Local search approach for the competitive facility

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Local search approach for the competitive facility location problem in mobile networks Ivan Davydov1 and Yury Kochetov2 and Stephan Dempe3 1

Sobolev Institute of Mathematics 4, pr. ak. Koptyga, Novosibirsk, Russia [email protected] 2

Novosibirsk State University 2, ul. Pirogova, Novosibirsk, Russia [email protected] 3

TU Bergakademie Freiberg Akademiestr. 6, Mittelbau, Freiberg, Germany [email protected]

ABSTRACT We present a new leader-follower facility location game for 5G high-speed networks. Two mobile operators, the leader and the follower, compete to attract customers of high-speed internet connection. The leader acts first by opening some base stations, anticipating that the follower will react by creating her own base stations and renting some leader’s stations. Each customer patronizes an operator with the highest speed of connection for him. Each operator aims to maximize own profit. We provide a formulation of this game as a bi-level nonlinear integer programming problem, where the leader and the follower maximize their own profits at the first and second levels, respectively and the customers move from one operator to another until the Nash equilibrium is reached. Local search heuristics for this problem are designed and tested on randomly generated and real data. Computational experiments show high efficiency of the approach. Keywords: Stackelberg game, bi-level programming, competitive location, cellular network. 2000 Mathematics Subject Classification: 68T20, 90B80, 90C11.

1

Introduction

The competitive facility location models have a long history starting from the pioneering studies of Hotteling (Hotteling, 1929) and von Stackelberg (Stackelberg, 1934). In duopoly models presented in (Stackelberg, 1934), decisions of two companies are made sequentially. The company that makes the initial decision is called as leader and another one is called as follower. We assume that the leader and the follower have complete information about aims and

resources of the companies and the customers behavior. Comprehensive review for current results in this field can be found in (Kress and Pesch, 2012), (Dempe, Kalashnikov, Perez-Valdes and Kalashnykova, 2015). The first competitive facility location model for the cognitive radio network is presented in (Iellamo, Alekseeva, Chen, Coupechoux and Kochetov, 2015). It is assumed that each customer patronizes the company with the best signal quality from its nearest base station. Each customer makes own decision independently from other customers, as in the most facility location models (Karakitsiou, 2015). In this work we propose a new model for customer behavior. We assume that customers are mobile. They experience and evaluate the quality of connection over the whole network. Thus, the choice of the company (mobile operator) is made upon the average quality of service (Davydov, Coupechoux and Iellamo, 2016). In other words, we assume that customers can move from one operator to another until the average quality of service will be the same and customers reach the Nash equilibrium. We present a new competitive facility location model with three levels of the decision making: – the leader makes own decision to maximize the total profit at the first level; – the follower maximizes own profit having complete information about the leader’s decision; – each customer patronizes an operator (the leader or the follower) and they reach Nash equilibrium in terms of the average quality of service. In order to guarantee the Nash equilibrium for customers we introduce a nonlinear equation for the market share of both operators. We show that this equation has exactly one root with appropriate properties. We include this equation into the follower problem and get the bi-level nonlinear mixed integer problem. Objective function and one of the constraints of the follower problem are nonlinear. For the leader problem we apply local search heuristics: Stochastic Tabu Search and Variable Neighborhood Search. For a given solution of the leader, we apply local search heuristic again to find the best reaction of the follower. In order to get rid of nonlinear constraint we solve the follower problem for each root of the quadratic equation separately and select one of them. We consider both cooperative and non-cooperative behavior of the operators and introduce corresponding auxiliary problems. Computational results for the randomly generated benchmarks and real data for districts of Paris illustrate the efficiency of the approach. Our main contribution in this work is to propose a tri-level competitive facility location model, provide a well posed tri-level mixed integer nonlinear formulation and suggest an effective local search approach to tackle this problem. The paper is organized as follows. Section 2 presents the detailed description of the model and customers behavior. Section 3 introduces the exact mathematical formulation of the problem. We provide two variants of the model for cooperative and noncooperative operators, respectively. For both cases we design local search methods in Sections 4 and 5. Computational results are discussed in Section 6. Finally, Section 7 concludes the paper and shows some lines for future research.

