The Single Depot Multiple Set Orienteering Problem

Ravi Kant, Abhishek Mishra and Siddharth Sharma

Department of CSIS, Birla Institute of Technology and Science, Pilani, 333031, India

Keywords:

The Orienteering Problem, The Set Orienteering Problem.

Abstract:

In this article, we present the single Depot multiple Set Orienteering Problem (sDmSOP), a new variant of the

classical Set Orienteering Problem (SOP). A signiﬁcant feature of sDmSOP is the presence of many travelers

who set off from the same depot and return there at the end of their journey. The objective of the problem

is to maximize the proﬁt while remaining within the budget; hence the challenge at hand involves searching

multiple paths among the mutually clustered sets for travelers. A set’s proﬁt can be collected with a single

node visit only. Supply-chain management, the bus delivery problem, etc., are just a few examples where

the sDmSOP has proven useful. By simulating the instances of the Generalized Traveling Salesman Problem

(GTSP) using GAMS 37.1.0, we determine the optimal proﬁt for GTSP instances for some small and medium

instances which follow the triangular and symmetric properties. We ﬁnd that the use of multiple travelers

is beneﬁcial for both service providers and customers, as it allows service providers to offer their services

to customers at a lower cost because the service provider gets a signiﬁcant amount of proﬁt using multiple

travelers.

1 INTRODUCTION

Due to their practical implications, NP-hard routing

problems with proﬁts have gained interest. Speciﬁ-

cally, (Archetti and Speranza, 2015) studied arc rout-

ing problems, whereas (Archetti et al., 2014) and (Gu-

nawan et al., 2016) focused on node routing problems.

Proﬁt in Arc routing problems is associated with arcs,

while in node routing problems, it is associated with

nodes. Since the node routing problem has real-world

applications in supply chain management, the bus de-

livery problem, and smart city waste management, it

is more important to research. (Golden et al., 1987)

proposed the Orienteering Problem (OP), which is

now widely known as a classical example of a node

routing problem. The goal of the OP is to maximize

proﬁt by making as many node visits as possible uti-

lizing a ﬁxed budget and a single traveler. There is a

proﬁt to be earned from each node in the OP, but it

can only be earned once.

A new variant of the OP, termed the Set Orien-

teering Problem (SOP), was recently suggested by

(Archetti et al., 2018), in which the nodes are grouped

into mutually exclusive clusters, and the proﬁt asso-

ciated with a cluster can only be achieved if at least

one node is visited by the traveler. The problem is

based on the revenue generated by clusters as opposed

to individual nodes. (Archetti et al., 2018), (P

eni

cka

et al., 2019), and (Carrabs, 2021) proposed the so-

lutions for the SOP by combining the Lin-Kernighan

heuristic given by (Lin and Kernighan, 1973) and the

Tabu-search meta-heuristic given by (Glover and La-

guna, 1998), the Variable Neighbourhood Search, and

the Biased random-key genetic algorithm (BRKGA),

respectively. It is shown that BRKGA achieves better

results in terms of time.

This article focuses on a generalized version of

the Set Orienteering Problem, which we refer to as

the single Depot multiple Set Orienteering Problem

(sDmSOP), in which customers (vertices) are grouped

in mutually disjoint sets, and the proﬁt is associated

with each set. In sDmSOP, multiple travelers be-

gin and end their journey at a ﬁxed depot in order

to collect the maximum proﬁt by visiting as many

sets as possible exactly once within the given budget.

While the research into the SOP is signiﬁcant because

of the practical applications proposed by the afore-

mentioned authors, many real-world situations can-

not be simulated with a single traveler, such as the

well-known bus delivery problem and reliable sup-

ply where more than one distributor is required. It

was discovered that the SOP could be applied to the

supply chain by grouping customers from different

chains together; however, in sDmSOP, we must also

account for the situation in which more than one trav-

eler is required to provide the services to the supply

Kant, R., Mishra, A. and Sharma, S.

The Single Depot Multiple Set Orienteering Problem.

