Essay: A declarative hybrid approach to the modelling and optimization of supply chain problems

This paper presents a declarative hybrid approach to modelling, solving and optimization of the supply chain problems using transformation. Two environments (mathematical programming (MP) and constraint logic programming (CLP)) were integrated. This integration, hybridization as well as an adequate multi-dimensional transformation of the problem helped significantly reduce the combinatorial problem. The following dimensions were considered in the transformation: data, properties and the structure of the problem, and the properties of modelling and solving the CLP environment. The MP in operations research context and declarative CLP, in which constraints are treated in different ways and different methods are implemented, were combined to use the strengths of both. This approach is particularly important for the decision and optimization models with an objective function and many discrete decision variables added up in multiple constraints, common in supply chain, manufacturing and logistic problems. The presented approach is compared with classical mathematical programming on the same data sets. For illustrative models, its use allowed finding optimal solutions eight to one hundred times faster and reducing the size of the combinatorial problem to a significant extent. The analysed case includes modelling and optimization of the hybrid model, which in addition to linear and integer constraints includes also logical and symbolic constraints. Introduction of additional constraints weakens the structure of the model and excludes the use of the MP-based classical approach.
Keywords: supply chain, optimization, declarative paradigm, constraint logic programming, mathematical programming.
1. Introduction
A supply chain (SC) may be considered an integrated process in which a group of several organizations, such as suppliers, producers, distributors and retailers, work together to acquire raw materials with a view to converting them into end products which they distribute to retailers (Beamon 1998). Simultaneously considering supply chain production, distribution and transport planning problems greatly advances the efficiency of the all processes. Supply chain (SC) optimization involves making decisions for proper organization, and all these processes of supply chains which are vital for retaining the competitive edge of companies in a global economy and fast moving and rapidly changing circumstances and needs. These problems are often very large and complex due to the large number of facilities of the supply chain such as the number of plants, warehouses and retailers, and due to complex interactions and constraints among these facilities such as the modes of transport, or relocation of warehouses, the different nature of the demand, and resource/capacity constraints.
Thus, the nature of the problems is characterized by a large number of constraints and discrete variables.
This is confirmed by a number of optimization models, the review of which is presented in (Huang et al. 2003, Mula et al. 2010). Such a structure of decision and optimization problems in SC is causing considerable problems in the application of operations research (OR) methods, mathematical programming (MP) in particular, to both modeling and optimization.
For this reason, the major motivation behind this study was to develop an alternative approach to SC problem modeling and optimization. It was assumed that such an approach should be highly effective in optimization and far more flexible in problem modelling than OR methods.
An important contribution of the presented approach is to propose a declarative hybrid implementation platform that supports the hybrid modeling, hybrid optimization, multi-dimensional transformation and analysis of decision problems in the supply chain. In this platform two environments are hybridized, constraint logic programming (CLP) and mathematical programming (MP), in which constraints are treated in different ways and different methods are implemented to use the strengths of both for solving complex and constrained problems. The new combined approach proposed is no worse than either of its components. A vast majority of cases is significantly improved in both the modeling and optimization.
A declarative constraint logic programming with MP-library as a hybrid system was chosen as the best structure for the implementation of this approach.
The rest of the paper is organized as follows: Section 2 describes our motivation and analyses the state of the art in this domain. Section 3 gives the concept of the novel constraint logic programming approach with MP-library and implementation platform. The optimization models as the illustrative examples are described in Section 4. Computational examples and tests of the implementation platform are presented in Section 5. The discussion on possible extensions of the proposed approach and conclusions is included in Section 6.
2. Materials and Methods
Constraints are logical relations between variables, each variable taking values from a specific domain. Thus a constraint restricts the possible values that a variable can take, i.e. it represents some partial information about the variables of interest. Constraints are:
‘ declarative, they specify a relationship between entities (variables) without determining a specific computational procedure;
‘ may specify partial information, i.e., the constraint need not uniquely specify the values of its variables,
‘ additive, we are interested in the conjunction of constraints and not in the order in which they are imposed;
‘ rarely independent, normally constraints share variables.
