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International Journal of Geographical Information Science Publication details, including instructions for authors and subscription information: http://www.tandfonline.com/loi/tgis20
An efficient method for single-facility location and path-connecting problems in a cell map a
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c
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K. Y. Chang , C. M. Su , G. E. Jan & C. P. Chen a
Department of Merchant Marine, National Taiwan Ocean University, Keelung, Taiwan b
Department of Transportation Science, National Taiwan Ocean University, Keelung, Taiwan c
Department of Electrical Engineering, National Taipei University, Taipei, Taiwan Published online: 02 Sep 2013.
To cite this article: K. Y. Chang, C. M. Su, G. E. Jan & C. P. Chen (2013) An efficient method for single-facility location and path-connecting problems in a cell map, International Journal of Geographical Information Science, 27:10, 2060-2076, DOI: 10.1080/13658816.2013.820830 To link to this article: http://dx.doi.org/10.1080/13658816.2013.820830
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International Journal of Geographical Information Science, 2013 Vol. 27, No. 10, 2060–2076, http://dx.doi.org/10.1080/13658816.2013.820830
An efficient method for single-facility location and path-connecting problems in a cell map K.Y. Changa *, C.M. Sub , G.E. Janc and C.P. Chena a
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Department of Merchant Marine, National Taiwan Ocean University, Keelung, Taiwan; Department of Transportation Science, National Taiwan Ocean University, Keelung, Taiwan; c Department of Electrical Engineering, National Taipei University, Taipei, Taiwan (Received 26 October 2012; accepted 26 June 2013) This article presents an efficient method for simultaneously finding both the Weber cell and optimal connective paths in a grid. As numerous barriers of arbitrary shape and weighted regions are distributed in the cell map of this research, the problem scenario is similar to working out a real-life facility location selection and path-routing problems in a geographical map. In this study, the Weber problem of finding a single-facility location from an accumulation cost table is generated by a grid wave propagation method (higher-geometry maze router). After finding the Weber point (cell), optimal connective paths with minimum total weighted cost are backtracked between the Weber location cell and the demand cells. This new computation algorithm with linear time and space complexity can be integrated as a spatial analytical function within GIS. Keywords: facility location problem; maze router; Weber point; weighted region
1. Introduction The facility location problem can be divided into the continuous one and the discrete other, both of which are part of applications in supply chain management (Wolf 2011, 2012). The facility location problem, an economic problem of great practical importance, is that of finding the locations of one or more facilities, such as an industrial plant or a warehouse, to minimize the cost (or maximize the profit) of satisfying the demand for a commodity. Within practical applications, we are usually interested in placing a facility with respect to a set of existing locations, the so-called demand sites. For the special case where there is exactly one point to be located in the Euclidean plane, the problem is known as the classical Weber problem. Weber studied the problem of selecting the location of a warehouse such that the total travel distance between the warehouse and a set of spatially distributed customers is minimized. In a formal way, the classic Weber problem can be stated as follows. Given a set of n demand points {x1 , x2 , . . . , xn } in the Euclidean plane, further, let d(xi , X ) denote the Euclidean distance between two points xi and X . The point that minin d(xi , X ), i.e. the sum of distances between the demand points xi and X , is mizes f (X ) = i=1
termed the Weber point. A review of the Weber problem can be found, e.g., in Drezner and Hamacher (2002).
