A Constraint Satisfaction Problems Based Scalable Framework to

Address Large-Scale Realistic Scheduling and Routing Problems

Liwen Zhang

, Sara Maqrot

, Florent Mouysset

and Christophe Bortolaso

Research and Technological Innovation Department, Berger-Levrault, 64 Rue Jean Rostand, Lab

ege, France

{ﬁrstname.lastname}@berger-levrault.com

Keywords:

Constraint Satisfaction Problem, Scheduling and Routing Problem, Scalable Framework, OptaPlanner,

Meta-Heuristic.

Abstract:

Scheduling and routing solutions for organizational staff , along with decision-making support for timetabling,

have become increasingly complex. This paper addresses the challenges associated with realistic large-scale

generic routing and scheduling problems with a multi-day horizon. We introduce a 2-level scalable framework

featuring a scalable use-case adapter and a scalable optimizer. In optimizer, a Scheduling and Routing Problem

(SRP) model and a conﬁgurable constraint system are implemented using OptaPlanner. In the experimentation

section, we present two real-life use cases in Spain and France, involving up to 481 activities to be performed

by 18 staff members over 4 weeks. These scenarios are submitted to our model under different meta-heuristic

conﬁgurations. The results demonstrate the achievement of high-quality optimized solutions within a short

computing time of just 8 minutes. Additionally, a detailed investigation is conducted to interpret the scores of

optimized solutions in an understandable manner.

1 INTRODUCTION

In the industrial ﬁeld, there is a wide diversity of busi-

ness areas that require the optimization of schedul-

ing and routing solutions, considering numerous

business-oriented objectives and strategic rules. Con-

sequently, the generation of scheduling and routing

solutions for staff in a given organization, even pro-

viding the decision-making support for timetabling

decision makers in these organizations, has become

extremely complex. This complexity is evident when

it involves a large number of constraints from various

use cases, such as the well-known Vehicle Routing

Problem and its variants (Caceres-Cruz et al., 2014).

When the large-scale problem has to be ﬁgured

out (e.g. 500 activities to schedule within 5 weeks),

we observe that there is a real need for better mod-

eling and solving the scheduling and routing prob-

lems. Since demands vary and the response capac-

ities provided by each service-delivery organization

are no less varied, there is a strong pressure on these

organizations to regularly manage their service ﬂows.

https://orcid.org/0009-0003-4692-1956

https://orcid.org/0000-0003-2261-4081

https://orcid.org/0009-0009-1220-3933

https://orcid.org/0000-0002-6635-9345

A scalable scheduling and routing framework along

with the Human Machine Interface (HMI) of an op-

erational product, supports decision-makers in cap-

turing all the necessary knowledge to address op-

erational scheduling and routing challenges. This

framework helps describe the problem by identify-

ing key requirements and gathering necessary infor-

mation from all stakeholders. As the volume of infor-

mation can become signiﬁcant, integrating this frame-

work into the organization’s information system may

make sense.

In this paper, we address the complexity of

Scheduling and Routing Problems (SRP) in opera-

tional organizations by presenting a scalable two-level

framework designed for real-world SRP. Our contri-

bution is threefold: 1) we investigate the key con-

cepts of SRP over various time horizons (from one

day to multi-month schedules). We start by drawing

from relevant content in the literature to facilitate the

modeling phase, the second level of our framework.

2) Using data processed by Adapter (the ﬁrst level

of our framework), we implement an OptaPlanner-

based SRP model, featuring a conﬁgurable constraint

system based on the Constraint Satisfaction Problem

(CSP) paradigm. This allows us to efﬁciently solve

SRP through various meta-heuristic algorithms em-

bedded in OptaPlanner. As far as we know, no stud-

Zhang, L., Maqrot, S., Mouysset, F. and Bortolaso, C.

A Constraint Satisfaction Problems Based Scalable Framework to Address Large-Scale Realistic Scheduling and Routing Problems.

DOI: 10.5220/0013109600003893

Paper published under CC license (CC BY-NC-ND 4.0)

In Proceedings of the 14th International Conference on Operations Research and Enterpr ise Systems (ICORES 2025), pages 45-56

ISBN: 978-989-758-732-0; ISSN: 2184-4372

ies have used an OptaPlanner-oriented optimizer to

address real-life SRP that simultaneously optimize

the density of activities assigned at the beginning of

the scheduling horizon and balance the weekly work-

loads of staff members. This is further combined

with traditional constraints such as respecting time

windows and staff unavailability, all within a multi-

day scheduling horizon. 3) For experimental val-

idation, we apply the framework to two real-word

use cases of the Preventive Maintenance Scheduling

and Routing Problem (PMSRP), one in Spain and the

other in France. To the best of our knowledge, an

OptaPlanner-based CSP model for tackling PMSRP

has not been addressed in the literature. We evaluate

and compare the solution quality generated by differ-

ent meta-heuristic algorithms embedded in OptaPlan-

ner.

