Integer Variables Research Articles

Multi-objective optimization of inter-story isolated buildings using metaheuristic and derivative-free algorithms

Inter-story isolation (ISI) is a passive control technique that involves the placement of seismic isolation devices between stories. Multiple isolation layers installed at different story levels can reduce the response of the isolation devices, while maintaining control of the primary building. A multi-objective optimization (MOO) study is conducted in this paper to select the optimum stories of installation and the effective stiffness and damping properties of specified number of isolation layers. The maximum standard deviations of inter-story drift ratio (ISDR) and isolation drift, derived from random vibration analysis (RVA) of a shear-building model are used as the objective functions of the optimization problem, thus accounting for the seismic responses of both the primary building and the isolation devices. To solve the resulting mixed-integer problem, the combinations of integer variables are fully enumerated and a continuous-variable sub-problem is solved for each combination. Four metaheuristic and one derivative-free algorithms are comparatively assessed for solving the continuous-variable multi-objective sub-problems corresponding to the combinations of the integer variables. Along with the optimal trade-offs between the response objectives, multi-objective optimization reveals trends in response with varying isolation configuration as well as the enhancement of the ISI building performance with increasing number of isolation layers.

Slicing Resource Allocation for eMBB and URLLC in 5G RAN

This paper investigates the network slicing in the virtualized wireless network. We consider a downlink orthogonal frequency division multiple access system in which physical resources of base stations are virtualized and divided into enhanced mobile broadband (eMBB) and ultrareliable low latency communication (URLLC) slices. We take the network slicing technology to solve the problems of network spectral efficiency and URLLC reliability. A mixed-integer programming problem is formulated by maximizing the spectral efficiency of the system in the constraint of users’ requirements for two slices, i.e., the requirement of the eMBB slice and the requirement of the URLLC slice with a high probability for each user. By transforming and relaxing integer variables, the original problem is approximated to a convex optimization problem. Then, we combine the objective function and the constraint conditions through dual variables to form an augmented Lagrangian function, and the optimal solution of this function is the upper bound of the original problem. In addition, we propose a resource allocation algorithm that allocates the network slicing by applying the Powell–Hestenes–Rockafellar method and the branch and bound method, obtaining the optimal solution. The simulation results show that the proposed resource allocation algorithm can significantly improve the spectral efficiency of the system and URLLC reliability, compared with the adaptive particle swarm optimization (APSO), the equal power allocation (EPA), and the equal subcarrier allocation (ESA) algorithm. Furthermore, we analyze the spectral efficiency of the proposed algorithm with the users’ requirements change of two slices and get better spectral efficiency performance.

A Hybrid Framework for Simultaneous Process and Solvent Optimization of Continuous Anti-Solvent Crystallization with Distillation for Solvent Recycling

Anti-solvent crystallization is frequently applied in pharmaceutical processes for the separation and purification of intermediate compounds and active ingredients. The selection of optimal solvent types is important to improve the economic performance and sustainability of the process, but is challenged by the discrete nature and large number of possible solvent combinations and the inherent relations between solvent selection and optimal process design. A computational framework is presented for the simultaneous solvent selection and optimization for a continuous process involving crystallization and distillation for recycling of the anti-solvent. The method is based on the perturbed-chain statistical associated fluid theory (PC-SAFT) equation of state to predict relevant thermodynamic properties of mixtures within the process. Alternative process configurations were represented by a superstructure. Due to the high nonlinearity of the thermodynamic models and rigorous models for distillation, the resulting mixed-integer nonlinear programming (MINLP) problem is difficult to solve by state-of-the-art solvers. Therefore, a continuous mapping method was adopted to relax the integer variables related to solvent selection, which makes the scale of the problem formulation independent of the number of solvents under consideration. Furthermore, a genetic algorithm was used to optimize the integer variables related to the superstructure. The hybrid stochastic and deterministic optimization framework converts the original MINLP problem into a nonlinear programming (NLP) problem, which is computationally more tractable. The successful application of the proposed method was demonstrated by a case study on the continuous anti-solvent crystallization of paracetamol.

