Traveling Salesman Problem with Hotel Selection: Comparative Study of the Alternative Mathematical Formulations

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Traveling Traveling Salesman Salesman Problem Problem with with Hotel Hotel Selection: Selection: Comparative Comparative Study of the Alternative Mathematical Formulations Study of the Alternative Mathematical Formulations a Başkent a

Cemal Aykut Gencelaa, Barış Keçeciaa* Cemal Aykut Gencel , Barış Keçeci *

University, Department of Industrial Engineering, Ankara, 06790, Turkey. Başkent University, Department of Industrial Engineering, Ankara, 06790, Turkey.

Abstract Abstract Tour scheduling has a central importance in logistics in service-based industries. Many logistics and distribution planning Tour scheduling a central in Traveling logistics inSalesman service-based industries. Many logistics andisdistribution planning problems can be has modeled as a importance variant of the Problem (TSP). One such variation the TSP with Hotel problems can be modeled variant of the Traveling Salesman Problem (TSP). such variation thea TSP withofHotel Selection (TSPHS) in whichasaasalesman needs to visit a group of cities in a way thatOne the cities are visitedisvia number trips Selection (TSPHS) in which salesman needs to visit a group of in in a way thatEach the cities visited number trips whose lengths cannot exceedathe maximum length a salesman cancities cover a day. of theare trips startsvia anda ends at aofhotel. whose lengths cannot exceed the maximum length a salesman can cover in a day. Each of the trips starts and ends at a hotel. Moreover, the hotel location where the salesperson ends one day has to be the same as the starting hotel for the next day. The Moreover, the hotel whereproblems the salesperson one daypractical has to beapplications the same asinthe startingThis hotel for the next day.ofThe TSPHS is one of thelocation node routing and hasends interesting real-life. study constitutes the TSPHS is one the node routing problemswhich and has practical applications in real-life. This study constitutes the comparison of of mathematical formulations areinteresting already exist in the literature with the formulations which haveofnew comparison of mathematical formulations which are already exist in the literature with the formulations which have new adaptations for sub-tour elimination constraints. adaptations for sub-tour elimination constraints. © 2019 The Authors. Published by Elsevier B.V. © 2019 2019 The The Authors. Authors. Published Published by by Elsevier Elsevier B.V. Ltd. © This This is is an an open open access access article article under under the the CC CC BY-NC-ND BY-NC-ND license license (https://creativecommons.org/licenses/by-nc-nd/4.0/) (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an and openpeer access articleunder underthe theresponsibility CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection review of ICPR25 International Scientific &Scientific Advisory&and Organizing committee Peer-review under responsibility of the scientific committee of the ICPR25 International Advisory and Organizing Selection peer review under the responsibility of ICPR25 International Scientific & Advisory and Organizing committee members committeeand members members Keywords: Service-based industries; logistics; traveling salesman problem; hotel selection; mathematical formulations Keywords: Service-based industries; logistics; traveling salesman problem; hotel selection; mathematical formulations

1. Introduction 1. Introduction The TSPHS is originally introduced by Vansteenwegen [1]. It is an extension of the classical TSP. The main The TSPHS is originally introduced by Vansteenwegen [1]. It is an extension of the classical TSP. The main difference between these two problems is, in TSPHS a mobilized entity (salesman) cannot visit all locations difference between these two problems is, in TSPHS a mobilized entity (salesman) cannot visit all locations

* Corresponding author. Tel.: +90-312-2466666-2097; fax: +90-312-2466660. * Corresponding Tel.: +90-312-2466666-2097; fax: +90-312-2466660. E-mail address:author. [email protected] E-mail address: [email protected] 2351-9789 © 2019 The Authors. Published by Elsevier B.V. 2351-9789 © 2019 Thearticle Authors. Published by Elsevier B.V. This is an open access under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) This is an and openpeer access article under CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Selection review under the the responsibility of ICPR25 International Scientific & Advisory and Organizing committee members Selection and peer review under the responsibility of ICPR25 International Scientific & Advisory and Organizing committee members 2351-9789 © 2019 The Authors. Published by Elsevier Ltd. This is an open access article under the CC BY-NC-ND license (https://creativecommons.org/licenses/by-nc-nd/4.0/) Peer-review under responsibility of the scientific committee of the ICPR25 International Scientific & Advisory and Organizing committee members 10.1016/j.promfg.2020.01.270

