# Path Optimization along Buoys Based on the Shortest Path Tree with Uncertain Atmospheric and Oceanographic Data

Authors: Han Xue; Tian Chai
Journal: Computational Intelligence and Neuroscience
 (2021)

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## Abstract

In order to design an optimized route for ships in line with economic benefits; avoid bad weather; reduce unnecessary detours; shorten the navigation time; and achieve the purpose of safety, fuel saving, and punctual arrival, this paper takes the navigation mark as the node of the tree, takes the connection of the adjacent navigation marks as the tree path, and divides the distance of the adjacent buoys by the ship’s speed as the path cost. The speed calculation collects the current hydrometeorological data such as wind and wave data, uses Aertssen’s deceleration formula to adjust the speed, and improves Dijkstra to find the shortest path. In the experiment, two routes from Dongdu to Xiamen Gang Kou are compared under bad weather conditions. Route 1 is with 5.877 m/s average wind speed, 0.860 m/s wave speed, and total distance 34717 m. Route 2 is with 8.503 m/s average wind speed, 1.429 m/s wave speed, and total distance 30223 m. The calculated ship speed travelling in route 1 is 12.243 km, and its travelling time is 1.53 h. The calculated ship speed travelling in route 2 is 10.523 km, and its travelling time is 1.55 h. Although the total distance of route 1 is longer, it takes less time for ships to travel in route 1. The experimental results verify the effectiveness of the navigation algorithm based on the shortest path tree of uncertain weather maps.


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## Body

## 1. Introduction

Ships are often affected by a variety of different marine meteorological factors. The ship route selection is often not satisfactory. Therefore, it is of great significance and economic effect to design an optimized route for ships and avoid bad weather, so as to shorten the voyage time and achieve the purpose of safety, fuel saving, and punctual arrival at the port.In the research of the shortest path tree, Li proposed a graph-based decomposition dynamic shortest path tree algorithm [1]. An actual network topology was used to calculate the convergence time of static algorithm and various dynamic algorithms when the weight of a single edge changed. Based on the dynamic shortest path tree algorithm, Wang proposed an efficient shortest path tree maintenance algorithm. Through statistical analysis of the network operation in a certain period of time, the algorithm obtained the edges with frequent changes in weights and dealt with the edges with frequent changes in weights [2]. Through the statistical analysis of the operation of the network in a certain period of time, the edge with frequent weight changed, so as to avoid adding it to the shortest path tree and reducing the update times of the shortest path tree. Dai improved the reliability of the shortest path tree by constantly replacing the edges with high probability and then obtained the most reliable shortest path tree [3]. A new calculation method was proposed to calculate the reliability of the shortest path tree. Yang proposed unstable edge and statistical unstable edge [4]. It could make the constructed path tree more stable on the dynamic network and spend less operation to update the shortest path tree.In the study of meteorological routes, Wen proposed an intelligent hybrid algorithm to design the objective function according to the navigation requirements [5]. The objective function and fitness function were designed according to the navigation requirements and multiple constraints. Li proposed the optimization of ship meteorological route and built a weighted punishment fitness evaluation model based on the influence factors of weather and sea state [6]. By optimizing the route and speed, the weather routes under various user preferences could be obtained. Zhang et al. optimized the dynamic optimal meteorological route according to the medium and short-term forecast [7]. The route according to the medium and short-term forecast data was dynamically selected. Using risk analysis theory and technology, Sun can objectively and quantitatively simulate and evaluate the safety of the routes that may reach the target sea area within two days [8]. The quantitative evaluation of the route where ships will arrive in two days was carried out.In the navigation research of navigation marks to navigation, Sang obtained the navigation mark in the route obtained from the database [9]. The third-party radar chart data and the route radar map and the third-party radar map were superimposed to form the route radar map. The meteorological data on the route weather radar map is updated in real time [10]. The proposed route weather radar chart showed a visual platform for passengers to query the route weather. Lin and Zhong proposed an intelligent optimization design of ship routes based on the information data of ports, channels, meteorology, and hydrology [11]. The database of intelligent routes was developed to provide ships with instant information of marine environment.Due to the uncertainty and inaccuracy of environmental data, fuzzy logic is a good tool to deal with environmental data. Cacciola et al. derived a representative model of the multivariate relationships and predicted in order to organize traffic in advance from the estimated parameters [12]. Cacciola et al. provided exhaustive details on how to build a fuzzy system [13]. Postorino and Versaci explained in detail how to easily structure a bank of fuzzy rules [14]. Structuring fuzzy or neuro-fuzzy rule banks was capable of modeling the problem of uncertainty and imprecision present in the data.In this paper, the navigation mark is taken as the node of the tree, the connecting line of the navigation mark in the channel is taken as the path of the tree, and the distance of the adjacent beacon divided by the speed of the ship is taken as the path cost. The speed calculation collects the current hydrometeorological data such as wind and wave data, uses Aertssen’s deceleration formula to adjust the speed, and improves Dijkstra to find the shortest path.


