TY - JOUR
AU - Bai,, Quan
AB - Abstract This paper addresses the issue of the wireless mobile robot deployment for the ad hoc network establishment in disaster environments, which aims to maximize the important locations covered by the established ad hoc network so as to improve the performance of task allocation. In many disaster environments, the number of wireless mobile robots usually is much less than the number of important locations in the environment so that maximizing the important locations covered by the established ad hoc network is the primary objective of wireless mobile robot deployment approaches. To maximize the coverage of important locations, most of the current approaches were developed based on greedy algorithms. Due to the myopia of greedy algorithms, these approaches can only maximize the coverage of important locations of each wireless mobile robot rather than the whole network. To this end, two mathematical programming-based wireless mobile robot deployment approaches are proposed for ad hoc network establishment in disaster environments. The proposed approach can create suitable deployment locations for all wireless mobile robots in a disaster environment. The experimental results demonstrate that ad hoc networks established by the proposed approaches can cover more important locations in a disaster environment than those established by greedy algorithm-based approaches. 1. INTRODUCTION Nowadays, disasters throughout the world such as Indian Ocean tsunami, 2008 Sichuan earthquake, the hurricane Katrina etc. have become important social and political concerns [1]. In these disasters, due to the destruction of local infrastructures and the blocking of roads, Wireless mobile Robots (WRs) have played an important role in disaster rescues. They can enter the places, where people cannot go to search for important locations (ILs) (e.g. locations of victims, fires, etc.), and move to suitable locations to establish ad hoc networks. The established ad hoc networks can continuously adjust their locations according to the requirements of the rescue in disaster environments. In disaster environments, three major types of ad hoc networks are usually established for different purposes, which are (i) the ad hoc sensor networks [2], established to collect and transfer the information of ILs (e.g. the status of victims, the size of the burning areas, etc.), so as to support the decision making for the disaster rescue; (ii) the ad hoc communication networks [3], established to recover the communication among first responders in disaster environments, where local communication infrastructures are destructed; and (iii) the ad hoc beacon networks [4], established to locate and mark ILs in an environment, so as to improve the efficiency and effectiveness of the disaster rescue. Due to the wireless communication and mobility capabilities of WRs, the established ad hoc networks have the following characteristics: low infrastructure dependence [5], low expenses [6], quick deployment [7], quick adaptability [8] and scalability [9]. Although the purposes for establishing ad hoc networks in disaster environments are different, the primary objectives of these ad hoc networks are the same, i.e. to deploy WRs (e.g. bases, relays and beacons) at suitable locations, so as to maximize the number of ILs covered by the networks. Therefore, in this paper, we intend to propose approaches to find suitable deployment locations for WRs to establish the ad hoc network for the disaster rescue to achieve the above objectives. In disaster environments, there are three main challenging issues to be considered during the WR deployment, which are (1) Limited resources of WRs: Since WRs enter a disaster environment with only limited resources (e.g. battery, portable wireless communication devices, etc.) and their sensing and communication distance is limited, they can only cover a limited number of ILs in an environment; (2) Limited number of WRs: Due to blocked roads and distributed transport facilities, only a small number of WRs can enter the environment to establish the ad hoc network, which might be much less than the number of ILs in an environment; and (3) Ubiquitous obstacles in the environment: The ubiquitous obstacles caused by collapsed buildings in disaster environments can interrupt wireless signals and hinder search paths, which make many locations in a disaster environment hard to or worthless to be covered by WRs. In recent 20 years, with the development of wireless technologies, many ad hoc network establishment approaches have been proposed from different perspectives [10–14]. A number of sensor or relay deployment approaches were developed for the establishment of sensor networks [15, 16]. Most of these sensor networks are established for the monitoring of general environments so that the energy and connectivities of sensors are their primary concerns during the sensor or relay deployment. A number of WR deployment approaches [17, 18] were proposed for the establishment of ad hoc networks in disaster environments. Most of these approaches used similar settings of sensor networks, in which WRs were assumed to be small, cheap and portable and the number of them was unlimited, thus these approaches mainly focused on maximizing the coverage of areas in a disaster environment. Even if some of approaches [3] aim to maximize the coverage of ILs in a disaster environment, they are still based on the assumption of having enough WRs to cover all ILs in the environment. Based on this assumption, these WR deployment approaches were developed based on greedy algorithms [3, 19, 20], which can quickly create deployment locations for WRs in a disaster environment. However, greedy algorithm-based approaches are myopic, because they establish an ad hoc network through deploying WRs one after another, so the new WRs are deployed at the locations only aiming to maximize the number of additional ILs covered by the network without adjusting locations of the deployed WRs. Thus, ad hoc networks established by greedy algorithm-based approaches can only maximize the coverage of ILs of the new deployed WR rather than the whole network. If the number of WRs is not enough to cover all ILs in a disaster environment, this kind of approaches cannot work well. Against this background, the motivation of our research is to remove the assumption of having enough WRs to cover all ILs and to deploy a limited number of WRs in a disaster environment at suitable locations to establish an ad hoc network. The established ad hoc network should be able to cover the maximum ILs in the environment. This problem is termed as the