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(customers) at once without giving any break (break at a hotel location). This is due to the fact that the salesman has limited working hours. This sequence of customers, which starts and ends at a hotel location, is called as “trip”. A “tour” is merged with a sequence of connected trips that covers all customers. The final hotel of a trip must be the initial hotel of the next trip. And the starting point of the first trip and the ending point of the last trip have to be the same hotel location, which eventually means the large tour has to have same initial and final hotels given in advance. The primary objective of the TSPHS is to minimize the number of trips (each has total travel time less than or equal to the maximum time budget), while the secondary objective is to minimize the total travel time of a tour. Fig.1 shows an example solution of the problem.

Customer Hotel Start-End Fig. 1. An example solution of problem.

It is noteworthy to present a number of applications which can be modeled as a TSPHS whereas a salesman needs several days to visit all customers: A multi-day tourist visit to a certain region can be the most straightforward real world example of the problem; a multi-day trip planning problem for truck drivers in which every day trip should start and end on an appropriate parking space where the driver will rest and refuel the truck; a mailman who wants to split his round (tour) into a number of connected sub-rounds (trips) in order to lighten his bag; a routing problem of an electric vehicle whose maximal duration is restricted by the battery charge and at the end of each trip the vehicle has to find a station to recharge. Table 1 provides a brief overview of related literature on TSPHS. Table 1. Literature review for TSPHS. Title

Authors

Year

Model/Method

The travelling salesperson problem with hotel selection [1]

P. Vansteenwegen, W. Souffriau, K. Sörensen

2012

MILP formulation, Two heuristics based on neighborhood search

A memetic algorithm for the travelling salesperson problem with hotel selection [2]

Marco Castro, Kenneth Sorensen, Pieter Vansteenwegen, Peter Goos

2013

MILP formulation, Memetic algorithm

A fast metaheuristic for the travelling salesperson problem with hotel selection [3]

Marco Castro, Kenneth Sörensen, Pieter Vansteenwegen, Peter Goos

2015

Set partitioning formulation, Order-first split second method

A Variable Neighborhood Search Heuristic for the Traveling Salesman Problem with Hotel Selection [4]

Marques M. Sousa, Luiz Satoru Ochi, Igor Machado Coelho, Luciana Brugiolo Gonçalves

2015

Variable neighborhood search based heuristic

Solution of Traveling Salesman Problem with Hotel Selection in the Framework of MILP-Tropical Optimization [5]

Mohammadreza Radmanesh, Manish Kumary, Alireza Nematiz, Mohammad Sarim

2016

MILP-Tropical mathematics formulation, Pure rational and safe floating-point approaches both utilize branch-and-bound

A hybrid dynamic programming and memetic algorithm to the Traveling Salesman Problem with Hotel Selection [6]

Yongliang Lu, Una Benlic, Qinghua Wu

2018

Dynamic programming, Infeasible local search

A complete literature review related with TSPHS can be found in [2] and [6]. The remainder of this paper is organized as follows. Section 2 provides the mathematical definition of the problem. Section 3 presents and details



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the mathematical formulations. Section 4 is dedicated to the computational results and comparisons of the formulations, followed by conclusions in Section 5. 2. Problem definition The TSPHS can be defined on a complete graph G  (V , A) where V is the set of vertices and A  {(k , l ) | k , l  V , k  l} is the set of arcs. There are s available hotels (which are indexed through k  1,..., s ) and n customers (which are indexed through k  ( s  1),..., ( s  n) ) in the set V . Each customer k is assigned a service or visiting time Tk where Tk  0, k  1,..., s . The time ckl needed to travel between locations k and l is known for all ( k , l )  A . The available time for each trip d  1,..., m is limited to C . The goal of TSPHS is to minimize the number of connected trips ( m ) required to visit n customers, and as well as to minimize the total

