行政院國家科學委員會專題研究計畫 成果報告
回程取貨車輛路線問題之巨集啟發式解法研究
計畫類別： 個別型計畫
計畫編號： NSC93-2211-E-216-009-
執行期間： 93 年 08 月 01 日至 94 年 07 月 31 日 執行單位： 中華大學運輸科技與物流管理學系
計畫主持人： 卓裕仁 共同主持人： 韓復華
計畫參與人員： 王生德 朱佑旌
報告類型： 精簡報告
處理方式： 本計畫可公開查詢
中 華 民 國 94 年 10 月 17 日
行政院國家科學委員會補助專題研究計畫成果報告
回程取貨車輛路線問題之巨集啟發式解法研究
計畫類別： 個別型計畫 □ 整合型計畫 計畫編號：NSC93－2211－E－216－009－
執行期間： 93 年 8 月 1 日至 94 年 7 月 31 日
計畫主持人：卓裕仁 共同主持人：韓復華
計畫參與人員：王生德、朱佑旌
成果報告類型(依經費核定清單規定繳交)：精簡報告 □完整報告
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□赴國外出差或研習心得報告一份
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□出席國際學術會議心得報告及發表之論文各一份
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□涉及專利或其他智慧財產權，□一年□二年後可公開查詢
執行單位：中華大學運輸科技與物流管理系
摘要
回程取貨車輛路線問題(Vehicle Routing Problem with Backhauls, VRPB)是車輛路線問題 (Vehicle Routing Problem, VRP)的延伸，在物流配送實務上有很高的應用價值。由於 VRPB 屬 於解題複雜度很高的 NP-hard 問題，現有文獻所探討的求解方法尚不多見。
本研究結合門檻接受法(Threshold Accepting, TA)與傳統啟發式方法，設計一套可求解 VRPB 之巨集啟發式解法，並藉由自行產生例題進行測試，以分析此 TA_VRPB 巨集啟發式 解法之解題績效。TA_VRPB 方法之執行架構包括：起始解構建模組、鄰域搜尋模組和門檻接 受模組。在起始解構建模組方面，本研究提出了一種平行式的鄰近點法；鄰域搜尋模組包括：
路線內 2_opt 節線交換、路線間 1_0、1_1 節點交換及 S_S 節線交換等四種方法；門檻接受模 組則設計了三種 TA 執行架構（TA1、TA2、TA3）。
本研究經由修改 Solomon 的 VRPTW 國際標竿例題產生了 81 個 VRPB 測試例題，並以 C++語言撰寫執行 TA_VRPB 的電腦程式，然後透過三階段的實驗設計進行例題測試與數值 結果分析。結果發現：(1) 本研究提出之平行式鄰近點法比傳統的循序式鄰近點法具有更佳 的解題能力；(2) TA2 架構的解題績效優於 TA1 與 TA3 架構；(3) 在目標值方面，平均車輛數 從 9.88 降至 6.74，平均路線距離從 923.82 降至 727.44。由上可知，整個門檻接受法確實能提 高解題精確度且執行時間短，顯示 TA 在處理回程取貨車輛路線問題上具有不錯的應用潛力。
關鍵字：回程取貨車輛路線問題、巨集啟發式解法、門檻接受法、平行式鄰近點法。
Abstract
The Vehicle Routing Problem with Backhauls (VRPB), which simultaneously considers the operations of delivery and pickup, is a variant of the classical Vehicle Routing Problem (VRP).
Successful application of the VRPB to the real-world distribution will improve the performance of logistics.
Due that the computational complexity of VRPB is NP-hard, this paper develops a meta-heuristic approach, which integrates the Threshold Accepting (TA) meta-strategy with the implementation techniques of modified Nearest Neighbor and Exchange procedures, to solve the VRPB. The main framework of the proposed TA_VRPB meta-heuristic consists of initial solution constructor (ISC) module, neighborhood search (NS) module and Threshold Accepting (TA) module.
Thereare81 instances,modified from abank ofSolomon’sVRPTW benchmark instances, generated for testing and analyzing the performance of the proposed meta-heuristics. Three phases of experiments are designed and related parameters are set. Numerical results imply the following conclusions: (1) proposed Parallel Nearest Neighbor procedure is superior to the traditional Sequential Nearest Neighbor procedure; (2) among three TA frameworks, TA2 experiences better performance than others; and (3) the average fleet size is reduced from 9.88 to 6.74 and the average routing distance is diminished from 923.82 to 727.44. In sum, the proposed TA meta-heuristic actually provides an efficient and robust tool for VRPB applications.
Keywords: Vehicle Routing Problem with Backhauls, Meta-heuristics, Threshold Accepting, Parallel Nearest Neighbor.
一、前言
物流配送實務講求作業績效與資源有效利用。以國內某汽車路線貨運公司新竹營業所之 每日營運作業流程為例，在配送與集貨兩項作業中，外務員(兼司機)於上午將到站貨物配送 給客戶後，即直接原車(空車)在外至其他客戶處收取交運貨物，然後再回到營業所將貨物卸 下與分類（王春分等人, 2000）。這種利用回程時取貨的物流配送作業型態最常發生在食品雜 貨業上；對食品雜貨業來說，超級市場和零售店即屬於送貨需求點，而食品雜貨的供應商則 為取貨需求點。美國自 1982 年以來，食品雜貨業由於對回程空車的利用，增加車輛使用率，
估計每年在運輸成本上節省了一千六百億美元（Daniel 等人, 1988）。上述回程取貨型態的物 流配送作業即可應用「回程取貨車輛路線問題 (Vehicle Routing Problem with Backhauls, VRPB)」的數學模式來求解，以提升實務業者的作業效率、降低營運成本。
典型的車輛路線問題(Vehicle Routing Problem, VRP)乃是由同一車種、固定容量的車隊，
從單一場站出發，服務完一群需求量已知的顧客後返回中心場站；目標在使車輛路線的總運 輸成本最小化。而所謂的 VRPB，係屬於 VRP 的一種衍生問題型態。VRPB 將顧客需求點分 為兩部分，一為中心場站到顧客的送貨點(Linehauls)，此一部分是將貨物從中心場站運送給顧 客；另一為顧客到場站的取貨點(Backhauls)，由客戶需求點收取貨物運回中心場站。VRPB 問題假設車輛必須先服務完所有送貨點顧客，然後才能開始服務取貨點顧客，如此可避免重 新整理車廂貨物的排列和浪費時間。
二、研究目的