A Simple Method for Finding Optimal Paths of Hot and Cold Streams inside Shell and Tube Heat Exchangers to Reduce Pumping Cost in Heat Exchanger Network Problems

In this paper, a simple method is presented for the synthesis and retrofit of heat exchanger networks (HENs) by considering pressure drop as well as finding proper path of streams inside heat exchangers (HEs) to reduce the pumping cost of network. Generally, HEN problems lead to MINLP models which have convergence difficulties due to the existence of both continuous and integer variables. In this study, instead of solving these variables simultaneously, a combination of Genetic Algorithm (GA) with Quasi Linear Programming (QLP) and Integer Linear Programming (ILP) was used for solving the problem. GA was used to find optimal HENs structure and streams paths, whereas continuous variables were solved by QLP. For the retrofit of HENs, a modified ILP model was used. Results show that the proposed method has the ability to reduce the cost of annual pumping due to considering optimal paths for streams in the HEs compared to the literature.

Efficient Solution of Distributed MILP in Control of Networked Systems

This paper considers the distributed solution of Mixed-Integer Linear Programming (MILP) problems, a class of problems which is of interest, e.g., in optimization-based control of networked systems involving hybrid dynamics. For a larger number of subsystems, the high combinatorial complexity arising from the integer variables usually prohibits the use of centralized solution schemes, and thus requires distributed computational approaches. The proposed approach is inspired by results based on the Shapley-Folkman-Starr theorem, but it relaxes some conservative assumptions in order to enhance the computational efficiency. Numerical experiments for different MILP problems confirm the advantage of the proposed method with respect to computation times.

Energy-Efficient Channel Estimation

We investigate the energy-efficient channel estimation in wireless networks, where one pilot is inserted for every several data slots to estimate the channel coefficients. Both the channel state information estimation error and the time-varying model are considered. We first formulate the energy-efficient channel estimation problem into a mixed integer nonlinear programming (MINLP) problem, where the variables include the transmitted power and the number of data slots. Due to the NP-hard nature, we degenerate the MINLP problem into a series of non-concave optimization problems without integer variables. Then we solve these problems using successive convex approximation, geometric programming, and the Dinkelbach algorithm to obtain a point satisfying the Karush-Kuhn-Tucker (KKT) conditions. Furthermore, we develop a low-complexity sub-optimal scheme through binary variable relaxation to obtain a solution, and the convergence point satisfies the KKT conditions of the relaxed non-concave problem. Simulation results demonstrate the convergence and effectiveness of our proposed schemes.

Application of particle swarm optimisation for coverage estimation in software testing

A Markov approach for test case generation and code coverage estimation using particle swarm optimisation technique is proposed. Initially, the DD-graph is taken from control flow graph of the software code by joining decision to decision. The DD-graph identifies the sequences of independent paths using c-uses and p-uses based on set theory approach and compared to cyclomatic complexity. Automatic test cases are generated and the nature of test cases are integer, float and Boolean variables. Using this initial test suite, the code coverage summary is generated using gcov code coverage analysis tool, the branch probability percentage is considered as TPM values with respect to each branch in the DD-graph. Path coverage is used as a fitness function which is the product of node coverage and TPM values. Iterate this algorithm until reaches 100% code coverage among each independent test path. The randomness of the proposed approach is compared to genetic algorithm.

Application of particle swarm optimisation for coverage estimation in software testing

A Markov approach for test case generation and code coverage estimation using particle swarm optimisation technique is proposed. Initially, the DD-graph is taken from control flow graph of the software code by joining decision to decision. The DD-graph identifies the sequences of independent paths using c-uses and p-uses based on set theory approach and compared to cyclomatic complexity. Automatic test cases are generated and the nature of test cases are integer, float and Boolean variables. Using this initial test suite, the code coverage summary is generated using gcov code coverage analysis tool, the branch probability percentage is considered as TPM values with respect to each branch in the DD-graph. Path coverage is used as a fitness function which is the product of node coverage and TPM values. Iterate this algorithm until reaches 100% code coverage among each independent test path. The randomness of the proposed approach is compared to genetic algorithm.

