Modular Multi-Objective Genetic Algorithm for Large Scale Bi-level Problems University of Trieste Stefano Costanzo Supervisor: L. Castelli Co-Supervisor: A. Turco > Abstract Genetic Algorithms are search heuristics that mimic the process of natural selection. They belong to the larger class of evolutionary algorithms. This kind of optimization algorithms are well-suited for complex real application problems that do not have an exact solution. Furthermore, GAs can be integrated with the Pareto Dominance Criteria to solve multi-objective problems. The proposed Multi-Objective Genetic Algorithm for Structured Inputs, MOGASI, incorporates modules and operators of both general purpose and custom GA. It is designed for an efficient solving of generic problems and those with a specific data structure. It has been tested and its efficiency confirmed on different multi-objective optimization problems against other genetic algorithms, well-known in literature. The module-based structure of this algorithm has been represented in form of a BPMN workflow in the framework of a regional European project, which opens up new possibilites for the definition of engineering and scientific workflows. The MOGASI algorithm has been successfully used for the solution of a real-world problem, i.e. the application of a peak-load pricing policy in the management of the European air traffic with the goal of reducing congestion, minimizing delays and maintaining air navigation service providers’ revenues within a specified threshold. Results show that significant improvements in traffic distribution can be achieved through this simple charges mod- ulation scheme. 1. Genetic Approach The concept of modularity has been intensively studied in evolutionary computation as a way of improving the innovativeness and scalability of the evolutionary search. Many authors suggest that evolutionary algorithms are suitable for being as- sembled progressively to enhance the optimization process. In our vision, modularity is the ability to freely combine and reuse algorithm components. If we consider the classic genetic algorithm (GA) phases as assemblable modules, in theory such modules could be chosen from a wide variety of specialized GAs discussed in literature. MOGASI (Multi-Ob- jective Genetic Algorithm for Structured Inputs) was construed having precisely this idea in mind. The proposed decomposition has two aspects. The first one consists in splitting the GA phases in communicating sep- arate modules, so that each block would remain independent from the specific implementation of the other modules relying only on the data produced and shared by those modules. The second aspect consists in the identification of tar- get data structures based on decision variable characteristics, which are grouped in macro-categories and handled by specific strategies. A pre-processing phase has been implemented in MOGASI to reduce the global problem complexity. Pre-processing methods include a number of simplification, elimination and shrinking methods discussed in literature, aimed in particular at simplifying constraints and narrowing the bounds imposed on the decision variables. Further- more, since during the optimization standard genetic operators may break the structure imposed by the encoding of structured decision variables, a generic repair mechanism (FIXER) has been introduced for correcting such structure violations and increasing the possibility that only feasible solutions are generate. MOGASI has been designed for high-performance on both general problems and those with specific structures. Its behavior was as- sessed based on test taken from the famous Deb library. The goal was to identify a better Pareto front with respect to other compara- ble genetic algorithm approaches, in particular MOGA-II (Multi-Ob- jective Genetic Algorithm) and NSGA-II (Non-Dominated Sorting Al- gorithm). The Inverted Generational Distance (IGD) was selected as the performance metric due to its ability to measure the distance of the found non-dominated front from a reference sample of the real Pareto front and to collect information regarding the accura- cy and the uniformity of the computed front (lower IGD values are preferred). The charts on the left clearly show the MOGASI superior performance on 20,000 design evaluations in two multi-objective minimization problems. 2. BPMN Decomposition Engineering and scientific workflows are widely used, but there is currently no accepted standard for their definition. In an effort to provide a standard approximation to work- flow representation in engineering applications, Comin et al. propose the use of a well-defined standard from the area of business processes, BPMN 2.0, as a formal representation of optimization workflows, which can be used for both pro- cess representation and model execution. BPMN workflows can be easily used for GA decomposition. The workflow on the right has been developed and imple- mented in collaboration with ESTECO SpA in form of a pro- totype in the framework of the regional European project titled “Collaborative Methodology for the Definition of Engi- neering Design Processes”. A Genetic Algorithm (GA) is decomposed into its main sub-procedures: offspring solution generation, evaluation, selection and stopping condition. It would be possible to represent these phases with call activi- ties and to provide a BPMN model also for their inner structures, but this will be addressed in future works. The aim of this process is to show that it is possible to define the building blocks of any optimization algorithm, which opens the door to the possibility of customizing and hybridizing optimization algorithms depending on the specific application problem. 