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We present novel mathematical models for inventory management within a reverse logistics system. Technological advancements, sustainability initiatives, and evolving customer behaviours have significantly increased the demand for repaired products. Our models account for varying demand levels for newly produced and repaired items. To optimize overall costs with constrained scenarios, we formulated mixed integer programming problems. Solution procedures for the proposed problems are introduced, and the accuracy of these solutions has been validated through numerical experiments. Additionally, we address the cost of waste disposal as an environmental concern. This paper develops a multiobjective mathematical model and provides an algorithm for the Pareto solution. Various scalarization techniques are utilized to identify the Pareto front, and a comparison of these techniques is presented., Comment: 24 pages, 16 figures

This paper is concerned with determining the shortest path for a pursuer aiming to intercept a moving target travelling at a constant speed. We introduce an intuitive and efficient mathematical model outlined as an optimal control problem to address this challenge. The proposed model is based on Dubins path where we concatenate two possible paths: a left-circular curve or a right-circular curve followed by a straight line. We develop and explore this model, providing a comprehensive geometric interpretation, and design an algorithm tailored to implement the proposed mathematical approach efficiently. Extensive numerical experiments involving diverse target positions highlight the strength of the model. The method exhibits a remarkably high convergence rate in finding solutions. For experiment purposes, we utilized the modelling software AMPL, employing a range of solvers to solve the problem. Subsequently, we simulated the obtained solutions using MATLAB, demonstrating the efficiency of the model in intercepting a moving target. The proposed model distinguishes itself by employing fewer parameters and making fewer assumptions, setting the model simplifies the complexities, and thus, makes it easier for experts to design optimal path plans.

Locomotion is essential for living cells. It enables bacteria and algae to explore space for food, cancer to spread, and immune system to fight infections. Motile cells display trajectories of intriguing complexity, from regular (e.g. circular, helical, and so on) to irregular motions (run-tumble), the origin of which has remained elusive for over a century. This dynamics versatility is conventionally attributed to the shape asymmetry of the motile entity, to the suspending media, and/or to stochastic regulation. We propose here a universal approach highlighting that these movements are generic, occurring for a large class of cells and artificial microswimmers, without the need of invoking shape asymmetry nor stochasticity, but are encoded in their inherent nonlinear evolution. We show, in particular, that for a circular and spherical particle moving in a simple fluid, circular, helical and chaotic motions (akin to a persistent random walk) emerge naturally in different regions of parameter space. This establishes the operating principles for complex trajectories manifestation of motile systems, and offers a new vision with minimal ingredients. The reduced evolution equations based on symmetries are consistent with those derived for a model of an autophoretic particle including diffusion, emission/absorption at the particle surface and hydrodynamics, and provide qualitative and quantitative agreement., Comment: 11 pages, 3 figures

Autonomous locomotion is a ubiquitous phenomenon in biology and in physics of active systems at microscopic scale. This includes prokaryotic, eukaryotic cells (crawling and swimming) and artificial swimmers. An outstanding feature is the ability of these entities to follow complex trajectories, ranging from straight, curved (circular, helical...), to random-like ones. The non-straight nature of these trajectories is often explained as a consequence of the asymmetry of the particle or the medium in which it moves, or due to the presence of bounding walls, etc... Here, we show that straight, circular and helical trajectories emerge naturally in the absence of asymmetry of the swimmer or that of suspending medium. Our first proof is based on general considerations, without referring to an explicit form of a model. We show that these three trajectories correspond to self-congruent solutions. Self-congruency means that the states of the system at different moments of time can be made identical by an appropriate combination of rotation and translation of the coordinate space. We show that these solutions are exhibited by spherically symmetric particles as a result of a series of pitchfork bifurcations as the activity is increased. Self-congruent dynamics in one and two dimensions are analyzed as well. Finally, we present a simple explicit nonlinear exactly solvable model of fully isotropic phoretic particle that shows the transitions from a non-motile state to straight motion to circular motion to helical motion as a series of spontaneous symmetry-breaking bifurcations. Whether a system exhibits or not a given trajectory only depends on the numerical values of parameters entering the model, while asymmetry of swimmer shape, or anisotropy of the suspending medium , or influence of bounding walls are not necessary., Comment: 23 pages, 6 figures

We consider a 2D model of an autophoretic particle in which the particle has a circular shape and emits/absorbs a solute that diffuses and is advected by the suspending fluid. Beyond a certain emission/absorption rate (characterized by a dimensionless P\'eclet number, $Pe$) the particle is known to undergo a bifurcation from a non motile to a motile state, with different trajectories, going from a straight to circular and to a chaotic motion by progressively increasing $Pe$. From the full model involving solute diffusion and advection, we derive a reduced closed model which involves only two time-dependent amplitudes $C_1(t)$ and $C_2(t)$ corresponding to the first two Fourier modes of the solute concentration field. This model consists of two coupled nonlinear ordinary differential equations for $C_1$ and $C_2$ and presents several great advantages:(i) the straight and circular motions can be handled fully analytically, (ii) complex motions such as chaos can be analyzed numerically very efficiently in comparison to the numerically expensive full model involving partial differential equations, (iii) the reduced model has a universal form dictated only by symmetries, (iv) the model can be extended to higher Fourier modes. The derivation method is exemplified for a 2D model, for simplicity, but can easily be extended to 3D, not only for the presently selected phoretic model, but also for any model in which chemical activity triggers locomotion. A typical example can be found, for example, in the field of cell motility involving acto-myosin kinetics. This strategy offers an interesting way to cope with swimmers on the basis of ordinary differential equations, allowing for analytical tractability and efficient numerical treatment., Comment: 13 pages, 2 figures

We propose new mathematical models of inventory management in a reverse logistics system. The proposed models extend the model introduced by Nahmias and Rivera with the assumption that the demand for newly produced and repaired (remanufacturing) items are not the same. We derive two mathematical models and formulate unconstrained and constrained optimization problems to optimize the holding cost. We also introduce the solution procedures of the proposed problems. The exactness of the proposed solutions has been tested by numerical experiments. Nowadays, it is an essential commitment for industries to reduce greenhouse gas (GHG) emissions as well as energy consumption during the production and remanufacturing processes. This paper also extends along this line of research, and therewith develops a three-objective mathematical model and provides an algorithm to obtain the Pareto solution., Comment: Submitted

Multi-objective integer or mixed-integer programming problems typically have disconnected feasible domains, making the task of constructing an approximation of the Pareto front challenging. The present paper shows that certain algorithms which were originally devised for continuous problems can be successfully adapted to approximate the Pareto front for integer, and mixed-integer, multi-objective optimization problems. Relationships amongst various scalarization techniques are established to motivate the choice of a particular scalarization in these algorithms. The proposed algorithms are tested by means of two-, three- and four-objective integer and mixed-integer problems, and comparisons are made. In particular, a new four-objective algorithm is used to solve a rocket injector design problem with a discrete variable, which is a challenging mixed-integer programming problem., Comment: 32 pages