Proceedings of the 8th WSEAS Int. Conf. on ARTIFICIAL INTELLIGENCE, KNOWLEDGE ENGINEERING & DATA BASES (AIKED '09)
Model Optimisation for Complex Systems using Fuzzy Networks Theory NEDYALKO PETROV Faculty of Computer Systems and Control Technical University of Sofia 8, Kliment Ohridski Street, Sofia 1000 BULGARIA
[email protected],
[email protected]
ALEXANDER GEGOV School of Computing University of Portsmouth Buckingham Building, Portsmouth PO1 3HE UNITED KINGDOM
[email protected] http://userweb.port.ac.uk/~gegova/
Abstract: - This paper presents an application of the novel theory of fuzzy networks for optimising models of systems characterised by uncertainty, non-linearity, modular structure and interactions. The application of the theory is demonstrated for retail price models in the context of converting a multiple rule base fuzzy system (MRBFS) into an equivalent single rule base fuzzy system (SRBFS) by linguistic composition of the individual rule bases. During the conversion process, the transparency of the MRBFS is fully preserved while its accuracy is improved to a level comparable with the accuracy of the SRBFS. This improvement is achieved by increasing the number of linguistic terms for the intermediate variable connecting the individual rule bases. Key-Words: - Complex systems, fuzzy systems, fuzzy networks, rule bases, rule based systems, composition models, retail pricing, multiple rule base fuzzy systems, single rule base systems. In contrast, a fuzzy network represents a white box type of model that takes into account the interactions among subsystems [4]. This capability can bring potential advantages in modelling complex systems. In this context, the paper demonstrates the application of the fuzzy networks theory for optimising the accuracy of a multiple rule base fuzzy system (MRBFS) while preserving its transparency.
1 Introduction A complex system is commonly defined as a system composed of interconnected parts (subsystems) that as a whole exhibit one or more properties not obvious from the properties of the individual parts (subsystems). They are usually described by a number of features such as uncertainty, non-linearity, modular structure and interactions [1], [2]. These features often present a serious challenge to the modelling of such process. In this context, fuzzy systems have already proved themselves as a powerful tool for dealing with nonprobabilistic uncertainty [7]. The most common cause for this type of uncertainty can be data that is in some way incomplete or ambiguous. At the same time, the implementation of fuzzy logic by means of fuzzy systems helps with the tackling of non-linearity [3], [5], [7]. In this sense, the rule base of a fuzzy system is usually capable of representing quite well strongly non-linear functions in complex processes which usually can not be dealt with by other types of mathematical models. However, in spite of the relative success of fuzzy systems in capturing uncertainty and non-linearity, there are other features of complexity, which can not be taken into account [3], [4], [5]. For example, the interactions among subsystems and the high dimensionality in terms of large number of inputs may lead to the deterioration of fuzzy models. This deterioration can be attributed to the ‘black box’ nature of fuzzy systems which consider only the inputs to and the outputs from a complex process rather than any interactions.
ISSN: 1790-5109
2 Types of Fuzzy Systems The most common fuzzy system consists of a single rule base (SRB) whereby the associated fuzzy model is of a black box type [3], [7], In this case, a SRB fuzzy system (SRBFS) deals with all process inputs at the same time while not taking into account the interactions and the structure of the system. Another less common fuzzy system consists of multiple rule bases (MRB) whereby the associated fuzzy model is of a white box type [3], [5], [8], [9]. In this case, the MRB fuzzy system (MRBFS) deals with process inputs sequentially in time while taking into account the interactions and the dimensionality of the system. A MRBFS is usually derived by inferential decomposition of a SRBFS such that all individual rule bases are subject to fuzzification, inference and defuzzification [3], [5]. The latter is usually accompanied with some transformation error that accumulates for every single subsystem and results in decreased accuracy of the whole MRBFS. The most uncommon fuzzy system (called fuzzy network) consists of networked rule bases (NRB)
116
ISBN: 978-960-474-051-2
Proceedings of the 8th WSEAS Int. Conf. on ARTIFICIAL INTELLIGENCE, KNOWLEDGE ENGINEERING & DATA BASES (AIKED '09)
intermediate variable. In this case, the resultant SRBFS has the same inputs as the inputs to the first operand SRBFS and the same outputs as the outputs from the second operand SRBFS, whereas the intermediate variable does not appear in the resultant SRBFS.
