Single Linear Programming Problem Research Articles

Incorporating local uncertainty management into distribution system planning: An adaptive robust optimization approach

As the penetration of renewable energy sources (RES) significantly increases, their inherent uncertainty has brought about a paradigm shift in distribution system planning and operation. The existing planning methodologies usually consider the uncertainty during the investment and day-ahead stages while neglecting the interactive power fluctuations during the intraday stage, resulting in planning schemes with a risk of inadequate regional stability. In light of this, this paper proposes a novel adaptive robust distribution system planning approach that integrates local uncertainty management (LUM) to provide a comprehensive and refined consideration of uncertainty. First, we introduce the concept of LUM that can effectively address the deviations between predicted and actual power of RES and load. Second, we propose the adaptive robust planning model for the distribution system, in which the investment and day-ahead operation are considered in the first stage, while LUM is considered in the second stage to maintain the local stability of interactive power. The proposed model can actively manage uncertainties of RES and load demand, reaching the optimal balance between the investment cost, operational cost, and LUM cost. Finally, by employing the column-and-constraint generation (C&CG), the adaptive robust planning model is decoupled into a master min problem and a slaver max-min problem, after which the max-min problem is converted into a single mixed-integer linear programming (MILP) problem through the strong duality theory and big-M method. The effectiveness of the proposed model is demonstrated using a real-world distribution system located in Fujian Province, China. The simulation results indicate that the proposed model achieves a quantitative decrease of approximately 10% in total cost compared to deterministic planning. Sensitivity analysis is also conducted to reveal the trade-off between economic cost and robustness level.

Discretised Flux Balance Analysis for Reaction–Diffusion Simulation of Single-Cell Metabolism

Metabolites have to diffuse within the sub-cellular compartments they occupy to specific locations where enzymes are, so reactions could occur. Conventional flux balance analysis (FBA), a method based on linear programming that is commonly used to model metabolism, implicitly assumes that all enzymatic reactions are not diffusion-limited though that may not always be the case. In this work, we have developed a spatial method that implements FBA on a grid-based system, to enable the exploration of diffusion effects on metabolism. Specifically, the method discretises a living cell into a two-dimensional grid, represents the metabolic reactions in each grid element as well as the diffusion of metabolites to and from neighbouring elements, and simulates the system as a single linear programming problem. We varied the number of rows and columns in the grid to simulate different cell shapes, and the method was able to capture diffusion effects at different shapes. We then used the method to simulate heterogeneous enzyme distribution, which suggested a theoretical effect on variability at the population level. We propose the use of this method, and its future extensions, to explore how spatiotemporal organisation of sub-cellular compartments and the molecules within could affect cell behaviour.

Active Model Discrimination for Piecewise Affine Inclusion Systems

This paper proposes a novel set-membership active model discrimination (AMD) algorithm for actively separating/discriminating among a set of piecewise Affine inclusion systems with bounded noise. Specifically, to overcome the difficulties of dealing the integer variables in the lower/inner level of the associated bilevel optimization problem that stem from the mapping of the piecewise inclusions and subregions, we propose an alternative reformulation that moves the integer variables into the higher/outer level. This reformulation allows us to leverage Karush-Kuhn-Tucker (KKT) conditions to obtain an equivalent single level mixed-integer linear programming (MILP) problem. Moreover, in contrast to standard AMD algorithms that strictly enforces model separation, we propose a slight modification/extension that separates as many models as possible when a strict separation of all models is not possible.

Fuzzy harmonic mean technique for solving fully fuzzy multi-objective transportation problem

In this research article, an algorithm is developed using a fuzzy harmonic mean technique to solve a fully fuzzy multi-objective transportation problem (FFMOTP). First, the FFMOTP will be formalized mathematically, and then by using fuzzy arithmetic, it will be decomposed into three-level (lower, middle, & upper) multi-objective linear programming problems. After then, by using the fuzzy harmonic mean as a tool, the three-level multi-objective linear programming problems are converted into three-level single objective linear programming problems. Further, these single objective linear programming problems for all the three levels are solved to give the Combine fuzzy optimal solution. In the end, two numerical examples are solved to elaborate our proposed algorithm completely and compare the result obtained by our proposed algorithm with the results obtained using the other methods.

