Discrete Time Horizon Research Articles

Scheduling the repair and replacement of individual components in operating systems: a bi-objective mathematical optimization model

Preventive maintenance (PM) is performed so that failure is avoided while corrective maintenance is performed after a failure has occurred in order to restore the system back to an operational state. This research aims at scheduling PM activities for a multi-component system within a finite time horizon. We consider a setting with two stakeholders, being the system operator and the maintenance workshop, and two different contract types governing their joint activities, namely an availability contract and a turn-around time contract. Components in the systems that are to be maintained are sent to the maintenance workshop, which needs to schedule and perform all maintenance activities while at the same time satisfying the contract and not exceeding the workshop capacity. Our modelling is based on a mixed-binary linear optimization model of a PM scheduling problem with so-called interval costs over a finite and discretized time horizon. We enhance this scheduling model with the flow of individual components through the maintenance workshop, including stocks of spare components, both those components that need repair and the repaired ones. The resulting scheduling model is then utilized in the optimization of two main contracts, namely maximizing the availability of repaired (or new) components and minimizing the deviation from the contracted turn-around times for the components in the maintenance loop. Each of these objectives is combined with the objective to minimize the costs for maintenance of the operating system, leading to two bi-objective optimization problems. We analyse the two contracting forms between the stakeholders by studying and comparing the Pareto fronts resulting from different parameter settings, regarding minimum allowed stock levels and investments in repair capacity of the workshop. Our bi-objective mixed-binary linear optimization model is able to capture important properties of the results from the contracting forms as well as to show that, in our setting, an availability contract performs better than a turn-around time contract in terms of tractability.

Scenario-based Distributionally Robust Optimization for the Stochastic Inventory Routing Problem

We consider a class of the inventory routing problem in a discrete and finite time horizon, where the demand for homogeneous products at retail stores is uncertain and varies across different scenarios. The supplier is required to determine the times to visit retailers, the replenishment quantities to each retailer, and the routing of a vehicle so as to minimize the sum of stockout, holding, and transportation costs. We propose a scenario-based distributionally robust optimization framework to tackle this problem. We transform the distributionally robust optimization model into a mixed-integer problem, which can be solved efficiently by our proposed algorithm. We adopt a warm-start procedure that utilizes the solution to the nominal model in our methodological framework. Then we apply a Tabu search algorithm, integrated with column generation, to solve a set-partitioning-like integer linear programming model so that a better route set can be identified. By doing so, a large-scale scenario-based distributionally robust optimization model can be solved. We conduct a case study of a fuel company and construct realistic instances to demonstrate the performance of our proposed method. Computational results suggest that the model taking into account various scenarios is more effective when random demands can be classified; the model with a linear decision rule outperforms a non-adaptive model; and the model with the route set identified by an improved algorithm can deliver a better solution than the original route set.

$$\varepsilon $$-Nash Equilibria of a Multi-player Nonzero-Sum Dynkin Game in Discrete Time

We study the infinite horizon discrete time N-player nonzero-sum Dynkin game ( $$N\ge 2$$ ) with stopping times as strategies (or pure strategies). We prove existence of an $$\varepsilon $$ -Nash equilibrium point for the game by presenting a constructive algorithm. One of the main features is that the payoffs of the players depend on the set of players that stop at the termination stage which is the minimal stage in which at least one player stops. The existence result is extended to the case of a nonzero-sum game with finite horizon. Finally, the algorithm is illustrated by two explicit examples in the specific case of finite horizon.

The discrete time break scheduling problem under fatigue and no preemption: solution methods and impact of work regulations

We address the discrete time break scheduling problem with no preemption when workers' fatigue impacts their productivity. We propose a Mixed Integer Linear Programming model to solve the one break problem to optimality, using a lexicographic approach where the production amount is maximised first, and then the break length over a discrete time horizon. We develop a Variable Neighbourhood Search algorithm to solve the multiple break problem. In addition to proposing efficient solution methods to the problem, our incentive is to assess the impact on the production amount and on workers' welfare of rest break regulations laid down in legislation or collective agreements. We conducted an extensive simulation study to represent a wide range of workers' profiles defined in terms of fatigability and recovery speed. Simulation results show that regulations slightly affect the production amount whereas they allow for large improvements of workers' welfare as long as breaks are optimised as a second objective. The production amount is also shown to be quite sensitive to the break timing. Finally, multiple breaks can improve the production amount and workers' welfare in many situations, which questions the widespread belief that endowing workers with a single short break would optimise the production amount.

