Analysis of the Hiring Cost Impact with a Bi-objective Model for the

Multi-depot Open Location Routing Problem

Joel-Novi Rodríguez-Escoto

, Samuel Nucamendi-Guillén

and Elias Olivares-Benitez

Facultad de Ingeniería, Universidad Panamericana, Álvaro del Portillo 49, Zapopan, México

Keywords: Location-routing Problem, Multi-objective Optimization, Exact Methods, AUGMECON2, Goal

Programming, Heterogeneous Fleets.

Abstract: This paper investigates the effect of the hiring cost over transportation cost and the capacity utilization for the

vehicles used. This analysis is conducted on a multi-depot open location-routing problem. The problem

consists of determining the optimal number of depots to open, as well as the design of the open routes in order

to satisfy the demand for all of the customers while seeking the best trade-off between the total traveling and

opening cost. To solve the problem, we propose a bi-objective mixed-integer linear model, which is solved

using two different approaches: the augmented epsilon constraint 2 (AUGMECON2) method and the

weighting revised multi-choice goal programming (WRMCGP) method. Both approaches are implemented,

solving benchmark instances and comparing the quality of the Pareto fronts in terms of multi-objective metrics.

Accordingly, the results indicate that AUGMECON2 performs better than WRMCGP concerning the quality

of the Pareto Front and the elapsed CPU time, for instances with a homogeneous fleet. However, the

WRMCGP reported the best solution time in the heterogeneous instances. In summary, considering

heterogeneous fleets, the results demonstrate that the hiring cost can be reduced up to 85%, with 73% more

vehicle utilization on average.

1 INTRODUCTION

The vehicle routing problem (VRP) is one of the most

studied problems in Operations Research due to the

diverse applications where it can be implemented. Its

practical relevance and theoretical complexity make

this topic so attractive, as this provides solutions to

several kinds of logistics and transportation problems

(Elshaer & Awad, 2020; Irnich et al., 2014; Kardar et

al., 2011). A particular variant of the VRP, known as

the multi-depot location-routing problem

(MDOLRP), determines the number of depots to

open, the location of those depots, and the routes'

design simultaneously. The problem studied in this

work is inspired by the situation faced by a Mexican

company, which imports material from suppliers in

the USA. The firm agrees with a third-party company

an exclusive contract transport the raw material. This

translates into high costs. The reason for the high cost

comes from the supplier agreement since it requires

https://orcid.org/0000-0002-8687-6809

https://orcid.org/0000-0003-4169-3395

https://orcid.org/0000-0001-7943-3869

to contract a homogeneous fleet. The company is

interested in analyzing the hiring of different

transportation suppliers and considering a

heterogeneous fleet. Given the specific context, the

MDOLRP can be used to solve the problem. The

purpose is to determine the impact of hiring costs over

the total cost. In the single-objective version of this

problem (Nucamendi-Guillén et al., 2020), the

authors focused on minimizing the total incurred cost

expressed as the sum of the traveling and vehicle’s

hiring cost. However, the study did not evaluate the

effect of the hiring cost over the traveling cost. The

purpose of this work is to conduct a bi-objective

analysis to show if reducing the number of vehicles

used forces to create routes that increase the cost

significantly.

The present study follows two objectives: (i) to

solve the bi-objective problem model with exact

methods, and (ii) to solve and compare the

performance of each solution method based on the

Rodríguez-Escoto, J., Nucamendi-Guillén, S. and Olivares-Benitez, E.

Analysis of the Hiring Cost Impact with a Bi-objective Model for the Multi-depot Open Location Routing Problem.

DOI: 10.5220/0010266604250432

In Proceedings of the 10th International Conference on Operations Research and Enterprise Systems (ICORES 2021), pages 425-432

ISBN: 978-989-758-485-5

425

quality of the Pareto fronts, using multi-objective

metrics. In addition, the analysis of the costs between

homogeneous and heterogeneous fleets is conducted.