## 2. Shortest Path Tree of Uncertain Weather Map

### 2.1. Aertssen’s Deceleration Method

Aertssen proposed the allowable speed of cargo ship based on propeller idling, deck wave, bottom beating, and cargo moving [15]:(1)UwsUs=1−mLpp−n,where Uws is the allowable speed, Us is the still water velocity, Lpp is the length between the two columns, and m and n are the stall estimation formula coefficients. Table 1 shows the coefficient of ship stall estimation formula.Table 1
Coefficient of ship stall estimation formula.
Beaufort scale5678Head seam9132136n0.020.060.110.18Bow seam7101423n0.020.050.080.12Beam seam3.55.07.010.0n0.010.030.050.07Following seam1247n00.010.020.03


### 2.2. Dijkstra Shortest Path

Dijkstra algorithm is a typical single source shortest path algorithm, which is used to calculate the shortest path from one node to all other nodes [16]. It extends from the starting point to the outer layer until it reaches the end point. In the adjacency matrix of a given graph, the shortest path is recorded by traversing all the paths of the known graph, and then the replacement is continuously judged in the loop. Firstly, the set of all points is divided into two parts: one side has been traversed, and the other side has not been traversed. Firstly, the initial value is given to infinity, and then all nodes are traversed to find the shortest path with the smallest weight in the traversal path. The shortest path can be obtained through constant modification. (Algorithm 1)Algorithm 1: Dijkstra shortest path algorithm.
Input: graphG = (V, E), edge E, length l, vertex V.Output: for all nodeu reachable from s, dist(u) is set to the distance from s to u.for allu belongs to Vdist(u) = infinity;prev(u) = null;enddist(s) = 0;H = make queue(V); using dist values as keyswhileH is not emptyu = delete min(V);for all edges(u, v) belongs to Eif dist(v) > dist(u) + l(u, v)dist(v) = dist(u) + l(u, v)prev(v) = u;decrease key(H, v);endendendThe computation cost of the algorithm is O(VlogV).


### 2.3. Proposed Algorithm

In this algorithm, the navigation mark is used as the node of the tree, the connecting line of the navigation mark in the channel is taken as the path of the tree, and the distance of the adjacent navigation mark divided by the ship speed is the path cost. Speed calculation collects current hydrometeorological data such as wind and wave data and adjusts the speed by Aertssen’s deceleration formula.The specific algorithm is described as follows:Step 1: take the navigation mark as the vertex setV of the tree, and take the connecting line of the adjacent navigation marks of the channel as the path edge set E of the tree. A nonnegative weight simple connected undirected graph G = <V, E> is constructed. D is the adjacency matrix of graph G. The starting point is P0.Step 2: assign initial valuesD [i] = inf (infinity) and mark [i] = 0.Step 3: read the hydrometeorological data such as wind and wave of the current channel.Step 4: according to formula (1), calculate the speed and time cost of ships passing through each path, where D is the adjacency matrix of graph G.Step 5: ifv makes d [J] smaller, that is, d [v] + L [v] [J] < d [J], update d [J], that is, d [J] = D [v] + L [v] [J] and update path, that is, path [J] = v.Step 6: select the nodes with the minimum path value and put them into the set.Step 7: if the remaining node set is empty, end the output structure; otherwise, go to step 3.


