Multicriteria correlation preference information (MCCPI) with nonadditivity index for decision aiding

Abstract

MCCPI (Multiple Criteria Correlation Preference Information) is a kind of 2 dimensional decision preference information obtained by pairwise comparison on the importance and interaction of decision criteria. In this paper, we introduce the nonadditivity index to replace the Shapley simultaneous interaction index and construct an undated MCCPI based decision scheme. We firstly propose a diagram to help decision maker obtain the nonadditivity index type MCCPI, then establish transform equations to normalize them into desired capacity and finally adopt a random generation MCCPI based comprehensive decision aid algorithm to explore the dominance relationships and creditable ranking orders of all decision alternatives. An illustrative example is also given to demonstrate the feasibility and effectiveness of the proposed decision scheme. It’s shown that based on some good properties of nonadditivity index in practice, the updated MCCPI model can deal with the internal interaction among decision criteria with relatively less model construction and calculation effort.

Keywords

Multiple criteria decision analysis capacity fuzzy measure nonadditivity index interaction index

1 Introduction

Compared with the probability additive measure, the capacity [7], also called fuzzy measure [25], nonadditive measure [3], has a major merit in representing various interaction situations among multiple correlative criteria. This merit of capacity basically roots from its monotonicity property with respect to set inclusion, or more explicitly, the nonadditivity with respect to disjoint subsets. It is commonly accepted that superadditive (subadditive) capacities mean positive (negative) interactions among decision criteria. The combination of capacity and Choquet integral [7] is the most widely adopted pattern for dealing with and aggregating the interdependent multiple criteria associated decision information.

The capacity and Choquet integral based decision pattern usually begins with decision maker’s preference information on importance and interaction of decision criteria and ends with overall evaluations (Choquet integrals) or a consistent ranking order on all alternatives. The decision preference information can be generally represented by a special type of capacity, e.g., k-additive capacity [11] only concerns at most k-order interaction and neglects the higher ones; p-symmetric capacity [23] represents decision with p groups of anonymous referees or indifferent criteria, k-order representative capacity only uses upper or lower k-order subsets to define the multicriteria decision context [29] and so on [5]. The preference information can also be represented in detail in terms of an equivalent transform of capacity, e.g., capacity value itself just represents the importance of criteria subset, the Shapley interaction index [10, 16] describes the simultaneous interaction among criteria, the nonadditivity index [3 , 28] shows interaction phenomenon of criteria associated with the nonadditivity. The preference information can be further implicitly represented through some partial orders on some decision alternatives in terms of the Choquet integral or other type of nonlinear integral [2, 26]. Usually, these three types of representation approaches for preference information are given in terms of interval ranges or comparison forms [12]. With them, many capacity identification methods, such as least-squares approach [12 , 22], the maximum split method [21, 22], the least absolute deviation method [1], the maximum entropy approach [16 , 20], the compromise method [31] and so on [4], are proposed and applied to obtain the most satisfactory capacity, by which the final overall evaluations or ranking order can be naturally generated.

Lately, a special form of preference information, called the MCCPI (Multiple Criteria Correlation Preference Information), has been introduced and studied [30], which is basically a set of 2-dimensional (importance, interaction) pairwise comparison preference information and can be regarded as a natural generation of the 1-dimensional (only importance) pairwise comparison information obtained by another well-known decision method, AHP (Analytic Hierarchy Process) [24]. The MCCPI can be obtained through an aid tool called the refined diamond diagram, which holistically composes all the situations of the overall importance and the simultaneous interaction between two criteria. With MCCPI, some optimization models, like the least squares based nonlinear programming model and the least absolute deviation based linear programming model, are established to get the most desired 2-additive or ordinal capacity for closely catering to the decision maker’s preference and judgments [30, 32]. Furthermore, a random generation based comprehensive decision aid algorithm with the Shapley importance and interaction indices is proposed to show the whole view of allowable dominance relationships among all decision alternatives [32].

Although the Shapley simultaneous interaction index, along with other types of probabilistic simultaneous interaction indices [10, 15], e.g., Möbius representation [6] and Banzhaf simultaneous interaction index, has some outstanding axiomatic characteristics, but meanwhile it has some shortcomings for multiple criteria decision making [3 , 28]. For example, with 2, 3, 4 and 5 criteria, it ranges in [-1, 1], [-2, 1], [-3, 3] and [-4, 6]. This changeable range for different cardinalities brings difficulties in directly comparing these indices values. Superadditive capacity can have a negative simultaneous value, i.e., these indices values do not precisely reflect the interaction kind associated with nonadditivity and disobey some commons sense in decision context. Furthermore, the interaction of criteria subset that includes as long as one dummy criterion will always be zero, regardless of the kind or degree of the interaction existed among other non-dummy criteria, which also conflicts with people’s intuition.

