A nonlinear interval number programming method for uncertain optimization problems

In this paper, a linear fractional programming problem with interval coefficients in both objective function and constraints is solved. The uncertain objective function is transformed into two deterministic objective functions. The two objective functions are converted into a single objective problem through the linear combination method. Through a modified possibility degree, the uncertain inequality and equality constraints are transformed into deterministic inequality constraints. The existing results are concerning the qualitative and quantitative analysis of basic notions in parametric linear fractional programming problem. These notions are the set of feasible parameters, the solvability set and the stability set of the first kind. A numerical example is investigated to demonstrate the effectiveness of the present approach.

In Chapter 8 we described a sampling based procedure for stochastic multiobjective mathematical programming problems. In this chapter we will present a procedure which also uses sampling but in a more structured approach. The method presented here is an interactive procedure based on Box’s complex search. It requires the decision maker to provide progressive articulation of preferences. This method has an advantage over the interactive methods described in Chapter 8 in that the decision maker’s underlying value function need not be explicitly specified. Also, the problem may have nonlinear objective functions and nonlinear constraints as well as objective function coefficients that are random variables.

Distribution system planning through a generalized Benders decomposition approach

Geometric programming is now a well-established branch of optimization theory which has its origin in the analysis of posynomial programs. Geometric programming transforms a mathematical program with nonlinear objective function and nonlinear inequality constraints into a dual problem with nonlinear objective function and linear constraints. Although the dual problem is potentially simpler to solve, there are certain computational difficulties to be overcome. The gradient of the dual objective function is not defined for components whose values are zero. Moreover, certain dual variables may be constrained to be zero (geometric programming degeneracy).To resolve these problems, a means to find a solution in the relative interior of a set of linear equalities and inequalities is developed. It is then applied to the analysis of dual geometric programs.

We propose a procedure to determine optimal magnet systems in the framework of the nondestructive evaluation technique Lorentz force eddy-current testing (LET). The underlying optimization problem is clearly defined considering the problem specificity of nondestructive testing scenarios. The quantities involved are classified as design variables, and system and scaling parameters to provide a high level of generality. The objective function is defined as the absolute defect response signal (ADS) of the Lorentz force resulting from an inclusion inside the object under test. Associated constraints are defined according to the applied force sensor technology. A numerical procedure based on the finite-element method is proposed to evaluate the nonlinear objective and constraint functions, and the method of sequential quadratic programming is applied to determine unconstrained and constrained optimal magnet designs. Consequently, we propose a new magnet design based on the Halbach principle in combination with high saturation magnetization iron–cobalt alloys. The proposed magnet system outperforms currently available cylindrical magnets in terms of weight and performance. The corresponding defect response signal is increased up to 180% in the case of small defects located close to the surface of the specimen. The combination of active and passive magnetic materials provides an increase of the ADS by 15% compared with the magnet designs that are built solely from permanent magnet material. The proposed procedure provides a highly adaptive optimization strategy in the framework of LET and proposes new magnet systems with inherently improved characteristics.

The novel industrial manufacturing sector inevitably faces problems of uncertainty in various aspects such as raw material availability, human resource availability, processing capability and constraints and limitations imposed by the marketing department. These problems have to be solved by a methodology which takes care of such unexpected information. As the analyst faces this man made chaotic and due to natural disaster problems, the decision maker and the implementer have to work collaboratively with the analyst for taking up a decision on an innovative strategy for implementation. Such complex problems of vagueness and uncertainty can be handled by the hybrid evolutionary intelligence algorithms. In this chapter, a new hybrid evolutionary optimization based methodology using a specific non-linear membership function, named as modified S-curve membership function, is proposed. The modified S-curve membership function is first formulated and its flexibility in taking up vagueness in parameters is established by an analytical approach. This membership function is applied for its useful performance through industrial production problems by employing hybrid evolutionary optimization algorithms. The novelty and the originality of this non-linear S-curve membership function are further established using a real life industrial production planning of an industrial manufacturing sector. The unit produces 8 products using 8 raw materials, mixed in various proportions by 9 different processes under 29 constraints. This complex problem has a cubic non-linear objective function. Comprehensive solutions to a non-linear real world objective function are achieved thus establishing the usefulness of the realistic membership function for decision making in industrial production planning.

In this paper a robust optimization problem with uncertain objective function is considered. The uncertainty is modeled by specifying a scenario set, containing a finite number of objective function coefficients, called scenarios. Additional knowledge in scenario set can be represented by using a mass function defined on the power set of scenarios. This mass function defines a belief function, which in turn induces a family of probability distributions in scenario set. One can then use a generalized Hurwicz criterion, i.e. a convex combination of the upper and lower expectations, to solve the uncertain problem. Recently, possibility theory has been applied to extend the model of uncertainty based on belief functions. Namely, belief function can be induced by a random fuzzy set. In this paper we show how this generalized model can be applied to robust optimization.

