Search

Many applications in data analysis study whether two categorical variables are independent using a function of the entries of their contingency table. Often, the categories of the variables, associated with the rows and columns of the table, are grouped, yielding a less granular representation of the categorical variables. The purpose of this is to attain reasonable sample sizes in the cells of the table and, more importantly, to incorporate expert knowledge on the allowable groupings. However, it is known that the conclusions on independence depend, in general, on the chosen granularity, as in the Simpson paradox. In this paper we propose a methodology to, for a given contingency table and a fixed granularity, find a clustered table with the highest χ2 statistic. Repeating this procedure for different values of the granularity, we can either identify an extreme grouping, namely the largest granularity for which the statistical dependence is still detected, or conclude that it does not exist and that the two variables are dependent regardless of the size of the clustered table. For this problem, we propose an assignment mathematical formulation and a set partitioning one. Our approach is flexible enough to include constraints on the desirable structure of the clusters, such as must-link or cannot-link constraints on the categories that can, or cannot, be merged together, and ensure reasonable sample sizes in the cells of the clustered table from which trustful statistical conclusions can be derived. We illustrate the usefulness of our methodology using a dataset of a medical study. Open Access funding provided thanks to the CRUE-CSIC agreement with Springer Nature. This research has been financed in part by research projects EC H2020 MSCA RISE NeEDS (Grant agreement ID: 822214), FQM-329, P18-FR-2369 and US-1381178 (Junta de Andalucía, with FEDER Funds), PID2019-110886RB-I00 and PID2019-104901RB-I00 (funded by MCIN/AEI/10.13039/501100011033). This support is gratefully acknowledged.

In power systems, Economic Power dispatch Problem (EPP) is an influential optimization problem which is a highly non-convex and non-linear optimization problem. In the current study, a novel version of Differential Evolution (NDE) is used to solve this particular problem. NDE algorithm enhances local and global search capability along with efficient utilization of time and space by making use of two elite features: selfadaptive control parameter and single population structure. The combined effect of these concepts improves the performance of Differential Evolution (DE) without compromising on quality of the solution and balances the exploitation and exploration capabilities of DE. The efficiency of NDE is validated by evaluating on three benchmark cases of the power system problem having constraints such as power balance and power generation along with nonsmooth cost function and is compared with other optimization algorithms. The Numerical outcomes uncovered that NDE performed well for all the benchmark cases and maintained a trade-off between convergence rate and efficiency.

Given the considerable complexity of process plants, there has been a great deal of research focused on aiding the design of plant layout through mathematical optimisation, i.e. optimising the positioning of the equipment in the plant for space and cost efficiency. Recently, the use of modular approaches within the construction industry, whereby work is performed off-site before being assembled on-site, has become a popular and powerful way of reducing build schedules and costs. Modular approaches have many other real applications where items must be packed to minimise the connections between them (e.g. piping, wiring, modular office and factory layouts) and consider the modular layout of the system. In this paper, we provide a formulation of the problem that, in addition to the standard layout problem, considers a modular block layout to allow modular construction and transportation of the plant. The problem is represented as a directed network, with the aim to pack the items into predefined containers and minimise the rectilinear distance between the connected items. We propose mixed-integer linear programming (MILP) models for the 2-dimensional and 3-dimensional problems and solve them using the state-of-the-art mathematical programming solver, Gurobi. Because of the combinatorial nature of the problem, solutions involving a large number of items may not converge and a suboptimal solution must be considered. However, our results suggest that even in the case of optimising a large number of items, the suboptimal solutions found after a reasonable number of iterations where deemed, by a domain expert, to be a good enough starting point to continue the design process, especially in the early concept phase.

The selfish routing of network flow often considers a single objective, namely travel time or travel distance, and optimisation models are often guided by the principle of user equilibrium (UE). A more challenging approach is to consider multiple objectives simultaneously, as for example distance, travel time and pollution. In this paper we present a bi-criteria problem whereby the road users selfish objective of minimising their travel time is at odds with the objective of minimising the overall fuel consumption of all road users. The approach taken is to manipulate “free” parameters, namely speed limits, in an attempt to coerce the road users into behaviour which helps the latter objective. Motivated by the work done on the Price of Anarchy (PoA) into classifying the suboptimality of equilibrium with respect to the minimum total travel time we look to classify the equilibrium solutions with respect to a weighted sum model of the minimum total travel time and overall fuel consumption. Our results show that small changes to these “free” parameters can result in solutions which Pareto dominate other solutions. Whilst our measure for the suboptimality of equilibrium solutions can highlight the inefficiency of a network configuration itself, it does not allow insight into the overall quality of the solution when compared with other network configurations.

