INTRODUCTION
Setting of controllable input variables to meet a required specification of
quality characteristic (or response variable) in a process is one of the common
problems in the process quality control. But generally there are more than one
quality characteristics in the process and the experimenter attempts to optimize
all of them simultaneously. Since response variables are different in some properties
such as scale, measurement unit, type of optimality and their preferences, there
are different approaches in model building and optimization of MRS problems.
Moreover, optimizing the response surfaces considering dispersion effect (variance
of responses) as objective function will increase reliability and robustness.
In this study, robust design and multiresponse optimization have been analyzed
by MADM methods. Some earlier works in multiresponse optimization: Derringer
and Suich (1980) applied a desirability function to optimize multiresponse
problems in a static experiment. Castillo et al. (1996)
demonstrated the use of modified desirability functions for optimizing the multiresponse
problem. Layne (1995) presented a procedure that simultaneously
considers three functions, the weighted loss function, the desirability function
and a distance function, to determine the optimum parameter combination. Khuri
and Conlon (1981) proposed a procedure based on a polynomial regression
model to simultaneously optimize several responses. Logothetis
and Haigh (1988) also optimized a five response process by utilizing the
multiple regression technique and the linear programming approach. Pignatiello
(1993) utilized a variance component and a squared deviationfromtarget
to form an expected loss function to optimize a multiple response problem. This
method is difficult to implement. The first reason is that a cost matrix must
be initially obtained and the second reason is that it needs more experimental
data. Chapman (19951996) proposed a cooptimization
approach, which uses a composite response. Leon (19961997)
presented a method, which is based on the notions of a standardized loss function
with specification limits, to optimize a multiresponse problem. However, only
the nominalthe best (NTB) characteristic is suitable for this approach, which
may limit the capability of this approach. Ames et al.
(1997) presented a quality loss function approach in response surface models
to deal with a multiresponse problem. The basic strategy is to describe the
response surfaces with experimentally derived polynomials, which can be combined
into a single loss function by using known or desired targets. Next, minimizing
the loss function with respect to process inputs locates the best operating
conditions. Lai and Chang (1994) proposed a fuzzy multiresponse
optimization procedure to search for an appropriate combination of process parameter
settings. A strategy of optimizing the most possible response values and minimizing
the deviation from the most possible values is used when it considers not only
the most possible value, but also the imprecision of the predicted responses.
Hsieh (2006) used neural networks to estimate relation
between control variables and responses. Tong et al.
(1997) developed a MultiResponse Signal toNoise (MRSN) ratio, which integrates
the quality loss for all responses to solve the multiresponse problem. The
conventional Taguchi method can be applied based on MSRN. The optimum factor/level
combination can be obtained. Su and Tong (1997) also
proposed a principle component analysis approach to perform the optimization
of the multiresponse problem. Initially, the quality loss of each response
is standardized; principle component analysis is then applied to transform the
primary quality responses into fewer quality responses. Finally, the optimum
parameter combination can be obtained by maximizing the summation standardized
quality loss. Tong and Su (1997) proposed a procedure,
which applied fuzzy set theory to MultipleAttribute DecisionMaking (MADM)
to optimize a multiresponses problem. Tong et al.
(2007) use VIKOR methods in converting Taguchi criteria to single response
and then find regression model and related optimal setting. Kezemzadeh (2008)
proposed a general framework for multiresponse optimization problems based on
goal programming. Their study proposes a general framework in MRS problems according
to some existing study and some types of related decision makers and attempts
to aggregate all of characteristics in one approach. Amiri
et al. (2008) used genetic algorithm to find best solution of Multiresponse
problem in fuzzy environment. Their proposed method was combination of simulation
approach, fuzzy goal programming and genetic algorithm. In viewpoint of multiple
objective decision making, Bashiri et al. (2009)
have proposed an approach in which Global Criterion (GC) have been applied to
aggregate multiresponse surfaces in simulation model of probabilistic inventory
model.
MULTIPLE RESPONSE SURFACE
RSM: Response Surface Methodology (RSM) is a collection of statistical
and mathematical techniques useful for developing, improving and optimizing
processes. RSM also has the ability to produce an approximate function using
a smaller amount of data (Yeh, 2003). However, most previous
applications based on RSM have only dealt with a singleresponse problem and
multiresponse problems have received only limited attention. In today’s
complex manufacturing processes, call for simultaneous optimization of several
quality characteristics rather than optimizing one response at a time. Studies
have shown that the optimal factor settings for one performance characteristic
are not necessarily compatible with those of other performance characteristics.
In more general situations we might consider finding compromising conditions
on the input variables that are somewhat favorable to all responses (Koksoy,
2007). More details on RSM, related designs and optimization of response
surfaces are given by Kleijnen (2007, 2008),
Myers and Montgomery (2002).
Multiple response optimization: Nowadays in most industrial applications,
there isn’t just one response and the analysts try to find operating
status that satisfies all quality characteristics simultaneously. Three
steps in these problems are:
• 
Data collection and analysis 
• 
Model building and verification 
• 
Optimization 
Dual Response Surface (DRS) is one of MRS problems in which the mean and variance
of a special quality characteristic is estimated by a polynomial surface and
in the optimization stage, both variance and mean are optimized simultaneously.
The characteristics of MRS problems which it is necessary to attend to the model
building and optimization stages are: different importance of responses, different
measuring units, different scales and magnitude, different types of optimality,
different direction, different preferences of responses and also different types
of decision makers (Kazemzadeh, 2008). MRO problems have
been studied in several areas at different aspects. We can categorize all viewpoints
in the literature into three general categories:
Desirability viewpoints: In this category, researchers try aggregate
information of each responses and get one response. Then optimization
is performed on single objective called desirability function.
Priority based (a.k.a classic optimization) methods: Some cases
have responses with different importance, in such problem, we must consider
most important response for optimization and if solutions weren’t
