The Complexity and Expressive Power of Limit Datalog

Motivated by applications in declarative data analysis, in this article, we study Datalog Z —an extension of Datalog with stratified negation and arithmetic functions over integers. This language is known to be undecidable, so we present the fragment of limit Datalog Z programs, which is powerful enough to naturally capture many important data analysis tasks. In limit Datalog Z , all intensional predicates with a numeric argument are limit predicates that keep maximal or minimal bounds on numeric values. We show that reasoning in limit Datalog Z is decidable if a linearity condition restricting the use of multiplication is satisfied. In particular, limit-linear Datalog Z is complete for Δ 2 EXP and captures Δ 2 P over ordered datasets in the sense of descriptive complexity. We also provide a comprehensive study of several fragments of limit-linear Datalog Z . We show that semi-positive limit-linear programs (i.e., programs where negation is allowed only in front of extensional atoms) capture coNP over ordered datasets; furthermore, reasoning becomes coNEXP-complete in combined and coNP-complete in data complexity, where the lower bounds hold already for negation-free programs. In order to satisfy the requirements of data-intensive applications, we also propose an additional stability requirement, which causes the complexity of reasoning to drop to EXP in combined and to P in data complexity, thus obtaining the same bounds as for usual Datalog. Finally, we compare our formalisms with the languages underpinning existing Datalog-based approaches for data analysis and show that core fragments of these languages can be encoded as limit programs; this allows us to transfer decidability and complexity upper bounds from limit programs to other formalisms. Therefore, our article provides a unified logical framework for declarative data analysis which can be used as a basis for understanding the impact on expressive power and computational complexity of the key constructs available in existing languages.

Consistency Cuts for Dantzig-Wolfe Reformulations

A New Family of Valid-Inequalities for Dantzig-Wolfe Reformulation of Mixed Integer Linear Programs In “Consistency Cuts for Dantzig-Wolfe Reformulation,” Jens Vinther Clausen, Richard Lusby, and Stefan Ropke present a new family of valid inequalities to be applied to Dantzig-Wolfe reformulations with binary linking variables. They show that, for Dantzig-Wolfe reformulations of mixed integer linear programs that satisfy certain properties, it is enough to solve the linear programming relaxation of the Dantzig-Wolfe reformulation with all consistency cuts to obtain integer solutions. An example of this is the temporal knapsack problem; the effectiveness of the cuts is tested on a set of 200 instances of this problem, and the results are state-of-the-art solution times. For problems that do not satisfy these conditions, the cuts can still be used in a branch-and-cut-and-price framework. In order to show this, the cuts are applied to a set of generic mixed linear integer programs from the online library MIPLIB. These tests show the applicability of the cuts in general.

Measuring efficiency of a recycling production system with imprecise data

<p style='text-indent:20px;'>Resources scarcity and environmental degradation have made sustainable resource utilization and environmental protection worldwide. A circular economy system considers economic production activities as closed-loop feedback cycles in which resources are used sustainably and cyclically. Improving the eco-efficiency of the circular economy system has both theoretical value and practical meaning. In this work, the efficiency measurement model of the circular economy system with imprecise data based on network data envelopment analysis is proposed. The two-level mathematical programming approach is employed for measuring the system and process efficiencies. The lower and upper bounds of the efficiencies scores are calculated by transformed conventional one-level linear programs so that the existing solution methods can be applied. The proposed method is applied to assess the circular economy system of EU countries. Our results show that most countries have large difference among fuzzy efficiencies between the production efficiency and recycling efficiency stages, which reveals the source that causes the low efficiency of the circular economy system.</p>

A multi-objective integrated procurement, production, and distribution problem of supply chain network under fuzziness uncertainties