Graph planning gives rise to fundamental algorithmic questions such as shortest path, traveling salesman problem, etc. A classical problem in discrete planning is to consider a weighted graph and construct a path that maximizes the sum of weights for a given time horizon $T$. However, in many scenarios, the time horizon is not fixed, but the stopping time is chosen according to some distribution such that the expected stopping time is $T$. If the stopping time distribution is not known, then to ensure robustness, the distribution is chosen by an adversary, to represent the worst-case scenario. A stationary plan for every vertex always chooses the same outgoing edge. For fixed horizon or fixed stopping-time distribution, stationary plans are not sufficient for optimality. Quite surprisingly we show that when an adversary chooses the stopping-time distribution with expected stopping time $T$, then stationary plans are sufficient. While computing optimal stationary plans for fixed horizon is NP-complete, we show that computing optimal stationary plans under adversarial stopping-time distribution can be achieved in polynomial time. Consequently, our polynomial-time algorithm for adversarial stopping time also computes an optimal plan among all possible plans.

With the advent of the era of massive data, the increase of storage demand has far exceeded current storage capacity. DNA molecules provide a reliable solution for big data storage by virtue of their large capacity, high density, and long-term stability. To reduce errors in storing procedures, constructing a sufficient set of constraint encoding is critical for achieving DNA storage. A new version of the Marine Predator algorithm (called QRSS-MPA is proposed in this paper to increase the lower limit of the coding set while satisfying the specific combination of constraints. In order to demonstrate the effectiveness of the improvement, the classical CEC-05 test function is used to test and compare the mean, variance, scalability, and significance. In terms of storage, the lower limit of construction is compared with previous works, and the result is found to be significantly improved. In order to prevent the emergence of a secondary structure that leads to sequencing failure, we give a more stringent lower bound for the constraint coding set, which is of great significance for reducing the error rate of DNA storage amidst its rapid development.

We introduce and study a class of optimization problems we call replenishment problems with fixed turnover times: a very natural model that has received little attention in the literature. Clients with capacity for storing a certain commodity are located at various places; at each client the commodity depletes within a certain time, the turnover time, which is constant but can vary between locations. Clients should never run empty. The natural feature that makes this problem interesting is that we may schedule a replenishment (well) before a client becomes empty, but then the next replenishment will be due earlier also. This added workload needs to be balanced against the cost of routing vehicles to do the replenishments. In this paper, we focus on the aspect of minimizing routing costs. However, the framework of recurring tasks, in which the next job of a task must be done within a fixed amount of time after the previous one is much more general and gives an adequate model for many practical situations. Note that our problem has an infinite time horizon. However, it can be fully characterized by a compact input, containing only the location of each client and a turnover time. This makes determining its computational complexity highly challenging and indeed it remains essentially unresolved. We study the problem for two objectives: min–avg minimizes the average tour cost and min–max minimizes the maximum tour cost over all days. For min–max we derive a logarithmic factor approximation for the problem on general metrics and a 6-approximation for the problem on trees, for which we have a proof of NP-hardness. For min–avg we present a logarithmic factor approximation on general metrics, a 2-approximation for trees, and a pseudopolynomial time algorithm for the line. Many intriguing problems remain open.

This paper proposes a novel Stackelberg game-oriented optimal control approach to address the bounded constraint-following control problem for uncertain mechanical systems. First, the uncertainties (possibly fast time-varying) in the system are assumed to be bounded with an unknown boundary which lies in a specified fuzzy evidence number. In practical engineering, bounded system performance is always demanded, such as the inequality constraint. A diffeomorphism transformation approach is proposed to transform the constrained system into a restructured one satisfying the bounded constraint. Second, we propose an adaptive robust control oriented by the constraint-following control to render the restructured system to follow the specified constraints accurately with deterministic performance (guaranteeing uniform boundedness and uniform ultimate boundedness). The self-adjusting adaptive law (leakage-type) can compensate for the uncertainties and avoid overcompensation. Third, a Stackelberg game-oriented optimization approach is proposed to obtain the optimal control parameters based on the fuzzy evidence theory. In the optimization approach, the two control parameters, are considered as two players with respective cost functions related to system performance and control cost. Furthermore, the optimization problem is solved by obtaining the Stackelberg strategy, which is proved to exist in analytic form. Ultimately, the permanent magnet synchronous linear motor system simulation is presented to show the design process and the excellent performance of the proposed optimal control scheme.

