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Cluster analysis aims to categorize data objects into cohesive groups based on their intrinsic characteristics, often modeled by probability distributions. This paper presents a novel Mathematical Programming-Dynamic Programming (MP-DP) clustering method developed by the authors, applied to datasets characterized by exponential, right-triangular, and uniform distributions. The MP-DP technique optimizes cluster partitions by leveraging the probability distributions inherent in the data. We conducted a comparative evaluation to assess the performance of MP-DP against four established clustering methodologies: K-Means, Fuzzy C-Means, expectation-maximization, and Genie++ hierarchical clustering. Results from extensive simulations and real-world datasets consistently demonstrate the superior efficacy of MP-DP in achieving optimal clustering outcomes. Specifically, MP-DP excels in handling diverse data distributions and effectively mitigating the effects of noise and uncertainty, thereby enhancing clustering accuracy and reliability. This study highlights the significant advancement offered by MP-DP in clustering research. It underscores the method’s potential for applications across various domains, such as healthcare, environmental monitoring, and manufacturing, where robust and efficient data clustering is essential for insightful data analysis and decision-making.

This manuscript demonstrates robust optimality conditions, Wolfe and Mond–Weir type robust dual models for a robust mathematical programming problem involving vanishing constraints (RMPVC). Further, the theorems of duality are examined based on the concept of generalized higher order invexity and strict invexity that establish relations between the primal and the Wolfe type robust dual problems. In addition, the duality results for a Mond–Weir type robust dual problem based on the concept of generalized higher order pseudoinvex, strict pseudoinvex and quasiinvex functions are also studied. Furthermore, numerical examples are provided to validate robust optimality conditions and duality theorems of Wolfe and Mond–Weir type dual problems. [ABSTRACT FROM AUTHOR]

We consider the constrained optimization problem for a smooth function defined on a Banach space with smooth constraints of equality and inequality type. We show that for this problem, under the known sufficient second-order optimality conditions, the set of Lagrange multipliers can be replaced by a smaller set. [ABSTRACT FROM AUTHOR]

Digital microfluidic biochip (DMFB) has attracted attention in the biochemical and medical industries. In particular, a microelectrode dot array (MEDA) biochip, which is composed of a two-dimensional microelectrode array, enables to realize fine-grained manipulation such as dilution, mixing, sensing, and so on in real-time. Unlike existing DMFB biochips, a MEDA architecture allows microelectrodes to control a certain volume of droplet in a fine-grained manner and can vary droplet volume and shape in such a way that it efficiently conducts synthesis and manipulation of droplets. There have been many works in order to improve the efficiency of synthesis of MEDA biochips; however, the synthesis, especially droplet routing, has never considered the shape-dependent velocity of droplets. In this paper, we propose the droplet routing techniques for MEDA biochips with shape-dependent velocity of droplets. The proposed techniques take the advantage of variant velocities of droplets dependent on the shapes and aim to reduce the overall routing time of a droplet from a source to a destination. Simulation results confirm that the proposed techniques can shorten the routing time by 80% compared to the state-of-the-art techniques.

Introduction. As early as 1744, the great Leonhard Euler noted that nothing at all took place in the universe in which some rule of maximum or minimum did not appear [12]. Great many today’s scientific and engineering problems faced by humankind are of optimization nature. There exist many different methods developed to solve optimization problems, the number of these methods is estimated to be in the hundreds and continues to grow. A number of approaches to classify optimization methods based on various criteria (e.g. the type of optimization strategy or the type of solution obtained) are proposed, narrower classifications of methods solving specific types of optimization problems (e.g. combinatorial optimization problems or nonlinear programming problems) are also in use. Total number of known optimization method classes amounts to several hundreds. At the same time, methods falling into classes far from each other may often have many common properties and can be reduced to each other by rethinking certain characteristics. In view of the above, the pressing task of the modern science is to develop a general approach to classify optimization methods based on the disclosure of the involved search strategy basic principles, and to systematize existing optimization methods. The purpose is to show that genetic algorithms, usually classified as metaheuristic, population-based, simulation, etc., are inherently the stochastic numerical methods of direct search. Results. Alternative statements of optimization problem are given. An overview of existing classifications of optimization problems and basic methods to solve them is provided. The heart of optimization method classification into symbolic (analytical) and numerical ones is described. It is shown that a genetic algorithm scheme can be represented as a scheme of numerical method of direct search. A method to reduce a given optimization problem to a problem solvable by a genetic algorithm is described, and the class of problems that can be solved by genetic algorithms is outlined. Conclusions. Taking into account the existence of a great number of methods solving optimization problems and approaches to classify them it is necessary to work out a unified approach for optimization method classification and systematization. Reducing the class of genetic algorithms to numerical methods of direct search is the first step in this direction.

