A multilevel approach to ubiquitous modeling and solving constraints in combinatorial optimization problems in production and distribution
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Abstract
Constraints, although ubiquitous in production and distribution planning, scheduling and control, often lead to inconsistencies in the decisionmaking process. The constraintbased modeling helps circumvent many organizationimpacting issues. To address this, we developed a multilevel approach to the modeling and solving of combinatorial optimization problems. It is versatile and effective owing to the use of multilevel presolving and multiple paradigms, such as constraint programming, logic programming, mathematical programming and fuzzy logic, for their complementary strengths. The capability of this framework and its advantage over mathematical programming alone or over hybrid frameworks is shown in the illustrative example, in which combinatorial optimization is used as a benchmark to prove the effectiveness of the proposed approach. Knowledge of the problem is stored in the form of facts.
Keywords
Constraint logic programming Mathematical programming Constraint satisfaction problem ubiquitous modeling and solving constraints Presolving Hybrid methods Manufacturing and distribution1 Introduction
Constraints are ubiquitous in various areas of production and distribution. They may correspond to materials, technologies, resources, time, capacity, interoperation transport, etc. (production) or to available storage capacity, selection of distributors, transport duration, number of transport means, their types, etc. (distribution). Proper constraint modeling and solving will have a considerable impact on productiondistribution planning and control and will allow decisionmakers to detect difficult situations early enough to handle them safely. If the constraints are modeled and solved, fully or partially, at the moment they appear in a given area or for a given problem, but not collectively in subsequent stages, the global decision or optimization model will be simpler and solving it will take less time. We can thus talk about ubiquitous modeling and solving of constraints (UMSC) in these problems.
The concept of UMSC can be the basis for creating decision support systems and process optimization and control at various levels. Planning, scheduling and resource allocation problems are usually modeled using OR (operation research) approaches such as mathematical programming (MP) methods, with linear programming (LP) used at the strategic or tactical level (aggregated level). More detailed planning, scheduling and resource allocation require integer and binary decision variables and to this end, MIP (Mixed Integer Programming), IP (Integer Programming) or MILP (Mixed Integer Linear Programming) Schrijver [17] models have to be employed. The resulting models are very complex. To solve them, substantial computing effort is necessary. Real problems may become NPhard problems. In the 1990s, constraint programmingbased environments appeared Rossi et al. [16]; Benhamou et al. [3]; Tsang [25]; Apt [1] and their implementations to production and distribution problems were reported Liess and Michelon [12]; Rocha and Ramos [15]; Bocewicz and Banaszak [4]. Enormous flexibility, easiness and the range of constraint modeling gave them definite advantage over ORbased environments. But only at the modeling stage – CPbased environments turned out to be less effective at the solving stage, in particular, in solving combinatorial optimization problems. Their effectiveness often depends on the structure of modeled constraints and data, or on decision variable domain range. This issue does not exist in the MPbased environments, where modelsolving methods are independent of both constraint structure and data. As for the UMSC concept, the approach that combines multiple paradigms such as CP constraint programming, LP logic programming, MPmathematical programming, FLfuzzy logic, etc. is the best solution.
The paper is organized as follows. Section 2 describes the constraintbased methods. The multilevel approach to ubiquitous constraints modeling and solving is presented in Section 3. The illustrative example and computational results of the proposed approach are described in Sections 4 and 5. Final remarks and possible future development and applications are shown in Section 6.
2 Constraintdriven programming, mathematical programming and integrated methods – basic principles
For the ubiquitous modeling and solving of different types of constraints, the most effective and most flexible are constraintdriven methods and environments based mostly on constraint satisfaction problems (CSPs) Tsang [25]; Apt [1]. A constraint satisfaction problem (CSP) is defined by a set of decision variables, X _{1},X _{2},...,X _{ n }, and a set of constraints, C _{1},C _{2},...,C _{ m }. Each decision variable X _{ i } has a nonempty domain D ^{ X i } of possible values. Each constraint C _{ i } involves some subset of the decision variables and specifies the feasible combinations of values for that subset. A state of the problem is defined by an assignment of values to some or all of the decision variables, {X _{ i } = v _{ i },X _{ j } = v _{ j } , ...}. An assignment that does not violate any constraints is called a consistent or legal assignment. A complete assignment is one in which every variable is mentioned, and a solution to a CSP is a complete assignment that satisfies all the constraints. Some CSPs also require a solution that maximizes an objective function than we speak about COP (Constraint Optimization Problem).
