Journal of Industrial Engineering, University of Tehran, Special Issue, ٢٠١١, PP. ١٣-٢٣ ١٣
* Corresponding author: Tel: +٩٨- ٢١- ۵۵٢٧٧۵۶١ Fax: +٩٨- ٢١- ۵۵٢٧٧۴٠٠ Email: bashiri @shahed.ac.ir
Metaheuristic Based Multiple Response Process Optimization
Mahdi Bashiri*١, Reza B. Kazemzadeh٢, Anthony C. Atkinson٣ and Hossein Karimi١
١Department of Industrial Engineering, Shahed University, Tehran, Iran
٢Department of Industrial Engineering, Tarbiat Modares University, Tehran, Iran
٣ Department of Statistics of LSE, London , England (Received ۶ October ٢٠١٠, Accepted ٢٩ June ٢٠١١)
Abstract
The simultaneous optimization of multiple responses is an important problem in the design of industrial processes in order to achieve improved quality. In this paper, we present a new metaheuristic approach including Simulated Annealing and Particle Swarm Optimization to optimize all responses simultaneously. For the purpose of illustration and comparison, the proposed approach is applied to two problems taken from the literature. The results of our study show that the proposed approach outperforms the other approaches and can find better solutions.
Finally, in both cases, we present the results of a sensitivity analysis incorporating experimental design.
Keywords: Multiple response optimization, Simulated annealing, Particle swarm optimization, Desirability function
Introduction
Response Surface Methodology (RSM) has extensive applications in industrial settings. It is a collection of techniques for finding the relationship between a response (y) and input variables (x١,x٢,...,xn). The purpose of the experimenter often is to find the optimal setting of the input variables to maximize (or minimize) the response. In RSM, the input variables are transformed into coded dimensionless variables.
A standard experimental design in RSM is the Central Composite Design (CCD), used to find the relationship between response and input variables. The various techniques used in RSM are described by Box and Draper (١٩٨٧) [١], Khuri and Cornell (١٩٩۶) [٢] and Myers and Montgomery (٢٠٠٢) [٣].
In some applications there is more than one process or product response. The selection of optimal settings of the input variables with simultaneous consideration of multiple responses is called a Multi Response Surface (MRS) problem. There are typically three stages in the solution of such problems: experimental design and data collection, model building and optimization.
After the ١st and ٢nd stages we write the model as follows:
m j
x f
yj j j , ١,٢,..., , (١)
where yj is jth of m responses, fj x is a function relating the jth response to the input variables and j is random error.
This paper presents an approach for simultaneous optimization of all the responses in MRS problems by the use of the two metaheuristics: Simulated Annealing and Particle Swarm Optimization. The paper is organized as follows. The next section reviews current approaches to MRS problems. The third section contains our approach and the new algorithm. In the succeeding section we present two examples solved using our approach and compare our solutions with those obtained from other approaches. Conclusions are in the last section.
Main approaches to MRS problems Given a model of each response, a basic and simple approach to MRS problems is the
١۴ Journal of Industrial Engineering, University of Tehran, Special Issue, ٢٠١١
use of response contour plots, determining the optimal solution by visual inspection.
However, unless both the numbers of responses and input variables are small, this method is inefficient and should not be used.
Some approaches to MRS problems aggregate all responses in a single objective form which is then optimized. Examples are the priority based approach [۴], desirability functions [۵] and the loss function [۶].
In the priority based approach, the decision maker selects the most important of the responses as an objective function and uses the desired values of the other responses as constraints; there is no simultaneous optimization of all responses.
In the desirability function approach, all responses are transformed to a scale-free value between ٠ and ١ using the desirability function djfor the jth response. The computed desirability for each response is combined to construct an overall desirability, which is then optimized.
Derringer (١٩٩۴) proposed a weighted geometric mean for the overall desirability function [٧]. Kim and Lin (١٩٩٨, ٢٠٠٠ and ٢٠٠٢) suggested maximizing the lowestdj, as overall desirability value of the responses [٨, ٩, ١٠]. The loss function approach attempts to minimize the costs associated with the distances of the responses from their targets namely:
T x y C T x y x y
L ( ) ( ) ( ) , (٢)
Here y(x) is the vector of responses, x is the vector of input variables, T is the target vector of the responses and C is the cost matrix containing the relative importance of each response. See Vining (١٩٩٨) and Ko and Kim (٢٠٠۵) [١١, ١٢].
