A Hybrid Method for Modeling and Solving Supply Chain Optimization Problems with Soft and Logical Constraints
This paper presents a hybrid method for modeling and solving supply chain optimization problems with soft, hard, and logical constraints. Ability to implement soft and logical constraints is a very important functionality for supply chain optimization models. Such constraints are particularly useful for modeling problems resulting from commercial agreements, contracts, competition, technology, safety, and environmental conditions. Two programming and solving environments, mathematical programming (MP) and constraint logic programming (CLP), were combined in the hybrid method. This integration, hybridization, and the adequate multidimensional transformation of the problem (as a presolving method) helped to substantially reduce the search space of combinatorial models for supply chain optimization problems. The operation research MP and declarative CLP, where constraints are modeled in different ways and different solving procedures are implemented, were linked together to use the strengths of both. This approach is particularly important for the decision and combinatorial optimization models with the objective function and constraints, there are many decision variables, and these are summed (common in manufacturing, supply chain management, project management, and logistic problems). The system with Eplex library was proposed to implement a hybrid method. Additionally, the proposed hybrid transformed model is compared with the MILP-Mixed Integer Linear Programming model on the same data instances. 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.
A supply chain (SC) is a system of producers, distributors, suppliers networks, customers, retailers, activities, resources, and information involved in moving services and/or products from producers to customers. All members of the SC are working together to achieve this goal . Supply chain management (SCM) covers planning, control, and management of all activities and resources in production, distribution, and transportation.
Simultaneous consideration of distribution, production, and transport planning and control problems greatly advances the effectiveness of the processes and operations of all of these problems. These problems are characterized by high complexity due to the large number of different activities of the SC (plants, vehicles, distribution centers, retailers, etc.) and due to many interactions and different restrictions among these activities (i.e., capacity, modes of transportation, relocation of distribution centers, nature of demand, balance, environmental, etc.). For modeling these restrictions it is usually sufficient to use linear, integer, or binary constraints.
In contrast, soft and logical constraints are better suited to modeling issues related to trade agreements, marketing, contracts, competition, security, finance, and so forth.
Thus, the nature of the SC problems is characterized by a number of constraints and discrete decision variables, as confirmed by a variety of optimization models, the review of which is presented in [2, 3]. The structure of decision and optimization problems in a SC poses a real challenge in terms of mathematical programming (MP) application to both their modeling and their optimization. Very often discussed problems become overconstrained (the problem where solution does not exist, i.e., valuation of variables that satisfies all the constraints).
For this reason, the main motivation behind this study was to develop an alternative approach, highly effective in optimization and far more flexible in problem modeling than mathematical programming methods, especially when modeling logical and soft constraints.
A contribution of this paper is to propose a declarative method for hybrid modeling and solving of the supply chain optimization problems using multidimensional transformation as a presolving method. Moreover, a hybrid transformed combinatorial model with soft, hard, and logical constraints for supply chain optimization problems has been described. In addition, a declarative hybrid implementation framework has been presented, in which mathematical programming (MP) and constraint logic programming (CLP) are integrated.
A declarative constraint logic programming environment with MP-library (Eplex) as a declarative hybrid framework was chosen for the implementation of this method and model.
The rest of this paper is organized as follows: Section 2 describes material and methods. Section 3 gives the main assumptions of the declarative method with MP-library and describes implementation framework. The optimization models (MILP, hybrid, and hybrid transformed), computational examples, and efficiency tests of the declarative implementation framework are shown in Section 4. The summary and discussion on possible extensions are included in Section 5.
The use of constraint-based methods for the modeling of decision and optimization problems that contain constraints appears to be natural and effective owing to their character and properties. Constraints reflect the logical relationships between decision variables. Each variable can take values within a given domain—constraints restrict the ranges of those values. The properties of constraints include the following:(i)Declarativity. Define the relationships between variables; no computation algorithm is needed.(ii)Additivity. The set of constraints, their context, and conjunction are important, not their sequence in adding to each other.(iii)Dependence. Constraints often share decision variables, commonly in the problems of supply chain management, production, transport, and logistics.(iv)Domain Solution. Constraints rarely specify the values of decision variables—the domains within which the variables can take on values are specified more often.