For constraints that bind more than two decision variables, constraint propagation effectiveness decreases significantly and the number of backtrackings increases rapidly Sitek and Wikarek [20]. This feature makes the CSPbased environment, such as CP (Constraint Programming) and CLP (Constraint Logic Programming) less effective in solving complex problems, such as planning, scheduling or resource allocation. The structure of the constraints has no effect on the effectiveness of the MP methods, in contrast to a large number of integer decision variables, which, when present, reduce it substantially. Both MP and CLP involve decision variables and constraints. However, the types of the decision variables and constraints that are used and the way the constraints are solved are different in the two approaches Bockmayr and Kasper [5]; Hooker [11]; Barth and Bockmayr [2]. MP methods take into account only linear constraints (equations and inequalities) which include binary, integer and continuous decision variables. In the constraintdriven approach, the programming language is richer in terms of the types of constraints. In addition to linear equations and inequalities, there are various other constraints: nonlinear, disequalities, and symbolic (disjunctive, exclude, cumulative, alldifferent, profile etc.) Rossi et al. [16]. Moreover, CLP is less effective in combinatorial optimization problems Escudero et al. [9] that often occur in planning, scheduling, and control in manufacturing and distribution.
To sum up, in mathematical programming, the sets of constraints (equations) describe the problem but do not indicate how to solve it. In constraint programming, each constraint invokes a procedure that screens out unacceptable solutions. The most important elements that decide the effectiveness of mathematical programming include: relaxation, tools for filtering and duality theory Schrijver [17]. Relaxation methods in MP tend to be more effective when constraints and/or objective functions contain many decision variables. In general, MP relies on numerical calculation, which increases its effectiveness. In contrast, constraintbased environments (CP/CLP) may fail when constraints contain many decision variables. This follows from the fact that these environments (CP/CLP) are based on logic processing and the constraints do not propagate well. Moreover, CP/CLP are often insufficient for finding optimal solutions due to lack of relaxation technology and numerical calculation. However, the CP/CLP environments are a more powerful modeling language with which any type of constraints can be easily modeled and they use the structure within a problem (horizontal structure), thus contrary to MP with real vertical structure (a model is independent from data). Also, the CP/CLP environments have built in methods for removing infeasible values from variable domains (filtering, domain consistency, constraint propagation, etc). It is clear that both approaches (MP and CP/CLP) are complementary in many aspects and areas.

Double modeling  use both CLP and MP models and exchange information while solving.

Searchinference duality  view CLP and MP methods as special cases of a search/inference duality.

Decomposition  decompose problems into a CLP part and an MP part using a Benders scheme (Benders decomposition integrates two solution methods: one that solves the master problem, and one that solves the subproblem).
In all our previous papers dealing with hybridization Sitek and Wikarek [20, 21, 22], the problem was modeled as a whole at one level, the model was transformed and on the basis of the posttransformation model, the ultimate problem in the form of MILP was generated. Even though the hybrid approach made it possible to solve larger problems and to reduce optimization time relative to mathematical programming methods, its effectiveness in these two areas was insufficient for SSCM (Sustainable Supply Chain Management) problems. This is how the idea of developing a multilevel approach implementing the UMSC concept appeared. The proposed solution was especially effective in solving discrete optimization problems for SSCM, as shown in the form of comparative analysis with the hybrid and MP methods in Fig. 13a, b and c. The contribution of this study is in the modification of the original hybrid approach and its extension to the multilevel form represented by a modified model transformed for the SSCM problem. This approach is able to use the ubiquitous modeling and solving constrains for combinatorial optimization problems in manufacturing and distribution problems. The proposed approach integrates CP, CLP, MP and Fuzzy Logic (FL). It is an extension of the hybrid approach but differs from it in two aspects. The problem has to be modeled in the form of a set of CSPs and new methodology of constraint solving (e.g. multiple presolving) has to be employed. In addition, the models in this paper, in the form of CSPs and CSPs^{T} before and after transformation, were presented for SSCM problem for the first time. The corresponding constraints have a new, mostly binary, structure.
