A Particle Swarm Optimization Algorithm Based on Time-Space Weight for Helicopter Maritime Search and Rescue Decision-Making

One of the important problems to be solved in maritime search and rescue (MSAR) is decision-making, and the premise of it is determining the mission area for search and rescue unit. To solve the problem that classical cellular iterative search (CIS) algorithm is easy to fall into local optimal solution when determining the mission area, the particle swarm optimization algorithm based on time-space weight (TS-PSO) is proposed in this paper. This algorithm summarizes the optimization objectives and constraint conditions of the MSAR mission area planning according to search theory, carries out the parametric modeling of mission area legitimately and obtains the global optimal solution by continuous exploration in the parameter definition domain. On this basis, by analyzing the time-space weight of drift prediction data, the optimization results are further improved. Finally, through the case simulation analysis, it can be seen that the TS-PSO algorithm can effectively make up for the deficiency of the CIS algorithm and further improve the success probability of optimal MSAR mission area.


I. INTRODUCTION
In helicopter maritime search and rescue (MSAR) mission, it is of great significance for reducing the loss of life and property that distressed facilities, vessels and people are found and rescued efficiently, the decisive factor of which is the accurate positioning of optimal mission area. According to the search theory [1]- [3], it can be known that the probability of success (POS), which is calculated from the probability of coverage (POC) and the probability of detection (POD), can be used to judge the pros and cons of MSAR mission area. Then the POS value can be optimized to the highest by parameter adjustment. The determination of optimal mission area is conductive for search and rescue unit (SARU) to find search and rescue target (SART) efficiently, thereby achieving a higher mission success rate. Therefore, the mission area planning is a primary concern of decision-making issues.
The associate editor coordinating the review of this manuscript and approving it for publication was Rosario Pecora .
In recent years, many scholars have done massive researches on the MSAR mission area planning, and a lot of MSAR decision-making systems have been also widely used, such as SARMAP [4], SAROPS [5] and CASP [6], etc. The cellular iterative search (CIS) algorithm is a common algorithm in mission area planning, which can find the optimal mission area under current conditions by obtaining drift prediction data through integrating GIS and marine environment database. In addition, the improvement and optimization of CIS algorithm is one of the important research directions in this issue. For example, Xiao [7] established the optimal searching rectangle allocation algorithm, which used folding seeking method to allocate searching rectangles under constraints and optimizes POS value to the maximum. Agbissoh et al. [8], further used CIS algorithm to plan mission area, which obtained the rectangular region with the highest POS value by iteratively extending based on grid partition of given predicted discrete points and using the grid with the highest POC value as a benchmark.
At the same time, heuristic algorithms [9], [10] like particle swarm optimization algorithm (PSO) [11] are also widely applied in MSAR. Rafferty and McGookin [12], designed an autonomous air-sea search and rescue system, which planned the helicopter search path in a given region based on PSO algorithm, and obtained a better path than random search proved by simulation analysis. Lv et al. [13], proposed an improved PSO algorithm to solve the searching problem of discrete maritime static targets, where the sub-region importance evaluation function was established in order to maximize the POS value of plan within a certain time. Based on PSO algorithm, Liu et al. [14], developed the continuousdiscrete PSO (CDPSO) algorithm to generate multi-agent shape formation, and illustrated the application of the algorithm in conjunction with the MH370 search and rescue process. Ye et al. [15], provided a simulation-based multi-agent PSO (SA-PSO) algorithm, which supported marine oil spill decision-making by integrated simulation and optimization of response device allocation and process control. Sánchez-García et al. [16], proposed a distributed and dynamic PSO algorithm for UAV networks (dPSO-U), which took the victims movements and the communication among UAVs into consideration, to generate trajectories for drone formations in large-scale disaster scenarios. Simulation results showed that this algorithm could find target faster. Wang et al. [17], proposed various path planning algorithms based on distributed particle swarm optimization (DPSO) for UAV swarms, which were MDC-DPSO, FCO-DPSO, ACE-PSO, and the simulation results showed that these algorithms could effectively meet the requirements of detection time and accuracy. These studies provide feasible methods for exploring the global optimal solution of mission area. However, there is no previous study about the application of PSO algorithm in helicopter MSAR mission area planning, which will be a useful way for decision-maker to solve this problem.
Through in-depth study, it can be seen that the classical CIS algorithm, which is the main algorithm in mission area planning, can obtain more optimal mission area. However, the insufficient analysis of SART drift prediction data and the inflexibility of grid partition when explore optimal mission area often leads to local optimal solutions. Therefore, aiming at these shortcomings of it, a particle swarm optimization algorithm based on space-time weight (TS-PSO) is proposed in this paper to solve this problem in helicopter MSAR mission area planning. According to the classical search theory, the optimization objectives and constraint conditions are determined, and the global optimal solution of parameterized mission area is explored by means of particle swarm optimization algorithm. Then the time-space weight of drift prediction data is defined, which will further increase the POS value of optimal mission area. Meanwhile, in order to study the optimization effects of adding time-space (TS) weight, the TS-CIS algorithm is further put forward by combining it with CIS algorithm. Finally, through a case simulation analysis, the optimization results of CIS algorithm, PSO algorithm, TS-CIS algorithm and TS-PSO algorithm in different time intervals are compared. The simulation results illustrate that the TS-PSO algorithm can explore the global optimal solution, and the average optimization result is significantly improved, which has high practical value.
The rest of this paper is organized as follows. The optimization objective and constraint condition of MSAR mission area planning and the specific application of PSO algorithm in this problem are given in Section II. Then the definition of the TS weight is given in Section III, and the TS-CIS algorithm and the TS-PSO algorithm are further put forward. In order to illustrate the improvement effects of the proposed algorithms, the helicopter MSAR decision support system is designed in Section IV, and the effects are verified via many casesimulation analyses. The last section concludes this paper and offers the future work.