One of the main objectives in MRS problems is robustness in product or process, reaching the specified mean with minimum variance. Chiao and Hamada (٢٠٠١) propose a quality measure which is the probability that m component responses are simultaneously meeting their respective specification (S) or the proportion of conformance [١٣]. They proposed it to
incorporate robustness into these problems.
The objective function can be written
S Y
max p , (٣)
where Y is the vector of responses and S is the specification region depending on values
j j u
l , which are the lower and upper limits of theith response
j j m j
u l
S ,
١ . (۴) For the optimization stage, Del Castillo and Montgomery (١٩٩٣) solved the problem by using the generalized reduced gradient (GRG) algorithm, which is available in software packages such as Microsoft Excel [١۴]. Del Castillo et al. (١٩٩۶) used a gradient-based optimization approach by modifying the desirability function to be everywhere differentiable [١۵]. In a latter study Tong and Xu (٢٠٠٢) used a goal programming approach to find the optimal solution [١۶].
When the number of responses (or objectives) and constraints increase, the probability of finding a local instead of global optimum is increased and, in these cases, metaheuristic approaches can be helpful for finding the global optimum [١٧].
Ortiz et al. (٢٠٠۴) developed a multiple- response solution technique using a GA in conjunction with an unconstrained desirability function [١٨]. Some other recent works on multi-response optimization problems are as follows;
Tong et al. (١٩٩٧) developed a multi- response signal to-noise (MRSN) ratio, which integrates the quality loss for all responses to solve the multi-response problem [١٩]. Tong et al. (٢٠٠۵) also consider the correlation of responses and use PCA and TOPSIS methods to find the best variable setting [٢٠]. Kun-Lin Hsieh (٢٠٠۶) used neural networks to estimate the relationship between control variables and responses [٢١]. Tong, et al. (٢٠٠٧) use VIKOR methods in converting Taguchi criteria to single responses and then derive a regression model and the related optimal setting [٢٢]. Kazemzadeh et al., (٢٠٠٨) proposed a general framework for multi- response optimization problems based on
Metaheuristic Based Multiple ….. ١۵
goal programming and compared some existing methods [٢٣]. They attempted to aggregate all characteristics into one approach, including the priorities of certain types of decision makers. Bashiri and Hejazi (٢٠٠٩) used Multiple Attribute Decision Making (MADM) methods such as VIKOR, PROMETHEE II, ELECTRE III and TOPSIS in converting multiple responses to a single response in order to analyze data from robust experimental designs [٢۴].
Particle Swarm Optimization
Particle Swarm Optimization (PSO) is a significant member of swarm intelligence techniques. It was proposed by Eberhart and Kennedy (١٩٩۵) as an optimization method [٢۵]. PSO is a population based search algorithm founded on the simulation of the social behavior of bees, birds or a school of fishes. Each individual within the swarm is represented by a vector in multidimensional search space. This vector has one assigned vector that determines the next movement of the particle and is called the velocity vector.
The PSO algorithm determines how to update the velocity of a particle. Each particle updates its velocity based on the current velocity, the best position (p_best) it has explored so far and on the global best position (g_best) explored by the swarm [٢۶, ٢٧, ٢٨]. Movement of each particle is shown in Figure ١, and it is based on equations (۵), (۶). Equation (۵) shows that the velocity vector is updated by the global best position, personal best position and current position of each particle. Equation (۶) shows that each particle moves with its own velocity.
Figure ١: Individual particle movement
)) ( _
( .
)) ( _
( . ) ( . ١) (
٢
١
t x best g rand b
t x best p rand b t v w t v
i
i i
i (۵)
) ( ) ( ١)
(t x t v t
xi i i (۶)
Here i indexes the particle, t is the index of iteration,vi is the vector of velocity, xi is the position, w is the weight of the current velocity, b١ is the weight of the difference between personal best and current position,
b٢ is the weight of the difference between global best and current position and randis used for randomization.
This approach is widely used in recent research such as that of Wang and Liu (٢٠١٠) who applied a chaotic PSO to assembly sequence planning [٢٩]. Yang and Lin (٢٠١٠) took advantage of PSO to solve a serial multi-echelon inventory model [٣٠].
Lin et al. (٢٠١٠) base an efficient job-shop scheduling algorithm on particle swarm optimization [٣١]. Our proposed metaheuristic approach for MRS problems, based on PSO, is described in the following section.
