G D Kostina

I O Chudaykin

S A Kostina

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Academic Research International   Vol. 5(3)  May  2014 
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Copyright © 2014 SAVAP International                                                                            ISSN: 2223-9944,  eISSN: 2223-9553 
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Reduction of a Multi Criteria Optimization Problem to a Single 
Criterion Task 
G. D. Kostina1, S. A. Kostina2, I. O. Chudaykin3 
Moscow Institute of Electronic Technology, Technical University,  
RUSSIA. 
2 
chudsvetlana@yandex.ru  
ABSTRACT 
This paper is devoted to a problem of a multi criteria optimization of systems. A 
variant of the linear weighted sum scalarization is described for the reduction of 
discrete multi criteria optimization problems to a single criterion task. This variant 
uses the utility concept to transform each of multiple criteria to a uniform numeric 
form. The reduction is done by defining a generalized criterion as a linear sum of 
weighted utility values associated with original individual criteria of given 
optimization problem. The use of utility values instead of actual values of criteria 
considerably simplifies the aggregation of multiple criteria that in general can be of 
different types and ranges of values. Created single generalized criterion can be used 
to determine the best alternative among the given set of alternatives by a single 
criterion optimization. The detailed algorithm is given that implements all operations 
to calculate such a criterion and to determine the optimal solution of the problem. 
Keywords: Multi criteria optimization, Linear weighted sum scalarization method, 
Utility concept 
INTRODUCTION 
In the design of systems of different types we are usually interested in finding a design 
variant that represents the best choice subject to some restrictions. Typically such a variant is 
obtained with the use of an appropriate optimization method, on the base of a set of 
predefined criteria.  
With more than one criterion in the search for the best variant of the system, we have a multi 
criteria optimization problem. Theoretical aspects of solution of such problems are given in  
works of M. Ehrgott (2005), M. Emmerich and A. Deutz (2006), J. Klapka, P. Pinos, and V. 
Sevcik (2013), to mention only a few. Examples of practical use of methods of multi criteria 
optimization are described, in particular, in publications of A. Aleti, et al (2013) and Xin-She 
Yang (2011). As a rule, these methods require complex algorithms and, in general, they allow 
to find only approximate solutions. 
There are known attempts to solve multi criteria optimization problems with the use of the 
particle swarm optimization (PSO) method (R. Roy, et al (2012), M. Thamari, et al (2013), 
M. Nobile, et al (2012)). However, the PSO method itself has serious shortcomings and 
cannot be considered as a universal optimization technique (S. Kaur, et al (2013)). In 
particular, it usually works only on real numbers. To apply such a technique to a discrete 
optimization problem, it is necessary to map the discrete search space to a continuous domain 
and, after the optimization is performed, to demap the result back into the original discrete 
space. 
Academic Research International   Vol. 5(3)  May  2014 
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Copyright © 2014 SAVAP International                                                                            ISSN: 2223-9944,  eISSN: 2223-9553 
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There are methods to transform the multi criteria optimization problem into a single criterion 
problem. One known approach of this kind is epsilon-constrained method, in which one of 
given original criteria is chosen as an objective function to be optimized, with the remaining 
criteria considered as constraints (M. Emmerich and A. Deutz (2006)). Another known 
approach is related to a weighted sum scalarization, where a single objective function is an 
aggregation of weighted original criteria (M. Ehrgott (2005)). 
In this paper, a variant of the linear weighted sum scalarization is described for the reduction 
of discrete multi criteria optimization problems to a single criterion task. The reduction is 
done by defining a single generalized criterion as a linear sum of utility values associated 
with multiple original criteria of the given optimization problem. The use of utility values 
instead of actual values of criteria considerably simplifies the aggregation of multiple criteria 
that in general can be of different types with different ranges of their values. The single 
generalized criterion defined in this way can be used to determine the best design alternative 
among the given set of alternatives by single criterion optimization. It is assumed further that 
the set of alternatives and the number of criteria, for all alternatives, are finite. 
MULTI CRITERIA OPTIMIZATION 
Let Q1, Q2, …, Qm   be individual criteria chosen for evaluation of a number of possible 
design alternatives of some system. With these criteria, Q= (Q1, Q2, …, Qm) is a 
multidimensional objective function. Assume that the number of alternatives is finite, with 
alternatives A1, A2, …, An. Formally, the system being optimized can be represented as the 
following matrix of alternatives: 
 ………… (1) 
In this matrix, rows with numbers 1, 2, …,n  represent possible alternatives A1, A2, …, An, 
and columns with numbers 1, 2, …, m correspond to individual criteria. Each element qij in 
the matrix is the value of criterion Qij in alternative Aij. In general, criteria can be of different 
types, so that each alternative is represented as a record Ai = (qi1,  qi2,  …, qim). It assumed 
also that each criteria has values in some range, so that  
Qj,min<= Qj<= Qj,max, ……………….. (2) 
Where Qj,min and Qj,max are minimal and maximal values of criterion Qj for all alternatives, 
with j = 1, 2, …, m. 
The problem of multi criteria optimization here is the choice of such alternative Ai, for which 
we have  
maxQj, j = 1, 2, …, m, ……………… (3) 
Subject to restrictions (2). However, as was stated in the introduction, to find the best 
alternative in this case requires the use of complex optimization procedure. We shall show 
now how the the problem can be reduced to a single criterion task. 
REDUCTION OF MULTI CRITERIA OPTIMIZATION PROBLEM TO SINGLE 
CRITERION TASK 
The simplest reduction method is to define a single generalized criterion as the following 
linear aggregation: 
Academic Research International   Vol. 5(3)  May  2014 
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Copyright © 2014 SAVAP International                                                                            ISSN: 2223-9944,  eISSN: 2223-9553 
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S =   ∑ pjQj, ……………………. (4) 
wherepj   is a positive weight of criterion   Qj. 
In this case, each alternative Ai can be represented by the sum  
Si = ∑