In view of these disadvantages with existing interaction indices, we recently proposed the concept of nonadditivity index [27] to explicitly express the interaction degree and kind associated with nonadditivity, which has an uniform range, consistent with nonadditivity, moderate compromise with the effect of dummy. Albeit a type of internal interaction index, the nonadditivity index, also a one-to-one linear mapping of capacity, can be served as a good alternative of simultaneous interaction index in multiple criteria decision making in consideration of its advantages in practice and easy understanding in decision context.

In this paper, we first construct a different diagram to aid the decision maker to explicitly represent nonadditivity index form MCCPI of every pair of decision criteria, where the relative importance and interaction between two criteria are both divided into nine categories and result into 68 intersection regions for the choice of pairwise comparison. Then by adopting the least square divergence principle as well as least absolute deviation principle, we establish some optimization models to transform the MCCPI of multiple criteria into the ordinary or special types of capacities with the consideration of the boundary and monotonicity condition and other specific and complex preference constraints on 3 or more decision criteria. Finally, to reveal the whole view of dominance situation between all decision criteria with the given acceptable intersection regions, a random simulation based comprehensive decision aid algorithm is proposed to generate all the three types dominance relationships of all pairs of alternatives together with credibility frequencies and finally to achieve the most creditable ranking order of all decision alternatives.

This paper is organized as follows. After the introduction, we collect some basic knowledge about the capacity, Shapley simultaneous interaction index and nonadditivity index in Section 2. In Section 3, we briefly show the Shapley interaction based MCCPI and its decision making scheme and steps. In Section 4, we present some tools and rules to obtain the nonadditivity index based MCCPI. Sections 5 and 6 are for the identification models to derive the capacities directly from the MCCPI and the random generation based comprehensive decision aid algorithm, respectively. Section 7 provides an empirical analysis of the updated decision scheme. Finally, we conclude the paper in Section 8.

2 Capacity and its interaction indices

Let N = {1, 2, …, n}, n ≥ 2, be the decision criteria set, $P (N)$ its power set, and |S| the cardinality of subset S ⊆ N.

Definition 1. [7 , 25] A capacity on N is a set function $μ : P (N) \to [0, 1]$ such that

μ (∅) =0, μ (N) =1 ; (boundary condition)

∀A, B ⊆ N, A ⊆ B implies μ (A) ≤ μ (B). (monotonicity condition)

The capacity value of a decision subset is usually regarded as its importance to the decision problem, the minimum and maximum importances are normalized as 0 and 1, respectively. The monotonicity of capacity value with respect to set inclusion means a new criterion’s join can not decrease the importance of original coalition. An explicit representation of monotonicity is the nonadditivity with respect to disjoint subsets, which leads to a common sense of interaction among multiple criteria.

Definition 2. [12 , 27] A capacity μ on N is said to be ★-additive, if $\begin{matrix} μ (A \cup B) \overset{★}{=} & μ (A) + μ (B), \forall A, B \subseteq N, A, \\ B \neq \emptyset, A \cap B = \emptyset . \end{matrix}$ where " $\overset{★}{=}$ " stands for "= (resp . ≥ , ≤ , > , and <)", "★-additive" stands for "additive (resp. superadditive, subadditive, strict superadditive, and strict subadditive)".

It is commonly accepted that an additive (superadditive, subadditive) capacity means that decision criteria are mutually independent (complementary, substitutive), or their interactions are all zero (nonnegative, nonpositive).

The interaction phenomenon could be represented by the Shapley simultaneous interaction index.

Definition 3. [10–12 , 14] Let μ be a capacity on N, the Shapley interaction index of μ is defined as

$\begin{matrix} I_{μ} (A) = & \sum_{B \subseteq N ∖ A} \frac{1}{| N | - | A | + 1} {(\begin{matrix} | N | - | A | \\ | B | \end{matrix})}^{- 1} \\ \sum_{C \subseteq A} {(- 1)}^{| A ∖ C |} μ (C \cup B), A \subseteq N, \end{matrix}$ (1)

The Shapley interaction index, as well as the probabilistic simultaneous interaction indices, has many outstanding axiomatic characteristics [10, 16] but fails to represent the interaction kind and degree associated with nonadditivity [27, 28].

Definition 4. [27] Let μ be a capacity on N, the nonadditivity index of μ is defined as $n_{μ} (A) = μ (A) - \frac{1}{2^{| A | - 1} - 1} \sum_{C \subset A} μ (C), A \subseteq N .$ (2)

The nonadditivity index has the following properties:

If μ on N is ★-additive, then n_μ (A) $\overset{★}{=} 0$ , ∀A ⊆ N, |A|≥2.

