Super- and subadditive constructions of aggregation functions

Information Fusion 34 (2017) 49–54

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Information Fusion journal homepage: www.elsevier.com/locate/inffus

Super- and subadditive constructions of aggregation functions Alexandra Šipošová a,∗, Ladislav Šipeky a, Fabio Rindone b, Salvatore Greco c,d, Radko Mesiar a,e a

Department of Mathematics and Descriptive geometry, Faculty of Civil Engineering, Slovak University of Technology, Radlinského 11, 810 05 Bratislava, Slovakia b Department of Economics and Business, University of Catania, 95029 Catania, Italy c Department of Economics and Business, University of Catania, Corso, Italia, 55, 95129 Catania, Italy d University of Portsmouth, Portsmouth Business School, Centre of Operations Research and Logistics (CORL), Richmond Building, Portland Street, Portsmouth PO1 3DE, United Kingdom e University of Ostrava, IRAFM, 30.dubna 22, Ostrava, Czech Republic

a r t i c l e

i n f o

Article history: Received 15 January 2016 Revised 16 May 2016 Accepted 25 June 2016 Available online 27 June 2016 Keywords: Aggregation function Subadditive transformation Superadditive transformation Decomposition integral

a b s t r a c t Two construction methods for aggregation functions based on a restricted a priori known decomposition set and decomposition weighing function are introduced and studied. The outgoing aggregation functions are either superadditive or subadditive. Several examples, including illustrative figures, show the potential of the introduced construction methods. Our approach generalizes several known constructions and optimization methods, including decomposition and superdecomposition integrals. We present also an economic applications of the introduced concepts.

1. Introduction Aggregation functions play an important role in many domains where an n-dimensional input representation is represented by a single value. For more information and details we recommend monographs [1,5] . Recall that for n ∈ N a monotone function A: [0, 1]n → [0, 1] is called an aggregation function whenever it satisfies two boundary conditions A(0, ..., 0 ) = A(0 ) = 0 and A(1, ..., 1 ) = A(1 ) = 1. Observe that we will not consider the usual convention A(x ) = x for 1-dimensional aggregation functions. Note also that, in general, some other interval I can be considered instead of the unit interval [0, 1]. However, our results related to [0, 1] domain can be easily generalized to the domain I. In several practical situations, the aggregation function A is not known on its full domain [0, 1]n , but only on a subdomain H ⊆ [0, 1]n . More often the boundary condition A(1 ) = 1 is not important, i.e., A and λA gives the same information for the user, independently of λ ∈ ]0, ∞[. This is, e.g., the case when A is considered as a utility function. The above facts have inspired us ∗

Corresponding author. E-mail addresses: [email protected] (A. Šipošová), [email protected] (L. Šipeky), [email protected] (F. Rindone), [email protected] (S. Greco), [email protected] (R. Mesiar). http://dx.doi.org/10.1016/j.inffus.2016.06.006 1566-2535/© 2016 Elsevier B.V. All rights reserved.

© 2016 Elsevier B.V. All rights reserved.

to introduce two construction methods for aggregation functions when only a partial information is known. Our approach was motivated by the ideas from [6,7] dealing with superadditive and subadditive transformations of aggregation functions on [0, ∞[. Recall that a function F: [0, ∞[n → [0, ∞[ is called superadditive (subadditive) whenever, for any x, y ∈ [0, ∞[n , it holds F (x + y ) ≥ F (x ) + F (y ) (F (x + y ) ≤ F (x ) + F (y )). F is additive if and only if it is both superadditive and subadditive, i.e., F (x + y ) = F (x ) + F (y ). If F is defined on some subdomain In ∈ [0, ∞[n ; then the above inequalities (equalities) are considered for x, y ∈ In such that also x + y ∈ In . Our contribution is organized as follows. In Section 2, based on a decomposition set H and weighing function B, we introduce superadditive and subadditive functions B∗ and B∗ , and the related aggregation functions AH,B and AH,B , including two motivating examples and some preliminary results. In Section 3, we exemplify the functions B∗ and B∗ for several decomposition pairs (H, B ) and show the link of our constructions to decomposition and superdecomposition integrals [9,10]. In Section 4 we present an economic application showing how the introduced concepts permits to define and measure the utilization rate of the production capacity of a firm. Finally, some concluding remarks are added.