雖然 VRPB 在實務上有很廣泛的應用，然而現有文獻探討 VRPB 的求解方法尚不多見。
由於 VRPB 屬於解題複雜度很高的 NP-hard 問題，其求解所需時間會隨節點數的遞增而呈指 數性的成長，因此實務應用時多採用啟發式(heuristics)或巨集啟發式(meta-heuristics)方法進行 求解。基於以上認知，本研究之主要目的乃在透過對目前文獻發表的各種 VRPB 求解方法進 行回顧與分析，了解 VRPB 的解題特性與關鍵因素，並發展一套績效良好且適合求解 VRPB 的巨集啟發式演算法。
本研究主要內容包括：(1) 蒐集並回顧有關 VRPB 問題與求解方法之文獻，以了解 VRPB 的問題求解特性與關鍵；(2) 以 Solomon（1983）所提出之 VRPTW 國際標竿例題為基礎進行 修改，以建立 VRPB 測試題庫；(3) 結合改良式的鄰近點(Nearest Neighbor)法、鄰域搜尋 (Neighborhood Search)法、門檻接受(Threshold Accepting, TA)法，設計求解 VRPB 之巨集啟發 式解法架構；(4) 以 C++語言撰寫上述巨集啟發式解法之電腦程式，並進行例題測試，以瞭 解該巨集啟發式方法應用於 VRPB 的解題特性與執行績效。
三、文獻探討
國內外有關 VRPB 各種求解方法之文獻並不多，茲將其重點彙整於表 1。由表 1 可知，
VRPB 求解方法主要可分成數學規劃法與啟發式方法兩部分，但以啟發式方法為主要趨勢，
近年來並朝向使用巨集啟發式方法之應用。此外，為測試各種求解方法之績效，學者多以 VRPTW 國際標竿題庫為基礎，設計出不同的 VRPB 測試例題。而從各文獻方法的測試結果 可發現：VRPB 的求解品質受送貨點顧客與取貨點顧客間的相對位置關係影響甚鉅。
表 1、VRPB 求解方法彙整
年代 作者 使用方法 限制條件 測試例題
1984 Deif & Bodin 。節省法 。車容量限制
。路線長度限制
。取貨比例：
10%、20%、50%
1989 Goetschalckx &
Jacobs-Blecha
。空間填滿曲線法
。K-中位法
。貪心法
。車容量限制 。取貨比例：
25%、50%
1997 Toth & Vigo 。整數線性規劃模式
。拉氏下界
。車容量限制 。取貨比例：
20%、34%、50%
1999 Toth & Vigo 。先分群再排路線
。拉式鬆弛法
。車容量限制 。取貨比例：
20%、34%、50%
2000 Osman & Wassan
。節省插入法
。節省指派法
。禁制搜尋法
。車容量限制 。取貨比例：
25%、50%；
20%、34%、50%
2002 Wade & Salhi
。貪心插入法 。車容量限制 。取貨比例：
25%、50%；
20%、34%、50%
四、研究方法
(一) 巨集啟發式解法架構
本研究採用門檻接受法(TA)結合改良式的鄰近點法與傳統鄰域搜尋法，構建一套巨集啟 發式解法，並將之命名為 TA_VRPB。整個 TA_VRPB 的設計理念在於藉由 TA 可接受劣於現 解之鄰解的機制，以便使搜尋過程能夠脫離局部最佳解的束縛，且 TA 採用確定性的接受法 則，執行方式較為簡單。
整套 TA_VRPB 解題架構係依循 GIDS 法的概念，分為三個程序：第一個程序為起始解 構建模組(ISC module)，第二個程序為強調深度搜尋的鄰域搜尋模組(NS module)，第三個程 序為強調包容與廣度搜尋的門檻接受模組(TA module)。如圖 1 所示，起始解構建程序根據傳 統的循序式鄰近點法(Sequential Nearest Neighbor, SNN)，提出改良的平行式鄰近點法(Parallel NN, PNN)以產生起始可行解；接著進行鄰域搜尋改善程序，分別以路線內交換改善模組和路 線間交換改善模組，以搜尋局部最佳解。其中，1_0_b、1_1_b、S_S_b 分別代表各交換法的 交換策略為最佳改善(best-improvement)策略，1_0_f、1_1_f、S_S_f 代表各交換法的交換策略 為首先改善(first-improvement)策略；最後，門檻接受程序則針對鄰域搜尋模組構建的結果，
使用 TA 演算法進行搜尋以跳脫局部最佳解的束縛，本研究共提出 TA1、TA2、TA3 三種執行 方式。
圖 1、TA_VRPB 解題架構
(1) 起始解構建模組
以往求解 VRP 相關問題的文獻多使用節省法來構建起始解，並以路線成本節省為主要考 量。本研究認為傳統的鄰近點法(Nearest Neighbor, NN)僅以距離為考慮因素，較節省法容易執 行且解題速度快；然而傳統鄰近點法採取循序的方式來構建路線，求得解的精確度較差。有 鑑於此，本研究提出了改良式的鄰近點法來做為起始解的構建方法，將節省法中採用的平行 式構建路線機制納入 NN 法中，稱之為平行式鄰近點法(PNN)。所謂循序的概念意指：一條路 線構建完畢之後再構建另一條新的路線，直到所有需求點皆已納入所有路線為止；而平行的 概念意指：同時構建多條路線，直到所有需求點皆已納入所有路線內為止。
(2) 鄰域搜尋模組
起始解模組將起始解構建完成後，接著進入鄰域搜尋模組，除了採用路線內 2_opt 節線 交換、路線間 1_0 節點交換、1_1 節點交換等方法外，更依據先送貨後取貨的特質提出 S_S(segment-segment)整段交換法。其中，1_0 節點交換模組同時具備減少車輛數和路線成本 的功能；其餘交換模組主要功能皆為降低路線成本。
本研究路線內改善模組採用 2_opt 交換法，交換的策略採用首先改善(first-improve)策略。
在車容量、時間窗及先送貨後取貨的限制下，經由路線內節線交換機制，以達成縮短總路線 成本的目標；在路線間交換的階段，採用 1_0 節點交換法、1_1 節點交換法；和本研究提出 的 S_S 整段交換法。交換的策略則採用最佳改善(best-improve)策略；在車輛容量及先送貨後 貨的限制下，經由不同路線間需求點交換的機制，以達成縮短總路線成本的目標。
(3) 門檻接受模組
本研究在門檻接受模組，以前述之路線間交換法為核心工具，但為節省運算時間，皆採
SNN, PNN
用首先改善之執行方式且不考慮路線內 2_opt 交換法。在控制參數方面，包括起始門檻值(T0)、
門檻數列長度(K)及門檻數列收斂型態。門檻數列在門檻數列長度(K)期間，自起始門檻值(T0)
逐漸收斂到 0，本研究門檻數列收斂型態採用線性遞減門檻數列。為了探討 TA 的改善效率，
本研究設計了三種執行模組，TA1、TA2 的執行過程為串聯的模式，TA3 的執行過程類似並 聯的概念。
(二) 測試例題產生與實驗設計
由文獻回顧可知，大多數的文獻皆根據 Solomon (1983)所建立的 VRPTW 測試題庫為基 礎，先不考慮時間窗之參數限制，然後將需求點分成送貨點顧客及取貨點顧客兩部分，並設 定取貨點顧客的百分比（例如：10%、30%、50%），以產生 VRPB 測試例題。設定取貨點顧 客百分比之目的乃在於分析取貨點顧客比例對求解績效之影響；至於取貨點顧客的選取，則 採隨機選取方式。
Solomon 的 VRPTW 測試例題共有 56 題，依各需求點的散佈型態分成：均勻隨機(R)、
聚落(C)、混合(RC)三類；各類之下又分成兩種型態：車輛容量小、路線最大長度短，車輛容 量大、路線最大長度長。因此，共分成 R1、R2、C1、C2、RC1 及 RC2 等六小類，各小類下 有若干例題，其需求點座標不變，僅各需求點的時間窗有所不同。
本研究亦以 Solomon 的題庫為基礎，挑選 R101、C101、RC101 三個例題，先不考慮時 間窗之參數限制，並分別選擇前 25、50 及 100 個顧客；然後設定取貨點顧客的百分比為 10%、
30%、50%，並採隨機方式選取取貨點顧客，將需求點分成送貨點顧客及取貨點顧客兩部分；
每種情境各產生三題，總共產生了 81 個測試例題。
本研究針對方法與模組之組合方式、控制參數之設定，共設計了三個階段的測試實驗。
實驗一首先針對兩種起始解法(SNN 與 PNN)進行測試，以了解改良式方法之適用性與解題效 果；實驗二根據三種路線間交換法（1_0、1_1、S_S）之特性，設計了兩種鄰域搜尋組合並配 合 PNN 起始解方法，來測試三種 TA 執行方式(TA1、TA2、TA3)的解題績效；實驗三則針對 不同的 TA 控制參數數值範圍進行其敏感度分析。
在 TA 的控制參數方面，有兩個控制參數：起始門檻值(T0)與執行次數(K)，並據以構成
一個門檻數列長度{Tk}。三種 TA 執行方式與參數設定範圍分別為：T0= 0.01、0.05、0.1、0.2、
0.3 與 0.5；TA1 和 TA3 之 K 設為 10，TA2 設為 5。本研究採用的是直線遞減型的門檻數列，
其門檻值比率自 T0逐漸下降至 0，並根據 VRPB 之問題特性，提出下列的門檻數列計算公式：