M.R.S.A & M.R.P.S.O approach After fitting m response functions to the experimental data, the estimated responses are,
) ,..., ,
ˆj fj(x١ x٢ xn
y , j ١,..., m . (٧)
It is necessary to aggregate all responses into a single function. In the proposed method we use a desirability function-based approach for this aggregation. To find both an optimal setting of the input variables and a robust design, we need a second criterion in our optimization process. By assuming that the response vector Y(m١) has a multivariate normal distribution with mean and variance , respectively, the joint probability density function of Y(m١)is:
Y Y
Y f
T m
١ ٢ ١/ ٢ /
١٢ exp
٢ ,
; .
(٨)
Here ١, ٢,..., n is the mean vector of
responses and is their variance-covariance matrix. Now we can compute the criterion
S Y
p for each setting of the input
Current
Personal Best Position
Global Best Position New Position Velocity Vector
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variables). The larger the value of this criterion, the more robust is the design.
After computing the criterion for each setting of the input variables, we use the criterion value as a second responseypr. The fitted model for this response is
) ,..., ,
ˆpr fpr (x١ x٢ xn
y . (٩)
Finally the overall function to aggregate all responses will be:
m p
j
p j
j d
ND
١/ ١
١ ١
min .
x ,
(١٠) In (١٠) j is the weight of the jth response to be defined by the decision maker (DM) and
djis the desirability value which can be computed from the functions below for STB (smaller the better), NTB (nominally the best) and LTB (larger the better) types of responses.
j j
j j j j
j j j
j j STB
j
l y
u y l l
u y u
u y d
) ( ١ ˆ
) ( ) ˆ
( ˆ
) ( ٠ ˆ
x x x
x
(١١)
j j
j j j j
j j j
j j LTB
j
u y
u y l l
u l y
l y d
) ˆ ( ١
) ˆ ( )
ˆ (
) ˆ ( ٠
x x x
x
(١٢)
j j j j
j j j
j j j j
j j j
j j j j NTB
j
u y T T
u T y
T y l l
T y T
u y or l y d
) ˆ( )
ˆ( ١
) ( ) ˆ
ˆ( ١
) ˆ( ) ˆ( ٠
x x x x
x x
(١٣)
Here Tj is the target value of the jth response which is of NTB type, to be defined by the producer according to production requirements. Equation (١٠) can be written as:
m p
j
p pr pr p j
j d y
ND
١/ ١
١ ˆ ١
min ,
. x (١۴)
In (١۴) yˆpr is the response that is the probability of conforming to the specification region defined in (٩). It is of
LTB type response (yˆpr dpr), with pr the weight of this response.
At this stage, we need to optimize (١۴) and find the optimal value. Our proposed approach is illustrated in Figure ٢. We used MATLABfor the calculations.
Some characteristics of the proposed approach are:
١) It is based on the desirability function;
٢) It pays attention to product or process robustness;
٣) MRSA (Multiple Response Simulated Annealing) and MRPSO (Multiple Response Particle Swarm Optimization) do not guarantee that we find the optimal setting of the input variables. However, but the probability of getting trapped at a local optimum is very small;
۴) The proposed approach has no restriction on the number of input or response variables.
Further, additional quality criteria can be added at the model building stage.
In the next section we present the results of using MRSA & MRPSO in two examples from the literature and compare our results with other methods used for the same examples.
Comparison of Proposed Metaheuristic approach with other approaches
١. Example of GMA welding process
This example concerns Gas Metal Arc Welding (GMAW). Correia et al. (٢٠٠۴) study optimization of a GMA welding process [٣٢]. They used RSM to find the optimal setting of the process combined with a Genetic Algorithm (GA) for numerical optimization. They make comparisons with two other methods.
The GMA welding process establishes an electric arc between a continuous metal filler electrode and the weld pool. The process variables are reference voltage (T), wire feed speed (F) and welding speed (S), which we called x١,x٢ andx٣ in our general formulation. The response variables (y) are deposition efficiency (dexp), penetration (pexp), width (Wexp) and reinforcement (Rexp).
Target values at the optimum for these responses are in Table ٣.
Metaheuristic Based Multiple ….. ١٧
Figure ٢: Overview of MRSA & MRPSO proposed algorithm in multiple response optimization
Correia et al. ٢٠٠۴ used a CCD (Central Composite Design), the values of the four responses from which are in Table ٢ [٣٢].
