Competitive facility location under attrition

In this paper we address the possibility that in a competitive facility location model, one of the existing competing facilities will go out of business. We find the best location for a new facility protecting against such a possibility. Four commonly used decision rules (optimistic, pessimistic, minimax regret, and expected value) are analyzed and optimally solved within a given relative accuracy. The results of extensive computational experiments are reported. Special upper bounds, that may be a basis for other optimization problems, are designed. They are much tighter than existing lower bounds. The number if iterations is reduced by a factor close to 5000, and consequently run times were improved by about the same factor. The largest instance of 10 existing competing facilities and 20,000 demand points was solved in less than one second by each of the four decision criteria. The idea of possible scenarios can be investigated by other models in future research.


Introduction
A decision maker is faced with uncertainty about future conditions.There are several possible scenarios that are termed "states of the world".A list of options is available and the decision maker needs to select one of the options without knowing for sure which state of the world will take place.There are four widely used decision analysis rules (optimistic, pessimistic, minimax regret and expected value).For details see, for example, Winston and Goldberg (2004), Winston and Albright 39 Page 2 of 19 (2016), Goodwin and Wright (2014), Lawrence and Pasternack (2002), and the discussion in Sect. 2.
We apply these decision analysis rules to the location of a new competing facility among p existing competing facilities with future uncertainty.There are p + 1 states of the world (scenarios): (1) one scenario is that all existing facilities remain in the market, (2) each one of the existing facilities (p scenarios) will no longer exist once the new facility enters the market and its location has already been determined.The same analysis and solution approaches can be applied to possibilities that more than one facility will fail.The number of options (selection of the location for the new facility) is infinite.Any point in the area is an option for the location of the new facility.In brick and mortar competing facilities, such as shopping malls, it can be very costly to move the facility to a better location once it is established, and is thus not included in the list of options.Our goal is to find a good location for the new facility that will protect against such uncertainties.
Any model for estimating the market share captured at each location can be applied.We selected for demonstration purposes the gravity model (Huff 1964(Huff , 1966;;Reilly 1931) which is commonly used for estimating the market share captured (for example, Aboolian et al. 2007aAboolian et al. , b, 2009;;Bell et al. 1998;Fernández et al. 2007;Hodgson 1981;Sáiz et al. 2009).The specific gravity model is detailed in Sect.3.For a review of various competitive models see Drezner (2022).
There are a few papers that apply decision analysis rules (mainly the minimax regret rule) to location models.For example, Averbakh and Berman (2000) applied the minimax regret rule to the median location problem and Puerto et al. (2009) applied it to the single-facility ordered median location problem.Daskin et al. (1997) investigated a probabilistic minimax regret model.Drezner and Drezner (2002) and Drezner (2009) incorporated future market conditions, such as changes in future buying power distribution, into the gravity model.Future market conditions were analyzed as a set of possible scenarios.The minimax regret rule is applied.
This paper is organized as follows.In Sect. 2 the four decision analysis rules are reviewed and analyzed, and in Sect. 3 the gravity model used to estimate the captured market share by facilities is detailed.In Sect. 4 the solution procedures designed for finding the optimal solution within a given relative accuracy  > 0 for the four decision rules models are constructed.Extensive computational experiments are reported in Sect.5, and we summarize and discuss the contribution of the paper in Sect.6.

Decision analysis rules
There are three commonly used decision analysis rules that do not require probability estimates for each state of the world, and one (expected value) that does.The three that do not require probability estimates are: (1) hoping for the best possible outcome (optimistic), (2) protecting against the worst possible outcome (pessimistic), and (3) minimizing the maximum gap (regret) for each state of the world between the profit at the selected option and the maximum possible profit for that state of the world (minimax regret).

Optimistic rule:
The location that gives the largest market share among all states of the world is selected.

