Robust Solution Approach Research Articles

A practical and robust approach for solving the multi‐compartment vehicle routing problem under demand uncertainty using machine learning

AbstractThe multi‐compartment vehicle routing problem is an extension of the vehicle routing problem and consists of designing a set of routes to perform the collection or delivery of different product types from customers with minimal costs. The product types are incompatible with each other and must be transported separately in multiple compartments. In practice, several uncertainties such as uncertainty in demands can arise, where the exact demand of customers is not known at the time of planning. To deal with these uncertainties, decision‐makers have to rely on robust solutions. A solution is considered robust when it can resist perturbations in every possible demand scenario as much as possible. In the day to day business of most logistic companies, historical data about each customer can be stored and used to make intelligent decisions regarding the expected demands. In this article, we propose an adaptive large neighborhood search for solving the robust multi‐compartment vehicle routing problem under demand uncertainty and present a robust solution approach for the problem in practical settings by employing machine learning. We show that by using our approach, the solutions obtained have lower recourse costs and have a lower gap between expected and actual costs, which is a favorable outcome to have in practice.

An Optimization Model for the Student-to-Project Supervisor Assignment Problem-The Case of an Engineering Department

Purpose. Empirical studies on the topic of assigning university project students to supervisors are currently underexplored. Such studies are critical to success of both the students and the university. Whilst extant research on this topic has contributed to an understanding of student assignments, what appears to be missing is application of a comprehensive framework to inform formulation and validation of a robust solution approach that takes account of both student and supervisor preferences, to optimize a real-life student-to-project supervisor assignment problem. Methodology. Questionnaire and interview surveys with project coordinators, project supervisors, head of department and students were conducted to identify factors surrounding the student-to-project supervisor assignment, through a case study approach in a university department offering engineering degree programs. This study not only develops a framework to understand an effective student-to-project supervisor assignment decision but also applies it in practice, through a case study in a University department offering engineering degree programs. An integer linear programming model was developed and implemented in an optimization software to optimize the student-to-project supervisor assignment, using data from the case study. Findings. Using OpenSolver, validated model results show improvements in matching both students and project supervisors’ preferences, whilst complying with supervisors’ workloads. These results also reveal an improvement in minimizing the project coordinator’s time in doing the assignment by introducing a standardized approach that concurrently considers all variables in a consistent manner. Originality. The contribution lies in: (1) development of a robust framework for student-to-supervisor assignments, (2) explicit consideration of contextual factors that recognize different assignment scenarios, (3) identification of feedback loops to recognize not only the need for continuous improvement in student-to-supervisor assignments but also links to performance in final year projects, (4) unique insights to guide project coordinators in relation to an efficient, effective, comprehensive, and standardized approach to the student-to-project supervisor assignment, and (5) a deeper understanding of a comprehensive range of factors that play a role in student-to-project supervisor assignments in higher education institutions.

Parametric model-order-reduction development for unsteady convection

A time-averaged error indicator with POD-hGreedy is developed to drive parametric model order reduction (pMOR) for 2D unsteady natural convection in a high-aspect ratio slot parameterized with the Prandtl number, Rayleigh number, and slot angle with respect to the gravity. The error indicator is extended to accommodate the energy equation and Leray regularization. Despite being two-dimensional and laminar, the target flow regime presents several challenges: 1) there is a bifurcation in the angle parameter space; 2) the solution can be multivalued, even at steady state; and 3) the solution exhibits spatio-temporal chaos at several points in the parameter space. The authors explore several reduced-order models (ROMs) and demonstrate that Leray-regularized Galerkin ROMs provide a robust solution approach for this class of flows. They further demonstrate that error-indicated pMOR can efficiently predict several QOIs, such as mean flow, mean Nusselt number and mean turbulent kinetic energy, even in the presence of a bifurcation. Finally, they show that spatio-temporal chaos can lead to lack of reproducibility in both the full-order model and the reduced-order model and that the variance in the full-order model provides a lower bound on the pMOR error in these cases.

