Multi-objective Vehicle Routing Problems are mainly used in three ways [135]: (i) to extend classical academic problems in order to improve their practical application (while never losing sight of the initial objective), (ii) to generalise classic problems, and (iii) to study real-life cases in which the objectives have been clearly identified by the decision-maker and are dedicated to a specific practical problem or application. In any of the contexts cited above, in addition to the intrinsic economic cost of routing, e.g. number of vehicles, travel distance and delivery time, a number of objectives has been considered to be optimised, among which are the following:
• Service level. This objective is related to the minimisation of the longest route length [45, 177, 185, 231].
• Constraints. This objective regards the minimisation of the number or the extent of violated constraints [197, 16, 34].
• Workload imbalance. Here one aims at minimising the difference of the work- load between the longest and the shortest routes [133,134,187].
• Security. There are problems which concern that the risks of accidents is minimised [112, 168].
• Accessibility. This is a dual objective, which corresponds to maximising the coverage of a geographical area while minimising the number of mobile facilities [75, 74].
• Geography. Sometimes it is required that customers in the same region are serviced by the same vehicle [248].
The algorithm that will be presented in Chapter 5can be extended to deal with any number of these objectives, however, as stated earlier, simulation results will only
be presented for the standard number of routes f1(R) in (3.4) and travel distance
f2(R) in (3.5), plus the delivery time f3(R) in (3.14). The optimisation of further
objectives will be suggested as a direction of future research.
3.4.1 Overview of metaheuristic approaches
Multi-objective VRPs account for less research than the two variants surveyed ear- lier. Some of these studies are described here.
Rahoual et al. [197] tackled the VRPTW with a Genetic Algorithm (GA) based on the first version of the NSGA [223], taking into account the minimisation of the number of routes, the travel distance, and the penalties associated to the violated constraints. The constraints considered were the capacity, distance, and duration limits, in addition to the time windows. This approach considers a randomly gene- rated initial population, single-point crossover [57], and a mutation operator which consists in changing the position of a customer from one vehicle to another using one out of of five different procedures, the choice of which to run is made randomly. The authors presented results to instances in the Solomon’s benchmark categories C1 and R1, however, many of them present violated constraints, which means that the solutions obtained are infeasible.
Murata and Itai [177] proposed a two-fold multi-objective Evolutionary Algorithm (EA), based on NSGA-II [67], for solving a class of VRP with normal (NDP) and high (HDP) demands, considering the minimisation of the number of vehicles and the maximum routing time, i.e. the route with maximum duration. This algorithm generates a random initial population, uses the cycle crossover [181], and two muta- tion operators: one of them modifies the assignment of customers to routes and the other reverses the order in which customers are visited in a route. In the first stage they solved the NDP and use the resulting solutions to initialise the optimisation of
the HDP. They also propose the ratio of the same route similarity measure in order to evaluate the similitude between solutions to both problems.
Jozefowiez et al. [134, 136] addressed the CVRP with Route Balancing, in which the total travel distance and the difference between the longest and shortest routes lengths are to be minimised. They implemented a parallel enhanced version of NSGA-II [67], which considered two crossover operators, namely the RBX of Potvin and Bengio [192] and the Split function of Prins [195], and a mutation based on 2-opt local search.
Ombuki et al. [182] considered VRPTW as a bi-objective optimisation problem, where the number of vehicles and the travel distance are to be minimised, and used a GA for solving it. In this approach 90% of the initial population are randomly generated solutions and the remaining 10% are solutions generated with a greedy procedure based on the nearest neighbour method. The authors introduced the best
cost route crossover (BCRC), which aims at simultaneously minimising the number
of vehicles and travel distance while checking feasibility, and proposed the constrai-
ned route reversal mutation, which purpose is to invert a sequence of customers.
Tan et al. [228] proposed a hybrid multi-objective EA for solving the VRPTW re- garding the minimisation of the number of routes and the travel distance. This approach starts with a randomly generated initial population, which is then submit- ted to the designed route-exchange crossover and to a multi-mode mutation that consists in three operations, namely partial swap, split route, and merge routes, from which only one is executed. Additionally, the λ-interchange, and the proposed
intra route and shortest pf local search heuristics were implemented, which were
executed every 50 generations for all individuals in the population. This analysis found that, despite categories C1 and C2 have positively correlating objectives, the
majority of theSolomon’s benchmark instances in categories R1, R2, RC1 and RC2 have conflicting objectives, as was previously noticed from Table 3.5.