3 A multilevel approach to ubiquitous modeling and solving constraints for combinatorial optimization
The problem is modeled as a set of subproblems in the form of CSPs for particular areas of production, distribution, storage, etc. (Level 1).
All CSPs are modeled using the set of CLP predicates. This set includes the predicates dedicated to a given CSP, e.g., predicates implementing individual constraints, general predicates used in each modeled problem, such as those creating lists of data from the sets of facts, and incorporated predicates (disjunctive(), exclude(), cumulative(), alldifferent() etc. ). The modeling process uses the set of facts as an information layer for the problem.
The CSP, standard or variant (DCSP, FCSP etc.) is then presolved (Level 2), using the methods which reduce the decision variable domains and can transform the variables and constraints. Constraint propagation, problem transformation or both methods combined into one are used as presolving methods (Section 3.1). In the next step, the constraints (financial, environmental or transport constraints) that bind the subproblems are modeled (Level 3). They are also presolved (Level 4). Now the additional and auxiliary constraints are modeled. These do not result directly from the structure of the problem but from the data instances, the user’s specific requirements or possibility of increasing the effectiveness of the solution (Level 5). The next step involves presolving the model and generating the ultimate MP model. Finally, the MP model is optimized using the MPbased environment (Level 6). For illustrative example (Section 4) the MP model takes the form of MILP (Mixed Integer Linear Programming) model.
The multilevel architecture allows modeling complex problems through the parallel modeling of subproblems at a given level and related constraints at subsequent levels. This will result in the multiple use of presolving thus increasing its effectiveness.
3.1 Presolving
Presolving methods used at various levels constitute an important element of the proposed approach (Fig. 4). Presolving eliminates redundant information from the problem formulation while simultaneously trying to simplify and strengthen the formulation. It can be very effective and is often essential for solving instances. Especially for integer programming problems, fast and effective presolving methods are very important. Here, presolving may have a form of standard constraint propagation, problem transformation or a combination of these two methods. Constraint propagation is one of the CSP algorithm methods Rossi et al. [16]; Apt [1] whereas the transformation is the author’s concept introduced, to varying degrees and in varying forms, to CP/MP environments hybridization Sitek and Wikarek [20, 22]; Sitek [18]; Wikarek [26].
Likewise, the analysis of the remaining facts may result in the reduction of other decision variables of the modeled problem. In the transformed model, there will be much less decision variables than the initial one. Due to the multilevel hybrid approach, the reduction information will be transferred to the next level, resulting in further reductions in both decision variables and constraints. The application of presolving in practice is shown in (Section 4).
3.2 Multilevel model
This way of modeling makes it very easy for a given class/type of problems, where changing sets of fact instances is all that has to be done. Modeling becomes more complicated in the case of a new problem class/type with new constraint types. These new constraints have to be modeled in the form of CLP predicates (Section 4).
For the illustrative example, completely new constraint formalization is proposed, resulting from the application of the new multilevel architecture of the model. As a result, a large part of constraints had a binary form (Fig. 2), which is especially beneficial for the effectiveness of constraints propagation and transformation.
3.3 Innovativeness presented approach
Introducing presolving methods (constraint propagation and transformation) at different levels of problem modeling (Fig. 4) and mathematical programming solving methods is an important innovation of the proposed approach. Extremely timeconsuming and generating a lot of backtrackings labeling is eliminated from the general search algorithm for solving CSPs (Fig. 3). Ubiquitous use of presolving methods in various forms and manners contributes substantially to the reduction, aggregation and transformation of decision variables and constraints. Thus the ”lean” CSPs become the basis for generating a noticeably smaller MILP model.
For the solution of the generated model, we use mathematical programming methods and techniques instead of CLP, which is possible to avoid a lot of backtracking labeling techniques (extremely timeconsuming). You can also skip other CLP techniques such as variable ordering, forward checking etc.
