**7. Appendix A**

In this appendix, we apply the conventional ANN formulation ([1] ,[6] ,[7]) and ANFIS [8] to case 'B' of the adopted case study. It is noteworthy that strict application of the conventional ANN or ANFIS to this case study is not possible because the desired system output is unknown (it is not possible to evaluate the objective function except by using the robot simulator). Therefore, the numerical BFGS has been used, as before, in the training phase with the same objective function defined in Eq. (9). As said earlier, with the conventional ANN formulation the output of the ANN described in Eq. (1) can be interpreted as fuzzy rules of the form:

$$\begin{aligned} \text{SLANCV} \quad \mathbf{x}\_i^p &> - \begin{pmatrix} b\_j \ \end{pmatrix} \Big/ \text{w}\_{liij} \quad \text{then} \quad o\_{jk} = w\_{jk} \\ \mathbf{i} &= \mathbf{1}\_i \mathbf{2}, \dots, \mathbf{N}\_i \end{aligned}$$

 At the weights/biases level, robust optimization can be used to develop robust rules. Rules are robust when they are reachable from different weight initializations (i.e. they are not sensitive to a particular initialization) and lead to acceptable performance when

We believe that merging symbolic AI (logic) with non-symbolic AI (ANNs) through our

1. The resulting learning system is transparent to the user and its reliability can be easily assessed. A suitable strategy has been outlined for improving its reliability based on

3. The logic-based approach to optimization can be less prune to local minima trapping. 4. The approach is applicable to a broad class of engineering problems where the corresponding correct output to a certain example input is not necessarily available (but a means of assessing the fitness of the output is available through simulation or

We do not claim that the proposed approach can outperform existing approaches in all problems, however, we can certainly claim that we offered researchers, a framework truly worthy of investigation for complex optimization, control and design problems. The best approach will always remain problem-dependent which is the charm and challenge of

After thanking Allah almighty for giving the authors the stimulus required to complete this work, the first author would like to thank her students for their valuable inquiries during a summer course on ANNs that she gave. Their eagerness to understand, queries and continuous criticisms helped her a lot to better formulate her thoughts and served as a valuable encouragement, especially during the early part of this work. It is true that, at many occasions, we learn from our students even more than we teach them! The second

In this appendix, we apply the conventional ANN formulation ([1] ,[6] ,[7]) and ANFIS [8] to case 'B' of the adopted case study. It is noteworthy that strict application of the conventional ANN or ANFIS to this case study is not possible because the desired system output is unknown (it is not possible to evaluate the objective function except by using the robot simulator). Therefore, the numerical BFGS has been used, as before, in the training phase with the same objective function defined in Eq. (9). As said earlier, with the conventional ANN formulation

the output of the ANN described in Eq. (1) can be interpreted as fuzzy rules of the form:

*p*

 / / 1,2,...

*SLANCV x > b N w then o = w i = ,N*

*i j i hij jk jk i*

new proposed framework can achieve the following advantages:

2. The system is robust to noisy inputs and disturbances.

subject to small perturbations.

the analysis of the extracted rules.

author acknowledges the support of EJUST.

practical measurements).

engineering optimization.

**6. Acknowledgement** 

**7. Appendix A** 

**5. Conclusion** 

The fact that the antecedent of a rule depends on both the bias of the corresponding hidden neuron and the weights from inputs-to this hidden neuron makes it difficult the use of known inputs constraints in weights/biases initialization. Thus, we are forced to use small random weights and biases initially, train the ANN, extract the rules and then re-train in case some of the rules yield un-plausible antecedents. For our case study, 3 out of the 7 rules had un-plausible antecedents. For example, one of the rules stated:

$$\begin{aligned} \text{SLANCV} \quad v \prec -1.1959689, \quad w \rhd 0.1675315, \quad v\_{nf} - v\_{nb} \succ 0.8878251, \quad a\_{nf} - a\_{nb} \succ 0.1353403, \\ \rho \prec -2.7918388, \quad a \prec -0.2060677, \\ \text{decrease} \quad v \quad by \quad v\_{\text{inc}} = -6.6637447, \quad \text{decrease} \quad a \quad by \quad a\_{\text{inc}} = -5.8271221 \end{aligned}$$

Clearly comparing *ρ* to a negative threshold is not logical.

