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In this work, we propose a new Gaussian process (GP) regression framework that enforces the physical constraints in a probabilistic manner. Specifically, we focus on inequality and monotonicity constraints. This GP model is trained by the quantum-inspired Hamiltonian Monte Carlo (QHMC) algorithm, which is an efficient way to sample from a broad class of distributions by allowing a particle to have a random mass matrix with a probability distribution. Integrating the QHMC into the inequality and monotonicity constrained GP regression in the probabilistic sense, our approach enhances the accuracy and reduces the variance in the resulting GP model. Additionally, the probabilistic aspect of the method leads to reduced computational expenses and execution time. Further, we present an adaptive learning algorithm that guides the selection of constraint locations. The accuracy and efficiency of the method are demonstrated in estimating the hyperparameter of high-dimensional GP models under noisy conditions, reconstructing the sparsely observed state of a steady state heat transport problem, and learning a conservative tracer distribution from sparse tracer concentration measurements.

In this paper, we describe a branch-and-price algorithm for the personalized nurse scheduling problem. The variants that appear in the literature involve a large number of constraints that can be hard or soft, meaning that they can be violated at the price of a penalty. We capture the diversity of the constraints on individual schedules by seven generic constraints characterized by lower and upper bounds on a given quantity. The core of the column generation procedure is in the identification of individual schedules with minimum reduced cost. For this, we solve a shortest path problem with resource constraints (SPPRC) where several generic constraints are modeled as resource constraints. We then describe dominance rules adapted to the presence of both upper and lower bounds on the resources and leverage soft constraints to improve the dominance. We also describe several acceleration techniques for the solution of the SPPRC, and branching rules that fit the specificities of the problem. Our numerical experiments are based on the instances of three benchmarks of the literature including those of the two international nurse rostering competitions (INRC-I and INRC-II). Their objective is threefold: assess the dominance rules and the acceleration techniques, investigate the capacity of the algorithm to find provable optimal solutions of instances that are still open, and conduct a comparison with best published results. The most noticeable conclusion is that the improved solution of the SPPRC allows to solve optimally all the INRC-II instances where a four-week planning horizon is considered and 40% of the eight-week instances. History: Accepted by Andrea Lodi, Area Editor for Design & Analysis of Algorithms–Discrete. Supplemental Material: The online appendix is available at https://doi.org/10.1287/ijoc.2023.0019. [ABSTRACT FROM AUTHOR]

Based on the hierarchical control structure, optimization and control problems of large‐scale and multivariate plants are solved sequentially. The economic performance of the plant plays an essential role in the plant‐wide modern industry. The optimal operating conditions will change as the economic criteria changes throughout the operation of the plant as the result of variations in raw material prices, product prices, production demand, market fluctuations, disturbances, and so forth. In reality, soft constraints are frequently used to denote the production requirements of various operating conditions. In order to improve economic performance and guarantee feasibility for the entire plant operation, a novel economic model predictive control (EMPC) strategy is proposed to control the constrained multi‐variable process system with varying economic performance criteria under soft constraints. By incorporating the transient steady‐state and two categories of slack variables for soft constraints, a modified economic performance index is optimized to cope with the changing criteria. In addition, a contractive constraint is added to the closed‐loop system to guarantee stability for non‐dissipative stage costs. This approach ensures recursive feasibility and asymptotic stability. The effectiveness of the proposed method is demonstrated by numerical examples and the fluid catalytic cracking unit (FCCU) process. [ABSTRACT FROM AUTHOR]

Semi-supervised clustering typically relies on both labeled and unlabeled data to guide the learning process towards the optimal data partition and to prevent falling into local minima. However, researchers' efforts made to improve existing semi-supervised clustering approaches are relatively scarce compared to the contributions made to enhance the state-of-the-art fully unsupervised clustering approaches. In this paper, we propose a novel semi-supervised deep clustering approach, named Soft Constrained Deep Clustering (SC-DEC), that aims to address the limitations exhibited by existing semi-supervised clustering approaches. Specifically, the proposed approach leverages a deep neural network architecture and generates fuzzy membership degrees that better reflect the true partition of the data. In particular, the proposed approach uses side-information and formulates it as a set of soft pairwise constraints to supervise the machine learning process. This supervision information is expressed using rather relaxed constraints named "should-link" constraints. Such constraints determine whether the pairs of data instances should be assigned to the same or different cluster(s). In fact, the clustering task was formulated as an optimization problem via the minimization of a novel objective function. Moreover, the proposed approach's performance was assessed via extensive experiments using benchmark datasets. Furthermore, the proposed approach was compared to relevant state-of-the-art clustering algorithms, and the obtained results demonstrate the impact of using minimal previous knowledge about the data in improving the overall clustering performance. [ABSTRACT FROM AUTHOR]