These properties and the character of constraints make them a natural form of knowledge and information modeling, particularly adequate in such areas as production, logistics, transport, or distribution, where they become resource, time, environment, transport, or finance constraints of linear, nonlinear, logical, or integer type. An optimization problem of SCM connects the issues of storage, production, distribution, and transportation taking account of conditions relating to time, environment, and finance.
Therefore, we claim that constraint-based environments are a natural way of modeling the problem with constraints [4–8]. These environments usually use the declarative programming paradigm which specifies the conditions the solution has to satisfy (what is to be achieved) and not the detailed algorithm (how it should be achieved). The constraint satisfaction problem (CSP) is the fundamental concept of constraint-based declarative environments [4, 9].
CSP is a mathematical problem described as a set of objects whose condition must satisfy a set of constraints.
Formally, CSP is described by a set of decision variables and set of constraints . Each decision variable has a domain of possible variables. Each constraint involves some subset of the decision variables and determines the combination of values for this subset. A state of the CSP is described by an assignment of values to decision variables. An assignment that meets all constraints is named a consistent/legal assignment.
These techniques allow one to quickly find a domain solution or conclude that domains are contradictory.
CSPs are the basis for constraint programming (CP)/constraint logic programming (CLP) environments .
Several modifications of the classic model of CSP have been proposed: flexible CSPs (relaxing assumptions that each solution must satisfy all constraints, including FuzzyCSP), dynamic CSPs (a sequence of the static CSPs), and decentralized CSPs .
Problem representation as a CSP offers several advantages. First, it is closer to a natural presentation of a problem (constraints over variables). Second, there is no requirement for presenting the solving method/algorithm. Finally, its effectiveness for yes/no decision problems is higher than that of traditional methods.
The use of a declarative method, for example, CLP, offers better decision and optimization modeling possibilities compared to the MP-based approach. Advantages and shortcomings of both (MP and CLP) are widely reported in the literature [12, 13] including our own experience [14, 15].
In MP environments, the number of modeling constraint types is modest, including only linear and integer constraints. CLP environments make use of a wider range of constraints; in addition to those named above, nonlinear, logical, or predefined symbolic (cumulative, alldifferent, disjunctive, etc.) constraints are embedded. Different types of constraints require different computational effort in both environments. MP-based approaches are effective in solving linear constraints and optimization but poor at handling integer (binary) constraints. Logical constraints are hard to model in the MP-based environments, where a number of linear and integer constraints (common in SCM problems) have to be contained. In CLP-based environments, integer and binary (domain integrity) and logical and nonlinear constraints are easy to model and solve. However, linear constraints, which add up many decision variables, increase the computational effort and extend the solving time, especially when an optimization problem is being solved. Yet, this constraint type is common in production, distribution, and transportation problems, thus for SCM optimization.
Therefore, MP-based environments are effective in solving linear constraints and optimization. In contrast, the CP-based environments are effective for modeling and solving integer and logical constrains and finding feasible solutions.
Each issue of problem modeling and solving can be considered in three areas (dimensions). The first concerns the proposed method for both modeling and solving. The second relates to the structure and nature of the problem (types of constraints, the decision or optimization problem, etc.). The last area refers to the data instances. Each of the presented environments uses these areas in a different way (Figure 1). CLP mainly uses the data instances and slightly less the structure of the problem. Due to the declarative nature, external methods or algorithms are not required. The CLP model is already the method. MP mainly focuses on the method and then the structure and properties of the problem. The data instances are outside of MP model. Therefore, hybrid method allows maximum exploration of all these three areas (Figure 1).
The majority of supply chain optimization problems are formulated as MP-based models (MIP-Mixed Integer Programming; MILP, Mixed Integer Linear Programming; IP, Integer Programming; and MOOP, multiobjective optimization programming) [2, 6, 7].
