Development of a soft computing based framework for engineering design optimisation with quantitative and qualitative search spaces. Victor Oduguwa1, Rajkumar Roy1 and Didier Farrugia2 1
Department of Enterprise Integration, School of Industrial and Manufacturing Science, Cranfield University, Cranfield, Bedford, MK43 0AL, UK
[email protected] and
[email protected] 2 Corus R, D and T, Swinden Technology Center, Rotherham, S60 3AR, UK
ABSTRACT Most real world engineering design optimisation approaches reported in the literature aim to find the best set of solutions using computationally expensive quantitative (QT) models without considering the related qualitative (QL) effect of the design problem simultaneously. Although the QT models provide various detailed information about the design problem, unfortunately, these approaches can result in unrealistic design solutions. This paper presents a soft computing based integrated design optimisation framework of QT and QL search spaces using meta-models (Design of Experiment, DoE). The proposed approach is applied to multi-objective rod-rolling problem with promising results. The paper concludes with a detailed discussion on the relevant issues of integrated QT and QL design strategy for design optimisation problems outlining its strengths and challenges. Keywords: Multi-objective optimisation, quantitative and qualitative search space, soft computing application, rolling system design. 1. INTRODUCTION Models used in engineering design optimisation are mostly based on quantitative information. This information is based on a principle of numerical reckoning and is the ability to express the behaviour of a phenomenon in numbers using numerical models. The quantitative models (QT) are a partial representation of the design problem. There are aspects of a physical system that are less understood and ambiguous or cannot be modelled using a numerical framework; these are termed as qualitative. Qualitative (QL) models include abstractions of the relations in a numerical model, the causal dependencies between its variables and explicit representations of the modelling assumptions used to describe a physical system. This paper presents a novel multi-objective design optimisation framework to deal with QT and QL models or search spaces simultaneously.
This integrated approach is expected to identify more realistic design solutions. Optimisation of real life (industrial) design problems often requires computationally expensive QT models, such as Finite Element Analysis (FEA) or Computational Fluid Dynamics (CFD) models. QL models can also become computationally expensive depending on the number of rules used in the model. The models become even more expensive if the design involves a very large number of variables (large scale). The integrated design optimisation framework presented in this paper uses meta-modelling (using design of experiment, DoE [24]) approach to develop computationally inexpensive and approximate QT models for the optimisation. The QL models are developed as Fuzzy Expert Systems with selected variables. Fuzzy logic is extensively used in decision support systems where inference is based on imprecise and incomplete information [45, 33]. An existing multi-objective Genetic Algorithm (GA) is used in the integrated optimisation framework. Genetic Algorithms (GAs) are a robust optimisation technique for engineering design problems [12, 31]. The next two sections report related research in Fuzzy Logic application in engineering optimisation and approximation strategies in design optimisation problems. The fourth section introduces the concept of multi-objective optimisation and mentions different popular techniques used. Section five presents an integrated optimisation framework to deal with the QT and QL search spaces together. The next section demonstrates an application of the framework to a case study on rod rolling design optimisation. The seventh section describes a solution strategy for the rod design followed by case study results and discussion in section eight. Finally, the paper identifies the future research directions and outlines the main conclusions of the research.
2.
FUZZY LOGIC AND ENGINEERING OPTIMISATION There are several fuzzy-based approaches developed to incorporate QL knowledge into design. A discussion on non-fuzzy approaches to deal with QL knowledge is presented in Oduguwa [28]. Fuzzy set theory has been used as a mathematical formulation to represent QL knowledge in decision making [3]. Two of the earlier works dealing with optimization of fuzzy systems are by Tanaka et al. [42], and Zimmerman [47]. Since then several variations of fuzzy based approaches have been reported in the literature. Fuzzy-optimisation-based approaches tend to fuzzify the classical theories in an attempt to deal with the imprecision and vagueness in human reasoning regarding design variable preferences, constraints and goals. Approaches based on fuzzy mathematical programming include: fuzzy goal programming, flexible programming, fuzzy multiobjective optimisation, possibilistic programming with fuzzy preference operators and fuzzy linear programming. Antonsson [2] also developed a fuzzybased approach referred to as Method of Imprecision for engineering design problems where designers are allowed to express their preference over a range of design values. Most of these fuzzy-based approaches fundamentally fuzzify the elements (constraints, goals or design variables) of an underlying mathematical formulation and do not combine the QL evaluation within the optimisation search. Fuzzy Genetic Algorithms (FGA) manages problems in an imprecise environment. It combines fuzzy concepts with genetic algorithms. Approaches using fuzzy fitness evaluation function for the GA chromosomes have been reported in the literature [8, 19]. Hsu et al. [13] adopted a fuzzy optimization algorithm for determining the optimal gap openings of the programming points in the blow moulding process and in fuzzy controlled simulation optimisation. Medaglia et al. [21] proposed an approach that incorporated QL knowledge into the optimization strategy. Approaches based on fuzzy fitness however, offer more realistic solutions since they reflect the fuzzy nature of the problem and they provide more insights. They are capable of dealing with the linguistic expression that is present in the qualitative descriptions of goals, constraints, and preferences made by humans. Roy [34] developed a design optimisation framework where both QT and QL types of criteria are handled separately. The paper identified multiple ‘good’ design solutions based on the principal QT criteria and later evaluated them individually based on the other QL criteria. However, none of the reported applications combined both the QT and QL objectives simultaneously in the objective space.
3. APPROXIMATION STRATEGIES IN DESIGN OPTIMISATION Approximate models are built to simplify the representation of a complex behaviour. There are various reasons why analysts develop approximate models, it could be to speed up the numerical solution process, or in some cases it is the only way to explain a complex phenomenon. Engineering design problems are characterised by large complex features that hinder the use of conventional optimisation techniques. Most of these problems rely on the output of expensive computational simulations. The problem is compounded with the use of GAs as the optimisation engine, since GAs require a large number of fitness evaluations to find acceptable solutions. There are a number of approximation techniques reported in the literature that help to develop computationally inexpensive models, such as: response surface methodology [23], kriging [35], artificial neural networks [40] and inductive learning [22]. The techniques are essentially used to develop QT models of design linking the design variables with design evaluation(s). Response surface method (RSM) is one of the most popular approaches and is the focus of this paper. RSM can be used to create smooth approximations of the response data. Several authors [1, 39] have shown from comparative studies that the RSM and the Kriging approximations yield comparable results and neither really outperforms the other. Oduguwa [28] presents a description of different approximation techniques used in engineering design.
