Probability of unique integer solution to a system of linear equations

We consider a system of m linear equations in n variables Ax = d and give necessary and sufficient conditions for the existence of a unique solution to the system that is integer: x ∈ {−1, 1}ⁿ. We achieve this by reformulating the problem as a linear program and deriving necessary and sufficient conditions for the integer solution to be the unique primal optimal solution. We show that as long as m is larger than n/2, then the linear programming reformulation succeeds for most instances, but if m is less than n/2, the reformulation fails on most instances. We also demonstrate that these predictions match the empirical performance of the linear programming formulation to very high accuracy.     

Scheduling of nonresumable jobs and flexible maintenance activities on a single machine to minimize makespan

This study addresses a single machine scheduling problem with periodic maintenance, where the machine is assumed to be stopped periodically for maintenance for a constant time w during the scheduling period. Meanwhile, the maintenance period [u, v] is assumed to have been previously arranged and the time w is assumed not to exceed the available maintenance period [u, v] (i.e. w ? v − u). The time u(v) is the earliest (latest) time at which the machine starts (stops) its maintenance. The objective is to minimize the makespan. Two mixed binary integer programming (BIP) models are provided for deriving the optimal solution. Additionally, an efficient heuristic is proposed for finding the near-optimal solution for large-sized problems. Finally, computational results are provided to demonstrate the efficiency of the models and the effectiveness of the heuristics. The mixed BIP model can optimally solve up to 100-job instances, while the average percentage error of the heuristic is below 1%.     

The p-median problem under uncertainty

Consider the need to currently locate p facilities but it is possible that up to q additional facilities will have to be located in the future. There are known probabilities that 0 ? r ? q facilities will need to be located. The p-median problem under uncertainty is to find the location of p facilities such that the expected value of the objective function in the future is minimized. The problem is formulated on a graph, properties of it are proven, an integer programming formulation is constructed, and heuristic algorithms are suggested for its solution. The heuristic algorithms are modified to reduce the run time by about two orders of magnitude with minimal effect on the quality of the solution. Optimal solutions for many problems are found effectively by CPLEX. Computational results using the heuristic algorithms are presented.     

A bicriteria flowshop scheduling with a learning effect

In many situations, a worker’s ability improves as a result of repeating the same or similar tasks; this phenomenon is known as the learning effect. In this paper the learning effect is considered in a two-machine flowshop. The objective is to find a sequence that minimizes a weighted sum of total completion time and makespan. Total completion time and makespan are widely used performance measures in scheduling literature. To solve this scheduling problem, an integer programming model with n² + 6n variables and 7n constraints where n is the number of jobs is formulated. Because of the lengthy computing time and high computing complexity of the integer programming model, the problem with up to 30 jobs can be solved. A heuristic algorithm and a tabu search based heuristic algorithm are presented to solve large size problems. Experimental results show that the proposed heuristic methods can solve this problem with up to 300 jobs rapidly. According to the best of our knowledge, no work exists on the bicriteria flowshop with a learning effect.     

Factorization of singular integer matrices

It is well known that a singular integer matrix can be factorized into a product of integer idempotent matrices. In this paper, we prove that every n  × n (n > 2) singular integer matrix can be written as a product of 3n + 1 integer idempotent matrices. This theorem has some application in the field of synthesizing VLSI arrays and systolic arrays.     

Constraint programming for computing non-stationary (R, S) inventory policies

This paper proposes a constraint programming model for computing the finite horizon single-item inventory problem with stochastic demands in discrete time periods with service-level constraints under the non-stationary version of the “periodic review, order-up-to-level” policy (i.e., non-stationary (R, S) or, simply (R_n, S_n)). It is observed that the modeling process is more natural and the required number of variables is smaller compared to the MIP formulation of the same problem. The computational tests show that the CP approach is more tractable than the conventional MIP formulation. Two different domain reduction methods are proposed to improve the computational performance of solution algorithms. The numerical experiments confirmed the effectiveness of these methods.     

Approximate methods for convex minimization problems with series–parallel structure

Consider a problem of minimizing a separable, strictly convex, monotone and differentiable function on a convex polyhedron generated by a system of m linear inequalities. The problem has a series–parallel structure, with the variables divided serially into n disjoint subsets, whose elements are considered in parallel. This special structure is exploited in two algorithms proposed here for the approximate solution of the problem. The first algorithm solves at most min{m, ν − n + 1} subproblems; each subproblem has exactly one equality constraint and at most n variables. The second algorithm solves a dynamically generated sequence of subproblems; each subproblem has at most ν − n + 1 equality constraints, where ν is the total number of variables. To solve these subproblems both algorithms use the authors’ Projected Newton Bracketing method for linearly constrained convex minimization, in conjunction with the steepest descent method. We report the results of numerical experiments for both algorithms.     