II. MISSION AREA PLANNING WITH PSO ALGORITHM
As claimed by international aeronautical and maritime search and rescue manual [18], when a maritime distress warning is received, the chief thing that decision-makers should do is to obtain the distress location and time to predict drift trajectory [19]- [21]. When an object floats on the water surface without power, its drift condition is mainly affected by environmental factors such as surface wind and flow and its own factors such as immersion ratio and ballast condition [22]. Meanwhile, the calculation of wind-induced drift and flow-induced drift are influenced by uncertain factors. Therefore, the Monte Carlo method [23]- [25] is generally used in drift prediction and the Lagrange method [26]- [28] is used to constantly track the change of position. In this paper, drift prediction data provided by XiaMen LanHaitian Information Technology CO., LTD [29] is the basis of subsequent calculations. And its format is similar to the data provided by a lot of authoritative institutions [30], which is conducive to the practical application of the algorithm to be presented in following description.
After obtaining the drift prediction data, the decisionmakers will filter the data within the required time interval based on SARU capabilities and then plan the mission area, which is a key link in formulating a helicopter MSAR response plan [31]. In order to find the optimal mission area, it is necessary to calculate the POS value in current mission area on the basis of the optimal search theory. Then, according to the boundary information of drift prediction data, the optimal mission area with the highest POS value can be searched by exploring the region parameters using optimization algorithm. It can be seen from the above argument that when determining the helicopter MSAR mission area, drift prediction is the data basis, probability calculation is its optimization objective, and the data boundary constrains the exploration of mission area. In addition, due to the rapid response capacity of helicopter, the mission area tends to be not very large. As a result, the longitude and latitude can be converted from spherical coordinate system to plane coordinate system, which is usually represented by the Mercator coordinates, within the margin of error.

A. OPTIMIZATION OBJECTIVE
The values of POC, POD and POS for a certain SARU in a mission area can be calculated based on the search theory [32]. When exploring optimal mission area, the optimization objective must make the POS value as large as possible, which needs to be comprehensively calculated from the POC of SART and the POD of SARU. In addition, parallel line search pattern is most commonly used in helicopter MSAR mission, as a result, the mission area discussed in this paper is a rectangular region.

1) POC CALCULATION
The location distribution of SART at different times can be found based on the drift prediction data. Because the helicopter MSAR mission is generally performed within a certain period of time, the prediction data from multiple consecutive prediction time points will be selected for analysis, which is usually expressed as discrete points. The POC value is always calculated based on the ratio of the number of points contained in the current mission area to the number of all points. Therefore, when the multiple discrete point sets corresponding to multiple time points, which is defined by DD filtrate = {S i |i ∈ [a, b]}, need to be considered, the POC value can be calculated as follows.
where a is the starting point of time, b is the ending point of time, R i = n(S i ) n(S i ) is the coverage ratio of each set, and S ij is the set of drift prediction points in S ij which is contained in the mission area.

2) POD CALCULATION
When the SARU searches in a certain rectangular region without knowing the specific information of targets, such as the location and movement, the random search method can be used. Then the POD value can be calculated as follows.
where W is the sweep width of SARU, V is the search speed of SARU, T is the search time of SARU, A is the area of the rectangular region. The POD value directly reflects the response capability of SARU to find the SART, which is mainly determined by its own flight performance and airborne equipment. According to the search theory set up by Koopman [33], [34], the search effort of SARU can be expressed as Z = W · V · T , while C = Z A can be taken to measure the effective coverage of SARU, which are the important references for decisionmakers.