Three-way decision is consistent with human cognitive processes and applied to different information systems. However, since the real world is more imprecision, hesitation and ambiguity, experts can select interval fuzzy information to represent their evaluations. In the evaluation process, experts sometimes may not guarantee the interval fuzzy information satisfy the constraint between membership and non-membership and further impact make decisions. Interval-valued q-rung Orthopair fuzzy set (IVq-ROFS) are proposed to improve the fault-tolerant ability and ensure the experts' evaluation make sense. In this case, this paper generalizes IVq-ROFS to three-way decision, which consists of the construction of three-way decision and the aggregation of interval continuity of IVq-ROFS. For considering the aggregation of interval continuity of IVq-ROFS, we firstly investigate the IVq-ROFS integral to aggregate continuous IVq-ROFSs by developing simplified interval-valued q-rung Orthopair fuzzy set (SIVq-ROFS). Particularly, in order to obtain integrable region, we design $\varepsilon$ -amendment to amend integral region of IVq-ROFS for removing unmeaning point in advance. To simplify the calculation, we further utilize Taylor series of the integral function to replace the complicated integral function. Secondly, a new equivalence class method is constructed to support three-way decision. Thirdly, with the aid of the integral of IVq-ROFS and equivalence class method, we availably fuse the loss functions of the equivalence class and obtain the classifying rules of three-way decision by comparing the expected losses. Finally, we elaborate three-way decision by using a case of medical diagnosis about nephropathy and prove the validity of the investigated model.

This work is concerned with the design of a two-step distributed state estimation scheme for large-scale systems in the presence of unknown-but-bounded disturbances and noise. The set-membership approach is employed to construct a compact set containing the states consistent with system measurements and bounded noise and disturbances. The tightened feasible region is then provided to a moving horizon estimator that determines the optimal state estimates. Partitioning of the overall problem and coordination of the resulting subproblems are achieved using decomposition of the optimality conditions and community detection. The proposed strategy is tested on a case study based on a reactor–separator system widely used in the literature. Its performance is compared to those of centralized and distributed (without set-membership) implementations, allowing to highlight its effectiveness.

The design of fuzzy neural network (FNN) has long been a challenging problem since most methods rely on approximation error to train FNN, which may easily occur overfitting phenomenon to degrade the generalization performance. To improve the generalization performance, a fuzzy neural network with multi-objective optimization algorithm (MOO-FNN) is proposed in this paper. First, the multi-level learning objectives are designed around the generalization performance to guide the training process of FNN. Then, the method utilizes the approximation error, the structure complexity, and the output smoothness indicators instead of a single indicator to improve the evaluation accuracy of generalization performance. Second, a MOO algorithm with continuous-discrete variables is developed to optimize FNN. Then, MOO is able to use a novel particle update method to adjust both the structure and parameters rather than adjust them separately, thereby achieving suitable generalization performance of FNN. Third, the convergence of MOO-FNN is analyzed in detail to guarantee its successful applications. Finally, the experimental studies of MOO-FNN have been performed on model identification of nonlinear systems to verify the effectiveness. The results illustrate that MOO-FNN has a significant improvement over some state-of-the-art algorithms.

Faced with any decision-making problems with fuzzy multi-criteria information, decision-makers generally set a minimum requirement on each criterion for satisfying their own preferences, thereby forming a fuzzy set on the criterion universe, which is called the criterion fuzzy set. From the perspective of realistic decision, the final decision of all alternatives should be determined according to this criterion fuzzy set. In view of this, this paper proposes the criterion-oriented three-way ranking and clustering strategies, which can solve the qualitative clustering and ranking problems of all alternatives from the perspective of criterion fuzzy sets. Firstly, we define a criterion fuzzy set as a criterion-oriented fuzzy concept and propose a criterionoriented relative risk loss model and discuss related properties accordingly. Meanwhile, we use the generalized fuzzy rough lower (upper) approximation to estimate the absolute (relative) conditional probability between the binary fuzzy class of the alternative and the criterion-oriented fuzzy concept. Then, a criterion-oriented absolute (relative) three-way clustering strategy is proposed, which can perform qualitative analysis on all alternatives. Furthermore, based on the three-way semantics and global cost function, we recommend a criterion-oriented absolute (relative) three-way ranking strategy, which can rank all alternatives. Finally, through numerical example, comparative analysis and sensitivity analysis, we test the feasibility and effectiveness of the proposed criterion-oriented three-way ranking and clustering strategies.

