Deterministic Flow Research Articles

Optimization Problem for Probabilistic Time Intervals of Quasi-Deterministic Output and Self-Similar Input Data Packet Flow in Telecommunication Networks

Introduction. When managing traffic at the packet level in modern telecommunication networks, it is proposed to use methods that transform a self-similar stochastic packet flow into a quasi-deterministic one. To do this, it is required to apply complex probabilistic laws of distribution of self-similar flows. From the literature, methods of balancing the network load are known, which, with the problem indicated above, contribute to increasing the efficiency of telecommunication systems. However, there is no strictly mathematical solution to find out the optimal probabilistic characteristics of the output flow, based on the input flow. The presented research is intended to fill this gap. Its objective is to create a method for determining the optimal probabilistic characteristics of the packet flow, using the minimum value of the proximity measure of the self-similar input and quasi-deterministic output flows.Materials and Methods. To solve the research problem, the parameters of the output flow distribution were selected so that the approximation function was close to 𝛿𝛿-function. The Kullback-Leibler divergence was used as a proximity measure of the input and output distributions of time intervals. Methods of set theory, metric spaces, multidimensional optimization, and teletraffic were used. The solution algorithm included minimization of the Kullback-Leibler divergence and the limit passage to 𝛿𝛿-function.Results. A probability distribution is shown — an approximation of 𝛿𝛿-function, which maintains the equality of time intervals of a quasi-deterministic output packet flow. A method for transforming a self-similar input flow into a quasideterministic output flow is presented. The Kullback–Leibler divergence was used as a measure of their proximity. The minimum of the Kullback-Leibler divergence between the input and output flows with a normal distribution was achieved in the case of equality of the mathematical expectations of these flows. Using the passage to the limit, it has been established that time interval T between packets of the quasi-deterministic output flow must be equal to the mathematical expectation of the time intervals between packets of the input self-similar flow. To obtain a quasi-deterministic flow, the passage to the limit is performed for the found value of the mathematical expectation at σ → 0.Discussion and Conclusion. The application of this method will reduce the negative impact of self-similarity of network traffic on the efficiency of the telecommunication network. The use of quasi-deterministic flows makes it possible to predict the load of network resources, which can be the basis for improving the quality of user service. Two difficulties associated with calculations and practical implementation of the solution are eliminated. Firstly, it is difficult to use the delta function as a function of the output flow distribution density. Secondly, there are no ideal deterministic flows in the operation of telecommunication networks. The proposed method has great potential in the design and optimization of communication networks.

Laminar Flow Infrared Spectroelectrochemistry.

In this work, we advance lab-on-chip electrochemistry and spectroscopy by combining these capabilities onto a single platform, thereby achieving mid-infrared spectroelectrochemistry (SEC) for the first time. The key feature of this technique is the use of deterministic laminar flow patterns to precisely transport a reacted solution from upstream electrodes to a downstream spectral detection region. Laminar flow spectroelectrochemistry (LF-SEC) is therefore a completely new approach, which derives its distinction and advantage over traditional SEC by physically separating electrode and attenuated total reflection (ATR) elements. As such, these functional elements retain optimal properties, such as inert, highly conductive electrodes and a bare ATR element for sensitive Fourier transform infrared (FTIR) spectroscopy. By combining ATR-FTIR with a scanning aperture system, LF-SEC provides the additional advantage of spectroscopically monitoring reactions at individual electrodes. The LF-SEC system design is first optimized through a series of targeted experiments using a ferricyanide/ferrocyanide redox pair to validate electrochemical functionality, undertake spectroscopic calibration, optimize experimental parameters, and finally validate the quantitative relationship between FTIR results and the reaction rate under galvanostatic control. After optimization, we demonstrate the technique by monitoring the oxidation of the therapeutic compound ascorbic acid (vitamin C) in the presence of biomolecular interference from a molecule with an overlapping oxidation potential. We find that molecular availability causes the reaction to switch between reaction pathways, which we could finely monitor using LF-SEC. This work opens the door to future developments that take advantage of the microfluidic reactor setup, with benefits ranging from portability to high-throughput studies under precise reaction conditions.