travel time of the tour, such that a) every trip should start and end in one of the available hotels, b) a trip should start in the hotel where the previous trip ended, c) the starting point of the first trip and the end point of the final trip are assumed to be k  1 (This starting and ending hotel can also be used as an intermediate hotel during the tour), d) the travel time of each trip must not exceed C . Since there is no limit on the number of visits to a hotel, a solution to the TSPHS is not necessarily a single cycle. TSPHS is a generalization of the TSP, and hence it is also in the NP-hard category. Since the TSP is in the heart of many routing problems, TSPHS also shares some similarities with several other routing problems dealing with multiple-route planning; including multiple TSP, vehicle routing problem (VRP), multi-depot and multi-trip VRP and etc. [6] 3. Mathematical formulations In this section we introduce the mathematical models for the TSPHS. The first model proposed for the problem is the one by [1]. It is stated in [1] that the mathematical model of TSPHS with lexicographical ordering of the objectives is non-linear. But however for a fixed number of trips, the TSPHS can be formulated as a mixed-integer linear programming (MILP) formulation. The optimal solution of the TSPHS with lexicographically ordered objectives can then be found by solving this formulation for increasing values of number of trips until a feasible solution is found. This MILP formulation is called as VSS in this paper. The second model is proposed by [2] and thereafter it is called as CVSS. CVSS is the modification of the VSS in two ways: 1) a weighted objective function is used which minimizes both the number of trips and the total distance; and 2) the Miller–Tucker–Zemlin (MTZ) subtour elimination constraints were replaced by the much more efficient Dantzig–Fulkerson–Johnson (DFJ) constraints. But since the DFJ constraints have exponential number of constraints, instead of DFJ, we use the MTZ constraints in CVSS to be able to model and solve it in a conventional solver. Next, we add the sub-tour elimination constraints (proposed for TSP) of Gavish and Graves [7], which is mentioned in [8] to the VSS and CVSS, respectively. And hence we obtain the fourth and fifth formulations called as VSS-GG and CVSS-GG. The last mathematical model is the adaptation of Fox–Gavish–Graves formulation [9], which is also mentioned in [8] to the TSPHS. And the fifth model is called as FGG in this paper. The formulations VSS, CSVG, VSS-GG and CSVG-GG 3 2 4 have O(n ) number of decision variables and O(n ) number of constraints, while the FGG formulation has O(n ) 2 number of decision variables and O(n ) number of constraints.

3.1. VSS formulation The formulation VSS is given below. In this formulation the binary decision variable xkld is equal to 1 in case the customer/hotel pair (k , l ) is visited in trip d and 0 otherwise. The auxiliary variable ui indicates the position of customer i in a trip.

min

m sn sn

 x d 1 k 1 l 1 k l

kld (ckl

 Tl )

(1)

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Subject to, s n

x l 2

1l1

1

sn

x k 2

1

k1m

s sn

 x

 1, d  1,..., m

(4)

kld

 1, d  1,..., m

(5)

s s n

 x sn

 k 1

xkhd 

m s n

 x d 1 k 1

sn

 k 1

kld

xkhd 

sn sn

 x k 1 l 1

(3)

kld

l 1 k 1

k 1 l 1

(2)

sn

x l 1

hl ( d 1) ,

m  2; d  1,..., (m  1); h  1,..., s

 1, l  ( s  1),..., ( s  n) sn

x l 1

kld (ckl

hld ,

d  1,..., m; h  ( s  1),..., ( s  n)

 Tl )  C , d  1,..., m

(6)

(7)

(8)

(9)

m   ui  u j  1  n1  xijd , i  (s  1),...,(s  n); j  (s  1),...,(s  n)  d 1 

(10)

xkld {0,1}, d  1,..., m; k  1,..., ( s  n); l  1,..., ( s  n)

(11)

ui  0, i  ( s  1),..., ( s  n )

(12)



The objective function (1) minimizes the total travel duration. With the help of constraints (2) and (3), the tour starts and ends in the starting hotel 1. Constraints (4) and (5) guarantee each trip starts and ends in one of the available hotels. Constraint (6) ensures that, if a trip ends in a hotel then the next trip starts in the same hotel. With the help of constraint (7) every customer is visited once and constraint (8) enables the trip continuity. Constraint (9) is the trip time restriction and the last constraint (10) prevents the sub-tours. (11) and (12) are the integrality and non-negativity constraints on the decision variables.