The function they used for aggregating the responses is;
) , ( )
(
) ( )
) ( (
٢ exp ٢
exp
٢ exp ٢
exp
t t t
t
t t t
t
R i r cr R W
i w cw W
D i d cd D
P i p cp P i Of
where Of(i) is the aggregated function value for the ith experiment in their RSM and
GA method, cp, cd, cw and cr are weights of the responses ( ١, ٢, ٣ and ۴ in our MRSA method) and Pt, Dt, Wt, Rt are target values for penetration, deposition efficiency, width and reinforcement respectively (Tj,j ١,...,۴ in our proposed method).
Alsopexp(i),dexp(i), wexp(i) and rexp(i) are the predicted values for each response (yˆj(x) in our method). The ranges of settings for input variables are given in Table ١. The experimental settings, as well as the values of each response, are in Table ٢.
١٨ Journal of Industrial Engineering, University of Tehran, Special Issue, ٢٠١١
Table ١: Range of setting for input variables in welding process example
Parameter Range
Reference Voltage ,T (V) ٣۶.٠-٣٩.١ Wire feed speed, F (m/min) ٣.٩-٩.٧ Welding speed, S (cm/min) ۵٠-٧٠
Table ٢: Experiments and results in welding process example
Factor Response
T F S Pe De We Re
٣۶.٨ ۵.۴ ۵۵.٠ ٣.۵ ٨١.٨ ۵.١ ١.٢
٣٨.۴ ۵.۴ ۵۵.٠ ٣.٨ ٧٩.۶ ۴.٧ ١.٣
٣۶.٨ ٨.٣ ۵۵.٠ ۵.١ ٩١.٣ ٩.١ ١.٨
٣٨.۴ ٨.٣ ۵۵.٠ ۵.٣ ٩٠.۴ ٩.١ ١.۶
٣۶.٨ ۵.۴ ۶۵.٠ ٣.٠ ٧٩.۴ ۴.٨ ١.٠
٣٨.۴ ۵.۴ ۶۵.٠ ٣.١ ٧٩.۶ ۴.۶ ١.۵
٣۶.٨ ٨.٣ ۶۵.٠ ۴.۵ ٧٠.٩ ٧.٠ ١.٣ ٣٨.۴ ٨.٣ ۶۵.٠ ۴.۵ ۶٩.۶ ۶.٨ ١.١ ٣۶.٠ ۶.٩ ۶٠.٠ ٣.۵ ٨۴.٢ ۶.٧ ٢.٠ ٣٩.١ ۶.٩ ۶٠.٠ ٣.۵ ٨٨.٣ ۶.٢ ١.٨ ٣٧.٧ ٣.٩ ۶٠.٠ ٣.١ ٧٩.٧ ٣.٨ ١.٠ ٣٧.٧ ٩.٧ ۶٠.٠ ۵.۵ ٩٠.٨ ٧.٨ ٢.٠ ٣٧.٧ ۶.٩ ۵١.۶ ۴.٣ ٨٨.٠ ۶.١ ١.٠ ٣٧.٧ ۶.٩ ۶٨.۴ ٣.٨ ۶٩.٠ ۴.١ ٠.٨
٣٧.٧ ۶.٩ ۶٠.٠ ۴.٠ ٧۴.٣ ۵.۵ ١.٣
٣٧.٧ ۶.٩ ۶٠.٠ ٣.٩ ٧٨.٠ ۵.۶ ١.٣ ٣٧.٧ ۶.٩ ۶٠.٠ ۴.٠ ٧٢.١ ۶.٢ ١.٢ ٣٧.٧ ۶.٩ ۶٠.٠ ۴.٠ ٧٣.٣ ۶.٣ ١.٢ ٣٧.٧ ۶.٩ ۶٠.٠ ٣.٨ ٧٣.۵ ۶.٢ ١.٣ ٣٧.٧ ۶.٩ ۶٠.٠ ٣.۴ ٧۶.٣ ۵.٩ ١.۵
The fitted response functions with coded variables are:
٢ ٢ ٢ ٠.۵٧ ٠.۴٣ ٢۶
٠. ۴٩ ٠. ٣۵ ١. ٨١ ٣.
