Fault-tolerant extension of Vehicle Routing Problem solutions
4.5 The eciency of the fault-tolerant extension
4.5.2 Eciency analysis of the proposed algorithm on the Solomon instancesSolomon instances
The impact of the proposed algorithm was also investigated by using standardized VRP test instances. Solomon instances [163] were chosen for the test; this data set is a common test set in the VRP literature.
The Solomon instances
M.M. Solomon generated 6 sets of VRP-TW instances (R1, R2, C1, C2, RC1, and RC2). Each problem has one depot and 100 customers with well-determined loca-tions, demands, time windows, and service times. This thesis does not deal with time windows and service times; only the location data and the demands dened by the instances are used. The6groups of the problems dier - besides the time-related data - in the customers' locations' relation - except groups R1 and R2 - ("R" stands for randomized and "C" denotes clustered placement). Another dierence between the 6 instance groups is the capacity of the vehicles (in groups R1, C1, and RC1, each vehicle has a capacity of 200, in C2, the capacity is700, nally, in groups R2 and RC2, the value of the capacity is 1000). Instances of a group dier only in the time-related parameters; the customers' locations in a group are identical.
For example, the RC202 instance (that is an element of the RC2 instance group) contains a mix of clustered and randomly generated customers and allows long routes to be created. The number of customers is 100, and the vehicle capacity is 1000. More specic data (e.g., locations, demands) can be found in [163]. The location of the100 customers (small circles) and the depot (marked with the largest circle) can be seen in Figure 4.7.
Figure 4.7. Location of the depot and the 100 customers of Solomon's RC202 in-stance.
The results of the proposed method for the Solomon instances
First, the results of the proposed algorithm are detailed for Solomon's RC202 in-stance, then the results for all the six groups of Solomon's instances (without time-specic data) are summarized.
ExecutingStep1−Step4of the algorithm of subsection 4.4.3 for the RC202 instance results in the initial route set illustrated in Figure 4.8.
Figure 4.8. The route set, obtained for Solomon's RC202 instance.
This route set is composed of 10 routes. This means that the algorithm has to deal with 1024 dierently directed route compositions. Among the obtained solutions, the minimal averaged route change cost has a value of53.29, the maximal value is65.38, the average value of the route change costs is58.22, and the standard variance of the 1024 cases is 2.22. This means that, when an unexpected event occurs, the extra cost of the best solution - the cost originating from helping the broken-down vehicle - is81.5% of the extra cost of the worst solution and 91.5% of the extra cost of the average of all the cases. The obtained values - completed with the results got for the other instance groups of Solomon - are presented in Table 4.7.
The algorithm was tested on all types of Solomon's instances [163]. The result of the fault-tolerant route planning algorithm on the groups of Solomon instances is summarized in Table 4.7. The columns of the table contain the minimal and the maximal rcc values (see 4.17), the standard variance and the mean of the rcc values for the dierent route direction compositions, the ratio of the best and the worst case, and the ratio of the best and the average case per Solomon instance groups. The histograms of the averaged minimalrccof the possible route direction compositions for each instance group can be found in Appendix T.
The results show that the application of the algorithm improves the eciency of route execution in all cases. Of course, the improvement is the largest - even 10%-15% - compared to the worst possible response to the unexpected event. However, the algorithm also ensures a 4%-8% improvement compared to the average in the case of a vehicle breakdown. The results show that the improvement is bigger when there are clustered customer locations and is smaller for random locations. Its reason
Table 4.7. The eciency of the algorithm for the dierent Solomon instance groups.
Name of the Solomon instance group
C1 C2 R1 R2 RC1 RC2
The smallest averaged
route change cost; 42.17 49.79 62.39 62.39 54.91 53.29
the result of the algorithm The highest averaged
route change cost; 47.27 59.40 67.41 67.41 64.41 65.38
the worst possible choice How much (in %) the obtained
solution is better related to the 10.8% 16.2% 7.4% 7.4% 14.7% 18.5%
worst possible case The mean averaged
route change cost 44.68 54.48 65.05 65.05 59.39 58.22
The standard variance of
averaged route change costs 1.14 2.03 0.82 0.82 1.75 2.22
How much (in %) the obtained
solution is better related 5.6% 8.6% 4.1% 4.1% 7.5% 8.5%
to the average
can be that clustered customers determine more strictly the route that they should be assigned to, and an inecient selection of a helper vehicle causes longer extra routes than in the case of random - thus, not so strictly separable - locations.
4.6 The contribution of this chapter
The contribution of this chapter is threefold: (1) it proposes an algorithm that nds the route direction composition of any solution of an arbitrary Capacitated Single Depot Vehicle Routing Problem without Time Windows, that tolerates the best the breakdown of one vehicle, on average, assuming the constraints of subsection 4.4.1, (2) it gives an algorithm that determines the helper vehicle and the route modica-tions for the cases when a vehicle of the eet breaks down during route execution and (3) it determines the eciency of the proposed fault-tolerant extension on a simple example and the well-known VRP benchmark problems of Solomon.
When a vehicle breaks down, the proposed method counts on minimizing the extra cost of the breakdown by applying a greedy method: inserting the remaining cus-tomers into the closest route (keeping to the constraint of using exactly one helper vehicle). The total replanning of the routes - which is the common method found in literature - is NP-hard.
For the investigated problems, the proposed algorithm ensures 4%-8% improvement compared to the average in the case of a vehicle breakdown. These problems include both random and clustered customer locations. The average eciency improves
when the customers are clustered.
Due to the variations in the number of customers, computation times dier highly.
The calculation of the results in the simple example presented in subsection 4.5.1 took 0.487 seconds, while for Solomon's instances, it took 81 seconds on average using an HP Compaq tc4200 tablet PC with Intel Pentium M 2.00 GHz processor and 1 GB RAM. Because these calculations have to be carried out before the route execution during the route planning process, the delay in computation does not inuence the logistic system's performance.