Trade-offs analysis of sustainability dimensions using integer-valued data envelopment analysis

Conducting an in-depth exploration of trade-offs between sustainability aspects is a notable matter of taking decisions. Furthermore, there are many real world investigations that trade-offs and sustainability should be dealt with in the presence of desirable and undesirable materials while some of them accept integer amounts. Therefore, this study addresses trade-offs of sustainability dimensions when undesirable and integer-valued measures are presented. For this purpose, approaches based upon data envelopment analysis (DEA) are proposed. To explain, DEA models are introduced to calculate individual and group marginal rates of substitution and also directional marginal rates of substitution when integer and undesirable variables are observed. These procedures are applied to calculate trade-offs between different sustainability dimensions, including economic, environmental and social ones. The applications of ports and industrial parks are provided to clarify the approaches appeared in this study. The results derived from the proposed strategies show the usefulness and validity of them.

Beam Selection and Power Allocation for Massive Connectivity in Millimeter Wave NOMA Systems

Massive connectivity is one of the killer use cases for fifth generation (5G) wireless systems over the millimeter wave (mmWave) band. Due to the sparse nature of mmWave channel, several users may choose the same beam as the strongest one, leading to severe intra-beam interference (intra-BI). Although intra-BI can be mitigated through digital precoding, the required number of RF chains for fulfilling massive connectivity becomes large. By applying non-orthogonal multiple access (NOMA) and properly allocating transmit power to each beam and user, the sum rate can be maximized using a smaller number of RF chains. However, the power allocated to each user depends on that allocated to other users within the same beam. Since the joint beam selection and power allocation problem contains a non-convex objective function with a large set of coupled and mixed integer variables, direct solutions may not exist particularly when both the number of users and antennas are large. In this work, we decompose the aforementioned problem into two subproblems, where the beam selection sub-problem is first solved under an equal power allocation. The outcome of beam selection is then used as the input to the power allocation sub-problem. For both subproblems, we develop numerous efficient algorithms to find the active beam set and the user power allocation, respectively. The notions of embedded methods from machine learning and that of intelligent searching from metaheuristics are adopted in our work as the key ingredient for algorithm constructions. Numerical results demonstrate that in the region of large user population, the proposed beam selection and power allocation algorithms can effectively improve the sum rate using a less number of RF chains in comparison with some existing solutions.

Fuzzy reliability redundancy optimisation with signed distance method for defuzzification using genetic algorithm

Consideration of impreciseness is more realistic for modelling of physical phenomena. This impreciseness can be considered in several ways like, interval/stochastic/fuzzy or mixture of these. In this work, we have taken for optimising of the system reliability of a redundancy allocation problem formulated from a complex network system with imprecise parameters in the form of trapezoidal fuzzy numbers (TrFN). The signed distance method has been used to defuzzify the fuzzy values. Then big-M penalty technique is used to transform the problem to unconstrained optimisation problem. To solve these problems, we have implemented the real coded elitist genetic algorithm (RCEGA) for integer variables with tournament selection, intermediate crossover and one neighbourhood mutation. For illustration, the five link bridge network system has been solved and the results have been presented.

Fault Diagnosis of Backward Conflict-Free Petri Nets by Generalized Markings

This paper addresses the fault diagnosis problem in discrete event systems using backward conflict-free Petri nets, where two different transitions have no common output places. The notion of generalized markings is employed, where the number of tokens in a place can be negative. Based on the special structures of backward conflict-free Petri nets and the generalized marking corresponding to an observed transition sequence, an integer linear programming (ILP) problem is constructed where the numbers of unknown integer variables and constraints are independent of the length of the observed transition sequence. A diagnosis algorithm is proposed by associating two different objective functions with the ILP problem and solving it. The computational efficiency of the proposed algorithm depends on the net structure only, regardless of the length of an observed transition sequence.

