Scalable preference disaggregation: A multiple criteria sorting approach based on the MapReduce framework Jiapeng Liu November 22, 2018 School of Management Xi’an Jiaotong University P.R. China 1
Introduction • Multiple criteria sorting (MCS) is the practice of assigning a set of alternatives evaluated on multiple criteria to predefined and preference-ordered categories. 2
Introduction • In this study, we assume that the preference information is composed of a set of assignment examples on reference alternatives. 3
Introduction • In this study, we assume that the preference information is composed of a set of assignment examples on reference alternatives. • Many approaches have been proposed to deal with MCS problems based on this indirect preference information. • methods motivated by value functions, such as the UTADIS method and its variants, and the MHDIS method • methods based on outranking relations, such as the ELECTRI Tri-B methods, ELECTRI Tri-C methods, ELECTRI-based methods, and PROMETHEE-based methods • rule induction-oriented procedures, such as the DRSA method and its extensions • techniques incorporating the weighted Euclidean distance 3
Introduction • In the era of big data, developments in information technology have resulted in an explosive growth in data gathered from various fields. 4
Introduction • In the era of big data, developments in information technology have resulted in an explosive growth in data gathered from various fields. • This needs decision analysis methods to scale up well with the requirements of big data. 4
Introduction • In the era of big data, developments in information technology have resulted in an explosive growth in data gathered from various fields. • This needs decision analysis methods to scale up well with the requirements of big data. • However, it is challenging for existing MCS methods to deal with problems that contain a large set of alternatives and massive preference information. 4
Introduction • In the era of big data, developments in information technology have resulted in an explosive growth in data gathered from various fields. • This needs decision analysis methods to scale up well with the requirements of big data. • However, it is challenging for existing MCS methods to deal with problems that contain a large set of alternatives and massive preference information. • Traditional decision problems usually involve several dozens of alternatives. 4
Introduction • In the era of big data, developments in information technology have resulted in an explosive growth in data gathered from various fields. • This needs decision analysis methods to scale up well with the requirements of big data. • However, it is challenging for existing MCS methods to deal with problems that contain a large set of alternatives and massive preference information. • Traditional decision problems usually involve several dozens of alternatives. • These methods require the data to fit into the main memory, in which LP/IP solvers search for the optimal solution. 4
Introduction • In the era of big data, developments in information technology have resulted in an explosive growth in data gathered from various fields. • This needs decision analysis methods to scale up well with the requirements of big data. • However, it is challenging for existing MCS methods to deal with problems that contain a large set of alternatives and massive preference information. • Traditional decision problems usually involve several dozens of alternatives. • These methods require the data to fit into the main memory, in which LP/IP solvers search for the optimal solution. • This exceeds the processing capabilities of existing MCS methods in terms of the memory consumption and/or the computational time when dealing with huge amounts of data. 4
Introduction • Thus, new techniques must be introduced to redesign existing MCS methods so that they can scale up well with new storage and time requirements. 5
Introduction • Thus, new techniques must be introduced to redesign existing MCS methods so that they can scale up well with new storage and time requirements. • In this study, we propose a new approach based on the MapReduce framework, in order to address the MCS problem with a large set of alternatives and massive preference information. 5