*Corresponding author. Email:
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Facility location problems in the presence of barriers have more practical relevance than general facility location problems. Military zones, mountains, lakes, and existing facilities are examples of barriers where neither the transit through the regions nor the placement of facilities within the regions is allowed (Canbolat and Wesolowsky 2010). The single-facility location problem with barriers (Aneja and Parlar 1994, Klamroth 2002), also called Forbidden Weber Problem (Katz and Cooper 1981), was extensively studied by many researchers, thereby applying methods of operations research (Mitchell 1993, Butt and Cavalier 1996, Dearing et al. 2005, Frießs et al. 2005, Canbolat and Wesolowsky 2010). An application of facility location in the continuous plane was presented by Murray and Tong (2007), who discussed coverage optimization of siren siting in Ohio. GIS applications of other location models have also been studied (Akella et al. 2010, Murray 2010, Delmelle et al. 2012). Recently, spatial analytical methods for continuous surface in different geographic phenomena have been discussed (Yao and Murray 2013). In some regions, travelling costs may increase significantly when passing through areas of heavy traffic, rough roads, etc. Such regions, which are termed weighted regions in this article, however, are different from barriers, because the transit through them is not forbidden. In the previous study of the grid plane, an optimal grid-positioning location was developed to approximate a single-facility location (Plastria 1991); however, barriers and weighted regions are not included in his article. In application of the GIS spatial analyst (Murray 2010), a simple router and accumulation method to find a site (single-facility location with barriers) and the deriving paths are rectilinear or Manhattan distance. To improve the simple router (Lee 1961), Jan et al. (2005) proposed the higher-geometry maze router. Their method has the same time and space complexity as Lee’s algorithm. The proposed Weber problem with barriers and weighted regions is significantly more complicated than the previous Weber problem works. Once the positive weights for sources (demand cells) and the locations of demand cells are given, the algorithm runs the higher-geometry maze router to obtain a cost-accumulated table for the Weber cell and find optimal paths between the Weber cell and sources in the cell map. To avoid multiple direction changes for little gain in travelling cost, path routing requires a turn penalty in the area of numerous barriers (Szlapczynski 2006). In this article, in terms of the facility location problem with barriers and weighted regions, the range of this research application is wider than the previous ones. The concept of cost accumulation is applied to determine an optimal facility location within the cell map. Furthermore, once the Weber cell has been found, optimal connective paths with minimum total cost (or distance) are determined between the Weber cell and the given sources (demand cells), based on the higher-geometry maze router. The connective paths are no longer rectilinear and can be integrated as a spatial analytical function. The advantage of this proposed method is that it can find the concurrent available potential facility location (which is another area with relative minimum cell value) with the connective paths. This approach is also very efficient, and the computation time is quite reasonable for modern computers. The remainder of this article is organized as follows. Section 2 gives the mathematical description of Weber problem and the Cell-Map-Weber-Problem in combination with an overview of necessary data structures and the higher-geometry maze router. Section 3 describes the efficient method for finding the optimal facility location of the Weber cell and its path planning with demands. Section 4 shows the examples of the results by computer simulation of the Weber problem with different shapes of barriers and various weighted regions. Conclusions are presented in Section 5.
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2. Problem formulation and review of the higher-geometry maze router
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This article presents an algorithm for finding the optimal facility location within a cell map that contains barriers of arbitrary form as well as weighted regions. While barriers are regions, where neither the transit through the regions nor the placement of facilities within the regions is allowed, weighted regions may or may not be passable, depending on the computed cost values of regional weights. Since the start cell of the wave propagation is called ‘source’, ‘source’ means the demand cell for the facility location problem in the rest of this article. 2.1. Formulation of the Weber problem The Weber and path-connecting problems are to find the optimal location of a single facility for sources in the plane and determine the route to and from the facility and each source at a minimum total transportation cost. In the remainder of the article, the Cell-Map-WeberProblem with higher-geometry maze router including some extensions concerning barriers and weighted regions will be discussed. In contrast to the original Weber problem, which is defined in the Euclidean plane, the Cell-Map-Weber-Problem is defined in a cell map with the individual cells being arranged in m rows and n columns. The Weber problem is to find the optimal location of a single facility to serve a given number of demand nodes which are located in the Euclidean plane in such a way that the total transportation costs are a minimum. The Cell-Map-Weber-Problem constitutes the analogue for a cell map in which the individual cells are arranged in m rows and n columns; in the present article, however, some extensions concerning barriers and weighted regions will be discussed. Before an algorithm that may be applied for solving this problem, even for the case that region with heterogeneous costs have to be handled, some basics requirements have to be discussed. The value f (X ) of a location X ∈ R2 for the chosen facility depends on the weight of demands wl and the location of sources. The Weber objective function seeks to minimize the sum of all transportation costs between the Weber cell and sources. The mathematical model of this problem is formulated as follows: min f (X ) =
