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The rectilinear distance minisum problem with minimum distance constraints:
Lmarion
0966-8349(95)ooo&1
Science, Vol. 3, No. 3, pp. 203-215, 1995 Copyright @ 1996 Elsevier Scmce Ltd Printed in Gnat Britain. All rights reserved 0966.8349196 $15.00 + 0.00
THE RECTILINEAR DISTANCE MINISUM PROBLEM MINIMUM DISTANCE CONSTRAINTS:
WITH
J. BRIMBERG Department of Engineering Management, Royal Military College of Canada, Kingston, Ontario K7K 5LO and
G. 0. WESOLOWSKY Management Science and Information Systems, MGD School of Business, McMaster University, Hamilton, Ontario L8S 4M4, Canada Abstract-This paper describes a mathematical model for locating a single facility on a continuous plane, which considers transportation (or service) costs between the facility and a set of demand points as well as social costs arising from the undesirable characteristics of the facility. The transportation costs are given by a standard minisum objective function, while the social costs appear implicitly in the form of lower bound constraints on the distances between the facility and the demand points. The model is analyzed under the assumption that distances are measured by the rectilinear norm, and an efficient branch-and-bound algorithm is derived to solve this case. Keywords:
Facility location, minisum, branch-and-bound.
1. INTRODUCTION
Many facilities which provide a service are termed “undesirable” or “obnoxious” because they possess characteristics which have an adverse effect on property values, may cause lower quality of life through pollution, or may pose a serious danger to the individuals living nearby. Examples of obnoxious facilities include nuclear power plants, garbage dump sites, mega-airports, and chemical plants. Erkut and Neuman (1989) review the operations research literature on the location of undesirable facilities, and provide a classification scheme for this type of problem. One of the criteria in their classification scheme deals with the nature of the objective function. For the location of a single obnoxious facility, the most frequently objective is the maximization of the distance between the new facility and the nearest demand point, the consumer or population center. This is referred to as the maximin criterion. A less popular objective, the maxisum, involves the maximization of a weighted sum of distances between the facility and the demand points. The individual weights reflect the relative importance of the demand points. In both cases, the tendency would be to push the undesirable facility as far away as possible so that a finite region must be specified in which to locate it. For an extensive list of references dealing with maximin and maxisum criteria for locating single and multiple facilities, the reader is referred to Erkut and Neuman (1989). Our model uses the rectilinear norm, also referred to as rectangular, Manhattan or city-block distance, to calculate travel distances between pairs of points on the plane. The rectilinear norm is applicable in many settings, such as urban areas where travel often roughly follows a rectangular grid. 203
204
.I. BRIMBERG
and G. 0. WESOLOWSKY
Examples of the concept that “obnoxiousness” is a function of rectilinear distance include Drezner and Wesolowsky (1983), Mehrez et al. (1986), and Giannikos (1993). Essentially, rectilinear distances are approximations of grid distances and hence, taken literally, one would have to assume that obnoxiousness is proportional to distances in a grid and that the spread of this undesirability is in the form of expanding diamonds, that is, squares rotated by 45 degrees that represent iso-distance contours. The undesirability of having a high-crime district may, for example, be measured by speed of transportation links rather than distance. In this case grid travel and hence rectilinear distances could be appropriate. Actual pollution patterns or other “iso-obnoxiousness” patterns may vary. They may follow a circular pattern if there are no obstacles: for example, where there is direct radiation in an open space. This means that a model with Euclidean distances should be used. However, airborne pollution, for example, could follow very complex patterns due to air currents and obstacles, and then Euclidean distances would not be, even approximately, appropriate. A further complication is that it is often the perceived obnoxiousness, rather than physically measured obnoxiousness, that is relevant. The perception of the undesirability of closeness to a dumpsite for a resident, for example, may be highly subjective and not dependent directly on any objectivemeasurement. Perceived obnoxiousness is very difficult to measure in terms of distances only. The value of using rectilinear distances in an obnoxiousness model is primarily as an approximation which permits the solution of a complex problem. While this distance norm may not be exactly the right one in a particular application, in all likelihood neither are any other relatively simple norms. This norm is also often used as an approximation of more general distance functions in order to make location problems more tractable; see, e.g. Love et al. (1988). A key assumption in the above-mentioned maximization models is that the costs, real or perceived, pertaining to the undesirable aspects of the new facility far outweigh the service costs. For example, the costs associated with living near a garbage dump are often considered to be much larger than the costs of transporting the garbage from customers to the dump. However, there are other situations which are not as clear-cut, such as the location of electrical power utilities. Here the costs of transmission, including loss of power, are important issues which are totally unrepresented by maximin and maxisum criteria. In this paper, we present a model formulation for locating a single facility on a continuous plane which attempts to incorporate both types of costs, i.e. the service (or transportation) costs and those due to the obnoxious nature of the facility. The model has a standard minisum objective to minimize transportation costs. Meanwhile, the “social” costs are included implicitly in the form of constraints forcing the location of the facility to be outside a specified forbidden region around each demand point. This type of formulation has been considered before only in a discrete setting by Moon and Chaudhry (1984). Examples of location in the presence of forbidden regions were discussed in Buchanan (1988), Hamacher and Nickel (1992), and Buchanan and Wesolowsky (1993). Analytical results for our model are presented in Sections 3 and 4. These properties are employed in the following section to outline an efficient “branch-and-bound” type of algorithm for the rectilinear distance minisum problem with infeasible regions. The principle used is to decompose the problem and in fact, the solution space, into cells, in each of which the problem becomes equivalent to a linear programming problem. This follows an idea introduced by Drezner and Wesolowsky (1983) for the single facility rectilinear distance maximin problem, and since used in several subsequent papers: for example, Mehrez et al.
