- Email: [email protected]
On the constrained OWA aggregation problem with single constraint
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ScienceDirect Fuzzy Sets and Systems ••• (••••) •••–••• www.elsevier.com/locate/fss
On the constrained OWA aggregation problem with single constraint Lucian Coroianu a,∗ , Robert Fullér b,c a Department of Mathematics and Computer Science, University of Oradea, Romania b Department of Informatics, Széchenyi István University, Gy˝or, Hungary c Institute of Applied Mathematics Óbuda University, Budapest, Hungary
Received 10 September 2016; received in revised form 19 April 2017; accepted 21 April 2017
Abstract In this note we present a simple proof for the constrained OWA aggregation problem with a single constraint but with variable coefficients. © 2017 Elsevier B.V. All rights reserved. Keywords: Ordered weighted average operators (OWA operators); Constrained OWA aggregation problem
1. Introduction Starting with the remarkable work of Yager (see [11]), OWA operators (that is, ordered weighted average operators) have known an impressive development certified by their effective use in important research fields such as: decision making, risk analysis, regression analysis and others (see, e.g., [5,9,12–14]). A citation-based survey of the literature in this topic can be found in [3]. A particular topic which gained the interest of many researchers is the one which deals with optimization problems associated to OWA operators. The first paper where this problem is investigated is [10]. The idea is to maximize the ordered weighted operator with respect to linear inequality constraints. This problem is called (see [10]) a constraint OWA aggregation problem. In the same paper [10], this problem is reduced to a linear programming optimization problem by introducing new variables, obtaining a mixed integer linear programming problem. In general, this method is quite complex even in the simplest case when we have one restriction. In paper [2], the exact solution is provided for the special case when we have one restriction and all coefficients in the constraint are equal to one. Besides, there are also other important contributions dedicated to optimization problems associated with OWA operators (see, e.g. [1,4,6–9]). An effective and fast algorithm for the exact solution of the general problem addressed in [10] is still an open problem. In this contribution we extend the result of paper [2] as this time the coefficients in the constraint are variable. In this way, the case of a single constrained OWA aggregation problem is completely solved. The issue with this general case is the fact that there is no any evident method to reduce the * Corresponding author.
E-mail addresses: [email protected] (L. Coroianu), [email protected] (R. Fullér). http://dx.doi.org/10.1016/j.fss.2017.04.013 0165-0114/© 2017 Elsevier B.V. All rights reserved.
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study to the special case when the coefficients in the constraint are all equal to 1. The natural approach would be to substitute the variables in order to obtain a restriction where all coefficients are equal to 1. But then of course we completely loose control on the objective function as the substitutions on the constraint cannot be introduced in the objective function. This also explains why almost all examples in the literature on the constraint OWA aggregation problem consider only the case when all coefficients are equal to 1. Instead, in order to obtain the exact solution, first, we will associate to the constrained OWA aggregation problem a linear program which will be solved using its dual. The idea of using the dual is not new in optimization problems related to OWA operators. It was previously used in paper [2] for the special case when all the coefficients in the constraint are equal to 1, but in [8] too, for an optimization problem with decreasing weights. In our approach, in the most interesting case where the objective function has a finite maximum, first we will prove that it suffices to study constrained OWA aggregation problems where the coefficients in the constraint form a nondecreasing sequence. This simplification of the problem is decisive because the solution of the general problem will be obtained from a problem where the sequence of the coefficients is nondecreasing by using a suitable permutation to rearrange the order of the coefficients. Now, the special problem where the coefficients form a nondecreasing sequence will be transformed into a linear program and the exact solution of this problem will be obtained by finding the exact solution of its dual linear program. From here, we will easily obtain the solution of the initial constrained OWA aggregation problem. The basics about the constrained OWA aggregation problem are given in the next section. Section 3 is divided in two subsections. In the first one we provide the exact solution (when it exists) of the OWA aggregation problem with a single constraint where the coefficients are arbitrary reals. We will prove that actually it suffices to study only three types of problems, and only one of them needs a more detailed investigation. As for the two remaining problems, either we have a trivial solution or, the supremum of the objective function is ∞ when the feasible set is nonempty. In the second subsection we propose concrete examples where, applying the main results of the paper, we find the exact solution for some constrained OWA aggregation problems. This time, the coefficients in the constraint are not all equal to 1, and we believe that this is the most important improvement with respect to other works in this topic. Note that in one example we actually consider an objective function having n + 1 variables. Still, we can find the exact solution for any particular value of n. The results are summarized in the conclusions section, where we also acknowledge that the general case of constrained OWA aggregation problem, where we have an arbitrary number of constraints, is still an open problem in respect to the exact solution. 2. Formulation of the problem We recall here some well-known facts on the Ordered Weighted Averaging (OWA). Consider the vector w = (w1 , . . . , wn )T , where w1 + . . . + wn = 1 and 0 ≤ wi ≤ 1, for all i ∈ {1, . . . , n}. The quantities w1 , . . . , wn are called weights. Then, consider the mapping F : Rn → [0, 1], F (x1 , . . . , xn ) =
n
wi y i ,
i=1
where yi is the i-th largest element of the bag {x1 , . . . , xn }. Next, suppose that A is a matrix of type (m, n) with real entries and suppose that b ∈ Rm . The problem max F (x1 , . . . , xn ) subject to Ax ≤ b, x ≥ 0, is called a constrained OWA aggregation problem (see [10]). So far, exact analytical solution is known (see [2]) only for the special case of constrained OWA aggregation problem max F (x1 , . . . , xn ) subject to x1 + . . . + xn ≤ 1, x ≥ 0.
