Multiobjective Optimization of Bioethanol Production via Hydrolysis Using Hopfield-Enhanced Differential Evolution

Introduction

In recent times, emerging technologies in engineering optimization frequently present themselves in multi-objective (MO) settings (Eschenauer et al., 1990; Statnikov and Matusov 1995). Strategies in MO optimization can be simply classified into two groups. First being methods that use the concept of Pareto-optimality to trace out the non-dominated solutions at the Pareto curve, for instance in; Strength Pareto Evolutionary Algorithm (SPEA) (Zitzler and Thiele, 1998) and Non-dominated Sorting Genetic Algorithm II (NSGA-II) by Deb et al., (2002). The second group of methods is known as the scalarization/weighted approaches. During the application of these methods, the objective functions are aggregated into a single weighted function which is then resolved for a finite series of scalar (weight) values. Some established scalarization techniques include the Weighted Sum method (Fishburn, 1967; Triantaphyllou, 2000), Goal Programming (Luyben and Floudas, 1994) and Normal-Boundary Intersection method (NBI) (Das and Dennis, 1998). Using these techniques, the scalars are used to consign relative trade-offs to the objectives during the aggregation procedure. Hence, alternative near-optimal solutions are generated for various values of the scalars. See Eschenauer et al., (1990), Sandgren, (1994) and Statnikov and Matusov (1995) for detail investigations and explanations on MO techniques in engineering optimization.

In MO optimization problems, determining a highly efficient set of solutions can be a very daunting process. Some headway regarding this issue (revolving around concepts such as diversity and convergence) has been proposed in the last years. These ideas were then used as indicators to evaluate the solution set produced by the optimization algorithm (Grosan, 2003). Such assessments were then used to benchmark the algorithm’s performance. These concepts unfortunately could not absolutely state and rank the superiority of solution sets produced by an algorithm against other such sets by other algorithms. The only known concept that can be used generally for the overall ranking of solution sets is the idea of ‘Pareto-dominance’. The Hypervolume Indicator (HVI) (Zitzler et al., 2008) is a set measure reflecting the volume enclosed by a Pareto front approximation and a reference set (Knowles and Corne, 2003; Igel et al., 2007; Emmerich et al., 2005). The HVI thus guarantees strict monotonicity regarding Pareto dominance (Fleischer, 2003; Zitzler et al., 2003). This makes the ranking of solution sets and hence algorithms possible for any given MO problem.