Linear Fractional Programming Research Articles

Solving Triangular Fully Fuzzy Linear Fractional Programming Problem via Parametric Approach

Solving Triangular Fully Fuzzy Linear Fractional Programming Problem via Parametric Approach

Solving Multi-Objective Linear Fractional Programming Problems by Novel Methods

In this paper, we suggested two new techniques for transforming multi-objective linear fractional programming problems into single-objective linear fractional programming problems (SOLFPP) and then solving them by methods of SOLFPP. Illustrative numerical examples are presented for demonstration purposes. Also, the obtained results are compared with another method studied previously.

Resource Allocation Policies for Battery Constrained Energy Harvesting Communication Systems With Co-Channel Interference

This paper studies resource allocation policies for energy harvesting (EH) multi-user multiple input single output (MU- MISO) communication systems. The multi-antenna EH base station (BS) is equipped with a limited-capacity battery. Though employing the multi-antenna at the BS provides the channel diversity, this leads to the co-channel interference which makes the resource allocation problem hard to solve. For this challenging scenario, we first consider off-line policies based on full channel state information (CSI) and energy arrival information (EAI) to obtain the best performance for any feasible resource allocation policies. We propose an iterative algorithm using generalized linear fractional programming to obtain an optimal policy. To achieve a low-complexity sub-optimal policy, we propose another iterative algorithm using the successive convex approximation. Based on the off-line policies, we develop on-line policies in which only statistical CSI and EAI are available. The complexity of the proposed policies is derived. Finally, simulation results evaluate the performance of the proposed approaches and show that the proposed polices outperform the benchmark.

Optimal flow and capacity allocation in multiple joint quickest paths of directed networks

Shared vertices or edges in joint paths bring difficulties to flow routing and scheduling with delay requirements in networks with consideration of both edge lengths and capacities since flow along the different paths will encounter each other in capacitated edges with time dislocation. For an amount of flow, the quickest path problem (QPP) presents a good link for path lengths and capacities with the transmission time of the flow. Extended from the QPP, we propose an edge-path form traffic model for an amount of flow through multiple joint paths with different lengths in one-source one-sink directed capacitated networks. Then, an optimization model for minimum transmission time within feasible traffic constrained by edge capacities is constructed. We then derived the vertex–edge form of the optimization model from the edge-path form. The proposed optimization models in both forms are proved to be linear fractional programming problems, which can be solved in polynomial time. A routing algorithm based on the solution of the vertex–edge form optimization is developed combined with the DFS-based route-searching method. The proposed model and algorithm could be applicable in the real-time operation and management of practical network systems.

Modify Symmetric Fuzzy Approach to Solve the Multi-Objective Linear Fractional Programming Problem

The property of fuzzy sets is approached as an instrument for the construction and finding of the value of the multi-objective linear fractional programming problem (MOLFPP), which is one of the systems of decision problems that are covered by fuzzy dealings. The paper introduces an approach to convert and solve such a problem by modifying the symmetric fuzzy approach, suggesting an algorithm, and demonstrating how the fuzzy linear fractional programming problem (FLFPP) can be answered without raising the arithmetic potency. Also, it introduces a technique that uses an optimal mean to convert MOLFPP to a single LFPP by modifying the symmetric fuzzy approach. A numeric sample is provided to clarify the qualification of the suggested approach and compare the results with other techniques, which are solved by using a computer application to test the algorithm of the above method, indicating that the results obtained by the fuzzy environment are promising.

Efficient Ranking Function Methods for Fully Fuzzy Linear Fractional Programming problems via Life Problems

In this paper, we propose two new ranking function algorithms to solve fully fuzzy linear fractional programming (FFLFP) problems, where the coefficients of the objective function and constraints are considered to be triangular fuzzy numbers (TrFN) s. The notion of a ranking function is an efficient approach when you want to work on TrFNs. The fuzzy values are converted to crisp values by using the suggested ranking function procedure. Charnes and Cooper’s method transforms linear fractional programming (LFP) problems into linear programming (LP) problems. The suggested ranking functions methods' applicability to actual problems of daily life, which take data from a company as an example, is shown, and it presents decision-making and exact error with the main value problem. The study aims to find an efficient solution to the FFLFP problem, and the simplex method is employed to determine the efficient optimal solution to the original FFLFP problem.

