to the difficulty in assuming a particular point of reference, for determining pairwise barrier distances to embed in the objective function. Unlike the Weber problem, where existing facilities are known to be present and the aim is to locate a single new facility to minimise the actual weighted distance between new and existing facilities [9], the SLPP carries the proposition that all the temporary facilities are new facilities, with no presence of existing temporary facilities at the onset of the construction project. As a result, not all demand points are well known from the start of the SLPP, to be used to construct the convex hull of existing facilities, and from which the optimum location of new facilities can be analytically derived [3]. The space-discretised layout together with the shortest path distance matrix [Dmn ]m, n ∈ L for all pairs of cells’ centroids are provided as input to the space-discretised SLP models presented in Section 4. In the next section, we introduce the discrete SLPP models for which the input provided by the pre-processing step is supplied. 4. Space-discretised SLPP The non-convex continuous space defined in G-SLP means that the tracking of a reasonable estimate of the barrier distance function, between the temporary facilities and between temporary and permanent facilities, is difficult to achieve. We note that the space-discretised version of the SLPP has been frequently adopted in the literature [8,25,26,30,38], though exact techniques adopted were limited to solving 4 facilities [11] and meta-heuristics were reported to solve instances of up to 17 facilities [20]. We also note that we are not aware of any attempt in the literature to incorporate the barriers within the objective function of the discrete SLP models. In this section, we present two space-discretised models of the SLPP, namely a single-cover discrete SLP model, SD-SLP, and a multi-cover discrete SLP model, MD-SLP. The difference between these two models is mainly in terms of the allowable coverage

40

A.W.A. Hammad et al. / Computers and Operations Research 82 (2017) 36–51

Fig. 2. Transport network represented as L1-visibility graph.

of locations by the temporary facilities. In SD-SLP only a single location is assigned to each facility, and the facility has to be positioned within the interior of the location assigned. In contrast, Model MD-SLP allows facilities to cover more than a single location, hence their positions are not constrained to be within the boundaries of the assigned location. Due to the approximation of the distance function, the space-discretised MIP models presented in this section can be represented using mixed integer linear programming (MILP). The definition of Model SD-SLP is presented next.

x xi − 0.5W i ≥ (lm − 0.5Wm )zim

∀i ∈ F , ∀m ∈ L

(23)

∀i ∈ F , ∀m ∈ L

x xi + 0.5W i ≤ (lm + 0.5Wm )zim + WS (1 − zim )

(24) y yi − 0.5H i ≥ ( lm − 0.5Hm )zim

∀i ∈ F , ∀m ∈ L

y yi + 0.5H i ≤ ( lm + 0.5Hm )zim + HS (1 − zim )

(25)

∀i ∈ F , ∀m ∈ L (26)

4.1. Single-cover model A similar version to Model SD-SLP has been adopted previously in the literature – see for instance [11,16] – though effectively modelling the impact of barriers on the inter-facility distance estimate has generally been ignored. We introduce assignment variables to match temporary facilities with a set of discrete locations, L, obtained after applying a specific space-discretisation approach. Each discretised location on the construction site can accommodate a certain number of facilities, depending on the area of the location. Fitting a facility in any location entails satisfying the problem’s feasibility requirements. In line with current work in the literature, we assume that in the SD-SLP, each resulting discrete location can accommodate at least the largest of all temporary facilities. The pre-computed distance between the discrete locations, accounting for the presence of the barriers, is given by Dmn . In order to filter out facilities that cannot fit within the boundaries of the assigned location, a check is carried out to identify locations where the dimensions of a chosen facility are within the limits imposed by the boundaries of that respective location. Specifically, for each i ∈ F, we define: Feasi ≡ {m ∈ L: Wi ≤ Wm ∧ Hi ≤ Hm ∧ Hi ≤ Wm ∧ Wi ≤ Hm }. Let the integer variable zim equal 1 if facility i is assigned to location m and 0 otherwise. Model SD-SLP, defined by Eq. (21), Eqs. (3)–(7), Eqs. (17)–(20), and Eqs. (22)–(27), is a space-discretised representation of the SLPP.

minimise

 

Ti j Dmn zim z jn

(21)

i, j∈F m,n∈L i= j

subject to Eqs. (3)–(7), Eqs. (17)–(20),

 m∈F easi

zim = 1

∀i ∈ F

(22)

zim ∈ {0, 1}

∀i ∈ F , ∀m ∈ L

(27)

In particular, the objective function, Eq. (21) is still a measure of inter-facility interaction, as presented by the objective function of G-SLP, though this time we incorporate the distance function as an L1-distance metric between locations [38]. Hence, Eq. (21) is designed to minimise the total transport cost between the locations, where it is assumed that the distance between facilities is approximated by the distance between the discretised locations, to which the facilities are assigned, on the planar region of the construction site, accounting for the present barriers. Eq. (22) specifies that each temporary facility is to be assigned to a location from the set Feasi .The facility rotation constraints and the facility overlap constraints, as defined earlier in Eqs. (3)–(7), are included in SD-SLP. Eqs. (23)–(26) define the boundary constraints, specifying that facilities must be fully positioned within the location to which they are assigned. The domains of the variables employed in SD-SLP are specified in Eqs. (17)–(20) and (27). Barrier constraints, Eqs. (8)–(12), are not incorporated in SD-SLP since only single coverage of the convex locations generated through the discretisation scheme presented earlier, by the facilities is permitted. This means that the barrier constraints are implicitly enforced due to having the facilities enclosed within the boundaries of the locations. We note that SD-SLP in its current format is non-linear, and this is due to the following: 1) the quadratic term, zim zjn , in the objective function, Eq. (21); 2) the bi-linear terms in Eqs. (5)–(6); and 3) the piecewise linear absolute value function, present in Eqs. (5)–(6). For demonstration purposes and to maintain the brevity of the discussion, we illustrate next how the quadratic term in the objective function and the bilinear term in Eq. (5) can be linearised; the same approach is adopted for Eq. (6). To linearise the quadratic term in Eq. (21), we introduce the auxiliary non-negative continuous decision zim jn variable, and

A.W.A. Hammad et al. / Computers and Operations Research 82 (2017) 36–51

the following set of constraints, Eqs. (28)–(30) – for a detailed description of these reformulations see [16].

zim jn ≤ zim

∀i, j ∈ F , ∀m, n ∈ L : i < j

(28)

zim jn ≤ z jn

∀i, j ∈ F , ∀m, n ∈ L : i < j

(29)

4.2. Multi-cover model In this section, a novel SLP model, named MD-SLP, is proposed where multiple coverage of discretised locations by facilities is permitted. Parameters and variables included in MD-SLP have been introduced previously, either in G-SLP or in SD-SLP. Model MD-SLP is defined as follows:

minimise zim jn ≥ zim + zim − 1

∀i, j ∈ F , ∀m, n ∈ L : i < j

(30)

The term zim zjn is thus replaced with zim jn in Eq. (21). To linearise the bilinear term in Eq. (5), we introduce the non-negative continuous auxiliary variable γixj , along with the following constraints, Eqs. (31)–(33):

γixj ≤ WS μxi j ∀i, j ∈ F : i < j

(31)

γixj ≤ W i + W j ∀i, j ∈ F : i < j

(32)







W i + W j − γixj ≤ WS 1 − μxi j



∀i, j ∈ F : i < j

(33)

where γixj = μxi j (W i + W j ) In order to linearise the absolute value function, the binary y auxiliary variables τixj and τi j , and the non-negative continuous auxiliary variables xi j andyi j are introduced, along with the following equations, Eqs. (34)–(43):

xi j ≥ x i − x j

∀i, j ∈ F : i < j

xi j ≤ xi − x j + 2Ws τixj

xi j ≥ x j − x i

∀i, j ∈ F : i < j

∀i, j ∈ F : i < j 

xi j ≤ x j − xi + 2Ws 1 − τixj

yi j ≥ y i − y j



∀i, j ∈ F : i < j

∀i, j ∈ F : i < j 

xi j ≥ 0.5(W i + W j )μxi j yi j ≥ 0.5(H i + H j )μyi j



(37)

(38)

∀i, j ∈ F : i < j

yi j ≤ y j − yi + 2Hs 1 − τiyj

(35)

(36)

∀i, j ∈ F : i < j

yi j ≤ yi − y j + 2Hs τiyj

yi j ≥ y j − y i

(34)

(39)

(40)

∀i, j ∈ F : i < j

∀i, j ∈ F : i < j ∀i, j ∈ F : i < j

(41)

(42)

(43)

where xi j = |xi − x j | and yi j = |yi − y j |. For the remainder of this paper, we will be presenting the non-linearised formulations of the space-discretised MIP models to ensure clarity. Linearisation techniques similar to the ones discussed above are used to produce the final MILP models.