Kl L
wl · d(xlk , xl(k+1) ).qlk
l=1 k=1
s.t. X ∈ 2 wl ≥ 0 x1l = x2l = ... = xLl = X ∗ qlk > 0 where l is set of sources (or paths), l = 1, 2, . . . , L; k is number for segment for each path, k = 1, 2, . . . , Kl ; wl is positive weight of demand for source; qlk is regional weight of path l for k-piecewise Euclidean line; d (xlk , xl(k+1) ) is piecewise Euclidean distance for k segment of path l; and x1k1 , x2k2 , ..., xlkl are the last points of path l representing each source cell. For the generalization of the Weber problem without barriers, the minimum total travL f (X ) = wl · d(xl , X ), where d(xl , X ∗ ) reflects the Euclidean distance elling cost is min ∗ X
l=1
between a source cell xl and the Weber cell X ∗ . For the same objective function, d(xl , X ∗ ) represents the shortest path (several segments of Euclidean line) between a source cell xl and the Weber cell X ∗ (Bischoff and Klamroth 2007) for the Weber problem with
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barriers. For the Weber problem with barriers and weighted regions, f (X ∗ ) minimizes the sum of all transportation costs, where the path obtained by the higher-geometry maze router, d(xlk , xl(k+1) ) ∗ qlk , represents the distance cost that is piecewise distance segments of Euclidean line multiples from a source cell to the Weber cell, with or without regional weight. In the following subsections, we present the data structures which are necessary to describe the higher-geometry maze router.
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2.2. Data structures The required data structures for the higher-geometry maze router (also called the grid wave propagation method) include a cell map, buckets, and linked lists. 2.2.1. Cell map A grid plane is divided into a cell map with m rows and n columns. Each cell requires at least three parameters for its characterization, namely RW (Regional Weight), AT (Arrival Time), and Vis (Visited). The first parameter, RW , is used to indicate whether a cell is located in a barrier, in a weighted region, or in the so-called free space. AT stores the travel time from the source cell of a path to the current cell. For all cells, the initial values of AT are set to infinity. The third parameter, Vis, is a Boolean variable specifying whether all neighbors of a specific cell have already been visited or not. For all cells, the initial values of Vis are set to false. Figure 1 shows a cell map with barriers being represented by black cells, weighted regions illustrated by gray cells, and the free space visualized by white cells. In addition, the initial values for the three parameters RW, AT, and Vis are specified. 2.2.2. Buckets and linked lists The buckets ß consist of a sequence of initially empty queues denoted by continuous integers in the range of 0 to AT max , where AT max is the maximum AT value and less than N. Each queue is a bucket LLn that stores a series of cell indices, where n is both the index and header value of bucket LLn , and n≤ATi,j < n +1 for the ATi,j values of the cells in bucket LLn . In addition, a temporary list TL is used in the algorithm to store the indices (i, j) of visited cells until all cells in the current bucket have updated their neighboring cells.
Figure 1. A cell map.
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Figure 2. Illustration of the 4-geometry maze router. (a) A 4-geometry neighborhood. (b) Expansion result of 4-geometry.
2.3. A higher-geometry maze router A cell map stores individual cells which are arranged in m rows and n columns, and specifies, in addition, which cells are elements of barriers, of weighted regions, and of the free space, respectively. Each cell represents a pixel in the cell map. Thus, there should be no significant distortion between the original geometry map and the raster chart; the size and the shape of the cells is exactly the same as the pixels. The λ-geometry allows edges with angles equal to iπ /λ; this is for all i and λ = 2, 4, 8, and ∞, corresponding to rectilinear, 45◦ , 22.5◦ , and Euclidean geometries, respectively (Sherwani 1999, Jan et al. 2005, 2008). The methods proposed in this article are based on the 4-geometry maze router in order to reduce the amount of computation required. Unlike other graph models, this takes advantage of the nature√of raster or grid planes and √ limited weights in λ-geometry (1 only √ in 2-geometry; 1 and 2 only in 4-geometry; 1, 2, and 5 only in 8-geometry, etc). However, the proposed methods work in λ-geometry and the running time is O(λN) for the λ-geometry maze router, where N is the number of cells in the grid √ plane. In a 4-geometry neighborhood, each cell has eight neighbors at a distance of 1 or 2 units from the center cell. Thus, movement along diagonal directions is possible in a cell plane with barriers and weighted regions that has a source cell S and a destination cell D, as depicted in Figure 2a. After execution of the 4-geometry maze router, the AT values of all cells are obtained as shown in Figure 2b. The 4-geometry maze router is introduced as follows: Function 1: 4-geometry maze router (Si,jl , RWi,j ) Step 0: Initialization For each cell Ci,j (RWi,j , ATi,j , Visi,j ) in an m × n grid plane, the initial RWi,j value is 1 if the cell Ci,j is in the free space or ∞ if Ci,j is in the barrier. If the parameter RW of any cell has a finite value between 1 and ∞, the cell belongs to one of the weighted regions. ATi,j = ∞ and Visi,j = false for all cells, 0 ≤ i ≤ m–1, 0 ≤ j ≤ n–1. The initial value of index is 0. Step 1: Input the coordinates of a given source, Si,jl , where l is the source number. If RWi,j of the source cell is not equal to ∞, then update ATi,j = 0, else return the error message ‘The source cell is in the barrier’ and terminate. Step 2: Compute the number of required buckets. Step 2.1: Determine √ the required index number (buckets) of the linked list LLindex . index_no = ( 2× RW max )/10 + 2, where RW max is the maximum RW in the weighted regions. Step 3: Compute the time of arrival between the source cell and the remaining cells.