Reliability analysis of a grid-connected hybrid renewable energy system using hybrid Monte-Carlo and Newton Raphson methods

The integration of renewable energy sources into the power grid is essential for sustainable development, yet it presents significant dependability challenges, particularly in terms of reliability, stability, and robustness due to the inherent variability of these sources. This research introduces a novel hybrid methodology that combines Monte Carlo simulation with Newton-Raphson power flow analysis to enhance the reliability assessment of grid-connected hybrid renewable energy systems. This innovative approach uniquely addresses the limitations of existing methodologies by merging the probabilistic handling of uncertainties with precise deterministic power flow analysis. Our hybrid method significantly reduces the Loss of Load Expectation (LOLE) to 5 h per year and the Loss of Load Energy Expectation (LOEE) to 200 MWh per year, outperforming traditional methods which typically report LOLEs of 2020 h/year and LOEEs of 10001000 MWh/year. Additionally, the hybrid method achieves a reduction in power losses to 1.2%, showcasing its superior efficiency compared to the 2.5% losses seen with standalone Monte Carlo methods. Real-time validation using the IEEE-30 bus model further confirms the practical applicability and robustness of our approach, making it a pivotal tool for enhancing grid stability and optimizing renewable energy integration. This research not only advances the methodology for reliability assessment but also sets a new standard for balancing accuracy and computational efficiency in energy system management. The implications of this work are far-reaching, offering significant contributions to both grid reliability and the sustainable management of renewable energy resources.

Invariant Gibbs measure for a Schrödinger equation with exponential nonlinearity

We investigate the invariance of the Gibbs measure for the fractional Schrödinger equation of exponential type (expNLS) i∂tu+(−Δ)α2u=2γβeβ|u|2u on d-dimensional compact Riemannian manifolds Md, for a dispersion parameter α>d, some coupling constant β>0, and γ≠0. (i) We first study the construction of the Gibbs measure for (expNLS). We prove that in the defocusing case γ>0, the measure is well-defined in the whole regime α>d and β>0 (Theorem 1.1 (i)), while in the focusing case γ<0 its partition function is always infinite for any α>d and β>0, even with a mass cut-off of arbitrary small size (Theorem 1.1 (ii)). (ii) We then study the dynamics (expNLS) with random initial data of low regularity. We first use a compactness argument to prove weak invariance of the Gibbs measure, in the defocusing case γ>0, in the whole regime α>d and β>0 (Theorem 1.4 (i)). In the large dispersion regime α>2d, we can improve this result by constructing a local deterministic flow for (expNLS) for any β>0 and γ≠0. Using the Gibbs measure, we prove that solutions are almost surely global for any β,γ>0, and that the Gibbs measure is invariant (Theorem 1.4 (ii)). (iii) Finally, in the particular case d=1 and M=T, we are able to exploit some probabilistic multilinear smoothing effects to build a global probabilistic flow for (expNLS) for 1+22<α≤2 for any β,γ>0 (Theorem 1.6).

Optimal power flow method with consideration of uncertainty sources of renewable energy and demand response

Optimal power flow (OPF) calculation methods are important for the power system operation and mainly focus on the deterministic power flow calculation, neglecting the impact of demand response on online security calculation of power systems with renewable energy sources. Therefore, this paper proposes an OPF calculation method that considers the uncertainties of wind power, photovoltaic (PV) power generation and demand-side response. Firstly, the research focuses on the renewable energy grid, considering the uncertainties of wind power and PV power generation, and establishes uncertainty models for wind power and PV output. Secondly, based on cloud model theory, an uncertainty model for demand response is established. According to the established models, an efficient OPF model is constructed with a linearized submodels considering multiple uncertainties. By testing on the IEEE 30-bus system as a typical example, we found the effectiveness and superiority of the proposed OPF calculation method can benefit the power system economic operation and demand side resource utilization.

Lorenz Attractors: Exploring its properties and the application value of chaos theories

Since Lorenz published Deterministic Nonperiodic Flow in 1963, Lorenz attractors and their equations have occupied an important position in the fields of mathematics, physics, meteorology and so on. Lorenz attractors reveal the aperiodic behavior and sensitivity to initial conditions in deterministic systems, which have attracted great attention in the scientific community. This paper deeply analyzes its characteristics, formation mechanism and performance in chaotic systems, and shows that it produces aperiodic behavior patterns in deterministic systems and is extremely sensitive to small changes in initial conditions, providing a new perspective for understanding the complexity and diversity of nature. Lorenz attractors are widely used in chaos theory, providing tools for chaos research and ideas for solving practical problems. It has shown its application potential in the field of meteorological prediction. It is expected to stimulate researchers interest in Lorenz attractors and chaos phenomena, promote the in-depth application and development of chaos theory in more fields, continue to reveal the mysteries of nature, and lead a new chapter in scientific exploration.

