Compact Formulation Research Articles

A numerical study of initial flow past an impulsively started rotationally oscillating circular cylinder using a transformation-free HOC scheme

The initial development of the two dimensional viscous, incompressible flow induced by an impulsively started circular cylinder which performs time dependent sinusoidal rotational oscillations about its axis is investigated numerically. The investigation is based on the solutions of stream function-vorticity formulation of Navier-Stokes equations on non-uniform polar grids using higher order compact formulation. The numerical method is validated by comparing the computed results with existing experimental and numerical results for Reynolds numbers Re = 150 and 500. The effects of forced oscillation frequency f and peak rotation rate αm on the early development of the flow structure in the near wake region are discussed. Results are given for the initial development with time of the flow structure at the rear of the cylinder at Re = 200. The details of the formation, movement, closure points, and strengths of the vortices behind the cylinder are presented. The velocity profiles at different locations and vorticity profiles at the surface of the cylinder are also shown. The effect of increase in αm on the timing of the formation of the vortices, the closed wake length, and the thickness of the boundary layer is investigated.

The vehicle routing problem with service level constraints

We consider a vehicle routing problem which seeks to minimize cost subject to service level constraints on several groups of deliveries. This problem captures some essential challenges faced by a logistics provider which operates transportation services for a limited number of partners and should respect contractual obligations on service levels. The problem also generalizes several important classes of vehicle routing problems with profits. To solve it, we propose a compact mathematical formulation, a branch-and-price algorithm, and a hybrid genetic algorithm with population management, which relies on problem-tailored solution representation, crossover and local search operators, as well as an adaptive penalization mechanism establishing a good balance between service levels and costs. Our computational experiments show that the proposed heuristic returns very high-quality solutions for this difficult problem, matches all optimal solutions found for small and medium-scale benchmark instances, and improves upon existing algorithms for two important special cases: the vehicle routing problem with private fleet and common carrier, and the capacitated profitable tour problem. The branch-and-price algorithm also produces new optimal solutions for all three problems.

On recoverable and two-stage robust selection problems with budgeted uncertainty

In this paper, the problem of selecting p out of n available items is discussed, such that their total cost is minimized. We assume that the item costs are not known exactly, but stem from a set of possible outcomes modeled through budgeted uncertainty sets, i.e., the interval uncertainty sets with an additional linear (budget) constraint, in their discrete and continuous variants. Robust recoverable and two-stage models of this selection problem are analyzed through an in-depth discussion of variables at their optimal values. Polynomial algorithms for both models under continuous budgeted uncertainty are proposed. In the case of discrete budgeted uncertainty, compact mixed integer formulations are constructed and some approximation algorithms are proposed. Polynomial combinatorial algorithms for the adversarial and incremental problems (the special cases of the considered robust models) under both discrete and continuous budgeted uncertainty are constructed.

The sport teams grouping problem

The sport teams grouping problem (STGP) concerns the assignment of sport teams to round-robin tournaments. The objective is to minimize the total travel distance of the participating teams while simultaneously respecting fairness constraints. The STGP is an NP-Hard combinatorial optimization problem highly relevant in practice. This paper investigates the performance of some complimentary optimization approaches to the STGP. Three integer programming formulations are presented and thoroughly analyzed: two compact formulations and another with an exponential number of variables, for which a branch-and-price algorithm is proposed. Additionally, a meta-heuristic method is applied to quickly generate feasible high-quality solutions for a set of real-world instances. By combining the different approaches’ results, solutions within 1.7% of the optimum values were produced for all feasible instances. Additionally, to support further research, the considered STGP instances and corresponding solutions files were shared online.