Pessimistic rule:
For each location we find the lowest market share among all states of the world.The location with the highest of these lowest market shares is selected.
Minimax Regret: For each state of the world we find the largest market share among all locations.The regret for a particular location and a particular state of the world is the difference between the largest market share for this state of the world and the market share captured by the selected location.The regret value for a specific location is the largest regret among all states of the world for this location.The location with the smallest regret value is selected.
Suppose that the probability of each state of the world can be reliably estimated and thus assumed given.The expected value rule is based on the given list of probabilities.

Expected Value:
The expected value for each location is calculated by the given probabilities for each state of the world.The location with the highest expected value is selected.

Properties of the decision rules for competitive location models
In the computational experiments we applied p + 1 states of the world: no facility failing, or one of the p facilities failing.If we also consider the possibility that two competing facilities may go out of business, we get 1 + p + p(p−1) 2 states of the world which will just increase the size of the problem.As long as the market share can be calculated for each state of the world, it can be solved the same way for reasonable size p.Including all possible facility removals, including removal of all p facilities, leads to 2 p states of the world which is practically solvable for p ≤ 10 .However, considering more than p + 1 states of the world may complicate the estimation of the probabilities for each state of the world.For example, when removing more than one facility there may be exogenous forces, such as a recession, that lead to a correlation between the probabilities of facilities going out of business.It is also possible that the new facility is part of a chain and some of the existing facilities belong to the same chain.The objective is to maximize the market share captured by the whole chain.Such a situation does not affect the analysis or solution approaches.In the rest of the paper we assume that no facility failing is always a possibility, and there are p additional states of the world (in the computational experiments p = p).
The market share at location X, when no facility goes out of business, is defined as M 0 (X) (Huff 1964(Huff , 1966;;Reilly 1931).The maximum market share is captured 39 Page 4 of 19 at location X = X 0 .When a combination of facilities 1 ≤ j ≤ p go out of business, the market share captured by a new facility at location X is defined as M j (X) and its best location is X = X j .All these market share values can be calculated by any of the available competitive location models.The location X j that maximizes M j (X) for j = 0, 1, … , p can be found within a relative accuracy  > 0 by Big Triangle Small Triangle (BTST, Drezner and Suzuki 2004), or Big Square Small Square (BSSS, Hansen et al. 1981).See also Sect.4.1 for details of the modified BSSS algorithm that was specially designed for the models analyzed and tested in this paper.
Proof Removing competing facilities reduces competition and the market share captured by the new facility at location X cannot decrease regardless of the competitive model used. ◻ Proof By Lemma 1 and the optimality of Proof By Lemma 2 M 0 (X 0 ) cannot be the maximum market share (it may tie it).Therefore, the maximum market share is the maximum among M j (X j ) for j = 1, … , p .Any location X cannot increase the market share for any M j (X j ) because M j (X j ) is the maximum market share for this state of the world.◻ Property 2 By the pessimistic rule the best location is X 0 with the objective value M 0 (X 0 ).
Proof At location X 0 , M 0 (X 0 ) ≤ M j (X 0 ) for j = 1, … , p by Lemma 1.Therefore, the minimum value of the objective function at X 0 for all states of the world is M 0 (X 0 ) .For any location X, by the maximality of X 0 , M 0 (X) ≤ M 0 (X 0 ) so the minimum value of all states of the world cannot exceed M 0 (X 0 ) for any location X. ◻ Property 3 By the minimax regret rule the best location X minimizes max 0≤j≤p M j (X j ) − M j (X) .
Proof The regret of selecting location X when facility j (or several facilities) go out of business (including the case j = 0 ) is M j (X j ) − M j (X) .The maximum regret for all the possible states of the world at location X is max 0≤j≤p M j (X j ) − M j (X) .By the minimax regret rule the objective is to minimize the maximum regret by selecting the best location X. ◻ In summary, by the optimistic rule we "hope" that the facility (or facilities) that go out of business is the combination that will increase the market share the most for the new facility.By the pessimistic rule we protect against the possibility that no competing facility will fail.These are extreme cases.The minimax regret rule seems to be the most sensible to apply when no probabilities for each state of the world can be reliably determined.
For the first three decision rules we do not need an estimate of the probabilities for each state of the world.Assume that a set of probabilities  0 ,  1 , … ,  p is given such that ∑ p j=0  j = 1 .The following property follows directly the definition of the expected value.
Property 4 By the expected value rule the best location is the X that maximizes ∑ p j=0  j M j (X).