Fig. 14 illustrates a typical ANFIS architecture for the case of a 2 inputs 1, 2 *x x* single output, 2 rules example. *Aij* is the membership function of the ith input in the jth rule. The DOF of a rule is computed using the 'Product' operator, i.e. it is the product of the output of the membership functions of a certain rule (Layer 2). NORM units (Layer 3) divides the individual DOF of each rule by the sum of DOF of all rules to produce a normalized degree of firing *wj* . Layer 4 computes the consequent of each rule j for each output k, *jk f* as a function of the inputs 1 *Ni jk ijk jk i f = p +r* . The overall system output is computed as a

weighted sum of the different rules consequents ( 1 *Nh j k jk j= <sup>f</sup> = w <sup>f</sup>* , *Nh* is the number of rules

which equals 2 in Fig. 14). Similarly, to be able to compare our approach to ANFIS [8], we use the same block diagram given in Fig. 5 but replacing the typical feed-forward ANN with an ANFIS. The ANFIS formulation does not impose restrictions on membership function's choices. Therefore, sigmoid membership functions have been chosen, for the purpose of comparison with our approach. In this case, the membership function of the jth rule takes the form:

$$A\_{ij} = \text{sig}\left(a\_{ij}\left(\mathbf{x}\_i - c\_{ij}\right)\right)$$

Our approach can be viewed as a modified ANFIS system with the *'Product*' operator replaced by the 'ior' operator and with 0 *ijk p =* . As indicated by the results (Fig. 15), these modifications enhance the performance considerably. ANFIS training involves the estimation of the parameters , *ijk jk p r* for each rule contributing to the output as well as the membership functions parameters *ij ij a ,c* of each rule. The extracted rules after training are as follows:

It is clear from Fig. 15, that both the conventional ANN and ANFIS produce inferior results to those obtained using the proposed approach (refer to Fig. 7). Thus, the proposed modifications to conventional ANNs succeeded in producing an improved ANFIS system capable of outperforming both conventional ANNs and ANFIS for problems where the

desired ANN output is not known a priori (like in the path tracking case study).

$$\begin{array}{llll} & \text{if} & \text{v} > 1.205344 \text{ and} & \text{ow} > -0.0968983 \text{ and} & (\text{v}\_{\text{ref}} - \text{v}\_{\text{orb}}) > 0.0373195 \\ \text{and} & (\text{w}\_{\text{ref}} - \text{w}\_{\text{rb}}) < 0.1962357 \text{ and} & \rho > 1.2538755 \text{ and} & \alpha < 0.1218786 \\ \text{then} & \text{v}\_{\text{inc}} = 0.5642881 \text{v} & + & -0.2429051 \text{\*} \text{w} + -0.2429051 \text{\*} (\text{v}\_{\text{ref}} - \text{v}\_{\text{orb}}) + \\ & & 0.3081844 \text{\*} (\text{w}\_{\text{ref}} - \text{w}\_{\text{rb}}) & + & -0.6377921 \text{\*} \text{.} \\ & & 0.2372806 \text{\*} \text{.} & + & 0.3039157 \text{ +} - 0.7144687, \\ \omega\_{\text{inc}} = 0.6927095 \text{\*} \text{v} + 0.7480350 \text{\*} \text{.} & + & 0.7480350 \text{\*} (\text{v}\_{\text{ref}} - \text{v}\_{\text{rb}}) & + \\ & & 0.2289957 \text{\*} (\text{w}\_{\text{ref}} - \text{w}\_{\text{rb}}) & + & 0.5583952 \text{\*} \text{.} \\ & & - 0.4286660 \text{\*} \text{.} & + & 0.2080972 \text{ --} - 0.4508324 \end{array}$$

It is clear from Fig. 15, that both the conventional ANN and ANFIS produce inferior results to those obtained using the proposed approach (refer to Fig. 7). Thus, the proposed modifications to conventional ANNs succeeded in producing an improved ANFIS system capable of outperforming both conventional ANNs and ANFIS for problems where the desired ANN output is not known a priori (like in the path tracking case study).

