This is to certify that the thesis work entitled "Optinialitji Analysis on the Basis of Rectangular (Manhattan) Distance Measure for Maximin LHDs Obtained by the Iterated Local Search heuristics Approach" was carried out by Md. In this research work, the main objective is to study the optimality of the maximum LHD obtained by ILS approach regarding Manhattan distance measurement.
LIST OF TABLES
Background
In the literature, the optimal criterion for maximin LHDs is defined in different ways [Grosso et al. Here, the heuristic approach of Iterated Local Search (ILS) will be considered to find the optimal (maximin) LHDs [Grosso et al.
Literature Review .1 Experimental Designs
- Optimal Criteria and Approaches
- Distance Measure
Liefvendahl and Stocki (2006) also compared the performance of CP and genetic algorithms for optimal LHDs. Li and Wu (1997) considered a class of column-wise Pair-wise (CP) algorithms in the context of building oversaturated optimal models.
Goals of the Thesis
It is noted that the maximum LHDs obtained by ILS approach are optimal on the basis of Euclidean distance measure. Implementation of the ILS approach to find out the optimal LHDs with respect to maximum optimal criteria in Euclidean distance measure.
Structure of the Thesis
Introduction
Definition of Distance Function (Metric)
Note that forp = I we get the taxi norm (Manhattan), forp = 2 we get the Euclidean norm, and asp approaches oo the p-norm approaches the infinity norm or the maximum norm. In mathematics, the Euclidean distance or the Euclidean metric is the "ordinary" distance between two points that one would measure with a ruler, and is given by the Pythagorean formula. The Euclidean norm is by far the most commonly used norm on, but there are other norms on this vector space, as will be shown below.
The Euclidean distance between points p and q is the length of the line segment that connects them (p, q).
Squared Euclidean Distance
The standard Euclidean distance can be squared to place progressively greater weight on objects further apart.
Minkowski Distance
- Definition
The Minkowski distance can also be seen as a multiple of the moving average of the component-wise differences between two points P and Q. Note that a circle is a set of points with a fixed distance, called the radius, from a point that the center.
Chebyshev Distance
- Formal Definition
- Taxicab Distance Versus Euclidean Distance
The taxi distance depends on the rotation of the coordinate system, but does not depend on its reflection about a coordinate axis or its translation. As the size of the city blocks decreases, the points become more numerous and become a rotated square in continuous taxi geometry. A sphere formed using the Chcbyshev distance as a metric is a cube with each face perpendicular to one of the coordinate axes, but a sphere formed using the Manhattan distance is an octahedron: these are double polyhedra, but among cubes only the square (and 1-dimensional line segment) are self-dual polytopes. The Chebyshev distance refers to the L metric or norm.
Taxicab geometry, considered by Hermann Minkowski in the 19th century, is a form of geometry in which the usual distance fi.inction or metric of Euclidean geometry is replaced by a new metric in which the distance between two points is the sum of the absolute differences of their Cartesian coordinates.
Hamming Distance
- Special Properties
- Applications
The Hamming distance is named after Richard Hamming, who introduced it in 1950 in his fundamental article on Hamming codes Error detection and error correction codes. For comparing sequences of different lengths, or sequences that require not only substitutions but also insertions or deletions, as expected, a more advanced metric such as the Levenshtein distance is more suitable. 2 Orthogonal modulation uses the Hamming distance, while phase modulation uses the Lee distance.
On a grid like a chessboard, the Hamming distance is the minimum number of moves it will take a rook to move from one cell to another.
Levenshtein Distance
- Applications
Hamming bit weight analysis is used in several disciplines, including information theory, coding theory, and cryptography. If the strings are the same size, the Hamming distance is an upper bound on the Levenshtein distance. The Levenshtein distance between two sets is not greater than the sum of their Levenshtein distances from the third set (triangle inequality).
The Levenshtein distance can also be calculated between two longer strings, but the cost to calculate it, which is roughly proportional to the product of the two string lengths, mnakes.
Lee Distance
Introduction
Iterated Local Search
Iterative local search is a metaheuristic designed to introduce another problem-specific local search as if it were a black box. This allows Iterated Local Search to maintain a more general structure than other metaheuristics currently in practice. The local search applied to the initial solution gives the starting point s' of the walk in the set S'.
The perturbation scheme takes a locally optimal solution, s', and produces another solution from which to start the local search in the next iteration.
Maximin Latin Hypercube Designs
ILS has many of the desirable features of a metaheuristic: it is simple, easy to implement, robust, and highly effective. How effective this approach turns out to be depends largely on the choice of local research, concerns and acceptance criteria. This dichotomy is important because the optimization of the algorithm can be done progressively, and thus the ILS can be kept at any desired level of simplicity.