A Closest Benders Cut Selection Scheme for Accelerating the Benders Decomposition Algorithm

The Benders decomposition algorithm often shows poor convergence. To improve the convergence of the Benders decomposition algorithm. Recently, it was proposed the use of feasibility cuts closest to a solution in the set defined by all feasibility cuts. We extend this feasibility cut selection scheme to a new cut selection scheme for optimality cuts and propose a new Benders separation framework that a single linear programming problem can solve. We show that optimality cuts generated by this scheme are Pareto optimal when some conditions are satisfied. Theoretical connections to the existing Benders cut generation methods are also identified. Extensive computational experiments on the multiple classes of benchmark problems demonstrate that the proposed algorithm improves the convergence speed and computational time. Summary of Contribution: The Benders decomposition algorithm is one of the most widely used algorithms in operations research. However, the Benders decomposition algorithm often shows poor convergence for some optimization problems. In this paper, to improve the convergence of the Benders decomposition algorithm, we propose a unified closest Benders cut generation scheme. We give theoretical properties of the proposed Benders cuts, including Pareto optimality and facet-defining conditions. Also, we conducted extensive computational tests on various instances, such as network design and expansion problems. The results show the effectiveness of the closest Benders cut compared with existing algorithms and Cplex.

On the solution of robust transmission expansion planning using duality theorem under polyhedral uncertainty set

The solution techniques of tri-level robust optimization for large-scale power systems continue to remain as a great challenge. Existing techniques rely on using either Karush-Kuhn-Tucker (KKT) conditions or primal-dual (PD) formulation which leads to introduce new binary variables or bilinear terms. In doing so, a bilevel model is obtained which cannot be solved by KKT or PD due to the presence of binary variables and thus it is required to be solved using decomposition-based methods such as Benders decomposition or column-and-constraint generation algorithm. This paper aims at proposing a solution technique to deal with the robust transmission expansion planning problem where the lower level problem is replaced by its dual problem. The advantage of dualizing the lower level problem is that all the uncertain parameters are integrated into the objective function which can be further recast as a linear constraint. Afterwards, the polyhedron uncertainty set is handled using duality theory. Since neither binary variables nor bilinear terms are introduced in this approach, the lower level can be substituted using PD formulation rendering a single level mixed-integer linear programming problem. Numerical experiments on two test systems verify that the proposed model is superior compared with the existing techniques in terms of computational burden.

Fuzzy solution of fully fuzzy multi-objective linear fractional programming problems

In this study, we present a novel method for solving fully fuzzy multi-objective linear fractional programming problems without transforming to equivalent crisp problems. First, we calculate the fuzzy optimal value for each fractional objective function and then we convert the fully fuzzy multi-objective linear fractional programming problem to a single objective fuzzy linear fractional programming problem and find its fuzzy optimal solution which inturn yields a fuzzy Pareto optimal solution for the given fully fuzzy multi-objective linear fractional programming problem. To demonstrate the proposed strategy, a numerical example is provided.

Bayesian persuasion in unlinked games

Originating from Kamenica and Gentzkow (Am Econ Rev 101(6):2590–2615, 2011), we analyze multi-receiver Bayesian persuasion games with heterogeneous beliefs without strategic interactions among receivers, which we call unlinked. We show that given the receivers’ best-responses, the sender’s rationalizable strategies are obtained from a single linear programming problem.

Combined optimization for retrofitting of heat recovery and thermal energy supply in industrial systems

Predicting the outcome of possible changes in interlinked industrial energy systems is hard, especially in retrofit scenarios. This leads to severe uncertainties when making investment decisions. In this paper, a new combined optimization approach is presented that aims to support decision-making in these cases. Our approach links models for the optimal design of supply systems and heat recovery systems with operational constraints and is specifically designed for retrofit applications. It is formulated as a single combined mixed-integer linear programming (MILP) problem. The approach is applied in a case study representing a typical industrial process, where the supply system and the heat recovery are adapted. The optimal solution shows a cost-effective way for a transition to more efficient use of energy and an increased share of renewable sources.

Efficient and sustainable closed-loop supply chain network design: A two-stage stochastic formulation with a hybrid solution methodology

In recent years, consumers and legislators have pushed companies to design their supply chain networks to consider environmental and social impacts as an important performance outcome. Due to the role of resource utilization as a key component of logistics network design, another primary goal of design is ensuring available scarce resources are used as efficiently as possible across all facilities. To address efficiency issues in a sustainable closed-loop supply chain network, a stochastic integrated multi-objective mixed integer nonlinear programming model is developed in this paper, in which sustainability outcomes as well as efficiency of facility resource utilization are considered in the design of a sustainable supply chain network. In doing so, efficiency is assessed using a bi-objective output-oriented data envelopment analysis model. A hybrid three-step solution methodology is presented that creates a linear form of the original mixed integer nonlinear programming problem using piecewise McCormick envelopes approach. In the second step, an aggregated single objective programming model is derived by exploiting the multi-choice goal programming. Finally, a Lagrangian relaxation algorithm is developed to effectively solve the latter stochastic single objective mixed integer linear programming problem. The application of the proposed approach is investigated with data drawn from a case study in the electronics industry. This case study illustrates how firms may balance sustainability and efficiency in the supply chain network design problem. Further, it demonstrates the integration of efficiency results in improving economic aspects of sustainability as well as social responsibility outcomes, but also highlights the trade-offs that exist between efficiency and environmental impacts.