}
~ 1 ,
1) ) (
( )
{ ( 0 0
for k K
K k T K
R N
X T C
T
k k
(1)
解題績效之分析包括兩項指標：(1) 解題精度：第一目標為車輛數，第二目標為路線成 本；一般常使用與「真確最佳解」或「已知最佳解」之成本誤差(%)為比較基準；但本研究所 產生的 VRPB 例題，並無真確最佳解或已知最佳解，故僅能比較不同階段的改善情形。因此，
本研究將從所有求得的解當中挑選出暫優解當作比較基準；(2)解題速度：一般使用電腦運算 所需的 CPU 時間。由於各電腦的系統、配備、並不相同，所使用的程式語言亦不一致，因此
五、結果與討論
(一) 測試結果分析
(1) 起始解建構模組之測試結果
表 2 為 SNN 與 PNN 兩種方法在所有 81 個測試例題的平均結果比較。由表 2 可知，SNN 之平均車輛數為 7.86 輛，平均路線成本為 1075.65；PNN 所求得的平均車輛數為 9.88 輛，平 均路線成本為 923.82。此測試結果與以往文獻報導關於循序式節省法及平行式節省法之比較 相似，即循序式方法在車輛數目標上表現優於平行式方法，而在路線成本目標方面則以平行 式方法表現較佳。
表 2、起始解建構模組測試結果彙整 指標
方法 平均車輛數 平均路線成本
SNN 7.86 1075.65
PNN 9.88 923.82
總平均 8.87 999.73
(2) 門檻接受模組之測試結果
由於 1_0、1_1、S_S 三種交換法中，僅 1_0 具有可以減少車輛數的功能，因此本研究設 計了兩種鄰域搜尋的執行順序：N1(1_1→S_S→1_0)與 N2(1_0→1_1→S_S)。N1 表示先執行 路線成本改善再執行車輛數改善，而 N2 則表示先執行車輛數改善再執行路線成本改善。門 檻接受模組之測試結果彙整如表 3 所示，其中，改善幅度(%)係以表 2 的 PNN 測試結果平均 值（車輛數 9.88、路線成本 923.82）為基礎進行計算；三種 TA 執行架構的起始門檻值 T0皆 設定為 0.1，TA1 和 TA3 之 K 設為 10，TA2 設為 5。
表 3、門檻接受模組測試結果彙整
PNN_N1 PNN_N2 總平均
門檻接受模組
目標值 改善(%) 目標值 改善(%) 目標值 改善(%)
車輛數 6.74 35.86 8.86 10.34 7.99 19.12
TA1 路線成本 727.44 21.26 878.23 4.93 802.83 13.10
車輛數 8.53 13.70 8.83 10.61 8.68 12.15
TA2 路線成本 866.84 6.17 876.77 5.09 871.80 5.63
車輛數 8.78 11.17 8.86 10.34 8.82 10.76
TA3 路線成本 877.65 5.00 878.11 4.95 877.88 4.97
車輛數 7.88 20.25 8.85 10.43 8.36 15.34
總平均 路線成本 834.95 9.62 877.70 4.99 856.33 7.31
從表 3 之結果可發現：(1) 三種 TA 執行架構中，以 TA1 的解題表現最佳，平均車輛數 為 7.99、平均路線成本為 802.83，皆優於其他兩種架構；(2) 兩種鄰域搜尋組合中，以 N1 組 合的解題表現較佳，平均車輛數為 7.88、平均路線成本為 834.95，皆優於 N2 組合；(3) 所有 模組組合中，以 TA1 搭配 PNN_N1 的解題表現最佳，平均車輛數為 6.74，改善幅度為 35.86%，
平均路線成本為 727.44，改善幅度為 21.26%，績效特別顯著；(4) 就整體績效而言，總平均 車輛數自 9.88 降至 8.36，改善幅度為 15.34%，總平均路線成本自 923.82 降至 856.33，改善 幅度為 7.31%。
(3) TA 控制參數之測試結果
TA 的參數設定包括起始門檻值(T0)與門檻數列長度(K)，表 4 針對 T0設定值之測試結果 進行彙整。從表 4 可知：當 T0值為 0.01 至 0.50 時，其解題績效並無任何改變，顯示 TA_VRPB 的方法架構對控制參數並不敏感，這也顯示 TA_VRPB 是一個穩定的方法。
表 4、起始門檻(T0)參數測試結果彙整
T0 0.01 0.05 0.10 0.20 0.30 0.50
車輛數 6.74 6.74 6.74 6.74 6.74 6.74 路線成本 727.44 727.44 727.44 727.44 727.44 727.44
(二) 結論與建議
本研究嘗試將門檻接受法應用於求解 VRPB，整套啟發式方法包括起始解構建、鄰域搜 尋與門檻接受三大模組。起始解以傳統循序式鄰近點法(SNN)為基礎，據以提出改良的平行 式鄰近點法(PNN)；鄰域搜尋包括，2-opt 節線交換法、1_0 節線交換法、1_1 節線交換法及 S_S 整段交換法，其中 S_S 為本研究所設計之新方法；最後再搭配門檻接受法設計三種不同 的執行模式加強搜尋改善效果。為檢視本研究提出 TA_VRPB 之解題績效，透過修改 Solomon VRPTW 測試題庫產生了 81 個 VRPB 例題並進行求解測試，所得結果彙整如下：
1. 本研究提出之平行式鄰近點法(PNN)在路線成本目標方面表現優於傳統的循序式鄰近點 法(SNN)，但是在車輛數目標方面則不及 SNN；此結果顯示這兩種方法可能具有很高的互 補性。
2. 在鄰域搜尋改善方面，1_0 節點交換法同時具備減少車輛數和降低路線成本的功能；鄰域 搜尋方法的組合，則以先執行 2-opt、1_1、S_S 等僅能改善路線成本的方法後，最後再執 行 1_0 以改善車輛數的順序(N1)為佳。
3. 三種 TA 執行架構中，以 TA1 的改善效果最佳。所有模組組合則以 TA1 搭配 PNN_N1 的 表現最佳，平均車輛數 6.74（改善幅度 35.86%），平均路線成本 727.44（改善幅度 21.26%）。 4. TA_VRPB 的解題績效不受起始門檻值(T0)的數值範圍影響，由此可知，TA 是一個有效、
穩健(Robust)的巨集啟發式方法。
後續的研究方向可考慮：
1. 由於 SNN 與 PNN 具有互補的效果，可考慮利用權重的方式將兩種方法整合成一個起始解 構建法，以擷取其優點。
2. 在車輛數目標的改善方面，本研究只應用了 1_0 節點交換法，略嫌不足，後續研究可參考 多車種車輛路線問題的求解方法，設計其他能改善車輛數的方法，以增進整體求解績效。
3. 本研究僅應用門檻接受法(TA)求解 VRPB，未來可嘗試結合其他的巨集啟發式方法，例如：
禁制搜尋法(Tabu Search, TS)或螞蟻演算法(Ant Colony System, ACS)。
4. 可嘗試將本研究發展的平行式鄰近點法與 TA_VRPB 架構應用在其他複雜的車輛路線問 題，例如：時窗限制回程取貨車輛路線問題(VRPBTW)、同時收送貨車輛路線問題(PDVRP) 等。
參考文獻
王春分、吳柏嫻、林伶芝(2000)，市區內零擔貨物運送路線問題之研究 - 以新竹貨運在新竹 市區之配送為例，畢業專題報告，中華大學交通與物流管理學系畢業專題報告。
申生元(1999)，時窗限制車輛途程問題，交通大學工業工程與管理研究所博士論文。
卓裕仁(2001)，以巨集啟發式方法求解多車種與週期性車輛路線問題之研究，交通大學運輸 工程與管理學系所博士論文。
Christofides,N.& S.Eilon (1969),“An AlgorithmsforVehicleDispatching Problem,”
Operations Research Quarterly, Vol.20, pp.568-589.
Daniel, O., B. Golden & A. Edward (1988),“VehicleRouting with Backhauls:Models,Algorithms, and CaseStudies,”in B.Golden and A.Assad (eds.),
Vehicle Routing: Methods and Studies,
North-Holland, Amsterdam, pp.127-147.Deif, I. & L. Bodin (1984),“Extension ofthe Clarke-Wright algorithms for solving the Vehicle Routing with Backhauling,”in A.Kidder(ed.),
Proceedings of the Babson Conference on Software Uses in Transportation and Logistics Management, Babson Park, MA, pp.75-96.