4. MULTI-OBJECTIVE OPTIMISATION Most real world problems are characterised by several non-commensurable and conflicting objectives. Multiobjective optimisation seeks to minimise the n components f(x) = (f1(x),…, fn(x)), of a possibly nonlinear vector function f of a decision variable x in the search space. Each of these objectives has a different optimal solution. There is no unique (Utopian) solution to a multi-objective problem but a set of non-dominated solutions referred to as Pareto-optimal set. A solution to this class of problem is Pareto-optimal if from a point in the design space, the value of any other solution cannot be improved without deteriorating at least one of the others. The objective for a complex multi-objective optimisation problem is to find different solutions close and well distributed on the true Pareto-optimal front. The conditions for a solution to become dominated with respect to another solution are described as follows. For a problem having more than one objective function (say, fj, where j = 1,…., M and M > 1), a solution x(1) is said to dominate solution x(2) if the following conditions are satisfied:
a) The solution fj(x(1)) is no worse than fj(x(2)) for all j = 1, 2,…., M objectives. b) The solution x(1) is strictly better than x(2) in at least one objective.
This model is a second-order polynomial, which includes main effects, interaction effects and purely quadratic effects.
Several genetic-algorithm-based multi-objective optimisation techniques have been reported in the literature. The main thrust of such algorithms is to produce a spread of multiple optimal solutions rather than a single optimal solution. VEGA [36] is one of the first, since then possibilities of improving computational efficiency and solution accuracy have been explored by other researchers [11,48,18,44,9,43]. The genetic algorithm based multi-objective optimisation technique, NSGAII [9], is adopted in this study since it is one of the most popular multi-objective evolutionary algorithms. 5. THE SOFT COMPUTING FRAMEWORK The research proposes to use RSM approximation technique to reduce computational expense of QT models, capture knowledge about QL aspects using a fuzzy expert system. Then both QT and QL types of criteria are used as fitness functions for a genetic algorithm based multi objective optimisation approach. The multi-objective optimisation assumes a functional relationship to use QT and QL fitnesses together. Oduguwa et al. [28] have presented a mathematical justification for the functional relationship between QT and QL and identified the conditions under which this is true. Fuzzy logic is adopted because it allows engineers to incorporate expert knowledge directly into the process. The multi-objective algorithm is used to explore the functional relationships and hence the interaction between the QT and QL knowledge. 5.1. QT Model Building This section describes the main steps involved in deriving the QT model using a Design of Experiments (DoE) method. DoE involves making several designs at once and investigating the joint effects of changes on a response variable. The DoE method using regression analysis, also known as analysis of variance (ANOVA), is used to develop meta-models. Figure shows the steps involved in building the models. The controllable variables of the problem are solicited from the domain experts. The Design of Experiments method is used to identify a suitable sample space depending on the affordable number of FEA (experimental) runs and the required model accuracy [39]. Each sample point represents a vector of decision variables to be used as inputs to the simulation experiments. The data set and the results obtained from the experiment can then be fitted to quadratic regression response surface models (quantitative models) using a least squares method of the type shown in equation 1.
Figure 1: QT Model Building
y ( x) 0
k
k
x i i
i1
i 1
2 ij xi
i
ij xi x j
(1)
j 1
where, ε is a random error and x is a vector of k system factors, (x1, x2,…..xk) ́ using least square estimation. In this study the DoE method is used to develop approximate models which act as surrogate for expensive computational FE simulations. This approach reduces the computational cost of the FE runs since the meta-models are used to represent the design problem. The QT model development process is shown in Figure 1. The next section presents steps to build QL model. 5.2 QL Model Building This section presents a modelling strategy based on combining the features of RSM and fuzzy logic modelling methods. In a broader context, the main aim for developing fuzzy QL models from statistical data set is to improve information efficiency through the combination of QL and QT information. This hybrid combination of methodologies introduces some structure in the use of QL information. Here, the modelling strategy maps a structured design variable space unto an
unstructured QL information space. The fuzzy modelling method is then used to develop a generalised fuzzy model based system. This section presents two novel approaches that enable the development of a QL fuzzy model from a statistical data set and discusses the motivations for developing the approaches. Variable Reduction Approach An important consideration when using the fuzzy-based approach for optimisation problems is the influence of the problem dimensionality on the number of rules. As the number of variables increases, the number of rules required in the fuzzy rule base also grows exponentially. This can make QL evaluation computationally expensive for optimisation purposes. This section presents a simple and novel approach for reducing the number of design variables used as inputs prior to fuzzy model development. The basic principle is based on approximating the variables into a smaller number of response variables using RSM approximation method (Figure 2). The response variables are then used as fuzzified input variables in the fuzzy expert system module (QL model).