3) POS CALCULATION
When the POC value and the POD value are calculated, the POS value can be calculated as follows.
The POS value is a significant indicator for formulating and optimizing MSAR response plans, which means that the higher it is, the greater the probability that SART is successfully found and rescued by SARU in the mission area will be. It can be figured out from the calculation formulas of POS, POC and POD that when a specific SARU is selected to perform MSAR mission, the POC value increases and the POD value decreases with the area of rectangular region being larger, and vice versa. As a result, the maximum POS value can be found according to the parameter change of rectangular region, which provides a theoretical basis for the subsequent optimization algorithm of mission area.

B. CONSTRAINT CONDITION
In the process of exploring optimal mission area, it is necessary to calculate some important information of discrete points corresponding to the drift prediction data, namely the minimum convex boundary (MCB) and the minimum enclosing rectangle (MER), which are the constraint conditions of optimization algorithm. The MCB and MER can be calculated as shown in Fig. 1 and Fig.2 when the drift prediction data is provided. It must be illustrated that multiple initial position of drift prediction calculation should be selected within a certain range near the alarm position, because the exact position of SART cannot be obtained accurately. In order to differentiate these data, the points with different color represent the simulation results of different initial positions.

1) MCB CALCULATION
The MCB of discrete points should be a convex polygon composed of boundary points. Therefore, MCB calculation means to find the boundary point set that makes up the convex polygon. The detailed processes are shown below: Step 1: put all points in the Cartesian coordinate system and find the point with the largest ordinate and the smallest abscissa. Take this point as the initial point P 0 (x 0 , y 0 ) and enter it into the boundary point set; Step 2: assign (x 0 − 1, y 0 ) to P s and (x 0 , y 0 ) to P e ; Step 3: calculate the base vector V base = (P e − P s ), which is the basis of each iteration; Step 4: taking P e as the rotation center and V base as the starting vector, scan other points in clockwise direction to find certain point P next with the smallest rotation angle; Step 5: judge whether the point P next is P 0 . If not, put P next into the boundary set and make P s = P e , P e = P next , then return to Step 3. Otherwise, continue to execute Step 6; Step 6: output the closed curve that is connected with boundary points, which is the MCB.
The algorithm can find all vertices C MCB [n] of the MCB by continuously scanning and updating the reference vector when the discrete point set C input [n] is given. The flowchart and pseudocode of MCB are shown as Fig.3.

2) MER CALCULATION
Rotating Calipers method [35] is an algorithm that uses convexity to generate MER of any convex polygon. After the MCB of a calculated discrete point set is obtained, the MER of it can be calculated by this algorithm. The detailed processes are shown as follows: Step 1: randomly select one edge of MCB as the starting boundary B start and define it as the base boundary B base ; Step 2: find the farthest point among the other points and make a line L parallel parallel to base boundary; Step 3: create two closest lines L a vertical and L b vertical perpendicular to base boundary, and ensure that all points are located between them. Then, calculate the four intersections P 1 , P 2 , P 3 , P 4 of B base , L parallel , L a vertical and L b vertical ; Step 4: generate a rectangle R base using these four points in step 3 and calculate its area A base . Then store R base and A base as key-value pairs into the set S RA ;  Step 5: find another boundary B next of MCB adjacent to the base boundary in clockwise direction. If B next = B start , make B base = B next and return to Step 2. Otherwise, execute Step 6; Step 6: find the rectangle in set S RA with smallest area, which is MER of the scatter.
The algorithm can find all rectangle vertices C MER [4] of MCB boundaries and its area, then the rectangle with the smallest area can be found. The flowchart and pseudocode of MER are shown as Fig.4. VOLUME 8, 2020 C. PSO ALGORITHM Based on the optimization objective and constrain conditions mentioned above, the mission area can be planned using PSO algorithm. Considering that the mission area discussed in this paper is rectangular region, it can be defined by five parameters.
where cLon represents the longitude of the rectangle center; cLat represents the latitude of the rectangle center, maL represents the length of the rectangle, maW represents the width of the rectangle, maA represents the angle between the perpendicular of long side and the positive direction of horizontal axis. As a result, to determine the mission area means to determine these 5 parameters. In addition, according to the optimal search theory, the optimization objective is maximizing POS value of mission area with constraint conditions, which can be expressed in (5).
where A MCB represents the inner region of MCB, L MER represents the length of MER, W MER represents the width of the MER.
It should be noted that the optimized POS value only shows relativeness among algorithms when comparing the results of them, which means it cannot represent the actual success probability of MSAR mission execution when discussed separately.

