The most difficult solution transformation-related process in solving the MLE response selection problem evolves around efficiently transitioning the search through the state subsets of the decision space with the use of inter-state solution transformations. In particular, it is important to acknowledge that the size of the subsets of the decision space associated with the states of solutions increases exponentially with the sizes of the states. In other words, considerably more solutions reside in decision subspaces associated with higher states. More importantly, however, it is also critical to acknowledge that this is not necessarily true in exclusive respect of the feasible decision space.
Configuring such a process is particularly challenging given that no research in this respect could be uncovered in the literature. Nevertheless, employing the fundamental structure of discrete probability spaces, three functional parametric configurations are proposed next, namely (1) an exponential growth distribution approach, (2) a binomial distribution approach and (3) a states clusteringapproach. Overall, the proposed configurations are particularly useful considering that they are generic enough to be applicable to any MLE response selection problem instance, and are not constrained by the size of the set of VOIs or the size of the set of MLE resources. Additionally, these configurations are expected to be able to provide a full range of transformations across all states, while simultaneously promoting exploitation and exploration of the search space.
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To be more specific, the first configuration approach above is best suited for solving MLE response selection problems in which the feasible decision space grows exponentially with the size of the states, and in which the number of states is not large. The second approach, on the other hand, is best suited for solving problems in which the highest states do not monopolise a significantly high proportion of the entire feasible decision space or, analogously, if these states contain a significant proportion of infeasible solutions within their respective domain subspaces (which may, in turn, waste valuable computational budget by generating too many infeasible solutions during the search process). Lastly, the third approach is believed to generally be most effective (in comparison to the other two approaches) when a large number of states is present, regardless of the nature of the underlying state subspaces.
The exponential growth distribution approach
Provided that an inter-state transformation has been triggered, it is required to establish a range of probabilities associated with entering any other state given that the string is currently in a certain state. Let ˜nτ ≤ nτ represent the number of VOIs in the reduced string of the current
solution. The proposed probability of performing a transformation that moves the search to a solution in state s ∈ {1, . . . , ˜nτ}, provided that it currently is in state s0 ∈ {1, . . . , ˜nτ} with
s6= s0, is given by Ps0s= PS 1 +Pn˜τ s=2αsβ− αs0β −1 , if s = 1 PSαsβ Pn˜τ s=2αsβ −1 , if s0= 1 PSαsβ 1 +Pn˜τ s=2αsβ− αs0β −1 , otherwise, (7.6)
where α, β ∈ <+are respectively linear and exponential user-defined parameters tuned according
to preferences in respect of the expected proportion of the search spent amongst the states. Here, larger values of α and β increase the expected proportion of the search process spent searching in subspaces containing solutions corresponding to larger states, with β being more sensitive to incremental changes as a result of its exponential nature. This approach therefore requires two- user defined parameters. The combined use of these two parameters results in a greater variety of inter-state configurations, and therefore in a greater degree of flexibility, in comparison to the other two approaches. Moreover, using this approach, it is easy for the user to simply choose one fixed configuration of α and β, and use it over multiple time stages (i.e. the user is most certainly not required to redefine these parameters at the start of every search process). It is possible to demonstrate that
X s∈{1,...,˜nτ} s6=s0 Ps0s PS = 1, s0 ∈ {1, . . . , ˜nτ},
which is a necessary identity for a feasible configuration of this solution transformation process. Based on (7.6), the probability of moving into any other state provided that the solution is currently in a certain state may be modelled as an ergodic Markov chain, as illustrated in Figure 7.13, with transition probability matrix
s0 s 0 P12 . . . P1˜nτ P21 0 . . . P2˜nτ .. . ... ... ... Pn˜τ1 Pn˜τ2 . . . 0
and steady-state probabilities " 1 1 +Pn˜τ s=2αsβ α(2β) 1 +P˜nτ s=2αsβ . . . α(˜n β τ) 1 +P˜nτ s=2αsβ # .
According to the law of large numbers [52], the expected proportion of the search spent in each state therefore approaches that of the entries in the vector above as the number of algorithmic iterations approaches infinity. Again, it is possible to demonstrate that the sum of these entries add up to 1. These steady-state probabilities may, however, not accurately reflect the desired search repartition if the number of iterations performed by the algorithm is too small and the number of states is too large.
1 2 P21 P12 . . . n˜τ P˜nτ1 P1˜nτ Pn˜τ2 P2˜nτ
Figure 7.13: Markov chain of inter-state transition probabilities.
The binomial distribution approach
A random variable U ∼ (˜n, p) is governed by a binomial probability distribution based on ˜n trials with success probability p if
P (U = u) = n!˜ u!(˜n− u)!p
u(1− p)˜n−u, u∈ {0, 1, . . . , ˜n}.
Here, the distribution parameter ˜n once again represents the maximum number of VOIs that may be visited in a reduced solution string, while the user-defined parameter p is responsible for steering the expected proportion of the search spent in every state. This approach therefore only requires one user-defined parameter, where the search process is expected to spend more time investigating higher-state solutions whenever a higher success probability parameter is employed. Furthermore, the probability of performing a transformation that moves the search to a solution in state u∈ {0, 1, . . . , ˜n}, provided that it currently is in state u0 ∈ {0, 1, . . . , ˜n} with u 6= u0, is given by Pu0u= PS P (U = u) 1− P (U = u0) , where X u∈{0,1,...,˜n} u6=u0 Pu0u PS = 1, u0 ∈ {0, 1, . . . , ˜n}.
Because state 0 contains only one solution, which may easily be evaluated a priori, it is noted that it would be impractical to allocate a computational budget to access this state. If P (U = 0) is very small, this issue is trivial. If this probability is not insignificant, however (such as in
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scenarios where the maximum number of states ˜n is small and the steering parameter p is not set too high), then an alternative inter-state transformation approach should be employed, or the distribution should be recalibrated accordingly so as to prevent access to state 0 within the underlying Markov chain.
Similarly to the first approach, the parameter p need not be redefined by the user at the start of every run of the search process. It may, for example, be pre-set as a function of the number of VOIs and MLE resources available at the beginning of a problem instance. Moreover, the sum of all ideal quantities of VOI interceptions (P
s∈ZNsτ), as provided by the various decision
entities (see§4.3), may also serve as a guideline as to the choice of an appropriate value for this parameter.
The states clustering approach
Due to the nature of the binomial distribution, states that are located far away from the mean are unlikely to be visited during the search process2, and this is more pronounced when a large number of states is present. Consequently, the decision maker may prefer to adopt a states clustering approach as a remedying alternative, in which groups of states are regarded as the outcomes of a certain binomial (or similarly applicable) distribution. Then, upon selection of a certain outcome, any state within the cluster represented by this outcome is entered with a certain probability, based on the number of states within the cluster as well as the cardinalities of these states. In this configuration, states which are far away from the mean are therefore allowed to be visited more frequently during the search process than in the previous approach, while states which are close to the mean value are not favoured as intensively. The states clustering approach may also be merged with the structure of the exponential growth approach if necessary.