41

 

Ti j Dmn zim z jn

(44)

i, j∈F m,n∈L i= j

subject to Eqs. (3)–(20), Eq. (22), Eq. (27)

  |xi − lmx | + yi − lmy  ≤ max {Wm , Hm } + (Ws + Hs )(1 − zim ) ∀i ∈ F , ∀m ∈ L (45) The objective function, Eq. (44), in MD-SLP, has the same interpretation as the objective function of SD-SLP. Eq. (45) requires that the distance between the centroid of a facility and the location to which it is assigned through Eq. (22), not exceed the maximum dimension of the location. Even though Eq. (22) requires the facilities be assigned to only a single location, the definition of Eq. (45) in Model MD-SLP permits the facilities to cover multiple locations. Multiple location coverage is achieved by relaxing the restriction that facilities are required to be fully enclosed within the assigned location. In Model MD-SLP, the design constraints are borrowed from Model G-SLP, in which facilities are required to be non-overlapping and to avoid travel barriers. This is different to the boundary constraints, Eqs. (23)–(26) in SD-SLP, which impose that facilities must be fully enclosed within their assigned location. To linearise Model MD-SLP, the bilinear terms and absolute value functions in the objective function and/or the constraints are treated using the linearisation approach presented in Section 4.1. In the next section we present an analysis of the properties of Models SD-SLP and MD-SLP. 4.3. Analysis of space discretised models Both Models SD-SLP and MD-SLP solve a discrete version of the continuous SLPP. In this section we analyse the theoretical properties of both models. As mentioned previously, the main difference between Models SD-SLP and MD-SLP is in the requirement of having the facilities entirely positioned within the space delineated by the boundaries of the locations in SD-SLP, which is not the case in MD-SLP. In Model SD-SLP the discrete locations implicitly account for barrier and site boundary constraints. This is a consequence of the space discretisation approach employed to construct the discrete locations (See Section 3). On the other hand, Model MD-SLP explicitly accounts for the barriers and site boundary constraints but offers more layout flexibility by allowing facilities to cover more than a single discrete location. Hence, we observe that any facility arrangement which is feasible in SD-SLP is also feasible in MD-SLP, but not the converse. We display this though an illustrative figure, Fig. 3. Given three temporary facilities, F1, F2 and F3, that interact with one another and with building B1, it is required to place them all as close as possible to one another. The figure clearly shows that this arrangement, which is feasible in Model MD-SLP, is infeasible in Model SD-SLP, due to violation of the location boundary constraints (Eqs. (23)–(26)). Based on the above argument, we state the following proposition: Proposition 1. For a given space discretisation, Model MD-SLP is a relaxation of Model SD-SLP.

42

A.W.A. Hammad et al. / Computers and Operations Research 82 (2017) 36–51

of Model SD-SLP we have: y x zim = 1 ⇒ xi + 0.5W i ≤ lm + 0.5ε and yi + 0.5H i ≤ lm + 0.5ε y x ⇒ xi + 0.5W i + yi + 0.5H i ≤ lm + lm +ε y x = xi + yi ≤ lm + lm + ε − 0.5W i − 0.5H i

y

x +y −l ≤ From Eq. (45) imposed in MD-SLP, we have:xi − lm i m

ε ⇔ xi + yi ≤ lmx + lmy + ε.

x + l + ε > l x + l + ε − 0.5W − 0.5H . Thus, It is clear that lm i i m m m Eq. (45) defines a broader region, in comparison to the boundary constraints defined in Model SD-SLP for the same cell edge size ε. This shows that PSD−SLP ⊆ PMD−SLP . Since both Models SD-SLP and MD-SLP have the same objective function, ζ MD − SLP ≤ ζ SD − SLP always holds and, for a given space discretisation, MD-SLP is a relaxation of SD-SLP.  y

y

5. Cutting plane algorithm Fig. 3. Infeasible facility arrangement for Model SD-SLP but not for Model MD-SLP.

Proof: Without any loss of generality, consider a spacediscretised SLPP where space is discretised using a uniform square grid of cell edge size ε , as specified in the approach presented in Section 3 – the same result holds for any space-discretisation pattern. Let ζ SD − SLP and ζ MD − SLP be the optimal values of the objectives of Models SD-SLP and MD-SLP, respectively. Also, let PSD−SLP and PMD−SLP define the feasible regions of SD-SLP and MD-SLP, respectively. Let x be a multi-dimensional vector of appropriate dimension, we have: PSD−SLP ≡ {x such that Eqs. (3)–(7), Eqs. (17)–(20), Eqs. (22)– (27) are satisfied} PMD−SLP ≡ {x such that Eqs. (3)–(20), Eq. (22), Eq. (27), Eq. (45) are satisfied} Since both models are solved for the same number of locations, the decision variables in both models are the same. The only difference in the feasible regions of PMD−SLP and PSD−SLP is that Model SD-SLP requires location boundary constraints (Eqs. (23)–(26)), while Model MD-SLP requires barrier and site boundary constraints (Eqs. (8)–(16)) as well as linking constraints (Eq. (45)), which link facilities to the discretisation grid to approximate pairwise facility distances. We need to show that PSD−SLP ⊆ PMD−SLP . Since, by construction, the discrete locations resulting from the space-discretisation procedure satisfy the site boundary and barrier constraints, Eqs. (8)–(16), any facility fully enclosed within a discrete location, as imposed by Eqs. (23)–(26), also verifies these constraints. This shows that the solution space delimited by constraints Eqs. (23)–(26) is contained within that delineated by constraints Eqs. (8)–(16). We display this through the following. Given we assume our barriers are rectangular in shape, cells are arranged around the four sides of the barrier boundaries. We analyse the cells within the region of the construction site, R, defined by HS ≤ R ≤ yb for a given barrier, b ∈ B; a similar analysis for cells in regions on the remaining sides of all other barriers follows. For a given facility i ∈ F, Model MD-SLP requires that yi ≥ yb + 0.5H i , and yi + 0.5W i ≤ HS hold, which would be equivalent to Eq. (25) in Model SD-SLP. In addition, Eqs. (23), (24), (26) are required to hold in Model SD-SLP, restricting the location of the facility to within the boundaries of the assigned cell. No such restriction is imposed in Model MD-SLP. This is clearly a relaxed requirement, thus the space delineated by Eqs. (23)–Eq. (26) is contained within that of Eqs. (8)–(16) Assume that the optimal solution of Model SD-SLP gives zim = 1 x and and, without any loss of generality, further assume that xi ≥ lm y yi ≥ lm (a similar analysis holds for other positons of the centroid of facility i in location m). From the location boundary constraints

In this section, we propose a novel cutting plane algorithm, dubbed the location decomposition algorithm (LDA), to solve the space-discretised SLPP. We choose to decompose SD-SLP, due to its mathematical structure, which unlike Model MD-SLP, offers a favourable context for a cutting plane algorithm due to its implicit representation of barrier and site boundary constraints within the space discretisation pattern. The proposed algorithm works from a relaxed formulation of SD-SLP and iteratively adds constraints until feasibility is recovered. The LDA is composed of two MILP formulations: a relaxed Master model, namely R-SLP, which allocates facilities to locations, m ∈ L; and a feasibility sub-problem, FeasLocm , which ensures a feasible design at each discrete location m ∈ L. First, Model R-SLP is solved and an initial assignment of facilities to locations is provided. Cuts are generated if conditions on area requirements of the assigned facilities are not met. The algorithm loops back to solve R-SLP, and the resulting outcome, if no area condition is breached, is passed on to FeasLocm , to check for feasibility within each location. If a location is found to be infeasible, due to the violation of the facility overlap or the boundary conditions, a cut is once again added to Model R-SLP, to prevent this infeasible assignment in the future. The process is then repeated until no infeasible assignments are found. We next introduce the MIP models used within the decomposition algorithm, present the decomposition algorithm and then we prove that the LDA is exact for the space-discretised SLPP. 5.1. Relaxed SD-SLP formulation We propose to relax the rotation, overlap and boundary constraints, represented by Eqs. (3)–(7) & Eqs. (23)–(26) in Model SD-SLP. The decision variables ri , xi , yi , W i , H i ∀i ∈ F and μxi j , μyi j ∀i, j ∈ F : i < j can then be dropped and we name the resulting formulation R-SLP, i.e. R-SLP is comprised of Eq. (21), (22) and Eq. (27). R-SLP then becomes the master problem in our decomposition approach. Solving this relaxed model results in an assignment of facilities to locations such that the transport cost between locations is minimised. To check and impose feasibility, we iteratively generate feasibility cuts which are added to Model R-SLP. Specifically, for any location at which feasibility is violated, a feasibility cut of the form:



zi,lη ≤ Uη

∀η

(46)

i∈Fη

where η is the cut index, lη is the current location at cut η, Fη = {i ∈ F : zilη = 1} is the set of facilities currently assigned to lη and Uη = |Fη | − 1 is an upper bound on the assignment of the combination of facilities at this location; is generated and included in Model R-SLP.

