模糊鲁棒优化的样本方法与确定等价分析

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本文标题"Sampling Method for Robust Fuzzy Optimization"聚焦于模糊鲁棒最优化领域的创新研究。作者孙楚仁，来自上海国际贸易学院松江区的一所国际商学院，通过将模糊性融入传统的鲁棒优化问题中，提出了一个新的最优化模型，即鲁棒模糊优化问题。这个模型的核心在于处理不确定参数的模糊特性，使得决策过程更为适应实际环境中的不确定性。首先，文章从理论角度出发，对提出的鲁棒模糊优化模型进行了深入分析。作者证明了这一模型实际上可以转化为一个非线性优化问题，这为理解和求解该问题提供了明确的数学基础。在这个过程中，作者特别关注了该模型可行集的特性，探讨了其结构和约束条件，这对于理解和评估解决方案的有效性至关重要。接下来，文章的核心部分是提出了一种针对鲁棒模糊凸优化的抽样方法。这里，作者强调了当问题被抽样后，原本可能复杂的优化问题可以转变为可求解的多项式问题，这对于实际应用中的计算效率提升具有重要意义。在抽样策略中，作者还详细分析了可行性的概率，并给出了确保特定可行性概率所需的最小样本量，这对于控制误差和提高抽样策略的有效性有着定量的支持。关键词方面，"Robust fuzzy optimization"、"Probability of feasibility"、"Deterministic equivalence"以及"Fuzzy convex set"等都揭示了文章的主要研究内容和技术路线。此外，文章还涉及到了AMS分类号90C34和90C90，这表明它属于数学优化理论中的多个子领域，展示了其在学术界的严谨性和专业性。这篇首发论文不仅引入了鲁棒模糊优化的新概念，而且提供了一种有效的抽样方法，以解决此类问题中的不确定性。这对于理解和处理实际决策问题中的模糊性和不确定因素具有重要的实践价值。

option[21], robust multi-stage investment [4, 49], power supply[46], truss topology design[1],

supply chain management[9, 16], Other application of this methodology can be found in

[48, 20], etc. For complete survey of robust optimization, we refer the author to [12].

Though robust methodology is much useful, there a r e some drawbacks for t his method-

ology. Firstly, to use robust optimization methodology, one needs to know exactly the

uncertainty set U before modelling, which is generally impossible, while exactness of the

uncertainty set aﬀects the robust solutions terribly. In practice, one can only know that

a uncertain parameter belongs to some set fuzzily. Secondly, robust optimization method-

ology is too pessimistic. It considers all possible uncertainty equally, while in practice,

some uncertainty will just happen with small probability, which may inﬂuences robust so-

lutions and values vastly. Moreover, it’s impossible for one to pretend against all po ssible

uncertainty in practice because uncertainty have multiple origins[2]. These two drawbacks

require us to generalize robust optimization methodology.

We extend robust optimization methodology via introducing fuzziness to the uncer-

tainty set U considering the fact that it can not be known exactly. We do so also because

fuzziness is used in various ﬁelds when modelling since Zedah [34] posed t he concepts of

fuzzy numbers and fuzzy set. Currently, the concept of fuzzy programming is posed and

it has been applied in various kinds of real problems, such as water allocation, facility

location, etc. In reality, data is always considered to be fuzzy instead o f speciﬁc. In our

generalization of robust optimization methodology, the uncertain parameter s is assumed

to belong to an uncertainty set fuzzily, correspondingly, the constraints are not pro hibited

to hold exactly, instead, they a r e just requested to hold vaguely. By dosing so, we get

a new kind of optimization problems, called robust fuzzy optimization problems in the

sequel. A robust fuzzy optimization problem can be formulated as follows.

min c

x (1.3)

s.t. g(x, s) ∈

S, ∀s ∈

where

S and

U a r e two fuzzy sets.

Obviously, as s is a fuzzy number, g(x, s) is a f uzzy vector. In the above formulation,

for any fuzzy number s ∈

U, the solution x should satisfy that g(x, s) is in the fuzzy set

U, which embodies robustness. Also, we consider the data s to be fuzzy, which incarnates

fuzziness. Though the terminology ”robust fuzzy” appeared already in control theory[30],

robust design[29], clustering[31], in which we emphasize that the idea adopted in [30] is

something like the one posed in this paper, which introduces the way to keep a fuzzy

PID control structure in another fuzzy PID structure(essentially robust fuzziness), the

idea of robust fuzziness in this article is posed in a more general formulation, which is a

generalization of [14].

The structure of the rest ar t icle is as follows. In section 2, we come to introduce some

basic concepts of fuzzy sets and membership functions. In section 3, we make a basic

assumption for problem (1.3), and then show that robust fuzzy optimization problems

are generally diﬃcult. We design a sampling algorithm to solve the robust fuzzy convex

optimization problems in section 4. Finally, in section 5, future r esearch on this topic is

remarked.

2 Preliminaries

In this section, we introduce some basic concepts and results on fuzzy sets and mem-

bership functions. All these concepts and results can be fo und in [10, 11, 2 8], etc.

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