(1)
For the general case, although it can be transformed into a mixed integer linear programming problem (see [10]), we still miss a method that proves to be effective in all cases. In this paper, we will solve completely the constrained OWA aggregation problem max F (x1 , . . . , xn ) subject to α1 x1 + . . . + αn xn ≤ β, x ≥ 0,
(2)
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where the only additional assumption is that wi > 0, for all i ∈ {1, . . . , n}. Moreover, we will provide an exact analytical solution for the cases when the maximum is finite. It is pretty obvious that in general, problem (2) can be transformed into one of the following constrained OWA aggregation problems max F (x1 , . . . , xn ) subject to α1 x1 + . . . + αn xn ≤ 0, x ≥ 0,
(3)
max F (x1 , . . . , xn ) subject to α1 x1 + . . . + αn xn ≥ 1, x ≥ 0,
(4)
max F (x1 , . . . , xn ) subject to α1 x1 + . . . + αn xn ≤ 1, x ≥ 0.
(5)
or
For this reason, in what follows, we will focus only on the three problems from above. 3. Solution of the problem 3.1. Main theoretical results Let us start with problem (3). Clearly, if αi > 0 for all i ∈ {1, . . . , n}, then the unique optimum point is x ∗ = (0, . . . , 0). Then, if there exists i0 ∈ {1, . . . , n} such that αi0 ≤ 0, then the problem (3) has no solution as the function is unbounded in the feasible region. This can be summarized in the following proposition. Proposition 1. Consider problem (3). Then: (i) if αi > 0 for all i ∈ {1, . . . , n}, then x ∗ = (0, . . . , 0) is the unique solution and hence the maximum value is 0; (ii) if there exists i0 ∈ {1, . . . , n} such that αi0 ≤ 0, then the problem has no solution and, the function F is unbounded on the feasible set, that is, the supremum of F over the feasible set is ∞. Let us move now towards problem (4). The proof of the following proposition is just a simple exercise. Proposition 2. Consider problem (4). Then: (i) if there exists i0 ∈ {1, . . . , n} such that αi > 0, then the problem has no solution and, the function F is unbounded on the feasible set with supremum equal to ∞; (ii) if αi ≤ 0 for all i ∈ {1, . . . , n}, then the problem has no solution as the feasible set is the empty set. Now, we deal with the most important case, that is, problem (5). First, we observe that in the case when there exists i0 ∈ {1, . . . , n} such that αi0 ≤ 0, then F is unbounded on the feasible set, hence, the supremum of F over the feasible set is ∞. It remains to discuss the case when αi > 0 for all i ∈ {1, . . . , n}. We will solve problem (5) without imposing other restrictions on the coefficients on the constraint, but, it will serve to our purposes to begin with the particular case when α1 ≤ . . . ≤ αn . Suppose that x1∗ , . . . , xn∗ is a solution of (5). Let us note that we must have α1 x1∗ + . . . + αn xn∗ = 1. Indeed, if α1 x1∗ + . . . + αn xn∗ < 1, then let ε > 0 be sufficiently small, so that α1 (x1∗ + ε) + . . . + αn (xn∗ + ε) < 1. It is immediate that F (x1∗ + ε, . . . , xn∗ + ε) > F (x1∗ , . . . , xn∗ ) and this contradicts the fact that x1∗ , . . . , xn∗ is a solution of (5). Next, if Sn denotes the set of all permutations of the set {1, . . . , n} and if σ ∈ Sn , then we necessarily have α1 xσ∗1 + . . . + αn xσ∗n ≥ 1. Indeed, if for some σ we would have α1 xσ∗1 + . . . + αn xσ∗n < 1 then again, let ε > 0 be sufficiently small, so that α1 (xσ∗1 + ε) + . . . + αn (xσ∗n + ε) < 1. But this easily implies F (xσ∗1 + ε, . . . , xσ∗n + ε) > F (x1∗ , . . . , xn∗ ), a contradiction. It readily follows that actually we have min{α1 xσ∗1 + . . . + αn xσ∗n : σ ∈ Sn } = 1. Now, it is well-known that if a1 ≤ . . . ≤ an and b1 ≥ . . . ≥ bn , then a1 b1 + . . . + an bn ≤ a1 bσ1 + . . . + an bσn , for all σ ∈ Sn . Therefore, since α1 ≤ . . . ≤ αn , we get
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min{α1 xσ∗1 + . . . + αn xσ∗n : σ ∈ Sn } = α1 y1∗ + . . . + αn yn∗ , where yi∗ is the i-th largest element of the bag {x1∗ , . . . , xn∗ }. Thus, we get α1 y1∗ + . . . + αn yn∗ = 1. This implies that F (x1∗ , . . . , xn∗ ) ≤ max{F (y1 , . . . , yn ) : α1 y1 + . . . αn yn = 1, y1 ≥ y2 ≥ . . . ≥ yn ≥ 0}. But obviously the converse inequality also holds, and therefore, in order to find a solution of problem (5), it suffices to solve the problem max F (x1 , . . . , xn ) subject to α1 x1 + . . . + αn xn = 1, x1 ≥ x2 . . . ≥ xn ≥ 0.