This article focuses on the L2-L/H optimization control issue for a family of nonlinear plants by Takagi-Sugeno (T-S) fuzzy approach with actuator failure. First, considering unmeasurable system states, sufficient criteria for devising fuzzy imperfect premise matching dynamic output feedback controller to maintain asymptotic stability while guaranteeing a mixed performance for T-S fuzzy systems are provided. Therewith, in the light of feasible areas of dynamic output feedback controller membership functions (MFs), a new MFs online learning policy using gradient descent algorithm is proposed to learn the real-time values of MFs so as to acquire a better L2-L/H control effect. Different from the traditional method using imperfect premise matching scheme, under the proposed optimization algorithm, the trajectory of mixed performance index is lowered effectively. Afterward, a sufficient criterion is presented for assuring the convergence of the error of cost function. Lastly, the superiority of this online optimization learning policy is confirmed via simulations.

A metering roundabout where traffic is controlled by traffic lights with phase times influenced by queue detector occupancy might be the solution to enhance performance when there are unbalanced traffic flows at roundabouts. There have, however, been minimal studies on how the distance of the queue detector from the stop line affects signal phase time durations and the queuing lengths. This research, therefore, seeks to develop a Cuckoo Search/Local search Algorithm using parameters such as arrival volumes, conflicting volumes, detector distance and phase time to investigate the relationship of signal setting, detector location and queuing formulations. Also, some additional statistical tests were performed for the fitness of the data. In order to conduct solid model calibrations and validations, model output data was compared against the AIMSUN model. The results from the analyses demonstrated that the queue detector distance can affect phase time durations and vehicle queuing lengths on the controlling approach as well as queuing lengths on the metered approach. This study showed that, based on the study for the Old Belair Road roundabout, the total queue length (controlling + metered) will be minimized from 689 to 499 m by the proposed methods when the detector is relocated at 210 m from the roundabout stop line, giving longer phase green times and resulting in decreased intersection queuing lengths.

In this paper, we extend the AENP and the GMC criterion proposed by Zhang et al. (2008) to the case of nonregular orthogonal designs. A G-AENP and correspondingly a G-GMC criterion are proposed. The confounding frequency vector (CFV) and the generalized wordlength pattern (GWLP), as the base of MGA and GMA criteria, are shown to be functions of the G-AENP. Some optimal properties of G-GMC designs are obtained. At the last, we give an efficient algorithm for finding optimal designs and tabulate some G-GMC designs with 16- and 18-run for application and comparison with GMA and MGA designs.

We study the optimal sustainable harvesting of a population that lives in a random environment. The novelty of our setting is that we maximize the asymptotic harvesting yield, both in an expected value and almost sure sense, for a large class of harvesting strategies and unstructured population models. We prove under relatively weak assumptions that there exists a unique optimal harvesting strategy characterized by an optimal threshold below which the population is maintained at all times by utilizing a local time push-type policy. We also discuss, through Abelian limits, how our results are related to the optimal harvesting strategies when one maximizes the expected cumulative present value of the harvesting yield and establish a simple connection and ordering between the values and optimal boundaries. Finally, we explicitly characterize the optimal harvesting strategies in two different cases, one of which is the celebrated stochastic Verhulst Pearl logistic model of population growth.

We obtain the optimal execution strategy for two sequential trades in the presence of a transient price impact. We first present a novel and general solution method for the case of a single trade (a metaorder) that is executed as a sequence of sub-trades (child orders). We then analyze the case of two sequential metaorders, including the case where the size and direction of the second metaorder are uncertain at the time the first metaorder is initiated. We obtain the optimal execution strategy under two different cost functions. First, we minimize the total cost when each metaorder is benchmarked to the price at its initiation, the total separate costs approach widely used by practitioners. Although simple, we show that optimizing total separate costs can lead to a significant understatement of the real costs of trading whilst also adversely impacting order scheduling. We overcome these issues by introducing a new cost function that splits the second metaorder into two parts, one that is predictable when the first metaorder is initiated and a residual that is not. The predictable and residual parts of the second metaorder are benchmarked using the initiation prices of the first and second metaorders, respectively. We prove existence of an optimal execution trajectory for linear instantaneous price impact and positive definite decay, and derive the explicit form of the minimizer in the special case of exponentially decaying impact. Various numerical examples are included for illustration, each of which admits a unique optimal solution, however uniqueness in general remains unproven.