Due to the properties and character of the models of supply chain optimization and a large number of integer decision variables, MP-based models and methods can only be used to solve problems of small size. Another weak point of these models is that only integer, linear, and binary constraints can be applied. In fact, issues related to the supply chain optimization are often nonlinear, logical, and so forth.
For the above reasons, the problems were formulated in a new way with new hybrid transformed models.
The most important objectives behind this research were to develop hybrid optimization models and methods for supply chain optimization problems using a declarative hybrid method with multidimensional transformation.
Hybridization results from the fact that what is difficult to model and solve in CLP can be easily modeled and solved in MP and vice versa. The best programming framework for the implementation of the hybrid models and methods is a declarative CLP environment with MP-library. Moreover, such declarative hybrid models and methods allow the use of all areas (dimensions) of the problem (methods, structure, and data) to solve it (Figure 1). Finally, this method allows the multidimensional transformation of the problem (Section 4.3), which reduces its size and significantly affects the speed of finding a solution. Among other things, the presented method differs from the integrative approaches CP/MP known from the literature [12, 13, 16, 17] in the use of multidimensional transformation as an integral part of the declarative hybrid method.
3. The Concept and Implementation of the Declarative Hybrid Method
Based on the characteristics of the CLP and MP environments, their mutual complementarity, the properties of supply chain optimization problems, and previous experiences with hybridization [14, 15] we have proposed the concept of a declarative method for modeling and optimization. The main assumptions of the proposed concept are as follows:(i)The problem is modeled by a set of facts and a set of constraints (integer, linear, logical, etc.).(ii)Models can be formulated in the form of MIP/MILP/IP models or hybrid models.(iii)Constraints may take two forms: soft and hard.(iv)The problem is modeled by CLP-based predicates, a far more flexible technique than the MP-based procedures and formulas.(v)Multidimensional transforming of the model as a presolving method is done by CLP-based predicates (Section 3.1).(vi)Optimization is done through MP-based procedures.
The scheme of the implementation framework for declarative hybrid method is presented in Figure 2.
From a variety of declarative constraint programming environments for the implementation of the framework, software [4, 18] was selected. is a high-level modeling, control, and declarative programming language with interfaces to third-party solvers. was used to model the problem, transform it, and search for a domain solution by constraint propagation (Figure 2, Table 1). This solution was then the basis for the final MILP model, developed in the Eplex library  of the environment. The Eplex library allows MIP/IP/MILP problems to be modeled in environment and optimized by an external mathematical programming solver.
3.1. The Concept of Multidimensional Transformation of the Problem
Due to the nature of models in SC/SCM (a large number of integer decision variables, summation of many decision variables in the objective function and constraints), the constraint propagation efficiency decreases significantly.
Constraint propagation  is an important component of constraint logic programming affecting the efficiency and effectiveness of the search solution. For these reasons, studies into more effective way of constraint propagation have been performed [14, 15]. The result of this study was to propose a new multidimensional transformation of the problem as a presolving method. Transformation is this element of this hybrid method which distinguishes it from the MP and CLP integration methods reported in the literature [13, 20, 21]. This method proposes a multidimensional transformation of a problem , which involves decision variables and constraints . The following dimensions are considered : data (D), properties and the structure of the problem (S), and the properties of the modeling and solving CLP environment (CLP). Table 1 describes the symbols used in the transformation.The transformation , performed in the CLP environment, results in the following (Section 4.3):(i)The reduction in the total number of and changing the nature of decision variables attained through aggregation and elimination.(ii)The reduction in the number of constraints resulting from the problem properties.(iii)The introduction of additional constraints that limit the solution search space without affecting the problem (Table 6).(iv)The reduction of the solution search time.
4. Illustrative Examples
The proposed method has been verified on two SCM optimization illustrative examples. First example was formulated as MILP model based on  overconstraints (2)–(27). This model was used to compare the DHIF relative to the classical MP solver for the same data sets [22, 23]. Then the hybrid model (HM) (1)–(30) was implemented and solved only in DHIF because the declarative and logical character of constraints (28)–(30) weakens the structure of the model. In addition, these constraints are difficult to implement in the MP-based environment and cause the increase in the number of decision variables and the number of constraints and extend the solution search time beyond acceptable limits.