Function Approximation (QT)
Design Variables
xi, i = 1…n
Fuzzy Reasoning (QL)
Membership Grade, Domain Value
yj,=f(x); j = 1…k
The rule generation module starts by initially mapping the input fuzzy set to an output fuzzy set for all the response values in the structured data set. Then for each input fuzzy set, the data set is sorted to identify relationship amongst the combination of the input fuzzy set and the output fuzzy set. The input fuzzy set is modified for all data points not showing consistency with the overall data structure and the procedure is iterated until all the data points shows consistency in terms of relationship of both input and output fuzzy sets. The list of rules unique in terms of input and output fuzzy structure is then identified to form the generalised fuzzy rule base. The fuzzy variables and the rules are used in a standard fuzzy expert system to form the QL model. The expert system uses mean of a maxima type defuzzification technique [28]. Table 1: Procedure for Generating Fuzzy Rules from Numerical Data Set Number of data , number of fuzzy sets , number of input fuzzy set Step 1: Map fuzzy sets for i : = 1 to do Map input fuzzy sets() onto output fuzzy set Step 2: Map fuzzy partitions onto the data set for i : = 1 to do sort input fuzzy set to correlate with output fuzzy set while samples data do not correlate, modify input fuzzy sets Step 3: Identify rules into groups that correlates with a category in the fuzzy output Step 4: Aggregate rules by selecting a member from each group
Figure 2: Approximating Variables for Fuzzy Inputs Generating Fuzzy Rules from Numerical Data This section presents a novel approach for generating fuzzy rules from numerical data. Since the values of the approximated response variables are derived from statistical principles, the nature of the variable space and its partition is not always intuitive to the experts. These present difficulties in constructing the fuzzy sets and fuzzy rules. In order to develop the fuzzy rule base it is important to ensure the appropriate correlation of the variable partitions (QT information) with the consequent variable (QL information) in a way that is consistent with the overall fuzzy model. Therefore, this section presents a novel approach for generating fuzzy rules from numerical data. The basic solution strategy is to ensure the consistency of the fuzzified input fuzzy set with the output fuzzy sets for all the sampled data set. This solution strategy is illustrated in Figure 3. Table 1 also shows a step-by-step procedure for generating fuzzy rules from numeric data.
5.3 Integrated Optimisation with QT and QL models The optimisation module as shown in Figure 4 is based on genetic algorithms (GA) integrated with a fuzzy reasoning module. The multi-objective optimisation framework uses the NSGA II to explore functional relationship between the QT and QL models. The QT models adopted in the study are response surface models obtained from DoE. While the QL models are the fuzzy models of the design. The fuzzy evaluation module embedded into the NSGAII outputs two results, the membership function value and the defuzzified domain value. These outputs represent the approximate QL evaluation of the design problem and it is used to assign a fitness value to the Q L aspect of the individual member of the population. The QL fitness evaluation also takes into account the membership grade to ensure that membership grade below a selected threshold is penalised using the penalty function method.
fitness value based on the QT and QL models. The multiobjective ranking mechanisms then perform a nondomination ranking procedure on each member where they are assigned a ranking value based on their location in the objective of parameter space.
Figure 3: Framework for Generating Fuzzy Rules from Numeric Data
6. APPLICATION TO ROD ROLLING PROBLEM The proposed approach is illustrated using a rod rolling design problem. In complex hot rolling of rods, Finite Element (FE) analysis is often used to study the effect of roll design and complex thermo-mechanical interactions during high temperature rolling on key properties such as shape, microstructure, rolling loads and torque[17, 25, 37]. FE models have been used to develop a detailed understanding of the rolling process at meso-scale level. Although FE techniques allow an entire rolling sequence to be studied, it is still time consuming (mostly in 3D) [16, 45, 20], despite improvement in both hardware and software to use them as embedded into a general optimizer for optimizing roll design sequences. There are very few cases reported in the literature where soft computing techniques have been used to solve design optimisation of the rolling problem. Several authors, have applied fuzzy reasoning, for roll pass design [32,15,38] and genetic algorithms in other related metal forming problems [14,6,26,27,5]. In all of these applications, none has considered QL evaluation simultaneously within a design optimisation framework. It is therefore necessary to develop a new low cost methodology for optimisation based not only on FE responses, but also on the engineer’s qualitative knowledge for solving complex engineering design problems.
Figure 4: Optimisation for Integrated QT and QL Search Space Solution of each member of the population is based on a ranking mechanism that considers the fitness value from the QT and QL models. In order to select the fittest member of the population, each individual is ranked based on the Pareto dominance criteria stated in section 4. Individual members of the population are assigned a
Figure 5: Rolling Process
6.1. Rod Rolling Process A schematic layout of a rod rolling process is shown in Figure 5. The process is a continuous manufacturing process whereby a square billet (dimension ranging from 100mm to 150mm) referred to as the stock is deformed into a rod size ranging between 5mm to 12mm. The rolling operation is a high speed, high production process in which a pair of rolls rotates at the same peripheral speed in opposite directions. The stock is continuously deformed by passing it through a series of high rolling mill stands. During the rolling process, the stock undergoes changes in the mechanical, geometrical and thermal characteristics.
used to define region of interest, six variables identified and their operating range specified. Table 2 shows the factors and factor levels used in simulations. A small composite design described in section 5.1 is applied to sample the design space problem.
Design of the rolling system involves consideration of the mechanical, thermal and thermo-mechanical behaviour of the process [41], and the optimisation of roll pass design [10]. Modelling of the rolling process is used to predict mill parameters (roll separating force, torque) and deformation characteristics such as the lateral spread and the evolution of metallurgical properties. These predictions are obtained using design variables related to the rolls and stock such as geometrical and material characteristics: temperature, friction etc. This study is interested in the geometry of the rod and the associated deformation load. This case study focuses on maximising the roundness of the rod profile and the minimisation of the associated deformation load.
Geometrical parameters: Expert domain knowledge is used to select the geometrical parameters and the region of interest defined according to the ranges shown in Table 2. These six factors are varied in the simulation runs. Other parameters such as roll speed and friction are kept constant to make the simulations comparable. A summary of these geometrical parameters is shown in Table 3.
Process conditions used in experiment The choice of geometrical parameters and material properties is discussed in this section. The choice of parameters is driven by the need to mimic the real design problem experienced on the plant in the study. The results obtained can then be validated using existing domain knowledge.
