Unleashing Constraint Optimisation Problem solving in Big Data environments

Journal of Computational Science 45 (2020) 101180 Contents lists available at ScienceDirect Journal of Computational Science journa l h om epage: www.elsevier.com/locate/jocs Unleashing Constraint Optimisation Problem solving in Big Data environments Álvaro Valencia-Parraa, Ángel Jesús Varela-Vacaa,∗, Luisa Parodyb, María Teresa Gómez-Lópeza,1 aDpto. Lenguajes y Sistemas Informáticos, Universidad de Sevilla, IDEA Research group, Spain bDpto. Métodos Cuantitativos, Universidad Loyola Andalucía, IDEA Research group, Spain a r t i c l e i n f o Article history: Received 25 November 2019 Received in revised form 2 June 2020 Accepted 18 June 2020 Available online 24 June 2020 Keywords: Big Data Optimisation problem Constraint programming Distributed data Heterogeneous data format a b s t r a c t The application of the optimisation problems in the daily decisions of companies is able to be used for finding the best management according to the necessities of the organisations. However, optimisation problems imply a high computational complexity, increased by the current necessity to include a massive quantity of data (Big Data), for the creation of optimisation problems to customise products and services for their clients. The irruption of Big Data technologies can be a challenge but also an important mechanism to tackle the computational difficulties of optimisation problems, and the possibility to distribute the problem performance. In this paper, we propose a solution that lets the query of a data set supported by Big Data technologies that imply the resolution of Constraint Optimisation Problem (COP). This proposal enables to: (1) model COPs whose input data are obtained from distributed and heterogeneous data; (2) facilitate the integration of different data sources to create the COPs; and, (3) solve the optimisation problems in a distributed way, to improve the performance. It is done by means of a framework and supported by a tool capable of modelling, solving and querying the results of optimisation problems. The tool integrates the Big Data technologies and commercial solvers of constraint programming. The suitability of the proposal and the development have been evaluated with real data sets whose computational study and results are included and discussed. © 2020 Elsevier B.V. All rights reserved. 1. Introduction The use of optimisation problems let organisations manage the resources, time, and cost of their processes. However, the necessity to create customised products according to the profiles of the clients implies the creation of thousands of optimisation problems. Moreover, the incorporation of more interesting data, frequently provided by multiple systems and services [33], will make companies more competitive. Moreover, the integration of more complex data implies the resolution of optimisation problems that could suppose a computationally complex task, further an extra effort to integrate heterogeneous data format provided by different sources. COPs are frequently used to model and solve optimisation problems since they include among their advantages: the ability to model ∗Corresponding author. E-mail addresses: [email protected] (Á. Valencia-Parra), [email protected] (Á.J. Varela-Vaca), [email protected] (L. Parody), [email protected] (M.T. Gómez-López). 1http://www.idea.us.es/ problems declaratively, regardless of how it will be solved; their applicability to many real-life examples of industry, so its versatility and power are empirically demonstrated; and the existence of a significant amount of commercial tools that solve the constraint satisfaction problems locally. The resolution time of the problem can be compromised for both the complexity of the constraints and the size of the data set. Thereby, in the current scenarios, the information involved in a COP is not always centralised. The exponential growth of the data produced and stored and the distribution of the information have promoted the use of Big Data paradigm [25,31], producing difficulties that have not been adapted to COPs [41]. This new situation can produce that, the data and constraints that model their relations are distributed among different subsystems (also known as nodes) that constitute the global model. These models can be computationally highly complex since they can have distributed data or constraints. This interruption of Big Data into COPs can be seen as a new challenge, but also as a new opportunity to solve distributed problems and data. Before the Big Data burst into the scene, Distributed Constraint Optimisation Problems (DCOPs) [21,11] were defined to solve COPs where neither their constraints nor data were in a single system. https://doi.org/10.1016/j.jocs.2020.101180 1877-7503/© 2020 Elsevier B.V. All rights reserved. 2 Á. Valencia-Parra, Á.J. Varela-Vaca, L. Parody et al. / Journal of Computational Science 45 (2020) 101180 DCOP paradigm proposes a set of algorithms to solve a set of COPs that share variables. The resolution of each of them must be aware of the rest of the constraints that restrict the values of the variables. For this reason, DCOPs are focused on the communication between the distributed nodes that must synchronise the share variables to obtain a correct result of every constraint, variable and in each node. However, DCOP proposals do not provide a mechanism for a better distribution of the DCOPs for the optimal resolution of them. In this context, our proposal is based on the creation and distribution of isolated COPs in a Big Data infrastructure, whose partial resolution provides information to solve a greater problem utilising the MapReduce paradigm [10]. Furthermore, none of the algorithms proposed to solve DCOP have been presented as a practical and appropriate solution to actual environments, even less when the data used is characterised by its high variability, velocity, and volume (i.e., Big Data). Although the existence of the problem was formulated in our previous works [35,36], to the best of our knowledge, there are no solutions that adequate the COPs