Application of fuzzy multi-objective linear programming to aggregate production planning

This study develops a fuzzy multi-objective linear programming (FMOLP) model for solving the multi-product aggregate production planning (APP) decision problem in a fuzzy environment. The proposed model attempts to minimize total production costs, carrying and backordering costs and rates of changes in labor levels considering inventory level, labor levels, capacity, warehouse space and the time value of money. A numerical example demonstrates the feasibility of applying the proposed model to APP problem. Its advantages are also discussed. The proposed model yields a compromise solution and the decision maker's overall levels of satisfaction. In particular, in contrast to other APP models, several significant characteristics of the proposed model are presented.     

A two-phase possibilistic linear programming methodology for multi-objective supplier evaluation and order allocation problems

In this study, an integration of the analytic hierarchy process (AHP) and a multi-objective possibilistic linear programming (MOPLP) technique is developed to account for all tangible, intangible, quantitative, and qualitative factors which are used to evaluate and select suppliers and to define the optimum order quantities assigned to each. A multi-objective linear programming technique is first employed to solve the problem. To model the uncertainties encountered in the integrated supplier evaluation and order allocation methodology, fuzzy theory is adopted. Hence, possibilistic linear programming (PLP) is proposed for solving the problem, as it is believed to be the best approach for absorbing the imprecise nature of the real world. In the supplier evaluation phase, environmental criteria are also considered.     

The use of a fuzzy multi-objective linear programming for solving a multi-objective single-machine scheduling problem

This paper develops a fuzzy multi-objective linear programming (FMOLP) model for solving a multi-objective single-machine scheduling problem. The proposed model attempts to minimize the total weighted tardiness and makespan simultaneously. In this problem, a proposed FMOLP method is applied with respect to the overall acceptable degree of the decision maker (DM) satisfaction. A number of numerical examples are solved to show the effectiveness of the proposed approach. The related results are compared with the Wang and Liang's approach. These computational results show that the proposed FMOLP model achieves lower objective functions and higher satisfaction degrees.     

Application of fuzzy goal programming approach to multi-objective linear fractional inventory model

In this paper, we propose a model and solution approach for a multi-item inventory problem without shortages. The proposed model is formulated as a fractional multi-objective optimisation problem along with three constraints: budget constraint, space constraint and budgetary constraint on ordering cost of each item. The proposed inventory model becomes a multiple criteria decision-making (MCDM) problem in fuzzy environment. This model is solved by multi-objective fuzzy goal programming (MOFGP) approach. A numerical example is given to illustrate the proposed model.     

Supplier selection using fuzzy AHP and fuzzy multi-objective linear programming for developing low carbon supply chain

Environmental sustainability of a supply chain depends on the purchasing strategy of the supply chain members. Most of the earlier models have focused on cost, quality, lead time, etc. issues but not given enough importance to carbon emission for supplier evaluation. Recently, there is a growing pressure on supply chain members for reducing the carbon emission of their supply chain. This study presents an integrated approach for selecting the appropriate supplier in the supply chain, addressing the carbon emission issue, using fuzzy-AHP and fuzzy multi-objective linear programming. Fuzzy AHP (FAHP) is applied first for analyzing the weights of the multiple factors. The considered factors are cost, quality rejection percentage, late delivery percentage, green house gas emission and demand. These weights of the multiple factors are used in fuzzy multi-objective linear programming for supplier selection and quota allocation. An illustration with a data set from a realistic situation is presented to demonstrate the effectiveness of the proposed model. The proposed approach can handle realistic situation when there is information vagueness related to inputs.     

