Goal Function Value Research Articles

Extensive Study of Modern Approaches Used in Identifying the Buckingham Potential

This paper presents two mathematical approaches of estimating unknown parameters of the Buckingham potential. Two functional forms of the potential was investigated owing to the complexities and difficulties arising from the original Buckingham potential. These complexities are due to the combination of both powers and an exponent in mathematical representation of the potential. Hence, we decided to consider approximating the parameters of the simpler form, widely known as the modified Buckingham potential. The method of least squares (differential and integral methods) were used to estimate the potential parameters and the calculation of the optimized values of the constructed goal function were included in the paper. The calculated estimates showed reasonable agreement with experimental datasets of copper atoms over a large vicinity of the interatomic potential distance. This conclusion was reached through the graphical analysis of potential energy curves and through the error between the respective curves. Optimized goal function values were used to compare the efficiency of the methods.

Optimal interatomic potentials using modified method of least squares: Optimal form of interatomic potentials

Abstract The problem of optimization of interatomic potentials is formulated and solved by means of generalization of the Morse, Kaxiras–Pandey, and Rydberg potentials. The interatomic potentials are treated as solutions of some second-order ordinary differential equations which will be classified and analyzed. The most appropriate analytic form of the understudied potentials will be proposed based on a one-dimensional search for the parameter, γ \gamma , which is the power of the interatomic distance, r r . The optimal analytic form will also be proposed for metals such as gold, copper, aluminium, titanium, and the silver–copper alloy. The method of least squares will be used to estimate the potential parameters. Phenomenological potentials such as the classical Rydberg, classical Morse, generalized Morse, Kaxiras–Pandey, and classical Lennard–Jones will be studied, and new potentials based on the combination of some of the aforementioned potentials will also be proposed. Metrics such as the goal function values, will be used to identify which optimal value of the parameter, γ \gamma , is most appropriate to introduce into the preferred interatomic potential for interaction between atoms.

Sequential path-equilibration algorithm for highly accurate traffic flow assignment in an urban road network

Nowadays, the issue of traffic flow assignment has become an interdisciplinary topic that concerns multiple research areas and branches of science. This work is focussed on the mathematical and computational aspects of the equilibrium traffic assignment problem in the case when route flows are considered to be decision variables. Firstly, we obtain a fixed-point mapping with the explicit contraction operator, which is proven to equilibrate the journey times on feasible routes between a single origin-destination pair of nodes with the quadratic rate. Remarkable that from mathematical perspectives, the developed operator generalizes most path-equilibration operators already exploited by researchers. Secondly, we use the obtained fixed-point procedure to run the sequential path-equilibration algorithm for traffic flow assignment on well-known test urban road networks with arc-additive travel time functions. Our computational results appear to demonstrate higher accuracy of user-equilibrium traffic assignment solutions than the best ones known to us. In other words, developed within this paper sequential path-equilibration algorithm leads to solutions with less goal function values compared to the best solutions for today, according to our knowledge.

A Decision Making Model Based on Intuitionistic Multiplicative Preference Relations With Acceptable Consistency

Two categories of hesitancy often exist in an intuitionistic multiplicative preference relation (IMPR). One is inconsistency among three or more preferences in an IMPR, and the other is hesitancy of a non-diagonal intuitionistic multiplicative judgment itself. Capturing such hesitancy plays an important role for deriving priority weights of IMPRs and measuring inconsistency degrees of IMPRs. This article first analyzes a recent consistency definition of IMPRs and reveals its deficiency. A normalized intuitionistic multiplicative weight vector is introduced and utilized as a benchmark of different intuitionistic multiplicative weight vectors with equivalency. A new notion of consistent IMPRs is then proposed by building the link between an IMPR and its normalized intuitionistic multiplicative weights. In order to find the closest IMPR with consistency to an original IMPR, the paper establishes a minimization model, in which the hesitancy difference between the two IMPRs is regarded as a constraint and the optimal goal function value is used to determine an inconsistency index of the original IMPR. The minimization model is further transformed into a least square model and its analytical solution is discovered by the Lagrangian multiplier method. Based on the analytical solution, an intuitionistic multiplicative judgment based index is devised to measure inconsistency of IMPRs, and an approach with checking acceptable consistency is developed for multi-criteria decision making with IMPRs. The presented models are illustrated and validated by three numerical examples including a hierarchical multi-criteria decision making problem.

Equivalent instances of the simple plant location problem

In this paper we deal with a pseudo-Boolean representation of the simple plant location problem. We define instances of this problem that are equivalent, in the sense that each feasible solution has the same goal function value in all such instances. We further define a collection of polytopes whose union describes the set of instances equivalent to a given instance. We use the concept of equivalence to develop a method by which we can extend the set of instances that we can solve using our knowledge of polynomially solvable special cases.

Multicriteria optimization of landscapes using GIS-based functional assessments

This paper introduces a new way of using GIS to support decision making in the planning process and to develop regional guidelines. The method of `multicriteria optimization' helps new methodological standards to be established for integrating the various results of functional landscape ecology assessments of the type usually carried out in ecological planning, and enables the overall comparison of competing aims. This technique allows different aims in a geographical region to be quantified and takes into account different weightings of scenarios. Different functional assessments (soil erosion hazards caused by water flow, groundwater regeneration, water discharge regulation) and an assessment of the agricultural production function were carried out using GIS for a test site measuring 48 km 2 in a glacial end moraine area to the north of Leipzig (Saxony, Germany). The results were presented in the form of ordinal assessment classes which express tendencies. On the basis of these assessment results, aims for the calculation of an optimal land-use pattern were defined and weighted in different scenarios. Multicriteria optimization calculates optimal compromises with regard to these aims which can be measured and compared with goal function values. Although the method cannot replace decision making by the planner, which is often difficult and can generally not be quantitatively verified, it can help to make the planning process more objective.

A Monte-Carlo approach for 0–1 programming problems

A two-phase global random search procedure for solving some computationally intractable discrete optimization problems is proposed. Guarantees for quality of random search results are derived from analysis of non-asymptotic order statistics and distribution-free intervals that are obtainable in this way: the confidence interval for a quantile of given order, or the tolerance interval for the parent distribution of goal function values. It has been shown that results, related to the multiconstrained 0–1 knapsack problem, within a few percentage from the true optimal solution can be obtained.

Solving linear programming problem with non-deterministic constraints using fuzzy numbers arithmetic

The concept of solving the linear programming problem with non-deterministic constraints is presented in this paper. It is assumed that the right-hand sides of the constraints are fuzzy numbers, and the problem is solved for their mean values. Furthermore, a fuzzy solution of the problem is specified for the optimal basis of the linear programming problem with mean values. A solution feasibility measure is introduced. When the measure is too small, another feasible solution of the problem for the mean values is chosen, which is non-optimal and which corrects the feasibility of the fuzzy problem with simultaneous deterioration of the goal function value.