maximum ILs coverage problem (MILCP). To handle the MILCP, in 2015, a WRs deployment approach was developed by authors for the ad hoc network establishment in disaster environments and the preliminary results were published as a short paper in IAT 2015 conference [21]. This paper is extended from the previous work by proving the NP-hardness and the non-linear of the MILCP, comparing the greedy and non-greedy algorithm-based approaches, revising the formulation of the proposed approaches and adding an extension of the quadratic programming formulation to handle the MILCP with ILs in multiple important levels, new experiments, related work and corresponding analysis. In this paper, two mathematical programming-based WR deployment approaches are proposed for the ad hoc network establishment. In particular, a linear programming (LP)-based WR deployment approach is proposed for the MILCP, which is adjusted from a current WR deployment approach and can find suitable deployment locations for WRs to maximize the ILs covered by the established ad hoc network; and a quadratic programming (QP) formulation of the MILCP is proposed, which can create the same deployment locations for WRs as the LP-based WR deployment approach. In addition, the QP-based approach can be extended to handle the MILCP in many real-life situations in disaster environments. The rest of the paper is organized as follows. The problem description, definitions and characteristics of the MILCP are given in Section 2. Non-greedy algorithm-based approaches and greedy algorithm-based approaches are compared in Section 3. The LP-based approach is introduced in detail in Section 4. The QP formulation of the MILCP and its extension are introduced in detail in Section 5. The experiments are demonstrated and the results are analysed in Section 6. The related work is introduced in Section 7. The paper is concluded and the future work is outlined in Section 8. 2. PROBLEM DESCRIPTION AND DEFINITIONS Let D be a 2D disaster environment, which is divided into a number of equivalent size square locations, where ILs indicate the locations having rescue tasks (i.e. saving victims, extinguishing fires, etc.) and an area indicates a number of adjacent locations, so an area many contain one or more ILs. At a time T ⁠, M number of ILs are distributed in D ⁠. Let ILSet={IL1,IL2,IL3,…,ILM} represent all ILs, where ILi represents the ith IL and 1≤i≤M ⁠. At the same time, N WRs are going to establish the ad hoc network in D ⁠. Let <WR1,WR2,WR3,…,WRN> represent all WRs, where WRj represents the jth WR and 1≤j≤N ⁠. In addition, the number of ILs is much more than the number of WRs (i.e. M≫N ⁠). The IL and WR are formally defined by Definitions 1 and2, respectively. Definition 1 An important location (⁠ ILi ⁠) can be defined as a two-tuple ILi=<ILNo,ILoci> ⁠, where ILNois the ID of ILiand ILociis the location of ILi ⁠. Definition 2 A wireless mobile robot (⁠ WRj ⁠) can be defined as a two-tuple WRj=<WRNo,WLocj> ⁠, where WRNois the ID of WRjand WLocjis the deployment location of WRj ⁠. In this paper, the distance between two locations in D is calculated by Euclidean distance [23, 22]. Since WRs relies on wireless technologies, the sensing and communication distances of WRs are limited. We assume that the sensing and communication distances of all WRs in an environment are same and equal to r so that if the distance between locations of an IL (e.g. ILi ⁠) and a WR (e.g. WRj ⁠) is less than or equals to r (i.e. Dis(ILoci,WLocj)≤r ⁠, refer to Definition 1 and 2), we say that ILi is covered by WRj ⁠. The objective of the MILCP is to deploy all WRs in a disaster environment to establish an ad hoc network (i.e. ANet ⁠) so as to maximize the number of ILs covered by the deployed WRs in ANet ⁠. The objective value objval of ANet can be calculated by Equation (1). objval=argmaxWLocj∑∀ILi∈ILSetCover(ILi), (1) where Cover(ILi) is a boolean function. If ILi is covered by at least one deployed WR in ANet ⁠, Cover(ILi)=1 ⁠, otherwise, Cover(ILi)=0 ⁠; The value of Objval is an integer number between 0 and M ⁠, where 0 and M represent that ANet does not and does achieve the objective of the MILCP, respectively. 2.1. The NP-hardness of the MILCP In this section, the NP-hardness of the MILCP are proven. Theorem 2.1 The MILCP is an NP-hard problem. Proof To prove the NP-hardness of the MILCP, we first introduce a well-studied NP-hard problem, which is termed as the minimum WR deployment problem (MWRDP) [3]. The MWRDP can be described as that given M number of ILs in an environment and the sensing and communication distances of WRs (i.e. r ⁠), how to find the minimum number of deployment locations for WRs, at which all ILs in the environment can be covered by at least one WR. Zhu et al. [20] have proved that the MWRDP is an NP-hard problem so that if the solution of the MILCP can be reduced to solve the MWRDP in polynomial time, the MILCP is NP-hardness. For an environment with M ILs, we assume that there is a solution of the MILCP. The solution should have a variable N to indicate the maximum number of WRs. At the beginning, we can set N as 1. Based on the solution of the MILCP, the deployment locations of N WRs that can cover the maximum number of ILs (i.e. M′ ⁠) in the environment, can be found. Then, by comparing M′ with M ⁠, there could be two kinds of situations, i.e. M′=M and M′<M ⁠. In the first situation, the solution of the MILCP is the solution of the MWRDP, while in the second situation, we can increase N by 1 and use the solution of the MILCP to create the deployment locations of WRs again. The above process can be continuously conducted until M′=M (i.e. the first situation). Then, the solution of the MILCP with N WRs is the solution of the MWRDP. Therefore, no matter in which situation, the MILCP is an NP-hard problem.