Francesca Maggioni

Analyzing the quality of the expected value solution in stochastic programming

Stochastic programs are usually hard to solve when applied to real-world problems; a common approach is to consider the simpler deterministic program in which random parameters are replaced by their expected values, with a loss in terms of quality of the solution. The Value of the Stochastic Solution—VSS—is normally used to measure the importance of using a stochastic model. But what if VSS is large, or expected to be large, but we cannot solve the relevant stochastic program? Shall we just give up? In this paper we investigate very simple methods for studying structural similarities and differences between the stochastic solution and its deterministic counterpart. The aim of the methods is to find out, even when VSS is large, if the deterministic solution carries useful information for the stochastic case. It turns out that a large VSS does not necessarily imply that the deterministic solution is useless for the stochastic setting. Measures of the structure and upgradeability of the deterministic solution such as the loss using the skeleton solution and the loss of upgrading the deterministic solution will be introduced and basic inequalities in relation to the standard VSS are presented and tested on different cases.

A stochastic model for the daily coordination of pumped storage hydro plants and wind power plants

We propose a stochastic model for the daily operation scheduling of a generation system including pumped storage hydro plants and wind power plants, where the uncertainty is represented by the hourly wind power production. In order to assess the value of the stochastic modeling, we discuss two case studies: in the former the scenario tree is built so as to include both low and high wind power production scenarios, in the latter the scenario tree is built on historical wind speed data covering a time span of one and a half year. The Value of the Stochastic Solution, computed by a modified new procedure, shows that in scenarios with low wind power production the stochastic solution allows the producer to obtain a profit which is greater than the one associated to the deterministic solution. In-sample stability of the optimal function values for increasing number of scenarios is reported.

Velocity, energy, and helicity of vortex knots and unknots.

In this paper we determine the velocity, the energy, and estimate writhe and twist helicity contributions of vortex filaments in the shape of torus knots and unknots (as toroidal and poloidal coils) in a perfect fluid. Calculations are performed by numerical integration of the Biot-Savart law. Vortex complexity is parametrized by the winding number w given by the ratio of the number of meridian wraps to that of longitudinal wraps. We find that for w<1 vortex knots and toroidal coils move faster and carry more energy than a reference vortex ring of same size and circulation, whereas for w>1 knots and poloidal coils have approximately same speed and energy of the reference vortex ring. Helicity is dominated by writhe contributions. Finally, we confirm the stabilizing effect of the Biot-Savart law for all knots and unknots tested, found to be structurally stable over a distance of several diameters. Our results also apply to quantized vortices in superfluid 4He .

Stochastic optimization models for a single-sink transportation problem

In this paper, we study a single-sink transportation problem in which the production capacity of the suppliers and the demand of the single customer are stochastic. Shipments are performed by capacitated vehicles, which have to be booked in advance, before the realization of the production capacity and the demand. Once the production capacity and the demand are revealed, there is an option to cancel some of the booked vehicles against a cancellation fee; if the quantity shipped from the suppliers using the booked vehicles is not enough to satisfy the demand, the residual quantity is purchased from an external company. The problem is to determine the number of vehicles to book in order to minimize the total cost. We formulate a two-stage and a multistage stochastic mixed integer linear programming models to solve this problem and test them on a real case provided by Italcementi, the primary Italian cement producer and the fifth largest cement producer in the world. We test the influence of different scenario-tree structures on the solutions of the problem, as well as sensitivity of the results with respect to the cancellation fee.

Bounds in Multistage Linear Stochastic Programming

Multistage stochastic programs, which involve sequences of decisions over time, are usually hard to solve in realistically sized problems. Providing bounds for optimal solution may help in evaluating whether it is worth the additional computations for the stochastic program vs. simplified approaches. In this paper we generalize measures from the two-stage case, based on different levels of available information, to the multistage stochastic programming problems. A set of theorems providing chains of inequalities among the new quantities are proved. Numerical results on a case study related to a simple transportation problem illustrate the described relationships.

