Super Solutions of Random Instances of Satisfiability

2015 ◽  
pp. 314-325
Author(s):  
Peng Zhang ◽  
Yong Gao
Keyword(s):  
Random Instances

scholarly journals On the NP-Isomorphism Problem with Respect to Random Instances

1995 ◽  
Vol 50 (1) ◽  
pp. 151-164 ◽  
Author(s):  
J. Wang ◽  
J. Belanger
Keyword(s):  
Isomorphism Problem ◽  
Random Instances

A Model for Integrated Inventory and Assortment Planning

2021 ◽  
Author(s):  
Victor Martínez-de-Albéniz ◽  
Sumit Kunnumkal
Keyword(s):  
Closed Form Solution ◽  
Optimal Solution ◽  
Integer Program ◽  
Form Solution ◽  
Assortment Planning ◽  
Dimensional Parameter ◽  
Planning Decisions ◽  
Fixed Point Equation ◽  
Point Equation ◽  
Random Instances

Integrating inventory and assortment planning decisions is a challenging task that requires comparing the value of demand expansion through broader choice for consumers with the value of higher in-stock availability. We develop a stockout-based substitution model for trading off these values in a setting with inventory replenishment, a feature missing in the literature. Using the closed form solution for the single-product case, we develop an accurate approximation for the multiproduct case. This approximated formulation allows us to optimize inventory decisions by solving a fractional integer program with a fixed point equation constraint. When products have equal margins, we solve the integer program exactly by bisection over a one-dimensional parameter. In contrast, when products have different margins, we propose a fractional relaxation that we can also solve by bisection and that results in near-optimal solutions. Overall, our approach provides solutions within 0.1% of the optimal policy and finds the optimal solution in 80% of the random instances we generate. This paper was accepted by David Simchi-Levi, optimization.

scholarly journals Solving the one-warehouse N-retailers problem with stochastic demand: An inter-ratio policies approach

2021 ◽  
pp. 131-142
Author(s):  
Gabriela Chavarro ◽  
Matthaus Fresen ◽  
Esneyder Rafael González ◽  
David Barrera Ferro ◽  
Héctor López-Ospina
Keyword(s):  
Cost Savings ◽  
Function Evaluation ◽  
Normal Density ◽  
Single Product ◽  
Total Cost ◽  
Stochastic Version ◽  
Normal Density Function ◽  
Stochastic Solution ◽  
Random Instances ◽  
The One

In this paper, we consider a two-echelon supply chain in which one warehouse provides a single product to N retailers, using integer-ratio policies. Deterministic version of the problem has been widely studied. However, this assumption can lead to inaccurate and ineffective decisions. In this research, we tackle the stochastic version of two-echelon inventory system by designing an extension of a well-known heuristic. This research considers customer demands as following a normal density function. A set of 240 random instances was generated and used in evaluating both the deterministic and stochastic solution approaches. Due to the nature of the objective function, evaluation was carried out via Monte Carlo simulation. For variable demand settings, computational experiments shows that: i) the use of average demand to define the inventory policy implies an underestimation of the total cost and ii) the newly proposed method offers cost savings.

scholarly journals An Improved Differential Evolution Solution for Software Project Scheduling Problem

2015 ◽  
Vol 2015 ◽  
pp. 1-9 ◽  
Author(s):  
A. C. Biju ◽  
T. Aruldoss Albert Victoire ◽  
Kumaresan Mohanasundaram
Keyword(s):  
Differential Evolution ◽  
Project Scheduling ◽  
Optimization Technique ◽  
Solution Procedure ◽  
Solution Technique ◽  
Software Project ◽  
Scheduling Problem ◽  
Np Hard Problem ◽  
Random Instances ◽  
Project Scheduling Problem