-1 ≤ n_μ (A) ≤1, ∀A ⊆ N.

n_μ (A) =1 ⇔ $μ (S) = ɛ_{A}^{+} (S)$ , S ⊆ A, and n_μ (A) = -1 ⇔ $μ (S) = ɛ_{A}^{-} (S), S \cap A \neq \emptyset$ , where $ɛ_{A}^{+}$ , is a capacity such that $ɛ_{A}^{+} (B) = 1$ iff B ⊇ A, B ⊆ N, and $ɛ_{A}^{+} (B) = 0$ otherwise; $ɛ_{A}^{-}$ , is a capacity such that $ɛ_{A}^{-} (B) = 1$ iff B∩ A ≠ ∅, and $ɛ_{A}^{-} (B) = 0$ otherwise.

This means nonadditivity index is consistent with nonadditivity, with uniform range, and gets its extremely interaction cases with

ɛ_{A}^{+}

and

ɛ_{A}^{-}

Same to the Shapley interaction index, the nonadditivity index is also a linear representation (one to one mapping) of the capacity [28]. A capacity μ on N can be represented through nonadditivity index n_μ as

$\begin{matrix} μ (A) = & n_{μ} (A) + \sum_{C \subset A} \sum_{i = | C |}^{| A | - 1} \prod_{j = i}^{| A | - 1} \frac{| A | - j}{2^{j} - 1} n_{μ} (C), \\ \forall A \subseteq N . \end{matrix}$ (3)

The Choquet integral is a generally adopted form of nonlinear integral to aggregate the capacity based decision information and also has many brilliant aggregation function properties [3].

Definition 5. For a given x ∈ [0, 1] ⁿ, the discrete Choquet integral of x with respect to capacity μ on N is defined as: $C (x) = \sum_{i = 1}^{n} (x_{(i)} - x_{(i - 1)}) μ ({(i), \dots, (n)}),$ where x_(.) is a non-decreasing permutation induced by x_i, i = 1, …, n, i.e., x₍₁₎ ≤ … ≤ x_(n), and x₍₀₎ = 0 by convention.

3 Shapley interaction index based MCCPI and its decision scheme

Suppose criteria set N = {i, j}, we can have $\begin{matrix} 0 \leq I_{μ} ({i}), I_{μ} ({j}) \leq 1, I_{μ} ({i}) + I_{μ} ({j}) = 1, \\ | I_{μ} ({i, j}) | \leq 2 min (I_{μ} ({i}), I_{μ} ({j}) = 2 min (I_{μ} ({i}), \\ 1 - I_{μ} ({i})), \end{matrix}$ (4)

which can be reflected by the points in a diamond diagram [30, 32], see Fig. 1 for its refined version, where the segment OA or the interval [0, 1] as well as the segment FF′ is divided into some sub segments.

Fig. 1

Refined diamond diagram.

Specifically, if the point’s first coordinate falls into [0.00, 0.125) (resp [0.125, 0.25), [0.25, 0.375), [0.375, 0.50), [0.50, 0.50], (0.50, 0.625], (0.625, 0.75], (0.75, 0.875], (0.875, 1]), then it means the criterion i is extremely less (resp. very strongly less, strongly less, slightly less, equally, slightly more, strongly more, very strongly more, and extremely more) important than j. If the point belongs to sub-diamond OBAB′ (resp. OBAC, OCAD, ODAE, OEAF, OB′AC′, OC′AD′, OD′AE′, and OE′AF′), then the partial interaction between two criteria is almost zero (resp. slightly positive, strongly positive, very strongly positive, extremely positive, slightly negative, strongly negative, very strongly negative, and extremely negative), where the points O = (0.00, 0.00), A = (1.00, 0.00), B = (0.50, 0.05), B′ = (0.50, - 0.05), C = (0.50, 0.25), E = (0.50, 0.75), F = (0.50, 1.00), C′ = (0.50, - 0.25), D′ = (0.50, - 0.50), E′ = (0.50, - 0.75), and F′ = (0.50, - 1.00).

By using the above 9 × 9 = 81 refined segments in the diamond, decision maker can get the relative importance and partial interaction of all pairs of decision criteria, denoted as $I_{i}^{ij}$ , $I_{j}^{ij}$ and $I_{ij}^{P}$ , where i, j ∈ N and i < j. Then we can construct the following group of equations to bridge the MCCPI to the Shapley interaction index of the most desired capacity [30]: $\begin{matrix} I_{i}^{ij} & = \frac{I_{μ} ({i})}{I_{μ} ({i}) + I_{μ} ({j})} \\ I_{ij}^{P} & = \frac{I_{μ} ({i, j})}{I_{μ} ({i}) + I_{μ} ({j})} \end{matrix}$ (5)

By introducing the distance measure and the boundary and monotonicity conditions, we can construct a capacity identification model as follows:

$\begin{matrix} min & z = \sum_{i, j \in N, i \neq j} (D (I_{i}^{ij}, \frac{I_{μ} (i)}{I_{μ} ({i}) + I_{μ} ({j})}) + D (I_{ij}^{P}, \frac{I_{μ} ({i, j})}{I_{μ} ({i}) + I_{μ} ({j})})) \\ s . t . & Boundary and monotonicity conditions of the capacity, \\ Specific conditions of special types of capacity (if needed), \\ Preference on higher order {subsets}^{'} importance and interaction, \end{matrix}$ (6)

where D (x, y) is a distance between x and y, the optimal capacity can be ordinary or any particular type of capacity, and the higher order preference can be given in comparison or interval form as in works [8 , 30].

