Maren Urselmann

A hybrid evolutionary algorithm for solving two-stage stochastic integer programs in chemical batch scheduling

Abstract This contribution deals with the solution of two-stage stochastic integer programs with discrete scenarios (2-SIPs) that arise in chemical batch scheduling under uncertainty. Since the number of integer variables in the second-stage increases linearly with the number of scenarios considered, the real world applications usually give rise to large scale deterministic equivalent mixed-integer linear programs (MILPs) which cannot be solved easily without incorporating decomposition methods or problem specific knowledge. In this paper a new hybrid algorithm is proposed to solve 2-SIPs based on stage decomposition: an evolutionary algorithm performs the search on the first-stage variables while the second-stage subproblems are solved by mixed-integer programming. The algorithm is tested for a real-world scheduling problem with uncertainties in the demands and in the production capacity. Numerical experiments have shown, that the new algorithm is robust and superior to state-of-the-art solvers if good solutions are needed in short CPU-times.

Engineered versus standard evolutionary algorithms: A case study in batch scheduling with recourse

An engineered evolutionary algorithm for a realistic chemical batch scheduling problem with uncertain data is developed systematically. The problem is formulated as a two stage stochastic integer program with discrete scenarios. The model is solved by a stage decomposition-based hybrid algorithm using an evolutionary algorithm combined with mixed-integer programming. Earlier experiments with a standard evolutionary algorithm led to the hypothesis that the constrained search space is not covered well such that in some cases the population converges to a subset of the solution space which does not include the best known solution. An efficient engineered evolutionary algorithm is developed which is shown to cover the feasible set significantly better such that a high quality feasible schedule can be generated comparatively fast. As the hierarchical structure of the case study is typical for many batch scheduling problems, some general principles may be postulated from the experience gained here.

Design of memetic algorithms for the efficient optimization of chemical process synthesis problems with structural restrictions

Abstract In Urselmann et al., 2011a , Urselmann et al., 2011b we presented a memetic algorithm (MA) for the design optimization of reactive distillation columns. The MA is a combination of a problem-specific evolutionary algorithm (EA) that optimizes the design variables and a mathematical programming (MP) method that solves the continuous sub-problems with fixed discrete decisions which are proposed by the EA to local optimality. In comparison to the usual superstructure formulation, the search space of the MA is significantly reduced without excluding feasible solutions. The algorithm computes many different local optima and can handle structural restrictions and discontinuous cost functions. In this contribution, a systematic procedure to modify the MA to solve more complex design problems is described and demonstrated using the example of a reactive distillation column with an optional side- or pre-reactor with structural restrictions on the number of streams. New concepts to handle connected and optional unit operations are proposed.

Design of problem-specific evolutionary algorithm/mixed-integer programming hybrids: two-stage stochastic integer programming applied to chemical batch scheduling

Engineering optimization often deals with large, mixed-integer search spaces with a rigid structure due to the presence of a large number of constraints. Metaheuristics, such as evolutionary algorithms (EAs), are frequently suggested as solution algorithms in such cases. In order to exploit the full potential of these algorithms, it is important to choose an adequate representation of the search space and to integrate expert-knowledge into the stochastic search operators, without adding unnecessary bias to the search. Moreover, hybridisation with mathematical programming techniques such as mixed-integer programming (MIP) based on a problem decomposition can be considered for improving algorithmic performance. In order to design problem-specific EAs it is desirable to have a set of design guidelines that specify properties of search operators and representations. Recently, a set of guidelines has been proposed that gives rise to so-called Metric-based EAs (MBEAs). Extended by the minimal moves mutation they allow for a generalization of EA with self-adaptive mutation strength in discrete search spaces. In this article, a problem-specific EA for process engineering task is designed, following the MBEA guidelines and minimal moves mutation. On the background of the application, the usefulness of the design framework is discussed, and further extensions and corrections proposed. As a case-study, a two-stage stochastic programming problem in chemical batch process scheduling is considered. The algorithm design problem can be viewed as the choice of a hierarchical decision structure, where on different layers of the decision process symmetries and similarities can be exploited for the design of minimal moves. After a discussion of the design approach and its instantiation for the case-study, the resulting problem-specific EA/MIP is compared to a straightforward application of a canonical EA/MIP and to a monolithic mathematical programming algorithm. In view of the results the benefits of customising the EA are discussed.

