有什么车辆调度算法的最新研究，比如用强化学习的方法？
https://www.zhihu.com/question/312332312
策略算法工程师之路-图优化算法(一)(二分图&最小费用最大流)
https://zhuanlan.zhihu.com/p/103825713
模拟退火（SA）算法求解容量受限的车辆路径（CVRP）问题
https://zhuanlan.zhihu.com/p/136992849
世界冠军之路：菜鸟车辆路径规划求解引擎研发历程
https://zhuanlan.zhihu.com/p/61939417
VRP相关知识整理
https://www.cnblogs.com/shankun/p/Vehicle_Routing_Problem.html
vrp算法有哪些？
https://www.zhihu.com/question/319111079
GD -> SGD -> ADAM
VRPPD(Vehicle Routing Problem with Pickup and Delivery):具有提货和交付的车辆路径规划问题。有一组车辆需要从仓库或起始点出发，分别访问一系列提货地点以收集货物，然后将货物交付到相应的交付地点，最终返回仓库或结束点。问题的目标通常是最小化总行驶距离、时间或成本，或者在满足一定的约束条件下，找到一组最佳的路线方案。
VRP : vehicle routing problems
CVRP：Capacitated Vehicle Routing Problem，有容量约束的车辆路径问题
SEC： Subtour Elimination Constraints
VRPB：VRP with Backhauls
VRPSPD：VRP with Simultaneous Pickup and Delivery
VRPDDP：VRP with Divisible Deliveries and Pickups
VSP：Vehicle Scheduling Problems
MVCTP：Multi-Vehicle Covering Tour Problem
PDP：Pickup-and-Delivery Problem
DARP：Dial-a-Ride Problem
PVRP：Periodic VRP
IRP： Inventory Routing Problems
SDVRP：Split Delivery VRP
2E-VRP：2-Echelon VRP
PTP：Profitable Tour Problem
CPTP：Capacitated PTP
TOP：Team Orienteering Problem
OP：Orienteering Problem
PCVRP：Prize-Collecting VRP
PCTSP：Prize-Collecting TSP
VRPPC：VRP with Private fleet and Common carrier
MV-TPP：Multiple Vehicle Traveling Purchaser Problem
PPVRP：Pallet-Packing VRP
VRPC：VRP with Compartments
DCVRP：Distance-constrained CVRP
VRPM：VRP with Multiple use of vehicles
MTVRP：Multi-Trip VRP
VRPTW：VRP with Time Windows
VRPSTW：VRP with Soft Time Windows
MDVRP：Multi(ple) Depot VRP
HFVRP：Heterogeneous or mixed Fleet VRP
TTRP：Truck-and-Trailer Routing Problem
VRPTT：VRP with Trailers and Transshipments
VRPMS：VRP with Multiple Synchronization constraints
OVRP：Open VRP
SCVRP：symmetric CVRP
ACVRP：asymmetric CVRP
AP：Assignment Problem
SST：Shortest Spanning Tree
GSEC：Generalized Subtour Elimination Constraints
CCC：Capacity Cut Constraints
SP：Set Partitioning
SC：Set Covering
CG：Column Generation
GrVRP：Graphical Vehicle Routing Problem
问题描述：顶点0表示仓库，N={1,2,…,n}表示客户集，车队集用K={1,2,…,|K|}表示，
q
i
q_i
qi表示第i个客户的需求，为客户集
S
⊆
N
S \subseteq N
S⊆N服务的小车开始于仓库0,只访问客户集S中的点一次，，然后返回到仓库0,用
c
i
j
c_{ij}
cij表示小车从点i到点j的运行成本,Q表示小车的容量
点集
V
=
{
0
}
∪
N
=
{
1
,
2
,
.
.
.