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A. Šipošová et al. / Information Fusion 34 (2017) 49–54

2. Super- and subadditive constructions of aggregation functions Fix n ∈ N = {1, 2, . . .}. In what follows, an arbitrary subset H of [0, 1]n such that 0 ∈ H will be called a decomposition set. A function B : H → [0, 1], not identically equal to zero, with B(0 ) = 0 and such that B(x) ≤ B(y) whenever x ≤ y for x, y ∈ H, will be called a decomposition weighing function. For any subset S⊆[0, ∞[ of nonnegative real values, we will denote by inf S the greatest lower bound of S, and by sup S the smallest upper bound. If S is unbounded then sup S = ∞ by convention. Moreover, the convention that inf ∅ = ∞ and sup ∅ = 0 will be also considered. Although a decomposition weighing function is defined only on H which, in the extreme case, may consist besides 0 just of a single point, one may introduce its transformation to the entire unit n-cube [0, 1]n by letting



B∗ (x ) = inf

k 

(i )

B (y )

(i ) k

| k ∈ N, (y )i=1 ∈ H ;

i=1

k

k 



y

B∗ (x )= sup

≥x

(1)

i=1

 and

(i )

k 

B ( y (i ) )

| k ∈ N, (y(i) )ki=1 ∈ Hk ;

i=1

k 

 y(i ) ≤x .

i=1

(2) B∗

1]n

Observe that, in general, B∗ and are mappings from [0, → [0, ∞]. The pair (H, B ) will be called subadmissible if B∗ (1) ∈ ]0, ∞[, and superadmissible if B∗ (1) ∈ ]0, ∞[. The set of all subadmissible and superadmissible pairs will be denoted simply by Subn and Supern , respectively. For any subadmissible (superadmissible) pair (H, B ) we may introduce normalized versions of the transformation of B introduced above by letting

AH,B : [0, 1]n → [0, 1]; x → B∗ (x )/B∗ (1 )

(3)

and

AH,B : [0, 1]n → [0, 1]; x → B∗ (x )/B∗ (1 ),

a set of securities H ⊂ n+ is available. In this context B : H → + is a price function. Fix a vector of outcomes x = [x1 , . . . , xn ] ∈ n+ . A super-replication portfolio ([2]) is a set of securities y(i ) ∈ H, i = k (i ) ≥ x. Among all the super-replication 1, . . . , k, such that i=1 y portfolios, one economic operators look for that one with the minimum price which is given by B∗ (x). One can suppose that all outcomes of considered securities can be normalized so that they take value in [0, 1], and one has B: [0, 1]n → [0, 1]. Also the prices can be normalized in the interval [0, 1]. In fact, in this context the maximal attainable vector of outcomes is 1 having B∗ (1) as minimal price of the super replication portfolio. Therefore the prices of portfolio x ∈ [0, 1]n in the considered financial market is given by AH,B = B∗ (x )/B∗ (1 ). Quite expectedly, the introduced functions B∗ and B∗ as well as their normalized versions AH,B and AH,B , are subadditive and superadditive, respectively: Proposition 1. If (H, B ) is a subadmissible pair, then AH,B is a subadditive aggregation function. Analogously, if (H, B ) is a superadmissible pair, then AH,B is a superadditive aggregation function. Proof. Because of subadmissibility and superadmissibility assumptions, the functions AH,B and AH,B are well defined. Monotonicity of both AH,B and AH,B follow from the monotonicity and nonnegativity of decomposition weighing functions. Clearly, AH,B (0 ) = 0 (AH,B (0 ) = 0 ) and AH,B (1 ) = 1 (AH,B (1 ) = 1 ). It remains to prove sub- and superadditivity, and it is clearly sufficient to do this for B∗ and B∗ . The proof that B∗ and B∗ are subadditive and superadditive is given in Propositions 3 and 2 , respectively, of [7] . For arbitrary x, y ∈ [0, 1]n let (x¯ (i ) )ki=1 and (y¯ ( j ) )j=1 be a kk ¯ (i ) ≥ x and tuple and an -tuple of vectors in H for which i=1 x  ( j ) ≥ y. Since, by the choice of these k- and -tuples, the ¯ y j=1 sum of the vectors in the (k +  )-tuple (x¯ (1 ) , . . . , x¯ (k ) , y¯ (1 ) , . . . , y¯ ( ) ) is at least x + y, it follows by the definition of B∗ that

