In this study, we investigate an appointment sequencing and scheduling problem with heterogeneous user delay tolerances under service time uncertainty. We aim to capture the delay tolerance effect with heterogeneity, in an operationally effective and computationally tractable fashion, for the appointment scheduling problem. To this end, we first propose a Tolerance-Aware Delay (TAD) index that incorporates explicitly the user tolerance information in delay evaluation. We show that the TAD index enjoys decision-theoretical rationale in terms of Tolerance sensitivity, monotonicity, and convexity and positive homogeneity, which enables it to incorporate the frequency and intensity of delays over the tolerance in a coherent manner. Specifically, the convexity of TAD index ensures a tractable modeling of the collective delay dissatisfaction in the appointment scheduling problem. Using the TAD index, we then develop an appointment model with known empirical service time distribution that minimizes the overall tolerance-aware delays of all users. We analyze the impact of delay tolerance on the sequence and schedule decisions and show that the resultant TAD appointment model can be reformulated as a mixed-integer linear program (MILP). Furthermore, we extend the TAD appointment model by considering service time ambiguity. In particular, we encode into the TAD index a moment ambiguity set and a Wasserstein ambiguity set, respectively. The former captures effectively the correlation among service times across positions and user types, whereas the latter captures directly the service time data information. We show that both of the resultant TAD models under ambiguity can be reformulated as polynomial-sized, mixed-integer conic programs (MICPs). Finally, we compare our TAD models with some existing counterpart approaches and the current practice using synthetic data and a case of real hospital data, respectively. Our results demonstrate the effectiveness of the TAD appointment models in capturing the user delay tolerance with heterogeneity and mitigating the worst-case delays. History: Accepted by Pascal Van Hentenryck, Area Editor for Computational Modeling: Methods & Analysis. Funding: S. Wang was supported by the National Natural Science Foundation of China [Grants 71922020, 72171221, and 71988101, entitled “Econometric Modeling and Economic Policy Studies”], the Fundamental Research Funds for the Central Universities [Grant UCAS-E2ET0808X2], and the Major Program of National Natural Science Foundation of China [Grant 72192843]. S. Wang was also supported by a grant from MOE Social Science Laboratory of Digital Economic Forecasts and Policy Simulation at UCAS. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2023.0025 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2023.0025 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

We demonstrate a compact, cost-effective snapshot spectral imaging system named Aperture Diffraction Imaging Spectrometer (ADIS), which consists only of an imaging lens with an ultra-thin orthogonal aperture mask and a mosaic filter sensor, requiring no additional physical footprint compared to common RGB cameras. Then we introduce a new optical design that each point in the object space is multiplexed to discrete encoding locations on the mosaic filter sensor by diffraction-based spatial-spectral projection engineering generated from the orthogonal mask. The orthogonal projection is uniformly accepted to obtain a weakly calibration-dependent data form to enhance modulation robustness. Meanwhile, the Cascade Shift-Shuffle Spectral Transformer (CSST) with strong perception of the diffraction degeneration is designed to solve a sparsity-constrained inverse problem, realizing the volume reconstruction from 2D measurements with Large amount of aliasing. Our system is evaluated by elaborating the imaging optical theory and reconstruction algorithm with demonstrating the experimental imaging under a single exposure. Ultimately, we achieve the sub-super-pixel spatial resolution and high spectral resolution imaging. The code will be available at: https://github.com/Krito-ex/CSST.