Dueck, G. & T.Scheuer(1990),“Threshold Accepting:aGeneralPurpose Optimization Algorithm Appearing Superior to Simulated Annealing,” Journal of Computational Physics, Vol.90, pp.161-175.
Dumamel, C., J. Potvin, J.Y. & J. Rousseau (1997), “A tabu search heuristic for the vehicle routing problem with backhauls and time windows,”Transportation Science, Vol. 31, pp.49-59.
Gélinas, S., M. Desrochers, J. Desrosiers & M. Solomon (1995), “A new branching strategy for time constrained routing problems with application to backhauling,”Annals of Operations
Research, Vol. 61, pp.91-109.
Goetschalckx, M. & C. Jocabs-Blencha(1989),“TheVehicleRouting Problem with Backhauls,”
European Journal of Operational Research, Vol.42, pp.39-51.
Kontoravdis, G. & J. Bard (1995), “A GRASP for the vehicle routing problem with time windows,”
ORSA Journal of Computing, Vol. 7, pp.10-23.
Lin,S.(1965),“ComputerSolution oftheTraveling Salesman Problem,”
Bell System Technology Journal, Vol.11, pp.2245-2269.
Osman, I. (1993),“Metastrategy Simulation Annealing and Tabu Search AlgorithmsfortheVehicle Routing and Scheduling Problem with Time Windows,” European Journal of Operational
Research, Vol.41, pp.421-451.
Potvin J., C. Duhamel & F.Guertin (1996),“A GeneticAlgorithms for Vehicle Routing Problem with Backhauling,”
Applied Intelligence, Vol.6, pp.345-355.
Solomon, M. (1983), Vehicle routing and Scheduling with Time Window Constraints: Models and
Algorithms, Ph.D. Dissertation, University of PeSNNylvania.
Thangiah, R., J. Potvin & T.Sun (1996),“HeuristicApproach to Vehicle Routing with Backhauls and TimeWindows,”
Computers and Operations Research, Vol. 23, No. 11, pp.1043-1057.
Toth, P. & D. Vigo (1997), “An Exact Algorithms for the Vehicle Routing Problem with Backhauls,”
Transportation Science, Vol.31, No.4, pp.372-385.
Toth, P. & D. Vigo (1999),“A HeuristicAlgorithm fortheSymmetricand Asymmetric Vehicle Routing Problem with Backhauls,” European Journal of Operational Research, Vol.113, pp.528-543.
Wade, A. & S. Salhi (2002),“An investigation into anew classofvehiclerouting problem with backhauls,”
Omega, Vol.30, pp.479-487.
Yano, T., L. Chan & K. Richter (1987),“VehicleRouting atQuality Stores,”
Interfaces, Vol.17,
No.2, pp.52-63.計畫成果自評
一、本計畫之研究內容及項目，皆與原計畫書之內容一致，相符程度達 100%，並能達成預期 目標。
原計畫書內容 成果報告內容
研 究 目 的
發展一套績效良好且適合求解 VRPB 的巨 集啟發式演算法，並透過對目前文獻發表 的各種 VRPB 求解方法進行回顧與分析，
了解 VRPB 的解題特性與關鍵因素
透過對目前文獻發表的各種 VRPB 求解方 法進行回顧與分析，了解 VRPB 的解題特 性與關鍵因素，並發展一套績效良好且適 合求解 VRPB 的巨集啟發式演算法
研 究 內 容
1. 蒐集國內外文獻中已發表的 VRPB 例題 或修改其他 VRP 例題，以建立 VRPB 測 試題庫
2. 結合多起始解建構、深度化包容搜尋、
廣度化擾動搜尋等策略群組，設計求解 VRPB 之巨集啟發式解法架構
3. 以 C 語言在電腦上撰寫演算法程式，並 進行例題測試，藉以瞭解巨集啟發式方 法應用於 VRPB 測試例題的解題特性與 執行績效
1. 蒐集並回顧有關 VRPB 問題與求解方法 之文獻，以了解 VRPB 的問題求解特性 與關鍵
2. 以 Solomon（1983）所提出之 VRPTW 國際標竿例題為基礎進行修改，以建立 VRPB 測試題庫
3. 結合改良式的鄰近點(Nearest Neighbor) 法、鄰域搜尋(Neighborhood Search)法、
門檻接受(Threshold Accepting, TA)法，
設計求解 VRPB 之巨集啟發式解法架構 4. 以 C++語言撰寫上述巨集啟發式解法之 電腦程式，並進行例題測試，以瞭解該 巨集啟發式方法應用於 VRPB 的解題特 性與執行績效
研 究 成 果
1. 完成相關文獻之蒐集與回顧，整理出各 種巨集啟發式方法所運用到的巨集策 略，並了解各巨集策略的解題成效與適 用情形，以強化 GIDS 方法的解題能力 2. 完成 VRPB 問題之特性與解題關鍵因素
分析，並建立對應的巨集策略與求解機 制
3. 完成 VRPB 測試例題建立與實驗設計，
並進行系統化 GIDS 應用測試，分析其 解題績效
4. 完成期末報告之撰寫，並將研究成果發 表於國際或國內學術期刊上
1. 完成相關文獻之蒐集與回顧（詳見本報
告第三節），並發表一篇相關論文（詳見
計畫成果自評第二項）
2. 完成並建立對應 VRPB 問題特性的巨集 策略與 TA_VRPB 求解機制（詳見本報 告第四節），並據以強化 GIDS 方法的解 題能力
3. 完成 VRPB 測試例題建立與實驗設計，
並進行系統化 GIDS 應用測試，分析其 解題績效（詳見本報告第五節）
4. 完成期末報告撰寫，並已發表研究成果 於國際學術期刊上（詳見計畫成果自評 第二項）
二、研究成果之學術或應用價值
本計畫的研究成果將有助於國內從事巨集啟發式方法或車輛路線相關問題求解之研究人 員參考，進而提升相關領域在理論上的解題績效。在實務的應用價值方面，國內的公路路線
貨運業者（大榮貨運、新竹貨運），快遞業者（宅急便、統一速捷），甚至郵局的包裹與快遞
業務，皆可應用本研究發展的回程取貨車輛路線問題模式與求解方法，來改善物流配送作業 之效率、降低物流成本。
對於參與本研究之人員與研究生，可建立其在相關領域的專業知識與技能，並藉由研究 成果的發表與交流，達到知識分享之目的。如下所列，本計畫相關之成果已發表了三篇學術 論文，並有一篇正在投稿中；本計畫的經費與研究內容也協助了一位研究生完成碩士論文，
以及一位正在進行碩士論文的研究生。
(1) Cho, Y.J. and Wang, S.D., “A Threshold Accepting Meta-heuristic for the Vehicle Routing Problem with Backhauls and Time Windows,”Journal of the Eastern Asia Society for
Transportation Studies, Vol.6, pp.3022-3037, 2005.（全文詳見附件）
(2) 卓裕仁、王生德，「時窗限制回程取貨車輛路線問題之巨集啟發式解法設計與測試」，第一
屆台灣作業研究學會學術研討會暨 2004 年科技與管理學術研討會，國立台北科技大學(台 北)，作業研究與管理科學類，188 至 194 頁(光碟)，民國 93 年 11 月。（全文詳見附件）
(3) 卓裕仁、王生德，「回程取貨車輛路線問題(VRPB)之解法回顧」，中華民國運輸學會第十
八屆學術研討會，國立交通大學(新竹)，運輸網路類，編號 1413(光碟)，民國 92 年 12 月。
(4) 卓裕仁、王生德、朱佑旌，「以巨集啟發式方法求解回程取貨車輛相關路線問題之研究」，
運輸計畫季刊（投稿中）。
(5) 王生德，以巨集啟發式方法求解時窗限制回程取貨車輛路線問題(VRPBTW)之研究，中華
大學科技管理研究所運輸科技與物流管理組碩士論文，民國 93 年 6 月。
(6) 朱佑旌，巨集啟發式解法應用於時窗限制多車種回程取貨車輛路線問題之研究，中華大學
運輸科技與物流管理學系暨研究所碩士論文（進行中）。
三、本計畫的研究成果相當適合在學術期刊上發表，但是因尚未開發完整的電腦輔助決策系 統與使用者介面，故不適合申請專利。演算法方面已於學術期刊上發表，屬於個人智慧 財產權，歡迎引用。
四、本研究主要發現車輛路線相關問題的種類相當多，而且隨著限制條件的不同，必須設計 符合其問題特性的求解方法。以本研究的對象 VRPB 而言，由於有「先送貨後取貨」的 順序限制，因此傳統的路線內 2-opt 節線交換即無法發揮效用。此外，若再加上「時間窗」
限制的話，則其鄰域搜尋改善更加困難。這些問題皆需要再投入人力與時間進行更深入 與廣泛的研究，也是本人後續的重要研究課題。
可供推廣之研發成果資料表
□ 可申請專利 可技術移轉 日期：94 年 10 月 15 日
國科會補助計畫
計畫名稱：回程取貨車輛路線問題之巨集啟發式解法研究 計畫主持人：卓裕仁
計畫編號：NSC-93-2211-E-216-009 學門領域：土木工程(交通)
技術/創作名稱
TA_VRPB 巨集啟發式方法發明人/創作人
卓裕仁 中文：TA_VRPB 為一可用於求解 VRPB 問題之巨集啟發式方法，共包含 三個求解程序：起始解構建(ISC)模組、鄰域搜尋(NS)模組、門檻接 受(TA)模組。
技術說明
英文：可利用之產業 及 可開發之產品
1. 可利用之產業：公路路線貨運業、快遞業者、郵局及其他以配 送為主的物流業者。
2. 可開發之產品：物流配送車輛路線派遣之決策支援系統
技術特點
TA_VRPB 採用巨集啟發式方法技術，可以在很快的時間內求得精 確度不錯的近似解。
推廣及運用的價值
可提升業者車輛路線派遣之績效，並有效減少運輸成本。
附件：已發表論文二篇
A THRESHOLD ACCEPTING META-HEURISTIC FOR THE VEHICLE ROUTING PROBLEM WITH BACKHAULS AND TIME WINDOWS
Yuh-Jen CHO Sheng-Der WANG
Assistant Professor Master (In Army)
Department of Transportation Technology Institute of Technology Management and Logistics Management Chung Hua University
Chung Hua University 707, Sec. 2, Wu Fu Rd., Hsin Chu, 707, Sec. 2, Wu Fu Rd., Hsin Chu, 300, Taiwan, R.O.C.
