A New Methodology for Solving Multiobjective Chance-Constrained Problems: An Application on IoT Systems

This study presents a newly developed methodology to transform the chance-constrained problem into a deterministic problem and then solving this multiobjective deterministic problem with the proposed method. Chance-constrained problem contains independent gamma random variables that are denoted as a i j . Two methods are proposed to obtain the deterministic equivalent of chance-constrained problem. The first of the methods is directly based on using the distribution, and the second consists of normalizing probabilistic constraints using Lyapunov’s central limit theorem. An algorithm which uses the Global Criterion Method is developed to solve the multiobjective deterministic equivalent of chance-constrained problem. The methodology is applied to a real-life engineering problem that consists of an IoT device and its data sending process. Using Lyapunov’s central limit theorem for large numbers of random variables is found to be more appropriate.

Analysis of the Dynamical Behaviour of a Two-Dimensional Coupled Ecosystem with Stochastic Parameters

Because the two-dimensional coupled ecosystem has perfect symmetry, the dynamical behavior of symmetric dynamical system is discussed. The analysis of the dynamical behavior of a two-dimensional coupled ecosystem with stochastic parameters is explored in this paper. Firstly, a two-dimensional coupled ecosystem with stochastic parameters is established, it is transformed into a deterministic equivalent system by orthogonal polynomial approximation. Then, analysis of the dynamical behaviour of equivalently deterministic coupled ecosystems is performed using stability theory. At last, we analyzed the dynamical behaviour of non-trivial points by means of the mathematics analysis method and found the influence of random parameters on asymptotic stability in coupled ecosystem is prominent. The dynamical behaviour analysis results were verified by numerical simulation.

Welfare-maximizing transmission capacity expansion under uncertainty

We apply the JuDGE optimization package to a multistage stochastic leader–follower model that determines a transmission capacity expansion plan to maximize expected social welfare of consumers and producers who act as Cournot oligopolists in each time period. The problem is formulated as a large-scale mixed integer programme and applied to a 5-bus instance over scenario trees of varying size. The computational effort required by JuDGE is compared with solving the deterministic equivalent mixed integer programme using a state-of-the-art integer programming package. This article is part of the theme issue ‘The mathematics of energy systems’.

An Extensive Operational Law for Monotone Functions of LR Fuzzy Intervals with Applications to Fuzzy Optimization

The operational law put forward by Zhou et al. on strictly monotone functions with regard to regular LR fuzzy numbers makes a valuable push to the development of fuzzy set theory. However, its applicable conditions are confined to strictly monotone functions and regular LR fuzzy numbers, which restricts its application in practice to a certain degree. In this paper, we propose an extensive operational law that generalizes the one proposed by Zhou et al. to apply to monotone (but not necessarily strictly monotone) functions with regard to regular LR fuzzy intervals (LR-FIs), of which regular fuzzy numbers can be regarded as particular cases. By means of the extensive operational law, the inverse credibility distributions (ICDs) of monotone functions regarding regular LR-FIs can be calculated efficiently and effectively. Moreover, the extensive operational law has a wider range of applications, which can deal with the situations hard to be handled by the original operational law. Subsequently, based on the extensive operational law, the computational formulae for expected values (EVs) of LR-FIs and monotone functions with regard to regular LR-FIs are presented. Furthermore, the proposed operational law is also applied to dispose fuzzy optimization problems with regular LR-FIs, for which a solution strategy is provided, where the fuzzy programming is converted to a deterministic equivalent first and then a newly-devised solution algorithm is utilized. Finally, the proposed solution strategy is applied to a purchasing planning problem, whose performances are evaluated by comparing with the traditional fuzzy simulation-based genetic algorithm. Experimental results indicate that our method is much more efficient, yielding high-quality solutions within a short time.

Modelling and analysis of uncertain hub maximal covering location problem in the presence of partial coverage

The hub maximal covering location problem aims to find the best locations for hubs so as to maximize the total flows covered by predetermined number of hubs. Generally, this problem is defined in the framework of binary coverage. However, there are many real-life cases in which the binary coverage assumption may yield unexpected decisions. Thus, the partial coverage is considered by stipulating that the coverage of an origin-destination pair is determined by a non-increasing decay function. Moreover, as this problem contains strategic decisions in long range, the precise information about the parameters such as travel times may not be obtained in advance. Therefore, we present uncertain hub maximal covering location models with partial coverage in which the travel times are depicted as uncertain variables. Specifically, the partial coverage parameter is introduced in uncertain environment and the expected value of partial coverage parameter is further derived and simplified with specific decay functions. Expected value model and chance constrained programming model are respectively proposed and transformed to their deterministic equivalent forms. Finally, a greedy variable neighborhood search heuristic is presented and the efficiency of the proposed models is evaluated through computational experiments.