Testing copositivity via mixed–integer linear programming

We describe a simple method to test if a given matrix is copositive by solving a single mixed-integer linear programming (MILP) problem. This methodology requires no special coding to implement and takes advantage of the computational power of modern MILP solvers. Numerical experiments demonstrate that the method is robust and efficient.

Optimal scheduling of gas and electricity consumption in a smart home with a hybrid gas boiler and electric heating system

Space and water heating are two major components of household energy consumption, and a number of work has been done to combine multiple residential heating technologies to develop hybrid heating solutions that are more flexible, efficient and economical. This paper studied and modelled a smart home with a novel hybrid heating system, which consists of a wall-hung gas boiler (for both space and water heating) that is connected to a water tank with an auxiliary electric resistance heater (for water heating), and auxiliary split air conditioners (for space heating). A multi-objective optimization was further formulated to optimally schedule the gas and electricity consumption of a smart home equipped with the hybrid heating system as well as an onsite photovoltaic (PV) system, a battery energy storage system, an electric vehicle and various types of flexile loads. Two objectives, i.e. the minimization of operating energy cost (including both electricity and gas payment) and the maximization of thermal comfort, were considered. The optimization problem was finally transformed to a single objective mixed integer linear programming problem that can be solved by existing optimization solvers. Simulation results verified the operational performance of the studied hybrid heating system and the effectiveness of the proposed scheduling method. Sensitivity analysis was conducted considering various levels of electricity/gas prices, onsite PV capacity, battery capacity and outdoor air temperature, and it shows that in all the scenarios the operating energy cost of the smart home with the hybrid heating system was always lower than that with the pure electricity or pure gas heating systems. The performance under various uncertainties and the benefits of adopting model predictive control were also demonstrated. It is noted that this paper only focuses on the optimal scheduling and operation costs of the hybrid heating system. Optimal configuration and sizing of the heating system with the initial investment (i.e. capital expenditures) considered are identified as research directions in the future.

Finding all solution sets of piecewise-linear interval equations using an integer programming solver

This paper presents an efficient and easily implementable method for finding all solution sets of piecewise-linear interval equations using integer programming. In this method, the problem of finding all solution sets is formulated as a mixed integer programming problem, and it is solved by a high-performance integer programming solver such as CPLEX. It is shown that all solution sets are obtained by solving a single mixed integer programming problem and related linear programming problems. Numerical examples are given to confirm the effectiveness of the proposed method.

A diurnal flux balance model of Synechocystis sp. PCC 6803 metabolism.

Phototrophic organisms such as cyanobacteria utilize the sun’s energy to convert atmospheric carbon dioxide into organic carbon, resulting in diurnal variations in the cell’s metabolism. Flux balance analysis is a widely accepted constraint-based optimization tool for analyzing growth and metabolism, but it is generally used in a time-invariant manner with no provisions for sequestering different biomass components at different time periods. Here we present CycleSyn, a periodic model of Synechocystis sp. PCC 6803 metabolism that spans a 12-hr light/12-hr dark cycle by segmenting it into 12 Time Point Models (TPMs) with a uniform duration of two hours. The developed framework allows for the flow of metabolites across TPMs while inventorying metabolite levels and only allowing for the utilization of currently or previously produced compounds. The 12 TPMs allow for the incorporation of time-dependent constraints that capture the cyclic nature of cellular processes. Imposing bounds on reactions informed by temporally-segmented transcriptomic data enables simulation of phototrophic growth as a single linear programming (LP) problem. The solution provides the time varying reaction fluxes over a 24-hour cycle and the accumulation/consumption of metabolites. The diurnal rhythm of metabolic gene expression driven by the circadian clock and its metabolic consequences is explored. Predicted flux and metabolite pools are in line with published studies regarding the temporal organization of phototrophic growth in Synechocystis PCC 6803 paving the way for constructing time-resolved genome-scale models (GSMs) for organisms with a circadian clock. In addition, the metabolic reorganization that would be required to enable Synechocystis PCC 6803 to temporally separate photosynthesis from oxygen-sensitive nitrogen fixation is also explored using the developed model formalism.

