Z − : the set of delivery nodes, Z − = { 1 , 2 , 3 , ⋯ , n } . All nodes here represent the turbines, which need kinds of maintenance based on the results of condition monitoring.

Z + : the set of pick up nodes, Z + = { n + 1 , n + 2 , ⋯ , 2 n } , which is the same nodes as Z − but is in return process.

Z = Z − ∪ Z + .

N: the set of all the nodes; N = Z ∪ [ 0 , 2 n + 1 ] , in which 0 is the harbor in delivery process while 2 n + 1 is the harbor in pickup process.

V: the set of service vessels.

T: the set of days in the planning period; T = { 1 , 2 , ⋯ } represents the length of the period.

C i P E : the penalty cost per day for the delaying maintenance task on turbine i, which is related the difference between preventive maintenance and corrective maintenance cost, and additional production loss due to delayed maintenance.

C v i j : the traveling cost of vessel v from node i to j.

T v i j : the time (hours) for vessel v traversing arc ( i , j ) .

T i M : the time needed for performing the maintenance task on turbine i; T 0 M = T 2 n + 1 M = 0 .

L i : the weight of spare parts and equipment for maintenance on turbine i.

P i : the required personnel number for maintenance on turbine i.

T v d M A X : the maximum working hours on day d for vessel v, which is used as the weather limitation for different vessels.

L v M A X : the load capacity of vessel v.

P v M A X : the personnel capacity of vessel v.

T i L A T E : the latest day to perform the maintenance task on turbine i without incurring a penalty.

x v i j d = { 1 ,         vessel v travelsfromnode i tonode j onmaintenanceday d 0 ,           otherwise

y i : the number of delayed days for maintenance task on turbine i. y i = 0 if the preventive maintenance action is carried out before the turbine is down.

t v i d : the time at which vessel v visits turbine i on maintenance day d.

k v i d : the total load weight on vessel v just after it leaves node i on maintenance day d.

q v i d : the total personnel number on vessel v just after it leaves node i on maintenance day d.

min { ∑ v ∈ V ∑ i ∈ N ∑ j ∈ N ∑ d ∈ T C v i j x v i j d + ∑ i ∈ Z − C i P E y i } (1)

Constraints:

∑ j ∈ N ∑ v ∈ V ∑ d ∈ T x v i j d = 1 ,       ∀ i ∈ Z , (2)

∑ i ∈ N x v 0 i d = 1 ,       ∀ v ∈ V , d ∈ T , (3)

∑ j ∈ N x v j i d = ∑ j ∈ N x v i j d ,       ∀ v ∈ V , d ∈ T , i ∈ N , (4)

∑ i ∈ N x v i ( 2 n + 1 ) d = 1 ,       ∀ v ∈ V , d ∈ T , (5)

∑ j ∈ N x v j i d = ∑ j ∈ N x v ( n + i ) j d ,       ∀ v ∈ V , d ∈ T , i ∈ Z − , (6)

∑ v ∈ V ∑ d ∈ T x v i ( i + n ) d = 1 ,       ∀ i ∈ Z V , (7)

t v ( n + i ) d − t v i d ≥ T i M ,       ∀ i ∈ Z − , v ∈ V , d ∈ T , (8)

∑ j ∈ N ∑ v ∈ V ∑ d ∈ T ( d ⋅ x v i j d − y i ) ≤ T j L A T E ,       ∀ i ∈ Z − , (9)

( t v i d + T i j − t v j d ) x v i j d ≤ 0 ,       ∀ i , j ∈ N , v ∈ V , d ∈ T , (10)

∑ i ∈ Z − ∑ j ∈ N L i x i j v d ≤ L v M A X ,       ∀ v ∈ V , d ∈ T , (11)

( k v i d − L j − k v j d ) x v i j d = 0 ,       ∀ i ∈ Z − , j ∈ N , v ∈ V , d ∈ T , (12)

( k v i d − k v j d ) x v i j d = 0 ,       ∀ i ∈ N \ Z − , j ∈ N , v ∈ V , d ∈ T , (13)

( q v i d − P j − q v j d ) x v i j d = 0 ,       ∀ i ∈ Z − , j ∈ N , v ∈ V , d ∈ T , (14)

( q v i d + P j − q v j d ) x v i j d = 0 ,       ∀ i ∈ Z + , j ∈ N , v ∈ V , d ∈ T , (15)

0 ≤ k v i d ≤ L v M A X ,       ∀ i ∈ N , v ∈ V , d ∈ T , (16)

0 ≤ q v i d ≤ P v M A X ,       ∀ i ∈ N , v ∈ V , d ∈ T , (17)

t v ( 2 n + 1 ) ≤ T v d M A X ,       ∀ v ∈ V , d ∈ T , (18)

t v 0 d = 0 ,       ∀ v ∈ V , d ∈ T , (19)

y i ≥ 0 ,       ∀ i ∈ Z − (20)

The objective function Equation (1) minimizes the sum of the sailing cost and the penalty cost associated with not performing maintenance within time window. Other fixed costs, such as personnel cost, are not presented in this function since they will happen anyway and the costs are fixed which would not affect the optimizing process and result. The means of the constraints are listed as follows [33] :

1) Equation (2) ensures that each OWT is visited only once for delivery and once for pick up.