This paper proposes a differential evolution (DE) method for the software project scheduling problem (SPSP). The interest on finding a more efficient solution technique for SPSP is always a topic of interest due to the fact of ever growing challenges faced by the software industry. The curse of dimensionality is introduced in the scheduling problem by ever increasing software assignments and the number of staff who handles it. Thus the SPSP is a class of NP-hard problem, which requires a rigorous solution procedure which guarantees a reasonably better solution. Differential evolution is a direct search stochastic optimization technique that is fairly fast and reasonably robust. It is also capable of handling nondifferentiable, nonlinear, and multimodal objective functions like SPSP. This paper proposes a refined DE where a new mutation mechanism is introduced. The superiority of the proposed method is experimented and demonstrated by solving the SPSP on 50 random instances and the results are compared with some of the techniques in the literature.

Chapter 6. Incomplete Algorithms

2021 ◽  
Author(s):  
Henry Kautz ◽  
Ashish Sabharwal ◽  
Bart Selman
Keyword(s):  
Local Search ◽  
Greedy Algorithms ◽  
Lagrangian Method ◽  
Stochastic Local Search ◽  
Local Minima ◽  
Backtracking Search ◽  
Sat Solvers ◽  
Random Instances ◽  
Adaptive Noise ◽  
Key Techniques

Research on incomplete algorithms for satisfiability testing lead to some of the first scalable SAT solvers in the early 1990’s. Unlike systematic solvers often based on an exhaustive branching and backtracking search, incomplete methods are generally based on stochastic local search. On problems from a variety of domains, such incomplete methods for SAT can significantly outperform DPLL-based methods. While the early greedy algorithms already showed promise, especially on random instances, the introduction of randomization and so-called uphill moves during the search significantly extended the reach of incomplete algorithms for SAT. This chapter discusses such algorithms, along with a few key techniques that helped boost their performance such as focusing on variables appearing in currently unsatisfied clauses, devising methods to efficiently pull the search out of local minima through clause re-weighting, and adaptive noise mechanisms. The chapter also briefly discusses a formal foundation for some of the techniques based on the discrete Lagrangian method.

Towards a Predictive Computational Complexity Theory for Periodically Specified Problems: A Survey

2005 ◽  
Author(s):  
Harry B. Hunt III ◽  
Madhav V. Marathe
Keyword(s):  
Statistical Mechanics ◽  
Vlsi Design ◽  
Regular Structure ◽  
Combinatorial Problems ◽  
Practical Applications ◽  
Combinatorial Objects ◽  
Graph Theoretic ◽  
Computer Scientists ◽  
Random Instances ◽  
Problem Instances

The preceding chapters in this volume have documented the substantial recent progress towards understanding the complexity of randomly specified combinatorial problems. This improved understanding has been obtained by combining concepts and ideas from theoretical computer science and discrete mathematics with those developed in statistical mechanics. Techniques such as the cavity method and the replica method, primarily developed by the statistical mechanics community to understand physical phenomena, have yielded important insights into the intrinsic difficulty of solving combinatorial problems when instances are chosen randomly. These insights have ultimately led to the development of efficient algorithms for some of the problems. A potential weakness of these results is their reliance on random instances. Although the typical probability distributions used on the set of instances make the mathematical results tractable, such instances do not, in general, capture the realistic instances that arise in practice. This is because practical applications of graph theory and combinatorial optimization in CAD systems, mechanical engineering, VLSI design, transportation networks, and software engineering involve processing large but regular objects constructed in a systematic manner from smaller and more manageable components. Consequently, the resulting graphs or logical formulas have a regular structure, and are defined systematically in terms of smaller graphs or formulas. It is not unusual for computer scientists and physicists interested in worst-case complexity to study problem instances with regular structure, such as lattice-like or tree-like instances. Motivated by this, we discuss periodic specifications as a method for specifying regular instances. Extensions of the basic formalism that give rise to locally random but globally structured instances are also discussed. These instances provide one method of producing random instances that might capture the structured aspect of practical instances. The specifications also yield methods for constructing hard instances of satisfiability and various graph theoretic problems, important for testing the computational efficiency of algorithms that solve such problems. Periodic specifications are a mechanism for succinctly specifying combinatorial objects with highly regular repetitive substructure. In the past, researchers have also used the term dynamic to refer to such objects specified using periodic specifications (see, for example, Orlin [419], Cohen and Megiddo [103], Kosaraju and Sullivan [347], and Hoppe and Tardos [260]).