whereby the associated fuzzy model is also of a white box type [3], [4]. In this case, a NRB fuzzy system (NRBFS) deals with process inputs sequentially in both time and space while taking into account the interactions and the structure of the system. A SRBFS can be derived from a NRBFS system by linguistic composition of all individual rule bases such that fuzzification, inference and defuzzification is applied only once to the equivalent rule base of the SRBFS. Overall, NRBFSs represent a novel extension to both SRBFSs and MRBFSs [3], [4]. In particular, NRBFSs provide a bridge between SRBFSs and MRBFSs that facilitates their use. Also, the linguistic composition approach in NRBFSs is based on physical considerations as opposed to the decomposition approach in MRBFSs, which is based on mathematical considerations.
i1
z
z
.
RB2
o2
=
i1
RB
o2
Fig.1. Horizontal merging of two rule bases Vertical merging is a binary operation that can be applied to a pair of parallel SRBFS, i.e. systems located in the same layer of a MRBFS. This operation merges the operand SRBFSs from the pair into a single resultant SRBFS [3], [5]. When Boolean matrixes are used as formal models for representing the rule bases of the operand SRBFSs, the vertical merging operation is like an expansion of the first operand matrix along its rows and columns. In particular, the resultant matrix is obtained by expanding each non-zero element from the first operand matrix to a block that is the same as the second operand matrix and by expanding each zero element from the first operand matrix to a zero block of the same dimension as the second operand matrix. In this case, the inputs to the resultant SRBFS represent the union of the inputs to the operand SRBFSs, whereas the outputs from the resultant SRBFS represent the union of the outputs from the operand SRBFS. This operation can always be applied due to the ability to concatenate the inputs and the outputs of any two parallel SRBFS.
3 Linguistic Composition The novel theory of fuzzy networks introduces formal models for presentation of fuzzy systems such as if-then rules and integer tables, Boolean matrixes and binary relations grid and interconnection structures, incidence and adjacency matrixes, and block schemes and topological expressions [5]. It also presents techniques for formal manipulation of fuzzy rule bases based on several basic operations. Some of them are to be found in mathematics and are therefore well known, whereas others are quite novel in terms of the underlying theory and have been introduced recently [3], [5]. The operations use Boolean matrixes or binary relations for representation of the individual rule bases in order to facilitate the manipulation in the context of the linguistic composition approach. Two of the proposed in the theory operations – horizontal merging of rule base systems and vertical merging of rule base systems – are used in this paper for converting a multiple rule base fuzzy system (MRBFS) into an equivalent single rule base fuzzy system (SRBFS). Horizontal merging is a binary operation that can be applied to a pair of sequential SRBFSs, i.e. systems residing at different levels within the same layer of a MRBFS. This operation merges the operand SRBFSs into a single resultant SRBFS [3], [5]. When Boolean matrixes are used as formal models for representing the rule bases of the operand SRBFSs, the horizontal merging operation is identical with Boolean matrix multiplication. The latter is similar to convex matrix multiplication, whereby each arithmetic multiplication is replaced by a ‘minimum’ operation and each arithmetic addition is replaced by a ‘maximum’ operation. Therefore, this operation can be applied only when all the outputs from the first SRBFS are fed forward as inputs to the second SRBFS in the form of an
ISSN: 1790-5109
RB1
i1
RB1
o1
i1 =
i2
RB2
o2
o1 RB
i2
o2
Fig.2. Vertical merging of two rule bases These two operations have been implemented in MATLAB® environment for the objective of this paper. In order to work with Boolean matrixes as formal models of rule bases, two additional MATLAB functions that convert the integer representation of a rule base into a Boolean matrix form and then back from a Boolean matrix form into an integer form, are implemented. The result of the first function is a Boolean matrix with rows that represent all possible permutations of the linguistic terms of the inputs from the integer rule base form sorted in ascending order. The columns of that matrix represent all possible permutations of the linguistic terms of the outputs from the integer rule base form sorted in ascending order. An element of the
117
ISBN: 978-960-474-051-2