Different points in the acceptable segments of same MCCPI may lead to different optimal capacities, and at larger chance the different overall evaluations and final ranking orders of decision alternatives. In order to show the whole dominance situation and get the most creditable ranking order, a random generation MCCPI based comprehensive decision aid algorithm has been proposed, see reference [32], whose main idea will be shown and succeeded in algorithm 1 in this paper.

As mentioned previously, compared to Shapley interaction index, nonadditivity has advantages in representing decision maker’s preference information. It’s better to combine the nonadditivity index with MCCPI as well as its comprehensive decision aid algorithm. We’ll finish these tasks in the following sections.

4 The nonadditivity index type MCCPI

MCCPI obtains decision maker’s preference information by pairwise comparison of decision criteria through a 2-dimension perspective, i.e., the relative importance and partial interaction. If adopt nonadditivity index as an intermediary to show this type of 2-dimensional preference information, the Fig. 2 can be taken as an identification aid tool.

Fig. 2

The diagram of nonadditivity index type MCCPI.

Suppose the decision criteria set only consists of two elements N = {i, j}, then $\begin{matrix} 0 \leq μ {i} \leq 1, \\ 0 \leq μ {j} \leq 1, \\ μ {i, j} = 1 . \end{matrix}$ (7) By adopting a coordinate system where horizontal axis represents value of μ {i} and vertical axis μ {j}, we can represent Eq. 7 into a diagram to get the nonadditivity index type MCCPI. That is, points in rectangle ODHD′ show all the cases of capacity values of i and j as well as their interactions.

First, the relative importance between criteria i and j is represented by their capacity values. For example, point H = (1, 1) stands for μ {i} =1, μ {j} =1 and means criterion i is as much important as criterion j; point D = (1, 0) stands for μ {i} =1, μ {j} =0 and means criterion i is extremely more important than criterion j; point D′ = (0, 1) stands for μ {i} =0, μ {j} =1 and means criterion i is extremely less important than criterion j. Adopting the nine scale of relative importance similar in AHP method, we can take

the points in line OH to show two criteria i and j have same importance;

the points in ▵OGH, exclusive of points in line OH, to show that criterion i is slightly more important than criterion j, where G = (1, 0.75);

the points in ▵OFG, exclusive of points in line OG, to show that criterion i is fairly more important than criterion j, where F = (1, 0.5);

the points in ▵OEF, exclusive of points in line OF, to show that criterion i is strongly more important than criterion j, where E = (1, 0.25);

the points in ▵ODE, exclusive of points in line OE, to show that criterion i is extremely more important than criterion j, where D = (1, 0);

the points in ▵OHG′, exclusive of points in line OH, to show that criterion i is slightly less important than criterion j, where G′ = (0.75, 1);

the points in ▵OG′F′, exclusive of points in line OG′, to show that criterion i is fairly less important than criterion j, where F′ = (0.5, 1);

the points in ▵OF′E′, exclusive of points in line OF′, to show that criterion i is strongly less important than criterion j, where E′ = (0.25, 1);

the points in ▵OE′D′, exclusive of points in line OE′, to show that criterion i is extremely less important than criterion j, where D′ = (0, 1).

Second, the partial interaction among two criteria i and j can be represented by the nonadditivity index. For example, in Fig. 2, the nonadditivity index of point H is: $n_{μ} (H) = μ {i, j} - μ {i} - μ {j} = 1 - 1 - 1 = - 1 .$ Similarly, $n_{μ} (D) = n_{μ} (D^{'}) = 0, n_{μ} (C) = 0.25, n_{μ} (O) = 1 .$ where C = (0.75, 0), O = (0, 0). Actually, we can take

the points of ▵GHG′, exclusive of points in line GG′, to represent the extremely negative interaction existing between criteria i and j;

the points of trapezoid FGG′F′, exclusive of points in line FF′, to represent the strongly negative interaction existing between criteria i and j;

the points of trapezoid EFF′E′, exclusive of points in line EE′, to represent the fairly negative interaction existing between criteria i and j;

the points of trapezoid DEE′D′, exclusive of points in line DD′, to represent the slightly negative interaction existing between criteria i and j;

the points of line DD′ to represent criteria i and j have no interaction.

the points of trapezoid CDD′C′, exclusive of points in line DD′, to represent the slightly positive interaction existing between criteria i and j, where C = (0.75, 0), C′ = (0, 0.75);

the points of trapezoid BCC′B′, exclusive of points in line CC′, to represent the fairly positive interaction existing between criteria i and j, where B = (0.50, 0), B′ = (0, 0.50);

the points of trapezoid ABB′A′, exclusive of points in line BB′, to represent the strongly positive interaction existing between criteria i and j, where A = (0.25, 0), A′ = (0, 0.25);

the points of ▵OAA′, exclusive of points in line AA′, to represent the extremely positive interaction existing between criteria i and j, where O = (0, 0).