B∗ ( x + y ) ≤

k 

B(x¯ (i ) ) +

i=1

(4)

where in both cases 1 ∈ [0, 1]n denotes the all-one vector. Let us give two economic examples of possible applications of normalized subadmissible and superadmissible normalized transformations AH,B and AH,B . Example 1. Let us suppose that function B is a production function (see e.g., [4,11]) related to a given product so that from the vector of input quantities x = [x1 , . . . , xn ] ∈ n+ the quantity B(x ) ∈  is obtained. More precisely, one can imagine that there is a set of admissible input vectors H ⊆ n+ , so that, in fact, one can imagine the production function as mapping from H to + . One can also suppose that the input quantities are normalized so that x ∈ [0, 1]n and, consequently, H ⊆ [0, 1]n . Also the output can be normalized in the interval [0, 1]. Considering that it could be possible to get a greater output by splitting the production related to a vector of input x = [x1 , . . . , xn ] ∈ [0, 1] in the family of vector of inputs y(i ) ∈   Hk , i = 1, . . . , k with ki=1 y(i ) ≤ x obtaining as output ki=1 B(y(i ) ), by means of the superadditive transformation we get that the maximal output is given by B∗ (1). Therefore, the normalized production function related to basic production function B and to the set of admissible input vectors H is given by AH,B = B∗ (x )/B∗ (1 ). Example 2. Let us consider a financial market (see e.g., [3]) where uncertainty is represented by a set of states S = {s1 , . . . , sn }. States from S are exhaustive and mutually exclusive so that only one state will be true. In this context each vector x = [x1 , . . . , xn ] ∈ n+ can be considered as a feasible security that pays an outcome xi , i = 1, . . . , n, if the state si is revealed true. Suppose that on the market

 

B(y¯ ( j ) ) .

j=1

Now, it is evident that B∗ (x + y ) ≤ B∗ (x ) + B∗ (y ). Similarly, for any x, y ∈ [0, 1]n let (x¯ (i ) )ki=1 and (y¯ ( j ) )j=1 be a  k-tuple and an -tuple of vectors in H for which ki=1 x¯ (i ) ≤ x and  ¯ ( j ) ≤ y. By the choice of these k- and -tuples, the sum of j=1 y

the vectors in the (k +  )-tuple (x¯ (1 ) , . . . , x¯ (k ) , y¯ (1 ) , . . . , y¯ ( ) ) is this time at most x + y, and so from the definition of B∗ we have

B∗ ( x + y ) ≥

k  i=1

B(x¯ (i ) ) +

 

B(y¯ ( j ) ) .

j=1

B∗ ( x

Again, it is evident that + y ) ≥ B∗ (x ) + B∗ (y ). This implies suband superadditivity of B∗ and B∗ and completes the proof.  We illustrate our proposals in the next simple example. Let n = 1 and consider a trivial decomposition system H = {0, 1/t } for some fixed positive integer t. Further, let B be a decomposition weighing function defined by B(0 ) = 0 and B(1/t ) = b > 0. Obviously, B∗ (0 ) = 0. For any x ∈ ]0, 1], letting k = tx (the ceiling of tx) we have x ∈](k − 1 )/t , k/t ], so that B∗ (x ) = kb and hence B∗ (1 ) = tb; it follows that AH,B (x ) = B∗ (x )/B∗ (1 ) = t x/t , which is a subadditive aggregation function. By the same token, letting  = tx (the floor of tx) we have x ∈ [/t, ( + 1 )/t[, so that B∗ (x ) = b, B∗ (1 ) = tb, and AH,B (x ) = B∗ (x )/B∗ (1 ) = t x/t , which is a superadditive aggregation function. Proposition 2. If (H, B ) is a subadmissible pair, then AH,B = B if and only if H = [0, 1]n and B is subadditive, with B(1 ) = 1. Analogously, if (H, B ) is a superadmissible pair, then AH,B = B if and only if H = [0, 1]n and B is superadditive, with B(1 ) = 1.