The remainder of this paper is structured as fol-

lows. Section 2 presents a review of the literature

related to our study. In Section 3, we describe our

proposed scalable framework designed to address the

various SRP. Section 4 provides illustrative computa-

tional experiments and discussion. Finally, some fu-

ture researches are suggested in Section 5.

2 LITERATURE REVIEW

Originally, the Traveling Salesman Problem (TSP) is

a commonly known class in all routing and scheduling

problems. According to (Gutin and Punnen, 2006),

there are many variants of TSP. These authors de-

scribe a set of at least ten variants of this basic formu-

lation, and the main differences arise from the way

objective functions and constraints are approached

within different contexts. For example, minimizing

the total visit cost following a Hamiltonian path in the

Messenger problem (J

unger et al., 1995). (Dantzig

and Ramser, 1959) argue that the family of TSP prob-

lems can be considered as a subclass of the VRP, and

then VRP is identiﬁed by (Bektas, 2006) as a “multi-

TSP”. Some variants of the VRP have been exten-

sively studied in recent years. With aging populations

and an increasing life expectancy, Home Health Care

(HHC) has become a common healthcare delivery

manner in Europe. Therefore, HHCRSP is a new VRP

application that considers the variability of business

constraints, along with the diversity of the HHC orga-

nization (Di Mascolo et al., 2021). Furthermore, with

the fourth industrial revolution underway, driven by

Internet of Things (IoT) technology, there is a grow-

ing interest in another widely recognized VRP appli-

cation, the PMSRP, particularly in complex scenarios

involving distributed systems where maintenance as-

sets are geographically dispersed (Rashidnejad et al.,

2018).

Vehicle Routing Problem (VRP) is a well-known

class within Scheduling and Routing Problems (SRP).

By deﬁnition, the primary objective of a VRP is to op-

timize the delivery routes of a ﬂeet of vehicles from

a depot to various customer sites, all while accom-

modating a variety of constraints that add complex-

ity to the planning process. These constraints, often

speciﬁc to different business ﬁelds, include temporal

restrictions, which are especially critical in schedul-

ing and routing. The offer-demand paradigm high-

lights the importance of aligning both customer and

staff satisfaction with respect to time management.

This involves synchronizing individual schedules to

balance the needs and preferences of both parties.

The decision-maker must ensure that the services are

carried out without compromising quality and within

an acceptable operating duration. Frequently, a time

window (referring to a temporal interval specifying

possible starting times for a service) characterizes ei-

ther the customer’s availability or their preferred ser-

vice time (Ibaraki et al., 2005). For normal services,

a temporal tolerance is possible. Hence, two types of

time windows are retained in the literature (Dekhici

et al., 2019): the hard (or ﬁxed) time window or the

soft (or ﬂexible) time window. In (Rest and Hirsch,

2016), the authors focused on the working hours of

the staff. Over a weekly horizon, for instance, if a

staff signs a 25-hour-per-week contract, a maximum

daily working time may also inﬂuence the deﬁnition

of their schedule. A time interval can be applied to in-

dicate the earliest start time and the latest ﬁnish time

of the staff’s daily working rounds (Di Gaspero and

Urli, 2014). To reduce fatigue and increase work ef-

ﬁciency, a daily lunch break is considered in (Coelho

et al., 2016).

Decision-makers must deﬁne directions within

their decisions, specifying the objectives to be

achieved or the risk to be avoided. A frequently used

minimization criterion in this context is travel time

(Dekhici et al., 2019), where total transportation costs

are minimised in (Nguyen et al., 2022). When staff

work capacities are exceeded, overtime costs become

a signiﬁcant concern. (Carello and Lanzarone, 2014)

aim to reduce the overtime costs of daily workers.

To ensure staff satisfaction, workloads must be bal-

anced. (Yalc¸ında

g et al., 2016) apply utilization rates

to achieve personal workload balance. These rates

compare actual workload to theoretical work capac-

ity. In (Quintanilla et al., 2020), the authors focus

on balancing the number of visits performed by each

staff.

The choice of resolution methods varies depend-

ICORES 2025 - 14th International Conference on Operations Research and Enterprise Systems

Figure 1: Overview of SRP framework architecture.

ing on the mathematical nature of the target problem.

