Fuzzy Linear Programming Approach Research Articles

A dual-inexact fuzzy stochastic model for water resources management and non-point source pollution mitigation under multiple uncertainties

Abstract. In this research, a dual-inexact fuzzy stochastic programming (DIFSP) method was developed for supporting the planning of water and farmland use management system considering the non-point source pollution mitigation under uncertainty. The random boundary interval (RBI) was incorporated into DIFSP through integrating fuzzy linear programming (FLP) and chance-constrained programming (CCP) approaches within an interval linear programming (ILP) framework. This developed method could effectively tackle the uncertainties expressed as intervals and fuzzy sets. Moreover, the lower and upper bounds of RBI are continuous random variables, and the correlation existing between the lower and upper bounds can be tackled in RBI through the joint probability distribution function. And thus the subjectivity of decision making is greatly reduced, enhancing the stability and robustness of obtained solutions. The proposed method was then applied to solve a water and farmland use planning model (WFUPM) with non-point source pollution mitigation. The generated results could provide decision makers with detailed water supply–demand schemes involving diversified water-related activities under preferred satisfaction degrees. These useful solutions could allow more in-depth analyses of the trade-offs between humans and environment, as well as those between system optimality and reliability. In addition, comparative analyses on the solutions obtained from ICCP (Interval chance-constraints programming) and DIFSP demonstrated the higher application of this developed approach for supporting the water and farmland use system planning.

Solid waste management under uncertainty: a generalized fuzzy linear programming approach

In this study, a generalized fuzzy linear programming (GFLP) method is developed to identify optimal waste-flow-allocation schemes under uncertainty. A stepwise interactive algorithm (SIA) is advanced to solve the GFLP model and generate solutions expressed as fuzzy sets. This solution method can handle fuzzy sets with known membership functions, regardless of the shapes of these functions. Moreover, solutions expressed as fuzzy sets can also be obtained through SIA. The developed method is applied to a case study of waste allocation planning problem under uncertainty. The results indicate that reasonable solutions can be obtained for planning waste allocation practices. Compared with interval solutions derived from interval linear programming method, the fuzzy solutions obtained through GFLP can provide more information. Therefore, the decision-makers can make tradeoffs between system stability and plausibility and thus identify desired policies for solid waste planning under uncertainty.

An Interval-Parameter Fuzzy Linear Programming Approach for Accounting and Planning of Energy-Environmental Management Systems

In this paper, an energy systems planning (ESPM) model was developed for supporting the planning of an energy management system where the environmental protection was considered under uncertainty. This model was solved based on the interval-parameter fuzzy linear programming, which incorporated the fuzzy linear programming and interval-parameter linear programming into a general optimization framework. Through the conduction of this study, (a) complexities regarding energy activities and environmental conservation in energy management systems could be identified and analyzed, (b) uncertainties of parameters in energy man-agement systems could be incorporated into the planning process on the basis of interval numbers and fuzzy sets, (c) the interval form solutions could enhance the adaptation of ESPM to uncertain conditions, (d) the trade-off between energy activities and environmental protection measures as well as that between optimality and reliability of energy sys- tems couldbe reflected through the introduction of satisfaction degrees, and (e) desirable cost-minimized schemes for energy activitiessuch as energy resources allocation, technological development, facility capacity expansion and pollutant emission control could be provided for decision makers.

Development of stock portfolio optimization application program using fuzzy linear programming

Investing in the stock area is one of the popular category of investment nowadays. To get the maximum profit in stock trading, user usually creates a portfolio. In this paper the author present the Fuzzy Linear Programming approach to optimize the stock proportion in the Markowitz’s portfolio selection model. Computer based program with PHP programming language are made to facilitate and improve the accuracy in the calculation of the optimal allocation of the stock portfolio selection. The computer program are able to passively analyze 45 best stocks contained in the Indonesia Stock Exchange.