(6)
Thus, any solution of problem (6) is a solution of problem (5). But all solutions of problem (5) can be derived from the solutions of problem (6). Indeed, suppose that (x1∗ , . . . , xn∗ ) is a solution of (5) such that xi∗ < xj∗ for some i, j ∈ {1, . . . , n}, i < j . Let then (t1∗ , . . . , tn∗ ) ∈ Rn be such that tk∗ = xk∗ , if k ∈ {1, . . . , n} {i, j }, ti∗ = xj∗ and tj∗ = xi∗ . As we know, α1 x1∗ + . . . + αn xn∗ = 1 and α1 t1∗ + . . . + αn tn∗ ≥ 1. But, by simple calculations we get n
αk xk∗
k=1
−
n
αk tk∗ = (xj∗ − xi∗ )(αj − αi )
k=1
and taking into account all the hypotheses, it follows that necessarily αi = αj . Therefore, any solution of (5) is the permutation of a solution (x1∗ , . . . , xn∗ ) of (6), which corresponds to some σ ∈ Sn , with the property that, if for some i, j ∈ {1, . . . , n}, i < j , we have xσ∗i > xσ∗j , then αi = αj . This also means that, if α1 < α2 < . . . < αn , then (x1∗ , . . . , xn∗ ) is a solution of problem (5), if and only if (x1∗ , . . . , xn∗ ) is a solution of problem (6). In addition, any permutation of (x1∗ , . . . , xn∗ ) will not be a solution of problem (5) (the maximum is achieved but the constraint will not be satisfied). Lets us now focus on finding the solution of problem (6). First, note that it makes no difference if we replace the condition α1 x1 + . . . + αn xn = 1, with the condition α1 x1 + . . . + αn xn ≤ 1 (this was already proved at the beginning of this section). Therefore, we may consider problem (6) as a primal linear program which has its dual as min t1 subject to α1 t1 − t2 ≥ w1 , α2 t1 + t2 − t3 ≥ w2 , . . . , αn t1 + tn − tn+1 ≥ wn ,
(7)
where tk ≥ 0, for all k ∈ {1, . . . , n + 1}. We will repeat now the reasoning used in [2]. Summing up the first k inequalities from above, k = 1, n, we get t1 ≥
w1 + . . . + wk + tk+1 , k = 1, n α1 + . . . + αk
This implies that t1 ≥
w1 +...+wk ∗ α1 +...+αk ∗
, where k ∗ ∈ {1, . . . , n} is such that
w1 + . . . + wk ∗ w 1 + . . . + wk = max : k ∈ {1, . . . , n} . α1 + . . . + αk ∗ α1 + . . . + αk This easily implies that (t1∗ , . . . , tn∗ ) is a solution of (7), where w 1 + . . . + wk ∗ , α1 + . . . + αk ∗ w 1 + . . . + wk ∗ ∗ tk+1 = t1 − (α1 + . . . + αk ) , k = 1, n. α1 + . . . + αk t1∗ =
From the duality theorem, if there exists (x1∗ , . . . , xn∗ ) ∈ Rn that satisfies the constraints in (6) and such that F (x1∗ , . . . , xn∗ ) = t1∗ , then (x1∗ , . . . , xn∗ ) is actually a solution of (6) and hence, a solution of (5). Such solution obviously exists as we can take
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1 , α1 + . . . + αk ∗ xk∗∗ +1 = . . . = xn∗ = 0. x1∗ = . . . = xk∗∗ =
Therefore, we just found an optimal solution to problem (5). Now, we can easily deduce the solution for the general case. The following theorem summarizes the results obtained just above. We give the theorem in the general setting of problem (5) as its proof immediately follows from the results obtained just above. Theorem 3. Consider problem (5). Then: (i) if there exists i0 ∈ {1, . . . , n} such that αi0 ≤ 0, then F is unbounded on the feasible set and its supremum over the feasible set is ∞; (ii) if αi > 0, i ∈ {1, . . . , n}, then taking (any) σ ∈ Sn with the property that ασ1 ≤ ασ2 ≤ . . . ≤ ασn , and k ∗ ∈ {1, . . . , n}, such that w 1 + . . . + wk ∗ w1 + . . . + w k = max : k ∈ {1, . . . , n} , ασ1 + . . . + ασk∗ ασ1 + . . . + ασk then (x1∗ , . . . , xn∗ ) is an optimal solution of problem (5), where xσ∗1 = . . . = xσ∗k∗ =
1 , ασ1 + . . . + ασk∗
xσ∗k∗ +1 = . . . = xσ∗n = 0.
In particular, if 0 < α1 ≤ α2 ≤ . . . ≤ αn , and k ∗ ∈ {1, . . . , n} is such that w 1 + . . . + wk ∗ w 1 + . . . + wk = max : k ∈ {1, . . . , n} , α1 + . . . + αk ∗ α1 + . . . + αk then (x1∗ , . . . , xn∗ ) is a solution of (5), where 1 , α1 + . . . + αk ∗ xk∗∗ +1 = . . . = xn∗ = 0. x1∗ = . . . = xk∗∗ =
We easily notice that assertion (ii) of the above theorem generalizes the main result in [2], where the particular case α1 = . . . = αn = 1 was considered. 3.2. Examples Theorem 3 has the important advantage that it provides an exact solution to problem (5), whereas, other methods transform the problem into a linear programming task which may be quite complex in providing the solution. For example, by the method used in [10], considering F a function in n variables, problem (5) is transformed into a linear program with n2 variables and also with increased number of constraints. Clearly, this method is not always effective. To our knowledge, there are no examples in the literature where one tries to find a solution for a problem of the type (5), unless we deal with the particular case when α1 = . . . = αn = 1. This is why, we will present two examples where such requirement does not hold. Although, the main results are totally independent of the requirement w1 + . . . + wn = 1, for conformity with the general theory, in our examples this requirement will be satisfied. Example 1. Consider the problem 1 2 3 1 y1 + y2 + y3 + y4 → max, 10 5 10 5 subject to
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x1 + 3x2 + 2x3 + x4 ≤ 1, x ≥ 0. Considering the permutation 1 2 3 4 σ= , 1 4 3 2 by simple inspection, we get that w1 + . . . + wk max : k ∈ {1, . . . , 4} ασ1 + . . . + ασk is achieved for k ∗ = 2. We also note that
1 ασ1 +ασ2
= 12 . Applying Theorem 3, (ii), it readily follows that
an optimal solution for our problem. In addition, the maximum value is
1 1 2 , 0, 0, 2
is
3 20 .