A.W.A. Hammad et al. / Computers and Operations Research 82 (2017) 36–51

Two conditions may prompt the generation of cuts in the algorithm. First, whenever the area of a location is found to be too small to fit the assigned facilities, a cut is produced to prevent this assignment. Second, cuts are generated whenever the set of facilities assigned to a location is found to violate boundary or overlap constraints. This may occur if the total area of the facilities assigned to a location does not exceed the location’s area but facilities’ shapes cannot be arranged without breaching design requirements. These two verifications are detailed next.

43

Algorithm 1 Location Decomposition Algorithm

5.2. Cut generation by area violation Let Cm denote the set of all facilities assigned to location m after solving R-SLP. For any location m ∈ L with |Cm | > 1, a check is carried out on the sum of areas of facilities in Cm . If the area of location m, Aream , is greater than sum of area of facilities assigned  in Cm , i∈Cm Areai , a cut of the form Eq. (46) is enforced in R-SLP, where lη = m, Fη = Cm and Uη = |Cm | − 1. Observe that this cut can be extended to all locations, with an area smaller than or equal to that of m. This then ensures that the algorithm does not iterate over locations for which the optimisation results will be deemed infeasible, thus reducing the total number of iterations within the algorithm. If the area of the location does not exceed that of the assigned facilities, we need to check the feasibility of the design therein. 5.3. Cut generation by design violation For each location in which the relaxed assignment does not violate the area, we solve a MIP to determine the existence of a feasible arrangement of facilities at this location. We call these feasibility sub-problems FeasLocm . Let CPm ≡ {i, j ∈ Cm : i < j} be the set of all pairs of temporary facilities assigned to location m. ModelFeasLocm is summarised in Eqs. (47)–(64)



minimise

(i, j ) ∈ C Pm

μxi j +μyi j

(47)

x xi + 0.5W i ≤ lm + 0.5Wm y yi − 0.5H i ≥ lm − 0.5Hm y yi + 0.5H i ≤ lm + 0.5Hm

subject to

∀m ∈ L, ∀i ∈ Cm

(59)

∀m ∈ L, ∀i ∈ Cm

(60)

∀m ∈ L, ∀i ∈ Cm

W i = ri Hi + (1 − ri )Wi

∀i ∈ Cm

(48)

H i = (1 − ri )Hi + riWi

∀i ∈ Cm

(49)

μxi j , μyi j ∈ {0, 1} ∀(i, j ) ∈ C Pm

(50)

W i , H i , xi , yi ≥ 0

  xi − x j  ≥ 0.5(W i + W j )μxi j   yi − y j  ≥ 0.5(H i + H j )μy

∀(i, j ) ∈ C Pm

ij

μxi j + μyi j ≥ 1

∀(i, j ) ∈ C Pm

















yi ≥ yb + 0.5H i yi ≤ yb − 0.5H i

xi ≥ xb + 0.5W i xi ≤ xb − 0.5W i x

∀(i, j ) ∈ C Pm

y

(51) (52)

β ib ∀i ∈ Cm , ∀b ∈ B

(53)

  β yib + 1 − β yib HS ∀i ∈ Cm , ∀b ∈ B

(54)

x

β ib ∀i ∈ Cm , ∀b ∈ B

(55)

  β xib + 1 − β xib WS ∀i ∈ Cm , ∀b ∈ B

(56)

y

β ib + β xib + β ib + β yib ≥ 1 ∀i ∈ Cm , ∀b ∈ B x xi − 0.5W i ≥ lm − 0.5Wm

∀m ∈ L, ∀i ∈ Cm

(57) (58)

ri ∈ {0, 1}

∀i ∈ F

(61) (62)

∀i ∈ Cm

(63) (64)

If there exists no solution satisfying Eqs. (48)–(64), the arrangement of facilities would be deemed infeasible and a cut of the form Eq. (46) is generated. Even though in FeasLocm we are only interested in the existence of a feasible layout of facilities inside the locations to which they have been assigned by R-SLP, after extensive testing it was observed that incorporating the objective  y function min (i, j )∈C Pm μxi j +μi j into FeasLocm resulted in a faster convergence of the MILP solver (see Section 6 for details on the experimental framework). As in the SD-SLP model, the absolute value functions and any bi-linear terms are linearised through use of binary and auxiliary variables. The cut generation procedure is repeated over all locations with two or more assigned facilities. The algorithm terminates whenever the number of conflict-free locations is equal to the number of available locations. The pseudo-code of the LDA is displayed in Algorithm 1. To complement the explanation of the pseudo-code, and to demonstrate the interplay of the master and sub-problems in LDA, we provide Fig. 4 as an illustrative example. In this example, we assume that the frequency parameter is set such that a single value,

44

A.W.A. Hammad et al. / Computers and Operations Research 82 (2017) 36–51

Fig. 4. Example illustrating interplay of master and sub-problem of LDA.

representative of high travel trips, is assigned between facility F1 and Barrier B1, and Facility F2 and Barrier B2, and between F1 and F2. The first step in LDA involves solving the Master Problem, R-SLP, resulting in an assignment of Facilities F1 and F2 to the same Location, L1, Step 2. This is the assignment which produces the lowest transportation cost. However, even though the area constraint may not be violated, FeasLocm is infeasible when solved for such an assignment due to the violation of overlap and boundary constraints at location L1, Step 3. As a result, and as shown in Step 4, six cuts are generated, one for each of the discrete locations. These cuts specify that at any location, a maximum of 1 facility can be assigned. The cuts are added to R-SLP, which is re-solved, resulting in the optimum layout shown in Step 5. Since the number of available locations is equal to the number of locations for which FeasLocm and the area condition are not violated, the algorithm terminates. The following propositions show that Algorithm 1. is exact for the space-discretised SLPP and that it converges in a finite number of iterations. Proposition 2. LDA finds the optimal solution of the spacediscretised SLPP as represented by Model SD-SLP (correctness). Proof: Let ζ R − SLP and ζ SD − SLP be the optimal values of the objectives of R-SLP and SD-SLP, respectively. Since initially Model R-SLP only has binary variables and assignment constraints which are also present in Model SD-SLP, the feasible region of R-SLP contains that of SD-SLP and thus ζ R − SLP ≤ ζ SD − SLP . Adding cuts to R-SLP only removes infeasible solutions of SD-SLP. Thus, ζ R − SLP remains lower than ζ SD − SLP and converges to ζ SD − SLP when no infeasible locations are found. Hence, the solution provided by Algorithm 1. upon termination is feasible for Model SD-SLP.  Proposition 3. LDA converges in a finite number of steps (finiteness) Proof: At any location, the maximum number of feasibility cuts |F | |F | is ( ) = 2|F | − |F | − 1 and the total number of feasibility i=2 i cuts is upper bounded by |L|(2|F| − |F| − 1). Since there is a finite

number of locations and facilities, the maximum number of iterations is finite.  Propositions 2 and 3 show that the solutions obtained by the LDA satisfy all imposed constraints and optimality conditions for the space-discretised SLPP, and that the algorithm converges in a finite number of steps. The LDA is therefore an exact optimisation method for Model SD-SLP. In the next section we examine the performance of the proposed discrete SLPP models, where we also benchmark LDA against SD-SLP for large scale instances. 6. Computational experiments Extensive computational experiments are carried out to assess the performance of the models and approaches presented in this paper. The computational test experiments implemented in this section are grouped as follows: Category 1: The quality of the solution of discrete SLPP, depending on the space discretisation scheme assumed, is examined in this category. The aim of experiments in this category is to compare both MILP Models for the space-discretised SLPP, i.e. SD-SLP and MD-SLP. For both models, we specifically measure the impact of the discretisation on the approximation of the barrier distance as pairwise location distances, instead of using the actual distance between facility centroids. To ensure a comprehensive analysis, we consider several levels of discretisation — represented by the size of the grids adopted — tested on realistic instances of the SLPP. The experiments aim to quantify the trade-off between barrier distance accuracy and computational performance. Test instances with 5 and 10 temporary facilities are used in this category. Category 2: The performance of Model SD-SLP versus the LDA approach is contrasted using randomly generated realistic instances with 10 to 20 temporary facilities. The objective in this category of tests is to assess the applicability of the proposed exact cutting plane algorithm to large instances, typically intractable using pure MIP implementation, such as directly solving Model SD-SLP. A detailed algorithmic assessment of the performance of the LDA is also carried out.