A memetic algorithm for global optimization in chemical process synthesis

Design optimization problems in chemical engineering and in many other engineering domains are characterized by the presence of a large number of discrete and continuous decision variables, complex nonlinear models that restrict the search space, nonlinear cost functions, and the presence of many local optima. The classical approach to such problems are mixed integer nonlinear program solvers that work on a superstructure formulation which explicitly represents all design alternatives. The structural decisions lead to a large number of discrete variables and an exponential increase in the computational effort. The mathematical programming (MP) methods which are usually employed to solve the continuous subproblems that arise by fixing the discrete variables provide only one local optimum which depends strongly on the initialization. Thus standard methods may not find the global optimum despite long computation times. In this contribution we introduce a memetic algorithm (MA) for the global optimization of a computational demanding real-world design problem from the chemical engineering domain. The MA overcomes the problem of getting stuck in local optima by the use of an evolution strategy (ES) which addresses the global optimization of the design decisions, whereas a robust MP solver is used to handle complex nonlinear constraints as well as to improve the individuals of the ES by performing a local search in continuous sub-spaces in an integrated fashion. The MA is discussed in detail, the novel decomposition of the problem class at hand is analyzed and the MA is tested for the example of the optimal design of a reactive distillation column with several thousand decision variables. The MA is the only algorithm that finds the global solution in reasonable computation times. The introduction of structural decisions and additional constraints and discontinuous penalty terms lead only to a moderate increase in the computational effort which demonstrates the potential of this class of memetic algorithms in real-world design optimization problems.

Selection of a DFO Method for the Efficient Solution of Continuous Constrained Sub-Problems within a Memetic Algorithm for Chemical Process Synthesis

In this contribution a derivative-free memetic algorithm (MA) for the design optimization of chemical processes is introduced. Design optimization problems are characterized by nonlinear cost functions and highly constrained and multi-modal search spaces. The MA is a combination of an evolution strategy that addresses the global optimization of discrete and continuous design decisions and a derivative-free optimization method (DFO) that performs a local optimization of the continuous sub-problems that remain after fixing the discrete decisions. The MA calls a process simulation software to simulate the design alternatives, i.e. the evaluation of the objective and the constraints is a black box. In this contribution, the focus lies on the selection of a suitable DFO solver for efficiently solving the continuous constrained sub-problems. Based on latin-hypercube samplings of sub-problems of two instances of a real-world case study, surrogate models for the objective and the constraints were generated. A set of DFO methods was tested and compared on the surrogate models. The method that showed the best performance was coupled to the MA which was then applied to the real-world case study.

Derivative-Free Chemical Process Synthesis by Memetic Algorithms Coupled to Aspen Plus Process Models

Abstract Design optimization problems of chemical processes are characterized by a large number of discrete and continuous design decisions, highly non-linear models and multi-modal continuous subspaces. In our previous work, we introduced a derivative-free memetic algorithm (MA) for design optimization which is a combination of an evolutionary algorithm (EA) and a derivative-free optimization (DFO) method. The EA addresses the global optimization of all design variables, whereas the DFO method locally optimizes the continuous sub-problems that arise by fixing the discrete variables with respect to design specifications. The MA calls the simulation software Aspen Plus to simulate the design alternatives. In this contribution, the MA is extended to consider two objectives. Therefore, the selection procedure of the MA is replaced by a multi-objective selection and the continuous optimization problem which is addressed by the DFO method is reformulated.

Flowsheet Optimization by Memetic Algorithms

Abstract In this contribution, the memetic algorithm (MA) proposed in [1] is extended to optimize a flowsheet problem that comprises a reactive distillation column with optional amounts of catalyst on the stages and an (optional) external reactor such that different degrees of integration can be considered. The MA consists of an evolution strategy and a mathematical programming solver. The focus of this paper is on the influence of the presence of structural decisions which are represented as discrete variables in the optimization problem on the computational efficiency of the solution method. The introduction of discrete variables may result in an exponential increase of the computational effort needed for the solution by MINLP techniques. The results of the MA are compared to those obtained using commercially available MINLP solvers.

Evolutionäre Algorithmen zur Echtzeitoptimierung von Produktionsplänen für Chargenprozesse (Evolutionary Algorithms for the Online Optimization of Batch Production Schedules)

Die Echtzeitoptimierung von Produktionsplänen für Chargenprozesse ist gekennzeichnet durch kritische Antwortzeiten und unsichere Daten. Unsicherheitsbehaftete Entscheidungsprozesse auf gleitenden Zeithorizonten können durch Modelle mit gemischt-ganzzahliger Kompensation abgebildet werden. Der Beitrag vergleicht zwei hybridisierte evolutionäre Algorithmen für eine Fallstudie aus dieser Problemklasse. The real-time optimization of batch production schedules is characterized by critical response times and uncertain data. Decision processes under uncertainty on moving horizons can be represented by models with mixed-integer compensation. This contribution compares two hybridized evolutionary algorithms for a case study which belongs to this class of problems.

Generic vs. Engineered evolutionary algorithms in batch scheduling with recourse

Abstract This paper considers a case study of a batch chemical scheduling problem on a moving horizon with significant uncertainties in demand. The scheduling problem is represented as a two-stage stochastic integer program and solved by a stage-decomposition based hybrid algorithm with an evolutionary algorithm for the first-stage and mathematical programming for the second-stage. We describe an engineered evolutionary algorithm with systematic inclusion of process knowledge versus a generic evolutionary algorithm. The former exploits the hierarchical structure of operation, batching and scheduling decisions in the solution space representation and the mutation operator. Comparative numerical experiments show that the coverage of the feasible search space is significantly improved and the convergence to good solutions is faster.