The existing three-way decision-making methods cannot effectively handle incomplete multi-attribute decision-making problems in real life, it is necessary to explore an effective three-way multi-attribute decision-making model in incomplete fuzzy decision systems. First, we consider the preference of decision-makers for each alternative and introduce the concept of pre-decisions, thus an incomplete fuzzy decision system is obtained. Then, the weighted conditional probabilities are calculated with the aid of the defined similarity relation. Subsequently, we introduce the notion of relative utility functions and then present an approach to determine the relative utility function values. Afterwards, we construct a three-way decision model in incomplete fuzzy decision systems and apply it to the modeling of incomplete multi-attribute decision-making problems. Our study not only enriches three-way decision and multi-attribute decision-making theories, but also provides a new perspective for realistic incomplete multi-attribute decision-making problems. At last, the results of comparative and experimental analyses demonstrate the validity, stability and superiority of our proposed model.

In recent years, grey models based on fractional-order accumulation and/or derivatives have attracted considerable research interest because they offer better performance in handling limited samples with uncertainty than integer-order grey models; however, there remains room for improvement. This paper considers a more flexible and general structure for the fractional grey model by incorporating a generalized fractional-order derivative (GFOD) that complies by memory effects, resulting in the development of a generalized fractional grey model (denoted as GFGM(1,1)). Specifically, we comprehensively analyse the modelling mechanism of the proposed GFGM(1,1) model, involving model parameter estimation and time response function derivation, and discuss the link between the proposed approach and existing special cases. Then, to further improve the efficacy of the proposed approach, four mainstream metaheuristic algorithms are employed to ascertain the orders of fractional accumulation and derivatives. Finally, we carry out a series of simulation studies and a real-world application case to demonstrate the applicability and advantage of the our approach. The numerical results show that GFGM(1,1) outperforms other benchmarks, and some significant insights are obtained from the numerical experiments.

This article addresses the event-triggered consensus problem for Takagi–Sugeno fuzzy fractional-order multiagent systems with switching topologies. First, to effectively avoid the frequent communication among agents, a state-based event-triggered consensus strategy is designed, which uses the local information from neighboring agents at sampling moments. Then, several sufficient conditions, which rely on the fractional derivative number and time delay information, are presented to guarantee the consensus of fractional-order multiagent systems based on Takagi–Sugeno fuzzy model. Moreover, Zeno behaviors are precluded by proving that the interval length of the two consecutive event-triggering moments for each agent is greater than a positive constant. Finally, some numerical examples are presented, which not only demonstrated the rationality of our proposed consensus protocol but also shown that the presented consensus method based on the designed event-triggered control protocol has the advantage for avoiding communication congestion compared to the existing results.

This paper investigates the stochastic stability analysis and stabilization problems for discrete-time Takagi-Sugeno fuzzy semi-Markov jump systems with upper-bounded sojourn time. The fuzzy rules can be different for different system modes. Consequently, the membership functions for fuzzy rules are dependent on the system modes. Allowing for the fact that semi-Markov kernel (SMK) are difficult to fully obtain in practice, the elements in the SMK of the underlying systems are deemed to be partly known, which is more general than both semi-Markov jump systems with completely available semi-Markov kernel and Markov jump systems with unknown transition probabilities. Afterwards, the stability and stabilization conditions are established by part of the known SMK information and then by all the known SMK information. In the end, the validity and the superiority of our proposed theoretical results are exemplified via a single-link robot arm and a truck-trailer model.

Karunanayaka and Tang (2017) introduced a class of compromise designs for estimating main effects under the baseline parameterization. Their approach is to add some runs to the basic one-factor-at-a-time design, and its implementation requires computer search except for the case of adding one run where a theoretical result is available. The reliance on computer search only allowed them to find the limited results from adding up to four runs to the basic one-factor-at-a-time design. In this paper, we provide a systematic construction of compromise designs, without computer search and without restriction on the run size and the number of factors. Closed-form expressions of the bias and efficiency criteria are obtained for the new designs. Both theoretical and empirical studies show that our designs outperform those of Karunanayaka and Tang (2017) saving small-sized problems.

This paper is concerned with the problem of computing the approximate preinverses of a fuzzy matrix with max-product composition. Employing the weighted $L_1$ distance, a variety of evaluation functions are defined according to the preselected weights, and then an analytical method is proposed to determine all approximate preinverses of equal quality by minimizing the preselected evaluation function. Since only the approximate preinverses with fewer or preferably no zeros are acceptable for many practical applications, a criterion for selecting weights is established to ensure that the determined approximate preinverses are desirable. Some examples are given to illustrate our results.