The rectilinear distance minisum problem
205
(1986) and Giannikos (1993). The special properties of this particular problem are used to develop an efficient method for pruning cells and also to make the solution in each cell very simple. A sample problem is given to illustrate in detail the workings of the algorithm.
2. THE MODEL
We consider the following location problem in the plane:
minimize W(X) = i
wid(x, 4
(1)
i=l
subject to
d(x, ai) 2 ri
for
i = 1,. . . , n.
In (1) n is the number of demand points or customers; ai = (ali, uZi) is the known fixed location of customer i; x = (x1, x2) is the unknown location of the new facility; d(x, ai) is a distance function which gives the travel distance between any two point locations x, UiE R2; wi is a specified nonnegative weight which converts the distance travelled from customer i to the new facility into a transportation cost; and ri > 0 is a specified lower bound on the distance separating customer i from the new facility, i = 1,. . . , n. For our application, distances will be measured by the rectilinear norm, so that
d(X,
Ui) =
1X1 - Uli ( + IX* - U2i 1,V X, UiE [w2.
It is well known that the objective function Nfx) is a convex function of x for any distance norm d(x;). Furthermore, for the rectilinear norm, the unconstrained problem is easily solved as two one-dimensional median problems due to the separability of Win the x1 and x2 coordinates; see Love et al. (1988). An early paper which uses the piecewise linearity of W(x) for rectilinear distances is Bindshedler and Moore (1961). Another well known and useful property is that an optimal solution x* can always be found at (qi, u,J for some i andj; graphically, this means that if horizontal and vertical lines are drawn through each demand point, an optimal solution will be found at one of the intersections. Also, the unconstrained problem can be formulated as a linear programming problem; see Love et al. (1988). The difficulty in solving the problem (1) arises from the fact that each constraint d(x, ai) 2 ri defines a nonconvex feasible region. Thus, problem (1) cannot be expressed as a linear program. Furthermore, any general descent algorithm is only guaranteed to reach a local optimum. In order to solve (l), we use a “trick” employed by Drezner and Wesolowsky (1983). The basic procedure is to divide the plane into at most (n + 1)2 rectangular cells by drawing horizontal and vertical lines through each demand point. As we shall see, problem (1) is thereby converted into a series of two-variable linear programming problems, one in each cell.
J. BRIMBERG
206
and G. 0. WESOLOWSKY
Fig. 1. An example problem.
Refer to Fig. 1 for an illustration of the problem. Here, we have five demand points jnZ=l5). The feasi bl e region for locating a new facility defined by constraint set d(x, ai) >, li, ,+.*, 57 is given by the entire plane less the areas enclosed by the diamond contours d(x, ai) = ri centered at each demand point. Note that the plane is divided into (5 + 1)2 = 36 rectangular cells, and that the feasible region in any single cell is a convex polygon. 3. PRELIMINARY
ANALYSIS
Problem (1) has the following form for rectilinear distances: minimize i
Wi( 1X 1 -alil
(2)
+ IxZ-aZ?il)
i=l
subject to 1Xl -
ali
1 + 1x2 - Ll2i 1b
ri, i = 19..
.Y
FL
Alternatively, by using the sign function, if u>v ifu
wX(xl - a,,)S(x,,
ali) +
tx2
-
a2i)S(x2,
a*i))
(3)
The rectilinear distance minisum problem
207
R=Qlk
RCQZk
=lk
=2k
cell
k
=4k
‘3k
R-41(
RCQ3k
Fig. 2. Labelling for cell k.
subject to (X1 - ali) (Xl, ali) + (X2 - Uzi) 6 (X2, a*i) B ri, i = 1,. . .) n. Now, consider any one of the rectangular cells formed by the horizontal and vertical lines through the set of demand points. Assume, without loss of generality, that this is a closed cell k as shown in Fig. 2. There are, therefore, four disjoint “rectangular corner quadrants (RCQ)” around each cell, labelled counterclockwise as RCQ,,, RCQ,,, RCQ,,, and RCQ,,. Let each RCQ include its defining lines. Therefore, it is evident that any demand point Ui will fall into exactly one of these quadrants. It then follows that inside each cell, the values of 6 in Problem (3) will remain constant and can be determined, thereby transforming (3) into a linear programming problem. Note that the borders of the rectangle must also be included in the constraint set in order to define the cell. As was shown in previous work on the maximin problem (Mehrez et al., 1986; Appa and Giannikos, 1994; Brimberg and Mehrez, 1994), the special properties of the problem being considered may make actual linear programming unnecessary. Our minisum problem with its particular constraints has especially “nice” properties that make the solution within a cell very easy. 4. PROPERTIES
OF THE PROBLEM
As noted above, the infeasible regions are given by diamonds, that is, squares rotated 45 degrees, centered at the demand points ui, i = 1,. . , n. For convenience, the “sides” of constraint diamond i centered at a, are labelled counterclockwise: DS,,, DS,,, DS,,, and DS,,, as shown in Fig. 3. The following property is fairly self evident from visualization: Property
1. Consider rectangular cell k, where k E { 1,. . . , M} with M < (n + 1)’ denoting the total number of cells. Let a, fall in RCQ,, where i E { 1, . . . , n} and t E { 1,. . , 4). Then,
208
J. BRIMBERG and G. 0. WESOLOWSKY
Fig. 3. Labelling for a constraint diamond.
at most one side of constraint diamond i will intersect cell k, and furthermore this side can only be D$. The following property is a direct result of the linear programming optimization within cell k.
formulation
for
Property 2. The feasible region for locating a new facility in cell k is either the null set or a convex polygon. In the latter case, an optimal solution in cell k can be found at the minimum-valued vertex of the polygon.
