Types Of Uncertainty Sets Research Articles

Robust liner ship routing and scheduling schemes under uncertain weather and ocean conditions

Since sailing along the Northern Sea Route (NSR) through the Arctic Ocean comes into reality, commercial shipping developments including opening new liner services along this route have been put on the agenda. However, uncertainties within weather and ocean conditions in this region, especially during temperature-varying seasons, may cause failure to on-time arrivals, resulting unignorable or even unaffordable monetary losses. As a result, these uncertainties will inevitably affect cargo shippers’ willingness of choosing this being-explored route, even if on average its shipping time is much shorter than that of the traditional Asia-Europe shipping lines. In this context, the present paper describes a liner ship routing and scheduling problem considering schedule-sensitive demand and late-arrival penalty for operating incoming NSR shipping lines in the future. By using two types of uncertainty sets, bounded and budget-bounded uncertainty sets, we construct and solve two robust counterparts of this problem. The deterministic model and bounded robust model can be seen as special cases of the budget-bounded robust model when the uncertainty budget is set to null and full, respectively. A multifaceted case study of planning liner shipping routes and schedules along the NSR is conducted to validate the efficacy and efficiency of the proposed models and identify the effects of uncertainty bounds and budgets on the solution performance. The following facts and conclusions are revealed from the optimization results of the case study: The budget-bounded robust model functions the best in the worst condition; low revenue rate and high demand sensitivity are the major factors for excluding a candidate port from the route; the order of a visited port in the route decides the weight of its uncertainty budget on the entire shipping line; setting the uncertainty budget in a decelerative way will enhance the scheduling robustness of solutions.

The robust bilevel continuous knapsack problem with uncertain coefficients in the follower’s objective

We consider a bilevel continuous knapsack problem where the leader controls the capacity of the knapsack and the follower chooses an optimal packing according to his own profits, which may differ from those of the leader. To this bilevel problem, we add uncertainty in a natural way, assuming that the leader does not have full knowledge about the follower’s problem. More precisely, adopting the robust optimization approach and assuming that the follower’s profits belong to a given uncertainty set, our aim is to compute a solution that optimizes the worst-case follower’s reaction from the leader’s perspective. By investigating the complexity of this problem with respect to different types of uncertainty sets, we make first steps towards better understanding the combination of bilevel optimization and robust combinatorial optimization. We show that the problem can be solved in polynomial time for both discrete and interval uncertainty, but that the same problem becomes NP-hard when each coefficient can independently assume only a finite number of values. In particular, this demonstrates that replacing uncertainty sets by their convex hulls may change the problem significantly, in contrast to the situation in classical single-level robust optimization. For general polytopal uncertainty, the problem again turns out to be NP-hard, and the same is true for ellipsoidal uncertainty even in the uncorrelated case. All presented hardness results already apply to the evaluation of the leader’s objective function.

Two-Stage Distributionally Robust Optimization for a Two-Allocation p-Hub Median Problem

This paper presents a novel two-stage distributionally robust optimization model of the two-allocation p-hub median problem with different hub link scales. With the objective of minimizing overall costs of building and operating the hub network, the choices of hub locations and hub link scales are decided in the first stage, while the optimal flows are determined in the second stage once the uncertain demands have been realized. Before establishing the hub network, we just have partial distribution information about the uncertain flow demands, which can be described by a given perturbation set based on the historical information. Due to the ambiguous distributions leading to a computationally intractable model, we reformulate the proposed model into the tractable robust counterpart forms under two types of uncertainty sets (Box[Formula: see text]ellipsoidal perturbation set and Generalized ellipsoidal perturbation set). Finally, to demonstrate the effectiveness and applicability for our model, we conduct a case study for the express network system in the Beijing–Tianjin–Hebei region.

Consensus Modeling with Asymmetric Cost Based on Data-Driven Robust Optimization

The robust optimization method has progressively become a research hot spot as a valuable means for dealing with parameter uncertainty in optimization problems. Based on the asymmetric cost consensus model, this paper considers the uncertainties of the experts’ unit adjustment costs under the background of group decision making. At the same time, four uncertain level parameters are introduced. For three types of minimum cost consensus models with direction restrictions, including MCCM-DC, $$\varepsilon $$ -MCCM-DC and threshold-based (TB)-MCCM-DC, the robust cost consensus models corresponding to four types of uncertainty sets (Box set, Ellipsoid set, Polyhedron set and Interval-Polyhedron set) are established. Sensitivity analysis is carried out under different parameter conditions to determine the robustness of the solutions obtained from robust optimization models. The robust optimization models are then compared to the minimum cost models for consensus. The example results show that the Interval-Polyhedron set’s robust models have the smallest total costs and strongest robustness. Decision makers can choose the combination of uncertainty sets and uncertain levels according to their risk preferences to minimize the total cost. Finally, in order to reduce the conservatism of the classical robust optimization method, the pricing information of the new product MACUBE 550 is used to build a data-driven robust optimization model. Ellipsoid uncertainty set is proved to better trade-off the average performance and robust performance through different measurement indicators. Therefore, the uncertainty set can be selected according to the needs of the group.

A mixed-integer conic programming formulation for computing the flexibility index under multivariate gaussian uncertainty

We present a methodology for computing the flexibility index when uncertainty is characterized using multivariate Gaussian random variables. Our approach computes the flexibility index by solving a mixed-integer conic program (MICP). This methodology directly characterizes ellipsoidal sets to capture correlations in contrast to previous methodologies that employ approximations. We also show that, under a Gaussian representation, the flexibility index can be used to obtain a lower bound for the so-called stochastic flexibility index (i.e., the probability of having feasible operation). Our results also show that the methodology can be generalized to capture different types of uncertainty sets.