to data supported by Big Data infrastructure, dealing with large amounts of data, using the distributed node to parallelise the computation, getting results quickly, and managing a variety of data efficiently. As aforementioned, we identified the necessity of the COP resolution within different Big Data scenarios in [35,36]. In those previous work, we focused on describing the scenarios and their characteristics, then a solution was prototyped. However, the previous work lacks formalisation, implementation and evaluation with real data was done. In order to fulfil those gaps, we propose a solution which enables to define, execute and solve Constraint Optimisation Problems with Distributed Data in Big Data environment. The main contributions are four main: •Formalisation of Constraint Optimisation Problems with Distributed (Big) Data: An optimisation problem must be modelled describing the constraints that relate the data and the objective function to be optimised. When the data is distributed and the resolution of the problems depends on this data, the description of the model must include the distributed data than can have heterogeneous formats and come from Big Data environments. We formalise COP models that include these aspects. •Integration and transformation of heterogeneous data formats: The heterogeneity of the data formats makes necessary data preparation by means of a transformation into a unified format to apply the latter analysis. This task is especially relevant in our proposal since the mapping between input data to the optimisation problem must be defined, at the same time that the output obtained must be recovered. •Enable the optimal evaluation of queries over data provided by the resolution of Optimisation Problems: The resolution of various COPs can provide useful information, for example, to ascertain the most suitable products for each provider. However, to select the most appropriate for a specific organisation, these partial solutions obtained from the evaluation of the COPs must be queried. Our proposal is focused on the analysis of the queries over the COP data output to avoid, when it is possible, the resolution of the COPs, minimising the evaluation time. •Provide an integral framework to support the process: To cover the previous proposals, it is necessary a leap of advances in technologies that support the integration of the Big Data infrastructures with constraint optimisation solvers. This new technology is proposed as a new component in the Big Data ecosystem to enable the management of a great amount of data and the creation and distribution of several COPs, whose output data can be later queried. Unfortunately, there are no technological solutions that enable the application of optimisation problems with Big Data characteristics in real scenarios. This lack of tools means that these challenges need an extra effort to be solved. In order to tackle our proposal, FAst BIg cOstraint LAboratory (FABIOLA) framework is proposed. FABIOLA is a new Hadoop-based component [3] in the Big Data ecosystem. FABIOLA enables the modelling of optimisation problems whose heterogeneous format of the data, the amount of data to deal with, and the required velocity in the resolution of the problems forces a Big Data solution. The FABIOLA framework also provides the necessary techniques to solve COPs in a distributed way, either to obtain a result by optimising time and resources or because the nature of the problem is distributed. FABIOLA prototype was presented in previous works [35,36], thereby, this paper can be seen as an extension of the previous ones. However, certain challenges tackled here were not tackled in that previous works: how to transform the data for its integration with other distributed data; how to optimise the data obtained from the optimisation by means of a query language that reduces the COPs evaluation, and; how an integral solution could support the whole process providing an implemented tool used for the community as a new component of the Big Data ecosystem, and; the empirical evaluation of the approach with real data. The remainder of the paper is organised as follows: Section 2 formalises the modelling of the optimisation problems in Big Data environments. Section 3 details the different phases of the proposal that are illustrated with a running example to make the problem understandable. Section 4 analyses various operators for processing the results obtained from the resolution of the optimisation problems. Section 5 details the proposed architecture and the methodology necessary to support FABIOLA framework and the developed infrastructure. Section 6 presents the formalisation applied to a real case with a high volume of data. This section also shows a computational, statistical, and analytical study of our proposal. Section 7 includes the main related work. Finally, conclusions are drawn and future work is proposed in Section 8. 2. Overview of the proposal: Optimisation Problems within Big Data scenario The creation of a COPDD is a complex task, especially when the quantity of data involved, and the number of COPs is very high. This is why this creation must be recommendable automated. For this reason, our proposal integrates a set of components that facilitate both the creation and the distribution of the COPs for their resolution. Definition 1. A COPDD is defined by the tuple DS, COP, DM, DMapDS →DM, DMapDM →COP → DSoutput, where: •DS is the Data Set used as input data to find out the optimal solution of the problem. •COP is the Constraint Optimisation Problem. •DM is the Data Model that describes the relationship between the attributes of the DS and the COP. •DMapDS →DM and DMapDM →COP are the Data Mapping (DMap) that represents the relation between the attributes of the DS and the attributes of the DM, and the attributes of the DM and the variables used in the COP, respectively. •DSoutput is a data set obtained from the resolution of an optimisation problem according to the attributes defined in the DM and their corresponding values. Each element of the tuple of the COPDD plays a role to create and solve the COPs in an efficient way, as shown in Fig. 1. They are combined to create a framework that integrates the four phases of the Á. Valencia-Parra, Á.J. Varela-Vaca, L. Parody et al. / Journal of Computational Science 45 (2020) 101180 3 Fig. 1. Overview of the approach. COPDD creation: (1) data set preparation; (2) COP description; (3) mapping among the data sets and the COP, and; (4) data querying optimising the performance. Some definitions have been included to clarify the phases of the proposal, each of them is detailed in the following sections. 