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3.2. CSVG formulation The formulation CSVG is given below. In this formulation the binary decision variable x kld and the auxiliary variable ui are the same as defined in VSS formulation. Additionally, yd is the binary variable which is equal to 1 if at least one customer/hotel is visited in trip d .

min

m sn sn



xkld (c kl  Tl )  M

d 1 k 1 l 1 k l

m

y d 1

d

(13)

Subject to (10) – (12) and, s

s

 x

kld

 0, d  1,..., m

(14)

kld

 1, l  ( s  1),..., ( s  n)

(15)

k 1 l 1

m s n

 x d 1 k 1

sn

 k 1

xkhd 

x

s sn

 x

kld

k 1 l 1 k l

sn s

 x

kld

k 1 l 1 k l

sn sn

 x s n l 2

1l1

k 2 sn

 k 1

h  ( s  1),..., ( s  n); d  1,..., m

k1d

(16)

 yd , d  1,..., m

(17)

 yd , d  1,..., m

(18)

 Tl )  C , d  1,..., m

1

sn

x

hld ,

l 1

kld (ckl

k 1 l 1 k l

x

sn

(20)

 yd  y( d 1) , d  1,..., (m  1)

xkhd  yd 

(19)

sn

x l 1

hl ( d 1)

 y( d 1) , h  1,..., s; d  1,..., (m  1)

(21)

(22)

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 k 1

xkhd 

sn

x l 1

hl ( d 1)

 1  y( d 1) , h  1,..., s; d  1,..., (m  1)

(23)

xkld  yd , d  1,..., m; k  1,..., ( s  n); l  1,..., ( s  n); k  l

(24)

yd  y(d 1) , d  1,...,(m  1)

(25)

yd  {0,1}, d  1,..., m

(26)

The objective function (13) lexicographically minimizes the total distance and the number of trips where M in the second term is very big constant. Constraint (14) does not exist in the original paper of Castro et al. [2]. This constraint is added, kindly not modifying rest of the formulation structure, to avoid passing between any hotels inside any trip. Constraint (15) ensures that every customer is visited only once, and constraint (16) ensures the connectivity of each trip. Constraints (17) and (18) guarantee each trip starts and ends in one of the available hotels. Constraint (19) restricts the length of a trip. Constraints (20) and (21) ensure that the tour starts and ends in the starting hotel 1. Constraints (22) and (23) guarantee that, if a trip ends in a hotel then the next trip starts at the same hotel. Constraint (24) holds the if-then condition where in a trip if at least one customer/hotel is visited then the corresponding variable for that trip takes the value 1, which means it is being used. Constraint (25) ensures that the trips are performed on sequential days, starting on day 1. And (26) is the integrality constraint. 3.3. VSS-GG formulation The formulation VSS-GG is given below. In this formulation the sub-tour elimination constraint (10) is replaced with the constraints (27) and (28). The auxiliary decision variable g kl is a continuous flow-type variable which shows the order of visited arc (k , l ) on a trip. min (1) Subject to (2) – (9), (11) and, sn

 k 1 k h

g kh 

g kl  n

sn

g l 1 l h

m

x d 1

kld

hl

 1, h  ( s  1),..., ( s  n)

 1, k  1,..., ( s  n); l  1,..., ( s  n); k  l

g kl  0, k  1,..., ( s  n); l  1,..., ( s  n)

(27)

(28) (29)

Constraint (29) is a flow type sub-tour elimination constraint. (28) is the bounding constraint for the auxiliary variables and g kl is equal to 0 in case the variable x kld is equal to 0. (29) is the non-negativity constraint on the decision variables. 3.4. CSVG-GG formulation The formulation CSVG-GG is given below. In this formulation the sub-tour elimination constraint (10) is replaced with the constraints (27) and (28). The auxiliary decision variable g kl is defined same as in the VSS-GG formulation.

