### Min-Max vs. Min-Sum Vehicle Routing: A Worst

```L. Bertazzi, B. Golden, and X. Wang
Route 2014
Denmark
June 2014
1
Introduction

In the min-sum VRP, the objective is to minimize the total
cost incurred over all the routes

In the min-max VRP, the objective is to minimize the
maximum cost incurred by any one of the routes

Suppose we have computer code that solves the min-sum
VRP, how poorly can it do on the min-max VRP?

Suppose we have computer code that solves the min-max
VRP, how poorly can it do on the min-sum VRP?
2
Introduction
 Applications of the min-max objective
 Disaster relief efforts

Serve all victims as soon as possible
 Computer networks
 Minimize maximum latency between a server and a client
 Balance amount of work among drivers and/or across a time
horizon
3
An Instance of the VRP
The min-max solution
The min-sum solution
Max # vehicles = 2
total cost
 6  2 5  10 .47
min-max cost  3  5  5.24
total cost
 6  3 2  10 .24
min-max cost  4  2 2  6.83
4
Motivation behind our Worst-Case
Study
 Observation: The min-max solution has a slightly
higher (2.2%) total cost, but it has a much smaller
(23.3%) min-max cost
 Also, the routes are better balanced
 Is this always the case?
 What is the worst-case ratio of the cost of the longest
route in the min-sum VRP to the cost of the longest
route in the min-max VRP?
 What is the worst-case ratio of the total cost of the
min-max VRP to the total cost of the min-sum VRP?
5
Variants of the VRP Studied
 Capacitated VRP with infinitely many vehicles (CVRP_INF)
 Capacitated VRP with a finite number of vehicles (CVRP_k)
 Multiple TSP (MTSP_k)
 Service time VRP with a finite number of vehicles (SVRP_k)
6
CVRP_INF
 Capacitated VRP with an infinite number of vehicles


rMM
:
the cost of the longest route of the optimal min-max solution


MS
r
:
the cost of the longest route of the optimal min-sum solution


z MM
:

z
the total cost of the optimal min-max solution

: the total cost of the optimal min-sum solution
MS
 The superscript denotes the variant
7
A Preview of Things to Come
 For each variant, we present worst-case bounds
 In addition, we show instances that demonstrate that
the worst-case bounds are tight
8
CVRP_INF

MS
r

MM
r

# customers: n =1+1/ε
capacity = 1/ε

rMM
2

MS
r
 2  n  21   
1  
9
CVRP_INF
z

MM
z

MS

# customers: n =1+1/ε
capacity = 1/ε
z

MM
 21  1  

zMS
 4  n  2
 5
10
CVRP_k
 Capacitated VRP with at most k vehicles available

k 
k 
k 
k 
rMS
 zMS
 zMM
 krMM
k  k 
 rMS
rMM  k

k 
k 
k 
k 
zMM
 krMM
 krMS
 kzMS
k 
k 
 z MM
z MS
k
11
CVRP_k
k 
k 
k 
rMS rMM  k
rMM  2  2
rMS  2k  k 1
k 
12
CVRP_k
k 
k 
zMM zMS  k
k 
zMM
 2k
k 
zMS
 2  k 1
13
MTSP_k
 Multiple TSP with k vehicles
 The customers just have to be visited
 Exactly k routes have to be defined
M  M 
M 
M 
M 
M 

r
rMM  k
r

z

z

kr
 MS
MS
MS
MM
MM

M 
M 
M 
M 
M  M 
zMM
 krMM
 krMS
 kzMS
 zMM
zMS  k
14
MTSP_k
M 
rMM  2
M 
M 
rMS rMM  k
rMS  2  k  12   
M 
 2k  k  1
15
MTSP_k
M 
zMM
 2k
M 
M 
zMM zMS  k
M 
zMS
 2  2k  1  k  1
 2  3 k  1
16
SVRP_k
 Service time VRP with at most k vehicles
 Cost of a route = travel time + service time
 Routing of the min-sum solution is not affected by
service times
 Routing of the min-max solution may be affected by
service times
 r S   z S   z S   kr S   r S  r S   k
MS
MS
MM
MM
MS
MM
17
SVRP_k
Min-max solution without service
times
Min-max solution with service
times
18
SVRP_k
S 
S 
rMS rMM  k
S 
rMM
 2  2t  2
S 
rMS
 2k  2kt  k 1
19
SVRP _k
S 
S 
S 
S 
S 
S 
z

kr

kr

kz

z
z
 The bound MM
MM
MS
MS
MM
MS  k is
still valid, but no longer tight
 We prove the tight bound
S 
S 
zMM
 kzMS
 k 1T, where T = total service time
in our paper
20
SVRP_k
S 
zMM  kzMS  k 1T
S 
z MM  2k  T
S 
zMS  2   k 1  T
S 
21
A Summary
Ratio of the cost of the
longest route
CVRP_INF
CVRP_k
MTSP_k
SVRP_k

MS
r

MM
r
Ratio of the total cost
 z

MM
z

MS

k 
k 
M 
M 
zMM zMS  k
S 
S 
zMM  kzMS  k 1T
zMS zMM  k
rMS rMM  k
rMS rMM  k
k 
k 
M 
M 
zMM zMS  k
S 
S 
22
Conclusions
 If your true objective is min-max, don’t use the min-
sum solution
 If your true objective is min-sum, don’t use the min-
max solution
23
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