2.1. Running example To understand the proposal, a simple running example is used, although a real and more complex case study to compute the evaluation is given in Section 6. The running example is about a planning problem [9], that describes a company that manufactures two types of products: a standard product A, and a more sophisticated product B. Management charges a certain price for each unit of product A and a price for the unit of product B, whose profits are pAand pB respectively. Therefore, the profit of the company comes from the multiplication of the price by the number of units (i.e., qAand qB for products A and B) of each product. Manufacturing one unit of product A requires hAhours of labour and rmAunits of raw material. Similarly, for one unit of B, hBhours of labour and rmBunits of raw material are needed. Besides, the company’s policy indicates that the number of units manufactured of product A has to be, at least, the double of units than the produced for product B. At present, a nH of hours of labour and a nRM of units of raw material are available. The problem is focused on to maximise the company’s total revenue. Although every element of the COPDD will be described in the following sections, a brief approximation to the formalisation with the running example can be done at this point by identifying the Data Set (DS) and data to optimise: •DS: priceA, priceB, numUnitsA, numUnitsB, hoursA, hoursB, hoursLabour, rawA, rawB, unitsRawMaterial. •COP is defined to maximise the profit. 3. Phases of the proposal As presented in Fig. 1, four are the phases that our proposal supports to create and solve the COPDDs. Following sections describe each of them. 3.1. Data set preparation Data preparation is one of the most consuming processes in the Big Data pipeline [22,44]. One of the steps is the data transformation to fit the heterogeneous data formats into a single one for a later application of algorithms over all tuples. This type of problem is also tackled for the COPDD, that needs a DS with homogeneous tuples used as inputs. It implies that each input data (Dataa, Datab, . . ., Datan) must be transformed into a single one structure (DS), following the data transformation processes as detailed in [46]. Therefore, a heterogeneous data format is a set of data (Dataa, Datab, . . ., Datan) formed of a different set of attributes among them. Definition 2. A DS is a set of tuples formed of a set of attributes, {a1, a2, . . ., an}, where each tuple presents an assignment of values {val1, val2, . . ., valn} to each attribute. Some of the elements of the second row of Table 1 (cf., #ID, nH, nRM, hA, rmA, pA, hB, rmB, pB) presents the elements of the DS for the running example. 3.2. Constraint Optimisation Problem description A COP is a Constraint Satisfaction Problem (CSP) where an optimisation function determines the ‘best solution’ from the set of possibles. They come from Constraint Programming (CP) [39], an Artificial Intelligence (AI) discipline where a large number of problems and other areas of Computer Science can be seen as individual cases of CSP. Examples include scheduling, temporal reasoning, graph problems, and configuration problems. The basis of CP stands on the resolution of a CSP. Thereby, to understand correctly what is a COP, the CSP definition must be introduced before. Definition 3. A CSP represents a reasoning framework consisting of variables, domains and constraints V, D, C, where V is a set of n variables {v1, v2, . . ., vn} whose values are taken from finite domains {Dv1, Dv2, . . ., Dvm} respectively, and C is a set of constraints on their values. The constraint ck(xk1, . . ., xkm) is a predicate that is defined on the Cartesian product Dk1× . . . × Dkj. This predicate is true iff the value assignment of these variables satisfies the constraint ck. 4 Á. Valencia-Parra, Á.J. Varela-Vaca, L. Parody et al. / Journal of Computational Science 45 (2020) 101180 Table 1 Example of tuples for the running example. IN OUT OT #ID nH nRM hArmApAhBrmBpBprofit qAqBComp. #1 1500 400 3 4 5 4 6 10 570 58 28 Comp1 #2 900 200 1 2 3 2 3 4 300 100 0 Comp2 #3 1200 – 7 4 14 – 7 30 160 80 4640 Comp2 #4 – 300 2 3 9 4 6 11 900 100 0 Comp3 #5 1500 – – 1 8 – – 15 12000 1500 0 Comp4 Table 2 COP for the running example. Variables &domains: priceA, priceB, profit: Float; numUnitsA, numUnitsB, hoursA, hoursB: Integer; rawA, rawB, hoursLabour, unitsRawMaterial: Integer; Constraints: B i=A(hoursi∗ numUnitsi) ≤ hoursLabour B i=A(rawi∗ numUnitsi) ≤ unitsRawMaterial numUnitsA− 2 * numUnitsB≥ 0 profit =B i=A(pricei∗ numUnitsi) Optimisation function: Maximise(profit) The search for solutions for a CSP is based on the instantiation concept. An assignment of a variable, or instantiation, is a pair variable-value (x, a) which represents the assignment of the value a to the variable x. An instantiation of a set of variables is a tuple of ordered pairs, where each sorted pair (x, a) assigns the value a to the variable x. A tuple ((x1, a1), . . ., (xi, ai)) is consistent if it satisfies all the constraints formed by variables of the tuple. Several possible solutions can be found to satisfy a CSP, but sometimes only the more suitable wants to be obtained, one that optimises the solution. In those cases, a COP must be modelled. Definition 4. A COP is a CSP in which the solutions optimise (minimise or maximise) an objective function, f. For the example, for maximising the company’s total revenue can be formulated as shown in Table 2. The values of nH, nRM, hA, rmA, pA, hB, rmB, pBdepend on the company where the products are