In our case, constructing the feasible region in a cell is quite straightforward. Denote the corners of cell k by ctk, t = 1, , . . , 4, as shown in Fig. 2, and the subset of demand points in RCQtk as A,,. Note that some of the subsets Atk may be empty. We need to determine Atk = max {ri - d(~,~, ai)}, t = 1,. . . , 4, (4) %‘A,, and the demand point attkl corresponding to Atk. It follows that aIrk provides the furthest covering diamond edge or dominant constraint originating in RCQ,k. Thus, the remaining constraints d(X, ai) 2 Ti, VUi~ Atk, ai # UF,klr can be dropped. We refer to the leading edge given by the dominant constraint from RCQfk as ML,,. To determine if ML,, cuts the interior of cell k (see Fig. 4), the following necessary and sufficient condition is tested: 0 < A,, < d(c,+z,k, c,.J, t = 1, . . . , 4,
(5)
where index t + 4 = t, for all integer valued t. If this condition is satisfied, the corner point elk must be infeasible. All of the following conditions are necessary and sufficient for the feasible region to be non-empty: maxlo, A,J + max(& At+&
< &G+~,~, c,&, r = 1, 2
(6)
The rectilinear distance minisum problem
=2k
‘lk
=3k
‘IL
209
Fig. 4. Feasible region within a cell.
At/c+ AI+l,k~Pk-d(Cl+l,k,Ctk)rt=1,...,4
(7)
where Pk equals the length of the perimeter of cell k. Relation (6) ensures that the parallel edges MLtk, ML,,,, k do not overlap, while (7) verifies that adjacent edges ML,,, ML,+,,k do not project beyond each other in cell k. Furthermore, it is sufficient to verify that (6) is satisfied for t = 1, 2, and (7) is satisfied for te{l, . ..) 4) to guarantee that the feasible region in cell k will be non-empty. Conditions for testing the location of the ‘intersection point of two adjacent edges are readily derived. Suppose that two such edges ML,, and ML,, l,k cut cell k [condition (5)], and furthermore a feasible region exists within cell k [conditions (6), (7)]. Then, the intersection of ML,,, and ML,, I,k will be in cell k provided that
Otherwise these edges will cross a common side of cell k before they intersect with each other. It follows from relations (5HS) that the feasible region in any cell k can be constructed in a straightforward manner, and that this region, if it exists, is a convex polygon with at most eight vertices. It will require O(n) operations to determine the leading edges (ML,,) in cell k, after which the vertices of the polygon feasible region can be found independently of n. A further O(n) operations are needed to evaluate the objective function W(x) at each vertex. Since there are as many as (n + 1)’ cells to inspect, a total enumeration of vertices to find an optimal solution has complexity of O(n3). The following results provide criteria for selecting candidate cells in a branch-and-bound algorithm, and for rejecting cells and vertices which cannot be optimal, in order to reduce the total number of computations. Property 3. Consider any cell k. Let the optimal solution x* of the unconstrained minisum problem in (1) be located in RCQtk for some te (1, . . . , 4). Then, a lower bound on the objective function in cell k occurs at the corner point c,~. In other words,
210
.I. BRIMBERG
and G. 0. WESOLOWSKY
W(c,,) < min IV(x). xecellk
Proof: Due to the separability and convexity of W(x) in the x1 and x2 coordinates, it follows that w(c,k)
G w(c,-
1,
k)>w(c,k) G w(c,+ k). 1,
Furthermore, W(x) is a linear function within cell k. Therefore, the corner point cfk must provide a lower bound on wx) in cell k. 0 Applying the same reasoning gives the following extension. Property 4. Let cell k have a non-empty feasible region and x* E RCQ,,. Then an optimal solution xl, in cell k occurs at a vertex on MLtI, if ML,, cuts cell k and has a feasible interval inside the cell, or at the closest feasible vertex to ctk otherwise. Property 4 is an adaptation to rectilinear distances of the “visibility” property discussed by Hansen et al. (1985) for Euclidean distances. We now turn to a bounding mechanism which can be used to eliminate a large number of candidate cells at one time. Let the horizontal and vertical lines through a, be denoted by Hi and y respectively. The coordinates of x* are given by (a,,, as2) where r, s E {1, . . . , n>, since x* can always be found at an intersection point of the Hi and c. We obtain the following result. Property 5. Suppose Hi is above (resp. below) x*. Then, a lower bound of the objective function on the entire half-plane above (resp. below) Hi occurs at the point on Hi directly above (resp. below) x*, i.e. at y: = (a,,, ui2). Similary, when r/;.is to the right (resp left) of x*, a lower bound of W(x) on the entire half-plane to the right (resp. left) of v occurs at z* = (ail YasA. Proof: Consider an Hi above x* (a similar proof holds for the other cases). Choose any point x in the half-plane above H,, and let 2 denote the intersection point of the line segment from x to x* with Hi. Since Wis a convex function of x, W(g) < W(x). Using the separability property of Wit also follows that W(y*) < W(k). Hence, W(y*) < W(x), for all x in the half plane above Hi. 0
We are now ready to present a branch-and-bound algorithm for problem (1). The method can be viewed as laying a series of “tiles” outward from the center x*, until a further search is no longer required. Property 5 will be used to determine the order of the tile-laying. The lower bounds obtained by Properties 3 and 5 will allow us to fathom individual cells or entire half planes. Finally, Property 4 will reduce the search for an optimal solution in a cell from a possible eight vertices to at most two; P, and P, are such points in Fig. 4. 5. A SOLUTION ALGORITHM Step 1 (Initialization). Determine the unconstrained
optimum x*. If x* is a feasible solution of(l), stop (an optimal solution of the constrained problem is found). Set the value of the objective function for the current best candidate solution initially to infinity (IV’= co). Take the four adjacent cells with common corner point x* as the starting search rectangle (SR).