Rules-based heuristic approach for the U-shaped assembly line balancing problem

The type-2 U-shaped assembly line balancing problem is important for many just-in-time manufactures, but an efficient algorithm is not available at present. Thus, in this study, a novel heuristic approach based on multiple rules and an integer programming model is proposed to address this problem. In the proposed approach, three rules are systematically grouped together, i.e., task selection, task assignment, and task exchange rules. The sufficient conditions for implementing the exchange rules are proposed and proved. Thirteen small or medium scale benchmark issues comprising 63 instances were solved, where the computational results demonstrate the efficiency and effectiveness of the proposed method compared with integer programming. The computational results obtained for 18 examples comprising 121 instances demonstrate that the task exchange rules significantly improve the computational accuracy compared with the traditional heuristic. Finally, 30 new standard instances produced by a systematic data generation process were also solved effectively by the proposed approach. The proposed heuristic approach with multiple rules can provide a theoretical basis for other local search algorithms, especially for addressing issues such as the U-Shaped assembly line balancing problem.     

An O(mn log (nU)) time algorithm to solve the feasibility problem

The phase I maximum flow and most positive cut methods are used to solve the feasibility problem. Both of these methods take one maximum flow computation. Thus the feasibility problem can be solved using maximum flow algorithms. Let n and m be the number of nodes and arcs, respectively. In this paper, we present an algorithm to solve the feasibility problem with integer lower and upper bounds. The running time of our algorithm is O(mn log (nU)), where U is the value of maximum upper bound. Our algorithm improves the O(m² log (nU))-time algorithm in [12]. Hence the current algorithm improves the running time in [12] by a factor of n. Sleator and Goldberg’s algorithm is one of the well-known maximum flow algorithms, which runs in O(mn log n) time, see [5]. Under similarity assumption [11], our algorithm runs in O(mn log n) time, which is the running time of Sleator and Goldberg’s algorithm. The merit of our algorithm is that, in the case of infeasibility of the given network, it not only diagnoses infeasibility but also presents some information that is useful to modeler in estimating the maximum cost of adjusting the infeasible network.     

Bipanconnectivity of faulty hypercubes with minimum degree

In this paper, we consider the conditionally faulty hypercube Q_n with n ? 2 where each vertex of Q_n is incident with at least m fault-free edges, 2 ? m ? n − 1. We shall generalize the limitation m ? 2 in all previous results of edge-bipancyclicity. We also propose a new edge-fault-tolerant bipanconnectivity called k-edge-fault-tolerant bipanconnectivity. A bipartite graph is k-edge-fault-tolerant bipanconnected if G − F remains bipanconnected for any F ⊂ E(G) with ∣F∣ ? k. For every integer m, under the same hypothesis, we show that Q_n is (n − 2)-edge-fault-tolerant edge-bipancyclic and bipanconnected, and the results are optimal with respect to the number of edge faults tolerated. This not only improves some known results on edge-bipancyclicity and bipanconnectivity of hypercubes, but also simplifies the proof.     

The domination numbers of cylindrical grid graphs

Let γ(P_m □ C_n) denote the domination number of the cylindrical grid graph formed by the Cartesian product of the graphs P_m, the path of length m, m ? 2 and the graph C_n, the cycle of length n, n ? 3. In this paper, methods to find the domination numbers of graphs of the form P_m □ C_n with n ? 3 and m = 2, 3 and 4 are proposed. Moreover, bounds on domination numbers of the graphs P₅ □ C_n, n ? 3 are found. The methods that are used to prove that results readily lead to algorithms for finding minimum dominating sets of the above mentioned graphs.     

Enumerating K best paths in length order in DAGs

We address the problem of finding the K best paths connecting a given pair of nodes in a directed acyclic graph (DAG) with arbitrary lengths. One of the main results in this paper is the proof that a tree representing the kth shortest path is obtained by an arc exchange in one of the previous (k − 1) trees (each of which contains a previous best path). An O(m + K(n + log K)) time and O(K + m) space algorithm is designed to explicitly determine the K shortest paths in a DAG with n nodes and m arcs. The algorithm runs in O(m + Kn) time using O(K + m) space in DAGs with integer length arcs. Empirical results confirming the superior performance of the algorithm to others found in the literature for randomly generated graphs are reported.     

A combinatorial problem associated with nonograms

Associated with an m × n matrix with entries 0 or 1 are the m-vector of row sums and n-vector of column sums. In this article we study the set of all pairs of these row and column sums for fixed m and n. In particular, we give an algorithm for finding all such pairs for a given m and n.     