Penerapan algoritma Dinkelbach dan transformasi Charnes Cooper pada pemrograman fraksional linear di UD Bintang Furniture

Linear fractional programming is a special case of non-linear programming with an objective function consisting of the ratio of two linear functions. The problem can be solved using the Dinkelbach algorithm and the Charnes Cooper transformation. The essence of these two methods is to convert the problem of linear fractional programming into a linear programming problem which then provides an optimal value of each variable in its objective function. In this study, we will solve the problem of linear fractional programming at UD Bintang Furniture, that is to determine the optimal value of the comparison between profits and production costs of the company. The results show that the Dinkelbach algorithm method requires more iterations than Charnes Cooper's transformation. Despite this, both methods produce the same optimal value. 
 Keywords: Linear fractional programming, Dinkelbach algorithm, Charnes Cooper transformation. MSC2020: 90C32

Solving min–max linear fractional programs based on image space branch-and-bound scheme

Based on the image space branch-and-bound scheme, this paper presents a novel algorithm for globally solving the min–max linear fractional programs (MMLFP), which has many applications in management optimization, engineering optimization, economic investment and so on. For finding the global optimal solution of the MMLFP, by leading into new parameter, we firstly transform the MMLFP into an equivalent fractional problem (EFP). Next, by using convex hull and concave hull approximation of bilinear function, we construct the linear relaxation problem (LRP) for computing the lower bound of the global minimum value of the EFP in the image space branch-and-bound algorithm. By subsequently solving a series of linear relaxation problems and refining the initial image space rectangle, the proposed algorithm is globally convergent to the optimum solution of the EFP. In addition, we give the computational complexity analysis of the algorithm based on the exhaustive branching rule. Finally, computational comparisons show better computational performance of the algorithm.

An exact iterative algorithm to solve a linear fractional programming problem

An exact iterative algorithm to solve a linear fractional programming problem

An Application of Fully Intuitionistic Fuzzy Multi-objective Linear Fractional Programming Problem in E-education System

This article addresses a special class of non-linear programming problems, viz., linear fractional programming problems having multiple objectives. In solving the real-life linear fractional optimization problems, the ambiguity and hesitation in the decision are inherent and ever-present, therefore, it is perfectly viable to formulate and solve these optimization models using the intuitionistic fuzzy environment. The purpose of this study is to propose a simple and computationally efficient approach to obtain the solution of multiple objective linear fractional programming problems having all the decision variables and parameters expressed in terms of triangular intuitionistic fuzzy numbers. The proposed solution algorithm is primarily based on the goal programming approach, fuzzy-based linearization technique, and a membership function strategy. The original linear fractional programming problem is first converted to its equivalent deterministic/crisp multi-objective linear fractional optimization problem using the weighted goal programming methodology along with the linear membership technique to resolve the intuitionistic fuzzy constraints into the crisp one. Finally, the variable transformation technique for the under- and over-deviational variables of the goal programming model is employed to linearize all the fractions involved in the problem so as to convert the original problem to an equivalent linear optimization problem. Further, this linear programming problem can be solved using any available commercial packages. Moreover, a numerical illustration is provided to demonstrate the steps of the proposed technique followed by the analysis and solution of an E-education set-up problem. The discussion and comparisons of the practical case establish the relevancy and usefulness of the proposed model.

Managing irrigation water resources with economic benefit and energy consumption: an interval linear multi-objective fractional optimization model under multiple uncertainties

At present, water shortage has increasingly become one of the key factors restricting the sustainable development of agriculture. And during the process of agricultural production, the energy consumption has also been paid more and more attention. Therefore, in order to deal with the above problems, an improved interval linear multi-objective fractional programming model (ILMOFPM) was developed, which has the characteristic is that it not only focuses on the optimal allocation of the amount of irrigation water resources, but also on improving irrigation water resources use efficiency. To demonstrate its effectiveness and applicability, ILMOFPM was applied to a case study in Jinghuiqu irrigation district, Shaanxi Province, China. Maximum economic benefit per unit of water resource and minimum energy consumption per unit of water resource were selected as the planning objectives. A series of irrigation water resources optimal allocation schemes were obtained, which could give the decision makers the desired optimal scheme according to the actual situation. Based on the results of the established model, economic benefit and energy consumption generally decreased as the representative hydrological years changed from wet to dry, and the total crop planting area also showed a decreased trend. Moreover, a lot of water resources was saved by introducing crop evapotranspiration and effective precipitation in the developed model, 57.45%, 44.17%, 39.33% of total water resources consumption were saved under different representative hydrological years, respectively.