The rectilinear distance minisum problem
211
Fig. 5. Initial SR (search rectangle),
Step 2 (Testing candidate cells). For each new cell k added to SR calculate the lower bound W(c,,) (see Property 3). If W(c,,) > W’, discard the cell. Otherwise, determine whether a non-empty feasible region exists [conditions (6) and (7)]. If so, find the optimal solution x; in cell k using Property 4. From all new cells tested, retain the one with the lowest value W(x;). If W(x;) < W’, set W’= W(x;) and the current best candidate solution to XL. Step 3 (Branching). Determine the lower bounds W(y[), W(y$ sides of SR (see property 5). If
W(z,?J, W(z$ on the four
stop (the current best candidate is an optimal solution). Otherwise, expand SR by adding the row of cells along the side with the smallest lower bound. Return to Step 2 in order to examine new cells. Example. Consider the example given in Fig. 1, where wi = 1 for i = 1, . . . ,5. See Table 1. Table 1
1 2 3 4 5
2 4 5.5 7 8.25
3 4 3.75 6 2.25
2 1.5 2.25 1.5 I .25
212
J. BRIMBERG
and G. 0. WESOLOWSKY
Fig. 6. Second SR.
Fig. 7. Third SR.
The rectilinear distance minisum problem
213
Fig. 8. Fourth SR.
In Step 1 of the algorithm we get the unconstrained solution x* = (5.5,3.75), which is infeasible. We set IV’to co. The starting search rectangle SR is shown in Fig. 5. In Step 2, it is determined that these tiles do not have a feasible solution. The four bounds W(y,’ = (5.5, 3)) = 14.75, W(y,“:= (5.5, 4)) = 14.25, kV(z; = (4, 3.75)) = 15.5, W(zE = (7, 3.75)) = 15.5 are found in step 3 and are labelled in Fig. 5 next to the appropriate “sides” of the rectangle made up of the four tiles. The lowest bound is for the “upward” direction (14.25) and, hence, we extend the search rectangle that way and return to Step 2. We summarize the subsequent action of the algorithm with Figs 6-9. The new “upward” bound is 20.25; the tiles which have just been added contain feasible solutions, the best resulting in IV’= 17.25; see Fig. 6. The lowest bound (14.75) is now in the downward direction so that two tiles are added in this direction in Fig. 7. The result is a new bound of 17 for the downward direction and a new w’ of 16.25 for a new best candidate solution. There are now two lowest lower bounds of 15.5, and we arbitrarily choose to add two tiles in the “right” direction as in Fig. 8. The best feasible solution in the new tiles is again valued 16.25: in fact, at the same spot across the boundary. The lower bound on the right half-plane is 19.25. We now add tiles to the left as in Fig. 9, because the lower bound on that half-plane is 15.5. The new tiles do not contain a better feasible solution, and now min (W(yc = (5.5,2.25)), W(yi*,= (5.5, 6)), W(Z~ = (2, 3.75)) W(z,*,= (8.25, 3.75))) > w’(7, 3) = 16.25, stopping the algorithm with the best solution at (7, 3).
214
J. BRIMBERG and G. 0. WESOLOWSKY
Fig. 9. Final SR.
6. CONCLUSIONS
A minisum problem with infeasible regions was formulated; the problem is to locate a facility on a continuous plane in order to minimize a weighted sum of distances between the facility and a set of demand points subject to a minimum specified separation distance between the facility and each demand point. Several properties of the model are derived under the assumption of rectilinear distances. The properties are used to devise an efficient branch-and-bound algorithm. Acknowledgements-This research was supported, in part, by the Natural Sciences and Engineering Research Council of Canada and the Department of National Defence (Canada) Academic Research Program.
REFERENCES Appa, G. M. & Giannikos, I. (1994) Is linear programming necessary for single facility location with maximin of rectilinear distance? Journal of the Operational Research Society, 45, 97-107. Bindshedler, A. E. & Moore, J. M. (1961) Optimal location of new machines in existing plant layouts. 7he Journal ofIndustrial Engineering, XII, 41-48. Brimberg, 3. 8~ Mehrez, A. (1994) Multi-facility location using a maximin criterion and rectangular distances. Location Science, 2, 11-19. Buchanan, D. J. (1988) Interactive computer graphical approaches to some maximin and minimax location problems. Doctoral Thesis, McMaster University, Hamilton, Ontario, Canada. Buchanan, D. J. & Wesolowsky, G. 0. (1993) Locating a noxious facility with respect to several polygonal regions using asymmetric distances IIE Transactions, 25, 77-88. Drezner, Z. & Wesolowsky, G. 0. (1983) The location of an obnoxious facility with rectangular distances. Journal of Regional Science, 23, 241-248. Erkut, E. & Neuman, S. (1989)Analytical models for locating undesirable facilities. European Journal ofoperational Research, 40, 275-291.