2

System Model

In the following subsections we will briefly describe the details of the mathematical model.

2.1

Network Model

We consider two competitive operators, the leader and the follower, which we denote by O1 and O2 , respectively. They compete to serve customers by installing and configuring new 5G networks. We assume that both operators already have 4G networks, thus each of them owns a number of base stations with operating equipment. By S we denote the set of all sites, where the base stations can be installed. This set is made of three subsets: S = Sf ∪ S1o ∪ S2o , where

Sio is the set of sites having a 4G base station installed by Oi , i ∈ {1, 2}, and Sf is a set of free

sites for potential new installations. In the first phase, O1 chooses a subset of S1n for the installation of its new 5G base stations subject to a budget constraint. We assume that O2 does not allow O1 to use sites occupied by her old equipment in order to anticipate the worst case. Thus S1n ⊂ Sf ∪ S1o . In the second

phase, O2 is deploying its 5G network by choosing a set S2n for its 5G base stations. She

has the choice of all the sites in S, i.e. S2n ⊂ S. If a 5G base stations of O2 is placed on a

site in S1n , she will have to pay the rent price r to O1 . We assume that due to the competitive reasons the follower does not allow the leader to use her sites for installation as leader already have a benefit of acting first. The leader on the other hand may take an advantage from the sharing appropriately setting the rent price. At the third phase, customers make their choice of the operator. Let κ be the installation cost of a new base station. Let βκ with 0 < β < 1 be the installation cost when the site is already occupied by a base station. We assume that O1 has a total budget of K for installation. On the contrary, we suppose that the followers initial expenses are not limited. This assumption is made to allow leader to anticipate the worst case. Let λ be the traditional operational cost per unit of time for a single operator base station. Let (1 + α)λ with 0 < α < 1 be the traditional operation cost for a shared base station.

2.2

Propagation Model

For a customer located at υ we define the signal power from base station b as Pb (υ). The Signal to Interference plus Noise Ratio (SINR) of the considered customer in υ with respect to base station b is then given by: Pb (υ) , i=b Pi (υ) + N

γb (υ) = 

(2.1)

where N is the thermal noise power in the band. This value defines the real quality of the signal obtained by the customer due to interference between different base stations. Too many base stations concentrated in a relatively small area make it hard to distinguish between signals thus lowering the quality of the connection. It is assumed that customer in υ is covered by b if γb (υ) ≥ γmin for some threshold γmin . Customer in υ is served by b if it is covered and Pb (υ) ≥ Pi (υ) for all i = b. Note that at every location, customers can be served by at most one base station from each operator. For a customer located in υ and served by station b, the physical data rate achievable by this customer is denoted cb (υ), which is an increasing nonlinear function of γb (υ) with cb (υ) = 0 if γb (υ) < γmin . In our numerical experiments we use the following value of the physical

data rate cb (υ) = 8 · 108 log2 (1 + γb (υ)) which corresponds to the values obtained during the measurements in a dense city area.

2.3

Traffic Model

We assume there is a constant traffic demand in the network that operators will potentially serve. In every location υ, there is a demand λ(υ)/μ(υ), where λ(υ) is the arrival rate and 1/μ(υ) is the average file size. Note that this demand is statistical and can be shared by O1 and O2 or not served at all. Let assume that υ is covered by O1 and a proportion p1 (υ) of the demand is served by base station b from O1 . A proportion 1 − p1 (υ) of the demand is served by O2 if (υ) is covered by O2 . We focus in this paper on a specific case for p1 : if location υ is not covered by O1 then p1 (υ) = 0. Otherwise, p1 does not depend on the location and depends only on the overall relative quality of service in the network O1 compared to O2 . The idea behind this assumption is that customers are mobile and they choose their operator not only with respect to the quality of service at a particular location but rather to the average experienced quality. Then, the load created by customers in υ on base station b is p1 (υ)1b (υ), where 1b (υ) = λ(υ) μ(υ)c1b (υ)

and c1b (υ) is an increasing function of γ1b (υ) which denote the physical data rate in