Example 2. For some arbitrary n ∈ N, consider the problem n n k yk+1 → max, 2n k=0
subject to n+1
k · xk ≤ 1, x ≥ 0.
k=1
By Theorem 3, (ii), this problem can easily be solved for any given n. We just need to find ⎫ ⎧ k ⎪ ⎪ 1 n ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ 2n l l=0 ∗ : k ∈ {0, . . . , n} . k = arg max k+1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ l ⎭ ⎩ l=1
For example, if n = 7, then by simple calculations we get k ∗ = 4. Applying the conclusion of Theorem 3, (ii), it readily follows that x ∗ = (x1∗ , x2∗ , . . . , x8∗ ) is an optimal solution of our problem, where x1∗ = . . . = x5∗ =
1 ∗ , x = x7∗ = x8∗ = 0. 15 6
4. Conclusions In this note, we presented a simple method to obtain the exact solution of the constrained OWA aggregation problem with single constraint on arbitrary coefficients. Obviously, the method used in this paper also applies in the case where we have an arbitrary number of constraints and such that the matrix of the constraints is increasing on any of its rows. In such cases, the constrained OWA aggregation problem will become a linear programming problem and hence, it can be solved by using suitable algorithms for such programming tasks. For the general case, where the matrix of coefficients is not necessarily nondecreasing on any of its rows, there is still an open question the finding of a simple algorithm that would give us the exact solution. Up to now, the only algorithm that applies for the general case, is the one given by Yager in [10]. As we already mentioned in this paper, this algorithm is very difficult to implement when we have a large number of variables and constraints. What is sure, is that the method presented in this paper for the case of problems with single constraints cannot be extended inductively to the general case. For simplicity, suppose we have a constrained OWA aggregation problem with two constraints with arbitrary coefficients in both of them. The issue is that in general we cannot use a single permutation (as in the case with single constraint) to reduce the problem to a problem where the sequence of coefficients is nondecreasing in each of the constraints. Then, as in the case of linear or quadratic programs, separating the initial problem in two problems with single constraints will not give us in general the solution of the initial problem as we cannot relate the solution of the problems with single constraints to the solution of the initial problem.
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Acknowledgements The contribution of Lucian Coroianu was possible with the support of a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0861. References [1] B.S. Ahn, Parameterized OWA operator weights: an extreme point approach, Int. J. Approx. Reason. 51 (2010) 820–831. [2] C. Carlsson, R. Fullér, P. Majlender, A note on constrained OWA aggregation, Fuzzy Sets Syst. 139 (2003) 543–546. [3] A. Emrouznejad, M. Marra, Ordered weighted averaging operators 1988–2014: a citation-based literature survey, Int. J. Intell. Syst. 29 (2014) 994–1014. [4] R. Fullér, P. Majlender, On obtaining minimal variability OWA operator weights, Fuzzy Sets Syst. 136 (2003) 203–215. [5] J.M. Merigó, A.M. Gil-Lafuente, L-G. Zhou, H.-Y. Chen, Induced and linguistic generalized aggregation operators and their application in linguistic group decision making, Group Decis. Negot. 21 (2012) 531–549. [6] W. Ogryczak, P. Olender, On MILP models for the OWA optimization, J. Telecommun. Inf. Technol. 2 (2012) 5–12. [7] W. Ogryczak, P. Olender, Ordered median problem with demand distribution weights, Optim. Lett. 10 (2016) 1071–1086. ´ [8] W. Ogryczak, T. Sliwi´ nski, On efficient WOWA optimization for decision support under risk, Int. J. Approx. Reason. 50 (2009) 915–928. ´ [9] W. Ogryczak, T. Sliwi´ nski, On solving linear programs with the ordered weighted averaging objective, Eur. J. Oper. Res. 148 (2003) 80–91. [10] R.R. Yager, Constrained OWA aggregation, Fuzzy Sets Syst. 81 (1996) 89–101. [11] R.R. Yager, On ordered weighted averaging aggregation operators in multicriteria decision making, IEEE Trans. Syst. Man Cybern. 18 (1988) 183–190. [12] R.R. Yager, On the analytic representation of the Leximin ordering and its application to flexible constraint propagation, Eur. J. Oper. Res. 102 (1997) 176–192. [13] R.R. Yager, G. Beliakov, OWA operators in regression problems, IEEE Trans. Fuzzy Syst. 18 (2010) 106–113. [14] L. Zhou, H. Chen, J. Liu, Generalized multiple averaging operators and their applications to group decision making, Group Decis. Negot. 22 (2013) 331–358.