Die betekenis van die bepalings van die 1996 Grondwet: Die aanhef en hoofstuk 1

The Preamble and Chapter 1 This contribution is intended to be the first installment of a systematic interpretation of the Constitution of the Republic of South Africa 1996. 
 Due to the foundational and repetitive reference in the text to values, regard must constantly be had to those values when this Constitution is interpreted. 
 Even though the preamble does not contain positive norms, is an important interpretive source of the foundations of the Constitution. An important deviation from the preamble of the 1993 Constitution, is that the term Rechtsstaat ("constitutional state") is not employed. The introduction of this notion in South African law and its meaning in general is described. With reference to relevant dicta in recent constitutional cases, the Constitutional Principles in terms of which the 1996 Constitution was formulated and the text of the Constitution itself, it is argued that this is essentially a Rechtsstaat Constitution, but that the divergence in the range of constitutional values creates the danger of the constitutional state floundering in the waters of the social state. 
 Section 1, being the foundational provision, is not unamendable, but it is very tightly entrenched. The most profound values of the Constitution are set out in this compact formulation. The question is inevitably raised whether, where values have to compete for precedence in concrete circumstances, a hierarchy of values must be construed. An analysis of section 1 in the context of other relevant provisions of the Constitution reveals that human dignity is the primary nuclear value of the
 Constitution, supported by equality and freedom. Democracy, supremacy of the
 Constitution and the rule of law are structural and procedural values of the Constitution subordinate to the nuclear values and non-racialism and non-sexism are derived values. 
 How it is possible for a constitution to be superior law, as section 2 provides regarding the 1996 Constitution, is analysed against the background of the social contract theory. The weaknesses of this theory are exposed and it is argued that the force external to the Constitution that guarantees its primacy, is its practical legitimacy, i.e. sufficient support or acceptance of the authority of the Constitution by the citizenry. Section 2 is phrased in strong terms and means that no juridically relevant conduct, be it of a private or public law nature, can escape the test of constitutionality. 
 In the interpretation of section 3 the nature of citizenship and nationality is analysed with reference to international authorities and definitions of these concepts are developed. The legal implications of citizenship in the context of the Constitution are set out and the historical context of citizenship having been used in pre-constitutional times as an instrument for creating separate ethnic states, is described. The current post modern tendency in places to devalue citizenship is contrasted with the importance being attached to the notion in South Africa in the context of nation building and the employment of expatriates. 
 Regarding sections 4 and 5 the formal regulation of the national anthem and national flag is described. The national anthem may be amended by presidential proclamation, but changes to the national flag require an amendment of the Constitution. 
 Section 6, which deals with the complex language matter, protects linguistic diversity rather than the status of any languages. The Constitutional Court has determined that, although no express provision to this effect exists, individuals are entitled to use the language of their choice in their dealings and communications with the government. The state is required to promote "the indigenous languages of our people." This is interpreted to include the nine official indigenous African languages, Afrikaans, Khoi, Nama and San. In the determination of language policies Municipal Councils are required to take the language usage and preferences of the inhabitants into account and in the national and provincial at least two official languages must be used. Essential facts regarding language usage, demographic distribution, etc. must be taken into consideration for the determination of a language policy to conform to the Constitution.

High performance modelling of the transport of energetic particles for photon radiotherapy

This work consists of the validation of a new Grid Based Boltzmann Solver (GBBS) conceived for the description of the transport and energy deposition by energetic particles for radiotherapy purposes. The entropic closure and a compact mathematical formulation allow our code (M1) to calculate the delivered dose with an accuracy comparable to the Monte-Carlo (MC) codes with a computational time that is reduced to the order of few minutes without any special processing power requirement. A validation protocol with heterogeneity inserts has been defined for different photon sources. The comparison with the MC calculated depth-dose curves and transverse profiles of the beam at different depths shows an excellent accuracy of the M1 model.

Lifted polynomial size formulations for the homogeneous and heterogeneous vehicle routing problems

We propose compact formulations for the symmetric and asymmetric capacitated vehicle routing problems. These formulations are obtained by lifting, using the Reformulation–Linearization Technique, a novel polynomial size ordered path-based formulation. We show that the strongest proposed formulation is equivalent to the strongest multi-commodity flow formulation presented so far. In addition, we propose polynomial size valid inequalities that aim at further tightening the proposed formulations. Also, the tightest formulation is extended to model the fleet size and mix vehicle routing problem with fixed costs. We show that the new derived model is not comparable with a state-of-the-art polynomial length formulation. We present the results of computational experiments that demonstrate the tightness of the proposed formulations and the impact of the valid inequalities.