υ. The index 1 is here to recall that the SINR, so the physical data rate, and the load are computed in the network of O1 . This is important to specify, because in the rest of the paper station b is likely to be shared with O2 . We can now define the load of station b as: p1 ρ1b , where  ρ1b = A1b 1b (υ) and A1b is the serving area of b, i.e., the set of customer’s locations served by b in network O1 . Base station b is stable if p1 ρ1b < 1. We will consider only scenarios where

this condition is fulfilled.

2.4

Market sharing

Due to the assumption that customers are mobile and travel from one location to another during the day, we cannot apply the deterministic and probabilistic rules of customers behavior in our model (Karakitsiou, 2015). Thus, another approach is suggested. We assume that customers are the players of an evolutionary game. In this framework, the choice of a single customer does not influence the average throughput of an operator. An equilibrium is reached when average throughputs in both networks are the same. A similar rule was proposed in (Davydov et al., 2016). Let Aib be the area served by base station b ∈ Oi and Ai = ∪b∈Sin Aib be the over  all area served by operator Oi . Let A be the whole network area. Let Λi = b∈S n Aib pi λ(υ) i

be the global arrival rate in Oi , where pi is the probability for a customer to be with operator Oi .  Let Λ = A λ(υ) be the global demand over the entire network. A customer of Oi has a null throughput with probability

Λ−Λi Λ

and with probability ΛΛi an average throughput is defined as 1  pi λ(υ)(1 − pi ρib )cib (υ). (2.2) Λi n b∈Si Aib

As a result, the average throughput of each customer in Oi network is equal to the following 1  ti = pi λ(υ)(1 − pi ρib )cib (υ). (2.3) Λ n b∈Si Aib

Let us introduce the following notations 

I11 =

λ(υ)ρ1b c1b (υ),

I12 =

b∈S1n A1b



I21 =



λ(υ)c1b (υ),

(2.4)

λ(υ)c2b (υ).

(2.5)

b∈S1n A1b

λ(υ)ρ2b c2b (υ),

I22 =

b∈S2n A2b



b∈S2n A2b

We have Ii1 < Ii2 because ρib < 1. Now Λti = −p2i Ii1 + pi Ii2 , with p2 = 1 − p1 . We have t1 ≥ t2 iff f (p1 ) ≥ 0 where f (p1 ) = p21 (I21 − I11 ) + p1 (I12 − 2I21 + I22 ) + I21 − I22 .

(2.6)

The values of functions I11 and I12 can be calculated for a fixed solution of the leader, while the values of functions I21 and I22 are determined by the solution of the follower. Theorem 2.1. There exist at least one root p1 of f (p1 ) = 0 with 0 ≤ p1 ≤ 1 under the assumption that all the base stations are operating without overload. Proof. The p1 is derived from the equation, which equalize the values of throughputs in both networks. Let us note that, substituting p2 by 1 − p1 in the expression for the t2 we can observe that both are continuous functions for p1 ∈ (0, 1). Let us calculate the values of t1 and t2 at the   endpoints of this segment. We have t1 (0) = 0 while t1 (1) = Λ1 b∈S n A1b λ(υ)(1−ρ1b )c1b (υ) > 1   0. On the other hand, t2 (1) = 0 while t2 (0) = Λ1 b∈S n A2b λ(υ)(1 − ρ2b )c2b (υ) > 0. Thus, 2

there exists at least one intersection of continuous functions t1 (p1 ) and t2 (p1 ) in the segment p1 ∈ (0, 1). If we calculate the first derivative of t1 we get the following: t1 =

1  1  λ(υ)(1 − p1 ρ1b )c1b (υ) − p1 λ(υ)p1 ρ1b c1b (υ). Λ Λ n n b∈S1 A1b 

For t1 = 0 we have p1 =

2



n b∈S1

n b∈S1

b∈S1 A1b

 A1b



A1b

λ(υ)c1b (υ)