Do qubits dream of entangled sheep? Quantum measurement without classical output

Quantum mechanics is usually formulated with an implicit assumption that agents who can observe and interact with the world are external to it and have a classical memory. However, there is no accepted way to define the quantum–classical cut and no a priori reason to rule out fully quantum agents with coherent quantum memories. In this work, we introduce an entirely quantum notion of measurement, called a sensation, to account for quantum agents that experience the world through quantum sensors. Sensations eschew probabilities and instead describe a deterministic flow of quantum information. We quantify the information gain and disturbance of a sensation using concepts from quantum information theory and find that sensations always disturb at least as much as they inform. Viewing measurements as sensations could lead to a new understanding of quantum theory in general and to new results in the context of quantum networks.

Risk assessment for synthetic GICs: a quantitative framework for asset–liability management

AbstractThis study addresses a research gap in quantitative modeling framework and scenario analysis for the risk management of stable value fund wraps, a crucial segment of the U.S. financial market with over USD $400 billion in assets. In this paper, we present an asset–liability model that encompasses an innovative approach to modeling the assets of fixed-income funds coupled with a liability model backed by empirical analysis on a unique data set covering 80% of the stand-alone plan sponsor market, contrasting with models based solely on regular deterministic cash flows and interest rate differences. Our model identifies and analyzes two critical risk scenarios from the insurer’s perspective: inflationary and yield spike. Our approach demonstrates that the tail risk of wraps, used as an economic capital measure, is sensitive to characteristic parameters of the fund, such as the duration, portfolio composition and credit quality of assets. This finding significantly differs from U.S. regulatory approaches like the NAIC’s, which often result in a zero capital requirement. These findings reveal limitations in current actuarial risk and profitability metrics for U.S. insurers and argue that a more sophisticated risk model reproducing the two critical scenarios is necessary.

A Quantum-Safe Software-Defined Deterministic Internet of Things (IoT) with Hardware-Enforced Cyber-Security for Critical Infrastructures

The next-generation “Industrial Internet of Things” (IIoT) will support “Machine-to-Machine” (M2M) communications for smart Cyber-Physical-Systems and Industry 4.0, and require guaranteed cyber-security. This paper explores hardware-enforced cyber-security for critical infrastructures. It examines a quantum-safe “Software-Defined-Deterministic IIoT” (SDD-IIoT), with a new forwarding-plane (sub-layer-3a) for deterministic M2M traffic flows. A “Software-Defined Networking” (SDN) control plane controls many “Software-Defined-Deterministic Wide-Area Networks” (SDD-WANs), realized with FPGAs. The SDN control plane provides an “Admission-Control/Access-Control” system for network-bandwidth, using collaborating Artificial Intelligence (AI)-based “Zero Trust Architectures” (ZTAs). Hardware-enforced access-control eliminates all congestion, BufferBloat, and DoS/DDoS attacks, significantly reduces buffer-sizes, and supports ultra-reliable-low-latency communications in the forwarding-plane. The forwarding-plane can: (i) Encrypt/Authenticate M2M flows using quantum-safe ciphers, to withstand attacks by Quantum Computers; (ii) Implement “guaranteed intrusion detection systems” in FPGAs, to detect cyber-attacks embedded within billions of IIoT packets; (iii) Provide guaranteed immunity to external cyber-attacks, and exceptionally strong immunity to internal cyber-attacks; (iv) Save USD 100s of billions annually by exploiting FPGAs; and (v) Enable hybrid Classical-Quantum networks, by integrating a “quantum key distribution” (QKD) network with a classical forwarding plane with exceptionally strong cyber-security, determined by the computational hardness of cracking Symmetric Key Cryptography. Extensive experimental results for an SDD-WAN over the European Union are reported.

