An integrated approach to identify criteria interactions based on association rule and capacity in MCDA

Abstract

Criteria interaction is an inevitable factor to be considered, especially in complex multiple criteria decision analysis (MCDA) problems. With the help of capacity, criteria interactions could be modeled in the framework of MCDA. However, we cannot accurately understand the polarity and degree of criteria interactions for an MCDA problem in advance. To overcome this problem, we propose an unsupervised approach to identify and model criteria interactions based on association rules and 2-additive capacity. First, association rules between criteria are obtained to determine the polarity of criteria interactions. Then, through solving an optimization model, the degree of criteria interactions is modeled by 2-additive capacity. With the merits of identifying understandable interactions between criteria, decision makers could deal with complex MCDA problems in consideration with criteria interactions. The applicability and advantages of this approach are demonstrated by an urban sustainability assessment. The empirical study shows that positive correlations and negative interactions can be detected among three pillars of sustainability. City manager should focus on the improvement of the indicators that are positively interacted, because these indicators account for a large weight in aggregating overall sustainability performance and such interaction is not in line with sustainable development.

Appendix

(1) The three types of items of a criterion constitute a complete event group.

$$\begin{aligned} & a_{i}^{ + } + a_{i}^{ - } + a_{i}^{*} = {1} \\ & b_{j}^{ + } + b_{j}^{ - } + b_{j}^{*} = {1} \\ \end{aligned}$$

Proof

$$a_{i}^{ + } + a_{i}^{ - } + a_{i}^{*} = \frac{{|\{ R_{k} :c_{i}^{ + } \in R_{k} \} | + |\{ R_{k} :c_{i}^{ - } \in R_{k} \} | + |\{ R_{k} :c_{i}^{*} \in R_{k} \} |}}{|T|} = 1$$

$$b_{j}^{ + } + b_{j}^{ - } + b_{j}^{*} = \frac{{|\{ R_{k} :c_{j}^{ + } \in R_{k} \} | + |\{ R_{k} :c_{j}^{ - } \in R_{k} \} | + |\{ R_{k} :c_{j}^{*} \in R_{k} \} |}}{|T|} = 1$$

(2) The supports of none association rules with c_i (or c_j) for antecedent or consequent constitute a complete event group.

$$s_{1} + s_{2} + s_{3} + s_{4} + s_{5} + s_{6} + s_{7} + s_{8} + s_{9} = 1$$

Proof

We have known that a_i⁺+ a_i^-+ a_i^*=1

$$\begin{aligned} a_{i}^{ + } + a_{i}^{ - } + a_{i}^{*} & = \frac{{|\{ R_{k} :c_{i}^{ + } \in R_{k} ,c_{i}^{ - } \in R_{k} ,c_{i}^{*} \in R_{k} \} |}}{|T|} \\ & = \frac{{|\{ R_{k} :\{ c_{i}^{ + } ,c_{j}^{ + } \} \subseteq R_{k} ,\{ c_{i}^{ + } ,c_{j}^{ - } \} \subseteq R_{k} ,\{ c_{i}^{ + } ,c_{j}^{*} \} \subseteq R_{k} |}}{|T|} \\ & \quad + \frac{{|\{ R_{k} :\{ c_{i}^{ - } ,c_{j}^{ + } \} \subseteq R_{k} ,\{ c_{i}^{ - } ,c_{j}^{ - } \} \subseteq R_{k} ,\{ c_{i}^{ - } ,c_{j}^{*} \} \subseteq R_{k} |}}{|T|} \\ & \quad + \frac{{|\{ R_{k} :\{ c_{i}^{*} ,c_{j}^{ + } \} \subseteq R_{k} ,\{ c_{i}^{*} ,c_{j}^{ - } \} \subseteq R_{k} ,\{ c_{i}^{*} ,c_{j}^{*} \} \subseteq R_{k} |}}{|T|} \\ & { = }s_{1} + s_{2} + s_{3} + ... + s_{9} \\ \end{aligned}$$

(3) The antecedent support and consequent support can be decomposed into the support of the association rule.

$$\begin{aligned} a_{i}^{ + } \, = & \,s_{{1}} \, + \,s_{{2}} \, + \,s_{{3}} ;b_{j} .^{ + } \, = \,s_{{1}} \, + \,s_{{4}} \, + \,s_{{7}} \\ a_{i}^{ - } \, = & \,s_{{4}} \, + \,s_{{5}} \, + \,s_{{6}} ;b_{j} .^{ - } \, = \,s_{{2}} \, + \,s_{{5}} \, + \,s_{{8}} \\ a_{i}^{*} \, = & \,s_{{7}} \, + \,s_{{8}} \, + \,s_{{9}} ;b_{j} .^{*} \, = \,s_{{3}} \, + \,s_{{6}} \, + \,s_{{9}} \\ \end{aligned}$$

Proof

Take a_i⁺=s₁+ s₂+ s₃ as an example.

$$a_{i}^{ + } = \frac{{|\{ R_{k} :c_{j}^{ + } \in R_{k} \} |}}{|T|}$$

$$|\{ R_{k} :c_{i}^{ + } \in R_{k} \} | = |\{ R_{k} :\{ c_{i}^{ + } ,c_{j}^{ + } \} \subseteq R_{k} | + |\{ R_{k} :\{ c_{i}^{ + } ,c_{j}^{ - } \} \subseteq R_{k} | + |\{ R_{k} :\{ c_{i}^{ + } ,c_{j}^{*} \} \subseteq R_{k} |$$

