A New Algorithm to Rank Temporal Fuzzy Sets in

A New Algorithm to Rank Temporal Fuzzy Sets in Fuzzy Discrete Event Simulation. A. Anglani, A. Grieco, F. Nucci

Q. Semeraro, T. Tolio

Dipartimento di Ingegneria dell'Innovazione Università degli Studi di Lecce Via per Arnesano, Lecce, 73100 Italy

Dipartimento di Meccanica Politecnico di Milano Via Bonardi 9, Milano, 20133 Italy

Abstract- In this paper fuzzy set theory has been applied to discrete event simulation to model uncertainty in input data. Various approaches to fuzzy simulation have been proposed in literature, even if many problems are still to be solved. The key points are how to manage the simulation event list and how to update the fuzzy simulation clock. These two tasks are mainly based on the ranking algorithm. In the following, the classical algorithms have been applied to rank temporal fuzzy sets and the results have been compared with the ones obtained by the proposed ranking algorithm. The comparison has been performed by analyzing a simple case study in the manufacturing field. The results show how the new ranking algorithm can be very useful in a fuzzy simulation environment.

I.

INTRODUCTION

Simulation theory deals with process variability and leads to collect information about system dynamics. The study of production management problems is one of the most successful simulation applications. In this field, the simulation variables are of discrete type due to the intrinsic system properties. As fully demonstrated, system variability may be easily represented in a simulation model and useful information about system performance may be obtained. Up to now, in classical simulation approach, variability is only considered from the stochastic point of view, even if it may be defined in different ways on the basis of the available information and the relative certainty degree. If it is possible to describe system variability by means of statistical distributions, classical tools as the queuing theory, simulation and Petri nets are able to model production systems thoroughly and they may be considered exhaustive in the solution of management problems. Otherwise, if variability can not be described by statistical distributions, as a consequence of the lack of experimental data, innovative tools are necessary to estimate the production system performances. In this case, indeed, the hypotheses of classical tool are not verified and the corresponding approaches can not exploited. On one hand none of the available tools can manage non-stochastic uncertainty, on the other, the necessity to evaluate the performance is ever-increasing in current manufacturing production systems, especially when historical data can not be obtainable and only hazy knowledge about the system is available. In this paper an innovative approach to augment simulation theory applicability in the non-stochastic field is presented. This is performed by introducing fuzzy representation in simulation models. In discrete event simulation, different classes of information must be supplied in order to build the corresponding model (e.g. the system layout, the entity flow inside various processes, etc.). One of the most important classes concerns the occurrence time of events.

In fact, discrete event simulation models are based on events whose occurrence determines system state evolution. Discrete event simulators manage events in a proper list. The time advance steps are determined on the basis of the next event that will occur in the system and, consequently, the system state is updated. On the other hand, other events are inserted in the list as soon as the system state dynamically evolves. Furthermore, one of the main components is the simulation clock, in which the information on system current time is maintained. At each step the simulation clock is updated to the occurrence time of the first occurring event. Independently on simulation model complexity, the core mechanism in reproducing system evolution is the management of the event list and of the relative simulation clock. If no probability distribution is available to describe occurrence time events but the variable domain limits are known, useful information about system state evolution may be produced by fuzzy discrete event simulation. In fact, fuzzy set theory represents a good approach to deal with the ill-known definition in occurrence event times and fuzzy arithmetic operators may be usefully applied in the event list managing. In this case the variable domain is equivalent to the fuzzy set support range and a proper membership function must be set. This function represents the degree with which each value of the range belongs to the linguistic variable represented by the fuzzy set. By the previous mathematical representation, the event occurrence time is not represented by a single value sampled by a probabilistic distribution, but by a range of values, each one with a different membership level. This work focuses on the managing of the occurrence time events when variables are described in an uncertain way. The paper is organized as follows. In the second paragraph the application of fuzzy set theory to describe lack of knowledge in system parameters is illustrated with the relevant literature results in the application of fuzzy sets theory to discrete event simulation. A new approach to fuzzy simulation is shown in the third paragraph and some preliminary results are discussed in the following paragraphs. II. PREVIOUS RESEARCH Few studies have considered the possibility to bring up fuzzy set theory in the simulation of discrete event systems. In [1], the DEVS approach to model system dynamics has been modified by introducing fuzzy numbers. An interesting approach to fuzzy simulation has been presented in [2], in which the effects of an uncertain estimation of the processing times on the system performance have been studied. In [3], the authors deal with the problem of spurious state generation and study the correlation among different uncertain events.

