Genetic algorithms and Monte Carlo simulation for optimal plant design

Abstract

We present an approach to the optimal plant design (choice of system layout and components) under conflicting safety and economic constraints, based upon the coupling of a Monte Carlo evaluation of plant operation with a Genetic Algorithms-maximization procedure. The Monte Carlo simulation model provides a flexible tool, which enables one to describe relevant aspects of plant design and operation, such as standby modes and deteriorating repairs, not easily captured by analytical models. The effects of deteriorating repairs are described by means of a modified Brown–Proschan model of imperfect repair which accounts for the possibility of an increased proneness to failure of a component after a repair. The transitions of a component from standby to active, and vice versa, are simulated using a multiplicative correlation model. The genetic algorithms procedure is demanded to optimize a profit function which accounts for the plant safety and economic performance and which is evaluated, for each possible design, by the above Monte Carlo simulation.

In order to avoid an overwhelming use of computer time, for each potential solution proposed by the genetic algorithm, we perform only few hundreds Monte Carlo histories and, then, exploit the fact that during the genetic algorithm population evolution, the fit chromosomes appear repeatedly many times, so that the results for the solutions of interest (i.e. the best ones) attain statistical significance.

Introduction

When designing an industrial plant one must take into account the conflicting objectives of safe operation and economic service. This entails that an engineering analysis aiming at assessing the reliability, availability and safety levels of plant operation should be coupled with an economic analysis which estimates the associated costs of plant downtime, maintenance and repair.

In his work, the design engineer is faced with several choices concerning the type and number of components to be used and their assembly configuration. The just mentioned conflict between reliability/availability objectives and economic costs entails finding suitable approaches for identifying optimal solutions to the design problem. In the literature, such approaches included gradient methods, dynamic programming, integer programming, mixed integer and non-linear programming, and heuristics. A review of the work in this field can be found in [1], [2]. In particular, the redundancy allocation problem has been studied extensively [3], [4], [5], [6] but generally the configurations considered did not include the k-out-of-n: G (G=good) redundancies, actually largely adopted in many real systems.

From a theoretical point of view, the design problem amounts to the maximization of a properly defined function which accounts for the system safety and economic characteristics. When the plant is complex and the model of its behaviour is sufficiently realistic, the plant function is a complicate multivariate, non-linear function, which cannot be put explicitly in analytical form. This aspect of the problem has three consequences:

• for a desired plant output, there exist many possible design alternatives so that the search space for the optimization algorithm becomes tremendously large;

• for a given plant design, the evaluation of its associated safety and economic performance is not feasible through analytical methods;

• classical optimization methods encounter enormous difficulties in the maximization problem.

In this paper, we present an approach, which couples the Monte Carlo simulation method for the evaluation of plant safety and economic performance and the genetic algorithms for determining the optimal system design.

Genetic algorithms are computational tools founded on a direct analogy with the physical evolution of species [9]. An initial population of potential solutions (chromosomes), in the form of coded bit strings, is created and allowed to evolve over successive generations, through mating, crossover and mutation. These operations allow the algorithm to explore the state space in a very efficient manner. Genetic algorithms have been used to solve several engineering problems and are particularly effective for combinatorial optimization problems with large, complex search spaces. Recently, their application to the reliability field is receiving increasing attention [10], [11], [12], [13].

In our work, a potential solution proposed by an individual of the GA population consists of a choice of the types of components and their assembly layout. In the correspondence of each proposed configuration, the Monte Carlo method provides a flexible simulation tool capable of accounting, in a quite straightforward manner, for many realistic issues involved in plant reliability, such as the k-out-of-n: G schemes, and for many realistic economic issues such as the costs and revenues of an ageing plant [7], [8]. The objective function used to measure the goodness (fitness) of the proposed solution is the net profit of system operation for a given mission time. In each MC trial, this profit is obtained by computing the service revenue and by subtracting from this quantity all the costs associated with the system implementation and operation, i.e. component purchase and repair costs, system downtime costs, accident costs to restore external environmental conditions and refund from losses in case of an accident. The objective function so defined, accounts implicitly for possible availability and reliability constraints through the system downtime and accident costs, respectively. Explicit reliability constraints could have also been taken into account in a straightforward manner. Then, the problem becomes a search in the system configuration space of that configuration which maximizes the objective function.

The paper is organized as follows. In the next section we further define the problem of plant design and specify all the relevant issues, which come into play. Section 3 describes in detail the safety and economic constraints considered in our plant model and introduces the appropriate cost function. Section 4 briefly introduces the basic concepts of genetic algorithms (Monte Carlo simulation is considered a rather well known methodology, thus not deserving extensive treatment). In Section 5, the optimization algorithm based on the combination of Monte Carlo simulation and genetic algorithms is described. In Section 6, the algorithm is firstly applied for validation purposes to a simple system whose objective function can be computed analytically and whose best configuration can be found by inspection. A more extensive application is, then, performed for the optimization of a plant with similar characteristics such as those of a shale oil plant described in the literature [14]. In this case, the problem is formulated in such a way that the number of possible alternative design solutions is of the order of 1.5×10⁵. We conclude the paper with some general conclusions and remarks regarding the problem.