Selected resolution methods are highly tailored to

speciﬁc types of routing and scheduling problems. In

Operations Research (OR) related research, two main

approaches are explored: exact methods and approx-

imate methods. Exact methods face challenges when

addressing real-life use cases due to the sensitivity of

their Integer Linear Programming (ILP) based mathe-

matical models to problem size. Several mathematical

programming solvers, such as CPLEX and Gurobi,

employ exact methods for solution search. To illus-

trate the size of the problem that can be handled with

exact methods, in (Zhang et al., 2021), CPLEX is

used to solve the daily Home Health Care Scheduling

and Routing Problem (HHCSRP) based on ILP, with

a size of up to 61 activities per day. However, when

the size of the problem exceeds 61 activities, with the

constraints considered in the model, CPLEX cannot

ﬁnd the optimal solution within the predeﬁned max-

imum computation time of 10 hours. In such cases,

approximate methods are valuable for solving large-

size problems with numerous constraints in various

business contexts. These methods, based on meta-

heuristic algorithms, provide reliable alternatives to

exact methods thanks to their capacity to efﬁciently

explore vast search spaces within reasonable compu-

tation times. These approximate algorithms incorpo-

rate mathematical models for problem-solving, either

directly or through Constraint Programming model-

ing. They are then coupled with meta-heuristic al-

gorithms such as Simulated Annealing (Kirkpatrick

et al., 1983) or Tabu Search (Glover and Laguna,

1998) to generate near-optimal solutions. Although

few articles mention the use of approximate solvers

for large-scale scheduling and routing problems, ex-

amples such as (Smirnov and Shilov, 2010) highlight

the application of approximated methods embedded

in the Choco solver to address real-life dynamic logis-

tics problems. Furthermore, in (Zhang et al., 2023),

OptaPlanner is used to tackle the daily HHCSRP with

a size of up to 130 activities per day.

In conclusion, addressing scheduling and routing

challenges involves managing the trade-off between

problem complexity (diverse constraints and decision

variables) and computational time for solution gen-

eration. Despite the approximation methods existing

in previous research, the resolution of real-world in-

dustrial problems remains a persistent challenge due

to their signiﬁcant scale, diversity, and dynamic con-

straints in various application scenarios.

3 SCALABLE FRAMEWORK TO

ADDRESS REAL-WORLD

SCHEDULING AND ROUTING

PROBLEMS

Drawing on insights from the literature review, we

present a scalable framework to address the chal-

lenges related to computational time and scheduling

requirements in real-world SRP.

3.1 Architecture Overview

Our optimization framework consists of two mod-

ules, the (Adapter) and the (Optimizer), which to-

gether form a consistent architecture. This framework

provides a generative solution to the SRP. Figure 1

presents an overview of the linear process that tra-

A Constraint Satisfaction Problems Based Scalable Framework to Address Large-Scale Realistic Scheduling and Routing Problems

Figure 2: Illustration of three-level process in Optimizer of the scalable framework.

verses the two modules. At the end of this process, the

scheduling and routing solution is obtained in a well-

formatted JSON ﬁle, which can be used to display the

optimized solution via a calendar or map view of a

web application.

The ﬁrst module (Adapter) is responsible for an-

alyzing web requests from various business prod-

ucts related to different scheduling problems. It per-

forms data structure adaptation and conversion to

the generic API of the second module (Optimizer).

The API deﬁned in this module allows users to ac-

tivate or deactivate embedded constraints as needed.

This module can include different adapters for vari-

ous use cases from related products, such as the PM-

SRP adapter for the Preventive Maintenance Schedul-

ing and Routing Problems. Additionally, this module

is scalable to meet the scheduling and routing require-

ments of new products.

In the second module (Optimizer), the previously

translated data from the corresponding adapter is used

to optimize solutions with either an open-source or a

commercial solver. Currently, only one open-source

AI solver, OptaPlanner, is developed to perform SRP

optimization, following a three-level process, as illus-

trated in Figure 2:

1. Modeling. Based on the CSP paradigm, as shown

in Figure 3, an SRP model is constructed. Each

concept is deﬁned with its necessary parameters,

and the relationships between these concepts are

speciﬁed. Importantly, the decision variables (an-

notated by @ShadowVariable or @PlanningList-

Variable) are indicated in the concepts annotated

by @PlanningEntity. This part is detailed in Sec-

tion 3.2.1. Note that the attribute names shown in

the ﬁgure after “ ” correspond to the annotation

used in Section 3.2.1.

2. Constraint Formulation. Based on the SRP

model, a series of constraints is pre-formulated.

These constraints can be activated or deactivated

based on feedback from the adapters. The formu-

lation of these constraints is detailed in Section

3.2.2.

3. Resolution. The SRP is resolved using the ap-

proximate methods embedded in OptaPlanner, as

detailed in Section 3.2.3.

It is important to note that this module is also scal-

able, allowing for the integration of new solvers to

benchmark against OptaPlanner’s performance. This

three-level process is illustrated in Figure 2.

OptaPlanner is selected as the ﬁrst solver devel-

oped in this module due to its open-source nature

and its intuitive annotation system for constructing

CSP-based models. The OptaPlanner annotation sys-

tem is detailed in our previous research work (Zhang

et al., 2023). Additionally, the numerous embedded

construction heuristics and local search-based meta-

heuristics enable us to select the appropriate combina-

tion of algorithms to tackle various problems. Finally,

OptaPlanner is fully written in Java, which facilitates

its integration into our architecture, driven by Spring

Boot on the backend.

