Standard Linear Programming Problem Research Articles

Sensitivity Analysis for Fuzzy Linear Programming Problem by Mehar Method

Numerous manufacturing companies face significant financial setbacks due to excessive raw material costs, low profits, and no taxes on imported goods. The production companies also suffer from uncertainty concerning expenses, production levels, sales figures, and the accessibility of raw materials. Therefore, in our study, we developed an innovative method to design a fuzzy linear programming model customised for the Doura Refinery of the Middle Refineries Company. We emphasised five essential products: white oil, naphtha, light puffs, heavy Jet, and liquid gas. The main goal of our approach was to achieve optimal decision-making by identifying the most favourable production quantities and evaluating the corresponding net income derived from product sales. Following the Mehar method, we employed a methodology that converts the fuzzy linear programming problem into a standard linear programming problem. Subsequently, we conducted a sensitivity analysis, focusing only on the impact of fuzzy cost changes, and successfully derived optimal solutions for this case. Therefore, we aim to resolve the challenges of fuzzy linear programming and explore how

Primal-Dual Path-Following Methods and the Trust-Region Updating Strategy for Linear Programming with Noisy Data

In this article, we consider the primal-dual path-following method and the trust-region updating strategy for the standard linear programming problem. For the rank-deficient problem with the small noisy data, we also give the preprocessing method based on the QR decomposition with column pivoting. Then, we prove the global convergence of the new method when the initial point is strictly primal-dual feasible. Finally, for some rank-deficient problems with or without the small noisy data from the NETLIB collection, we compare it with other two popular interior-point methods, i.e. the subroutine pathfollow.m and the built-in subroutine linprog.m of the MATLAB environment. Numerical results show that the new method is more robust than the other two methods for the rank-deficient problem with the small noise data.

Pool trading model within a local energy community considering flexible loads, photovoltaic generation and energy storage systems

This paper presents a pool trading model within a local energy community considering home energy management systems (HEMSs) and other consumers. A transparent mechanism for market clearing is proposed to incentivise active prosumers to trade their surplus energy within a rule-based pool market in the local energy community. A price-based demand response program (PBDRP) is considered to increase the consumers’ willingness to modify their consumption. The mathematical optimization problem is a standard mixed-integer linear programming (MILP) problem to allow for rapid assessment of the trading market for real energy communities which have a considerable number of consumers. This allows for novel energy trading strategies amongst different clients in the model and for the integration of a pool energy trading model at the level of the local energy community. The objective function of the energy community is to minimize the overall bills of all participants while fulfilling their demands. Two different scenarios have been evaluated, independent and integrated operation modes, to show the impacts of coordination amongst different end-users. Results show that through cooperation, end-users in the local energy community market can reduce the total electricity bill. This is shown in a 16.63% cost reduction in the independent operation and a 21.38% reduction in the integrated case. Revenues for active consumers under coordination increased compared to independent operation of the HEMS.

Clarification of the Formalization Process for Models of Economic Dynamics in the Study of the Development of Organizational and Technological Systems

The article substantiates the creation and practical development of organizational and technological systems in high-tech production on the basis of the main theory and solution of linear programming problems. The main problem when using models of economic dynamics in creating this type of system is the large dimension of the emerging optimization problems, which prevents the direct use of known computational procedures for solving linear programming problems. To do this, we consider an extreme problem for direct dynamic analysis of the Leontief model using the standard linear programming problem. At the same time, we should not forget about the problem of determining the target functional, the choice of which prevents the choice of the optimal trajectory of development of the simulated system. Due to these difficulties, the need for quality research methods of optimal trajectories to obtain theoretical conclusions for making scientifically sound planning decisions is determined. Theorems on highways describe the qualities of invariance of optimal trajectories of extreme dynamic problems in relation to the purpose of the target functional and to the choice of the optimal trajectory. The theoretical generalizations of Samuelson’s main theory were used in a consulting project on planned studies of the introduction of organizational and technological system for the production of high-tech products at a machine-building enterprise. Keywords system, technology, technology, highway, model, trajectory, programming, functional, vector, function, matrix.

