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During the last few years we have witnessed impressive developments in the area of stochastic local search techniques for intelligent optimization and Reactive Search Optimization. In order to handle the complexity, in the framework of stochastic local search optimization, learning and optimization has been deeply interconnected through interaction with the decision maker via the visualization approach of the online graphs. Consequently a number of complex optimization problems, in particular multiobjective optimization problems, arising in widely different contexts have been effectively treated within the general framework of RSO. In solving real-life multiobjective optimization problems often most emphasis are spent on finding the complete Pareto-optimal set and less on decision-making. However the com-plete task of multiobjective optimization is considered as a combined task of optimization and decision-making. In this paper, we suggest an interactive procedure which will involve the decision-maker in the optimization process helping to choose a single solution at the end. Our proposed method works on the basis of Reactive Search Optimization (RSO) algorithms and available software architecture packages. The procedure is further compared with the excising novel method of Interactive Multiobjective Optimization and Decision-Making, using Evolutionary method (I-MODE). In order to evaluate the effectiveness of both methods the well-known study case of welded beam design problem is reconsidered.

In the modern-day of optimal design and decision-making, optimization plays the main role [1,2]. Yet most studies in the past concentrated in finding the optimum corresponding to only a single design objective. However in the real-life design problems there are numerous objectives to be considered at once.

Efficient multiobjective optimization algorithms facilitate the decision makers to consider more than one conflicting goals simultaneously. Some examples of such algorithms and potantial applications could be found in [3-7].

Within the known approachs to solving complicated optimal design problems there are different ideologies and considerations in which any decision-making task would find a fine balance among them.

According to [8,9] the general form of the Multiobjective optimization problems is stated as; Minimize , Subjected to, where is a vector of n decision variables; is the feasible region and is specified as a set of constraints on the decision variables; is made of m objective functions subjected to be minimization. Objective vectors are images of decision vectors written as. Yet an objective vector is considered optimal if none of its components can be improved without worsening at least one of the others. An objective vector is said to dominate, denoted as, if for all k and there exist at least one h that. A point is Pareto optimal if there is no other such that dominates.

The task of optimal design, described above, is devided into two parts: 1) An optimization procedure to discover trad-off conflicting goals of design; 2) A decision-making process to choose a single preferred solution from them. Although both processes of optimization and decision-making are considered as two joint tasks, yet they are often treated as a couple of independent activities. For instance evolutionary multiobjective optimization (EMO) algorithms [10,11] have mostly concentrated on the optimization aspects, developing efficient methodologies of finding a set of Pareto-optimal solutions in a problem. However finding a set of trade-off optimal solutions is just half the process of optimal design. This has been the reason why EMO researchers were looking to find ways to efficiently integrate both optimization and decision making tasks in a user-friendly and flexible way [

On the other hand, in multiple criteria decision making (MCDM) algorithms [13,14] often the single optimal solution is chosen by collecting the decision-maker preferences where multiobjective optimization and decision making tasks are combined for obtaining a point by point search approach [13,15,16]. In addition in multiobjective optimization and decision-making, the final obtained solutions must be as close to the true optimal solution as possible and the solution must satisfy the preference information. Towards such a task, an interactive analysis tool to try various preferences to arrive at a solution is essential. This fact has motivated some researches to carry on the important task of integration between multiobjective optimization and multiple criteria decisionmaking [10,12,17,18]. Although in multiobjective optimization, interactions with a decision-maker can come either during the optimization process, such as in the interactive EA optimization [

According to [

In the second way, a MCDM procedure is integrated within an EMO to find a preferred set of Pareto-optimal solutions. In this way, the search is concentrated on an important region of the Pareto-optimal front. This allows the optimization task to value the preferences of the DM. The research and publications on interactive evolutionary algorithms (EA) and applications are numerous. The researches in the field and various problem domains in which an EA simulation is carried out by the involvement of the DM reviewed by Takagi [

In [

Visualization is an effective approach in the operations research and mathematical programming applications to explore optimal solutions, and to summarize the results into an insight, instead of numbers [29,30]. Fortunately during past few years, it has been a huge development in combinatorial optimization, machine learning, intelligent optimization, and reactive search optimization (RSO) [8,9,21], which have moved the advanced visualization methods forward [

Concerning solving the MCDM problems, utilizing RSO, the final user is not distracted by technical details, instead concentrates on using his expertise and informed choice among the large number of possibilities. Algorithms with self-tuning capabilities like RSO make life simpler for the final user. And to doing so the novel approach of RSO is to integrat the machine learning techniques, artificial intelligence, reinforcement learning and active or query learning into search heuristics. According to the original literature [

The aim of local search is to find the minimum of a combinatorial optimization function f, so called fitness function, on a set of discrete possible input values X. To doing so the focus would be on a local search, hinting at RSO with internal self-tuning mechanisms, and BrainComputer Optimization (BCO), with a DM in an interacttive problem-solving loop. Accordingly in this context the basic problem-solving strategy would consists of starting from an initial tentative solution modifying the optimization function. According to [

.