A.W.A. Hammad et al. / Computers and Operations Research 82 (2017) 36–51

45

Fig. 5. Scenario generation from realistic base design, using various grid discretisation schemes.

Category 3: The tests in this category seek to assess the impact of two measures, found to be influential, on the computational performance of the LDA: 1) uniformity of the frequency parameter; and 2) the area ratio factor, defined as the ratio of the total facility area to the total available layout area. Test instances with 16 temporary facilities are used in this category. In all the generated instances of Category 1 and 2 tests, the dimensions of the facilities are randomly produced from a uniform distribution, with set lower and upper bounds of 5 m and 25 m respectively, derived from realistic construction projects involving various temporary facilities. In Category 3 we set two upper bounds on the facility dimensions, depending on the tests conducted, namely 25 m and 50 m. We next detail the results of the computational experiments carried out in each of the categories presented. 6.1. Benchmark SD-SLP vs MD-SLP In this section we examine and contrast the computational performance of SD-SLP and MD-SLP, where we focus on the solution quality and solve time of both MIPs, relative to the discretisation scheme employed. A base design, representative of a realistic construction project layout is selected, and is shown in Fig. 5. The construction site has dimensions of 300 × 300 m and contains three buildings, B1, B2 and B3, with respective centroids of (75, 220), (245, 227.5) and (187.5, 55). The space discretisation scheme proposed above has the flexibility to work with any number of barriers. Our choice of 3 barriers is merely to represent a reasonable construction site layout. The models and the analysis are therefore still applicable for any arbitrary number of barriers. We note that as the barrier number increases, the available number of locations for placing the temporary facilities will decrease. We superimpose a grid structure, as explained in Section 3, on the proposed base design, with the size of the grid determined by the set value of ε . We adopt 4 values for ε in order to obtain 4 space-discretised patterns, referred to as scenarios. An illustrative representation of each scenario is displayed in Fig. 5. In Scenario

1, the region available for assigning temporary facilities to is divided into 57 cells. In Scenario 2, a total of 9 cells are available. Scenario 3 produces 2 cells, while Scenario 4, with the coarsest discretisation, yields a single discretised space. The space discretisation scheme is implemented in Python and is used to pre-process the base design so that a pairwise distance matrix for each scenario is produced. Model SD-SLP and MD-SLP are implemented using AMPL [14] where CPLEX 12.6.1 is employed as the MIP solver, with a time limit set to 600 s. All test problems are solved on a PC with 3.4 GHz CPU and 16 GB RAM. Our initial computational tests in this section are conducted with |F| = 10. We then reduce the number of facilities to 5 to highlight the difference between both models at a very fine discretisation. The pairwise travel frequency parameters for each instance in all tests are derived from an exponential distribution, with a rate parameter of 2359, fitted to data corresponding to recorded travel between facilities from actual construction projects. In accordance with the literature, the interaction amongst facilities in SLPP happens to be large for a small set of the temporary facilities to be positioned [15]. For each facility number considered, a total of 25 computational experiments are conducted, where for each value of ε , 5 instances of the design of Fig. 5 are tested. Results highlighting the performance of SD-SLP and MD-SLP for |F| = 10 are given in Table 1, while Table 2 is for tests carried out on instances with |F| = 5. For each model, each table shows the objective function value, OF, the percentage deviation from true transport cost, TC, the percentage deviation from true barrier distance, TD, and the total run time. We note that TC accounts for the frequency parameter between facilities, while TD is purely based on the difference in value between the actual and approximated distances only, with no inclusion of the frequency parameter. For instances that are not solved to optimality we also report, in the OF column, the relative gap between the best integer objective and the objective of the best node that is remaining. In order to calculate the true cost, we construct the L1-visibility graph of the site layout obtained after solving each of the respective models, as described in Section 3. For a

46

A.W.A. Hammad et al. / Computers and Operations Research 82 (2017) 36–51

Table 1 Impact of grid size of discretisation on performance of SD-SLP and MD-SLP for |F| = 10. SD-SLP

MD-SLP

SD-SLP vs MD-SLP

Instance

ε

OF [% gap]

TC (%)

TD (%)

Run time (s)

OF [% gap]

TC (%)

TD (%)

Run time(s)

OF

 Run time

1

25 50 75 100 25 50 75 100 25 50 75 100 25 50 75 100 25 50 75 100

1.65E + 07 [32] 1.86E + 07 4.41E + 07 4.62E + 07 1.75E + 07 [45] 1.66E + 07 6.62E + 07 7.71E + 07 1.48E + 07 [42] 1.70E + 07 6.36E + 07 7.45E + 07 1.38E + 07 [37] 1.61E + 07 6.00E + 07 7.03E + 07 1.78E + 07 [31] 1.87E + 07 7.93E + 07 9.19E + 07

18.5 36.9 30.4 37.2 16.5 32.0 23.0 21.4 18.9 38.3 22.4 25.9 33.7 32.9 25.2 23.1 23.8 33.9 25.2 22.5

17.9 43.6 48.3 56.2 16.6 39.5 47.9 54.5 35.3 40.3 45.8 51.2 40.8 41.5 48.6 52.6 34.7 39.8 47.4 51.1

600 15.1 0.6 0.1 600 12 0.1 0 600 7.9 0.1 0 600 9.1 0.2 0.1 600 9.4 0.1 0

2.37E + 07 [67] 1.86E + 07 4.41E + 07 4.62E + 07 1.47E + 07[76] 1.66E + 07 6.62E + 07 7.71E + 07 1.37E + 07 [78] 1.70E + 07 6.36E + 07 7.45E + 07 1.31E + 07[71] 1.61E + 07 6.00E + 07 7.03E + 07 1.57E + 07 [79] 1.87E + 07 7.93E + 07 9.19E + 07

21.5 40.3 29.1 37.3 46.3 40.4 23.1 8.4 23.2 49.8 24.8 13.6 22.7 38.2 28.7 10.3 21.5 35.2 19.1 9.7

19.0 48.8 50.3 50.4 14.0 38.6 50.2 51.4 29.1 44.9 47.2 50.1 17.1 38.8 48.2 50.4 15.1 37.8 43.9 50.2

600 7 0.2 0.1 600 1.7 0.3 0 600 6.4 0.2 0 600 7.6 0.3 0 600 7.8 0.2 0

−7.24E + 07 0.0 0E + 0 0 0.0 0E + 0 0 0.0 0E + 0 0 2.79E + 06 0.0 0E + 0 0 0.0 0E + 0 0 0.0 0E + 0 0 1.10E + 06 0.0 0E + 0 0 0.0 0E + 0 0 0.0 0E + 0 0 6.64E + 05 0.0 0E + 0 0 0.0 0E + 0 0 0.0 0E + 0 0 2.10E + 06 0.0 0E + 0 0 0.0 0E + 0 0 0.0 0E + 0 0

0 8.3 −0.1 −0.1 0 10.3 −0.2 0 0 1.5 −0.1 0 0 1.5 −0.2 0 0 1.6 −0.1 0

2

3

4

5

Table 2 Impact of grid size of discretisation on performance of SD-SLP and MD-SLP for |F| = 5. SD-SLP

MD-SLP

SD-SLP vs MD-SLP

Instance

ε

OF [% gap]

TC (%)

TD (%)

Run time (s)

OF [% gap]

TC (%)

TD (%)

Run time(s)

OF

 Run time

1

25 50 75 100 25 50 75 100 25 50 75 100 25 50 75 100 25 50 75 100

1.54E + 07 1.65E + 07 1.66E + 07 2.10E + 07 1.62E + 07 [21] 1.71E + 07 1.68E + 07 2.15E + 07 1.58E + 07 1.32E + 07 1.32E + 07 2.10E + 07 1.68E + 07 1.31E + 07 1.30E + 07 2.07E + 07 1.43E + 07 1.33E + 07 1.34E + 07 2.14E + 07

16.5 29.9 30.2 38.7 16.3 12.3 35.6 37.1 16.2 26.4 44.6 39.4 16.6 29.8 48.3 38.0 16.6 24.2 35.4 32.4

