In this paper we study an actual problem proposed by an agricultural cooperative devoted to harve... more In this paper we study an actual problem proposed by an agricultural cooperative devoted to harvesting corn and grass. The cooperative uses harvesters for harvesting the crop and trucks for carrying it from the smallholdings to the landowners’ silos. The goal is to minimize the total working time of the machinery. Therefore, the cooperative needs to plan both the harvesters and trucks routing. This routing problem simultaneously incorporates the following characteristics: time windows, nested decisions, processing times required to service each facility and the fact that facilities must be visited in clusters. A binary integer linear programming model is proposed to solve this problem. However, since approaches dealing directly with such formulation lead to considerable computation times, we propose a heuristic alternative solution approach for the problem. The heuristic is applied to the case of the cooperative “Os Irmandiños” with a large number of landowners and smallholdings. We report on extensive computational tests to show that the proposed heuristic approach can solve large problems effectively in reasonable computing time.
In this paper we introduce the notion of $m$-irreducibility that extends the standard concept of ... more In this paper we introduce the notion of $m$-irreducibility that extends the standard concept of irreducibility of a numerical semigroup when the multiplicity is fixed. We analyze the structure of
the set of $m$-irreducible numerical semigroups, we give some properties of these numerical semigroups and we present algorithms to compute the decomposition of a numerical semigroup with multiplicity $m$ into $m$-irreducible numerical semigroups.
In this paper we present a mathematical programming formulation for the ω-invariant of a numerica... more In this paper we present a mathematical programming formulation for the ω-invariant of a numerical semigroup for each of its minimal generators which is an useful index in commutative algebra (in particular in factorization theory) to analyze the primality of the elements in the semigroup. The model consists of solving a problem of optimizing a linear function over the efficient set of a multiobjective linear integer program. We offer a methodology to solve this problem and we provide some computational experiments to show the efficiency of the proposed algorithm.
In this paper we present a procedure to build the set of irreducible numerical semigroups with a ... more In this paper we present a procedure to build the set of irreducible numerical semigroups with a fixed Frobenius number. The construction gives us a rooted tree structure for this set. Furthermore, by using the notion of Kunz-coordinates vector we translate the problem of finding such a tree into the problem of manipulating $0-1$ vectors with as many component as the Frobenius number.
This paper presents a model for the expansion of transportation networks incorporating specific r... more This paper presents a model for the expansion of transportation networks incorporating specific requirements about population coverage, budget constraints, intermediate goals and origin–destination flows, among others. The model is applicable to the current expansion project of the Spanish high-speed railway network that has been proposed by the Spanish Government under the program Strategic Planning of Infrastructure and Transport (see source [c]).
Our approach looks for solutions that may be used as additional information in the decision-making process of any network expansion. We report on the application of this methodology to the Spanish railway network and on a computational experience based on simulated data varying the number of cities and time horizons which proves the efficiency of the proposed algorithm.
Arithmetical invariants---such as sets of lengths, catenary and tame degrees---describe the non-u... more Arithmetical invariants---such as sets of lengths, catenary and tame degrees---describe the non-uniqueness of factorizations in atomic monoids. We study these arithmetical invariants by the monoid of relations and by presentations of the involved monoids. The abstract results will be applied to numerical monoids and to Krull monoids.
This paper presents a new methodology to count the number of numerical semigroups of given genus ... more This paper presents a new methodology to count the number of numerical semigroups of given genus or Frobenius number. We apply generating function tools to the bounded polyhedron that classifies the semigroups with given genus (or Frobenius number) and multiplicity. First, we give theoretical results about the polynomial-time complexity of counting the number of these semigroups. We also illustrate the methodology analyzing the cases of multiplicity 3 and 4 where some formulas for the number of numerical semigroups for any genus and Frobenius number are obtained.
Multiobjective discrete programming is a well-known family of optimization problems with a large ... more Multiobjective discrete programming is a well-known family of optimization problems with a large spectrum of applications. The linear case has been tackled by many authors during the past few years. However, the polynomial case has not been studied in detail due to its theoretical and computational difficulties. This paper presents an algebraic approach for solving these problems. We propose a methodology based on transforming the polynomial optimization problem to the problem of solving one or more systems of polynomial equations and we use certain Gröbner bases to solve these systems. Different transformations give different methodologies that are theoretically stated and compared by some computational tests via the algorithms that they induce.