The fuzzy shortest path problem provides the shortest way to the decision-maker having least possible distance from source to destination. Niroomand et al. (Oper Res 17:395–411, 2017) recently advanced a method for solving the fuzzy shortest path problem. They divided the problem into two sub-problems and solved them separately. They asserted that their method always meets a unique upper and lower bound on the fuzzy shortest distance from source to destination for each α . The proposed study focuses on a significant omission in Niroomand et al. method. The flaws in their approach stem from not clearly revealing the solution concept for the shortest path problem. The flaws of their approach are addressed in this study, and new approaches are proposed to overcome these flaws. The proposed approaches use the complete ranking method to solve the fuzzy shortest path problem. The proposed approaches ensure that the fuzzy shortest distance is equal among all possible shortest paths. The proposed research is carried out using numerical examples of fuzzy shortest path problems. The computational results of proposed approaches are compared to existing methods. [ABSTRACT FROM AUTHOR]

In this article, we formulate a new mixed-type dual problem for a mathematical program with vanishing constraints. The presented dual problem does not involve the index set, however, the dual models contain the calculations of index sets, which makes it difficult to solve these models from an algorithm point of view. The weak, strong and strict converse duality theorems are discussed in order to establish the relationships between the mathematical program with vanishing constraints and its mixed type dual under generalized convexity. To validate the results, a non-trivial example is discussed. Our dual model unifies the dual models discussed in [Q. Hu, J. Wang, Y. Chen, New dualities for mathematical programs with vanishing constraints, Annals of Operations Research, 287 (2020) 233-255]. [ABSTRACT FROM AUTHOR]

With the increasing demand for fast, accurate, and reliable biological sensor systems, miniaturized systems have been aimed at droplet-based sensor systems and have been promising. A micro-electrode dot array (MEDA) biochip, which is one kind of the miniaturized systems for biochemical protocols such as dispensing, dilutions, mixing, and so on, has become widespread due to enabling dynamical control of the droplets in microfluidic manipulations. In MEDA biochips, the electrowetting-on-dielectric (EWOD) technique stands out since it can actuate droplets with nano/picoliter volumes. Microelectrode cells on MEDA actuate multiple droplets simultaneously to route locations for the purpose of the biochemical operations. Taking advantage of the feature, droplets are often routed in parallel to achieve high-throughput outcomes. Regarding parallel manipulation of multiple droplets, however, the droplets are known to be initially placed at a distant position to avoid undesirable mixing. The droplets thus result in traveling a long way for a manipulation, and the required biochip size for routing is also enlarged. This paper proposes a routing method for droplets to reduce the biochip size on a MEDA biochip with the allowance of splitting during routing operations. We mathematically derive the routing problem, and the experiments demonstrate that our proposal can significantly reduce the biochip size by 70.8% on average, compared to the state-of-the-art method. [ABSTRACT FROM AUTHOR]

This paper introduces a coordination model that copes with the conflicts between a supplier and a retailer in a two-stage supply chain. In this model, the supplier aims to maximise his/her profit, while the retailer aims to maximise his/her expected profit under a stochastic demand. To solve this model, an -coordination solution to this model is introduced, which implies the supplier and the retailer both will make a same concession in their profits to cooperate with each other. Further, we show that each feasible solution to this model is an -coordination solution for a sufficiently large value , while the -optimum coordination solution gives the minimum concession for the supplier and the retailer. Besides, it is proved that the -optimum coordination solution can be obtained by solving a mathematical programming problem. At last, a numerical example and sensitivity analysis are given to show the performance of the proposed method. [ABSTRACT FROM AUTHOR]

