On Idle Energy Consumption Minimization in Production: Industrial Example and Mathematical Model

This paper, inspired by a real production process of steel hardening, investigates a scheduling problem to minimize the idle energy consumption of machines. The energy minimization is achieved by switching a machine to some power-saving mode when it is idle. For the steel hardening process, the mode of the machine (i.e., furnace) can be associated with its inner temperature. Contrary to the recent methods, which consider only a small number of machine modes, the temperature in the furnace can be changed continuously, and so an infinite number of the power-saving modes must be considered to achieve the highest possible savings. To model the machine modes efficiently, we use the concept of the energy function, which was originally introduced in the domain of embedded systems but has yet to take roots in the domain of production research. The energy function is illustrated with several application examples from the literature. Afterward, it is integrated into a mathematical model of a scheduling problem with parallel identical machines and jobs characterized by release times, deadlines, and processing times. Numerical experiments show that the proposed model outperforms a reference model adapted from the literature.Comment: Accepted to 9th International Conference on Operations Research and Enterprise Systems (ICORES 2020

Data-driven Algorithm for Scheduling with Total Tardiness

In this paper, we investigate the use of deep learning for solving a classical NP-Hard single machine scheduling problem where the criterion is to minimize the total tardiness. Instead of designing an end-to-end machine learning model, we utilize well known decomposition of the problem and we enhance it with a data-driven approach. We have designed a regressor containing a deep neural network that learns and predicts the criterion of a given set of jobs. The network acts as a polynomial-time estimator of the criterion that is used in a single-pass scheduling algorithm based on Lawler's decomposition theorem. Essentially, the regressor guides the algorithm to select the best position for each job. The experimental results show that our data-driven approach can efficiently generalize information from the training phase to significantly larger instances (up to 350 jobs) where it achieves an optimality gap of about 0.5%, which is four times less than the gap of the state-of-the-art NBR heuristic

Finding an optimal Nash equilibrium to the multi-agent project scheduling problem

30 pagesInternational audienceCooperative projects involve a set of self-interested contractors, each in charge of a part of the project. Each contractor may control the duration of his activities, which can be shorten up to an incompressible limit, by gathering extra resources at a given cost. In this context the resulting project makespan depends on all contractors' decisions. The client of the project is interested in a short project makespan. As an incentive, the client offers a daily reward to be shared among the contractors in order to complete the project earlier than expected. In practice, either the reward sharing policy results from an upfront agreement or payments are freely allocated by the client himself. Each contractor is only interested in the maximization of his own profit, and behaves accordingly. This paper addresses the problem of finding a Nash equilibrium and a sharing policy that minimize such project makespan while ensuring its local stability. We explain how the resulting problem, which is NP-hard, can be modeled and solved with mixed integer linear programming (MILP). A computational analysis on large instances proves the effectiveness of our approach. Useful insights are also derived from an empirical investigation of the influence of reward sharing policy, for a better understanding of how a project customer should make the most of his funds in such project management context

Nash equilibria for the multi-agent project scheduling problem with controllable processing times

This paper considers a project scheduling environment in which the activities of the project network are partitioned among a set of agents. Activity durations are controllable, i.e., every agent is allowed to shorten the duration of its activities, incurring a crashing cost. If the project makespan is reduced with respect to its normal value, a reward is offered to the agents and each agent receives a given ratio of the total reward. Agents want to maximize their profit. Assuming a complete knowledge of the agents’ parameters and of the activity network, this problem is modeled as a noncooperative game and Nash equilibria are analyzed.We characterize Nash equilibria in terms of the existence of certain types of cuts on the project network. We show that finding one Nash equilibrium is easy, while finding a Nash strategy that minimizes the project makespan is NP-hard in the strong sense. The particular case where each activity belongs to a different agent is also studied

Price of anarchy and price of stability in multi-agent project scheduling

International audienceWe consider a project scheduling environment in which the activities are partitioned among a set of agents. The owner of each activity can decide its length, which is linearly related to its cost within a minimum (crash) and a maximum (normal) length. For each day the project makespan is reduced with respect to its normal value, a reward is offered to the agents, and each agent receives a given ratio of the reward. As in classical game theory, we assume that the agents’ parameters are common knowledge. We study the Nash equilibria of the corresponding non-cooperative game as a desired state where no agent is motivated to change his/her decision. Regarding project makespan as an overall measure of efficiency, here we consider the worst and the best Nash equilibria (i.e., for which makespan is maximum and, respectively, minimum among Nash equilibria). We show that the problem of finding the worst Nash equilibrium is NP-hard (finding the best Nash equilibrium is already known to be strongly NP-hard), and propose an ILP formulation for its computation. We then investigate the values of the price of anarchy and the price of stability in a large sample of realistic size problems and get useful insights for the project owner