多個低階決策單位二階層規劃應用之研究－以預算分配為例

Abstract

過去許多學者們為解決當時社會上所發生的一些經濟與管理問題，促使了數學規劃相關理論的發展與突破，而層級結構的規劃模式可以清楚描繪出組織體系之間的決策模式與運作過程，促使多層級規劃的概念逐漸展開。而二階層規劃係為解決二個層級結構間分權式決策的尋優問題，可視為一個多層級規劃的特殊型式，在上階層之決策者稱為領導者，而在下階層之決策者稱為追隨者。通常在真實的環境中，在下階層結構中會出現超過一個以上的追隨者，這種型式的層級結構則被稱為多個低階決策單位之二階層規劃。此種規劃所產生的問題，係上階層領導者的決策不僅會受到這些下階層追隨者的決策影響，同時也會被這些下階層決策單位之間的相互關係所影響。 因此，在本篇論文中，試圖建構一種屬於一個高階決策與多個低階決策系統的二階層預算分配模型，並針對此預算分配模型，分別探討多個低階決策系統之間合作、非合作與部分合作變數所衍生的關連性問題。而本文新設計的二階層預算分配運作機制，係上階層領導者從所有下階層決策單位提出的全部計畫中，挑選出能為組織創造最大價值的計畫，並給予合理的預算支援，惟上階層領導者在尋優的決策過程中，應同時考量並兼顧下階層各單位的良性競爭與均衡發展，以避免資源的錯置或不當浪費。 本論文最後將採取兩個階段來解決上述所建構多個低階決策單位之二階層預算分配的尋優問題。第一階段係上階層領導者運用改良式資料包絡法，從全部的計畫中初步篩選出具有效率的計畫，此為預算分配前之重要決策程序；第二階段則運用灰關聯方法處理多個低階決策單位之間的關係，並發展出一種啟發式演算法，試圖求解有關合作、非合作與部分合作之決策變數所衍生的二階層預算分配問題，以獲得上、下兩個階層所有的決策者均可接受的可行有效解，俾提供上階層領導者作出最適當的決策，而這個新發展的演算法比過去的典型演算方式更為簡單容易。

The bilevel programming (BLP) problem can be viewed as an uncooperative, two-person game in unbalanced economic markets. The BLP problem is a special case of multilevel programming (MLP) problems with a two-level structure. A decision maker at the upper level is known as the leader, and, at the lower level, is known as the follower. Usually, in a real world situation, there is more than one follower in the lower level; this type of the hierarchical structure is called a bilevel multi-follower (BLMF) decision-making model. Therefore, the leader’s decision will be affected not only by the reactions of the followers, but also by the relationships among the followers. In this thesis, the budget allocation model is a bilevel decision-making system with one single upper level decision maker and multiple lower level decision-making units. There are two types of BLMF models for the budget allocation that has been developed; one is a classical module that uses the uncooperative variable, and the other is a new module with partial cooperative variables. In the new bilevel budget allocation models, the upper level chooses the better projects from multiple proposals to maximize the value of the lower level projects and to minimize the ratio of the funding differences among the divisions. The budget distribution problems are solved using two-stage methods. In stage one, a new generalized data envelopment analysis (GDEA), an improvement of data envelopment analysis (DEA), is developed. It is an important procedure of distribution to guarantee the quality of the proposals from the upper level decision maker. In the next stage, the grey relational analysis and a new heuristic algorithm take advantage of solving this problem and present a feasible solution of this particular model. The algorithm is efficient, and solutions are acceptable for real world situations. It is simpler than the classical solution methods are.