2) Equations (3) and (5) ensure that each vessel leaves and returns the harbor only once every day.

3) Equations (4) and (6) ensure flow conservation at each node.

4) Equation (7) means that if the vessel needs to present during the maintenance operation on 1 OWT, it will only leave the OWT when the operation is completed.

5) Equation (8) is precedence constraints which force the pickup is not done before completing the maintenance operation on the same OWT.

6) Equation (9) is soft constraints, which require that the maintenance task is performed within the preferred time.

7) Equation (10) keeps the travelling time compatibility of each vessel.

8) Equation (11) ensures the service vessels are not overloaded.

9) Equation (12) expresses the compatibility requirements between routes and vessel loads.

10) Equation (13) ensures that no extra load added when the vessels pick up from OWTs.

11) Equations (14) and (15) describe the compatibility requirements between routes and personnel number on the vessels.

12) Equations (16) and (17) guarantee that neither of load or personnel number exceeding the vessel limitations.

13) Equation (18) imposes a maximal working time of the service vessels on each day.

14) Equation (19) means the time is counted from the vessels leaving the harbor.

15) Equation (20) set the delayed maintenance day to be non-negative.

Ant colonies can accomplish complex tasks that far exceed the individual capabilities of a single ant [34] . ACO is a meta-heuristic technique which is inspired by the foraging behavior of the ant species [35] . The ACO model was applied firstly to solve the Travelling Salesman Problem (TSP). The 2 main phases of the algorithm constitute the solution construction and the pheromone update. For TSP, m ants concurrently build a tour and select cities randomly at the beginning of the tour construction. At each construction step, ant k decides which city to visit next according to a random proportional rule. The probability with which ant k, currently at city i, chooses to go to city j is [34] [36] :

p i j k = [ τ i j ] α [ η i j ] β ∑ l ∈ N i k [ τ i j ] α [ η i j ] β ′ ,       if   j ∈ N i k (21)

where τ i j is the pheromone deposited on a r c ( i , j ) , η i j = 1 / d i j , which represents the visibility of city j towards city i which is inversely proportional to the distance d i j , α and β are 2 parameters which determine the relative influence of the pheromone trail and the heuristic information, and N i k is the set of cities that ant k has not visited yet [34] .

The pheromone trails are updated after tours constructing by evaporating at a constant rate and accumulating with new deposits:

τ i j ← ( 1 − ρ ) τ i j + ∑ k = 1 m Δ τ i j k ,       ∀ ( i , k ) ∈ L (22)

where 0 < ρ ≤ 1 is the pheromone evaporation rate and Δ τ i j k is the amount of pheromone that ant k deposits on the arcs it has visited, defined as follows:

Δ τ i j k = { 1 / C k         if a r c ( i , j ) belongstoT k 0                         otherwise (23)

where C k is the length of the tour T k built by ant k. By using this rule, the probability increases that forthcoming ants will use this arc.

ACO is a meta-heuristic technique which is a very good algorithm for solving optimization problem typically Travelling Salesman Problem (TSP) as mentioned above. In classical TSP problem, there are many cities and only 1 salesman. If there are 2 or more salesmen to travel all these cities and each city can and only can be traveled once, how to solve this Multi-TSP problem? The RSOMF problem may have 2 or more vessels which is very similar with Multi-TSP problem. This section describes the principle of Multi-ACO which is a modification from basic ACO as described in Section 3.1.

The idea of Multi-ACO is evolved from classical ACO. The procedure of classical ACO Algorithm solving TSP problem can be written as Figure 1. Multi-ACO has 2 or more groups with the same number of ants and each group has its own pheromone (group 1, group 2 … and pheromone1, pheromone 2 … accordingly). The procedure of the algorithm can be written as Figure 2 assuming that there are n salesmen, i.e. n groups of ants.

The relevant functions used in ACO and Multi-ACO algorithms have been described in section 3.1. In each iteration of Multi-ACO, the ants with same index (k) in different groups visit nodes (cities) according to their own probabilities using roulette wheel selection principle. After all nodes are visited by ants with same index (k) for all groups, the pheromones are updated for all the groups, and then move to next index ( k + 1 ). After all nodes are visited by ants of all groups, the iteration number increases by 1 until maximum iterations or other termination criterion. Then the best routes of n groups with the same index are recorded as the best route.