λ(υ)ρ1b c1b (υ)

as the unique bent point. Now we can repeat

the same procedure with t2 . Thus we have t2 =

 1  (2 λ(υ)(1 − p1 )ρ2b c2b (υ) − λ(υ)c2b (υ)). Λ n n b∈S2 A2b

b∈S2 A2b



Equality t2 = 0 gives us p1 = 1 −

2



n b∈S2

n b∈S2





A2b A2b

λ(υ)c2b (υ)

λ(υ)ρ2b c2b (υ)

as a unique bent point. Thus both

functions have only one possible bent in the segment p1 ∈ [0, 1] which leads to exactly one intersection point. This proves the theorem.

3

Problem Formulation

As mentioned in the introduction, we are interested in the problem of strategic base station placement where two operators enter the market at different times (a leader and a follower), deploy their stations on possible candidate sites so as to maximize their profits. We assume that the revenues of both operators are proportional to their market share, i.e. P1 = p1 C, and P2 = p2 C = (1 − p1 )C where C is the total capacity of the market. The objective function is

the revenue minus the operational costs. In this section, we model this problem as a tri-level optimization problem. Let us introduce two groups of decision variables. The first group is the leader’s variables:



if the leader deploys station on a site j ∈ Sf ∪ S1o

1

xj =

0, otherwise, 

xij =

1

if the location i is served from leader’s station j

0, otherwise.

The second group is the follower’s variables:  1 if the follower deploys station on a site j ∈ S yj = 0, otherwise,  yij =

1

if the location i is served from follower’s station j

0, otherwise.

Let us also recall the main notations form Section 2: -r and λ defines the sharing and maintenance prices, respectively; -Pib provides the signal power from base station b to customers located at point i; -γmin gives the minimum threshold value for SINR to cover the customers. Now the competitive location problem can be written as the following tri-level mixed integer nonlinear programming model: max (p∗1 C +

xj ,xij



rj xj yj∗ +

j∈Sf

subject to

 j∈S1o



rj yj∗ −

κxj +

j∈Sf

Pib xb ≥ γmin

 j∈Sf ∪S1o ,j=b

 j∈Sf ∪S1o



[λxj (1 − yj∗ ) + (1 + α)λxj yj∗ ])

βκxj ≤ K

(3.1)

(3.2)

j∈S1o

Pij xj + γmin N − Γ(1 − xib ) ∀b ∈ Sf ∪ S1o , i ∈ I

(3.3)

Pib xb ≥ Pij xj − Γ(1 − xib ) ∀b, j ∈ Sf ∪ S1o , j = b, i ∈ I

(3.4)

xij ≤ xj ∀i ∈ I, j ∈ Sf ∪ S1o

(3.5)

where y ∗ is the optimal solution of the follower problem max((1 − p∗1 )C −

yi ,yij

subject to Pib yb ≥ γmin

 j∈S1o



rj y j −

 j∈Sf

rj xj yj −



λyj (1 − xj ))

(3.6)

j∈Sf ∪S2o

Pij yj + γmin N − Γ(1 − yib ) ∀b ∈ S, i ∈ I

(3.7)

j∈S,j=b

Pib yb ≥ Pij yj − Γ(1 − yib ) ∀b, j ∈ S, i ∈ I

(3.8)

yij ≤ yj ∀i ∈ I, j ∈ S

(3.9)

and p∗1 is derived from the following set of constraints p21 (I21 (y) − I11 (x)) + p1 (I12 (x) − 2I21 (y) + I22 (y)) + I21 (y) − I22 (y) = 0 0 ≤ p1 ≤ 1.

