A hybrid intelligent system for multiobjective decision making problems

https://doi.org/10.1016/j.cie.2006.06.011Get rights and content

Abstract

In attempt to solve multiobjective problems, many mathematical and stochastic methods have been developed. The methods operate based on the structured model of the problem. But most of the real-world problems are unstructured or semi-structured in objectives or constraints that caused lag of application of these traditional approaches in such problems. In this paper, a systematic design is introduced for such real multiobjective problems using hybrid intelligent system to cover ill-structured situations. Specially, fuzzy rule bases and neural networks are used in this systematic design and the developed hybrid system is established on noninferior region with the ability of mapping between objective space and solution space. The proof-of-principle results obtained on three test problems suggest that the proposed system can be extended to higher dimensional and more difficult multiobjective problems. A number of suggestions for extensions and application of the system is also discussed.

Introduction

Many real problems involve simultaneous optimization of several incommensurable and often competing objectives (Zeleny, 1998). Generally, there is no single optimal solution but rather a set of alternative solutions. These solutions are called noninferior in the wider sense that no other solutions in the search space are superior to them when all objectives are considered (Cohon, 1987).

To find such solutions, many classical algorithms are developed that define a substitute problem, reducing the vector objective function to a scalar objective function. Using such a substitute, a compromise solution is found subject to specific conditions (Hwang & Yoon, 1981). Weighting method, constraint method and Nise method are sample of approaches used these classical algorithms (Cohon, 1987). But generally, these methods can not find multiple noninferior solutions simultaneously. Moreover, if some of objectives are noisy or have discontinuous variables, these methods may not work effectively. The methods are very sensitive to the weights and generally expansive as they require knowledge of priorities. In summarize, classical approaches to handle multiobjective problems are inadequate and inconvenient to use (Srinivas & Deb, 1994).

On the other hand, the stochastic based methods specifically evolutionary algorithms (MOEA) are widely used in multiobjective solution process (Jones, Mirrazavi, & Tamiz, 2002). Popularity, parallel processing and flexibility of these methods are the main reasons of this extensive utilization (Jaszkiewicz, 2004). But unfortunately while some of these methods are known as prior or progressive articulation most of posterior methods also require some initial factors such as tournament size, Pareto spread and sharing value (Coello, Van Veldhuizen, & Lamont, 2002). Assigning appropriate values of these factors is generally difficult as it relates to the shape and separation of a given problem (Zitzler et al., 2003). In addition the searching techniques used these factors are very sensitive and in some cases (i.e. fitness function) restrict the scope of the problem into two or three objectives (Van Veldhauizen & Lamont, 2000). However, both classic and stochastic approaches would not support unstructured or semi-structured problems. The approaches are founded on specified mathematical models and vagueness, imprecision and incompleteness can not be supported with these approaches.

In contrast, the intelligent systems are traditionally used in such ill-structured cases of real-world problems (Gholamian & Fatemi Ghomi, 2005). Specially, various neural networks and rule-based systems are developed in a wide range of ill-structured optimization problems. However, in multiobjective problems the applications are restricted to auxiliary of other implementation methods.

Neural networks (NN) have been extensively used in the last four decades in multiobjective problems and various designed networks are utilized to obtain noninferior points. Extended Hopfield model (Balicki, Kitowski, & Stateczny, 1998), Kohonen network (McMullen, 2001) and feedforward networks (Shimizu, Tanaka, & Kawada, 2004) are samples of developed networks. Feedforward networks are also used aside with classical methods (i.e. IWPT method) (Sun, Stam, & Steuer, 2000), Meta-heuristics (Liong, Khu, & Chan, 2001) and fuzzy logic (Gen, Ida, & Kobuchi, 1998). But most of applications are restricted to special conditions and few NN based methods have found considerable success in general view.

The rule-based systems are used along with Meta-heuristics and even classical approaches. For example Nabrzyski and Weglarz (1997) introduced a rule-based system aside with a Tabu search method to control Tabu list, choose neighbors, define aspiration level and change attributes. EESA (Jwo, Liu, Liu, & Hsiao, 1995) and PAMUCII (Coelho & Bouillard, 2004) are other samples of such rule-based applications. In applications with classical methods for example Poulos, Rigatos, Tzafestas, and Koukos (2001) used fuzzy rule-based system to control incremental changes of weights in a weighting method for a warehouse multiobjective problem. As another work, Agrell, Stam, and Fischer (2004) developed a rule-based decision support system aside with Tchebycheff interactive method to provide an agro-ecological and economic assessment of various types of land uses. Also, Rasmy, Lee, Abd El-Wahed, Ragab, and El-Sherbiny (2002) developed fuzzy expert system to extract priorities and aspiration levels based on linguistic preferences and then convert multiobjective problems to equivalent goal programming model.

Fuzzy rule bases are also used to extract unstructured objective functions of decision maker (Carlsson & Fuller, 2000). The fuzzy rules are developed with linguistic variables in antecedent and linguistic objectives in consequents and then Tsukamoto’s fuzzy reasoning method is used to determine crisp mathematical structure of objective functions. Similarly, Sugeno’s fuzzy reasoning method with fuzzy decision making method are used as MANFIS network, which is an extension of the ANFIS network for fuzzy multiobjective problems to extract related single objective model and then this model is optimized via genetic algorithm (Cheng, Cheng, & Lee, 2002).

As shown in all applications, the rule bases are used with marginal roles. In contrast, in this study the rule bases are used with an original role. Meanwhile neural networks are also used in developing process or as complementary of the system.

The rest of the paper is organized as follows. Section 2 introduces some basic definitions and fundamental concepts. Section 3 describes the stages of system development in detail. The paper is then convoyed with numerical examples in Section 4. In this section three test problems are introduced and the systematic design is performed for these problems. Finally, discussions and conclusions are devoted in Section 5.

Section snippets

Basic concepts

Let fi(x) (i = 1, 2,  ,P): Ω  Λ be objective functions to be minimized where x = (x1,  ,xn) is a feasible solution from some universes Ω. Then the standard multiobjective problem is formulated as follows.

Max z = F(x) = (f1(x),  ,fp(x))

Subject to:gi(x)0,i=1,2,,m,x=(x1,,xn)Ω,z=(z1,,zp)Λ.A multiobjective problem thus consists of n decision variables, m constraints and p objectives of which any or all of the objective functions may be linear or nonlinear. The evaluation function maps decision variables (x =

System development

The fuzzy inferencing systems are also defined in various manners; however two important types of fuzzy inferencing systems are Sugeno-type and Mamdani-type system (Jang & Sun, 1997). These two types of inference systems vary somewhat in the way outputs are determined. In Sugeno-type inference, the consequences are linear combination of the inputs and the output is the weighted linear combination of the consequents; while Mamdani system infers using implication and aggregation (fuzzy)

Numerical experiments

In this section, three benchmark test examples are provided to clarify the performance of system development for multiobjective problems. Two first problems are unconstrained and so the Sugeno inference system is developed for them while the third problem is a constrained problem and so Mamdani inference system is developed for it. The examples are developed by MATLABInc used under Pentium IV personal computer.

Conclusions

Even though two objectives are used in problems presented in this paper, more objectives and more dimensional variables can be handled with the system. Moreover, the objectives do not require being continuous or smooth and more complex ones could also be used. In fact, the system is not dependent to mathematical structure of the model and so unlike traditional approaches, formalization of the model would not affect in the process of system development. As recommendations for future studies, the

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