8 OWTs are selected from illustration in Figure 3 randomly as the first case. The parameters of these turbines are shown in Table 2. Time window in the table means that the maintenance action must be taken before the number of days passed or the penalty cost will be applied due to do the corrective maintenance instead of predictive maintenance. Penalty cost per hour may be varied but using fixed number for simplifying. Load and personnel requirements are weight of spare parts and the number of employees needed to carry out the maintenance action. Task duration means that the number of hours needed to fix the problem of the offshore wind turbines. Table 3 shows the maximum hours the service vessels can work mainly based on the weather forecasting and condition of vessels. The maximum working hours may be different or same for different service vessels. For distinguishing, we assume that the maximum working hours for each day are different.

The process of the program can be seen in Figure 2. There are 2 groups of ants and each of them represents a vessel. The routing of each group of ants represents the routing and scheduling of maintenance for 1 vessel. Based on the relevant research [38] [39] about the tuning of ACO and several trials, the number of ants in each group should be approximately the number of nodes the ants be visited which are the number of offshore turbines in this case. Therefore, the parameters of Multi-ACO are set as: number of ants of each group is 10, the maximum iteration is 200, important coefficient of pheromone α and β are set as 1 and 5 respectively, and the pheromone evaporation coefficient is 0.1.

The results of maintenance scheduling and routing with 8 offshore turbines are shown in Table 4. It is clear that only vessel of SWATH is used for maintenance service for the 8 turbines due to that it is cheaper and faster than Vessel Smit Bronco. In this table, D means delivery process while P means pick up (return) process. The final objective value is 2436.0 and its changing with the iteration is shown in Figure 4.

Figure 5 shows the illustration of scheduling and routing for 8 offshore wind turbines. At first glance, the routing seems not the optimal one for pick up processes in Figure 5(b) and Figure 5(d). However, if we consider that T61 in Figure 5(b) will be the last one to be completed the maintenance action and thus the personnel in this turbine should be the last ones to be picked up, and the same apply to T27 in Figure 5(d).

In order to examine the Multi-ACO performance for a large number turbines’ wind farm, a new offshore wind farm with 36 turbines that are selected from illustration in Figure 3 which are all turbines with kinds of service requirements. The information of 2 vessels is the same as shown in Table 1 and the maximum working hours for each day is the same as Table 3. The conditions and parameters of 36 turbines are not presented because of too much data. This would not influence the target of the case, which is to test the performance the Multi-ACO performance for large number of turbines.

The parameters of Multi-ACO changes because of the increasing the number of wind turbine. The number of ants of each group is set as the same number of wind turbines, i.e. 36, and the maximum iteration is set as 300. The results are shown in Table 5 and Figure 6. The vessel SWATH and vessel Smit Bronco visit and repair, replacement 19 turbines and 17 turbines respectively. Vessel SWATH need 4 days to visit and maintain all these 19 turbines, and it needs 5.0581, 5.2515, 5.2674, and 6.6055 hours for each day which are less than that of the maximum working hours of this vessel in Table 3. Vessel Smit Bronco need 3 days to visit and maintain 17 turbines, and it needs 6.9796, 7.1976 and 7.1590 hours for each day which are also less than that of maximum working hours of this vessel in Table 3. The objective value of fitness function of Equation (1) is 11,391.89. Figure 7 and Figure 8 show the details illustration of the routing and scheduling of these 2 vessels.

It is easy to see that vessel SWATH visit more turbines than vessel Smit Bronco even using longer time (1 more day). The reason is that the more turbines visited by vessel SWATH, the better (lower cost) of the routing and scheduling as long as the turbines can be visited (for repairing and replacement) within the time windows. It is common to get this type solution by Multi-ACO, which is one of the benefits comparing to scheduling and routing manually.

This section compares the proposed algorithm with existing commercial software Xpress® Optimization. Xpress® Optimization is a commercial optimization solver developed by FICO for linear programming (LP), mixed integer linear programming (MILP), convex quadratic programming (QP), convex quadratically constrained quadratic programming (QCQP), second-order cone programming (SOCP) and their mixed integer counterparts. Xpress includes a general purpose non-linear solver, Xpress NonLinear, including a successive linear programming algorithm (SLP, first-order method), and Artelys Knitro (second-order methods) [40] [41] . The software uses branch-and-bound algorithm to solve the NP-hard problem [33] [42] . Same cases are used as in [33] and the results between Xpress® and Multi-ACO are compared in Table 6.

For simplicity, “0” in Table 6 represents the node “Harbor”, the first time the turbine number presents means the vessel delivery the personnel to this turbine, and the second time of the same number means the vessel come again to pick up the personnel to harbor. We can see from Table 6 that both methods can solve the problem effectively. However, almost all the results of Multi-ACO are better than existing commercial software, which shows the advantage of proposed algorithm.