Giannikos, 1.(1993) Locating multiple obnoxious facilities in the plane under the rectilinear metric. Paper presented to ISOLDE VI: International Symposium on Locational Decisions, June 16-24, Lesvos and Chios, Greece.
The rectilinear distance minisum problem
215
Hansen, P., Peeters, D., Richard, D. & Thisse, J.-F. (1985) The minisum and minimax location problems revisited. Operations Research,
33, 1251-1265.
Hamacher, H. W. & Nickel, S. (1992) Restricted planar location problems and applications. Preprint no. 227, Universitiit Kaiserlautern, Fachbereich Mathematik, Erwin-Schriidinger-Strasse, D-6750 Kaiserlautern. Love, R. F., Morris, J. G. & Wesolowsky, G. 0. (1988) Facilities location: Mode/s and methods. Amsterdam: North-Holland. Mehrez, A., Sinuany-Stern, Z. & Stulman, A. (1986) An enhancement of the Drezner-Wesolowsky algorithm for single facility location with maximin of rectilinear distance. Journal of the Operational Research Society, 37, 971-977.
Moon, I. D. & Chaudhry, S. S. (1984) An analysis of network location problems with distance constraints. Management
Science, 30, 290-307.
0966-8349(95)ooo&1
Science, Vol. 3, No. 3, pp. 203-215, 1995 Copyright @ 1996 Elsevier Scmce Ltd Printed in Gnat Britain. All rights reserved 0966.8349196 $15.00 + 0.00
THE RECTILINEAR DISTANCE MINISUM PROBLEM MINIMUM DISTANCE CONSTRAINTS:
WITH
J. BRIMBERG Department of Engineering Management, Royal Military College of Canada, Kingston, Ontario K7K 5LO and
G. 0. WESOLOWSKY Management Science and Information Systems, MGD School of Business, McMaster University, Hamilton, Ontario L8S 4M4, Canada Abstract-This paper describes a mathematical model for locating a single facility on a continuous plane, which considers transportation (or service) costs between the facility and a set of demand points as well as social costs arising from the undesirable characteristics of the facility. The transportation costs are given by a standard minisum objective function, while the social costs appear implicitly in the form of lower bound constraints on the distances between the facility and the demand points. The model is analyzed under the assumption that distances are measured by the rectilinear norm, and an efficient branch-and-bound algorithm is derived to solve this case. Keywords:
Facility location, minisum, branch-and-bound.
1. INTRODUCTION
Many facilities which provide a service are termed “undesirable” or “obnoxious” because they possess characteristics which have an adverse effect on property values, may cause lower quality of life through pollution, or may pose a serious danger to the individuals living nearby. Examples of obnoxious facilities include nuclear power plants, garbage dump sites, mega-airports, and chemical plants. Erkut and Neuman (1989) review the operations research literature on the location of undesirable facilities, and provide a classification scheme for this type of problem. One of the criteria in their classification scheme deals with the nature of the objective function. For the location of a single obnoxious facility, the most frequently objective is the maximization of the distance between the new facility and the nearest demand point, the consumer or population center. This is referred to as the maximin criterion. A less popular objective, the maxisum, involves the maximization of a weighted sum of distances between the facility and the demand points. The individual weights reflect the relative importance of the demand points. In both cases, the tendency would be to push the undesirable facility as far away as possible so that a finite region must be specified in which to locate it. For an extensive list of references dealing with maximin and maxisum criteria for locating single and multiple facilities, the reader is referred to Erkut and Neuman (1989). Our model uses the rectilinear norm, also referred to as rectangular, Manhattan or city-block distance, to calculate travel distances between pairs of points on the plane. The rectilinear norm is applicable in many settings, such as urban areas where travel often roughly follows a rectangular grid. 203
204
.I. BRIMBERG
and G. 0. WESOLOWSKY
Examples of the concept that “obnoxiousness” is a function of rectilinear distance include Drezner and Wesolowsky (1983), Mehrez et al. (1986), and Giannikos (1993). Essentially, rectilinear distances are approximations of grid distances and hence, taken literally, one would have to assume that obnoxiousness is proportional to distances in a grid and that the spread of this undesirability is in the form of expanding diamonds, that is, squares rotated by 45 degrees that represent iso-distance contours. The undesirability of having a high-crime district may, for example, be measured by speed of transportation links rather than distance. In this case grid travel and hence rectilinear distances could be appropriate. Actual pollution patterns or other “iso-obnoxiousness” patterns may vary. They may follow a circular pattern if there are no obstacles: for example, where there is direct radiation in an open space. This means that a model with Euclidean distances should be used. However, airborne pollution, for example, could follow very complex patterns due to air currents and obstacles, and then Euclidean distances would not be, even approximately, appropriate. A further complication is that it is often the perceived obnoxiousness, rather than physically measured obnoxiousness, that is relevant. The perception of the undesirability of closeness to a dumpsite for a resident, for example, may be highly subjective and not dependent directly on any objectivemeasurement. Perceived obnoxiousness is very difficult to measure in terms of distances only. The value of using rectilinear distances in an obnoxiousness model is primarily as an approximation which permits the solution of a complex problem. While this distance norm may not be exactly the right one in a particular application, in all likelihood neither are any other relatively simple norms. This norm is also often used as an approximation of more general distance functions in order to make location problems more tractable; see, e.g. Love et al. (1988). A key assumption in the above-mentioned maximization models is that the costs, real or perceived, pertaining to the undesirable aspects of the new facility far outweigh the service costs. For example, the costs associated with living near a garbage dump are often considered to be much larger than the costs of transporting the garbage from customers to the dump. However, there are other situations which are not as clear-cut, such as the location of electrical power utilities. Here the costs of transmission, including loss of power, are important issues which are totally unrepresented by maximin and maxisum criteria. In this paper, we present a model formulation for locating a single facility on a continuous plane which attempts to incorporate both types of costs, i.e. the service (or transportation) costs and those due to the obnoxious nature of the facility. The model has a standard minisum objective to minimize transportation costs. Meanwhile, the “social” costs are included implicitly in the form of constraints forcing the location of the facility to be outside a specified forbidden region around each demand point. This type of formulation has been considered before only in a discrete setting by Moon and Chaudhry (1984). Examples of location in the presence of forbidden regions were discussed in Buchanan (1988), Hamacher and Nickel (1992), and Buchanan and Wesolowsky (1993). Analytical results for our model are presented in Sections 3 and 4. These properties are employed in the following section to outline an efficient “branch-and-bound” type of algorithm for the rectilinear distance minisum problem with infeasible regions. The principle used is to decompose the problem and in fact, the solution space, into cells, in each of which the problem becomes equivalent to a linear programming problem. This follows an idea introduced by Drezner and Wesolowsky (1983) for the single facility rectilinear distance maximin problem, and since used in several subsequent papers: for example, Mehrez et al.