This paper presents a new methodology for solving multiobjective integer linear programs (MOILP) ... more This paper presents a new methodology for solving multiobjective integer linear programs (MOILP) using tools from algebraic geometry. We introduce the concept of partial Gröbner basis for a family of multiobjective programs where the right-hand side varies. This new structure extends the notion of Gröbner basis for the single objective case to the case of multiple objectives, i.e., when there is a partial ordering instead of a total ordering over the feasible vectors. The main property of these bases is that the partial reduction of the integer elements in the kernel of the constraint matrix by the different blocks of the basis is zero. This property allows us to prove that this new construction is a test family for a family of multiobjective programs. An algorithm “á la Buchberger” is developed to compute partial Gröbner bases, and two different approaches are derived, using this methodology, for computing the entire set of Pareto-optimal solutions of any MOILP problem. Some examples illustrate the application of the algorithm, and computational experiments are reported on several families of problems.
This paper presents a new complexity result for solving multiobjective integer programming proble... more This paper presents a new complexity result for solving multiobjective integer programming problems. We prove that encoding the entire set of nondominated solutions of the problem in a short sum of rational functions is polynomially doable, when the dimension of the decision space is fixed. This result extends a previous result presented in De Loera et al. (INFORMS J. Comput. 21(1):39–48, 2009) in that there the number of the objective functions is assumed to be fixed whereas ours allows this number to vary.
In this paper we study an actual problem proposed by an agricultural cooperative devoted to harve... more In this paper we study an actual problem proposed by an agricultural cooperative devoted to harvesting corn and grass. The cooperative uses harvesters for harvesting the crop and trucks for carrying it from the smallholdings to the landowners’ silos. The goal is to minimize the total working time of the machinery. Therefore, the cooperative needs to plan both the harvesters and trucks routing. This routing problem simultaneously incorporates the following characteristics: time windows, nested decisions, processing times required to service each facility and the fact that facilities must be visited in clusters. A binary integer linear programming model is proposed to solve this problem. However, since approaches dealing directly with such formulation lead to considerable computation times, we propose a heuristic alternative solution approach for the problem. The heuristic is applied to the case of the cooperative “Os Irmandiños” with a large number of landowners and smallholdings. We report on extensive computational tests to show that the proposed heuristic approach can solve large problems effectively in reasonable computing time.
In this paper we introduce the notion of $m$-irreducibility that extends the standard concept of ... more In this paper we introduce the notion of $m$-irreducibility that extends the standard concept of irreducibility of a numerical semigroup when the multiplicity is fixed. We analyze the structure of
the set of $m$-irreducible numerical semigroups, we give some properties of these numerical semigroups and we present algorithms to compute the decomposition of a numerical semigroup with multiplicity $m$ into $m$-irreducible numerical semigroups.
In this paper we present a mathematical programming formulation for the ω-invariant of a numerica... more In this paper we present a mathematical programming formulation for the ω-invariant of a numerical semigroup for each of its minimal generators which is an useful index in commutative algebra (in particular in factorization theory) to analyze the primality of the elements in the semigroup. The model consists of solving a problem of optimizing a linear function over the efficient set of a multiobjective linear integer program. We offer a methodology to solve this problem and we provide some computational experiments to show the efficiency of the proposed algorithm.
In this paper we present a procedure to build the set of irreducible numerical semigroups with a ... more In this paper we present a procedure to build the set of irreducible numerical semigroups with a fixed Frobenius number. The construction gives us a rooted tree structure for this set. Furthermore, by using the notion of Kunz-coordinates vector we translate the problem of finding such a tree into the problem of manipulating $0-1$ vectors with as many component as the Frobenius number.
This paper presents a model for the expansion of transportation networks incorporating specific r... more This paper presents a model for the expansion of transportation networks incorporating specific requirements about population coverage, budget constraints, intermediate goals and origin–destination flows, among others. The model is applicable to the current expansion project of the Spanish high-speed railway network that has been proposed by the Spanish Government under the program Strategic Planning of Infrastructure and Transport (see source [c]).