Thus constraints are a natural medium and form for all (researchers, practitioners, professionals and end-users) to express problems in many fields, especially in logistic, transport, manufacturing, scheduling, distribution, supply chain etc. In the above problems, there are resource, capacity, time, transportation, environmental, etc. constraints.
We strongly believe that the constraint-based environment (Apt and Wallace 2006, Sitek and Wikarek 2008, Rossi et al. 2006) offers a very good framework for representing the knowledge and information needed for the decision support and optimization in supply chain problems.
The central issue for a constraint-based environment is a constraint satisfaction problem (CSP) (Apt and Wallace 2006, Buscemi and Montanari 2008). Constraint satisfaction problem is the mathematical problem defined as a set of elements whose state must satisfy a number of constraints. Constraint satisfaction originated in the field of artificial intelligence in the 1970s, during the 1990s, embedding of constraints into a programming language was developed.
Formally, a constraint satisfaction problem is defined as a triple (X,D,C), where X is a set of variables, D is a domain of values, and C is a set of constraints. Every constraint is in turn a pair (t,R) (usually represented as a matrix), where t is an n-tuple of variables and R is an n-ary relation on D. An evaluation of the variables is a function from the set of variables to the domain of values, v:X’D. An evaluation v satisfies constraint ((x1,’,xn),R) if (v(x1),..v(xn))’R. A solution is an evaluation that satisfies all constraints.
Constraint satisfaction problems (CSPs) on finite domains are typically solved using a form of search. The most widely used techniques include variants of backtracking (Kondrak and Beek 1997), constraint propagation, and local search. Constraint propagation embeds any reasoning that consists in explicitly forbidding values or combinations of values for some decision variables of a problem because a given subset of its constraints cannot be satisfied otherwise (Rossi et al. 2006).
These techniques allow one to quickly find a domain solution or conclude that domains are contradictory.
There are two reasons, which encourage such a constraint representation. The first reason is that it is closer to the natural problem statement, since variables represent entities and the constraints do not have to be translated, just stated over the entities they concern. The second is that CSP methods are in many cases more efficient in solving such problems than other methods.
CSPs are frequently used in constraint programming. Constraint programming is the use of constraints as a programming language to encode and solve problems. Constraint logic programming (CLP) is a form of constraint programming (CP), in which logic programming is extended to include concepts from constraint satisfaction. A constraint logic program is a logic program that contains constraints in the body of clauses (predicates). Constraints can also be present in the goal (Rossi et al. 2006). Constraint logic programming (CLP) is a new class of declarative programming languages whose primitive operations are based on constraints (e.g. constraint solving and constraint entailment). CLP languages naturally combine constraint propagation with nondeterministic choices. As a consequence, they are particularly appropriate for solving a variety of combinatorial search problems, using the global search paradigm.
The declarative approach and the use of logic programming provide incomparably greater possibilities for decision problems modeling than the pervasive approach based on mathematical programming. Unfortunately, discrete optimization is not a strong suit of these environments.
Based on (Apt and Wallace 2006, Rossi et al. 2006, Bocewicz and Banaszak, Dang et al. 2013, Relich 2014), and previous work (Sitek and Wikarek 2008, Sitek and Wikarek 2012), we observed some advantages and disadvantages of these environments. An integrated approach of constraint logic programming (CLP) and mathematical programming (MP) can help to solve optimization problems that are intractable with either of the two methods alone (Milano and Wallace 2010, Achterberg et al. 2008, Bockmayr and Kasper 2004). In both MP and CLP, there is a group of constraints that can be solved with ease and a group of constraints that are difficult to solve. Both MP and finite domain CP/CLP involve variables and constraints. However, the types of the variables and constraints that are used, and the way the constraints are solved, are different in the two approaches (Bockmayr and Kasper 2004).
MP relies completely on linear equations and inequalities in integer variables, i.e., there are only two types of constraints: linear arithmetic (linear equations or inequalities) and integrity (stating that the variables have to take their values in the integer numbers). In finite domain CP/CLP, the constraint language is richer. In addition to linear equations and inequalities, there are various other constraints: disequalities, nonlinear, symbolic (alldifferent, disjunctive, cumulative etc.) (Rossi et al. 2006).
Integrity constraints are difficult to solve using mathematical programming methods and often the real problems of MP make them NP-hard.