Decision systems for solving real-world combinatorial problems must be able to report infeasibility in such a way that users can understand the reasons behind it, and determine how to modify the problem to restore feasibility. Current methods mainly focus on reporting one or more subsets of the problem constraints that cause infeasibility. Methods that also show users how to restore feasibility tend to be less flexible and/or problem-dependent. We describe a problem-independent approach to feasibility restoration that combines existing techniques from the literature in novel ways to yield meaningful, useful, practical, and flexible user support. We evaluated the resulting framework on three real-world applications and conducted a qualitative expert user study with participants from different application domains. [ABSTRACT FROM AUTHOR]

The problem of scheduling final exams is a type of educational planning in which the goal is to find a suitable schedule for the exams of an educational institution in a way that makes optimal use of available resources. The purpose of this study is to find and solve a suitable mathematical model for the problem of scheduling final exams. The model presented in this research is integer linear programming with zero variables, which has obtained a justified and satisfactory solution. In addition to observing the educational rules, such as the limited period for holding exams, non-conflicts of each student's exams, optimal use of classroom capacity, etc., this model tries to minimize the penalties due to the proximity of consecutive exams of students. Also, the Ferdowsi University of Mashhad is considered the case study, and the model is tested on its data. Finally, to evaluate the accuracy of the model and the quality of the answers, the results of solving the proposed mathematical model are compared with the university examination schedule, which is prepared manually. Computational results show the superiority of the proposed model in terms of using an optimal number of available classes, non-interfere of exams that have common students, reducing the number of students who take several exams in one day, and minimizing the number of students whose exams are smaller than one day. [ABSTRACT FROM AUTHOR]

Ecological networks such as plant–pollinator systems and food webs vary in space and time. This variability includes fluctuations in global properties such as the total number and intensity of interactions in the network but also in the number and intensity of local (i.e. node level) species interactions.Fluctuations of species' properties can significantly affect higher‐order network features, for example, robustness and nestedness, and should therefore be taken into account in null models for pattern detection and hypothesis testing.In ecological research, classical null models treat node‐level properties as 'hard' constraints that cannot fluctuate. Here, we review and synthesize a set of maximum‐entropy methods that allow for fluctuating ('soft') constraints, offering a new addition to the classical toolkit of the ecologist. We illustrate the methods with some practical examples, pointing to currently available open‐source computer codes. We clarify how this approach can be used by experimental ecologists to detect non‐random patterns with null models that not only rewire, but also redistribute interaction strengths by allowing fluctuations in the enforced constraints.Explicit modelling of interspecific heterogeneity through local (i.e. species level) fluctuations of topological and quantitative constraints offers a statistically robust and expanded (e.g. including weighted links) set of tools to understand the assembly and resilience of ecological networks. [ABSTRACT FROM AUTHOR]

Semi-supervised clustering typically relies on both labeled and unlabeled data to guide the learning process towards the optimal data partition and to prevent falling into local minima. However, researchers’ efforts made to improve existing semi-supervised clustering approaches are relatively scarce compared to the contributions made to enhance the state-of-the-art fully unsupervised clustering approaches. In this paper, we propose a novel semi-supervised deep clustering approach, named Soft Constrained Deep Clustering (SC-DEC), that aims to address the limitations exhibited by existing semi-supervised clustering approaches. Specifically, the proposed approach leverages a deep neural network architecture and generates fuzzy membership degrees that better reflect the true partition of the data. In particular, the proposed approach uses side-information and formulates it as a set of soft pairwise constraints to supervise the machine learning process. This supervision information is expressed using rather relaxed constraints named “should-link” constraints. Such constraints determine whether the pairs of data instances should be assigned to the same or different cluster(s). In fact, the clustering task was formulated as an optimization problem via the minimization of a novel objective function. Moreover, the proposed approach’s performance was assessed via extensive experiments using benchmark datasets. Furthermore, the proposed approach was compared to relevant state-of-the-art clustering algorithms, and the obtained results demonstrate the impact of using minimal previous knowledge about the data in improving the overall clustering performance.