A branch‐and‐cut algorithm for the Team Orienteering Problem

AbstractThe Team Orienteering Problem aims at maximizing the total amount of profit collected by a fleet of vehicles while not exceeding a predefined travel time limit on each vehicle. In the last years, several exact methods based on different mathematical formulations were proposed. In this paper, we present a new two‐index formulation with a polynomial number of variables and constraints. This compact formulation, reinforced by connectivity constraints, was solved by means of a branch‐and‐cut algorithm. The total number of instances solved to optimality is 327 of 387 benchmark instances, 26 more than any previous method. Moreover, 24 not previously solved instances were closed to optimality.

Pickup and delivery problem with time windows: A new compact two-index formulation

We propose a formulation for the pickup and delivery problem with time windows, based on a novel modeling strategy that allows the assignment of vehicles to routes explicitly in two-index flow formulations. It leads to an effective compact formulation that can benefit OR practitioners interested in solving the problem by general-purpose optimization software. Computational experiments indicate that the proposed formulation has interesting features and best overall performance in relation to other compact formulations.

High-order implicit finite difference schemes for the two-dimensional Poisson equation

In this paper, a new family of high-order finite difference schemes is proposed to solve the two-dimensional Poisson equation by implicit finite difference formulas of (2M+1) operator points. The implicit formulation is obtained from Taylor series expansion and wave plane theory analysis, and it is constructed from a few modifications to the standard finite difference schemes. The approximations achieve (2M+4)-order accuracy for the inner grid points and up to eighth-order accuracy for the boundary grid points. Using a Successive Over-Relaxation method, the high-order implicit schemes have faster convergence as M is increased, compensating the additional computation of more operator points. Thus, the proposed solver results in an attractive method, easy to implement, with higher order accuracy but nearly the same computation cost as those of explicit or compact formulation. In addition, particular case M=1 yields a new compact finite difference schemes of sixth-order of accuracy. Numerical experiments are presented to verify the feasibility of the proposed method and the high accuracy of these difference schemes.

Nonperturbative Casimir effect and monopoles: Compact Abelian gauge theory in two spatial dimensions

We demonstrate that Casimir forces associated with zero-point fluctuations of quantum vacuum may be substantially affected by the presence of dynamical topological defects. In order to illustrate this nonperturbative effect we study the Casimir interactions between dielectric wires in a compact formulation of Abelian gauge theory in two spatial dimensions. The model possesses topological defects, instanton-like monopoles, which are known to be responsible for nonperturbative generation of a mass gap and for a linear confinement of electrically charged probes. Despite the model has no matter fields, the Casimir energy depends on the value of the gauge coupling constant. We show, both analytically and numerically, that in the strong coupling regime the Abelian monopoles make the Casimir forces short-ranged. Simultaneously, their presence increases the interaction strength between the wires at short distances for certain range of values of the gauge coupling. The wires suppress monopole density in the space between them compared to the density outside the wires. In the weak coupling regime the monopoles become dilute and the Casimir potential reduces to a known theoretical result which does not depend on the gauge coupling.

Primal Heuristics for Branch and Price: The Assets of Diving Methods

Primal heuristics have become essential components in mixed integer programming (MIP) solvers. Extending MIP-based heuristics, our study outlines generic procedures to build primal solutions in the context of a branch-and-price approach and reports on their performance. Our heuristic decisions carry on variables of the Dantzig–Wolfe reformulation, the motivation being to take advantage of a tighter linear programming relaxation than that of the original compact formulation and to benefit from the combinatorial structure embedded in these variables. We focus on the so-called diving methods that use reoptimization after each linear programming rounding. We explore combinations with diversification-intensification paradigms such as limited discrepancy search, sub-MIP, local branching, and strong branching. The dynamic generation of variables inherent to a column generation approach requires specific adaptation of heuristic paradigms. We manage to use simple strategies to get around these technical issues. Our numerical results on generalized assignment, cutting stock, and vertex-coloring problems set new benchmarks, highlighting the performance of diving heuristics as generic procedures in a column generation context and producing better solutions than state-of-the-art specialized heuristics in some cases.