## 2.1. Aertssen’s Deceleration Method

Aertssen proposed the allowable speed of cargo ship based on propeller idling, deck wave, bottom beating, and cargo moving [15]:(1)UwsUs=1−mLpp−n,where Uws is the allowable speed, Us is the still water velocity, Lpp is the length between the two columns, and m and n are the stall estimation formula coefficients. Table 1 shows the coefficient of ship stall estimation formula.Table 1
Coefficient of ship stall estimation formula.
Beaufort scale5678Head seam9132136n0.020.060.110.18Bow seam7101423n0.020.050.080.12Beam seam3.55.07.010.0n0.010.030.050.07Following seam1247n00.010.020.03


## 2.2. Dijkstra Shortest Path

Dijkstra algorithm is a typical single source shortest path algorithm, which is used to calculate the shortest path from one node to all other nodes [16]. It extends from the starting point to the outer layer until it reaches the end point. In the adjacency matrix of a given graph, the shortest path is recorded by traversing all the paths of the known graph, and then the replacement is continuously judged in the loop. Firstly, the set of all points is divided into two parts: one side has been traversed, and the other side has not been traversed. Firstly, the initial value is given to infinity, and then all nodes are traversed to find the shortest path with the smallest weight in the traversal path. The shortest path can be obtained through constant modification. (Algorithm 1)Algorithm 1: Dijkstra shortest path algorithm.
Input: graphG = (V, E), edge E, length l, vertex V.Output: for all nodeu reachable from s, dist(u) is set to the distance from s to u.for allu belongs to Vdist(u) = infinity;prev(u) = null;enddist(s) = 0;H = make queue(V); using dist values as keyswhileH is not emptyu = delete min(V);for all edges(u, v) belongs to Eif dist(v) > dist(u) + l(u, v)dist(v) = dist(u) + l(u, v)prev(v) = u;decrease key(H, v);endendendThe computation cost of the algorithm is O(VlogV).


## 2.3. Proposed Algorithm

In this algorithm, the navigation mark is used as the node of the tree, the connecting line of the navigation mark in the channel is taken as the path of the tree, and the distance of the adjacent navigation mark divided by the ship speed is the path cost. Speed calculation collects current hydrometeorological data such as wind and wave data and adjusts the speed by Aertssen’s deceleration formula.The specific algorithm is described as follows:Step 1: take the navigation mark as the vertex setV of the tree, and take the connecting line of the adjacent navigation marks of the channel as the path edge set E of the tree. A nonnegative weight simple connected undirected graph G = <V, E> is constructed. D is the adjacency matrix of graph G. The starting point is P0.Step 2: assign initial valuesD [i] = inf (infinity) and mark [i] = 0.Step 3: read the hydrometeorological data such as wind and wave of the current channel.Step 4: according to formula (1), calculate the speed and time cost of ships passing through each path, where D is the adjacency matrix of graph G.Step 5: ifv makes d [J] smaller, that is, d [v] + L [v] [J] < d [J], update d [J], that is, d [J] = D [v] + L [v] [J] and update path, that is, path [J] = v.Step 6: select the nodes with the minimum path value and put them into the set.Step 7: if the remaining node set is empty, end the output structure; otherwise, go to step 3.