With the increasing demand for fast, accurate, and reliable biological sensor systems, miniaturized systems have been aimed at droplet-based sensor systems and have been promising. A micro-electrode dot array (MEDA) biochip, which is one kind of the miniaturized systems for biochemical protocols such as dispensing, dilutions, mixing, and so on, has become widespread due to enabling dynamical control of the droplets in microfluidic manipulations. In MEDA biochips, the electrowetting-on-dielectric (EWOD) technique stands out since it can actuate droplets with nano/picoliter volumes. Microelectrode cells on MEDA actuate multiple droplets simultaneously to route locations for the purpose of the biochemical operations. Taking advantage of the feature, droplets are often routed in parallel to achieve high-throughput outcomes. Regarding parallel manipulation of multiple droplets, however, the droplets are known to be initially placed at a distant position to avoid undesirable mixing. The droplets thus result in traveling a long way for a manipulation, and the required biochip size for routing is also enlarged. This paper proposes a routing method for droplets to reduce the biochip size on a MEDA biochip with the allowance of splitting during routing operations. We mathematically derive the routing problem, and the experiments demonstrate that our proposal can significantly reduce the biochip size by 70.8% on average, compared to the state-of-the-art method.

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In last few years, lots of researchers have proposed different methods to solve the constrained matrix games with fuzzy payoffs. In this paper, it has been shown that the mathematical programming problem of constrained matrix games with fuzzy payoffs, considered by researchers, is mathematically invalid and hence the method, proposed by researchers to obtain the complete solution (minimum expected gain of Player I, maximum expected loss of Player II and their corresponding optimal strategies) of constrained matrix games with fuzzy payoffs by solving the mathematical programming problem with fuzzy payoffs, are also invalid. Further, in the present paper, a new method has been proposed to find the complete solution of matrix games with fuzzy payoffs. To illustrate the proposed method, some existing numerical problems of constrained matrix games with fuzzy payoffs have been solved by the proposed method. [ABSTRACT FROM AUTHOR]

Summary: This R package determines optimal stratification of univariate populations under stratified sampling designs using a parametric‐based method. It determines the optimum strata boundaries (OSB), optimum sample sizes (OSS) and multiple other quantities for the study variable, y, using the best‐fit probability density function of a study variable available from survey data. The method requires the parameters and other characteristics of the distribution of the study variable to be known, either from available data or from a hypothetical distribution if the data are not available. In the implementation, the problem of determining the OSB is formulated as a mathematical programming problem and solved by using a dynamic programming technique. If the data of the population (i.e. the study variable) are available to the surveyor, the method estimates its best‐fit distribution and determines the OSB and OSS under Neyman allocation, directly. When the dataset is not available, stratification is made based on the assumption that the values of the study variable, y, are available as hypothetical realisations of proxy values of y from past/recent surveys. Thus, it requires certain distributional assumptions about the study variable. At present, the package handles stratification for the populations where the study variable follows a continuous distribution: namely, Pareto, Triangular, Right‐triangular, Weibull, Gamma, Exponential, Uniform, Normal, Lognormal and Cauchy distributions. In this paper, applications of major functionalities in the package are illustrated with a number of real/simulated as well as some hypothetical populations. [ABSTRACT FROM AUTHOR]

This study investigates optimal wind power generator bidding strategies in the real‐time electricity market. The goal is to maximise its operating profit by determining the optimal amount of wind power to bid in the real‐time market. A bi‐level stochastic optimisation model is proposed in which the upper‐level problem is minimising the negative profit of wind power producers, while the low‐level problem clears the real‐time market. The uncertainties in the wind power production, thermal power, hydro power, demand, and energy storage are considered in the stochastic model. This study utilises a mathematical programming problem with equilibrium constraints (MPEC) and Karush–Kuhn–Tucker (KKT) conditions to transform the bi‐level problem into an equivalent single‐level mixed‐integer linear problem (MILP). Case studies obtained with the IEEE RTS‐24 Bus system demonstrate the effectiveness of the proposed model and the effect of scenarios on the wind power producer's profit. [ABSTRACT FROM AUTHOR]