Proceedings of the 8th WSEAS Int. Conf. on ARTIFICIAL INTELLIGENCE, KNOWLEDGE ENGINEERING & DATA BASES (AIKED '09)
that run sequentially in time. This model is considered for comparison purposes. All the models are based on two interconnected formulas provided by experts and are given by Equations (1)-(2) [6]:
resultant matrix is set to 1 if it reflects an existing mapping from an input onto an output permutation, or to 0 otherwise. [3]. The result of the second function is a rule base represented in the form of an integer table. For a fuzzy system with m inputs and n outputs, the first m columns of the table represent the linguistic terms of the inputs to the system and the next n columns represent the linguistic terms of the outputs from the system. The latter integer table is used for creating a fuzzy system in the Fuzzy Logic Toolbox™ for MATLAB®. The next section describes the procurement phase of the retail pricing process and discusses three fuzzy models of this process as a case study for demonstrating an accuracy optimizing method presented in section 5.
(1)
mc = pmc (est / 50)
(2)
where the maximum cost (mc) depends on the ‘provisional maximum cost’ (pmc) and the ‘expected sell through’ (est), whereas the ‘provisional maximum cost’ depends on the ‘expected selling price’ (esp) and the ‘margin’ (m). ‘Provisional maximum cost’ is an approximation of the ‘maximum cost,’ the ‘expected sell through’ is the percentage of a product to be sold, the ‘expected selling price’ is the price at which the product is expected to be sold and the ‘margin’ is the difference between the selling price and the cost of a product, divided by the product selling price.
4 Retail Pricing Process The formation of product prices in the retail industry is a typical complex process. It usually consists of a certain number of interacting subprocesses which are characterised by non-probabilistic uncertainty. The subprocesses represent different stages in the price formation process, whereby the associated uncertainty is related to the incomplete, ambiguous and contradictory nature of the information available about the factors that determine the price. In this context, the determination of product prices in the retail industry appears to be a challenging task that could make the difference between success and failure in the business. The most influential factor in the determination of a product price is usually the maximum cost that the retailer can afford to pay to the manufacturer or the trader for the delivery of a product in order to make a profit. Three fuzzy models for modelling the maximum cost for a product are considered. The first model represents a MRBFS. It has very good transparency and can deal with process inputs sequentially in time [5]. However, this model is expected to be the worst in terms of accuracy, because of the fuzzification – defuzzification error accumulated for every single subsystem in the MRBFS. The second model is obtained from the first model and represents a NRBFS. It has very good transparency and can deal with process inputs sequentially in time [5]. This model is expected to have better accuracy than the first model, because of the single fuzzification, inference and defuzzification sequence applied to its resultant rule base. The third model represents a SRBFS. It is expected to be the best model in terms of accuracy [5]. However, this model is not transparent and it can not deal with process inputs sequentially in time [5]. Therefore, it is usually not suitable for modelling of complex processes
ISSN: 1790-5109
pmc = esp (1 - m / 100)
4.1 MRBFS Model The first model represents a MRBFS structured as a hierarchical fuzzy system [5], [6], [8], [9]. It is based on inferential composition of two interacting single rule base fuzzy subsystems (SRBFSS) based on Equation (1) and Equation (2), respectively. In this case, the composition is applied to the subsystems such that each of them is subject to fuzzification, inference and defuzzification. The model is illustrated in Fig.3. z
i1 i2
RB1
RB2
o
i3
Fig.3. MRBFS model The first SRBFSS has two inputs and one output. The inputs are presented by five linguistic terms each, as shown in Fig.4. The output from this subsystem is an intermediate variable presented by eleven linguistic terms as shown in Fig.5. The second SRBFSS has also two inputs and one output. The inputs are presented by eleven and five linguistic terms, respectively, as already shown in Fig.5 and Fig.4. The first input is the same as the output from the first SRBFSS, i.e. the intermediate variable. The output from this subsystem is presented by eleven linguistic terms, as already shown in Fig.5.