Stochastic optimal control over unreliable communication links

In this paper LQG control over unreliable communication links is derived. That is to say, the communication channels between the controller and the actuators and between the sensors and the controller are unreliable. This is of growing importance as networked control systems and use of wireless communication in control are becoming increasingly common. The problem of how to optimize LQG control in this case is examined in the situation where communication between the components is done with acknowledgments. Previous solutions to finite horizon discrete time hold-input LQG control for this case do not fully utilize the available information. Here a new solution is presented which resolves this limitation. The solution is linear and covers communication channels subject to both packet losses and delays. The new control scheme is compared with previous solutions for LQG control in simulations, which demonstrates that a significant improvement in the cost can be achieved by fully utilizing the available information.

Simultaneous scheduling of replacement and repair of common components in operating systems

In order for a system to stay operational, its components need maintenance. We consider two stakeholders—a system operator and a maintenance workshop—and a contract governing their joint activities. Components in the operating systems that are to be maintained are sent to the maintenance workshop, which should perform all maintenance activities on time in order to satisfy the contract. The maintained components are then sent back to be used in the operating systems. Our modeling of this system-of-systems includes stocks of damaged and repaired components, the workshop scheduling, and the planning of preventive maintenance for the operating systems. Our modeling is based on a mixed-binary linear optimization (MBLP) model of a preventive maintenance scheduling problem with so-called interval costs over a finite and discretized time horizon. We generalize and extend this model with the flow of components through the workshop, including the stocks of spare components. The resulting scheduling model—a mixed-integer optimization (MILP) model—is then utilized to optimize the main contract in a bi-objective setting: maximizing the availability of repaired (or new) components and minimizing the costs of maintaining the operating systems over the time horizon. We analyze the main contract and briefly discuss a turn-around time contract. Our results concern the effect of our modeling on the levels of the stocks of components over time, in particular minimizing the risk for lack of spare components.

An Effective Matheuristic for the Multivehicle Inventory Routing Problem

This study considers the multivehicle inventory routing problem in which a supplier has to build a distribution plan over a discrete time horizon to replenish a set of customers that faces a given demand. Transportation costs as well as inventory costs at the supplier and at the customers are to be minimized. A matheuristic algorithm is proposed that is based on sequentially solving different mixed integer linear programs. The algorithm merges the advantage of being easy to design and implement, as it is mainly based on the problem formulation, with the benefit of providing high-quality solutions. A computational study is performed on benchmark test instances by comparing the results with the ones obtained from previous algorithms proposed in the literature. The results show that the matheuristic algorithm outperforms the existing heuristic algorithms and finds a significant number of new best solutions in both small and large instances.

Lower and upper bounds for scheduling energy-consuming tasks with storage resources and piecewise linear costs

This paper considers the problem of scheduling a set of time- and energy-constrained preemptive tasks on a discrete time horizon. At each time period, the total energy required by the tasks that are in process can be provided by two energy sources: a reversible one and a non-reversible one. The non-reversible energy source can provide an unlimited amount of energy for a given period but at the expense of a time-dependent piecewise linear cost. The reversible energy source is a storage resource. The goal is to schedule each task preemptively inside its time window and to dispatch the required energy to the sources at each time period, while satisfying the reversible source capacity constraints and minimizing the total cost. We propose a mixed integer linear program of pseudo-polynomial size to solve this NP-hard problem. Acknowledging the limits of this model for problem instances of modest size, we propose an iterative decomposition matheuristic to compute an upper bound. The method relies on an efficient branch-and-price method or on a local search procedure to solve the scheduling problem without storage. The energy source allocation problem for a fixed schedule can in turn be solved efficiently by dynamic programming as a particular lot-sizing problem. We also propose a lower bound obtained by solving the linear programming relaxation of a new extended formulation by column generation. Experimental results show the quality of the bounds compared to the ones obtained using mixed integer linear program.