Normally supportive sublattices of crystallographic space groups.

The tabulation of normal subgroups of 3D crystallographic space groups that are themselves 3D crystallographic space groups (csg's) is an ambitious goal, but would have a variety of applications. For convenience, such subgroups are referred to as `csg-normal' while normal subgroups of the crystallographic point group (cpg) of a crystallographic space group are referred to as `cpg-normal'. The point group of a csg-normal subgroup must be a cpg-normal subgroup. The present work takes a significant step towards that goal by tabulating the translational subgroups (a.k.a. sublattices) that are capable of supporting csg-normal subgroups. Two necessary conditions are identified on the relative sublattice basis that must be met in order for the sublattice to support csg-normal subgroups: one depends on the operations of the point group of the space group, while the other depends on the operations of the cpg-normal subgroup. Sublattices that meet these conditions are referred to as `normally supportive'. For each cpg-normal subgroup (excluding the identity subgroup 1) of each of the arithmetic crystal classes of 3D space groups, all of the normally supportive sublattices have been tabulated in symbolic form, such that most of the entries in the table contain one or more integer variables of infinite range; thus it could be more accurately described as a table of the infinite families of normally supportive sublattices. For a given pair of cpg-normal subgroup and normally supportive sublattice, csg-normal subgroups of the space groups of the parent arithmetic crystal class can be constructed via group extension, though in general such a pair does not guarantee the existence of a corresponding csg-normal subgroup.

OPTIMASI PENJADWALAN WAKTU KERJA MENGGUNAKAN INTEGER PROGRAMMING

Scheduling workers is one of the problems faced by every company. The regulations set by the company, the availability of the number of workers, and the division of labor are the determining factors in the scheduling system. This worker scheduling problem can be modeled as an Integer Programming problem. Integer Programming is an optimization technique with linear objective functions, linear constraint functions, and integer variables. This paper discusses the formulation of worker scheduling problems in the form of Integer Programming with workers in companies engaged in the production of Crumb Rubber with the objective function of minimizing the number of workers employed. The next model is implemented using the help of LINGO 11.0 software. The implementation results show that the model is able to produce optimal employee schedules.

A three-dimensional group search optimization approach for simultaneous planning of distributed generation units and distribution network reconfiguration

This paper deals with the simultaneous distributed generation (DG) planning and distribution network reconfiguration (SDGNR) issue. The problem is formulized as an optimization model which includes three types of variables, i.e. DGs location as the integer variables, DGs operating point as the continuous ones and switches open\\close state as the binary variables. A new approach entitled three-dimensional group search optimization (3D-GSO) method is also introduced to cope with such a problem. The proposed method is a general optimization scheme applicable to all types of optimization problems which deal with an integer, continuous, and binary variables at the same time. The revised approach is successfully applied to the SDGNR problem with the objective of total loss reduction in power distribution systems. Power flow criteria, as well as operation constraints, are all together accommodated in the process of optimization. Five different scenarios at three load levels are also considered to cover all possible conditions. The validity of the proposed 3D-GSO approach in handling SDGNR problem is assured through comprehensive simulation studies on 33-bus and 69-bus test systems.

Fast feasibility check of the multi-material vertical alignment problem in road design

When building a road, it is critical to select a vertical alignment which ensures design and safety constraints. Finding such a vertical alignment is not necessarily a feasible problem, and the models describing it generally involve a large number of variables and constraints. This paper is dedicated to rapidly proving the feasibility or the infeasibility of a Mixed Integer Linear Program (MILP) modeling the vertical alignment problem. To do so, we take advantage of the particular structure of the MILP, and we prove that only a few of the MILP’s constraints determine the feasibility of the problem. In addition, we propose a method to build a feasible solution to the MILP that does not involve integer variables. This enables time saving to proving the feasibility of the vertical alignment problem and to find a feasible vertical alignment, as emphasized by numerical results. It is on average 75 times faster to prove the feasibility and 10 times faster to build a feasible solution.