unique, then find best solution by comparing status of alternative solution
in next important responses and foresaid steps is repeated till considering
all responses or finding unique optimal solution. Most popular methods
in this group are: utility function method, global criterion method, bounded
objective function method and lexicographic method that generate a set
of Pareto optimal solution.
Loss function: In this category, based on loss function (represented
by Taguchi) all responses value are aggregated and convert to single one.
There were wide range of researches, have been studied to develop and
generalize taguchi loss function with respect to special trait of its
cases.
APPLICATION OF MADM METHODS
Overview of MADM methods: Multiple attribute decision making methods
are developed for selecting, ranking or rating (or sometimes categorizing) several
alternative according to have several attribute which be maximized, minimized
or reach a goal. Some of MADM methods use comparison of alternative for each
attribute and collect these results to make a best decision such as ELECTREs,
PROMETHEEs and some algorithms try to find a solution with maximum similarity
to ideal and maximum dissimilarity to non ideal solution such as TOPSIS and
VIKOR (Opricovic and Tzeng, 2004). In this study we focus
on PROMETHEE II, VIKOR, TOPSIS and ELECTRE III to analyze special design of
experiments.
MADM methods which used in this study are ELECTRE III, PROMETHEE II,
VIKOR and TOPSIS. Application of these methods helps us to product one
score from several responses representing aggregate score of each experiment.
Summary of each algorithms are shown below:
ELECTRE III: The ELECTRE III method, like every outranking method,
is based on the axiom of partial comparability, according to which preferences
are simulated with the use of four binary relations: indifference (I);
heavy preference (P); light preference (Q) and noncomparability (R).
Furthermore, the thresholds of preference (p), indifference (q) and veto
(v) have been introduced, so that relations are not expressed mistakenly
due to differences that are less important. The multicriteria model can
be described as following: assuming that A is the finite group of n possible
alternative solutions and m the number of the evaluation criteria (j =
1, 2,..., m). If it is assumed that the objective functions of all criteria
should be maximized, the concordance matrix is defined with the elements:
where, w_{j }and c_{j} (a, b) are weight of jth criteria
and preference level of a into b (a, b are alternatives).
where, p_{j} and q_{j} are the preference and indifference
thresholds, respectively, which can either be constants or functions of
e.g., the criteria performances. a_{bj }is performance of alternativebon
jth criteria. The discordance matrix can be calculated as long as the
veto threshold (v_{j} ) has been defined; then credibility matrix
is built from concordance and discordance matrixes.
The determination of the hierarchy rank is achieved by calculating the superiority
ratio for each alternative. This ratio is calculated from the credibility matrix
and is the fraction of the elements’ sum of every alternative’s line
(Q), to the sum of the elements of the alternative’s respective column
(W). The numerator represents the total dominance of the specific alternative
over the rest and the denominator the dominance of the remaining alternatives
over the former. Therefore this fraction is the score of each alternative which
be maximized (Papadopoulos and Karagiannidis, 2006).
TOPSIS: TOPSIS (technique for order preference by similarity to an ideal
solution) method is presented by Chen and Hwang (1992),
with reference to Hwang and Yoon (1981). The basic principle
is that the chosen alternative should have the shortest distance from the ideal
solution and the farthest distance from the negativeideal solution. The TOPSIS
procedure consists of the following steps:
• 
Calculate the normalized decision matrix. The normalized
value r_{ij} is calculated as: 
where, g_{ij} is performance of alternativei on jth critera.
• 
Calculate the weighted normalized decision matrix. The
weighted normalized value v_{ij} is calculated as: 
v_{ij} = w_{i}r_{ij ;}
j = 1,..., m; i = 1,... ,n 
where, w_{i} is the weight of the ith attribute or criterion
and Σ_{i}w_{i} = 1.
• 
Determine the ideal and negativeideal solution 
A* = {v_{1}*,…, v_{n}*} = {max_{j}
(v_{ij})i ∈ I’ , min_{j} (v_{ij})i
∈ I"}
A^{} = {v_{1}^{},…,v_{n}^{}}
= {min_{j}(v_{ij})i ∈ I’ , max_{j}(v_{ij})i
∈ I"} 
where I’ is associated with benefit criteria and I" is associated
with cost criteria. Also we can convert cost criteria to benefit (y_{ij}
= x_{i}*–x_{ij}, where, x_{i}* is worst
value in ith column) and then continue steps.
• 
Calculate the separation measures, using the ndimensional
euclidean distance. The separation of each alternative from the ideal
solution is given as: 
Similarly, the separation from the negative ideal solution is given as:
• 
Calculate the relative closeness to the ideal solution.
The relative closeness of the alternative a_{j} with respect
to A^{*} is defined as: 
C_{j}^{*} = D_{j}¯/(Dj^{*
}+D_{j}¯) 
(9) 
VIKOR methods: The VIKOR method was developed for multicriteria
optimization of complex systems. It determines the compromise rankinglist,
the compromise solution and the weight stability intervals for preference
stability of the compromise solution obtained with the initial (given)
weights. This method focuses on ranking and selecting from a set of alternatives
in the presence of conflicting criteria. It introduces the multicriteria
ranking index based on the particular measure of closeness to the ideal
solution. Assuming that each alternative is evaluated according to each
criterion function, the compromise ranking could be performed by comparing
the measure of closeness to the ideal alternative. The multicriteria measure
for compromise ranking is developed from the L_{p}metric used
as an aggregating function in a compromise programming method.
The VIKOR method includes the following steps.
• 
Determine the normalized decision matrix similar to
TOPSIS 
• 
Determine the ideal and negativeideal solutions. The
ideal solution A* and the negative ideal solution A¯ are determined
similar to TOPSIS 
• 
Calculate the utility measure and the regret measure.
The utility measure and the regret measure for each alternative are
given as: 
where, S_{i} and R_{i} represent the utility measure
and the regret measure, respectively and w_{j} is the weight of
the jth criterion.
• 
Calculate the VIKOR index. The VIKOR index can be expressed
as follows: 
Where:
Q_{i} 
= 
ith alternative VIKOR value 
i 
= 
1,…,m 
S* 
= 
MinS_{i} 
S¯ 
= 
MaxS_{i} 
R* 
= 
MinR_{i} 
R¯ 
= 
MaxR_{i} 
v 
= 
Weight of the maximum group utility (and is usually set to 0.5) 
• 
Rank the order of preference. The alternative with the smallest
VIKOR value is determined to be the best solution (Tong
et al., 2007) 
PROMETHEE II: One other MADM studied in this research is PROMETHEE was
presented in 1982 and was developed in 1985, 1994. PROMETHEE is the outranking
method and use preference functions to shows alternative differences (Albadvi
et al., 2007). PROMETHEE II is summarized in following steps:
• 
Collecting alternative and weighted criteria 
• 
Determination of Preference Function for each criteria. See Chou (2007) for more information about preference
functions. Figure 1 shows two usual types of preference
function. 
d_{j}(a, b) = g_{aj}–g_{bj}, (according
to Fig. 1 and type of H(d))