In brief, in Fig. 2, the relative importance of two criteria is defined by the ratio of their capacity with the critical values of 4, 2, 4/3, 1, 3/4, 1/2 and 1/4 to show the cases ranging from extremely more important to extremely less important; the interaction between two criteria is classified by the nonadditivity index of their union set with the critical values of -0.75, -0.5, -0.25, 0, 0.25, 0.5 and 0.75 to represent the cases ranging from extremely negative to extremely positive. One can see that, in Fig. 2, there are 69 types of 2-dimensional preference information of two criteria, e.g., the point X=(0.5, 0.8) falls into the intersection of the ▵OG′F′ with the trapezoid EFF′E′ and is used to show criterion i is strongly less important that criterion j and their interaction is fairly negative (actually their nonadditivity index is -0.3). From Fig. 2, one can also find that if two criteria are relatively equally important, they can possess more negative interaction cases. The more asymmetric of their weights, the less cases of negative interactions, e.g., in the line OD as well as line OD′, two criteria have no negative interaction cases.

5 MCCPI based capacity identification method

Suppose the nonadditivity index type MCCPI of decision criteria pair {i, j} ⊆ N is given as $(μ_{ij}^{p} ({i}), μ_{ij}^{p} ({j}), n_{ij}^{p} ({i, j})),$ where $μ_{ij}^{p} ({i})$ is relative importance of criterion i, $n_{ij}^{p} ({i, j})$ the nonadditivity index, in the pairwise comparison of these two criteria by using the Fig. 2. That is, $n_{ij}^{p} ({i, j}) = 1 - μ_{ij}^{p} ({i}) - μ_{ij}^{p} ({j}) .$ Further assume the capacity mostly caters the above MCCPI as μ, we can construct the following group of equations to bridge nonadditivity index type MCCPI and capacity μ: $\begin{matrix} μ ({i}) & = μ_{ij}^{p} ({i}) μ ({i, j}) \\ μ ({j}) & = μ_{ij}^{p} ({j}) μ ({i, j}) \\ n_{μ} ({i, j}) & = n_{ij}^{p} ({i, j}) μ ({i, j}) \end{matrix}$ (8) By introducing the distance measure, we can construct the capacity identification model as follows:

$\begin{matrix} min z = \sum_{i, j \in N, i < j} & D (μ ({i}), μ_{ij}^{p} ({i}) μ ({i, j})) + D (μ ({j}), μ_{ij}^{p} ({j}) μ ({i, j})) \\ + D (n_{μ} ({i, j}), n_{ij}^{p} ({i, j}) μ ({i, j})) \\ s . t . & Boundary and monotonicity conditions of the capacity, \\ Specific conditions of special types of capacity (if needed), \\ Preference on higher order {subsets}^{'} importance and interaction, \end{matrix}$ (9) where D (x, y) is a distance between x and y, e.g., the quadratic or absolute distance form.

The objective of model (9) is to minimize the divergence between two hand sides of Eq. (8), i.e., the deviation between the MCCPI and the optimal capacity. If adopt the absolute distance, we have a chance to present the above model into the multiple goals linear programming by introducing the positive and negative deviation variables. According to model (9), we can construct the following linear programming:

$\begin{matrix} min z & = \sum_{i, j \in N, i < j} d_{ij}^{+} + d_{ij}^{-} + e_{ij}^{+} + e_{ij}^{-} + f_{ij}^{+} + f_{ij}^{-} \\ s . t . & Boundary and monotonicity conditions of the capacity, \\ Specific conditions of special types of capacity (if needed), \\ Preference on higher order {subsets}^{'} importance and interaction, \\ μ ({i}) - μ_{ij}^{p} ({i}) μ ({i, j}) - d_{ij}^{+} + d_{ij}^{-} = 0, i, j \in N, i < j, \\ μ ({j}) - μ_{ij}^{p} ({j}) μ ({i, j}) - e_{ij}^{+} + e_{ij}^{-} = 0, i, j \in N, i < j, \\ n_{μ} ({i, j}) - n_{ij}^{p} ({i, j}) μ ({i, j}) - f_{ij}^{+} + f_{ij}^{-} = 0, i, j \in N, i < j, \end{matrix}$ (10) where $d_{ij}^{+}, d_{ij}^{-}, e_{ij}^{+}, e_{ij}^{-}, f_{ij}^{+}$ and $f_{ij}^{-} \geq 0$ .