A. Šipošová et al. / Information Fusion 34 (2017) 49–54

Proof. Since the functions AH,B and AH,B are defined on [0, 1]n and are sub- and superadditive by Proposition 1, the conditions AH,B = B and AH,B = B imply that H = [0, 1]n and B is subadditive and superadditive, respectively, and B(1 ) = 1. Conversely, if H = [0, 1]n and B is subadditive, then B∗ = B by [7] and since 1 = B(1 ) = B∗ (1 ), we have AH,B = B. The proof for the superadditive case is similar and therefore omitted.  Observe that on the space of subadmissible pairs Subn we have a natural partial order Sub defined by

(H1 , B1 ) Sub (H2 , B2 ) if and only if H1 ⊇ H2 and B1 |H2 ≤ B2 . (5) Similarly, on the space of superadmissible pairs Supern we have a natural partial order Super defined by

(H1 , B1 ) Super (H2 , B2 ) if and only if H1 ⊆ H2 and B1 ≤B2 |H1 . (6) This allows us to compare the values of the corresponding aggregation functions as follows. Proposition 3. Let (H1 , B1 ), (H2 , B2 ) ∈ Subn and (H1 , B1 ) Sub (H2 , B2 ). If (B1 )∗ (1 ) = (B2 )∗ (1 ), then AH1 ,B1 ≤ AH2 ,B2 . Analogously, if (H1 , B1 ), (H2 , B2 ) ∈ Supern are such that (H1 , B1 ) Super (H2 , B2 ) and B∗1 (1 ) = B∗2 (1 ), then AH1 ,B1 ≤ AH2 ,B2 . Proof. For a fixed x ∈ [0, 1]n , let (x¯ (i ) )ki=1 be a k-tuple of veck ¯ (i ) ≥ x. Due to (H1 , B1 ) Sub (H2 , B2 ) tors in H2 such that i=1 x k k ( i ) ¯ ) ≤ i=1 B2 (x¯ (i ) ), and thus also (B1 )∗ (x) ≤ it holds i=1 B1 (x (B2 )∗ (x). Since we have assumed that (B1 )∗ (1 ) = (B2 )∗ (1 ), it follows that

AH1 ,B1 (x ) = (B1 )∗ (x )/(B1 )∗ (1 ) ≤ (B2 )∗ (x )/(B2 )∗ (1 ) = AH2 ,B2 (x ) . k

Similarly, if (x¯ (i ) )ki=1 is a k-tuple of vectors in H1 such that

x¯ (i ) ≤ x, the assumption (H1 , B1 ) Super (H2 , B2 ) ensures that  ¯ (i ) ) ≤ ki=1 B2 (x¯ (i ) ), and thus also B∗1 (x ) ≤ B∗2 (x ). Now, B i=1 1 (x ∗ ∗ from B1 (1 ) = B2 (1 ) it follows that ik=1

AH1 ,B1 (x ) = B∗1 (x )/B∗1 (1 ) ≤ B∗1 (x )/B∗1 (1 ) = AH2 ,B2 (x ) , which completes the proof.



Remark. The above result will not be valid in general if, say, in the Supern case, the assumption B∗1 (1 ) = B∗2 (1 ) is dropped. To see this, for n = 1, H1 = {0, 1/2}, H2 = {0, 1/2, 1}, B1 (1/2 ) = 1 and B2 (1/2 ) = 1, B2 (1 ) = 4, so that B∗1 (1 ) = 2 and B∗2 (1 ) = 4. It is then easy to see that, for example, AH1 ,B1 (1/2 ) = 1/2, while AH2 ,B2 (1/2 ) = 1/4, violating the inequality AH1 ,B1 ≤ AH2 ,B2 . We continue with an auxiliary result in dimension 1. Proposition 4. Let H ⊆ [0, 1] be a decomposition set and let B : H → [0, 1] be a decomposition weighing function of dimension 1. Then, (a) (H, B ) ∈ Sub1 if and only if inf{B(x )/x | x ∈ H \ {0}} > 0, and (b) (H, B ) ∈ Super1 if and only if sup{B(x )/x | x ∈ H \ {0}} < ∞. Proof. We have (H, B ) ∈ Sub1 if and only if B∗ (1) > 0, and (H, B ) ∈ Super1 if and only if B∗ (1) < ∞. We begin by proving that inf{B(x )/x | x ∈ H} > 0 implies B∗ (1) > 0. Suppose that inf{B(x )/x | x ∈ H} = b > 0. This means that B(x) ≥ bx for every x ∈ n H. Thus, for every n-tuple x1 , x2 , . . . , xn ∈ H such that i=1 xi ≥ 1 n n we have i=1 B(xi ) ≥ b i=1 xi ≥ b, so that B∗ (1) ≥ b > 0. Similarly, assume that sup{B(x )/x | x ∈ H} = b < ∞. This means that B(x) ≤ bx for every x ∈ H. Thus, for every n-tuple x1 , x2 , . . . , xn ∈ H with n n n ∗ i=1 xi ≤ 1 we have i=1 B (xi ) ≤ b i=1 xi ≤ b, implying that B (1) ≤ b < ∞.