Fig. 1: 
Two types of preference function 
• 
Calculation of total preference of a in respect of b
as: 
• 
Calculation of positive and negative flow: 
• 
Find net flow that shows power of each alternatives
and can be considered as alternative score. 
Ø (a) = ø^{+}
(a)–ø¯ (a) 
(16) 
PROPOSED METHOD
In single response surface, regression model is built from relation between
controllable variables and response values but in multiresponse surface,
regression model must be fitted for each response individually. In most
methods (such as desirability function and weighted sum) combination of
several response applied to find single response. The aim of this study
is to use scores of MADM methods as one response instead of several responses
achieved from experiments. For this purpose we use Φ_{i},
Q_{i}, S_{i} and C_{i }for aggregating several
response data. Figure 2 shows summary of proposed methods.
There’s one important notation in combination of goal programming approach
and MADM tools that in most of MADM methods there are just two groups of objectives:
Benefit criteria, which should be maximized and Cost criteria which should be
minimized; but in GP problem, it’s desired that performance values are
equal to specific target value. The studied case in this study is selected from
Kazemzadeh , (2008) that use goal programming to find optimal controllable
variable setting. This method is more applicable and reliable than other MRO
methods. For instance in comparison with desirability function and loss function,
Since this method is based on decision making tools, the analyst can use other
assumptions which defined and applied in other decision making science such
as group decision making, fuzzy decision making, qualitative data, etc. When
classic optimization methods such as goal programming, global criterion optimization
and bounded objective applied in MRS optimization, the analysts had to fit response
surface for each response variable and then find Pareto solution. Since fitting
the regression functions have an error, foresaid method increases family error
and decrease reliability, but the proposed method aggregates multiple response
variables (given from experiments) to calculate only one aggregation value firstly
and then fit only one regression function which simply optimized by single objective
optimization methods.

