Web-based Application for Multi-Objective Optimization in ... - CiteSeerX

Web-based Application for Multi-Objective Optimization in Process Systems Yoshiaki SHIMIZUa, Jae-Kyu Yooa and Yasutsugu TANAKAa a

Department of Production Systems Engineering, Toyohashi University of Technology, Toyohashi 441-8580, Japan Abstract Recently, multi-objective optimization (MOP) has been highly required to deal with complex and global decision environment toward agile and flexible manufacturing. To facilitate its wide application, we developed a novel method named MOON2 (Multi-Objective optimization with value function mode led by Neural Network) as a Web-based application. By that, everyone can engage in MOP readily and easily regardless of knowledge about MOP and computer configuration of users. In this paper, we introduce MOON2R (MOON2 using Radial Basis Function (RBF) networks) that has more flexible modeling ability of value function. After outlining the solution procedure of MOON2R, the proposed system configuration will be explained with an illustration. Keywords multi-objective optimization, Internet, RBF network, pair comparison 1. INTRODUCTION Multi-objective optimization (MOP) is increasingly interested in supporting agile and flexible manufacturing in complex and global decision environment, and expected to solve various problems in chemical engineering[1,2]. To avoid stiffness and shortcomings of the conventional methods, we proposed a novel prior articulation method named MOON2[3], and implemented its algorithm as a Web-based application. It was realized as a client-sever architecture through common gateway interface (CGI) so that everyone can use the system regardless of his/her own computation environment. To facilitate its wide application, in this paper, we have improved the modeling procedure of value function using RBF networks (RBFN)[4]. After outlining MOON2R, configuration and usage of the Web-based application system will be shown illustratively. 2. SOLUTION PROCEDURE THROUGH MOON2R The problem concerned here will be described generally as follows. (p.1)

Min f(x) = {f1(x), f2(x),…, fN(x)}

subject to x ∈ X,

where x denotes a decision variable vector, X a feasible region, and f an objective function vector some elements of which conflict and are incommensurable with each other.

Generally speaking, MOP can be classified into the prior articulation method and the interactive one. However, conventional methods of MOP have both advantages and disadvantages over the other. For example, since the former derives a value function separately from the searching process, decision maker (DM) will not be worried about the tedious interactions during the searching process as will be in the later. On the other hand, though the later can articulate elaborately the attainability among the conflicting objectives, the former will pay little attention on that. Consequently, the derived solution may be far from the best compromise of DM. In contrast to it, MOON2 and MOON2R can not only resolve these problems but also handle any kinds of problem, i.e., linear programs, non-linear programs, integer programs, and mixed-integer programs under multi-objectives by incorporating with proper optimization methods. 2.1. Identification of value function using neural networks First we need identify a value function that integrates each objective function into an overall one. For this purpose, we adopted a neural network (NN) due to its superior ability of the nonlinear modeling. Particularly, RBFN whose structure is shown in Fig.1 has some advantages compared with the back propagation network, i.e., small computational load, easy dynamic adaptation against incremental operations[4]. Learning of the network is carried out to minimize the squared sum of the difference between training data yi (i=1,.., p) and output from RBFN with respect to weights w j ( j = 1, L , m) , i.e.,

p

m

Min ∑ ( y i − ∑ w j h j ( x i )) 2 + j =1

j =1

m

∑λ j =1

j

w 2j

where λi (i = 1,L , m) denotes regularization pareameter. Such training data is gathered through pair comparisons regarding the relative preference of DM among the trial solutions. That is, DM is asked to reply which he/she likes, and how much it is between every pair of the trial solutions. Just like AHP (Analytic Hierarchy Process, Saaty)[5], such responses will be taken place by using the linguistic statements, and then transformed into the score as shown in Table 1. After doing such pair comparisons over k trial solutions1, we can obtain a pair comparison matrix whose i-j element aij represents the degree of preference of f j compared with f i (Refer Fig.4 appeared in the later). After all, the pair comparison matrix provides totally k2 training data for RBFN. The m

g (x )

w w11 h1 ( x)

x1

wjj w

‥‥‥ h j (x)

x2

∑w j =1

j

hj

Output: y

wmh( x) = exp − ( x − c)  

wm

r

2

2

   

‥‥‥ hm (x) Hidden layer

‥‥‥

xn

Input: x

Table 1 Conversion table Linguistic statements Equally Moderately Strongly Demonstrably Extremely

1 3 5 7 9

Intermediate judgment 2,4,6,8 between the two adjacent

Fig.1. Basic Structure of RBF 1

aij

Under mild conditions, total number of comparison is limited to k(k-1)/2.