51

Conversely, suppose that inf{B(x )/x | x ∈ H} = 0. If B(z ) = 0 for some z ∈ H, then we clearly have B∗ (1 ) = 0. We therefore may assume that B(x) = 0 for every x ∈ H. Since B is non-decreasing and positive, the equality inf{B(x )/x | x ∈ H} = 0 holds for such a B if and only if there is a sequence x1 , x2 , . . . , xn , . . . of elements of H such that limn→∞ xn = 0 and limn→∞ B(xn )/xn = 0. This means that for every arbitrarily small ε > 0 there exists an nε such that for every n ≥ nε we have B(xn ) ≤ ε xn . Let m = 1/xn ; note that m − 1 < 1/xn ≤ m. Since mxn ≥ 1, for n ≥ nε we have

B∗ (1 ) ≤ mB(xn ) ≤ ε mxn < ε (1 + 1/xn )xn = ε (1 + xn ) ≤ 2ε which means that B∗ (1) < 2ε for every ε > 0 and hence B∗ (1 ) = 0. It remains to prove the converse in the supremum case. Suppose that sup{B(x )/x | x ∈ H} = ∞; since B is non-decreasing, the only way the supremum attains the value of infinity is that there is a sequence x1 , x2 , . . . , xn , . . . of elements of H such that limn→∞ xn = 0 and limn→∞ B(xn )/xn = ∞. This means that for every arbitrarily large k > 0 there exists an nk such that for every n ≥ nk we have xn ≤ 1/2 and B(xn ) ≥ kxn . This time let m = 1/xn ; note that m ≤ 1/xn < m + 1. Since mxn ≤ 1, for n ≥ nk we have

B∗ (1 ) ≥ mB(xn ) ≥ kmxn > k(1/xn − 1 )xn = k(1 − xn ) ≥ k/2 which means that B∗ (1) > k/2 for arbitrarily large k > 0 and so B∗ (1 ) = 0. The proof is complete.  Based on Proposition 4, we prove the following general result for any dimension. For a vector x = (x1 , . . . , xn ) ∈ [0, 1]n we define max(x ) = max{xi | 1 ≤ i ≤ n} and let (x )i = xi for 1 ≤ i ≤ n. Proposition 5. Let H ⊂ [0, 1]n be a decomposition set and let B : H → [0, 1] be a decomposition weighing function of dimension n ≥ 1. Then, (a) (H, B ) ∈ Subn if and only if for each  i ∈ {1, ..., n} there is an x ∈ H such that (x)i > 0 and inf B(x(x)) | x ∈ H, (x )i > 0 > 0 for some i ∈ {1, . . . , n};

(b) (H, B ) ∈ Supern if and only if sup

i



B (x ) max(x )

 |x ∈ H \ {0} < ∞.

Proof. For any i ∈ {1, 2, . . . , n} let Hi be the set of all z ∈ ]0, 1] such that (x )i = z for some x ∈ H. Also, for any i ∈ {1, 2, . . . , n} such that Hi = ∅ and for every z ∈ Hi we let Bi (z ) = inf{B(x ) | x ∈ H, (x )i = z}. Now, for (a), we have (H, B ) ∈ Subn if and only if B∗ (1) > 0, which is equivalent to the existence of an i ∈ {1, 2, . . . , n} such that (Bi )∗ (1) > 0. Since Bi is a one-dimensional weighing function for Hi , by part (a) of Proposition 4 the condition (Bi )∗ (1) > 0 is equivalent to inf{Bi (x )/x | x ∈ Hi } > 0. By definition of Bi the last condition is equivalent to inf{B(x )/(x )i | x ∈ H, (x )i > 0} > 0, which proves (a). For (b), let H0 be the set of all z ∈ ]0, 1] for which there exists an x ∈ H such that max(x ) = z, and let B˜(z ) = sup{B(x ) | max(x ) = z}. We now have (H, B ) ∈ Supern if and only if B∗ (1) < ∞, which, by definition of B˜, happens if and only if (B˜ )∗ (1 ) < ∞. Since B˜ is a one-dimensional weighing function for H0 , by part (b) of Proposition 4 the condition (B˜ )∗ (1 ) < ∞ is equivalent to sup{B˜(x )/x | x ∈ H0 } < ∞. Invoking the definition of B˜ again, the last condition is equivalent to sup{B(x )/ max(x ) |x ∈ H} < ∞, proving (b).  3. Examples In this section we will present examples of functions B∗ and B∗ for specific decomposition sets and related decomposition weighting functions, as well as their links to some well known optimization and construction methods. Example 1. Let H ={(0, 0), (0.1, 0.1)} and B(y )= y1 , where y1 is the first coordinate of y. Values of B∗ and B∗ are depicted in Figs. 1