300, Taiwan, R.O.C. Fax: +886-3-5186545
Fax: +886-3-5186545 E-mail: m9103066@chu.edu.tw E-mail: yjcho@chu.edu.tw
Abstract: This paper presents a meta-heuristic, which is based on the Threshold Accepting combined with modified Nearest Neighbor and Exchange procedures, to solve the Vehicle Routing Problem with Backhauls and Time Windows (VRPBTW). The VRPBTW assumes that trucks initially start from the depot, deliver goods to the linehaul customers, successively pickup goods from the backhaul consumers, and finally return to depot. Eighty-one instances are generated to identify the performance of the proposed meta-heuristic. Four experiments are designed and related parameters are set. Numerical results imply the following conclusions: (1) modified Nearest Neighbor procedures are superior to the traditional Nearest Neighbor; (2) among three TA frameworks, TA2 experiences better performance than others;
and (3) the average fleet size is reduced from 18.54 to 15.18 and the average routing time is diminished from 1770.83 to 1267.09. In sum, the proposed TA meta-heuristic actually provides an efficient and robust tool for VRPBTW applications.
Key Words: meta-heuristic, threshold accepting, vehicle routing problem, backhauls, time windows
1. INTRODUCTION
The Vehicle Routing Problem with Backhauls and Time Windows (VRPBTW), which simultaneously considers the operational sequence of delivery and pickup as well as the service time interval of customers, is an extensive variant of the classical Vehicle Routing Problem (VRP). The VRPBTW involves two different subsets of customers known as linehauls and backhauls. A linehaul customer requires a given quantity of goods from a central depot, whereas a given quantity of goods is collected from a backhaul customer and brought back to the depot. Moreover, the backhauls must be visited after the linehauls in each route. Additionally, each customer must be serviced within a specified time interval (or time window). The lower and upper bounds of the time window define the earliest and latest time for the beginning of service at the customer. Hence, a vehicle is not allowed to begin service at a customer location after its time window’s upper bound. Furthermore, a waiting time is incurred if a vehicle reaches a customer before the lower bound. Each customer also has a specified maneuvering time which is the time spent by the vehicle to load or unload the goods.
The total route time of a vehicle is the sum of travel time (which is proportional to the distance traveled), waiting time and maneuvering time. The first objective of the VRPBTW is to minimize the fleet size (i.e., number of vehicles). Then, for the same fleet size, minimize
and the time windows of each customer.
The backhauling operation of trucks comes from practical considerations. Since most vehicles are rear-loaded, it is often inconvenient to rearrange onboard delivery loads to accommodate new pick-up loads. This precedence constraint is frequently adopted in order to reduce empty travel and to save cost for the LTL (Less than Truck-Load) transport carriers and home delivery companies. Examples of vehicle routing problems with backhauls are mostly found in land-based distribution systems (e.g., grocery industry, where linehaul customers are supermarkets and backhaul customers are grocery suppliers). Successful application of the VRPBTW to the real-world cases will improve the performance of distribution. For example, Yano et al. (1987) describe an interesting application for Quality Stores, a Michigan-based retail chain of approximately 40 stores with one centralized distribution center located near Toledo, OH. Quality Stores owns or leases 11 trucks, which are nearly identical with respect to weight and volume of shipments. This fleet is used to deliver goods to stores and pickup goods from nearby suppliers. Quality Stores has found that the savings realized from using its fleet rather than common carriers to backhaul goods from suppliers located near its stores helped it to save US$450,000 in 1986.
Although many different heuristic and exact algorithms may be found in the literature for vehicle routing problems with backhauls (VRPB) or for vehicle routing problems with time windows (VRPTW), only a few recent papers are devoted to the VRPBTW. Due that the computational complexity of VRPBTW is NP-hard (Thangiah et al., 1996), efficient heuristic approaches are eagerly necessary for solving the VRPBTW. In this article, we present a meta-heuristic approach, which integrates the Threshold Accepting (TA) meta-strategy with the implementation techniques of traditional local search algorithms, to solve the VRPBTW.
This paper is organized as follows. Section 2 presents a mathematical definition and formulation of the VRPBTW, then, surveys the existing solution methods of the VRPBTW and the concepts of the TA approach. Section 3 describes the details of TA meta-heuristic for the VRPBTW. Computational results are reported in Section 4 on 81 newly created VRPBTW instances with 100 customers. Finally, Section 5 concludes the paper.
2. PROBLE DEFINITION AND LITERATURE REVIEW 2.1 Mathematical Formulation of VRPBTW
Let G = (N, A) be a complete undirected graph with node set N and arc set A = {(i, j)| i, j N}.
The node set is partitioned into N = {{0}, L, B}, where 0 is the depot, L is the set of linehaul customers, L = {1, …, n}, and B is the set of backhaul customers, B = {n+1, …, n+m}. Each node i N is associated with: (1) a non negative quantity ai or bi of goods to be delivered or picked up; (2) a time window [ei, li], where eiand liare the lower and upper bound of the time window, respectively (and where e0 is the earliest start time of each vehicle from the depot and l0the latest return time); and (3) a maneuvering time sjfor loading or unloading the goods (with s0 = 0). Finally, a symmetric traveling time matrix T = [tij] satisfying the triangle inequality is defined on E.
Given a fleet of identical vehicles, V = {1, …, v}, each with capacity d, the VRPBTW then consists of finding a set of vehicle routes with minimal costs (fleet size and routing time), originating from and terminating at the depot, such that: (1) each vehicle services one route;
(2) each customer node i is visited exactly once; (3) the quantity of goods on-board never exceeds the vehicle capacity; (4) the linehaul customers precede the backhauls on each vehicle route; (5) the start time on each vehicle route is greater than or equal to the time
window’s lower bound at depot; (6) the return time on each vehicle route is less than or equal to the time window’s upper bound at depot; (7) the time of beginning of service at each node i is less than or equal to the time window’s upper bound li; and (8) the time of beginning of service at each vertex i is greater than or equal to the time window’s lower bound ei. The mathematical formulation of VRPBTW, which is modified from Goetschalckx and Jacobs-Belcha (1989), can be described as follows, where xijk = 1 if arc (i, j) is traversed by vehicle k, 0 otherwise; uik= 1 if linehaul customer i is served by vehicle k, 0 otherwise; vik= 1 if backhaul customer i is served by vehicle k, 0 otherwise; ti= real variable to denote the time starting to serve customer i; and T is a extremely large value.
