為了和亞馬遜電子商務抗衡，沃瑪爾超市想出了一個物流策略：安排他
們的員工在下班以後，送貨物到客戶手上。而這個策略成為我們的研究動
機，將此物流的策略轉換成一個數學問題進行求解。
首先，我們將整個策略轉換成一個路徑規劃的問題，並根據三種不同
的目的擁有不同的目標式。此問題可以辦轉換成如下：在一個無向圖中，
有三種點，分別為代表超市本身的根r、一個代表員工家點的集合S和一個
代表客戶點的U。至於我們的目的，希望找到最多|S|條路徑，使得某種成
本(根據不同目的的成本)最小化。當然的，這jSj條路徑能包含所有在這個
圖上的點；而每一條路近的起始點都是r，然後結束在某個si屬於S。
在第一個模型中，我們討論目標式最小化所有路徑的成本，這樣使得公
司的利益能最大化。而此模型可以被轉換成是一個單出發點多條終端漢米
爾路徑問題，而此問題擁有一個5/3的近似解。
第二個模型，我們客戶的角度出發，滿足客戶的需求感。我們的目標式
定為最小化擁有最大失望值的客戶；失望值為客戶收到貨的實際時間減去
客戶直接到超市取貨的直線時間。首先我們證明此問題即使在樹上也是一
個NP困難的問題，並可轉換成在樹上的公車問題進行求解。
在最後一個模型中，我們從員工的角度出發，讓員工能夠公平地擁有下
班時間。我們將目標式定為最小化擁有最大失望值的員工回家時間。我們
證明此問題依然是個NP困難的問題，並且設計一個11倍的近似演算法。

To compete with Amazon's e-business, WalMart came up with a new
concept of logistics which asks their employees to deliver goods on their way home after work. This motivates us to study the route planning problem with such real applications.
We consider this problem to be a vehicle routing problem from three perspectives. This problem is formulated as follows: Given an edge-weighted graph G = (V;E), V = r and S and U, with one node r in G representing the Walmart retail store and each of the other employee nodes si in S representing the house location of every employee, the goal is to nd at most |S| paths
that traverse all the nodes in U, i.e. all the customer nodes, such that the total shipping cost is minimized, with three different objective functions.
First, on the company side, the objective is to minimize the total distance cost of all the trials; for example, minimizing the total gas consumption. The problem can be simply transformed into SDMTHPP (the single-depot multiple-terminal Hamiltonian path problem), which can be approximated within a 5/3 ratio.
Second, from the perspective of customers, we consider the minimax
regret version of this problem to fulfil the satisfaction of customers. That is, the objective is to minimize the maximum regret of all customers, where the regret of a customer is defined to be the cost of a path to get there minus the shortest distance between the start node and the customer node. We show the NP-hardness result of this vehicle routing problem even in trees, and can be transformed into the School Bus Problem with Regret Minimization in trees.
Finally, from the perspective of employees, we balance every employee's delivery time after their busy daytime work. Thus, the goal is to minimize the difference between the distances of all trials, i.e the minimax regret of all the employees. We also show the NP-hardness result of this vehicle routing problem even in trees, and design an 11-approximation algorithms for solving the problem in trees.

1 Introduction 6
1.1 Related Work . . . . . . . . . . . . . . . . . . . . . 6
1.2 Contribution . . . . . . . . . . . . . . . . . . . . . 8
2 Preliminary 9
2.1 Input and Notation . . . . . . . . . . . . . . . . . . 10
2.2 Problem Denfition . . . . . . . . . . . . . . . . . . 12
2.3 Different Perspectives . . . . . . . . . . . . . . . . 12
3 Path model 13
3.1 Problem Formulation . . . . . . . . . . . . . . . . . . 13
3.2 SDMTHPP . . . . . . . . . . . . . . . . . . . . . . . . 14
4 Regret model 15
4.1 Notation and Observation . . . . . . . . . . . . . .. .15
4.2 Customer Model . . . . . . . . . . . . . . . . . . . . 16
4.2.1 Problem Formulation . . . . . . . . . . . . . . . . .16
4.2.2 Transformation . . . . . . . . . . . . . . . . . . . 18
4.3 Employee Model . . . . . . . . . . . . . . . . . . . . 19
4.3.1 Problem Formulation . . . . . . . . . . . . . . . . .19
4.3.2 Ideas and Challenge . . . . . . . . . . . . . . . . .21
4.3.3 Algorithm . . . . . . . . . . . . . . . . . . . . . .22
4.3.4 Example and Implement . . . . . . . . . . . . . . . .29

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