Dealing with controversial information is an important issue in several application contexts. Formal argumentation enables reasoning on arguments for and against a claim to decide on an outcome. Dung's abstract Argumentation Framework (AF) has emerged as a central formalism in argument-based reasoning. Key aspects of the success and popularity of Dung's framework include its simplicity and expressiveness. Integrity constraints help to express domain knowledge in a compact and natural way, thus keeping easy the modeling task even for problems that otherwise would be hard to encode within an AF. In this paper, we first explore two intuitive semantics based on Kleene and Lukasiewicz logics, respectively, for AF augmented with (strong) constraints—the resulting argumentation framework is called Constrained AF (CAF). Then, we propose a new argumentation framework called Weak constrained AF (WAF) that enhances CAF with weak constraints. Intuitively, these constraints can be used to find "optimal" solutions to problems defined through CAF. We provide a detailed complexity analysis of CAF and WAF, showing that strong constraints do not increase the expressive power of AF in most cases, while weak constraints systematically increase the expressive power of CAF (and AF) under several well-known argumentation semantics. [ABSTRACT FROM AUTHOR]

The alignment stability and controllability deteriorate considerably when the air spring vibration isolation system for the high power density main engine (HPDME-ASVIS) is in a special running state of high power and high output torque. The coupling characteristics among the alignment components are obtained by analyzing the mechanical characteristics of HPDME-ASVIS. A control response model is established, which can predict the alignment state and the vertical displacement of air springs after k times of inflation/deflation. By solving the linear programming problem, the alignment controllability judgment models are constructed for two conditions, which are under hard constraints and soft constraints, respectively. Furthermore, alignment reconfigurability is described and an autonomous control method is established. The experimental results confirm that high output torque will cause large and serious coupling of horizontal offset, self-rotation angle, and vertical deformation of air springs. Meanwhile, hey cannot reach the given control precision range. The traditional control method under hard constraints will result in a state of oscillation and may bring many reliability problems to HPDME-ASVIS. The alignment reconfigurability analysis can judge the alignment control influence degree based on different torque accurately. The autonomous control can use the control parameters autonomous adjustment and the spatial configuration to realize the alignment control reconfigurability for two different alignment control fault states. The stability of the alignment control system in HPDME-ASVIS can be improved observably, which is of great significance to the engineering application of the high power density main engine (HPDME) in the field of marine propulsion technology. [ABSTRACT FROM AUTHOR]

Selecting a subset of candidates with various attributes under fairness constraints has been attracting considerable attention from the AI community, with applications ranging from school admissions to committee selections. The fairness constraints are usually captured by absolute upper bounds and/or lower bounds on the number of selected candidates in specific attributes. In many scenarios, however, the total number of selected candidates is not predetermined. It is, therefore, more natural to express these fairness constraints in terms of proportions of the final selection size. In this paper, we study the proportional candidate selection problem, where the goal is to select a subset of candidates with maximum cardinality while meeting certain proportional fairness constraints. We first analyze the computational complexity of the problem and show strong inapproximability results. Next, we investigate the algorithmic aspects of the problem in two directions. First, by treating the proportional fairness constraints as soft constraints, we devise two polynomial-time algorithms that could return (near) optimal solutions with bounded violations on each fairness constraint. Second, we design an exact algorithm with a fast running time in practice. Simulations based on both synthetic and publicly available data confirm the effectiveness and efficiency of our proposed algorithms. [ABSTRACT FROM AUTHOR]

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The alignment stability and controllability deteriorate considerably when the air spring vibration isolation system for the high power density main engine (HPDME-ASVIS) is in a special running state of high power and high output torque. The coupling characteristics among the alignment components are obtained by analyzing the mechanical characteristics of HPDME-ASVIS. A control response model is established, which can predict the alignment state and the vertical displacement of air springs after k times of inflation/deflation. By solving the linear programming problem, the alignment controllability judgment models are constructed for two conditions, which are under hard constraints and soft constraints, respectively. Furthermore, alignment reconfigurability is described and an autonomous control method is established. The experimental results confirm that high output torque will cause large and serious coupling of horizontal offset, self-rotation angle, and vertical deformation of air springs. Meanwhile, hey cannot reach the given control precision range. The traditional control method under hard constraints will result in a state of oscillation and may bring many reliability problems to HPDME-ASVIS. The alignment reconfigurability analysis can judge the alignment control influence degree based on different torque accurately. The autonomous control can use the control parameters autonomous adjustment and the spatial configuration to realize the alignment control reconfigurability for two different alignment control fault states. The stability of the alignment control system in HPDME-ASVIS can be improved observably, which is of great significance to the engineering application of the high power density main engine (HPDME) in the field of marine propulsion technology.