,
n
}
V=\{0\} \cup N = \{1,2,...,n\}
V={0}∪N={1,2,...,n}
无向边集
E
=
{
e
=
{
i
,
j
}
=
{
j
,
i
}
:
i
,
j
∈
V
,
i
≠
j
}
E=\{e=\{i, j\}=\{j, i\}:i, j \in V, i \ne j\}
E={e={i,j}={j,i}:i,j∈V,i=j},
c
i
j
c_{ij}
cij其中
i
,
j
∈
V
i, j \in V
i,j∈V
有向边集
A
=
{
(
i
,
j
)
∈
V
×
V
:
i
≠
j
}
A=\{(i, j) \in V \times V: i \ne j\}
A={(i,j)∈V×V:i=j}
无向图
G
=
(
V
,
E
,
c
i
j
,
q
i
)
G=(V,E, c_{ij}, q_i)
G=(V,E,cij,qi)
有向图
G
=
(
V
,
A
,
c
i
j
,
q
i
)
G=(V,A, c_{ij}, q_i)
G=(V,A,cij,qi)
q
0
=
0
q_0=0
q0=0
路线序列
r
=
(
i
0
,
i
1
,
i
2
,
.
.
.
,
i
s
,
i
s
+
1
)
,
i
0
=
i
s
+
1
=
0
，
S
=
{
i
1
,
i
2
,
.
.
.
,
i
s
}
⊆
N
r=(i_0,i_1,i_2,...,i_s,i_{s+1}),i_0=i_{s+1}=0，S=\{i_1,i_2,...,i_s\} \subseteq N
r=(i0,i1,i2,...,is,is+1),i0=is+1=0，S={i1,i2,...,is}⊆N,路由成本
c
(
r
)
=
∑
p
=
0
s
c
i
p
,
i
p
+
1
c(r)=\sum_{p=0}^s{c_{i_p,i_{p+1}}}
c(r)=∑p=0scip,ip+1,容量约束
q
(
S
)
=
∑
i
∈
S
q
i
≤
Q
q(S)=\sum_{i \in S}q_i \le Q
q(S)=∑i∈Sqi≤Q
无向图割集
δ
(
S
)
=
{
{
i
,
j
}
∈
E
:
i
∈
S
,
j
∉
S
}
\delta(S)=\{\{i,j\} \in E:i \in S, j \notin S\}
δ(S)={{i,j}∈E:i∈S,j∈/S}
有向图割集
δ
−
(
S
)
=
{
{
i
,
j
}
∈
A
,
i
∉
S
,
j
∈
S
}
,
δ
+
(
S
)
=
{
{
i
,
j
}
∈
A
,
i
∈
S
,
j
∉
S
}
\delta^-(S)=\{\{i,j\} \in A, i \notin S, j \in S\}, \delta^+(S)=\{\{i,j\} \in A, i \in S, j \notin S\}
δ−(S)={{i,j}∈A,i∈/S,j∈S},δ+(S)={{i,j}∈A,i∈S,j∈/S}
r
(
S
)
r(S)
r(S)表示服务于子客户集S需要的最小车辆路线数
紧凑形式
目标：
(
1.1
)
m
i
n
i
m
i
z
e
∑
(
i
,
j
)
∈
A
c
i
j
x
i
j
(1.1) minimize \sum_{(i, j) \in A} c_{ij}x_{ij}
(1.1)minimize∑(i,j)∈Acijxij
约束：
(
1.2
)
∑
j
∈
δ
−
(
i
)
x
i
j
=
1
,
∀
i
∈
N
(1.2) \sum_{j \in \delta^-(i)} x_{ij} = 1, \forall i \in N
(1.2)∑j∈δ−(i)xij=1,∀i∈N
∑
i
∈
δ
−
(
j
)
x
i
j
=
1
,
∀
j
∈
N
\sum_{i \in \delta^-(j)} x_{ij} = 1, \forall j \in N
∑i∈δ−(j)xij=1,∀j∈N
(
1.3
)
∑
j
∈
δ
+
(
0
)
x
0
j
=
∣
K
∣
(1.3) \sum_{j \in \delta^+(0)} x_{0j} = |K|
(1.3)∑j∈δ+(0)x0j=∣K∣
(
1.4
)
∑
(
i
,
j
)
∈
δ
+
(
S
)
x
i
j
≥
r
(
S
)
(1.4) \sum_{(i,j) \in \delta^+(S)} x_{ij} \ge r(S)
(1.4)∑(i,j)∈δ+(S)xij≥r(S)
(
1.5
)
x
i
j
∈
{
0
,
1
}
(1.5) x_{ij} \in \{0,1\}