We extend the notion of globalized robustness to consider distributional information beyond the support of the ambiguous probability distribution. We propose the globalized distributionally robust counterpart that disallows any (respectively, allows limited) constraint violation for distributions residing (respectively, not residing) in the ambiguity set. By varying its inputs, our proposal recovers several existing perceptions of parameter uncertainty. Focusing on the type 1 Wasserstein distance, we show that the globalized distributionally robust counterpart has an insightful interpretation in terms of shadow price of globalized robustness, and it can be seamlessly integrated with many popular optimization models under uncertainty without incurring any extra computational cost. Such computational attractiveness also holds for other ambiguity sets, including the ones based on probability metric, optimal transport, ϕ-divergences, or moment conditions, as well as the event-wise ambiguity set. Numerical studies on an adaptive network lot-sizing problem demonstrate the modeling flexibility of our proposal and its emphases on globalized robustness to constraint violation. History: Antonio Frangioni, Area Editor for Design & Analysis of Algorithms—Continuous. Funding: Z. Chen was supported by [General Research Fund Grant 9043424, NSFC/RGC Joint Research Scheme N_CityU105/21] from the Hong Kong Research Grants Council. S. Wang was supported by the National Natural Science Foundation of China [Grants 71922020, 72171221, and 71988101, entitled “Econometric Modeling and Economic Policy Studies”] and the Fundamental Research Funds for the Central Universities [Grant UCAS-E2ET0808X2]. Supplemental Material: The software that supports the findings of this study is available within the paper and its Supplemental Information ( https://pubsonline.informs.org/doi/suppl/10.1287/ijoc.2022.0274 ) as well as from the IJOC GitHub software repository ( https://github.com/INFORMSJoC/2022.0274 ). The complete IJOC Software and Data Repository is available at https://informsjoc.github.io/ .

This work focuses on a broad class of facility location problems in the context of adaptive robust stochastic optimization under the state-dependent demand uncertainty. The demand is assumed to be significantly affected by related state information, such as the seasonal or socio-economic information. In particular, a state-wise ambiguity set is adopted for modeling the distributional uncertainty associated with the demand in different states. The conditional distributional characteristics in each state are described by a support, as well as by mean and dispersion measures, which are assumed to be conic representable. A robust sensitivity analysis is performed, in which, on the one hand, we analyze the impact of the change in ambiguity-set parameters (e.g., state probabilities, mean value abounds, and dispersion bounds in different states) onto the optimal worst-case expected total cost using the ambiguity dual variables. On the other hand, we analyze the impact of the change in location design onto the worst-case expected second-stage cost and show that the sensitivity bounds are fully described as the worst-case expected shadow-capacity cost. As for the solution approach, we propose a nested Benders decomposition algorithm for solving the model exactly, which leverages the subgradients of the worst-case expected second-stage cost at the location decisions formed insightfully by the associated worst-case distributions. The nested Benders decomposition approach ensures a finite-step convergence, which can also be regarded as an extension of the classic L-shaped algorithm for two-stage stochastic programming to our state-wise, robust stochastic facility location problem with conic representable ambiguity. Finally, the results of a series of numerical experiments are presented that justify the value of the state-wise distributional information incorporated in our robust stochastic facility location model, the robustness of the model, and the performance of the exact solution approach.

In this paper, we consider a redundancy allocation problem for a series parallel system with uncertain component lifetimes that minimizes system costs while safeguarding system reliability over a given threshold level. We consider mixed redundancy strategies of cold standby and active redundancy with multiple types of components. We address lifetime uncertainty in the framework of distributionally robust optimization. In particular, we assume the probability distributions of the component lifetimes are not exactly known with only limited distributional information (e.g., mean, dispersion, and support) being available. We protect the worst-case system reliability constraint over all the possible component lifetime distributions that are consistent with the given distributional characteristics. The proposed modeling framework enjoys computationally attractive structures. The evaluation of the worst-case system reliability in our redundancy allocation problem can be transformed into a linear program, and the resulting overall redundancy allocation optimization problem can be cast as a mixed integer linear program that does not induce any additional integer variables (other than original allocation variables). In addition, the extreme joint distribution of component lifetimes can be efficiently recovered by solving a linear program. Our modeling framework can also be extended to incorporate the startup failures and common-cause failures for cold standbys and active parallels, respectively, to cater to more computationally complex settings. Finally, the computational experiments positively demonstrate the performance of the proposed approach in protecting system reliability.

In this paper, we consider a generic class of adaptive optimization problems under uncertainty, and develop a data-driven paradigm of adaptive probabilistic robust optimization (APRO) in a robust and computationally tractable manner. The paradigm comprises two phases: 1) bilayer information granulation (IG), which involves the data-mining techniques and nested decomposition of convex sets that establish and restructure the knowledge from data and 2) robustization and optimization over the restructured knowledge by the IG, which forms the APRO model. The tradeoff between the solution optimality and the robustness of the resulting data-driven APRO model can be achieved by adjusting the number of clusters and the number of nested decomposition units of the IG process. We draw the connections of the APRO model with the stochastic programming and the regular robust optimization models, respectively, and show that the APRO model can be regarded as a generalized version of both models. We show that the APRO model can be transformed into the second-order conic programming which is computationally tractable and can be solved efficiently by the off-the-shelf solvers. Furthermore, the model can be extended by robustizing the probability parameters. Finally, an application on two-stage facility location planning is presented, and the computational results demonstrate the performance and the insights of using the data-driven APRO models.