1) PARTICLE DEFINITION a: POSITION AND VALUE RANGE OF PARTICLE
According to the parameterized definition of the mission area, the positions of particles obtained by each iteration can be expressed as follows.
It can be seen from (3) that the POS value of the mission area is determined by the POC and POD value, which will be influenced by coupling effect of cLon, cLat, maL, maW and maA. Therefore, the fundamental work of PSO algorithm is to define the value range of each parameter.
The ranges shown in (7) are mainly determined based on the following analyses.
a. X i cLon and X i cLat determine the central location of the mission area. According to the analysis of optimization objective, the evolution trend of optimal mission area is to contain mor points and to maximize the coverage ratio of SARU. The first trend makes optimal mission area as large as possible, and the second makes it as small as possible. In conclusion, points defined by X i cLon and X i cLat are generally within MCB. b. X i maL and X i maW determine the size and shape of the mission area. When considering only SART and ignoring the change of POD value, the optimal mission area is MER under the condition that the area is as small as possible while the POC value is as large as possible. When considering only SARU and ignoring the change of POC value, it is obvious that the smaller the mission area is, the greater the coverage ratio will be. Assuming that the SARU just completes a parallel line search in MER, then according to the correspondence between coverage ratio and POD value, the ratio of search effort to mission area should not be greater than 5 when the POD value is no more than 0.99, which means L MER ·W MER X i maL ·X i maW ≤ 5. Therefore, under the condition of balancing the accuracy and computational efficiency as much as possible, the side length of square mission area is taken as the lower bound of these two parameters.
c. X i maA represents the angle between the vertical direction of the long side and the positive longitude direction. Therefore, each mission area corresponds to two angle values, and the value range should be [0, π]. However, if the range is shortened compulsorily, the PSO algorithm may cannot effectively cover all feasible solutions in iterative process. As a result, the range of X i maA is extended to [0, 2π].

b: VELOCITIES OF THE PARTICLES
According to the basic definition of PSO algorithm, the velocities of particles in each iteration can be expressed as follow.
The value of V i ma will be calculated according to the global optimal position and local optimal position, and it will be used for updating X i ma . As a result, its range is not limited. However, when the updated X i ma is beyond its range, the particle position should be performed out-of-range processing in order to restrict V i ma indirectly. These contents will be described in the following discussion.
2) PARAMETER UPDATING For each particle, the updated formula of position and velocity can be obtained with the basis theories of PSO algorithm [36], [37], which is shown as follows.
where: V i ma is the velocity of particles at the i th iteration, X i ma is the position of particles at the i th iteration, ω represents the inertia factor for increasing the rate of convergence, D i global = X i gbest − X i ma is the vector from current particle's position to the global optimal position, D i particle = X i pbest − X i ma is the vector from current particle's position to its local optimal position, c g represents the global learning factor, c p represents the local learning factor, r i g , r i p are five-dimension vectors and the value in each dimension is a random number with u (0, 1).
The value of ω determines both global optimization ability and local optimization ability. When it increases, the global optimization ability is strong and the local optimization ability is weak. While when it decreases, the relationship is reversed. However, the better optimal result can be obtained by dynamic ω, which could be changed linearly or dynamically according to certain measure function during the process of PSO algorithm exploration. Currently, the linear decreasing weight (LDW) [38] strategy as follows is widely applied.
where G k represents the maximum number of iterations, ω ini represents the initial inertia weight, ω end represents the maximum inertia weight. These three parameters have little correlation and can be processed independently. When these parameters calculated in i th iteration is out of their ranges, the nearest boundary value will be taken. In another words, the minimum boundary value is taken when the parameter value is less than the minimum value, and the maximum boundary value is taken when the parameter value is greater than the maximum value.

4) OPTIMAL MISSION AREA ITERATIVE EXPLORATION
The optimal mission area can be explored iteratively after the above preparations. The termination condition of iterative exploration is that when the set number of iterations is reached or the variation of global optimal fitness is less than ε in continuous n times. The detailed processes of PSO algorithm are expressed as follows.
Step 1: initialize the position of each particle according to parameter range, and define the velocity as 0; Step 2: calculate the fitness of each particle and initialize the local optimum and global optimum; Step 3: update the velocity, position and fitness; Step 4: calculate current local optimal fitness value and local optimal solution; Step 5: calculate current global optimal fitness value and global optimal solution; Step 6: judge whether the termination conditions are satisfied. If not, return to Step 3. Otherwise, continue to execute Step 7; Step 7: output current global optimal solution as the optimal mission area of the PSO algorithm.
The process to search the optimal mission area of the discrete points with PSO algorithm is shown in Fig.5, where a) shows the initial mission area, b) shows the solution after 5 th iterations, c) shows the solution after 10 th iterations, and d) shows the optimal mission area. In each iteration, the regions with green boundary are mission areas corresponding to each particle, while the regions with blue boundary is the global optimal mission area in current iteration.