An Algorithm for the Generalized Assignment Problem with Special Ordered Sets

The generalized assignment problem (GAP), the 0–1 integer programming (IP) problem of assigning a set of n items to a set of m knapsacks, where each item must be assigned to exactly one knapsack and there are constraints on the availability of resources for item assignment, has been further generalized recently to include cases where items may be shared by a pair of adjacent knapsacks. This problem is termed the generalized assignment problem with special ordered sets of type 2 (GAPS2). For reasonably large values of m and n the NP-hard combinatorial problem GAPS2 becomes intractable for standard IP software, hence there is a need for the development of heuristic algorithms to solve such problems. It will be shown how a heuristic algorithm developed previously for the GAP problem can be modified and extended to solve GAPS2. Encouraging results, in terms of speed and accuracy, have been achieved.     

Comparative studies on dynamic programming and integer programming approaches for concave cost production/inventory control problems

This paper is concerned with classical concave cost multi-echelon production/inventory control problems studied by W. Zangwill and others. It is well known that the problem with m production steps and n time periods can be solved by a dynamic programming algorithm in O(n ⁴ m) steps, which is considered as the fastest algorithm for solving this class of problems. In this paper, we will show that an alternative 0–1 integer programming approach can solve the same problem much faster particularly when n is large and the number of 0–1 integer variables is relatively few. This class of problems include, among others problem with set-up cost function and piecewise linear cost function with fewer linear pieces. The new approach can solve problems with mixed concave/convex cost functions, which cannot be solved by dynamic programming algorithms.     

A recursive algorithm for constructing complicated Dixon matrices

Whether the determinant of the Dixon matrix equals zero or not is used for determining if a system of n + 1 polynomial equations in n variables has a common root, and is a very efficient quantifier elimination approach too. But for a complicated polynomial system, it is not easy to construct the Dixon matrix. In this paper, a recursive algorithm to construct the Dixon matrix is proposed by which some problems that cannot be tackled by other methods can be solved on the same computer platform. A dynamic programming algorithm based on the recursive formula is developed and compared for speed and efficiency to the recursive algorithm.     

Bound reduction using pairs of linear inequalities

We describe a procedure to reduce variable bounds in mixed integer nonlinear programming (MINLP) as well as mixed integer linear programming (MILP) problems. The procedure works by combining pairs of inequalities of a linear programming (LP) relaxation of the problem. This bound reduction procedure extends the feasibility based bound reduction technique on linear functions, used in MINLP and MILP. However, it can also be seen as a special case of optimality based bound reduction, a method to infer variable bounds from an LP relaxation of the problem. For an LP relaxation with m constraints and n variables, there are O(m ²) pairs of constraints, and a naïve implementation of our bound reduction scheme has complexity O(n ³) for each pair. Therefore, its overall complexity O(m ² n ³) can be prohibitive for relatively large problems. We have developed a more efficient procedure that has complexity O(m ² n ²), and embedded it in two Open-Source solvers: one for MINLP and one for MILP. We provide computational results which substantiate the usefulness of this bound reduction technique for several instances.     

An active index algorithm for the nearest point problem in a polyhedral cone

We consider the problem of finding the nearest point in a polyhedral cone C={x∈R ⁿ :D x≤0} to a given point b∈R ⁿ , where D∈R ^m×n . This problem can be formulated as a convex quadratic programming problem with special structure. We study the structure of this problem and its relationship with the nearest point problem in a pos cone through the concept of polar cones. We then use this relationship to design an efficient algorithm for solving the problem, and carry out computational experiments to evaluate its effectiveness. Our computational results show that our proposed algorithm is more efficient than other existing algorithms for solving this problem.     

A class of primal affine scaling algorithms

We obtain a class of primal affine scaling algorithms which generalize some known algorithms. This class, depending on a r-parameter, is constructed through a family of metrics generated by −r power, r ? 1, of the diagonal iterate vector matrix. We prove the so-called weak convergence of the primal class for nondegenerate linearly constrained convex programming. We observe the computational performance of the class of primal affine scaling algorithms, accomplishing tests with linear programs from the NETLIB library and with some quadratic programming problems described in the Maros and Mészáros repository.     

Reliability of a consecutive (r, s)-out-of-(m, n):F lattice system with conditions on the number of failed components in the system

A consecutive(r, s)-out-of-(m, n):F lattice system which is defined as a two-dimensional version of a consecutive k-out-of-n:F system is used as a reliability evaluation model for a sensor system, an X-ray diagnostic system, a pattern search system, etc. This system consists of m × n components arranged like an (m, n) matrix and fails iff the system has an (r, s) submatrix that contains all failed components. In this paper we deal a combined model of a k-out-of-mn:F and a consecutive (r, s)-out-of-(m, n):F lattice system. Namely, the system has one more condition of system down, that is the total number of failed components, in addition to that of a consecutive (r, s)-out-of-(m, n):F lattice system. We present a method to obtain reliability of the system. The proposed method obtains the reliability by using a combinatorial equation that does not depend on the system size. Some numerical examples are presented to show the relationship between component reliability and system reliability.