The remainder of the paper is organized as

follows. Section 2 presents the literature review of the

MDOLRP. Section 3 describes the multi-objective

optimization model, the solution methods, and the

characteristics of the set of instances. Section 4

reports the results obtained, including the comparison

with the multi-objective metrics and the hiring cost

analysis. Finally, the concluding remarks of this work

are presented in section 5.

2 LITERATURE REVIEW

The location-routing problem (LRP) is a

generalization of the VRP in which the optimal

number of open depots and optimal design of the

routes are simultaneously determined (Wu et al.,

2002). As a generalization of the VRP, the LRP is

considered as an NP-hard problem. Because of this, it

is difficult to obtain optimal solutions to large

instances in reasonable computational time, justifying

the use of metaheuristics (Adhi et al., 2019; Rabbani

et al., 2017). Nevertheless, small instances can be

solved with exact methods, with standard

computational capacity and enough time to solve

(Braekers et al., 2016; Ramos et al., 2014).

In the LRP, the selection of the depot represents a

strategical decision, while the design of the routes is

an operational decision. However, additional features

can affect the decision of location and routing, for

instance, to have a limit in the number of depots to

open or when facilities have limited capacity

(Schneider & Drexl, 2017). Finally, the characteristic

of considering open routes denote that the vehicle is

not required to return to the depot after visiting the

last customer, which is common when a third party

executes the distribution since they assume the cost

of the "empty" return (Braekers et al., 2016).

The single objective approach of the location-

routing problem usually requires minimizing a

combination, sometimes weighted, of fixed and

variable costs. Differently, the multi-objective

approach optimizes conflicting objectives, for

instance, minimizing the travel cost and maximizing

the level of service (Drexl & Schneider, 2015;

Tavakkoli-Moghaddam et al., 2010). When the

conflicting objectives belong to different dimensions,

a normalization process is performed before

implementing the model.

One of the methods commonly used is weighted

goal programming (WGP), which has been

previously applied in multi-objective location-routing

problems (Zhao & Verter, 2015; Rabbani et al., 2017;

Asefi et al., 2019; Yousefi et al., 2017).

In the specific case of bi-objective problems, the

Epsilon-constraint method arises as an approach to be

applied in multi-objective routing problems

(Kabadurmus et al., 2019; Toro et al., 2017; Arango

González et al., 2017). An improved version of the ε-

constraint, the augmented epsilon-constraint method

(AUGMECON) has gained researchers' attention to

solve multi-objective routing problems (Wang et al.,

2018; Amini et al., 2019).

The literature review illustrates the tendency to

solve realistic routing problems via exact multi-

objective methods, from the weighting method,

weighting goal programming, to epsilon-constraint

based method. These techniques optimize objectives

that belong to different nature and in quantity, even if

they are conflicting. Even when, in general, the WGP

and its variations are frequently used to solve multi-

objective routing problems, they present

complications when solving large-scale instances,

justifying the use of methods as the ones proposed in

the present work, which are enough to solve and

analyze the bi-objective location-routing problem.

3 MATHEMATICAL MODEL AND

SOLUTION METHODS

This work provides a detailed analysis of the bi-

objective version of the MDOLRP (Nucamendi-

Guillén et al., 2020), following two different

strategies. The details and characteristics of the

problem, the assumptions, and the model formulation

are presented next. The MDOLRP model is

functional to analyze the impact of hiring cost in

minimizing the total cost. For practical purposes, the

following assumptions are made:

 The routes generated should be finished in the

manufacturing company;

 The return cost of the supplier transport to the

depot is considered on the hiring cost. It means

the open routes;

 The demand is deterministic and quantified in

pallets;

 The traveled distance is translated into

monetary terms.

