Computational experiments for the stochastic maintenance routing problem

Extensive computational experiments have been carried out to evaluate the performance of the proposed stochastic model for the maintenance routing problem under uncertainty (SMRP). The set of data used for the SMRP is similar to the one used for the MRP in Section 4. However, the data has been adapted in order to be suitable for the SMRP. Here, the experiments on the stochastic model are performed on the West Gabbard case study only. The specification of the vessels used in the model is based on Table 1. The travel time per km of each vessel follows a normal distribution

with mean µ = 1.71 and standard deviation σ = 0.7. We also assume that the required transfer time for technicians and equipment from a vessel to a turbine follows a normal distribution with mean 11 minutes and standard deviation 4 minutes, N (11, 4). In the Monte Carlo simulation, the vessel penalty cost (˘c) is set to e650 per hour to penalise a vessel that returns after the weather/time window (12 hours) which is almost the same as the downtime cost per hour. The example of the turbines data is similar to the ones in Table 2 where the maintenance time is also assumed to follow a normal distribution. The mean repair time is set to the values of the third column of Table 2 whereas the standard deviation is set to 2 and 3 hours for PM and CM activities respectively.

6.1. Computational Results for SMRP

In this study, three scenarios have been considered by varying the value of θ to 50%, 70% and 90%. In other words, the experiments have been performed using the 50%, 70% and 90% quantiles to obtain the expected total maintenance cost. The experimental results are presented in Table 5, which is divided into three sections where the results with 50%, 70% and 90% quantiles (θ) are provided. The table presents the breakdown cost together with turbine and vessel penalty costs which refer to penalty costs for not maintaining a turbine and for not returning to the O&M base within the weather window, respectively. The table also shows the CPU time required to solve each instance, which increases with the number of turbines to maintain. The table also reveals that the total maintenance cost increases with the number of turbines. The transportation cost is relatively small when compared to the total downtime cost. When a higher value of θ is used, the total penalty cost increases. This is reasonable as the value of θ affects the expected maintenance duration and travel time. When the value of θ is high, the maintenance duration and travel times are long.

Table 6 presents a summary of the computational results for the MRP and SMRP. The results on the Deterministic MRP column in the table are obtained by solving the deterministic MRP using the proposed LNS. In the deterministic MRP, the stochastic parameters are approximated by their mean values. The table shows the results of the SMRP when the 50%, 70% and 90% quantiles are used. The difference (%) between the deterministic MRP and the stochastic MRP solutions is also provided. It can be noted that diff (%) increases when the value of θ increases. Here, 50%, 70% and 90% quantiles yield approximately a difference of 3.5%, 7.9% and 15.6% respectively. The total maintenance costs produced by the proposed stochastic model are slightly higher than the one produced by the deterministic MRP model. However, the solutions on the stochastic model are more realistic as uncertain conditions are considered.

Table 5: The computational results for the stochastic maintenance routing

|V | |I| Total cost broken-down Cost (%)

CPU (s) Transport Preventive Corrective Turbine penalty Vessel penalty

Simulation-based Optimisation method (θ = 50%) 2

6 30 027.60 907.73 25 185.70 3 934.17 0.00 0.00 150

7 35 308.98 674.99 31 404.21 3 229.79 0.00 0.00 299

8 41 086.07 780.21 37 346.03 2 959.83 0.00 0.00 466

3

9 43 661.13 1 102.10 37 380.11 5 178.92 0.00 0.00 1 279

10 48 448.20 955.72 41 207.59 6 284.88 0.00 0.00 2 110

11 53 972.59 1 072.50 42 893.42 5 779.77 0.00 4 226.91 2 600 4

12 56 885.58 1 185.98 47 362.77 6 920.80 0.00 1 416.03 4 990

13 61 974.01 1 319.91 51 749.86 8 904.24 0.00 0.00 4 748

14 67 355.56 1 274.63 54 637.42 10 092.07 0.00 1 351.44 4 879 Simulation-based Optimisation method (θ = 70%)

2

6 31 213.38 456.60 29 086.66 1 670.12 0.00 0.00 149

7 37 071.54 819.79 33 192.60 3 059.15 0.00 0.00 283

8 42 929.75 777.76 38 156.90 3 995.09 0.00 0.00 378

3

9 45 266.08 926.68 38 777.36 5 562.04 0.00 0.00 1 115

10 50 674.51 1 533.88 40 483.37 6 228.96 0.00 2 428.31 2 108

11 56 295.47 675.21 49 665.62 5 627.05 0.00 327.58 2 584

4

12 59 288.21 1 258.92 49 030.36 8 590.94 0.00 407.99 4 992

13 64 454.13 1 298.33 51 169.74 11 014.84 0.00 971.22 4 768 14 69 992.38 1 490.12 55 582.22 10 405.79 0.00 2 514.25 4 863 Simulation-based Optimisation method (θ = 90%)