52

A. Šipošová et al. / Information Fusion 34 (2017) 49–54

Fig. 1. B∗ from Example 1.

Fig. 3. B∗ from Example 2.

Fig. 2. B∗ from Example 1.

k (i ) appearing in expression and2, respectively. Observe that i=1 y (1) and (2) always have the form k(0.1, 0.1) for k ∈ {0, 1, ..., 10}, because the only vector that can be used for summation is (0.1, 0.1). This explains the shape of the graphs in Figs. 1 and 2. Example 2. Let H = { ( 0, 0 ), ( 0.8, 0.3 ), ( 0.2, 0.7 )} and let B(0.8, 0.3 ) = 0.8, B(0.2, 0.7 ) = 0.6. It can be shown that in this case we have B∗ (1, 1 ) = B∗ (1, 1 ) = 1.4. The corresponding values of B∗ and B∗ are depicted in Figs. 3 and4, respectively.  Example 3. Let H = {(0, 0 ), (0.2, 0.3 ), (0.5, 0.7)} and let B = be ∗ the product. A schematic description of B is in Fig. 5.

Fig. 4. B∗ from Example 2.

Example 4. In this example we will use a segment for H by letting H = {(x, y )| x ∈ [0.1, 1], y = 0.1 − x} ∪ {(0, 0 )}. The weighing function is defined as follows B : H \ {(0, 0 )} → 0.05. The function B∗ is depicted in Fig. 6(2D) and in Fig. 7(3D). For a finite universe X = {1, . . . , n}, any non-empty set C ⊆ 2X \ {∅} is called a collection, and any set G of collections is called a decomposition system. A set function μ: 2X → [0, 1] which is monotone and satisfies the boundary conditions μ(∅ ) = 0 and μ(X ) = 1 is called a capacity. Lehrer in [9] introduced the decomposition integral IG,m : [0, 1]n → [0, ∞] as follows:



IG,m (x ) = sup

k  i=1

ai μ ( Ei )| ( )

Ei ki=1

∈ G, a1 , . . . , ak ≥ 0,

k 



ai 1 Ei ≤ x

i=1

(7)

Fig. 5. B∗ from Example 3.

A. Šipošová et al. / Information Fusion 34 (2017) 49–54

53

Table 1 Production procedures. Procedure

α

β

γ

P1 P2 P3

3 1 2

1 3 2

3 2 2

Proposition 7. Let G = {C j | j ∈ J} be a decomposition system and μ a capacity on X = {1, . . . , n}. Then IG,μ = min{(BC j ,μ )∗ | j ∈ J}, i.e., the decomposition integral is just the minimal value of newly introduced functionals (BC j ,μ )∗ . 4. An economic application Firm ABC produces the article γ using factor α and factor β . There are three possible production procedures shown in Table 1. Observe that the production function relating the input quantity of factors α and β with the output quantity of product γ can be interpreted as an aggregation function B, and the three production procedures P1, P2 and P3 can be interpreted as the partial knowledge of the aggregation function B on the subdomain H = {(3, 1 ), (1, 3 ), (2, 2 )}. This means B(3, 1 ) = 3, B(1, 3 ) = 2, B(2, 2 ) = 2. The manager of firm ABC wants to maximize the production of article γ under the constraint of the available quantities of factors α and β , denoted by xα and xβ , respectively. Therefore the quantity of article γ produced is given by

Fig. 6. Contour lines of B∗ from Example 4.