(1.5)xij∈{0,1}
浓缩形式
目标
(
1.1
）
m
i
n
i
m
i
z
e
c
T
x
(1.1） minimize c^T x
(1.1）minimizecTx
约束
(
1.2
)
x
(
δ
+
(
i
)
)
=
1
,
x
(
δ
−
(
j
)
)
=
1
,
∀
i
∈
N
,
∀
j
∈
N
(1.2) x(\delta^+(i)) = 1,x(\delta^-(j)) = 1, \forall i \in N, \forall j \in N
(1.2)x(δ+(i))=1,x(δ−(j))=1,∀i∈N,∀j∈N
(
1.3
)
x
(
δ
+
(
0
)
)
=
∣
K
∣
(1.3) x(\delta^+(0)) = |K|
(1.3)x(δ+(0))=∣K∣
(
1.4
)
x
(
δ
+
(
S
)
)
≥
r
(
S
)
(1.4) x(\delta^+(S)) \ge r(S)
(1.4)x(δ+(S))≥r(S)
(
1.5
)
x
a
∈
{
0
,
1
}
,
a
∈
A
(1.5) x_a \in \{0,1\}, a \in A
(1.5)xa∈{0,1},a∈A
目标
(
1.1
）
m
i
n
i
m
i
z
e
c
T
x
(1.1） minimize c^T x
(1.1）minimizecTx
约束
(
1.2
)
x
(
δ
(
i
)
)
=
2
∀
i
∈
N
(1.2) x(\delta^(i)) = 2 \forall i \in N
(1.2)x(δ(i))=2∀i∈N
(
1.3
)
x
(
δ
(
0
)
)
=
2
∣
K
∣
(1.3) x(\delta^(0)) = 2|K|
(1.3)x(δ(0))=2∣K∣
(
1.4
)
x
(
δ
(
S
)
)
≥
2
r
(
S
)
(1.4) x(\delta^(S)) \ge 2r(S)
(1.4)x(δ(S))≥2r(S)
(
1.5
)
x
e
∈
{
0
,
1
,
2
}
,
∀
e
∈
δ
(
0
)
,
x
e
∈
{
0
,
1
}
∀
e
∈
E
−
δ
(
0
)
(1.5) x_e \in \{0,1,2\}, \forall e \in \delta(0), x_e \in \{0, 1\} \forall e \in E - \delta(0)
(1.5)xe∈{0,1,2},∀e∈δ(0),xe∈{0,1}∀e∈E−δ(0)
● NRP(node routing problem)
● ARP(Arc Routing Problems)
● Delivery and Collection
● Simple Visits and Vehicle Scheduling
● Alternative and Indirect Services
● Point-to-Point Transportation
● Repeated Supply
● Non-split and Split Services
● Combined Shipment and Multi-modal Service
● Routing with Profits and Service Selection
● Dynamic and Stochastic Routing
● Loading
● Route Length
● Multiple Use of Vehicles
● Time Windows and Scheduling Aspects
● Multiple Depot VRP
● Heterogeneous or mixed Fleet VRP
● Routing of Trucks and Trailers
● balancing constraints
● inter-route resource constraints
● Synchronization constraints
(1)Task synchronization
(2) Operation synchronization
(3) Movement synchronization
(4)Load synchronization
(5)Resource synchronization
● Single Objective Optimization
● Hierarchical Objectives
● Multi-criteria Optimization
● Branch-and-Bound
● column generation
● Branch-and-Cut