built. Table 1 shows some possible scenarios, where the output data (i.e., profit, qA, qB) is obtained according to each input data (per tuple). Every input data can be instantiated (e.g., tuples # 1 and # 2), and the output indicates the resulting profit and how many units of each product produces (cf., qAand qB). It is also possible that some input values were unknown (cf., the value ‘–’ in the table). In that case, the COP tries to find values for these variables to optimise the variable profit (cf., tuples # 3, # 4 and # 5). In this example, we can observe that there is another attribute, such as Company (cf., Comp.), that is not part of and does not influence the resolution of the optimisation problem. However, this kind of variables can be helpful to answer later queries, such as, Which is the manufacturer with the highest profit? Which is the manufacturer that produces more products B? If there are several tuples of input data, several are the possible optimal solutions that can be found. Thereby, the created COP can be seen as a meta-COP which is partially instantiated for each tuple of the DS. How to map the attributes of DS with the inputs of the COP is described is explained in the next subsections. 3.3. The mapping between the DS and COP In order to link the DS with the COP, it is necessary to define an independent DM which allow users to define how the data from the DS is used as input of the COPs, and the resultant data obtained by solving the COPs. Definition 5. A DM is a set of attributes, {a1, a2, . . ., an}, which is divided into three disjointed groups: Input (IN), Output (OUT), and Others (OT) attributes. IN ∩ OUT = ∅ ∧ IN ∩ OT = ∅ ∧ OT ∩ OUT = ∅(1) These sets of attributes are defined as: •IN: specific attributes from the DS that will be linked to the input variables of the COP. •OUT: attributes that describe the output data of the COP. The values of these attributes are obtained from the COP resolution. •OT: specific attributes form the DS that describe additional information and that can be used to make further queries combining them with output attributes. These attributes are unrelated to the COP since they lack influence in its resolution. The DM enables the separation of the DS and the COP model in such a way that the COP model can be applied to many data sets. However, to relate the DM and attributes of the DS and the variables of the COP, we need to introduce another concept, the DMap. Following definitions determine how the DS attributes and the COPs inputs are mapped through the DM. Definition 6. DMapDS →DM is the relation between the attributes of DS and the IN and OT attributes of the DM, formed of a list of attributes {a i, . . ., a m} ⊆ DS where a subset of attributes are related to IN, {a i, . . ., a m} → {IN}, and another subset of attributes are related to OT, {a h, . . ., a g} → {OT}. DMapDS→DM : DS → {IN, OT}, (2) ∀a i∈ DS : ∃ix∈ IN|a i= ix∧ oh∈ OT|a i= oh, |IN| ≥ 1 (3) Definition 7. DMapDM →COP is the relation between the attributes of the data model DM and the variables of the COP, V. It is described by a list of attributes {ik, . . ., in, outg, . . ., outl} where a subset of attributes from IN are related to the input variables of the COP, {ik, . . ., in} → {V}, and another subset of attributes from OUT are related to the output variables of the COP, {outg, . . ., outl} → {V}. DMapDM→V: {IN, OUT} → V, |V| ≥ 2 (4) ∀ik∈ IN : ∃vj∈ V|ik= vj∧ vh∈ V|ik= vh(5) ∀outg∈ OUT : ∃vi∈ V|outg= vi∧ vj∈ V|outg= vj, (6) ∀(ik, outg)|ik∈ IN, outg∈ OUT : vt∈ V|ik= vt∧ outg= vt(7) Following the running example, the DS in Table 1 is remarked as IN, OT, and OUT according to the DM definition and to represent the DMap. To illustrate how the Data Mapping is carried out, once the transformation has been performed. Fig. 2 represents the mapping between the DS and the COP through the DM. A first DMap determines the relation between the attributes of the DS and the DM. The second DMap relates the DM and the variables of the COPs [46]. Applied to the running example, the attributes of DS (nH, nRM, hA, rmA, pA, hB, rmB, pB, Comp.) are mapped into attributes of the DM as labelled as IN and OT in Table 2. For instance, nH from the DS is mapped as nH IN’s attribute. Similarly, nH IN’s attribute is mapped as hoursLabour variable of the COP. Á. Valencia-Parra, Á.J. Varela-Vaca, L. Parody et al. / Journal of Computational Science 45 (2020) 101180 5 Fig. 2. Data mapping applied to the example. Fig. 3. Process for computing COPDD for distributed data. Regarding the constraints, the COP can be modelled employing a set of fixed constraints such as seen in the example of Table 2. Therefore, each tuple represents a possible assignment of data to the variables of the COP, which can be solved (i.e., OUT) independently. However, in other types of problems, the constraints are built dynamically due to the constrained-relation existing between the values of the attributes of the DM. Thus, each tuple produces a customised COP. The solution of these customised COPs could be influenced by a criterion regarding the IN attributes, for instance, to order the COPs based on descending criteria of pAattribute. This 6 Á. Valencia-Parra, Á.J. Varela-Vaca, L. Parody et al. / Journal of Computational Science 45 (2020) 101180 Fig. 4. Illustration for independent resolution of COPDDs. criterion establishes the solution of ID tuples: # 3, # 4, # 5, # 1, and # 2. Thus, COP for the tuple # 3 will be solved in the first place, the COP for the tuple # 4 in the second place, and so on. 3.4. Data querying The DSoutput contains every DS tuple combined with the outputs (OUT) obtained after the distributed COPs evaluations. Later analysis of this data should be relevant. Following the running example, to know the maximum profit, the minimum or the average. Classical query operators can be applied to DSoutput, but it is necessary to analyse which types of attributes (IN, OUT and OT) can be involved in each operator. For this reason, we revisit the classical relational algebra [42] employing its unary operators (i.e., selection, projection and aggregation) that can be applied to DSoutput. •Selection (). When the selection operator is applied to a DSoutput, depending on the types of attributes involved in the ϕ, the selection can be applied before and/or after the resolution of the COPDD. In our proposal, only IN and OT attributes can be involved. •Projection (). When the projection operator is applied to a DSoutput, the attributes involved {a1, a2, . . ., an}⊂ {IN, OUT, OT}. For this reason, the projection is applied after the optimisation problem resolutions, since OUT attributes can be included in the projection. •Aggregation (). There are five aggregation functions (f), i.e., SUM, COUNT, AVG, MAX, and MIN. The application of each function returns a number (Integer or Float) and can be applied in IN, OUT and OT attributes. These three operators can be combined and nested (thanks to the closure property of the relational algebra) to query a DSoutput. In following section, we propose to overtake part of the query evaluation to the COPs resolution to reduce the number of optimisation problems to solve, and take the advantages of the distributed and parallel evaluation of the COPs, as it is detailed in the following subsections. 4. Optimising data set query for solving COPs The presented operators can involve every tuple, however, some data analysis done by querying could not entail every tuple, such as ’to obtain the maximum profit when nH is greater than 1, 200’. In this section, we analyse how they can be solved to reduce the computation complexity of obtaining and querying DSoutput, proposing the systematic process shown in Fig. 3 applied before the COPDDs evaluation, since the order in which the operators are applied affects the time consuming of the distributed resolution of the optimisation problems drastically, being crucial how the tasks to solve a query are scheduled [43] in Big Data ecosystems. The application of these steps avoids the resolution of COPs that do not fit the selection operator or those that do not correspond with the attribute to aggregate. The first step is to analyse whether a selection operator is included. It might reduce the input data involved in the COPDDs. For instance, in the running example of the previous section, let be only optimised the products whose rmAhave a value greater than 2, the tuples # 2 and # 5 might not be included in the computation of the COPDDs. If there is a selection operator, the application of the filtering must be developed, as it is carried out in classical data. In the current proposal, only selections over input and other types of attributes can be applied. The second step is related to the aggregation operator. If an aggregation operator is included, how the information is managed depends on the types of attributes and the aggregation functions (i.e., SUM, COUNT, AVG, MAX, and MIN). IN and OT are managed as usual data applying the aggregation operator or relational algebra, meanwhile OUT for MIN or MAX function will be used to improve the optimisation of the COPs to reduce the search space of the variables. Therefore, the Computation of independent COPDD or Computation COPDD with reduction will be carried out depending on whether an aggregation operator over OUT attributes have been used or not. Both types of COPDDs are explained in the following subsections. The third step is related to the projection operator, where a new DS is created as a subset of attributes of the DSoutput. This new DS is the result of the projection operator and the solution to the problem. An example is to obtain the profit but not being necessary to return qAand qB. 4.1. Computation of independent COPDD When operators are applied to OUT attributes, every COPs can be solved individually, being or not in distributed nodes. Thereby, the DS described in the formalisation can be divided into different and distributed nodes. DS can be represented as a set of data sets, ds1, ds2, . . ., dsn, each of them located in an independent node. Each tuple p within a dsi∈ {ds1, ds2, . . ., dsn} represents a problem to be optimised. Each COP depends on the values of the DM for each tuple. It is similar to use each tuple p to instantiate the COP, creating a COPpin which the values are taken from p. The group of COPs generated for each tuple of all DS conforms a workgroup, as we defined in Fig. 4. A particularity of this scenario is that the COPs within the workgroup can be computed independently without the application of any operator. Thus, the workgroup can be seen as a unique task Á. Valencia-Parra, Á.J. Varela-Vaca, L. Parody et al. / Journal of Computational Science 45 (2020) 101180 7 Fig. 5. Illustration for reduction based on aggregation. Fig. 6. Architecture and methodology of FABIOLA. that needs to be computed in a distributed way. As commented, in the running example can be to obtain the average of the profit for all the products, being necessary to compute all the COPs to obtain the profit values before obtaining the average. The calculation of the profit for each COP is unrelated to any other COP. Therefore, the computation of the COPs can be performed independently at any order, because all the COPs must be solved. 4.2. Computation of COPDD with reduction When an aggregation operator is applied to obtain the MAX or MIN value of an OUT attribute, the evaluation of a COPpcan use solutions obtained from the resolution of previous COPs of the same DS (dsi). Thus, each tuple conforms a COPpin which constraints and values are taken directly from p. There are possible values from the same DS that might be shared from one solved COPito another unsolved COPjto improve the search for solutions reducing the search space. Thus, the domain of variables of the (unsolved) COPjmight be reduced to enhance the search space. Following the running example, the objective function could be to determine the maximum profit with the minimum value of units of A (cf., qA). In this case, assuming the solution of the COP, for instance, for the tuple # 1 the values of profit and qA(cf., Table 1) could be used to reduce the domains of the variables profit and numUnitsAin the unsolved COPs since a greater value of numUnitsAand a less value of profit