ˆ F S T F S
ype
FS F
T S F
yˆde ٧۴.٣٢ ٢.٧٩ ١٠.۶۵ ١١.٠۴٧ ٢ ١٠.١٠ ٢ ١٩.۴ FS T
S F
yˆwe ۵.٩٣ ٢.۵٩ ١.١۵ ٠.٧٧ ٢ ١.٩٩
FS TF
S F T S F yre
۴٩ ٠. ۴۵ ٠.
۵٨ ٠. ٢ ٠. ۵٩ ٠. ١٩۶ ٠. ٣٧ ٠. ٢٨ ١.
ˆ ٢ ٢ ٢
In this study the preferences for all responses are not equal but the priorities are:
٠.۵, ٠.٣, ٠.١ and ٠.١ for penetration, deposition, width and reinforcement respectively. The target values and maximum and minimum values (lj,uj) for each response are reported in Table ٣.
We solved the welding process problem by MRSA ( =٠.٩٩, T=١٠, T٠=٠.٠٠١ and p=١) and MRPSO (number of particles=۴٠,
W=۴.۵, B١=١.۵, B٢=٢.۵). The methods and results are shown in Table ۴, which also includes a comparison with the GA and RSM methods.
If we take weights of ٠.۵, ٠.٣, ٠.١ and ٠.١ for the response preferences the ND (non desirability) values for RSM, GA, MRPSO and MRSA are ٠.۶۵, ٠.٧١, ٠.١٢ and ٠.٣٢ respectively; the MRSA and MRPSO methods outperform GA and RSM in the GMAW problem. Note that there is no attention to Robustness in the RSM and GA methods so that in calculating the MRSA and MRPSO for this comparison it has been assumed that pr ٠.
Table ٣ : Target values, limits and types of responses in the welding process example
Penetration Deposition Width Reinforcement Target value
۵.٣ ١٠٠% ٨.۵ ١.۵
Maximum
۵.۵ ١٠٠ ٩.١ ٢
Minimum
٣ ۶۵ ۴.١ ٠.٨
Type NTB LTB NTB NTB
As described above, interest in the MRSA and MRPSO methods, is in minimizing the overall non-desirability by using of equation (١٠). The final results for the controllable factors (T, F and S) in MRSA are -٠.٢٨٨۴, ٠.۴٢٨٩ and -٠.٩٠٩٨ and for MRPSO -٠.٣١٩۴, ٠.۶٠٠٢, and -٠.٠۶۴٨ respectively. We can see that MRSA has the better result when compared to the MRPSO and the other approaches in this example.
For the proposed MRSA and MRPSO approaches, a sensitivity analysis was made using a factorial design. Figure ٣ illustrates the results of the factorial design analysis of the MRSA approach as main effects plots. It shows that the best settings for the parameters are Alpha=٠.٩٩, T=١٠ and T٠=٠.
The main effects plots from the same analysis for the MRPSO are in Figure ۴. The results show that the best parameter settings for this algorithm are: number of particles=۴٠, W=۴.۵, B١=١.۵, B٢=٢.۵.
Metaheuristic Based Multiple ….. ١٩
Table ۴: Computed optimal values for each response with four methods in welding process example
response Target
RSM GA MRSA MRPSO