If the search space is given by binary strings with a given length, the moves can be those changing the individual bits, and therefore L is equal to the string length M. The successor of the current point is a point in the neighborhood with a lower value of the function f to be minimized. If no neighbor has this property, i.e. if the configuration is a local minimizer, the search stops [

[

IMPROVING-NEIGHBOR returns an improving element in the neighborhood. Here `the local search works very effectively. This is mainly because most combinatorial optimization problems have a very rich internal structure relating the configuration X and the f value [

In problem-solving methods of Stochastic Local Search, where the free parameters are tuned through a feedback loop, the user is considered as a crucial learning component in which different options are developed and tested until acceptable results are obtained. As suggested in [

In the RSO approach of problem solving the brain-computer interaction is simplified. This is done via learning-optimizing process which is basically the insertion of the machine learning component into the solution algorithm. The strengths of RSO are associated to the human brain characteristics i.e. learning from the past experience, learning on the job, rapid analysis of alternatives, ability to cope with incomplete information, quick adaptation to new situations and events [8,9]. The term of intelligent optimization in RSO refers to the online and offline schemes based on the use of memory, adaptation, incremental development of models, experimental algorithmics applied to optimization, intelligent tuning and design of heuristics. In this context with the aid of advanced visualization tools implemented within the software architecture packages the true meaning of numbers, and the conveyed information, are considered for better solutions. Therefore the integration of visualization and automated problem solving and optimization would be the center of attention.

During the process of solving the real-world problems exploring the search space, utilizing RSO, many alternative solutions are tested and as the result adequate patterns and regularities appear. While exploring, the human brain quickly learns and drives future decisions based on the previous observations and searching alternatives. For the reason of rapidly exploiting the most promising solutions the online machine learning techniques are inserted into the optimization engine of RSO [

A number of complex optimization problems arising in widely different contexts and applications have been effectively treated by the general framework of RSO. This include the real-life applications, computer science and operations research community combinatorial tasks, applications in the area of neural networks related to machine learning and continuous optimization tasks. In the following we summarize some applications in real-life enginering application areas which are the main interests of this research. In the area of electric power distribution there have been reported a series of real-world applications [

Grapheur and LIONsolver [8,9,22,23,59] are two implementaions of RSO. The software implements a strong interface between a generic optimization algorithm and DM. While optimizing the systems produce different solutions, the DM is pursuing conflicting goals, and tradeoff policies represented on the multi-dimensional graphs [21,23]. During multi-dimensional graphs visualization in these software packages, it is possible to call user-specific routines associated with visualized items. This is intended as the starting point for interactive optimization or problem solving attempts, where the user specifies a routine to be called to get information about a specific solution. These implementions of RSO are based on a three-tier model, independent from the optimization algorithm, effective and flexible software architecture for integrating problem-solving and optimization schemes into the integrated engineering design processes and optimal design, modeling, and decision-making.

For solving problems with a high level of complexity, modeling the true nature of the problem is of importance and essential. For this reason a considerable amount of efforts is made in modeling the MOO problems in Scilab (available in the appendix) which later will be integrated into optimizer package. Here, as an alternative to the previous approaches [17,18,28] the robust and interactive MOO algorithm of RSO is proposed in order to efficiently optimize all the design objectives at once in which couldn’t be completely considered in the previous attempts. In this framework the quality of the design, similar to the previous research workflows, is measured using a set of certain functions, then an optimization algorithm is applied in order to optimize the function to improve the quality of the solution. Once the problem is modeled in Scilab it is integrated to the optimizer via advanced interfaces to the RSO algorithm and its brain-computer evolutionary multiobjective optimization implementations and visualization. In this framework the application of learning and intelligent optimization and reactive business intelligence approaches in improving the process of such complex optimization problems is accomplished. Furthermore the problem could be further treated by reducing the dimensionality and the dataset size, multidimensional scaling, clustering and visualization tools.

The Scilab file contains a string definition, i.e. g_name, inluding a short, mnemonic name for the model as well as two 8-bit integers, i.e. g_dimension and g_range, defining the number of input and output variables of the model. Additionally the file has two real-valued arrays; i.e. g_min and g_max, containing the minimum and maximum values allowed for each of the input and output variables. The following is a simple definition of a function that can be understood by utilized software package [

g_name = “ZDT1”;

g _dimension = int8(2);

g _range = int8(2);

g _min = [0, 0, 0, 0];

g _max = [1, 1, 1, 1];

g _names = [“x1”, “x2”, “f1”, “f2”];

function f = g_function(x)

f1_x1 = x(1)

g_x2 = 1 + 9 * x(2)

h = 1 - sqrt(f1_x1 / g_x2)

f = [ 1 - f1_x1, 1 - g_x2 * h ]

endfunction;

The problem of welded beam design [

As it is shown in the above figure the beam is welded on another beam carrying a certain load P. The problem is well studied as a single objective optimization problem [

end of the beam. The mathematical formulation of the problem is given as;

