The Obnoxious Competitive Facility Location Model

In this paper we propose a new competitive location model that considers the possible negative impact generated by competing facilities (such as cannabis dispensaries) on surrounding communities. The facilities cannot be located too close to the communities. Therefore, when distances are Euclidean, the facilities must be located at a point outside a set of circles centered at the communities. After formulating the model, a specially designed efficient algorithm that solves the single facility location problem within a given relative accuracy of optimality is constructed. A total of 128 instances are solved in a relatively short time. The largest instance of 10 existing competing facilities and 20,000 demand points was solved in less than 15 min of computer time. This new model opens avenues for future research by designing similar new models. Also, the algorithm designed in this paper can be applied to solving other location problems with outside of a set of circles constraints.


Introduction
Location models seek the best location for one or more facilities that maximize or minimize an objective function (Francis et al. 1992;Love et al. 1988).The objective of competitive facility location models (Drezner 2022) is to maximize the buying power (market share) attracted to the new facilities.Obnoxious facilities location models (Church and Drezner 2022) wish to locate the facilities far from communities because the facilities generate a negative impact on surrounding communities.Motivated by the observation that some competitive facilities may also be obnoxious, i.e. generate some negative impact, we propose a new model for locating facilities that is based on the idea of combining the characteristics of both competitive and obnoxious facility location models.Church and Drezner (2022) provided a thorough review of obnoxious facilities location models.Early papers include Austin et al. (1970); Fielding (1970); Mumphrey et al. (1971); Mumphrey and Wolpert (1973); Austin (1974); Wolpert (1976).Among their examples of obnoxious facilities they list adult entertainment clubs and cannabis dispensaries.Such facilities, in addition to being obnoxious so that people do not wish to be close to them, also compete for customers.Other examples for this type of facilities include small stadiums or other venues that compete for customers but generate a lot of traffic and noise when events are held.Massage parlors, gun shops, and check cashing stores may draw some criminal elements.In these examples, one wishes that children will not be able to get to the facilities by walking.It is possible that establishments like schools should be at a distance from the obnoxious facilities but they generate no demand for the products offered by the facilities.Shopping malls generate a lot of traffic and commotion and it is not desirable to reside very close to them.
One of the early contributions to obnoxious location models is Orloff (1977).He analyzed the location of fire stations and suggested that fire stations should not be located too close to a community but should not be located farther than a certain distance from it.There is a ring around each demand point where a fire station should be located.Fire stations get their revenue from servicing customers, so they are "competing" for customers.They are more likely to be called by a customer if he is closer to them.Church and Drezner (2022) provide complete details of the Orloff (1977) paper.
We extend existing approaches for solving competitive location models to incorporate the obnoxious nature of some facilities.For example, the ring idea of Orloff (1977) can be applied as a modification of a recent competitive location model (Drezner et al. 2011;Drezner et al. 2012).Drezner et al. (2011) proposed that each competing facility has a "sphere of influence" determined by its attractiveness level.More attractive facilities have a larger sphere of influence.The buying power spent by a customer in the sphere of influence of several facilities is divided among the competing facilities.The buying power of a customer located outside all the spheres of influence is lost.Formulating the obnoxious competitive facility location based on this model just requires to disallow the location of the new facility inside an inner circle while attracting customers within an outer circle.We can require that the radius of the inner circle is proportional to the outer circle because the negative impact of the facility is increasing when its attractiveness increases.More attractive facilities usually generate more nuisance.
In the vast competitive location research literature (for a recent review see Drezner (2022), as far as we know no one has investigated the obnoxious competitive facilities location problem.It is implicitly assumed that customers are more likely to patronize a closer competing facility.We introduce the obnoxious effect that some competing facilities may have so they cannot be too close to the communities.

3
The Obnoxious Competitive Facility Location Model In this paper we illustrate the implementation of the obnoxious competitive idea by applying it on the gravity model (Huff 1964(Huff , 1966)), which is the most widely used competitive facilities location model.The same solution procedures can be applied to any competitive model.See the discussion following Eq.(3) in Section 3. In Section 2 we briefly describe the gravity model.In Section 3 the proposed model is formulated and illustrated by an example problem.A special algorithm, based on the BSSS (Hansen et al. 1981) algorithm, which is suitable for solving a problem within a given relative accuracy from optimality, is designed and detailed in Section 4. It can be applied also to other location models that restrict the location of the facility to outside of a set of circles.In Section 5 extensive computational experiments solving 128 instances yield good results.The paper is summarized in Section 6 and ideas for future research are suggested.

