Integer Second-order Cone Programming Research Articles

On efficiency and the Jain’s fairness index in integer assignment problems

Given two sets of objects, the integer assignment problem consists of assigning objects of one set to objects in the other set. Traditionally, the goal of this problem is to find an assignment that minimizes or maximizes a measure of efficiency, such as maximization of utility or minimization of cost. Lately, the interest in incorporating a measure of fairness in addition to efficiency has gained importance. This paper studies how to incorporate these two criteria in an integer assignment, using the Jain’s index as a measure of fairness. The original formulation of the assignment problem with this index involves a non-concave function, which renders a non-linear non-convex problem, usually hard to solve. To this aim, we develop two reformulations, where one is based on a quadratic objective function and the other one is based on integer second-order cone programming. We explore the performance of these reformulations in instances of real-world data derived from an application of assigning personnel to projects, and also in instances of randomly generated data. In terms of solution quality, all formulations prove to be effective in finding solutions capturing both efficiency and fairness criteria, with some slight differences depending on the type of instance. In terms of solving time, however, the performances of the formulations differ considerably. In particular, the integer quadratic approach proves to be much faster in finding optimal solutions.

Day-Ahead Operation of an Urban Energy System Considering Traffic Flows and Peak Shaving

With the increasing penetration of new-type loads such as electric vehicles and hydrogen fuel vehicles in urban power grids, the peak-to-valley load difference increases sharply, and a multi-energy coordination model is desirable. This article proposes a day-ahead operation model of an urban energy system considering traffic flows and peak shaving, which can positively contribute to multi-energy complement and low-carbon emission. The proposed model minimizes the total cost of electricity and gas by optimizing the charging and discharging strategies of energy storage, in which the output of the wind turbine and energy management of the energy hub are adaptively adjusted. The urban energy system is represented by a second-order cone (SOC) energy flow model, and hence, the optimization problem is modeled as a mixed integer SOC programming (MISOCP). Finally, test results on an integrated urban energy network indicate that the energy storage and multi-energy coordination can alleviate the peak load cutting and valley filling. The relationship between urban grid operation cost and peak-valley difference is also discussed. The maximum utilization of renewable energy sources using gasoline vehicles has been presented in this study to illustrate cost and emission reductions for a sustainable integrated electricity and transportation infrastructure.

Distributionally robust single machine scheduling with risk aversion

This paper presents a distributionally robust (DR) optimization model for the single machine scheduling problem (SMSP) with random job processing time (JPT). To the best of our knowledge, it is the first time a DR optimization approach is applied to production scheduling problems in the literature. Unlike traditional stochastic programming models, which require an exact distribution, the presented DR-SMSP model needs only the mean-covariance information of JPT. Its aim is to find an optimal job sequence by minimizing the worst-case Conditional Value-at-Risk (Robust CVaR) of the job sequence’s total flow time. We give an explicit expression of Robust CVaR, and decompose the DR-SMSP into an assignment problem and an integer second-order cone programming (I-SOCP) problem. To efficiently solve the I-SOCP problem with uncorrelated JPT, we propose three novel Cauchy-relaxation algorithms. The effectiveness and efficiency of these algorithms are evaluated by comparing them to a CPLEX solver, and robustness of the optimal job sequence is verified via comprehensive simulation experiments. In addition, the impact of confidence levels of CVaR on the tradeoff between optimality and robustness is investigated from both theoretical and practical perspectives. Our results convincingly show that the DR-SMSP model is able to enhance the robustness of the optimal job sequence and achieve risk reduction with a small sacrifice on the optimality of the mean value. Through the simulation experiments, we have also been able to identify the strength of each of the proposed algorithms.