Minimize:
m n
j v
k
x
jk1 1 0 (1)
Minimize:
m
n i
m n
j v
k
t
ijx
ijk0 0 1
(2)
Subject to
11
v
k
u
iki = 1 ~ n
(3)1
1
v
k
v
iki = n+1 ~ n+m
(4)d u a
i ikn i
1
k = 1 ~ v
(5)d v b
i ikm n
n i
1
k = 1 ~ v
(6)
v if j , n n m
n if j
x u
jk ijk jk
m n
i 0 1~
~ 1
0
; k = 1 ~ v (7)
v if i n n m
n if i
x u
ik ijk ik
m n
j
1 ~
~ 0
0
; k = 1 ~ v (8)
1
0 0, 1
n i
m n
n j
x
ijkk = 1 ~ v
(9)T x t
s t
t
j
i
i
ij ( 1
ijk) i, j = 1 ~ n+m; k = 1 ~ v
(10)i i
i
t l
e
i = 1 ~ n+m
(11)1 0
;
0 1
u or
u
k ik i = 0 ~ n; k = 1 ~ v
(12)1 0
;
0
1 v or
v
k
ik i = 0, n+1 ~ n+m; k = 1 ~ v
(13)1 0 or
x
ijk
;t
i 0i, j = 0 ~ n+m; k = 1 ~ v
(14) Equations (1) and (2) are objective functions which minimize the fleet size and the routing time respectively. Constraint sets (3) and (4) separately state that each linehaul customer andindict that the aggregated demand for delivery and pick-up in vehicle k could not exceed the capacity of vehicle. Constraint sets (7) and (8) commonly describe the relationships of flow conservation. Constraint set (9) states that precedence of linehaul and backhaul in each vehicle k, where backhaul customers must be served after linehaul customers. Constraint set (10), which calculates the earliest starting time to serve customer j, guarantees that sub-tour could be breaking. Equation set (11) is the time window constraint of serving customer i.
Constraint sets (12), (13) and (14) define the domain of decision variables respectively.
2.2 Existing Methods for VRPBTW
In this sub-section, we review the exact algorithm, heuristics, and meta-heuristics for solving the VRPBTW. Gélinas et al. (1995) proposed an exact algorithm, based on a column generation technique for solving a set partitioning formulation of the VRPBTW. This algorithm found optimal solutions to different problems, with up to 100 customers, derived from those found in Solomon’s (1983) VRPTW test set.
A parallel insertion heuristic, Push-Forward Insertion Heuristic (PFIH), for the VRPBTW is proposed in Kontoravdis and Bard (1995). This heuristic uses an efficient method for inserting customers into the routes, and was applied to problems with up to 100 customers and 3 vehicles. Thangiah et al. (1996) propose a heuristic based on the insertion procedure of Kontoravdis and Bard (1995) to obtain initial solutions to the VRPBTW. Then, the initial solutions are improved through the application of -interchanges and 2-opt* exchanges, which previously developed for the VRPTW only. This two-phase heuristic was used to solve problems of size 25, 50 and 100 (Gélinas et al., 1995), for which the optimal solution is known in most cases. In addition, the heuristic was tested on 24 newly created problems of size 250 and 500.
Potvin et al. (1996) present a Genetic Algorithm (GA) which is combined with a greedy route construction heuristic. The greedy heuristic inserts the customers one by one into the routes, using a fixed ordering of customers. The genetic algorithm is used to find good orderings for the insertion heuristic. In Duhamel and Potvin (1997), a Tabu Search (TS) meta-heuristic is proposed to solve the VRPBTW. The TS includes a greedy insertion heuristic from Kontoravdis and Bard (1995) for constructing an initial feasible solution, and a tabu search procedure based on three different neighborhood search heuristics: extended 2-opt, Or-opt and Swap. Shen (1999) proposed a two-stage meta-heuristic, RNETS, based on a route neighborhood structure to solve the VRPBTW. In the first stage, a nested parallel method is used to construct initial feasible routes from the lower bound direction. Then, those routes are improved by local search in the stage II.
Table 1 summarizes the previous works. Most researchers adopt the insertion-based heuristics to construct feasible routes firstly. Several methods execute the local search heuristics to improve the initial solution. Meta-heuristics, such as Tabu Search (TS) and Genetic Algorithm (GA), present good performance on solving the VRPBTW. On the other hand, the testing instances almost are modified from the Solomon’s VRPTW benchmark instances (Solomon, 1987). The size of the instances with known exact solutions is up to 100 customers.
2.3 TA Meta-heuristic
The Threshold Accepting (TA), a variant of the well-known Simulated Annealing (SA), was introduced by Dueck and Scheurer (1990). Both TA and SA belong to the generic search methods which would allow the overall search move through some bad solutions when bypass the trap of a local optimum. The essential difference between TA and SA consists of the different acceptance rules. TA adopts a deterministic acceptance criterion to choose a neighbor
solution and does not require the generation of random numbers and exponential functions for which SA does.
Table 1. The Existing Solution Methods for VRPBTW
Year Authors Methods Size of Instances
1995 Kontoravdis and Bard Parallel Insertion (PFIH) 25, 50, 100 1995 Gélinas et al. Column Generation 25, 50, 100 1996 Thangiah et al. Parallel Insertion
Local Search
25, 50, 100, 250, 500 1996 Potvin et al. Greedy Insertion
Genetic Algorithm
25, 50, 100 1997 Duhamel and Potvin Greedy Insertion
Local Search Tabu Search
25, 50, 100
1999 Shen Route Neighborhood Local Search
100
We explain the concept and procedure of TA more precise. TA is an iterative and convergent meta-heuristic, which guides the subordinate heuristic to search. For a cost-minimized problem, without loss of generalization, a feasible solution is as the current configuration, named Xcurrent, with value of cost function C(Xcurrent
).TA firstly generates a non-increasing
series of threshold values {Tk}. Then, TA guides the neighborhood search process moving from the current configuration to a new configuration, said as Xnew, which is not much worse than the incumbent. As shown in Figure 1, TA accepts the new configuration only if C(Xnew) is
less than C(Xcurrent) + T
k, at iteration k. While the kth iteration of neighborhood search completed, TA starts the next iteration with threshold Tk+1. TA repeats the previous procedure until all of the thresholds in the series are adopted.Figure 1. Concept of Threshold Accepting
Even simpler structured than SA, TA has been referenced less often. In Dueck and Scheurer (1990), TA was utilized to solve the Traveling Salesman Problem (TSP) of 442 cities and yielded very-near-to-optimum routes. Han and Cho (2000) presented several modifications on parameters setting and implementation of TA, and applied them to solve TSP and VRP (Vehicle Routing Problem). Moreover, Han and Cho (1999) combined TA with GDA (Great Deluge Algorithm) and FFS (Flip-Flop Search) to form a GIDS (Generic Intensification and
Fleet Size and Mix Vehicle Routing Problem and the Periodic Vehicle Routing Problem (Han and Cho, 2002).
3. TA META-HEURISTIC FOR VRPBTW
In this article, we propose a TA approach for solving the VRPBTW, named as TA_VRPBTW.
The main framework of the proposed TA_VRPBTW meta-heuristic, as shown in Figure 2, consists of Initial Solution Constructor (ISC) module, Local Search (LS) module and Threshold Accepting (TA) module. In the ISC module, we modified the traditional Nearest Neighbor algorithm by considering the influences of time windows and the parallel implementation. In the LS module, we adopt 2_opt intra-route arc exchange, (1_0) and (1_1) inter-route node interchange, and (S_S) inter-route arc exchange procedures. Additionally, three TA implemental structures are designed in the TA module.