D. COMPARISON OF THE CIS ALGORITHM AND THE PSO ALGORITHM
This paper takes a helicopter MSAR mission of searching for the missing people fallen off a sunken fishing boat on July 2, 2018 as reference. The initial drift prediction locations is located a circular region, the central position is (E118 • 30', N023 • 22') and the radius is 5km. In addition, the probabilities of the initial locations follow a uniform distribution. Ten initial locations are randomly selected in this region and the drift prediction data within one day is calculated. Assuming that the simulation start time is 0 and the data is predicted every one hour, then there are 24 groups of data, which is used as the time index for filtering.
All drift prediction data are divided into 4 time-intervals to be discussed. The optimization results of CIS algorithm and PSO algorithm are analyzed as shown in Fig.6. As the theoretical definition of CIS algorithm, its results are mainly related to the grid quantity, while the results of PSO are primarily influenced by the particle quantity. As a result, the grid quantity within [50,1000] and the particle quantity within [5,100] are respectively taken for analyses. In order to discuss the variation pattern clearly, the data is fitted to f (x) = a · x b + c with 95% confidence interval and represented by a solid line.
According to the data in Table. 1 and the mathematical properties of the fitted curve, c is the extreme value of the corresponding curve. Therefore, for the CIS algorithm, c value is the POS value of optimal mission area when grids quantity is positive infinity, while for the PSO algorithm, c value is the POS value of the optimal mission area when particles quantity is positive infinity. Consequently, the extreme value can be taken as an objective scale for the calculation results of these two algorithms.
It can be seen from above that the optimization results of the PSO algorithm are obviously better than that of the CIS algorithm when exploring the optimal mission area in these four periods. The main reason is that the PSO algorithm takes the parameterized definition of mission area as basis, VOLUME 8, 2020  which is able to explore any combination conditions of all dimensions within the value range of parameters. In addition, the value of each dimension can be equal to any value in its range according to the velocity and position updating process of the PSO algorithm, which can be regarded as continuous optimization process. Therefore, the PSO algorithm makes up for the shortcoming that the CIS algorithm is often trapped in the local optimal solution, and can make the optimization result converge to global optimal solution as much as possible.

III. MISSION AREA PLANNING BASED ON TS WEIGHT A. TIME-SPACE WEIGHT DEFINITION
In actual helicopter MSAR missions, the distress position reported by SART is generally inaccurate due to systematic errors. As a result, the initial position of drift prediction can be considered in a primary area (PA) which follows a certain two-dimensional probability distribution. At the same time, because of the accumulated error of drift prediction algorithm, the data accuracy decreases gradually with prediction time increasing. Therefore, it is necessary to solve these problems before planning MSAR mission area, which is called the TS weight issue in this paper.

1) TIME WEIGHT
Assuming that the drift prediction result includes data from N t time points, and the reliability of data at i th predicted time after the initial simulation time will be C i = aa i , where aa is the accuracy of drift prediction algorithm. If the mission area will be optimized based on the drift prediction data in the period of time from a to b, the time weight W i T of drift prediction data can be expressed as follows.
where C total = b a C i , i is each predicted time, a, b mean integers relative to the starting time of drift prediction.

2) SPACE WEIGHT
Assuming that the probability distribution function of actual initial location in PA is F (lon, lat) and the probability density function is f (lon, lat), then the specific relationship is shown as follows.
F (lon, lat) = PA f (lon, lat)d lat d lon (12) When N s initial positions are selected randomly, the probability density of j th initial position is P j s , and the space weight W j s of drift prediction data can be expressed as follows.
where P j s = f lon j , lat j , lon j , lat j ∈ PA, j represents each initial position, P total = N s 1 P j s represents the sum of all probability density.

3) TS WEIGHT
Assuming that the discrete point set of j th initial position at i th predicted time is S ij , then the TS weight W ij TS of it can be expressed as follows.
represents the sum of time weights and space weights.
Furthermore, the weighted data set filtered by the capability of SARU can be expressed as DD filtrate through drift prediction and TS weight calculation. It represents the corresponding relationship between each discrete point set S ij and its TS weight W ij TS .
where b − a + 1 = N t .

B. ALGORITHM IMPROVEMENT BASED ON TS WEIGHT
Through the above detailed analysis of these specific characteristics of drift prediction data, it can be figured out that the TS weight further considers the uncertainty of data calculated by drift prediction algorithm. With these worthy considerations, the predicted position probability of SART is closer to the actual distribution to some extent. The calculation method of POC is reorganized on this basis, in addition, the TS-CIS algorithm and TS-PSO algorithm are further proposed in the following parts.