Thus,

$$\begin{gathered} a_{i}^{ + } = \frac{{|\{ R_{k} :c_{i}^{ + } \in R_{k} \} | + |\{ R_{k} :\{ c_{i}^{ + } ,c_{j}^{ + } \} \subseteq R_{k} | + |\{ R_{k} :\{ c_{i}^{ + } ,c_{j}^{ - } \} \subseteq R_{k} | + |\{ R_{k} :\{ c_{i}^{ + } ,c_{j}^{*} \} \subseteq R_{k} |}}{|T} \hfill \\ \, = support(c_{i}^{ + } \Rightarrow c_{j}^{ + } ) + support(c_{i}^{ + } \Rightarrow c_{j}^{ - } ) + support(c_{i}^{ + } \Rightarrow c_{j}^{*} ) \hfill \\ { = }s_{1} + s_{2} + s_{3} \hfill \\ \end{gathered}$$

(4) The support of association rules is less than or equal to antecedent support and consequent support.

$$\begin{aligned} & s_{{1}} \, \le \,a_{i}^{ + } ,s_{{2}} \, \le \,a_{i}^{ + } ,s_{{3}} \, \le \,a_{i}^{ + } ;s_{{1}} \, \le \,b_{j}^{ + } ,s_{{2}} \, \le \,b_{j}^{ - } ,s_{{3}} \, \le \,b_{j}^{*} \\ & s_{{4}} \, \le \,a_{i}^{ - } ,s_{{5}} \, \le \,a_{i}^{ - } ,s_{{6}} \, \le \,a_{i}^{ - } ;s_{{4}} \, \le \,b_{j}^{ + } ,s_{{5}} \, \le \,b_{j}^{ - } ,s_{{6}} \, \le \,b_{j}^{*} \\ & s_{{7}} \, \le \,a_{i}^{*} ,s_{{8}} \, \le \,a_{i}^{*} ,s_{{9}} \, \le \,a_{i}^{*} ;s_{{7}} \, \le \,b_{j}^{ + } ,s_{{8}} \, \le \,b_{j}^{ - } ,s_{{9}} \, \le \,b_{j}^{*} \\ \end{aligned}$$

Proof

Take s₁ ≤ a₁ as an example.

We have known that $|\{ R_{k} :\{ c_{i}^{ + } ,c_{j}^{ + } \} \subseteq R_{k}\} | \le |\{ R_{k} :c_{i}^{ + } \in R_{k} \} |$ Thus,

$$s_{1} = \frac{{|\{ R_{k} :\{ c_{i}^{ + } ,c_{j}^{ + } \} \subseteq R_{k}\} |}}{|T|} \le \frac{{|\{ R_{k} :c_{i}^{ + } \in R_{k} \} |}}{|T|} = a_{i}^{ + }$$

(5) The probability that the types of items related to c_i or c_j appearing in the transaction set lies in the interval [0, 1].

$$\begin{aligned} & 0\, \le \,a_{i}^{ + } \, + \,b_{j}^{ + } - s_{{1}} \, \le \,{1},0\, \le \,a_{i}^{ - } \, + \,b_{j}^{ + } - s_{{4}} \, \le \,{1},0\, \le \,a_{i}^{*} \, + \,b_{j} .^{ + } - s_{{7}} \, \le \,{1} \\ & 0\, \le \,a_{i}^{ + } \, + \,b_{j}^{ - } - s_{{2}} \, \le \,{1},0\, \le \,a_{i}^{ - } \, + \,b_{j}^{ - } - s_{{5}} \, \le \,{1},0\, \le \,a_{i}^{*} \, + \,b_{j} .^{ - } - s_{{8}} \, \le \,{1} \\ & 0\, \le \,a_{i}^{ + } \, + \,b_{j}^{*} - s_{{3}} \, \le \,{1},0\, \le \,a_{i}^{ - } \, + \,b_{j}^{*} - s_{{6}} \, \le \,{1},0\, \le \,a_{i}^{*} \, + \,b_{j} .^{*} - s_{{9}} \, \le \,{1} \\ \end{aligned}$$

Proof

Take 0 ≤ a₁ + b₁-s₁ ≤ 1 as an example.

As we known from Definition 7 that $a_{i}^{ + } = \frac{{|\{ R_{k} :c_{i}^{ + } \in R_{k} \} |}}{|T|}$, $b_{j}^{ + } = \frac{{|\{ R_{k} :c_{j}^{ + } \in R_{k} \} |}}{|T|}$, $s_{1} = \frac{{|\{ R_{k} :\{ c_{i}^{ + } ,c_{j}^{ + } \} \subseteq R_{k} \} |}}{|T|}$.

Define that $C_{i}^{ + } = \{ R_{k} :c_{i}^{ + } \in R_{k} \}$, $C_{j}^{ + } = \{ R_{k} :c_{j}^{ + } \in R_{k} \}$.

Then, $P(C_{i}^{ + } \cup C_{j}^{ + } ) = P(C_{i}^{ + } ) + P(C_{j}^{ + } ) - P(C_{i}^{ + } \cap C_{j}^{ + } ) = = a_{i}^{ + } + b_{j}^{ + } - s_{1}$

$$0 \le P(C_{i}^{ + } \cup C_{j}^{ + } ) \le 1$$

Thus,

$$0 \le a_{i}^{ + } + b_{j}^{ + } - s_{1} \le 1$$