In this study, a generalized fuzzy linear programming (GFLP) method is developed for dealing with uncertainties expressed as fuzzy sets. The feasibility of fuzzy solutions of the GFLP problem is investigated. A stepwise interactive algorithm (SIA) based on the idea of design of experiment is then advanced to solve the GFLP problem. This SIA method was implemented through (i) discretizing membership grade of fuzzy parameters into a finite number of α-cut levels, (ii) converting the GFLP model into an interval linear programming (ILP) submodel under every α-cut level, (iii) solving the ILP submodels through an interactive algorithm and obtaining the associated interval solutions, (iv) acquiring the membership functions of fuzzy solutions through statistical regression methods. A simple numerical example is then proposed to illustrate the solution process of the GFLP model through SIA. A comparison between the solutions obtained though SIA and Monte Carlo method is finally conducted to demonstrate the robustness of the SIA method. The results indicate that the membership functions for decision variables and objective function are reasonable and robust.
Management of environmental problems is a priority for socio-economic sustainable development throughout the world. Amounts of factors should be considered in environmental management, leading to great complexity in actual decision making. Systems analysis techniques have been widely applied to handle above environmental management issues. The systems analysis techniques can be classified into: systems engineering models and systems assessment tools; moreover, the systems engineering models, which involve cost-benefit analysis (CBA), forecasting models (FMs), simulation models (SMs), optimization models (OMs), and integrated modeling system (IMS) can be seen as the core technologies [41]. Among the five models above, OM is a fairly useful tool for supporting effective environmental management [2] and [49]. Since 1970s, various optimization methods have been applied to environmental management issues, such as air quality management, solid waste management, and water resources management [12], [13], [14], [19], [20], [21], [22], [23] and [50]. As early as in 1992, Van Beek et al. [49] demonstrated how operational research could play an important role in solving environmental problems.
However, in many real-world environmental management problems, uncertainties exist in various system components and their interrelationships. For example, waste generation rate within a city is related to many socio-economic and environmental factors, and exhibits uncertain and dynamic features; the efficiency of a municipal wastewater treatment plant is affected by wastewater flow rate, and is uncertain in nature; regional air quality is mainly influenced by air pollutant emissions within this area, which also present uncertain characteristics [11]. Such uncertainties can lead to increased complexities in the related optimization efforts. Simply ignoring these uncertainties is considered undesired as it may result in inferior or wrong decisions [40], [44] and [51]. Therefore, inexact optimization methods are desired for supporting environmental management under uncertainty.
The paper is organized as follows: in the next section, previous work on inexact optimization methods will be reviewed. In Section 3, a generalized fuzzy linear programming method and the related computational procedures will be introduced and investigated. An illustrative example is given in Section 4. Short conclusions are made in Section 5.
In this study, a generalized fuzzy linear programming (GFLP) method was developed to deal with uncertainties expressed as fuzzy sets. As an extension of conditional fuzzy linear programming (FLP) method, the developed GFLP method allows all parameters to be expressed as fuzzy sets, and generate fuzzy solutions. A stepwise interactive algorithm (SIA) has been advanced to solve the GFLP problem. This algorithm is proposed based on the idea of design of experiment and the interactive algorithm proposed by Huang et al. [19], [20], [21], [22] and [23] for interval-parameter linear programming problems. This SIA method is implemented through (i) discretizing membership grade of fuzzy parameters in GFLP into a finite number of α-cut levels, (ii) converting the GFLP model into an interval linear programming (ILP) submodel under every α-cut level; (iii) solving the ILP submodels through an interactive algorithm and obtaining the associated interval solutions, (iv) acquiring the membership functions of fuzzy solutions through statistical regression methods.
(2)
An illustrative case of the GFLP problem was proposed to present the solution process of SIA. Six α-cut levels (i.e. 1, 0.9, 0.7, 0.5, 0.3, 0) were selected to cut the fuzzy parameters in the GFLP model, leading to six ILP submodels and six groups of fuzzy interval solutions. The membership functions of decision variables and the objective function were then approximated through polynomial regression methods. To demonstrate the robustness of the SIA method, a comparison between the solutions obtained though SIA and Monte Carlo (MC) method was conducted. The MC method was applied to generate random α-cut levels from the unit interval [0, 1]. Then the related ILP submodels were formulated and the associated interval solutions were obtained. Afterwards, another group of solutions under these α-cut levels were obtained through the membership functions generated by SIA. The comparison indicated that the membership functions generated from SIA are reasonable and robust.
(3)
Although the developed GFLP method and its solving algorithm (i.e. SIA) were illustrated through a simple numerical example, the results indicate that this method maintains several advantages in dealing with uncertainties expressed as fuzzy sets. The developed GFLP method can deal with all kinds of fuzzy numbers with known membership functions, regardless whether or not these membership functions are linear. It allows uncertainties to be directly communicated into the optimization process, and generates the fuzzy number solutions. The developed SIA method would not lead to large computational requirement which is much meaningful for the GFLP method to be applied to practical management problems.
(4)
The developed GFLP in this study mainly focuses on linear constraints and objective function. Consequently, further study on nonlinear optimization problems will be required. Besides, GFLP will also confront difficulties in dealing with other kinds of uncertainties such as interval and random parameters. Therefore, further research is needed to introduce other uncertainty quantification methods into the GFLP framework to improve its applicability to deal with multiple uncertainties extensively involved in many practical problems.
