Abstract
Entropy has continuously arisen as one of the pivotal issues in optimization, mainly in portfolios, as an indicator of risk measurement. Aiming to simplify operations and extending applications of entropy in the field of LR fuzzy interval theory, this paper first proposes calculation formulas for the entropy of function via the inverse credibility distribution to directly calculate the entropy of linear function or simple nonlinear function of LR fuzzy intervals. Subsequently, to deal with the entropy of complicated nonlinear function, two novel simulation algorithms are separately designed by combining the uniform discretization process and the numerical integration process with the proposed calculation formulas. Compared to the existing simulation algorithms, the numerical results show that the advantage of the algorithms is well displayed in terms of stability, accuracy, and speed. On the whole, the simplified calculation formulas and the effective simulation algorithms proposed in this paper provide a powerful tool for the LR fuzzy interval theory, especially in entropy optimization.
Highlights
Entropy, which made its debut in 1948, was designed by Shannon [1] to measure the randomness or uncertainty of a random phenomenon
With regard to the situation when f is a complicated nonlinear function, two simulation algorithms are designed to approximate the entropy of function on the basis of the presented calculation formulas, including the uniform discretization algorithm (UDA) referring to the method of generating the exact membership degree of each function value proposed by Miao et al [34] and the numerical integration algorithm (NIA) based on the operational law of inverse credibility distribution (ICD) proposed by Zhou et al [35]
It follows immediately from Theorems 1 and 2. Through this calculation formula for entropy of function, the crisp integration can be obtained by substituting the ICDs of LR fuzzy intervals (i.e., Φi−1, see Equation (9) in Definition 6) into Equation (34), which avoids the difficulty of calculating the entropy of function via the credibility measure Cr and the function S in Equation (33)
Summary
Entropy, which made its debut in 1948, was designed by Shannon [1] to measure the randomness or uncertainty of a random phenomenon. Liu and Zhang [15] proposed a general multi-period fuzzy portfolio optimization model with return demand which involved the entropy based on possibility measure to maximize both terminal and cumulative diversification. Yin et al [16] provided a possibility-based robust design optimization framework in which the entropy of the fuzzy system response was regarded as the variability index for the uncertain structural-acoustic system. Our paper improves the existing work and reduces the computation complexity via an explicit expression for both linear and nonlinear functions involving different types of fuzzy variables
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