Writhing and coiling of closed filaments

The kinematics of writhing and coiling of circular filaments is here analysed by new equations that govern the evolution of curves generated by epicycloids and hypocycloids. We show how efficiency of coil formation and compaction depend on writhing rates, relative bending, torsion and mean twist energy. We demonstrate that for coiling formation hypocycloid evolution achieves higher writhing rates, but in terms of deformation energy, the epicycloid evolution is much more effective. We also show how the occurrence and multiple appearance of inflexional configurations determine coil formation. Compaction and packing rate are also briefly examined. These results are fundamental and provide useful information for physical applications and for modelling natural phenomena, including relaxation of magnetic fields in the solar corona, magnetic dynamos in astrophysical flows, tertiary folding of macromolecules in chemical-physics, and DNA packing in cell biology.

Bounds and Approximations for Multistage Stochastic Programs

Consider (typically large) multistage stochastic programs, which are defined on scenario trees as the basic data structure. It is well known that the computational complexity of the solution depends on the size of the tree, which itself increases typically exponentially fast with its height, i.e., the number of decision stages. For this reason approximations which replace the problem by a simpler one and allow bounding the optimal value are of great importance. In this paper we study several methods to obtain lower and upper bounds for multistage stochastic programs and we demonstrate their use in a multistage inventory problem.

Monotonic bounds in multistage mixed-integer stochastic programming

Multistage stochastic programs bring computational complexity which may increase exponentially with the size of the scenario tree in real case problems. For this reason approximation techniques which replace the problem by a simpler one and provide lower and upper bounds to the optimal value are very useful. In this paper we provide monotonic lower and upper bounds for the optimal objective value of a multistage stochastic program. These results also apply to stochastic multistage mixed integer linear programs. Chains of inequalities among the new quantities are provided in relation to the optimal objective value, the wait-and-see solution and the expected result of using the expected value solution. The computational complexity of the proposed lower and upper bounds is discussed and an algorithmic procedure to use them is provided. Numerical results on a real case transportation problem are presented.

On the groundstate energy of tight knots

New results on the groundstate energy of tight, magnetic knots are presented. Magnetic knots are defined as tubular embeddings of the magnetic field in an ideal, perfectly conducting, incompressible fluid. An orthogonal, curvilinear coordinate system is introduced and the magnetic energy is determined by the poloidal and toroidal components of the magnetic field. Standard minimization of the magnetic energy is carried out under the usual assumptions of volume- and flux-preserving flow, with the additional constraints that the tube cross section remains circular and that the knot length (ropelength) is independent from internal field twist (framing). Under these constraints the minimum energy is determined analytically by a new, exact expression, function of ropelength and framing. Groundstate energy levels of tight knots are determined from ropelength data obtained by the SONO tightening algorithm. Results for torus knots are compared with previous work, and the groundstate energy spectrum of the first prime knots — up to 10 crossings — is presented and analysed in detail. These results demonstrate that ropelength and framing determine the spectrum of magnetic knots in tight configuration.New results on the groundstate energy of tight, magnetic knots are presented. Magnetic knots are defined as tubular embeddings of the magnetic field in an ideal, perfectly conducting, incompressible fluid. An orthogonal, curvilinear coordinate system is introduced and the magnetic energy is determined by the poloidal and toroidal components of the magnetic field. Standard minimization of the magnetic energy is carried out under the usual assumptions of volume- and flux-preserving flow, with the additional constraints that the tube cross section remains circular and that the knot length (ropelength) is independent from internal field twist (framing). Under these constraints the minimum energy is determined analytically by a new, exact expression, function of ropelength and framing. Groundstate energy levels of tight knots are determined from ropelength data obtained by the SONO tightening algorithm. Results for torus knots are compared with previous work, and the groundstate energy spectrum of the first prime knots — up to 10 crossings — is presented and analysed in detail. These results demonstrate that ropelength and framing determine the spectrum of magnetic knots in tight configuration.

Solution Approaches for the Stochastic Capacitated Traveling Salesmen Location Problem with Recourse

A facility has to be located in a given area to serve a given number of customers. The position of the customers is not known. The service to the customers is carried out by several traveling salesmen. Each of them has a capacity in terms of the maximum number of customers that can be served in any tour. The aim was to determine the service zone (in a shape of a circle) that minimizes the expected cost of the traveled routes. The center of the circle is the location of the facility. Once the position of the customers is revealed, the customers located outside the service zone are served with a recourse action at a greater unit cost. For this problem, we compare the performance of two different solution approaches. The first is based on a heuristic proposed for the Capacitated Traveling Salesman Problem and the second on the optimal solution of a stochastic second-order cone formulation with an approximate objective function.