Signomial Geometric Programming Research Articles

Nonlinear fuzzy fractional signomial programming problem: A fuzzy geometric programming solution approach

Fuzzy fractional signomial programming problem is a relatively new optimization problem. In real world problems, some variables may vacillate because of various reasons. To tackle these vacillating variables, vagueness is considered in form of fuzzy sets. In this paper, a nonlinear fuzzy fractional signomial programming problem is considered with all its coefficients in objective functions as well as constraints are fuzzy numbers. Two solution approaches are developed based on signomial geometric programming comprising nearest interval approximation with parametric interval valued functions and fuzzy α-cut with min–max approach. To demonstrate the proposed methods, two illustrative numerical examples are solved and the results are comparatively discussed showing its feasibility and effectiveness.

Signomial Geometric Programming Approach to Solve Non-Linear Fractional Programming Problems

Signomial Geometric Programming Approach to Solve Non-Linear Fractional Programming Problems

The maximal angle between 5×5 positive semidefinite and 5×5 nonnegative matrices

The paper is devoted to the study of the maximal angle between the $5\times 5$ semidefinite matrix cone and $5\times 5$ nonnegative matrix cone. A signomial geometric programming problem is formulated in the process to find the maximal angle. Instead of using an optimization problem solver to solve the problem numerically, the method of Lagrange Multipliers is used to solve the signomial geometric program, and therefore, to find the maximal angle between these two cones.

Online Optimal Control for Wireless Cooperative Transmission by Ambient RF Powered Sensors

Ambient radio frequency (RF) energy harvesting technology provides a hopeful solution to power the low-power electronic system. Nevertheless, the extremely low power density and randomness of ambient RF energy pose a significant obstacle to the application of ambient RF energy harvesting technology. Aiming at wireless transmission by ambient RF-powered sensors, this paper presents a cooperative communication technology by combining ambient backscattering and energy beamforming to significantly decrease the transmission energy consumption of each sensor. For wireless transmission via this cooperative communication technology, a novel optimization problem is formulated to maximize the long-term time-averaged sensed data size, which jointly takes into account sensing scheduling, transmission power allocation and the control of sensed data size. Moreover, we employ techniques of Lyapunov drift and Lyapunov optimization to propose an online optimal control algorithm, which achieves the optimal solution to the proposed problem without requiring a-priori knowledge of harvested energy and wireless channels. By exploiting mixed integer programming (MIP) and signomial geometric programming (SGP) that cause the proposed problem difficult to solve, a novel Sequential Convex Approximation algorithm is further introduced into the online optimal control algorithm. Simulation results indicate the proposed online algorithm offers a dramatic performance gain over alternative online algorithms.

Global optimisation for a developed price discrimination model: A signomial geometric programming-based approach

This paper presents a price discrimination model for a manufacturer who acts in two different markets. In order to have a fair price discrimination model and compare monopoly and competitive markets, it is assumed that there is no competitor in the first market (monopoly market) and there is a strong competitor in the other market (competitive market). The manufacturer’s objective is to maximise the total benefit in both markets. The decision variables are selling price, lot size, marketing expenditure, customer service cost, flexibility and reliability of production process, set-up costs, and quality of products. The proposed model in this paper is a signomial geometric programming problem which is difficult to solve and find the globally optimal solution. So, this signomial model is converted to a posynomial geometric type and using an iterative method, the globally optimal solution is found. To illustrate the capability of the proposed model, a numerical example is solved and the sensitivity analysis is implemented under different conditions.

Modified signomial geometric programming (MSGP) and its applications

A "signomial" is a mathematical function, contains one or more independent variables. Richard J. Duffin and Elmor L. Peterson introduced the term "signomial". Signomial geometric programming (SGP) optimization technique often provides a much better mathematical result of real-world nonlinear optimization problems. In this research paper, we have proposed unconstrained and constrained signomial geometric programming (SGP) problem with positive or negative integral degree of difficulty. Here a modified form of signomial geometric programming (MSGP) has been developed and some theorems have been derived. Finally, these are illustrated by proper examples and applications.