3. Bi-level Central Scheme Peak-load pricing (PLP), a two-tariff charging scheme commonly used in public transport and utilities, has been tested on the European Air Traffic Management (ATM) system as a means for reduc- ing airspace congestion. In particular, a centralised approach to PLP with a Central Planner (CP) set- ting out peak and off-peak tariffs on entire Europe- an network is proposed. In the first phase congested airspace sectors and their peak and off-peak hours are identified. In a second phase the CP assesses and imposes tariffs to reduce the overall shift on the network, where shift is intended as difference between the actual and the scheduled flight departure time. Such differentiated charges should guarantee that Air Navigation Service Pro- viders (ANSPs) are able to recover their operational costs (the so called Revenue Neutrality condition) while inducing the Airspace Users (AUs) to route their traffic in a configuration that the network is able to sustain. The interaction be- tween CP and AUs is modelled as a Stackelberg game, where the actions of the leader affect the choices of the follower. In this context the CP is the leader, setting the peak and off-peak tariffs, and the AUs are the followers, making a routing choice based on those tariffs. This scheme has been formulated as a bilevel linear programming problem. The leader level is tackled by MOGASI and has the peak and off-peak tariffs as variables for each ANSP and two ob- jectives: minimization of the total shift time for all flights and minimization of the maximum difference in the revenue neutrality of every ANSP. The follower level is tackled by a specially developed exact solver, the objective of which is to perform each flight at minimum total cost. The optimization problem at leader-level has been constrained by a maxi- mum allowed violation of the average capacity of all sectors and of the revenue neutrality of 20%. The formulation of the problem as bi-level multi-objective optimisation was effective, but the presence of a large number of solutions required a post-processing analysis. The Parallel Coordinates chart on the following figure depicts multivar- iate data sets and solutions in predefined ranges represented by vertical lines, whereas each solution is represented by a coloured polyline. Four representative Pareto solutions have been chosen. Variables shown on the chart: • TGS - Total sum of Global Shift over all flights • ACV - Average Capacity Violation • RNV - Maximum Revenue Neutrality Violation • nCB - Number of Capacity Breaches (number of flights violating the capacities) • nCV - Number of Capacity Violations (number of (sector/airport, hour) pairs that violate their nominal capacity levels) The results of the optimization driven by MOGASI are compared with the baseline solution presented as the red line. It exhibits the highest TGS while the revenue neutrality is perfectly matched for each ANSP. The most desirable solution of all is represented by the green line as it significantly improves TGS and keeps the revenue neutrality violation at around the 10% with respect to the baseline solution. The presented solutions show that the modulation of en-route charges indeed redistributes the traffic under each point of view. A range of interesting alternative solutions has been obtained with the applied heuristics. The presented PhD research work has been financed by ESTECO S.p.A.
Modular Multi-Objective Genetic Algorithm for Large Scale Bi-level Problems
Apr 13, 2017
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Modular Multi-Objective GeneticAlgorithm for Large Scale Bi-level Problems
University of Trieste
Stefano CostanzoSupervisor: L. CastelliCo-Supervisor: A. Turco
> AbstractGenetic Algorithms are search heuristics that mimic the process of natural selection. They belong to the larger class of evolutionary algorithms. This kind of optimization algorithms are well-suited for complex real application problems that do not have an exact solution. Furthermore, GAs can be integrated with the Pareto Dominance Criteria to solve multi-objective problems.
The proposed Multi-Objective Genetic Algorithm for Structured Inputs, MOGASI, incorporates modules and operators of both general purpose and custom GA. It is designed for an efficient solving of generic problems and those with a specific data structure. It has been tested and its efficiency confirmed on different multi-objective optimization problems against other genetic algorithms, well-known in literature. The module-based structure of this algorithm has been represented in form of a BPMN workflow in the framework of a regional European project, which opens up new possibilites for the definition of engineering and scientific workflows.
The MOGASI algorithm has been successfully used for the solution of a real-world problem, i.e. the application of a peak-load pricing policy in the management of the European air traffic with the goal of reducing congestion, minimizing delays and maintaining air navigation service providers’ revenues within a specified threshold. Results show that significant improvements in traffic distribution can be achieved through this simple charges mod-ulation scheme.
1. Genetic ApproachThe concept of modularity has been intensively studied in evolutionary computation as a way of improving the innovativeness and scalability of the evolutionary search. Many authors suggest that evolutionary algorithms are suitable for being as-
sembled progressively to enhance the optimization process. In our vision, modularity is the ability to freely combine and reuse algorithm components. If we consider the classic genetic algorithm (GA) phases as assemblable modules, in theory such modules could be chosen from a wide variety of specialized GAs discussed in literature. MOGASI (Multi-Ob-jective Genetic Algorithm for Structured Inputs) was construed having precisely this idea in mind.
The proposed decomposition has two aspects. The first one consists in splitting the GA phases in communicating sep-arate modules, so that each block would remain independent from the specific implementation of the other modules relying only on the data produced and shared by those modules. The second aspect consists in the identification of tar-get data structures based on decision variable characteristics, which are grouped in macro-categories and handled by specific strategies. A pre-processing phase has been implemented in MOGASI to reduce the global problem complexity. Pre-processing methods include a number of simplification, elimination and shrinking methods discussed in literature, aimed in particular at simplifying constraints and narrowing the bounds imposed on the decision variables. Further-more, since during the optimization standard genetic operators may break the structure imposed by the encoding of structured decision variables, a generic repair mechanism (FIXER) has been introduced for correcting such structure violations and increasing the possibility that only feasible solutions are generate.