The gravity model
There are many models for estimating the captured market share M(X) by a new facility at location X (for a review of such models see Drezner 2022).All the proposed formulations and solution approaches for finding the best location for a new competing facility can be implemented for any model as long as M(X) can be calculated for each location X.There are n demand points in the area, and p existing competing facilities.We opted to apply the commonly used gravity model (Huff 1964(Huff , 1966) ) which is based on the following definitions.
B i is the buying power at demand point i for i = 1, … , n, A j is the attractiveness level of facility j for j = 1, … , p, A is the attractiveness level of the new facility, is the distance between demand point i and existing facility j, is the distance between demand point i and location X, is the parameter of the distance decay function, The estimated market share captured by the new facility located at X when all j facilities operate is according to the traditional gravity model: The gravity model was originally proposed by Reilly (1931).He suggested that a community between two large metropolitan areas will divide their patronage proportionally to the size of the metropolitan area and inversely proportional to the square of the distance from them.This means a power decay of 1 d 2 which imitates gravitational force where the size of the metropolitan area is its mass, hence the name the gravity model.In the competitive facilities location literature, the gravity model has been the focus of many research papers (for example, Aboolian et al. 2007a;Bell et al. 1998;Drezner 1994a, b;Drezner et al. 1998;Huff 1964Huff , 1966;;Jain and Mahajan 1979, to name a few).The most widely used decay functions are the power decay f (d, ) = 1 d (originally proposed by Huff 1964Huff , 1966, as a generalization of , and the exponential decay f (d, ) = e − d (proposed by Hodgson 1981; Wilson  1976).Drezner (2006) found that the exponential decay function fits real data of shopping malls better than power decay.Some modifications to the basic gravity model were recently proposed.Drezner et al. (2020) suggested to replace the attractiveness multiplier A j by a facility depend- ent distance decay parameter j .Drezner et al. (2018) suggested that attractiveness levels are stochastic and not constants.Not all customers have the same perception of the attractiveness level.Fernández et al. (2019) assumed that a customer only patronizes those facilities for which they feel that the attractiveness is greater than or equal to a threshold value.Drezner et al. (2022) observed that ignoring "time spent in facilities" may bias market share estimates in a competitive environment.All distances are modified by adding the same extra distance.

The gravity model variant used in this paper
In the computational experiments we applied the distance decay function f (d, ) = 1 d 2 with the distance modified by the distance correction suggested in Drezner and Drezner (1997).The demand points are actually areas with many customers in each area, not all of them are at the same distance from the facilities.Drezner and Drezner (1997) suggested to replace d 2 with d 2 + 0.24S where S is the area of a "demand point".We experimented with demand points distributed in a 10 by 10 square.We assume that the area of demand point i is proportional to its buying power B i and they cover the whole square.The area of the square is 100.Therefore, the area of demand point i is Consequently, formula (1) for the calculation of the market share by a new facility located at X is: ( . (2) does not depend on X and is a constant for each i.We define it as i and the expression for M 0 (X) is simplified to: where the constants i need to be calculated only once for each demand point when the set of competing facilities is given.If facility j is removed from the set of competing facilities, A and B i remain the same, and i is changed to i − , and similarly if more than one facility fails.