2

6 33 644.74 703.42 30 264.86 2 676.46 0.00 0.00 135

7 39 704.57 797.28 33 713.16 5 194.12 0.00 0.00 259

8 45 915.91 532.06 40 206.29 5 045.07 0.00 132.49 468

3

9 48 571.13 940.70 40 243.69 6 575.84 0.00 810.90 1 210

10 54 164.92 954.56 43 787.25 6 492.25 0.00 2 930.87 2 079

11 59 904.69 879.97 47 633.21 9 482.00 0.00 1 909.52 2 615

4

12 63 039.36 1 698.43 52 125.27 9 215.67 0.00 0.00 5 102

13 68 306.38 1 381.41 54 316.12 11 207.97 0.00 1 400.88 4 768

14 74 087.88 1 322.15 63 619.66 8 813.34 0.00 332.74 4 958

Table 7 shows the different routes for |V | = 3 and |I| = 9 generated by solving the deterministic MRP and stochastic MRP with θ= 50%, 70% and 90%. In the table, the text in red indicates the pick-up nodes as the the vessels will pick up technicians from the turbines. The table also presents the number of technicians in the vessels after leaving the nodes (turbines). According to the solutions obtained, all three vessels are used to maintain the turbines. The solutions generated also prioritise the broken down turbines (Turbines T8 and T9) where these turbines are visited first in order to minimise the total cost. The corrective maintenance (CM) task is needed for these turbines as they are not operating. Surprisingly, in the solution generated for SMRP with θ= 50%,

Table 6: Summary of the computational results for the MRP and SMRP

|V | |I| Deterministic MRP

Stochastic MRP (SMRP)

Quantile (50%) Quantile (70%) Quantile (90%) Total cost Diff.(%) Total cost Diff.(%) Total cost Diff.(%)

2

6 29 227.76 30 027.60 2.74 31 213.38 6.79 33 644.74 15.11 7 34 124.84 35 308.98 3.47 37 071.54 8.64 39 704.57 16.35 8 38 941.43 41 086.07 5.51 42 929.75 10.24 45 915.91 17.91

3

9 42 388.62 43 661.13 3.00 45 266.08 6.79 48 571.13 14.59 10 47 225.06 48 448.20 2.59 50 674.51 7.30 54 164.92 14.70 11 52 068.29 53 972.59 3.66 56 295.47 8.12 59 904.69 15.05

4

12 55 130.80 56 885.58 3.18 59 288.21 7.54 63 039.36 14.35 13 60 429.96 61 974.01 2.56 64 454.13 6.66 68 306.38 13.03 14 65 419.78 67 355.56 2.96 69 992.38 6.99 74 087.88 13.25

Average 3.29 7.67 14.93

Vessel 1 is expected to visit six turbines at the expense of a vessel penalty cost for the lateness to return to O&M base (approx. 1.6 hours). This is due to the fact that travel time and repair time are relatively small when θ= 50%. In this case, the vessel penalty cost is smaller than the travel cost of other vessels to visit the turbines that are allocated to Vessel 1. In the solutions produced for SMRP with θ= 70% and 90%, all vessels return to the O&M base within the weather window. It is interesting to observe that the solution with θ= 90% (the worst weather conditions) is relatively similar to the solution of deterministic MRP. Although the routes generated by these solutions are not the same, the number of turbines visited by each vessel along with the number of technicians required are quite similar.

6.2. Computational Results for SMRP considering the unexpected broken-down turbines

This subsection presents the computational results for the SMRP considering the unexpected turbine breakdowns. In this experiment, an offshore windfarm consists of 125 turbines (J ) and within one day 6 to 14 turbines need to be maintained. The first step is to determine the sampled turbines by executing Algorithm 5. The value of t_j, j ∈ J⁰ (the last days that the turbine has been maintained or repaired) is randomly generated between 1 and 45 days assuming that the turbine needs to be visited 6 to 8 times in one year. The failure rate of an offshore turbine and its components is taken from Carroll et al. (2016). The failure rate of the turbine (λ) is 8.273 failure per turbine per year, whereas the failure rate of its components together with the average number of

Table 7: The routes generated for instances with |V | = 3 and |I| = 9 Determistic MRP