Fuzzy linear programming approach to multiattribute decision making with multiple types of attribute values and incomplete weight information

In the classical Linear Programming Technique for Multidimensional Analysis of Preference (LINMAP), the decision maker (DM) gives the pair-wise comparisons of alternatives with crisp truth degree 0 or 1. However, in the real world, DM is not sure enough in all comparisons and can express his/her opinion with some fuzzy truth degree. Thus, DM's preferences are given through pair-wise comparisons of alternatives with fuzzy truth degrees, which may be represented as trapezoidal fuzzy numbers (TrFNs). Considered such fuzzy truth degrees, the aim of this paper is to develop a new fuzzy linear programming technique for solving multiattribute decision making (MADM) problems with multiple types of attribute values and incomplete weight information. In this method, TrFNs, real numbers, and intervals are used to represent the multiple types of decision information. The fuzzy consistency and inconsistency indices are defined as TrFNs due to the alternatives’ comparisons with fuzzy truth degrees. Hereby a new fuzzy linear programming model is constructed and solved by the possibility linear programming method with TrFNs developed in this paper. The fuzzy ideal solution (IS) and the attribute weights are then obtained. The distances of alternatives from the fuzzy IS can be calculated to determine their ranking order. The implementation process of the method proposed in this paper is illustrated with a strategy partner selection example. The comparison analyzes show that the method proposed in this paper generalizes the classical LINMAP, fuzzy LINMAP and possibility LINMAP.

Development of sustainable irrigation planning with multi-objective fuzzy linear programming for Ukai–Kakrapar irrigation project, Gujarat, India

Multi-objective fuzzy linear programming (MOFLP) approach is applied with four conflicting objectives, viz maximization of net benefits, employment generation, minimization of cost of cultivation and maximization of revenue generation from municipal and industrial supplies (M and I), on a water resources project (Ukai), Gujarat, India. The results from the model are reported for the most critical year (90% exceedance probability), critical year (85% exceedance probability), normal year (75% exceedance probability), and wet year (60% exceedance probability) inflow conditions. The degree of satisfaction of the proposed MOFLP model, considering all objectives together, for wet year, normal year, critical year and most critical year are found to be 0.527, 0.515, 0.50, and 0.46 respectively; and corresponding net irrigation benefits for different inflow conditions are computed as 10 611.91 Million Rs, 10 476.67 Million Rs, 8 311.0044 Million Rs, and 6 900.051 Million Rs, respectively. The proposed MOFLP model indicated that probable inflow corresponding to 75% dependability level is marginally sufficient to meet the requirement of the study area, and water availability becomes deficit in the command area for 85% dependability inflow condition. The optimized crop areas from the model, complying with the requirement of existing flood rules, and satisfying relevant conflicting objectives would help the decision makers in sustainable management of water resources in Ukai command area.

Two-Phase Multi Objective Fuzzy Linear Programming Approach for Sustainable Irrigation Planning

The objective of the present study is to develop the irrigation planning model and to apply the same in the form of Two-Phase Multi Objective Fuzzy Linear Programming (TPMOFLP) approach for crop planning in command area of Jayakwadi Project Stage I, Maharashtra State, India. The development of TPMOFLP model is on the basis of various Linear Programming (LP) models and Multi Objective Fuzzy Linear Programming (MOFLP) models, these models have been applied for maximization of the Net Benefits (NB), Crop production (CP), Employment Generation (EG) and Manure Utilization (MU) respectively. The significant increase in the value of level of satisfaction (λ) has been found from 0.58 to 0.65 by using the TPMOFLP approach as compare to that of MOFLP model based on maxmin approach. The two-phase approach solution provides NB = 1503.56 Million Rupees, CP = 335729.30 Tons, EG = 29.74 Million Man days and MU = 160233.70 Tons respectively. The proposed model will be helpful for the Decision Maker (DM) to take a decision under conflicting situation while planning for different conflicting objectives simultaneously and has potential to find out an integrated irrigation planning with prime consideration for economic, social and environmental issue.