16.7 27.3 37.0 38.7 16.7 26.1 39.0 39.5 16.5 23.5 36.3 38.1 16.6 23.0 35.7 47.0 16.6 23.4 32.7 35.4

543 3.5 0.1 0 600 6.3 0.1 0 486 154.9 0 0 345 146.5 0 0 367 50.3 0 0

1.44E + 07 1.65E + 07 1.66E + 07 2.10E + 07 1.51E + 07 [27] 1.71E + 07 1.68E + 07 2.15E + 07 9.40E + 06 1.19E + 07 1.32E + 07 2.10E + 07 9.28E + 06 1.17E + 07 1.30E + 07 2.07E + 07 9.47E + 06 1.21E + 07 1.34E + 07 2.14E + 07

26.3 27.1 31.6 36.3 25.1 27.5 29.4 37.0 31.2 32.1 33.7 36.1 31.1 31.7 33.3 36.1 31.3 32.2 32.4 36.1

19.0 22.9 27.6 35.2 19.4 19.8 20.4 37.8 19.0 19.4 28.1 53.2 19.0 40.6 51.9 57.6 19.0 19.2 19.3 30.7

330.3 7 0.2 0.1 600 4.9 0.1 0.1 338.7 1.2 0.1 0 246.8 1 0.1 0 294 0.9 0.1 0

9.29E + 05 0.0 0E + 0 0 0.0 0E + 0 0 0.0 0E + 0 0 1.09E + 06 0.0 0E + 0 0 0.0 0E + 0 0 0.0 0E + 0 0 6.43E + 06 1.37E + 06 0.0 0E + 0 0 0.0 0E + 0 0 7.53E + 06 1.36E + 06 0.0 0E + 0 0 0.0 0E + 0 0 4.80E + 06 1.18E + 06 0.0 0E + 0 0 0.0 0E + 0 0

330.3 7 0.2 0.1 600 4.9 0.1 0.1 338.7 1.2 0.1 0 246.8 1 0.1 0 294 0.9 0.1 0

2

3

4

5

direct contrast of both models, we also compute the difference in objective function values, OF = OF(SD) − OF(MD), and runtimes. We first describe the findings derived from Table 1. The following general relationships, applicable to both SD-SLP and MD-SLP, can be inferred. First, the run times of both models increase as the discretisation becomes finer. The fastest instances to solve are the ones where a discretisation of ε = 100 is adopted, with an associated solving time of around 0.02 s for both Models. The solve times increase dramatically as the discretisation becomes finer, reaching the limit imposed for instances solved with ε = 25 for both models. Second, as the discretisation becomes finer, the deviation in distance measure from true barrier distance, TD decreases. This can be attributed to the approximated distance approaching the true barrier distance function with smaller grid sizes. Third, instances where ε = 25 fail to solve to optimality within the specified time limit in both models. This is attributed to the highly combinatorial nature of both models when the number of locations is increased at ε = 25. We note that in these instances, the relative gap reported in the SD-SLP model is always less than

that reported by MD-SLP. Fourth, the deviation from true transport cost does not seem to be affected only by the closeness of the approximated barrier distance to the actual one. For example, in Instance 3, at ε = 75, when Model SD-SLP is solved, the deviation from true cost is measured at 22.4%. With a finer discretisation of ε = 50, the deviation increases to 38.3%. A similar pattern is seen in Model MD-SLP, when the same instance is solved at the latter discretisation scheme. This is a consequence of weighting the objective function with the frequency parameter. Even though a coarse approximation of barrier distance occurs when a very large discretisation scheme is adopted, the weighting specified between facilities within an instance influences the magnitude of the resulting objective function value and in turn also impacts the true cost computations. Hence, unlike the barrier distance computations, for finer discretisation grids, the gap between true and approximated cost will not always decrease, as the objective function value is also highly dependent on the travel frequency parameter. It is also noticed that the deviation between the approximated and true distance is not regularly equal in magnitude

A.W.A. Hammad et al. / Computers and Operations Research 82 (2017) 36–51

to the deviation in objective function value. Again this discrepancy is mainly attributed to the frequency parameter incorporated in the objective function of both models. Therefore, depending on the frequency parameter, even when distance is relatively small for a particular discretisation scheme, changing ε will alter the number of available locations, and so the allocation of facilities to locations will be modified. The frequency-weighted transportation cost is thus expected to change too. Unlike the distance measure, whose deviation from the true barrier distance is known to decrease with finer ε , a trend in objective value function deviation cannot be associated with the alteration in the discretisation scheme deployed. Contrasting both models, the objective function value of MDSLP is always the same (at least as good) as that of SD-SLP in all instances solved to optimality. We note that the OF column reports a figure greater than zero for the cases when the models are not solved to optimality within the specified time limit. Since OF is calculated by subtracting the OF value of MD-SLP from SD-SLP, for cases where an optimal solution can be yielded for both models, the objective function value of MD-SLP never falls below that of SD-SLP. This is in line with Proposition 1 above. The run times of both models seem to be comparable, with slightly faster solving times for Model MD-SLP. We also note that both models managed to obtain at least a feasible solution within the time limit imposed for all solved instances, though the relative gap in SD-SLP is lower than that of MD-SLP for all such instances not solved to optimality. The combinatorial nature of both models means that yielding an optimal solution for the instances where the discretisation is set at ε = 25 is not possible. In order to display the contrast in objective function value between Model SD-SLP and MD-SLP at fine discretisation patterns, we decrease the number of facilities to 5; Table 2 is produced as a result. The results from the table indicate that most instances associated with ε = 25 can be solved within the specified time limit for both models. Model MD-SLP now yields a lower objective function value at the fine discretisation of ε = 25 and ε = 50. This enforces Proposition 1. We also notice that the solve time of Model MD-SLP is better than that of Model SD-SLP and this can be attributed to the fewer number of constraints in Model MD-SLP in comparison to SD-SLP. In particular, SD-SLP requires the location boundary constraints to hold for each location. Model MD-SLP on the other hand has the site boundary and barrier constraints, which are lower in number compared to the location boundary constraints of SD-SLP. We can conclude the following from Tables 1 and 2: both models seem to display the same trend with regards to solvability. Model MD-SLP presents a slightly closer representation of the continuous SLP due to permitting multi-cover of locations by facilities when a grid is superimposed for estimating the barrier distance, compared to Model SD-SLP, as evident from the lower deviation in distance approximation for finer discretisation. Even though the results show that Model MD-SLP is slightly more effective at solving instances with large number of cells, along with its lower solving time and objective function value, Model SD-SLP has a more favoured mathematical structure. This is attributed to its implicit representation of barrier and site boundary constraints within discrete locations, which plays a central role in the proposed cutting-plane algorithm. 6.2. Performance of the LDA In this section, we highlight the main findings on the computational performance of our proposed exact decomposition method for solving large instances of space discretised SLPP. Scenario 2 associated with Model SD-SLP, highlighted in Section 6.1, is utilised for generating instances in this section. As in Section 6.1, the pairwise travel frequency parameter is derived from an