A power balancing method of distributed generation (DG) and electric vehicle charging is presented for minimizing operation costs of distribution systems with uncertainties, which includes the uncertainties in output power of DG and the randomness of charging power of electric vehicles (EV). A probability model is established for the uncertain characteristics of DG and electric vehicle charging. A multi‐state optimization coordination method for DG of renewable energy systems and electric vehicle charging in distribution systems based on quadratic rotation cone programming is presented to minimize the expected generation cost of generators in the main power grid, the expected operation cost of DG systems in distribution systems, and the expected social outage loss. An objective function maximizing the operation efficiency of distribution systems with DGs and EVs is proposed. Using quadratic rotated conic programming, the nonlinear objective function and constraint functions are transformed into a linear form. An IEEE 14‐node distribution system is used as a study example to illustrate adaptability of the proposed model and the feasibility of the proposed method. The simulation results show that the proposed method simplifies the original problem of the optimization problem and makes its solution faster, more stable, and better.

In recent years, many researchers in the field of fuzzy mathematics and operations research in the world are interested to solve the interval type-2 fuzzy linear programming (IT2FLP) problems. In IT2FLP problems all parameters, including coefficients in objective function, coefficients in the constraints and right hand sides of the constraints are interval type-2 fuzzy numbers (IT2FN). Hence, the IT2FLP model is one of the most important and the most appropriate models to interpret and determine the optimal decision. In this paper, a new ranking function method is proposed for solving these problems. Moreover, an example is used to study and describe the proposed method.

In this paper, we study general linear programs in which right hand sides are interval numbers. This model is relevant when uncertain and inaccurate factors make difficult the assignment of a single value to each right hand side. When objective function coefficients are interval numbers in a linear program, classical criteria coming from decision theory (like the worst case criterion) are usually applied to determine robust solutions. When the set of feasible solutions is uncertain, we identify a class of linear programs for which these classical approaches are no longer relevant. However, it is possible to compute the worst optimum solution. We study the complexity of this optimization problem when each right hand side is an interval number. Then, we exhibit some duality relationships between the worst optimum solution problem and the best optimum solution to the dual problem.

Linear bilevel programming with interval coefficients

A nonlinear fractional programming approach for environmental–economic power dispatch

Geometric Programming is a new technique developed to solve nonlinear engineering design problems including linear or nonlinear constraints. This paper illustrates the use of Geometric Programming in obtaining optimal design parameters for a class of welded beam structures. The procedure is illustrated through the solution of a particular welded beam design formulation. In G/P format the problem solved consists of 9 nonlinear constraints, 24 terms, 7 variables, with 16 degrees of difficulty and a nonlinear objective function. Geometric Programming is compared to several other solution techniques, and found to be very efficient. Computational experience suggests that other problems of this class may be solved with similar efficiency. The welded beam problem given is a real world design situation typical of many encountered in actual practice. The solution is given for the first time in this paper.

A global optimization method for topology optimization using a genetic algorithm is proposed in this paper. The genetic algorithm used in this paper is assisted by the Kriging surrogate model to reduce computational cost required for function evaluation. To validate the global topology optimization method in flow problems, this research works on two single-objective optimization problems, where the objective functions are to minimize pressure loss and to maximize heat transfer of flow channels, and the multi-objective optimization problem, which combines these two problems. The shape of flow channels is represented by the level set function, and the pressure loss and the temperature of the channels are evaluated by the Building-Cube Method (BCM), which is a Cartesian-mesh CFD approach. The proposed method resulted in an agreement with previous study in the single-objective problems in its topology, and achieved global exploration of non-dominated solutions in the multi-objective problem.

The traditional linear programming model is deterministic. The way that uncertainty is handled is to compute the range of optimality. After the optimal solution is obtained, typically by the simplex method, one considers the effect of varying each objective function coefficient, one at a time. This yields the range of optimality within which the decision variables remain constant. This sensitivity analysis is useful for helping the analyst get a sense for the problem. However, it is unrealistic because objective function coefficients tend not to stand still. They are typically profit contributions from products sold and are subject to randomly varying selling prices. In this paper, a realistic linear program is created for simultaneously randomizing the coefficients from any probability distribution. Furthermore, we present a novel approach for designing a copula of random objective function coefficients according to a specified rank correlation. The corresponding distribution of objective function values is created. This distribution is examined directly for central tendency, spread, skewness and extreme values for the purpose of risk analysis. This enables risk analysis and business analytics, emerging topics in education and preparation for the knowledge economy.