The rectilinear distance minisum problem
205
(1986) and Giannikos (1993). The special properties of this particular problem are used to develop an efficient method for pruning cells and also to make the solution in each cell very simple. A sample problem is given to illustrate in detail the workings of the algorithm.
2. THE MODEL
We consider the following location problem in the plane:
minimize W(X) = i
wid(x, 4
(1)
i=l
subject to
d(x, ai) 2 ri
for
i = 1,. . . , n.
In (1) n is the number of demand points or customers; ai = (ali, uZi) is the known fixed location of customer i; x = (x1, x2) is the unknown location of the new facility; d(x, ai) is a distance function which gives the travel distance between any two point locations x, UiE R2; wi is a specified nonnegative weight which converts the distance travelled from customer i to the new facility into a transportation cost; and ri > 0 is a specified lower bound on the distance separating customer i from the new facility, i = 1,. . . , n. For our application, distances will be measured by the rectilinear norm, so that
d(X,
Ui) =
1X1 - Uli ( + IX* - U2i 1,V X, UiE [w2.
It is well known that the objective function Nfx) is a convex function of x for any distance norm d(x;). Furthermore, for the rectilinear norm, the unconstrained problem is easily solved as two one-dimensional median problems due to the separability of Win the x1 and x2 coordinates; see Love et al. (1988). An early paper which uses the piecewise linearity of W(x) for rectilinear distances is Bindshedler and Moore (1961). Another well known and useful property is that an optimal solution x* can always be found at (qi, u,J for some i andj; graphically, this means that if horizontal and vertical lines are drawn through each demand point, an optimal solution will be found at one of the intersections. Also, the unconstrained problem can be formulated as a linear programming problem; see Love et al. (1988). The difficulty in solving the problem (1) arises from the fact that each constraint d(x, ai) 2 ri defines a nonconvex feasible region. Thus, problem (1) cannot be expressed as a linear program. Furthermore, any general descent algorithm is only guaranteed to reach a local optimum. In order to solve (l), we use a “trick” employed by Drezner and Wesolowsky (1983). The basic procedure is to divide the plane into at most (n + 1)2 rectangular cells by drawing horizontal and vertical lines through each demand point. As we shall see, problem (1) is thereby converted into a series of two-variable linear programming problems, one in each cell.
J. BRIMBERG
206
and G. 0. WESOLOWSKY
Fig. 1. An example problem.
Refer to Fig. 1 for an illustration of the problem. Here, we have five demand points jnZ=l5). The feasi bl e region for locating a new facility defined by constraint set d(x, ai) >, li, ,+.*, 57 is given by the entire plane less the areas enclosed by the diamond contours d(x, ai) = ri centered at each demand point. Note that the plane is divided into (5 + 1)2 = 36 rectangular cells, and that the feasible region in any single cell is a convex polygon. 3. PRELIMINARY
ANALYSIS
Problem (1) has the following form for rectilinear distances: minimize i
Wi( 1X 1 -alil
(2)
+ IxZ-aZ?il)
i=l
subject to 1Xl -
ali
1 + 1x2 - Ll2i 1b
ri, i = 19..
.Y
FL
Alternatively, by using the sign function, if u>v ifu
wX(xl - a,,)S(x,,
ali) +
tx2
-
a2i)S(x2,
a*i))
(3)
The rectilinear distance minisum problem
207
R=Qlk
RCQZk
=lk
=2k
cell
k
=4k
‘3k
R-41(
RCQ3k
Fig. 2. Labelling for cell k.