Fig. 2: 
Summary of proposed methods 
NUMERICAL EXAMPLE
Here, Pignatiello (1993), there are two response variables
(y_{1}, y_{2}) and three setting variables (x_{1}, x_{2},
x_{3}). A 2^{3} design with 4 replicates were used in this example.
Table 1: 
Experiments results 

*Average of replicate results, **SD of replicate results 
Table 2: 
Decision matrix 

*Cost criteria, **ELECTREIII parameters 
Table 3: 
Results of MADM methods 

The experimental data are shown in Table 1 that contains experiments
ID, response value for each replication according to related design. In this
example it is assumed that the target values of the responses (y_{1},
y_{2}) are 103 and 73, respectively.
Summary of proposed methods are as follows:
• 
Construction of decision matrix (Table
2) 
According to robust design, we consider standard deviations(s) and y_{1},
y_{2} as responses. Y new in Table 2 shows absolute
deviations of y_{i} and related target (103, 73 for y_{1},
y_{2}).
• 
Find alternative scores by MADM methods. Table
3 shows the results of MADM methods 
• 
Find regression model between controllable variables
and scores 
Table 4: 
Analysis of variance results for regression models by
MINITAB software 

Table 5: 
Results of MADM methods 

Table 4 shows that all regression model are meaningful
at α = 0.05 (p<α).
C =0.557500+0.03 X_{1}+0.2
X_{2}–0.105 X_{3}–0.0175 X_{1}X_{2}–0.0275
X_{1}X_{3}–0.0125 X_{2}X_{3} 
(17) 
S =1.08375+0.08375 X_{1}+0.29125
X2–0.29125 X_{3}+0.04125 X1X2–0.04125 X1X3–0.08375
X2X3 
(18) 
Q =0.529–0.025 X_{1}–0.3135
X_{2}+0.1545 X_{3}+0.01275 X_{1}X_{2}+0.04225
X_{1}X_{3}–2.5E04 X_{2}X_{3} 
(19) 
Φ =0.008775–0.050175 X_{1}+0.393475
X_{2}–0.287425 X_{3}+0.09122 X_{1}X_{2}+0.0221
X_{1}X_{3}+0.0229 X_{2}X_{3 } 
(20) 
• 
Optimizing fitted Function. Table 5
shows results of optimization 
For Example, optimization model for TOPSIS based regression function
is:
Optimize C 
= 
0.557500+0.03X_{1}+0.2X_{2}–0.105
X_{3}–0.0175 X_{1} X_{2}–0.0275X_{1}X_{3}–0.0125X_{2}X_{3} 
subject to:
1 ≤ x_{1}, x_{2}, x_{3}
≤ 1 
Only for VIKOR methods, the function must be minimized but for other
methods, the objective function must be maximized. Table
5 shows that, because of common approach, the all MADM method give
similar results.
CONCLUSION
In this study, usage of multiple criteria decision making in MRO problem
was represented. According to results (Table 3), optimal
solutions of proposed method for aforesaid MADM methods have similarity.
The proposed MADM based Methods are related to decision maker viewpoint
about important degree of responses. In the given results it is assumed
that the responsemeans are more important than the standard deviations.
Another advantage of this method is to consider standard deviation that
contributes to robust experimental design also because of fitting only
one response regression function, the proposed method decreases statistical
error. Since this method attempts to obtain one value from several responses;
it can be categorized in desirability function approach. Future Studies
on MRO problem can focus on other MADM methods application, fuzzy logic
issues and comparing between multiple criteria decision making tools.