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Step 3.1: Move the source cell into the temporary list TL and update the source cell’s Visi,j to true. Step 3.2: Remove the source cell from the TL into the bucket LL0 . Step 3.3: For each cell in the LLindex , update the AT i,j of its neighboring cells. Step 3.3.1: Remove the index of the first cell Ci,j from the front end of the LLindex and update this cell’s Visi,j to true. Step 3.3.2: Update the ATi’,j’ value of the 4-geometry neighbors of cell Ci,j . Case 1: If |i − i|2 + |j − j|2 = 1, then New_ATi ,j = ATi,j + 1 × (RWi,j + RWi ,j )/2. Case 2: If |i − i|2 + |j − j|2 = 2, √ then New_ATi ,j = ATi,j + 2 × (RWi,j + RWi ,j )/2. If New_ ATi’,j’ < ATi’,j’ , then ATi’,j’ = New_ ATi’,j’ . Step 3.3.3: Iterations If LLindex is not empty, then go back to steps of Step 3.3. Step 3.4: Move the cells’ indices in the TL into their corresponding buckets. Step 3.4.1: For all the indices in the TL, move (i, j) from TL into LLAT modindex_no . i,j
Step 3.4.2: If the TL is empty, then update the index value index = (index + 1) mod index_no Step 3.5: Iterations If rest of buckets is not empty, then repeat steps of Step 3.3. [Function of 4-geometry maze router]
3. Methods for Weber location and optimal path planning The proposed Weber location and optimal path-planning algorithm is presented in this section. The Weber location method developed in this study makes it possible to find the Weber cell when many barriers with arbitrary shapes and weighted regions exist in the cell map. Once the Weber cell has been found, the optimal connective paths between the Weber cell and the source cells are determined.
3.1. Grid wave propagation method The expansion of higher-geometry maze router is similar to grid wave propagation as shown in Figure 3a–e. The Weber cell is generated from an accumulation table by a grid wave propagation method. The accumulation table contains the sum of ATi,j values and can be generated from each source cell to the remainder cells, based on the 4-geometry maze router in the cell map. It is assumed that the demand weights of given sources in the cell map are known. The wave propagation scheme for the Weber location problem with given sources is illustrated in Figure 3. The initial conditions of given sources (expressed by S 1 , S 2 , S 3, and S 4 ) and barriers are shown in Figure 3a. Wave propagation from each source to the remainder cells with weights of demand (WD) is illustrated in Figure 3b–e, respectively. Figure 3f indicates the cell X∗ , which is the Weber location, i.e. it has the minimum total cost from the accumulation table.
3.2. The Weber location and optimal path-planning algorithm This section discusses the algorithm and provides an analysis of its correctness and performance. Two functions termed 4-geometry maze router (function for Weber cell and optimal path planning between the Weber cell and given sources) and Weber_cell_searching are defined with respect to a structured programming approach. The purpose of the second function, Weber_cell_searching, is to search for the Weber location. The input parameter for this function is named Total_ATi,j . The other function is used for calculating optimal
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Figure 3. Illustration of wave propagation for Weber location problem with given sources. (a) Initialization. (b) Wave propagation from S 1 (WD = 1). (c)Wave propagation from S 2 (WD = 2). (d) Wave propagation from S 3 (WD = 3). (e) Wave propagation from S 4 (WD = 3). (f) The minimum total cost is X∗ for all S.