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Available online at www.sciencedirect.com
ScienceDirect Fuzzy Sets and Systems ••• (••••) •••–••• www.elsevier.com/locate/fss
On the constrained OWA aggregation problem with single constraint Lucian Coroianu a,∗ , Robert Fullér b,c a Department of Mathematics and Computer Science, University of Oradea, Romania b Department of Informatics, Széchenyi István University, Gy˝or, Hungary c Institute of Applied Mathematics Óbuda University, Budapest, Hungary
Received 10 September 2016; received in revised form 19 April 2017; accepted 21 April 2017
Abstract In this note we present a simple proof for the constrained OWA aggregation problem with a single constraint but with variable coefficients. © 2017 Elsevier B.V. All rights reserved. Keywords: Ordered weighted average operators (OWA operators); Constrained OWA aggregation problem
1. Introduction Starting with the remarkable work of Yager (see [11]), OWA operators (that is, ordered weighted average operators) have known an impressive development certified by their effective use in important research fields such as: decision making, risk analysis, regression analysis and others (see, e.g., [5,9,12–14]). A citation-based survey of the literature in this topic can be found in [3]. A particular topic which gained the interest of many researchers is the one which deals with optimization problems associated to OWA operators. The first paper where this problem is investigated is [10]. The idea is to maximize the ordered weighted operator with respect to linear inequality constraints. This problem is called (see [10]) a constraint OWA aggregation problem. In the same paper [10], this problem is reduced to a linear programming optimization problem by introducing new variables, obtaining a mixed integer linear programming problem. In general, this method is quite complex even in the simplest case when we have one restriction. In paper [2], the exact solution is provided for the special case when we have one restriction and all coefficients in the constraint are equal to one. Besides, there are also other important contributions dedicated to optimization problems associated with OWA operators (see, e.g. [1,4,6–9]). An effective and fast algorithm for the exact solution of the general problem addressed in [10] is still an open problem. In this contribution we extend the result of paper [2] as this time the coefficients in the constraint are variable. In this way, the case of a single constrained OWA aggregation problem is completely solved. The issue with this general case is the fact that there is no any evident method to reduce the * Corresponding author.
E-mail addresses: [email protected] (L. Coroianu), [email protected] (R. Fullér). http://dx.doi.org/10.1016/j.fss.2017.04.013 0165-0114/© 2017 Elsevier B.V. All rights reserved.
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study to the special case when the coefficients in the constraint are all equal to 1. The natural approach would be to substitute the variables in order to obtain a restriction where all coefficients are equal to 1. But then of course we completely loose control on the objective function as the substitutions on the constraint cannot be introduced in the objective function. This also explains why almost all examples in the literature on the constraint OWA aggregation problem consider only the case when all coefficients are equal to 1. Instead, in order to obtain the exact solution, first, we will associate to the constrained OWA aggregation problem a linear program which will be solved using its dual. The idea of using the dual is not new in optimization problems related to OWA operators. It was previously used in paper [2] for the special case when all the coefficients in the constraint are equal to 1, but in [8] too, for an optimization problem with decreasing weights. In our approach, in the most interesting case where the objective function has a finite maximum, first we will prove that it suffices to study constrained OWA aggregation problems where the coefficients in the constraint form a nondecreasing sequence. This simplification of the problem is decisive because the solution of the general problem will be obtained from a problem where the sequence of the coefficients is nondecreasing by using a suitable permutation to rearrange the order of the coefficients. Now, the special problem where the coefficients form a nondecreasing sequence will be transformed into a linear program and the exact solution of this problem will be obtained by finding the exact solution of its dual linear program. From here, we will easily obtain the solution of the initial constrained OWA aggregation problem. The basics about the constrained OWA aggregation problem are given in the next section. Section 3 is divided in two subsections. In the first one we provide the exact solution (when it exists) of the OWA aggregation problem with a single constraint where the coefficients are arbitrary reals. We will prove that actually it suffices to study only three types of problems, and only one of them needs a more detailed investigation. As for the two remaining problems, either we have a trivial solution or, the supremum of the objective function is ∞ when the feasible set is nonempty. In the second subsection we propose concrete examples where, applying the main results of the paper, we find the exact solution for some constrained OWA aggregation problems. This time, the coefficients in the constraint are not all equal to 1, and we believe that this is the most important improvement with respect to other works in this topic. Note that in one example we actually consider an objective function having n + 1 variables. Still, we can find the exact solution for any particular value of n. The results are summarized in the conclusions section, where we also acknowledge that the general case of constrained OWA aggregation problem, where we have an arbitrary number of constraints, is still an open problem in respect to the exact solution. 2. Formulation of the problem We recall here some well-known facts on the Ordered Weighted Averaging (OWA). Consider the vector w = (w1 , . . . , wn )T , where w1 + . . . + wn = 1 and 0 ≤ wi ≤ 1, for all i ∈ {1, . . . , n}. The quantities w1 , . . . , wn are called weights. Then, consider the mapping F : Rn → [0, 1], F (x1 , . . . , xn ) =
n
wi y i ,
i=1
where yi is the i-th largest element of the bag {x1 , . . . , xn }. Next, suppose that A is a matrix of type (m, n) with real entries and suppose that b ∈ Rm . The problem max F (x1 , . . . , xn ) subject to Ax ≤ b, x ≥ 0, is called a constrained OWA aggregation problem (see [10]). So far, exact analytical solution is known (see [2]) only for the special case of constrained OWA aggregation problem max F (x1 , . . . , xn ) subject to x1 + . . . + xn ≤ 1, x ≥ 0.