Devising a method for identifying the model of multi-criteria expert estimation of alternatives

An approach to constructing mathematical models of individual multicriterial estimation was proposed based on information about the ordering relations established by the expert for a set of alternatives. Structural identification of the estimation model using the additive utility function of alternatives was performed within axiomatics of the multi-attribute utility theory (MAUT). A method of parametric identification of the model based on the ideas of the theory of comparative identification has been developed. To determine the model parameters, it was proposed to use the midpoint method that has resulted in the possibility of obtaining a uniform stable solution of the problem. It was shown that in this case, the problem of parametric identification of the estimation model can be reduced to a standard linear programming problem. The scalar multicriterial estimates of alternatives obtained on the basis of the synthesized mathematical model make it possible to compare them among themselves according to the degree of efficiency and, thus, choose "the best" or rank them. A significant advantage of the proposed approach is the ability to use only non-numerical information about the decisions already made by experts to solve the problem of identifying the model parameters. This enables partial reduction of the degree of expert’s subjective influence on the outcome of decision-making and reduces the cost of the expert estimation process. A method of verification of the estimation model based on the principles of cross-validation has been developed. The results of computer modeling were presented. They confirmed the effectiveness of using the proposed method of parametric model identification to solve problems related to automation of the process of intelligent decision making.

THE METHOD OF ALTERNATIVE RANKING FOR A COLLECTIVE EXPERT ESTIMATION PROCEDURE

Context. The actual problem of constructing a mathematical model of a collective multi-criteria expert estimation of alternatives, which is an integral part of the automation of the intellectual decision-making process, has been solved. Objective. The goal of the work is to develop a method for determining relative collective multi-criteria estimation of alternatives and their subsequent ranking based on information about personal preferences of experts. The object of research is the process of analysis and decision-making in multi-criteria conditions. The subject of the research are the methods of structural and parametric identification of the model of multi-criteria estimation of alternatives.Method. The paper proposes an approach to constructing a model of collective multi-criteria estimation of alternatives based on information about partial-order relationships established by experts on the set of available alternatives. A method for structural and parametric identification of a model of multi-criteria estimation, which based on the ideas of the theory of comparator identification is proposed. It is shown that the solution to the problem of choosing the structure of a model of optimal complexity should be carried out in the class of Kolmogorov-Gabors polynomial. To find the parameters of the estimation model, it is proposed to use a method that is based on the calculating of the Chebyshev point. It is shown that in this case, the parametric identification problem of the model can be reduced to the standard linear programming problem. The scalar collective multi-criteria estimates of alternatives obtained on the basis of the synthesized mathematical model make it possible to compare them with each other in terms of “quality” and, thus, select the “best” of them or rank them.Results. An approach has been developed to construct a mathematical model of collective multi-criteria expert estimation, on the basis of which it is possible to determine group generalized estimates of alternatives, as well as to rank them. The results of simulation modeling, which demonstrate the practical feasibility and effectiveness of the proposed approach are presented.Conclusions. A significant advantage of the approach is the ability to use only non-numerical information about the preferences of experts. This allows you to partially solve the problem of subjectivity of expert opinions in the process of decision-making and reduce the cost of a collective expert estimation of alternatives. The synthesized model of collective expert estimation can serve as the basis for solving the problems of estimating the quality of various projects, investment management, strategic planning, and the development of problem-oriented decision support systems. In the future, it is worth considering the possibility of supplementing the presented approach with the possibility of taking into account estimates of the qualitative composition and competence of individual experts, which are included in the group.