47

exponential distribution. AMPL is utilised for implementing Algorithm 1 [14] where CPLEX 12.6.1 is employed as the MIP solver. In the LDA, a time limit of 600 s is imposed on the combination of solve times of both R-SLP and FeasLocm . 6.2.1. Algorithm profile To examine the algorithmic behaviour of LDA, we test it on 1100 instances, generated by gradually incrementing the number of facilities, starting from |F| = 10 till we reach 20 facilities. Therefore, a total of 100 instances are initiated for each temporary facility number to be located. Table 3 summarises the minimum, average and maximum number of generated cuts and loops of iterations observed in the tests conducted on the realistic instances. From the results, we see that the minimum, average and maximum runtime of LDA increases with increasing number of facilities. This is natural since LDA is composed of MIP formulations which are sensitive to the increase in the number of variables and constraints associated with increasing the number of facilities. The average runtime remains lower than the time limit specified for all cases. We note that the greatest difference in runtimes is observed between cases involving 15 and 16 facilities, where a jump of 656% occurs. We also notice that beyond 15 facilities, the maximum runtime becomes equal to the maximum set time limit. Looking at the trend in the cuts generated in the algorithm, our results indicate that the average number of cuts is always increasing with number of facilities considered. The maximum number of cuts does not follow the increasing trend of the average number of cuts, since a sharp increase in the maximum number of cuts is noticed between |F| = 13 and |F| = 14, followed by a gradual decrease for cases involving more than 16 facilities, and then a slight increase once cases with 20 facilities are solved. In contrast, the minimum number of cuts generated remains the same for all tested instances. The average number of iterations appears to be correlated with the average number of cuts generated. The rate of increase of the number of iterations, as the number of facilities increases, is lower than the rate at which additional cuts are generated. This is because within the LDA several cuts will be produced within a single iteration, whenever the area condition is not satisfied or when FeasLocm is infeasible. Globally, instances with a small number of facilities are less likely to violate boundary and design constraints within an assigned location, whereas, as the number of facilities increases, it becomes harder to satisfy these constraints, hence leading to greater number of iterations in the algorithm and the generation of more cuts as a result. These tests also suggest that, from a certain number of facilities onwards, degenerate behaviour may be observed more frequently. This is potentially due to the topological aspects of some instances, e.g. a high number of similarly shaped facilities inducing a combinatorial explosion of the number of cuts. In order to grasp the full behaviour of the LDA, we examine the performance of the master and sub-problems embodying the algorithm, namely R-SLP and FeasLocm , in terms of solving times. Fig. 6 shows the average time spent in each of the aforementioned MIPs for the generated instances. It is evident from the figure that for all number of facilities, the average solve time of FeasLocm is much lower than that of R-SLP for all cases considered. This observation highlights the agility of the cut generating MIP formulations due to design violation within LDA. Unlike FeasLocm , the time taken to solve R-SLP increases dramatically as the number of facilities considered is increased. This is attributed to the increase in the size of the R-SLP resulting from the addition of generated cuts to it as constraints. Fig. 6 promotes the capability of LDA in solving large instances of the space-discretised SLPP in a reasonable amount of time (less than the imposed 600 s on average). We next investigate the behaviour of the LDA in light of two influencing

48

A.W.A. Hammad et al. / Computers and Operations Research 82 (2017) 36–51 Table 3 Effect of increasing number of facilities on performance of LDA∗ . |F|

Run time (s)

Cuts

No. of iterations

∗

Min Avg. Max Min Avg. Max Min Avg. Max

10

11

12

13

14

15

16

17

18

19

20

3.09 3.36 3.75 0 1 3 1 1 2

3.79 4.15 4.61 0 2 4 1 1 3

4.86 5.18 5.82 0 2 4 1 1 3

5.75 7.68 63.35 0 2 22 1 1 3

5.69 9.99 189.54 0 5 220 1 2 11

5.73 10.43 203.22 0 6 220 1 2 11

24.52 78.88 600 0 29 264 1 3 13

31.78 150.41 600 0 27 139 1 3 10

40.55 305.15 600 0 62 132 1 4 7

49.08 421.83 600 0 73 110 1 5 7

63.49 474.06 600 0 116 154 1 7 8

All test results are based on 100 instances for each facility number, where run time limit is set to 600 s

Fig. 6. Comparison of average solve times of R-SLP and FeasLoc within LDA. Notes: All statistics test results are based on 100 instances for each facility number, where run time limit is set to 600 s for each case.

factors: the uniformity of the frequency parameter embedded in the objective function and the ratio of the available area for layout to the total facility area.

Table 4 Impact of uniformity of frequency parameter on the performance of LDA∗ . Facility parameter

6.2.2. Influencing factors of LDA 6.2.2.1. Frequency parameter uniformity. Along with its effect on the magnitude of the objective function, as demonstrated in Section 6.1, we hypothesise that the topographic instance structure created by the embedded frequency parameter in the objective function will have a direct influence on the performance of LDA. This is perfectly in-agreement with the literature, where travel frequency parameter input to MIP layout models is known to contribute to the complexity of the instance of the combinatorial optimisation problem [15,35,42]. In this section we therefore attempt to capture the relationship between a single underlying characteristic of the frequency parameter and the extent to which LDA is impacted by such a structure. The aspect which we are interested in examining is the uniformity of the frequency parameter and its influence on the solvability of LDA. To assess the impact of the frequency parameter on the solution methodology, we use test instances generated from Scenario 2. In this benchmark, the number of facilities is fixed to 16. This number is chosen in accordance with the performance of LDA, as indicated in Table 3. We vary the dimensions of the facilities, while the frequency parameter among temporary facilities and between temporary and permanent facilities is fixed to 1. For the non-uniform case, the frequency parameter and the dimensions of the facilities are randomly sampled from realistic distributions. Hence a total of 200 experiments are conducted; 100 instances generated for each case. We stress the following point: realistically

Run time (s)

Cuts

No. of iterations

Min Avg. Max Min Avg. Max Min Avg. Max

Uniform

Non-uniform

17.67 78.23 600∗ 1.9 22 198 1 2 8

24.32 94.71 600∗ 0 29 264 1 3 13

∗ All test results are based on 100 instances, where |F| = 10 and run time limit is set to 600 s

speaking, the case where the SLPP is solved for completely uniform frequency parameters is highly unlikely to occur. However, it is our purpose in this section to highlight that the structure of the frequency parameter can impact the performance of LDA. Table 4 summarises the results obtained. From a computational perspective, the performance of the LDA is on average better when a fully uniform frequency parameter is adopted between all temporary facilities, compared to a non-uniform parameter. On average, the inclusion of a uniform frequency parameter results in a 22% decrease in the solving time of LDA, compared to cases where the frequency parameter is non-uniform. The same trend is seen in the number of cuts and number of iterations. Less effort is required to

A.W.A. Hammad et al. / Computers and Operations Research 82 (2017) 36–51 Table 5 Effect of facility to location ratio on performance of LDA∗ . Facility – to – location ratio

Run time (s)

Cuts

No. of iterations

% optimal solution

Min Avg. Max Min Avg. Max Min Avg. Max 100

Lowr

Mediumr

Highr

3.17 9.15 56.76 0 1 22 1 1 2 91

24.52 78.88 600 0 28 264 1 2 13 23.3

49.91 489.73 600 0 106 396 1 4 13

∗

All test results are based on 100 instances for each ratio category, where run time limit is set to 600 s

place facilities by the algorithm, when the frequency weighting has virtually no impact on steering the objective function direction. For a lot of the tested instances, non-uniform frequency parameters between the temporary facilities often lead to a combinatorial explosion in the number of alternatives available, since the weighting of the frequency parameter becomes influential in determining the direction for the algorithm to take to minimise the objective function. With a uniform frequency parameter set to 1 between permanent facilities and temporary facilities and among temporary facilities, the function being minimised is the distance function between facilities, and the actual positon given to the facilities relative to one another is not imperative so long as it is a packed formation that results, which groups the facilities into the least number of locations, hence reducing the objective function. We also found that the solvability of LDA is greatly improved if varying non-unity frequency figures are adopted between the permanent and temporary facilities. This directs the algorithm to focus only on those locations that are closest to the barriers for positioning the facilities in, thus reducing the feasible search space. 6.2.2.2. Area ratio. Another factor found to be influencing the difficulty of SLPP instances is the ratio,r, representing the sum of areas of the temporary facilities to the sum of areas of discretised locations. This is because of two main reasons: 1) the size of the search region is affected by the size of facilities measured with respect to the available area on a construction site; 2) an integral part of the algorithm relies on the generation of cuts whenever certain conditions, including the area size conditions between facilities and locations, are not met. As such an analysis is conducted to investigate the effect that this ratio can have on the solvability of space-discretised SLPP instances. A total of 100 instances are generated, using Scenario 2, with |F| = 16. The frequency parameter is kept fixed across all instances, while the facility dimensions are varied to produce different area ratios. We have imposed an upper bound of 50 m on the dimension of the facilities to ensure that all the instances generated can fit within the grid size associated with Scenario 2. The instances generated are grouped into three classes of ratios, derived based on actual conditions likely to be present in SLPP. In particular, an instance is deemed to hold a low ratio if its computed ratio happens to be less than or equal to 0.09, while instances where r ∈ (0.09, 0.25] are classified as having a medium area ratio. Finally, any instance where r ∈ (0.25, 0.4], is a representative for a high area ratio case. Ratios greater than 0.4 are not considered as it is not common in a SLPP to have such exaggerated ratios. Table 5 summarises the effect of the area ratio r on the performance of the LDA. We observe that the total number of instances that are solved to optimality decreases as r increases. We also

