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Interval games are an extension of cooperative coalitional games, in which players are assumed to face payoff uncertainty. Characteristic functions thus assign a closed interval, instead of a real number. In this paper, we first examine the notion of solution mapping, a solution concept applied to interval games, by comparing it with the existing solution concept called the interval solution concept. Then, we define a Shapley mapping as a specific form of the solution mapping. Finally, it is shown that the Shapley mapping can be characterized by two different axiomatizations, both of which employ interval game versions of standard axioms used in the traditional cooperative game analysis such as efficiency, symmetry, null player property, additivity and separability.

Interval games are an extension of cooperative coalitional games, in which players are assumed to face payoff uncertainty. Characteristic functions thus assign a closed interval instead of a real number. This study revisits two interval game versions of Shapley values (i.e., the interval Shapley value and the interval Shapley-like value) and characterizes them using an axiomatic approach. For the interval Shapley value, we show that the existing axiomatization can be generalized to a wider subclass of interval games called size monotonic games. For the interval Shapley-like value, we show that a standard axiomatization using Young's strong monotonicity holds on the whole class of interval games.

We address the robust vehicle routing problem with time windows (RVRPTW) under customer demand and travel time uncertainties. As presented thus far in the literature, robust counterparts of standard formulations have challenged general-purpose optimization solvers and specialized branch-and-cut methods. Hence, optimal solutions have been reported for small-scale instances only. Additionally, although the most successful methods for solving many variants of vehicle routing problems are based on the column generation technique, the RVRPTW has never been addressed by this type of method. In this paper, we introduce a novel robust counterpart model based on the well-known budgeted uncertainty set, which has advantageous features in comparison with other formulations and presents better overall performance when solved by commercial solvers. This model results from incorporating dynamic programming recursive equations into a standard deterministic formulation and does not require the classical dualization scheme typically used in robust optimization. In addition, we propose a branch-price-and-cut method based on a set partitioning formulation of the problem, which relies on a robust resource-constrained elementary shortest path problem to generate routes that are robust regarding both vehicle capacity and customer time windows. Computational experiments using Solomon’s instances show that the proposed approach is effective and able to obtain robust solutions within a reasonable running time. The results of an extensive Monte Carlo simulation indicate the relevance of obtaining robust routes for a more reliable decision-making process in real-life settings.

An interval parameter sensitivity analysis is developed to quantify the impact of simulation model parameters on the model outputs. This sensitivity analysis contains two main steps: the interval uncertainty propagation and the interval sensitivity index. The interval perturbation method is introduced to estimate the extreme bounds of model outputs according to the interval input parameters, which significantly reduces the computation cost of extensive Monte Carlo simulations. Since the output of the interval model are interval quantities, the traditional probabilistic sensitivity method and its sensitivity index are inappropriate as we only have the bounds of samples without inner data points. Hence, this work proposes an interval similarity operator based on the relative interval position operator, which is applicable to measure the variation of interval outputs. This interval sensitivity operator mainly quantifies the discrepancy between intervals based on six typical cases of the interval relative position. Finally, an academic case and a satellite structure case are analyzed to verify the feasibility and efficiency of the proposed method.

This paper is devoted to the construction of a nonintrusive interval uncertainty propagation approach for the response bounds evaluation of multibody systems. The motivation for this effort is twofold. First, the traditional methods using the Taylor inclusion function and interval arithmetic usually lead to the wrapping effect. Second, the real-life multibody dynamics models are mostly large systems, which are highly rigid, nonlinear, and discontinuous; however, many conventional, intrusive interval analysis methods are not suitable for such large, complicated multibody systems. To end these, a polynomial chaos inclusion function using Legendre orthogonal basis is presented for analyzing such multibody dynamics models with interval uncertainty, where the Galerkin projection method is adopted to compute the Legendre polynomial coefficients. The capacity of the Legendre polynomial inclusion function to alleviate the wrapping effect is proved by a mathematical example. Through sampling, the nonintrusive algorithm expresses the original multibody dynamics system with interval uncertainty as the deterministic differential algebraic equations, followed by calculation using the general numerical integration method. The response bounds at each time step are predicted using the truncated Legendre polynomial expansion. A benchmark test based on three methods is analyzed to demonstrate the effectiveness of this approach. Moreover, an artillery multibody dynamics model created in ADAMS/Solver can reproduce a suite of experimental results, and is then specifically investigated to illustrate the superiority of this method in large, complicated multibody dynamic systems.