1) POC CALCULATION WITH TS WEIGHT
Taking TS weight of drift prediction data into consideration, the POC value can be calculated as follows.
where DD coverage = S ij : W ij TS |S ij ∈ DD filtrate represents the contained data set and its corresponding TS weight, R ij = n(S ij ) n(S ij ) represents the coverage ratio of filtered data set S ij , S ij represents the subset of S ij which is covered by mission area. VOLUME 8, 2020

2) TS-CIS ALGORITHM
The POC value is usually calculated by classical CIS algorithm without TS weight, which considers the SART appearing with equal probability in the predicted position. Therefore, the TS-CIS algorithm is proposed in this paper on the basis of (16). The optimization procedure of this algorithm mainly includes two parts, grid partition and optimal mission area iterative extension.

a: GRID PARTITION
According to the basic process of the CIS algorithm, grid partition of the MER should be completed firstly in the TS-CIS algorithm, because it will influence the calculation of POC value. There are two main methods of grid partition, namely dividing with fixed grid length and fixed grid quantity, while both of them need to adjust the distribution of grid according to the actual situation. Suppose that nL is the actual number of long-side grids, nW is the actual number of short-side grids, L Grid is the length of actual grid long-side, W Grid is the length of actual grid short-side, C AG is the center position of the rectangle formed with all grids, N Grid is the actual grid quantity.
When grids are divided with fixed grid length L 0 Grid and W 0 Grid , the length and width of MER are generally not the integral multiples of grid length. The following relationship of nL, nW and C AG must be satisfied. (17) where C MER represents the position of the MER center. N Grid = nL · nW represents the actual grid quantity. When grids are divided with fixed grid quantity N 0 Grid , the length-width ratio of MER generally does not meet the requirement of grid quantity. The following relationship of nL, nW must be satisfied, by which L Grid = L MER nL and W Grid = W MER nW can be calculated.
It is necessary to illustrate that the specific way of grid partition must be selected according to actual conditions. The accuracy and efficiency of algorithm must be --integrated --considered integrally, and the appropriate way can be selected according to the range of drift prediction data. Meanwhile, a two-dimensional vector (RP L , RP W ) is used to represent the relative position of certain grid in all grids so that the following iterative extension process can be executed more conveniently. And the RP L means the position serial number in L MER direction, while RP W is the position serial number in W MER direction. After the grid partition is completed, the POC value of each grid can be calculated and the grid with the largest POC value will be selected as the initial region of the TS-CIS algorithm.

b: OPTIMAL MISSION AREA ITERATIVE EXTENSION
In the extension process, a rectangular region will be obtained in each iteration of the TS-CIS algorithm, because the optimal mission area, the MER and each grid are all rectangular areas. The detailed processes of optimal mission area iterative extension are expressed as follows.
Step 1: define the initial region as the base region R base and calculate the corresponding POD value and POS value.
Step 2: extend R base in four relative directions (−1, 0), (0, +1), (+1, 0) and (0, −1). Then four new rectangular regions R (−1,0) , R (0,+1) , R (+1,0) , R (0,−1) can be obtained; Step 3: calculate the corresponding POC, POD and POS value of each new rectangular region; Step 4: define the region with the maximum POS value as R MAX , and compare this POS value with that of R base . If the former is bigger, make R base = R MAX and return to Step 2. Otherwise, continue to execute Step 5; Step 5: output R base as the optimal mission area of TS-CIS algorithm.
The process to search the optimal mission area of the discrete points with TS-CIS algorithm is shown in Fig.7, where a) shows the initial region of TS-CIS, b) and c) give the results after the first and the seventh extension, and d) shows the optimal mission area obtained by TS-CIS algorithm. In each iteration, the red region is R base , the orange region is R (−1,0) , the green region is R (0,+1) , the blue region is R (+1,0) , the purple region is R (0,−1) .
It must be figured out that the MER and corresponding grid partition result limit the ability of CIS algorithm to explore optimal mission area. Under this influence, the value of maA must be the long side direction or the short side direction of MER, the value of maL and maW must be integral multiple of the grid side length. the value of cLon and cLat are both associated with the location of output R base and each grid. Therefore, through the comparative analysis of two algorithms, an important conclusion can be drawn that CIS algorithm can only find the local optimal solution, while PSO algorithm can obtain the global optimal solution, which further proves the comparative conclusions of these two algorithms from a theoretical point of view.

3) TS-PSO ALGORITHM
Combined with the discussion of PSO algorithm, the TS weight of drift prediction data will change the POC calculation method in each iteration, and consequently change the calculation result of particle fitness. Therefore, TS-PSO algorithm is further proposed in this paper. The main flow of the algorithm is the same as that described in detail in previous section, and only the calculation method of particle fitness is promoted. In this way, it is convenient to discuss the improvement effects of TS weight on different optimization algorithms results.