Finally, note that although the entire current overview is given in the context of addressing combinatorial optimization problems, in reality much of what is covered can be easily extended to continuous optimization problems.
Definition of LHD
For this reason, the search must be driven by other optimality criteria, which take into account other values in addition to D 1.
Optimality Criteria
Note that for large enough p, each term in the sum in (3.4) dominates all subsequent terms. A clear disadvantage of the Opt(Ø) criterion, if we are interested in maximin values (maximum D 1 value), is that LHDs with smaller (better) Ø1 can have worse (smaller) D, i.e. in this way the search in the solution space can be guided by some kind of heuristic function.
While the two criteria above are strictly related to maximin values and they will be widely used in the definition of approaches for the detection of niacinin solutions, for the sake of completeness, we also mention that also other optimality criteria, which are not necessarily related.
ILS Heuristic for Maximin LHD
In Table 3.1, we present some of them and also approaches for creating an optimal Latin hypercube design. Park 1994 A two-stage (exchange and Newton-type integrated mean square) error algorithm and entropy measures Morris and 1995 Simulated annealing. Liefvendahl and 2006 Columnwise-pairwise Minimum distance in Stocki and genetic algorithms Audze-Eglajs function Dam et al.
- Initialization (Is)
- Acceptance Rule
It is obscrvcd that the effect of CP-based local search and RP-based local search is not significant [Jamali (2009)]. Among the two types of local moves [Jamali (2009)], we considered Best Improve (BI) acceptance rule since there is no significant difference with respect to output (see [Jamali (2009)]). Basically, a perturbation is similar to a local move, but it must be somehow less local, or, more precisely, it is a move within a neighborhood larger than the one used in the local search.
Among the two types of perturbation operators, e.g. (i) Cyclic Order Exchange (COE) and (ii) Pairwise Crossover (PC) proposed in [Jamali (2009)], we consider COE. I) Cyclic Order Exchange (COE): Our first disruption procedure is the Cyclic Order Exchange (COE).
- Pairwise Crossover
Now we randomly choose two rows (points), say X2 and x5 and we randomly choose the column (component) rn = 4. Then, after the SCOE perturbation we get the following LHD X' (Eq. 3.8), note that bold levels indicate the values modified with respect to X). This is similar to biological crossover — we randomly pick two points (rows) and then swap randomly selected portions of them.
Here we present three variants of PC, namely Single Pair Crossover (SPC) and Multiple Pair Crossover (MPC). 2a) Single Pair Crossing (SPC): For SPC, we first randomly select two rows, say, x1 and x, i :Aj, in the current LHD X*; then we choose a component at random, I say.
Introduction
In this chapter, we discuss the optimality of the experimental results obtained by the ILS approach. Initially, we will present the optimal LHDs to show the performance of the ILS approach with respect to the Euclidean distance measure. Next, we will also briefly discuss the multicollinearity of the optimal LHDs obtained by the ILS approach.
Experimental Results and Discussion for Euclidean Measure
The experimental results of the ILS approach in relation to the Euclidean measure [Jarnali, 2009] are given in Table 4.3 and Table 4.4. The performance of the ILS approach in terms of maximizing LHDs in the L2 measure is outstanding compared to other approaches available in the literature. Apama (2012) also analyzed the performance of the ILS approach with respect to the multicollinearity of the optimal LHD measured in Euclidean distance.
We can conclude that the optimal LHDs selected by the ILS approach with respect to the Euclidean measure have poor multicollinearity, i.e.
Introduction
Experimental Results and Comparison for Manhattan Measure
It is noted in the table that although in MLH-ILS, the L2 distance measurement is considered, the c1 values of the MLH-ILS design are comparable to the other two models. Again in Table 5.2, it is observed that the MLH-SA, OMLH-MSA and OLH-Y designs are optimized with respect to the rectangular distance measure (L) while the proposed design - MLH-ILS is optimized with respect to the Euclidean distance measure (L 2). It is noted in the table that although the distance measure L2 is considered in the MLH-ILS design, the D1(J1)" and (LI) values of the MLE-l-ILS design are comparable with respect to the other three models.
It should be noted that few values are available in the literature to measure the Manhattan distance.
Experimental Study for Impact of Trials
Here we also observed that the impact of the study regarding the D1' values is not significant. It is noted that the abscissa of each figure indicates the number of trials, while ordinate indicates the D' values. Now in all Figures 5.2(a) - 5.2(g) except some N values it is observed that the impact of the trial on LHD with respect to D1 M values is not significant.
The tables show that the influence of the experiments is not significant even with regard to the values of DM1 and D 1 '2.
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