A Novel Parameter –free Optimization Algorithm for Solving Real Engineering Problems: Golden Ratio Optimization Method (GROM)

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1. Introduction

A new meta-heuristic optimization algorithm known as golden ratio optimization method (GROM) that is based on natural growth [1]. The golden ratio is often found in nature and even in the human body while many of the most often examples of the golden ratio have been  found in plants, animals, insects and other natural structures and man-made works of art, design etc [2].  The Golden ratio is also known as the Golden Section, Golden Mean, Divine Proportion, or the Greek letter Phi, the Golden Ratio is a special number that approximately equals 1.618 [3].

2. What is the Golden Ratio?

In 1202, mathematician Leonardo Fibonacci was among the first people who introduced this golden ratio which is a series of numbers in which a number except the first two numbers is equal to the sum of the two previous numbers [4]. In this series, the ratio of two consecutive numbers is almost the same for all the numbers and is known as golden ratio.

2.1. The Mathematical Side of the Golden Ratio

The mathematics of the golden ratio is relatively simple. A line is divided into two parts “a” and “b” so that the ratio of the larger section (a) to the smaller section (b) is equal to the ratio of the whole length (a + b) to the larger section. This results in the formula: a / b = (a + b) / a. The outcome of this formula is an irrational number often called the “golden number” or phi in mathematics. The golden number phi is approximately equal to 1.618. . Moreover, the ratio between two consecutive numbers is equal to 1.618 which is the golden ratio or φ

Fig 1: The Golden Ratio

2.2. The Golden Ratio and the Fibonacci sequence

The Fibonacci sequence is the sum of the two numbers before it. It goes: 0, 1, 1, 2, 3, 5, 8, 13, 21, and so on, to infinity. The Fibonacci sequence starts with the number 1 (or sometimes 0), and every number is the sum of the two preceding terms [5]. So the first numbers would be 1, 1, 2, 1 + 2 = 3, 2 + 3 = 5, 8, 13 … and so on. This series of numbers is directly related to the golden ratio. In mathematics, two quantities are in the golden ratio if their ratio is the same as the ratio of their sum to the larger of the two quantities.

Fig 2: The Fibonacci sequence

3. Inspiration of GROM The GROM algorithm is inspired by the golden ratio of plant and animal growth which is formulated by the well-known mathematician Fibonacci. The Fibonacci sequence can be observed in nature, and the rabbit population growth was evolved, and in the natural occurrences like leaf arrangements in plants [6]. The ratio itself comes from the Fibonacci sequence, a occurring sequence of numbers that can be found approximately, from the number pairs on rabbit. For example, the remains of ancient Egypt, Greece, and Rome are some of these examples. The ratio of the length of the Pyramid’s base to its height is equal to the golden ratio.

Fig 3: Inspiration of GROM

4. Examples of Golden Role in Real World

 The golden ratio is commonly found in nature, and it can also be used to achieve beauty, balance and harmony in art, design such as architecture, logos, UI Design and photography [7] .The golden ratio isn’t just some mathematical theory it is implemented all the time in the real world natural phenomenon.

Fig 4: Examples of Golden Ratio

5. Golden Ratio in Logo Design

When designing a logo, you can even imagine the Fibonacci sequence as a series of circles, and then rearrange them to form a grid as the foundation for your logo design. Designing the logo using the golden ratio is one of the most important techniques in the present [8]. Nowadays, many designers have mastered this technique and small firms can come up with a golden ratio logo design. The most complex form of logo in golden ratio Apple logo, each piece is drawn in a golden ratio [9]. Apple logo is already a simple and popular logo, but who would say that it is so complexly drawn. The scapular curvature uses a certain circle size from the golden ratio of circles.

Fig 5: Golden Ratio in Logo Design

6. Golden Ratio Optimization Method

The ‘‘φ number’’ is derived from the Fibonacci sequence series are as follows: 1, 1, 2, 3, 5, 8, 13, 21, 34, 55, 89, 144, 233, 377, 610, 987, 1597, 2584, and 4181. Except the first two numbers, the others are equal to the summation of the previous two numbers, but also the ratio of each two consecutive numbers is an amazing number close to 1.618, which is known as the ‘‘golden ratio.’’ And the more surprising thing is that any Fibonacci number can be obtained through a golden number as follows:

And the answer always comes in the form of an integer, exactly equivalent to the sum of the two previous members. For example;

When the golden number is used with 6 decimal, a response of 8.00000033 is obtained. The result with more accurate calculations will be closer to 8.

The fitness of this solution is obtained and compared with the worst solution. If the mean solution has better fitness compared to the worst solution, the worst one is replaced by the mean solution.

To perform the global search and then the local search during the implementation of the algorithm, Fibonacci’s formula is used as follows;

Moreover, in order to search the whole space of the problem, a random movement is also added to the new solution. Now, to update the solutions, the following equation is used:

Every single solution attempts to come close to the best answer and refrain from the worst solution. To do this, the golden ratio is also employed as follows for all the solutions.

GROM is simple method which is free from any parameter tuning. The results and comparison shows the surprising performance.

7. Flowchart of GROM

Fig 6: Flowchart of GROM

8. Applications of GROM

To evaluate the performance of the GROM in solving real engineering optimization problems, various applications are employed they are as follows,

  • Tension / Compression spring design [10].
  • Welded beam design [11].
  • Pressure vessel design [12].
  • Cantilever beam design problem [13].
  • Gear train design problem [14].
  • Optimal power flow [15].
Fig 7: Applications of GROM

9. Advantages of GROM

  • GROM is a parameter-free with sample implementation.
  • The volume of calculation in each iteration is not big, and GROM is a fast algorithm.
  • The solution updating is performed very well which results in fast convergence to the best final results.
  • The golden ratio causes fast convergence and reaching the best global answer.
  • GROM is a powerful algorithm in both local and global searches.
  • GROM is very robust and different trials reach almost the same results.
  • GROM has excellent performance in solving large-scale problems, discrete problems, actual engineering problems, and high constraints problems


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