The Gravity Model
The gravity model was proposed by Reilly (1931) and refined by Huff (1964Huff ( , 1966)).According to the gravity model, the probability that a customer patronizes a facility is proportional to its attractiveness and declines according to a distance decay function.The basic gravity model is based on p competing facilities and n demand points that exist in an area.One or more new competing facilities are planned to be located in the area.The objective is to maximize the captured market share by the new facilities by finding the best location(s) for them.A distance decay function f (d, ) with a parameter depending on the retail category is defined.In the original gravity model (Reilly 1931) it is assumed that the distance decay parallels gravity decay and thus f (d) = 1 d 2 .Huff (1964, 1966) suggested a decay function f (d, ) = 1 d for some  > 0 .Other distance decay functions include: exponential decay f (d, ) = e − d (Wilson 1976;Hodgson 1981), f (d) = e −1.705d 0.409 (that was applied to grocery stores by Bell et al. 1998), and a logit function (Drezner et al. 1998).Drezner (2006) collected data by interviewing customers patronizing shopping malls in Orange County, California.She found that exponential decay f (d, ) = e − d fits the data better than power decay f (d, ) = 1 d and concluded that this is the case for shopping malls in general.The decline in attracting customers is slower for shopping malls than, for example, grocery stores.This was also confirmed in Drezner et al. (2020).

Parameters of the Gravity Model
For locating one new competing facility, the following are the parameters for the formulation: is the buying power at demand point is the attractiveness level of facility j for j = 1, … , p, d ij is the distance between demand point i and facility j, f (d, ) is the distance decay function, is the parameter of the distance decay function which depends on the retail category, X = (x, y) is the location of a new competing facility, A is the attractiveness of the new competing facility, D i (X) is the distance between demand point i and location X.
According to the gravity model, the proportion P i (X) of the buying power at demand point i captured by a new facility located at X is: where the distance decay function f (d, ) is the same for all competing facilities in the same retail category.The market share M(X) is defined as the percentage of buying power attracted by the new facility located at X out of the total available buying power.The estimated market share captured by a new facility at location X is therefore: The optimization problem is to find the best location X that maximizes the captured market share M(X).There are some generalizations of the basic model (1).For example, it is possible that some of the existing facilities belong to the same chain as the new facility.In this case the objective is to maximize the market captured by the whole chain following the construction of the new facility.The sum of the expressions for the existing facilities belonging to the same chain is added to the numerator of (1).Some models seek to minimize the cannibalization of the market share of existing facilities that belong to the same chain as a second objective (for example, Mason and Milne 1994;Moorthy and Png 1992;Drezner 2011;Plastria 2005).

Formulation of the Proposed Model
In computational experiments (for example in Drezner 1994), the best location for a new facility for a distance decay f (d, ) = 1 d , tends to be at a demand point because a facility located at a demand point captures all the buying power at that demand point.Since the competing facility is obnoxious, we require a minimum distance R between the facility and all demand points.It is implicitly assumed that existing facilities are also farther than R from the demand points but it is not required for the analysis and the solution method.We have no control over the locations of existing facilities.The optimization problem is: (1) The Obnoxious Competitive Facility Location Model Formulation (3) can be modified to accommodate any competitive location model (for example, Hotelling 1929;Leonardi and Tadei 1984;Berman and Krass 1998;Drezner et al. 2011).The expression for M(X) in (3) should be changed to the way market shares are estimated by the other models.The proposed solution algorithm in Section 4 can be applied to any approach for calculating M(X).Once M(X) can be calculated for any location X, the subroutine that calculates M(X) can be programmed to calculate it for other approaches.
We can require a different distance R for different demand points.The solution methods are exactly the same, but for simplicity we apply the same R in the computational experiments to avoid an excessive set of tables.If we model each demand point as a circular area, centered at the demand point, and require that the obnoxious facility must be at least a certain distance from every point in the circle, then the distance R should be the radius of each circle plus an extra required distance.It is also possible that there is no demand at some demand points for the products sold at the facility (i.e.B i = 0 ) at some demand points.For example, schools have no demand for cannabis, but such demand points should be included in the constraints.