Robust Coordinated Transmission and Generation Expansion Planning Considering Ramping Requirements and Construction Periods

Two critical issues have arisen in transmission expansion planning with the rapid growth of wind power generation. First, severe power ramping events in daily operation due to the high variability of wind power generation pose great challenges to multi-year planning decision making. Second, the long construction periods of transmission lines may not be able to keep pace with the fast growing uncertainty due to the increasing integration of wind power generation. To address such issues, we propose a comprehensive robust planning model considering different resources, namely, transmission lines, generators, and FACTS devices. Various factors are taken into account, including flexibility requirements, construction period, and cost. We construct the hourly net load ramping uncertainty (HLRU) set to characterize the variation of hourly net load including wind power generation, and the annual net load duration curve uncertainty (LDCU) set for the uncertainty of normal annual net load duration curve. This results in a two-stage robust optimization model with two different types of uncertainty sets, which are decoupled into two different sets of subproblems to make the entire solution process tractable. Numerical simulations with real-world data show that the proposed model and solution method are effective in coordinating different flexible resources and rendering robust expansion planning strategies.

Robust solutions to box-constrained stochastic linear variational inequality problem

We present a new method for solving the box-constrained stochastic linear variational inequality problem with three special types of uncertainty sets. Most previous methods, such as the expected value and expected residual minimization, need the probability distribution information of the stochastic variables. In contrast, we give the robust reformulation and reformulate the problem as a quadratically constrained quadratic program or convex program with a conic quadratic inequality quadratic program, which is tractable in optimization theory.

Convex Relaxations of Chance Constrained AC Optimal Power Flow

High penetration of renewable energy sources and the increasing share of stochastic loads require the explicit representation of uncertainty in tools such as the optimal power flow (OPF). Current approaches follow either a linearized approach or an iterative approximation of non-linearities. This paper proposes a semidefinite relaxation of a chance constrained AC-OPF which is able to provide guarantees for global optimality. Using a piecewise affine policy, we can ensure tractability, accurately model large power deviations, and determine suitable corrective control policies for active power, reactive power, and voltage. We state a tractable formulation for two types of uncertainty sets. Using a scenario-based approach and making no prior assumptions about the probability distribution of the forecast errors, we obtain a robust formulation for a rectangular uncertainty set. Alternatively, assuming a Gaussian distribution of the forecast errors, we propose an analytical reformulation of the chance constraints suitable for semidefinite programming. We demonstrate the performance of our approach on the IEEE 24 and 118 bus system using realistic day-ahead forecast data and obtain tight near-global optimality guarantees.

An integrated supply chain configuration model and procurement management under uncertainty: A set-based robust optimization methodology

Supply chain configuration and supplier selection are two significant strategic decision making problems in supply chain management. In order to tackle the uncertain environment in these two problems, we propose an integrated mathematical programming model based on robust optimization theory. The framework of the proposed supply chain network involves a forward flow from heterogeneous capacitated suppliers offering price discounts to customers, and also a reverse flow from customer zones to disposal centers or production facilities. The objective function of the model is to minimize the total cost of the supply chain network by determining a set of best suppliers, order allocation to selected suppliers, the location of facilities in each layer and transportation quantities between them. Moreover, to measure the imprecise input parameters, according to three types of uncertainty sets, the robust counterparts of the model have been developed. Finally, to validate the robust model, the eventual optimal solutions are compared with the deterministic model, and numerical studies have been implemented along with sensitivity analysis.

Generalized light robustness and the trade-off between robustness and nominal quality

Robust optimization considers optimization problems with uncertainty in the data. The common data model assumes that the uncertainty can be represented by an uncertainty set. Classic robust optimization considers the solution under the worst case scenario. The resulting solutions are often too conservative, e.g. they have high costs compared to non-robust solutions. This is a reason for the development of less conservative robust models. In this paper we extract the basic idea of the concept of light robustness originally developed in Fischetti and Monaci (Robust and online large-scale optimization, volume 5868 of lecture note on computer science. Springer, Berlin, pp 61–84, 2009) for interval-based uncertainty sets and linear programs: fix a quality standard for the nominal solution and among all solutions satisfying this standard choose the most reliable one. We then use this idea in order to formulate the concept of light robustness for arbitrary optimization problems and arbitrary uncertainty sets. We call the resulting concept generalized light robustness. We analyze the concept and discuss its relation to other well-known robustness concepts such as strict robustness (Ben-Tal et al. in Robust optimization. Princeton University Press, Princeton, 2009), reliability (Ben-Tal and Nemirovski in Math Program A 88:411–424, 2000) or the approach of Bertsimas and Sim (Oper Res 52(1):35–53, 2004). We show that the light robust counterpart is computationally tractable for many different types of uncertainty sets, among them polyhedral or ellipsoidal uncertainty sets. We furthermore discuss the trade-off between robustness and nominal quality and show that non-dominated solutions with respect to nominal quality and robustness can be computed by the generalized light robustness approach.

Robust vehicle routing problem with deadlines and travel time/demand uncertainty

In this article, we investigate the vehicle routing problem with deadlines, whose goal is to satisfy the requirements of a given number of customers with minimum travel distances while respecting both of the deadlines of the customers and vehicle capacity. It is assumed that the travel time between any two customers and the demands of the customer are uncertain. Two types of uncertainty sets with adjustable parameters are considered for the possible realizations of travel time and demand. The robustness of a solution against the uncertain data can be achieved by making the solution feasible for any travel time and demand defined in the uncertainty sets. We propose a Dantzig-Wolfe decomposition approach, which enables the uncertainty of the data to be encapsulated in the column generation subproblem. A dynamic programming algorithm is proposed to solve the subproblem with data uncertainty. The results of computational experiments involving two well-known test problems show that the robustness of the solution can be greatly improved.