Resource Allocation and Basestation Placement in Downlink Cellular Networks Assisted by Multiple Wireless Powered UAVs

In this paper, we focus on a downlink cellular network, where multiple UAVs serve as aerial basestations to provide wireless connectivity to ground users through frequency division multi-access (FDMA) scheme. The UAVs are exclusively powered by a wireless charging station located on the ground following save-then-transmit protocol. In such a UAV-assisted cellular network, joint optimization for user association, resource allocation and basesation placement are investigated to maximize the downlink sum rate. The problem is formulated as a mixed integer optimization problem and is thus challenging to solve. We propose an efficient solution based on alternating optimization, by iteratively solving one of the three subproblems (i.e., user association, resource allocation and basesation placement) with the other two fixed. Specificly, user association is solved as a standard linear programming problem by relaxing the binary association indicators into continuous variables. For basestation placement and resource allocation, we resort to successive convex optimization technique, which iteratively solves a lower-bound problem. After iteratively solving the three subproblems, we further propose an algorithm based on penalty method and successive convex optimization to make the association indicators feasibly binary. We conduct comprehensive experiments for the optimal solution to the three subproblems with insightful results. We also show that the optimal downlink sum rate cannot be always enhanced by deploying more UAVs due to non-negligible tradeoff between energy/communication sources and co-channel interference. Moreover, the proposed solution outperforms a baseline strategy leveraged from an existing work, especially with favorable channel condition and sufficient frequency resources.

Optimization the shape of the Francis hydraulic turbine

The article presents one of the possible methods for optimizing the blade shape of a Francis radial-axial hydraulic turbine. The method for optimizing the shape of the turbine blade is based on the criterion of the maximum mechanical moment developed by the turbine. The blade shape optimization operation is conventionally divided into two stages. At the first stage of optimization of the blade shape, the analytical expression of the moment developed by the turbine is presented in a Taylor series by variable parameters − the coordinates of the vertices of the characteristic polyhedron of the median surface of the turbine blade. Adding the boundary conditions in the formulation of the optimization problem in the form of equalities − the contact of the median surface of the turbine blade with the turbine hub and rim, as well as conditions in the form of inequalities − the concavity of the greater part the median surface allows to reduce the problem of optimizing the blade shape to a standard linear programming problem. It is proposed to carry out 50−60 similar operations with small steps in the variables − the coordinates of the vertices of the characteristic polyhedron. Thus, it is necessary to move into the zone of optimal values of the coordinates of the vertices of the characteristic polyhedron of the median surface of the blade. At the second stage, it is proposed to continue the search for the optimal values of the coordinates of the vertices of the characteristic polyhedron of the median surface of the blade, applying for this purpose one of the most effective algorithms of genetic optimization.

Finite-time stability and stabilisation of singular discrete-time linear positive systems with time-varying delay

ABSTRACTIn this paper, the problem of finite-time stability and stabilisation for positive singular discrete-time linear systems with time-varying delay is investigated. We first present novel delay-dependent sufficient conditions for positivity and finite-time stability of the unforced systems. We then apply the obtained results to solve finite-time stabilisation problem of the considered systems. The sufficient conditions for the positivity and finite-time stabilisable of such systems are formulated in terms of a standard linear programming (LP) problem. Numerical examples are provided to illustrate the effectiveness and advantages of our results.