□ 3. NON-GREEDY ALGORITHM-BASED APPROACHES VS. GREEDY ALGORITHM-BASED APPROACHES To handle the WR deployment, the difference between greedy algorithm-based approaches and non-greedy algorithm-based approaches is demonstrated by the following example. Supposing there are a number of ILs distributed in a disaster environment and two WRs are going to be deployed to maximize the coverage of ILs, which are shown in Fig. 1. FIGURE 1. View largeDownload slide An example of WRs deployment. FIGURE 1. View largeDownload slide An example of WRs deployment. In Fig. 1, the crosses represent ILs in the environment, which form a diamond area. The longest and the highest distances of the diamond area are 16 and 8, respectively. Two points represent the two WRs and the two areas inside the dash lines represent their sensing and communication ranges, both of which are 5. The deployment locations that can maximize the number of ILs covered by the two WRs are described in Fig. 2a, in which all ILs in the disaster environment can be covered by WRs. However, in greedy algorithm-based approaches, the first WR must be deployed at the centre of the diamond area, which is described in Fig. 2b. In this situation, without adjusting the deployment location of the first WR, no matter how to deploy the second WR, it is impossible for the two WRs to cover all ILs in the environment. FIGURE 2. View largeDownload slide The deployment locations of WRs. (a) The optimal deployment of two WRs and (b) the greedy algorithm-based deployment of two WRs. FIGURE 2. View largeDownload slide The deployment locations of WRs. (a) The optimal deployment of two WRs and (b) the greedy algorithm-based deployment of two WRs. Theoretically, the maximum difference on the coverage of ILs between the greedy algorithm-based approaches and non-greedy algorithm-based approaches can reach 25% ⁠. Supposing there are four ILs arranged in a line and intervals between any two adjacent ILs are 10 and the two WRs employed in the last example are going to cover the four ILs. Apparently, the deployment locations of two WRs that can cover the four ILs are described in Fig. 3a. However, based on greedy algorithm-based approaches, if the first WR is deployed at the location to cover the two ILs in the middle, the two WRs can cover three ILs at most, which are described in Fig. 3b. In this situation, the difference on the ILs coverage between two kinds of approaches is 25% ⁠. FIGURE 3. View largeDownload slide The deployment locations of WRs. (a) The deployment based on non-greedy algorithm-based approaches and (b) The deployment based on greedy algorithm-based approaches. FIGURE 3. View largeDownload slide The deployment locations of WRs. (a) The deployment based on non-greedy algorithm-based approaches and (b) The deployment based on greedy algorithm-based approaches. From the above examples, it can be seen that in some circumstances, the coverage of ILs between the non-greedy algorithm-based approaches and the greedy algorithm-based approaches for all the situations have a significant difference. 4. THE LINEAR PROGRAMMING-BASED APPROACH FOR THE MILCP In order to create the suitable deployment locations for WRs in a disaster environment for the MILCP, a linear programming (LP)-based approach, which is adjusted from Guo et al.’s binary integer programming approach (i.e. Subsection 5.3. in [3]) for the MWRDP, is proposed. In the WR deployment problem, the MWRDP is one of the well-studied problems and can be handled by the LP. From the description of the MWRDP (i.e. Proof of Theorem 1 in Section 2.1), it can be seen that the MWRDP is also based on the assumption of having enough WRs to cover all ILs in an environment. The proposed LP-based approach can solve the MILCP by removing the assumption of the MWRDP. In this section, the finding of potential deployment locations (i.e. PLSet ⁠), the construction of the covering relationship matrix (i.e. β ⁠), the LP formulation of the MWRDP, and the proposed LP-based approach are introduced in detail in the following subsections. 4.1. The finding of potential deployment locations In a disaster environment, WRs can be deployed at any locations. However, most of these deployment locations are net very useful, at which WRs cannot cover or can only cover one or two ILs. In order to reduce the computational complexities in the following steps, the potential deployment locations should be found by removing these unuseful deployment locations. Since the coverage area of a WR is a circle, the problem of deploying WRs to cover a number of ILs is the same as the problem of using circles to cover scattered points in a plane. In papers [3, 24], authors have approved that if a fixed-size circle can cover the maximum number of scattered points in a 2D plane, there are at least two points on the border of the circle so that potential deployment locations of WRs can be found based on Theorem 4.1 described as follows. Theorem 4.1 Given two ILs (i.e. IL1and IL2 ⁠) and their locations (i.e. ILoc1and ILoc2 ⁠) in an environment and the sensing and communication distance of WRj (i.e. r ⁠), if the distance between ILoc1and ILoc2is less than or equals to 2r ⁠, potential deployment locations (i.e. PLock ⁠) can be found. If WRjis deployed at PLock ⁠, IL1and IL2are on the border of the sensing and communication range of WRj ⁠. Proof As shown in Fig. 4, the two crosses are locations of two ILs (i.e. IL1 at ILoc1 and IL2 at ILoc2 ⁠); the circle area inside the solid line is the sensing and communication range of WRj (i.e. the sensing and communication distance r ⁠); and the point is the potential deployment location (i.e. PLock ⁠), found based on ILoc1 and ILoc2 ⁠. Based on the definition of the circle, the relationship between coordinates of Loc1=(x1,y1) ⁠, Loc2=(x2,y2) and PLock=(xk,yk) are described as Equation (2). (x1−xk)2+(y1−yk)2=r2(x2−xk)2+(y2−yk)2=r2 (2) When (x1−x2)2+(y1−y2)2≤2r ⁠, xk and yk in Equation (2) have two solutions for each combination of two ILs.□ FIGURE 4. View largeDownload slide The potential deployment locations found based on Theorem 4.1. FIGURE 4. View largeDownload slide The potential deployment locations found based on Theorem 4.1. Theorem 4.1 is described as PLock=PLFound(ILi,ILi′,r) in this paper. Based on Theorem 4.1, all potential deployment locations can be found based on all ILs (i.e. ILSet ⁠) in a disaster environment, which is described in Algorithm 1. Algorithm 1 is explained as follows. At the beginning, PLSet is initialized to ∅ (Line 1). After that, for any two different ILs (i.e. ILi and ILi′ ⁠), if the Euclidean distance of their locations is not more than 2r (i.e. Dis(ILoci,ILoci′)≤2r ⁠), potential deployment locations (i.e. PLock ⁠) are found based on Theorem 4.1 and recorded in PLSet (Lines 2 to 6). After employing Algorithm 1, all potential deployment locations in a disaster environment are found and recorded as PLSet={PLoc1,PLoc2,PLoc3,…,PLocL} ⁠, where L is the number of potential deployment locations and PLock represents the kth potential deployment location. Algorithm 1 The finding of all potential deployment locations View Large Algorithm 1 The finding of all potential deployment locations View Large 4.2. The construction of the covering relationship matrix After finding all potential deployment locations PLSet ⁠, the covering relationships between ILs in ILSet and potential deployment locations in PLSet can be calculated and represented by an M×L coefficient matrix β ⁠, which is described in Equation (3). β=b1,1b1,2…b1,Lb2,1b2,2…b2,L…………bM,1bM,2…bM,L, (3) where M is the number of ILs in ILSet ⁠, L is the number of potential deployment locations in PLSet and each element bi,k in β is a binary value, if the ith IL (i.e. ILi ⁠) can be covered by the WR at the kth potential deployment location (i.e. PLock ⁠), bi,k=1 ⁠, otherwise, bi,k=0 ⁠. 