Tan et al. [229] slightly modified the previous approach for tackling the Truck and
Trailer VRP. In this variant, different types of vehicles are considered, which means
that some of them have certain limitations. The modified version of this algorithm considers the nearest neighbour density estimation technique in order to preserve population diversity.
Tan et al. [231] proposed a multi-objective evolutionary approach, based on the earlier approach of Tan et al. [228], for solving the SVRP in which the demand is the stochastic parameter. They considered, in addition to the minimisation of the number of routes and travel distance, the minimisation of driver remuneration, i.e. delivery time. An initial solution is generated so that it uses a random number of vehicles and approximately the same number of customers are serviced in each route. In this study, instead of the 2-opt and λ-interchange local search heuristics, two methods were implemented, namely the shortest path search and the which
directional search.
Ghoseiri and Ghannadpour [111] presented a study using a goal programming ap- proach for the formulation of the problem and an adapted a GA to solve it. Part of the initial population is initialised randomly and part is initialised by using the I1 heuristic and the λ-interchange local search. The authors introduced the best
cost-best route crossover (BCBRC), which selects a best route from each parent and
is very similar to the BCRC of Ombuki et al. [182] with minor differences, and the
sequenced based mutation (it is actually a recombination operator), which, given two
offspring solutions produced from the recombination phase, randomly selects an arc to break a route on each of the solutions and then make an exchange on the routes before and after the break points to produce two new offspring. Two local search
heuristics are incorporated in the GA, from which one is executed at the end of each generation for a portion of the population. To our best knowledge, this is the only multi-objective study which actually presents the overall Pareto approximation found to each instance of the Solomon’s benchmark set.
Pacheco and Mart´ı [185] addressed the problem of routing school buses, which consists of transporting a group of students from their homes to a school, by means of Tabu Search (TS). They considered the minimisation of the total number of buses while simultaneously minimising the maximum time that a student spends in the bus, that is the longest route. The problem is solved by considering both objective functions separately. Since the value of the first objective, the number of routes (or buses), is a discrete number (bounded by the number of locations), the authors followed the simple method that consists of minimising the second objective, the maximum length of a route, for each possible value of the number of routes. This algorithm utilises the two constructive methods proposed by Corber´an et al. [45] and the one by Fisher and Jaikumar [87] for generating initial solutions. Then, the TS employs a modified version of the CROSS exchange of Taillard et al. [226] to perform the local search.
Finally, Beham [16] proposed a TS approach for solving the VRPTW, and tested it on the instances of Homberger and Gehring [129], regarding the minimisation of the number of routes, the total travel distance, and the time constraint violation. In addition to the list of forbidden moves, two extra memories were used: a list of non-dominated solutions from previous neighbourhoods, which are used to restart the search process, and the archive containing the overall found non-dominated solutions. A solution can be added to the archive when it is not dominated by the solutions in the archive and this is not full. If the archive is full, the solution is added based on the result of the crowding distance [67]. This method starts by generating
an initial solution with the I1 heuristic, which is then improved by selecting one of the local search heuristics 2-opt, Or-opt, and λ-interchange.
3.4.2 Results from previous studies
With the exception of the study of Ghoseiri and Ghannadpour [111], the remaining multi-criteria studies mentioned above which tackled the Solomon’s instances, na- mely those of Rahoual et al. [197], Ombuki et al. [182], and Tan et al. [228], did not make their results available in a proper multi-objective manner. Instead, Ombuki et al. [182] reported their results for the solution with the smallest number of routes and for that with the shortest travel distance. Rahoual et al. [197] and Tan et al. [228] presented their results only for the solution with shortest travel distance. Following the statement above, and similarly to the single-objective case, results from these multi-objective studies are presented in two ways. The first considers the difference between their best results and the best-known, regarding both the solution with the smallest number of routes and that with the shortest travel distance. The average difference over the instances in each set category are shown in Table 3.11for the solutions with the smallest number of routes and in Table 3.12 for the solutions with the shortest travel distance. These tables have the same format as Tables 3.6 and 3.7.
In Table 3.11we see that the approach of Rahoual et al. [197] obtained the smallest difference in the number of routes (6.16%), however they only presented results for two categories. The EA of Tan et al. [228] obtained solutions which nearly doubled (13.93%) the number of routes from those obtained by Ombuki et al. [182] (7.19%), though they correspond to a saving in the travel distance (0.8%).