 pj qij. ……………………. (5) 
Unfortunately, this form of the single generalized criterion is not appropriate, since individual 
criteria can have different types and cannot be summed. In addition, this form does not reflect 
the extent of satisfaction corresponding to each individual criterion. 
To bypass the difficulty of aggregation of multiple criteria of different types and to take into 
account the aspect of satisfaction corresponding to each individual criterion, we express the 
single generalized criterion in the following modified form: 
W = ∑ pj Uj, …………………... (6) 
wherepj has the same meaning as in expression (4), and Uj is utility value of criterion Qj. 
Utility value represents the degree of satisfaction of each criterion  on a chosen uniform scale. 
For example, for the criterion “the price of car” we may accept, that utility value for the 
highest price is zero, while for a low price it is 100. Here zero corresponds to “not 
acceptable” and 100 to “completely accepted”. Utility values between zero and 100 
correspond to different levels of “accepted”.  
To use expression (6), original matrix of multi criteria alternatives (1) must be transformed 
into the following utility matrix 
 …………… (7) 
In this matrix, each element uij is utility value of criterion Qj in alternative Ai. It is important 
to note that all elements of utility matrix are assumed to have real values. Using this matrix, 
we can now express the resulting value of each alternative in the form 
Vi =   ∑

 pj uij …………………… (8) 
Calculating all values Vi and choosing the maximal of them, we determine the optimal of 
given alternatives. However, to perform cuch calculation it is necessary initially  to obtain 
values of elements of utility matrix (7) and to set weights pjof the original individual criteria. 
MAPPING MATRIX OF ALTERNATIVES TO UTILITY MATRIX 
As was stated above, values of different original criteria in matrix (1) have in general 
different type. In addition, some criteria can be discrete, while others can be continuous. But 
in any case, as was assumed, their values are restricted according to inequalities (2). 
A simple way to transform values of original criteria to their utility values is to use a linear 
transformation, making utility values lying in some fixed range (for example, in the range of 
nonnegative numeric values of (0, 100)). Such a transformation can be done according to 
expression  
uij = ajqij + bj, …………………...…… (9) 
Where aj and bj are scale coefficients corresponding to individual criterion Qj. 
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Copyright © 2014 SAVAP International                                                                            ISSN: 2223-9944,  eISSN: 2223-9553 
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To determine values of coefficients aj and bj in (9), it is necessary to choose limiting values  
(uij)min and (uij)max and associate them with the most preferable and least preferable values of 
each criterion Qj, j = 1, 2, …, m. Depending on criterion, value (uij)max can be associated with 
(Qj)min or (Qj)max.  
Suppose that, as an example that values of original criterion Qj are restricted as  
0.5 <= Qj<= 1.2 …………………...…… (10) 
Thus, (Qj)min= 0.5 and  (Qj)max= 1.2. Assume now, that the designer associates value 0.5 with 
utility (uij)min = 0  and value 1.2 with utility (uij)max = 100. In this case, coefficients ajand  bj in 
(9) can be found by solving linear equations 
(uij)min = 0      =  aj(Qj)min +  bj, 
(Qj)max = 1.2   = aj(Qj)max +  bj …………. (11) 
Solving this system of equations, we will have aj= 140 and bj = -70. Thus, for example, if 
original criterion has value qij = 1.0 for alternative Ai, then its utility value uij = 70. 
Note once again that the association of limiting values (uij)min and (uij)max with criterion 
values (Qj)min and  (Qj)max depends on the criterion. If, for example, criterion Qj is the price of 
a car, then it is natural to associate utility value (uij)min with  (Qj)max  and utility value (uij)max 
with (Qj)min.  
If some of the original criteria are not numeric (for examples, colors), then values of such 
criteria must be initially transformed into appropriate numerical values. 
The choice of weights pj in expression (8) is in general a non-formalized procedure.   It can 
be done by experts on the base of importance of original individual criteria. In such a choice, 
it is reasonable to ask experts to choose the weights in such a way that they are positive 
values satisfying, for example, the condition 
∑ pj = 1 …………………...………… (12) 
THE ALGORITHM 