Evaluation of Fuzzy Linear Programming Application in Energy Models

Fuzzy Linear Programming (FLP) approach is an effective approach that has not been attended extensively by research and academic centers in energy planning field up to now. It seems that this inattention roots in some unknown and uninvestigated grounds about this approach that should be researched before any sectoral, regional, national or international applications. In this paper the authors tried to investigate different factors influence FLP modeling. Some factors challenge FLP application in energy modeling field. Other factors strengthen it as a serious competitor for other strategies under uncertainty like Stochastic Programming and Minimax Regret Strategy. Unfamiliarity of energy modelers and planners with this method, no favor of academic circles, confusion of modelers and planners stemmed from plurality of fuzzification and defuzzification methods and the lack of effective and comprehensive softwares for solving FLP problems are some of the obstacles against FLP application in energy planning field. In contrast, variant methods for different problems simplicity, flexibility, and possibility of FLP application in current supply energy models are strong points that propound FLP as an effective approach. Some remedies had been proposed to overcome these obstacles. Designing a software is a vital step. GAMS environment help us for this designation. Fuzzy GAMS is an idea to expand GAMS capabilities to fuzzy optimization models.

Global supplier selection considering sustainability and carbon footprint issue: AHP multi-objective fuzzy linear programming approach

With growing concern about environmental protection, carbon emission has become an important factor for global supplier evaluation. Most of the earlier models have focused on cost, quality, geographical location, risk, etc., issues but not given enough importance to carbon emission for global supplier evaluation. Few studies have addressed sustainability but not succeeded to quantify carbon footprint for outsourcing. This paper quantifies the outsourcing carbon footprint that is used as objective function in multi-objective fuzzy linear programming for global supplier evaluation and quota allocation. As specific contribution, different carbon management factors and outsourcing carbon cap are introduced in the model. In multi-objective formulation, total cost, quality rejection, late delivered item, carbon footprint, and total value of purchasing (TVP) are considered as five objectives that have to be optimised simultaneously. AHP is used to calculate the performance of suppliers’ on qualitative criteria. ...

An integrated supply chain model for new products with imprecise production and supply under scenario dependent fuzzy random demand

In the present day business scenario, instant changes in market demand, different source of materials and manufacturing technologies force many companies to change their supply chain planning in order to tackle the real-world uncertainty. The purpose of this paper is to develop a multi-objective two-stage stochastic programming supply chain model that incorporates imprecise production rate and supplier capacity under scenario dependent fuzzy random demand associated with new product supply chains. The objectives are to maximise the supply chain profit, achieve desired service level and minimise financial risk. The proposed model allows simultaneous determination of optimum supply chain design, procurement and production quantities across the different plants, and trade-offs between inventory and transportation modes for both inbound and outbound logistics. Analogous to chance constraints, we have used the possibility measure to quantify the demand uncertainties and the model is solved using fuzzy linear programming approach. An illustration is presented to demonstrate the effectiveness of the proposed model. Sensitivity analysis is performed for maximisation of the supply chain profit with respect to different confidence level of service, risk and possibility measure. It is found that when one considers the service level and risk as robustness measure the variability in profit reduces.

Optimizing Human Diet Problem Based on Price and Taste Using Multi-Objective Fuzzy Linear Programming Approach

Low price and good taste of foods are regarded as two major factors for optimal human nutrition. Due to price fluctuations and taste diversity, these two factors cannot be certainly and determinately evaluated. This problem must be viewed from another perspective because of the uncertainty about the amount of nutrients per unit of foods and also diversity of people’s daily needs to receive them.This paper discusses human diet problem in fuzzy environment. The approach deals with multi-objective fuzzy linear programming problem using a fuzzy programming technique for its solution. By prescribing a diet merely based on crisp data, some ofthe realities are neglected. For the same reason, we dealt with human diet problem through fuzzy approach. Results indicated uncertainty about factors of nutrition diet -including taste and price, amount of nutrients and their intake- would affect diet quality, making the proposed diet more realistic.