4.3. The linear programming formulation of the MWRDP Based on potential deployment locations (i.e. PLSet ⁠, refer to Section 4.1) and the covering relationship matrix (i.e. β ⁠, refer to Section 4.2), the MWRDP can be formulated by the linear programming (LP) [3]. The variables of this LP formulation are represented by a binary vector X={x1,x2,x3,…,xL} ⁠, where xk is a binary value and 0 and 1 represent the kth potential deployment location is not chosen or is chosen as a deployment location for a WR, respectively. The LP formulation of the MWRDP is described in Equation (4). min∑i=1Lxks.t.xk∈{0,1}1≤k≤L∑k=1Lbi,k·xk≥11≤i≤M (4) Equation (4) is explained as follows. The objective of the LP formulation of the MWRDP is to minimize the number of chosen deployment locations (i.e. the first line of Equation (4)). Then, the value of each xk in X should be a binary value, which is either 0 or 1 (i.e. the second line of Equation (4)). Finally, the LP formulation of the MWRDP can ensure that each IL in the environment (i.e. ∀ILi∈ILSet ⁠) should be covered by at least one WR (i.e. the third line of Equation (4)). In this paper, the LP formulation of the MWRDP is described as LPForm(PLSet,β) ⁠, where ILSet is the set of ILs in the environment, and PLSet is the set of potential deployment locations and β is the M×L coefficient matrix, where M and L are the number of ILs in ILSet and the number of potential deployment locations in PLSet ⁠, respectively. 4.4. The proposed linear programming-based approach In this subsection, the LP formulation of the MWRDP is adjusted to solve the MILCP. The idea of this adjustment can be described as follows. Suppose that we employ the LP formulation of the MWRDP to create the minimum number of WRs (i.e. N′ ⁠) to cover all ILs in a disaster environment based on PLSet and β ⁠. Comparing N′ with N of the MILCP (i.e. the number of WRs in the environment), there are two situations: (1) N′≤N and (2) N′>N ⁠. In the first situation, the number of WRs in the environment is enough to cover all ILs in the environment, the deployment locations of WRs in the LP formulation of the MWRDP are also the solution for WRs in the MILCP. In the second situation, since the number of WRs in the environment is not enough to cover all ILs in the environment, we can repeat the following two steps, which are (1) eliminating a number of ILs from ILSet to obtain ILSet′ and reconstruct PLSet′ and β′ based on ILSet′ (refer to Sections 4.1 and 4.2 ); and (2) employing the LP formulation of the MWRDP to create a new minimum number of WRs N′ for ILSet′ ⁠, PLSet′ and β′ ⁠. With the increasing number of eliminated ILs from ILSet ⁠, the above two steps are repeated until N≥N′ ⁠. However, for the same number of eliminated ILs, if the eliminated ILs are different, different N′ created by the LP formulation of the MWRDP are various. In order to obtain the optimal deployment locations for WRs, for each number of eliminated ILs (i.e. en ⁠), all en-combinations of eliminated ILs should be calculated based on the LP formulation of the MWRDP. The proposed LP-based approach is described in Algorithm 2. Algorithm 2 The proposed LP-based approach View Large Algorithm 2 The proposed LP-based approach View Large Algorithm 2 is explained as follows. At the beginning, the number of eliminated ILs (i.e. en ⁠) is initialized to 0 (Line 1). Then, the LP formulation of the MWRDP is employed to create the solution (i.e. Nen′ and Xen′ ⁠) for ILSet ⁠, PLSet and β (Line 2). If the number of WRs in the MILCP is less than the minimum number of WRs to cover all ILs in the environment (i.e. N<Nen′ ⁠), we begin to eliminate ILs from ILSet and the number of eliminated ILs increases (i.e. en increases from 1 to M ⁠) (Lines 3–4). For each value of en ⁠, the following four steps are executed in order. Step 1: the temporary set Temp is initialized to ∅ (Line 5). Step 2: all en-combinations of ILs (i.e. CMen kinds of combinations) are calculated and recorded in ESeten={een,1,een,2,…,een,Q} ⁠, where een,q⊂ILSet and ∣een,q∣=en (Line 6). Step 3: for each combination of en number of ILs (i.e. een,q∈ESeten ⁠), ILSet is obtained from eliminating ILi∈een,q from ILSet ⁠, PLSet′ is reconstructed based on ILSet′ (refer to Section 4.1) and β′ is reconstructed based on ILSet′ and PLSet′ (refer to Section 4.2) (Lines 8–10). Then, the LP formulation of the MWRDP is employed to create solutions (i.e. Nen,q′ and Xen,q′ ⁠) for PLSet′ and β′ ⁠, which are recorded in Temp (Lines 11 and 12). Step 4: after creating the solutions for all en-combinations of ILs (i.e. ∀een,q∈ESeten ⁠), the solution with the minimum value of Nen,q′ is chosen as the final solution of en number of eliminated ILs and recorded in (Nen′,Xen′) ⁠) (Line 13). The above four steps (Lines 3–13) will be repeated until Nen′ in the optimal solution of en number of eliminated ILs is less than or equals to N (i.e. N≥Nen′ ⁠) (Line 14). Then, Xen′ contains the optimal deployment locations for N WRs in the MILCP (Line 15). 4.5. The computational complexity of the LP-based approach In this section, the computational complexity of the LP-based approach is analysed. In this analysis, we assume that there are M number of ILs, N number of WRs and L number of potential deployment locations in the environment. In addition, although the potential deployment locations are derived from ILs (refer to Section 4.1) and the number of potential deployment locations mainly depends on the distance between two ILs, and M and L do not have a direct relationship. To compare with the greedy algorithm-based approach, the computational complexity of the greedy algorithm-based approach is analysed at first. The greedy algorithm-based approach finds the deployment location for WRs only based on the number of covered ILs. For each WR, the approach recalculates the number of ILs that can be covered at each potential deployment location, the computational complexity of which is O(M×L) ⁠. Therefore, the total computational complexity of the greedy algorithm-based approach to deploy N number of WRs is O(N×M×L) ⁠. In the LP-based approach, the main computational consuming process is to calculate combinations of eliminated ILs (refer to Lines 6–13 in Algorithm 2). All combinations of all numbers of eliminated ILs need to be calculated for ∑en=0MMen times, and the computational complexity of which is O(M×M!) ⁠. For each combination of eliminated ILs, the LP-based approach needs to employ the LP to find the deployment location for N number of WRs, and the computational complexity of which is represented by O(LP) ⁠. The total computational complexity of the LP-based approach to deploy N number of WRs is O(M×M!)