(1)
For the general case, although it can be transformed into a mixed integer linear programming problem (see [10]), we still miss a method that proves to be effective in all cases. In this paper, we will solve completely the constrained OWA aggregation problem max F (x1 , . . . , xn ) subject to α1 x1 + . . . + αn xn ≤ β, x ≥ 0,
(2)
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where the only additional assumption is that wi > 0, for all i ∈ {1, . . . , n}. Moreover, we will provide an exact analytical solution for the cases when the maximum is finite. It is pretty obvious that in general, problem (2) can be transformed into one of the following constrained OWA aggregation problems max F (x1 , . . . , xn ) subject to α1 x1 + . . . + αn xn ≤ 0, x ≥ 0,
(3)
max F (x1 , . . . , xn ) subject to α1 x1 + . . . + αn xn ≥ 1, x ≥ 0,
(4)
max F (x1 , . . . , xn ) subject to α1 x1 + . . . + αn xn ≤ 1, x ≥ 0.
(5)
or
For this reason, in what follows, we will focus only on the three problems from above. 3. Solution of the problem 3.1. Main theoretical results Let us start with problem (3). Clearly, if αi > 0 for all i ∈ {1, . . . , n}, then the unique optimum point is x ∗ = (0, . . . , 0). Then, if there exists i0 ∈ {1, . . . , n} such that αi0 ≤ 0, then the problem (3) has no solution as the function is unbounded in the feasible region. This can be summarized in the following proposition. Proposition 1. Consider problem (3). Then: (i) if αi > 0 for all i ∈ {1, . . . , n}, then x ∗ = (0, . . . , 0) is the unique solution and hence the maximum value is 0; (ii) if there exists i0 ∈ {1, . . . , n} such that αi0 ≤ 0, then the problem has no solution and, the function F is unbounded on the feasible set, that is, the supremum of F over the feasible set is ∞. Let us move now towards problem (4). The proof of the following proposition is just a simple exercise. Proposition 2. Consider problem (4). Then: (i) if there exists i0 ∈ {1, . . . , n} such that αi > 0, then the problem has no solution and, the function F is unbounded on the feasible set with supremum equal to ∞; (ii) if αi ≤ 0 for all i ∈ {1, . . . , n}, then the problem has no solution as the feasible set is the empty set. Now, we deal with the most important case, that is, problem (5). First, we observe that in the case when there exists i0 ∈ {1, . . . , n} such that αi0 ≤ 0, then F is unbounded on the feasible set, hence, the supremum of F over the feasible set is ∞. It remains to discuss the case when αi > 0 for all i ∈ {1, . . . , n}. We will solve problem (5) without imposing other restrictions on the coefficients on the constraint, but, it will serve to our purposes to begin with the particular case when α1 ≤ . . . ≤ αn . Suppose that x1∗ , . . . , xn∗ is a solution of (5). Let us note that we must have α1 x1∗ + . . . + αn xn∗ = 1. Indeed, if α1 x1∗ + . . . + αn xn∗ < 1, then let ε > 0 be sufficiently small, so that α1 (x1∗ + ε) + . . . + αn (xn∗ + ε) < 1. It is immediate that F (x1∗ + ε, . . . , xn∗ + ε) > F (x1∗ , . . . , xn∗ ) and this contradicts the fact that x1∗ , . . . , xn∗ is a solution of (5). Next, if Sn denotes the set of all permutations of the set {1, . . . , n} and if σ ∈ Sn , then we necessarily have α1 xσ∗1 + . . . + αn xσ∗n ≥ 1. Indeed, if for some σ we would have α1 xσ∗1 + . . . + αn xσ∗n < 1 then again, let ε > 0 be sufficiently small, so that α1 (xσ∗1 + ε) + . . . + αn (xσ∗n + ε) < 1. But this easily implies F (xσ∗1 + ε, . . . , xσ∗n + ε) > F (x1∗ , . . . , xn∗ ), a contradiction. It readily follows that actually we have min{α1 xσ∗1 + . . . + αn xσ∗n : σ ∈ Sn } = 1. Now, it is well-known that if a1 ≤ . . . ≤ an and b1 ≥ . . . ≥ bn , then a1 b1 + . . . + an bn ≤ a1 bσ1 + . . . + an bσn , for all σ ∈ Sn . Therefore, since α1 ≤ . . . ≤ αn , we get
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min{α1 xσ∗1 + . . . + αn xσ∗n : σ ∈ Sn } = α1 y1∗ + . . . + αn yn∗ , where yi∗ is the i-th largest element of the bag {x1∗ , . . . , xn∗ }. Thus, we get α1 y1∗ + . . . + αn yn∗ = 1. This implies that F (x1∗ , . . . , xn∗ ) ≤ max{F (y1 , . . . , yn ) : α1 y1 + . . . αn yn = 1, y1 ≥ y2 ≥ . . . ≥ yn ≥ 0}. But obviously the converse inequality also holds, and therefore, in order to find a solution of problem (5), it suffices to solve the problem max F (x1 , . . . , xn ) subject to α1 x1 + . . . + αn xn = 1, x1 ≥ x2 . . . ≥ xn ≥ 0.