Applying fuzzy multi-objective linear programming to project management decisions with the interactive two-phase method

The aim of this paper is to develop an interactive two-phase method that can help the Project Manager (PM) with solving the fuzzy multi-objective decision problems. Therefore, in this paper, we first revisit the related papers and focus on how to develop an interactive two-phase method. Next, we establish to consider the imprecise nature of the data by fulfilling the possibilistic programming model, and we also assume that each objective work has a fuzzy goal. Finally, for reaching our objective, the detailed numerical example is presented to illustrate the feasibility of applying the proposed approach to PM decision problems at the end of this paper. Results show that our model can be applied as an effective tool. Furthermore, we believe that this approach can be applied to solve other multi-objective decision making problems.     

An interactive possibilistic programming approach for a multi-objective hub location problem: Economic and environmental design

This paper presents a new multi-objective mathematical model for a multi-modal hub location problem under a possibilistic-stochastic uncertainty. The presented model aims to minimize the total transportation and traffic noise pollution costs. Furthermore, it aims to minimize the maximum transportation time between origin-destination nodes to ensure a high probability of meeting the service guarantee. In order to cope with the uncertainties and the multi-objective model, we propose a two-phase approach, including fuzzy interactive multi-objective programming approach and an efficient method based on the Me measure. Due to the NP-hardness of the presented model, two meta-heuristic algorithms, namely hybrid differential evolution and hybrid imperialist competitive algorithm, are developed. Furthermore, a number of sensitivity analyses are provided to demonstrate the effectiveness of the presented model. Finally, the foregoing meta-heuristics are compared together through different comparison metrics.     

Comparative study of the applicability of fuzzy multi-objective linear programming models through cost-effective analysis for mold manufacturing

The mold-manufacturing process consists of prototype design, production, assembly, and testing. As products tend to vary, have short due dates, and life cycles, are highly precise and must be responsiveness to customers, production system planning is complex and the relationship between outsourcing capability and in-house capacity is crucial to mold-manufacturing. Differentiation of core operations vs. non-core operations in internal vs. external environments and time control are essential for mold manufacturing when planning production systems. To analyze the cost-effectiveness of capacity planning and its relationship to suppliers, this work applies a novel fuzzy multi-objective linear programming model. Considered factors are order quantity allocation, due dates, manufacturing quantity, capacity, defect rates, back-log, and the purchasing discount. The applicability of three fuzzy theories is assessed using total costs, punishment costs, and crashing costs. Implementation results demonstrate the potentials for cost-effective capacity planning and outsourcing, and identify the applicability of these fuzzy theories to a specific mold-manufacturing case.     

Solving fuzzy (stochastic) linear programming problems using superiority and inferiority measures

In this paper, the author presents a model to measure the superiority and inferiority of fuzzy numbers/fuzzy stochastic variables. Then, the new measures are used to convert the fuzzy (stochastic) linear program into the corresponding deterministic linear program. Numerical examples are provided to illustrate the effectiveness of the proposed method.     

灰色二层多目标线性规划问题及其解法

针对二层多目标线性规划问题,结合灰色系统的特性,提出了一般灰色二层多目标线性规划问题,并给出了模型的相关定义和定理.针对漂移型灰色二层多目标线性规划问题,提出一种具有全局收敛性质的求解算法.首先通过线性加权模理想点法把多目标转化为单目标;然后当可行域为非空紧集时,利用库恩塔克条件把双层转化为单层,再利用粒子群算法搜索单目标单层线性规划即可得到原问题的解;最后通过算例表明了该算法的有效性.     

A fuzzy linear programming model for risk evaluation in failure mode and effects analysis

In traditional approach, failure mode and effects analysis determines the risk priories of failure modes through the risk priority number which is determined by multiplication of three risk factors namely, failure occurrence (O), failure severity (S) and failure detection ability (D). In this approach, different weights of risk factors were not taken into consideration so that the three risk factors were assumed to have the same weight. This may not be realistic in real applications. In this paper we treat the risk factors as fuzzy variables and evaluate them using fuzzy linguistic terms and fuzzy ratings. As a result, fuzzy risk priority numbers (FRPNs) are proposed for prioritization of failure modes. The FRPNs are defined as fuzzy geometric means of the fuzzy ratings for O, S and D and can be computed using alpha-level sets and linear programming models. A numerical example is provided to examine the results of this model.     