An Example
To illustrate and motivate the optimization problem, and the solution approaches detailed in Section 4, we depict and discuss a specific example.In Fig. 1 the instance of n = 100 demand points and 10 existing facilities tested in Section 5 is depicted.The gray areas are circles of radius 8 centered at each demand point.If the minimum required distance between the new facility and each demand point is R = 8 , all gray areas are forbidden for the location of the new facility.Some of the gray areas outside the square of side 100 were partially truncated in the figure.We applied the distance decay function f (d, ) = 1 d 2 .The optimal location of the new facility is marked with a "1" when there are no distance constraints (the standard model), and by "2" for the optimal obnoxious location.The facilities' locations and market shares of these two cases are reported in Table 2 in Section 5.
The location "1" of the unconstrained problem is in the interior of five circles of radius 8 surrounding demand points.It is at distance 2.8 from the point closest to it, and at distance 7.6 from the demand point directly below it.Note that there are two demand points very close to one another and one of them is the closest to "1".It captures a market share of 11.975%. (3) The location "2" of the constrained problem is in a very small white area, hardly visible to the naked eye.It is at distance 8 from two demand points (it is located at the intersection point between two circles), and at a distance greater than 8 from all other demand points.It captures a market share of 11.115%.

Solution Algorithms
The unconstrained version of formulation (3) which is the standard non-obnoxious single competitive facility location problem was solved in Drezner and Drezner (2004) within a relative accuracy  > 0 from optimality by the global optimiza- tion algorithm Big Triangle Small Triangle (BTST, Drezner and Suzuki 2004).It can also be solved within a relative accuracy  > 0 by the Big Square Small Square (BSSS, Hansen et al. 1981) global optimization algorithm.Drezner et al. (2018) solved the Weber problem (Weber 1909;Church 2019;Wesolowsky 1993;Love et al. 1988;Drezner et al. 2002;Francis et al. 1992) outside a set of circles by the Big Arc Small Arc algorithm.However, the Weber problem is convex and if the optimal solution to the unconstrained problem is not feasible, the optimal solution to the constrained problem must be on the periphery of a circle.Since the objective function (2) is not convex, the obnoxious solution may not be located on a periphery of a circle.It is possible that the second best local optimum is located at a feasible point and all locations on feasible arcs of circles may be inferior.
The solution to the standard non-obnoxious problem must be in the convex hull of the demand points.By a theorem in Wendell and Hurter (1973), for any point outside the convex hull of the demand points, there is a point in the convex hull which The Obnoxious Competitive Facility Location Model is closer to all demand points.However, for the constrained problem (3), the optimal solution can be outside the convex hull.For example, if the distance R is large, it is possible that the whole convex hull is infeasible.For location problems where the feasible region is the convex hull of the demand points, BTST is more efficient than BSSS because in BTST the convex hull is triangulated by Delaunay triangulation (Aurenhammer et al. 2013;Okabe et al. 2000) and all triangles are feasible so that no feasibility verification is required.We propose to apply the BSSS algorithm (Hansen et al. 1981), modified to incorporate infeasible disks, for solving within a relative accuracy  > 0 the optimization problem (3).In order to implement the algorithm we need to establish lower and upper bounds in a square.
The proposed special BSSS algorithm can be applied to any two dimensional location problem with constraints limiting the location of the facility to outside a set of circles.For example see McGarvey and Cavalier (2003).A similar algorithm can be constructed for problems in three dimensions, with constraints limiting the location of the facility to be outside a set of spheres, by modifying the Big Cube Small Cube (BCSC, Schöbel and Scholz 2010) algorithm.

The BSSS Algorithm With Out of Circles Constraints
1.The list of squares consists of one square (the "big" square) enclosing the search area.The four vertices of the "big" square are feasible.2. Calculate the value of the objective function at the four vertices of the "big" square and set the lower bound LB to the best objective value.3. Find the maximum value of the objective function at the feasible points among the five points described in Section 4.1, and update LB if a better objective value is found.4. If the LB was updated in Step 3, scan all squares in the list and remove squares for which UB i < LB + . 5. Remove the "big" square from the list and create four "small" squares by perpendicular lines through the "big" square's center.6.The four "small" squares are evaluated in order.Select the first "small" square.
(a) If there is a circle for which all 4 vertices are in its interior, the whole square is inside that circle and thus there is no feasible point in that "small" square.Go to Step 6d.(b) Calculate an upper bound UB i in the "small" square disregarding feasibility.
If UB i < LB +  , go to Step 6d.(c) Add the "small" square to the list (d) If all 4 "small" squares were evaluated, go to Step 7. (e) Select the next "small" square and go to Step 6a.
7. If the list of squares is empty, stop with LB as the solution.8. Select the square in the list with the maximum UB i as the "big" square.9.If UB i < LB +  , stop with LB as the solution.Otherwise, go to Step 3. Notes 1.All the circles have the same radius R, and the 4 "small" squares have the same side s.If s > R √ 2 , the four vertices of each "small" square cannot all be in the interior of one circle.In such a case there is no need for Step 6a.