Considering nonstatistical uncertainties and/or insufficient historical data in security return forecasts, fuzzy set theory has been applied in the past decades to build portfolio selection models. Meanwhile, various risk measurements such as variance, entropy, and Value-at-Risk have been proposed in fuzzy environments to evaluate investment risks from different perspectives. Sharpe ratio, also known as the reward-to-variability ratio, which measures the risk premium per unit of the nonsystematic risk (asset deviation), has received great attention in modern portfolio theory. In this study, the Sharpe ratio in fuzzy environments is introduced, whereafter, a fuzzy Value-at-Risk ratio is proposed. Compared with Sharpe ratio, Value-at-Risk ratio is an index with dimensional knowledge that reflects the risk premium per unit of the systematic risk (the greatest loss under a given confidence level). On the basis of the two ratios, a multi-objective model is built to evaluate their joint impact on portfolio selection. Then, the proposed model is solved by a fuzzy simulation based multi-objective particle swarm optimization algorithm, where the global best of each iteration is determined by an improved dominance times based method. Finally, the algorithm superiority is justified via comparing with existing solvers on benchmark problems, and the model effectiveness is exemplified by using three case studies on portfolio selection.

In this study, we consider a portfolio-optimization-incorporated budget investment problem under managers' risk tolerance ambiguity. In order to capture the decision dynamics driven by the risk tolerance ambiguity, a two-stage adaptive optimization model is developed. The budget allocation is the first-stage decision, which is made before knowing each manager's actual risk tolerance level, and the portfolio selection conducted by each manager is the second-stage decision, which adapts to the manager's risk tolerance. We introduce the concept of risk-neutral budget threshold (RNBT) that is modeled by a fuzzy set granule, and upon which the ambiguous risk tolerance curve is constructed, which can realistically capture the managers' risk-averse and/or risk-seeking attitudes. Due to the (realistic) nonconvex/nonconcave structure of the risk tolerance curve, and the existence of the ambiguity, the resulting problem is essentially a nonconvex adaptive optimization problem under uncertainty. To achieve a robust modeling and an efficient solution, we first restructure and robustize the information of fuzzy RNBTs and then transform the developed model into a mixed integer linear programming (MILP), which can be handled efficiently by off-the-shelf mixed integer program solvers. Leveraging the derived MILP structure, we can use the Benders decomposition to further enhance the scalability of the model. Furthermore, some model extensions on robustizing the probability estimations are discussed. Finally, computational studies are performed to demonstrate the effectiveness and insights of the model.

Solving optimization problems under hybrid uncertainty bears a heavy computational burden. In this study, we propose a unified structured optimization approach, termed robust granular optimization (RGO), to tackle the optimization problems under hybrid manifold uncertainties in a computationally tractable manner. Essentially, the RGO can be regarded as a complementary fusion of granular computing and robust optimization techniques. The paradigm of RGO consists of three core phases: 1) uncertainty identification, 2) information granulation in which basic granular units (BGUs) are formed, and 3) robust optimization realized over the BGUs. Following the proposed paradigm, we develop two classes of RGO models for general single-stage and two-stage optimization problems with separable and higher order hybrid uncertainties, respectively. It is shown that both types RGO models can be equivalently transformed into linear programs or mixed integer linear programs that can be handled efficiently by off-the-shelf solvers. Furthermore, a target-based tradeoff model is developed to enhance the flexibility of the RGO models in balancing the granularity level (or robustness level) and the solution conservativeness. The tradeoff model can also be efficiently solved by a binary search algorithm. Finally, sufficient computational studies are presented, and comparisons with the existing approaches show that the RGO models can bring much higher computational efficiency and scalability without losing much optimality, and the RGO solutions exhibit a stronger resistance to the uncertainty.

