An exact solution for flow in a porous pipe

Summary An exact solution of the Navier-Stokes equations for flow in a porous pipe is presented. This solution allows the suction or injection at the wall to vary with axial distance and will provide new insight into flows through porous pipes.

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Resumé Une solution exacte d'équation de Navier-Stokes est présentée pour l'écoulement d'un liquide visqueux dans un tube perméable. Ce liquide est aspiré ou injecté avec une vélocité variable et la solution donne une nouvelle optique quant aux tubes poreuses.

     

An exact solution procedure for a cluster hub location problem

This paper proposes a new (MIP) model formulation and a new solution procedure for the hub network design problem under a non-restrictive policy introduced by Sung and Jin [Sung, C.S., Jin, H.W., 2001. Dual-based approach for a hub network design problem under non-restrictive policy. European Journal of Operational Research 132 (1), 88–105]. The model formulation contains significantly fewer variables so that optimal solutions for the LP-relaxation of the model can be determined for large instances using standard procedures for LP-models. Furthermore, the LP-relaxation provides very tight lower bounds. Computational results are given, which demonstrate that the new model formulation allows for solving much larger instances. It turned out that the new (exact) solution procedure, which utilises the new model formulation, is faster than the heuristic proposed by Sung and Jin (2001). It is also shown that the problem is np-hard.     

An exact similarity solution in radiation-gas-dynamics

We have found an exact similarity solution of the point explosion problem in the case when the total energy of the shock wave that is produced is not constant but decreases with time and when the loss due to radiation escape is significant. We have compared the results of our exact solution with those of exact numerical solutions of Elliot and Wang and have explained the cause why our solution differs from theirs in certain aspects.     

An exact solution of a scalar-tensor theory of gravitation

A class of exact solutions of the cylindrically symmetric space-time with two degrees of freedom corresponding to the vacuum field equations of a scalar-tensor theory, formally similar to Brans-Dicke theory, is obtained. The solutions possess wave-like character.     

An exact solution framework for a broad class of vehicle routing problems

This paper presents an exact solution framework for solving some variants of the vehicle routing problem (VRP) that can be modeled as set partitioning (SP) problems with additional constraints. The method consists in combining different dual ascent procedures to find a near optimal dual solution of the SP model. Then, a column-and-cut generation algorithm attempts to close the integrality gap left by the dual ascent procedures by adding valid inequalities to the SP formulation. The final dual solution is used to generate a reduced problem containing all optimal integer solutions that is solved by an integer programming solver. In this paper, we describe how this solution framework can be extended to solve different variants of the VRP by tailoring the different bounding procedures to deal with the constraints of the specific variant. We describe how this solution framework has been recently used to derive exact algorithms for a broad class of VRPs such as the capacitated VRP, the VRP with time windows, the pickup and delivery problem with time windows, all types of heterogeneous VRP including the multi depot VRP, and the period VRP. The computational results show that the exact algorithm derived for each of these VRP variants outperforms all other exact methods published so far and can solve several test instances that were previously unsolved.     

An exact solution of a quasilinear Fisher equation in cylindrical coordinates

Lie symmetry method is applied to analyse Fisher equation in cylindrical coordinates. Symmetry algebra is found and symmetry invariance is used to reduce the equation to a first-order ODE. The first-order ODE is further analysed to obtain exact solution of Fisher equation in explicit form.     

An exact solution framework for the minimum cost dominating tree problem

The minimum cost dominating tree problem is a recently introduced NP-hard problem, which consists of finding a tree of minimal cost in a given graph, such that for every node of the graph, the node or one of its neighbours is in the tree. We present an exact solution framework combining a primal–dual heuristic with a branch-and-cut approach based on a transformation of the problem into a Steiner arborescence problem with an additional constraint. The effectiveness of our approach is evaluated on testbeds proposed in literature containing instances with up to 500 nodes. Our framework manages to solve all but four instances from literature to proven optimality within 3 h (most of them in a few seconds). We provide optimal solution values for 69 instances from literature for which the optimal solution was previously unknown.     

An exact solution approach for the interdiction median problem with fortification

Systematic approaches to security investment decisions are crucial for improved homeland security. We present an optimization modeling approach for allocating protection resources among a system of facilities so that the disruptive effects of possible intentional attacks to the system are minimized. This paper is based upon the p-median service protocol for an operating set of p facilities. The primary objective is to identify the subset of q facilities which, when fortified, provides the best protection against the worst-case loss of r non-fortified facilities. This problem, known as the r-interdiction median problem with fortification (IMF), was first formulated as a mixed-integer program by Church and Scaparra [R.L. Church, M.P. Scaparra, Protecting critical assets: The r-interdiction median problem with fortification, Geographical Analysis 39 (2007) 129–146]. In this paper, we reformulate the IMF as a maximal covering problem with precedence constraints, which is amenable to a new solution approach. This new approach produces good approximations to the best fortification strategies. Furthermore, it provides upper and lower bounds that can be used to reduce the size of the original model. The reduced model can readily be solved to optimality by general-purpose MIP solvers. Computational results on two geographical data sets with different structural characteristics show the effectiveness of the proposed methodology for solving IMF instances of considerable size.     