On the other hand, regarding solutions with the shortest travel distance, we observe in Table 3.12 that the GA of Ombuki et al. [182] achieved the smallest difference
Author C1 C2 R1 R2 RC1 RC2 Total Rahoual et al. [197] 0.00 10.78 6.16 7.18 13.17 10.60 Ombuki et al. [182] 0.00 0.00 7.83 16.67 7.35 8.33 7.19 0.01 0.13 0.29 1.16 0.03 3.19 0.77 Tan et al. [228] 0.00 0.00 10.12 32.58 7.31 30.21 13.93 0.06 0.16 -1.70 0.93 -1.75 -2.85 -0.80 Ghoseiri and Ghannadpour [111] 0.00 0.00 9.75 31.82 10.81 16.67 12.26
0.00 0.28 1.68 9.92 0.90 4.35 3.10
Table 3.11: Average difference, grouped by instance category, between results from previous multi-objective studies and the best-known results, considering the solutions with the smallest number of routes toSolomon’s benchmark set.
Author C1 C2 R1 R2 RC1 RC2 Total Rahoual et al. [197] 0.00 -0.03 -0.02 7.18 16.26 12.37 Ombuki et al. [182] 0.00 0.00 1.06 -13.31 2.27 -3.57 -2.57 0.01 0.13 2.47 1.72 3.72 3.49 1.92 Tan et al. [228] 0.00 0.00 -0.64 -31.62 -2.66 -26.53 -10.52 0.06 0.16 1.04 8.49 1.25 8.58 3.33
Ghoseiri and Ghannadpour [111] 0.00 0.00 14.38 46.97 14.84 25.00 18.00
0.00 0.28 0.80 11.15 0.41 3.87 3.01
Table 3.12: Average difference, grouped by instance category, between results from previous multi-objective studies and the best-known results, considering the solutions with the shortest travel distance toSolomon’s benchmark set.
between their results and the best-known (1.92%), followed by the approach of Gho- seiri and Ghannadpour [111] (3.01%). In contrast to the single-objective proposals, these studies were not able to find the best-known solutions for all instances in ca- tegories C1 and C2, since the difference in the travel distance is not 0%, except for the algorithm of Ghoseiri and Ghannadpour [111] which obtained the best-known solutions only for instances in category C1.
The second way of presenting results is the traditionally found in the literature, which corresponds to the format of Tables3.8and 3.9. Results from previous multi- objective studies are shown in Table 3.13, which shows, for each author (row) and instance category (column), the average number of routes (upper figure) and the
Author C1 C2 R1 R2 RC1 RC Total Rahoual et al. [197] 10.00 12.90 12.60 887.78 1362.17 1487.00 Ombuki et al. [182] 10.00 3.00 12.67 3.09 12.38 3.50 427.00 (min R) 828.48 590.60 1212.58 956.73 1379.87 1148.66 57484.35 Ombuki et al. [182] 10.00 3.00 13.17 4.55 13.00 5.63 471.00 (min D) 828.48 590.60 1204.48 893.03 1384.95 1025.31 55740.33 Tan et al. [228] 10.00 3.00 12.92 3.55 12.38 4.25 441.00 828.91 590.81 1187.35 951.74 1355.37 1068.26 56293.06 Ghoseiri and Ghannadpour
[111] (min R)
10.00 3.00 12.92 3.45 12.75 3.75 439.00
828.38 591.49 1228.60 1033.53 1392.09 1162.40 58735.22 Ghoseiri and Ghannadpour
[111] (min D)
10.00 3.00 13.50 3.82 13.25 4.00 456.00
828.38 591.49 1217.03 1049.62 1384.3 1157.41 58671.12
Table 3.13: Best average results from previous multi-objective studies, grouped by instance category, for the Solomon’s benchmark set.
average travel distance (lower figure). Additionally, the last column presents, the total number of routes and total travel distance for all 56 instances. In this case, the studies of Ombuki et al. [182] and Ghoseiri and Ghannadpour [111] present two series of results, one corresponding to the solutions with the smallest number of routes (min R) and the other regarding solutions with the shortest travel distance (min D).
From these studies, we see that the one of Ombuki et al. [182] (min R) obtained the solutions with the smallest number of routes, in all categories and in total, though they have an increased travel distance when compared with the other approaches. The EA of Tan et al. [228] achieved solutions with the shortest travel distance to instances in categories R1 and RC1, while that of Ghoseiri and Ghannadpour [111] for category C1 and that of Ombuki et al. [182] (min D) for the remaining categories C2, R2 and RC2, and in total.
These studies will also be considered in Chapters 4 and 5, when evaluating the performance of the designed multi-objective EAs, in order to known how its perfor- mance compares with previous multi-objective approaches.