Smart Charging of Electric Vehicles Considering SOC-Dependent Maximum Charging Powers

The aim of this work is to schedule the charging of electric vehicles (EVs) at a single charging station such that the temporal availability of each EV as well as the maximum available power at the station are considered. The total costs for charging the vehicles should be minimized w.r.t. time-dependent electricity costs. A particular challenge investigated in this work is that the maximum power at which a vehicle can be charged is dependent on the current state of charge (SOC) of the vehicle. Such a consideration is particularly relevant in the case of fast charging. Considering this aspect for a discretized time horizon is not trivial, as the maximum charging power of an EV may also change in between time steps. To deal with this issue, we instead consider the energy by which an EV can be charged within a time step. For this purpose, we show how to derive the maximum charging energy in an exact as well as an approximate way. Moreover, we propose two methods for solving the scheduling problem. The first is a cutting plane method utilizing a convex hull of the, in general, nonconcave SOC–power curves. The second method is based on a piecewise linearization of the SOC–energy curve and is effectively solved by branch-and-cut. The proposed approaches are evaluated on benchmark instances, which are partly based on real-world data. To deal with EVs arriving at different times as well as charging costs changing over time, a model-based predictive control strategy is usually applied in such cases. Hence, we also experimentally evaluate the performance of our approaches for such a strategy. The results show that optimally solving problems with general piecewise linear maximum power functions requires high computation times. However, problems with concave, piecewise linear maximum charging power functions can efficiently be dealt with by means of linear programming. Approximating an EV’s maximum charging power with a concave function may result in practically infeasible solutions, due to vehicles potentially not reaching their specified target SOC. However, our results show that this error is negligible in practice.

Combinatorial Heuristics for Inventory Routing Problems

We consider the deterministic inventory routing problem over a discrete finite time horizon. Given clients on a metric, each with daily demands that must be delivered from a depot and holding costs over the planning horizon, an optimal solution selects a set of daily tours through a subset of clients to deliver all demands before they are due and minimizes the total holding and tour routing costs over the horizon. In the capacitated case, a limited number of vehicles are available, where each vehicle makes at most one trip per day. Each trip from the depot is allowed to carry a limited amount of supply to deliver. We develop fast heuristics for both cases by solving a family of prize-collecting Steiner tree instances. Computational experiments show our heuristics can find near-optimal solutions for both cases and substantially reduce the runtime compared with a pure mixed integer programming formulation approach.

Maximizing Cloud Revenue using Dynamic Pricing of Multiple Class Virtual Machines

The Infrastructure as a Service (IaaS) cloud industry that relies on leasing virtual machines (VMs) has significant portion of business values of finding the dynamic equilibrium between two conflicting phenomena: underutilization and surging congestion. Spot instance has been proposed as an elegant solution to overcome these challenges, with the ultimate goal to achieve greater profits. However, previous studies on recent spot pricing schemes reveal artificial pricing policies that do not comply with the dynamic nature of these phenomena. Motivated by these facts, this paper investigates dynamic pricing of stagnant resources in order to maximize cloud revenue. Specifically, our proposed approach manages multiple classes of virtual machines in order to achieve the maximum expected revenue within a finite discrete time horizon. For this sake, the proposed approach leverages the Markov decision processes with a number of properties under optimum controlling conditions that characterize a model's behaviour. Further, this approach applies approximate stochastic dynamic programming using linear programming to create a practical model. Experimental results confirm that this approach of dynamic pricing can scale up or down the price efficiently and effectively, according to the stagnant resources and the load thresholds. These results provide significant insights to maximizing the IaaS cloud revenue.

Accelerated Point-Wise Maximum Approach to Approximate Dynamic Programming

In this article, we describe an approximate dynamic programming (ADP) approach to compute lower bounds on the optimal value function for a discrete time, continuous space, and infinite horizon setting. The approach iteratively constructs a family of lower bounding approximate value functions by using the so-called Bellman inequality. The novelty of our approach is that, at each iteration, we aim to compute an approximate value function that maximizes the point-wise maximum taken with the family of approximate value functions computed thus far. This leads to a nonconvex objective, and we propose a gradient ascent algorithm to find stationary points by solving a sequence of convex optimization problems. We provide convergence guarantees for our algorithm and an interpretation for how the gradient computation relates to the state-relevance weighting parameter appearing in related ADP approaches. We demonstrate through numerical examples that, when compared to the existing approaches, the algorithm we propose computes tighter suboptimality bounds with comparable computation time.