objective values of every pair, say, f i and f j become the 2N inputs, and an i-j element aij one output. Using some test problems, we ascertain that a few typical value functions can be modeled correctly by a reasonable number of pair comparisons as long as the number of objective function is less equal to three[1]. By viewing thus trained RBFN as a function VRBF such that: {f i(x), f j(x)}∈R2N→aij∈R1, it should be noticed that the following relation holds. V RBF ( f i , f k ) = a ik > V RBF ( f j , f k ) = a jk ⇔ f i f f

j

(1)

Hence we can rank the preference of any trial solutions easily by the output from RBFN that is calculated by fixing one of the input vector at an appropriate reference, say f R. VRBF( f (x); f R) =a･R

(2)

Since the responses required for DM are simple and relative, his/her load in the tradeoff analysis is very small. 2.2. Incorporation with optimization methods Now the problem to be solved can be described as follows. (p.2) Max VRBF ( f ( x), f R )   subject to  x ∈ X

Since we can evaluate any solution from VRBF under multi-objectives once x is prescribed, we can apply the most appropriate optimization method for the problem under concern, i.e., nonlinear programming, direct search method, and even more meta-heuristic method like genetic algorithm, simulated annealing, tabu search, etc. Also we can verify the optimal solution of (p.2) locates on the Pareto optimal solution set as long as Eq.(1) holds[6]. If we try to use the algorithm that requires gradients of the objective function like nonlinear programs, we can calculate conveniently them by the following relation. ∂ V RBF ( f ( x ), f ∂x

R

)

 ∂ V RBF ( f ( x ), f  =     ∂f ( x ) 

R

)   ∂f ( x )         ∂x 

(3)

We can complete the above calculation by applying the numeric differentiation for the first term in R.H.S. of Eq.(3) while deriving the analytic form for the second. 3. IMPLEMENTATION AS WEB-BASED APPLICATION Due to the various reasons such as little knowledge about MOP, computer environment, etc., it is not necessarily easy for everyone to engage in MOP. To deal with such circumstances, we implemented MOON2R on the Internet as a client-server architecture that enables us to carry out MOP readi ly and effectively. Core of the system is divided into a few independent modules each of which is realized using the appropriate implementation tool. The optimizer module solves a single objective

optimization problem through incorporating the identified value function, specifying the problem as a Fortran programming format, and compiling it by Fortran compiler. Though only sequential quadratic programming (SQP) is implemented presently, various methods are possibly available as mentioned already (GA was applied elsewhere[1]). The identifier module provides a modeling process of the value function based on the neural network where a pair comparison is easily performed just by mouse click operation on the Web page. Moreover, the graphic module generates various graphical illustrations for easy understanding about the results. The user interface of the MOON2R system is a set of Web pages created dynamically during the solution process. The pages described with HTML (hypertext markup language) are viewed by users’ browser that is a client of the server computer. The server computer is responsible for data management and computation whereas the client takes care of input and output. That is, users are required to request a certain service and to input some parameters. In turn, they can receive the service through visual and/or sensible browser operation. In practice, the user interface is a program creating HTML pages and transferring information between the client and the server. The programs creating HTML pages are programmed using CGI programming languages named Ruby. As the role of CGI, every treatment is carried out on the server side, and any particular tasks are not assigned to the browser side (See Fig.2). Consequently users are not only free from the maintenance of the system such like update, issue, reinstall, etc. but also are regardless of their computation environment like operating system, configuration, performance, etc. Though there are several sites serving (single-objective) optimization library (e.g., http://www-neos.mcs.anl.gov/), none is known regarding MOP except for NIMBUS[7] (http://nimbus.math.jyu.fi/) so far. However, since NIMBUS belongs to an interactive method, it has the disadvantages mentioned already. On the other hand, the articulation process of MOON2R is separated from the searching process, DM can engage in the interaction at his/her own paces, and will not be worried about by the hurried/idle responses like the interactive methods. Also it should be noted that the required responses are simple and relative, and DM needs not any particular knowledge about the theory of MOP. Such easy usage, small load in the tradeoff analysis, and maintenance-free features will be expected to facilitate the decision making from a comprehensive point of view that should be required for agile and flexible problem-solving in chemical engineering. The URL of the system is http://scmoon2.tutpse.tut.ac.jp/cgi-bin/moon2v2/. (Presently Japanese version only ) Server