Approximating Min-Mean-Cycle for Low-Diameter Graphs in Near-Optimal Time and Memory

We revisit Min-Mean-Cycle, the classical problem of finding a cycle in a weighted directed graph with minimum mean weight. Despite an extensive algorithmic literature, previous work failed to achieve a near-linear runtime in the number of edges $m$. We propose an algorithm with near-linear runtime $\tilde{O}(m (W_{\max}/\epsilon)^2)$ for computing an $\epsilon$ additive approximation on graphs with polylogarithmic diameter and weights of magnitude at most $W_{\max}$. In particular, this is the first algorithm whose runtime scales in the number of vertices $n$ as $\tilde{O}(n^2)$ for the complete graph. Moreover---unconditionally on the diameter---the algorithm uses only $O(n)$ memory beyond reading the input, making it “memory-optimal". Our approach is based on solving a linear programming (LP) relaxation using entropic regularization, which reduces the LP to a Matrix Balancing problem---à la the popular reduction of Optimal Transport to Matrix Scaling. We then round the fractional LP solution using a variant of the classical Cycle-Canceling algorithm that is sped up to near-linear runtime at the expense of being approximate, and implemented in a memory-optimal manner. The algorithm is simple to implement and is competitive with the state-of-the-art methods in practice.

Optimal solution of neutrosophic linear fractional programming problems with mixed constraints

In this paper, linear fractional programming problems have been extended to neutrosophic sets, and the operations and functionality of these laws are studied. Moreover, the new algorithm is based on aggregation ranking function and arithmetic operations of triangular neutrosophic sets. Furthermore, in this paper, we take up a problem where the constraints are both equality and inequality neutrosophic triangular fuzzy number. Lead from genuine issue, a few numerical models are considered to survey the legitimacy, profitability and materialness of our technique. At last, some numerical trials alongside one contextual analysis are given to show the novel techniques are better than the current strategies.

Optimisation frameworks for integrated planning with allocation of transportation resources for industrial gas supply chains

This work addresses the integrated optimisation of production-distribution planning and allocation of transportation resources for industrial gas supply chains. The production-distribution planning decisions include the production plan, purchasing plan for both a liquefied product and raw material from external suppliers, distribution plan by railcars and trucks, and demand allocation. In contrast, the allocating decisions of transportation resources involve the number of trucks and railcars at each plant, depot, and third-party supplier. First, we propose a mixed-integer nonlinear programming (MINLP) model, and then the MINLP model is reformulated as a mixed-integer linear fractional programming (MILFP) model. Furthermore, we present a multi-objective optimisation (MOO) model as an alternative approach. As solution strategies, we adopt Dinkelbachs algorithm and the reformulation-linearisation method for the MILFP model, whereas the ε-constraint method is used for the MOO model. Finally, industry-relevant case studies illustrate the applicability and performance of the proposed models and solution methods.

Agricultural Water Optimal Allocation Using Minimum Cross-Entropy and Entropy-Weight-Based TOPSIS Method in Hetao Irrigation District, Northwest China

Affected by the temporal and spatial changes of natural resources, human activities, and social economic system policies, there are many uncertainties in the development, utilization, and management process of irrigation district agricultural water resources, which will increase the complexity of the use of irrigation district agricultural water resources. Decision makers find it challenging to cope with the complexity of fluctuating water supplies and demands that are critical for water resources’ allocation. In response to these issues, this paper presents an optimization modeling approach for agricultural water allocation at an irrigation district scale, considering the uncertainties of water supply and demand. The minimum cross-entropy method was used to estimate the parameters of hydrologic frequency distribution functions of water supply and demand, which are the basis for agricultural water resources’ optimal allocation and the evaluation of water resources’ carrying capacity in the Hetao Irrigation District. Interval Linear Fractional Programming was used to find water availability, shortage, and use efficiency in different irrigation areas of the Hetao Irrigation District (HID) under different scenarios. The denominator of fractional planning is the environmental goal, and the numerator is the economic goal; so, the objective function of fractional programming is the utility rate required in the post-optimization analysis. Future water availability and shortage scenarios are adopted consistent with the Representative Concentration Pathways’ (RCPs’) framework, and future water use scenarios are developed using the Shared Socioeconomic Pathways’ (SSPs’) framework. Results revealed that under SSP1, the annual water consumption increased from 30 billion m3 to 60 billion m3, almost doubling in Urad. The annual water consumption under SSP2 and SSP3 increased slightly, from 30 billion m3 to about 50 billion m3. The amount of water available for well irrigation in Urad decreased from 300 to 250 billion m3, while the amount of water available for canal irrigation in Urad remained at 270 billion m3 from 2010 s to 2030 s, only dropping to 240 billion m3 in 2040 s. The entropy-weight-based Technique for Order Preference by Similarity to an Ideal Solution (TOPSIS) method was applied to evaluate agricultural water resources’ allocation schemes because it can avoid the subjectivity of weight determination and can reflect the dynamic changing trend of irrigation district agricultural water resources’ carrying capacity. The approach is applicable to most regions, such as the Hetao Irrigation District in the Upper Yellow River Basi with limited precipitation, to determine water strategies under the changing environment.