A bivariate Chebyshev polynomials approach is proposed to estimate the dynamic response bounds of nonlinear systems with interval uncertainties. The existing collocation method directly searches the maximum and minimum values of the surrogate model in the entire interval space by the scanning method (SM). The presence of too many uncertain parameters will lead to expansive computational cost. To overcome this shortcoming, the dynamic response is decomposed by a bivariate function decomposition (BFD), established based on high-order Taylor expansion, into the sum of multiple univariate and bivariate response functions. The above univariate and bivariate functions are fitted using Chebyshev polynomials, and polynomial coefficients are obtained through one-dimensional (1D) and two-dimensional (2D) interpolation points. Thus, the solution of the nonlinear dynamic systems with uncertain parameters can be transformed into that of univariate and bivariate Chebyshev interval functions. The extremum values of the low-dimensional Chebyshev interval functions can be found by SM, and then the bounds of dynamic response are acquired by interval arithmetic. Since SM searches for extreme values only in 1D and 2D uncertain domains, the amount of calculation is reduced compared to searching the whole uncertain space. The efficiency, practicability and effectiveness of the proposed interval uncertainty analysis method are proved by three dynamic examples.

The parameter uncertainty has an important effect on the motion planning of overhead cranes, especially in relation to its industrial safety of production activities. Thus, a novel uncertain estimation-and-optimization strategy is proposed for motion planning of overhead cranes with uncertainty in this paper. The main work of this paper includes the following aspects. First, the overhead crane is simplified as a double pendulum model and the corresponding motion planning is described as an optimal control problem with uncertainty. Second, uncertainties are expressed as interval parameters where only the upper and lower bounds are required without probability information and a bounds estimation problem for optimal control with uncertainty is established; the solution contains all possible values. Third, the bounds estimation problem is solved by a surrogate model-based method, the motion trajectory intervals of overhead cranes are obtained. Fourth, in order to reduce the influence of uncertainty on the motion planning of overhead cranes, an optimization method is introduced to reduce the sensitivity to uncertainty. Finally, the numerical examples show that high accurate interval estimation results are obtained with a reasonable computational cost, and the sensitivity of motion trajectory to uncertainty is reduced obviously with the help of optimization. The proposed strategy provides a guidance for uncertain analysis and online controller design of overhead cranes.

With the increasingly severe energy supply and environmental pressures, high-end equipment is gradually adopted to reduce the carbon emissions of manufacturing industry which makes its low-carbon structural design a critical research hotspot. The best structural scheme can be got by multi-attribute decision-making (MADM) with design requirements. However, the decision-making attributes in the structural design of high-end equipment are too many at first and low-carbon attributes are seldom fully considered. Moreover, there are a large amount of related data with linguistic vagueness, interval uncertainty, and information incompleteness, which fail to be handled simultaneously. There, this paper proposes an integrated MADM method of low-carbon structural design for high-end equipment based on attribute reduction considering incomplete interval uncertainties. First, distribution reduction of low-carbon structural design is carried out to obtain the minimum attribute set and encompass low-carbon attributes comprehensively. Second, a collaborative filtering algorithm is utilized to complete the missing data in the subsequent design process. Third, interval rough numbers (IRNs) are integrated into DEMATEL-ANP (DANP) and multi-attribute border approximation area comparison (MABAC) to quickly rank the alternative schemes for high-end equipment and determine which is the best. The rationality and robustness of the proposed method are verified through the case study and comparative analysis of a hydraulic forming machine.

This paper explores the study of multi-choice multi-objective transportation problem (MCMTP) under the light of conic scalarizing function. MCMTP is a multi-objective transportation problem (MOTP) where the parameters such as cost, demand and supply are treated as multi-choice parameters. A general transformation procedure using binary variables is illustrated to reduce MCMTP into MOTP. Most of the MOTPs are solved by goal programming (GP) approach, but the solution of MOTP may not be satisfied all times by the decision maker when the objective functions of the proposed problem contains interval-valued aspiration levels. To overcome this difficulty, here we propose the approaches of revised multi-choice goal programming (RMCGP) and conic scalarizing function into the MOTP, and then we compare among the solutions. Two numerical examples are presented to show the feasibility and usefulness of our paper. The paper ends with a conclusion and an outlook on future studies.

We study the model of an integer linear program with binary variables within the framework of interval programming, which can be used to represent various optimization problems whose input data are affected by interval-valued uncertainty. In the considered model, the inexact or uncertain coefficients of the integer program can be independently perturbed within the given lower and upper bounds. In this paper, we characterize the main properties of 0–1 interval linear programs with respect to feasibility and optimality. Namely, we discuss the feasible and optimal solutions in the weak and in the strong sense, i.e. solutions feasible or optimal for some or for each choice of the interval data. We also address the problem of computing the best and the worst optimal value, as well as the related problem of describing the possibly disconnected set of all optimal values. Due to the dependency problem inherently present in interval programming, we study formulations involving both equations and inequality constraints. Moreover, we prove that for pure 0–1 interval linear programs the standard transformation of splitting an equation constraint into two opposite inequalities preserves the set of (weakly) optimal solutions, which is generally not true for interval linear programs with continuous variables.