3.1 Problem Definition

The following notation is used:

𝑛 ∶ 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟𝑠

𝑚 ∶ 𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑐𝑎𝑟𝑟𝑖𝑒𝑟𝑠

ICORES 2021 - 10th International Conference on Operations Research and Enterprise Systems

426

𝑄



∶ 𝑀𝑎𝑥𝑖𝑚𝑢𝑚 𝑐𝑎𝑝𝑎𝑐𝑖𝑡𝑦 𝑝𝑒𝑟 𝑎𝑛𝑦 𝑣𝑒ℎ𝑖𝑐𝑙𝑒

𝑤





∶ 𝐻𝑖𝑟𝑖𝑛𝑔 𝑐𝑜𝑠𝑡 𝑝𝑒𝑟 𝑣𝑒ℎ𝑖𝑐𝑙𝑒 𝑙 𝑝𝑒𝑟 𝑐𝑎𝑟𝑟𝑖𝑒𝑟 𝑖

𝐷





: 𝑇𝑟𝑎𝑛𝑠𝑝𝑜𝑟𝑡 𝑐𝑜𝑠𝑡 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑑𝑒𝑝𝑜𝑡 𝑖 𝑎𝑛𝑑 𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟 𝑗

𝐶





: 𝑇𝑟𝑎𝑛𝑠𝑝𝑜𝑟𝑡 𝑐𝑜𝑠𝑡 𝑏𝑒𝑡𝑤𝑒𝑒𝑛 𝑠𝑢𝑝𝑝𝑙𝑖𝑒𝑟𝑠 𝑖 𝑎𝑛𝑑 𝑗

Let 𝑃1,..,𝑛1, be the set that denotes the

nodes to visit. Let 𝑃𝑝1,..,𝑛, the node-set for the

suppliers (where n represents the number of suppliers

to serve) to collect whereas the node 𝑛1 denotes

the final node (i.e., the manufacturing plant). Let 𝐹

1,…,𝑚 be the set of potential carriers (where m

represents the number of carriers to contract), where

𝑅



1,…,𝑘



 denotes the set of vehicles per each

carrier, where 𝑘



represents the number of vehicles

per each carrier. The capacity and the hiring cost of

each vehicle 𝑙 belonging to carrier 𝑖 are denoted as 𝑄





and 𝑤





, respectively. The demand per supplier 𝑗 is

in 𝑑



. The transport cost is due to the next matrix:

depot 𝑖 to supplier 𝑗 is in 𝐷



, and supplier 𝑖 to

supplier 𝑗 is in 𝐶



Regarding the variable set, let 𝑜





be a binary

variable equal to 1, if the arc 𝑖,𝑗 is active to travel

from the depot 𝑖 and the first node 𝑗 using the

vehicle 𝑙, and equal to 0 otherwise. Let 𝑥



be a binary

variable equal to 1, if the arc 𝑖,𝑗 is traveled between

nodes 𝑖 and 𝑗, and equal to 0 otherwise. Let 𝑧





equal to 1, if vehicle 𝑙 is contracted from the carrier 𝑖,

and equal to 0 otherwise. Furthermore, let the

auxiliary variable 𝑣





be a continuous variable that

denotes the sum of the demands of the remaining

nodes of the route after departing from carrier 𝑖 using

vehicle 𝑙 when 𝑜





1. In addition, let the

auxiliary variable 𝑟



be a continuous variable that

denotes the sum of the demands of the remaining

nodes of the route after visiting supplier 𝑖 when 𝑥





Since this work aims to analyze the impact of the

hiring cost (𝐹2) above the traveling cost (𝐹1), the

objective is conflicted naturally, as proportionally

contrary to the problem. The bi-objective model

approach is addressed. The mathematical formulation

is:

𝐹1𝑚𝑖𝑛𝐷











𝑜











𝐶



𝑥















(1)

𝐹2𝑚𝑖𝑛𝑤





𝑧















(2)

Subject to:

𝑜











1,∀𝑖 ∈𝐹;𝑙∈𝑅



(3)

𝑜















𝑥









1,∀ 𝑗 ∈𝑃𝑝,

(4)

𝑥











1,∀𝑗 ∈𝑃𝑝,

(5)

𝑧





𝑜









∀𝑖 ∈𝐹;𝑙∈𝑅



(6)

