指導老師：林燦煌博士學生 : 劉芳怡

A problem generator-solver heuristic for vehicle routing with soft time windows 指導老師：林燦煌博士學生:劉芳怡

目的 • 考慮軟性時間窗的車輛排程問題 • 特別重要的是允許決策者(decision-makers)從logistics and marketing-sales兩邊，藉由適當的contract negotiations在客戶訂單的運送時間上，來決定最小的車隊size。 • 結合nearest-neighbor與problem generator

Problem formulation • 設定兩種群組的變數。 • 第一:the sequence in which vehicles visit customers

第二:當vehicle k 開始active就設為1，且至少要拜訪一個客戶。

(Penalty function) 1.cei、cli is the unit penalty。 2.每一客戶i的開始服務時間。

impose a maximum limit wtmax on the waiting time of a vehicle at any customer, to contain possible high levels of waiting times before customer service begins • aj:customer j開始服務時間。 • ai:customer i開始服務時間。 • si:service time • tij:travel time :if customer j follows customer i in the sequence visited by vehicle k

十 • Overall objective function for the VRPSTW include three components: route cost, vehicle activation cost and time window violation cost. tcij:customer i to customer j的距離乘上travel time tij。 :if customer j follows customer i in the sequence visited by vehicle k。 wk:車輛k活動的成本。 zk:if vehicle k is active。 Pi:customer i time window violation cost。

The effect of time window violations can be expressed in terms of the total average time window deviation per customer • The measure of (6) is critical since it provides an indication of the size of the time window violations.

Solution method • 利用a problem instance and a solution engine，來解VRPSTW問題。 • Problem generator--重複產生軟性時間窗的案例，產生不同客戶數的軟性時間窗案例及可允許的最大時間窗違反。 • Solution engine--利用nearest-neighbor heuristic(NNH)結合penalty function，來當作客戶選擇的標準。

The instance generator engine • the generator selects customer i that has the minimum violation,which is less than a tightness coefficient ε. • the generator engine allows violations for the first┌ nλ/100┐customers and selects customer j, which satisfies the property below, for time window fixing:

The solution engine • 在選擇customer j時有一個最低成本Cj。 • The cost Cjcan be mathematically expressed as: • bd,ba,bu,bp是權重，定義為所有選擇標準在每一metric的相關貢獻度。 • bd+ba+bu+bp=1 • bd, ba,bu, bp≧0. • NNH在implementation時，使用各種bd,ba,bu,bp值下去實驗。

define the last three sub-metrics of (8) as follows: 顧客j可被服務的最晚時間

The heuristic • Algorithm PGSH(problem generator solver heuristic)

使用一個參數γ來增加λ值

Computational results • Three metrics for each data set are reported: (a) the number of vehicles reached by PGSH (problem generator solver heuristic) (b) the percentage of non-violated time windows (TW) for each vehicle fleet size (c) the total average violation of time windows for each vehicle fleet size(TATWD).

7%的時間窗違反只要16輛車。 最佳解的17輛車是沒有時間窗違反。

沒有違反時間窗的百分比 車輛數從後面五個客戶看來PGSH有明顯的達到最佳解，且沒有違反時間窗。

The percentage of non-violated time windows 72 55 ε PGSH Balakrishnan 圖上橫座標55、72分別是文獻中Balakrishnan在16、17vehicles的最好解如果不管ε值，可以觀察到PGSH的解優於Balakrishnan

Original heuristic of Baker and Scaffer for hard case =客戶需求總合/車的容量

結論 • 本篇方法解的engine，是建立在nearest-neighbor heuristic，適當的應用在當時間窗違反(time window violations)時，會有一個懲罰值(penalty)。 • 這特別是為了fleet planning and contract negotiations，因為他可以允許決策者做出最好的權衡在時間窗擴張和車輛數之間。

指導老師：林燦煌博士學生 : 劉芳怡