Point coverage with heterogeneous sensor networks: A robust optimization approach under target location uncertainty

In this study, we consider the point coverage problem and aim to locate heterogeneous sensor networks under target uncertainty. Target location uncertainty phenomenon is commonly seen in some fields such as military and security, where only probabilistic information regarding target location is usually available via intelligence, historical data, and expert opinions. In these circumstances, possible scenarios are usually generated and several courses of action are developed to address these scenarios. Taking into account this real-life phenomenon as well as the realistic constraints associated with it, we firstly develop an integer nonlinear program for locating a heterogeneous sensor network with hub-spoke topology for a given target scenario. In this topology, wireless sensors constitute the lower level network, while the hubs constitute the upper level by collecting data from the sensors and fusioning the transmitted detection data. Secondly, we propose a linear approximation for the nonlinear model which provides computational efficiency for solving large size problem instances. Then, utilizing the p-robustness concept, we develop a robust counterpart of the deterministic formulation which accounts for multiple scenarios simultaneously and generates a compromise solution that ensures a predefined threshold for regret percentages of individual scenarios. For the first time in the point coverage literature, we define the heterogeneous sensor network location problem within target location uncertainty concept and propose an efficient robust optimization approach for solving it. Finally, we present an illustrative case study to show the applicability of the proposed robust solution approach.

A linearized energy hub operation model at the presence of uncertainties: An adaptive robust solution approach

A linearized energy hub operation model at the presence of uncertainties: An adaptive robust solution approach

From bidding strategy in smart grid toward integrated bidding strategy in smart multi-energy systems, an adaptive robust solution approach

The integration of different energy carriers may result in significant cross effects among multi-energy players in both decision making and energy provision perspectives. Accordingly, the consumption of other energy types, would be subject to an hourly-based pattern along with electric energy. Therefore, the proactive role of energy hub system (EHS), employed to represent the interactions between different energy carriers, can be developed not only for electricity market, but also for different types of energy. However, due to load/energy price forecast errors, EHSs might not be able to efficiently participate in day-ahead market as a prosumer. Therefore, a proper bidding strategy is required to effectively characterize the associated uncertainties. To do so, this paper proposes an adaptive robust integrated bidding strategy for EHS to participate in day-ahead energy markets. Accordingly, the EHS operator can bid in multiple energy markets at a time to maximize/minimize its benefit/operation costs. The proposed model has been developed as a min-max-min problem in the context of adaptive robust optimization, which is solved with a new solution consisting of bi-level decomposition, duality theory, primal cutting planes and a post-event analysis. A comprehensive case study has been also conducted to evaluate the performance of the proposed model.

Robust Solution Approach for the Dynamic and Stochastic Vehicle Routing Problem

The dynamic and stochastic vehicle routing problem (DSVRP) can be modelled as a stochastic program (SP). In a two-stage SP with recourse model, the first stage minimizes the a priori routing plan cost and the second stage minimizes the cost of corrective actions, performed to deal with changes in the inputs. To deal with the problem, approaches based either on stochastic modelling or on sampling can be applied. Sampling-based methods incorporate stochastic knowledge by generating scenarios set on realizations drawn from distributions. In this paper we proposed a robust solution approach for the capacitated DSVRP based on sampling strategies. We formulated the problem as a two-stage stochastic program model with recourse. In the first stage the a priori routing plan cost is minimized, whereas in the second stage the average of higher moments for the recourse cost calculated via a set of scenarios is minimized. The idea is to include higher moments in the second stage aiming to compute a robust a priori routing plan that minimizes transportation costs while permitting small changes in the demands without changing solution structure. Additionally, the approach allows managers to choose between optimality and robustness, that is, transportation costs and reconfiguration. The computational results on a generic dynamic benchmark dataset show that the robust routing plan can cover unmet demand while incurring little extra costs as compared to the preplanning. We observed that the plan of routes is more robust; that is, not only the expected real cost, but also the increment within the planned cost is lower.