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paths between the Weber cell and given sources. There are two input parameters for this l and RFMvi,j . A detailed description of these functions follows: function, Si,j Function 2: Weber_cell_searching (Total_ATi,j ) Step 1: Given an initial value of Total_ATi,j , representing accumulated ATi,j -values after executing the wave propagation scheme in the accumulation table. Mv = Total_ATi,j where Mv is the minimum value. Step 2: Calculate Mv. For j = 0 to n–1 do For i = 0 to m–1 do If (New_Total_ATi,j < Mv) then Mv = New_Total_ATi,j End {If} End {For} End {For} Step 3: The updated Mv value is stored in FMv, where FMv is the final minimum value that denotes the final result of Mv. FMv = Mv Step 4: Insert the coordinate (i, j) of the FMv into RFMvi,j and show this in the cell map. RFMvi,j is the recorded final minimum value and denotes the coordinate of the FMv (also called the Weber cell) that has the minimum total cost with given sources. [The function of Weber_cell_searching] Function 3: The optimal path planning between the Weber cell and given sources (Si,jl , RFMvi,j ) Step 1: Insert the coordinates of given sources, Si,jl , into the linked list of the source, LLsource . Let l be the number of the source. Step 2: For each source, retrieve the individual table of all ATi,j values for source l. Step 2.1: Show the coordinates of the Weber cell from RFMvi,j in the cell map. Step 2.2: Remove the coordinates of the first source from the front end of LLsource . Step 3: Backtracking Step 3.1: Trace optimal paths from the Weber cell by selecting one of the 4-geometry neighbors with the smallest arrival time value and repeating this selection step by step. Step 3.2: Once retrieve the store all ATi,j values, store the coordinate of the smallest ATi,j value of 4-geometry neighbors into the linked list of the path, LLlpath . Step 3.3: Check whether ATi,j = 0. Case 1: If ATi,j = 0, then break and go to Step 4. Case 2: If ATi,j = 0, then repeat Step 3.2. Step 4: Clear the coordinates of the process source, Si,jl , from the LLsource and all the ATi,j values in the cell map. Step 5: If LLsource is not empty, then repeat Step 2. Step 6: Display paths. Step 6.1: Remove the coordinates of the searched path from the LLlpath . Step 6.1.1: Mark the coordinates of the path found with a line in the cell map. Step 6.1.2: Clear the process source and check whether LLlpath is empty. Case 1: If LLlpath is empty, then go to Step 6.2. Case 2: If LLlpath is not empty, then repeat Step 6.1. Step 6.2: Check whether l = 0. l = l –1 Case 1: If l = 0, then the corresponding optimal paths are shown in the cell map for given sources. Case 2: If l > 0, then repeat Step 6.1. [The function of optimal path planning between the Weber cell and given sources]
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Algorithm: The Weber location and optimal path planning: Step 0: Initialization For each cell Ci,j (RW i,j , ATi,j , Visi,j ) in a define m × n grid plane, the initial RWi,j value is set to 1 if the cell Ci,j is in the free space or ∞ if it is in the barrier. If the parameter RW of any cell has a finite value between 1 and ∞, the cell belongs to one of the weighted regions. ATi,j = ∞ and Visi,j = false for all cells, 0 ≤ i ≤ m–1, 0 ≤ j ≤ n–1. The initial values of index and l are 0 and 1, respectively. Step 1: Input facility locations and weights, SW l , of given sources, Si,jl . Step 2: Call the 4-geometry maze router (Si,jl , RWi,j ) function and store the ATi,j values between the source cell and other cells in the cell plane. Step 3: Update all AT values and store them in the accumulation table. Total_ATi,j = ATi,j ; dlsSW 1 Step 4: Set l = l +1, then call the 4-geometry maze router (Si,jl , RWi,j ) function and store the ATi’,j’ values. Step 5: Update the Total ATi,j values and store them in the accumulation table. Total_ATi,j = Total_ATi,j + ATi,j × SW 1 Step 6: Run the sources one after another with function 1, then repeat Step 4. Otherwise, go to Step 7. Step 7: Call the Weber_cell_searching (Total_ATi,j ) function. Step 8: Call the function of optimal path planning between the Weber cell and given sources (Si,jl , RFMvi,j ). [The Weber location and optimal path-planning algorithm]