118
ISBN: 978-960-474-051-2
Proceedings of the 8th WSEAS Int. Conf. on ARTIFICIAL INTELLIGENCE, KNOWLEDGE ENGINEERING & DATA BASES (AIKED '09)
and are composed linguistically into an equivalent single rule base [3], [4]. An identity rule base (IRB) is added to the NRBFS model for the purpose of making the linguistic composition operations for vertical and horizontal merging compatible. The output from the IRB is identical to its input counterpart. The equivalent single rule base for the model is then subject to a single fuzzification, inference and defuzzification sequence. The NRBFS model is illustrated in Fig.7.
Fig.4. Linguistic terms for inputs i1, i2 and i3 of the MRBFS model
z
i1 RB1
i2
i3
Fig.5. Linguistic terms for intermediate variable z and output o of the MRBFS model
i3
IRB
Fig.7. NRBFS model The linguistic composition of the individual rule bases is performed according to the following formula:
For consistency, all inputs and outputs in the two fuzzy subsystems above are considered in the variation range [0, 100]. The first subsystem of the MRBFS model has 25 rules, whereas the second subsystem has 55 rules. The rules in these subsystems are derived using a simple variation of the data clustering approach. In this case, the derivation is done on the basis of the closest possible approximation of the data sets generated by the formulas from Equation (1) and Equation (2), respectively. The overall MRBFS model is simulated for all possible 125 permutations of the discrete crisp values of the inputs {0, 25, 50, 75, 100}. The results from the model simulation are shown in Fig.6, where the data is given in black (´+´ marker) and the model is given in blue (´x´ marker).
RB = (RB1 + RBID) * RB2
(3)
where RB1 is the rule base for the first subsystem, RB2 is the rule base for the second subsystem, RBID is an identity rule base, RB is the equivalent single rule base for the model, and the symbols ‘*’ and ‘+’ denote horizontal and vertical merging operations. The fuzzy model has three inputs and one output. The inputs are presented by five linguistic terms each, as already shown in Fig.4. The output is presented by eleven linguistic terms as already shown in Fig.5. The overall number of rules for the equivalent single rule base of the model is 125. For consistency, all inputs and outputs in the fuzzy model are considered in the variation range [0, 100]. The NRBFS model is simulated for all possible 125 permutations of the discrete crisp values of the inputs {0, 25, 50, 75, 100}. The results from the model simulation are shown in Fig.8, where the data is given in black (´+´ marker) and the model is given in blue (´x´ marker).