A Stable Analytic Model for Tilted-Coil Antennas in a Concentrically Cylindrical Multilayered Anisotropic Medium

A set of compact and stable formulations is presented to calculate the response of logging tools employing tilted-coil antennas in anisotropic concentrically cylindrical multilayered formation. The presented formulations can be used to investigate the effect of mandrel, borehole, and invasion. Our formulations are different from previous ones in three aspects. First, the tool mandrel can be cylindrically multilayered, and electrical parameters of each layer are arbitrary. That is to say, the mandrel is not limited to being perfectly metallic. Second, the formation can be either anisotropic or isotropic. Finally, the propagation of electromagnetic wave in an inhomogeneous cylindrical medium is described by the generalized reflection and transmission coefficients without the numerical overflow issue. Numerical results in comparison with those from other approaches have demonstrated the validity and stability of our new formulations.

Nodes Migration Scheduling of Access Networks

In this paper, we consider the nodes migration scheduling problem that aims to migrate nodes from an outdated access network to a new one. In order to avoid services’ interruption, a bridge is installed temporarily between the two networks and, as for the nodes, they are migrated, one at each time period, from the old to the new network. The objective is to find a migration’s sequence that minimizes the cost of the capacity to install on the bridge. We provide a compact formulation of polynomial size in order to solve the problem optimally. We reformulate it in such a way that the new model reduces drastically the branch-and-bound search tree. In addition, we give a lower bound based on a bi-partitioning problem that turns out to be close to the optimal solutions. We conduct experiments on a case study consisting of migrating eNodeBs of a 4G network from an access network to another. The computational experiments show the performance of our approach as the proposed model provided optimal solutions for up to 40 nodes.

Assortment Optimization Under a Single Transition Model

In this paper, we consider a Markov chain choice model with single transition. In this model, customers arrive at each product with a certain probability. If the arrived product is unavailable, then the seller can recommend a subset of available products to the customer and the customer will purchase one of the recommended products or choose not to purchase with certain transition probabilities. The distinguishing features of the model are that the seller can control which products to recommend depending on the arrived product and that each customer either purchases a product or leaves the market after one transition. We study the assortment optimization problem under this model. Particularly, we show that this problem is generally NP-Hard even if each product could only transit to at most two products. Despite the complexity of the problem, we provide polynomial time algorithms for several special cases, such as when the transition probabilities are homogeneous with respect to the starting point, or when each product can only transit to one other product. We also provide a tight performance bound for revenue-ordered assortments. In addition, we propose a compact mixed integer program formulation that can solve this problem of large size. Through extensive numerical experiments, we show that the proposed algorithms can solve the problem efficiently and the obtained assortments could significantly improve the revenue of the seller than under the Markov chain choice model.

On imposing connectivity constraints in integer programs

In many network applications, one searches for a connected subset of vertices that exhibits other desirable properties. To this end, this paper studies the connected subgraph polytope of a graph, which is the convex hull of subsets of vertices that induce a connected subgraph. Much of our work is devoted to the study of two nontrivial classes of valid inequalities. The first are the a, b-separator inequalities, which have been successfully used to enforce connectivity in previous computational studies. The second are the indegree inequalities, which have previously been shown to induce all nontrivial facets for trees. We determine the precise conditions under which these inequalities induce facets and when each class fully describes the connected subgraph polytope. Both classes of inequalities can be separated in polynomial time and admit compact extended formulations. However, while the a, b-separator inequalities can be lifted in linear time, it is NP-hard to lift the indegree inequalities.

The Traveling Purchaser Problem with time-dependent quantities

The deterministic Traveling Purchaser Problem looks for a tour visiting a subset of markets in order to satisfy a positive discrete demand for some products at minimum traveling and purchasing costs. In this paper, we assume that the quantities available in the markets for all the products are time-varying decreasing at a constant rate. We propose a compact mixed integer formulation for the problem, and strengthen it with the introduction of connectivity constraints. A new branching strategy and a primal heuristic enforcing the bounding operations have been embedded into a branch-and-cut framework. The branching rule exploits a simple valid inequality and the potential presence of necessary markets. The resulting method outperforms CPLEX 12.6 when used to solve the proposed model. The algorithms have been tested on standard TSPLIB instances, modified to include products and quantities that decrease at different rates of consumption.