Exact fuzzy optimal solution of fully fuzzy linear programming problems with unrestricted fuzzy variables

Kumar et al. (Appl. Math. Model. 35:817?C823, 2011) pointed out that there is no method in literature to find the exact fuzzy optimal solution of fully fuzzy linear programming (FFLP) problems and proposed a new method to find the fuzzy optimal solution of FFLP problems with equality constraints having non-negative fuzzy variables and unrestricted fuzzy coefficients. There may exist several FFLP problems with equality constraints in which no restriction can be applied on all or some of the fuzzy variables but due to the limitation of the existing method these types of problems can not be solved by using the existing method. In this paper a new method is proposed to find the exact fuzzy optimal solution of FFLP problems with equality constraints having non-negative fuzzy coefficients and unrestricted fuzzy variables. The proposed method can also be used to solve the FFLP problems with equality constraints having non-negative fuzzy variables and unrestricted fuzzy coefficients. To show the advantage of the proposed method over existing method the results of some FFLP problems with equality constraints, obtained by using the existing and proposed method, are compared. Also, to show the application of proposed method a real life problem is solved by using the proposed method.     

矿山产能分配的变量模糊线性规划模型

针对矿山资源开采过程中产能不确定的分配问题,引入了模糊结构元素理论。将产能用结构元表示,并利用结构元加权序将模糊数比较转化为单调函数比较,将含有模糊变量的线性规划问题等价转化为经典线性规划问题。以某矿山为例,建立矿山产能分配的变量模糊线性规划模型,并进行求解。结果表明：实现了将实际问题中的模糊事件进行精确表达,原问题的求解更简便。得到矿山产能取得最大可能利润时的可能分配。应用结构元加权序求解的线性规划模型优于结构元元序的。     

Oil-importing optimal decision considering country risk with extreme events: A multi-objective programming approach

From perspective of energy security, this study focuses on oil-importing optimal decision based on multi-objective programming approach. Different from other models, country risk is considered as the main objective to minimize risk exposure of importing disruption. What is more, this model connects emergency management with programming, and optimal decisions are solved under different scenarios of emergency, where one given kind of extreme events break out and impact exporting regions to different degrees. Specifically, two main steps are involved in the proposed methodology, including impact analysis of the extreme events and optimization programming under scenarios of emergency. The first step is to statistically analyze whether and to what extent the given extreme events impact country risk of oil-exporting sources. Secondly, a multi-objective programming model is formulated, and optimal decision is simulated under different scenarios with extreme events. For illustration, China's oil-importing optimization is performed to verify the practicability of the novel methodology. The experimental results suggest that wars in Middle East may significantly enhance country risk of Middle East; and China's oil-importing optimal plan should be changed correspondingly. This further indicates that the proposed methodology can be utilized as an effective tool to adjust oil-importing plan according to certain extreme events.     

Multiobjective fuzzy linear programming for sustainable irrigation planning: an Indian case study

 Multiobjective fuzzy linear programming (MOFLP) irrigation planning model is formulated for the evaluation of management strategy for the case study of Jayakwadi irrigation project, Maharashtra, India. Three conflicting objectives net benefits, agricultural production and labour employment are considered in the irrigation planning scenario. All three criteria are to be maximised and the last two are sustainability related. All three objective functions are quantified by linear membership functions in a fuzzy multi objective framework. It is observed from MOFLP solution that net benefits, agricultural production and labour employment are 2.031×10⁹ Rupees, 2.1186×10⁶ tons, 3.5858×10⁷ man-days respectively with degree of truth (λ) 0.5715. Analysis of results indicated that net benefits, agricultural production, labour employment have decreased by 4.13, 5.39 and 3.4% as compared to ideal values in the crisp linear programming (LP) model. Present address: Department of Civil Engineering, Birla Institute of Technology and Science, Pilani, Rajasthan, India E-mail: ksraju@bits-pilani.ac.in     