Fuzzy Linear Programming: A Modern Tool for Decision Making

The modern trend in industrial application problem deserves modeling of all relevant vague or fuzzy information involved in a real decision making problem. In the first part of the paper, some explanations on tri partite fuzzy linear programming approach and its importance have been given. In the second part, the usefulness of the proposed S-curve membership function is established using a real life industrial production planning of a chocolate manufacturing unit. The unit produces 8 products using 8 raw materials, mixed in various proportions by 9 different processes under 29 constraints. A solution to this problem establishes the usefulness of the suggested membership function for decision making in industrial production planning. Key words: Fuzzy linear programming, Satisfactory solution; Decision maker; Implementer; Analyst; Fuzzy constraint; Vagueness.

A generalized fuzzy linear programming approach for environmental management problem under uncertainty

In this study, a generalized fuzzy linear programming (GFLP) method was developed to deal with uncertainties expressed as fuzzy sets that exist in the constraints and objective function. A stepwise interactive algorithm (SIA) was advanced to solve GFLP model and generate solutions expressed as fuzzy sets. To demonstrate its application, the developed GFLP method was applied to a regional sulfur dioxide (SO2) control planning model to identify effective SO2 mitigation polices with a minimized system performance cost under uncertainty. The results were obtained to represent the amount of SO2 allocated to different control measures from different sources. Compared with the conventional interval-parameter linear programming (ILP) approach, the solutions obtained through GFLP were expressed as fuzzy sets, which can provide intervals for the decision variables and objective function, as well as related possibilities. Therefore, the decision makers can make a tradeoff between model stability and the plausibility based on solutions obtained through GFLP, and then identify desired policies for SO2-emission control under uncertainty. Implications: The GFLP method is effective in addressing air quality management planning associated with fuzzy parameter in the constraints and objective. The stepwise interactive algorithm proposed to solving the GFLP model would not lead to complicated intermediate submodels. Thus it would be much applicable for large-scale air quality management problems. The applications of the method can help decision makers in (i) generating allocation schemes for SO2 emissions; (ii) identifying the plausibility of each allocation alternative; and (iii) estimating the total costs of different SO2 emission control policies.

Solving a bi-objective cell formation problem with stochastic production quantities by a two-phase fuzzy linear programming approach

We propose a bi-objective cell formation problem with demand of products expressed in a number of probabilistic scenarios. To deal with the uncertain demand of products, a framework of two-stage stochastic programming model is presented. The proposed model considers minimizing the sum of the miscellaneous costs (machine constant cost, expected machine variable cost, cell fixed-charge cost, and expected intercell movement cost) and expected total cell loading variation. Because of conflicting objectives, we develop a two-phase fuzzy linear programming approach for solving bi-objective cell formation problem. To show the effectiveness of the proposed approach, numerical examples are solved and the results are compared with the two existing approaches in the literature. The computational results show that the proposed fuzzy method achieves lower objective functions as well as higher satisfaction degrees.

Optimizing Human Diet Problem with Fuzzy Price Using Fuzzy Linear Programming Approach

There are two factors that influence diet problem which are rates of diabetes and other diet-related diseases are growing and fluctuation of the food price. This paper discusses human diet problem with fuzzy price. The approach used linear programming with fuzzy objective coefficient using linear membership function. The result showed that the uncertainty of food prices slightly affected the budget of human diet.

Fuzzy Linear Programming Approach for Solving Fuzzy Transportation Problems with Transshipment

To find the fuzzy optimal solution of fuzzy transportation problems it is assumed that the direct route between a source and a destination is a minimum-cost route. However, in actual application, the minimum-cost route is not known a priori. In fact, the minimum-cost route from one source to another destination may well pass through another source first. In this paper, a new method is proposed to find the fuzzy optimal solution of fuzzy transportation problems with the following transshipment: (1) From a source to any another source, (2) from a destination to another destination, and (3) from a destination to any source. In the proposed method all the parameters are represented by trapezoidal fuzzy numbers. To illustrate the proposed method a fuzzy transportation problem with transshipment is solved. The proposed method is easy to understand and to apply for finding the fuzzy optimal solution of fuzzy transportation problems with transshipment occurring in real life situations.