A Framework for Bridging the Gap Between Symbolic and Non-Symbolic AI 47

Fig. 15. Shows the results for using conventional ANNs and ANFIS instead of the proposed formulation for case B. By comparing these results with those in Fig. 7 , the superiority of

[1] G J. Benitez, J. Castro, I. Requena, "Are Artificial ANNs Black Boxes?", IEEE Trans. on

[2] J. Espinosa, J. Vandewalle, V. Wertz, *Fuzzy Logic, Identification and Predictive Control* 

[4] R. Setiono, "Extracting M-of-N Rules from Trained ANNs", *IEEE Trans. on ANNs*, Vol. 11,

[5] S. Mitra, S. Pal, "Fuzzy Multi-Layer Perceptron, Inferencing and Rule Generation" , *IEEE* 

[6] H. Senousy, M. Abou-El Makarem, "New Reliable Neural-Based Automatic Diesel Fault

[7] I. Hamid, H. Senousy, M. Abou-Elmakarem, "An Improved Fuzzy Logic Controller For

Diagnosis Systems", International Conference on Mechanical Engineering and

Ship Steering Based on Ior Operator and Neural Rule Extraction", ICCES08,

*(Advances in Industrial Control)*, Springer-Verlag, London, 2005. [3] Simon Haykin, ANNs and Learning Machines, Prentice Hall, November 2008.

*Trans. on ANNs*, Vol. 6, No.1, January 1995. pp. 51-63.

Production MDP9, Cairo, Egypt, January 2008.

the proposed approach-thanks to Allah- is clear.

No.2, January 2000. pp. 510-519.

ANNs, September, 1997.

**8. References** 

Fig. 15. Shows the results for using conventional ANNs and ANFIS instead of the proposed formulation for case B. By comparing these results with those in Fig. 7 , the superiority of the proposed approach-thanks to Allah- is clear.

#### **8. References**

46 Recurrent Neural Networks and Soft Computing

Fig. 14. Architecture of a typical ANFIS system for a 2 inputs single output example


**3** 

*Iran* 

Tayebeh Hajjari

**Ranking Indices for Fuzzy Numbers** 

*Department of Mathematics, Firoozkooh Branch, Islamic Azad University, Firoozkooh,* 

Fuzzy set theory has been studied extensively over the past 30 years. Most of the early interest in fuzzy set theory pertained to representing uncertainty in human cognitive processes (see for example Zadeh (1965)). Fuzzy set theory is now applied to problems in engineering, business, medical and related health sciences, and the natural sciences. In an effort to gain a better understanding of the use of fuzzy set theory in production management research and to provide a basis for future research, a literature review of fuzzy set theory in production management has been conducted. While similar survey efforts have been undertaken for other topical areas, there is a need in production management for the same. Over the years there have been successful applications and implementations of fuzzy set theory in production management. Fuzzy set theory is being recognized as an important

Kaufmann and Gupta (1988) report that over 7,000 research papers, reports, monographs,

As evidenced by the large number of citations found, fuzzy set theory is an established and growing research discipline. The use of fuzzy set theory as a methodology for modeling and analyzing decision systems is of particular interest to researchers in production management due to fuzzy set theory's ability to quantitatively and qualitatively model problems which involve vagueness and imprecision. Karwowski and Evans (1986) identify the potential applications of fuzzy set theory to the following areas of production management: new product development, facilities location and layout, production scheduling and control, inventory management, quality and cost benefit analysis. Karwowski and Evans identify three key reasons why fuzzy set theory is relevant to production management research. First, imprecision and vagueness are inherent to the decision maker's mental model of the problem under study. Thus, the decision maker's experience and judgment may be used to complement established theories to foster a better understanding of the problem. Second, in the production management environment, the information required to formulate a model's objective, decision variables, constraints and parameters may be vague or not precisely measurable. Third, imprecision and vagueness as a result of personal bias and subjective opinion may further dampen the quality and quantity of available information. Hence, fuzzy set theory can be used to bridge modeling gaps in descriptive and prescriptive decision

and books on fuzzy set theory and applications have been published since 1965.

**1. Introduction** 

problem modeling and solution technique.

models in production management research.

Faculty of Engineering - Ain Shams University, Computer Engineering & Systems Department Cairo, EGYPT, November 25-27, 2008.