Decomposition methods for large-scale network expansion problems

Network expansion problems are a special class of multi-period network design problems in which arcs can be opened gradually in different time periods but can never be closed. Motivated by practical applications, we focus on cases where demand between origin-destination pairs expands over a discrete time horizon. Arc opening decisions are taken in every period, and once an arc is opened it can be used throughout the remaining horizon to route several commodities. Our model captures a key timing trade-off: the earlier an arc is opened, the more periods it can be used for, but its fixed cost is higher, since it accounts not only for construction but also for maintenance over the remaining horizon. An overview of practical applications indicates that this trade-off is relevant in various settings. For the capacitated variant, we develop an arc-based Lagrange relaxation, combined with local improvement heuristics. For uncapacitated problems, we develop four Benders decomposition formulations and show how taking advantage of the problem structure leads to enhanced algorithmic performance. We then utilize real-world and artificial networks to generate 1080 instances, with which we conduct a computational study. Our results demonstrate the efficiency of our algorithms. Notably, for uncapacitated problems we are able to solve instances with 2.5 million variables to optimality in less than two hours of computing time. Finally, we provide insights into how instance characteristics influence the multi-period structure of solutions.

Modelling and assessing the impact of the DSO remuneration strategy on its interaction with electricity users

This paper presents a simulation-based methodology for assessing the impact of employing different distribution system operator’s remuneration strategies on the economic sustainability of electrical distribution systems. The proposed methodology accounts for the uncertainties posed by the integration of distributed electricity generation resources, and the roll out of smart meters. The different remuneration strategies analysed in this paper include notably new distribution tariffs based on individual peak power consumption and time-dependent rates that are contingent on the time of energy consumption, both requiring smart meters to work. The distributed electricity generation resources are modelled through an optimisation framework and an investment decision process that gradually deploys household photovoltaic installations depending on their profitability and the electricity charges, including the distribution rates. The impact of the distribution system operator’s remuneration strategy is measured by an accurate modelling of the remuneration mechanism of this entity, which can adapt to various distribution tariff designs. We analyse this impact over a discrete time horizon. Our methodology is illustrated with several examples of distribution tariffs including old –based on energy consumption or on per-connection fees– as well as new –based on power consumption or time-of use fees– designs. Finally, we provide a comprehensive sensitivity analysis of the proposed simulation environment to the main parameters of the methodology.

Data-driven Clustering and Feature-based Retail Electricity Pricing with Smart Meters

We consider an electric utility company that serves retail electricity customers over a discrete-time horizon. In each period, the company observes the customers' consumption as well as high-dimensional features on customer characteristics and exogenous factors. These features exhibit three types of heterogeneity—over time, customers, or both. Based on the consumption and feature observations, the company can dynamically adjust the retail electricity price at the customer level. The consumption depends on the features: there is an underlying structure of clusters in the feature space, and the relationship between consumption and features is different in each cluster. Initially, the company knows neither the underlying cluster structure nor the corresponding consumption models. We measure the company's performance by its average regret, i.e., the profit loss per period per customer, relative to a clairvoyant who knows the underlying cluster structure and the consumption model in each cluster. We design a data-driven policy of joint spectral clustering and feature-based pricing, and show that its average regret converges to zero at the fastest achievable rate. We conduct case studies based on real-life smart meter data from Texas and simulation experiments. Relative to the company policy, our policy increases company profits by 146% over a three-month period.

Optimizing the investments in mobile networks and subscriber migrations for a telecommunication operator

AbstractWe consider the context of a telecommunication company that is at the same time an infrastructure operator and a service provider. When planning its network expansion, the company can leverage over its knowledge of the subscriber dynamic to better optimize the network dimensioning, therefore avoiding unnecessary costs. In this work, the network expansion represents the deployment and/or reinforcement of several technologies (e.g., 2G, 3G, 4G), assuming that subscribers to a given technology can be served by this technology or older ones. The operator can influence subscriber dynamic by subsidies. The planning is made over a discretized time horizon while some strategic guideline requirements are required at the end of the time horizon. Following classical models, we consider that the willingness of customers for shifting to a new technology follows an S‐shape piecewise constant function. We propose a mixed‐integer linear programming formulation, improved through several valid inequalities and a heuristic algorithm. We assess the formulation numerically on real instances.