TRANSPORTATION MANAGEMENT IN A DISTRIBUTED LOGISTIC CONSUMPTION SYSTEM UNDER UNCERTAINTY CONDITIONS

The problem of supply management in the supplier-to-consumer logistics transport system has been formed and solved. The novelty of the formulation of the problem consists in the integrated accounting of costs in the logistic system, which takes into account at the same time the cost of transporting products from suppliers to consumers, as well as the costs for each of the consumers to store the unsold product and losses due to possible shortages. The resulting optimization problem is no longer a standard linear programming problem. In addition, the work assumes that the solution of the problem should be sought taking into account the fact that the initial data of the problem are not deterministic. The analysis of traditional methods of describing the uncertainty of the source data. It is concluded that, given the rapidly changing conditions for the implementation of the delivery process in a distributed supplier-to-consumer system, it is advisable to move from a theoretical probability representation of the source data to their description in terms of fuzzy mathematics. At the same time, in particular, the fuzzy values of the demand for the delivered product for each consumer are determined by their membership functions. Distribution of supplies in the system is described by solving a mathematical programming problem with a nonlinear objective function and a set of linear constraints of the transport type. In forming the criterion, a technology is used to transform the membership functions of fuzzy parameters of the problem to its theoretical probabilistic counterparts – density distribution of demand values. The task is reduced to finding for each consumer the value of the ordered product, minimizing the average total cost of storing the unrealized product and losses from the deficit. The initial problem is reduced to solving a set of integral equations solved, in general, numerically. It is shown that in particular, important for practice, particular cases, this solution is achieved analytically. The paper states the insufficient adequacy of the traditionally used mathematical models for describing fuzzy parameters of the problem, in particular, the demand. Statistical processing of real data on demand shows that the parameters of the membership functions of the corresponding fuzzy numbers are themselves fuzzy numbers. Acceptable mathematical models of the corresponding fuzzy numbers are formulated in terms of bifuzzy mathematics. The relations describing the membership functions of the bifuzzy numbers are given. A formula is obtained for calculating the total losses to storage and from the deficit, taking into account the bifuzzy of demand. In this case, the initial task is reduced to finding the distribution of supplies, at which the maximum value of the total losses does not exceed the permissible value.

On Finite-Time Stability of Linear Positive Differential-Algebraic Delay Equations

This brief addresses finite-time stability of linear positive differential-algebraic equations with delay. Being different from the Lyapunov function method, delay-dependent sufficient conditions for positivity and finite-time stability of the system are established in terms of standard linear programming problems. A numerical example is given to illustrate the obtained results.

LP approach to exponential stabilization of singular linear positive time-delay systems via memory state feedback

This paper deals with the exponential stabilization problem by means of memory state feedback controller for linear singular positive systems with delay. By using system decomposition approach, singular systems theory and Lyapunov function method, we obtain new delay-dependent sufficient conditions for designing such controllers. The conditions are given in terms of standard linear programming (LP) problems, which can be solved by LP optimal toolbox. A numerical example is given to illustrate the effectiveness of the proposed method.

Weighted‐average ℓ1 filtering for switched positive systems

SummaryIn this paper, the filtering problem for a class of switched positive systems with dwell time is investigated. A novel weighted‐average technique is proposed for filter design such that the final estimate of the unmeasurable outputs of the considered system is more accurate than that of traditional approaches. The main contributions of this paper are summarized as follows: By exploiting the positivity and characteristics of switched positive systems with dwell time, a candidate linear copositive Lyapunov function, which is both quasi‐time‐dependent and mode‐dependent, is presented to establish the closed‐loop stability of the considered systems. Upon the established closed‐loop stability, less conservative bounded positive filters (both upper‐bound and lower‐bound filter) with ℓ1 disturbance attenuation performance are designed for the considered system. By introducing a proper weight, a weighted‐average approach, which is more general than the bounded filter design method, is proposed for filter design. The worst ℓ1 disturbance attenuation performance of the novel developed filter is evaluated. Both the bounded filters and the weighted‐average filter are designed by solving standard linear programming problems. A numerical example illustrates the effectiveness of the proposed approach. Copyright © 2017 John Wiley & Sons, Ltd.

An Improvement for Fuzzy Stochastic Goal Programming Problems

We examined the solution process for linear programming problems under a fuzzy and random environment to transform fuzzy stochastic goal programming problems into standard linear programming problems. A previous paper that revised the solution process with the lower-side attainment index motivated our work. In this paper, we worked on a revision for both-side attainment index to amend its definition and theorems. Two previous examples were used to examine and demonstrate our improvement over previous results. Our findings not only improve the previous paper with both-side attainment index, but also provide a theoretical extension from lower-side attainment index to the both-side attainment index.

A new method to generate the ground structure in truss topology optimization

ABSTRACTThe aim of this article is to provide an effective method to generate the ground structure in truss topology optimization. The core of this method is to place nodal points for the ground structure at the intersection of the first and third principal stress trajectories, which are obtained by solving the equivalent static problem in the design domain with a homogeneous isotropic material property. It is applicable to generate the ground structure for arbitrary regular and irregular geometric design domains. The proposed method is tested on some benchmark examples in truss topology optimization. The optimization model is a standard linear programming problem based on plastic design and solved by the interior point algorithm. Compared with other methods, the proposed method may use a well-defined ground structure with fewer nodes and bars, resulting in faster solution convergence, which shows it to be efficient.