(6)
Thus, any solution of problem (6) is a solution of problem (5). But all solutions of problem (5) can be derived from the solutions of problem (6). Indeed, suppose that (x1∗ , . . . , xn∗ ) is a solution of (5) such that xi∗ < xj∗ for some i, j ∈ {1, . . . , n}, i < j . Let then (t1∗ , . . . , tn∗ ) ∈ Rn be such that tk∗ = xk∗ , if k ∈ {1, . . . , n} {i, j }, ti∗ = xj∗ and tj∗ = xi∗ . As we know, α1 x1∗ + . . . + αn xn∗ = 1 and α1 t1∗ + . . . + αn tn∗ ≥ 1. But, by simple calculations we get n
αk xk∗
k=1
−
n
αk tk∗ = (xj∗ − xi∗ )(αj − αi )
k=1
and taking into account all the hypotheses, it follows that necessarily αi = αj . Therefore, any solution of (5) is the permutation of a solution (x1∗ , . . . , xn∗ ) of (6), which corresponds to some σ ∈ Sn , with the property that, if for some i, j ∈ {1, . . . , n}, i < j , we have xσ∗i > xσ∗j , then αi = αj . This also means that, if α1 < α2 < . . . < αn , then (x1∗ , . . . , xn∗ ) is a solution of problem (5), if and only if (x1∗ , . . . , xn∗ ) is a solution of problem (6). In addition, any permutation of (x1∗ , . . . , xn∗ ) will not be a solution of problem (5) (the maximum is achieved but the constraint will not be satisfied). Lets us now focus on finding the solution of problem (6). First, note that it makes no difference if we replace the condition α1 x1 + . . . + αn xn = 1, with the condition α1 x1 + . . . + αn xn ≤ 1 (this was already proved at the beginning of this section). Therefore, we may consider problem (6) as a primal linear program which has its dual as min t1 subject to α1 t1 − t2 ≥ w1 , α2 t1 + t2 − t3 ≥ w2 , . . . , αn t1 + tn − tn+1 ≥ wn ,
(7)
where tk ≥ 0, for all k ∈ {1, . . . , n + 1}. We will repeat now the reasoning used in [2]. Summing up the first k inequalities from above, k = 1, n, we get t1 ≥
w1 + . . . + wk + tk+1 , k = 1, n α1 + . . . + αk
This implies that t1 ≥
w1 +...+wk ∗ α1 +...+αk ∗
, where k ∗ ∈ {1, . . . , n} is such that
w1 + . . . + wk ∗ w 1 + . . . + wk = max : k ∈ {1, . . . , n} . α1 + . . . + αk ∗ α1 + . . . + αk This easily implies that (t1∗ , . . . , tn∗ ) is a solution of (7), where w 1 + . . . + wk ∗ , α1 + . . . + αk ∗ w 1 + . . . + wk ∗ ∗ tk+1 = t1 − (α1 + . . . + αk ) , k = 1, n. α1 + . . . + αk t1∗ =
From the duality theorem, if there exists (x1∗ , . . . , xn∗ ) ∈ Rn that satisfies the constraints in (6) and such that F (x1∗ , . . . , xn∗ ) = t1∗ , then (x1∗ , . . . , xn∗ ) is actually a solution of (6) and hence, a solution of (5). Such solution obviously exists as we can take
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1 , α1 + . . . + αk ∗ xk∗∗ +1 = . . . = xn∗ = 0. x1∗ = . . . = xk∗∗ =
Therefore, we just found an optimal solution to problem (5). Now, we can easily deduce the solution for the general case. The following theorem summarizes the results obtained just above. We give the theorem in the general setting of problem (5) as its proof immediately follows from the results obtained just above. Theorem 3. Consider problem (5). Then: (i) if there exists i0 ∈ {1, . . . , n} such that αi0 ≤ 0, then F is unbounded on the feasible set and its supremum over the feasible set is ∞; (ii) if αi > 0, i ∈ {1, . . . , n}, then taking (any) σ ∈ Sn with the property that ασ1 ≤ ασ2 ≤ . . . ≤ ασn , and k ∗ ∈ {1, . . . , n}, such that w 1 + . . . + wk ∗ w1 + . . . + w k = max : k ∈ {1, . . . , n} , ασ1 + . . . + ασk∗ ασ1 + . . . + ασk then (x1∗ , . . . , xn∗ ) is an optimal solution of problem (5), where xσ∗1 = . . . = xσ∗k∗ =
1 , ασ1 + . . . + ασk∗
xσ∗k∗ +1 = . . . = xσ∗n = 0.
In particular, if 0 < α1 ≤ α2 ≤ . . . ≤ αn , and k ∗ ∈ {1, . . . , n} is such that w 1 + . . . + wk ∗ w 1 + . . . + wk = max : k ∈ {1, . . . , n} , α1 + . . . + αk ∗ α1 + . . . + αk then (x1∗ , . . . , xn∗ ) is a solution of (5), where 1 , α1 + . . . + αk ∗ xk∗∗ +1 = . . . = xn∗ = 0. x1∗ = . . . = xk∗∗ =
We easily notice that assertion (ii) of the above theorem generalizes the main result in [2], where the particular case α1 = . . . = αn = 1 was considered. 3.2. Examples Theorem 3 has the important advantage that it provides an exact solution to problem (5), whereas, other methods transform the problem into a linear programming task which may be quite complex in providing the solution. For example, by the method used in [10], considering F a function in n variables, problem (5) is transformed into a linear program with n2 variables and also with increased number of constraints. Clearly, this method is not always effective. To our knowledge, there are no examples in the literature where one tries to find a solution for a problem of the type (5), unless we deal with the particular case when α1 = . . . = αn = 1. This is why, we will present two examples where such requirement does not hold. Although, the main results are totally independent of the requirement w1 + . . . + wn = 1, for conformity with the general theory, in our examples this requirement will be satisfied. Example 1. Consider the problem 1 2 3 1 y1 + y2 + y3 + y4 → max, 10 5 10 5 subject to
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x1 + 3x2 + 2x3 + x4 ≤ 1, x ≥ 0. Considering the permutation 1 2 3 4 σ= , 1 4 3 2 by simple inspection, we get that w1 + . . . + wk max : k ∈ {1, . . . , 4} ασ1 + . . . + ασk is achieved for k ∗ = 2. We also note that
1 ασ1 +ασ2
= 12 . Applying Theorem 3, (ii), it readily follows that
an optimal solution for our problem. In addition, the maximum value is
1 1 2 , 0, 0, 2
is
3 20 .
