Characterization of some aggregation functions stable for positive linear transformations.

*(English)*Zbl 0952.91020The authors consider the characterization of some classes of aggregation functions and aggregators (the family of aggregation function) often used in operation research, especially in multicriteria decision making problems. The examples of aggregation functions are: arithmetic mean, weighted arithmetic mean, minimum, maximum, etc.

The choice of proper aggregation function strongly depends on application. In order to obtain reasonable results the function must often fulfill some special conditions. The authors adopt an axiomatic approach in which they divide the properties of aggregation function into three categories: natural properties, stability properties and algebraic properties. In each group they provide several definitions describing some general features of aggregation function. They explore more deeply the features of stability for positive linear transformation and increasing monotonicity. The aggregation function \(M^{(m)}\) fulfills these conditions if: \[ M^{(m)} (rx_1+ t,\dots, rx_m+ t)= rM^{(m)} (x_1,\dots, x_m)+ t, \quad\text{for }r>0,\;t\in \mathbb{R}, \] \[ x_i'< x_i' \Rightarrow M^{(m)} (x_1,\dots, x_i,\dots, x_m)\leq M^{(m)} (x_1,\dots, x_i,\dots, x_m), \quad\text{for all }i= 1,\dots, n. \] The main advantage of the paper are several theorems which describe the family of all aggregation functions fulfilling three specific properties. The first two are increasing monotonicity and stability for positive linear transformation. The third property is one of the well known algebraic properties such as associativity, decomposability and bisymmetry. For each combinations of these properties the authors find all aggregators which fulfill them.

The paper is a theoretical one but the results obtained have valuable practical implications. The results can help the decision maker in choosing the proper aggregation function on the basis of some expected properties.

The choice of proper aggregation function strongly depends on application. In order to obtain reasonable results the function must often fulfill some special conditions. The authors adopt an axiomatic approach in which they divide the properties of aggregation function into three categories: natural properties, stability properties and algebraic properties. In each group they provide several definitions describing some general features of aggregation function. They explore more deeply the features of stability for positive linear transformation and increasing monotonicity. The aggregation function \(M^{(m)}\) fulfills these conditions if: \[ M^{(m)} (rx_1+ t,\dots, rx_m+ t)= rM^{(m)} (x_1,\dots, x_m)+ t, \quad\text{for }r>0,\;t\in \mathbb{R}, \] \[ x_i'< x_i' \Rightarrow M^{(m)} (x_1,\dots, x_i,\dots, x_m)\leq M^{(m)} (x_1,\dots, x_i,\dots, x_m), \quad\text{for all }i= 1,\dots, n. \] The main advantage of the paper are several theorems which describe the family of all aggregation functions fulfilling three specific properties. The first two are increasing monotonicity and stability for positive linear transformation. The third property is one of the well known algebraic properties such as associativity, decomposability and bisymmetry. For each combinations of these properties the authors find all aggregators which fulfill them.

The paper is a theoretical one but the results obtained have valuable practical implications. The results can help the decision maker in choosing the proper aggregation function on the basis of some expected properties.

Reviewer: Stefan Chanas (Wrocław)

##### MSC:

91B06 | Decision theory |

90C70 | Fuzzy and other nonstochastic uncertainty mathematical programming |

90C29 | Multi-objective and goal programming |

90B50 | Management decision making, including multiple objectives |

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\textit{J.-L. Marichal} et al., Fuzzy Sets Syst. 102, No. 2, 293--314 (1999; Zbl 0952.91020)

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