subject to (X1 - ali) (Xl, ali) + (X2 - Uzi) 6 (X2, a*i) B ri, i = 1,. . .) n. Now, consider any one of the rectangular cells formed by the horizontal and vertical lines through the set of demand points. Assume, without loss of generality, that this is a closed cell k as shown in Fig. 2. There are, therefore, four disjoint “rectangular corner quadrants (RCQ)” around each cell, labelled counterclockwise as RCQ,,, RCQ,,, RCQ,,, and RCQ,,. Let each RCQ include its defining lines. Therefore, it is evident that any demand point Ui will fall into exactly one of these quadrants. It then follows that inside each cell, the values of 6 in Problem (3) will remain constant and can be determined, thereby transforming (3) into a linear programming problem. Note that the borders of the rectangle must also be included in the constraint set in order to define the cell. As was shown in previous work on the maximin problem (Mehrez et al., 1986; Appa and Giannikos, 1994; Brimberg and Mehrez, 1994), the special properties of the problem being considered may make actual linear programming unnecessary. Our minisum problem with its particular constraints has especially “nice” properties that make the solution within a cell very easy. 4. PROPERTIES
OF THE PROBLEM
As noted above, the infeasible regions are given by diamonds, that is, squares rotated 45 degrees, centered at the demand points ui, i = 1,. . , n. For convenience, the “sides” of constraint diamond i centered at a, are labelled counterclockwise: DS,,, DS,,, DS,,, and DS,,, as shown in Fig. 3. The following property is fairly self evident from visualization: Property
1. Consider rectangular cell k, where k E { 1,. . . , M} with M < (n + 1)’ denoting the total number of cells. Let a, fall in RCQ,, where i E { 1, . . . , n} and t E { 1,. . , 4). Then,
208
J. BRIMBERG and G. 0. WESOLOWSKY
Fig. 3. Labelling for a constraint diamond.
at most one side of constraint diamond i will intersect cell k, and furthermore this side can only be D$. The following property is a direct result of the linear programming optimization within cell k.
formulation
for
Property 2. The feasible region for locating a new facility in cell k is either the null set or a convex polygon. In the latter case, an optimal solution in cell k can be found at the minimum-valued vertex of the polygon.
In our case, constructing the feasible region in a cell is quite straightforward. Denote the corners of cell k by ctk, t = 1, , . . , 4, as shown in Fig. 2, and the subset of demand points in RCQtk as A,,. Note that some of the subsets Atk may be empty. We need to determine Atk = max {ri - d(~,~, ai)}, t = 1,. . . , 4, (4) %‘A,, and the demand point attkl corresponding to Atk. It follows that aIrk provides the furthest covering diamond edge or dominant constraint originating in RCQ,k. Thus, the remaining constraints d(X, ai) 2 Ti, VUi~ Atk, ai # UF,klr can be dropped. We refer to the leading edge given by the dominant constraint from RCQfk as ML,,. To determine if ML,, cuts the interior of cell k (see Fig. 4), the following necessary and sufficient condition is tested: 0 < A,, < d(c,+z,k, c,.J, t = 1, . . . , 4,
(5)
where index t + 4 = t, for all integer valued t. If this condition is satisfied, the corner point elk must be infeasible. All of the following conditions are necessary and sufficient for the feasible region to be non-empty: maxlo, A,J + max(& At+&
< &G+~,~, c,&, r = 1, 2
(6)
The rectilinear distance minisum problem
=2k
‘lk
=3k
‘IL
209
Fig. 4. Feasible region within a cell.
At/c+ AI+l,k~Pk-d(Cl+l,k,Ctk)rt=1,...,4
(7)
where Pk equals the length of the perimeter of cell k. Relation (6) ensures that the parallel edges MLtk, ML,,,, k do not overlap, while (7) verifies that adjacent edges ML,,, ML,+,,k do not project beyond each other in cell k. Furthermore, it is sufficient to verify that (6) is satisfied for t = 1, 2, and (7) is satisfied for te{l, . ..) 4) to guarantee that the feasible region in cell k will be non-empty. Conditions for testing the location of the ‘intersection point of two adjacent edges are readily derived. Suppose that two such edges ML,, and ML,, l,k cut cell k [condition (5)], and furthermore a feasible region exists within cell k [conditions (6), (7)]. Then, the intersection of ML,,, and ML,, I,k will be in cell k provided that
Otherwise these edges will cross a common side of cell k before they intersect with each other. It follows from relations (5HS) that the feasible region in any cell k can be constructed in a straightforward manner, and that this region, if it exists, is a convex polygon with at most eight vertices. It will require O(n) operations to determine the leading edges (ML,,) in cell k, after which the vertices of the polygon feasible region can be found independently of n. A further O(n) operations are needed to evaluate the objective function W(x) at each vertex. Since there are as many as (n + 1)’ cells to inspect, a total enumeration of vertices to find an optimal solution has complexity of O(n3). The following results provide criteria for selecting candidate cells in a branch-and-bound algorithm, and for rejecting cells and vertices which cannot be optimal, in order to reduce the total number of computations. Property 3. Consider any cell k. Let the optimal solution x* of the unconstrained minisum problem in (1) be located in RCQtk for some te (1, . . . , 4). Then, a lower bound on the objective function in cell k occurs at the corner point c,~. In other words,
210
.I. BRIMBERG
and G. 0. WESOLOWSKY
W(c,,) < min IV(x). xecellk
Proof: Due to the separability and convexity of W(x) in the x1 and x2 coordinates, it follows that w(c,k)
G w(c,-
1,
k)>w(c,k) G w(c,+ k). 1,
Furthermore, W(x) is a linear function within cell k. Therefore, the corner point cfk must provide a lower bound on wx) in cell k. 0 Applying the same reasoning gives the following extension. Property 4. Let cell k have a non-empty feasible region and x* E RCQ,,. Then an optimal solution xl, in cell k occurs at a vertex on MLtI, if ML,, cuts cell k and has a feasible interval inside the cell, or at the closest feasible vertex to ctk otherwise. Property 4 is an adaptation to rectilinear distances of the “visibility” property discussed by Hansen et al. (1985) for Euclidean distances. We now turn to a bounding mechanism which can be used to eliminate a large number of candidate cells at one time. Let the horizontal and vertical lines through a, be denoted by Hi and y respectively. The coordinates of x* are given by (a,,, as2) where r, s E {1, . . . , n>, since x* can always be found at an intersection point of the Hi and c. We obtain the following result. Property 5. Suppose Hi is above (resp. below) x*. Then, a lower bound of the objective function on the entire half-plane above (resp. below) Hi occurs at the point on Hi directly above (resp. below) x*, i.e. at y: = (a,,, ui2). Similary, when r/;.is to the right (resp left) of x*, a lower bound of W(x) on the entire half-plane to the right (resp. left) of v occurs at z* = (ail YasA. Proof: Consider an Hi above x* (a similar proof holds for the other cases). Choose any point x in the half-plane above H,, and let 2 denote the intersection point of the line segment from x to x* with Hi. Since Wis a convex function of x, W(g) < W(x). Using the separability property of Wit also follows that W(y*) < W(k). Hence, W(y*) < W(x), for all x in the half plane above Hi. 0
We are now ready to present a branch-and-bound algorithm for problem (1). The method can be viewed as laying a series of “tiles” outward from the center x*, until a further search is no longer required. Property 5 will be used to determine the order of the tile-laying. The lower bounds obtained by Properties 3 and 5 will allow us to fathom individual cells or entire half planes. Finally, Property 4 will reduce the search for an optimal solution in a cell from a possible eight vertices to at most two; P, and P, are such points in Fig. 4. 5. A SOLUTION ALGORITHM Step 1 (Initialization). Determine the unconstrained
optimum x*. If x* is a feasible solution of(l), stop (an optimal solution of the constrained problem is found). Set the value of the objective function for the current best candidate solution initially to infinity (IV’= co). Take the four adjacent cells with common corner point x* as the starting search rectangle (SR).
