這篇論文在探討Flow shop排程問題，並使用模糊數表示處理所需時間及預計完成時間，引用模糊集合論中的可能率及必然率來表達提前完成及延後完成的概念，另外將使用不同組合的可能率及必然率來表達目標函數。基因演算法被提出來解決這些目標函數，另外將在擇優保留的階段使用一個基於最長共同子字串的想法來改良它，使得演算法可以在較短的迭代次數內就達到停止條件。另外，將CUDA技術引用進入演算法，數值實驗結果顯示在效率上，GPU上的CUDA程式將比在CPU上的傳統程式還要良好。

The flow shop scheduling problems with fuzzy processing times and fuzzy due dates are investigated in this paper. The concepts of earliness and tardiness are interpreted by using the concepts of possibility and necessity measures that were developed in fuzzy sets theory. And the objective function will be taken into account through the different combinations of possibility and necessity measures. The genetic algorithm will be invoked to tackle these objective functions. A new idea based on longest common substring will be introduced at the best-keeping step. This new algorithm reduces the number of generations needed to reach the stopping criterion. Also, we implement the algorithm on CUDA. The numerical experiments show that the performances of the CUDA program on GPU compare favorably to the traditional programs on CPU.

1 Introduction 2
2 Fuzzy Numbers 4
3 Objective Functions Based on Possibility and Necessity Measures 6
3.1 Earliness for fuzzy flow shop problems . . . . . . . . . . . . . . . . 7
3.2 Tardiness for fuzzy flow shop problems . . . . . . . . . . . . . . . . 8
3.3 Earliness and tardiness for fuzzy flow shop problems . . . . . . . . . 8
3.4 Formulas for the objective functions . . . . . . . . . . . . . . . . . . 10
4 Design of Genetic Algorithm 12
4.1 Original Genetic Algorithm . . . . . . . . . . . . . . . . . . . . . . 12
4.2 Longest Common Substring . . . . . . . . . . . . . . . . . . . . . . 14
4.3 Rearranging the crossover . . . . . . . . . . . . . . . . . . . . . . . 15
4.4 A new stopping criterion . . . . . . . . . . . . . . . . . . . . . . . . 16
5 Numerical Results 17
5.1 Test data construction . . . . . . . . . . . . . . . . . . . . . . . . . 17
5.2 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
6 Applying CUDA Framework 22
6.1 Generally work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
6.2 Modified merge sort . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
6.3 The kernel run-time . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
6.4 Memory problems . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24
6.5 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25
7 Conclusions 27

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