A generalized moving-boundary algorithm to predict the heat transfer rate of counterflow heat exchangers for any phase configuration

In this work, a novel and robust solution approach is presented that can be used to predict the steady-state thermal heat transfer rate for counterflow heat exchangers with any combination of single-phase and two-phase conditions within the heat exchanger. This methodology allows for multiple internal pinching points, as well as all permutations of subcooled liquid, two-phase and superheated vapor sections for the hot and cold fluids. A residual function based on the matching of the required and available thermal conductances in each section is derived, and Brent's method is then used to drive the residual to zero. Examples are presented for the application of this methodology to a water-heated n-Propane evaporator. The computational time required to execute the model for a simple case is on the order of one millisecond when the tabular interpolation methods of CoolProp are applied. Source code for the algorithm is provided in the Python programming language as an appendix.

Optimal retrofit of moment resisting frames using braces accounting for geometric nonlinearity and serviceability conditions

The paper proposes three novel optimal retrofitting strategies for moment resisting frames using cross braces to satisfy given serviceability limitations. The influences of 2nd-order geometric effects are included. The well-known “ground structure”method is used to generate an initial network of possible brace layouts and automatically determine the best one using an optimization algorithm. Of special note is the use of a constraint to impose a realistic design criterion, in which a brace can be selected only if its area lies within specified bound limits. This in effect eliminates members with non-practical (too small) sizes. The formulation is cast as a challenging instance of a mixed integer nonlinear programming problem, which accommodates simultaneously continuous and binary variables. A two-step optimization algorithm is proposed as an efficient and robust solution approach. The safety and integrity of the retrofitted structures are assessed using step-by-step nonlinear elastic analyses.

A case analysis of a sustainable food supply chain distribution system—A multi-objective approach

Sustainable supply chain management is a topical area which is continuing to grow and evolve. Within supply chains, downstream distribution from producers to customers plays a significant role in the environmental performance of production supply chains. With consumer consciousness growing in the area of sustainable food supply, food distribution needs to embrace and adapt to improve its environmental performance, while still remaining economically competitive. With a particular focus on the dairy industry, a robust solution approach is presented for the design of a capacitated distribution network for a two-layer supply chain involved in the distribution of milk in Ireland. In particular the green multi-objective optimisation model minimises CO2 emissions from transportation and total costs in the distribution chain. These distribution channels are analysed to ensure the non-dominated solutions are distributed along the Pareto fronts. A multi-attribute decision-making approach, TOPSIS, has been used to rank the realistic feasible transportation routes resulting from the trade-offs between total costs and CO2 emissions. The refined realistic solution space allows the decision-makers to geographically locate the sustainable transportation routes. In addition to geographical mapping the decision maker is also presented with a number of alternative analysed scenarios which forcibly open closed distribution routes to build resiliency into the solution approach. In terms of model performance, three separate GA based optimisers have been evaluated and reported upon. In the case presented NSGA-II was found to outperform its counterparts of MOGA-II and HYBRID.

Integrated low-carbon distribution system for the demand side of a product distribution supply chain: a DoE-guided MOPSO optimiser-based solution approach

This article contributes to distribution system literature on three inter-linked aspects viz. formulation of a novel integrated low-carbon/green distribution system for the demand side of a Supply Chain (SC) with a single product and multiple consumers, i.e. drop-off points, a novel and robust solution approach through a Design of Experiment (DoE)-guided Multiple-Objective Particle Swarm Optimisation (MOPSO) optimiser and exhaustive analysis of the solutions (i.e. prioritisation, ranking and scenario analysis). The total costs, CO2 emission and the traversed distances of the vehicles during transportation are optimised. The optimisation model for the strategic decision-making is formulated by effectively integrating the 0–1 mixed-integer programming with a green constraint based on Analytic Hierarchy Process. Due to the computationally NP-hard characteristic of the model, a systematic and technically robust DoE-guided solution approach is designed using a commercial solver – modeFRONTIER®. DoE guides the solution through the MOPSO optimiser in order to eliminate the un-realistic set of feasible and optimal solution sets. A popular multi-attribute decision-making approach, TOPSIS, evaluates the solutions found from the Pareto optimal solution space of the solver. Finally, decision-makers’ preferences are analysed for monitoring the changes in the controlling parameters with respect to the changes in the decisions. A scenario analysis of the events by considering alternative possible outcomes is also conducted. It is found that the implemented methodology successfully routes the vehicles with optimal costs and low-carbon emission thus contributing to greening the environment on the demand side of a SC network.