𝑣







𝑑



𝑜







,∀𝑖 ∈𝐹;𝑗∈𝑃𝑝; 𝑙∈𝑅



(7)

𝑣







𝑄





𝑜







,∀𝑖 ∈𝐹;𝑗∈𝑃𝑝; 𝑙∈𝑅



(8)

𝑟





𝑑



𝑥





,∀𝑖∈𝑃𝑝;𝑗∈𝑃𝑝; 𝑖𝑗,

(9)

𝑟



𝑄



𝑑



𝑥



,∀𝑖∈𝑃𝑝;𝑗∈𝑃𝑝; 𝑖𝑗,

(10)

𝑣















𝑟









𝑟











𝑑



,∀ ℎ ∈𝑃𝑝,

(11)

𝑜







∈



0,1



,∀𝑖 ∈𝐹;

𝑗

∈𝑃𝑝; 𝑙∈𝑅



(12)

𝑥



∈



0,1



,∀𝑖∈𝑃𝑝;

𝑗

∈𝑃,

(13)

𝑧





∈



0,1



,∀𝑖 ∈𝐹;𝑙∈𝑅



(14)

𝑣







0,∀𝑖 ∈𝐹;

𝑗

∈𝑃𝑝; 𝑙∈𝑅



(15)

𝑟



0,∀𝑖∈𝑃𝑝;

𝑗

∈𝑃.

(16)

The objective functions (1) and (2) minimizes the

sum of the transportation and contracting costs,

respectively. Constraints (3) ensure that, for each

vehicle, at most, one departing arc must be activated.

In contrast, the group of constraints (4) assure each

supplier node must be visited only once, either from

the depot or from other nodes. Constraints (5) ensure

that all of the routes end at node n+1. The constraints

(6) activates the carriers for the selected vehicles

(departing nodes). On the other hand, constraints (7)

ensure that the demand of each departing node must

be satisfied. Also, the constraints (8) ensure that the

cumulative demand of each route does not exceed the

capacity 𝑄





. Constraints (9) and (10) operate in the

same way as (7) and (8) but involving only the

supplier nodes. Constraints (11) avoid having sub-

tours by controlling demand. Finally, constraints

(12)-(16) denote the nature of the variables.

Since this study aims to analyze the behavior of

the vehicles' hiring and traveling cost over the total

cost, for both the case of the heterogeneous and

Analysis of the Hiring Cost Impact with a Bi-objective Model for the Multi-depot Open Location Routing Problem

427

homogeneous fleet, it is necessary to use the multi-

objective optimization method. This optimization

approach is useful to manage decision-making

problems involving two or more conflicting

objectives. Given the characteristics of the problem,

this procedure is followed.

3.2 Solution Methods

The computational experiments were executed using

both approaches, WRMCGP and AUGMECON2.

Figure 1 describes the implementation of WRMCGP.

Figure 2 exhibits the implementation of

AUGMECON2.

Figure 1: WRMCGP diagram.

4 COMPUTATIONAL

EXPERIENCE

4.1 Set of Instances

For computational experimentation, we considered

two different sets of instances. The first set belongs to

the instances used for the multi-depot vehicle routing

problem, proposed in (Cordeau et al., 1997) and

labeled P and Pr. The second set involves the

instances with a heterogeneous fleet (Wang & Wu,

2015). For the heterogeneous instances, different

opening costs per each depot were considered. In the

instances involving a homogeneous fleet, all the

vehicles have the same opening cost.

After implementing the model, only three

instances of each group could be entirely solved (six

in total). Tables 1 and 2 indicate the description of

those instances. In these tables, columns 1-5 indicate

the name of the instance, number of suppliers (n),

number of depots (m), number of vehicles (k), and

vehicles' hiring costs, respectively.

Figure 2: AUGMECON2 diagram.

Table 1: Description of the homogeneous instances.

Instance

name

n m k

Vehicle

acit

Hiring

cost

P_02 50 4 5 160 31.80

Pr_01 48 4 4 200 63.13

02 96 4 7 195 60.70

Table 2: Description of the heterogeneous instances.