① Request service

D2 = 80

③ Return result

Fmax x2

CGI program

D1 = 100

② Issue task WWW server application

④ Receive service

Client

x1

Browser

user

Fig.2. Scheme of task flow through CGI

l = 1000

Fig. 3. Beam form design problem

4. ILLUSTRATIVE DESCRIPTION OF USAGE As a demonstration of the Web-based MOON2R, we provide a bi-objective design problem regarding decision on the strength of material[8]. To grasp the whole idea and the solution procedure of MOON2R, this demonstration is most valuable. We also provide another entry of the Web page for the original user problem. Below we will explain about the demonstration of the example problem. Moving to the target Web-page, we will find a formulation of the problem. (p.3) f1( x ) =

f2( x ) =

[(

)

(

π 2 2 2 2 x1 D2 − x2 + (l − x1 ) D1 − x2 4

)]

64  1 1  3 l3   x − +  4 1 4 4 3πE  D2 − x2 4 D14 − x14  D1 − x1 

subject to g1 ( x) = 180 −

9.78×106 x1 4.096×107 − x2

4

≥0

(4)

g 2 ( x) = 75.2 − x2 ≥ 0

(5)

g 3 ( x) = x2 − 40 ≥ 0

(6)

g 4 ( x) = x1 ≥ 0

(7)

h1 ( x) = x1 − 5x2 = 0

(8)

where x1 and x2 denote the tip length of the beam and the interior diameter respectively as shown in Fig.2. Inequalities Eqs.(4)-(8) represent appropriate design conditions. Moreover, objective functions f1 and f2 represent volume of the beam and static compliance of the beam respectively. Then input page for problem description is provided to input the objective functions and the constraints under the format similar to Fortran language. After repeated processes of input and confirmation, a set of trial solutions for the pair comparisons is generated arbitrarily within the hull convex spanned by the utopia and the nadir solutions2. Now for every pair of the trial solutions, DM is required to make a pair comparison through mouse click of radio button indicator. After showing the pair-comparison matrix thus obtained (See Fig.4), and checking its inconsistency from the AHP theory, the training process of RBFN will start. Its training results are presented both numerically and schematically. The subsequent stages proceed as follows: select an appropriate optimization method (presently only SQP is available); input the initial guess of SQP for the optimization search; click start button. The result of the multi-objective optimization is shown schematically compared with the utopia and the nadir solutions (See Fig.5). If DM would desire further articulation, additional search might be taken place before a satisfying solution has been found. In this case, the same procedures will be repeated within a narrower searching space around the earlier solution to improve it.

2

For example, a utopia is composed of fi(xi*) whereas a nadir of minj fj(xi*), (i=1,…, N) where xi* is the optimal solution of the problem such that “max fi(x) subject to x∈X.”

5. CONCLUSION Introducing a novel and general approach for multi-objective optimization named MOON2R, in this paper, we have implemented its algorithm as a Web-based application. It is unnecessary for everyone to have any particular knowledge about MOP, and to prepare the particular computer environment. They need only a Web browser to submit their problem, and to indicate their subjective preference between the pair of trial solutions generated automatically by the system. Eventually, it can facilitate the decision making from a comprehensive point of view that should be required to pursue the sustainable development in process systems. An illustrative description outlines the proposed system and its usage. Further studies should be devoted to add various optimization methods as applied elsewhere[1,6] besides SQP, and to improve certain user services that enable us to save and manage their own problems. The security routine for usage is also important aspects left for the future studies. REFERENCES [1] Y.Shimizu, J. Chem. Engng. Japan, 32, (1999) 51. [2] V.Bhaskar, K.S.Gupta and K.A.Ray, Reviews in Chemical Engng., 16 (2000), 1. [3] Y.Shimizu and A.Kawada, Trans. Soc. Instrument Control Engrs., 38, 11 (2002) 974. [4] M.J.L.Orr, Introduction to Radial Basis Function Networks, http://www.cns.uk/people/mark.html, 1996. [5] T.L.Saaty, The Analytic Hierarchy Process, McGraw-Hill, New York, 1980. [6] Y.Shimizu and Y.Tanaka, “A Practical Method for Multi-Objective Scheduling through Soft Computing Approach,” Japanese Soc. Mech. Engrs Int. J., to be appeared. [7] K.Miettinen and M.M.Makela, , Computers & Operations Research, 27 (2000) 709. [8] A.Osyczka, Multicriterion Optimization in Engineering with Fortran Programs, John Willey & Sons, New York, 1984.

i＼j Futo Fnad uto

F Fnad F1 F2 F3 F4

1 1/9 1/4 1/5 1/3 1/8

9 1 6 5 7 2

F1

F2

F3

4 1/6 1 1/3 1 1/5

5 1/5 3 1 3 1/3

3 8 1/7 1/2 1 5 1/3 3 1 5 1/5 1

Fig. 4. Pair comparison matrix

F4

Fig. 5. Page representing a final result