Table 2: Factors and Factor Levels used in Simulations
22.2
Pass Rad 30.8
Roll Gap 6.0
Roll Radius 707.8
1071.885
36
20
28
5.5
600
1000
33
17
24
3.5
450
900
30 27.8
14 11.8
20 17.1
2.5 0.92
300 192.2
800 728.115
Level
Height
Width
1.72
38.2
1 0 -1 -1.72
Temp
6.2. Experimental Procedure The factors affecting the rod rolling design problem are solicited from the domain expert and can be categorised as; (a) geometrical parameters such as height, width, roll gap, roll radius. (b) Related metallurgical parameters such as strain values, stress components and bulk temperature, (c) process parameters such as friction, roll speed etc. The independent variables especially relevant to the present simulation are height (h), width (w), roll gap (rg), roll radius, pass radius and the rolling temperature. It is important to understand the overall effects and interactions of these parameters on load and roundness. Roll designers can use this knowledge to design the optimum required design solutions that satisfies the conflicting objectives. Existing knowledge is
a)Transverse Section
(b) Full View Figure 6: Finite Element Contour Plots of Rod Profile
Table 3: Parameters used in Simulation Study Geometrical parameters Pass Depth 20mm Arc Radius 64mm Height Factor Width (W) Factor Roll Gap (Rg) Factor Pass Radius Factor Roll Radius Factor Temperature Factor Roundness (R) Response Load (L) Response
Material specification 0.08% Carbon steel C 0.087, Si 0.003, Mn 0.34, P 0.025 S 0.02. Hot rolled and annealed Suzuki (4.22) Process Parameters Friction: 0.3 Roll Speed: 1000 m/s
Material specification The material specification used in the study is shown in Table 3. The specification is identical for all runs. Process parameters The same loading conditions applied in all the simulations so that the response could be obtained under similar conditions. Finite Element Analysis and Data Extraction The finite element runs performed using Abaqus version 6.2.2. The mesh is generated using Patran software. Stock deformation is assumed symmetrical across horizontal and vertical planes through the stock centre therefore a quarter symmetry of the stock is modelled. A contour plot of a typical run is shown in Figure 6. The PEEQ (equivalent plastic strain) plot shows the deformation characteristics of the stock. Results are taken in the steady state (SS) zone of the rod. The SS is defined as the region where the deformation characteristics are assumed to be uniform. The SS zone is identified by using a qualitative judgement to identify region along the rod (see Figure 6b) where the contour profiles are parallel. 7. ROD DESIGN SOLUTION STRATEGY The framework of the solution strategy adopted for the rod rolling design problem is shown in Figure 7. This consists of QT modelling using design of experiment (DoE) sampling technique, QL modelling using fuzzy modelling techniques and multi-objective optimisation using NSGAII technique. The DoE technique is used to sample the design space of the rod problem and the method of least square used to fit regression models. These regression models are from here on referred to as the QT models. Thirty-two (32) different input configurations were proposed for the investigation. These served as input conditions for the FE simulations [28]. The outputs are used as inputs to the fuzzy reasoning module. This approach is adopted for two reasons. The first,is because the experts had a better understanding of the rod deformation in terms the chosen
Figure 7: Framework of Solution Strategy responses rather than the design variables. For example, it is easier to evaluate the design solution in terms of stock area deformation ratio than by the design variables such as stock height (h1, h2). Second, there is a secondary advantage of compressing the number of rules required for QL modelling of the rod deformation behaviour. The expert’s QL evaluation of the solutions and their reasoning with the response variables of the DoE are used to develop the QL models of the fuzzy reasoning module. The fuzzy reasoning module is integrated with the optimisation module to evaluate the QL aspect of the design problem. The section that follows discusses the main parts of the rod design solution strategy. 7.1. Quantitative Modelling Due to the high computational cost of the FE runs, a lowcost small composite design (SCD) fractional sampling method is adopted to select the sample data for the model development. The general SCD is a special resolution III fraction of a 2k design augmented with axial points and runs in the centre of the design. These designs are capable of estimating a second-order model and mostly suitable for problems with high computational cost [24]. In this study, the SCD matrix adopted is shown in Figure 8. This design shows a two-level, 6 factor fractional design augmented with axial points (α) with a value of 1.719 and two centre points. FE simulations were performed using the design matrix D in Figure 8 to generate the input values for the FE runs. From the observation of the FE results, the following nondimensional design parameters were used to analyse the rod shape: (a) initial stock area/roll area (SAR), (b) form
factor (FF): ratio of height to width ratio of deformed stock (h2/w2) and (c) roll radius/material height ratio(RRMR) (R/h1). For each run, values of the measured output for the three responses and load recorded. QT models of the four responses (SAR, FF, and RRMR) were generated by fitting the second order model type shown in equation 1 (main effects, interaction effects and quadratic effects). The fit with the lowest sum of squares error (highest R2) was selected; this resulted in the following experimental models as predicted using ANOVA: SAR = – 1.976 + 0.1106 h1 – 0.00157 h12 + 0.184 w1 – 0.0012 w12 – 0.104 Pr + 0.0025 Pr2 + 0.0046 Rr – 2.708E-6 Rr2 + 11.24E-6 T + 0.0026 h1w1 – 5.728E-4 h1Pr – 1.0455E-4 h1Rr – 0.002 w1Pr – 0.0036 w1Rg – 1.207E-4 w1 T … (2)
w1Rg + 2.532E-4 w1T – 0.0011 PrRg – 1.695E-4 RgRr + 4.319E-5 RgT (3) RRMR = 6.155 – 0.375h1 + 0.0056h12 + 0.061Rr + 5.877E-5 h1Rg – 9.319E-4 h1Rr – 1.267E-5 RgRr (4) Load = -2023520.422 + 50112.96 h1 – 853.369 h12 + 35728.755w1 + 434.706 w12 – 39003.709 Pr + 604.149 Pr2 + 19369.967 Rg – 1271.041Rg2 + 578.474 Rr – 2.206 Rr2 + 2799.198 T – 1.039 T2 + 1135 h1w1 – 177.396 h1Pr – 305.971 w1Pr – 2075.8333 w1Rg + 57.274 w1Rr – 77.103 w1T + 417.083 PrRg + 14.183 PrT – 0.413 RrT (5) The coefficient of determination (R2) for each of these models is shown in Table 4. Additional 5 set of data randomly selected as validation points and used to verify the accuracy of the QT model. The root mean square error is the difference between the actual response from the FE analysis and the predicted value of the QT models. The finite element results from the validation data set are input into the response models and the RMS error values are obtained. The value obtained for the normalised RMS error for the SAR response model is 4.07%. This is an acceptable range for the type of application. Most of the model has high R2 value and low MSE.
No 1 2 3 4
Table 4:Model Diagnostis of RS Models Response R2 R2adj RMSE SAR FF RRMR Load
0.994 0.999 0.999 0.996
0.987 0.995 0.999 0.984
4.07% 1.84% 0.8% 3.55%
The response variables in equations 2, 3 and 4, were used as inputs to the fuzzy rules, while equation 4 is the load response variable of the quantitative model to be minimized.