Instance

name

n m k

Vehicles

Capacities

Hiring

costs

min max min max

Wa-W15O1 20 2 15 4 8 20 35

Wa-W15O2 24 3 15 4 5 15 20

Wa-W15O3 25 3 15 4 5 15 20

The formulation was coded using the optimization

language AMPL and solved using Gurobi 9.0.0 using

computational equipment with an Intel Core i7-

6600U @ 2.6GHz processor with 16 GB of RAM,

under Windows 10 as OS. The computational time

limit is set to 7200 seconds per iteration in the Pareto

Front.

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428

4.2 Multi-objective Metrics

To measure the performance of the method, we use

the following metrics:

 NOS/NPS:

The number of optimal solutions in the Pareto

Front evidences the best performance between

algorithms; a higher NPS value is preferred (Rayat et

al., 2017).

 Quality metric (QM):

This metric calculates the proportion between the

number of non-dominated Pareto Front solutions of

method A and the total non-dominated solutions from

the combined Pareto front (Rayat et al., 2017), as

shown in equation (17).

𝑄



𝐴





𝑁𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑛𝑜𝑛  𝑑𝑜𝑚𝑖𝑛𝑎𝑡𝑒𝑑 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠 𝑜𝑓 𝑚𝑒𝑡ℎ𝑜𝑑 𝐴

𝑇𝑜𝑡𝑎𝑙 𝑛𝑢𝑚𝑏𝑒𝑟 𝑜𝑓 𝑛𝑜𝑛

𝑑𝑜𝑚𝑖𝑛𝑎𝑡𝑒𝑑 𝑠𝑜𝑙𝑢𝑡𝑖𝑜𝑛𝑠

(17)

The quality of solutions is compared against each

other, and the higher value algorithm is desirable.

 RPOS:

The metric determines the ratio of Pareto-optimal

solutions in 𝑃



that are not dominated by any other

solutions in 𝑃, which is the union of the sets of the

Pareto-optimal solution, and it is calculated using

equation (18) (Altiparmak et al., 2006):

𝑅𝑃𝑂𝑆



𝑃







𝑃







𝑋

∈𝑃



|∃ 𝑌∈𝑃∶ 𝑌≺

𝑋



𝑃



(18)

Where

𝑌≺𝑋 means the solution 𝑋 is dominated

by the solution

𝑌, and these solutions 𝑋 are removed

from the solution set

𝑃



. The higher value, the better.

 Hyperarea ratio (HR):

The proportion of the generated area (HR) (21) is

calculated by dividing the Pareto front area (HA) for

each point between the total area (TA) (Zitzler &

Thiele, 1999), as shown in eq. (21). The area (HA) of

the Pareto Front can be defined as the product of the

difference between coordinate (𝐹1



, 𝐹2



) for each

solution 𝑆



and the (highest) maximum point (M), as

defined in eq. (19). Lastly, the total area (TA) is the

product of the difference between the coordinates

(𝑀, 𝐹1



), and (𝑀,𝐹2



) (20). Figure 3 shows an

example to calculate Hyperarea of a Pareto front.

𝐻𝐴  

𝑓





𝑓









∗



𝑓





𝑓





,

(19)

𝑇𝐴 



𝑓





𝑓





∗

𝑓





𝑓



,

(20)

𝐻𝑅  𝐻𝐴 𝑇𝐴

⁄

(21)

Figure 3: Hyperarea of the Pareto front.

4.3 Experimental Results

This section is devoted to reporting the values of the

metrics used to evaluate each method's efficiency

over each specific group of instances. First, the values

of the NPS and elapsed CPU time is shown. Then, the

values of the Q(A), RPOS, and HR are displayed.

Tables 3 and 4 report the value of NPS and the

CPU time (in seconds) for each solution approach

over each type of instance. These two metrics are first

considered since they quickly indicate the method's

performance in terms of quality and speed. The

columns are identified with an (A) for

AUGMECON2 and a (W) for WRMGCP.