than previously calculated are unacceptable. Despite reduction operations, the aggregation operator (i.e., ) is necessary since the results of all COPs must be combined to determine the maximum profit and the minimum qA. As defined in Fig. 3, the aggregation operation has to be applied in the output variables profit and numUnitsA. In this scenario, problems from the same DS might be related internally to each other by reduction operations but they are unrelated to problems from other DSs. Therefore, the COPs of each DS, dsi, conforms independent workgroups that can be solved without regarding other workgroups. Thus, each workgroup can be seen as a unique task that needs to be computed in a distributed way. In Fig. 5, two workgroups are depicted related to the DS, ds1and dsn, 8 Á. Valencia-Parra, Á.J. Varela-Vaca, L. Parody et al. / Journal of Computational Science 45 (2020) 101180 Fig. 7. Data mapping in FABIOLA-UI. wherein each workgroup reduction are illustrated as arrow transitions between the COPs and  represents the aggregation operator which is applied in this context. In Big Data environments, the resolution of a problem can be based on partial problem resolutions that are combined, which is known as MapReduce technique [10]. It can also be applied to the resolution of optimisation problems. For instance, the maximum value of an attribute, where the value of this is distributed in different workgroups, is the maximum value of the all maximum obtained for each workgroup. This implies to optimise a problem, with the optimisation of each workgroup, that has been solved in proposals such as Hive [45]. The question is, how this optimisation process can be affected when the partial resolution implies also the COP resolutions. The solution of all workgroups need to be combined (cf. Fig. 5) to generate a global solution as a new OUT. A way of combining needs to be defined over the OUT attributes of COPs. In order to carry out this, MapReduce for optimisation is defined as follows. Definition 8. A MapReduce for Optimisation Problems, MR, is defined as a combination function which establishes the maximum or minimum value of an OUTivariable, being OUTi∈ {out1, out2, . . ., outn} and establishes a way of the combination by means of the operator, OP, as minimise (MIN) or maximise (MAX). Each variable v ∈ {out1, out2, . . ., outn} involved in the aggregation operator for a COPiis bounded with a specific value obtained in previous solved COPs of the same DS, (dsj). This specific value is a maximum whether the MIN value wants to be obtained, or the minimum whether the MAX function is used in the aggregation. 5. FABIOLA: the technological approach FABIOLA is presented as a technological solution for the automatic creation of COPs with distributed Data. In this section, the needed infrastructure, including the architecture, and the implemented tool are detailed. FABIOLA as a technological solution is composed of two differentiated parts: (1) the infrastructure (i.e., back-end) for the computation, and; (2) a web interface (i.e., front-end) to enable the access to the components described in previous sections. The FABIOLA back-end and front-end components are depicted in Fig. 6. The FABIOLA front-end is divided into two components: •FABIOLA-UI component is a web client-side application which enables to setup the environment to compute COPDDs. It provides sections for the Data importer to load external data; the COP description; Data Transformation; the Problem configuration (i.e., Data Mapping), and the System Configuration in terms of solver, memory, timeout of execution, etc. •FABIOLA Dashboard is a control panel from where the user can use various reporting and querying components. The dashboard enables users an easy-querying and visualisation of the data and results in FABIOLA. FABIOLA-UI, on the left, provides a fixed menu which gives direct access to different parts of FABIOLA components such as the dashboard, DS importers, DS creation, COP models, and instances as shown in Fig. 7. The same figure presents the main view once entering FABIOLA-UI in the section of creating a new instance. The creation of a new instance in FABIOLA means the creation stepby-step of a new COPDD. The figure shows an example of how to carry out the data mapping between the DS (cf., DS schema) and the attributes of the DM. The user can choose by drag-and-drop which attributes from imported DS are IN and OT. FABIOLA-UI provides an editor for the description of the COP model in Constraint Optimisation Problem. Currently, FABIOLA-UI supports any ChocoSolver-based optimisation syntax. Although the description of the COP can be a difficult task, it is done once and can be applied to every DS. In a similar way than in data mapping between DS and DM, the DMap between DM and the variables of the COP is described. FABIOLA-UI provides customised interfaces where operators can be applied over the data and COP results. For instance, projection operations can be applied for the results such as the forms provided. In this case, it is described the attributes to projection, selection and Á. Valencia-Parra, Á.J. Varela-Vaca, L. Parody et al. / Journal of Computational Science 45 (2020) 101180 9 Fig. 8. Data visualisation of results. Fig. 9. Data visualisation aggregated by regions. 