In CP/CLP, domain constraints with integers are easy to solve. The system of such constraints can be solved over integer variables in polynomial time. The inequalities between many variables, general linear constraints, and symbolic constraints are difficult to solve, which makes real problems in CP/CLP NP-hard (Bockmayr and Kasper 2004). This type of constraints reduces the strength of constraint propagation. As a result, CP/CLP is incapable of finding even the first feasible solution.
Both environments use various layers of the problem (methods, the structure of the problem, data) in different ways. The MP-based approach focuses mainly on the methods of optimization and, to a lesser degree, on the structure of the problem (Fig. 1). The data, however, is completely outside the model. The same model without any changes can be solved for multiple instances of data. In the CP-based approach, due to its declarative nature, the methods are already in place. The data and structure of the problem are used for its modeling (Fig. 1).
The vast majority of decision-making and optimization models for the problems of production, logistics, supply chain are formulated in the form of mathematical programming (MIP-Mixed Integer Programming, MILP – Mixed Integer Linear Programming, IP – Integer Programming) (Mula et al. 2010). Due to the structure of these models (adding together discrete decision variables in the constraints and the objective function) and a large number of discrete decision variables (integer and binary), they can only be applied to small problems. Another weakness is that only linear constraints can be used. In practice, the issues related to the production, distribution and supply chain constraints are often logical, nonlinear, etc. For these reasons the problem was formulated in a new way.
Figure 1. Layers used in the solution of the problem (MP-based and CP-based).
The result behind this work was to create a novel hybrid approach and implementation declarative hybrid platform for supply chain decision problems modeling and optimization instead of using mathematical programming or constraint programming separately. It follows from the above that what is difficult to solve in one environment can be easy to solve in the other.
The best structure for the implementation of the above approach is a declarative CLP environment with MP-library as a hybrid system. Furthermore, such a declarative hybrid approach allows the use of all layers of the problem (data, structure, methods) to solve it (Fig. 1). Finally, it allows the multi-dimensional transformation of the problem (section 4.3) to such a form that can fully exploit the strengths of the constraint propagation and MP-based methods. Among other things, the presented approach differs from the known from the literature integrative approaches CP / MP (Milano and Wallace 2010, Achterberg et al. 2008, Hooker 2002, Jain and Grossmann 2001) by the use of multi-dimensional transformation as an integral part of the declarative hybrid platform.
3. The concept of the declarative hybrid approach with transformation
In our declarative approach to modelling and optimization supply chain problems we proposed the approach, where:
‘ knowledge related to the problem can be expressed as linear, logical and symbolic constraints;
‘ the decision and optimization models solved using the proposed approach can be formulated as a pure model of MP or a hybrid model;
‘ the problem is modelled in the constraint logic programming environment by CLP predicates, which is far more flexible than the mathematical programming environment;
‘ multidimensional transforming the optimization model to explore its structure and data has been introduced by CLP predicates (3.1);
‘ constrained domains of decision variables, new additional constraints and values for some variables are transferred from CLP into MP environment by CLP predicates;
‘ merging and final building of the model is performed by MP-based procedures;
‘ optimization is performed by MP-based procedures;
‘ optimal solution has been transferred back to the CLP environment.
The schema of the implementation platform for declarative approach is presented in Fig. 2. From a variety of tools for the implementation of the CP/CLP, ECLiPSe software (www.eclipse.org 2015, Apt and Wallace 2006) of constraint programming applications. ECLiPSe contains several constraint solver libraries, a high-level modelling and control language, interfaces to third-party solvers, an integrated development environment and interfaces for embedding into host environments. ECLiPSe was used to model the problem, transform it and search for a domain solution by constraint propagation (Fig. 2, Table 1). This solution was then the basis for the final MILP model, developed in the Eplex library (Shen and Schimpf 2005) of the ECLiPSe environment. Since ECLiPSe version 5.7, standalone Eplex have become the standard. The previous lib(eplex), which loads Eplex with the range bounds keeper and the lib(ic) variant have now been phased out, so users of these old variants must now move to using standalone Eplex. The Eplex library allows MP/MIP/MILP problems to be modeled in ECLiPSe, and solved (optimized) by an external MP solver.