Among the four constraints, g_{1} deals with the shear stress developed at the support location of the beam which is meant to be smaller than the allowable shear strength of the material (13,600 psi). The g_{2} guarantees that normal stress developed at the support location of the beam is smaller than the allowable yield strength of the material (30,000 psi). The g_{3} makes certain that thickness of the beam is not smaller than the weld thickness from the standpoint. The g_{4} keeps the allowable buckling load of the beam more than the applied load P for safe design. A violation of any of the above four constraints will make the design unacceptable. More on adjusting the constraints would be available in [22,23]. Additionaly on the stress and buckling terms calculated in [

In applying I-MODE framework to real-world design tasks of [22,23] in which real decision makers will be involved there exists a number of shortcomings which we needs further attention. I-MODE software implementtation can consider a maximum of three objectives due to limitation of visual representation of the Pareto-optimal solutions.

Here the RSO software architecture of LIONsolver [21, 23,59] helps the designer to become aware of the different posibilities and focus on his preferred solutions, within the boundary of constraints. Consequently the constraints are transformed into a penalty function which sums the absolute values of the violations of the constraints plus a large constant. Unless the two functions are scaled, the effect of deflection in the weighted sum will tend to be negligible, and most Pareto-optimal points will be in the area corresponding to the lowest cost. Therefore each function is devided by the estimated maximum value of each function in the input range [

By associating a multidimentional graph for an advanced visualization, available in

These observations can inspire a problem simplification, by fixing the height to its maximum value and by expressing the length as a function of depth, therefore eliminating two variables from consideration in the future explorations this optimal design problem.

Further,

The proposed method which is developed on the basis of Reactive Search Optimization algorithms is compaired with the existing novel method of Interactive Multiobjective Optimization and Decision-making, using Evolutionary method in solving engineering optimal design problems. In order to evaluate the effectiveness of both

methods the well-known study case of welded beam design problem is reconsidered. The preliminary tests of the software environment in the concrete context of optimal designing the welded beam design problem have shown the effectiveness of the approach in rapidly reaching a design preferred by the decision maker via advanced visualization tools and brain-computer novel interactions.

Further the utilization of the proposed software architecture for multiobjective optimization and decision-making, with a particular emphasis on supporting flexible visualization is discussed. The applicability of the software can be easily customized for different problems and usage contexts. For instance the geometrical optimization problems e.g. curves and surfaces [53,54], and skinning problem [

Authors would like to thank Professor Miklos Hoffmann for his technical supports, valuable time and critical advices. Additionally the financial support of Jönköping University is highly acknowledged.

Welded beam design problem implementation in Scilab [

g_name = “weldedBeam”;

g_dimension = int8(4);

g_range = int8(5);

g_min = [0.125, 0.1, 0.1, 0.125, 0.0, 0.0, 0.0, 0.0, 0.0];

g_max = [5.0, 10.0, 10.0, 5.0, 350.0, 0.05, 1.0, 1.0, 10000.0];

g_names = [“welding depth (h)”, “welding length (l)”, “height (t)”, “thickness (b)”, “fabrication cost (f1)”, “end deflection (f2)”, “f1 with penalty”, “f2 with penalty”, “Penalty”];

P = 6000.0; L = 14.0; E = 3.0e7;

deltaMax = 0.25;

G = 12.0e6;

tauMax = 13600.0;

sigmaMax = 30000.0;

function f=g_function(x)

h = x(1), l = x(2), t = x(3), b = x(4)

//objectives

f1 = 1.10471*h*h*l + 0.04811*t*b*(14.0+l)

f2 = 4*P*(L^3) / (E*b*t^3)

//constraints

Penalty = 0

tau1 = P/(sqrt(2)*h*l)

M = P * (L + 0.5*l)

R = sqrt(.25 * (l*l + (h+t)^2))

J = 2.0*(h*l/sqrt(2))*(l*l/12.0 + .25*(h+t)^2)

tau2 = M * R / J

tauX = sqrt(tau1*tau1 + ((tau1*tau2*l)/R) + tau2*tau2)

if tauX > tauMax then

Penalty = Penalty + (tauX-tauMax)/tauMax

end

sigmaX = 6.0*P*L/(b*t*t)

if sigmaX > sigmaMax then

Penalty = Penalty + (sigmaX-sigmaMax)/ tauMax

end

if h > b then

Penalty = Penalty + (h - b) / b

end

PcX = (4.013*sqrt(E*G*t*t*b^6/36.0)/(L*L)) * (1-t/(2*L)*sqrt(E/(4.0*G)))

if PcX < P then

Penalty = Penalty + (P - PcX) / P

end

if Penalty > 0 then

f1p = g_max(5) + Penalty

f2p = g_max(6) + Penalty

else

f1p = f1

f2p = f2

end

f = [f1, f2, f1p/g_max(5), f2p/g_max(6), Penalty]

endfunction;