Figure 2. Main Framework of TA_VRPBTW
3.1 Initial Solution Construction Module
The ISC module, which is modified from the Nearest Neighbor algorithm originally applied to the traditional VRP, aims to construct an initial solution of the VRPBTW. The original definition of nearest neighbor is really to indict the candidate node with the minimal distance or travel time (Christofides and Eilon, 1969). For the sake of introducing the factor of time window, we extend the concept of “nearest neighbor”to the following two new aspects.
●The first new aspect considers that from a specific node to some candidate node, which has the earliest lower bound of time window, is the nearest neighbor of this specific node.
● The second new aspect thinks that the nearest neighbor of a specific node is some candidate node whose waiting time is the minimal. Here, the waiting time means that the different between the lower bound of time window and the arrival time from the specific node to this candidate.
To distinguish between these aspects, we name the algorithm under original definition of nearest neighbor as NN1_S, the algorithm under the first new aspect as NN2_S, and the algorithm under the second aspect as NN3_S, where “S”means the algorithm sequentially construct routes. On the other hand, NN1_P, NN2_P and NN3_P separately present the algorithms concurrently construct several routes under above corresponding aspects.
Therefore, we totally design six Nearest Neighbor algorithms, i.e., NN1_S, NN2_S, NN3_S, NN1_P, NN2_P and NN3_P for the ISC module.
We explain these aspects more precise by an example shown in Figure 3. Suppose that node A is the last node of the constructing route, and the finished time (ft) to serve node A is 20.
Additionally, nodes B, C and D are candidates, which satisfy the capacity constraint if be
included to the constructing route. According to the original definition of nearest neighbor, we have to select node D as the next node of the constructing route because the travel time (t = 5) leaving from node A is the minimal among all candidates. Under the first new aspect, node C is the nearest neighbor of node A because the lower bound of time window of node C (i.e., 31) is the earliest among candidates. Furthermore, node B, whose waiting time (wt = 0) is the minimal among candidates, is the nearest neighbor of node A under the definition of second new aspect.
Figure 3. An Example to Explain the Aspects of Nearest Neighbor
3.2 Local Search Module
The LS module, which is based on the exchange procedures, aims to improve the initial solution generated by the ISC module. Classical exchange procedures (Lin, 1965), such as 2_opt, (1_0), can also be easily adapted to the VRPBTW. Hence, we adopt four exchange procedures: 2_opt intra-route arc exchange, (1_0) inter-route node exchange, (1_1) inter-route node exchange, and (S_S) inter-route arc exchange. Where, the (S_S) exchange is an extension of the 2_opt* exchange procedure (Thangiah et al., 1996; Duhamel and Potvin, 1997). In order o distinguish 2_opt* from traditional 2_opt, the name of (S_S) is used instead of the extended 2_opt*, where (S-S) means the “segment-to-segment”.
●The 2_opt exchange procedure deletes two nonadjacent arcs on the linehaul-segment (or backhaul-segment) in a route, and re-links the route feasibly by adding two new arcs.
●The (1_0) exchange procedure extracts a linehaul (or backhaul) from one route, and inserts this linehaul (or backhaul) between two linehauls (or backhauls) or the last linehaul and the first backhaul in another route.
●The (1_1) exchange procedure interchanges two nodes (linehaul or backhaul) in two different routes. The new locations for inserting these two nodes must satisfy the precedence constraint of linehaul and backhaul.
● The (S_S) exchange procedure initially breaks two routes, says A and B, into four segments by deleting one arc in each route. Then, the first node on the rear-segment of route A is linked to the last node on the head-segment of route B, and the first node on the rear-segment of route B is linked to the last node on the head-segment of route A. Figure 4 presents the above concept of the (S_S) exchange procedure.
In Thangiah et al. (1996), the 2_opt* exchange only considers that the breakpoint in a route is chosen between the last linehaul and the first backhaul nodes. In Duhamel and Potvin (1997), the 2_opt* exchange considers that the breakpoint must not only occur between the last linehaul and the first backhaul on each route, but also occur between two linehauls or two backhauls on both routes. In this paper, the (S_S) exchange considers more feasible breakpoints as follows. As shown in Figure 4, the route A can be broken between two linehauls (l-l), two backhauls (b-b), or the last linehaul and the first backhaul (l-b). In the first case, the feasible breakpoint in the other route B could occur between (l-l) and (l-b). In the second case, the feasible breakpoint in route B could be chosen between (b-b) and (l-b). Finally, in the (l-b) case, the breakpoint could occur at any location, i.e., (l-l), (b-b) and (l-b), in route B.
Figure 4. Concept of (S_S) Exchange
Once the generation of the neighborhood of the current solution is established, a strategy must be decided for the acceptance of a new solution in the neighborhood. There are two strategies have been adopted: (1) The first-improvement strategy selects the first solution met in the neighborhood that improves the objective value of the current solution; and (2) The best-improvement strategy checks all neighbor solutions and selects the solution that obtains the largest improvement in the objective value of the current solution. The best-improvement strategy is utilized to rule the selection of neighbor solution in the LS module. Moreover, all of the four exchange procedures need to satisfy the capacity and time window constraints of routes, too.
3.3 Threshold Accepting Module
The TA module, which guides the subordinate exchange procedures described in previous subsection, aims to intensify the process of search and escape from the captivity of the local optimum. As described in Subsection 2.2, the TA has to decide the times of iterations (K), and create a decreasing series of threshold values, {Tk}, whose length is equal to K. At each run of iteration, TA guides the local search procedure to accept new neighbors according to a specific threshold value Tk. In this study, we design three processes to implement the TA module, named as TA1, TA2 and TA3 (see Figure 5).
●The TA1 module, as shown in Figure 5(a), sequentially executes a set of LS procedures K times. At each execution of iteration, this set of LS procedures accepts new neighbors according to the same threshold Tk.
●The TA2 module, as shown in Figure 5(b), sequentially executes a set of LS procedures K
times. At each execution of iteration, this set of LS procedures accepts new neighbors according to the same threshold Tk, and finally executes the LS module once.
●The TA3 module, as shown in Figure 5(c), separately executes a set of LS procedures in a specific order. At each execution of LS procedure, which consists of K runs of iteration, the series of threshold values {Tk} is independently adopted to rule the acceptance of new neighbors.
Figure 5. Executive Framework of TA Modules
As mentioned previously, there are two strategies for selecting neighbors, i.e., the first-improvement strategy and the best-improvement strategy. In these TA modules, we choose the first-improvement strategy utilized to the LS procedures. Note that the LS module executed in the TA2 module still adopts the best-improvement strategy.
Furthermore, the decreasing pattern of the threshold series guides the generation of threshold values. As shown in Equation (15), a linear decreasing pattern which refers to the previous study of Han and Cho (2000), is proposed. Where, C(X0) is the routing time of initial solution,
R is the fleet size of initial solution, N is the number of customers, T
0 is a pre-defined initial threshold rate, and K is the times of iteration. The ratio of C(X0) to (N + R) means the average routing time of a pair of customers, which is considered as high relative to the exchange procedure.}
~ 1 ,
1) ) (
( )
{ ( 0 0
for k K
K k T K
R N
X T C
T
k k
(15)
4. EXPERIMENTAL ANALYSIS
There are various different versions and parameters setting which could be designed to implement a meta-heuristic procedure. An important issue is which one is superior and
case is appropriate for identifying the performance of a specific procedure. Therefore, we adopt a series of numerical experiments to analyze the average-case performance of the proposed TA_VRPBTW meta-heuristic procedure.
4.1 Testing Instances Bank
There are 81 instances, modified from a set of Solomon’s VRPTW benchmark instances, generated for testing and analyzing the performance of TA_VRPBTW. Solomon (1983) originally generated a set of 56 VRPTW instances, which are classified into three patterns of customers’geographical spread: randomized (R), clustered (C), and mixed of randomized and clustered (RC). All of the VRPTW instances comprise 100 customer nodes with corresponding (x, y) coordinates, volume of demand, lower bound and upper bound of required time window, and a fixed amount of 10 time units for maneuvering (e.g. unload or load the goods at each customer location). In these instances, the traveling times are equal to the corresponding Euclidean distances. Real, double-precision distances were used (i.e., neither rounding nor truncation took place during the computations). The overall routing time is the sum of traveling time, waiting time, and maneuvering time.