The logical constraints relate to various practical situations. In the hybrid model, three logical constraints were introduced: (28) which excludes concurrent transportation of selected products by different transport means, which eliminates the necessity of repackaging unacceptable by the customer, (29) which limits the number of participating distributors, for example, due to a limited number of available portable scanning devices for radioactivity, biological, or chemical contamination, and (30) which allows the simultaneous production, distribution, transportation, and storage of exclusively one of the set of two selected products items.
Table 2 shows indices, decision variables, and parameters used in the models. The exemplified structure of the SCM network for these models, composed of manufacturers, distributors, customers, and multimodal transport, is shown in Figure 3. The MILP and hybrid models for illustrative examples are the cost models that take into account other types of parameters, that is, time (duration of supply and service by distributor, etc.), the transport mode, distributor’s capacity, capacity of transport unit, and the spatial parameters (volume/capacity occupied by the product).
The principal assumptions used in the construction of the models are as follows:(i)Supply chain network has a structure as in Figure 3.(ii)Supply chain management process uses information that relates to production (cost, versatility capacity, etc.), product item (volume), resources (costs, versatility, capacity, etc.), transportation (cost, time, mode, and capacity), demand (product item, time), inventory (costs, capacity, time, and versatility), and so forth.(iii)Transportation is multimodal (a limited number of transportation units for each mode, several modes of transportation).(iv)The environmental aspects of use of transportation units in different modes are taken into consideration.(v)Transport batch contains various types of products.(vi)It includes fixed and variable costs of supply (which may take the form of a linear function).(vii)The models have integer, linear, and logical constraints (only in hybrid model).(viii)The objective function is a linear function of cost relating to the entire supply chain.
4.1. Objective Function of the Optimization Models
The objective function (1) determines the total cost of supply chain management. It consists of five components. The first component relates to the total environmental costs (using various means of transport units). The environmental costs are dependent on the environmental levy, which can depend on fuel consumption (gasoline, oil, or gas) and carbon-dioxide emissions  and, on the other hand, on the number of courses by transportation units. The second component determines the cost of the supply from the factories to the distributors. Next element is responsible for the costs of the supply from the distributors to the customer. Another element specifies the production cost of the product item by the given factory. The last element of the objective function (1) specifies the costs (fixed) associated with the activities and processes of the distributors involved in the supply (loading, unloading, packaging, ticketing, and so on.).
The formulation of the objective function in this way allows the comprehensive optimization of the all costs in various aspects of SCM. The above objective function is the same for all models in Section 4.
The model has the following constraints (2)–(27). Constraint (2) specifies that all supplies of product item d produced by the factory a to all distributors b using mode of transport e do not exceed the production capacity at factory a. Constraint (3) ensures the coverage of all customer/retailer c orders () for product d as a result of supply by distributors b. The balance of the products d flow through the distributor b is constraint (4). Possibility of supply by the distributor b depends on its technical capabilities (5). Delivery time conditions are met by constraint (6).
The set of constraints (12), (13), and (14) set values of decision variables , based on binary decision variables . The remaining set of constraints (17)–(27) results from the nature of the mathematical programming model.
Constraint (28) excludes transportation of selected product d by different transport means .
Constraint (29) ensures that a group of products cannot be stored in more than centers simultaneously. Constraint (30) excludes concurrent production, distribution, and transport of selected products and .
The constraints above result from the practical reasons, including sales, security, marketing, contracting, or technology. Logical constraints occur especially often in a situation of strong competition.
Constraints (28)–(30) can be repeatedly reused for different pairs of product d and for some of or all of factories a and distributors b. Such a logical constraint is extremely difficult to implement in mathematical programming model. It demands that each logical constraint be replaced by many linear constraints. This sometimes leads to the explosion of the number of constraints.
Only declarative methods based on CSP make easy implementation of constraints such as (28), (29), (30) possible. The introduction to model (1)–(27) logical constraints such as (28), (29), and (30) changes its nature as well as converting it into a hybrid model (HM). This model can only be considered in the DHIF.