3.2 Optimizer Description

3.2.1 Modeling

(A) Data Model

Deﬁnition 1: Ofﬁce is the central entity in our system,

denoted by O and deﬁned by the following parame-

ters:

• g: Planning granularity (in seconds), serving to

convert all time-related parameters into integer

slot format. E.g. if dur

= 1 and ω = 60, then

the operation time of activity a is 60 seconds.

• h: Number of days in scheduling horizon.

• p: Number of activities to be scheduled within h.

• q: Number of available staff members within h.

ICORES 2025 - 14th International Conference on Operations Research and Enterprise Systems

Figure 3: Overview of OptaPlanner-based SRP model.

• [bs

∗

,be

∗

]: Time window for a daily break.

Deﬁnition 2: Activity is the primary planning entity

in our model, representing client-requested activities,

denoted by A = {a

,..,a

] and deﬁned by :

• dur

: Operation time required for activity a, in-

cluding the time to park, where. dur

∈ [1,∞]

• ε

: Expected schedule day requested for activity

a (for horizon h > 1 ).

Deﬁnition 3: Staff-Day is a planning entity associated

with Staff and Day, representing the set of available

staff members for deployment on a given day within

the scheduling horizon. Each Staff-Day element cor-

responds to a single workday of a staff member.

• Staff is denoted as S = {s | s ∈ S, |S| = q}, each

element s ∈ S is characterized by :

– ξ

: Maximum daily workload.

– ς

: Maximum weekly workload.

• Day is denoted as D = {d | d ∈ D, |D| = h}.

• Staff-Day is denoted as SD = {sd | sd ∈

SD, |SD| = α

∗

} , where α

∗

= q · h represents the

total number of available staff members within the

daily horizon. Each sd ∈ SD is characterized by

the following parameters:

– [start

∗

,end

∗

]: Work time window for staff

member s on day d.

– ω

: Week index within the horizon.

– [start

,end

]: Unavailable time windows for

staff member s on day d.

Deﬁnition 4: Location is an entity associated with Of-

ﬁce, Activity and Staff-Day, as depicted in Figure 3,.

It determines their respective locations, which are es-

sential for calculating travel time. Location is denoted

by L and characterized by:

• travel

i j

: Travel time from i to j, with i, j ∈

[1, p] ∪ [1,q] ∪ {0}, where {0} represents the of-

ﬁce. travel

i j

is computed using the Origin-

Destination cost matrix solver from ESRI

con-

sidering real-life trafﬁc conditions.

(B) Decision Variables in Planning Entity

As illustrated in Figure 3, Activity A and Staff-Day

SD are annotated with @PlanningEntity, allowing for

changes during the solution search process. Decision

variables are the attributes within these entities that

can change values. For a comprehensive exploration

of the optimization model-oriented annotation system

in OptaPlanner, refer to (Zhang et al., 2023).

For each sd ∈ SD, there exists a planning list

variable denoted by β

, which serves as the pri-

mary variable containing multiple chained activities

a ∈ A = {a

,.., a

}. This variable allows each activ-

ity a ∈ β

to link directly to another, establishing an

order among the planned entities. The chain begins

at the start location of sd ∈ SD and ends with the last

element of the list β

The travel time is calculated between each pair of

elements in β

. However, the last travel time in the

list β

is calculated from the ﬁnal element in β

the end point of the daily round of s, which can be the

ofﬁce location or the domicile location of s.

The following requirements guide the construc-

tion of a chain in this model :

• Each chain is open-ended and does not form a

loop. Figure 4 illustrates an example of a correct

chain (Chain 1) and an incorrect looping chain

(Chain 3).

• A chain is linear, not a tree. Therefore, each sd ∈

SD has only one a ∈ β

at the end of the chain. In

Figure 4, we illustrate an incorrect tree structure-

based chain (Chain 4).

• sd ∈ SD with β

= ∅ is also considered a 0-chain.

ESRI: https://www.esri.com/en-us/home

A Constraint Satisfaction Problems Based Scalable Framework to Address Large-Scale Realistic Scheduling and Routing Problems

• Only one round is allowed for sd ∈ SD, corre-

sponding to β

= 1.