100
80
60 t u p t u o
o
RB2
40
20
100 0
0
20
40
60 80 input permutations
100
120
140 80
Fig.6. Simulation results for MRBFS model
60 t u p t u o
4.2 NRBFS Model
20
The second model represents a NRBFS and it can be obtained from the first model. In this case, the rule bases of the two SRBFSS of the MRBFS model are preserved
ISSN: 1790-5109
40
0
0
20
40
60 80 input permutations
100
120
140
Fig.8. Simulation results for NRBFS model
119
ISBN: 978-960-474-051-2
Proceedings of the 8th WSEAS Int. Conf. on ARTIFICIAL INTELLIGENCE, KNOWLEDGE ENGINEERING & DATA BASES (AIKED '09)
MAE = ( Σ | di – mi | ) / n
4.3 SRBFS Model The third model represents a conventional SRBFS. The associated fuzzy model for this system is based on functional composition of the two formulas from Equations (1)-(2) into a single formula given by Equation (4) [6]. mc = esp (1 – m / 100) (est / 50)
where di and mi, i=1,n denote the data point and the model output for the i-th input permutation, respectively. The transparency is estimated by subtracting the sum of identity rule bases and identity intermediate variables from the overall sum of subsystems and intermediate variables and then dividing the result by the sum of inputs and outputs [6]. The assumption here is that each subsystem or intermediate variable improves the transparency by taking into account connections. However, this is not the case for identity subsystems and identity intermediate variables which are usually used for representing identity mappings in NRBFS models for the purpose of linguistic composition. In particular, this sum is obtained by the formula for the transparency index (TI) given in Equation (6)
(4)
The fuzzy model has three inputs and one output. The inputs are presented by five identical linguistic terms each, as already shown in Fig.4. The output is presented by eleven linguistic terms as already shown in Fig.5. For consistency, all inputs and outputs in the fuzzy model are considered in the variation range [0, 100]. i1 i2
o
RB
i3
(5)
TI = (Nn + Nz – Nidn – Nidz) / ( Nx + Ny )
Fig.9. SRBFS model
(6)
where Nn is the number of subsystems, Nz is the number of intermediate variables, Nidn is the number of identity subsystems, Nidz is the number of identity intermediate variables, Nx is the number of inputs and Ny is the number of outputs. The comparative evaluation of the three fuzzy models with respect to accuracy and transparency is summarised in Table 1.
The SRBFS model has 125 rules which are derived using a variation of the data clustering approach. In this case, the derivation is done on the basis of the closest possible approximation of the data sets generated by the formula from Equation (4). The SRBFS model is simulated for all possible 125 permutations of the discrete crisp values of the inputs {0, 25, 50, 75, 100}. The results from the model simulation are shown in Fig.10, where the data is given in black (´+´ marker) and the model is given in blue (´x´ marker).
Table 1. Comparative evolution of retail price fuzzy models
100 90
Indicator / Model MAE TI
80 70 60 t u p t u o
50
SRBFS 2.86 0.25
MRBFS 5.57 0.75
NRBFS 3.64 0.75
40 30
The simulation results show that in terms of accuracy the SRBFS model is the best, the NRBFS model is slightly worse, whereas the MRBFS is the worst of all. As far as transparency is concerned, the SRBFS model is the worst, whereas the MRBFS model and the NRBFS are better and identical to each other.
20 10 0
0
20
40
60 80 input permutations
100
120
140
Fig.10. Simulation results for SRBFS model
4.4 Comparative Evaluation of Models The three fuzzy models are evaluated by quantitative metrics, based on two formal indicators – mean absolute error and transparency [6]. The mean absolute error (MAE) for the model output is obtained by taking the absolute value of the difference between each data point in the data sets generated by the formulas from Equations (1)-(2) and the output of the corresponding fuzzy model. This difference is then summed over all 125 points from the simulations presented in Fig.6, Fig.8 and Fig.10. Finally, the sum is divided by the overall number of data points, as shown in Equation (5).
ISSN: 1790-5109
5 Optimisation of Model Accuracy Theoretically, a NRBFS model of a complex system should be at least as accurate as a MRBFS model of the same complex system [5]. The latter is due the single fuzzification-inference-defuzzification sequence applied to its resultant rule base. In this context, the comparative evolution of the models discussed in section 4 shows that the MAE of the MRBFS model decreases from 5.57 to 3.64 when it is converted to a NRBFS. During this process the transparency of the initial MRBFS model is fully
120
ISBN: 978-960-474-051-2
Proceedings of the 8th WSEAS Int. Conf. on ARTIFICIAL INTELLIGENCE, KNOWLEDGE ENGINEERING & DATA BASES (AIKED '09)
preserved. Once converted to a NRBFS, the accuracy of a MRBFS model of a complex system can be improved further due to the Boolean matrix multiplication nature of the horizontal merging operation. The latter allows the number of linguistic terms for the intermediate variable (p) of any two participating in a horizontal merging operation individual rule bases to be increased, while preserving the overall number of rules in the resultant SRB for the model. In this context, the MEA of a NRBFS model is expected to generally decrease with increasing the number of linguistic terms for the intermediate variables connecting the individual rule bases. The latter is due to the decreased approximation error while defying the linguistic terms for the individual rule bases. The NRBFS model discussed in section 4 is simulated for p = 1,350 for the purpose of this paper. The simulation results for the first 50 variations of p are shown in Fig.11.