4.3. The Transformation of Illustrative Models
The possibility of transformation of the problem is an important feature of the proposed method. Multidimensional transformation (Section 3.1) was used in the illustrative examples as a presolving method. The representation of the problem was altered using S-structure of the problem and D-data. Our goal was the transformation of the problem to make a significant reduction in its size without changing the problem. It consisted in changing the representation of the problem. Any acceptable routes connecting factories, distributors, and customers were generated based on the set of facts relating to orders and supply chain structure.
Then, the specific values of parameters like manufacturers a, distributors b, products d, mode of transport form factory to distributor , mode of transport form distributor to customer , and customers c (Table 2) were assigned to each of the acceptable routes. Thus, only one parameter, (the volume of supplies), needs to be determined.
This means that six sets of parameters more were to be determined before the transformation. In this step the sets of decision variables and constraints were subject to change. Thereby, the obtained model after the transformation MILP_T – has different decision variables and different constraints than those in the MILP (1)–(27). Some of the decision variables are redundant; other variables are subject to aggregation. This results in a dramatic reduction in their number. Decision variables before and after the transformation are shown in Table 3(a). The transformation also reduces or eliminates some of the constraints of the model. Thus, constraints (4), (6), (12), (13), and (14), present in the MILP (1)–(27) and shown in Section 4.2, are redundant in the MILP_T. Balance constraint (4) is unnecessary because the route defines the specific distribution center. Only those routes that meet the time constraints are generated; therefore constraint (6) does not make sense. Binarity ensures whether or not the route occurs; thus constraint (12) is redundant. Reduction of certain variables also affects the reduction of constraints, hence the lack of constraints (13) and (14), in the model. Constraints before and after the transformation are shown in Table 3(b).
In the following step of the transformation, the properties of the CLP including constraint propagation were used to find, based on the domains (Tables 4 and 5), certain quantities and create a new set of additional constraints (Table 6). Therefore, transformed model (MILP_T) was extended to additional constraints –. These constraints affect the efficiency of the search for a solution by narrowing down the search area. It is obvious that the transformed model (MILP_T) can be extended from logical constraints (28)–(30) to a hybrid model (HM).
The proposed transformation improves the efficiency and effectiveness of the constraint propagation by reduction of the number of backtracks. It will be explicitly shown in the numerical experiments in Section 4.6.
4.4. Soft Constraints
Constraints of the transformed model can be satisfied in the hard or soft form. This is due to practical circumstances. For some of the constraints, it is difficult to satisfy in full (hard), due to limited resources, time, technology, and so forth or their fulfillment requires unacceptable computational effort.
In the highly competitive environment we need to know how much “costs” exceed a given constraint or fulfill it in part.
Such constraints can be satisfied in an approximate way, that is, in the soft form. In this case, how this affects the value of the objective function should be determined.
The softening of constraints in this approach is a process which contains the following: (a) changing their form and (b) introducing a modification to the objective function.
For this purpose, the three additional components to the objective function have been introduced as a penalty for violation of constraint . Table 7 shows parameters and decision variables necessary to modify the model with soft constraints.
4.5. Decision Support
The implementation of the presented models by DHIF allows the decision support in the following areas of the supply chain management (including but not limited to):(i)The cost optimization of the supply chain management (Table 8).(ii)The volume/capacity of distributor depots/storehouses and the investigation of their influence on the overall costs (Figure 4(a), Table 9).(iii)The selection of the capacity of transportation units and modes, the transport fleet number for specific entire costs (Table 10, Figure 4(b)).(iv)The selection of a set of routes for cost optimization (Figure 5(a)).(v)Implementation of logical constraints in deliveries that result from technology, security, marketing, competition, and sales reasons (Table 8, , , , and Figure 5(b)).(vi)Costs exceeding the available means of transport (soft constraints), Table 11.
Supporting this type of decision is extremely important in a highly competitive environment. Making optimal choices under existing constraints within an acceptable time gives you a competitive advantage.