An experimental approach to the optimization of the dosage of hydrogen peroxide for Fenton and photo-Fenton processes

The determination of the hydrogen peroxide dosage scheme that minimizes hydrogen peroxide consumption while meeting the specified treatment outcome is crucial for Fenton and photo-Fenton processes. The challenge is building a methodology that provides the optimal dosage profile. However, the lack of detailed dynamic models prevents exploiting model-based optimization methods that have proved successful in other applications. Thus, this work addresses this challenge by providing a problem formulation identifying and discussing objectives and constraints, and the nature of the optimal solution. From this point, the work presents a novel dosage model and a consequent methodology aimed at experimentally optimizing the dosage profile along a discretized time horizon following recipe optimization concepts. The approach is parallel to the numerical solution of the model-based optimization problem posed by hydrogen peroxide dosage. The proposed methodology is validated in the remediation of a Paracetamol (PCT) solution, and the obtained results are assessed and discussed in regard of the evolution of the concentration of hydrogen peroxide, the contaminant (PCT), and Total Organic Carbon (TOC). The concentration of dissolved oxygen (DO), which is also monitored, allows providing a more comprehensive explanation of the nature of the process.

An integrated multi-ship crane allocation in Beirut Port container terminal

This paper investigates the integration between the quay and yard sides for multiple berthing ships with transshipment containers. This paper is motivated by the experience of an operator at Beirut Port. An integer linear programming model is formulated to minimize the total number of cranes used in both quay and yard sides for all berthing ships with transshipment containers unloading during a finite and discretized time horizon. The number of containers to be unloaded is determined in each time period, by each quay crane, at each ship bay location, along with the designated storage location at the yard side. The number of yard cranes needed at each storage yard block is also determined over the time horizon. Major capacity, time, and spatial constraints related to transshipment operations are taken into consideration. One insight from our numerical results is that restricting resources at the yard side will lead to an increase in required cranes at the quay side, and vice versa, which confirm results in earlier literature on single ship. However, we argue, via several counter examples, that single-ship solutions are not easily adaptable to multi-ship situations, which justifies the purpose of integrated formulations such as ours.

A branch-and-cut algorithm for the inventory routing problem with pickups and deliveries

This paper addresses the Inventory Routing Problem with Pickups and Deliveries (IRP-PD). A single commodity has to be picked up from several origins and distributed to several destinations. The commodity is made available at the supplier depot and at the pickup customers, with known amounts in each period of a discretized time horizon. The commodity is consumed by the delivery customers, whose demand, in each period of the time horizon, is known. The vehicles start and end their routes at the supplier depot. The objective is to determine a collection and distribution plan minimizing the sum of routing and inventory costs, satisfying inventory and capacity constraints. A mixed integer linear programming model is presented for the IRP-PD. Valid inequalities are proposed that are adapted from the inventory routing, the vehicle routing and the lot-sizing problems literature. Moreover, a new set of valid inequalities, called interval inequalities, is presented. A branch-and-cut algorithm is designed and tested on a large set of instances. Computational results show that the proposed approach is able to solve to optimality 946 out of 1280 instances. Moreover, the approach outperforms the state-of-the art method for the single vehicle case, increasing the number of instances solved to optimality from 473 to 538 out of 640 instances tested, with 133 new best known solutions.

Bioeconomic modeling of seasonal fisheries

Seasonality or periodicity–biological, environmental, or economic–are fundamental properties of most marine fisheries. We propose a generic infinite horizon discrete time fisheries management model by modifying existing models and frameworks of analysis to reflect seasonal variation or more general multiperiodicity in parameters, variables, or functional forms. Our model captures such variations via repeated cycles of multiple intervals with differing characteristics. The framework offers a simple and intuitive set up of arbitrary periodicity and seasonality in any feature, which significantly increases model realism. Further, it distances itself from continuous time modeling approaches where uniqueness and solvability of periodic models generally are difficult to assert. In our setting, the governing equations for the time-dependent value function of the management optimization problem are shown to be equivalent to a high-dimensional contraction and hence ensure uniqueness and a feasible solution algorithm. We illustrate our approach using a simple example to demonstrate that accounting for seasonality in fisheries management can improve outcomes considerably. Our framework also provides for analysis of seasonal regulatory measures. Ultimately, our approach applies to renewable resource management more generally and to many infinite-horizon, discrete time optimization problems with periodic features.