49

notice from the results that the run times increase with increasing r. The difference between a case with a low ratio and one with a high ratio, in terms of the average run times, is around 481 s. A similar increasing trend holds for the average number of cuts and number of iterations. A reason behind this trend is that as the area ratio increases, it becomes more difficult to fit facilities in locations. Hence more cuts are required in the LDA to account for the violations of the area requirements or violations of feasibility constraints, resulting in an increase in the number of iterations. 6.2.3. Comparison of LDA and SD-SLP The objective of this benchmark is to compare the computational performance of the LDA with that of Model SD-SLP, commonly adopted to solve the SLPP in the literature. LDA and SDSLP are contrasted on 1100 instances, similar to the ones adopted in Section 6.2.1. The solve times obtained using the LDA and Model SD-SLP are displayed in the box plot of Fig. 7. For all solved instances, the mean solve time of the LDA appears to always be lower than that of SD-SLP. The highest rate of improvement in mean solve time of the LDA compared to SD-SLP is recorded for cases |F| = 14 and |F| = 15, where the mean of solve times obtained by LDA is 83% lower than that obtained by SD-SLP. Beyond 15 facilities the rate of improvement of mean values decreases with the number of facilities. For instances with 18 facilities, the LDA has a mean of solve times that is 49% lower than that of SD-SLP (300 s and 590 s respectively). Solving instances with 20 facilities, the mean of solve time of LDA is 21% lower than that of SD-SLP (474 s vs 595 s). We analyse some statistics provided by the box-plot figure for illustrating the variability in the processing time of both LDA and SD-SLP. In particular, the inter-quartile range (IQR) of the solve times of the LDA is small for a low number of facilities and starts to increase after 17 facilities are considered. Conversely, the IQR of solve times of SD-SLP grows faster compared to that of LDA, and starts to diminish rapidly for large facility numbers, eventually reaching almost zero for cases where |F| ≥ 18. The trend in the IQR of both solving methods becomes more apparent when we examine the lower quartile (Q1) and the upper quartile (Q3) of solve times. Q1 values of LDA are lower than those of SD-SLP for nearly all cases solved, while Q3 values of solve times of LDA are lower than those of SD-SLP for cases where |F| ≤ 17 are considered. Beyond 18 facilities Q3 values for both methods become equivalent. It is also observed from Fig. 7 that the median of run times of LDA increases at a much lower pace, in comparison to that of SD-SLP, for cases 10 ≤ |F| ≤ 20. Percentage wise, great improvements in the median of run times are noticed for cases that are solved via LDA compared to SD-SLP. In particular, LDA outperforms SD-SLP for all instances where cases 10 ≤ |F| ≤ 18 are solved. The greatest difference between the median of solve times of both methods occurs at case |F| = 18 where the median corresponding to LDA is 95% lower than that recorded for SD-SLP. After |F| = 15, SD-SLP starts to display unstable behaviour, where the median solve time drops slightly at |F| = 16, but then shoots to 600 s for other consecutive cases. This is indicative of the loss in tractability of SD-SLP for solving the NP-hard SLPP. The aforementioned statistics derived from the box plots of Fig. 7 indicate that the computational performance of LDA surpasses that of SD-SLP for all the instances tested, especially for the hardest of instances solved. In terms of the total number of instances that can be solved to optimality we always notice that for all cases tested, the LDA is able to solve more instances compared to SD-SLP, within the specified time limit. Viewing the methods adopted in the SLPP literature, an MIP similar to SD-SLP was deployed to solve the space discretised SLPP to exact optimality for cases involving only 4 facilities [11]. The maximum number of facilities for which the SLPP has been solved is re-

50

A.W.A. Hammad et al. / Computers and Operations Research 82 (2017) 36–51

Fig. 7. Box plot contrasting run times of LDA vs SD-SLP with increasing facility number for ε = 50.

ported in [1] as 16 facilities, though this is achieved through use of meta-heuristics. Our decomposition approach therefore provides a breakthrough in solving the discrete SLPP to exact optimality. 7. Conclusion This paper proposes several models for solving the SLPP on construction sites in the presence of barriers. Our main contributions can be summarised as follows: 1) we formulated a general MIP model to represent the continuous SLPP, taking into consideration the presence of travel barriers, through the inclusion of a distance oracle and barrier constraints to prevent barriers from overlapping with facilities (G-SLP); 2) we developed a novel MIP which allows multiple facility cover of locations (MD-SLP), while accounting for the presence of barriers in discrete SLPP; 3) we formalised a space discretisation scheme, to approximate the barrier distance function, through the use of the L1-visibility concept, and we quantified its impacts on the solution quality produced by SD-SLP and MD-SLP, while simultaneously contrasting the performance of both models. We therefore presented a comprehensive analysis to enumerate the effect of the pre-processing phase (space-discretisation, distance approximation) on the discrete models. Measuring the impact of space-discretisation on barrier distance approximation in both models showed that the finer the discretisation, the closer the approximated distance is to the actual barrier path length, however a trade-off exits between distance accuracy and running times; 4) though the proposed MD-SLP is a more accurate representation of the continuous SLP, the mathematical format of Model SD-SLP allows for the adoption of a decomposition approach since facilities are confined to defined locations. Given the favourable structure of SD-SLP, we developed a cutting plane algorithm for SD-SLP, which is able to solve large instances of the SLPP to exact optimality (LDA). The proposed methodology was tested by comparing the computational performance of the LDA with that of Model SD-SLP, to contrast the model commonly adopted in the literature with our proposed algorithm. We also assessed the impact of influencing factors onto the cutting plane algorithm’s performance. From the computational results presented it can be concluded that the LDA is very effective at solving reasonably sized site layout problems in a practical time frame. For the instances solved, it

was nearly always apparent that the LDA outperformed the model commonly adopted in the literature, namely SD-SLP. Further, the average number of cuts and iterations required within the LDA does not always increase with the number of facilities, suggesting a robust algorithmic behaviour. The structure of the frequency parameters was found to influence the performance of LDA, where a uniform frequency parameter was easier to solve, in comparison to a non-uniform one. Inclusion of varying frequency figures between permanent and temporary facilities was found to expedite the run times of LDA. Tests carried out on the effect of the facility to location area ratio on LDA computational performance revealed that higher ratios were associated with an increase in average computational time, average number of generated cuts and average number of iterations. Even though our formulations for the SLPP are based on integer-linear models, the LDA offers a versatile framework to integrate non-linear objective functions, such as noise pollution. Indeed, since the problem is decomposed, the relaxed Master problem, R-SLP, can be modified to minimise alternative objectives while the feasibility sub-problems remain linear MIPs. The solvability of the resulting decomposed approach would then be improved compared to addressing the problem through a MINLP. Future research could focus on the development of fast heuristic approaches. Specifically, approaches that decompose the SLPP to exploit particular structures of the created sub-problems could be implemented. Improving the approximation of the distance function within the space-discretisation approaches for nonconvex regions is a promising line of research. In particular, the more general case of polyhedral barriers should be further studied. Acknowledgement This project was funded through an Australian Government Research Training Program Scholarship. References [1] Abdel R. Electimize a new evolutionary algorithm for optimization with applications in construction engineering; 2011. [2] Anumba C, Bishop G. Importance of safety considerations in site layout and organization. Can J Civ Eng Can Genie Civ 1997;24:229–36.