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min (13) Subject to (11), (14) – (29). 3.5. FGG formulation The formulation FGG is given below. In this formulation the binary decision variable xkldu is equal to 1 in case the customer/hotel pair (k , l ) in trip d is visited in order u and 0 otherwise. There are neither auxiliary variable nor sub-tour elimination constraints used in this formulation. With the help of the definition of four indexed binary decision variable, the customers/hotels are directly scheduled on a tour without creating any sub-tour. In fact this formulation is inspired by the machine scheduling problem and it is the adaptation of scheduling model to the TSP. As in the VSS the number of trips is fixed and determined as a parameter.

min

m s  n s n s n

 x

kldu (ckl

d 1 k 1 l 1 u 1 k l

 Tl )

(30)

Subject to, sn

x l 2

1

1l11

(31)

sn sn

 x k  2 u 1

k1mu

1

s sn sn

 x

kldu

l 1 k 1 u 1 s sn sn

 x

kldu

k 1 l 1 u 1

sn sn

 x k 1 u 1



khdu

sn sn

 k 1 u 1

 x k 1 l 1 u 1

 1, d  1,..., m

(34)

sn

kldu

xkhdu 

sn sn sn

(33)

x

m sn sn

d 1 k 1 u 1

 1, d  1,..., m

l 1

 x

(32)

hl ( d 1)1 ,

m  2; d  1,..., (m  1); h  1,..., s

 1, l  ( s  1),..., ( s  n)

sn sn

 x l 1 u 1

kldu (ckl

hldu ,

d  1,..., m; h  ( s  1),..., ( s  n)

 Tl )  C, d  1,..., m

(35)

(36)

(37)

(38)

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 d 1 l 1 u 1 l h

uxhldu 

m sn sn

 ux d 1 k 1 u 1 k h

khdu

 1, h  ( s  1),..., ( s  n)

(39)

xkldu  {0,1}, d  1,..., m; k , l , u  1,..., ( s  n )

(40)

The objective function (30) minimizes the total tour time. With the help of (31) and (32), the tour starts and ends in the starting hotel 1. Constraints (33) and (34) guarantee each trip starts and ends in one of the available hotels. Constraint (35) ensures that, if a trip ends in a hotel then the next trip starts in the same hotel. Constraint (36) provides that every customer is visited once and constraint (37) enables the trip continuity. Constraint (38) restricts the trip time and the (39) orders the (k , l ) pairs sequentially in a trip. (40) is the integrality conditions for the binary decision variables. 4. Computational experiments All of five mathematical formulations are tested on a set of existing benchmark instances (SET1, SET2, SET3 and SET4). All these sets of instances are generated by Vansteenwegen et al. [1] starting from well-known vehicle routing problem and TSP benchmark instances. Preliminary studies show us that SET3 and SET4 are too large to be solved by the mathematical models. Hence, we use only the SET1 and SET2. SET1 contains 16 instances in which six of the instances (each has 100 customers) are created from the capacitated vehicle routing problem with time windows of Solomon [10]. The remaining 10 of the instances (involving between 48 and 288 customers) are created from the multi-depot vehicle routing problem with time windows of Cordeau et al. [11]. SET2 contains 52 smaller instances and is generated using 13 instances of SET1 by using only the first 10, 15, 30 and 40 customers of the original instances. Three instances are not included in SET2 because their optimal solution involves a single trip. Therefore, SET2 contains only 4x13=52 test problems. Mathematical models are coded in OPL and CPLEX 12.8 is used as a solver. In the solver parameter environment the working memory (workmem) is set to 1024 MB, the time limit (tilim) is set to 10800 sec., when the tree memory limit is reached the strategy for the node storage is set to “file and compressed” (nodefileind=3), the number of threads (threads) used up to by the parallel branch-and-cut algorithm is set to 4 and all remaining parameters are left as in their default values. All experiments are run on a personal computer with an Intel Core i7-4770 processor (3.40 GHz) and 8 GB of RAM. Table 2. Results obtained by SET1 instances. Instance Name

Best OBJ

c101 c201 pr01 pr02 pr03 pr04 pr05 pr06 pr07 pr08 pr09 pr10 r101 r201 rc101 rc201

9591.1 9559.9 1412.2 2548.8 3404.2 4215.3 4948.9 5960.9 2070.3 3367.7 4414.9 5932.0 1695.5 1642.8 1673.4 1642.7