• When [bs

∗

,be

∗

] is deﬁned within an Ofﬁce setting,

a chain can be viewed as a two-part sequence. The

ﬁrst part begins at sd ∈ SD and ends at a

∈ A

with a termination time ≥ bs

∗

. The second part

starts at a

∈ A, where a start time equal to be

∗

travel

, and ends at another point, a

∈ A. This

two-part chain is illustrated in Figure 4 as Chain

For each a ∈ A, four decision variables are de-

ﬁned. These secondary variables, referred to shadow

variables in the OptaPlanner ecosystem, depend on

the primary variable β

, sd ∈ SD. A shadow variable

is a planning variable whose value can be deduced

from the state of β

. The deﬁnitions of the four deci-

sion variables are as follow :

• next

: The next activity following activity a in

a staff’s daily round, with the domain next

,..., a

}

• prev

: The previous activity preceding activity a

in a staff’s daily round, with the domain prev

,..., a

}

• start

: The start time of activity a within a day,

with the domain start

= [0,86400/g]

• std

asd

: The assigned staff s for staff-Day sd for ac-

tivity a, with the domain of std

asd

= {s

,..., s

}

3.2.2 Constraint Formulation

All constraints stated in this section are enabled or

disabled based on the different context of the tar-

get SRP. In our model, there is no objective func-

tion to characterize solution quality. Instead, each

constraint is linked to a hard or soft score based on

its violation. Consequently, a solution is deﬁned by

a global score, where the hard part is denoted by

Λ =

∑

{hard scores for each constraint}, and the soft

part by Ω =

∑

{soft scores for each constraint}. The

presence of a hard score indicates an infeasible solu-

tion, while the goal is to maximize the soft score. The

higher the score, the higher the overall score, indicat-

ing a better solution.

Furthermore, a list of binary values is required

in the format {1,1,1,1,0,0,0,0,0}. This list indicates

that constraints (1)-(4) are activated, while constraints

(5)-(9) are deactivated. It represents the necessary

constraints for a target SRP, generated by the corre-

sponding adapter based on the request from the re-

lated product.

Constraint (1) ensures that all care services are as-

signed to a staff member during the day, thereby pro-

hibiting any missing activities in the target use case.

If std

asd

= ∅ ∨ start

= ∅ Then Λ − 1

∀a ∈ A,∀sd ∈ SD

(1)

Constraint (2) ensures that each activity is sched-

uled within the working time window of staff member

s on day d.

If (start

≥ end

∗

) ∨ (start

+ dur

> end

∗

)

∨ (start

< start

∗

) Then Λ − 1

∀a ∈ A,∀sd ∈ SD

(2)

Constraint (3) prohibits scheduling activities dur-

ing the designated unavailability of the staff member

s on day d.

If ((start

≤ start

) ∧ (start

+ dur

≥ end

))

∨ ((start

> start

) ∧ (start

+ dur

< end

))

∨ ((start

+ dur

≥ start

)

∧ (start

+ dur

≤ end

))

∨ ((start

≥ start

) ∧ (start

≤ end

)

Then Λ − 1 ∀a ∈ A,∀sd ∈ SD

(3)

Constraint (4) ensures that the total workload of

the staff member s does not exceed the maximum

daily limit ξ

∑

i∈β

∑

j∈β

∪std

asd

i̸= j, j=i+1

(travel

i j

+ dur

) > ξ

Then Λ− | ξ

−

∑

i∈β

∑

j∈β

∪std

asd

i̸= j, j=i+1

(travel

i j

+ dur

) |

∀a ∈ A,∀s ∈ S, ∀sd ∈ SD

(4)

Constraint (5) aims to align the overall workload

of the staff member s as closely as possible with the

weekly maximum limit ς

∑

i∈β

∑

j∈β

∗

∪std

asd

∪ω

=ω

∗

i̸= j, j=i+1

(travel

i j

+ dur

) > ς

Then Ω− | ς

−

∑

i∈β

∑

j∈β

∗

∪std

asd

∪ω

=ω

∗

i̸= j, j=i+1

(travel

i j

+ dur

) |

∀a ∈ A,∀s ∈ S, ∀sd, sd

∗

∈ SD

(5)

Constraint (6) ensures that activity a with ε

̸= ∅,

is scheduled as closely as possible to the expected

schedule day.

If ε

̸= ∅ Then Ω− | ε

− {d | d ∈ std

asd

} |

∀a ∈ A,∀sd ∈ SD

(6)

ICORES 2025 - 14th International Conference on Operations Research and Enterprise Systems

Figure 4: Illustration of the chain-based planning list variable β

, for sd ∈ SD.

Constraint (7) minimizes the total travel times of

all staff members, operating without any conditional

requirements.

Ω−

∑

i∈β

∑

j∈β

∪std

asd

i̸= j, j=i+1

travel

i j

∀a ∈ A,∀sd ∈ SD (7)

Constraint (8) optimizes the density of activities

assigned at the beginning of the scheduling horizon.

The main goal is to improve resource efﬁciency by

reducing inactivity periods among staff, thus creat-

ing additional time slots for the insertion of urgently

added activities. There is no conditional part for this

constraint.

Ω − {d | d ∈ std

asd

} a ∈ A,∀sd ∈ SD (8)

Constraint (9) promotes balance in the weekly

workloads of the staff members, considering the

travel times between assigned activities and staff un-

availability as needed.

