在公路設計時需考慮自然環境、社會狀態、交通量需求、行車安全性、經濟等方面的因素，以決定公路路線位置、幾何形狀與各項結構物或相關設施的配置。其中，幾何線形設計是公路設計的基礎，公路縱坡度設計更是幾何線形設計中重要的一環。公路縱坡度設計時會一併考慮附屬設施的設置，如在坡度陡峭的路段考慮爬坡車道的設置、在地形變化劇烈之路段考慮橋樑或隧道的興建等。然而，傳統上的公路設計較難系統化的考慮附屬設施設置與縱坡度相互之間的影響，往往需要由有經驗的工程師研擬少數幾個可行方案擇優採行。由於設計工作常需要作繁複、大量的計算，消耗大量的時間與人力，所以提昇幾何線形設計的效率與品質，對於整體公路工程建設具有重大的意義。

在過去的文獻中雖有針對考慮某一特定附屬設施所發展的模式，能以成本為考量做出最經濟的縱坡度設計，但這些模式彼此不完全相容，以至於無法同時考慮多種附屬設施，且若每次要考慮新的附屬設施就必須建構新的模式，則研究將變得沒有效率。因此本研究希望可以發展一套模式，此模式能夠加入佈設各種附屬設施的考量，同時考慮公路縱坡度與附屬設施的設置與否，以求得最佳化設計。

本研究利用混合整數規劃的方法，以總成本最小化為目標建構數學模式，所考慮的總成本為各設計樁成本之線性加總，各設計樁的成本包含挖方、填方、棄土、借土、擋土結構、邊坡處理、用地、附屬設施設置等費用。模式中考慮到的限制，除了符合公路設計規範外，尚需讓模式因為選擇佈設某一附屬設施，而使一些特定變數受到特定的限制，因此本研究建立一個選擇的機制來達到此項功能。在求解方面，是以門檻接受法為核心所發展的分層演算法，求解出公路縱坡度與附屬設施設置的最佳化設計。另外為了提高求解的效率，分別針對減少整數變數與減少混合整數問題求解次數，發展了變樁距求解與兩階段求解兩種求解策略。

最後自行以C語言撰寫程式，搭配線性規劃套裝軟體CPLEX 7.0所提供的函式庫，以求解本模式之問題，經由數個測試例驗證本研究的正確性，以及本研究的求解策略確實能大幅改善求解效率。

Many factors have to be taken into consideration to determine the alignment as well as auxiliary facilities of a highway segment. Some of the major factors are environment, situiation, traffic demand, safety, and economy. Geometric alignment is the base of highway design, and vertical alignment in turn plays a major role in geometric alignment. Auxiliary facilities are often deployed at the same time vertical alignments are designed. For example, climbing lanes can be used at places where the grade is steep, or bridges/tunnels can be deployed in a rolling terrain. However, designing the vertical alignment and the various auxiliary facilities at the same time is hard to be done systematically by engineers. Determining the vertical alignment of a highway remains an art and relies mostly on the judgement of experienced engineers. Due to the complexity of the problem, the highway design work often requires significant amount of labor and budget. Moreover, under time and financial pressure, human engineers often select from a handful of feasible designs without searching for the optimal design systematically. It is thus import to improve the efficiency and quality of vertical alignment design.

Literature in the past have proposed various models to optimize the vertical alignment and auxiliary facility concurrently. However, all these models are developed for one single auxiliary facility, and they are not fully compatible with each other. This makes it hard for the user to take multiple auxiliary facilities into consideration. Besides, each additional auxiliary facility will require one new model.

In this research, we develop a mixed integer programming model that simultaneously solves for an optimal highway vertical alignment and auxiliary facility deployment. Important cost items such as construction, cut/fill, earth borrow/dump, slope retaining structure, slope treatment, and land are taken into consideration. The model also accounts for the costs as well as design criteria of multiple auxiliary facilities. A set of variables and constraints built into the model enables it to optimally select from a set of given available auxiliary facilities, determine the location(s) of each facility, and enable/disable code requirements accordingly. The optimization goal is to minimize total cost. The model can be solved with a two-tiered heuristic. We developed two additional heuristics to further improve the efficiency. The variable station spacing heuristic reduces the number of integer variables, and the two stage heuristic seeks to cut down the number of mixed-integer optimization problem solved. The Threshold Accepting heuristic is center to all these models. Besides, we develop two strategies to improve the efficiency in solving.

The heuristics are implemented in the C language. The code calls the CPLEX 7.0 library as the solution engine. Several computational examples are provided to demonstrate the correctness of the model and efficiency of the heuristics.

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