Remark 1. In the above model, if it only consists of the boundary and monotonicity conditions and the MCCPI based goal constraints, then the extremely superadditive capacity $ɛ_{N}^{+}$ , i.e., the capacity values of all the proper subsets in N are all zero, is always the optimal solution. In order to avoid this situation, we had better to add a few of hard constraints such as the special type of capacity or some preference information on some subsets, or we can just add some positivity and negativity constraints on the pair criteria by using the following rule: if $n_{ij}^{p} ({i, j}) >$ (resp. <) 0, then we add n_μ ({i, j}) ≥ δ (resp. n_μ ({i, j}) ≤ - δ); if $n_{ij}^{p} ({i, j}) =$ 0, then we add n_μ ({i, j}) ≤ δ and n_μ ({i, j}) ≥ - δ, where δ > 0 is a threshold and can be set as 0.05 generally.

It should be pointed out that the boundary and monotonicity conditions are linear combination of capacity; most of constraints for special types of capacity, like k-additive capacity, p-symmetric capacity, k-order representative capacity can also be represented as linear constraints (see [1 , 29]); and preference on higher order subsets is usually given in terms of linear equivalent representation of capacity, e.g., Shapley interaction index and nonadditivity index, with comparison or interval forms, which is essentially the linear combination of capacity. Furthermore, the multiple goals linear programming of model (10) can take any linear equivalent representation (one-to-one mapping) of capacity as variable and the linear property of the model will not change. For example, by Eq. (3), we can use the nonadditivity index to construct the boundary and monotonicity conditions as follows [28]:

$\begin{matrix} n_{μ} (\emptyset) = 0, & n_{μ} (N) + \sum_{C \subset N} \sum_{i = | C |}^{n - 1} \prod_{j = i}^{n - 1} \frac{n - j}{2^{j} - 1} n_{μ} (C) = 1; \\ n_{μ} (A \cup {i}) & + \sum_{C \subset A \cup {i}} \sum_{i = | C |}^{| A |} \prod_{j = i}^{| A |} \frac{| A | + 1 - j}{2^{j} - 1} n_{μ} (C) - n_{μ} (A) \\ - \sum_{C \subset A} \sum_{i = | C |}^{| A | - 1} \prod_{j = i}^{| A | - 1} \frac{| A | - j}{2^{j} - 1} n_{μ} (C) \\ \geq 0, \forall i \in N, A \subseteq N \ {i} . \end{matrix}$

Also, the optimization identification methods can be rewritten in matrix representation or marginal contribution representation form, see e.g., [3, 27].

6 Random generation MCCPI based comprehensive decision aiding method

As mentioned before, one can also imagine that different representative points of given acceptable intersection of same MCCPI may lead to different overall evaluations and then the different ranking orders of decision alternatives. We can resort the random simulation method to sketch out a general view of allowable dominance relationships of all alternatives. Algorithm 1 is the comprehensive decision aid algorithm based on random generation of nonadditivity index type of MCCPI.

Remark 2. In algorithm 1, the randomly generation of points in acceptable intersection can be obtained by first randomly generate a value of the nonadditivity index in its acceptable interval, then randomly generate the ratio of relatively importance of two criteria in its acceptable interval, and finally get their capacity values. For example, if an acceptable region is the intersection of ▵OG′F′ with trapezoid EFF′E′, which means criterion i is less important that criterion j and their interaction is fairly negative. We need firstly to get a random value in interval [-0.5, -0.25) as their interaction index and a random value in interval [3/4 1) as the ratio of their capacity value and then calculate the capacity values of the two criteria.

7 Empirical analysis on an example

See the application example in References [30, 32]: a decision maker has to rank seven cars according to four criteria: price, acceleration time, maximum speed and consumption, denoted as criteria 1, 2, 3, and 4. These cars’ partial evaluations on four criteria are given in Table 1.

Table 1
The partial evaluations of the seven types of cars

Car A Car B Car C Car D Car E Car F Car G

Price 0.153 0.256 0.823 0.939 0.675 0.876 0.922

Acceleration 0.586 0.902 0.557 0.721 0.689 0.354 0.687

Max speed 0.483 0.854 0.493 0.638 0.472 0.277 0.748

Consumption 0.845 0.417 0.521 0.712 0.611 0.758 0.691

	Car A	Car B	Car C	Car D	Car E	Car F	Car G
Price	0.153	0.256	0.823	0.939	0.675	0.876	0.922
Acceleration	0.586	0.902	0.557	0.721	0.689	0.354	0.687
Max speed	0.483	0.854	0.493	0.638	0.472	0.277	0.748
Consumption	0.845	0.417	0.521	0.712	0.611	0.758	0.691

The decision maker gives the nonadditivity index type of MCCPI as follows:

Criterion 1 is strongly more important than 2 and their partial interaction is extremely positive;

Criterion 1 is strongly more important than 3 and their partial interaction is strongly positive;

Criterion 1 is slightly less important than 4 and their partial interaction is extremely negative;

Criterion 2 is strongly more important than 3 and their partial interaction is slightly negative;

Criterion 2 is slightly less important than 4 and their partial interaction is strongly negative;

Criterion 3 is strongly less important than 4 and their partial interaction is strongly positive.