The time and space complexity of the above algorithm is calculated using the following theorems: Theorem 1. The time complexity of the algorithm is O(lN). Proof: It is obvious that the initiation of the given sources is bounded by O(1) in Step 1. The time complexities of Steps 2 and 3 are bounded by O(N) (Jan et al. 2005) to the time complexity of Steps 4, 5, and 6 and are dominated by Step 2, which takes O(N) time. Each cell only expands one time to 8 cells in the function 1. Thus, the time complexity is O(lN), where N and l are the number of cells in the cell map and number of given sources, respectively, and l < N. The time complexity of Step 7 is bounded by O(N) since at most N cells need to be searched in each cell map. Once the 4-geometry maze router and the Weber_cell_searching have finished executing, the final step is finding the optimal paths from the Weber cell to given sources. The time complexity of this step is O(lN). Thus, the total time complexity of the proposed algorithm to obtain these optimal paths for the Weber location problem is O(lN). Theorem 2. The space complexity of the algorithm is O(lN). Proof: It is obvious that the required memory space of the 4-geometry maze router, O(N) (Jan et al. 2005), since eight different directions are applied in Step 2. The other steps are bounded by O(N) (Jan et al. 2005). The total required memory space is O(lN) for l sources. Thus, this algorithm has space complexity of O(lN), where l < N. The algorithm proposed in this article can also be explained by the steps of the procedure shown in Figure 4. It is assumed that three sources are set in the cell map: Source 1, Source 2, and Source 3. To begin with, the weight of Source 1 is given as 3. Expanding from Source 1, we obtain all AT values in the cell map by the wave propagation scheme. Next, the weight of Source 2 is given as 2. The same scheme is executed to obtain all AT values. The same process is repeated for Source 3 with a weight of 3. Furthermore, the Weber cell with the smallest AT value is found from a cost accumulation table, which is
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Figure 4. Steps of procedures for both Weber location and optimal path planning.
generated by accumulating AT values from all sources. Finally, the optimal paths between the Weber cell and given sources are found using the 4-geometry maze-routing algorithm. 4. Examples of Weber location problem with barriers and weighted regions To implement the proposed algorithm, some examples were tested using a computer simulation that imitates grid wave propagation for a solution of the Weber location problem. In these examples, a set of seven sources (Source 1, Source 2, . . . , Source 7) with demand are given in Table 1 and then the Weber cell and its optimal paths to sources with several existing polygonal barriers and weighted regions are searched.
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Table 1.
K.Y. Chang et al. Allocation of demands to the individual sources.
Source no.
1
2
3
4
5
6
7
Weight of demand
7
7
5
3
2
1
1
4.1. Generalization of Weber location problem Several examples of the Weber problem are discussed within the next sections. The problems are related to equal demands as well as unequal demands and cover location problems within a uniform cell map as well as in complicated environments with barriers and weighted regions 4.1.1. The Weber problem with equal demand For the Weber problem with equal demand, data structures are defined as cells of free space in the cell map, as in the classical Weber problem. This is illustrated in Figure 5 by a set of seven sources (expressed by Source 1, Source 2, . . . , Source 7) with the demand of all
Figure 5. Illustration of the classical Weber problem (all demands are set to 1). (a) The Weber cell. (b) The Weber cell and the shortest paths to sources.
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source cells are set to 1. The objective is to find a Weber cell as well as the optimal paths between it and the source cells. The results are shown in Figure 5a and b; the yellow cell represents the Weber cell found by the algorithm, and the red paths are the optimal paths obtained. 4.1.2. The Weber problem with unequal demand Now let us consider the Weber problem with unequal demands whereby the demands for the individual sources are listed in Table 1, except that sources now have weights. The results are shown in Figure 6. Figure 6a and b illustrates the Weber cell found and the shortest paths between this Weber cell and given sources. The yellow cell represents the Weber cell and the red paths are the shortest paths obtained. On comparison with Figure 5, the shift effect of the Weber cell can be seen; this is due to sources with weights. 4.2. The Weber location problem in a complicated environment To obtain experimental results for the Weber location problem in a complicated environment, we applied the proposed algorithm to different patterns, as shown in
Figure 6. Illustration of the classical Weber problem with unequal demand. (a) The Weber cell. (b) The Weber cell and the shortest paths to sources.