Optimal Solution of Multi-Choice Mathematical Programming Problem Using a New Technique

The study deals with the multi-choice mathematical programming problem, where the right hand side of the constraints is multi-choice in nature. However, the problem of multi-choice linear programming cannot be solved directly by standard linear or nonlinear programming techniques. The aim of this paper is to transform such problems to a standard mathematical linear programming problem. For each constraint, exactly one parameter value is selected out of a multiple number of parameter values. This process of selection can be established in different ways. In this paper, we present a new simple technique enabling us to handle such problem as a mixed integer linear programming problem and consequently solve them by using standard linear programming software. Our main aim depends on inserting a specific number of binary variables and using them to construct a linear combination which gives just one parameter among the multiple choice values for each choice of the values of the binary variables. A numerical example is presented to illustrate our analysis.

A Variant of the Dual Face Algorithm Using Gauss-Jordan Elimination for Linear Programming

Using Cholesky factorization, the dual face algorithm was described for solving standard Linear programming (LP) problems, as it would not be very suitable for sparse computations. To dodge this drawback, this paper presents a variant using Gauss-Jordan elimination for solving bounded-variable LP problems.

L₁-Norm Low-Rank Matrix Decomposition by Neural Networks and Mollifiers.

The L1-norm cost function of the low-rank approximation of the matrix with missing entries is not smooth, and also cannot be transformed into a standard linear or quadratic programming problem, and thus, the optimization of this cost function is still not well solved. To tackle this problem, first, a mollifier is used to smooth the cost function. High closeness of the smoothed function to the original one can be obtained by tuning the parameters contained in the mollifier. Next, a recurrent neural network is proposed to optimize the mollified function, which will converge to a local minimum. In addition, to boost the speed of the system, the mollifying process is implemented by a filtering procedure. The influence of two mollifier parameters is theoretically analyzed and experimentally confirmed, showing that one of the parameters is critical to computational efficiency and accuracy, while the other not. A large number of experiments on synthetic data show that the proposed method is competitive to the state-of-the-art methods. In particular, the experiments on large matrices and a real application in the structure from motion indicate that the memory requirement of the proposed algorithm is mild, making it suitable for real applications that often involve large-scale matrix decomposition.

Dynamic shortest path association for multiple object tracking in video sequence

Persistently tracking multiple objects in cluttered environments is very challenging. We present a tracking association approach based on the shortest path faster algorithm. We first formulate the multiple object tracking as an integer programming problem of the flow network. Under this framework, the integer assumption is relaxed to a standard linear programming problem. Therefore, the global optimal solution can quickly be obtained using the fast dynamic shortest path algorithm, which highlights the dynamic programming characteristic of the shortest path, thus faster, algorithm. The proposed method avoids the difficulties of integer programming; more importantly, it has a lower worst-case complexity than competing methods but a better tracking accuracy and robustness in complex environments. Simulation results show that our proposed algorithm takes less time than other methods and can operate in real time.

Robust assessment of collapse resistance of structures under uncertain loads based on Info-Gap model

The paper proposes a pair of novel mathematical programming based approaches which directly determine the worst collapse load limit in one case, and the best limit in the other case of rigid perfectly-plastic structures subjected to uncertain-but-bounded applied loads using an Info-Gap model. The methods take advantage of the important properties in which the worst collapse load limit defined for an uncertain static formulation is equivalent to the most favourable solution of the uncertain kinematic limit analysis problem, and vice versa for the best collapse load limit. The formulation for capturing the worst collapse load limit (robust worst case solution) takes the form as a mathematical program with equilibrium constraints that is processed using a penalty algorithm, whilst that for the best collapse load limit is a standard linear programming problem. The efficiency and robustness of the proposed schemes are evidenced from a number of successfully solved examples.