IV. CASE SIMULATION ANALYSIS
In order to comparatively analyze these two algorithms and apply them to actual mission more conveniently, the helicopter MSAR decision-making system has been designed based on above theoretical methods, the system interface is shown in Fig.8. The gray rectangle boxes respectively contain SART editing, SARU editing and drift prediction, which are not the focus of this paper, so they are not discussed. The green rectangle box represents the drift data processing, it mainly includes the methods described in constraint conditions and time-space weight of drift prediction data. The yellow rectangle boxes refer to the comparative analysis and validation, which mainly includes the methods described in TS-CIS algorithm and TS-PSO algorithm. The upper right of system interface is the display window of the algorithm execution results, in which the cyan solid convex polygon is MCB, the purple solid rectangle is MER, the blue rectangle is the optimal mission area obtained by TS-CIS algorithm, and the red rectangle is the optimal mission area obtained by TS-PSO algorithm.

A. ANALYSIS OF TS-CIS ALGORITHM
The final result of TS-CIS algorithm is closely related to grid division based on the description of it. However, more information is needed to determine the specific length of the grid based on fixed grid length. In order to improve the practicability, this paper chooses to divide the grid based on the fixed grid quantity. Therefore, the influence of the grid quantity on POS value, run-time and iteration of TS-CIS algorithm have been analyzed with or without TS weight. Take the grid quantity as [50,1000], and divide the prediction data within 1 day into 4 groups for discussion. Fig.9 shows the relationship between the grid quantity and the POS value, where the red scatter is the output data when TS weight is considered, and the blue scatter is the output data when TS weight is not considered. Take the fitting curve analysis method as reference, the data is fitted to f (x) = a · x b + c with 95% confidence interval, which is represented as a solid line with corresponding color. The fitting results are shown in Table.2.
It can be seen that the optimization result of TS-CIS algorithm is significantly higher when TS weight is considered than the case it is not considered. And the slope of the fitting curve decreases greatly when the grid quantity is greater than 300. In other words, the improvement effect of the grid quantity on the POS value is significantly reduced.
As shown in Fig.10, the red scatter is the result of considering the TS weight, and the blue scatter is the result of not    considering the TS weight. It can be seen that TS weight has little impact on the number of iterations, but when the number of grids is greater than 500, the run-time increases greatly. Therefore, considering the performance of TS-CIS algorithm, a better effect can be achieved when the grid quantity is about 500.

B. ANALYSIS OF TS-PSO ALGORITHM
The final result of TS-PSO algorithm is closely related to particle quantity based on the description of it. Therefore, the influence of the particle quantity on POS value, runtime and iteration of TS-CIS algorithm have been analyzed with or without TS weight. Meanwhile, different initial states of particles will lead to different optimal results, which is determined by basic theory of PSO algorithm. As a result, in the case of the same particle quantity, the TS-PSO algorithm runs independently 100 times to obtain the average influence law, and the stability can be analyzed. Fig.11 shows the relationship between the particle quantity and the POS value, where the red scatter is the output data when TS weight is considered, and the blue scatter is the output data when TS weight is not considered. The data is fitted to f (x) = a · x b + c with 95% confidence interval as well, which is represented as a solid line of the corresponding color. The fitting results are shown in Table.3.
It can be seen that the optimization result of TS-PSO algorithm is significantly higher when TS weight is considered than the case it is not considered. And the slope of the fitting curve decreases greatly when the particle quantity is greater than 20. In other words, the improvement effect of the particle quantity on the POS value can be neglected. It is important to illustrate that the success ratio is defined to discuss the stability of TS-PSO algorithm. Suppose that the j th calculation of i th particle quantity is POS j i , then the success ratio SR i can be obtained by equation shown as follows.
where max POS i represents the max value of POS when particle quantity is i, min POS i represents the min value of POS when particle quantity is i, CL ∈ (0, 1) represents the confidence level, n (·) represents the number of simulation times satisfying the internal conditions.
As shown in Fig.12, when the particle quantity is more than 30, each simulation will converge to less than 5% of the total error from all simulation results with more than 90% probability, which satisfies the stability in practical application. On the other hand, as shown in Fig.13, the inflection point of the iteration occurs when the particle quantity is more than 50, while the run-time tends to diverge when the particle quantity is more than 60. Therefore, considering the performance of TS-PSO, a better effect can be achieved in practical application when the particle quantity is greater than 50.

