A branch and bound method for the solution of multiparametric mixed integer linear programming problems

In this paper, we present a novel algorithm for the solution of multiparametric mixed integer linear programming (mp-MILP) problems that exhibit uncertain objective function coefficients and uncertain entries in the right-hand side constraint vector. The algorithmic procedure employs a branch and bound strategy that involves the solution of a multiparametric linear programming sub-problem at leaf nodes and appropriate comparison procedures to update the tree. McCormick relaxation procedures are employed to overcome the presence of bilinear terms in the model. The algorithm generates an envelope of parametric profiles, containing the optimal solution of the mp-MILP problem. The parameter space is partitioned into polyhedral convex critical regions. Two examples are presented to illustrate the steps of the proposed algorithm.     

Fuzzy reasoning in the investigation of seismic behavior

In this paper, we propose a fuzzy logic–based mathematical model of a sequence of earthquakes using fuzzy reasoning tools. We formed a set of fuzzy implications in order to study them, and we computed their deviation, so that we could compare them and conclude about the most accurate one. The compositional rule of inference was considered, which is based on the generalized modus ponens scheme. The new fuzzy methodology was used for each implication. The data required for the implications were obtained from the aftershocks of an earthquake with significant effects in a specific area. The magnitude of the aftershocks and their time difference from the main incident provided the values for the new fuzzy algorithm application. Two samples were selected relative with the seismic activity, which occurred the following days. The one sample consisted of 30 values, and the other sample from all the values found in the data archives of the National Observatory of Athens. Results were shown for both samples. So a mathematical technique, which reproduces the incidents using basic information and based on only two parameters, is developed for the simulation of a seismic sequence, which follows a strong earthquake.     

Linearly Constrained Global Optimization and Stochastic Differential Equations

A stochastic algorithm is proposed for the global optimization of nonconvex functions subject to linear constraints. Our method follows the trajectory of an appropriately defined Stochastic Differential Equation (SDE). The feasible set is assumed to be comprised of linear equality constraints, and possibly box constraints. Feasibility of the trajectory is achieved by projecting its dynamics onto the set defined by the linear equality constraints. A barrier term is used for the purpose of forcing the trajectory to stay within the box constraints. Using Laplace’s method we give a characterization of a probability measure (Π) that is defined on the set of global minima of the problem. We then study the transition density associated with the projected diffusion process and show that its weak limit is given by Π. Numerical experiments using standard test problems from the literature are reported. Our results suggest that the method is robust and applicable to large-scale problems.     

Possible observation of “string excitations” of a hole in a quantum antiferromagnet

We argue that recently reported high resolution angle-resolved photoelectron spectra from cuprates, where an anomalous high-energy dispersion was identified, reveal the internal structure of the hole quasiparticle in quantum antiferromagnets and more importantly it is evidence for the existence of “string-excitations” which validate early predictions based on the t–J model. Their energy–momentum dispersion, the manner in which the spectral weight is transfered to higher energy string excitations, and the vanishing of the quasiparticle spectral weight near the Γ point, are all in agreement with predictions without adjusting any parameters.     

On the Logistic Equation in the Complex Plane

The famous logistic differential equation is studied in the complex plane. The method used is based on a functional analytic technique which provides a unique solution of the ordinary differential equation (ODE) under consideration in H ₂(𝔻) or H ₁(𝔻) and gives rise to an equivalent difference equation for which a unique solution is established in ?₂ or ?₁. For the derivation of the solution of the logistic differential equation this discrete equivalent equation is used. The obtained solution is analytic in {z ∈ ?: |z| <T}, T > 0. Numerical experiments were also performed using the classical 4th order Runge–Kutta method. The obtained results were compared for real solutions as well as for solutions of the form y(t) = u(t) + iv(t), t ∈ ?. For t ∈ ? the solution derived by the present method, seems to have singularities, that is, points where it ceases to be analytic, in certain sectors of the complex plane. These sectors, depending on the values of the involved parameters, can move at different directions, join forming common sectors, or pass through each other and continue moving independently. Moreover, the real and imaginary part of the solution seem to exhibit oscillatory behavior near these sectors.     

Flexible long-term capacity planning in closed-loop supply chains with remanufacturing

We deal with long-term demand-driven capacity planning policies in the reverse channel of closed-loop supply chains (CLSCs) with remanufacturing, under high capacity acquisition cost coupled with uncertainty in actual demand, sales patterns, quality and timing of end-of-use product returns. The objective is to facilitate the decision-making when the management faces the dilemma of implementing either a strategy of early large-scale investments to benefit from economies of scale and capacity readiness, or a flexible strategy of low volume but more frequent capacity expansions. We consider a CLSC with two sequential product-types. We study the system’s response in terms of transient flows, actual/desired capacity level, capacity expansions/contractions and total supply chain profit, employing a simulation-based system dynamics optimization approach. Extensive numerical investigation covers a broad range of real-world remanufacturable products under alternative scenarios in relation to the market preference over product-types. The key findings propose flexible policies as improved alternatives to large-scale capacity expansions/contractions in terms of adaptability to the actual pattern of end-of-use product returns and involved risk in the investments’ turnover. Flexible policies are also proposed as practices to avoid overcapacity phenomena in collection and remanufacturing capacity and as robust policies to product demand. Their implementation is revealed to be even more important for the case of remanufacturing, when a high capacity acquisition unit-cost ratio (remanufacturing/collection) is coupled with strong economies of scale. Finally, results under different information sharing structures show changes in remanufacturing policies, thus justifying the importance of coordination between the decision-maker and the distributor.     

Global optimization of robust chance constrained problems

We propose a stochastic algorithm for the global optimization of chance constrained problems. We assume that the probability measure with which the constraints are evaluated is known only through its moments. The algorithm proceeds in two phases. In the first phase the probability distribution is (coarsely) discretized and solved to global optimality using a stochastic algorithm. We only assume that the stochastic algorithm exhibits a weak* convergence to a probability measure assigning all its mass to the discretized problem. A diffusion process is derived that has this convergence property. In the second phase, the discretization is improved by solving another nonlinear programming problem. It is shown that the algorithm converges to the solution of the original problem. We discuss the numerical performance of the algorithm and its application to process design.     

The impact of two-product joint lifecycles on capacity planning of remanufacturing networks

Capacity planning in the reverse channel of closed-loop supply chains (CLSCs) involves complex issues due to the different lifecycles of product offerings in combination with the variability regarding product usage time, quality level of used products and return patterns. (Georgiadis, P., Vlachos, D., Tagaras, G., 2006. The impact of product lifecycle on capacity planning of closed-loop supply chains with remanufacturing. Production and Operations Management 15; 514–527) developed a system dynamics (SD) model to study a CLSC with remanufacturing for a single product which incorporates a dynamic capacity modeling approach. We extend this SD model for two sequential product-types under two alternative scenarios regarding the market preferences over the product-types; in the first scenario, the market is considered showing no preferences, while in the second scenario, the demand over a product-type can be satisfied only by providing units of the specific type. We study how the joint lifecycles of two product-types, entry time of the second product-type to the market and used product return patterns affect the optimal policies regarding expansion and contraction of collection and remanufacturing capacities. The results of extensive numerical investigation are tested for their statistical significance using analysis of variance (ANOVA). In the first scenario, the results show that the system performs best when the two lifecycles form a trapezoid pattern for total demand while in the second scenario, when the two lifecycles form a triangular pattern.