This document contains the post-print pdf-version of the refereed paper: "A novel algorithm for fast representation of a Pareto front with adaptive resolution: Application to multi-objective optimization of a chemical reactor" by Ihab Hashem, Dries Telen, Philippe Nimmegeers, Filip Logist and Jan Van Impe which has been archived on the university repository Lirias (https://lirias.kuleuven.be/) of the Katholieke Universiteit Leuven.The content is identical to the content of the published paper, but without the final typesetting by the publisher.

Clustered microbial communities are omnipresent in the food industry, e.g., as colonies of microbial pathogens in/on food media or as biofilms on food processing surfaces. These clustered communities are often characterized by metabolic differentiation among their constituting cells as a result of heterogeneous environmental conditions in the cellular surroundings. This paper focuses on the role of metabolic differentiation due to oxygen gradients in the development of Escherichia coli cell communities, whereby low local oxygen concentrations lead to cellular secretion of weak acid products. For this reason, a metabolic model has been developed for the facultative anaerobe E. coli covering the range of aerobic, microaerobic, and anaerobic environmental conditions. This metabolic model is expressed as a multiparametric programming problem, in which the influence of low extracellular pH values and the presence of undissociated acid cell products in the environment has been taken into account. Furthermore, the developed metabolic model is incorporated in MICRODIMS, an in-house developed individual-based modeling framework to simulate microbial colony and biofilm dynamics. Two case studies have been elaborated using the MICRODIMS simulator: (i) biofilm growth on a substratum surface and (ii) submerged colony growth in a semi-solid mixed food product. In the first case study, the acidification of the biofilm environment and the emergence of typical biofilm morphologies have been observed, such as the mushroom-shaped structure of mature biofilms and the formation of cellular chains at the exterior surface of the biofilm. The simulations show that these morphological phenomena are respectively dependent on the initial affinity of pioneer cells for the substratum surface and the cell detachment process at the outer surface of the biofilm. In the second case study, a no-growth zone emerges in the colony center due to a local decline of the environmental pH. As a result, cellular growth in the submerged colony is limited to the colony periphery, implying a linear increase of the colony radius over time. MICRODIMS has been successfully used to reproduce complex dynamics of clustered microbial communities.

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The highly competitive nature of the chemical industry requires the optimisation of the design and exploitation of (bio-)chemical processes with respect to multiple, often conflicting objectives. Genetic algorithms are widely used in the context of multi-objective optimisation due to their overall straightforward implementation and numerous other advantages. NSGA-II, one of the current state-of-the-art algorithms in genetic multi-objective optimisation has, however, two major shortcomings, inherent to evolutionary algorithms: 1) the inability to distinguish between solutions based on their mutual trade-off and distribution; 2) a problem-irrelevant stopping criterion based on a maximum number of iterations. The former results in a Pareto front that contains redundant solutions. The latter results in an unnecessary high computation time. In this manuscript, a novel strategy is presented to overcome these shortcomings: t-domination. t-domination uses the concept of regions of practically insignificant trade-off (PIT-regions) to distinguish between solutions based on their trade-off. Two solutions that are located in each other’s PIT-regions are deemed insignificantly different and therefore one can be discarded. Additionally, extrapolating the concept of t-domination to two subsequent solution populations results in a problem-relevant stopping criterion. The novel algorithm is capable of generating a Pareto front with a trade-off-based solution resolution and displays a significant reduction in computation time in comparison to the original NSGA-II algorithm. The algorithm is illustrated on benchmark scalar case studies and a fed-batch reactor case study.