Los modelos de ecuaciones estructurales basados en la varianza

There are many discrete problems for which an exhaustive search is not possible. In this case, some element of searching the space becomes vital. These methods consider the problem where

X ⊆ Zⁿ

and often make use of X being an ordered set. The many issues of deterministic optimisation now become important to the problem, such as not finding the optimum.

Methods can largely be grouped into local and global optimality seekers.

One important class of algorithms, covering both local and global optimisation, are the Adaptive Random Search (ARS) algorithms, for which Hong et al. (2014), An-drad´ottir (2014) and Amaran et al. (2016) provide reviews. At iteration j these meth-ods sample a set of solutions, Xj, from X according to some distribution Fj( · |Mj), where Mj is the past information kept, such as the estimated objective value of some recently simulated solutions. The solutions in Xj are allocated replication budgets to estimate their performance or another value function, v(x). The solution with the best value is stored as the current solution, x^∗_j. The differences between algorithms come in the sampling rule (both the distribution and size of Xj), the information retained, Mj, and the value function v(x).

Global ARS keeps Fj( · |Mj) positive over much of X throughout the process, as it must balance the exploitation of a good region with the possibility that another

region may be better. Examples of this are the stochastic ruler (for example, see Nelson (2014)) and simulated annealing (for example, see Andrad´ottir (2014)), both of which take a single sample from Fj( · |Mj).

By reducing the target to local optimality, local ARS methods need not keep exploring the whole of X . Instead, methods like COMPASS (Hong and Nelson, 2006) and AHA (Xu et al., 2013) focus on a ‘most promising area’, X_j^∗, outside which Fj( · |Mj) is zero. These areas are designed to include points near the best current solution, x^∗_j, but excluding other points that have already been visited. COMPASS takes X_j^∗ to be all the points closer to x^∗_j than other visited points (by Euclidean distance). AHA uses the largest hyper-box containing x^∗_j with no other visited points in the interior of X_j^∗. Both of these examples have proven convergence properties in the sense that

Pr{x^∗_j is not locally optimal infinitely often} = 0.

Furthermore, these have useful stopping criteria where one can statistically test whether x^∗_j is locally optimal or not using a standard hypothesis test to help confidence in the finite-time results.

An approach that has grown in popularity over recent years is the use of Gaussian Random Fields. The SKOPE method proposed by Xu (2012) uses a stochastic kriging meta-model within AHA to select solutions ‘more likely’ to add improvement. At each iteration, a Latin Hypercube Design is combined with all previous observations to fit the meta-model over X_j^∗. Solutions are sampled uniformly from X_j^∗, but rather than simulating all of these, full simulation observations are only made at the solutions

with a high probability of being the best, assessed by Monte Carlo sampling from the conditional distribution defined by the meta-model. Results suggest this approach improves the performance of AHA. Salemi et al. (2019) goes further, proposing the direct use of a Gaussian Markov Random Field (GMRF) which simplifies the depen-dence structure by only allowing adjacent solutions to have non-zero relations in the model’s precision matrix. The GMRF parameters are fitted using a starting design, after which the Expected Improvement (accounting for all sources of uncertainty) is used to select the new point. This is proven to converge under mild conditions.

An alternative approach to a discrete problem is to use linear interpolation to make it a ‘continuous’ problem. One example of this is R-SPLINE (Wang et al., 2013). The interpolation allows gradient information to be used to produce a new point via a line search. This is followed by an evaluation of the solution’s neighbourhood in X .

Whilst the ARP can be formulated as an optimisation problem over X ⊂ Zⁿ, as it is for all IP formulations, there are difficulties in applying either ARS, GMRFs or interpolation methods. These primarily arise from the essentially categoric, rather than integer ordered, variables and combinatorial constraints associated with aircraft allocation, which mean that a single variable may not be independently varied whilst retaining a feasible solution.

In addition to these specialist simulation optimisation techniques, there are a range of heuristic methods. Many are inspired by metaheuristic methods in deterministic op-timisation, some of which are known to be appropriate for deterministic combinatorial optimisation problems. These include simulated annealing, evolutionary computing, ant colony optimisation and tabu search. The extension of these metaheuristics to

stochastic combinatorial optimisation problems is reviewed by Bianchi et al. (2009), including when the objective function must be estimated by simulation.

Many applications of simulation optimisation to discrete or combinatorial prob-lems use metaheuristics as the base algorithm. In a recent review of simulation op-timisation in semiconductor manufacturing, Ghasemi et al. (2018) list many papers applying genetic algorithms, particle swarm algorithms and tabu search. As part of their study of using symbiotic simulation for semiconductor manufacturing, Aydt et al. (2011) use an evolutionary algorithm to optimise tool operations. Can et al.

(2008) investigate the use of genetic algorithms for the Buffer Allocation Problem using various selection and combination methods. The authors find that whilst the algorithms generally perform well, they are sensitive to population size and the num-ber of ‘elite’ solutions. Lee et al. (2007) use a multi-objective genetic algorithm for retiming flights in an airline schedule to improve its robustness.

The primary problem with using metaheuristics is that they do not naturally account for uncertainty in the estimation of the objective function. Some imple-mentations just fix a replication number for evaluation throughout. This could be a considerable problem in scenarios where there are many sources of variability, such as semiconductor manufacturing (Ghasemi et al., 2018). Bianchi et al. (2009) discuss this issue and mention that the method used for comparing two solutions has a big impact on performance. If the uncertainty is not sufficiently controlled then the al-gorithms can produce very bad solutions. Methods for error control include adding statistical hypothesis tests before accepting a solution, growing the sample size with each iteration and adapting the sample size according to the results of hypothesis

tests. In the case of simulated annealing, Ball et al. (2018) propose a method to sequentially choose the number of replications at each solution whilst maintaining the core properties of the deterministic version of the algorithm. Other approaches include simheuristics (Juan et al., 2015), which aim to extend the metaheuristics to simulation optimisation in a different manner. These are discussed further in Section 2.3.1. The uncertainty issue is often not handled well in implementations of some metaheuristics, such as those commercially available. To aid this, Section 2.8 of Hong et al. (2014) considers three ways to improve the performance. Firstly, use a preliminary experi-ment to estimate the number of replications required to distinguish between a good and a bad solution. Secondly, restart the optimisation multiple times. Thirdly, use a R&S procedure to give some statistical confidence that the chosen solution is the best of those the algorithm found (Boesel et al., 2003).