## 3. Experiments

### 3.1. Instance Introduction

There are two routes from Dongdu to Xiamen Gang Kou. The first route is listed in Table2 with the navigation marks passed by.Table 2
The first route.
No.LongitudeLatitudeDistance between adjacent buoys (m)404118°04.0′E24°24.8′N1864Q2-1118°04.5′E24°23.9′N4364Q1-1118°06.4′E24°22.3′N8265106118°08.8′E24°18.4′N1781104118°09.7′E24°17.9′N1643102118°10.3′E24°17.2′N58978118°13.7′E24°16.5′N32876118°14.9′E24°15.1′N7616Xiamen Gang Kou118°17.6′E24°11.8′N—The second route is listed in Table3 with the navigation marks passed by.Table 3
The second route.
No.LongitudeLatitudeDistance (m)404118°04.0′E24°24.8′N1196402118°04.7′E24°24.7′N146122118°05.5′E24°24.4′N154120118°06.3′E24°24.8′N1854Q2118°06.4′E24°23.8′N179218118°07.0′E24°23.0′N250316118°08.0′E24°22.0′N147714-1118°08.0′E24°21.2′N162014118°08.9′E24°20.9′N424212118°10.7′E24°19.3′N1480Xiachen No. 7118°16.5′E24°13.3′N875Xiachen No. 8118°16.1′E24°13.6′N4182Xiamen Gang Kou118°17.6′E24°11.8′N—This algorithm is implemented on the cruise ship 16301 as an example. Table4 shows the basic information of the ship.Table 4
Ship information.
ParameterValueCall signBTTDIMO402657927Ship length32 mBreadth6 mShip nameCruise ship 16301MMSI413045130


### 3.2. Path Optimization under Good Weather Conditions

Under good weather conditions, the recommended route from Dongdu to Koumen is the second route as shown in Figure1.Figure 1
Recommended route from Dongdu to Koumen.Under good weather conditions, the wind speed curve is shown in Figure2 and the wave speed curve is shown in Figure 3 [17].Figure 2
Wind speed in good weather.
(a)(b)Figure 3
Wave speed in good weather.
(a)(b)Under good weather conditions, the calculated ship speed is shown in Figure4.Figure 4
Ship speed in good weather.


### 3.3. Path Optimization under Bad Weather Conditions

Under bad weather conditions, the recommended route from Dongdu to Koumen is the second route as shown in Figure5.Figure 5
Recommended route from Dongdu to Koumen under bad weather.Under bad weather conditions, the wind speed curve is shown in Figure6 and the wave speed curve is shown in Figure 7.Figure 6
Wind speed in good weather.
(a)(b)Figure 7
Wave speed in good weather.
(a)(b)Under good weather conditions, the calculated ship speed is shown in Figure8.Figure 8
Ship speed in good weather.


### 3.4. Comparison of Meteorological Routes

The calculation results under bad weather condition are compared in Table5.Table 5
Information of different routes under bad weather condition.
Avg. wind speed (m/s)Avg. wave speed (m/s)Avg. ship speed (km)Total distance (m)Travelling time (h)Route 15.8770.86012.243347171.53Route 28.5031.42910.523302231.55From Table5, we can see that, under bad weather condition, the wind and wave speeds in route 1 are slow. The ship navigation in route 1 is fast. The wind and wave speed in route 2 are fast. The ship navigation in route 1 is slow. Although the total distance of route 1 is longer, it takes less time for ships to travel in route 1.