Introduction • Thus, new techniques must be introduced to redesign existing MCS methods so that they can scale up well with new storage and time requirements. • In this study, we propose a new approach based on the MapReduce framework, in order to address the MCS problem with a large set of alternatives and massive preference information. • MapReduce is a popular parallel computing paradigm developed by Google Inc., which is designed to process large-scale data sets. 5
The proposed approach • The aim of this study is to classify a finite set of m alternatives A = { a 1 , a 2 , ..., a m } into p categories C = { C 1 , C 2 , ..., C p } , such that C h +1 is preferred to C h (denoted by C h +1 ≻ C h ), h = 1 , ..., p − 1. 6
The proposed approach • The aim of this study is to classify a finite set of m alternatives A = { a 1 , a 2 , ..., a m } into p categories C = { C 1 , C 2 , ..., C p } , such that C h +1 is preferred to C h (denoted by C h +1 ≻ C h ), h = 1 , ..., p − 1. • Such a classification decision is built on some preference information provided by the DM, which involves a set of assignment examples concerning a finite set of reference alternatives A R = { a ∗ , b ∗ , ... } . 6
The proposed approach • The aim of this study is to classify a finite set of m alternatives A = { a 1 , a 2 , ..., a m } into p categories C = { C 1 , C 2 , ..., C p } , such that C h +1 is preferred to C h (denoted by C h +1 ≻ C h ), h = 1 , ..., p − 1. • Such a classification decision is built on some preference information provided by the DM, which involves a set of assignment examples concerning a finite set of reference alternatives A R = { a ∗ , b ∗ , ... } . • An assignment example specifies the assignment of a reference alternative a ∗ ∈ A R to a category C ( a ∗ ) ∈ C . 6
The proposed approach • The aim of this study is to classify a finite set of m alternatives A = { a 1 , a 2 , ..., a m } into p categories C = { C 1 , C 2 , ..., C p } , such that C h +1 is preferred to C h (denoted by C h +1 ≻ C h ), h = 1 , ..., p − 1. • Such a classification decision is built on some preference information provided by the DM, which involves a set of assignment examples concerning a finite set of reference alternatives A R = { a ∗ , b ∗ , ... } . • An assignment example specifies the assignment of a reference alternative a ∗ ∈ A R to a category C ( a ∗ ) ∈ C . • All the alternatives a ∈ A ∪ A R are evaluated in terms of n criteria g 1 , g 2 , ..., g n . 6
The proposed approach • The aim of this study is to classify a finite set of m alternatives A = { a 1 , a 2 , ..., a m } into p categories C = { C 1 , C 2 , ..., C p } , such that C h +1 is preferred to C h (denoted by C h +1 ≻ C h ), h = 1 , ..., p − 1. • Such a classification decision is built on some preference information provided by the DM, which involves a set of assignment examples concerning a finite set of reference alternatives A R = { a ∗ , b ∗ , ... } . • An assignment example specifies the assignment of a reference alternative a ∗ ∈ A R to a category C ( a ∗ ) ∈ C . • All the alternatives a ∈ A ∪ A R are evaluated in terms of n criteria g 1 , g 2 , ..., g n . • The performance of a ∈ A ∪ A R on g j , j ∈ G = { 1 , ..., n } , is denoted by g j ( a ). 6
The proposed approach To perform the assignment of alternative a ∈ A ∪ A R , we shall use as the preference model an additive value function U of the following form: n � a ∈ A ∪ A R , U ( a ) = u j ( g j ( a )) , j =1 where U ( a ) is the comprehensive value of a , and u j ( g j ( a )), j = 1 , ..., n , are marginal value functions for each criterion. 7
The proposed approach Marginal value � ( ) u × j 3 u ( x ) j j 2 u ( x ) Actual marginal j j value function 1 u ( x ) j j Estimated piecewise-linear marginal value function � 0 g × Evaluation � ( ) a = x 0 x 1 x 2 b = 3 j x j j j j j j • In this study, a piecewise-linear function u j ( · ) is used to estimate the actual value function of criterion g j , j = 1 , ..., n . 8
The proposed approach • Defining the characteristic vector V ( a ) ∈ R γ of alternative a by T v 1 , ..., v 1 j ( a ) , ..., v γ j , ..., v 1 V ( a ) = 1 ( a ) , ..., v γ 1 1 ( a ) j ( a ) n ( a ) , ..., v γ n n ( a ) � �� � � �� � � �� � criterion g 1 criterion g n criterion g j and denote T ∆ u 1 , ..., ∆ u 1 j , ..., ∆ u γ j , ..., ∆ u 1 u = 1 , ..., ∆ u γ 1 n , ..., ∆ u γ n , 1 j n � �� � � �� � � �� � criterion g 1 criterion g n criterion g j we can compute comprehensive value U ( a ) as follows: U ( a ) = u T V ( a ) . 9
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