Optimal payment time and replenishment decisions for retailer’s inventory system under trade credit and carbon emission constraints

This study presents a multi-item inventory and pricing model by considering marketing, service activities, trade credit, carbon emissions, and the restrictions of production cost and storage space. In the proposed model, shortages are allowed and demand rate is a power function of service and marketing costs, and selling price. The main objective of this study is to optimize retailer’s payments time, service and marketing expenditure, and replenishment decisions in order to maximize retailer’s total profit and minimize carbon emissions, simultaneously. Model is developed in a fuzzy environment under carbon tax regulation when the length of credit period provided by supplier is less than or equal to the length of time in which no shortage happens. To solve the proposed model, we first transform the original problem into a multi-objective Signomial Geometric Programming (SGP) problem using fuzzy and hybrid parameters, which minimizes both the mean value and the total dispersion value of the objective function. Then a global optimization problem method has been used to solve the SGP problem. Efficiency of this algorithm is tested and compared with multi-objective genetic algorithm, multi-objective genetic algorithm with varying population, and hybrid heuristic algorithm. At the end, several numerical examples and sensitivity analysis are performed to demonstrate the application of the proposed model and solution procedure to obtain managerial insights.

An Energy Efficient Cooperative Communication Scheme in Ambient RF Powered Sensor Networks

The ambient radio frequency (RF) energy harvesting technology has recently been regarded as a potential solution for powering the wireless sensor networks (WSNs). However, the ultra-low power density of ambient RF energy is the main impediment to its further application. In this paper, we propose a novel energy efficient cooperative communication scheme (EECCS), which combines energy beamforming communication and ambient backscatter communication, to overcome the energy problem of WSNs powered by ambient RF energy harvesting. Moreover, to further reduce the energy consumption of nodes, we present an optimal resource allocation problem for EECCS. It can be formulated as a signomial geometric programming (SGP), which is nonconvex and NP-hard. We develop a Sequential Convex Approximation (SCA) algorithm for finding a solution of this SGP problem, which transforms the SGP problem into a sequence of geometric programming (GP) problems that are convex. The simulation results indicate that the EECCS can improve the energy efficiency of the ambient RF powered WSNs and maximize the total amount of data received by the sink.

An inventory model with credit, price and marketing dependent demand under permitted delayed payments and shortages: A signomial geometric programming approach

An inventory model with credit, price and marketing dependent demand under permitted delayed payments and shortages: A signomial geometric programming approach

An inventory model for non-instantaneous deteriorating items with credit period and carbon emission sensitive demand: a signomial geometric programming approach

ABSTRACTThis study considers sustainability issues and default risks in the context of joint inventory control and trade credit financing for Non-Instantaneous Deteriorating Items (NIDIs) in a supplier-retailer-customer system, in which the supplier offers an upstream full trade credit to the retailer and the retailer, in turn, provides a downstream partial trade credit to her/his customers. Demand rate is sensitive to the amount of carbon emissions, the length of the credit period suggested by the retailer to its customers and selling price. The main objective of this study is to optimize credit period, order quantity, and replenishment cycle time in order to maximize retailer’s total profit and minimize carbon emissions at the same time. For solving the proposed problem, we apply an approximation method to simplify the profit function and transform the problem into a constrained Signomial Geometric Programming (SGP) problem, then a global optimization approach is used for solving the model.Finally, a numerical illustration and sensitivity analysis are performed to demonstrate the problem and the solution procedure. Some useful managerial insights are obtained from computational results. The results show that the retailer will increase its total profit by considering trade credit policy and non-instantaneous deteriorating phenomenon.