After six times of 2-dimensional pairwise comparisons, the above MCCPI can be specifically represented as the points in Fig. 3, where the points (0.1, 0.05), (0.3, 0.1), (0.85, 0.95), (0.8, 0.3), (0.75, 0.8) and (0.1, 0.3) are adopted to represent the MCCPI of the criteria pairs (1, 2), (1, 3), (1, 4), (2, 3), (2, 4) and (3, 4), respectively.

Fig. 3

MCCPI of the four criteria.

According to model (10), we have the following linear programming: $\begin{matrix} min z & = d_{12}^{+} + d_{12}^{-} + \dots + d_{34}^{+} + d_{34}^{-} \\ + e_{12}^{+} + e_{12}^{-} + \dots + e_{34}^{+} + e_{34}^{-} \\ + f_{12}^{+} + f_{12}^{-} + \dots + f_{34}^{+} + f_{34}^{-} \\ s . t . & μ ({1}) - 0.1 μ ({1, 2}) - d_{12}^{+} + d_{12}^{-} = 0, \\ \dots \\ μ ({3}) - 0.1 μ ({3, 4}) - d_{34}^{+} + d_{34}^{-} = 0, \\ μ ({2}) - 0.05 μ ({1, 2}) - d_{12}^{+} + d_{12}^{-} = 0, \\ \dots \end{matrix}$ $\begin{matrix} μ ({4}) - 0.3 μ ({3, 4}) - d_{34}^{+} + d_{34}^{-} = 0, \\ n_{μ} ({1, 2}) - 0.85 μ ({1, 2}) - f_{12}^{+} + f_{12}^{-} = 0, \\ \dots \\ n_{μ} ({3, 4}) - 0.6 μ ({3, 4}) - f_{34}^{+} + f_{34}^{-} = 0, \\ n_{μ} ({1, 2}), n_{μ} ({1, 3}), n_{μ} ({3, 4}) \geq 0.05, \\ n_{μ} ({1, 4}), n_{μ} ({2, 3}), n_{μ} ({2, 4}) \leq - 0.05, \\ μ (\emptyset) = 0, \\ μ ({1, 2, 3, 4}) = 1, \\ μ (A \cup {i}) - μ (A) \geq 0, \forall A \subset N, \forall i \in N ∖ A, \end{matrix}$ (11) where $d_{ij}^{+}, d_{ij}^{-}, e_{ij}^{+}, e_{ij}^{-}, f_{ij}^{+}$ and $f_{ij}^{-} \geq 0$ , i, j ∈ {1, 2, 3, 4}, i < j.

Solving the above model, we have the optimal capacity and its nonadditivity index, as shown in Table 2.

Table 2

The optimal capacity and its nonadditivity index

A	μ (A)	n_μ (A)	A	μ (A)	n_μ (A)
∅	0	0	{2, 3}	0.05520446	-0.05000000
{1}	0.09479554	0.09479554	{2, 4}	0.10037175	-0.05520446
{2}	0.05520446	0.05520446	{3, 4}	0.37592937	0.22555762
{3}	0.05000000	0.05000000	{1, 2, 3}	1.00000000	0.46093556
{4}	0.10037175	0.10037175	{1, 2, 4}	1.00000000	0.51257745
{1, 2}	1.00000000	0.85000000	{1, 3, 4}	0.37592937	0.01105948
{1, 3}	0.36198885	0.21719331	{2, 3, 4}	0.37592937	0.13023544
{1, 4}	0.11152416	-0.08364312	{1, 2, 3, 4}	1.00000000	0.27753585

By the capacity in Table 2, we can calculate the Choquet integrals of all cars and have the following ranking order:

$\begin{matrix} Car D ≻ Car G ≻ Car E ≻ Car C ≻ Car F \\ ≻ Car B ≻ Car A . \end{matrix}$ (12)