Final value Difference (%)
Final value Difference (%)
Final value Difference(
%)
Final value Difference(%)
Penetration ۵.٣ ۵.۵ ٣.٨ ۵.۵ ٣.٨ ۵.٢٧ ٠.۵۶ ۴.٨٣ ٨.٨۶
Deposition ١٠٠ ٩٣.٨ ۶.٢ ٩٢.٢ ٧.٨ ٩۵.۵۶ ۴.۴۴ ٨٢.٢١ ١٧.٧٩
Width ٨.۵ ٨.٣ ٢.٣ ۶.۵ ٢٣.۵ ٨.٩٣ ٧.۵٩ ٧.٧١ ٩.٣٠
Reinforcemen
t ١.۵ ٢.٠ ٣٣.٣ ٢.٢ ۴۶.٧ ١.۴٧ ٢ ١.٧۵ ١۶.۶۶
Solution
Time(s) - - - ٠.١٧۵١٣٠ ٠.٢۵٠۵١۴
ND value ٠ ٠.۶۵٧۴ ٠.٧٠٩۴ ٠.١١٩٣ ٠.٣٢٠٨
٠.٩٩ ٠.٩٠ ٠.٨٠ ٠.٣٠ ٠.٢۵ ٠.٢٠ ٠.١۵
١٠ ۵ ١
٠.١٠٠ ٠.٠٠١
٠.٣٠ ٠.٢۵ ٠.٢٠ ٠.١۵
Alpha T
T٠
Main Effects Plot ( data means) for ND
Figure ٣: Main Effects plot of general Factorial Design of ND response according to the MRSA parameter settings for Example ١
۴٠ ٣٠ ٢٠ ١٠ ٠.۴۶٣
٠.۴۶٢
٠.۴۶١
٠.۴۶٠
۴.۵ ٣.۵
١.۵ ٠.۵
٠.۴۶٣
٠.۴۶٢
٠.۴۶١
٠.۴۶٠
٢.۵ ١.۵
NO_P W
B١ B٢
Main Effects Plot ( data means) for ND
Figure ۴: Main Effects plot of general Factorial Design of ND response according to the MRPSO parameter settings for Example ١
Table ۵: Experimental results with ۵ replications for wheel cover component example
run x١ x٢ x٣ x۴ x۵ x۶ x٧ replicate ١ replicate ٢ replicate ٣ replicate ۴ replicate ۵
y١ y٢ y١ y٢ y١ y٢ y١ y٢ y١ y٢
١ -١ -١ -١ -١ -١ -١ -١ ٧١١.٩ ٠.۵٩ ٧١٣.۴ ٠.۵٩ ٧١٢.٣ ٠.۴٧ ٧١٢.۴ ٠.٧١ ٧١١.٩ ٠.۶٣ ٢ -١ -١ -١ ١ ١ ١ ١ ٧٢۵ ٠.٧ ٧٢٠.١ ٠.٩١ ٧١١.٨ ١.١٣ ٧٢٣.٩ ٠.٧٩ ٧٢٠.٩ ٠.٧٨ ٣ -١ ١ ١ -١ -١ ١ ١ ٧١١.۶ ٠.۵۶ ٧١١.٧ ٠.۴۴ ٧١١.٣ ٠.۴۶ ٧١٢.١ ٠.۵٣ ٧١١.٧ ٠.۴۶ ۴ -١ ١ ١ ١ ١ -١ -١ ٧٣٣.٧ ١.۵ ٧٢۴.١ ١.۵۵ ٧٣٢ ١.٣٨ ٧٣٢.٧ ١.۴۵ ٧٣٣.٣ ١.۴۵ ۵ ١ -١ ١ -١ ١ -١ ١ ٧٢۵.۴ ١.٢۵ ٧٢١.۶ ١.٣۶ ٧٢٢.۶ ١.۵١ ٧٢٣.١ ١.٢٢ ٧٢١.١ ١.٢۵ ۶ ١ -١ ١ ١ -١ ١ -١ ٧٢٨.١ ١.١٧ ٧٢١.١ ٠.٩٧ ٧٢٢.٩ ٠.٩٨ ٧٢٣ ٠.٩٧ ٧١٩.٧ ٠.٧٣ ٧ ١ ١ -١ -١ ١ ١ -١ ٧٢۶.۶ ١.۵٢ ٧٣١.۴ ١.۵٨ ٧٣١.۴ ١.۶١ ٧٢٩.۶ ١.۴ ٧٣١.٣ ١.۵٧ ٨ ١ ١ -١ ١ -١ -١ ١ ٧١۴.٣ ٠.۵٧ ٧١۴.۴ ٠.۵١ ٧١٣.۶ ٠.۴۴ ٧١۶.٣ ٠.۴۴ ٧١۴.۶ ٠.۵۶
٢٠ Journal of Industrial Engineering, University of Tehran, Special Issue, ٢٠١١
٢. Wheel Cover Component Experiment
Harper, Kosbe, and Peyton (١٩٨٧) report an experiment for finding the optimum combination of injection molding parameters to minimize the imbalance of a plastic wheel cover component [٣٣]. The purpose of the experiment was to determine the effects of seven injection molding parameters (denoted asx١,x٢,...,x٧) on the quality characteristics of the component. These are measured by the total weight (y١ in grams) and the balance (y٢ in inch-ounces). Both y١ and y٢ are NTB-type responses with limits of (٧١٠, ٧١۵) and (٠.٣, ٠.۴) as the specification region, respectively. The results of the experiment are in Table۵.