'Full fusion' is not ineluctable during vesicular exocytosis of neurotransmitters by endocrine cells.

Vesicular exocytosis is an essential and ubiquitous process in neurons and endocrine cells by which neurotransmitters are released in synaptic clefts or extracellular fluids. It involves the fusion of a vesicle loaded with chemical messengers with the cell membrane through a nanometric fusion pore. In endocrine cells, unless it closes after some flickering ('Kiss-and-Run' events), this initial pore is supposed to expand exponentially, leading to a full integration of the vesicle membrane into the cell membrane-a stage called 'full fusion'. We report here a compact analytical formulation that allows precise measurements of the fusion pore expansion extent and rate to be extracted from individual amperometric spike time courses. These data definitively establish that, during release of catecholamines, fusion pores enlarge at most to approximately one-fifth of the radius of their parent vesicle, hence ruling out the ineluctability of 'full fusion'.

Liquisolid compacts of meloxicam: in-vitro and in-vivo evaluation

Background Meloxicam (MXM) is a poorly soluble drug and its low aqueous solubility leads to poor dissolution and bioavailability. Aim The aim of this study was to investigate the potential of a liquisolid system to improve the dissolution rate and the bioavailability of MXM. Materials and methods The liquisolid compacts were prepared using Avicel PH102 as a carrier material, Aerosil 200 as a coating material, Polyethylene glycol 400 as a liquid vehicle, and sodium starch glycolate as a superdisintegrating agent. The 15 liquisolid compact formulations were prepared by varying drug concentrations in the liquid vehicle. Attenuated total reflectance Fourier transform infrared spectroscopy, differential scanning calorimetry, and powder X-ray diffraction were used to investigate the physicochemical interaction and crystallinity of the drug in the liquisolid compact. MXM compacts were evaluated for uniformity of drug content, tablet hardness, friability, disintegration, and dissolution. An in-vivo study was carried out in male albino rats. The data were presented as mean±SD and were compared using the one-way ANOVA. A P-value less than 0.05 were considered to be significant. Results The liquisolid system of MXM was formulated successfully using Avicel PH102, Aerosil 200, and Polyethylene glycol 200. The results of attenuated total reflectance Fourier transform infrared spectroscopy, differential scanning calorimetry, and X-ray diffraction study indicated the existence of hydrogen bonding between drug and excipients and the complete amorphization of MXM. In-vitro evaluation parameters for the liquisolid compact were found to be within the acceptable limits. It was found that optimized liquisolid tablet formulation showed higher dissolution than the marketed tablet, with more than 80% drug release within 10 min. The pharmacokinetic data showed a higher bioavailability of liquisolid compact of MXM compared with the marketed product. Conclusion The liquisolid compact can be a promising alternative for the formulation of water-insoluble drug MXM with improved dissolution and bioavailability.

Visible light communication networks under ring and tree topology constraints

In this paper, we propose new mixed integer linear programming (MILP) models in order to maximize the total capacity of a wireless visible light communication (VLC) network subject to power and ring (or tree) topology constraints. VLC has appeared as a new promising technology as it allows to transmit data in significantly higher orders of magnitude than traditional radio frequency (RF) methods [7,11]. Moreover, it allows to establish wireless communications where RF technologies cannot operate in optimal conditions such as in airplanes, hospitals, nuclear areas, underground mining, underwater communications and disaster and traffic management, to name a few. Consequently, it is expected that most of the current existing protocols that use RF methods will incorporate this new technology in the near future. We propose exponential and compact polynomial formulations that we obtain from the classic traveling salesman and spanning tree problems in order to characterize the ring and tree topology constraints in our models. For each of the proposed models, we obtain robust formulations in the form of a MILP problem [6]. Finally, we adapt an exact iterative procedure to solve the exponential models to optimality [1]. For the deterministic and robust ring models, we solve instances with up to 500 and 300 nodes to optimality, respectively. Then, we further propose a new relaxed version of the algorithm that allows to compute tight near optimal solutions for all the exponential models with up to 500 nodes. In general, the two algorithms obtain solutions in significantly less CPU time when compared to the compact models.