Solution procedures
Location of one facility by any location model can be optimally solved within a relative accuracy  > 0 by the Big Square Small Square (BSSS, Hansen et al. 1981)  or Big Triangle Small Triangle (BTST, Drezner and Suzuki 2004) two dimensional global optimization algorithms.See also a general approach in Drezner (2007).We opted to apply the BSSS algorithm because in the computational experiments the demand points and facilities are located in a square.
The BSSS algorithm searches for the optimal solution in a square.It is a branch and bound algorithm.A list of squares is created and scanned.Branching is creating four "small" squares by partitioning a "big" square with perpendicular lines through its center.The algorithm terminates with a solution within a relative accuracy  > 0 once the upper bound for each square in the list is within the given relative accuracy of the best found solution.This means that the optimal value of the objective function cannot be higher than a relative accuracy  > 0 of the best found objec- tive.Let BF be the objective function of the best found solution during the search process, and UB max be the largest upper bound among all the squares in the list.The algorithm terminates when UB max ≤ BF(1 + ) .By definition, the optimal objective OPT, satisfies BF ≤ OPT ≤ UB max .Therefore, OPT ≤ BF(1 + ).

The solution algorithm
We modified the BSSS algorithm along the lines proposed in Drezner et al. (2023).As detailed in Drezner et al. (2023), in the original BSSS the value of the objective function is evaluated at the center of the square trying to improve the best found solution.In the modification, the value of the objective function is calculated at the vertices of the square rather than its center.When a square is partitioned into four small squares, the four small squares have 16 vertices but many of them overlap and the objective function needs to be evaluated only in five out of the 16 vertices.See Step 4 in the algorithm below.It is described for a maximization problem but can be easily modified for minimization.A relative accuracy  > 0 is given.For the 39 Page 8 of 19 relative accuracy to be meaningful, the objective function should be positive.For any square, the algorithm requires an expression for an upper bound for the value of the objective function at any point in that square.Derivation and calculation of the upper bound for each of the decision rules is detailed in Sect.4.2.

The modified BSSS algorithm
1. Initialization: the list of squares consists of one square that contains the demand points and facilities.The objective function is calculated at the four vertices of the square and the best one is defined as F. The point X * for which F was obtained is saved.The upper bound in the square, UB, is calculated and saved.2. The square in the list with the largest UB is selected.If UB < F(1 + ) , go to Step 8. 3. The selected square (the "big" one) is removed from the list and four "small" squares by perpendicular lines through the "big" square's center are created.4. The values of the objective function at five points (the center of the big square and the four centers of its sides), are evaluated and the values of F and X * are updated if a better solution is found.5.If the value of F was updated in Step 4, all the squares in the list are scanned and squares for which UB < F(1 + ) are removed from the list.6.For each of the four small squares: an upper bound UB in the square is calculated, and if UB ≥ F(1 + ) , the small square is added to the list and its UB saved.7. If the list of squares is not empty, go to Step 2. 8. Stop with X * as the solution point, and F as its objective.

