第五章 結論與建議
5.2 未來研究方向
(u L
本研究之演算法經數值範例測試求解後，演算法在 50 個顧客數以下具穩定之求 解品質，且與最佳解皆相當接近。
5.2 未來研究方向
本研究之演算法目前僅著重於解的品質，對於求解時間並未有很大的著墨，僅 設定在合理時間內求解，主要原因為研究目標之設定在於以演算法架構為主，參考 一已知的求解方法，根據問題特質修正解法內容來求解。在各個求解步驟裡，希望 皆採取較簡單且具有一定效果的方式，增加求解效率，並由問題特性著手，觀察求 解的過程，逐步測試多項不同的解題機制，而後定出演算法之最後架構。
本研究在相關參數部分，僅參考文獻之經驗法則，以可以穩定求解之目標為主，
未做深入探討，若能繼續深入搭配更細緻的解題機制，相信會有更佳的求解品質。
本研究演算法尚有許多可持續進行之議題，因此未來研究方向，將有以下幾點：
1. 部分解集空間的調整機制。如將一次調整加入或是刪減一個顧客，可修 改成調整多個；或是遞迴中設定不同 m 值的機制，來保留適合的車輛路 線巡迴組合作為解集合空間。
2. 可以更細緻的方式來修正，如參照挑選集合加入後可增加涵蓋多少原本 不涵蓋的顧客。
3. 參數部分的設定。可再多加嘗試不同的數值，來加快求解，在穩定求解 品質與速度上做權衡。
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文獻注釋
將一個貨物併裝組合視為一個集合，轉換為類似集合涵蓋問題(set covering problems, SCP)的數學規劃模式如下：
i ：貨物之編號，i= 1 ~ n ； I：所有貨物所成的集合。 涵蓋與集合分割問題(Set Partitioning Problem - SPP)之差別。限制式(2)使用不 等式而非等式是因為一般而言求解 SCP 會較 SPP 容易。而且，即使所求得之解違反 貨物僅能被涵蓋一次的限制，仍可於其後輕易藉由刪除併裝組合中重複之貨物，將
貨物修正為僅被涵蓋一次。限制式(3)為一般典型之 SCP 所沒有之限制式，其用途為
前述所求得之拉氏解與可行解之目標值可分別以L(u) 及 UB 來代表。有關拉式 鬆弛法遞迴運算過程中，拉式乘數之修正，則可參考(9)、(10)式之運算(Held &
Karp(1970)，式中t 代表第 t 次遞迴運算，UB 代表的是目前最佳的上限值。
除。於其過程中，所產生之新併裝組合亦將之加入於並裝組合空間之中。
（五）停止機制與演算法流程摘要
一般以拉式鬆弛法求解典型之集合涵蓋問題時，是針對以固定的集合空間，因 此當上下限值相等或差值在容忍範圍內時，即停止求解。但此演算法採用部分併裝 組合空間在遞迴運算的過程中加以調整的方式求解，就上下限值的意義上已經有所 不同，因此求解之停止機制也隨之改變。拉式鬆弛問題的目標式值L(u)，針對該次 迴圈之併裝組合空間，仍是一個有效的下限值，但是因為該併裝組合空間僅包含部 份的併裝組合，其並不是原來貨物併裝問題的有效下限值。至於針對SCP 所得的上
限值 UB，因求解併裝組合空間僅包含部份的併裝組合，因此可視為一個區域最佳
解。由於最佳解必小於或等於區域最佳解，所以此上限值對於原來貨物併裝問題來 說仍是一個有效的上限值。
由此可知，此研究之演算法無法利用上下限值夾擠得到確切的最佳解所在區 間，上下限值UB 值與 L(u)值僅能僅做為下次迴圈參數修正之用。所以在停止機制，
僅能利用「SCP 的解趨於穩定」，或是「當求解次數達到所設定次數」，來做為停止 條件。以後述之數據測試實驗為例，演算法由最初求解至產生近似解的過程，大約 需經過500 次的求解，且隨問題規模的加大，求解次數略有需要增加趨勢。有關整 個演算法之流程，如圖 3-1。
基於航空貨運計費機制的複雜性，航空貨運的併裝問題是一個相當困難的混合 整數規劃問題，處理較大規模的問題時，其求解時間過長並不能符合業界作業的需 求。此研究將該問題轉換成類似集合涵蓋問題的形式，再以拉式鬆弛法為基礎，發 展一遞迴性的啟發式演算法，數值測試之結果非常接近最佳解。
紀玟豪求解併裝問題之演算法流程圖
簡 歷
姓名：曾筠予 學歷：
國立交通大學運輸科技與管理所畢業 民國 94 年 6 月 國立交通大學科技管理輔所畢業 民國 94 年 6 月 國立交通大學運輸科技與管理學系畢業 民國 92 年 6 月 國立交通大學財務工程學程 民國 92 年 6 月 台北市立松山高級中學畢業 民國 88 年 6 月 著作：
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