Exploration of Recent Developments of Hybrid Nanofluids

Over the past two decades, research on hybrid nanofluids has developed at a breakneck pace. Hybrid nanofluids are the potential fluids that outperform conventional nanofluids heat transfer fluids regarding thermophysical characteristics and thermal performance. Hybrid nanofluids are traditionally prepared by emulsifying each nanoparticle into the base fluid independently or diffusing both particles as a composite form into the base fluid. Despite the popularity and broad application of hybrid nanofluids in industrial/manufacturing processes, the review article classifying the mathematical models and categorization of the various hybrid nanofluids is limited in the scientific community. In light of this, this study examines recent and past research articles on deterministic hybrid nanofluid flow problems by exploiting different categorizations and metrics (method to solve the differential equation, the geometry of choice and thermophysical effects that have been employed as well as the nano-particle types). Researchers will be able to use the findings of this study for a future research proposal. Essentially, it is to fill in the gaps in the literature, particularly regarding the mathematical model for hybrid nanofluid flow problems and determining the geometry, thermophysical properties, and nanoparticle type.

Robust Short-Term Operation of AC Power Network With Injection Uncertainties

With uncertain injections from Renewable Energy Sources (RESs) and loads, deterministic AC Optimal Power Flow (OPF) often fails to provide optimal setpoints of conventional generators. A computationally time-efficient, economical, and robust solution is essential for ACOPF with short-term injection uncertainties. Usually, applying Robust Optimization (RO) for conventional non-linear ACOPF results in computationally intractable Robust Counterpart (RC), which is undesirable as ACOPF is an operational problem. Hence, this paper proposes a single-stage non-integer non-recursive RC of ACOPF, using a dual transformation, for short-term injection uncertainties. The proposed RC is convex, tractable, and provides base-point active power generations and terminal voltage magnitudes (setpoints) of conventional generators that satisfy all constraints for all realizations of defined injection uncertainties. The non-linear impact of uncertainties on other variables is inherently modeled without using any affine policy. The proposed approach also includes the budget of uncertainty constraints for low conservatism of the obtained setpoints. Monte-Carlo Simulation (MCS) based participation factored AC power flows validate the robustness of the obtained setpoints on NESTA and case9241pegase systems for different injection uncertainties. Comparison with previous approaches indicates the efficacy of the proposed approach in terms of low operational cost and computation time.

Most Probable Flows for Kunita SDEs

We identify most probable flows for Kunita Brownian motions, i.e. stochastic flows with Eulerian noise and deterministic drifts. Such stochastic processes appear for example in fluid dynamics and shape analysis modelling coarse scale deterministic dynamics together with fine-grained noise. We treat this infinite dimensional problem by equipping the underlying domain with a Riemannian metric originating from the noise. The resulting most probable flows are compared with the non-perturbed deterministic flow, both analytically and experimentally by integrating the equations with various choice of noise structures.

Planning container inspection and repair: A case study

The aim of this paper is to introduce a real-life problem of optimal planning of container inspection and repair on several facilities over several time periods assuring effective supply of empty containers indispensable for seamless global maritime transportation. The containers are of several types and quality levels. The objective is to minimize the total holding, inspection, repair, transportation and rejection costs. We formulate a deterministic min-cost multi-commodity network flow problem and prove NP-hardness for two special cases. Values of the rejection costs are determined such that no container is rejected in an optimal solution if there exists a feasible solution with no container rejected. An algorithmic mechanism is proposed to support a negotiation of periodic container demands between the container provider and the container users with the aim to reduce the overall costs. Computer experiments with instances obtained by random deviation from industrial data demonstrate that the majority of instances with up to 4 facilities, 4 container types, 7 quality levels, and 7, 14 or 30 time periods can be solved with an acceptable quality using an academic version of CPLEX in two hours on a standard PC.

Dynamics of technology emergence in innovation networks.

To create the next innovative product, participants in science need to understand which existing technologies can be combined, what new science must be discovered, and what new technologies must be invented. Knowledge of these often arrives by means of expert consensus or popularity metrics, masking key information on how intellectual efforts accumulate into technological progress. To address this shortcoming, we first present a method to establish a mathematical link between technological evolution and complex networks: a path of events that narrates innovation bottlenecks. Next, we quantify the position and proximity of documents to these innovation paths. The result is an innovation network that more exhaustively captures deterministic knowledge flows with respect to a marketed innovative product. Our dataset, containing over three million biomedical citations, demonstrates the possibility of quantifying the accumulation, speed, and division of labour in innovation over a sixty-year time horizon. The significance of this study includes the (i) use of a purpose-generated dataset showing causal paths from research to development to product; (ii) analysis of the innovation process as a directed acyclic graph; (iii) comparison between calendar time and network time; (iv) ordering of science funders along technology lifecycles; (v) quantification of innovative activities' importance to an innovative outcome; and (vi) integration of publication, patent, clinical trial, regulatory data to study innovation holistically.