Table 3: NPS and computational time of optimality

homogeneous instances.

Instance

name

NPS CPU Time

(

sec

)

AW A W

P_02 5 8 1054 10373

Pr_01 9 6 34 44

Pr_02 5 3 3108 5023

Avera

e 6.33 5.67 1398 5146

It is evident that AUGMECON2 performs better

than WRMCGP, reporting denser fronts in general

and in the case of the homogeneous instances,

requiring 74.76 % less time on average than

WRMCGP. On the other hand, there is a difference of

65.69% on average in the heterogeneous instances in

favor of WRMCGP. Since the number of points in the

Front differs per method, a detailed analysis is

conducted to determine the variation between the

minimum and maximum values for the extreme

solutions of each Pareto Front. This analysis is shown

in section 4.4.

The next analysis consists of determining which

method produces better Pareto fronts with respect to

the remaining metrics Q(A), RPOS, and HR. Tables

5 and 6 show the values obtained for the

heterogeneous and homogeneous instances,

respectively. The columns are identified with an (A)

for AUGMECON2 and a (W) for WRMGCP.

Analysis of the Hiring Cost Impact with a Bi-objective Model for the Multi-depot Open Location Routing Problem

429

Table 4: NPS and computational time of optimality-solved

heterogeneous instances.

Instance

name

NPS CPU Time (sec)

A W A W

Wa-W15O1 6 5 611 276

Wa-W15O2 4 3 3126 1217

Wa-W15O3 3 3 996 129

Avera

e 4.33 3.67 1578 541

Table 5: Results of multi-objective metrics for the

homogeneous instances.

Instances

(

)

RPOS HR

A W A W A W

P_02 1 0.625 1 1 0.699 0.657

Pr_01 1 0.667 1 0.833 0.695 0.657

02 1 0.6 1 1 0.616 0.521

Avera

e 1 0.631 1 0.944 0.670 0.612

Table 6: Results of multi-objective metrics for the

heterogeneous instances.

Instances

(

)

RPOS HR

A W A W A W

Wa-W15O1 1 0.833 1 1 0.781 0.78

Wa-W15O2 1 0.75 1 1 0.318 0.258

Wa-W15O3 1 1 1 1 0.736 0.736

Avera

e 1 0.861 1 1 0.612 0.591

From tables 5 and 6, it can be observed that

AUGMECON2 performs better, improving by 36.9%

in quality, 2.56% in RPOS, and 8.66% in HR for

homogeneous instances and improving by 23.9% in

quality, 0.0% in RPOS, and 3.43% in HR in

heterogeneous instances. In general, AUGMECON2

outperforms WRMCGP, even when WRMCGP

presents a competitive computational time

performance in the heterogeneous instances.

In summary, it was demonstrated that the vehicles'

hiring costs play an essential role when the DM seeks

a better trade-off between the number of vehicles to

hire and the total distance traveled, representing a

metric of customer satisfaction. A detailed analysis is

presented next to better understand the impact of

hiring cost over travel distance and capacity

utilization.

4.4 Detailed Analysis of the Impact of

Vehicles' Hiring Cost

As mentioned before, a detailed analysis is conducted

to evaluate the impact of the hiring cost over the

transportation cost (in both cases) and vehicles'

capacity utilization. This study was performed in two

stages. First, we calculate the proportion of

improvement in the transportation cost (F1) and

hiring cost (F2) as the difference between the Front's

extreme points. Secondly, the capacity utilization per

vehicle was estimated for each extreme point. Figures

4 and 5 illustrate the calculation of these values for

the instances Pr_01 and Wa-W15O1, respectively.

Tables 7 and 8 summarize the percentage of

reduction in transportation and hiring cost over the

homogeneous and heterogeneous fleet instances,

respectively. In these tables, the first column indicates

the name of the instance. In contrast, columns two and

three reports the percentage of reduction (RED) in

transportation and increment (INC) in hiring cost,

respectively. Finally, columns five and six report the

minimum and maximum percentages of capacity

utilization, respectively.