Now we further test the random generation based comprehensive decision aid algorithm (1). We choose the random values from the intervals (2, 4], (2, 4], [3/4, 1), (2, 4], [3/4, 1), 1/4,1/2) to show the ratio of capacity values of pair criteria (1, 2), (1, 3), (1, 4), (2, 3), (2, 4) and (3, 4), and choose the random values from the intervals (0.75, 1], (0.5,0.75], [-1,-0.75), [-0.25,0), [-0.75,-0.50) and (0.5,0.75] to their partial interactions. By executing the algorithm 1 with $K = 100$ , we have the dominance relationships of all the cars and their frequencies as following:

d^≻ (A, B) =0.09, d^≺ (A, B) =0.91, d^≺ (A, C) =1, d^≺ (A, D) =1, d^≺ (A, E) =1, d^≻ (A, F) =0.04, d^≺ (A, F) =0.96, d^≺ (A, G) =1, d^≺ (B, C) =1, d^≺ (B, D) =1, d^≺ (B, E) =1, d^≺ (B, F) =1, d^≺ (B, G) =1, d^≺ (C, D) =1, d^≺ (C, E) =1, d^≻ (C, F) =1, d^≺ (C, G) =1, d^≻ (D, E) =1, d^≻ (D, F) =1, d^≻ (D, G) =0.92, d^≺ (D, G) =0.08, d^≻ (E, F) =1, d^≺ (E, G) =1, d^≺ (F, G) =1,

If set $K = 300$ , the following relationships’ frequencies have changed:

d^≻ (A, B) =0.08, d^≺ (A, B) =0.92, d^≻ (A, F) =0.03, d^≺ (A, F) =0.97, d^≻ (D, G) =0.88, d^≺ (D, G) =0.12.

If set $K = 500$ , the following relationships’ frequencies have changed:

d^≻ (A, B) =0.07, d^≺ (A, B) =0.93, d^≻ (A, F) =0.01, d^≺ (A, F) =0.99, d^≻ (D, G) =0.89, d^≺ (D, G) =0.11.

One can have a conclusion that the frequencies of all dominance relationships are relatively stable. By calculating the dominance scores of all cars, we can obtain the most credible overall ranking order with the generated MCCPI, which is same as that in Eq. (12). So, the car D is the best choice.

Remark 3. Here, we give a comparison analysis between the nonadditivity index based and Shapley interaction index based MCCPI methods. Theoretically, these two methods both aim to help decision maker to show his/her preference information about criteria in a relatively standard and normalized form. Based on different properties of nonadditivity index and Shapley interaction index, we can establish different aid diagrams to collect the different types of MCCPI data. Choosing which types of MCCPI depends on the decision maker’s consideration on interaction phenomenon among criteria. Main difference lies between nonadditivity index and Shapley simultaneous interaction index is that the former measures internal interaction situation among criteria and is also consistent with the nonadditivity properties whereas the latter describes comprehensive interaction situation and also reflects simultaneous changes on interaction trends. Furthermore, nonadditivity index type MCCPI can provide a intuitive and direct comparison among different cardinalities subsets since its uniform range property. From the computing perspective, taking the above illustrative application as an example and also comparing it with the result in [32], we can figure out that, with the similar input MCCPI data, both methods turn out the similar decision result, like the kinds of pairs of criteria and final ranking order of all alternatives. The main difference lying between two results is that the nonadditivity index based MCCPI method emphasis on the positive interaction among criteria, see that of criteria 1 and 2 in table 2, meanwhile the Shapley interaction index based MCCPI take a relatively good compromise between positive and negative interactions among criteria, see e.g., Table 3 in [32]. This difference highly comes from that nonadditivity index as an internal index involves less number of coefficients and easy to lead to the relatively extreme situation and Shapley interaction index as a comprehensive index involves more coefficients and can have a relatively even distribution on interaction of all subsets.

8 Conclusions

In this paper, we construct a nonadditivity index type MCCPI based multiple criteria decision aid scheme, which is an update version of the Shapely interaction index type MCCPI and its comprehensive decision algorithm. The nonadditivity index type MCCPI can be obtained by 2-dimensional comparison in terms of a rectangular diagram subdivided with different degrees of relative importance and partial interaction. On the grounds of a random generation of MCCPI, a random simulation algorithm is proposed to reveal the whole view of dominance relationships of all alternatives as well as the most creditable ranking order on them. The major advantage and convenience of this updated decision scheme root from the good properties and easy understanding of nonadditivity index. Since nonadditivity index is basically an internal interaction index and with less coefficients compared with some comprehensive interaction indices, the nonadditivity index type MCCPI method can lead to some extreme interaction situations in practice. The further research task will focus on more empirical studies of different MCCPI methods on some real decision problems and the rules and tools for MCCPI’s inconsistency recognition and adjustment as well as some corresponding decision aid software packages. Furthermore, some decision issues in traditional independent criteria context, like strategic weight manipulation [33], minimum, averaging and maximum ranking positions of decision alternatives [9, 19] also have potential necessities and feasibilities to be extended into the interactive criteria context with the capacity and nonlinear integral.

Footnotes

Acknowledgments

The work was supported by the National Natural Science Foundation of China (No. 71671096).

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