If ω

= ω

∗

Then Ω −



∑

i∈β

∑

j∈β

∪std

asd

i̸= j, j=i+1

(travel

i j

+ dur

| start

− end

−

∑

∗

∈β

∗

∑

∗

∈β

∗

∪

std

∗

̸= j

∗

, j

∗

(travel

∗

+ dur

∗

| start

∗

− end

∗



a,a

∗

∈ A,∀sd, s

∗

∈ SD

(9)

3.2.3 Resolution

The resolution of the problem in OptaPlanner begins

with the generation of an initial solution using con-

struction heuristics within a limited time frame while

aiming for the best possible quality. OptaPlanner sup-

ports ﬁve construction heuristics, including First Fit

and its variants, Weakest Fit and Strongest Fit (Bays,

1977), Allocate Entity from Queue (Semeria, 2001),

Regret Insertion (Diana and Dessouky, 2004), Alloca-

tion from pool (Lattner and Adve, 2002) and Scaling

construction heuristics (Katayama et al., 2009).

The second phase involves iteratively searching

for a solution that outperforms the current one us-

ing meta-heuristic algorithms. OptaPlanner’s conﬁg-

uration allows for the selection of optimization algo-

rithms and the deﬁnition of their intrinsic parameters.

The optimization process ends when predeﬁned stop-

ping criteria are met, typically deﬁned as reaching ei-

ther a maximum computational time limit or a spe-

ciﬁc number of iterations. In addition to construction

heuristics, OptaPlanner offers various types of local

search meta-heuristics:

1. Simple local search: Hill Climbing (Goldfeld

et al., 1966) and Late acceptance (Burke and

Bykov, 2017).

2. Meta-heuristics: Tabu Search (Glover

and Laguna, 1998), Simulated Annealing

(Van Laarhoven and Aarts, 1987), Great Deluge

(Dueck, 1993), and Variable Neighborhood

Descent (Gao et al., 2008).

4 TESTS

In experimental section, we aim to evaluate our target

problems using our framework in various conﬁgura-

tions. The conﬁgurations of our OptaPlanner-based

optimizer draw inspiration from our previous research

work (Zhang et al., 2023), using the First Fit con-

struction algorithm in conjunction with all the meta-

A Constraint Satisfaction Problems Based Scalable Framework to Address Large-Scale Realistic Scheduling and Routing Problems

Figure 5: Staff unavailabilities aligned with the day index within the planning horizon for the French use case.

heuristics embedded in OptaPlanner except for Tabu

Search. The choice of First Fit as the construction

heuristic for generating the initial solution is based

on the study conducted by (Macik, 2016), which pro-

vides an in-depth discussion of the conﬁguration of

the solver. This author, who examined the task as-

signment problem, demonstrated that First Fit is the

most effective construction heuristic. Similarly, in the

work of (Rios de Souza and Martins, 2020), First Fit

is also used to construct the initial solution, further

validating its effectiveness. Tabu Search is omitted

from the testing due to the complexity of our identiﬁ-

cation rule for the ID of each planning entity, which

limits the solver’s ability to execute the optimization

process effectively. Additionally, in line with (Zhang

et al., 2023), we utilize two termination criteria: either

when the global score in the form of HardSoftScore

reaches zero in both its hard and soft components,

or when the computation time reaches the predeﬁned

limit of 8 minutes. The choice of an 8-minute time

limit is based on client feedback.

4.1 Realistic Dataset Statement

The dataset was extracted from the database of an op-

erational product used by two Berger-Levrault clients:

one in Spain and the other in France. The Spanish

client (case ES) assigns staff to maintenance tasks on

water network infrastructures. Each staff member be-

gins and ends their daily work at the ofﬁce, with repair

points located within a city and a relatively short plan-

ning horizon. In contrast, the French client (case FR)

Table 1: Main data for Spanish Case and French Case.

Parameter Values of Case ES Values of Case FR

(O) g 60 60

(O) h 5 26

(O) p 90 481

(O) q 4 18

(O) [bs

∗

,be

∗

] [50400,55800] -

(A) dur

see Figure 6 see Figure 6

(A) ε

= 1, ε

2,ε

= 2, ε

3,ε

= 5, ε

= 4

(S) ξ

- ξ

= 32400

(S) ς

- ς

= 144000

(SD) [start

∗

,end

∗

] [32400,63000] [28800,70200]

(SD) ω

{0} {1,2, 3, 4}

(SD) [start

,end

] [32400,63000],

s = 1,d = 2

see Figure 5

assigns staff to perform maintenance on machines

at various client locations throughout France. Here,

each staff member begins and ends their daily work

at home, with a relatively long planning horizon. The

parameter settings for our model for the case of ES

are listed in column 2 of Table 1, while those for the

case of FR are described in column 3.