Another important novelty of the proposed approach is the ability to expand each of the subproblems modeled as CSPs without having to change and implement the whole problem as such Sitek and Wikarek [21, 22] This allows the use of a model with an architecture as presented in (Section 3.2). Knowledge of the problem is stored in the form of facts, which is possible due to the declarative approach – CLP. Facts are also transformed (Fig. 6) and become a data layer for the model (3.2) (Appendix A, B).
4 Illustrative example
CSPs for illustrative example
CSP  Description 

CSP0 presolving, provides binarity and integrality  CSP0 = (C = {1C0, 2C0)},X = {Zx_{p},X p _{ f,p },X r _{ y,p },X s _{ c,p },Tc_{c},Trc_{y},X b _{f,c,t},Y b _{ c,r,t }, 
Z b _{ r,y,t },X a _{ f,c,p,t },Y a _{ c,r,p,t },X k _{ f,c,p,t },Y k _{ c,r,p,t },Z k _{ j,y,p,t }}  
D = {D Z x0,D X p0,D X r0,D X s0,D T c0,D T r c0,D X b0,D Y b0,D Z b0,D X a0,D Y a0,D X k0,D Y k0,D Z k0})  
CSP1 for production  C S P1 = (C = {1C1, 2C1, 3C1)},X = {X p _{ f,p }},D = {D X p1}) 
CSP2 for recycling  C S P2 = (C = {1C2, 2C2, 3C2)},X = {X r _{ y,p }},D = {D X r2}) 
CSP3 for distribution centers  C S P3 = (C = {1C3, 2C3, 3C3, 4C3)},X = {X s _{ c,p },Tc_{c}},D = {D X s3,D T c3}) 
CSP4 for recycling centers  C S P4 = (C = {1C4, 2C4, 3C4, 4C4)},X = {X r _{ y,p },Trc_{y}},D = {D X r4,D T r c4}) 
CSP5 for transportation  C S P5 = (C = {1C5, 2C5, 3C5, 4C5)},X = {X b _{f,c,t},Y b _{ c,r,t },Z b _{ r,y,t }},D = {D X b5,D Y b5,D Z b5}) 
CSP6 allows to link CSP5 with CSP1 and CSP2  C S P6 = (C = {1C6, 2C6, 3C6, 4C6, 5C6, 6C6, 7C6)},X = {X b _{f,c,t},Y b _{ c,r,t },Z b _{ r,y,t },X a _{ f,c,p,t },Y a _{ c,r,p,t },Tc_{c},X k _{ f,c,p,t },Y k _{ c,r,p,t },Z k _{ r,y,k,t }},D = {D X b6,D Y b6,D Z b6,D X a6,D Y a6,D T c6,D X k6,D X k6,D Z k6}) 
CSP7 allows to link CSP1 with CSP2, CSP3 and CSP4  C S P7 = (C = {1C7, 2C7, 3C7, 4C7, 5C7)},X = {X k _{ f,c,p,t },Y k _{ c,r,p,t },Z k _{ r,y,p,t },X p _{ f,p },X s _{ c,p },X r _{ y,p }},D = {D X k7,D X k7,D Z k7,D X p7,D X s7,D X r7})n = 
All the experiments and studies were carried out for the same data instances and in the same computational environment. Data instances for CSPs are presented using a set of facts. The structure of the facts describing the illustrative example, the relationship between facts and keys attributes for facts are shown in Fig. 6. The schematic diagram follows that well known from the database design, ERD (Entity Relationship Diagram) Teorey et al. [24] .The same diagram (Fig. 6) shows the transformation of facts for the illustrative example which is a presolving element and refers to the model data layer (Table 1 and Appendix A, Appendix B). Some of the facts after transformation change their structure and some are eliminated (15 facts before and 10 after transformation). It has to be noted that the number of instances of each fact is also greatly reduced.
Appendix A and B list the facts and decision variables for particular CSPs and CSPs^{T}.
The implementation of the approach in Fig. 4, Section 3, which took the form of the framework, Fig. 7, was used to model and solve the illustrative example. Practical implementation of the UMSC concept in the framework form contains ubiquitous CSPs and ubiquitous presolving methods. Both elements are present at many levels (Fig. 7).