4.6. Computational Experiments and Analysis
A number of numerical experiments were conducted to verify and evaluate the proposed method, models, and framework. All the numerical examples refer to the supply chain with twenty customers , four distributors , five manufacturers , and four modes of transport . In the first phase of the experiments, the optimization was performed for ten examples, Pn1–Pn10, for the MILP model (1)–(27). The computational examples vary in the number of orders (N). The capacity available to the distributors and the number of transport units using the mode of transport e were the same for all the examples. The experiments were conducted to show the effectiveness and efficiency of the declarative hybrid method and implementation framework and their advantage over other methods. In order to prove this, model (1)–(27) was implemented in both the declarative hybrid implementation framework (DHIF) (MILP_T) and the mathematical programming environment (MILP). The experiments that follow were conducted to optimize examples Pn11–Pn20, which are the implementations of the hybrid model (HM) (1)–(30) in the DHIF. Examples Pn11–Pn20 were obtained from Pn1–Pn10 by the addition of logical constraints: (28) for Q3 series, (29) for Q2 series, and (30) for Q1 series. For example, Pn1–Pn3 the optimal solution for the MILP_T implementation in the DHIF was found faster than that for the MILP implementation in mathematical programming environment (Table 7). For illustrative models, the use of this method allows obtaining optimal solutions eight to one hundred times faster, with a significant reduction in the size of the combinatorial problem. For example, Pn4–Pn10 the optimal solution was found only for the MILP_T implementation within an acceptable time. Owing to the DHIF platform with the multidimensional transformation, the number of decision variables for the above examples was reduced even hundredfold and the number of constrains decreased at least 30-fold (Table 8). It is clear that optimization of the hybrid model (HM) was, due to its logical constraints, only possible using the declarative hybrid implementation framework. Also, the presented declarative hybrid method brought the expected results for this model. In further experiments, parameters such as number of transport units, distributor capacity were changed to observe the effect of this change on the optimal value of the objective function. Thus, Figure 4(a) shows the effect of capacity (distributor) on the objective function . Actual use of the distributor capacity () for particular distributors is shown in Table 9. In contrast, Table 10 and Figure 4(b) show the effect of the parameter on the objective function .
Table 11 shows the results of the optimization model MILP_T with the constraint on the number of means of transport in hard version and soft version . Computation times for the model with a soft constraint are slightly worse than the version with limited hard constraint. The differences are fully acceptable. You can also see the impact of the application of constraint on the objective function. And, most importantly, we managed to find the optimum solution, for example, in the model MILP_T with constraint .
In the highly competitive environment we are often forced to make quick and optimal decisions while assessing their costs. As it has been shown to support such decisions use of the presented approach and model is very effective.
The effectiveness of the proposed declarative hybrid method with multidimensional transformation is based on the reduction of a combinatorial optimization problem.
Thus, the application of this method practically to all the SCM optimization models ensures that the same or better solutions are found eight to more than one hundred times faster (very often optimal instead of feasible solutions). It needs to be highlighted that the proposed method not only enables faster solving of optimization problems but also enables faster solving of problems larger than those in [22, 25] (Table 8).
The proposed models and methods are highly recommended for all types of optimization problems in SCM or for similar problems where constraints include many discrete decision variables and their summation. Moreover, the proposed method allows successful modeling and solving of problems with logical constraints and therefore the DHIF implementations can be applied to area of logistics, transport, production, and scheduling or project management.
In addition to nonquestionable effectiveness and efficiency of the proposed method, it offers unlimited possibilities of optimization problem modeling. This method is also applicable to problems with soft constraints (Table 11).
The possibility of optimization models, which also have soft and logical constraints, is very important for practical applications in the highly competitive environment (in case of violation certain constraints and assessing the impact on the objective function).
Future studies of the proposed method and framework will focus on the following: (a) modeling problems with other logical and nonlinear constraints and so forth, (b) multiobjective optimization, and (c) introduction of fuzzy logic  and uncertainty to the models.
The authors declare that there is no conflict of interests regarding the publication of this paper.
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