Mathematical programming approaches for modeling a sustainable cropping pattern under uncertainty: a case study in Southern Iran

In recent years, the excessive and unreasonable use of chemicals, the occasional use of water, and the use of improper irrigation methods have created a worrying and unstable situation in developing countries’ agricultural activities. In the present study, the robust multi-objective fractional linear programming model (RMOLFP) was introduced to determine the sustainable optimal cropping pattern. This model was presented in the Gotvand irrigation and drainage network located in Khuzestan province, southern Iran, under two scenarios with and without considering the uncertainty to evaluate the ability of the model. The results showed that in the first scenario, the consumption of critical disruptive inputs of sustainable agriculture such as fertilizers and chemical pesticides decreased by 5.9% and 8.19%, respectively. On the other hand, the model's uncertainty condition was applied in the second scenario in which the increase in gross margin was reduced. There is a trade-off between protecting the optimization model against system uncertainty and gross margin. Finally, the ability of the proposed model to apply uncertainty conditions was verified by the Monte Carlo simulation method. The results of this simulation confirmed the use of the RMOLFP method in determining the sustainable optimal cropping pattern for the study area.

Mehar approach to solve fuzzy linear fractional minimal cost flow problems

To the best of author’s knowledge, only one approach is proposed in the literature to solve fuzzy linear fractional minimal cost flow problems (minimal cost flow problems in which each known arc cost is represented either by a triangular fuzzy number or a trapezoidal fuzzy number). In this paper, the mathematical incorrect assumptions, considered in the existing approach to solve fuzzy linear fractional minimal cost flow problems, are pointed out. Also, by generalizing an existing approach for solving fuzzy linear fractional programming problems, an approach (named as Mehar approach) is proposed to solve fuzzy linear fractional minimal cost flow problems. Furthermore, two numerical examples are solved to illustrate the proposed Mehar approach.

Tackling the fuzzy multi-objective linear fractional problem using a parametric approach

Finding efficient solutions for the multi-objective linear fractional programming problem (MOLFPP) is a challenging issue in optimization because more than one target has to be taken into account. For the problem, we face the concept of efficient solutions which is an infinite set especially when the objectives are in conflict. Since a classical method generally comes out with only one efficient solution, thus introducing new efficient approaches is helpful and beneficial for the decision makers to make their decisions according to more possibilities. In this paper, we aim to consider the MOLFPP with fuzzy coefficients (FMOLFPP) where the concept of α - cuts is utilized so as to transform the fuzzy numbers into closed intervals and rank the fuzzy numbers as well. Consequently, the fuzzy problem is changed into an interval valued multi-objective linear fractional programming problem (IV-MOLFPP). Subsequently, the IV-MOLFPP is further changed into linear programming problems (LPPs) using a parametric approach, weighted sum and max-min methods. It is demonstrated that the solution obtained is at least a weakly ɛ - efficient solution, where the value of ɛ helps a decision maker (DM) to make his decision appropriately i.e. DMs chose more likely the solutions with the lowest value of ɛ. Numerical examples are solved to illustrate the method and comparison are made to show the accuracy, and the reliability of the proposed solutions.

Applying Transformation Characteristics to Solve the Multi Objective Linear Fractional Programming Problems

Applying Transformation Characteristics to Solve the Multi Objective Linear Fractional Programming Problems

Towards Revenue-Driven Multi-User Online Task Offloading in Edge Computing

Mobile Edge Computing (MEC) has become an attractive solution to enhance the computing and storage capacity of mobile devices by leveraging available resources on edge nodes. In MEC, the arrivals of tasks are highly dynamic and are hard to predict precisely. It is of great importance yet very challenging to assign the tasks to edge nodes with guaranteed system performance. In this article, we aim to optimize the revenue earned by each edge node by optimally offloading tasks to the edge nodes. We formulate the revenue-driven online task offloading (ROTO) problem, which is proved to be NP-hard. We first relax ROTO to a linear fractional programming problem, for which we propose the Level Balanced Allocation (LBA) algorithm. We then show the performance guarantee of LBA through rigorous theoretical analysis, and present the LB-Rounding algorithm for ROTO using the primal-dual technique. The algorithm achieves an approximation ratio of <inline-formula><tex-math notation="LaTeX">$2(1+\xi)\ln (d+1)$</tex-math></inline-formula> with a considerable probability, where <inline-formula><tex-math notation="LaTeX">$d$</tex-math></inline-formula> is the maximum number of process slots of an edge node and <inline-formula><tex-math notation="LaTeX">$\xi$</tex-math></inline-formula> is a small constant. The performance of the proposed algorithm is validated through both trace-driven simulations and testbed experiments. Results show that our proposed scheme is more efficient compared to baseline algorithms.