B (xα , xβ ) = max ∗

k  i=1

B(yα(i ) , yβ(i ) )

k  i=1

| (yα(i) , yβ(i) )ki=1 ∈ Hk ; 

(yα(i) , yβ(i) )

≤ ( xα , xβ ) ,

which can be rewritten as

B∗ (xα , xβ ) = max{μ1 B(3, 1 ) + μ2 B(1, 3 ) + μ3 B(2, 2 )|μ1 (3, 1 )

+ μ2 ( 1 , 3 ) + μ3 ( 2 , 2 ) ≤ ( x α , x β ) , μ1 , μ2 , μ3 ∈ N } ,

∗

Fig. 7. 3D plot of B from Example 4.

Note that Lehrer’s approach covers several types of integrals, including the Choquet integral [8] (when G consists from all maximal chains in 2X ࢨ{∅}), the Shilkret integral [12] (when G consists from all singleton collections), and PAN-integral [14,15] (when G consists of all partitions of X). For a collection C = (Ei )ki=1 , denote HC = {ai 1Ei | ai ∈ [0, 1], Ei ∈ C }, and define BC ,μ : HC → [0, 1] by BC ,μ (ai 1Ei ) = ai μ(Ei ). Evidently, HC is a decomposition set and BC ,μ a weighing function, and thus (BC ,μ )∗ given by (2) is well defined. Now, the next results are immediate, compare also [6,7]. Proposition 6. Let G = {C j | j ∈ J} be a decomposition system and μ a capacity on X = {1, . . . , n}. Then IG,μ = max{(BC j ,μ )∗ | j ∈ J}, i.e., the decomposition integral is just the maximal value of newly introduced functionals (BC j ,μ )∗ . Similarly, when considering the superdecomposition integral IG,μ introduced in [10] by

 I

G,m

(x ) = in f

k 

ai μ ( Ei )| ( )

i=1

Ei ki=1

∈ G, a1 , . . . , ak ≥ 0,

k 

 ai 1 Ei ≥ x ,

i=1

(8) the next result is valid.

where μ1 , μ2 and μ3 are the number of events that production processes P1, P2 and P3 are activated, respectively. Each day the firm can process until 10 units of factor α and 10 units of factor β , which means that the maximal quantity of article γ that can be produced by firm ABC is B∗ (10, 10 ) = 12, obtained with μ1 = 2, μ2 = 2 and μ3 = 1. Each day the available quantity of factors α and β can be different so that different quantities of the article γ may be produced daily. Table 2 shows the unities of product γ produced with given quantities xα and xβ of factors α and β . The manager of firm ABC wants to determine each day the utilization rate of the production capacity defined as the ratio between the daily production and the maximal possible production, Table 2 Produced quantity of the article γ . x α \x β