DOI: 10.5220/0011681800003396

In Proceedings of the 12th International Conference on Operations Research and Enterprise Systems (ICORES 2023), pages 175-179

ISBN: 978-989-758-627-9; ISSN: 2184-4372

 2023 by SCITEPRESS – Science and Technology Publications, Lda. Under CC license (CC BY-NC-ND 4.0)

175

chain, and distributors can maximize their proﬁts by

offering these services to the chains that are most con-

venient for their customers to access.

The sDmSOP also has potential use in the well-

known school bus problem, in which the individuals

typically travel in clusters. Through the use of sDm-

SOP, the best route to pick up as many passengers

as possible from a given starting place can be deter-

mined.

This paper is categorized as follows: In section

2, We deﬁne and give a mathematical formulation of

the sDmSOP, the comparative results are discussed in

section 3, and section 4 contains the conclusion of the

paper.

2 PROBLEM DEFINITION

The sDmSOP is a generalization of SOP (Set Orien-

teering problem), so ﬁrst, we give a formal deﬁnition

of SOP.

The SOP can be formalized on an undirected com-

plete graph G(V, E) where V = {v

, ...., v

} is the set

of vertices and E = {e

i j

} is the set of edges, e

i j

de-

ﬁned as an edge between vertices v

and v

, moreover

a cost c

i j

≥ 0, is associated with each edge. The ver-

tices are partitioned into disjoint sets S = {s

, ...., s

}

such that their union contains all vertices of the graph.

The objective of the problem is to gain maximum

proﬁt by visiting the possible number of sets within a

distance constraint B with the predeﬁned starting and

ending depot. The proﬁt from a set can be collected if

only one vertex of a set is visited by a traveler.

In this paper, we propose the single Depot mul-

tiple Set Orienteering Problem (sDmSOP), a general

variant of the SOP.

2.1 Mathematical Formulation

To represent an integer linear programming formula-

tion for sDmSOP, we use some notations, which are

given as follows:

• Let v represent vertices and s represent sets

• t : {1, ..., m} (the different salesman).

• i, j : {1, ..., n} (the list of vertices).

• Assume the depot at vertex 1.

• Let c

i j

represent the edge weights.

• Let P

represent the proﬁt associated with a set S

• Let B represent a Budget.

We can then construct an Integer Linear Program

(ILP) formulation using the decision variables:

• x

ti j

: 1 if traveler t uses the edge (i, j) ∈ E and 0

otherwise.

• y

: 1 if vertex i is visited by traveler t and 0 oth-

erwise.

• z

: 1 if any vertex in set q is visited by traveler t

and 0 otherwise.

• u

i j

: ﬂow variable for the Sub-tour Elimination

Constraints (SECs).

The proposed mathematical formulation of sDmSOP:

maximize

∑

, (1)

subject to:

ti j

, y

, z

∈ {0, 1}, (2)

∑

ti j

i j

≤ B, (3)

∑

t1 j

= m =

∑

t j1

, ∀t, ∀ j, (4)

∑

∈V −{v

}

ti j

= y

t j

∀t, ∀ j, (5)

∑

∈V −{v

}

t ji

= y

t j

∀t, ∀ j, (6)

∑

∈S

= z

∀t, ∀q, (7)

∑

≤ 1 ∀q, (8)

0 ≤ u

i j

≤ (n −m)

∑

t=1

ti j

, ∀t, ∀i, (9)

∑

j∈V

i j

−

∑

j∈V−{1}

∑

t=1

, ∀t, ∀i | i ̸= j, i ∈ V − {1}.

(10)

The objective function (1) maximizes collected

proﬁts from the sets visited, constraints (2) deﬁne the

domain of the variables x

ti j

, y

, andz

. Constraint (3)

ensures that budget B is not exceeded. Constraint (4)

ensures that exactly m travelers start and end at depot

1. Equations (5) and (6) imply that the in-degree is

equal to the out-degree of a vertex except for the de-

pot. Constraint (7) ensures that a set S is visited by a

traveler t if any vertex in the set is visited and at most

one vertex can be visited per set. Constraint (8) im-

plies that no set can be visited by more than one trav-

elers while equations (9) and (10) are used to remove

the Sub-tours in the path. The Sub-tour Elimination

Constraints (SECs) are based on the Traveling Sales-

man Problem (TSP) model proposed by (Gavish and

ICORES 2023 - 12th International Conference on Operations Research and Enterprise Systems

176

Table 1: Comparison with optimal solutions on small instances with w < 0.5

Instance n t Pg w Opt.