In the context of robust optimization with information granules for distributional parameters, this paper investigates a two-stage waste-to-energy feedstock flow planning problem with uncertain capacity expansion costs. The objective is to minimize the worst-case overall loss in a mean-risk criterion where the risk is measured by a conditional value-at-risk operator. As a salient feature, an integrated uncertainty is considered which consists of not only the uncertainty in distribution shapes of the uncertain variables, but also the manifold uncertainties of the mean parameters. To tackle the robust optimization under such integrated uncertainty, we first discuss a distributional robust two-stage feedstock flow planning model with precise mean parameters that handles the uncertainty in distribution shape, and the model can be equivalently transformed into a linear program (LP). Furthermore, the precise-mean-based robust model is extended into the case of multifaceted uncertainty for mean-parameters that are allowed to assume intervals, historical-data-based probabilistic estimates, and/or human-knowledge-centric fuzzy set estimates, under different circumstances. These multifaceted uncertain mean-parameters are uniformly represented by using information granules, and a granular robust optimization model is then developed which maximizes the robustness of the solution within a shortfall tolerance, and realizes a tradeoff between the solution conservativeness and robustness. It is showed that the granular robust model is equivalent to solving a series of LPs and can be efficiently handled by a nested binary search algorithm. Finally, the computational study illustrates the model performance, solution analysis, and underlines a much higher scalability of the developed robust model compared to the stochastic programming approach.

Based on fuzzy value-at-risk (VaR), this paper proposes a new portfolio-selection model (PSM) called the VaR-based fuzzy PSM (VaR-FPSM). Compared with the existing FPSMs, the VaR can directly reflect the greatest loss of a selected case under a given confidence level. In this study, when the security returns are taken as trapezoidal, triangular, and Gaussian fuzzy numbers, several crisp equivalent models of the VaR-FPSM are derived, which can be handled by any linear programming solvers. In general situations, an improved particle swarm optimization algorithm on the basis of fuzzy simulation is designed to search for the approximate optimal solutions. To illustrate the proposed model and the behavior of the improved particle swarm optimization algorithm, two numerical examples are provided, and the results are discussed. Furthermore, the proposed algorithm is compared with some existing approaches to fuzzy portfolio selection, such as the genetic algorithm and simulated annealing.

Under the Value-at-Risk (VaR) criterion, this paper studies on the value of information and solution for two-stage fuzzy random optimization problems. First, the value of perfect information (VPI) in VaR criterion is discussed by studying the difference of the wait-and-see (WS) solution and the here-and-now (HN) solution to the two-stage fuzzy random programming with VaR criterion. Then, the value of fuzzy random solution (VFRS) in VaR is examined by investigating the difference of the HN solution and the random solution (RS), as well as the difference of HN solution and the expected value (EV) solution. Finally, a lower bound and an upper bound for the HN solution are derived.

The objective of this paper is to study facility-location problems in the presence of a hybrid uncertain environment involving both randomness and fuzziness. A two-stage fuzzy-random facility-location model with recourse (FR-FLMR) is developed in which both the demands and costs are assumed to be fuzzy-random variables. The bounds of the optimal objective value of the two-stage FR-FLMR are derived. As, in general, the fuzzy-random parameters of the FR-FLMR can be regarded as continuous fuzzy-random variables with an infinite number of realizations, the computation of the recourse requires solving infinite second-stage programming problems. Owing to this requirement, the recourse function cannot be determined analytically, and, hence, the model cannot benefit from the use of techniques of classical mathematical programming. In order to solve the location problems of this nature, we first develop a technique of fuzzy-random simulation to compute the recourse function. The convergence of such simulation scenarios is discussed. In the sequel, we propose a hybrid mutation-based binary ant-colony optimization (MBACO) approach to the two-stage FR-FLMR, which comprises the fuzzy-random simulation and the simplex algorithm. A numerical experiment illustrates the application of the hybrid MBACO algorithm. The comparison shows that the hybrid MBACO finds better solutions than the one using other discrete metaheuristic algorithms, such as binary particle-swarm optimization, genetic algorithm, and tabu search.