7 Conclusion This paper has demonstrated an application of the fuzzy networks theory for improving the accuracy of a MRBFS model of a complex system for retail pricing. However, the proposed in this paper two-step method can be used for improving the accuracy of any existing MRBFS model, while preserving its transparency. Moreover, the conception of the fuzzy networks theory and the presented optimisation method can be applied to any rule based model. The novel fuzzy networks theory facilitates the building of fuzzy models for complex processes. Some of the operations proposed in this theory can also be used for converting a SFBFS model in a more transparent and more flexible MRBFS model. The latter is a subject for future research.
Acknowledgements: The results presented in this
MAE
paper are obtained within a final year BSc project in the School of Computing, University of Portsmouth, in cooperation with the Technical University of Sofia. In this respect, the first author would like to thank the School of Computing at University of Portsmouth and the Faculty of Computer Systems and Control at the Technical University of Sofia for the provision of computing facilities and supervision support. References: [1] G. Ascia, V. Catania and M. Russo, VLSI Hardware Architecture for Complex Fuzzy Systems, IEEE Transactions on Fuzzy Systems, Vol.7, No.5, 1999, pp. 553-570. [2] M. Bucolo, L. Fortuna and M. La Rosa, Complex Dynamics through Fuzzy Chains, IEEE Transactions on Fuzzy Systems, Vol.12, No.3, 2004, pp. 289-255. [3] A. Gegov, Complexity Management in Fuzzy Systems, Springer, 2007 [4] A. Gegov, From Fuzzy Systems to Fuzzy Networks, WSEAS International Conference on Fuzzy Systems, Sofia, Bulgaria, 2008 [5] A.Gegov, Fuzzy Networks for Complex Systems, Book, Springer, 2011 (to appear) [6] A. Gegov, E. Gegov and P. Treleaven, Advanced Modelling of Retail Pricing by Fuzzy Networks, WSEAS International Conference on Fuzzy Systems, Sofia, Bulgaria, 2008 [7] T. Ross, Fuzzy Logic with Engineering Applications, Wiley, 2004 [8] L. Wang, Analysis and Design of Hierarchical Fuzzy Systems, IEEE Transactions on Fuzzy Systems, Vol.7, No.5, 1999, pp. 617-624. [9] X. Zeng and J. Keane, Approximation Capabilties of Hierarchical Fuzzy Systems, IEEE Transactions on Fuzzy Systems, Vol.13, No.5, 2005, pp. 659-672.
p
Fig.11. Model MEA variation with increasing the number of linguistic terms for the intermediate variable The results show that the MAE of the NRBFS model of the retail pricing process generally decreases to a level comparable with the MAE of the SRBFS model of the same process. The best accuracy achieved for the NRBFS model is identical to the accuracy of the SRBFS model and it is firstly achieved for p = 17. For p > 30 the MAE of the model varies in the range from 2.86 to 2.94 (0.08%), for p > 42 it varies in the range 2.86 to 2.87 (0.01%) and for p > 121 it equals 2.86. A two step algorithm for improving the accuracy of a MRBFS can be given as follows: 1. The MRBFS is converted to a NRBFS, according to the fuzzy networks theory. IRB are added to the model where it is necessary. 2. The number of linguistic terms for the intermediate variables is increased until the change of the accuracy for the model remains in a predefined range.
ISSN: 1790-5109
121
ISBN: 978-960-474-051-2