The fitted response functions for y١ and y٢
are:
٧ ۵
١
١ ٧٢٠.٧۶٣ ١.٨٧٣ ۵.٣١٨ ٣.۴٠٨
ˆ x x x
y
٧ ۵
١
٢ ٠.٩۶٧ ٠.١١٣ ٠.٣٢٨ ٠.١٧۴
ˆ x x x
y
If we assume that Y has a normal distribution with mean ( ١, ٢) and variance-covariance matrix , we can then find the probability of all responses being in their specification region for each run
S Y
p R where YR is the m dimensional response vector in the Rth run and S is the defined specification region.
The covariance matrix and mean responses for each run were computed from the results of Table ۵ and are given in Table ۶.
Table ۶: computed , in each run for wheel cover component example.
Run Variance-covariance Mean
١ ٠.٠٠١٨ ٠.٠٠٧۵٢
٠٠١٨ ٠. ٣٧٧ ٠.
۵٩٨ ٠.
٣٨ ٧١٢.
٢ ٠.٨٣٩٣۵ ٠.٠٢٨
٨٣٩٣۵ ٠. ٩٢٣ ٢۶.
٨۶٢ ٠.
٣۴ ٧٢٠.
٣ ٠.٠٠۵٢۵ ٠.٠٠٢٧ ٠٠۵٢۵ ٠. ٠٨٢ ٠.
۴٩ ٠.
۶٨ ٧١١.
۴ ٠.١۵٩۴۵ ٠.٠٠۴٠٣٠ ١۵٩۴۵ ٠. ٩٨٨ ١۵.
۴۶۶ ١.
١۶ ٧٣١.
۵ ٠.٠۴۴٨۵ ٠.٠١۴٣٧ ٠۴۴٨۵ ٠. ٨٠٣ ٢.
٣١٨ ١.
٧۶ ٧٢٢.
۶ ٠.۴٨٣٣۵ ٠.٠٢۴٣٨ ۴٨٣٣۵ ٠. ٧۴٢ ١١.
٩۶۴ ٠.
٠٨ ٧٢٣.
٧ ٠.٠٧٩۵۵ ٠.٠٠۶٨٣ ٠٧٩۵۵ ٠. ٣٢٨ ۴.
۵٣۶ ١.
٠۶ ٧٣٠.
٨ ٠.٠١۶۴۵ ٠.٠٠٣٩٣ ٠١۶۴۵ ٠. ٠٠٣ ١.
۵٠۴٠ ٠.
۶۴ ٧١۴.
S Y
p R is computed from (١۵). And the results are reported in Table٧.
٢ ٢ ١
١/
٢ ١/ ٧١۵ ١
٧١٠ ۴ ٠.
٣ ٠. ٢
dy dy e
S Y p
R R R R
R Y
Y
R
R (١۵)
Table ٧: Computed yˆpr
for each run of the wheel cover component example.
Run x۱ x۲ x۳ x۴ x۵ x۶ x۷ pY S
١ -١ -١ -١ -١ -١ -١ -١ ٠.٠١٠٩
٢ -١ -١ -١ ١ ١ ١ ١ ٢.٣٨E-۵١
٣ -١ ١ ١ -١ -١ ١ ١ ٠.٠۴١۵ ۴ -١ ١ ١ ١ ١ -١ -١ ٨.٨۶E-١۴٢ ۵ ١ -١ ١ -١ ١ -١ ١ ٢.٩٧E-٢۵
۶ ١ -١ ١ ١ -١ ١ -١ ١.١١E-٠۴
٧ ١ ١ -١ -١ ١ ١ -١ ۴.۶٠E-۴۴ ٨ ١ ١ -١ ١ -١ -١ ١ ٠.٠٢٠۵
Now pYR S is again treated as another response variable with its value depending on the settings of the factors. So we can fit a response surface to our new probability response. For the data of Table٧ the fitted response function is:
٧ ۶
۵ ۴
٣ ٢
١
٠٠۶۴ ٠. ٠٠١٣ ٠. ٠٠٩١ ٠. ٠٠۴ ٠.
٠٠١٣ ٠. ٠٠۶۴ ٠. ٠٠۴٠ ٠. ٠٠٩١ ٠.
x x
x x
x x
x ypr
In the MRSA & MRPSO approach we have used the data to find the probability and fitted a surface to predict the probability for any setting of the control variables. Chiao and Hamada (٢٠٠١), on the other hand, fitted response surfaces to the mean, standard deviation and correlation of the responses in order to compute the probability [١٣]. Also in our problem, finding the probability response is not our main objective; we need to optimize the probability response simultaneously with the other responses.