16 Á. Valencia-Parra, Á.J. Varela-Vaca, L. Parody et al. / Journal of Computational Science 45 (2020) 101180 Fig. 16. Each point, triangle and square is the Elapsed Real Time average for five executions. Black continuous line represents the overall Elapsed Real Time of the execution including aggregations and COP resolutions; black pointed line represents the Elapsed Real Time of the COP resolutions with domain reduction, and red line represents the Elapsed Real Time of the COP resolution without domain reduction. use various of them since each one is optimised for specific functionality. Although FABIOLA is not a new Big Data solution, it aims to be part of the Hadoop ecosystem. FABIOLA provides the necessary components to drive the solution of COPs with distributed data on a Hadoop-based architecture. In [54], Zhu et al. present various proposals that solve some of the challenges for optimisation problems in distributed information systems. The proposals focus on the optimisation of systems performance and three of them combine optimisation problems in big data environments. First, Zeng et al. [50] emphasised minimising the cost of renting cloud resources for executing MapReduce applications in public clouds, and propose a greedy-based scheduling algorithm to tackle this issue. Secondly, Zhou et al. [53] efficiently distribute data on different devices by considering the asymmetric characteristics among devices and data. To do that, they propose a preference model to quantitatively weight the storage performance imbalance when data are distributed on different devices. Finally, Zhao et al. [52] propose an algorithm to solve the data allocations under multiple constraints in polynomial time. However, these proposals are focused on performance optimisation and not on optimising the results obtained. On the other hand, Chen et al. [8] present a survey where challenges and opportunities of Big Data in data-intensive systems are identified. The authors identified the opportunity to improve decision-making focused on the customer, for instance, by developing customised products. In order to do that, the authors identified the techniques and technologies of Big Data. The authors highlight the need for the application of optimisation techniques to efficiently process a large volume of data within limited run times. Several optimisation techniques are mentioned (cf., Mathematical tools) however they missed out CP as an optimisation technique. CP presents a challenge in the scalability by solving some hard problems. However, CP has been successfully applied in different domains for solving optimisation problems, such as scheduling and planning. Although there exist several CP tools, such as IBM CPLEX Optimisation [23] and ChocoSolver [37] and solutions that create Constraint Problems according to the constraints extracted from databases [5,18,19,17,38], none of them provides an integrated solution for a Big Data solution. Big Data provides to CP a new perspective concerning the size and volume of data, and it is an excellent opportunity to exploit its possibilities to gain efficiency and optimisation in operational processes [34]. Additionally, Big Data tackles new challenges [16] dealing with automation of decision-making that involves several (millions) decision variables Á. Valencia-Parra, Á.J. Varela-Vaca, L. Parody et al. / Journal of Computational Science 45 (2020) 101180 17 Fig. 17. Memory consumption as the number of COPs scale progressively. Each point represents the average of the total of five executions (Kbytes). in optimisation of resource consumption, sustainability services, and finance. Nevertheless, the optimisation problems in CP need more flexibility and adaptability, since the exploration of heterogeneous, enormous and dynamic generation of data requires a quick adaptation of optimisation problem to more holistic solutions. DCOPs have been widely tackled in the literature [15,26,29,48,27,47]. Due to the complexity of the problems (NP-complete, [7,4]) most of the efforts have been done in a theoretical way with the proposition of models, algorithms, and strategies to solve this type of problems. Only ad-hoc applications [29], heuristic approaches [14,13] or simulators [15] have been developed for the community to solve those type of problems. The main challenges in the DCOP solvers are based on overcoming the massive use of memory needed to run the algorithms and to manage the highest number of messages exchanged between nodes. Currently, none approach of DCOPs applied to Big Data has been identified in the literature. To the best of our knowledge, FABIOLA is the first approach that applied Big Data in CP, although various have been the examples where the necessity to optimise problems with Big Data has been detected [51,24]. There are seminal works related to the application of Big Data in CP. Sunny-CP [1,2] is a portfolio of CP-solvers which can exploit the multicore capabilities of the machines to solve CSPs and COPs. However, this approach lacks applicability in the context of Big Data with multiple data sources with a large-scale amount of data. Cao et al. [6] study how to improve the solution of a largescale Integer Linear Programming (ILP) problems employing Big Data architecture. ILP is a discrete approach compared with CP. The authors applied ILP to solve traffic flow management. In [28], the authors propose the application of a CP-based scheduler for a Big Data architecture. On the other hand, there is an initiative to create a new language to adapt CP languages for Big Data applications [40], it is currently a very undeveloped approach with no continued development. Other approaches to Big Data and Optimisation techniques applied to real cases have been studied in [12]. For instance, the maritime logistics problem is modeled as a constraint problem and optimisation techniques have been applied although most of the approaches are focused on machine learning or logic programming techniques. 18 Á. Valencia-Parra, Á.J. Varela-Vaca, L. Parody et al. / Journal of Computational Science 45 (2020) 101180 8. Conclusions and future work Optimisation problems are found in several real-life scenarios. These optimisation problems become an extraordinary problem when the data involved are in a Big Data environment, which implies a massive quantity of information, that is also distributed and heterogeneous. FABIOLA has been developed to support the definition and resolution of COPs within Big Data scenarios. FABIOLA isolates the resolution of optimisation problems from where the data are and how the optimal outputs are found. Thanks to this isolation, the resolution of optimisation problems in Big Data environments has never been so simple as with FABIOLA. Internally, FABIOLA has been provided with a set of operators to query the optimal information obtained. In order to reduce the computational time and resources needed to solve the optimisation problems, the operators