$${{\mathcal {D}(\mathcal {C})}}$$ -optimization and robust global optimization

For solving global optimization problems with nonconvex feasible sets existing methods compute an approximate optimal solution which is not guaranteed to be close, within a given tolerance, to the actual optimal solution, nor even to be feasible. To overcome these limitations, a robust solution approach is proposed that can be applied to a wide class of problems called $${{\mathcal {D}(\mathcal {C})}}$$ -optimization problems. DC optimization and monotonic optimization are particular cases of $${{\mathcal {D}(\mathcal {C})}}$$ -optimization, so this class includes virtually every nonconvex global optimization problem of interest. The approach is a refinement and extension of an earlier version proposed for dc and monotonic optimization.

A robust solution approach for nonconvex quadratic programs with additional multiplicative constraints

In this paper, a robust algorithm is proposed for globally solving a nonconvex quadratic programming problem (P1) with several additional multiplicative constraints. To our knowledge, little progress has been made so far for globally solving problem (P1). The proposed algorithm is based on a robust solution approach that provides an essential ε -optimal solution. This solution is also stable under small perturbations of the constraints, and it turns out that such a robust optimal solution is adequately guaranteed to be feasible and to be close to the actual optimal solution. Convergence of the algorithm is shown and three solved sample problems are given to illustrate the robust stability and feasibility of the present algorithm.

Adaptive local discontinuous Galerkin approximation to Richards’ equation

We propose a spatially and temporally adaptive solution to Richards’ equation based upon a local discontinuous Galerkin approximation in space and a high-order, backward difference method in time. We cast our approach in terms of a general, decoupled adaption algorithm based upon operators. We define non-unique instances of all operators to result in an adaption method from within the general class of methods that is defined. We formally decouple the spatial adaption from the temporal adaption using a method of lines approach and limit the temporal truncation error so that the total error is dominated by the spatial component. We use a multiple grid approach to guide adaption and support the data structures. Spatial adaption decisions are based upon error and regularity indicators, which are economical to compute. The resultant methods are compared for two test problems. The results show that the proposed adaption methods are superior to methods that adapt only in time and that in cases in which the problem has sufficient smoothness, adapting the order of the elements in addition to the grid spacing can further improve the efficiency of this robust solution approach.

A Robust Solution Approach to the Dynamic Vehicle Scheduling Problem

This paper presents a solution approach to the dynamic vehicle scheduling problem. This approach consists of solving a sequence of optimization problems, where we take into account different scenarios for future travel times. We discuss the potential benefit of our approach compared to the traditional one, where the vehicle scheduling problem is solved only once for a whole period and the travel times are assumed to be fixed. Because in the multiple-depot case we cannot solve the problem exactly within reasonable computation time, we use a “cluster-reschedule” heuristic where we first assign trips to depots by solving the static problem and then solve dynamic single-depot problems. We use new mathematical formulations of these problems that allow fast solution by standard optimization software. Results of a computational study with real-life data are presented, in which we compare different variants of our approach and perform a sensitivity analysis with respect to deviations of the actual travel times from estimated ones.

Modified p-median approach for efficient GT cell formation

The objective of this paper is to propose the modified formulations over the classical p-median model for efficiently solving the cell formation (CF) problem with which production managers are faced in implementing group technology (GT) to manufacturing. In spite of its merits as the robust solution approach to GT cell formation, the classical p-median model is limited to small-size CF problems since it requires many binary variables. Necessity of many binary variables makes it difficult for cell designer to use the classical p-median model for solving even medium-sized CF problems, let alone large-size CF problems even though large-memory main frame computer and highly expensive commercial software are used. To resolve this sort of problems, we propose two modified p-median formulations which are even applicable to largest CF problems among the ones found in literature. Their main virtues are as follows. First, the modified p-median formulations encourage speedy implementation of the models. Secondly, the proposed formulations that are applied to moderately large-size CF problems with 30 or more machines can run effectively even on personal computer and education-purpose softwares such as the HYPER LINDO which are available at such a low cost compared with commercial software. Computational results from applying to moderately large-size CF problems with 30 to 50 machines show that the CF using the modified p-median formulations provides very promising implications as compared with the CF using the classical p-median model.