In this paper, an efficient and probabilistic complete planning algorithm called Composite-space RRT is presented to address motion planning with soft constraints for spherical wrist manipulators. Firstly, we propose a novel configuration space termed Composite Configuration Space ("Composite Space" for short), which is composed of the joint space and the task space. Then, collision-free paths are generated in the composite space by the Rapidly-exploring Random Trees (RRT) algorithm. Finally, the planned paths in the composite space are mapped into the corresponding joint-space paths by a local planner. As the analytical inverse kinematics (IK) of the spherical wrist is used in the local planner, the proposed Composite-space RRT algorithm is characterized by high efficiency and no numerical iteration. Moreover, this approach can effectively improve the smoothness of the end-effector orientation path. The effectiveness of the proposed algorithm is demonstrated on the Willow Garage's PR2 simulation platform with two typical orientation-constrained cases. [ABSTRACT FROM AUTHOR]

Doctor scheduling is a complex, costly and time-consuming exercise. This study develops a constraint satisfaction and penalty minimisation scheduling model for meeting ‘hard constraints’ and minimises the cost of violating ‘soft constraints’, i.e. the user inputs, the total number of doctors to be scheduled, the maximum penalty to be met, and the minimum number of doctors to be assigned per shift. The algorithm creates a schedule which checks against all the constraints. The total schedule penalty associated with the constraint violations should be less than or equal to the user input penalty. If this condition is met, the schedule gets produced as the final and near-optimal solution. The model is managed to create a near optimal schedule with the minimal rule violations. However, it is challenging to provide a schedule with no rule violations. Such a situation is shown by the amount of computational time required to create a zero-penalty schedule, hours or even days needed to create a zero-penalty schedule. The system creates a schedule for a short period (weekly schedule) to promote flexibility; however, such a system does not promote fairness. Fairness is achieved through a cyclic schedule with rotations equal to the total number of doctors being scheduled. The system is managed to create a streamlined and flexible working environment and helped to improve the quality of healthcare. An optimization protocol can be incorporated into the system to reduce the search space and get the best optimal schedule since it is possible to get many schedules under the same user-defined parameters.

Many industrial model predictive control applications, called non-square systems, have more variables to be controlled than manipulated variables available. At these cases, the control objectives are related to keep the controlled ones within a range, instead in reference value (setpoint). Assessing the model quality of these controllers is fundamental, however, most available MPC assessment methods are setpoint dependent, providing misleading results when these are unavailable. The methods proposed by Botelho et al. (2015, 2016a, 2016b) allow the model assessment of linear MPCs independently from the setpoints. Even though such methodologies have this advantage, there is not a procedure to assess this kind of controller. Therefore, this paper presents a guideline to apply Botelho et al. (2015, 2016a, 2016b) methods in non-square MPCs. The Shell Heavy Oil Process is used as a case study. The results demonstrate that the procedure is capable of estimate the effect of modeling problems and indicate the associated controlled variable, as well as whether the problem is due to a model-plant mismatch or unmeasured disturbance in non-square controllers. • A guideline for model assessment of MPCs is presented. • The focus is model assessment of non-square systems. • The methods proposed by Botelho et al. (2015, 2016a, 2016b) are used for the modeling errors quantification. • The Shell Heavy Oil Process is used as a case study. • The method was exhaustively tested considering several scenarios randomly generated. [ABSTRACT FROM AUTHOR]

This paper analyzes the single-source capacitated facility location problem (SSCFLP) with soft capacity constraints. Hence, the maximum capacity at each facility can be potentially exceed by incurring in a penalty cost, which increases with the constraint-violation gap. In some realistic scenarios, this penalty cost can be modelled as a piecewise function. As a result, the traditional cost-minimization objective becomes a non-smooth function that is difficult to optimise using exact methods. A mathematical model of this non-smooth SSCFLP is provided, and a biased-randomized iterated local search metaheuristic is proposed as a solving method. A set of computational experiments is run to illustrate our algorithm and test its efficiency. [ABSTRACT FROM AUTHOR]