The meta-heuristic algorithms have aroused great attention for controller optimization. However, most of them are inseparable from the explicit system models when addressing a constrained optimization problem (COP). In this paper, we propose a data-driven constrained bat algorithm via a gradient-based depth-first search (GDFS) strategy. In the proposed scheme, the GDFS strategy can predetermine a search space that satisfies some strict constraints (e.g., stability requirements) of the optimized system. Meanwhile, an improved boundary constraint handling method is proposed to limit the exploration process to the predetermined space. In this way, the proposed algorithm can solve the COP by utilizing experimental data from real scenes, thereby relieving the dependence on precisely modeling the complex system. Together with an ɛ-constraint-handling method, the bat algorithm is employed to seek the global optimum of the COP. The search performance is enhanced by the designed linear-varying elite layer-based local search and a social learning-based walk mechanism to dynamically balance exploration and exploitation. The convergence is ensured based on the criteria of the stochastic optimization algorithm. Experimental results on a servo drive system and benchmark test functions verify the effectiveness of the proposed algorithm.

This paper investigates the consensus problem for a class of multi-agent systems with P-one-sided Lipschitz nonlinearity. This nonlinearity offers a significant extension to application range of existing consensus control protocol design methods, which removes the assumption of the quadratic inner-boundedness. To solve this problem, a strongly connected graph topology is applied within the consensus protocol design framework. Moreover, the leader-following scenario is addressed under the graph topology. Under the above objectives, sufficient conditions are employed to yield consensus. Two examples are performed to illustrate the properties.

This paper introduces novel approaches to forecast pooling methods based on a nonparametric prior for a weight vector combining predictive densities. The first approach places a Dirichlet process prior on the weight vector and generalizes the static linear pool. The second approach uses a hierarchical Dirichlet process prior to allow the weight vector to follow an infinite hidden Markov chain. This generalizes dynamic prediction pools to the nonparametric setting. Efficient posterior simulation based on MCMC methods is also examined. Detailed applications to short-term interest rates, realized covariance matrices and asset pricing models demonstrate that the nonparametric pool forecasts well. The paper concludes with extensions and an application for calibrating and combining predictive densities.

Best-worst method (BWM) has attracted increasing attention. It has been generalized to different fuzzy environments and applied to various real-life decision problems. This paper develops a new intuitionistic fuzzy (IF) best-worst method (IFBWM) for multi-criteria decision-making (MCDM). When a decision maker (DM) makes comparisons, there may be some hesitancies. Thus, the reference comparisons are represented as intuitionistic fuzzy values (IFVs), the Best-to-Others vector and the Others-to-Worst vector are IF vectors. According to the multiplicative consistency of intuitionistic fuzzy preference relation, this paper gives the consistency equations and views them as IF equations. The derivation of optimal IF weights of criteria is formulated as an IF decision-making problem. Thereby, a mathematical programming model is constructed to assure that the derived optimal IF weights of criteria is a normalized IF weight vector. Depending on the risk preference of DM, four linear programming models are presented to obtain the optimal IF weights based on the constructed mathematical programming model for the optimistic DM, the pessimistic DM and the neutral DM, respectively. Furthermore, this paper investigates the process of improving the consistency. Several examples are demonstrated to show the application and effectiveness of the proposed IF BWM.

This paper investigates the robust beamforming design in a secrecy multiple-input single-output (MISO) network aided by the intelligent reflecting surface (IRS) with simultaneous wireless information and power transfer (SWIPT). Specifically, by considering that the energy receivers (ERs) are potential eavesdroppers (Eves) and imperfect channel state information (CSI) of the direct and cascaded channels can be obtained, we investigate the max-min fairness robust secrecy design. The objective is to maximize the minimum robust information rate among the legitimate information receivers (IRs). To solve the formulated non-convex design problem in bounded and probabilistic CSI error models, we utilize the alternating optimization (AO) and successive convex approximation (SCA) methods to obtain an approximate problem. Then, an iteration-based algorithm framework was proposed, where the unit modulus constraint (UMC) of the IRS is handled by the penalty dual decomposition (PDD) method. Moreover, a stochastic SCA method is proposed to handle the outage constrained design with statistical CSI. Finally, simulation results validate the promising performance of the proposed design.