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Figure 7. Illustration of the Weber location problem with barriers. (a) Initial map. (b) The Weber cell. (c) The Weber cell and optimal paths to sources.
Figures 7–9. In the Figures, the lavender, misty gray, and white cells represent the barriers (RWi,j = ∞), weight regions (1 < RWi,j < ∞), and free spaces (RWi,j = 1), respectively. It is impossible to pass through barriers; however, weight regions may be passed through depending on the total cost value of the path. Free spaces can always be passed through. Figure 7 illustrates an example of the Weber location problem with barriers. Figure 7a denotes the initial cell with barriers (Barrier 1 ∼ Barrier 4) set up arbitrarily, with Barrier 4 concave. The Weber cell found (the yellow cell) and its optimal collision-free paths (no
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Figure 8. Illustration of the Weber location problem with weighted regions. (a) Initial map. (b) The Weber cell. (c) The Weber cell and optimal paths to sources.
obstacle collides with the path) to the given sources (the red paths) are shown in Figure 7b and c, respectively. Another example is the effect of weighted regions in the cell map when searching for a Weber cell and finding optimal paths. Figure 8 shows the solved Weber location problem with weighted regions. Figure 8a denotes only the weighted regions in the initial cell (the numbers are the weight values), where weighted regions are assigned arbitrary weights of 2, 3, and 4, respectively. Figure 8b and c shows the solution with the Weber cell expressed
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Figure 9. Illustrations of Weber location problem with several polygonal barriers and weighted regions. (a) Initial map. (b) The Weber cell. (c) The Weber cell and optimal paths to sources.
by the yellow cell and its optimal paths to the given sources expressed by red paths. The different cost values of the environmental regions result in different traveling paths. Again, weighted regions may or may not be passable depending on their regional weights. A more complicated example with several complex polygonal barriers and weighted regions in the cell map is presented in Figure 9. As shown in Figure 9a, the cell map combines barriers (such as those in Figure 7) and weighted regions (such as those in Figure 8). Cells will be defined as barriers based on weight values. In Figure 9b and c, a yellow
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cell represents the desired Weber cell and red paths denote optimal collision-free paths, respectively. Computer simulation results show that the proposed methods are able to find a Weber cell and optimal paths with linear time and space complexity. A continuous space can be subdivided into a map of cells, for example 500 × 500 cells. The running time for a modern computer is close to real-time when dealing with simple computations of 500 × 500 cells. Also the proposed method is concurrent availability of the other potential facility locations and the connective paths. Thus, we can conclude from these simulations that our approach can be used to obtain optimal paths in a cell map with a complicated environment including concave barriers and weighted regions. 5. Conclusions This article presented the single Weber location and optimal path-connecting algorithm and its application to Weber location problems with several complicated polygonal barriers and weighted regions (convex and concave). This proposed method searches for the Weber cell and its optimal paths to the given sources based on the grid (cell)-based approach and the higher-geometry maze router. To minimize the number of direction changes while travelling a path away from the area of numerous barriers, a path routing with turn penalty is necessary to lay out the Weber facility location (Szlapczynski 2006). The concepts of grid wave propagation and cost accumulation were applied to compute cell AT values and determine a Weber cell. As preprocessing to construct a suitable search graph is not required, the method presented is very efficient. Time and space complexities are both O(lN). The selection of other potential facility locations and the connective paths are also available with this method. In the future, we plan to extend this algorithm to the multi-Weber location problem and the corresponding optimal connective paths in the continuous space with barriers and weighted regions. Thus, the model is significantly closer to real-life multi-facility location problems.
Acknowledgements We would like to thank the National Science Council of Taiwan, ROC, for its financial support and anonymous reviewers who provided many helpful comments on the draft of this article.
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