Platinum ratio search versus golden ratio search☆
Introduction
Many business and management problems can be viewed as optimal search problems. Examples include Hendrick et al. [1] who tried to find a better search method in the exploration of zinc ore, and Ladany [2] who tried to efficiently search for the optimal strategy of market segmentation and pricing for rooms of hotels, etc. Particularly, there is often the need to search for the location of the optimal point in a one-dimensional range. If the optimal point (optimum) is unique in the range (i.e., there is no other local optimum), it is a unimodal optimization problem.
All kinds of searches consume both time and money. It is an important issue in business and production operations especially when high-cost search experiments are necessary. For example, an oil or mining company is looking for oil or minerals in a belt-shaped geological area where the oil or mineral might exist at only one position. Which strategy minimizes the number of exploratory drillings? The answer to this question obviously reduces the cost for the company. A more specific expression of the problem is to search for the position of the optimum with the least number of search times (or experiments, observations), until the interval that contains the optimum is narrowed down to a given accuracy or tolerance. In this kind of situation, it is known that the golden ratio search is the best method to minimize the number of search experiments (Wilde and Beightler [3]). With the golden ratio search, the number of search times required to achieve certain accuracy can be strictly proved via induction.
The golden ratio is related to the Fibonacci numbers. It is 0.618 in value, the limiting ratio of the Fibonacci sequence . The Fibonacci numbers and the golden ratio are rarely a topic of academic research in business and management area. Yet, some exceptional studies are still found in management science. For example, Good [4] and Lipovetsky and Lootsma [5] separately discussed Fibonacci numbers and sectioning in the context of operational issues. Nahmias [6] and Jaraiedi and Zhuang [7] studied Fibonacci search (an alternative unimodal search method which yields the largest interval reduction with the given number of experiments), respectively, in inventory management and quality control issues. Zhang et al. [8] studied a job-scheduling problem with the golden ratio. Disney et al. [9] investigated a production and inventory control problem and found that the golden ratio minimized the sum of inventory and order variance over time.
Although the golden ratio search method is well known as the best choice for certain optimization problems, it by no means is the best in all respects. In this paper, instead of introducing or promoting golden ratio search, we will try to reveal the limitations of golden ratio search and to gain more insight to the search method from an economic viewpoint, specifically, with regard to the concern of cost (price). We will show that the golden ratio search is the best only in the sense of zero uncertainty, it is not economically the best in terms of the least expected cost. We will suggest the most economical search method with the “platinum” ratio of 0.55 for potential use in practice. Although our study does not have the mathematical rigor (as it is a simulation work), our findings are enough to question the dominance of the golden ratio search.
Section snippets
The search plan
For a unimodal optimization problem in range , we assume the objective function is continuous in to make sure every point in the range is available for sampling (experiment). Without the loss of generality, we consider a maximization problem, i.e., we want to search the value of , so that is maximized. (The minimization problem can also be considered.) The search procedure condenses, step by step, the range in which the maximum is located until the width of the
Prior information and the simulation
Before a search begins, the knowledge on the location of the optimum can be viewed probabilistically. If the location is known with certainty, there is no need to search at all. If there is no knowledge about the location of the optimum at all, we have no reason to suspect that the optimum is located in one part of the range than in any other part of the range. This implies that the likelihood of the optimum's location is uniformly distributed in the stated range. If prior information about the
On the search cost
The number of search times (observations) can be viewed as the search cost. It is necessary to distinguish between the realized search cost and the expected search cost. The realized search cost is the actual cost incurred when the search is finished. The expected search cost is the cost in expectation before the search starts, while the optimum's location can be anywhere in the range with a predefined probability. The expected cost can be estimated as the average of all the cost realizations
Results and conclusion
The search costs at different section ratios are shown in Figs. 1 and 2. The accuracy of search is set at 0.01. Fig. 1 is the mean cost (the expected cost) for various section ratios. Fig. 2 shows the maximum, the minimum, and the median search costs for various section ratios. It can be seen that at the golden ratio, 0.618, or its symmetric mirror, 0.382, the variation of the search cost is zero (see Fig. 2, where the maximum, minimum, and median search cost are all equal to 11). All the other
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This manuscript was processed by Area Editor Prof. B. Lev.