Fuzzy E.O.Q model with constant demand and shortages: A fuzzy signomial geometric programming (FSGP) approach

In this paper, a fuzzy economic order quantity (E.O.Q) model with shortages under fully backlogging and constant demand is formulated and solved. Here the model is solved by fuzzy signomial geometric programming (FSGP) technique. Fuzzy signomial geometric programming (FSGP) technique provides a powerful technique for solving many non-linear problems. Here we have proposed a new idea that is fuzzy modified signomial geometric programming (FMSGP) and some necessary theorems have been derived. Finally, these are illustrated by some numerical examples and applications.

An accelerated extended cutting plane approach with piecewise linear approximations for signomial geometric programming

This paper presents a global optimization approach for solving signomial geometric programming (SGP) problems. We employ an accelerated extended cutting plane (ECP) approach integrated with piecewise linear (PWL) approximations to solve the global optimization of SGP problems. In this approach, we separate the feasible regions determined by the constraints into convex and nonconvex ones in the logarithmic domain. In the nonconvex feasible regions, the corresponding constraint functions are converted into mixed integer linear constraints using PWL approximations, while the other constraints with convex feasible regions are handled by the ECP method. We also use pre-processed initial cuts and batched cuts to accelerate the proposed algorithm. Numerical results show that the proposed approach can solve the global optimization of SGP problems efficiently and effectively.

Energy Balance Based Lifetime Maximization in Wireless Sensor Networks Employing Joint Routing and Asynchronous Duty Cycle Scheduling Techniques

This paper introduces a novel energy balance based lifetime maximization issue in wireless sensor networks employing joint routing and asynchronous duty cycle scheduling techniques titled as EB-JRADCS problem. To formulate the EB-JRADCS problem a new asynchronous MAC protocol utilizing flooding of RTS and random sending of CTS named FRTS---RCTS is proposed. This protocol leads to new constraints called flow sharing (FS) constraints that joint the network lifetime maximization parameters including flow rate of information on any route and duty cycle of nodes. It is shown that the modeled EB-JRADCS problem can be expressed as a signomial geometric programming problem. Due to the complexity of solving the achieved problem, first it is converted into a simpler problem by relaxing FS constraints from equal to unequal form. Then the simplified problem is solved with the aid of a specific convexification method and the global optimum solution of the network lifetime is evaluated under various scenarios. The achieved optimum solution can be used as a benchmark for evaluating and comparing distributed and heuristic methods that aim to extend the network lifetime.

A Global Optimization Algorithm for Signomial Geometric Programming Problem

This paper presents a global optimization algorithm for solving the signomial geometric programming (SGP) problem. In the algorithm, by the straight forward algebraic manipulation of terms and by utilizing a transformation of variables, the initial nonconvex programming problem (SGP) is first converted into an equivalent monotonic optimization problem and then is reduced to a sequence of linear programming problems, based on the linearizing technique. To improve the computational efficiency of the algorithm, two range reduction operations are combined in the branch and bound procedure. The proposed algorithm is convergent to the global minimum of the (SGP) by means of the subsequent solutions of a series of relaxation linear programming problems. And finally, the numerical results are reported to vindicate the feasibility and effectiveness of the proposed method.

Optimization of Multi Plate Friction Clutch for Maximum Torque Transmitting Capacity Using Uniform Wear Theory

Optimization is the art of obtaining the best results under given circumstances. In design, construction and maintenance of any engineering system, the ultimate goal is either to minimize the effort required or to maximize the desired benefits. The various processes of optimization come under the subject Operation Research (O.R.). O.R. one of the branch of mathematics which is concerned which the application of scientific methods and techniques to design making problems and establishing optimal situations.In this project we aim to find the optimum design of a multi disc friction clutch through operations research techniques. The two methods used are stochastic programming and geometric programming.The existing design procedure treats design variables as deterministic ones. In actual practice due to manufacturing tolerances etc. design variables do not remain deterministic but become probabilistic. The probabilistic nature of these should be given proper consideration while designing. This project deals with an approach which takes into account the probabilistic nature of designs variables, as well as probability of satisfying constraint equations. Stochastic signomial geometric programming has been used for optimization.In this paper we aimed to solve the optimized result for the multi plate friction clutch using the uniform wear theory and obtained the optimum result of the inner radius and outer radius of the multi disc friction clutch.[7]