VSS CPU (sec.) 10803.29 10807.16 10805.80 -

OBJ 9821.80 1412.20 1643.70 -

CSVG GAP (%) 3.15 1.11 0.71 -

CPU (sec.) 1496.46 10802.37 10807.34 10805.33 -

OBJ 1412.20 2809.00 2078.20 1660.80 -

VSS-GG GAP (%) 0.15 12.08 1.92 1.82 -

CPU (sec.) 582.71 -

OBJ 1412.2 -

CSVG-GG GAP (%) 0.01 -

CPU (sec.) 418.08 10800.34 -

OBJ 1412.2 1758.8 -

GAP (%) 0.15 7.66 -



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Four performance indicators are used to compare the mathematical formulations. The optimal or best integer objective function (OBJ) value found so far within the given time limit, the solution time (CPU) where the optimal\best integer solution obtained, number of optimal solutions (#SOL), the percentage gap (GAP) between the best integer objective function value and the objective of the best node remaining in the branch-and-cut tree, which is calculated by |bestbound-bestinteger|/(1e-10+|bestinteger|). Table 2 shows the results of SET1 instances. In Table 2, the first column is the name of test problem, and the second column is the best known objective function value of the instance in the literature. The consecutive three columns are the CPU, OBJ and GAP values, respectively for the corresponding formulation. Results of the FGG formulation cannot be put in the table since it gives error in all instances and returns no result. From Table 2, it is obvious that all five formulations cannot solve big size (over 100 customers) instances. The GG sub-tour elimination constraints decrease the number of problems solved without any error by both VSS and CSVG formulations. But however, the GG constraints decrease the solution time in one instance. It is hard to give a generalization from these results. VSS cannot solve any instance optimally, while the others can solve only one instance within the 10800 sec time limit. Table 3 shows the results of SET2 instances. Because of the limited paper space, we give SET2 results only in abstract rather than in a whole table. In Table 3, formulations are given in the first column. Second column shows the performance indicators and on the first row as well as the best objective function value in the literature. The consecutive three columns give the maximum, average and minimum of each performance indicator. These values are calculated over 30 of 52 instances where all five formulations generate a result. Table 3. Results obtained by SET2 instances. Formulation

VSS

CSVG

VSS-GG

CSVG-GG

FGG

Performance Indicator

Max.

Best OBJ in Literature

1452.20

664.68

237.50

CPU (sec.)

10800.67

732.44

0.17

OBJ

1452.20

664.68

237.50

GAP (%)

2.47

0.12

0.00

Avg.

Min.

CPU (sec.)

30.97

3.32

0.23

OBJ

1452.20

664.68

237.50

GAP (%)

0.01

0.00

0.00

CPU (sec.)

5.01

0.48

0.09

OBJ

1452.20

664.68

237.50

GAP (%)

0.00

0.00

0.00

CPU (sec.)

132.60

10.20

0.22

OBJ

1452.20

664.68

237.50

GAP (%)

0.01

0.00

0.00

CPU (sec.)

958.52

71.77

0.14

OBJ

1452.30

664.68

237.50

GAP (%)

0.01

0.00

0.00

In Table 3, all five formulations give the same objective function value since we consider the instances where all formulations generate a result. In CPU time, the best performing formulation is the VSS-GG and the worst performing one is the VSS. In GAP percentage, the worst formulation is the VSS and all other formulations give almost near results according to each other. The GG sub-tour elimination constraints decrease the average solution time when VSS model is considered. On the contrary GG constraints negatively affect the average CPU time in CSVG model. Same results can be seen in the GAP value that GG sub-tour elimination constraints significantly reduce the average GAP value in VSS formulation while not in CSVG formulation. GAP results overlap with the CPU time results. Hence, the smaller the GAP value, the tighter the formulation and the shorter the solution time.