Example 2. For some arbitrary n ∈ N, consider the problem n n k yk+1 → max, 2n k=0
subject to n+1
k · xk ≤ 1, x ≥ 0.
k=1
By Theorem 3, (ii), this problem can easily be solved for any given n. We just need to find ⎫ ⎧ k ⎪ ⎪ 1 n ⎪ ⎪ ⎪ ⎪ ⎬ ⎨ 2n l l=0 ∗ : k ∈ {0, . . . , n} . k = arg max k+1 ⎪ ⎪ ⎪ ⎪ ⎪ ⎪ l ⎭ ⎩ l=1
For example, if n = 7, then by simple calculations we get k ∗ = 4. Applying the conclusion of Theorem 3, (ii), it readily follows that x ∗ = (x1∗ , x2∗ , . . . , x8∗ ) is an optimal solution of our problem, where x1∗ = . . . = x5∗ =
1 ∗ , x = x7∗ = x8∗ = 0. 15 6
4. Conclusions In this note, we presented a simple method to obtain the exact solution of the constrained OWA aggregation problem with single constraint on arbitrary coefficients. Obviously, the method used in this paper also applies in the case where we have an arbitrary number of constraints and such that the matrix of the constraints is increasing on any of its rows. In such cases, the constrained OWA aggregation problem will become a linear programming problem and hence, it can be solved by using suitable algorithms for such programming tasks. For the general case, where the matrix of coefficients is not necessarily nondecreasing on any of its rows, there is still an open question the finding of a simple algorithm that would give us the exact solution. Up to now, the only algorithm that applies for the general case, is the one given by Yager in [10]. As we already mentioned in this paper, this algorithm is very difficult to implement when we have a large number of variables and constraints. What is sure, is that the method presented in this paper for the case of problems with single constraints cannot be extended inductively to the general case. For simplicity, suppose we have a constrained OWA aggregation problem with two constraints with arbitrary coefficients in both of them. The issue is that in general we cannot use a single permutation (as in the case with single constraint) to reduce the problem to a problem where the sequence of coefficients is nondecreasing in each of the constraints. Then, as in the case of linear or quadratic programs, separating the initial problem in two problems with single constraints will not give us in general the solution of the initial problem as we cannot relate the solution of the problems with single constraints to the solution of the initial problem.
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Acknowledgements The contribution of Lucian Coroianu was possible with the support of a grant of the Romanian National Authority for Scientific Research, CNCS-UEFISCDI, project number PN-II-ID-PCE-2011-3-0861. References [1] B.S. Ahn, Parameterized OWA operator weights: an extreme point approach, Int. J. Approx. Reason. 51 (2010) 820–831. [2] C. Carlsson, R. Fullér, P. Majlender, A note on constrained OWA aggregation, Fuzzy Sets Syst. 139 (2003) 543–546. [3] A. Emrouznejad, M. Marra, Ordered weighted averaging operators 1988–2014: a citation-based literature survey, Int. J. Intell. Syst. 29 (2014) 994–1014. [4] R. Fullér, P. Majlender, On obtaining minimal variability OWA operator weights, Fuzzy Sets Syst. 136 (2003) 203–215. [5] J.M. Merigó, A.M. Gil-Lafuente, L-G. Zhou, H.-Y. Chen, Induced and linguistic generalized aggregation operators and their application in linguistic group decision making, Group Decis. Negot. 21 (2012) 531–549. [6] W. Ogryczak, P. Olender, On MILP models for the OWA optimization, J. Telecommun. Inf. Technol. 2 (2012) 5–12. [7] W. Ogryczak, P. Olender, Ordered median problem with demand distribution weights, Optim. Lett. 10 (2016) 1071–1086. ´ [8] W. Ogryczak, T. Sliwi´ nski, On efficient WOWA optimization for decision support under risk, Int. J. Approx. Reason. 50 (2009) 915–928. ´ [9] W. Ogryczak, T. Sliwi´ nski, On solving linear programs with the ordered weighted averaging objective, Eur. J. Oper. Res. 148 (2003) 80–91. [10] R.R. Yager, Constrained OWA aggregation, Fuzzy Sets Syst. 81 (1996) 89–101. [11] R.R. Yager, On ordered weighted averaging aggregation operators in multicriteria decision making, IEEE Trans. Syst. Man Cybern. 18 (1988) 183–190. [12] R.R. Yager, On the analytic representation of the Leximin ordering and its application to flexible constraint propagation, Eur. J. Oper. Res. 102 (1997) 176–192. [13] R.R. Yager, G. Beliakov, OWA operators in regression problems, IEEE Trans. Fuzzy Syst. 18 (2010) 106–113. [14] L. Zhou, H. Chen, J. Liu, Generalized multiple averaging operators and their applications to group decision making, Group Decis. Negot. 22 (2013) 331–358.