completing its mission provided that the travel distance and travel time to that patrol circuit are within certain autonomy levels. In addition, it was proposed in §5.2.3 that the idle MLE resource management operator should provide some form of input to the MLE response selection operator with respect to end-of-route assignment preferences. Such input may be configured as a set of preferred destinations associated with each active MLE resource, represented as sets containing one or more elements from the patrol circuit and base sets.
The proposed end-of-route assignment methodology for an active MLE resource henceforth con- sists of the following steps: For each patrol circuit in Vkτp , the estimated distance and time
autonomy level of MLE resource k are assessed if it were to travel to that patrol circuit after completing its mission. These autonomy levels are calculated by subtracting the expected dis- tance and time that the MLE resource takes to travel to, and service, all VOIs assigned to its visitation route as well as covering the last arc along its route (i.e. the arc linking the last VOI on its visitation route to the patrol circuit) from respectively its distance and time autonomy levels adkτ and atkτ of that MLE resource at the beginning of time stage τ . If there exists at least one candidate patrol circuit ρ∈ Vkτp that is both within the distance and time patrol autonomy threshold parameters Ad
kρ and Atkρ of that MLE resource, then the patrol circuit at the closest
distance from the MLE resource at the end of its mission is selected as its end-of-route vertex during time stage τ . If no such patrol circuit exists, then, for each base b∈ Vkτb (assuming that |Vb
kτ| ≥ 1), the estimated distance and time autonomy level of the MLE resource is assessed if
it were to travel to that base after completing its mission during time stage τ . If there exists at least one candidate base that is both within the distance and time autonomy levels of the MLE resource in question, then the base at the closest distance from the MLE resource at the end of its mission is selected as its end-of-route vertex. If no such base exists, then the route, and therefore, the entire solution, is classified as autonomy-constraint infeasible.
The reason for selecting end-of-route vertices based on spatial proximity from the last VOI in the visitation route of an MLE resource is motivated by the idea that the length of the last arc within the visitation route only impacts on travel costs (the travelling time associated with the last arc, on the other hand, has no impact on any of the objective function values). A pseudo- code description of the above end-of-route assignment procedure is given in Algorithm 7.1. Here, the term EndofRoutekτ refers to the end-of-route vertex assigned to MLE resource k during time stage τ . Furthermore, the parameters
˜ adkτ = adkτ− ηk tVe kτ(|Vkτe |)kτ − TVkτe (|Vkτe |)τ and ˜ atkτ = atkτ − tVe kτ(|Vkτe |)kτ − TVkτe (|Vkτe |)τ
are respectively the estimated distance and time autonomy levels of MLE resource k as soon as it has finished servicing the last VOI along its visitation route during time stage τ .
7.3 Solution representation and overall transformation process
Solutions to an MLE response selection problem instance should be encoded in very specific data formats which allow for the effective application of global and local search operations, easy evaluation of objective function values, end-of-route assignments and tests for solution feasibility. The various complex dynamic features associated with this problem, however, make it difficult to standardise these data formats to be used as part of an optimisation search process. Candidate solutions must be encoded in a suitably versatile format so as to be employed by different model components in a generic manner. In particular, the solution format should accommodateAlgorithm 7.1: End-of-route assignments procedure Input : Initial autonomy levels ad
kτ and atkτ, patrol autonomy thresholds Adkρ and Atkρ, and
model management input setsVkτp and Vkτb .
Output: End-of-route assignments for every active MLE resource during time stage τ . forall k∈ Vr do 1 Endof Routekτ ← ∅ 2 if |Vekτ| ≥ 1 then 3 if |Vkτp | ≥ 1 then 4 forall ρ∈ Vkτp do 5 Let ρdkτ = ˜adkτ − dVe kτ(|Vkτe |)ρkτ 6 Let ρt kτ = ˜atkτ − dVe kτ(|Vkτe |)ρkτ(ηk) −1 7
if there exists at least one ρ∈ Vkτp such that ρd
kτ ≥ Adkρ and ρtkτ ≥ Atkρ then 8
Endof Routekτ ← max{ρ, ρdkτ} 9 else if |Vb kτ| ≥ 1 then 10 forall b∈ Vkτb do 11 Let bd kτ = ˜adkτ− dVe kτ(|Vkτe |)bkτ 12 Let btkτ = ˜atkτ− dVe kτ(|Vkτe |)bkτ(ηk) −1 13
if there exists at least one b∈ Vb
kτ such thatbdkτ ≥ 0 and btkτ ≥ 0 then 14
Endof Routekτ ← max{b, bdkτ} 15
else
16
There exists no feasible end-of-route assignment for the current route
17
else
18
There exists no feasible end-of-route assignment for the current route
19
else
20
Go to step 11
7.3. Solution representation and overall transformation process 143
certain model management features in which subjective requirements are dictated by the MLE response selection operators on a temporal basis. The encoding scheme proposed for use in this dissertation is illustrated for a hypothetical problem instance in time stage τ0 with the following parameters: Vb = {B1, B2}, Vr = {a, b, c}, V0τr 0 = {0a, 0b, 0c}, Vτe0 = {1, 2, 3, 4, 5, 6, 7, 8} and Vp={P
1, P2, P3, P4}.