On-field Performance Evaluation in Soccer Based on Network Data Envelopment Analysis

In this paper, we present for the first time the on-field production process of soccer teams as a mixed (serial and parallel) structure two-stage network system. According to this, the first stage consists of two distinct sub-processes (offense and defense) that operate in parallel. These respectively use players’ offensive and defensive actions as inputs to produce two different intermediate measures, namely, goals scored and prevention of goals conceded. These, in turn, are the inputs of the second stage (points’ accumulation sub-process) that produces accumulated points. Furthermore, based on a two-stage network Data Envelopment Analysis (DEA) model, we estimate the offensive, defensive, and athletic efficiency of soccer teams during a league season. According to our proposed framework, these three different efficiency scores are provided (for each soccer team under evaluation) by a single linear programming problem. For this purpose, aggregate-over-games statistics from the 2013-14 Greek premier soccer league are used.

Multilevel Programming-Based Coordinated Cyber Physical Attacks and Countermeasures in Smart Grid

Since the Ukraine blackout in 2015, coordinated cyber-physical attacks (CCPAs) have been emerging and are used to mask line outages in the smart grid. In this paper, we investigate the features of CCPAs and constitute the mathematic formulation with respect to topologies and electric parameters of a power grid before and after attacks. With the objective of maximizing the number of overloaded lines, a bilevel programming model is developed to describe the interaction between the adversary and the control center. The most damaging CCPA can be determined by transforming the developed bilevel model to a single mixed-integer linear programming problem using the Karush-Kuhn-Tucker conditions. Based on the features of the bilevel model, the countermeasure is expressed as a trilevel model with one leader and multiple followers. The implicit enumeration-based searching strategy is proposed to solve the trilevel model to identify the protected meters. Both the implementation of CCPAs and the effectiveness of the developed countermeasure are verified on the modified IEEE 14-bus system.

IGDT‐based economic dispatch considering the uncertainty of wind and demand response

Integration of wind generation and demand response (DR) poses challenges for the power system operation due to their uncertain characteristics. In this study, an economic dispatching method considering the uncertainties of wind and DR is proposed. The method based on the information gap decision theory (IGDT) can be used to evaluate risks associated with uncertainties from risk averse (RA) and risk-seeking (RS) perspectives. The RA IGDT-based model can provide maximum tolerable robustness region for the required cost target. The RS IGDT-based model can help achieve the lowest operation costs with desired uncertainties. The proposed model is bi-level. The upper level subproblem aims to maximise (minimise) the allowable uncertainty level to satisfy the pre-determined cost target, while the lower level subproblem is to maximise (minimise) possible cost considering the uncertainties. The bi-level model is then transformed into a single level mixed integer linear programming problem that can be solved through commercial solves. Finally, the authors evaluate the performance of the IGDT-based approach by simulations on the modified 6-bus and IEEE 118-bus systems. The results show that the proposed approach can provide suggestions for system operators to make appropriate scheduling plan based on the expected cost targets and risk preferences.

Efficiency evaluation in two-stage data envelopment analysis under a fuzzy environment: A common-weights approach

Data envelopment analysis (DEA) has been genuinely known as an impeccable technique for efficiency measurement. In practice, since many production systems such as broadcasting companies, banking institutions and R&D organisations include two processes connected in series, we have need of utilising two-stage DEA models to identify the sources of inefficiency and explore in turn appropriate options for improving performance. The lack of the ability to generate the actual weights is not only an ongoing challenge in traditional DEA models, it can also have serious repercussion for the contemporary DEA models (e.g., two-stage DEA). This paper presents a common-weights method for two-stage structures that allows us to consider equality of opportunity in a fuzzy environment when evaluating the system efficiency and the component process efficiencies. The proposed approach first seeks upper bounds on factor weights and then determines a set of common weights by a single linear programming problem. We illustrate the developed approach with a data set taken from the literature.

A Linear Programming Approach for Minimizing a Linear Function Subject to Fuzzy Relational Inequalities With Addition–Min Composition

In this paper, we study an optimization problem of minimizing a linear function subject to fuzzy relational inequalities with the addition–min composition. This optimization setting has recently been proposed to model the network cogestion issue when a BitTorrent-like peer-to-peer file-sharing system is used for data transmission. In a 2014 paper, a pseudominimal index (PMI)-based approach was proposed to search for an optimal solution. It turns out that the PMI-based approach may require to solve several to many linear programming problems in order to get an optimal solution. In this paper, we point out that the feasible domain is indeed convex. And we only need to solve a single linear programming problem to generate an optimal solution for the original optimization problem. Furthermore, our approach could be extended to the case with a nonlinear continuous objective function.

A Novel Technique for Solving Multiobjective Fuzzy Linear Programming Problems

This study considers multiobjective fuzzy linear programming (MFLP) problems in which the coefficients in the objective functions are triangular fuzzy numbers. The study proposing a new technique to transform MFLP problems into the equivalent single fuzzy linear programming problem and then solving it via linear ranking function using the simplex method, supported by numerical example.