Figure 4: Percentage calculation of cost savings, for

instance Pr_01.

Table 7: Percentage of cost reduction and capacity

utilization for homogeneous instances.

Instance

name

RED INC

% Avg. of Capacity

utilization

T. cost H. cost

Min Max

P_02 5.76% 140.00% 37.36% 97.50%

Pr_01 8.57% 200.00% 27.38% 85.13%

Pr_02 2.78% 57.14% 56.88% 93.11%

Avera

e 5.70% 132.38% 40.54% 91.91%

The average percentage of increment cost for

homogeneous instances is 132.38% for hiring cost on

average, producing savings of around 5.70% on

average for transportation costs. On the other hand, if

we choose the minimum Hiring cost, this produces an

average increment of 6.12 % in transportation costs.

When analysing the capacity utilization, it raises

from 40.54% to 91.91%, which indicates that looking

for the best solution in distance tends to sub-utilize

the vehicles' capacity. Moreover, when the decision-

maker seeks for seizing the vehicle's capacity

utilization, the total traveled distance is worsened by

less than 10%, but producing increasing in hiring

costs up to two times more, which is significant since,

ICORES 2021 - 10th International Conference on Operations Research and Enterprise Systems

430

for these instances, the contracting costs represent up

to 60% of the total value (transportation + hiring

costs).

Table 8: Percentage of cost reduction and capacity

utilization in heterogeneous instances.

Instance

name

RED INC

% Avg. of Capacity

utilization

T. cost H. cost

Min Max

Wa-W15

15.58%

36.36%

72.74% 96.00%

11.02%

10.00%

92.86% 100%

7.42%

12.50%

87.62% 98.57%

Avera

e 11.34% 19.62% 84.41% 98.19%

In the instances with the heterogeneous fleet, it was

observed that savings in transportation costs raised to

11.34% while, for the hiring costs, the increment is

around 15.62%, on average. On the other hand, if we

choose the Hiring cost minimum, this provokes an

average increment of 12.95% in the transport cost. As

a conclusion, it can be observed that it is more rentable

to have different carriers (heterogeneous) to sacrifice

long travel and more vehicle utilized. When having

vehicles with different capacities, the model seeks a

better combination. It can also be confirmed when

observing the minimum and maximum values for

capacity utilization because vehicles have a

utilization of over 70% in the worst case.

Figure 5: Percentage calculation of cost savings for instance

Wa-W15O1.

In summary, considering that contracting costs

represent up to 40% of the total cost, we can initially

conclude that seizing vehicles utilization should be

preferred over total traveling distance for routing

problems involving hiring costs.

5 CONCLUSIONS AND FUTURE

RESEARCH

This work analyzes the impact of the hiring cost over

the transportation cost through a bi-objective model

for the MDOLRP, considering vehicles with a

homogeneous and heterogeneous fleet. The problem

was modeled using a bi-objective model and solved

using a commercial solver, testing literature instances

and obtaining optimality for instances up to 25

suppliers, 2 to 3 depots, and 15 vehicles for the

heterogeneous fleet and, in the case of the

homogeneous fleet, instances up to 96 suppliers, 4

depots, and 7 vehicles.

In terms of the methods used, AUGMECON2

outperformed WRMCGP. However, WRMCGP

performed faster in heterogeneous instances. In

addition, we demonstrate that in the heterogeneous

instances, the saving in hiring cost is significant by

maximizing the vehicles' capacity utilization without

significantly affecting transportation costs. In the case

of the homogeneous instances, the savings are less

substantial.

Future work involves the application of

metaheuristic algorithms to deal with large-scale

instances. In addition, the application of different

objectives as max time delivery, customer service

level, priority index, time windows, split delivery,

autonomous vehicles, and drone application can also

be interesting to the academic knowledge.

ACKNOWLEDGEMENTS

This research was supported by Universidad

Panamericana [grant number UP-CI-2020-GDL-07-

ING].

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