Thomas Tometzki

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.

Systematic Initialization Techniques for Hybrid Evolutionary Algorithms for Solving Two-Stage Stochastic Mixed-Integer Programs

This paper introduces new initialization approaches for evolutionary algorithms that solve two-stage stochastic mixed-integer problems. The two-stage stochastic mixed-integer programs are handled by a stage decomposition based hybrid algorithm where an evolutionary algorithm handles the first-stage decisions and mathematical programming handles the second-stage decisions. The population of the evolutionary algorithm is initialized by using solutions which are generated in a preprocessing step of the hybrid algorithm. This paper presents three different initialization approaches in which the two-stage stochastic mixed-integer program is exploited in order to obtain potentially good starting solutions for the evolutionary algorithm. In case of infeasible initializations, the population is driven toward feasibility by a penalty function. Comparisons of an evolutionary algorithm with a classical random initialization and the new initialization approaches for two real-world problems show that the new initialization approaches lead to high quality feasible solutions in significantly shorter computing times.

Hybrid evolutionary optimization of two-stage stochastic integer programming problems: An empirical investigation

In this contribution, we consider decision problems on a moving horizon with significant uncertainties in parameters. The information and decision structure on moving horizons enables recourse actions which correct the here-and-now decisions whenever the horizon is moved a step forward. This situation is reflected by a mixed-integer recourse model with a finite number of uncertainty scenarios in the form of a two-stage stochastic integer program. A stage decomposition-based hybrid evolutionary algorithm for two-stage stochastic integer programs is proposed that employs an evolutionary algorithm to determine the here-and-now decisions and a standard mathematical programming method to optimize the recourse decisions. An empirical investigation of the scale-up behavior of the algorithms with respect to the number of scenarios exhibits that the new hybrid algorithm generates good feasible solutions more quickly than a state of the art exact algorithm for problem instances with a high number of scenarios.

Multi-product batch scheduling with intermediate due dates using priced timed automata models

In this contribution, we discuss an extension of the earlier work on scheduling using reachability analysis of timed automata (TA) models, specifically addressing the problem of tardiness minimization. In the TA-based approach the resources, recipes and additional timing constraints are modeled independently as sets of priced timed automata. The sets of individual automata are synchronized by means of synchronization labels and are composed by parallel composition to form a global automaton. The global automaton has an initial location where no operations have been started and at least one target location where all operations that are required to produce the demanded quantities of end-products within the specified due dates have been finished. A cost-optimal symbolic reachability analysis is performed on the composed automaton to derive schedules with the objective of minimizing tardiness. The model formulation is extended to include release dates of the raw materials and due dates of the production orders. The meeting of due dates is modeled by causing additional costs (e.g. penalties for late delivery and storage costs for early production). The modeling approach and the performance of the approach are tested for two different case studies and the results are compared with that of a MILP formulation solved using the standard solver CPLEX. The numerical experiments demonstrate, that the TA-based approach is competitive compared to standard commercial solvers and good feasible solutions are obtained with considerably reduced computational effort.

Risk conscious solution of planning problems under uncertainty by hybrid multi-objective evolutionary algorithms

Abstract We consider the risk conscious solution of planning problems with uncertainties in the problem data. The problems are formulated as two-stage stochastic mixed-integer models in which some of the decisions (first-stage) have to be made under uncertainty and the remaining decisions (second-stage) can be made after the realization of the uncertain parameters. The uncertain model parameters are represented by a finite set of scenarios. The risk conscious optimization problem under uncertainty is solved by a stage decomposition approach using a multi-objective evolutionary algorithm which optimizes the expected scenario costs and the risk criterion with respect to the first-stage decisions. The second-stage scenario decisions are handled by mathematical programming. Results from numerical experiments for two real-world problems are shown.

Risk Management in Production Planning under Uncertainty by Multi-Objective Hybrid Evolutionary Algorithms

Abstract We consider production planning problems with uncertainties in the problem data. The optimization problems are formulated as two-stage stochastic mixed-integer models in which some of the decisions (first-stage) have to be made under uncertainty and the remaining decisions (second-stage) can be made after the realization of the uncertain parameters. The uncertain model parameters are represented by a finite set of scenarios. The production planning problem under uncertainty is solved by a stage decomposition approach using a multi-objective evolutionary algorithm which takes the expected scenario costs and a risk criterion into account to compute the first-stage variables. The second-stage scenario decisions are handled by mathematical programming. Results from numerical experiments for a multi-product batch plant are presented.

A hybrid multiple populations evolutionary algorithm for two-stage stochastic mixed-integer disjunctive programs

This article describes a hybrid multiple populations based evolutionary approach for disjunctive mathematical programs with uncertainties in the problem data. The problems are formulated as two-stage linear disjunctive programming problems which are solved by a stage decomposition based hybrid algorithm using multiple evolutionary algorithms to handle the disjunctive sets of the here-and-now (first stage) decisions and mathematical programming to handle the recourse (second stage) decisions. By an appropriate representation of the first-stage disjunctive solution space, the overall problem is decomposed into smaller subproblems without disjunctions. The resulting decomposed first-stage subproblems are solved independently by evolutionary algorithms, leading to parallel evolutions based on multiple populations. During the progress of the optimization, the number of subproblems is systematically reduced by comparing the current best global solution (upper bound) to lower bounds for the subproblems. This approach guaranties that the global optimal solution remains in the union of solution spaces of the remaining subproblems. A comparison of a classical evolutionary algorithm and the new multiple populations evolutionary algorithm for a real world batch scheduling problem shows that the new approach leads to a significantly improved coverage of the set of feasible solutions such that high quality feasible solutions can be generated faster.

OPTIMIZING HYBRID DYNAMIC PROCESSES BY EMBEDDING GENETIC ALGORITHMS INTO MPC

Abstract Optimizing the operation of processes with hybrid dynamics is challenging since the discrete dynamics (i.e. abrupt changes of states or inputs) introduces discontinuities with which gradient-based solvers often cannot cope very well. This contribution suggests a scheme that combines model predictive control (MPC) with genetic algorithms and embedded simulation of the hybrid dynamics. As demonstrated for the example of a chemical reactor, the genetic algorithm provides good results even if the prediction horizon includes points of discontinuities of the continuous dynamics.

Batch scheduling with intermediate due dates using timed automata models

Abstract In the process industries, the problem of scheduling multi-product batch plants to satisfy the demand for various end-products within specified due dates occurs frequently. In contrast to the state-of-the art approach of using mathematical model formulations to solve such scheduling problems, an alternative approach is to use reachability analysis for timed automata (TA). In this paper, we discuss an extension of our earlier work on scheduling using TA models where the problem of makespan minimization was addressed. We extend the formulation to the meeting of due dates, modelled as causing additional costs (e.g. penalties for late delivery and storage costs for early production). The proposed solution by reachability analysis of priced timed automata is tested on a case study to demonstrate its successful application.

Two-Layer Planning and Scheduling of Batch Production Processes Under Uncertainty

This paper proposes a new combined algorithm to solve planning and scheduling problems of batch production processes under uncertainty. For the planning problem under uncertainty, a two-stage stochastic integer program is formulated and solved by a stage-decomposition algorithm. To evaluate the feasibility of a production plan, a priced timed automata model is set up for the scheduling problem of the next planning period and solved using reachability analysis. If the targets from the planning layer cannot be met, a penalty term is returned to the planning layer which is used to re-evaluate the candidate production plans. The two-layer concept is applied to a multi-product batch plant demonstrator.