Vessel Nodes O&M T8 T4 T6 T5 T8 T4 T6 T5 O&M

V1 #Techs 12 8 6 4 0 4 6 8 12 12

Vessel Nodes O&M T9 T1 T3 T9 T1 T3 O&M

V2 #Techs 10 6 2 0 4 8 10 10

Vessel V3 Nodes O&M T2 T7 T2 T7 O&M

V3 #Techs 5 3 0 2 5 5

Stochastic MRP with 50% percentile

Vessel Nodes O&M T8 T2 T6 T4 T3 T8 T5 T2 T6 T4 T3 T5 O&M

V1 #Techs 12 8 6 4 2 0 4 0 2 4 6 8 12 12

Vessel Nodes O&M T9 T7 T9 T7 O&M

V2 #Techs 7 3 0 4 7 7

Vessel Nodes O&M T1 T1 O&M

V3 #Techs 4 0 4 4

Stochastic MRP with 70% percentile

Vessel Nodes O&M T9 T4 T1 T6 T9 T4 T1 T6 O&M

V1 #Techs 12 8 6 2 0 4 6 10 12 12

Vessel Nodes O&M T8 T2 T7 T3 T8 T2 T7 T3 O&M

V2 #Techs 11 7 5 2 0 4 6 9 11 11

Vessel Nodes O&M T5 T5 O&M

V3 #Techs 4 0 4 4

Stochastic MRP with 90% percentile

Vessel Nodes O&M T9 T5 T3 T2 T9 T5 T3 T2 O&M

V1 #Techs 12 8 4 2 0 4 8 10 12 12

Vessel Nodes O&M T8 T1 T7 T8 T1 T7 O&M

V2 #Techs 11 7 3 0 4 8 11 11

Vessel Nodes O&M T4 T6 T4 T6 O&M

V3 #Techs 4 2 0 2 4 4

technicians, repair time and maintenance cost (e) are given in Table 8. Note that we only consider the minor repair that needs less than 10 hours to repair the turbine.

In the Monte Carlo Simulation of Algorithm 5, the value of ¯T is set to 10,000 simulation runs.

The value of β is set to min{(|V | − 1), 2} where β turbines with the highest probability value is treated as the sampled turbines. Figure 4 presents the pareto chart of the probability of turbines that will break down generated for the instance with |V | = 3 and |I| = 9. In this case, Turbines T5 and T6 are considered as the sampled turbines and they are predicted to breakdown on that day. Table 9 reveals the experimental results for the SMRP considering the unexpected turbine breakdowns. The table provides the total maintenance costs for the a-priori route, route with

Table 8: The failure data for each component

C#1 C#2 C#3 C#4 C#5 C#6 C#7 C#8 C#9 C#10 C#11 C#12 C#13 C#14 C#15 C#16 C#17 C#18 C#19

λc 0.824 0.812 0.485 0.395 0.456 0.407 0.358 0.326 0.355 0.373 0.247 0.278 0.182 0.19 0.162 0.092 0.076 0.108 0.052

ρc 2.3 2 2.2 2.2 2.1 2 2.2 2.2 2.2 1.8 2.3 1.9 2.3 2.3 2.2 2.6 2.2 2.2 2.5

τc 9 5 7 8 9 4 5 4 8 2 8 4 10 5 5 5 7 7 7

e 1900 2400 3500 2500 1500 2000 2000 2300 2000 2400 2500 2000 1500 1300 3000 1100 5300 1200 2300

C#1: Pitch/Hyd C#6: Grease/Oil/Cooling Liq C#11: Sensors C#16: Tower/Fondation

C#2: Other components C#7: Electrical Components C#12: Pumps/Motors C#17: Power Supply/Converter C#3: Generator C#8: Contactor/Circuit Breaker/Relay C#13: Hub C#18: Service Item

C#4: Gearbox C#9: Controls C#14: Heaters/Coolers C#19: Transformeter

C#5: Blades C#10: Safety C#15: Yaw System

λc: failure rate of component ρc: # technicians needed τc: repair time (hours) e: maintenance cost in e

sampled turbines and optimised scenario without sampled turbines. The routes are generated by executing the proposed simulation-based optimisation method with 90%-quantile. According to the table, the unexpected broken down turbines (sampled turbines) have increased the maintenance cost by approximately 18% on average as the number of turbines that need to be visited increases.

The total maintenance cost is still 5.6% higher than the one of the a priori route although the sampled turbines have been removed from the routes. However, the routes based on the optimised scenario without sampled turbines are more robust as they consider the turbines that have a high chance of breakdown.