Generally, a valid fuzzy simulation model must verify two basic properties. The former asserts that the fuzzy simulation models must be able to reproduce every possible state evolution. The latter declares that impossible system evolutions must be discarded. As previously reported, in fuzzy simulation scheduled event occurrence times are not crisp but rather a set of possible values is available, each value with a different membership degree. At each time advance step there may be more than one possible evolution therefore great attention must be paid in the updating of the simulation clock. Indeed, in order to assure consistency to the system performance evaluation phase, the simulation clock update must not generate contradiction in the state evolution. For this reason, the core matter, in order to let the simulation model verify the first property, is related to fuzzy ranking algorithms. If event occurrence times are described by fuzzy sets, a fuzzy algorithm must be applied to select the next event that will occur in the system evolution. A relevant number of methods to rank fuzzy sets or fuzzy numbers are available in literature [4]. One of the most exploited is the Integral Value [5] algorithm. Briefly, assigned a temporal fuzzy set π(x), with unimodal membership function µ(x) representing the possibility distribution of event E occurrence, it is possible to define the quantities in (1), where gL(y) and gR(y) are the inverse functions of µL(x) and µR(x). The integral value is defined in (1). 1

1

0

0

I L = ∫ g L ( y)dy , I R = ∫ g R ( y)dy

I γ = γ I R + (1 − γ )I L

(1)

Once a value is assigned to the index γ, it is possible to determine the value of the integral value for each fuzzy sets and the ranking is performed on the calculated values. The ranking result varies according to the crisp parameter γ∈[0,1]. For this reason, the two approaches produce different results depending on the assigned parameter. In order to refine the way fuzzy sets are ranked, it is possible to replace the Integral Value with the Expected Existence Measures [6] operator. The expected existence measure of an event at time x , referred to as EEM in the following, is defined in (2). ∞  x  (2) EEM ( x ) =  ∫ µ ( x )dx   ∫ µ ( x )dx  .   −∞   −∞ Given the value x , the EEM measures the possibility of the proposition that event E had occurred on or before time x ∈ T . The ranking operator based on the EEM measures works in the following way. Firstly, a degree of confidence γ is assigned on the basis of the considered problem. For each fuzzy set, the corresponding x-axis value x is calculated such that EEM( x ) = γ. The ranking operation is performed by comparing the x quantities. Note that with both ranking algorithms, a crisp value replaces each fuzzy number and a deterministic ranking is performed. The main drawback in the application of these two methods to fuzzy simulation is the impossibility to generate every possible evolution. An illustrative example is reported in Fig. 1. Fuzzy

numbers π1 and π2 overlap, so both the two conditions π1π2 may occur. By means of the Integral Value and EEM algorithms, only the first condition is reproduced. For example, considering the EEM algorithm, the x values for fuzzy number π2 are always greater than the ones for fuzzy set π1, for any value of γ0. Integral Value

π1 EEM

γ0 π2 t

Fig. 1. Disadvantages of Integral Value and EEM In order to improve temporal fuzzy set ranking a heuristic method has been presented in [7]. In the new algorithm, given k fuzzy sets π1, π2, …, πk, the first operation is to assign a basic ranking between fuzzy sets (i = 1, …, k) by applying a user-set of comparison rules. For example, a basic ranking may be established by the following set of rules. 1. Cen(πi)