在求無限期非均質馬可夫決策過程（nonhomogeneous Markov decisinon processes）第一期的的最佳解時，我們通常要將它表示成有限期的動態規劃問題。動態規劃可以用合成函數型式表示，也可以用最常見的線性規劃型式表示。

　　合成函數型式在傳統上是一直被認為「中看而不中用」，動態規劃的教科書中，只有在開場白中，介紹一下這種簡潔、漂亮的數學型式，然後就被完全打入冷宮，認為線性規劃型式才是真正實用、真正能讓電腦去執行求解的型式。在一般期刊的文獻中甚至根本不提這種表示法，而是花大篇篇幅在它所衍生的線性規劃技術上作文章，最典型的例子是Bean, Hopp and Duenyas(1992) 在OR期刊所發表的論文。

　　本文將完全針對這個問題的合成函數型式，討論它的一些性質，我們可以利用這些性質，設計出一個非常簡單、有效率的演算法。

　　Hopp, Bean and Duenyas(1992) formulate a mixed integer program (MIP) to determine whether a finite time horizon is a forecast horizon in a nonhomogeneous Markov decision process(NMDP). Their formula are solved by complex Bender's decomposition In this thesis, we make an examination in details of the contraction property and affine mapping property of NMDP. By these properties we are relieved of the complex MIP formula and Bender's decomposition algorithm. The main contribution of the thesis is to show that it is not necessary to determine the optimal policies by running through the whole feasible solution space of their MIP problem. We only need to check a finite number of vertices at a polyhedral set shaped by the solution of the NMDP. The analysis shows insights into the NMDP and facilitate the prosess in determining the forecast horizon. Furthermore, this NMDP formulation is presented in the form of a simple dynamic function which is different from the linear program presented by Hopp, Bean and Duenyas.