Figure 4: Pareto chart of the probability of turbines that will breakdown (Instance with |V | = 3 and |I| = 9)

We also observe quite interesting results when considering the turbine breakdowns. Figure 5 shows the routes generated for the instance with |V | = 2 and |I = 8|. As illustrated on Figure 5a, in the a priori routes, all turbines can be maintained and visited by 2 vessels. However, once turbine T9 is considered as a sampled turbine that has high potential to breakdown, and turbine T6 that needs PM cannot be visited by Vessel 1 due to the lack of resources. This leads to the

Table 9: Experiment results for the SMRP considering the unexpected broken-down turbines (using 90%-quantile)

|V | |I|

A-priori Routes with sampled Optimized Scenario

route turbines without sampled turbines

Total Cost Total Cost Diff. (%) Total Cost Diff. (%) 2

6 33,644.74 39,527.85 17.4860 35,129.22 4.4122 7 39,704.57 44,659.93 12.4806 40,942.45 3.1177 8 45,915.91 52,336.98 13.9844 47,995.59 4.5293 3

9 48,571.13 60,902.75 25.3888 53,475.40 10.0971 10 54,164.92 65,095.14 20.1795 57,204.13 5.6110 11 59,904.69 72,544.47 21.0998 64,097.96 6.9999 4

12 63,039.36 73,186.63 16.0967 66,173.17 4.9712 13 68,306.38 79,507.95 16.3990 71,757.15 5.0519 14 74,087.88 85,754.08 15.7464 78,782.65 6.3368

Average 17.6512 5.6808

turbine penalty cost for not visiting turbine T6. However, the new routes illustrated in Figure 5c are more robust and flexible to accommodate turbine T9 that may breakdown.

Figure 5: Comparison of the generated routes for instance with |V | = 2 and |I| = 8

6.3. Sensitivity Analysis

A sensitivity analysis on the vessel penalty cost for late return vessels is carried out to provide useful information to decision makers. We evaluate the effect of the vessel penalty cost on the change in the total lateness (hours) and the routes generated for each vessel. In the previous experiments,

a vessel penalty cost of e650 per hour (˘c) is used which is based on the downtime cost. We vary this cost to e350 and e50 per hour. The former is determined based on the average personnel cost (for 12 technicians) per hour, whereas the latter indicates a relatively small vessel penalty cost.

The experiments are carried out on three problems. The first one is the problem with |V | = 2 and

|I| = 7 using θ = 50%, the second one |V | = 3 and |I| = 9 using θ = 70%, and the last one |V | = 4 and |I| = 14 using θ = 90%.

Figure 6 shows the total lateness for a vessel penalty cost (˘c) for a different problem. According to the figure, when a lower vessel penalty cost is used, a solution with a larger total lateness is generated. When ˘c is set smaller than the travel cost of a vessel per hour, a turbine tends to be visited by a vessel that has visited a turbine near to it at expense of a longer return time of this vessel. This may lead to a vessel penalty cost for the lateness for a vessel to return to O&M base.

However, this cost is still smaller than the travel cost for other vessel to visit that turbine. In other words, changing the vessel penalty cost per hour affects the turbine allocation configuration. For example, in the problem with |V | = 3 and |I| = 9 using θ = 70%, Vessels 1 and 2 visit 5 and 2 turbines respectively when ˘c= 50. If we increase the value of ˘c to e650, it changes the turbine allocation configuration where Vessels 1 and 2 visit 4 and 3 turbines, respectively. According to the results of these experiments, it is more realistic to set the value of ˘c toe650 as a balance turbine allocation to the vessels can be obtained while yielding the smallest lateness.

Hours

Figure 6: The total lateness for different scenarios

We also perform a sensitivity analysis on the maintenance/repair time where different statisti-cal distributions are used to approach this stochastic parameter. In the previous experiments, the normal distribution was used for the maintenance time. Here, we also apply exponential and log-normal distributions as these can also be used for this parameter in an offshore wind farm (Seyr and

Muskulus, 2016). The total maintenance cost is assessed by the change in the maintenance/repair time generated based on its distribution. The experiments are conducted on three problems using θ = 50% with |V | = 2 and |I| = 6, |V | = 3 and |I| = 10, and |V | = 4 and |I| = 12. The results of our experiments are presented in Figure 7, where the total maintenance cost based on the different distributions for the maintenance time is presented. The figure shows that for each problem, the use of normal distribution generates the lowest total maintenance cost followed by exponential and lognormal distributions. This indicates that a longer maintenance time will be generated when exponential or lognormal distributions are assumed.

Figure 7: The maintenance cost based on different distribution for the maintenance time