In the literature, solutions to a VRP instance are typically encoded as strings, which comprise substrings representing routes consisting of a subset of customers scheduled to be visited by a particular vehicle. The order in which customers are entered in such a substring is also the order in which the assigned vehicle visits them along its route. An example of such a solution string for the above hypothetical MLE response selection problem instance is presented in Figure 7.2. In the first route (substring), for instance, MLE resource a is scheduled to first visit VOI 2 from its departure point 0a, and then VOI 5, after which it is scheduled to relocate to base B1. In
terms of the decision variables of the model formulation of Chapter 4, this part of the solution associated with MLE resource a may be written as x0a2aτ0 = 1, x25aτ0 = 1, x5B1aτ0 = 1, and xijaτ0 = 0 otherwise. In addition, y2aτ0 = 1, y5aτ0 = 1, and yiaτ0 = 0 otherwise.
0a 2 5 B1 0b 3 6 7 B2 0c 8 4 1 P4
Figure 7.2: Example of a solution string encoding for the MLE response selection problem.
The string encoding scheme described above may be simplified by first removing the initial and end-of-route cells, and then inserting dummy cells containing zero elements instead as route delimiters, as shown in Figure 7.3. In this simplified string configuration, the sequence of integers between any two specific zeros therefore represents the order of the VOIs that have to be visited by a specific MLE resource in the set Vr. In the case of idle MLE resources, the respective
routes are, of course, left empty. This is represented by means of two consecutive zeros within the string.
2 5 0 3 6 7 0 8 4 1
Figure 7.3: Simplified solution string encoding with route delimiters corresponding to the expanded string encoding of Figure 7.2.
Although not indispensable, part of configuring a solution string involves accommodating the various complexities associated with the dynamic model constituents of the problem, which renders a solution highly sensitive to being ejected into the infeasible region of decision space whilst employing certain global search operators. This may create an excessive rate of infeasible neighbouring solutions generated during a stochastic solution transformation process, which may severely handicap the progression of the solution search process in the sense of wasting considerable amounts of time finding feasible transformations.
In particular, the use of VOI inclusion sets, as proposed in §5.2.2, suggests that any solution transformation resulting in the removal of one or more VOIs belonging to inclusion sets of their respective routes will generate infeasible neighbouring solutions. Even more extreme, the use of VOI ordered inclusion sets requires the VOI to be reinserted in exactly the right position within a specific route so as to maintain feasibility in that regard. Although not as extreme, the use of VOI exclusion sets, as proposed in§5.2.1, similarly implies that any solution transformation resulting in the insertion of one or more VOIs within a route belonging to the
corresponding exclusion sets will generate infeasible neighbouring solutions. Provided that the solution exploration transformations are performed at random, this problem is expected to be exacerbated as a result of three factors: (1) the number of VOIs belonging to VOI inclusion sets and VOI exclusion sets (where more is worse), (2) the number of scheduled active MLE resources (where fewer is worse), and (3) the probability of conducting an exploration transformation of the current solution (where a higher probability is worse). Furthermore, the combined assignment features presented in §5.3.1 may require certain cells (other than the dummy delimiter cells containing zero elements) to appear multiple times within specific routes and at specific positions within these routes.
One way of alleviating this shortcoming is to remove the VOIs belonging to VOI inclusion sets, VOI exclusion sets and combined assignments from the solution string, then to carry out the solution transformation process with respect to the reduced string, and finally to reinsert these VOIs strategically into feasible positions in such a way that they agree with the definitions implied by these sets1.
Another way of eliminating this shortcoming is to simply not consider VOIs belonging to such model management sets as candidate vertices to solution exploration transformation procedures. This method should preferably be employed whenever a relatively large number of VOIs is present, as reinserting those (even strategically) every time into a reduced string might be detrimental to the efficient exploitation of solutions mapped closely to or onto the Pareto front. Returning to the example above, suppose that the operator believes, based on his experience, that it is necessary to fix Iaτ0 = {2} and Ebτ0 = {4} as part of the MLE response selection operations conducted during the current time stage. Adopting the first alternative above, VOIs belonging to any of these sets may then temporarily be removed from the current string, as shown in Figure 7.4.
5 0 3 6 7 0 8 1
Figure 7.4: Solution string encoding after removing VOIs belonging to inclusion and exclusion sets.
In the solution methodology proposed in this section, the overall solution transformation process consists of three independent, successive sub-transformations performed on a reduced string. These sub-transformations processes are called general sub-transformations, reinsertion sub- transformations and end-of-route assignment sub-transformations. A general sub-transformation is first performed, during which a random reduced neighbouring string is generated from the current reduced string. In Figure 7.5, an inter-route sub-transformation is performed by re- moving VOI 3 and VOI 6 from the second substring (route) and reverse-inserting it into the first substring, while VOI 5 is removed from the first substring and inserted into the second substring.
6 3 0 7 5 0 8 1
Figure 7.5: A general sub-transformation performed on the reduced string encoding of Figure 7.4.
Following this sub-transformation, the VOIs belonging to inclusion and exclusion sets that were 1This does not mean that the generated neighbouring solution will always be feasible, as there are other