Global optimization of signomial geometric programming problems

This paper presents a global optimization approach for solving signomial geometric programming problems. In most cases nonconvex optimization problems with signomial parts are difficult, NP-hard problems to solve for global optimality. But some transformation and convexification strategies can be used to convert the original signomial geometric programming problem into a series of standard geometric programming problems that can be solved to reach a global solution. The tractability and effectiveness of the proposed successive convexification framework is demonstrated by seven numerical experiments. Some considerations are also presented to investigate the convergence properties of the algorithm and to give a performance comparison of our proposed approach and the current methods in terms of both computational efficiency and solution quality.

The discrete ellipsoid covering problem: A discrete geometric programming approach

The ellipsoid covering problem consists of covering an ellipsoid with spheres whose radii belong to a discrete set. The discrete nature of the radii of the spheres is one of the difficulties inherent to solving this problem. The other difficulty is to ensure that every point of the ellipsoid is covered by at least one sphere. Despite these difficulties, a good reason for studying this problem is its application in configuring gamma ray machines, used for treatment of brain tumors. This is a semi-infinite nonlinear discrete problem. To solve it, we present a weak version and model it as a discrete signomial geometric programming problem. To obtain convex constraints, we apply the condensation technique to approximate the model in combination with a continuous model applied to the discrete part. Thus, we use a primal dual interior point method to solve the problem. During model construction, we determine the smallest number of spheres that can be used to cover the ellipsoid using an auxiliary model that is similar to the model of the Knapsack problem. Finally, we present computational results for experiments.

A continuous strategy to solve a class of mixed optimization problems

This work presents a method to solve a class of discrete optimization problems, including linear, quadratic, convex, and discrete geometric programming problems. The methodology consists of inserting, in the original problem, additional geometric constraints where any viable solution is also discrete. Moreover, a strategy to solve signomial geometric programming problems is developed. Computational results are shown through some examples of facility location, machining economics and economic order quantity problems.

Optimal Incentive Pricing on Relaying Services for Maximizing Connection Availability in Multihop Cellular Networks

This paper investigates an incentive pricing problem for relaying services in multihop cellular networks. Providing incentives to encourage mobile nodes to relay data is a critical factor in building successful multihop cellular networks. Most existing approaches adopt fixed‐rate or location‐based pricing on rewarding packets forwarding. This study applies a mathematical programming model to determine an optimal incentive price for each intermediate node that provides relaying services. Under the obtained incentive price, the connection availability of the networks is maximized by using the same relaying costs as other pricing schemes. A signomial geometric programming problem is constructed, and a deterministic optimization approach is employed to solve the problem. Besides, quality‐of‐service constraints are added in the proposed model to mitigate the unfairness between connection availabilities of individual nodes. Computational results demonstrate that the proposed model obtains the optimal incentive price on relaying services to maximize connection availability of the networks.

Range reduction techniques for improving computational efficiency in global optimization of signomial geometric programming problems

Many global optimization approaches for solving signomial geometric programming problems are based on transformation techniques and piecewise linear approximations of the inverse transformations. Since using numerous break points in the linearization process leads to a significant increase in the computational burden for solving the reformulated problem, this study integrates the range reduction techniques in a global optimization algorithm for signomial geometric programming to improve computational efficiency. In the proposed algorithm, the non-convex geometric programming problem is first converted into a convex mixed-integer nonlinear programming problem by convexification and piecewise linearization techniques. Then, an optimization-based approach is used to reduce the range of each variable. Tightening variable bounds iteratively allows the proposed method to reach an approximate solution within an acceptable error by using fewer break points in the linearization process, therefore decreasing the required CPU time. Several numerical experiments are presented to demonstrate the advantages of the proposed method in terms of both computational efficiency and solution quality.