This section presents the algorithm that implements operations described in the previous 
sections. As was assumed in the introductory section, the number of alternatives n and the 
number of individual original criteria m are known, with the same number m for each 
alternative. We assume also that weights pj, j = 1, 2, …, m, are decided on outside the 
algorithm and are used by algorithm as input data. Values qij of elements of matrix (1) are 
also input data. If some original criteria are non-numeric, then it is assumed that they are 
transformed to a numeric form before being submitted to the algorithm. Thus, the algorithm 
will deal only with numeric values qij in (1). 
    The algorithm is presented as a sequence of numbered steps. These steps are as follows. 
[1] Set or input the number of original criteria m and the number of alternatives n. 
[2] Input values of original individual criteria qij for the matrix of alternatives (1). 
[3] Input limiting values (Qj)min and  (Qj)max of all criteria Qj, j = 1, 2, …, m. 
[4] Input limiting utility values (uij)min and (uij)max associated with criteria limits Qj)min 
and   (Qj)max of all criteria Qj, j = 1, 2, …, m.  
[5] Input values of weights pj,j = 1, 2, …, m. 
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Copyright © 2014 SAVAP International                                                                            ISSN: 2223-9944,  eISSN: 2223-9553 
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[6] Compute coefficients aj and bjfor all criteria   using  the system of linear equations 
(11). 
[7] Consider next alternative Ai, i = 1, 2, …,n. 
[8] Consider next original criterion Qj of this alternative, j = 1, 2, …,m. 
[9] Compute element (utility value) uij of matrix (7) using coefficients aj and bj found at 
step 6. 
[10] If not all criteria are considered, then return to Step 8. 
[11] Compute and save the generalized single criterion Vi of the current alternative.  
[12] If not all alternatives are considered, then return to Step 7. 
[13] Choose (Vi)max among values (V1, V2, …, Vn) as the criterion value representing the 
optimal alternative Ai, i {1, 2, …, n}. 
[14] Output information about the found optimal alternative (such as its index or name 
and value of its single criterion). 
[15] End. 
As was mentioned, the algorithm assumes that all elements of matrix of alternatives are 
numeric. If some criterion originally is not numeric, then it must be mapped to a numeric 
form. For example, criterion “color” can be mapped to numeric values in this way: white -> 
1, black -> 2, red -> 3, and so on.  
The main part of the algorithm is implemented as a double loop at steps from 7 to 10. The 
external loop corresponds to alternatives (rows of matrix of alternatives) and the internal loop 
to individual criteria (columns of matrix of alternatives).  
In addition to information output by the algorithm about the optimal alternative, it can be 
easily extended to output values of the generalized criterion for all other alternatives. This 
information can be useful for comparison of alternatives on the base of the generalized 
criterion. 
CONCLUSION 
A variant of a linear weighted sum scalarization is described to reduce a discrete multivariate 
optimization problem to a single criterion task. This variant uses the utility concept to 
transform multiple criteria of different types to a uniform numeric form and to obtain the 
single generalized criterion. The detailed algorithm is given that implements all operations 
necessary to compute such a criterion and to determine the optimal solution of the problem. 
 
  
Academic Research International   Vol. 5(3)  May  2014 
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Copyright © 2014 SAVAP International                                                                            ISSN: 2223-9944,  eISSN: 2223-9553 
www.savap.org.pk                                                    17                                www.journals.savap.org.pk 
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[6] Nobile, M., Besozzi, D., Cazzaniga, P., Mauri, G., & Pescini, D. (2012). A GPU-based 
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[7] Roy, R., Dehuri, S., & Cho, S. (2012). A Novel Particle Swarm Optimization 
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Reduction of a Multi Criteria Optimization Problem to a Single Criterion Task

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Reduction of a Multi Criteria Optimization Problem to a Single Criterion Task

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