Due to the extensive planning horizon and the

frequent unavailability of active staff members, Fig-

ure 5 presents detailed information regarding the pa-

rameter [start

,end

] for the case FR. It is im-

portant to note that the red and blue zones (repre-

senting weekends or holidays) indicate the unavail-

ability of speciﬁc staff members throughout the day

([start

,end

] = [28800,70200]). The orange zone

denotes the half-day inaccessibility for staff members

ICORES 2025 - 14th International Conference on Operations Research and Enterprise Systems

Table 2: Constraints activated for FR and ES cases.

Constraint# 1 2 3 4 5 6 7 8 9

Case ES 1 1 1 0 0 1 1 1 1

Case FR 1 1 1 1 1 0 1 0 1

Table 3: Comparison of solution scores: case ES.

SA HC LA GD VND

C. 1 0 0 0 0 0

C. 2 0 0 0 -6 0

C. 3 0 0 0 0 0

Λ 0 0 0 -6 0

C. 6 0 0 0 -1 0

C. 7 -341 -562 -1239 -1264 -532

C. 8 -179 -177 -174 -181 -164

C. 9 -244800 -257600 -195200 -3343600 -270000

Ω -245320 -258339 -196613 -3345045 -270696

([start

,end

] = [28800, 43200] ∨ [43200,70200]).

Finally, Figure 6 illustrates the duration distribu-

tions for all activities in two cases.

4.2 Numerical Result

After conducting several interviews to understand the

distinct scheduling requirements of the two clients,

Table 2 illustrates the activated constraints for each

case, with a value of 1 indicating that the correspond-

ing constraint is activated and 0 otherwise.

The test is carried out on a computer with an In-

tel(R) Core(TM) i7-7500U processor at 2.90 GHz and

16.0 GB of RAM under the Windows 10 operating

system. In both use cases, our initial evaluation in-

volves evaluating the hard component (Λ) and the soft

component (Ω) of the overall score for each solu-

tion generated under different meta-heuristic conﬁg-

urations in OptaPlanner : Simulated Annealing (SA),

Hill Climbing (HC), Lated Acceptance (LA), Great

Deluge (GD), and Variable Neighborhood Descent

(VND), while utilizing the same constructive algo-

rithm : First Fit. As demonstrated in Table 3 and Ta-

ble 4, we further detail the score generated by each

activated constraint.

Table 4: Comparison of solution scores: case FR.

SA HC LA GD VND

C. 1 0 0 0 0 0

C. 2 0 0 0 0 0

C. 3 0 0 0 0 0

C. 4 0 0 0 0 0

Λ 0 0 0 0 0

C. 5 -4850 -3552 -11448 -12458 -8055

C. 7 -95207 -89320 -112087 -113751 -87368

C. 9 -711408 -883310 -2180250 -3153722 -3130840

Ω -811465 -976182 -2303785 -3279931 -3226263

Examining Table 3 and Table 4, our initial conclu-

sions are:

• Except for the situation where case ES uses GD

for solution generation and fails to ﬁnd a feasible

solution due to its hard score Λ(caseES,GD) =

−6, all examined meta-heuristics successfully

ﬁnd feasible solutions for both cases within an

8-minute computing time, even when the size of

case FR is sufﬁciently large.

• In terms of solution quality for the two tested

cases, as indicated by the soft score Ω, the

best solution for case ES is achieved by LA

with Ω(caseES,LA) = −196613, while for case

FR, SA produces the optimal solution with

Ω(caseFR,SA) = −811465. This demonstrates

that the choice of meta-heuristic for solution gen-

eration depends on the speciﬁc use case.

• When analyzing the detailed scores for each ac-

tivated constraint, speciﬁcally C.9 for achieving

weekly workload balance, these values signiﬁ-

cantly inﬂuence the overall score of each solu-

tion. However, the interpretation of meaning be-

hind this score is unclear. Consequently, we will

introduce an indicator for this constraint later to

better demonstrate its beneﬁt.

For C.9, we assess the balance of workload for

each working week among all staff members who

have completed their assigned tasks. This is done

through the introduction of a new indicator, calcu-

lated using formula (10), which calculates the average

Mean Deviation (MD) of each staff member’s total

workload per day. The formula is represented as:

week

∑

∗

i=1

MD(workload

per working day)

∗

s ∈ S

(10)

Here, h

∗

refers to the number of working days in

the scheduling horizon, excluding weekends and holi-

days. week refers to the week number within the plan-

ning horizon for each use case.

For the case ES, the scheduling horizon comprises

only 5 days, equivalent to 1 week. Table 5 illustrates

the value M

week

for the solutions outlined in Table 3.

