近年來比值規劃己有一些進展，但比值和問題一直沒有重大突破。Freund 和 Jarre 證明了比值和問題本身是一個NP-complete問題。大部分解比值和問題的決定型演算法是採取branch-and-bound方法。但它們通常僅能解少數幾項和的比值和問題。在這篇文章裡，我們利用 Birbli， Fang and Sheu所提出的隨機演算法設計一個新的方法來解比值和問題。這個方法是借由解一系列單一比值問題來計算樣本點的函數值，然後利用隨機機制所建立的交互 ``電磁'力讓這些樣本點移動。在數值結果方面我們可以處理到十六個線性比值和而且結果也相當不錯。

　In spite of recent progress in fractional programming, the sum-of-ratios problem remains untoward. Freund and Jarre proved that this is an NP-complete problem. Most of the existing methods overcome the difficulty using the deterministic type of algorithms, particularly the branch-and-bound method. They can solve only the sum of a few ratios. In this paper, we propose a new approach using the stochastic search algorithm by Birbil, Fang and Sheu. It computes each sample point by solving a single ratio problem and then moves the sample points according to randomly controlled interacting ``electromagnetic' forces. To test, we perform numerical experiments on problems up to the sum of sixteen linear ratios and the results are very positive.

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