C. COMPARISON OF THE TS-CIS ALGORITHM AND THE TS-PSO ALGORITHM
The values of c in different situations are shown in Fig.14. It can be seen that the optimal POS value decreases with the increase of drift time, indicating that the shorter the response time of SARU is, and the higher the POS value of helicopter MSAR mission will be. This relationship satisfies the objective law of MSAR mission, which indirectly demonstrates the rationality of above two algorithms. The further analysis of the POS promotion degree can be seen in Table.4, which illustrates that the TS weight will promote the optimal POS value of each algorithm, while TS-PSO algorithm improves the classical CIS algorithm more obviously.

D. COMPARISON FOR DIFFERENT SARUS
The TS-PSO algorithm proposed in this paper solves the problem of helicopter MSAR mission area planning. In order to discuss its universality, the situation that helicopter, fixedwing aircraft and ship search and rescue drowning person  is discussed, and the search efforts of different SARUs are shown in Table.5. According to the results of the performance analysis mentioned above, the TS-CIS algorithm with a grid quantity of 500 and TS-PSO algorithm with a particle quantity of 50 are selected, and the single POS values of these SARUs in different time interval are shown in Table.6. It should be noted that when the actual search time is greater than the search effort of SARU, the latter shall prevail.
In order to illustrate the improvement effect of TS-PSO algorithm more clearly, the data of same case in different algorithms are further analyzed and shown in Table.

E. DISCUSSION
According to the above analysis, it can be seen that both the TS-PSO algorithm and the TS-CIS algorithm can effectively improve the optimization result of the classical CIS algorithm in the mission area planning problem, and the former has     higher stability, faster convergence speed and shorter runtime. However, the proposed algorithms have some limitations and disadvantages. The drift prediction is the basis of this algorithm, so the optimization result is highly dependent on the accuracy of it, that is, the more accurate the drift prediction data is, the more reliable the optimization result is. In addition, the time weight can only be calculated when the accuracy of drift prediction algorithm is known, while the space weight can only be calculated when the probability distribution of the initial position is known. Therefore, the application of the TS weight will be greatly affected in the decision-making support of actual helicopter MSAR mission.

V. CONCLUSION
1. Aiming at the problem that the CIS algorithm is easy to fall into the local optimal solution when determining MSAR mission area, the TS-PSO algorithm is proposed in this paper. It can fully explore the definition domain of mission area parameters on the basis of analyzing the TS weight of drift prediction data, and then the global and greater optimal solution will be obtained.
2. The theoretical calculation process and optimization termination criterion of the TS-CIS and the TS-PSO are given. It can be seen that the input parameter of the former is grid distribution, and that of the latter is particle quantity. In addition, according to the theoretical characteristics of these two algorithms, the TS-CIS algorithm only calculates the local optimal solution due to the limitation of MER, while the TS-PSO algorithm can obtain the global optimal solution.
3. Through the analysis of the examples, it can be seen that the TS weight can improve the optimization results of both the TS-CIS algorithm and the TS-PSO algorithm, but the improvement effect is small. In comparison, the TS-PSO algorithm can improve the optimization result of the TS-CIS algorithm in all cases, and the improvement effect is relatively obvious. Therefore, it can be illustrated that the TS-PSO algorithm can not only explore the global optimal solution in the domain of parameterized mission area, but also its optimization result is significantly higher than the CIS algorithm, which proves the effectiveness and practicability of the TS-PSO algorithm sufficiently. 4. Moreover, there are many further works can be done. This paper discusses the application of the TS-PSO algorithm in the MSAR rectangular mission area planning, which corresponds to the area-datum search pattern. Then it can also be applied to the planning problem of point-datum search pattern and line-datum search pattern taking the parameterization method of rectangular mission area as a reference. At the same time, in order to illustrate the problem more intuitive, the single-helicopter-single-flight search process is discussed in this paper. However, the TS-PSO algorithm can also be extended to multi-helicopter-multi-flight search process according to the basic principle of algorithm and the data format of drift prediction. Another important problem is that the initial state of particle swarm has a great influence on optimization result and convergence rate, therefore, the relationship between them can be studied through design of experiment (DoE). These issues will be continuously carried out in the follow-up researches. Meanwhile, since the search theory can also be applied to land search and rescue (LSAR) [39] field, the TS-PSO algorithm mentioned in this paper and the follow-up scientific research achievements can also be applied to this field.
In this paper, according to the general procedure of PSO algorithm, a lot of improvement measures of the TS-PSO algorithm on the CIS algorithm are discussed from two perspectives, the search theory and the nature of drift prediction data. Then the improvement effect is adequately proved by comparison of theoretical calculation processes and case simulation analysis. Thereby the optimal mission area of different SARU in different time intervals can be obtained by this algorithm, and then the helicopter MSAR decision support can be made by comparing POS values. The TS-PSO algorithm provides a new theoretical basis for the field of MSAR mission area planning, and the heuristic algorithm represented by this algorithm can also be applied in this field and other related fields.