## 3.1. Instance Introduction

There are two routes from Dongdu to Xiamen Gang Kou. The first route is listed in Table2 with the navigation marks passed by.Table 2
The first route.
No.LongitudeLatitudeDistance between adjacent buoys (m)404118°04.0′E24°24.8′N1864Q2-1118°04.5′E24°23.9′N4364Q1-1118°06.4′E24°22.3′N8265106118°08.8′E24°18.4′N1781104118°09.7′E24°17.9′N1643102118°10.3′E24°17.2′N58978118°13.7′E24°16.5′N32876118°14.9′E24°15.1′N7616Xiamen Gang Kou118°17.6′E24°11.8′N—The second route is listed in Table3 with the navigation marks passed by.Table 3
The second route.
No.LongitudeLatitudeDistance (m)404118°04.0′E24°24.8′N1196402118°04.7′E24°24.7′N146122118°05.5′E24°24.4′N154120118°06.3′E24°24.8′N1854Q2118°06.4′E24°23.8′N179218118°07.0′E24°23.0′N250316118°08.0′E24°22.0′N147714-1118°08.0′E24°21.2′N162014118°08.9′E24°20.9′N424212118°10.7′E24°19.3′N1480Xiachen No. 7118°16.5′E24°13.3′N875Xiachen No. 8118°16.1′E24°13.6′N4182Xiamen Gang Kou118°17.6′E24°11.8′N—This algorithm is implemented on the cruise ship 16301 as an example. Table4 shows the basic information of the ship.Table 4
Ship information.
ParameterValueCall signBTTDIMO402657927Ship length32 mBreadth6 mShip nameCruise ship 16301MMSI413045130


## 3.2. Path Optimization under Good Weather Conditions

Under good weather conditions, the recommended route from Dongdu to Koumen is the second route as shown in Figure1.Figure 1
Recommended route from Dongdu to Koumen.Under good weather conditions, the wind speed curve is shown in Figure2 and the wave speed curve is shown in Figure 3 [17].Figure 2
Wind speed in good weather.
(a)(b)Figure 3
Wave speed in good weather.
(a)(b)Under good weather conditions, the calculated ship speed is shown in Figure4.Figure 4
Ship speed in good weather.


## 3.3. Path Optimization under Bad Weather Conditions

Under bad weather conditions, the recommended route from Dongdu to Koumen is the second route as shown in Figure5.Figure 5
Recommended route from Dongdu to Koumen under bad weather.Under bad weather conditions, the wind speed curve is shown in Figure6 and the wave speed curve is shown in Figure 7.Figure 6
Wind speed in good weather.
(a)(b)Figure 7
Wave speed in good weather.
(a)(b)Under good weather conditions, the calculated ship speed is shown in Figure8.Figure 8
Ship speed in good weather.


## 3.4. Comparison of Meteorological Routes

The calculation results under bad weather condition are compared in Table5.Table 5
Information of different routes under bad weather condition.
Avg. wind speed (m/s)Avg. wave speed (m/s)Avg. ship speed (km)Total distance (m)Travelling time (h)Route 15.8770.86012.243347171.53Route 28.5031.42910.523302231.55From Table5, we can see that, under bad weather condition, the wind and wave speeds in route 1 are slow. The ship navigation in route 1 is fast. The wind and wave speed in route 2 are fast. The ship navigation in route 1 is slow. Although the total distance of route 1 is longer, it takes less time for ships to travel in route 1.


## 4. Conclusion

In this paper, the navigation mark is taken as the node of the tree, the connecting line of the navigation mark in the channel is taken as the path of the tree, and the distance of the adjacent beacon divided by the speed of the ship is taken as the path cost. The speed calculation collects the current hydrometeorological data such as wind and wave data, uses Aertssen’s deceleration formula to adjust the speed, and improves Dijkstra to find the shortest path. In the experiment, two routes from Dongdu to Xiamen Gang Kou are compared under bad weather conditions. Route 1 is with 5.877 m/s average wind speed, 0.860 m/s wave speed, and total distance 34717 m. Route 2 is with 8.503 m/s average wind speed, 1.429 m/s wave speed, and total distance 30223 m. The calculated ship speed travelling in route 1 is 12.243 km and its travelling time is 1.53 h. The calculated ship speed travelling in route 2 is 10.523 km and its travelling time is 1.55 h. Although the total distance of route 1 is longer, it takes less time for ships to travel in route 1. The experimental results verify the effectiveness of the navigation algorithm based on the shortest path tree of uncertain weather maps.This paper will continue to optimize the algorithm in the future. Fuzzy logic will be used to model the uncertainty of data. Tide height and foggy weather conditions will be considered. Dijkstra algorithm will be improved.



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*Source: 6663446-2021-02-28.xml*