An exact model for cell formation in group technology

Despite the long history of the cell formation problem (CF) and availability of dozens of approaches, very few of them explicitly optimize the objective of cell formation. These scarce approaches usually lead to intractable formulations that can be solved only heuristically for practical instances. In contrast, we show that CF can be explicitly modelled via the minimum multicut problem and solved to optimality in practice (for moderately sized instances). We consider several real-world constraints that can be included into the proposed formulations and provide experimental results with real manufacturing data.     

An exact static solution approach for the service parts end-of-life inventory problem

This paper studies the spare parts end-of-life inventory problem that happens after the discontinuation of part production. A final ordering quantity is set such that the service process is sustained until all service obligations expire. Also, the price erosion of substitutable or new generation products over time makes it economically justifiable to consider switching to an alternative service policy for repair such as swapping the old product with a new one. This requires the joint optimization of the final order quantity and the time to switch from repair to an alternative service policy. To the best of our knowledge, the problem has not been optimally solved yet either in its static or dynamic formulation. In the current paper, we solve its static version as a bi-level optimization problem. We investigate the convexity of the objective function and give a computationally efficient algorithm to find an exact optimal solution up to any given numerical error level ??>?0. We illustrate our approach on some numerical examples and compare our results with earlier works on this problem.     

Onsager approximation and asymptotically exact solution of the Ashkin-Teller model

N. E. Bauman Higher Technical College, Moscow. Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 83, No. 1, pp. 83–91, April, 1990.     

On an exact solution of master equations for the model of reversible growth

The set of master equations for the monomer quasi-chemical reversible-growth model in a heterogeneous open medium material is studied. The exact solution to the master equations is obtained for the case where the velocity constants for the growth and decay reactions are linear in the particle number. It is shown that the model under consideration is canonically invariant w.r.t. the Polya distribution with a time-dependent mean size of the clusters.Translated from Teoreticheskaya i Matematicheskaya Fizika, Vol. 108, No. 2, pp. 327–336, August, 1996.     

An exact solution of a molecular field equation in the theory of ferromagnetism

The theory of complex variables is used to develop an exact closed-form solution of a transcendental equation basic to the molecular field theory of ferromagnetism. The analysis yields analytical expressions, in terms of elementary quadratures, for the reduced magnetization as it depends on the temperature and magnetic field.

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Zusammenfassung Im Rahmen der Theorie komplexer Variabler wird für eine, der Molekularfeldtheorie des Ferromagnetismus' zugrunde liegenden transzendenten Gleichung, eine exakte Lösung in geschlossener Form entwickelt. Die Rechnung liefert analytische Ausdrücke, in Form elementarer Quadraturen, welche die reduzierte Magnetisierung als Funktion der Temperatur und des Magnetfeldes beschreiben.

     

An analytical solution for a nonlinear time-delay model in biology

In this paper, the homotopy analysis method is applied to develop a analytic approach for nonlinear differential equations with time-delay. A nonlinear model in biology is used as an example to show the basic ideas of this analytic approach. Different from other analytic techniques, the homotopy analysis method provides a simple way to ensure the convergence of the solution series, so that one can always get accurate approximations. A new discontinuous function is defined so as to express the piecewise continuous solutions of time-delay differential equations in a way convenient for symbolic computations. It is found that the time-delay has a great influence on the solution of the time-delay nonlinear differential equation. This approach has general meanings and can be applied to solve other nonlinear problems with time-delay.     

An approximate solution approach for a scenario-based capital budgeting model

Real options techniques such as contingent claims analysis and dynamic programming can be used for project evaluation when the project develops stochastically over time and the decision to invest into this project can be postponed. Following that perspective, Meier et al. (Oper Res 49(2):196–2 06, 2001) presented a scenario based model that captures risk uncertainty and managerial flexibility, maximizing the time-varying of a portfolio of investment options. However, the corresponding linear integer program turns out to be quite intractable even for a small number of projects and time periods. In this paper, we propose a heuristic approach based on an alternative scenario based model involving a much less number of variables. The new approach allows the determination of reasonable quality approximate solutions with huge reductions on the computational times required for solving large size instances.