×O(LP) ⁠, which is much more than that of the greedy algorithm-based approach. However, in most of situations, the WRs do not need to calculated through all combinations of all numbers of eliminated ILs and the calculation of the LP-based approach ends when N≥Nen′ ⁠. Generally, the computational complexity of the LP-based approach is not stable, which mainly depends on the difference between the number of WRs in the environment (i.e. N ⁠) and the minimum number of WRs that can cover all ILs in the environment (i.e. N′ ⁠, the solution of the LP-formulation of the MWRDP). If the difference between N and N′ is small, the LP-based approach finds the solution for the MILCP with less computations than the greedy algorithm-based approach and vice versa. 5. THE QUADRATIC PROGRAMMING FORMULATION OF THE MILCP Due to the unstable computational complexity of the LP-based approach for the MILCP, sometimes, it is not efficient to employ the LP-based approach to create the deployment locations for WRs in a disaster environment. To this end, a quadratic programming (QP) formulation of the MILCP is proposed. The basic principle of the QP formulation of the MILCP is described as follows. All potential deployment locations that can cover a number of ILs are found and recorded in PLSet (i.e. refer to Section 4.1). The covering relationships between ILs in ILSet and potential deployment locations PLSet are calculated and recorded in an M×L coefficient matrix β (i.e. the covering relationship matrix, refer to Section 4.2), where M and L are the number of ILs and the number of potential deployment locations in the environment, respectively. The duplicated covering relationships between any two potential deployment locations are calculated and recorded in an L×L upper triangle matrix α (i.e. the duplicated covering matrix), where L is the number of potential deployment locations in the environment. The QP formulation of the MILCP are constructed based on PLSet ⁠, β and α ⁠. In this section, the construction of the duplicated covering matrix (i.e. α ⁠) and the QP formulation of the MILCP are introduced in detail in the following subsections. 5.1. The construction of the duplicated covering matrix In this subsection, the duplicated covering relationships between WRs at two potential deployment locations in PLSet can be calculated and represented by an L×L upper triangular matrix α ⁠, which is described in Equation (5). α=a1,1a1,2…a1,L0a2,2…a2,L…………00…aL,L, (5) where L is the number of potential deployment locations, each element ak1,k2 in α is a positive integer value to represent the number of duplicated ILs covered by WRs at the (k1)th and the (k2)th potential deployment locations. The procedure of constructing α is described in Algorithm 3. Algorithm 3 is explained as follows. For each potential deployment location (i.e. PLock ⁠), the set of covered ILs (i.e. Covk ⁠) by the WR at PLock is initialized to ∅ (Lines 1–2). For each IL (i.e. ILi ⁠), if ILi can be covered by the WR at PLock ⁠, ILi is included into Covk (Lines 3–5). For any two potential deployment locations (i.e. PLock1 and PLock2 ⁠), the number of duplicated ILs between PLock1 and PLock2 (i.e. ak1,k2 ⁠) is initialized to 0 (Line 9–11). After that, if k1 is less than k2 ⁠, for each duplicated IL (i.e. ILi ⁠) covered by WRs at PLock1 and PLock2 ⁠, ak1,k2 is increased by 1 (Lines 12–15). Finally, to avoid the repetitive computation of duplicated ILs, ILi is only kept in the set of covered ILs at PLock1 (i.e. Covk1 ⁠) and is eliminated from the set of covered ILs at PLock2 (i.e. Covk2 ⁠) (Line 16). Algorithm 3 The procedure of constructing α View Large Algorithm 3 The procedure of constructing α View Large 5.2. The quadratic programming formulation of the MILCP Based on potential deployment locations (i.e. PLSet ⁠), the covering relationship matrix β and the duplicated covering matrix α ⁠, the MILCP can be formulated by quadratic programming (QP). Same as the variables in the LP formulation of the MWRDP (refer to Section 4.3), the variables of this QP formulation are represented by a binary vector X={x1,x2,x3,…,xL} as well, where xk is a binary value and 0 and 1 represent that the kth potential deployment location is not chosen or is chosen as a deployment location for a WR, respectively. The QP formulation of the MILCP is described in Equation (6). max∑k=1L(xk∑i=1Mbi,k)−∑k1=1L∑k2=1Lxk1xk2ak1,k2s.t.∑k=1Lxk=Nxk∈{0,1}1≤k≤L∑k=1Lbi,kxk≥01≤i≤M (6) Equation (6) is explained as follows. The objective of the QP formulation is to choose N deployment locations from all potential deployment locations (i.e. PLSet ⁠) so as to maximize the number of ILs covered by all WRs in an ad hoc network, which is calculated from the difference between the sum of ILs covered by each chosen deployment location (i.e. ∑k=1L(xk∑i=1Mbi,k) ⁠) and the sum of duplicated ILs covered by any two chosen deployment locations (i.e. ∑k1=1L∑k2=1Lxk1xk2ak1,k2 ⁠) (i.e. the first line of Equation (6)). The number of chosen deployment locations should equal to N (The second line of Equation (6)). In addition, the value of each xk in X should be a binary value, which is either 0 or 1 (i.e. the third line of Equation (6)). Finally, each IL in TSet may be or may not be covered by WRs (i.e. the fourth line of Equation (6)). 5.3. The computational complexity of the QP formulation of the MILCP In this section, the computational complexity of the QP formulation of the MILCP is analysed. The same as the analysis of the computational complexity of the LP-based approach (i.e. refer to Section 4.5), there are also M number of ILs, N number of WRs and L number of potential deployment locations in the environment. The total computational complexity of the greedy algorithm-based approach is O(N×M×L) (i.e. refer to Section 4.5). In the QP formulation, first, to construct the M×L coefficient matrix β ⁠, the WRs need to calculate for M×L times, the computational complexity of which is O(M×L) ⁠. Then, to construct the L×L adjacent matrix α ⁠, the WRs need to calculate for L(L+1)2 times, the computational complexity of which is O(L2) ⁠. In addition, the QP formulation employ the QP to find the deployment location for WRs, the computational complexity of which is represented by O(QP) ⁠. The total computational complexity of the QP formulation to deploy N number of WRs is O(L×(M+L))+O(QP) ⁠. In experiments, it can be found that the computational complexity to solve the QP is not high. Therefore, it can be seen that the computational complexity of the QP formulation is similar to that of the greedy algorithm-based approach. In addition, since the computational complexity of the QP formulation of the MILCP is only related to the number of ILs and potential deployment locations (i.e. M and L ⁠), comparing to the LP-based approach, the computational complexity of the QP formulation is stable. 