Our approach looks for solutions that may be used as additional information in the decision-making process of any network expansion. We report on the application of this methodology to the Spanish railway network and on a computational experience based on simulated data varying the number of cities and time horizons which proves the efficiency of the proposed algorithm.
Arithmetical invariants---such as sets of lengths, catenary and tame degrees---describe the non-u... more Arithmetical invariants---such as sets of lengths, catenary and tame degrees---describe the non-uniqueness of factorizations in atomic monoids. We study these arithmetical invariants by the monoid of relations and by presentations of the involved monoids. The abstract results will be applied to numerical monoids and to Krull monoids.
This paper presents a new methodology to count the number of numerical semigroups of given genus ... more This paper presents a new methodology to count the number of numerical semigroups of given genus or Frobenius number. We apply generating function tools to the bounded polyhedron that classifies the semigroups with given genus (or Frobenius number) and multiplicity. First, we give theoretical results about the polynomial-time complexity of counting the number of these semigroups. We also illustrate the methodology analyzing the cases of multiplicity 3 and 4 where some formulas for the number of numerical semigroups for any genus and Frobenius number are obtained.
Multiobjective discrete programming is a well-known family of optimization problems with a large ... more Multiobjective discrete programming is a well-known family of optimization problems with a large spectrum of applications. The linear case has been tackled by many authors during the past few years. However, the polynomial case has not been studied in detail due to its theoretical and computational difficulties. This paper presents an algebraic approach for solving these problems. We propose a methodology based on transforming the polynomial optimization problem to the problem of solving one or more systems of polynomial equations and we use certain Gröbner bases to solve these systems. Different transformations give different methodologies that are theoretically stated and compared by some computational tests via the algorithms that they induce.
This paper presents a new methodology for solving multiobjective integer linear programs (MOILP) ... more This paper presents a new methodology for solving multiobjective integer linear programs (MOILP) using tools from algebraic geometry. We introduce the concept of partial Gröbner basis for a family of multiobjective programs where the right-hand side varies. This new structure extends the notion of Gröbner basis for the single objective case to the case of multiple objectives, i.e., when there is a partial ordering instead of a total ordering over the feasible vectors. The main property of these bases is that the partial reduction of the integer elements in the kernel of the constraint matrix by the different blocks of the basis is zero. This property allows us to prove that this new construction is a test family for a family of multiobjective programs. An algorithm “á la Buchberger” is developed to compute partial Gröbner bases, and two different approaches are derived, using this methodology, for computing the entire set of Pareto-optimal solutions of any MOILP problem. Some examples illustrate the application of the algorithm, and computational experiments are reported on several families of problems.
This paper presents a new complexity result for solving multiobjective integer programming proble... more This paper presents a new complexity result for solving multiobjective integer programming problems. We prove that encoding the entire set of nondominated solutions of the problem in a short sum of rational functions is polynomially doable, when the dimension of the decision space is fixed. This result extends a previous result presented in De Loera et al. (INFORMS J. Comput. 21(1):39–48, 2009) in that there the number of the objective functions is assumed to be fixed whereas ours allows this number to vary.
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Papers by Victor Blanco
the set of $m$-irreducible numerical semigroups, we give some properties of these numerical semigroups and we present algorithms to compute the decomposition of a numerical semigroup with multiplicity $m$ into $m$-irreducible numerical semigroups.
Our approach looks for solutions that may be used as additional information in the decision-making process of any network expansion. We report on the application of this methodology to the Spanish railway network and on a computational experience based on simulated data varying the number of cities and time horizons which proves the efficiency of the proposed algorithm.
the set of $m$-irreducible numerical semigroups, we give some properties of these numerical semigroups and we present algorithms to compute the decomposition of a numerical semigroup with multiplicity $m$ into $m$-irreducible numerical semigroups.
Our approach looks for solutions that may be used as additional information in the decision-making process of any network expansion. We report on the application of this methodology to the Spanish railway network and on a computational experience based on simulated data varying the number of cities and time horizons which proves the efficiency of the proposed algorithm.