CSPs^{T} for illustrative example
CSP  Description 

CSP^{T}  \(CSP^{T}=(C=\{2Ta,2Tb,3T,4T,5Ta,5Tb,6T,7T,8T,9T,10T,11T,21Ta, 21Tb,21Tc,22T,23T,24T\}),X=\{X_{f,p,c,r,t1,t2}^{T} ,Xb_{f.c.t}^{T} ,Yb_{c,r,t}^{T} ,T{c_{c}^{T}} ,Tr{c_{c}^{T}} ,Zr_{y,p,t}^{T} \}, D=\{D_{X^{T}}^{0} ,D_{Xb^{T}}^{0} ,D_{Yb^{T}}^{0} ,D_{Tc^{T}}^{0} ,D_{Trc^{T}}^{0} ,D_{Zr^{T}}^{0} \})\) 
CSP8^{T} ensures the exclusion storage, transport and manufacturing  \(CSP8^{T}=(C=\{26T,27T,28T\}),X=\{X_{f,p,c,r,d1,d2}^{T} ,Zr_{r,y,k,d}^{T} \},D=\{D_{X^{T}}^{8} ,D_{Zr^{T}}^{8} \})\) 
CSP9^{T}  C S P9^{ T } = (C = {12T, 13T, 14T, 15T, 16T, 17T, 18T, 19T, 20T},X = {X f,p,c,r,d1,d2T,X b f,c,tT,Y b c,r,t T},D = {D X ^{ T }9,D X b ^{ T }9,D Y b ^{ T }9}) 
An alternative approach consists of modeling the whole problem in the form of an MILP model that covers all the areas and all the constraints and solves the problem using the MP solver. This approach is less effective computationally, as only small size problems can be solved within acceptable time (Appendix E), with nonlinear and logical constraints excluded.
The tool of choice for the implementation of the framework (Fig. 7) was ECL^{i} PS ^{e} Eclipse [8] which is an opensource software system for the costeffective development and deployment of constraint programming applications. MPbased environment in the implementation framework was EPLEX built in ECL^{i} PS ^{e} MPsolver. ECL^{i} PS ^{e} was used to implement the levels of the framework: Level 1, Level 2, Level 3, Level 4 and Level 5. EPLEX was used for implementing Level 6 and for solving.
5 Computational experiments for illustrative example
6 Conclusions
The paper provides a robust modeling and solving method to combinatorial optimization for production and distribution problems. This method, based on the concept of UMSC in multilevel architecture, is an extension to the hybrid approach, discussed in previous studies Sitek and Wikarek [20], Sitek and Wikarek [21], both in the area of modeling (modeling subproblems as separate CSPs, linked CSPs, logical CSPs and ancillary CSPs) and solving (with multiple use of presolving). The efficiency of the proposed method was examined through a set of computational experiments for illustrative example (optimization of SSCM). The experiments were also performed for the hybrid approach and mathematical programming. The outcome of the proposed method includes (a) flexible modeling and introduction of many binary constraints to the model, (b) a significant reduction in solution space, which allows solving larger problems in shorter times compared with hybrid approach and mathematical programming and (c) a possibility of independent modification and extension of the model CSPs.
The multilevel architecture is suitable for problems such as ubiquitous manufacturing, supply chain management, distribution, vehicle routing etc., because it allows the implementation of constraints and presolving of the constraints in all areas and at all levels of the problem. This property is very important as it makes it possible to early detect constraint infeasibility in each subproblem. The use of the paradigms of constraint logic programming and mathematical programming in multilevel architecture takes advantage of the synergy of individual paradigms, which is clear in numerical experiments (Section 5 Fig. 11a and b, Appendix E Tables 10, 11, 12, 13, 14 and 15).
Further studies will focus on the implementation of other productiondistribution models with the use of the proposed approach (Nielsen at al. [13]; Grzybowska and Gajšek [10]). The proposed method can be extended to modeling and solving soft and fuzzy constraints and multiobjective optimization. In the further extension of the method MP solvers will be replaced with heuristics and metaheuristics for solving industrial size problems. In future, a comparative analysis between the proposed approach and the pure CLPbased approach will be carried out. Advanced CLP techniques like different variable ordering techniques, GAC (GeneralizedArcConsistency) for nconstraints will be tested in the context of modeling and solving distribution problems.
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