The rectilinear distance minisum problem
211
Fig. 5. Initial SR (search rectangle),
Step 2 (Testing candidate cells). For each new cell k added to SR calculate the lower bound W(c,,) (see Property 3). If W(c,,) > W’, discard the cell. Otherwise, determine whether a non-empty feasible region exists [conditions (6) and (7)]. If so, find the optimal solution x; in cell k using Property 4. From all new cells tested, retain the one with the lowest value W(x;). If W(x;) < W’, set W’= W(x;) and the current best candidate solution to XL. Step 3 (Branching). Determine the lower bounds W(y[), W(y$ sides of SR (see property 5). If
W(z,?J, W(z$ on the four
stop (the current best candidate is an optimal solution). Otherwise, expand SR by adding the row of cells along the side with the smallest lower bound. Return to Step 2 in order to examine new cells. Example. Consider the example given in Fig. 1, where wi = 1 for i = 1, . . . ,5. See Table 1. Table 1
1 2 3 4 5
2 4 5.5 7 8.25
3 4 3.75 6 2.25
2 1.5 2.25 1.5 I .25
212
J. BRIMBERG
and G. 0. WESOLOWSKY
Fig. 6. Second SR.
Fig. 7. Third SR.
The rectilinear distance minisum problem
213
Fig. 8. Fourth SR.
In Step 1 of the algorithm we get the unconstrained solution x* = (5.5,3.75), which is infeasible. We set IV’to co. The starting search rectangle SR is shown in Fig. 5. In Step 2, it is determined that these tiles do not have a feasible solution. The four bounds W(y,’ = (5.5, 3)) = 14.75, W(y,“:= (5.5, 4)) = 14.25, kV(z; = (4, 3.75)) = 15.5, W(zE = (7, 3.75)) = 15.5 are found in step 3 and are labelled in Fig. 5 next to the appropriate “sides” of the rectangle made up of the four tiles. The lowest bound is for the “upward” direction (14.25) and, hence, we extend the search rectangle that way and return to Step 2. We summarize the subsequent action of the algorithm with Figs 6-9. The new “upward” bound is 20.25; the tiles which have just been added contain feasible solutions, the best resulting in IV’= 17.25; see Fig. 6. The lowest bound (14.75) is now in the downward direction so that two tiles are added in this direction in Fig. 7. The result is a new bound of 17 for the downward direction and a new w’ of 16.25 for a new best candidate solution. There are now two lowest lower bounds of 15.5, and we arbitrarily choose to add two tiles in the “right” direction as in Fig. 8. The best feasible solution in the new tiles is again valued 16.25: in fact, at the same spot across the boundary. The lower bound on the right half-plane is 19.25. We now add tiles to the left as in Fig. 9, because the lower bound on that half-plane is 15.5. The new tiles do not contain a better feasible solution, and now min (W(yc = (5.5,2.25)), W(yi*,= (5.5, 6)), W(Z~ = (2, 3.75)) W(z,*,= (8.25, 3.75))) > w’(7, 3) = 16.25, stopping the algorithm with the best solution at (7, 3).
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J. BRIMBERG and G. 0. WESOLOWSKY
Fig. 9. Final SR.
6. CONCLUSIONS
A minisum problem with infeasible regions was formulated; the problem is to locate a facility on a continuous plane in order to minimize a weighted sum of distances between the facility and a set of demand points subject to a minimum specified separation distance between the facility and each demand point. Several properties of the model are derived under the assumption of rectilinear distances. The properties are used to devise an efficient branch-and-bound algorithm. Acknowledgements-This research was supported, in part, by the Natural Sciences and Engineering Research Council of Canada and the Department of National Defence (Canada) Academic Research Program.
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Giannikos, 1.(1993) Locating multiple obnoxious facilities in the plane under the rectilinear metric. Paper presented to ISOLDE VI: International Symposium on Locational Decisions, June 16-24, Lesvos and Chios, Greece.
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