Necessary and sufficient symmetries in Event-Chain Monte Carlo with generalized flows and application to hard dimers.

Event-Chain Monte Carlo (ECMC) methods generate continuous-time and non-reversible Markov processes, which often display significant accelerations compared to their reversible counterparts. However, their generalization to any system may appear less straightforward. In this work, our aim is to distinctly define the essential symmetries that such ECMC algorithms must adhere to, differentiating between necessary and sufficient conditions. This exploration intends to delineate the balance between requirements that could be overly limiting in broad applications and those that are fundamentally essential. To do so, we build on the recent analytical description of such methods as generating piecewise deterministic Markov processes. Therefore, starting with translational flows, we establish the necessary rotational invariance of the probability flows, along with determining the minimum event rate. This rate is identified with the corresponding infinitesimal Metropolis rejection rate. Obeying such conditions ensures the correct invariance for any ECMC scheme. Subsequently, we extend these findings to encompass schemes involving deterministic flows that are more general than mere translational ones. Specifically, we define two classes of interest of general flows: the ideal and uniform-ideal ones. They, respectively, suppress or reduce the event rates. From there, we implement a comprehensive non-reversible sampling of a system of hard dimers by introducing rotational flows, which are uniform-ideal. This implementation results in a speed-up of up to ∼3 compared to the state-of-the-art ECMC/Metropolis hybrid scheme.

Data-Driven Carbon Footprint Management of Electric Vehicles and Emission Abatement in Electricity Networks

Electric vehicles (EVs) are believed to play an important role in mitigating carbon emissions in the transportation sector. However, EVs may still cause carbon emissions in the power sector if they are charged by electricity generated from burning fossil fuels like coal. Researchers have focused on managing emissions on the power generation side. However, the underlying driver of carbon emissions is the demand of consumers. To this end, a probabilistic carbon footprint management strategy is proposed for EVs in this paper. First, the conventional deterministic carbon emission flow model is extended to a probabilistic one (PCEF) to track the carbon footprint of EVs considering various uncertainties based on non-intrusive load monitoring (NILM) and the two-point estimation method (2PEM). Second, a stochastic chance-constrained carbon footprint management model for EV charging is proposed to address the carbon obligation allocation of EVs from the perspective of consumption and provide a technical basis for demand-driven stimulation to reduce carbon emissions. Third, a solution methodology is proposed to solve the formulated chance-constrained problem based on nonparametric Bayesian modeling and inference. The proposed model and methodologies are verified on the IEEE 39-bus system. The feasibility of the proposed PCEF model is validated. According to the simulation result, the computation speed of the proposed PCEF model is improved from 3456.8 seconds to 1341.3 seconds compared with the Monte Carlo simulation by sacrificing the accuracy within 2%. Besides, the proposed emission control strategy can realize a better emission control performance compared with the other state-of-art works. Compared with the base case, the total emission is reduced by 82 tons annually, and the emission reduction rate is increased by nearly twice.

Dynamic Inventory Management Model for Equipment Replacement with Probabilistic Distribution of Failures

The article deals with the issues of inventory management modeling of components to ensure the prompt replacement of deep-water pump installations for the oilfield industry in case of equipment failure. The review of literary sources on the subject of the study and the problem statement is given. The rationale for choosing a mathematical model for problem solution is given. The task of dynamic inventory management is to provide the necessary stock with minimal temporary losses from forced downtime at minimal component storage and supply cost for repair or replacement. The planning horizon is divided into several time periods where optimal amount of inventory is to be ensured. It is indicated that when planning stocks in the case of a large number of pieces of equipment operating under different conditions, it is advisable to use a probabilistic approach. An event distribution function is set for probabilistic flow of equipment shutdown. The problem is solved for each segment defining the margin value by means of the Wilson model for deterministic flow of events. The results of simulation modeling on the shutdowns of electric drive centrifugal pump installations at the wells of the Vyngapurovskoye field are presented. A histogram of equipment shutdown frequencies for various time intervals is given. An exponential best approximation function is constructed to approximate the number of stops from installation time to equipment stop, the function coefficients and the standard approximation error are calculated. The characteristic curve of the relative proportion of the operating equipment requiring replacement, as well as the planned number of installations supplied for equipment replacement, with time is given. The dynamics of safety margin changes, which ensures the guaranteed replacement of failed equipment on time with respect to time, is presented.