(3.10)

(3.11)

Objective function (3.1) and constraints (3.2)-(3.5) corresponds to the leader’s problem and controls variables {xj }, {xij }. Objective function (3.6)and constraints (3.7)-(3.9) formalizes the follower’s problem and controls variables {yj }, {yij }. Constraints (3.10)-(3.11) describe the customers behavior. The objective functions (3.1) and (3.6) can be understood as the total profit obtained, respectively by the leader and the follower, computed as the difference between the expected revenue from customers served and the operational costs for the stations installed. The sharing payment gives additional profit to the leader, and reduces the gain of the follower. Constraint (3.2) limits the maximum number of stations that the leader can afford to install due to it’s budget. Constraints (3.3) are the SINR conditions for a location to be covered. When xib = 1, the expression boils down to the SINR condition with respect to the SINR threshold γmin . Whenever xib = 0 then the condition is always fulfilled because of the large value of constant Γ. Constraints (3.4) combined with (3.3) state that the location satisfying the minimal SINR constraint is served by a base station providing the most powerful signal. Similarly to (3.4) and (3.3), constraints (3.7) and (3.8) guarantee that location satisfying the minimal SINR constraint is served by a base station providing the most powerful signal in the follower network. Constraints (3.5) and (3.9) state that a service is possible only if a station is installed. On the third step of the decision process, customers behavior is determined according to the quadratic equation (3.10) together with essential constraints on the market share value (3.11). Although the distribution of the customers is derived from the solution of an optimization problem, they minimize the difference of the average quality of service they are experience in both networks, in this particular case an equality condition on the throughputs (3.10) might be incorporated into the follower’s problem. This will lead to a bi-level formulation with nonlinear constraint on the lower level of the problem.

3.1

Follower’s behavior

The problem (3.1)-(3.11) is to find a solution that maximizes the leader’s profit. Nevertheless, we should note that the uniqueness of the follower’s solution in this formulation is not guaranteed. Different optimal solutions to the follower problem may lead to different objective function values of the leader’s problem. Thus, we deal with an ill-posed problem (Beresnev and Melnikov, 2016). In order to turn the problem into a well posed one, following the common practice (Dempe, 2002), we consider two possible types of follower’s behavior: • Cooperative behavior (optimistic case) among the set of optimal solutions the follower always selects one, which provides the maximal profit to the leader.

• Non-cooperative behavior (pessimistic case) the follower always selects the solution, which gives the minimal profit to the leader. Let us consider leader’s solution x0 = (x0j , x0ij ) and let F ∗ (x0 ) be the optimal follower’s profit. In order to find a cooperative and a noncooperative follower’s solutions (both provide the same profit F ∗ for the follower) we have to solve the auxiliary problems. Cooperative solution is derived from the following model:    sj x0j yj + rj y j − [λx0j (1 − yj ) + (1 + α)λx0j yj ]) max(p1 C + y

j∈S1o

j∈Sf

(3.12)

j∈Sf ∪S1o

subject to (3.7)-(3.11) and y((1 − p1 )C −





rj y j −

j∈S1o

rj x0j yj −

j∈Sf

 j∈Sf ∪S2o

λyj (1 − x0j )) ≥ F ∗ (x0 ).

In the non-cooperative case, (3.12) should be replaced with    rj x0j yj + rj y j − [λx0j (1 − yj ) + (1 + α)λx0j yj ]) min(p1 C + y

j∈Sf

j∈S1o

(3.13)

(3.14)

j∈Sf ∪S1o

subject to the same set of constraints. The optimal solution of the leader, depending on the follower’s behavior, can be derived from the formulations presented below. In the cooperative case we have the following objective function max max (p∗1 C + ∗ ∗





(3.15)

In the noncooperative case the objective function transforms as follows    ∗ ∗ ∗ (p C + r x y + r y − [λxj (1 − yj∗ ) + (1 + α)λxj yj∗ ]). max min j j j 1 j j ∗ ∗