5.4. The extension of the quadratic programming formulation of the MILCP From Equation (6), it can be seen that the number of ILs covered by the established ad hoc network can be calculated based on potential deployment locations PLSet ⁠, the covering relationship matrix β and the duplicated covering matrix α ⁠. Based on this, the QP formulation of the MILCP can be extended to calculate the numbers of multiple kinds of locations, such as less important locations, worthless locations, etc. covered by the established network. If WRs assign each kind of covered locations a value to represent its contribution to the established network, the utility of the established ad hoc network can be calculated and maximized by the QP. Based on this consideration, the extension of the QP formulation of the MILCP is proposed. The basic principle of the extension of the QP formulation of the MICLP is described as follows. All C kinds of locations and their contribution values to the utility of the established ad hoc network are identified and recorded in CSet={LSet1,LSet2,…,LSetC} and {λ1,λ2,…,λC} ⁠, respectively. All potential deployment locations that can deploy WRs are found and recorded in PLSet ⁠. The covering relationship matrices (i.e. {β1,β2,….,βC} ⁠) are constructed based on all kinds of locations in CSet and potential deployment locations PLSet ⁠. The duplicated covering matrices (i.e. {α1,α2,…,αC} ⁠) are constructed based on all kinds of locations in CSet and potential deployment locations PLSet ⁠. The QP formulation for all kinds of locations are constructed based on PLSet ⁠, {β1,β2,…,βC} and α1,α2,…,αC ⁠. In the proposed QP-based approach, the utility of an established ad hoc network (i.e. ANet ⁠) can be calculated as Equation (7) Utility(ANet)=∑LSetc∈CSetλcNum(Loci∈LSetc), (7) where Utility(ANet) is the utility of the established ad hoc network; LSetc is the set of the cth kind of locations in the disaster environment; λc is the contribution value of the cth kind of location to the utility of the established ad hoc network; and Num(Loci∈LSetc) is the number of the cth kind of locations covered by the established ad hoc network, which can be calculated based on PLSet ⁠, βc and αc ⁠. The QP formulation for the proposed QP-based approach can be described as follows max∑LSetc∈CSetλc∑k=1Lxk∑i=1Mbi,k,c−∑k1=1L∑k2=1Lxk1xk2ak1,k2,cs.t.∑k=1Lxk=Nxk∈{0,1}1≤k≤L∑k=1Lbi,k,cxk≥0Loci∈LSetc∈CSet (8) Equation (6) is explained as follows. The objective function is to choose N deployment locations from all potential deployment locations (i.e. PLSet ⁠) so as to maximize the utility of the established ad hoc network, which is calculated from the contribution values of all kinds of locations and the numbers of there kinds of locations covered by the established ad hoc network (i.e. the first and second lines of Equation (8)). The number of chosen deployment locations should equal to N (i.e. the third line of Equation (8)). In addition, the value of each xk in X should be a binary value, which is either 0 or 1 (i.e. the fourth line of Equation (8)). Finally, each location in CSet may be or may not be covered by WRs (i.e. the fifth line of Equation (8)). 6. EXPERIMENTS AND ANALYSIS As far as we know, there is no special platform for the ad hoc network establishment in disaster environments. In addition, it is hard to simulate all situations in disaster environments. Therefore, in the experiments, we mainly prove that the ad hoc network established by the LP-based approach (represented by LP) and the QP formulation of MILCP (represented by QP) have better performance than those established by the greedy algorithm-based approaches. Based on this consideration, the experiments are designed as follows. Two experiments (i.e. Experiments 1 and 2) are conducted in Matlab2014a [25] to evaluate the performance of LP and QP. Experiment 1 is to evaluate the optimization of the deployment locations created by LP and QP. The benchmark approach of Experiment 1 is the greedy algorithm-based relay deployment approach proposed by Guo et al. [3] (represented by GA). Experiment 2 is to compare computational complexities of LP and QP. 6.1. The experimental settings During the experiment, we found that compared with the greedy algorithm-based approaches, the improvement of LP and QP on the number of covered ILs is mainly related to the number of WRs, the distribution of ILs, etc, but is not related to the number of ILs. Therefore, in two experiments, 20 ILs are randomly distributed in a 25×25 grids disaster environment. The side length of a grid is used as a unit of distance. A number of WRs are going to be deployed to the environment. The sensing distances of all WRs are five units of distance. The distribution of ILs in the environment is illustrated in Fig. 5. FIGURE 5. View largeDownload slide The distribution of ILs in the environment. FIGURE 5. View largeDownload slide The distribution of ILs in the environment. In Fig. 5, the crosses represent the locations of ILs in the environment. Based on the LP formulation of the MWRDP, the minimum number of WRs that can cover all ILs in the environment is 6. The settings of two experiments are shown in Table 1. TABLE 1. The experimental settings. Parameters Values Size of the disaster environment 25×25 grids Number of ILs 20 Number of WRs 1–5 The sensing distance of WRs 5 units of distance Parameters Values Size of the disaster environment 25×25 grids Number of ILs 20 Number of WRs 1–5 The sensing distance of WRs 5 units of distance View Large TABLE 1. The experimental settings. Parameters Values Size of the disaster environment 25×25 grids Number of ILs 20 Number of WRs 1–5 The sensing distance of WRs 5 units of distance Parameters Values Size of the disaster environment 25×25 grids Number of ILs 20 Number of WRs 1–5 The sensing distance of WRs 5 units of distance View Large In Experiment 1, GA, LP and QP are employed to create deployment locations for WRs in the environment, respectively, when there are 1–5 WRs (i.e. N ⁠) in the environment. The maximum numbers of ILs covered by all WRs in the network are used to be the indicator of Experiment 1. In Experiment 2, LP and QP are employed to create deployment locations for WRs in the environment, respectively, when there are 1–5 WRs (i.e. N ⁠) in the environment. The number of computations to create the deployment locations for WRs is used as the indicator of Experiment 2. 