Linear, integer separable and fuzzy programming problems: a united approach towards automatic reformulation

For mathematical programming (MP) to have greater impact as a decision tool, MP software systems must offer suitable support in terms of model communication and modelling techniques. In this paper modelling techniques that allow logical restrictions to be modelled in integer programming terms are described and their implications discussed. In addition it is demonstrated that many classes of non-linearities which are not variable separable may be after suitable algebraic manipulation put in a variable separable form. The methods of reformulating the fuzzy linear programming problem as a Max-Min problem is also introduced. It is shown that analysis of bounds plays a key role in the following four important contexts: model reduction, reformulation of logical restrictions as 0-1 mixed integer programs, reformulation of nonlinear programs as variable separable programs and reformulation of fuzzy linear programs. It is observed that as well as incorporating an interface between the modeller and the optimiser there is a need to make available to the modeller software facilities which support the model reformulation techniques described here

Developing an Overbooking Fuzzy-Based Mathematical Optimization Model for Multi-Leg Flights

Overbooking is one of the most vital revenue management practices that is used in the airline industry. Identification of an overbooking level is a challenging task due to the uncertainties associated with external factors, such as demand for tickets, and inappropriate overbooking levels which may cause revenue losses as well as loss of reputation and customer loyalty. Therefore, the aim of this paper is to propose a fuzzy linear programming model and Genetic Algorithms (GAs) to maximize the overall revenue of a large-scale multi-leg flight network by minimizing the number of empty seats and the number of denied passengers. A fuzzy logic technique is used for modeling the fuzzy demand on overbooking flight tickets and a metaheuristics-based GA technique is adopted to solve large-scale multi-leg flights problem. As part of model verification, the proposed GA is applied to solve a small multi-leg flight linear programming model with a fuzzified demand factor. In addition, experimentation with large-scale problems with different input parameters’ settings such as penalty rate, show-up rate and demand level are also conducted to understand the behavior of the developed model. The validation results show that the proposed GA produces almost identical results to those in a small-scale multi-leg flight problem. In addition, the performance of the large-scale multi-leg flight network represented by a number of KPIs including total booking, denied passengers and net-overbooking profit towards changing these input parameters will also be revealed

A STOCHASTIC SIMULATION-BASED HYBRID INTERVAL FUZZY PROGRAMMING APPROACH FOR OPTIMIZING THE TREATMENT OF RECOVERED OILY WATER

In this paper, a stochastic simulation-based hybrid interval fuzzy programming (SHIFP) approach is developed to aid the decision-making process by solving fuzzy linear optimization problems. Fuzzy set theory, probability theory, and interval analysis are integrated to take into account the effect of imprecise information, subjective judgment, and variable environmental conditions. A case study related to oily water treatment during offshore oil spill clean-up operations is conducted to demonstrate the applicability of the proposed approach. The results suggest that producing a random sequence of triangular fuzzy numbers in a given interval is equivalent to a normal distribution when using the centroid defuzzification method. It also shows that the defuzzified optimal solutions follow the normal distribution and range from 3,000-3,700 tons, given the budget constraint (CAD 110,000-150,000). The normality seems to be able to propagate throughout the optimization process, yet this interesting finding deserves more in-depth study and needs more rigorous mathematical proof to validate its applicability and feasibility. In addition, the optimal decision variables can be categorized into several groups with different probability such that decision makers can wisely allocate limited resources with higher confidence in a short period of time. This study is expected to advise the industries and authorities on how to distribute resources and maximize the treatment efficiency of oily water in a short period of time, particularly in the context of harsh environments

Applications of fuzzy theories to multi-objective system optimization

Most of the computer aided design techniques developed so far deal with the optimization of a single objective function over the feasible design space. However, there often exist several engineering design problems which require a simultaneous consideration of several objective functions. This work presents several techniques of multiobjective optimization. In addition, a new formulation, based on fuzzy theories, is also introduced for the solution of multiobjective system optimization problems. The fuzzy formulation is useful in dealing with systems which are described imprecisely using fuzzy terms such as, 'sufficiently large', 'very strong', or 'satisfactory'. The proposed theory translates the imprecise linguistic statements and multiple objectives into equivalent crisp mathematical statements using fuzzy logic. The effectiveness of all the methodologies and theories presented is illustrated by formulating and solving two different engineering design problems. The first one involves the flight trajectory optimization and the main rotor design of helicopters. The second one is concerned with the integrated kinematic-dynamic synthesis of planar mechanisms. The use and effectiveness of nonlinear membership functions in fuzzy formulation is also demonstrated. The numerical results indicate that the fuzzy formulation could yield results which are qualitatively different from those provided by the crisp formulation. It is felt that the fuzzy formulation will handle real life design problems on a more rational basis

A note on the effectiveness of some de-fuzzification measures in a fuzzy pure factors portfolio

There are several methods to convert fuzzy or stochastic LP to conventional LP models. In this simple paper we evaluate the effectiveness of three proposed methods, using a numerical example from a pure factors portfolio.: fuzzy; stochastic; linear programming; pure factors portfolio