A fuzzy linear programming approach to layout design of dynamic cellular manufacturing systems with route selection and cell reconfiguration

This paper presents a novel integer non-linear programming model for the layout design of dynamic cellular manufacturing systems (DCMS) under an uncertain environment. A novel aspect of this model is concurrently making the interrelated cell formation and intracell layout decisions in a dynamic and uncertain environment. Other compromising aspects are: considering single-row layout of equal area facilities to locate machines in each cell, machine relocation based on locations assigned to a machine during two successive periods, calculation of material handling cost based on the distance between the locations assigned to machines, and presenting the existence of uncertainty in the model parameters by fuzzy numbers. Such an integrated model with an extensive coverage of important manufacturing features has not been proposed before and incorporates several design features including intracell layout, operation sequence, operation time, alternative process routing, duplicate machines, machine capacity, route selection, production volume of parts and cell reconfiguration. The uncertainty stems from part demand fluctuation and machine capacity. Linearization procedures are used to transform the proposed non-linear programming model into a linearized formulation. Finally, a new fuzzy linear programming approach which can consider the fuzziness of whole parameters in a mathematical model is developed to solve the linearized model. A comprehensive example is solved by the Lingo software to verify the performance of the proposed model and developed fuzzy approach. Also, the computational results show that the proposed model to some extent overcomes common disadvantages in the existing dynamic cell formation models that have not yet considered layout problems and fuzzy issues.

A fuzzy multicriteria generation expansion planning model

In this paper, we present a fuzzy multi-objective model for power generation expansion planning problem with uncertainty in objective functions and demand balance constraints. The purpose of the fuzzy generation expansion model is to determine the optimal generation amount in existing and new generation units and the optimal transmission network, given that decision maker may tolerate some violation in constraints and objective functions. A fuzzy linear programming approach is utilised to transform the multiobjective fuzzy generation expansion model into a linear program. To illustrate the proposed model, a Mexican electric power system case is studied. A comprehensive sensitivity analysis on the impact of different objective functions, environmental impact of different fuel types and changes in demand is done to illustrate the capability of the proposed model.

Irrigation Planning Under Uncertainty—A Multi Objective Fuzzy Linear Programming Approach

The problem of irrigation planning becomes more complex by considering an uncertainty. The uncertainties can be tackled by formulating the problem of irrigation planning as Fuzzy Linear Programming (FLP). FLP models can incorporate the scenario of real world problem. In the present study, Multi Objective Fuzzy Linear Programming (MOFLP) irrigation planning model is formulated for deriving the optimal cropping pattern plan for the case study of Jayakwadi project in the Godavari river sub basin in Maharashtra State, India. Four conflicting objectives are considered such as Net Benefits (NB), Crop/Yield Production (CP), Employment Generation/Labour Requirement (EG) and Manure Utilization (MU). Four different cases are considered to incorporate the uncertainty in MOFLP model. To include the uncertainty in irrigation planning problem only objectives are taken as fuzzy and constraints are crisp in nature in Case-I. To consider the uncertainty involved in availability of resources, in Case-II the stipulations are fuzzy. The technological coefficients are fuzzy in Case-III. The Case-IV includes both technological coefficients and stipulations fuzzy. The level of satisfaction (λ) works out to be 0.58, 0.50, 0.50 and 0.28 respectively for Case-I to IV. The results obtained in Case-IV are more realistic and promising as it involves the uncertainty in technological coefficients and stipulations simultaneously.

Application of fuzzy sets to manufacturing/distribution planning decisions in supply chains

In the real-world manufacturing/distribution planning decision (MDPD) integration problems in supply chains, the environmental coefficients and parameters are normally imprecise due to incomplete and/or unavailable information. This work presents a fuzzy linear programming approach based on the possibility theory. It applies this approach to solve multi-product and multi-time period MDPD problems with imprecise goals and forecast demand by considering the time value of money of related operating cost categories. The proposed approach attempts to minimize the total manufacturing and distribution costs by considering the levels of inventory, subcontracting and backordering, the available machine capacity and labor levels at each source, forecast demand and available warehouse space at each destination. This study utilizes an industrial case study to demonstrate the feasibility of applying the proposed approach to practical MDPD problems. The primary contribution of this paper is a fuzzy mathematical programming methodology for solving the MDPD integration problems in uncertain environments.