6.2. The experimental results and analysis of Experiment 1 The experimental results of Experiment 1 about the maximum number of ILs covered by the ad hoc network are illustrated in Fig. 6. FIGURE 6. View largeDownload slide The maximum number of ILs covered by WRs. FIGURE 6. View largeDownload slide The maximum number of ILs covered by WRs. In Fig. 6, the X-axis represents the number of WRs (i.e. N ⁠) in the environment, while the Y-axis represents the number of ILs covered by WRs in the environment. From Fig. 6, it can be seen that the numbers of ILs covered by the ad hoc network established by LP (i.e. the grey bars) and QP (i.e. the black bars) are always the same. We can say that QP can create the same deployment locations for WRs as LP. When there is only 1 WR in the environment (i.e. N=1 ⁠), the maximum numbers of ILs covered by the ad hoc network established by GA (i.e. the white bar), LP (i.e. the grey bar) and QP (i.e. the black bar) are same. This is because that the GA aims to maximize the number of ILs covered by the new deployed WR (i.e. the only WR). Both LP and QP aim to maximize the number of ILs covered by all WRs in the environment (i.e. the only WR) so that when N=1 ⁠, the objectives of the GA, LP and QP are the same. However, with the increase of the number of WRs in the environment, the drawback of the GA begins to appear, because it can only maximize the number of ILs covered by the new deployed WR rather than all WRs in the established ad hoc network. Different from GA, LP and QP create deployment locations for all WRs in the environment in one time so that LP and QP can create the sutiable deployment locations for WRs to maximize the number of ILs covered by all WRs in the environment. Therefore, when there are 4 WRs in the environment (i.e. N=4 ⁠), the deployment locations of WRs created by GA (i.e. the white bar) can only cover 16 ILs, while the deployment locations of WRs created by LP (i.e. the grey bar) and QP (i.e. the black bar) can cover 18 ILs. The difference between the numbers of ILs covered by WRs deployed by the greedy algorithm-based approach (GA) and programming-based approaches (LP and QP) is 2, which is 10% of all ILs in the environment. The number of ILs in this experiment is small. If the greedy algorithm-based approach (GA) and programming-based approaches (LP and QP) are employed to deployed WRs in a large disaster environment with thousands of ILs, the difference between the maximum numbers of ILs covered by WRs deployed based on the greedy algorithm-based approach (GA) and programming-based approaches (LP and QP) could be very big. Through the experiments, we found that the performance (i.e. the IL coverage of WRs) in both the greedy algorithm-based approach (GA) and the programming-based approaches was significantly affected by the distribution of ILs. Although the potential deployment locations are calculated from the distance between ILs, they could not reflect the distribution of ILs. Therefore, the performance of all approaches has much uncertainties, which is hard to calculate. However, we also found that the performance of the programming-based approaches was often better than that of the greedy algorithm-based approach (i.e. at least the same as the greedy algorithm-based approach). The uncertainties only refer to how much improvement of the programming-based approach can be achieved than the greedy algorithm-based approach. 6.3. The experimental results and analysis of Experiment 2 From the environmental setting shown in Table 1, there are 20 ILs in the environment (i.e. M=20 ⁠). Based on the finding of potential deployment locations (refer to Section 4.1), 90 number of potential deployment locations are found in the environment (i.e. L=90 ⁠), which is much less than the number of deployment locations in the environment (⁠ 25×25=625 grids). The experimental results of Experiment 2 are illustrated in Fig. 7. FIGURE 7. View largeDownload slide The number of computations used by LP and QP. FIGURE 7. View largeDownload slide The number of computations used by LP and QP. In Fig. 7, the X-axis represents the number of WRs (i.e. N ⁠) in the environment, while the Y-axis represents the number of computation used to create deployment locations for WRs. From Fig. 7, it can be seen that the number of computations used by LP (i.e. the grey bars) is unstable, which is related to the number of WRs in the environment (i.e. N ⁠). With the increase of N ⁠, the number of computations used by LP decreased sharply, which is 1 026 875, 431 909, 21 699, 1350 and 20, when N is 1, 2, 3, 4 and 5, respectively (i.e. For clearness, the maximum of Y-axis is only 40 000). This is because that when the number of WRs is small, more combinations of ILs (i.e. ESeten ⁠, refer to Algorithm 2) should be eliminated from the environment until the approach can find deployment locations for WRs to cover all remaining ILs in the environment. With the increase of the eliminated ILs, the combinations of ILs increase exponentially. Different from LP, the number of computations used by QP (i.e. the black bars) is stable. This is because that the most of computation used by QP is to construct the L×L upper triangle matrix α (see Section 2.1), which is only related with the number of ILs (i.e. M ⁠) and the number of potential deployment locations (i.e. L ⁠) in the environment rather than the combinations of ILs and the number of WRs (i.e. N ⁠). The number of computations used by QP to construct α are constant, which is L(2M+L+1)2=45×131=5895 ⁠, which is less that the number of computations used by LP in some of situations. 6.4. Summary The results of two experiments show us that the LP-based approach and the QP formulation have the following three features. The LP-based approach can create suitable deployment locations for WRs in a disaster environment. The ad hoc network established by the LP-based approach can cover more ILs in the environment than those established by the greedy algorithm-based approaches. The QP formulation of the MILCP can create the same deployment locations for WRs as the LP-based approach. The computational complexity of the LP-based approach is various in from situation to situation, while the computational complexity of the QP formulation of the MILCP is stable in all situations. 