Figure 8: 26-1 Small Composite Design Matrix FF = 11.109 – 0.190 h1 +0.0022 h12 – 0.525 w1 + 0.0077 w12 + 0.0022 Pr2 + 0.176 Rg + 0.00561 Rg2 – 0.0035 Rr + 5.191E-6 Rr2 – 0.0061 T + 9.765E-7 T2 – 8.722E-4 h1w1 + 0.0011 h1Pr + 7.015E-5 h1Rr – 0.0061 w1Rg – 4.1E-5
(a) Fuzzy Set SAR Figure 9: Membership Functions for Rolling Variables
(b) Fuzzy Set RRMR
visually classifying the data set into clusters. Where each cluster in the data space forms a fuzzy set and the degree of overlap in the fuzzy set represents the fuzziness in the clusters. It is important to point out at this initial stage, the classification exercise acts as a guide to identify the potential fuzzy sets and the range of each fuzzy set. The final features of these fuzzy sets is determined after the correlation of the combination fuzzy sets with the rule base shows consistency in model behaviour for all experimental values in the structured data set.
(c) Fuzzy Set FF Figure 9: Membership Functions for Rolling Variables (contd.) 7.2. Qualitative Modelling QL evaluation of the design solutions is performed by the experts to determine the suitability of each design. A structured interview technique is conducted with three experts, facilitated by a questionnaire, to capture the necessary knowledge. Fuzzy models are developed using the framework detailed in previous section. These formulated the response variables of equation 2, 3 and 4 as the antecedent part of the rules, and modelled the expert’s reasoning of the FE outputs as the consequent part of the rules. A total of nine rules are developed. The antecedent parts of the fuzzy rules are developed by fuzzifying three response variables from equations 2, 3 and 4. Using the framework in section 5.3, for each of the response variables, the fuzzy sets shown in Figure 9 are created with triangular membership functions and their corresponding linguistic labels low, average and high. It is not possible to use expert knowledge in this case to derive the properties of the fuzzy sets since such behaviours are not intuitive to the experts. Instead, the data set is used to create the fuzzy sets since the data captures the intrinsic characteristics of the design problem (Appendix A). It should be noted that the expert knowledge is not applicable at this stage to develop the fuzzy inputs. However, such intuitive knowledge is used in defining the fuzzy outputs. Hence the fundamental rationale for using FS in solving the RSD is still preserved. The features of the input fuzzy sets are initially determined by exploring the nature of the scatter in the data set for each input variable. This is synonymous to
Figure 10: Classification of FE Rod Shape Profiles The consequent part of the fuzzy rule is developed to represent the expert’s QL evaluation of the roundness of the rod profile. This is achieved by initially classifying the FE output of the rod profiles into five main categories as shown in Figure 10. These five categories are then formulated into the following five fuzzy sets as shown in Figure 11 with bell shaped membership functions. The corresponding linguistic labels are: Elliptical (E), Fairly Elliptical (FE), Flat Round (FLTR), Fairly Round (FR), and Round (R).The membership function of each fuzzy set and the degree of overlap are identified using the engineer’s interpretation of the FE outputs. These represented the way experts reason about the roundness of the rod profile. The initial classification and relation resulted to ten aggregated rules. Further fine tuning reduced the number of rules to nine (Table 5) and improved the distribution of the each rule in the over all fuzzy space. The fine tuning process involved modifying the FF and SAR fuzzy set as shown in Table A.1 in Appendix A. The effect of changing the input fuzzy set in shown in Table A.2 in the appendix. The table identifies details of rule generation after two iterations. It identifies the frequency of each rule in the rule table (APD) and how each rule is represented in the actual rule set of the fuzzy model (APR). By comparing details from the initial and final rule generation process, it is clear that the final rule set
and its structure is better than the initial iteration. The ratio APD/APR also expresses how well distributed each fuzzy set is in the fuzzy space. It can be seen that the final has an overall higher distribution average compared to the initial stage. Table A.3 in the Appendix A presents the rule identification from a set 30 examples.
cumulative effect of the other rule influences the determination of the rod profile. Ordering of the fuzzy antecedent rules are arranged such that truth function converges towards the form factor (ff) value since the form factor has the most significant effect on the rod profile.
The rules are then validated with three experts [28, 29, 30]. Aggregation of the outputs showed fuzzy model accuracy within 92%. The error in the fuzzy model is attributed to the combination of the error in the Q T models and the error due to the global approximation of the QL models. However, this level of accuracy is considered good enough for the experimental study.
The fuzzy sets, input, output fuzzy variables and fuzzy rule base all constitute the qualitative model that is used within the optimisation framework to evaluate the qualitative aspect (roundness) of the design problem. These fuzzy sets are then converted into a scalar value by the mean of maxima method of defuzzification, in the final step of the fuzzy inference cycle.