Establishing an upper bound
The BSSS or BTST algorithms for maximization require an upper bound for the best value in a square or a triangle, referred to in this section as a polygon.Drezner and Drezner (2004) solved optimally the gravity model with a power decay function by BTST.They tested two upper bounds.One simple upper bound, which can be used for any decreasing function of the distance, is to replace each distance d i (X) from a demand point to X in the polygon by the shortest possible distance to that polygon.The objective function evaluated by the shortest distances is an upper bound.A more complicated upper bound designed in Drezner and Drezner (2004) was more effective.Drezner (2007) suggested a general upper bound for functions which are a sum of terms.Each term is a function of the distance between a demand point and location X, and each term can be represented as a difference between two convex functions of the distance.Note that functions may not be convex by the coordinates (x, y) but can be convex as functions of the distance.
Let a i be the shortest squared distance between demand point i and the polygon, and f i (a i ) is defined by (3).We propose two upper bounds.The first upper bound, which applies to any declining function of the distance, (similar to the one proposed in Drezner and Drezner 2004) is: because the captured market share declines by the distance and the maximum possible value for each demand point is at the point with the minimum possible distance.The second upper bound derived below takes longer to calculate, but is much more effective.The solution algorithm runs more than 1,000 times faster.See Sect.4.2.2.
The function M(X) by ( 3) is not convex.As a function of the distance z, each term is AB i A+ B i i + i z 2 which can be represented as a difference between two convex functions of z (was actually done in Drezner 2007).We propose a similar approach by defining the function by the square of the distance rather than the distance: This function is convex in z in the segment a i ≤ z ≤ b i where a i ≥ 0 is the minimum squared distance to the polygon and b i > a i is the maximum squared distance to it and is much more efficient than the bound established in Drezner (2007).
The formulas for the minimum squared distance to a square centered at (x 0 , y 0 ) and a side 2s is (Drezner et al. 2023): and the maximum squared distance is: The line connecting (a i , f i (a i )) and where Proof The Lemma follows the convexity of f i (z) .◻ In conclusion, M(X) by ( 3) satisfies: The upper bound UB 2 is defined by the minimum value of ∑ n i=1 i d 2 i (X) in the poly- gon.To complete the derivation for UB 2 we find (4) in the polygon.Recall that  i > 0 .Problem (9) in the plane not restricted to a poly- gon is the well known Weber problem (Church 2019;Weber 1909) with squared Euclidean distances.The solution point to this problem is the center of gravity (x c , y c ) of all weighted points: Consider the term . For x c + Δ the increase in the value of the term is The value of In conclusion, where Δx 2 + Δy 2 is the minimum squared distance between the center of gravity (x c , y c ) and the polygon.For a triangle a formula for calculating it is available in Drezner and Drezner (2004), and for a square it is calculated by expression (5).

Calculating UB 2
Calculate the five different sums in Eq. ( 12) by evaluating all the terms.
1. Set all sums to 0. 2. For each demand point i: (a) Calculate a i and b i by Eqs. ( 5) and ( 6).
Add the appropriate term to each sum.
3. Find the center of gravity by Eq. ( 10).All the required sums are available.4. Find Δx 2 + Δy 2 for the center of gravity by Eq. ( 5). 5. Calculate UB 2 by ( 12). (10) Note that there is no need to save any value for each demand point and only one loop over all the demand points is required.If facility j is removed, a i and b i do not change but the functions in Step 2b are calculated with facility j missing, and i recalculated.We define UB 1 and UB 2 when facility j is missing as UB 1 (j) and UB 2 (j ) ( j = 0 if no facility is missing).

Proof
The parameter a i is the shortest squared distance to the polygon.Therefore, d 2 i (X) ≥ a i for any point X in the polygon, and in particular to the solution point of (9).◻ It is very unlikely that UB 2 (j) = UB 1 (j) .It can happen if the shortest squared dis- tance to the polygon for all demand points is to the same point.For example, in case of squares, we can have a large square that contains all demand points, and a little square at the corner of the large square, and no demand point is in the two strips originating at the little square along two sides of the large square.All shortest distances are to the vertex of the little square which is in the interior of the large square.In this example, the shortest distance from the center of gravity to the little square is also to the same vertex.

Comparing the two upper bounds
We compared the two upper bounds for solving the test instances detailed in Sect. 5. We solved problem (2) without any competing facilities failing, for various values of the number of demand points.The performance of the BSSS algorithm (Sect.4.1.1)is summarized in Table 1 applying the two upper bounds.We report the maximum length of the list of squares in the BSSS algorithm, the total number of squares that were evaluated during the procedure (an upper bound calculated for each), and the run time in seconds.
The upper bound UB 2 (0) performed much better than UB 1 (0) .UB 2 (0) also sig- nificantly outperformed the bound proposed in Drezner (2007).The largest problem reported in Drezner (2007) had n = 10,000 points and required an average run time of 681 s compared with UB 2 (0) which required 0.11 s on a faster com- puter ( UB 1 (0) required 567 s).However, the ratio of over 6000 may be some- what exaggerated but it is definitely over 1000 times faster.The total run time of UB 2 (0) is about 5000 times faster and the number of squares is miniscule com- pared with the numbers for UB 1 (0) .The results for market shares and coordinates (not reported) agree to seven significant digits.We tried = 10 −6 for UB 1 (0) and the program needed to terminate because we assigned 2,000,000 spaces for the list of squares, and the algorithm reached this list size.On the other hand, we tried UB 2 (0) with = 10 −8 and total run time increased to just 0.60 s which is about 30% higher.Similar performance was observed for UB 1 (j) and UB 2 (j) for 39 Page 12 of 19 j > 0 .In the computational experiments we applied UB 2 (j) and do not report the number of squares or the run times because they are negligible.