ILP

Sol. Time Gap (%)

11berlin52 52 2 g1 0.2 37 37 28.875 0.00

11berlin52 52 2 g1 0.3 43 43 51.046 0.00

11berlin52 52 2 g1 0.4 47 47 36.172 0.00

11berlin52 52 2 g2 0.2 1729 1729 36.188 0.00

11berlin52 52 2 g2 0.3 2090 2090 105.047 0.00

11berlin52 52 2 g2 0.4 2284 2284 96.828 0.00

11berlin52 52 3 g1 0.2 43 43 50.094 0.00

11berlin52 52 3 g1 0.3 48 48 8452.922 0.00

11berlin52 52 3 g1 0.4 51 51 1.578 0.00

11berlin52 52 3 g2 0.2 2090 2090 117.297 0.00

11berlin52 52 3 g2 0.3 2338 2338 627.5 0.00

11berlin52 52 3 g2 0.4 2508 2508 6.516 0.00

11eil51 51 2 g1 0.2 28 28 34.313 0.00

11eil51 51 2 g1 0.3 39 39 85.703 0.00

11eil51 51 2 g1 0.4 46 46 433.641 0.00

11eil51 51 2 g2 0.2 1376 1376 29.797 0.00

11eil51 51 2 g2 0.3 1911 1911 97.297 0.00

11eil51 51 2 g2 0.4 2272 2272 238.563 0.00

11eil51 51 3 g1 0.2 39 39 85.25 0.00

11eil51 51 3 g1 0.3 48 48 203.828 0.00

11eil51 51 3 g1 0.4 50 50 5.437 0.00

11eil51 51 3 g2 0.2 1911 1911 98.5 0.00

11eil51 51 3 g2 0.3 2421 2421 213.594 0.00

11eil51 51 3 g2 0.4 2475 2475 2.187 0.00

16eil76 76 2 g1 0.2 40 40 1909.547 0.00

16eil76 76 2 g1 0.3 59 59 3125.797 0.00

16eil76 76 2 g1 0.4 70 70 36372.547 0.00

16eil76 76 2 g2 0.2 2144 2144 10217.375 0.00

16eil76 76 2 g2 0.3 3090 3090 3397.734 0.00

16eil76 76 2 g2 0.4 3550 3550 43317.578 0.00

16eil76 76 3 g1 0.2 59 59 3089.171 0.00

16eil76 76 3 g1 0.3 72 72 1916444.594 0.00

16eil76 76 3 g1 0.4 75 75 8.468 0.00

16eil76 76 3 g2 0.2 3090 3090 3318.11 0.00

16eil76 76 3 g2 0.3 3632 3632 383533.094 0.00

16eil76 76 3 g2 0.4 3700 3700 52.172 0.00

Graves, 1978) and assessed for the Asymmetric Trav-

eling Salesman Problem by (

Oncan et al., 2009). In

the above ILP formulation equations (1)-(10) attempt

to ﬁnd out the optimal path with maximization of the

proﬁt using permutation of the sets and the vertices

which are to be visited in the speciﬁc set.

3 COMPARATIVE RESULTS

In this section, we simulate the mathematical model

using GAMS 37.1.0 on GTSP instances and give the

performance of comparative results using multiple

travelers. The simulation is done on the windows 10

platform with an i7-6400 CPU @3.4Ghz processor

with 32GB of RAM.

In section 3.1, we describe how the instances are

generated for sDmSOP, and the simulation results are

shown in section 3.2.

3.1 Test Instances

To analyze the comparative results of the above for-

mulation, the Generalized Traveling Salesman Prob-

lem (GTSP) instances suggested by (Noon, 1988) are

used. The branch and cut method proposed by (Fis-

The Single Depot Multiple Set Orienteering Problem

177

Table 2: Comparison with optimal solutions on medium instances with w < 0.5

Instance n t Pg w Best Possible Sol.