Figure 11: Membership Functions for Roundness Table 5: Fuzzy Rule Base Rule No
SAR
RRMR
FF
Roundne ss
1
Average
High
Low
R
2
High
Low
Low
FLTR
3
High
High
Low
FLTR
4
Average
Low
High
FE
5
Average
Low
Average
FR
6
Low
High
High
E
7
Average
High
High
E
8
Low
Low
High
E
9
Average
Low
Low
R
p1 sar p 2 rrmr
(6) p 3 ff 2 2 The compensatory weighted mean operator of the form shown in Equation 6 is used to combine the fuzzy sets in the antecedent part of the rule. This approach is adopted in order to allow for a weaker and less sensitive relationship among propositions when their truth-values are widely separated. This approach also ensures that the
7.3. Optimisation Module The aim of this module is to solve a rod design optimisation problem. The design problem consists of two cardinal objectives: to minimise the load using the QT model and to maximise the roundness using QL models of the rod profile. The multi-objective optimisation problem is formulated as shown below: Minimise Load f1(x) = P Maximise Roundness f2(x) = x7 Subject to P >0 x8 > 0.5 where; P = -2023520.422 + 50112.96x1 – 853.369x12 + 35728.755 x2 + 434.706x22 – 39003.709x3 + 604.149x32 + 19369.967x4 – 1271.041x42 + 578.474x5 – 2.206x52 + 2799.198 x6 – 1.039 x62 + 1135 x1x2 – 177.396x1x3 – 305.971x2x3 – 2075.8333x2x4 + 57.274x2x5 – 77.103 x2x6 +417.083 x3x4 + 14.183x3x6 – 0.413 x5x6 x1 is the height, x2, is the width, x3 is the pass radius, x4 is the roll gap, x5 is the roll radius and x6 is the temperature. QL is the qualitative formulation of fuzzy variables SAR, FF and RMMR expressed in terms of fuzzy sets and fuzzy rules shown in the previous section, x7 is the defuzzified scalar value of the fuzzy model and x8 is the associated membership grade. NSGAII [9] is adopted in the study as one of the most popular multi-objective GA. NSGAII is used to rank each member of the population in terms of the fitness from the QT model and the QL model using the nondomination criteria described in section 4. Roundness is represented by two values: a scalar domain value from the fuzzy expert system and its corresponding membership grade. Solutions having membership grade below a chosen threshold (0.5 in this study) are considered inappropriate for the rod problem and are penalised using the penalty function method. Similarly, fitness from the QT model expresses the load due to rod deformation; this needs to be minimised. The two criteria (QT and QL) contradict each other.
near optimal solutions superimposed on the random search space of 5000 points in each case.
350 Re gion 3 300 Region 6 Re gion 4
Nature of the Search Space and Optimal Solutions
Regi on 5 200 Re gion 1
100 50 0 0
0.2
0.4
0.6
0.8
1
Roundness
Figure 12: Qualitative and Quantitative Search Space
8. TEST RESULTS AND DISCUSSION The NSGAII is used to locate good solutions for the optimisation problem stated in section 7.3 by evaluating each member of the population using the models shown in the equation. For the multi-variable encoding problem, each variable is encoded first in binary form, and then linked together as a chain to form the chromosome. Design variables encoded for the rolling problem and variable bounds are estimated for feasible designs are shown in Table B.1 in Appendix B. The table also shows the output variables. The total resulting bit number of the chromosome is 68. The NSGA-II based algorithm has solved a design optimisation problem to minimise deformation load and maximise roundness of the rod shape. Experiments are carried out using the proposed algorithm for both objectives to illustrate how the algorithm deals with the QT and QL search space problem. The proposed algorithm is implemented for the test problem using C++ code on a Pentium 4 PC. The performances of the proposed solution algorithm for different values of crossover and mutation probabilities are first investigated. Ten independent GA runs are performed in each case using a different random initial population. The crowded tournament selection operator is used to select new offspring. Tests are carried out with the following parameters: population of size 100 for 500 generation with a crossover probability of 0.8 and a mutation probability of 0.05, and tournament selection with size 3. These results form the typical set obtained from 10 runs with different random number seeds. In most of the cases examined six out of ten runs obtained similar results. The following section reports on results carried for tests with the following membership grades 0.3, 0.5, 0.7. The influence of different membership grades on the search space is studied to explore how these affect the solutions achieved. Results shown in Figures 13-15, where obtained for the NSGA-II parameters outlined above. The figures show the set of
Figure 12 indicates the nature of the search space from the random sampling. This shows the nature of the search space as discontinuous with the solution almost equally spread amongst the six discrete regions. Figure 12 also shows that solutions tend to concentrate in regions close to the core of each consequent fuzzy set. This is expected since the core regions act as landmarks of the knowledge for formulating the fuzzy model. Figure 13-15 shows a filtered search space when compared to Figure 12. This suggests that x8 acts as a filter, removing solutions less than the set threshold value. For the design problem in this section, the greater the threshold value, the higher number of solutions removed. This implies that a significant proportion of the solution lies below the 0.7 threshold. This explains the differences in the figures. It is interesting to note that there is a dramatic reduction of region 4 (Figure 12) in Figures 13-15, as the membership grade threshold increases. This is because solutions lying in this region are composed of two fuzzy sets that have a small degree of overlap. Since the relationship between these fuzzy sets is weak, it is likely that solutions obtained from the composition of the fuzzy sets also bear a low membership grade. 350 Random Points
Optimal Solutions
300 250 200 Load (KN)
Regi on 2
150
150 100 50 0 0
0.2
0.4
0.6
0.8
1
Rou ndness
Figure 13: NSGA-II Results on P3 (x8= 0.3) Random Points Se lecte d O ptimal Solutions
350
O ptimal Solutions FE data se t
300 250 200 Load (KN)
Load (KN)
250
150
5
100 50 1 0 0
3
4
2 0.2
0.4
0.6
0.8
Roundne ss
Figure 14: NSGA-II Results on P3 (x8= 0.5)
1
fronts are all equally good, further higher level criteria could be applied to select a suitable solution for the problem. For example, solutions within a desired range could be preferable.
350 Random Points
Opt imal Solut ions
300
Load (KN)
250 200 150 100 50 0 0
0.2
0.4
0.6
0.8
1
Roundness
Figure 15: NSGA-II Results on P3 (x8 = 0.7)
Solutions shown in Table 6 show how designers have the visibility and the opportunity to select more suitable solutions for the given problem. For example, solutions number 1 and 2 has a load value of 179.5KN and 138.5KN respectively with very similar roundness value. This suggests that solution 2 could be most preferable in terms of the load factor since it has a lower deformation load compared to solution 1. Solutions 1 to 4 have very similar roundness but different load values. Since these four solutions have similar roundness values, they fall into the same class of fuzzy set ‘round’ with the same QL evaluation. This suggests that the design is less sensitive to changes in load. Research Issues of Integrated QT and QL Design Strategy in Design Optimisation This section critically evaluates the influence of QL evaluation in the integrated search space in terms of the nature of the problem, impact on the search algorithm and the effect on the solutions achieved. For each of these concepts, the advantages and drawbacks of the integrated design strategy are discussed.