Initially, we chose three VRPTW instances, R101, C101 and RC101, from the Solomon’s benchmark instances. Then, the first 25, 50 and 100 customers were extracted from the original VRPTW instances to individually construct the base of VRPBTW instances in three different sizes. Sequentially, the VRPBTW instances were created by randomly assigning 30, 50 and 70% of backhaul customers in each base. Therefore, 27 kinds of scenarios based on three patterns of geographical spread (R101, C101 and RC101), three sizes of instance (25, 50 and 100 customers), and three ratios of backhaul customers (30, 50 and 70%) are available.
We randomly generated three instances in each scenario, and finally built a bank of 81 VRPBTW instances. Note that the capacity of vehicle for every instance is equal to 400 units.
4.2 Experimental Designs
As described in Section 3, there are three kinds of modules, i.e., ISC, LS and TA combined in the TA_VRPBTW. Where, ISC module first constructs an initial feasible solution, and then, this initial solution is improved by LS and TA modules. The implementation of these three modules is sequential. Hence, four phases of experiments are designed to identify the average-case performance of various modules and overall TA_VRPBTW. The quality of solution is represented in minimizing two objectives of the fleet size and the routing time respectively.
●Experiment I. Six nearest neighbor procedures for ISC module are separately tested on the 81 VRPBTW instances. The purpose of Experiment I is to identify the effects of new proposed aspects of nearest neighbor and parallel construction strategy.
● Experiment II. Based on six initial solutions constructed by ISC module, four exchange-base procedures for LS module are separately tested. In Experiment II, we want to evaluate the performance of individual exchange procedure.
●Experiment III. According to the results of Experiment II, we combine effective exchange procedures to a sequential execution. Experiment III aims to find out the best combination of these exchange procedures.
●Experiment IV. Based on the results of Experiments III, three executive framework of TA module are tested. In Experiment IV, several parameters for implementing TA module are set, and the performance of the overall TA_VRPBTW is finally identified. As shown in
Equation (15), the value of parameters T0and K must be set.
Table 2 summarizes the procedures and parameters setting in four phases of experiments.
Even though the performance of 2_opt is inferior to the other exchange procedures in Experiment II, we design six combinations of (1_0), (1_1) and (S_S), in which 2_opt intra-route exchange procedure is still executed before these three inter-route exchange procedures. Because that TA2 module executes more times of exchange procedures than other TA modules in each run, we set the times of iteration, K = 5 for TA2 and K = 10 for TA1 and TA3 to balance their CPU execution time.
Table 2. Symbols of Procedures and Parameters for Experiments
LS Module Parameters
ISC
Module Separation Combination
TA
Module
T
0K
* NN1_SNN2_S NN3_S NN1_P NN2_P NN3_P
2_opt (1_0) (1_1) (S_S)
2_opt + (1_0) + (1_1) + (S_S) 2_opt + (1_0) + (S_S) + (1_1) 2_opt + (1_1) + (1_0) + (S_S) 2_opt + (1_1) + (S_S) + (1_0) 2_opt + (S_S) + (1_0) + (1_1) 2_opt + (S_S) + (1_1) + (1_0)
TA1 TA2 TA3
0.01 0.05 0.10 0.15 0.20
5 10
* K = 5 is specific to the TA2 module, and K = 10 is set for TA1 and TA3 modules.
4.3 Computational Results
The proposed TA_VRPBTW meta-heuristic was coded by the Borland C++ programming language and tested on the bank of instances mentioned in Subsection 4.1. The numerical experiments were executed on a personal computer with Pentium III 932MHz CPU, 384MB RAM and Windows XP.
Table 3 presents the numerical results of Experiment I which tests the six nearest neighbor procedures, i.e., three aspects of nearest neighbor implemented by sequential or parallel construction strategies, on 81 VRPBTW instances. Among six nearest neighbor procedures, NN2_P (the earliest lower bound of time window and parallel construction) performs superior than other procedures in both of average fleet size (13.98) and average routing time (1427.68).
Dramatically, NN2_S (the earliest lower bound of time window and sequential construction) obtains the worst results in both of objectives; average fleet size is 24.56 and average routing time is 2104.53. Consider that the average performance of three aspects of nearest neighbor, NN3 (the minimal waiting time) generates the least number of vehicles (fleet size is 15.27), and NN1 (the minimal traveling time) expends the least routing time, 1609.10. On the other hand, the parallel construction strategy experiences better results (average fleet size is 17.17, and average routing time is 1655.55) than the sequential construction strategy does (19.80 for fleet size, and 1909.79 for routing time). Summarily, the overall performances of ISC module (six nearest neighbor procedures) are 18.49 vehicles for average fleet size and 1782.67 units for average routing time. It is apparent that the proposed new aspects of nearest neighbor (earliest lower bound of time window and minimal waiting time) and parallel construction strategy are effective for creating good initial solutions.
Table 4 shows the computational results of Experiment II on individually testing four exchange procedures, i.e., 2_opt intra-route exchange, and (1_0), (1_1) and (S_S) inter-route exchanges. All of the exchange procedures are executed to improve the 6 initial solutions constructed by above six nearest neighbor procedures respectively. Judging from Table 4,
(1549.73). Note that (1_0) exchange is the only one procedure which is capable of reducing the vehicles of fleet (percentage of improvement is 7.90%). On the other hand, all of four exchange procedures generate smaller routing time, in the sequence of (1_0), (1_1), (S_S) and 2_opt, than initial solutions. The percentages of improvement in routing time are significantly different between exchange procedures (from 13.07%, 10.55% and 2.70% to 0.84%). Notice that the inferior effect on diminishing routing time of 2_opt is possibly due to the strict limitations of precedence and time window between the customers in the same route. Though that 2_opt performs poorly, we still adopt it in the following two experiments by executed after the ISC module done.
Table 3. Results of Experiment I for Six Nearest Neighbor Procedures Average Fleet Size Average Routing Time Nearest
Neighbor Sequential Parallel Average Sequential Parallel Average NN1 19.55 22.29 20.92 1631.52 1586.68 1609.10 NN2 24.56 13.98 19.27 2104.53 1427.68 1766.11 NN3 15.28 15.25 15.27 1993.33 1952.28 1972.80 Average 19.80 17.17 18.49 1909.79 1655.55 1782.67
St-Dev.† 2.91 182.42
†St-Dev. denotes the standard deviation of the objective values.
Table 4. Results of Experiment II for Individual Exchange Procedure Procedures Average
Fleet Size
Improvement (%)
Average Routing Time
Improvement (%)
ISC Module 18.49 --- 1782.67 ---
2_opt 18.49 0.00% 1767.74 0.84%
(1_0) 17.03 7.90% 1549.73 13.07%
(1_1) 18.49 0.00% 1594.67 10.55%
(S_S) 18.49 0.00% 1734.58 2.70%
In Experiment III, six combinations of (1_0), (1_1) and (S_S) inter-route exchange procedures are tested. Similar to Experiment II, all of the combinations are executed to improve six initial solutions respectively. As shown in Table 5, Combinations (1_1)+(S_S)+(1_0) and (1_1)+(1_0)+(S_S) gain the least average fleet size, 16.58 and 16.60 separately. Additionally, Combination (1_0)+(S_S)+(1_1) obtains the least routing time (1421.45) followed by Combinations (1_1)+(S_S)+(1_0) and (1_1)+(1_0)+(S_S). The total averages of fleet size and routing time are 16.82 and 1446.57, as well as the corresponding average percentages of improvement are 9.06% and 18.85%. Such a result indicts that these exchange procedures of the LS module perform well in routing time than in fleet size.
Moreover, even though the differences of performances between various combinations are insignificant, Combinations (1_1)+(S_S)+(1_0) and (1_1)+(1_0)+(S_S) seem to perform superior to others in both of objectives.
In order to identify the effects of LS module on improving different initial solutions, Table 6 summarizes the performances of ISC module and LS module according to different initial solutions. Each initial solution gains improvements on average fleet size and average routing time in varied degree. Interestingly, inferior initial solutions seem to obtain more