ILP

Sol. Time Gap (%)

21lin105 105 2 g1 0.2 82.817211 49 7200.828 40.83

21lin105 105 2 g1 0.4 104 84 7204.031 19.23

21lin105 105 3 g1 0.2 104 104 306.531 0.00

21lin105 105 3 g1 0.4 93 75 7200.812 19.35

30ch150 150 2 g1 0.2 94.106272 43 7201.218 54.3

30ch150 150 2 g1 0.4 139 110 7201.36 26.16

30ch150 150 3 g1 0.2 125.32 80 7201.687 36.16

30ch150 150 3 g1 0.4 149 140 7202.359 6.04

40d198 198 2 g1 0.2 162.32 50 7202.766 69.19

40d198 198 2 g1 0.4 197 169 7203.609 14.21

40d198 198 3 g1 0.2 197 130 7201.719 34.01

40d198 198 3 g1 0.4 19 171 7206.047 13.19

chetti et al., 1997) for the Symmetric Generalized

Traveling Salesman Problem was used for executing

the formulation. We modiﬁed the GTSP for the single

depots for the need of our formulation as follows:

1. Transfer the depot vertex from the non-depot set

to the depot set.

2. Sort the non-depot sets in ascending order of the

number of vertices in the set.

3. Iterate over the list, and if there is an empty set,

ﬁnd the ﬁrst vertex from a non-empty set with a

size greater than one and put it into the empty set

found.

This algorithm generates sets that satisfy the con-

straints of our problem.

The proﬁt is assigned to the sets using g

and g

schemes used by (P

eni

cka et al., 2019). In the g

scheme, the proﬁt of a set is equal to the number of

vertices. Whereas in the g

scheme for each vertex

numbered k, the proﬁt assigned is (1 + 714 × k)mod

100, and the proﬁt assigned to a set is the sum of prof-

its of the vertices in the set. In each case, the depot

set is assigned a proﬁt of 0.

3.2 Computational Results

It is not feasible to solve the large instances of GTSP

using GAMS 37.1.0, so we put the following criteria

for Table 1 and Table 2 for the simulation:

1. Table 1: threads = 1 on small GTSP instances

2. Table 2: threads = 1 with two hours of CPU time

on large instances.

here threads=1 means the system is using only one

thread to solve the instance. Results are presented in

Table 1 and Table 2. The organization of Table 1 and

Table 2 are as follows: The ﬁrst ﬁve columns rep-

resent the GTSP instance name, number of vertices

(n), number of travelers (t), the rule to generate the

proﬁt (P

) and the value of w. Budget (B) is calcu-

lated as w × t × T

max

, where T

max

is the solution of

the GTSP instance, Opt. column represents the best

possible solution given by GAMS, and the last three

columns represent the solution, time, and relative gap

for ILP respectively. There is no CPU time restriction

for Table 1; we run all the instances of table 1 till we

ﬁnd the optimal solution, while each instance of Ta-

ble 2 is run for two hours because if we try to solve

the large instances till we ﬁnd the optimal solution,

GAMS does run out of memory. Only one instance

of 21lin105 is solved optimally with three travelers

by GAMS in 306.531 seconds. The results presented

in Table 1 and Table 2 show that the service provider

gets the better proﬁt for multiple travelers for the same

instance.

4 CONCLUSION

In this paper, we introduced a new variant of the

Set Orienteering Problem, which has a single depot

and multiple travelers with all the travelers associated

with that particular depot; the objective of this prob-

lem is to gain maximum proﬁt out of mutually ex-

clusive sets by using multiple travelers with a ﬁxed

starting and ending depot within a ﬁxed budget B. A

ﬁxed proﬁt is associated with each cluster, calculated

using two rules named g

and g

, and the proﬁt can

only be gained if any traveler visits exactly one node

of a set. The sDmSOP is an extension of the single

depot multiple traveling salesman problem; it has an

application in the supply chain, where a distributor

has one service point from which the distributor can

supply the products to the retailers and can gain the

maximum proﬁt out of it while giving a better price

of the product to the retailers within a given budget.

ICORES 2023 - 12th International Conference on Operations Research and Enterprise Systems

178

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