MOGASI has been designed for high-performance on both general problems and those with specific structures. Its behavior was as-sessed based on test taken from the famous Deb library. The goal was to identify a better Pareto front with respect to other compara-ble genetic algorithm approaches, in particular MOGA-II (Multi-Ob-jective Genetic Algorithm) and NSGA-II (Non-Dominated Sorting Al-gorithm). The Inverted Generational Distance (IGD) was selected as the performance metric due to its ability to measure the distance of the found non-dominated front from a reference sample of the real Pareto front and to collect information regarding the accura-cy and the uniformity of the computed front (lower IGD values are preferred). The charts on the left clearly show the MOGASI superior performance on 20,000 design evaluations in two multi-objective minimization problems.
2. BPMN DecompositionEngineering and scientific workflows are widely used, but there is currently no accepted standard for their definition. In an effort to provide a standard approximation to work-flow representation in engineering applications, Comin et al. propose the use of a well-defined standard from the area of business processes, BPMN 2.0, as a formal representation of optimization workflows, which can be used for both pro-cess representation and model execution. BPMN workflows can be easily used for GA decomposition.
The workflow on the right has been developed and imple-mented in collaboration with ESTECO SpA in form of a pro-totype in the framework of the regional European project titled “Collaborative Methodology for the Definition of Engi-neering Design Processes”. A Genetic Algorithm (GA) is decomposed into its main sub-procedures: offspring solution generation, evaluation, selection and stopping condition. It would be possible to represent these phases with call activi-ties and to provide a BPMN model also for their inner structures, but this will be addressed in future works. The aim of this process is to show that it is possible to define the building blocks of any optimization algorithm, which opens the door to the possibility of customizing and hybridizing optimization algorithms depending on the specific application problem.
3. Bi-level Central SchemePeak-load pricing (PLP), a two-tariff charging scheme commonly used in public transport and utilities, has been tested on the European Air Traffic Management (ATM) system as a means for reduc-ing airspace congestion. In particular, a centralised approach to PLP with a Central Planner (CP) set-ting out peak and off-peak tariffs on entire Europe-an network is proposed. In the first phase congested airspace sectors and their peak and off-peak hours are identified. In a second phase the CP assesses and imposes tariffs to reduce the overall shift on the network, where shift is intended as difference between the actual and the scheduled flight departure time. Such differentiated charges should guarantee that Air Navigation Service Pro-viders (ANSPs) are able to recover their operational costs (the so called Revenue Neutrality condition) while inducing the Airspace Users (AUs) to route their traffic in a configuration that the network is able to sustain. The interaction be-tween CP and AUs is modelled as a Stackelberg game, where the actions of the leader affect the choices of the follower. In this context the CP is the leader, setting the peak and off-peak tariffs, and the AUs are the followers, making a routing choice based on those tariffs. This scheme has been formulated as a bilevel linear programming problem.
The leader level is tackled by MOGASI and has the peak and off-peak tariffs as variables for each ANSP and two ob-jectives: minimization of the total shift time for all flights and minimization of the maximum difference in the revenue neutrality of every ANSP. The follower level is tackled by a specially developed exact solver, the objective of which is to perform each flight at minimum total cost. The optimization problem at leader-level has been constrained by a maxi-mum allowed violation of the average capacity of all sectors and of the revenue neutrality of 20%.
The formulation of the problem as bi-level multi-objective optimisation was effective, but the presence of a large number of solutions required a post-processing analysis. The Parallel Coordinates chart on the following figure depicts multivar-iate data sets and solutions in predefined ranges represented by vertical lines, whereas each solution is represented by a coloured polyline. Four representative Pareto solutions have been chosen.
Variables shown on the chart:
• TGS - Total sum of Global Shift over all flights• ACV - Average Capacity Violation• RNV - Maximum Revenue Neutrality Violation• nCB - Number of Capacity Breaches (number of flights violating the capacities)• nCV - Number of Capacity Violations (number of (sector/airport, hour) pairs that violate their nominal capacity levels)
The results of the optimization driven by MOGASI are compared with the baseline solution presented as the red line. It exhibits the highest TGS while the revenue neutrality is perfectly matched for each ANSP. The most desirable solution of all is represented by the green line as it significantly improves TGS and keeps the revenue neutrality violation at around the 10% with respect to the baseline solution.The presented solutions show that the modulation of en-route charges indeed redistributes the traffic under each point of view. A range of interesting alternative solutions has been obtained with the applied heuristics.
The presented PhD research work has been financed by ESTECO S.p.A.
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