(3.16)

j∈Sf

j∈S1o

rj yj∗ −



[λxj (1 − yj∗ ) + (1 + α)λxj yj∗ ]),

xj ,xij yi ,yij

rj xj yj∗ +

j∈Sf ∪S1o

and the same set of constraints (3.2)-(3.11)

xj ,xij yi ,yij

4

j∈Sf

j∈S1o

j∈Sf ∪S1o

Variable neighborhood search

Although the constraints of the problem are linear, due to realistic model of customers behavior this is not the case for the goal function. Latter fact makes it hard to apply a broad variety of approaches, which works well with linear integer programming problems. In this study in order to tackle the problem we propose a double level metaheuristic based on the Variable Neighborhood Search framework (VNS), which performs well on similar problems (Davydov, Kochetov, Mladenovic and Urosevic, 2014; Khmelev and Kochetov, 2015). VNS method was proposed by Nenad Mladenovic and Pierre Hansen in (Mladenovic and Hansen, 1997). The method is based on the original local search scheme that lets one ”travel” from one local optimum to another looking for a global one, avoiding local optimum traps. The main feature that allows to avoid local optima traps is the exploitation of different neighborhoods which is realized in a shaking procedure. We use well-known Flip and Swap neighborhoods as basic to explore the search space. Flip neighborhood is the set of solutions obtained by flipping the assignment

of one variable. Swap neighborhood is the set of solutions obtained by interchanging the assignments of two variables. In order to decrease the time consumption we exploit the idea of randomized neighborhoods. This feature allows to avoid looping, significantly reduces the time per iteration, and usually improves search efficiency (Davydov et al., 2014; Melnikov, 2014). We denote by Swapq the part q, 0 ≤ q ≤ 1 of the Swap neighborhood chosen at random. Each element of Swap is included into Swapq with probability q, independently from each other. F lipq neighborhood is defined in the same way, but with the different value of parameter q. Thus, during the local search process only a part of the neighborhood is explored. We apply the stochastic tabu search (STS) from (Davydov et al., 2016) in order to tackle the lower level problem. Follower problem is the one to be solved a lot of times during the search process, as it provides an estimation on the quality of leader’s solution. On the one hand high quality solutions provide better estimation on the leaders goal function, but requires a lot of time to obtain. On the other hand, it is usually enough to make fast approximate evaluation which then would be refined at the incumbent update step if necessary. The method’s general scheme looks as follows. On each iteration a probabilistic procedure Shake(X, k) is applied to the leaders’ current solution X = (xj , xij ). This procedure replaces k randomly chosen facilities of the leader by other randomly chosen facilities. We apply a local improvement procedure to the resulting solution X  with respect to F lipq and Swapq probabilistic neighborhoods. The resulting local optimum X  is compared to the current solution X. If the new local optimum is better than the previous one, we replace the current solution with the new local optimum. Otherwise, we change the parameter k and go to the next iteration. The pseudocode of the heuristic is presented in Algorithm 1. The initial solution is chosen at random. The randomization parameters for the neighborhoods are set to be sufficiently small. The method stops after a given number of iterations or after a certain amount of computation time. Due to enormous amount or running time needed to obtain an optimal solution of the follower problem with commercial solvers, in this study at Line 9 we apply tabu search again yet with another set of parameters giving it more time in order to refine the solution of the followers problem. Although the experimental study showed high stability and efficiency of the inner-loop heuristic, this approach does not guarantee us a feasible solution for the leader’s problem as the followers solution and thus the goal function value are obtained with an approximation.

4.1

Optimal solution for the follower problem

Nonlinearity, arising on the third level leads to significant difficulties in calculation of the exact value of the goal function for the feasible solution of the leader. Among the number of different commercial solvers none was able to provide any reasonable estimations for the followers solution in a general case. This may be explained by the fact that in the general model of the followers problem the first term of the goal function is derived from a quadratic equation, thus cannot be expressed explicitly in straightforward manner. This in turn leads to nonlinear constraints, which help to derive the correct root through Vieta’s theorem. The latter fact makes it almost impossible for solvers to provide appropriate bounds in order to estimate the deviation from an optimum. While it’s hardly possible to overcome this obstacle, during the computational

Algorithm 1 VNS Metaheuristic 1: Initialize; Read input data; Generate initial solution of the leader X0 = (x0j , x0ij ), put X ∗ := X0 , k := 1. 2:

repeat

3:

Shaking step: select random solution X  from the k-Swap neighborhood

4:

Generate the flip& swap probabilistic neighborhood Nq (X  )

5:

for each X  ∈ N (X  ) do

6:

Apply Stochastic TS-heuristic to solve the follower problem for X  ; Denote the best found solution as Y 

7:

Calculate the goal function value ω for the leader both for optimistic (3.15) and pessimistic (3.16) scenario

8: 9: 10:

if ω(X  , Y  ) > ω(X ∗ , Y ∗ ) then Refine the solution to find the exact/approx solution Y  of the follower problem if ω(X  , Y  ) > ω(X ∗ , Y ∗ ) then Update the incumbent (X ∗ , Y ∗ ) := (X  , Y  ); X := X  ;k := 1

11: 12:

else X := X 

13: 14: 15: 16: 17: 18: 19:

end if else k := min(k + 1, km ax) end if end for until Given time limit is exceeded

experiment we are able to solve the problem with both roots, and choose the correct one in the processing step.

5

Experimental studies

Presented VNS approach has been implemented in C# environment (as well as TabuSearch from (Davydov et al., 2016)) and tested on randomly generated and real data instances. All tests were carried out on a PC with IntelCore i5 CPU (3.2Ghz), 4 Gb RAM. On the refinement step for the follower problem we incorporate the solutions provided by the DICOPT solver (assembled in GAMS IDE package). Although DICOPT does not guarantee to provide an exact solution for a general MINLP problem, it is able to proof the optimality in some cases.

5.1

Instances

In order to conduct the numerical experiments we used generated instances from (Davydov et al., 2016). This benchmark consist of 10 sets of instances with different number of customer locations (20, 40, ..,200). All locations are chosen with the uniform distribution over the square area. The number of sites is set to 1/4 of number of customers locations. The demand data were generated at random and normalized in order to satisfy the constraints on overload.

Second set of instances concerns the real data. We used the information from the internetplatform (Cartoradio, n.d.) in order to derive the data about the location of the customers and positions of existing base stations in the city of Paris. The geometric centers of the blocks are assumed to be customer locations. Traffic demand in each location is set to be proportional to the population of the block and then normalized in order to fulfill the condition on the overload of the base station. Two out of four major operators were assumed to be the leader and the follower, while the positions of base stations of another two were used to represent free sites. 20 instances were created, one for each district of Paris. There are about 100 customer locations in each instance, while number of occupied and free sites varies from 6 and 4 to 20 and 20, respectively. The other parameters in both sets of instances were set as follows: total capacity of the market is set to C = 3500. The sharing price was assumed to be unique for all base stations and equal to rb = 250. The price for installation of new base station for the leader was set to κ = 100. The discount coefficient, which is applied if the leader upgrade his old base station was set to β = 0.5. Installation budget of the leader was set to the values, which allows to occupy no more than a half of free sites. During the computational experiments the initial solution of the leader was generated at random, subject to the budget constraints. The neighborhood randomization parameter was set to q = 0.05 for Swap neighborhood and q = 0.2 for Flip neighborhood in heuristic approaches. In order to accelerate the search process while exploring the neighborhood on the upper level of the problem we use the solution of the followers problem obtained on the previous iteration as a starting point.

5.2

Results for the Follower problem

In the first experimental study our aim was to provide a comparison between heuristic and exact approach for the follower problem. As it was mentioned in the description of the model, nonlinearity which comes from the customers behavior brings significant troubles for MINLP solvers. If the value of p1 is modeled through (3.10) together with Viet theorem then none of GAMS built-in solvers is able to provide non-trivial solution in a reasonable amount of time. The latter is not the case if we choose one root of the quadratic formula to represent the value of p1 . In this case the nonlinear terms would appear only in the expression of the goal function, while constraints would be linear. For this model the DICOPT solver appeared to be able to find the solution of high quality for all sets of instances. Although it is possible to run the solver twice, both with plus and minus roots, it was noted during the test process that the plus root always gave a correct result. Table 1 provides the results of this comparison on the instances with real data. We use the best known leaders solution as an input here. STS heuristic stop criterion is set to 5 seconds (line 6 of VNS-heuristic pseudocode). We observe that within 5 seconds STS is able to find the best known solution for all instances in every run. DICOPT is able to find the same solutions, but on the average it takes more than one minute to solve the problem. Although it is possible to provide the initial solution to the solver in general, in practice it is turned out to be useless. During the test process we tried to