In this example we had two responses with separate functions to which we added another response. It is necessary to optimize all three responses simultaneously.
We used the MRSA with ٠.٩٩ as a cooling ratio, T=١٠, T٠=٠.٠٠١ and the MRPSO algorithm with the number of particles=۴٠, W=۴.۵, B١=٠.۵, B٢=١.۵and p=١. For both responses j ١. The computed optimal values for input and response variables are
Metaheuristic Based Multiple ….. ٢١
given in Table ٨. In order to find the best parameter settings for both algorithms we again used a factorial design with the parameters selected according to the results of this analysis. The main effects plots for the parameters of the algorithm are in Figures ۵ and ۶.
Table ٨: Comparison of MRSA and Chiao&Hamada (C&H) method for wheel cover
component example
C&H method*
MRSA method
MRPSO method
ND ١.٨٩٧٠ ١.٠٨٠٣ ١.٠٠٠٢
Improvement of overall result (%)
٠% ۴٢.٠۵% ۴٧.٢٧%
١*
x -١ -٠.٨۴٧٩ -٠.۶٠٩٣
٢*
x +١ ٠.٢٠٧٩ -٠.٧١٩۴
*
x٣ ٠ -٠.٣٣٧٣ ٠.٨۴٩٢
۴*
x -١ -٠.٠٧۶۴ ٠.٠٢١٣
۵*
x -١ -٠.٩٩٧۶ -٠.٩٣٩٢
۶*
x ٠ ٠.٨۶٠٧ -٠.٢٧٢۵
٧*
x +١ ٠.٣٧۴۶ ٠.۶٢۴٢
١*
y ٧١٠.١۶۴٠ ٧١٢.۵٩٢٩ ٧١٢.۴٩٩۶
٢*
y ٠.٣۵٢٠ ٠.۴٧٨٨ ٠.۴٨١۵
*
ypr ٠.٠٣٩٠ ٠.٠٢۶٣ ٠.٠٢٠١
Solution
Time(s) - ٠.١٩٢٢٧٩ ٠.٣١٠٩٨٨
*-Chiao and Hamada, (٢٠٠١)
٠.٩٩ ٠.٩٠ ٠.٨٠ ١.٠٨٣ ١.٠٨٢ ١.٠٨١ ١.٠٨٠ ١.٠٧٩
١٠ ۵ ١
٠.١٠٠ ٠.٠٠١
١.٠٨٣ ١.٠٨٢ ١.٠٨١ ١.٠٨٠ ١.٠٧٩
Alpha T
T٠
Main Effects Plot (data means) for ND
Figure ۵: Effects plot of general Factorial Design of ND response according to the MRSA parameter
settings in Wheel Cover Component Experiment
۴٠ ٣٠ ٢٠ ١٠ ١.٠٠۴٠ ١.٠٠٣۵
١.٠٠٣٠
۴.۵ ٣.۵
١.۵ ٠.۵
١.٠٠۴٠ ١.٠٠٣۵
١.٠٠٣٠
٢.۵ ١.۵
Nu m b e r o f p art icle W
b١ b٢
M a in Ef f e ct s P lot ( da t a me ans) f or N D
Figure ۶: Effects plot of general Factorial Design of ND response according to the MRPSO parameter settings in Wheel Cover Component
Experiment
From Table ٨ it can be seen that the aggregated function (ND) of the MRSA and MRPSO methods is lower than that for the C&H method. Also, for this example, the MRPSO method has better results than the MRSA approach. The computational time for both algorithms is reasonable, with that for MRSA lower than that for MRPSO.
Conclusion
In this paper, we have proposed an approach to improve quality and to achieve robust quality by the use of the statistical design of experiments with multiple responses and metaheuristic optimization methods that we have called MRSA &
MRPSO. The proposed approach uses the desirability value of all responses and also their proportion of conformance in meeting their specifications. The MRSA & MRPSO approaches we have developed are flexible in the number of input variables, number of response variables and their respective weights. The methods can easily be applied to different optimal process setting problems.
The results show that the proposed algorithms have an improved objective function value and so outperform previous ones.
Future research will investigate the combination of other methods, such as the loss function approach, with the proposed method as well as the solution of MRS problems by other metaheuristic algorithms.
We also hope to report in future on the use of these methods in practical examples.
٢٢ Journal of Industrial Engineering, University of Tehran, Special Issue, ٢٠١١
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