Figure 16: NSGA II Pareto Solution Plot Experimental Results The trade-off solutions between roundness and load located in the optimal region by the NSGA II optimisation algorithm are shown in Figure 16. Despite the complexity of the problem, NSGAII is able to find solutions in the optimal region of the design space. Nondominated solutions are obtained from the experimental runs. The Pareto optimal solution plot shows the spread of the optimal solutions in the two dimensions. Table 6: Selected Solutions
Table 6 also shows a selection of optimal solutions and their variable values from the experimental runs. It demonstrates the diversity of the vectors of the decision variables in the parameter space. Since solutions on these
For real-life optimisation problems such as the rolling process, the search space can be quite complex due to high dimensionality from large number of parameters [31]. This becomes more evident when a multi-pass model is developed. This implies exponential increase in interactions among those parameters, which result in ‘curse of dimensionality’ [4]. Incorporating QL in the search space offers a number of advantages. It provides opportunity to deal with the complexity. Here some of the uncertainty in the problem can be addressed through approximate QL models to improve the problem definition. Since the problem is represented in natural language, the limitations in transforming design problem knowledge into numerical representation are avoided. This reduces bias errors and should offer more robust way of incorporating knowledge into the problem space. It is also possible to develop models for concepts that are mathematically ill-defined and to represent these models in the search space. It also offers opportunity to decompose the large problem into manageable submodels and to combine concepts from different perspectives. This avoids information loss and leads to efficient use of information [46]. However QL evaluation search space also offers some drawbacks. The imprecise nature of QL evaluation can result in ambiguity. Consider a situation where irrespective of the search point the Q L evaluation is indifferent about the objective value. This can introduce ‘deception problem’ for the GAs. There
could also be a tendency to ignore the fact that the defuzzified scalar value from the QL model is semantically imprecise. This issue can result in selecting inappropriate trade-off solutions. The QL model is developed to take advantage of the tolerance of imprecision with which human reasons. Therefore, the output of the model (scalar value) should be treated as an imprecise approximation of human reasoning. In modelling QL evaluation, with current methods of aggregating fuzzy sets it is difficult to understand the semantic equivalence of the final defuzzified fuzzy set with the real problem. For example with mean of maxima method of defuzzification, the final fuzzy set determines the overall evaluation of the aggregated fuzzy sets. It is difficult to comprehend how a single point from aggregated fuzzy sets could possibly represent the underlying knowledge of the final set. This presents problems for the designers since it is difficult to understand the reasoning process that led to the solution. The impact of the QL evaluation in evolutionary search algorithm is explored in this section. The impact of QL evaluation in the proposed approach on the search algorithm is considered minimal since QL information is first transformed into cardinal information with the subsequent use of quantitative multi-criteria methods. The evolutionary search algorithm provides some advantages. The functional relationship between QT and QL obtained from exploring the trade-off solutions could facilitate the discovery of new knowledge. This can provide better insight into the problem and encourage triangulation of QT model with human understanding of the problem. However the nature of the integrated QT and QL search space could significantly influence the search space. Interlacing the discrete nature of QL evaluations, with the smooth nature of the response surface models results in complex search space. This is evident in the QT and QL search space problem (Figure 12) adopted in this study. Such a mixed discretecontinuous search presents difficulties for the evolutionary search algorithm. Another interesting feature that can influence the search is the measurement scale of the QT models and granularity of QL models. While measurement scale implies how much measurable information can be provided by the measurement scale, granularity of the QL model is the ability to represent the same physical system at increasingly finer level of detail. Since behaviour of a model abstracted at one level of measurement scale may have a different meaning at another level [7], it is imperative to ensure that the models have semantically equivalent scales. While an approximate model may be useful in solving a particular problem, its prediction do not necessarily agree with more detailed model with which it is associated. Therefore models with different scales can result to
invalid solutions. This aspect of the search needs to be investigated in the future. The solution obtained from the integrated design strategy is expected to deliver more realistic solutions to the overall problem. The basic assumption here is that efficient use of information can result to better problem definition which in turn delivers a more realistic solution assuming an adequate search technique is adopted. This should provide a spectrum of solutions based on different levels of desirability that give more insight into the problem. This can also be very useful in challenging existing mental models and in confirming what is already known about the problem. However there are user related problems with solutions obtained from such paradigm. The solution strategy is inherently stochastic in nature (GA), and designers lack understanding of fuzzy logic reasoning mechanism. Designers can be sceptical about the results obtained due to the black box nature of the optimisation. Traditionally designers use standard sensitivity analysis method to explore design behaviour for each criteria of evaluation. This can be very frustrating if this method is used on a fuzzy model and the reasoning mechanism can not provide any explanation for the results achieved. The analysis in this section shows that the integrated QT and QL design paradigm can offer a low cost solution strategy in comparison to conventional approaches to deal with complex engineering design problems. This has the potential to deliver multiple optimal solutions that considers QT and QL issues simultaneously for different classes of problems. However in order to realise these benefits there are various challenges emanating from some of the issues explored in this sections. These challenges are discussed in the following section. Challenges of QT and QL in Integrated Search Paradigm There are several challenges that can inhibit the wider application of the integrated design strategy for design problem. Some of these are outlined below. Application to real-life problems could present scalability issues. The increasing problem size due to the need to add more problem features to the problem could prove difficult. The computational cost required to generate QL models when simulating is exponential with increasing number of variables. Developing QL model that represents a broad range of physical phenomena from different perspective at a level which allows useful and verifiable inference to be drawn can be computationally expensive and nontrivial. Developing search procedures for QL and QT objectives greater than two is non-trivial. Higher number of QL objectives has the tendency to increase
the fragmentation in the search space. This is largely due to the discreteness in the QL search space. 9. FUTURE RESEARCH DIRECTIONS Future researches are required to address the issues outlined in the previous paragraph. This section briefly describes possible research directions. More studies are required on scalability to understand features that influence increased complexity of the problem as the problem size increases. The impact on the search space and the additional computational efforts required need to be explored. This study could influence efficient algorithm development to improve the adoption of such strategies for real world application. Comparative study is required to evaluate the performance of different types of integrated strategies. This should provide understanding on the efficiency and relevance of various strategies in relation to the problem scenarios. The fragmentation due to QL search space needs to be studied to develop algorithms to address the challenges from such features. 10. CONCLUSIONS The algorithmic engineering design optimisation problem is normally conducted in a QT environment without considering the related QL issues. In some cases, this could mean ignoring important issues related to the design problem. This paper has presented a soft computing framework for dealing with QT and QL models within multi-objective optimisation. The proposed solution approach is based on design of experiment methods, fuzzy logic, and multi-objective genetic algorithm techniques. The approach is applied to a two-objective rod rolling problem. The results show that functional relationships exist between QT and QL models and these can be solved simultaneously within a multi-objective optimisation framework. The paper also presented a critical evaluation of the integrated design strategy for real life design problems. Finally it is demonstrated that the proposed solution approach can be used to solve real world problems taking into account the related QL evaluation of the design problem. This research also identified the challenges and future work in optimising within an integrated QT and QL search space. ACKNOWLEDGMENTS The authors wish to acknowledge the support of the Engineering and Physical Sciences Research Council (EPSRC) and Corus R, D and T, Swinden Technology Centre, UK. Contribution from different members of Decision Engineering team in the Enterprise Integration department is also acknowledged.