Multivariate curve resolution (MCR) is a powerful tool in chemometrics that has been involved in the solution of many analytical problems. The introduction of partial or incomplete knowledge of reference values as known-value constraints in an MCR model can considerably reduce the extent of rotational ambiguity for all components. Known-value constraints can provide enough information for MCR methods to perform both the identification and quantitative analysis of first-order data sets. In practice, in addition to noise and non-ideal behavior, limitations in the reference methods or procedures cause deviation in measured known values. It is shown that deviation in the measured known values, when used as known-value constraints, may result in considerable quantification errors in MCR results and can challenge identification analysis. This contribution investigates the importance and effect of soft known-value constraints on the accuracy of MCR solutions. The influence of noise levels, the amount of deviation of known values from true values, and the interaction of these two factors were evaluated with simulated data. An illustration using soft known-value constraints is given for a batch reaction experiment. In multivariate curve resolution (MCR), the use of reference values as known-value constraints can considerably reduce the extent of rotational ambiguity. Because measured "known values" are always subject to experimental error, this paper investigates the importance and effect of "soft" known-value constraints on the accuracy of MCR solutions. The shaded parts show the area of feasible solutions (AFS) after applying the soft known-value constraints in addition to the non-negativity constraints. The points indicate the position of the MCR-ALS results after applying hard non-negativity and hard known-value constraints. Image 1 • The use of "soft" known-value constraints on multivariate curve resolution (MCR) results was studied. • Measured "known values" used as constraints in MCR methods are subject to experimental error and can lead to poor solutions. • The use of "soft" known-value constraints can accommodate uncertainty in reference values resulting in improved quantification in MCR. • A new Matlab code was written that calculates the range of feasible MCR solutions under soft known-value constraints. • The influence of noise levels, the amount of deviation of known values from true values, and the interaction of these two factors were investigated. [ABSTRACT FROM AUTHOR]

This paper presents a novel pulsatile Zone Model Predictive Control (pZMPC) for glycemic control in type 1 diabetic patients, which is an extension of the one presented in literature. Its main characteristics are (i) the explicit inclusion of a time-varying insulin on board constraint to promote a non-zero insulin delivery after a standard bolus infusion and to increase the postprandial system controllability, and (ii) the softening of this constraint to ensure closed-loop stability for an enlarged domain of attraction. Additionally, configuration methods are proposed for the insulin on board constraint based on carb counting regression models and glycemia rate of change information. In-silico results in the FDA-approved UVA/Padova simulator, under over/underestimation of the meal amounts, and under insulin sensitivity variability, show a superiority of the proposed glucose control over postprandial periods. [Display omitted] • A time-varying less restrictive insulin on board constraint was formulated. • A configuration method based on a carb-counting regression model. • Adaptive setting rooted in glycemia rate of change information. • Softening of this constraint while keeping closed-loop stability guarantee. [ABSTRACT FROM AUTHOR]

A quadratic program (QP) with soft inequality constraints with both linear and quadratic costs on constraint violation can be solved with the dual gradient method (GM) or the dual fast gradient method (FGM). The treatment of the constraint violation influences the efficiency and usefulness of the algorithm. We improve on the classical way of extending the QP: our novel contribution is that we obtain the solution to the soft-constrained QP without explicitly introducing slack variables. This approach is more efficient than solving the extended QP with GM or FGM and results in a similar algorithm than if the soft constraints were replaced with hard ones. The approach is intended for applications in model predictive control with fast system dynamics, where QPs of this type are solved at every sampling time in the millisecond range. [ABSTRACT FROM AUTHOR]