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Cemal Aykut Gencel et al. / Procedia Manufacturing 39 (2019) 1699–1708 Cemal Aykut Gencel; Baris Kececi / Procedia Manufacturing 00 (2019) 000–000

It is noteworthy to say that VSS, CSVG, VSS-GG, CSVG-GG and FGG formulations cannot find even an integer feasible solution to 3, 1, 3, 2 and 17 of 52 instances, respectively. Moreover, the number of optimal solutions (#SOL) found by VSS, CSVG, VSS-GG, CSVG-GG and FGG are 43, 46, 46, 46 and 30, respectively. 5. Conclusion We presented a comparison between existing mathematical models in the literature and adapted mathematical formulations for the Traveling Salesman Problem with Hotel Selection (TSPHS), an NP-hard problem that arises from several interesting real-world applications. The strength of a mathematical model depends on its LP (or linear) relaxation. Given two formulations both having a minimization type objective function, it would not be wrong to say that the one with the bigger LP relaxation value is the better. The VSS and CSVG models are in the category of Miller-Tucker-Zemlin (MTZ) based formulations. The VSS-GG and CSVG-GG are single commodity flow formulations. And the FGG is time dependent formulation. Theoretically, the single commodity based formulations are stronger than MTZ based formulations, since the projected polyhedron of the former is a proper subset of the latter. Computational assessments on 2 sets of 68 benchmark instances revealed that CSVG, VSS-GG and CSVG-GG outperform in terms of the number of optimal solutions obtained. The VSS-GG is able to find optimal solutions faster in small-sized instances, while the CSVG-GG is quicker in big-sized instances. For this purpose it would be more appropriate to use VSS-GG formulation where the decision makers need to act quickly relatively in small decision problems. On the other hand, the decision team might prefer to use CSVG-GG in bigger decision problems. CSVG and CSVG-GG formulations can be used for the instances where the number of trips is not known. With this feature, these formulations are more suitable for planning in strategic level where the number of trips in a problem could be considered as a long-term resource. Mathematical formulations can also be used to generate starting feasible solutions for the heuristic, mat-heuristic, meta-heuristic or exact algorithms. In this regard VSS or VSS-GG formulations can be used to generate initial feasible solutions since they give better relaxation values. The bigger problem instances need some powerful solution algorithms to be solved. Clearly, there is scope for further research in this interesting topic. References [1] P. Vansteenwegen, W. Souffriau, K. Sörensen, The travelling salesperson problem with hotel selection, Journal of the Operational Research Society, 63(2) (2012) 207-217. [2] M. Castro, K. Sörensen, P. Vansteenwegen, P. Goos, A memetic algorithm for the travelling salesperson problem with hotel selection, Computers & Operations Research, 40(7) (2013) 1716-1728. [3] M. Castro, K. Sörensen, P. Vansteenwegen, P. Goos, A fast metaheuristic for the travelling salesperson problem with hotel selection, 4OR, 13 (2015) 15-34. [4] M. M. Sousa, L. S. Ochi, I. M. Coelho, L. B. Gonçalves, A Variable Neighborhood Search Heuristic for the Traveling Salesman Problem with Hotel Selection, In: Conferencia Latino Americana de Informatica, (2015) 1-12. [5] M. Radmanesh, M. Kumary, A. Nematiz, M. Sarim, Solution of Traveling Salesman Problem with Hotel Selection in the Framework of MILP-Tropical Optimization, In: American Control Conference, (2016) 5593-5598. [6] Y. Lu, U. Benlic, Q. Wu, A hybrid dynamic programming and memetic algorithm to the Traveling Salesman Problem with Hotel Selection, Computers and Operations Research, 90 (2018) 193-207. [7] B. Gavish, S. C. Graves, The travelling salesman problem and related problems, (1978). [8] T. Öncan, İ. K. Altınel, G. Laporte, A comparative analysis of several asymmetric traveling salesman problem formulations, Computers & Operations Research, 36(3) (2009) 637-654. [9] K. R. Fox, B. Gavish, S. C. Graves, An n-constraint formulation of the (time-dependent) traveling salesman problem, Operations Research, 28(4) (1980) 1018-1021. [10] M. M. Solomon, Algorithms for the vehicle routing and scheduling problems with time window constraints, Operations Research, 35(2) (1987) 254-65. [11] J. F. Cordeau, G. Laporte, A. Mercier, A unified tabu search heuristic for vehicle routing problems with time windows, Journal of the Operational Research Society, 52(8) (2001) 928-36.