Upper bound for minimax regret
In order to evaluate the minimax regret objective we first need to find X j and M j (X j ) for 0 ≤ j ≤ p , which is the maximum possible market share captured by the new facility if facility j is removed.The minimax regret objective is to find the location X = (x, y) for the new facility that This is a minimization problem and the BSSS subroutine we programmed deals with a maximization.Rather than re-code the program we converted (13) to a maximization problem.Since the function of the maximization problem has to be positive we define M * = max 0≤j≤p M j (X j ) , which is a known constant once all M j (X j ) are cal- culated.Problem ( 13) is equivalent to: and M * − M j (X j ) + M j (X) ≥ M j (X) > 0 as required.We find the solution to: The solution point is the location for the minimax regret problem, and the objective function of the minimax regret is M * − M.

Upper bound for expected value
The objective for expected value for a list of probabilities 0 , 1 , … , , p is: To obtain the upper bound we simply replace M j (X) by UB 2 (j).

Computational experiments
The BSSS procedure 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.In the experiments we used a relative accuracy of = 10 −5 .

The test instances
Test instances that can be easily replicated for future comparisons are generated by a pseudo random number generator applied in many recent papers (for example, Kalczynski et al. 2022).It is based on the pseudo random number generator proposed by Law and Kelton (1991).Integer starting seed r 1 and multiplier , which are odd numbers not divisible by 5, are selected.Note that r 1 can be an even number not end- ing with a zero.We used = 314, 227 .The sequence is generated by the following rule for k ≥ 1: For demand points with coordinates between 0 and 10, a coordinate is r k 100,000 .The x-coordinates were generated by r 1 = 105,673 , and the y-coordinates were generated by r 1 = 123,461 .Up to n = 20,000 demand points were generated.The buying pow- ers B k were generated between 1 and 5 by r 1 = 329,453 and B k = 1 + 4 r k 1,000,000 .We also generated 10 competing facilities by r 1 = 444,939 and r 1 = 526,987 for the x and y coordinates, respectively.The attractiveness levels of competing facilities were ⌋ × 1,000,000 39 Page 14 of 19 generated by r 1 = 251,637 , and 1,000,000 .The new facility has an attractiveness of A = 1.
The parameters of the competing facilities are reported in Table 2 and their locations depicted in Fig. 1.In Table 2 we report the locations of the competing facilities and their attractiveness levels.The market share captured presently by the competing facilities before the new facility is located are reported for n = 100, 1000, 10,000 , and 20,000 demand points.For example, facility #1 is located at (4.44939, 5.26987), and has an attractiveness level of 2.006548.It presently captures 9.237% of the available demand when 100 demand points exist in the area.Facility #8 captures the smallest market share compared to the other facilities for any n because it is located near the boundary (see also Fig. 1).