Instance

Table 1: Follower market share: STS vs DICOPT Number STS DICOPT DICOPT of customers

(p1 )

(p1 )

t(sec)

1

108

0.416

0.416

57

2

86

0.432

0.432

216

3

96

0.486

0.486

112

4

113

0.458

0.458

131

5

97

0.421

0.421

264

6

95

0.456

0.456

273

7

103

0.468

0.468

59

8

97

0.418

0.418

168

9

89

0.435

0.435

180

10

88

0.447

0.447

57

use the best solution, found with STS, as an initial solution for the DICOPT, but the computation time was not reduced even for a second. Thus, we can see from the results, that STS is able to find the solutions of the same quality as the DICOPT solver, but clearly outperform it in terms of computational time. We observe similar results both on real data and on randomly generated instances. We also note that during all tests we did not encountered difficulties with multiple optimal solutions for the follower problem on any instance. Both optimistic and pessimistic scenarios coincide for these instances.

5.3

Results for the Leader problem

In Table 2 we compare the VNS approach, given in this paper with the Bi-level Tabu Search (BTS) scheme from (Davydov et al., 2016). While we are not able to provide any evaluations on global optimality of the solutions provided by both approaches, during this experiment we recognize that both approaches give us quite similar, not to say equal results. This in turn makes us believe that these solutions are not far from being optimal. During the first run we provided each algorithm 30 minutes of computational time for each instance. Time limit for STS was set to 3 sec. Under these conditions both algorithms were able to find the best known solution on 30 out of 30 instances. Under more strict limits (5 minutes of overall calculation, 0,5 sec of the inner problem) both approaches are still capable to reach the best known solution, although some deviations occur. Table 2 presents the results of this test on the real data instances. Columns 5 and 6 present the time in seconds when the best solution was found. As it comes from the result of the test, the problem appears to be easy enough for the local search approaches. Within a reasonable amount of time both approaches are able to find the best known solution. Such results might be explained by the nature of the goal function. Its structure provides a very rugged landscape which is known to be a good help for a local optimization approaches.

Instance

Table 2: Leader market share: VNS vs BTS Number VNS BTS VNS

BTS

of customers

(p1 )

(p1 )

t(sec)

t(sec)

1

108

0.415

0.416

82

139

2

86

0.429

0.426

42

56

3

96

0.486

0.486

91

68

4

113

0.458

0.458

276

118

5

97

0.411

0.414

131

198

6

95

0.456

0.434

187

251

7

103

0.468

0.451

261

198

8

97

0.418

0.418

48

75

9

89

0.429

0.425

217

288

10

88

0.447

0.442

249

187

6

Conclusions

We have considered new bi-level competitive base stations location problem with sharing. We have proposed a mathematical model for this problem under both optimistic and pessimistic scenario. Local search heuristics for this problem were designed and tested on randomly generated and real data. Computational experiments show high efficiency of the suggested approach. As a possible direction of further research it is interesting to include the sharing price to the set of leader variables. If this value is assumed to be equal for all base stations then one can find the optimal sharing price through a number of evaluations using the dichotomy search. In general case, if sharing prices may be different for each site, new algorithms, similar to (Panin, Pashchenko and Plyasunov, 2014), (Kochetov, Panin and Plyasunov, 2015) may be developed.

Acknowledgment The first author is supported by RFBR (project 16-31-00377). The second author is supported by the Ministry of Science and Education of the Russian Federation under the 5-100 Excellence Programme.

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