The random generation of fuzzy measures under complex linear constraints holds significance in various fields, including optimization solutions, machine learning, decision making, and property investigation. However, most existing random generation methods primarily focus on addressing the monotonicity and normalization conditions inherent in the construction of fuzzy measures, rather than the linear constraints that are crucial for representing special families of fuzzy measures and additional preference information. In this paper, we present two categories of methods to address the generation of linearly constrained fuzzy measures using linear programming models. These methods enable a comprehensive exploration and coverage of the entire feasible convex domain. The first category involves randomly selecting a subset and assigning measure values within the allowable range under given linear constraints. The second category utilizes convex combinations of constrained extreme fuzzy measures and vertex fuzzy measures. Then we employ some indices of fuzzy measures, objective functions, and distances to domain boundaries to evaluate the coverage performance of these methods across the entire feasible domain. We further provide enhancement techniques to improve the coverage ratios. Finally, we discuss and demonstrate potential applications of these generation methods in practical scenarios.

When dealing with a fuzzy measure on$n$elements, verifying satisfaction of the monotonicity conditions typically requires performing$n2^{n-1}$comparisons on measure values, while checking the convexity conditions involves$\binom{n}{2} 2^{n-2}$comparisons among marginal contributions. The exponential computation required for these checks in fuzzy measure optimization models often leads heuristic algorithms into numerous challenging situations. In this contribution, we propose efficient comparison algorithms based on sorting methods, linear extensions of fuzzy measures, and partial orders on set pairs of marginal contributions. With the aid of these algorithms, the computational complexity is substantially reduced to a linear level on average. Our numerical experiments confirm the significant benefit when it comes to scenarios with large values of$n$, (e.g.,$n>10$), allowing us to apply these methods to problems that were previously intractable.

Our study focuses on using fuzzy measures and integrals in optimization with interacting decision variables. We optimize piecewise linear objectives through the use of discrete Choquet integral, while s.t. a set of linear constraints. These optimization models are relevant in scenarios where decision variables interact in complex ways, with potential synergies or redundancies among them. Fuzzy measures, which are typically nonadditive and nonmodular set functions, provide a solid mathematical foundation for modeling such interactions. By extending traditional linear objectives, the discrete Choquet integral is a versatile tool for addressing these optimization problems. We decompose fuzzy measures with two opposite kinds of interactions, into submodular and supermodular parts, which accordingly transfer into the convex and concave components of optimized objectives. We customize the difference of convex algorithm for our specific objective functions that involve positive and negative interactions, and investigate its numerical scalability.

Random generation of fuzzy measures can be viewed as assigning$2^{n}$ordered random values from the unit interval to the linear extension of power-sets of inputs. Several recently proposed methods for constructing linear extensions have high computational cost. We propose a numerically efficient approach that directly obtains the linear extensions from convex combinations of 0-1 fuzzy measures, reducing the computational complexity to$O(n2^{n})$. The resulting algorithm is very short but is effective in generating a broad range of fuzzy measures as demonstrated by experiments. The full C++ code is presented.

Integrals defined with respect to fuzzy measures (capacities) are powerful tools in multicriteria decision making. Monotonicity is a basic property of capacity, which means that the marginal contribution of any single criterion to any subset of criteria is always nonnegative. In this paper, we present the capacity-based decision making theory in terms of marginal contributions, which provides an alternative perspective to this widely used decision making strategy. We construct the marginal contribution representations of the equivalent transformations of capacities, some particular capacities, three types of nonlinear integrals, and discuss the capacity identification methods. We also introduce some new concepts and representations, such as nonadditivity and nonmodularity indices, 0 to 1 variables-based linear constraints of k-maxitive capacity, a special representation of the Choquet integral and pan integral. We discuss constraints on marginal contributions which ensure supermodularity of capacities. Finally, an illustrative example is given to show the use of marginal contribution presentation in capacity-based decision making methods.

The probabilistic simultaneous interaction index has been widely adopted to measure the interaction among the decision criteria. However, this type of indices sometimes fails to reflect the kind of interaction associated with the nonadditivity of a fuzzy measure (capacity). For example, any simultaneous interaction index of the universal set of criteria w.r.t. a strictly superadditive capacity is not always positive. The main reason is that the simultaneous interaction index generalizes the notion of value by replacing the marginal contribution of a single criterion with the marginal simultaneous interaction of criteria subset. In this paper, we reform the generalization process and replace the marginal contribution with the marginal bipartition interaction, which can better reflect the kind of interaction associated with the nonadditivity, for example, superadditivity, subadditivity, strict-superadditivity, or strict-subadditivity. We construct a family of probabilistic bipartition interaction indices of subsets of criteria and study its properties. We discuss the issue of capacity identification based on the bipartition interaction index and demonstrate that the new type of interaction index can be adopted as a feasible alternative to describing the interaction phenomenon among the decision criteria.

Nonadditive robust ordinal regression (NAROR) is a widely adopted approach to analyze and reveal the dominance relationships among all decision alternatives based on nonadditive measures, called capacities. In this paper, we first investigate some advantages of the nonadditivity index as an explicit interaction index, as compared with the traditional probabilistic simultaneous interaction indices, and show that nonadditivity index can serve as an equivalent representation of a capacity. Then we enhance the NAROR method by using nonadditivity index as well as multiple-goal linear programming, where the former is used to replace the traditional interaction index to more naturally represent the decision maker's preferences, and the latter aims to replace the 0 to 1 mixed integer programming to enhance the ability to detect and adjust contradictory and redundant preference information. The updated NAROR's steps are constructed and discussed in detail and illustrated with a practical example.

The essential role of the particular families of capacities and the capacity identification methods is to help the decision maker to deal with the exponential complexity inherent in the construction process of the capacity. The 2-additive capacities appear to be the most popular among the particular families of capacities since they permit to model interactions between criteria while preserving simplicity. Besides the preference with respect to the decision criteria, most of the capacity identification methods also need to provide the desired overall evaluations of the decision alternatives in the learning set, which is a time-consuming task for the decision maker. In this paper, we propose some models to identify 2-additive capacities only from a kind of refined preference information with respect to the decision criteria called the multicriteria correlation preference information (MCCPI). The MCCPI is a group of 2-D preference information which can be described and obtained by the refined diamond diagram. The common principle of the proposed models is to minimize the different kinds of deviations between the MCCPI and the most desired 2-additive capacity(ies). A multicriteria decision making example is presented to show the feasibility of the proposed methods, and a 2-D scale of the MCCPI is also introduced in the further discussion of the illustrative example.