基于结构元矿山产能分配模糊线性规划模型

在矿山产能分配中,矿产品的大约利润、生产所需大约资源及提供生产的大约资源均为模糊数,综合模糊数学与模糊线性规划特点,建立基于结构元素的矿山产能分配的模糊线性规划预测模型。运用模糊结构元理论,将模型中用三角模糊数表达的模糊数用结构元表示,使模型中的参数的隶属函数得到解析表达,进而利用结构元加权序理论,将模型转为经典线性规划模型,实现对实际问题中的模糊事件的精确表达;且约束条件数目相对较少,简化了原模型求解,进而得到某矿山产能取得最大可能利润时的分配,可为决策者提供更精确信息。     

Extremist vs. centrist decision behavior: quasi-convex utility functions for interactive multi-objective linear programming problems

This paper presents the fundamental theory and algorithms for identifying the most preferred alternative for a decision maker (DM) having a non-centrist (or extremist) preferential behavior. The DM is requested to respond to a set of questions in the form of paired comparison of alternatives. The approach is different than other methods that consider the centrist preferential behavior.In this paper, an interactive approach is presented to solve the multiple objective linear programming (MOLP) problem. The DM's underlying preferential function is represented by a quasi-convex value (utility) function, which is to be maximized. The method presented in this paper solves MOLP problems with quasi-convex value (utility) functions by using paired comparison of alternatives in the objective space. From the mathematical point of view, maximizing a quasi-convex (or a convex) function over a convex set is considered a difficult problem to solve, while solutions for quasi-concave (or concave) functions are currently available. We prove that our proposed approach converges to the most preferred alternative.We demonstrate that the most preferred alternative is an extreme point of the MOLP problem, and we develop an interactive method that guarantees obtaining the global most preferred alternative for the MOLP problem. This method requires only a finite number of pivoting operations using a simplex-based method, and it asks only a limited number of paired comparison questions of alternatives in the objective space. We develop a branch and bound algorithm that extends a tree of solutions at each iteration until the MOLP problem is solved. At each iteration, the decision maker has to identify the most preferred alternatives from a given subset of efficient alternatives that are adjacent extreme points to the current basis. Through the branch and bound algorithm, without asking many questions from the decision maker, all branches of the tree are implicitly enumerated until the most preferred alternative is obtained. An example is provided to show the details of the algorithm. Some computational experiments are also presented.Scope and purposeThis paper presents the fundamental theory, algorithm, and examples for identifying the most preferred alternative (solution) for a decision maker (DM) having a non-centrist (or extremist) preferential behavior for Multiple Objective Linear Programming (MOLP) problems. The DM is requested to respond to a set of questions in the form of paired comparison of alternatives.Although widely applied, Linear Programming is limited to a single objective function. In many real world situations, DMs are faced with multiple objective problems in that several competing and conflicting objectives have to be considered. For these problems, there exist many alternatives that are feasible and acceptable. However, the DM is interested in finding “the most preferred alternative”. In the past three decades, many methods have been developed for solving MOLP problems.One class of these methods is called “interactive”, in which the DM responds to a set of questions interactively so that his/her most preferred alternative can be obtained. In most of these methods, the value (utility) function (that presents the DM's preference) is assumed to be linear or additive, concave, pseudo-concave, or quasi-concave. However, for MOLP problems, there has not been any effort to recognize and solve the quasi-convex utility functions, which are among the most difficult class of problems to solve. The quasi-convex class of utility functions represents an extremist preferential behavior, while the other aforementioned methods (such as quasi-concave) represent a conservative behavioral preference. It is shown that the method converges to the optimal (the most preferred) alternative. The approach is computationally feasible for moderately sized problems.