在存貨管理的決策過程中，決策者經常面對「過與不及」的問題，因此必須以預估之需求量作為訂購決策的依據，以達到降低存貨成本及最大化銷售利潤之目標。
    本論文提出隨機單期需求訂購批量模式(報童問題)之推廣與應用，以報童模型為基礎，融入模糊理論的推演，加以延伸並應用於財務領域中現金管理與遠期合約管理之議題。
本論文在考慮需求不確定下，構建了三個延伸報童模式。第三章主要目的是利用具有混同資料之機率模糊集合加以構建模糊報童模型，藉以分析總成本最低之最適訂購政策。首先，傳統報童問題中隨機性需求將被清楚定義，接著提出相對應之模糊分配函數以探討模糊觀點下之最適訂購政策，並以假設性之範例配合指數分配函數加以說明模型內涵，透過模型分析與解模糊化後，進一步比較模糊模式與一般傳統模式在最適訂購量與總成本之差異。第四章則是延續第三章之架構，利用模糊觀點推導出模糊積分定理，進而構建出模糊報童問題之一般公式，並將其運用在單期現金管理計畫上。第五章則是將傳統報童問題之單一決策變數(訂購量) 擴充為二個決策變數(訂購時機與訂購量)，並結合價格折扣與預購策略於傳統報童模型中，另外沿用過去需求分配未知的求解方法，融入多階段決策準則，構建出較實際之報童問題來決定最佳訂購時機與最適訂購量使得期望利潤最大化，此結果可作為遠期合約管理的參考。
    藉由嚴謹及具體之數學推導，說明模糊報童模型為傳統報童問題之一種延伸。研究結果指出在特定之分配函數下，模糊方法對不確定之需求的推估比使用單點估計之結果為佳。亦說明了在不確定的環境中，過去資料無法充分預測實際需求的現象。另外，本論文也具體解釋隨時間變動的預測誤差對預購時機與預購數量的影響。據此，本論文之主要貢獻乃透過理論分析，構建出較符合實際情況之隨機單期存貨模型，並強化傳統報童問題的實用性。

In the decision-making process of inventory management, the decision makers often face the dilemma of overage and underage, so they therefore must adjust the order quantity in accordance with real demand and reduce inventory tied up unnecessarily in the system without diminishing profit or increasing cost.  

This thesis puts forward some extensions and applications for newsvendor problem. Based on the basic framework of newsvendor model, the fuzzy set theory is introduced to deal with the topics of cash management and forward contract management. This thesis is formulated in three extended newsvendor models under considering the uncertain demand and ordering timing. In Chapter 3, the approach of probabilistic fuzzy set is utilized to construct the fuzzy newsvendor model with hybrid data and to analyze the optimal ordering policy so that the total cost is minimized. First of all, the randomness of demand will be defined clearly in the classical newsvendor problem, and then a corresponding fuzzy distribution function is derived from the crisp case to solve the optimal ordering policy in the fuzzy sense. Finally, a supposing example will be collocated with the exponential distribute function in order to explain the implication of the fuzzy model. After defuzzification, the difference of optimal order quantity and minimum total cost between the fuzzy model and crisp model are further compared. Chapter 4 is an extended case following the basic framework of Chapter 3 in which the fuzzy integral will be deduced to construct the general formula of the fuzzy newsvendor problem, and it is further applied to the single-period cash management. In Chapter 5, the single decision variable (i.e. order quantity) of the classical newsvendor model are expanded as two decision variables (i.e. ordering timing and order quantity), and the relation of price discounts and preorder policy are incorporated into the model. To continue using the distribution-free approach, a more realistic newsvendor problem will be constructed to determine the optimal ordering timing and quantity by means of the multistage decision making criterion so that the newsvendor’s profit is maximized. Moreover, this result can also be regarded as the reference in managing the forward contracts. 

Through conscientious and concrete mathematical analyses, this thesis may explain that the fuzzy newsvendor models are some reasonable extensions of the crisp newsvendor models. The results can also explain the phenomenon that the past data can not be used to fully predict the actual demand in the uncertain environment. Additionally, this thesis has stated how the time-variant forecasted errors could influence on the newsvendor’s preorder policy. All the proposed models are accompanied with some numerical examples to illustrate the topics. In summary, the main contributions of this thesis are to construct a more realistic stochastic single-period inventory model and to enhance the practicability of classical newsvendor models.