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Appendix A
Low
Average
High
SAR
RRMR
Form Factor
0.6225 0.6559 0.6871 0.7353 0.7753 0.8115 0.8558 0.8729 0.8831 0.8832 0.9004 0.9094 0.9938 1.0162 1.0162 1.0162 1.0162 1.0162 1.0162 1.0602 1.0741 1.1057 1.1326 1.1872 1.2522 1.3154 1.3204 1.3492 1.4362 1.5334
0.1154 0.12 0.12 0.12 0.12 0.1392 0.144 0.144 0.144 0.144 0.165 0.165 0.165 0.165 0.165 0.165 0.165 0.165 0.165 0.165 0.1908 0.2 0.2 0.2 0.2 0.24 0.24 0.24 0.24 0.2893
0.8390 0.8814 0.8952 0.9181 0.9887 0.9969 1.0213 1.1128 1.1175 1.1477 1.1549 1.1773 1.1872 1.2031 1.2061 1.2065 1.2108 1.2242 1.2556 1.3202 1.3694 1.4312 1.4406 1.4871 1.4993 1.7305 1.7696 1.7789 1.7875 1.8727
Figure A.1: Classification of Input Data Set of Roundness
Low
Average
High
Table A.1: Fuzzy Set used in Fine Tuning SAR
FF
Initial
Final
Initial
Final
0.623 0.656 0.687 0.735 0.775
0.623 0.656 0.687 0.735 0.775
0.839 0.881 0.895 0.918 0.989
0.839 0.881 0.895 0.918 0.989
0.812 0.856
0.812 0.856
0.997 1.021
0.997 1.021
0.873 0.883
0.873 0.883
1.113 1.118
1.113 1.118
0.883 0.900
0.883 0.900
1.148 1.155
1.148 1.155
0.909 0.994 1.016 1.016 1.016 1.016 1.016 1.016 1.060 1.074 1.106 1.133 1.187
0.909 0.994 1.016 1.016 1.016 1.016 1.016 1.016 1.060 1.074 1.106 1.133 1.187
1.177 1.187 1.203 1.206 1.206 1.211 1.224 1.256 1.320 1.369 1.431 1.441 1.487
1.177 1.187 1.203 1.206 1.206 1.211 1.224 1.256 1.320 1.369 1.431 1.441 1.487
Table A.2: Data Details for Analysing Rule Generation
Fuzzy set E FE FLTR FR R
Number of appearance in the data set (APD) 7 5 6 7 5 30
Initial Number of appearance in APD/APR aggregated rule set (APR) 2 3.5 2 2.5 2 3 1 7 3 1.6 10
17.6
Final Number of appearance in aggregated rule set (APR) 3 1 2 1 2 9
APD/APR 2.3 5 3 7 2.5 19.8
Table A.3: Identification of Fuzzy Rules for Roundness Runs
SAR
RRMR
Form Factor
Roundness
SAR
RRMR
Form Factor
Rule No
1
1.06
0.24
1.148
R
Average
High
Low
2
1.252
0.144
0.881
FLTR
High
Low
Low
1 2
3
1.32
0.24
1.113
FLTR
High
High
Low
3
4
0.812
0.144
1.431
FE
Average
Low
High
4
5
1.133
0.12
1.187
FR
Average
Low
Average
5
6
0.687
0.24
1.788
E
Low
High
High
6
7
1.074
0.2
1.021
R
Average
High
Low
1
8
1.533
0.144
0.839
FLTR
High
Low
Low
2
9
0.856
0.24
1.731
E
Average
High
High
7
10
0.735
0.2
1.499
E
Low
High
High
6
11
1.315
0.2
0.997
FLTR
High
High
Low
3
12
0.994
0.144
1.369
FE
Average
Low
High
4
13
0.775
0.12
1.77
E
Low
Low
High
8
14
0.623
0.12
1.779
E
Low
Low
High
8
15
0.909
0.12
1.118
R
Average
Low
Low
9
16
0.9
0.2
1.487
E
Average
High
High
7
17
0.883
0.139
1.32
FE
Average
Low
High
4
18
1.106
0.191
1.155
R
Average
High
Low
1
19
0.656
0.165
1.873
E
Low
Low
High
8
20
1.349
0.165
0.895
FLTR
High
Low
Low
2
21
1.436
0.165
0.918
FLTR
High
Low
Low
2
22
0.873
0.165
1.224
FR
Average
Low
Average
5
23
1.187
0.165
0.989
R
Average
Low
Low
1
24
0.883
0.165
1.441
FE
Average
Low
High
4
25
1.016
0.289
1.256
E
Average
High
High
7
26
1.016
0.115
1.177
FR
Average
Low
Average
5
27
1.016
0.165
1.211
FR
Average
Low
Average
5
28
1.016
0.165
1.203
FR
Average
Low
Average
5
29
1.016
0.165
1.206
FR
Average
Low
Average
5
30
1.016
0.165
1.206
FR
Average
Low
Average
5
Appendix B
Table B.1: Design Details of P3 Rolling Design Problem
Outputs
Design Variable
Binary bits
Design Variable Bounds
Height (h)
10
27.84