A branch-and-Benders-cut algorithm for a bi-objective stochastic facility location problem

In many real-world optimization problems, more than one objective plays a role and input parameters are subject to uncertainty. In this paper, motivated by applications in disaster relief and public facility location, we model and solve a bi-objective stochastic facility location problem. The considered objectives are cost and covered demand, where the demand at the different population centers is uncertain but its probability distribution is known. The latter information is used to produce a set of scenarios. In order to solve the underlying optimization problem, we apply a Benders’ type decomposition approach which is known as the L-shaped method for stochastic programming and we embed it into a recently developed branch-and-bound framework for bi-objective integer optimization. We analyze and compare different cut generation schemes and we show how they affect lower bound set computations, so as to identify the best performing approach. Finally, we compare the branch-and-Benders-cut approach to a straight-forward branch-and-bound implementation based on the deterministic equivalent formulation.

A column-and-constraint generation algorithm for two-stage stochastic programming problems

This paper presents a column-and-constraint generation algorithm for two-stage stochastic programming problems. A distinctive feature of the algorithm is that it does not assume fixed recourse and as a consequence the values and dimensions of the recourse matrix can be uncertain. The proposed algorithm contains multi-cut (partial) Benders decomposition and the deterministic equivalent model as special cases and can be used to trade-off computational speed and memory requirements. The algorithm outperforms multi-cut (partial) Benders decomposition in computational time and the deterministic equivalent model in memory requirements for a maintenance location routing problem. In addition, for instances with a large number of scenarios, the algorithm outperforms the deterministic equivalent model in both computational time and memory requirements. Furthermore, we present an adaptive relative tolerance for instances for which the solution time of the master problem is the bottleneck and the slave problems can be solved relatively efficiently. The adaptive relative tolerance is large in early iterations and converges to zero for the final iteration(s) of the algorithm. The combination of this relative adaptive tolerance with the proposed algorithm decreases the computational time of our instances even further.

Energy Efficiency Optimization in Massive MIMO Secure Multicast Transmission

Herein, we focus on energy efficiency optimization for massive multiple-input multiple-output (MIMO) downlink secure multicast transmission exploiting statistical channel state information (CSI). Privacy engineering in the field of communication is a hot issue under study. The common signal transmitted by the base station is multicast transmitted to multiple legitimate user terminals in our system, but an eavesdropper might eavesdrop this signal. To achieve the energy efficiency utility–privacy trade-off of multicast transmission, we set up the problem of maximizing the energy efficiency which is defined as the ratio of the secure transmit rate to the power consumption. To simplify the formulated nonconvex problem, we use a lower bound of the secure multicast rate as the molecule of the design objective. We then obtain the eigenvector of the optimal transmit covariance matrix into a closed-form, simplifying the matrix-valued multicast transmission strategy problem into a power allocation problem in the beam domain. By utilizing the Minorize-Maximize method, an iterative algorithm is proposed to decompose the secure energy efficiency optimization problem into a sequence of iterative fractional programming subproblems. By using Dinkelbach’s transform, each subproblem becomes an iterative problem with the concave objective function, and it can be solved by classical convex optimization. We guarantee the convergence of the two-level iterative algorithm that we propose. Besides, we reduce the computational complexity of the algorithm by substituting the design objective with its deterministic equivalent. The numerical results show that the approach we propose performs well compared with the conventional methods.

A Stochastic Programming Model for Service Scheduling with Uncertain Demand: An Application in Open-Access Clinic Scheduling

This paper addresses a scheduling problem which handles urgent tasks along withexisting schedules. The uncertainty in this problem comes from random situations ofexisting schedules and arrival of upcoming urgent tasks. To deal with the uncertainty,this paper proposes a stochastic integer programming (SIP) based aggregated onlinescheduling method. The method is illustrated through a study case from the outpatientclinic block-wise scheduling system which is under a hybrid scheduling policycombining regular far-in-advance policy and the open-access policy. The COVID-19pandemic brings more challenges for the healthcare system including the fluctuationsof serving time, and increasing urgent requests which this paper is designed for. TheSIP model designed in the method can easily accommodate uncertainties of theproblems, such as: no-shows, cancellations and punctuality of previously scheduledpatients as well as random arrival and preference of new patients. To solve the SIPmodel, the deterministic equivalent problem formulations are solved using theproposed bound-based sampling method.

Risk-Averse Optimization for Resilience Enhancement Under Uncertainty

With the growth of complexity and extent, large scale interconnected network systems, e.g., transportation networks or infrastructure networks, become more vulnerable towards external disturbances. Hence, managing potential disruptive events during design, operating, and recovery phase of an engineered system therefore improving the system’s resilience is an important yet challenging task. In order to ensure system resilience after the occurrence of failure events, this study proposes a mixed integer linear programming (MILP) based restoration framework using heterogenous dispatchable agents. Scenario based stochastic optimization (SO) technique is adopted to deal with the inherent uncertainties imposed on the recovery process from the nature. Moreover, different from conventional SO using deterministic equivalent formulations, additional risk measure is implemented for this study because of the temporal sparsity of the decision making in applications such as the recovery from extreme events. The resulting restoration framework involves with a large-scale MILP problem and thus an adequate decompaction technique, i.e., modified Langragian Relaxation, is also proposed in order to achieve tractable time complexity. Case study results based on the IEEE 37-buses test feeder demonstrate the benefits of using the proposed framework for resilience improvement as well as the advantages of adopting SO formulations.