(104e) A Priori Objective Dimensionality Reduction for Many-Objective Optimization
AIChE Annual Meeting
2022
2022 Annual Meeting
Computing and Systems Technology Division
Advances in Mixed-Integer Optimization and Optimization with Logistics Applications
Monday, November 14, 2022 - 1:42pm to 2:00pm
Analyzing the sustainability of a chemical process design, supply chain, or control scheme is an increasingly important aspect of decision-making in chemical industries. Sustainability inherently requires considering a large number of objectives beyond the traditional economic scope. These additional sustainability objectives come from the environmental and social impacts of a decision including carbon emissions, water usage, job creation, social equity, and so on. Scalarization techniques such as the weighted-sum and epsilon constraint methods rigorously produce the Pareto frontier of solutions for a multi-objective optimization problem which is a valuable decision-making tool. However, for problems with more than three objectives, called many-objective optimzation (MOO) problems, rigorous scalarization methods scale poorly and heuristic evolutionary methods are often applied. In sustainability studies the number of objectives can quickly make producing a rigorous Pareto frontier a computationally expensive task. Traditionally, this has been mitigated in sustainability studies by lumping objectives into the three pillars of sustainability: economics, environmental, and social. Unfortunately, this approach inherently results in objective trade-off information loss. As the number of objectives grows, conflict will inevitably occur between objectives within the same category. As an intuitive example occurring within the social pillar, consider that in a supply chain as more jobs are created by building more facilities throughout the region, more people suffer from worsened air quality due to closer proximity to chemical facilities. By grouping these objectives together within the social categorical objective this trade-off information is lost. A more systematic approach is to generate set of Pareto-optimal solutions and determines objective correlation using principal component analysis1. A major drawback of this method include that it requires generation of a set of Pareto-optimal solutions. As a result it may be susceptible to imposed bias from the set of generated solutions and changes in parameters will necessitate a new set of Pareto-optimal solutions.
In this work we propose an a priori method for determination of objective groups without requiring pre-generation of a set of solutions. The method utilizes the underlying problem structure of linear MOO problems. Linear MOO problems can be represented as a tripartite graph with nodes corresponding to objectives, variables, and constraints and edges linking objective and constraint nodes to variables included in their formulation. Exploiting this structure entails taking the objective and constraint coefficients and determining the relative importance of the shared variables towards reaching the optimal solution. Normalized coefficient vectors for each objective are used in calculating the direct link strength. Additionally, objectives that are linked through constraints have their secondary strength determined by examining the directions objective functions want to move along the hyperplanes of inequality constraints. A weighted sum is used to combine primary and secondary correlation strengths and these values are used as the edge weights in the objective space graph. Community detection is then used to identify groups of objectives that are most strongly correlated2. We also propose a metric to quantify the information lost through the grouping of objectives. This metric is used to compare the groups our method identifies with traditional categorical objective combination.
The methods developed are applied to a case study based on previously published work on a renewable-powered distributed ammonia supply chain3. The published works explored economic objectives and a multi-objective study of the tradeoff between economics and carbon emissions. Our work considers capital cost and operating cost as separate objectives as well as including water usage as an objective. The proposed method identifies a correlated relationship between operating cost and emissions which goes against the conventional objective grouping method where these correlated objectives would be in separate economic and environmental groups. The proposed information loss metric is used to verify that the pair identified by the algorithm reduces the lost information, producing a more useful decision-making tool.
In this work we propose an a priori method for determination of objective groups without requiring pre-generation of a set of solutions. The method utilizes the underlying problem structure of linear MOO problems. Linear MOO problems can be represented as a tripartite graph with nodes corresponding to objectives, variables, and constraints and edges linking objective and constraint nodes to variables included in their formulation. Exploiting this structure entails taking the objective and constraint coefficients and determining the relative importance of the shared variables towards reaching the optimal solution. Normalized coefficient vectors for each objective are used in calculating the direct link strength. Additionally, objectives that are linked through constraints have their secondary strength determined by examining the directions objective functions want to move along the hyperplanes of inequality constraints. A weighted sum is used to combine primary and secondary correlation strengths and these values are used as the edge weights in the objective space graph. Community detection is then used to identify groups of objectives that are most strongly correlated2. We also propose a metric to quantify the information lost through the grouping of objectives. This metric is used to compare the groups our method identifies with traditional categorical objective combination.
The methods developed are applied to a case study based on previously published work on a renewable-powered distributed ammonia supply chain3. The published works explored economic objectives and a multi-objective study of the tradeoff between economics and carbon emissions. Our work considers capital cost and operating cost as separate objectives as well as including water usage as an objective. The proposed method identifies a correlated relationship between operating cost and emissions which goes against the conventional objective grouping method where these correlated objectives would be in separate economic and environmental groups. The proposed information loss metric is used to verify that the pair identified by the algorithm reduces the lost information, producing a more useful decision-making tool.
1 Saxena, D.K., Duro, J.A., Tiwari, A., Deb, K., Zhang, Q. Objective Reduction in Many-Objective Optimization: Linear and Nonlinear Algorithms. (2013) IEEE Transactions on Evolutionary Computation., 17(1), 77-99.
2 Traag, V.A., Waltman, L., van Eck, N.J. From Louvain to Leiden: guaranteeing well-connected communities. (2019) Scientific Reports., 9, 5233.
3 Palys, M.J., Allman, A., Daoutidis, P. Exploring the Benefits of Modular Renewable-Powered Ammonia Production: A Supply Chain Optimization Study. (2019) Ind. Eng. Chem. Res.., 58(15), 5898-5908.