Another option in
Step 2 is to evaluate, in addition to the 4 vertices, all feasible intersection points between pairs of circles and apply the maximum value of the objective function among these intersection points as LB (see also Section 4.3).
In the computational experiments run times remain about the same when this option was tried.

A Lower Bound in a Square
An objective value at any feasible point in a square, can serve as a lower bound for that square.We do not always have a lower bound in a square.For example, if the whole square is infeasible, there is no lower bound in that square.The lower bound is used only in Step 3 of the algorithm to possibly improve the best feasible objective value LB found so far.As a lower bound in a square, if there is one, we find the best value of the objective function at up to four feasible vertices of the square.When a "big" square is partitioned into four "small" squares, the value of the objective function for all "small" squares is calculated at 16 vertices.Four of the 16 vertices are vertices of the "big" square that were already evaluated, and there is no need to re-evaluate them.Many vertices of the small squares coincide.Because of duplication, only five points need to be evaluated: the center of the "big" square and the centers of its four sides.Let the center of the "big" square be at (a, b) and its side be 2s.The five points are (a, b), (a, b ± s) , and (a ± s, b) .This approach reduces the run time for evaluating lower bounds by about 70%.

An Upper Bound in a Square
The distance decay function f (D, ) is, by definition, a decreasing function of D. Therefore, each P i (X) in Eq. ( 1) is a decreasing function of D i (X) .For each demand point i with B i > 0 , we find the shortest distance S i to any points in the square, whether feasible or not.Substituting S i for D i (X) for calculating P i (X) in Eq. ( 2) yields an upper bound in a square.Note that for functions where an upper bound requires the substitution of the largest distance in the objective function, the largest distance is the largest distance to the four vertices (whether feasible or not).
The minimum distance between a point X = (x, y) and a square is found as follows.Let the center of the square be at (a, b) and its side be 2s.There are 9 "zones" defined by the square and depicted in Fig. 2. Zone 1 is inside of the square where the shortest distance to all the points in the square is zero.Zones 2-5 touch a side of the square and the shortest distance is to a point on that side.Zones 6-9 touch a vertex of the square and the shortest distance is to that vertex.The shortest distance d min between a point (x, y) and all the points in the square, after evaluating the expressions for each of the 9 zones, can be expressed as: 1 3 The Obnoxious Competitive Facility Location Model

A Simple Heuristic
We found in the experiments that many of the optimal solutions (see Section 5.4.1) are at a feasible intersection point between two circles centered at demand points with radius R. Let the centers of the circles be at (x 1 , y 1 ) and (x 2 , y 2 ) and the radii of the two circles are both R. The distance between the centers of the two circles is d ≤ 2R .The two intersection points are: All pairs of demand points are evaluated and the distance between the centers d found.If d > 2R the pair is skipped.Otherwise, the two intersection points are found by (5).Feasibility of each intersection point is evaluated, and the value of the objective function at feasible intersection points calculated.The largest of these values is the result of the heuristic. (4) � Fig. 2 The nine zones In the example problem in Fig. 1 there are 309 feasible intersection points and one of them is indeed the optimal solution.The intersection point can be calculated exactly by (5).

Computational Experiments
The modified BSSS procedure that accommodates out of circles constraints, was coded in FORTRAN using double precision arithmetic and was compiled by an Intel 11.1 FORTRAN compiler using one thread with no parallel processing.The program was run on a desktop with the Intel i7-6700 3.4GHz CPU processor and 16GB RAM.Problems with up to n = 20, 000 points were solved in reasonable run time within a relative accuracy = 10 −5 .Fernández et al. (2007) solved a similar prob- lem by a branch and bound algorithm and experimented on instances of n = 50 or 100 demand points and relative accuracy = 10 −2 or 10 −4 .