Analytical Determination of the Proximity of Two Right-Circular Cylinders in Space

2016 ◽  
Vol 8 (4) ◽  
Author(s):  
Saurav Agarwal ◽  
Rangaprasad Arun Srivatsan ◽  
Sandipan Bandyopadhyay
Keyword(s):  
Polynomial Equation ◽  
General Purpose ◽  
Circular Cylinders ◽  
Analytical Determination ◽  
Closest Pair ◽  
Shortest Distance ◽  
Special Cases ◽  
Random Instances ◽  
Geometric Primitives

This paper presents a novel analytical formulation for identifying the closest pair of points lying on two arbitrary cylinders in space, and subsequently the distance between them. Each cylinder is decomposed into four geometric primitives. It is shown that the original problem reduces to the computation of the shortest distance between five distinct combinations of these primitives. Four of these subproblems are solved in closed form, while the remaining one requires the solution of an eight-degree polynomial equation. The analytical nature of the formulation and solution allows the identification of all the special cases, e.g., positive-dimensional solutions, and the curve of intersection when the cylinders interfere. The symbolic precomputation of the results leads to a fast numerical implementation, capable of solving the problem in about 50 μs (averaged over 1 × 106 random instances of the most general case) on a standard PC. The numerical results are verified by repeating all the calculations in a general-purpose commercial cad software. The algorithm has significant potential for applications in the various aspects of robotics and mechanisms, as their links can be modeled easily and compactly as cylinders. This makes tasks such as path planning, determination of the collision-free workspace, etc., computationally easier.

scholarly journals A Sustainable Multimodal Transport System: The Two-Echelon Location-Routing Problem with Consolidation in the Euro–China Expressway

2019 ◽  
Vol 11 (19) ◽  
pp. 5486 ◽  
Author(s):  
Lu ◽  
Lang ◽  
Yu ◽  
Li
Keyword(s):  
Transport System ◽  
Emission Reduction ◽  
Large Scale ◽  
Differential Evolution Algorithm ◽  
Location Routing Problem ◽  
Routing Problem ◽  
Location Routing ◽  
Multimodal Transport ◽  
Random Instances ◽  
Evolution Algorithm

Sustainable development of transport systems is a common topic of concern and effort in multiple countries, in which reducing carbon emissions is one of the core goals. Multimodal transport is an effective way to achieve carbon emission reduction and to efficiently utilize transport resources. The intercontinental transport system, represented by the Euro–China Expressway, is a prominent exploration that has recently received attention, which promotes the sustainable development of transport between countries and carbon emission reduction. In the intercontinental multimodal transport system, the reasonable connection of roads and railways, especially the optimization of consolidation, is an important link which affects the system's carbon emissions. This paper focuses on the consolidation of sustainable multimodal transport and summarizes the multimodal transport two-echelon location-routing problem with consolidation (MT-2E-LRP-C). We aim to solve multimodal consolidation optimization problem, especially locations of multimodal station, by routing of highway and railway. We propose a two-layer mixed integer linear problem (MILP) model, which highlights the consolidation of roads and railways, focuses on road and rail transport connections, and optimizes road routes and railway schemes. To validate the MT-2E-LRP-C model, we design a series of random instances for different quantities of nodes. In order to solve large-scale instances and realistic transport problems, we propose a hybrid differential evolution algorithm, which decomposes the problem into a railway layer and a highway layer for heuristic algorithm solving. Furthermore, the MILP model and algorithm are tested by small-scale random instances, and the hybrid differential evolution algorithm is solved for the large-scale random instances. Finally, we solve the realist instance from the Euro–China Expressway to develop instructive conclusions.

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