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

0 0 0 0 0 0 0 0 0 0 0

0 0 0 3 3 3 3 3 3 3 3

0 0 2 3 3 3 6 6 6 6 6

0 2 2 3 3 5 6 6 6 9 9

0 2 2 3 5 5 6 6 8 9 9

0 2 2 4 5 5 6 8 8 9 9

0 2 4 4 5 5 7 8 8 9 11

0 2 4 4 5 7 7 8 8 10 11

0 2 4 4 6 7 7 8 10 10 11

0 2 4 4 6 7 7 9 10 10 11

0 2 4 4 6 7 9 9 10 10 12

54

A. Šipošová et al. / Information Fusion 34 (2017) 49–54 Table 3 Utilization rate of the production capacity. x α \x β

0

1

2

3

4

5

6

7

8

9

10

0 1 2 3 4 5 6 7 8 9 10

0 0 0 0 0 0 0 0 0 0 0

0 0 0 0.25 0.25 0.25 0.25 0.25 0.25 0.25 0.25

0 0 0.167 0.25 0.25 0.25 0.5 0.5 0.5 0.5 0.5

0 0.167 0.167 0.25 0.25 0.417 0.5 0.5 0.5 0.750 0.750

0 0.167 0.167 0.25 0.417 0.417 0.5 0.5 0.667 0.750 0.750

0 0.167 0.167 0.333 0.417 0.417 0.5 0.75 0.667 0.750 0.750

0 0.167 0.333 0.333 0.417 0.417 0.583 0.75 0.667 0.750 0.917

0 0.167 0.333 0.333 0.417 0.583 0.583 0.75 0.667 0.833 0.917

0 0.167 0.333 0.333 0.5 0.583 0.583 0.75 0.833 0.833 0.917

0 0.167 0.333 0.333 0.5 0.583 0.583 0.75 0.833 0.833 0.917

0 0.167 0.333 0.333 0.5 0.583 0.75 0.75 0.833 0.833 1

that is B∗ (xα , xβ )/B∗ (10, 10 ) = AH,B (xα , xβ ). Table 3 shows the utilization rate of the production capacity depending on the available quantities of factors α and β . 5. Concluding remarks We have introduced two methods of constructing aggregation functions on [0, 1] in situation when only a partial information is available. We have exemplified the superadditive functions B∗ and the subadditive functions B∗ , having in mind that the related aggregation functions AH,B and AH,B are easily obtained by normalization of B∗ and B∗ , respectively. Besides this, we have shown the fact, that our approach can be seen as a generalization of decomposition and superdecomposition integrals [9,10]. In a similar way, one can show the fact, that the linear programming optimization problems can be formulated in the language of B∗ functionals (when we maximize the profit) and B∗ functionals (when we minimize the costs). Note that we can introduce weighing function B and the decomposition set H related to [0, ∞[ interval instead of [0, 1] interval, and then no normalization is needed (compare also the Examples 1 and 2). More details about the links of the linear programming and B∗ , B∗ functionals can be found in [13]. Moreover, we have also presented an economic application of the introduced concepts showing the usefulness of B∗ in the definition of the utilization rate of the production capacity of a firm. We expect applications of our approach in economics, social sciences, etc., and especially in multicriteria decision support. Acknowledgement The first, the second and the fifth author acknowledge support from the project VEGA 1/0420/15 and APVV 14-0013. The fifth

author was supported also by the NPU II project LQ 1602. Research of the fourth author was supported by the “Programma Operativo Nazionale” Ricerca Competitività “2007–2013” within the project “PON04a2 E SINERGREEN-RES-NOVAE” and FIR Università di Catania, “New developments in Multiple Criteria Decision Aiding (MCDA) and their application to territorial competitiveness”. References [1] G. Beliakov, A. Pradera, T. Calvo, Aggregation Functions: A Guide for Practitioners, Springer, Berlin, 2007. [2] B. Bensaid, J.P. Lesne, J. Scheinkman, Derivative asset pricing with transaction costs1, Math. Finance 2 (2) (1992) 63–86. [3] D. Darrell, Dynamic Asset Pricing Theory, Princeton University Press, 2010. [4] M. Fuss, D. McFadden, Production Economics: A Dual Approach to Theory and Applications, Applications of the Theory of Production, Vol. 2, Elsevier, 2014. [5] M. Grabisch, J.-L. Marichal, R. Mesiar, E. Pap, Aggregation Functions (Encyklopedia of Mathematics and its Applications), Cambridge University Press, 2009. [6] S. Greco, R. Mesiar, F. Rindone, L. Šipeky, Decomposition approaches to integration without a measure, Fuzzy Set. Syst. 287 (2016) (2016) 37–47. [7] S. Greco, R. Mesiar, F. Rindone, L. Šipeky, Superadditive and subadditive transformations of integrals and aggregation functions, Fuzzy Set. Syst. 291 (2016b) 40–53. [8] G. Choquet, Theory of capacities, Ann. Inst. Fourier 5 (1953) 131–295. [9] E. Lehrer, A new integral for capacities, Econ. Theory 39 (1) (2009) 157–176. Springer [10] R. Mesiar, J. Li, E. Pap, Superdecomposition integrals, Fuzzy Set. Syst. 259 (2015) 3–11. [11] R.W. Shepherd, Theory of Cost and Production Functions, Princeton University Press, 2015. [12] N. Shilkret, Maxitive measure and integration, Indagat. Math. (Proc.) 74 (1971) 109–116. Elsevier. [13] L. Šipeky, A. Šiposová, Constraint super- and sub-additive transformations. 3 p., in: Proc. Conf. CITCEP 2015, Krakow, 2015. To appear [14] Q. Yang, The pan-integral on the fuzzy measure space, Fuzzy Math. 3 (1985) 107–114. [15] Z. Wang, G. J.Klir, Generalized Measure Theory, Springer, New York, 2009.