A.W.A. Hammad et al. / Computers and Operations Research 82 (2017) 36–51 [3] Bischoff M, Fleischmann T, Klamroth K. The multi-facility location–allocation problem with polyhedral barriers. Comput Oper Res 2009;36:1376–92 Selected papers presented at the Tenth International Symposium on Locational Decisions (ISOLDE X). doi:10.1016/j.cor.2008.02.014. [4] Butt SE, Cavalier TM. An efficient algorithm for facility location in the presence of forbidden regions. Eur J Oper Res 1996;90:56–70. doi:10.1016/ 0377- 2217(94)00297- 5. [5] Cakı r O, Wesolowsky GO. Planar expropriation problem with non-rigid rectangular facilities. Comput Oper. Res 2011;38:75–89 Project Management and Scheduling. doi:10.1016/j.cor.2010.01.008. [6] Canbolat MS, Wesolowsky GO. The rectilinear distance weber problem in the presence of a probabilistic line barrier. Eur J Oper Res 2010;202:114–21. doi:10. 1016/j.ejor.2009.04.023. [7] Chen Y, Jiang Y, Wahab MIM, Long X. The facility layout problem in nonrectangular logistics parks with split lines. Expert Syst Appl 2015;42:7768–80. doi:10.1016/j.eswa.2015.06.009. [8] Dearing PM, Klamroth K, Jr RS. Planar location problems with block distance and barriers. Ann Oper Res 2005;136:117–43. doi:10.1007/s10479- 005- 2042- 4. [9] Drezner Z, Klamroth K, Schöbel A, Wesolowsky GO. The weber problem. Facility location: applications and theory. Springer Science & Business Media; 2002. [10] Drira A, Pierreval H, Hajri-Gabouj S. Facility layout problems: a survey. Annu Rev Control 2007;31:255–67. doi:10.1016/j.arcontrol.20 07.04.0 01. [11] Easa S, Hossain K. New mathematical optimization model for construction site layout. J Constr Eng Manag 2008;134:653–62. doi:10.1061/(ASCE) 0733-9364(2008)134:8(653). [12] El-Rayes K, Khalafallah A. Trade-off between safety and cost in planning construction site layouts. J Constr Eng Manag 2005;131:1186–95. doi:10.1061/ (ASCE)0733-9364(2005)131:11(1186). [13] Floyd RW. Algorithm 97: shortest path. Commun ACM 1962;5 345–.. doi:10. 1145/367766.368168. [14] Fourer R, Gay D, Kernighan B. Ampl. Boyd & fraser; 1993. [15] Hammad A, Akbarnezhad A, Rey D, Waller S. A computational method for estimating travel frequencies in site layout planning. J Constr Eng Manag 2015;4015102. doi:10.1061/(ASCE)CO.1943-7862.0 0 01086. [16] Hammad AWA, Akbarnezhad A, Rey D. A multi-objective mixed integer nonlinear programming model for construction site layout planning to minimise noise pollution and transport costs. Autom Constr 2016;61:73–85. doi:10.1016/ j.autcon.2015.10.010. [17] Hassan MM, Hogg GL. On constructing a block layout by graph theory. Int J Prod Res 1991;29:1263–78. doi:10.1080/00207549108930132. [18] Holt ASJ. Principles of construction safety. John Wiley & Sons; 2008. [19] Jr ARM, Liu W-H. New Tabu search heuristics for the dynamic facility layout problem. Int J Prod Res 2012;50:867–78. doi:10.1080/00207543.2010.545446. [20] Khalafallah A, Abdel-Raheem M. Electimize: new evolutionary algorithm for optimization with application in construction engineering. J Comput Civ Eng. 2011;25:192–201. doi:10.1061/(ASCE)CP.1943-5487.0 0 0 0 080. [21] Khalafallah A, El-Rayes K. Automated multi-objective optimization system for airport site layouts. Autom Constr 2011;20:313–20. doi:10.1016/j.autcon.2010. 11.001. [22] Klamroth K. A reduction result for location problems with polyhedral barriers. Eur J Oper Res 2001;130:486–97. doi:10.1016/S0377- 2217(99)00399- 9. [23] Kusiak A, Heragu SS. The facility layout problem. Eur J Oper Res 1987;29:229– 51. doi:10.1016/0377- 2217(87)90238- 4. [24] Lam KC, Tang CM, Lee WC. Application of the entropy technique and genetic algorithms to construction site layout planning of medium-size projects. Constr Manag Econ 2005;23:127–45. doi:10.1080/0144619042000202834. [25] Larson RC, Sadiq G. Facility locations with the manhattan metric in the presence of barriers to travel. Oper Res 1983;31:652–69. doi:10.1287/opre.31.4.652. [26] Li H, Love PE. Genetic search for solving construction site-level unequalarea facility layout problems. Autom Constr 20 0 0;9:217–26. doi:10.1016/ S0926-5805(99)0 0 0 06-0. [27] McGarvey RG, Cavalier TM. A global optimal approach to facility location in the presence of forbidden regions. Comput Ind Eng 2003;45:1–15. doi:10.1016/ S0360-8352(03)0 0 028-7. [28] McKendall AR Jr, Hakobyan A. Heuristics for the dynamic facility layout problem with unequal-area departments. Eur J Oper Res 2010;201:171–82. doi:10. 1016/j.ejor.2009.02.028.

51

[29] McKendall AR Jr, Shang J, Kuppusamy S. Simulated annealing heuristics for the dynamic facility layout problem. Comput Oper Res 2006;33:2431–44. doi:10. 1016/j.cor.2005.02.021. [30] Miyagawa M. Rectilinear distance to a facility in the presence of a square barrier. Ann Oper Res 2012;196:443–58. doi:10.1007/s10479- 012- 1063- z. [31] Moore JM. Facilities design with graph theory and strings. Omega 4; 1976. [32] Ning X, Lam KC. Cost–safety trade-off in unequal-area construction site layout planning. Autom Constr 2013;32:96–103. doi:10.1016/j.autcon.2013.01.011. [33] Ning X, Lam K-C, Lam MC-K. Dynamic construction site layout planning using max-min ant system. Autom Constr 2010;19:55–65. doi:10.1016/j.autcon.2009. 09.002. [34] Osman HM, Georgy ME, Ibrahim ME. A hybrid CAD-based construction site layout planning system using genetic algorithms. Autom Constr 2003;12:749– 64. doi:10.1016/S0926-5805(03)0 0 058-X. [35] Rossin DF, Springer MC, Klein BD. New complexity measures for the facility layout problem: an empirical study using traditional and neural network analysis. Comput Ind Eng 1999;36:585–602. doi:10.1016/S0360-8352(99)00153-9. [36] Sadeghpour F, Andayesh M. The constructs of site layout modeling: an overview. Can J Civ Eng 2015;42:199–212. doi:10.1139/cjce- 2014- 0303. [37] S¸ ahin R, Ertog˘ ral K, Türkbey O. A simulated annealing heuristic for the dynamic layout problem with budget constraint. Comput Ind Eng 2010;59:308– 13. doi:10.1016/j.cie.2010.04.013. [38] Savas¸ S, Batta R, Nagi R. Finite-size facility placement in the presence of barriers to rectilinear travel. Oper Res 2002;50:1018–31. doi:10.1287/opre.50.6.1018. 356. [39] Scholz D, Jaehn F, Junker A. Extensions to STaTS for practical applications of the facility layout problem. Eur J Oper Res 2010;204:463–72. doi:10.1016/j.ejor. 2009.11.012. [40] Singh SP, Sharma RRK. A review of different approaches to the facility layout problems. Int J Adv Manuf Technol 2006;30:425–33. doi:10.1007/ s0 0170-0 05-0 087-9. [41] Sirinaovakul B. An analysis of computer-aided facility layout techniques. Int J Comput Integr Manuf 1996;9:260–4. doi:10.1080/095119296131544. [42] Smith-Miles K, Lopes L. Measuring instance difficulty for combinatorial optimization problems. Comput Oper Res 2012;39:875–89. doi:10.1016/j.cor.2011. 07.006. [43] Tam CM, Tong TK, Leung AW, Chiu GW. Site layout planning using nonstructural fuzzy decision support system. J Constr Eng Manag 2002;128:220–31. [44] Tompkins JA, White JA, Bozer YA, Tanchoco JMA. Facilities planning. John Wiley & Sons; 2010. [45] Warshall S. A theorem on boolean matrices. J ACM 1962;9:11–12. doi:10.1145/ 321105.321107. [46] Wong C, Fung I, Tam C. Comparison of using mixed-integer programming and genetic algorithms for construction site facility layout planning. J Constr Eng Manag 2010;136:1116–28. doi:10.1061/(ASCE)CO.1943-7862.0 0 0 0214. [47] Xu J, Song X. Multi-objective dynamic layout problem for temporary construction facilities with unequal-area departments under fuzzy random environment. Knowl Based Syst 2015;81:30–45. doi:10.1016/j.knosys.2015.02.001. [48] Yahya M, Saka MP. Construction site layout planning using multi-objective artificial bee colony algorithm with Levy flights. Autom Constr 2014;38:14–29. doi:10.1016/j.autcon.2013.11.001. [49] Zhang H, Wang JY. Particle swarm optimization for construction site unequal-area layout. J Constr Eng Manag 2008;134:739–48. doi:10.1061/(ASCE) 0733-9364(2008)134:9(739). [50] Li Z, Anson M, Li G. A procedure for quantitatively evaluating site layout alternatives. Constr Manag. Econ 2001;19:459–67. doi:10.1080/ 01446190110044825. [51] Zouein P, Harmanani H, Hajar A. Genetic algorithm for solving site layout problem with unequal-size and constrained facilities. J Comput Civ Eng 2002;16:143–51. doi:10.1061/(ASCE)0887-3801(2002)16:2(143). [52] Zouein P, Tommelein I. Dynamic layout planning using a hybrid incremental solution method. J Constr Eng Manag 1999;125:400–8. doi:10.1061/(ASCE) 0733-9364(1999)125:6(400).