are applied to enrich the capabilities of the resolution of COPs by using the MapReduce operations. Besides, our solution facilitates the COPs modelling through a user interface, FABIOLAUI, that guides the final users through the application. FABIOLA has been evaluated in real scenarios where the necessity of improving the billing of thousands of customers by using COPs is essential. FABIOLA demonstrated to be a scalable and adaptable solution that improves trivial and stand-alone solutions. However, the most important results are the incredible improvement regarding the performance of execution and resource consumption in optimisation problems. The resolution of optimisation problems in Big Data environments has never been as simple to model and fast to solve as with FABIOLA. As future work, it would be interesting to extend FABIOLA to support the adaptation of various constraint solvers in the architecture. On the other hand, FABIOLA can be extended in many directions, for instance, related to the inclusion of new operators to optimise the queries. The possibilities of extends FABIOLA in many directions related to the operators and other scenarios such as distributed optimisation problems are desirable. Authors’ contribution All the authors are responsible for the concept of the paper, the results presented and the writing. All the authors have approved the final content of the manuscript. No potential conflict of interest was reported by the authors. Conflict of interest All the authors are responsible for the concept of the paper, the results presented and the writing. All the authors have approved the final content of the manuscript. No potential conflict of interest was reported by the authors. Acknowledgements This research was partially supported by Ministry of Science and Technology of Spain with projects ECLIPSE (RTI2018-094283- B-C33) and by Junta de Andalucía with METAMORFOSIS projects; the European Regional Development Fund (ERDF/FEDER); and by the University of Seville with VI Plan Propio de Investigación y Transferencia (VI PPIT-US). References [1] R. Amadini, M. Gabbrielli, J. Mauro, A multicore tool for constraint solving, Buenos Aires, Argentina, July 25–31, in: Proceedings of the Twenty-Fourth International Joint Conference on Artificial Intelligence, IJCAI 2015, 2015, 2015, pp. 232–238. 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Sci. 26 (2018) 409–418, http://dx.doi.org/10. 1016/j.jocs.2017.01.003. [54] X. Zhu, L.T. Yang, H. Jiang, P. Thulasiraman, B.D. Martino, Optimization in distributed information systems, J. Comput. Sci. 26 (2018) 305–306, http://dx. doi.org/10.1016/j.jocs.2018.04.020. Álvaro Valencia-Parra is a computer science Ph.D. student at the University of Seville. His research areas include the improvement of data transformation and data quality fields in Big Data paradigms. His goal is to improve the way in which final users deal with data preparation and specific scenarios in which configuring a Big Data pipeline might be tricky. For this purpose, he is working in the improvement of the data transformation processes by designing Domain-Specific Languages for very specific purposes, user interfaces, and semi-automatic approaches in order to assist users in these tasks. Angel J. Varela-Vaca received the B.S. degree in Computer Engineering at the University of Seville (Spain) and graduated in July 2008. M.Sc. on Software Engineering and Technology (2009) and obtained his PhD with honors at the University of Seville (2013). Angel is currently working as Assistant Professor at Languages and System Informatics Department at the Universidad Sevilla and belongs to the Idea Research Group. Angel has and leaded various private projects and participated in several public research projects and he has published several impact papers. He was nominated as a member of Program Committees such as ISD 2016, BPM Workshops 2017, SIMPDA 2018. He has been reviewer for international journals such as Journal of Supercomputing, International Journal of Management Science and Engineering Management Multimedia Tools and Applications, Human-Centric Computational and Information Sciences, Mathematical Methods in Applied Sciences among others. Luisa Parody studied computer engineering (including a minor in systems engineering) at the Universidad de Sevilla (Spain) and graduated with honours in July 2009. She then earned an M.Sc. degree in software engineering and technology (2010) and obtained her international PhD with honours at the Universidad Sevilla (2014). Since 2018, she has been working as an associate professor in Dto. Método Cuantitativos at the Universidad Loyola. She belongs to the IDEA Research Group and has participated in several private and public research projects and has published several high-impact papers. María Teresa Gómez-López is a Lecturer at the University of Seville and the head of the IDEA Research Group. Her research areas include Business Processes and Data management, and the things that disturb his thoughts are how to improve the business process models including better decisions and enriching the model with Data Perspectives. She has led several private and public research projects and has published several impact papers, among others in Information and Software Technology, Information Systems, Information & Software Technology, or Data & Knowledge Engineering. She was nominated as a member of Program Committees, such as ER 2018, BPM 2017, ISD 2017, ICIQ 2016, PHM 2016, IAwDQ 2013 or CLEIS 2012. She has been reviewing for international journals, such as International Journal of Data and Information Quality, Journal of Systems and Software, Artificial Intelligence in Medicine; Business & Information Systems Engineering Journal or Information Science. She has given keynotes or was invited speaker at the IV Workshop on Data & Artifact Centric BPM in Innsbruck, 5th International Workshop on Decision Mining & Modeling for Business Processes BPM in Barcelona, the X National Conference of BPM in Madrid, in the 28th IBIMA Conference, and in the biannual International Summer School on Fault Diagnosis of Complex Systems.