Leonard J. Jowers

Simulating continuous fuzzy systems

In previous studies we first concentrated on utilizing crisp simu- lation to produce discrete event fuzzy systems simulations. Then we extended this research to the simulation of continuous fuzzy systems models. In this pa- per we continue our study of continuous fuzzy systems using crisp continuous simulation. Consider a crisp continuous system whose evolution depends on dierential equations. Such a system contains a number of parameters that must be estimated. Usually point estimates are computed and used in the model. However these point estimates typically have uncertainty associated with them. We propose to incorporate uncertainty by using fuzzy numbers as estimates of these unknown parameters. Fuzzy parameters convert the crisp system into a fuzzy system. Trajectories describing the behavior of the system become fuzzy curves. We will employ crisp continuous simulation to estimate these fuzzy trajectories. Three examples are discussed.

Fuzzy Integer Programming

We did not discuss fuzzy integers in Chapter 2 so we will do this in the next section. Then in Section 24.3 we look at a standard integer programming problem and then fuzzify it. We plan to use our fuzzy Monte Carlo method in Section 24.4 to obtain (approximate) solutions to this fuzzy integer programming problem. There have been a few papers on fuzzy integers and optimization problems and we refer the reader to ([1]-[5]).

Fuzzy Linear Regression I

This chapter is a continuation of Chapter 11 and is based on [1]. We wish to use our Monte Carlo method to get approximate solutions for crisp numbers a i , 0 ≤ i ≤ m, to the fuzzy linear regression model

Fuzzy Max-Flow Problem

\overline{Y}=a_0+ a_1\overline{X}_1+...+a_m \overline{X}_m,

Fuzzy Shortest Path Problem

for \(\overline{X}_i\), 1 ≤ i ≤ m, triangular fuzzy numbers and \(\overline{Y}\) a triangular fuzzy number. The fuzzy linear regression model in equation (14.1) has been previously studied in [2]-[6]. In this model the independent variables \(\overline{X}_i\) will be given triangular fuzzy numbers, the dependent variable \(\overline{Y}\) will be a given triangular fuzzy number, so the best way to fit the model to the data is to use real numbers for the a i . If a i is also a triangular fuzzy number, then \(a_i\overline{X}_i\) will be a triangular shaped fuzzy number and the right side of equation (14.1) is a triangular shaped fuzzy number which is used to approximate \(\overline{Y}\) a triangular fuzzy number. If the a i are real numbers the right side of equation (14.1) will be a triangular fuzzy number which is better to use to approximate a triangular fuzzy number \(\overline{Y}\). The data will be \(((\overline{X}_{1l},...,\overline{X}_{ml}),\overline{Y}_l)\), 1 ≤ l ≤ n, for the \(\overline{X}_{il}=(x_{il1}/x_{il2}/x_{il3})\) triangular fuzzy numbers and \(\overline{Y}_l=(y_{l1}/y_{l2}/y_{l3})\) triangular fuzzy numbers. Given the data the objective is to find the “best” a j , 0 ≤ j ≤ m. We propose to employ our Monte Carlo methods to approximate the “best” values for the a j , j = 0,1,...,m.

Fully Fuzzified Linear Programming I

In this chapter we use fuzzy Monte Carlo methods to get approximate optimal fuzzy mixed strategies for fuzzy two-person zero-sum games. In the next section we briefly review the results for crisp two-person zero-sum games. Then in Section 15.3 we fuzzify the games and define optimal fuzzy values for the players and optimal fuzzy mixed strategies. In the fourth section we introduce our fuzzy Monte Carlo method and use it on an example problem to generate approximate solutions. The last section contains our conclusions and suggestions for future research. Our fuzzy Monte Carlo method will be programmed in MATLAB [6]. This chapter is based on [1].

Random Fuzzy Numbers and Vectors

If we take the “Fuzzy Min-Cost Capacitated Network” problem in Chapter 17 and delete the costs we get the “Fuzzy Max-Flow Problem” in this chapter. This is exactly what we will do. The crisp problem is outlined in the next section. Then in Section 19.3 we fuzzify the problem and plan to apply our fuzzy Monte Carlo method to get approximate solutions.

Inventory Control: Known Demand

PERT stands for “Project Evaluation and Review Technique”. A project defines a combination of interrelated activities (jobs) that must be completed in a certain order before the entire project can be completed. The project that we will concentrate on in this chapter is shown in Figure 26.1.