Clustering of Floating Tracers in a Random Velocity Field Modulated by an Ellipsoidal Vortex Flow

The influence of a background vortex flow on the clustering of floating tracers is addressed. The vortex flow considered is induced by an ellipsoidal vortex evolving in a deformation. The system exhibits various vortex motion regimes: (1) a steady state, (2) oscillation and (3) rotation of the ellipsoidal vortex core. The latter two induce an unsteady velocity field for the tracer, thus leading to irregular (chaotic) tracer motion. Superimposing a stochastic divergent velocity field onto the deterministic vortex flow allows us to observe significantly different tracer evolution. An ellipsoidal vortex has ellipsoidal symmetry, and the tracer’s trajectories exhibit the same symmetry inside the vortex. Outside the vortex, the external deformation flow symmetry dominates. Diffusion scattering and chaotic advection give tracers the opportunity to leave the region of ellipsoidal symmetry and form a picture of shear flow symmetry. We use the method of characteristics to integrate the floating tracer density evolution equation and the Euler Ito scheme for obtaining the floating tracer trajectories with a random velocity field. The cluster area and cluster mass from the statistical topography are used as the quantitative diagnostics of a floating tracer’s clustering. For the case of a steady ellipsoidal vortex embedded into the deformation flow with a random velocity field component, we found that the clustering characteristics were weakened by the steady vortex. For the cases of an unsteady ellipsoidal vortex, we observed clustering in the floating tracer density field if the contribution of the divergent component was greater than or equal to that of the rotational (nondivergent) component. Even when the initial floating tracer patch was set on the boundary of the oscillating ellipsoidal vortex, we observed the formation of clusters. In the case of a rotating ellipsoidal vortex, we also observed pronounced clustering. Thus, we argue that unsteady ellipsoidal vortex regimes (oscillation and rotation), which induce chaotic motion of the nearby passive tracer’s trajectories, are still conducive to clustering of floating tracers observed in the density field, despite the intense deformation introduced by strain and shear.

Advanced optimization models for the location of charging stations in e-mobility

For a reduction in environmental pollution and dependency on petroleum, electric vehicles (EV) present an advantageous alternative to traditionally fossil-fuel powered automobiles. Rapid growth in the number of EVs requires an urgent need to develop an adequate charging station infrastructure to stimulate and facilitate their usage. Due to restricted investments in the development of a sufficient infrastructure, locations have to be chosen deliberately. In this paper, two extensions considering different objectives and further constraints to the deterministic flow refuelling location problem (DFRLP), described by Vries and Duijzer (Omega 69:102–114, 2017), are introduced. First, our research shows that location-dependent construction costs significantly influence the charging infrastructure obtained. Tests for different cost scenarios are carried out and policy implications are discussed. The original DFRLP assumes an unlimited capacity, meaning it is always possible to refuel all EVs at all charging stations, where they stop. Hence, for a more realistic modelling, we assume a limited capacity of charging stations in our second extension. Finally, both extensions are evaluated using benchmark instances based on test instances from the literature.

A time-expanded network design model for staff allocation in mail centres

We consider a staff allocation problem at a sequential sorting facility. In this facility, staff need to be assigned to work areas, through which commodities flow sequentially to be processed. Assigning staff optimally involves a trade-off between several different objectives, such as minimising the overall number of workers, as well as having fewer changes in the staff levels over time. While optimising for these, many operational requirements need to be met, including minimum processing volumes, correct ordering/processing of the commodities, and not exceeding staff resource constraints. We develop a deterministic time-expanded network flow model to solve the staff allocation problem. The model addresses the problem at a more granular timescale and with more operational constraints than previously used models. We use a lexicographical approach to deal with the multiple objectives. To demonstrate the model’s value, we apply it to a staff problem of a UK mail centre, showing that in the majority of cases, our model improves on current staffing practises on both objectives. We also show how the model performs in several different scenarios, including increasing total mail volumes and changing the proportion of letters and parcels to be sorted.