Contents

List of Table	                                   IV
List of Figure	                                    V
Chapter 1.  Introduction	                           1
1.1.  Motivation and objective   1
1.2.  Organization of the dissertation   	3
Chapter 2.  Historical Review on Newsvendor Models 	  6
2.1.  Extensions and applications of crisp newsvendor models     6
2.1.1.  Extensions to different states about demand and related parameters 	  6
2.1.2.  Related applications in practice 	  8
2.2.  Historical review on fuzzy newsvendor models 	  9
Chapter 3.  Incorporating Probabilistic Fuzzy Sets into the Newsvendor Model with Hybrid Data                   11
3.1.  Exordium  	 11
3.2.  Difference between randomness and fuzziness 	 12
3.3.  Constructing a fuzzy newsvendor model   	 13
3.3.1.  Crisp case of newsvendor model   	 14
3.3.2.  Using hybrid data to estimate fuzzy demand during a single period	   14
3.3.3.  Estimate total cost based on probabilistic fuzzy set with fuzzy demand    16
3.3.4.  The case of the exponential distribution demand 17
3.4.  Numerical example  	 19
3.5.  Discussions    22
3.5.1.  Fuzzy case with the fuzzification of probabilistic fuzzy set in Section 3.3.3     23
3.5.2.  Fuzzy case of using triangular fuzzy number to fuzzify p.d.f. as   in Section 3.3.4   24
Chapter 4.  A Fuzzy Newsvendor Model for Cash Management 25
4.1.  Exordium   	 25
4.2.  Preliminaries  	 26
4.3.  Constructing a fuzzy cash management model    32
4.3.1.  Crisp case of stochastic cash management model  32
4.3.2.  Estimate total cost based on fuzzy integral with fuzzy probability distribution   and fuzzy cash demand  33
4.3.3.  Using exponential distribution as an example of Formula 4.1 to derive the main theorems    36
4.4.  Numerical example   39
4.5.  Discussions   44
4.5.1.  Using triangular fuzzy number   to fuzzify p.d.f. as   in Section 4.3.3    44
4.5.2.  The reason for the application of metric   in Definition 4.6   44
Chapter 5.  An Extended Newsvendor Model with Time-Variant Ordering Decisions	                                      46
5.1.  Exordium    46
5.2.  Notations and assumptions    48
5.2.1.  Notations  48
5.2.2.  Assumptions  	 49
5.3.  Model formulation    50
5.3.1.  The extended newsvendor model based on a single-stage decision making process    51
5.3.2.  Model derivation   53
5.3.3.  Determining the optimal ordering policy    54
5.4.  Multistage decision making logic   56
5.5.  Numerical example    58
5.6.  Sensitivity analysis   58
Chapter 6.  Conclusions	                             60
6.1.  Results   60
6.2.  Future studies   62
Bibliography	64
Appendix A	71
Appendix B	74



List of Tables
3.1. Crisp optimal solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
3.2. Optimal solutions for fuzzy case ( 2 1 ∆ ≤ ∆ ) . . . . . . . . . . . . . . . . . . . . . . 21
4.1. Optimal solutions for crisp case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 40
4.2. Optimal solutions for fuzzy case ( 2 1 ∆ ≤ ∆ ) . . . . . . . . . . . . . . . . . . . . . . . . 40
4.3. Optimal solutions for fuzzy case ( 1 2 ∆ < ∆ ) . . . . . . . . . . . . . . . . . . . . . . . . 41
5.1. The optimal solutions of the single-stage decision making process under
different k . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
5.2. The effects of changing parameters on the decision variables under
) 2 3 ( ) 1 ( 2
1 2 m k k − > + α α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59


List of Figures
3.1. Probability distribution of D . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
4.1.1. Graph of ) , ; ( *
2 ∆ ∆ θ R (scenario: 1 = θ ) . . . . . . . . . . . . . . . . . . . . . . . . . 42
4.1.2. Graph of ) , ; ( *
2 ∆ ∆ θ R (scenario: 6 . 0 = θ ) . . . . . . . . . . . . . . . . . . . . . . . 42
4.2.1. Graph of ( ) θ θ ); , ; ( *
2
* * ∆ ∆ Γ R (scenario: 1 = θ ) . . . . . . . . . . . . . . . . . . . 43
4.2.2. Graph of ( ) θ θ ); , ; ( *
2
* * ∆ ∆ Γ R (scenario: 6 . 0 = θ ) . . . . . . . . . . . . . . . . . 43
5.1. The decision time point and the ordering timing in a multistage decision
process . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

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