Real processes with heat exchange have usually complex behaviour and are energy intensive. In practical applications, the process variables are always bounded, and it is suitable to include these boundaries into the controller design. The soft-constrained robust model predictive controller has been designed to improve the control performance and energy consumption in comparison with the robust model predictive control with only hard constraints. Experimental application of soft-constrained robust model predictive control (SCR MPC) for a laboratory plate heat exchanger is presented in this paper. The plate heat exchanger is a non-linear process with asymmetric dynamics and is modelled as a system with parametric uncertainties. The controlled variable is the temperature of the heated fluid at the outlet of the heat exchanger and the manipulated variable is the volumetric flow rate of the heating fluid. The actuator is a peristaltic pump and the influence of the linear and non-linear actuator characteristics on the control performance is also investigated. The set-point tracking using SCR MPC is studied for the laboratory plate heat exchanger in an extensive case study. The experimental results confirmed the improvement of the control responses and reduction of energy consumption by introducing the soft constraints into MPC design. • The complex behaviour of the heat exchanger was modelled by parametric uncertainty. • The linear and non-linear characteristic of actuator were considered. • Soft-constrained robust MPC with integral action was investigated. • Experimental results of the laboratory case study were analysed. • Designed robust MPC improved control performance and energy savings. [ABSTRACT FROM AUTHOR]

This study examines how to improve the accuracy of auto parking path tracking control; therefore, a linear model predictive control with softening constraints path tracking control strategy is proposed. Firstly, a linear time-varying predictive model of vehicle is established, and the future state of the vehicle can be predicted. The designed objective function fully considers the deviation between the predictor variable and the reference variable. Also, the relaxation factors are added to the optimization process, and the control increment of each cycle is calculated by the quadratic programming. Through rolling optimization and feedback correction, all kinds of deviations in the control process can be corrected in time. Then, the Simulink/CarSim simulation is carried out jointly. Furthermore, the path tracking simulation based on proportion–integration–differentiation control and no control is also carried out to compare with the model predictive control. Finally, a real vehicle test is carried out for model predictive control algorithm, and a comparative experiment based on proportion–integration–differentiation control and no control is carried out. [ABSTRACT FROM AUTHOR]

In this study, the authors propose a stabilising data‐based model predictive controller for systems subject to constraints in which the prediction model is inferred from experimental data of the plant using a machine learning technique. The inference method is a modification of the kinky inference tailored for model predictive control. In particular, the modified method has a lower computational effort and provides smoother predictions than the original method. The controller formulation considers soft constraints in the outputs, hard constraints in the inputs and guarantees closed‐loop robust stability as well as performance by means of the use of different control and prediction horizons and a weighted terminal cost. Under the assumption that the model of the system is Hölder continuous, they prove that the closed‐loop system is input‐to‐state stable with respect to the estimation errors. The results are demonstrated in a case study of a continuously stirred‐tank reactor. [ABSTRACT FROM AUTHOR]

The search for compounds exhibiting desired physical and chemical properties is an essential, yet complex problem in the chemical, petrochemical, and pharmaceutical industries. During the formulation of this optimization-based design problem two tasks must be taken into consideration: the automated generation of feasible molecular structures and the estimation of macroscopic properties based on the resultant structures. For this structural characteristic-based property prediction task numerous methods are available. However, the inverse problem, the design of a chemical compound exhibiting a set of desired properties from a given set of fragments is not so well studied. Since in general design problems molecular structures exhibiting several and sometimes conflicting properties should be optimized, we proposed a methodology based on the modification of the multi-objective Non-dominated Sorting Genetic Algorithm-II (NSGA-II). The originally huge chemical search space is conveniently described by the Joback estimation method. The efficiency of the algorithm was enhanced by soft and hard structural constraints, which expedite the search for feasible molecules. These constraints are related to the number of available groups (fragments), the octet rule and the validity of the branches in the molecule. These constraints are also used to introduce a special genetic operator that improves the individuals of the populations to ensure the estimation of the properties is based on only reliable structures. The applicability of the proposed method is tested on several benchmark problems.

In the classical team orienteering problem (TOP), a fixed fleet of vehicles is employed, each of them with a limited driving range. The manager has to decide about the subset of customers to visit, as well as the visiting order (routes). Each customer offers a different reward, which is gathered the first time that it is visited. The goal is then to maximize the total reward collected without exceeding the driving range constraint. This paper analyzes a more realistic version of the TOP in which the driving range limitation is considered as a soft constraint: every time that this range is exceeded, a penalty cost is triggered. This cost is modeled as a piece-wise function, which depends on factors such as the distance of the vehicle to the destination depot. As a result, the traditional reward-maximization objective becomes a non-smooth function. In addition, a second objective, regarding the design of balanced routing plans, is considered as well. A mathematical model for this non-smooth and bi-objective TOP is provided, and a biased-randomized algorithm is proposed as a solving approach.