These ﬁndings conﬁrm that the use of LA in solution

generation achieves the optimal balance in weekly

workload for all staff members. This is reﬂected

in M(caseES,LA)

= 4.18, with a gap of = 4.18%

compared to the theoretical best value of 0. We ob-

serve that the weekly workload balance for GD seems

to be quite far from other values when compared to

other methods (e.g. : M(caseES,GD)

= 58.32 vs.

A Constraint Satisfaction Problems Based Scalable Framework to Address Large-Scale Realistic Scheduling and Routing Problems

Figure 6: Activities’ duration distributions for two cases.

Table 5: Weekly workload balance analysis: case ES.

SA HC LA GD VND

5.15 5.55 4.18 58.32 5.46

M(caseES,LA)

= 4.18 ). This difference arises be-

cause the ﬁnal solution score generated by GD is

reached after just 2 minutes of computation, with no

further improvement observed over the remaining 6

minutes. This indicates that GD, when used for so-

lution generation in our model with the given con-

straints of this use-case, results in being entrapped in

local optima.

For the case FR, the scheduling horizon covers 26

days, equivalent to 4 weeks. As shown in Figure 7,

the results conﬁrm that the use of SA in solution gen-

eration achieves the best balance of weekly workload

for all staff members. SA consistently produces the

lowest M

week

over 4 weeks, with respective gaps com-

pared to the theoretical best value of 0:

• gap

week1

= 16.73%

• gap

week2

= 25.73%

• gap

week3

= 11.83%

• gap

week4

= 26.64%

In conclusion, given the numerous staff unavail-

abilities within their workloads, especially for weeks

2 and 4, we observe a signiﬁcant decentralization of

staff unavailabilities, as shown in Figure 5. Con-

sequently, the optimized solution is less effective in

achieving an optimal weekly workload balance.

5 CONCLUSIONS, DISCUSSIONS

AND FUTURE WORKS

This paper introduces a two-level scalable optimiza-

tion framework designed to address large-scale, real-

world problems with a multi-day horizon. The Op-

timizer is currently conceptualized and implemented

using OptaPlanner, an open-source AI solver based on

Constraint Satisfaction Problems (CSP). To address

diverse scheduling requirements, we propose a scal-

able Adapter to parameterize the conﬁgurable con-

straint system within our Optimizer. In the exper-

imental section, we present two real-life use cases

subjected to our OptaPlanner-based Scheduling and

Routing Problem (SRP) model under different conﬁg-

urations. The results demonstrate the achievement of

high-quality optimized solutions within a short com-

puting time. In addition, a detailed investigation is

conducted to interpret the scores of optimized solu-

tions in an understandable manner.

For future work, ﬁrstly, in terms of experimen-

tation, we plan to expand the experimental results

presented in this paper. This would provide more

comprehensive evidence of the framework’s scalabil-

ity and adaptability across various scenarios. Incor-

porating additional case studies or presenting more

detailed performance metrics under varying condi-

tions (e.g., staff size, scheduling complexity) would

enhance the ﬁndings and enable clients to evaluate

the framework’s robustness across a broader spec-

trum of real-world situations. A deeper analysis of

the experimental outcomes is necessary, focusing on

why certain conﬁgurations performed better or worse

ICORES 2025 - 14th International Conference on Operations Research and Enterprise Systems

Figure 7: Weekly workload balance analysis : case FR.

and linking these ﬁndings to practical implications.

Moreover, a critical reﬂection on the limitations of

the study (e.g., computational time constraints, spe-

ciﬁc constraints that may limit applicability) will lend

transparency and allow for a more balanced interpre-

tation of the results.

Next, we will conduct additional interviews with

other clients to explore their scheduling requirements.

Additionally, assigning weights to each constraint to

reﬂect their importance for different clients will im-

prove our conﬁgurable constraint system, allowing it

to effectively meet additional SRP requirements.

As this work is ongoing, we also plan to integrate

additional solvers into our Optimizer, enabling a com-

parative evaluation of the solutions generated by dif-

ferent solvers. This analysis will help establish the

framework’s competitive advantage and highlight its

unique beneﬁts. Such a comparison will provide users

with a clearer understanding of the framework’s effec-

tiveness relative to existing solutions.

Furthermore, our current resolution approach pre-

sented in this article focuses on combinatorial opti-

mization theories and does not account for uncertain-

ties. Although our approach can efﬁciently generate

near-optimal solutions, addressing operational uncer-

tainties using stochastic formulations of SRP is chal-

lenging due to the complexity of the problem. To

overcome this limitation, we are considering integrat-

ing our proposed approach with algorithms based on

a multi-agent system. This new direction will en-

able posterior sensitivity analysis by utilizing a multi-

agent system in conjunction with solutions generated

by the optimization process. During this analytical

phase, we can introduce various uncertain variables,

supplementing those already identiﬁed in our ﬁeld ob-

servations, such as uncertainties related to travel du-

ration or staff availability.

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