A new class of fuzzy stochastic optimization models - two-stage fuzzy stochastic programming with value-at-risk (VaR) criteria is established in this paper. An approximation algorithm is proposed to compute the VaR by combining discretization method of fuzzy variable, random simulation technique and bisection method. The convergence theorem of the approximation algorithm is also proved. To solve the two-stage fuzzy stochastic programming problems with VaR criteria, we integrate the approximation algorithm, neural network (NN) and particle swarm optimization (PSO) algorithm, and hence produce a hybrid PSO algorithm to search for the optimal solution. A numerical example is provided to illustrate the designed hybrid PSO algorithm.

The objective of this paper is to study facility location problems under a hybrid uncertain environment involving randomness and fuzziness. A two-stage fuzzy random facility location model with recourse is developed in which the demands and the costs are assumed to be fuzzy random variables. As in general the fuzzy random parameters in the model can be regarded as continuous fuzzy random variables with infinite realizations, the computation of the recourse requires solving infinite second-stage programming problems. Owing to this fact, the recourse function cannot be calculated analytically, which implies that the model cannot benefit from the use of methods of classical mathematical programming. In order to solve the location problems of this nature, we first develop techniques of fuzzy random simulation. In the sequel, by combining the fuzzy random simulation, simplex algorithm and binary particle swarm optimization (BPSO), a hybrid algorithm is proposed to solve the two-stage fuzzy random facility location model. Finally, an illustrative numerical example is provided.

In this paper, using value-at-risk, a new fuzzy portfolio selection model named VaR-FPSM is proposed. The value-at-risk is the measure of risk, which describes the greatest loss of an investment with some confidence level. When security returns are same kind of fuzzy variable, we derive two crisp equivalent forms of the VaR-FPSM. Furthermore, in general situations, we designed a fuzzy simulation based particle swarm optimization (PSO) algorithm to find an approximately optimal result. To illustrate the proposed model and hybrid PSO algorithm, a numerical example is provided and some discussions on the results are given.

In real-world regression analysis, statistical data may be linguistically imprecise or vague. Given the co-existence of stochastic and fuzzy uncertainty, real data cannot be characterized by using only the formalism of random variables. In order to address regression problems in the presence of such hybrid uncertain data, fuzzy random variables are introduced in this study to serve as an integral component of regression models. A new class of fuzzy regression models that is based on fuzzy random data is built, and is called the confidence-interval-based fuzzy random regression model (CI-FRRM). First, a general fuzzy regression model for fuzzy random data is introduced. Then, using expectations and variances of fuzzy random variables, sigma-confidence intervals are constructed for fuzzy random input-output data. The CI-FRRM is established based on the sigma-confidence intervals. The proposed regression model gives rise to a nonlinear programming problem that consists of fuzzy numbers or interval numbers. Since sign changes in the fuzzy coefficients modify the entire programming structure of the solution process, the inherent dynamic nonlinearity of this optimization makes it difficult to exploit the techniques of linear programming or classical nonlinear programming. Therefore, we resort to some heuristics. Finally, an illustrative example is provided.

Reducing risks in location decisions when coping with imprecise information is critical in supply chain management so as to increase competitiveness and profitability. In this paper, a two-stage fuzzy facility location problem with value-at-risk (VaR), called VaR-FFLP, is proposed, which results in a two-stage fuzzy zero-one integer programming problem. Some properties of the VaR-FFLP, including the value of perfect information (VPI), the value of fuzzy solution (VFS), and the bounds of the fuzzy solution, are discussed. Since the fuzzy parameters of the location problem are represented in the form of continuous fuzzy variables, the determination of VaR is inherently an infinite-dimensional optimization problem that cannot be solved analytically. Therefore, a method based on the discretization of the fuzzy variables is proposed to approximate the VaR. The approximation approach converts the original problem into a finite-dimensional optimization problem. A pertinent convergence theorem for the approximation approach is proved. Subsequently, by combining the simplex algorithm, the approximation approach, and a mechanism of genotype-phenotype-mutation-based binary particle swarm optimization (GPM-BPSO), a hybrid GPM-BPSO algorithm is being exploited to solve the VaR-FFLP. A numerical example illustrates the effectiveness of the hybrid GPM-BPSO algorithm and shows its enhanced performance in comparison with the results obtained by other approaches using genetic algorithm (GA), tabu search (TS), and Boolean BPSO (B-BPSO).