7. RELATED WORK With the development of wireless technologies, ad hoc networks have played an important role in our lifes, [26–30] . Some approaches were proposed for the topology control in the ad hoc network. George et al. proposed an ad hoc and sensor network architecture to enable sensor to collaboratively collect and transfer the high volumes of accurate data in disaster environments [31]. In their architecture, an energy-efficient cut detection technique was proposed to manage the topology of the ad hoc network. Tan et al. proposed a topology control model for competitive nodes [32], which integrated the mobility models to construct small-world networks and the game-theoretic approaches to allocate resources. Xu et al. [33] also employed the non-cooperative game to control the topology of the ad hoc network so as to achieve the energy-efficient connectivity. Many sensor deployment approaches have also been proposed. Shimoda and Gyoda [34] studied the connectivity of terminals and their energy consumption in ad hoc networks under different communication scenarios. Ozturk et al. [35] employed the artificial bee colony algorithm and the probabilistic detection model to deploy sensors and establish the ad hoc network. Tu et al. proposed a flocking based deployment approach for sensors, which can uniformly deploy sensors around a target of interest in a decentralized manner [36]. Khoufi et al. [37] proposed a projection based sensor deployment approach to cover irregular areas. After that, Li et al. proposed a back-tracking sensor deployment approach to establish the sensor network and can fully cover an unknown area [38] Although many approaches have been proposed for the ad hoc network, their settings, objectives and assumptions are different with our approaches, so they cannot be employed to solve the MILCP. Yang et al. proposed a relay placement approach for sensor networks [16], which considers the two important issues in long-term sensor networks: the energy and the connectivity (i.e. coverage). In their approach, the deployment locations of relays are found through a trade-off between the longest lifetime of relays and the maximum coverage of sensors. However, in disaster environments, the ad hoc network is temporarily established for disaster rescues so that the energy of the WRs is not the primary concern when finding deployment locations. Unlike Yang et al.’s approach, our approach focuses on maximizing the coverage of ILs by the deployed WRs with limited sensing distance. Zhu et al. proposed a relay deployment approach for indoor sensor networks [20], with the consideration that walls can reduce the sensing range of relays. In Zhu et al.’s approach, the greedy algorithm-based method is employed in order to deploy relays with irregular sensing ranges. Most disaster environments are outdoors and there are not many obstacles, which reduce the sensing range of WRs. In addition, as explained in Section 1, greedy algorithm-based approaches are hard to maximize the number of ILs covered by WRs in an ad hoc network. Unlike Zhu et al.’s approach, the sensing ranges of WRs are considered as a circle in our approach. Reich et al. proposed an approach for establishing robot-sensor networks for search and rescue [18, 39], which can guide mobile sensor robots (i.e. WRs of this paper) to search and cover stationary tasks (i.e. ILs of this paper) in disaster environments. In Reich et al.’s approach, the deployment of mobile sensor robots only relies on a series of hierarchical behaviours, which cannot enable the deployed mobile sensor robots to cover the maximum number of stationary tasks in an environment. Unlike Reich et al.’s approach, in our approach, the deployment locations of WRs in disaster environment are created based on the mathematical programming, which can maximize the number of ILs covered by the established ad hoc network. Guo et al. proposed three dynamic relay deployment approaches for wireless networks in disaster environments [3], in which the deployment locations of relays (i.e. WRs of this paper) are at the locations with the maximum number of first responders (i.e. ILs of this paper) in the environment. However, the problem handled by Guo et al.’s approaches is the MWRDP rather than the MILCP. In addition, their three approaches are greedy algorithm-based approaches, which might limit their applications in real-life disaster environments. Unlike Guo et al.’s approaches, our approach can find the optimal deployment locations for WRs in the MILCP. 8. CONCLUSION AND FUTURE WORK In this paper, two WR deployment approaches were proposed based on the mathematical programming, which can maximize the ILs covered by the ad hoc network established by WRs. The proposed approaches can overcome the limitation of greedy algorithm-based approaches and find suitable deployment locations for all WRs in a disaster environment. The experimental results demonstrate that WRs in the proposed approaches can cover about 10% more important locations than that of in greedy algorithm-based approaches in most disaster environments. Theoretically, the coverage of important locations in the proposed approaches can reach to 25% more than that of in greedy algorithm-based approaches in some disaster environments. Although the proposed WR deployment approaches have a better performance than the greedy algorithm-based approaches on the WR deployment in disaster environments. There are some limitations in the proposed approaches. Solve these limitations to improve the applicability of the proposed approaches in real disaster environments in our main research directions in the future. First, in our approaches, the locations of WRs are described and calculated by two-dimensional coordinates, without consideration of the heights of locations, currently. In the future, by considering the height, the three-dimensional coordinates will be used to described the locations of WRs and calculated the distance between them. In addition, the proposed approaches can help WRs to find the suitable deployment locations and establish the ad hoc network in disaster environments. However, many special conditions in disaster environments are not considered currently by the proposed approaches, which might limit their applications in real disaster environments. In the future, we will consider more special conditions into the proposed approaches. 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TI - Two Mathematical Programming-Based Approaches for Wireless Mobile Robot Deployment in Disaster Environments
JO - The Computer Journal
DO - 10.1093/comjnl/bxz010
DA - 2019-06-01
UR - https://www.deepdyve.com/lp/oxford-university-press/two-mathematical-programming-based-approaches-for-wireless-mobile-xeh0AVVT0t
SP - 905
VL - 62
IS - 6
DP - DeepDyve
ER -