Summary of results
We experimented with nine instances for number of demand points between 100 and 20,000.For each n we solved 11 instances: no competing facility going out of business ( j = 0 ), and facility 1 ≤ j ≤ 10 going out of business.The upper bound UB 2 (j) is applied.
The optimal locations of the new facility in all cases are detailed in Table 3 (reporting also the market share captured by the new facility), and are depicted in Fig. 1 for n = 20,000 .The optimal location for the new facility following a removal of each existing facility are depicted in Fig. 1.For example, if facility #3 fails and leaves the market, the optimal location for the new facility is near the present location of facility #3.Some of the locations, including the optimistic, pessimistic, minimax regret, and expected value are concentrated in a small area which is zoomed in on the right for better details.We do not report run times because they are negligible, as reported in Table 1.The total run time for solving all 99 instances was only 3.62 s.The results do not change much as the value of n increases.This is because the results are governed by the locations of the competing facilities and the demand points are randomly spread in the square with random buying power for each.The pessimistic solution is, by Lemma 2, always at the location of the "no-fail" j = 0 solution.When facilities 1, 3, 4, 7, 10 go out of business, the "hole" created between surrounding competing facilities that remain in business attracts the new facility because more buying power becomes available than is available near facility 6 shown in the zoom-in area in Fig. 1.Facility 6 belongs to this group of facilities.The best location for the new facility is relatively close to the location of the failed facility.On the other hand, the hole generated by removal of facilities 2, 5, 8, 9 does not contain enough buying power and the most buying power is captured near the j = 0 (pessimistic) location.
It turns out that the removal of Facility 6 leaves a big "hole" which causes the new facility to locate nearby because there is a significant market share to be captured in that region once facility 6 goes out of business.Consequently, for all nine values of n, the optimistic location is obtained when facility j = 6 goes out of business.
The results for the minimax regret problem are reported in Table 4.For each number of demand points, we report the minimax regret optimal location X , and the regret value at this location.Note that there are only 9 instances because all values of j are incorporated in the objective function.We also report which of the 11 states of the world, j, had the maximum regret, and the captured market share M j ( X) at the minimax regret location.We compare it with M j (X j ) which is the maximum possible market share when facility j fails, which is also reported in Table 3.Note that the regret value is M j (X j ) − M j ( X) .Run times are very short because UB 2 (j) is applied.Since these values need to be calculated for each state of the world rather than only one state of the world, calculating each value of the objective function or the upper bound takes about 11 times longer than calculating them for a single state of the world reported in Tables 2 and 3.However, the total number of squares is 6529 compared with 8113 in Table 2, and the effort for managing the list of squares is lower.
In Table 5, the results for the expected value criterion are reported for 0 = 0.0, 0.1, … , 1.0 .For each value of 0 we assumed equal probabilities for each facility failing and applied j = 1− 0 p for j = 1, … , p .Run times are very short so we opted not to report them.All 99 instances were solved in a total time of 17.83 s which is an average of 0.18 s per instance with a maximum time of 0.66 s which means that each of the 99 instances was solved in no more than 0.66 s.The optimal locations for the facilities do not change by much as 0 increases.For 0 = 1 no facility fails and the location is at the j = 0 location which is the pessimistic loca- tion.The trajectory of the locations for n = 20,000 is depicted in Fig. 1 and since it is so short, it is shown only in the zoom-in figure on the right side.

Conclusions
We investigated the competitive facility location problem which is finding the best location of a new competing facility among a set of existing facilities that maximizes the market share captured by the new facility.There exists uncertainty about  the future of existing facilities.It is possible that one of the existing competing facilities will go out of business.We find the best location for a new facility which protects against such a possibility.Four commonly used decision rules (optimistic, pessimistic, minimax regret, and expected value) are analyzed and optimally solved within a given relative accuracy by a specially designed branch and bound algorithm based on the Big Square Small Square procedure (Hansen et al. 1981).Special upper bounds that are needed for the branch and bound procedure are designed.Upper bounds based on the same idea may assist in developing upper bounds for other optimization problems.Results of extensive computational experiments are reported and analyzed.The largest instance of 10 existing competing facilities and 20,000 demand points was solved in less than one second of computer time by each of the four decision criteria.As future research we propose to model similar possible scenarios for other models, and apply the proposed techniques (such as the upper bounds) proposed in this paper.

Table 1
Solving the test problems by BSSS † Maximum number of squares in the list ‡ Total number of generated squares

Table 2
The parameters of the competing facilities

Table 3
Results after facility failing

Table 4
Minimax Regret Results

Table 5
Expected value results