The Test Problems
Test problems that can be easily replicated for future comparisons are generated by a pseudo-random number generator applied in many recent papers (examples are given in Church et al. 2022).It is based on the pseudo-random number generator proposed by Law et al. (1991).Odd integer starting seed r 1 and a multiplier (not divisible by 5) are selected.The sequence is generated by the following rule for k ≥ 1: where ⌊x⌋ is x rounded down.
To create a sequence of values in a range (a, b), the value of the k th variable is a + r k 1,000,000 (b − a) .For our test problems we used = 314, 227 .We generated x i in the range (0, 100), and r 1 = 105, 673 .For y i the range is (0, 100) and r 1 = 123, 461 .For B i the range is (1, 2) and r 1 = 329, 453 .For existing facilities the coordinates were generated in (0, 100) with x j generated by r 1 = 444, 939 , and y j by r 1 = 526, 987 .Their attractiveness in the range (1, 3) was generated by r 1 = 753, 127 .The attractiveness of the new facility is 2.
For the minimum required distance we used a parameter D, which is the minimum distance required for n = 100 , with values of 1, 2, 3, 5, 8, 10, and 15.The R value for a given For these test problems, the facilities and demand points are located in a square of side 100 but the location of the new facility can be outside this square.An extended square, with each side moved outward by the minimum required distance R in each direction, contains all the demand points, facilities, and the "gray" areas.The optimal location is inside the extended square.
The Obnoxious Competitive Facility Location Model

Properties of the Test Problems
In Table 1 some properties of the test problems for n = 100, … , 20, 000 are listed.The total buying power at all demand points is depicted in the second column.We then show the percentage of the feasible area (inside the square of 100 by 100) as a function of the required distance R from the demand points.For clarification see the white (feasible) areas in Fig. 1.In the figure R = D = 8 , and the total area of the white regions inside the square is 13.31% of the square's area.

The Models
In Model 1 we use the distance decay function f (d, ) = 1 d 2 .In Model 2 we use the distance decay function f (d, ) = e −0.1d .For each model we selected one value of to avoid an excessive number is very easy, and requires very little computer run time, to apply different values of .Solving optimally the largest instance takes less than 15 min of computer time as reported in Tables 2-5.

Discussion of the Results
The results are quite consistent for both Model 1 (Tables 2 and 3) and Model 2 (Tables 4 and 5).The optimal market share declines very little for small values of D, because the infeasible area is very small, especially for large values of n.In most instances the optimal location does not change much as D increases.For small values of n (for example n = 100 ) the changes are more significant for both models as D increases.In Fig. 3 the graph of the value of the market share as a function of 0 ≤ D ≤ 20 for the example problem ( n = 100 by Model 1) is depicted.Calculating all 201 points in the graph took about 3.5 min of computer time.There is a trade-off between the market share captured and the required distance R. Practitioners can consider what value of D to apply so that the market share loss does not exceed a desired percentage.For example, up to R = D = 4 insignificant market share is lost.If it is acceptable to reduce the market share to 11%, the new facility can be located at the R = D = 8.1 optimal location.
By Table 1, for n = 100 and D = 10 , the feasible area is only about 5% of the square's area and thus the best feasible location is restricted to a small area.For n = 20, 000 the feasible area is only 2.46% of the square's area.However, there are many "white" areas all over the square so even though the total area is smaller, there is a "tiny" area near any location including the optimal non-constrained one.For D = 15 , which is an extreme value, there is very little feasible space in the square and a significant decrease of market share is observed.The optimal locations for D = 15 and n = 100, 200 are outside the 100 by 100 square for Model 1 (Table 2), and for n = 100, 200, 500 they are outside for Model 2 (Table 4).
The trajectory of the optimal location is depicted for n = 100 and 0 ≤ D ≤ 20 in Fig. 4. The 201 points on the trajectory where calculated in about 3.5 min.For D = 0 the location is at the optimal location of the non-obnoxious solution, at the start of the trajectory near (70,18).It then moves to the right and gets to its rightmost point for The Obnoxious Competitive Facility Location Model D = 8.2 .It "jumps" to the left, stays near (55, 20), and jumps to near (72, 30) for D = 9 .At D = 11.7 it jumps to the leftmost point and stays near it until D = 12.2 where it "jumps" outside the square to near (70, -4), and moves down on the perpendicular bisector between the two demand points located at about (60, 5) and (80, 1).Between D = 12.3 and D = 20 the location is on the intersection points between the two circles centered at these two demand points and can be found by the intersection heuristic.Many papers solving other location problems with constraints, (for example, Drezner et al. 2018), suggest first to find the solution to the "simple" unconstrained model and if the optimal solution is feasible, there is no need to solve the "complicated" constrained problem.One would expect that the simple model without constraints will be solved more efficiently than the "complicated" obnoxious model.However, checking Tables 2-5, the "complicated" obnoxious model requires about the same run time as the "simple" model for small values of D, but run times decline significantly as the minimum required distance increases.Therefore, if the "simple" problem is solved first, then: (i) if the solution is feasible there is no gain, and even a loss for larger values of D; (ii) if the solution to the simple problem is not feasible, the run time is at least doubled for any value of D because the "complicated" problem needs to be solved on top.The reduction in run time especially for large values of D results from many small squares that are located inside one circle.Such squares are eliminated from the list in Step 6a of the modified BSSS algorithm.On the other hand, the upper bound in a square is the same for the unconstrained and constrained problems with any value of D. For smaller values of D the lower bounds are about the same so the gap between the lower and upper bounds are about the same.
This confirms the observation "Less is More" that was reported in many recent papers (for example, Brimberg et al. 2017;Mladenović et al. 2016Mladenović et al. , 2020)): simpler formulations may result in less efficient performance.The Less is More principle is summarized in Mladenović et al. (2022).