In this paper, we devoted a design under uncertainty of a four-echelon supply chain network including multiple suppliers, multiple plants, multiple distributors and multiple customers. The proposed model is a bi-objective mixed integer linear programming which considers several constraints and aims to minimize the total costs including the procurement, production, storage and distribution costs as well as to maximize on-time deliveries (OTD). To bring the model closer to real-world planning problems, the objective function coefficients (e.g. procurement cost, production cost, inventory holding and transport costs) and other parameters (e.g., demand, production capacity and safety stock level), are all considered triangular fuzzy numbers. Besides, a hybrid mathematical model-based on credibility approach is constructed for the problem, i.e., expected value and chance constrained models. Moreover, to build the crisp equivalent model, we use different property of the credibility measure. The resulted crisp equivalent model is a bi-objective mixed integer linear programs (BOMILP). To transform this crisp BOMILP into a single objective mixed integer linear programs (MILP) model, we apply three different aggregation functions. Finally, numerical results are reported for a real case study to demonstrate the efficiency and applicability of the proposed model.

Novel Formulations and Logic-Based Benders Decomposition for the Integrated Parallel Machine Scheduling and Location Problem

We investigate the discrete parallel machine scheduling and location problem, which consists of locating multiple machines to a set of candidate locations, assigning jobs from different locations to the located machines, and sequencing the assigned jobs. The objective is to minimize the maximum completion time of all jobs, that is, the makespan. Though the problem is of theoretical significance with a wide range of practical applications, it has not been well studied as reported in the literature. For this problem, we first propose three new mixed-integer linear programs that outperform state-of-the-art formulations. Then, we develop a new logic-based Benders decomposition algorithm for practical-sized instances, which splits the problem into a master problem that determines machine locations and job assignments to machines and a subproblem that sequences jobs on each machine. The master problem is solved by a branch-and-cut procedure that operates on a single search tree. Once an incumbent solution to the master problem is found, the subproblem is solved to generate cuts that are dynamically added to the master problem. A generic no-good cut is first proposed, which is later improved by some strengthening techniques. Two optimality cuts are also developed based on optimality conditions of the subproblem and improved by strengthening techniques. Numerical results on small-sized instances show that the proposed formulations outperform state-of-the-art ones. Computational results on 1,400 benchmark instances with up to 300 jobs, 50 machines, and 300 locations demonstrate the effectiveness and efficiency of the algorithm compared with current approaches. Summary of Contribution: This paper employs operations research methods and computing techniques to address an NP-hard combinatorial optimization problem: the parallel discrete machine scheduling and location problem. The problem is of practical significance but has not been well studied in the literature. For the problem, we formulate three novel mixed-integer linear programs that outperform state-of-the-art formulations and develop a new logic-based Benders decomposition algorithm. Extensive computational experiments on 1,400 benchmark instances with up to 300 jobs, 50 machines, and 300 locations are conducted to evaluate the performance of the proposed models and algorithms.

A Branch-and-Bound Algorithm for Polymatrix Games ϵ-Proper Nash Equilibria Computation

When several Nash equilibria exist in the game, decision-makers need to refine their choices based on some refinement concepts. To this aim, the notion of a ϵ-proper equilibria set for polymatrix games is used to develop 0–1 mixed linear programs and compute ϵ-proper Nash equilibria. A Branch-and-Bound exact arithmetics algorithm is proposed. Experimental results are provided on polymatrix games randomly generated with different sizes and densities.

Diameters of Cocircuit Graphs of Oriented Matroids: An Update

Oriented matroids are combinatorial structures that generalize point configurations, vector configurations, hyperplane arrangements, polyhedra, linear programs, and directed graphs. Oriented matroids have played a key role in combinatorics, computational geometry, and optimization. This paper surveys prior work and presents an update on the search for bounds on the diameter of the cocircuit graph of an oriented matroid. The motivation for our investigations is the complexity of the simplex method and the criss-cross method. We review the diameter problem and show the diameter bounds of general oriented matroids reduce to those of uniform oriented matroids. We give the latest exact bounds for oriented matroids of low rank and low corank, and for all oriented matroids with up to nine elements (this part required a large computer-based proof). For arbitrary oriented matroids, we present an improvement to a quadratic bound of Finschi. Our discussion highlights an old conjecture that states a linear bound for the diameter is possible. On the positive side, we show the conjecture is true for oriented matroids of low rank and low corank, and, verified with computers, for all oriented matroids with up to nine elements. On the negative side, our computer search showed two natural strengthenings of the main conjecture are false.