Sampling has evolved into a universally accepted approach for gathering information and data mining as it is widely accepted that a reasonably modest-sized sample can sufficiently characterize a much larger population. In stratified sampling designs, the whole population is divided into homogeneous strata in order to achieve higher precision in the estimation. This paper proposes an efficient method of constructing optimum stratum boundaries (OSB) and determining optimum sample size (OSS) for the survey variable. The survey variable may not be available in practice since the variable of interest is unavailable prior to conducting the survey. Thus, the method is based on the auxiliary variable which is usually readily available from past surveys. To illustrate the application as an example using a real data, the auxiliary variable considered for this problem follows Weibull distribution. The stratification problem is formulated as a Mathematical Programming Problem (MPP) that seeks minimization of the variance of the estimated population parameter under Neyman allocation. The solution procedure employs the dynamic programming technique, which results in substantial gains in the precision of the estimates of the population characteristics. [ABSTRACT FROM AUTHOR]

Demand response (DR) can play an important role when dealing with the increasing variability of renewables in power system. This study proposes a bi‐level market model for wind‐integrated electricity market, where the DR requirement is paired with the wind profile to deal with wind variability. At the upper level, an electricity market operator aims to minimise the day‐ahead operation cost considering plausible wind generation scenarios. At the lower level, the DR exchange operator aims to maximise social welfare by trading aggregated DR among several aggregators. The solution at this level determines the optimal DR amount and price setting for each aggregator. The DR from the flexible loads is modelled from the end‐users' perspective considering their willingness parameter. The market model is formulated as a bi‐level optimisation problem using Lagrangian relaxation with Karush–Kuhn–Tucker optimality conditions. The effectiveness of the proposed scheme is demonstrated on sample 4‐bus and IEEE 24‐bus systems. Different scenarios such as high, medium and low levels of wind and DR are investigated. The high‐wind low‐DR scenario leads to minimum operation cost and is least inconvenient for end users as it sets the DR at a minimum level while keeping the higher levels of wind generation. [ABSTRACT FROM AUTHOR]

In this paper, an efficient computational method for numerical solution of infinite horizon optimal control problems is presented. In the proposed method, transformed Legendre spectral scheme is utilized to transcribe the problem to a mathematical programming problem which can be solved by the well-developed optimization algorithms. The main advantages of the present method are obtaining good results and high rate of convergence, while the infinite horizon problem is solved on the original time interval of the problem without transforming it to a finite one. Numerical results of three examples are presented and efficiency of the method is reported. [ABSTRACT FROM AUTHOR]

Summary: This paper deals with the numerical solution of optimal control problems with multiple delays in both state and control variables. A direct approach based on a hybrid of block‐pulse functions and Lagrange interpolating polynomials is used to convert the original problem into a mathematical programming one. The resulting optimization problem is then solved numerically by the Lagrange multipliers method. The operational matrix of delay for the presented framework is derived. This matrix plays an imperative role to transfer information between 2 consecutive switching points. Furthermore, 2 upper bounds on the error with respect to the L2‐norm and infinity norm are established. Several optimal control problems containing multiple delays are carried out to illustrate the various aspects of the proposed approach. The simulation results are compared with either analytical or numerical solutions available in the literature. [ABSTRACT FROM AUTHOR]

In this paper, we introduce two mathematical programming models of the resident scheduling problem (RSP). These were designed to solve the RSP in rotation and night-shift scheduling of residents at a hospital in Japan. Automatic scheduling is designed to support management in challenges such as increasing the satisfaction of residents and reducing the scheduling workload. To achieve this, we placed weighted penalties on soft constraints and used objective functions to automatically generate the schedules. The results improved the management of resident scheduling, and comparisons with schedules drawn up by hand confirmed the superiority of those produced by the systems. Following some fine-tuning based on user feedback, our night-shift scheduling system was put into use at Aichi Medical University Hospital in January 2016. [ABSTRACT FROM AUTHOR]