Testing the Intersection Heuristic
We also tested the intersection heuristic in Section 4.3 on all 112 instances with D > 0 .Total run time for all instances is less than 17 s! (or about an average of 0.15 s per instance).Note that for n = 20, 000 there are about 200 million pairs of circles to evaluate.In 39 instances the optimal solution was found, and in additional 33 The Obnoxious Competitive Facility Location Model instances it was less than 0.01% below the optimum.For n = 20, 000 and D = 1 , there are no intersection points between circles and thus the heuristic failed.The minimum distance between all pairs of points for n = 20, 000 is 0.162 (inside a square of a side 100).The minimum required distance for D = 1 is R = 10 √ 20,000 = 0.0707 .The distance between any pair of points is more than 2R.We have 20,000 small circles each centered at a demand point and not even one pair of  circles intersect.The intersection heuristic performs quite well for larger required distances, but if a close to optimal solution is sought, the algorithm proposed in this paper can be applied in a reasonable computer time.

Conclusions
We proposed an obnoxious competitive location model.The competing facilities also produce nuisance to nearby communities and therefore should be located farther than a minimum required distance from each community.This leads to an optimization model where the facility is located outside a set of circles centered at the communities.
A modified BSSS algorithm (based on the BSSS algorithm Hansen et al. 1981) that incorporates the requirement that the location must be outside a set of circles is constructed.The modified BSSS algorithm finds the optimal location for a facility within a given relative accuracy  > 0 of the optimal value of the objec- tive function.This algorithm can be applied to any location problem with the outside of circles constraints.The procedure is illustrated by solving the problem based on the gravity competitive location model (Huff 1964(Huff , 1966)).The same procedure can also be applied to any competitive location model.The objective function can have any shape as long as the probability of a customer patronizing a facility declines as the distance increases.This property is assumed in any competitive location model.The captured market share does not need to have a closed form expression and may even be discontinuous.The Obnoxious Competitive Facility Location Model The procedure was tested on a set of 128 instances.The solution algorithm is very efficient.The largest instance of 20,000 demand points and 10 existing facilities is solved in less than 15 min of computer time.
There are many possibilities for follow-up papers.Multiple facility location problems can be solved by specially designed heuristic or meta-heuristic algorithms.A cooperative obnoxious facilities model (Drezner et al. 2020) that considers the cumulative negative impact of the facilities is an appropriate model for locating multiple facilities.For two or three new competing competitive facilities.A variant of the Big Cube Small Cube (BCSC, Schöbel and Scholz 2010) can be constructed.The procedures designed in this paper can be implemented for solving other location models with an obnoxious component which require a location outside a set of circles.
Author Contributions Tammy Drezner came up with the idea, literature review and most of the writing.Zvi Drezner and Dawit Zerom did the analysis and the computational experiments.
Funding No funding was received.

Fig. 3
Fig. 3 Market share as a function of D for the example problem

Fig. 4
Fig.4The trajectory of the optimal location for the example problem

Table 1
Properties of the test problemsPercent of the feasible area in the Square

Table 5
Results for model 2 (larger problems) * R = 10D √ n a Maximum number of squares in the list b Run time in seconds