Efficient sensitivity based approach for AC transmission expansion planning

AC transmission expansion planning (TEP) is a complex problem to solve due to its large scale, mixed-integer, non-linear and non-convex characteristics. This paper presents an efficient sensitivity based approach which employs the primal-dual interior point method (PDIPM), for tackling integer variables in AC TEP. PDIPM's Hessian is used to evaluate the sensitivities of the objective function with respect to the line candidates, which are then used to find a feasible AC TEP solution. The proposed approach solves this non-convex mixed integer non-linear programming problem, without resorting to any use of relaxations/approximations in the AC TEP model. Also, the need and superiority of the AC TEP over the conventional DC TEP is established by comparing both using the proposed approach. Solutions of AC TEP with reactive power planning and an integrated multistage AC TEP are also obtained to demonstrate the performance of the proposed approach. Superior solution quality and lesser computation time of the proposed approach are confirmed by comparing the same with that of the branch and bound/cut approach, constructive heuristic algorithm, merit function-based approach, conic programming and particle swarm optimisation. The studies performed for the various test cases prove the efficacy and computational efficiency of the proposed approach.

Multiaircraft Optimal 4D Trajectory Planning Using Logical Constraints

This paper studies the trajectory planning problem for multiple aircraft with logical constraints in disjunctive form which arise in modeling passage through waypoints, distance-based and time-based separation constraints, decision-making processes, conflict resolution policies, no-fly zones, or obstacle or storm avoidance. Enforcing separation between aircraft, passage through waypoints, and obstacle avoidance is especially demanding in terms of modeling efforts. Indeed, in general, separation constraints require the introduction of auxiliary integer variables in the model; for passage constraints, a multiphase optimal control approach is used, and for obstacle avoidance constraints, geometric approximations of the obstacles are introduced. Multiple phases increase model complexity, and the presence of integer variables in the model has the drawback of combinatorial complexity of the corresponding mixed-integer optimal control problem. In this paper, an embedding approach is employed to transform logical constraints in disjunctive form into inequality and equality constraints which involve only continuous auxiliary variables. In this way, the optimal control problem with logical constraints is converted into a smooth optimal control problem which is solved using traditional techniques, thereby reducing the computational complexity of finding the solution. The effectiveness of the approach is demonstrated through several numerical experiments by computing the optimal trajectories of multiple aircraft in converging and intersecting arrival routes with time-based separation constraints, distance-based separation constraints, and operational constraints.

Dispatch of reactive power compensators in transmission grids

The phase-out of conventional power plants reduces the reactive power reserve from synchronous generators in transmission grids. Operational concepts need to be adapted to integrate reactive power compensators such as STATCOMs into voltage control schemes. This paper presents a novel two-stage optimization method for determining reactive power schedules, distinguishing manual and automatized control variables. The first stage optimizes the reactive power dispatch iteratively for the considered period using successive linear programming based on continuous variables due to relaxed integrality conditions. Afterwards, an embedded genetic algorithm re-discretizes integer variables. The second stage determines pareto-optimal reactive power schedules based on grid losses and switching frequency. Therefore, points of time are modeled as nodes and loss energies between two time-coupled points of time are modeled as arcs in a directed acyclic graph. The investigated scenario indicates that a high number of switching actions for passive compensators, such as shunt capacitors or reactors, is not necessary. Furthermore, it becomes apparent that the automated control of flexible devices like STATCOMs enables a reduction of active power losses.

P.504 Variation in cannabinoid 1 receptor and GABRA6 genes in interaction with recent stress influences distinct suicide risk-related phenotypes

In this work, an efficient formulation for the aggregation of multiple identical generating units for the mixed integer linear programming unit commitment problem is presented. To this end, as already done in literature identical units are clustered with a single integer variable defining the number of currently committed units. To correctly model the feasible aggregated ramp speed, a shifting generation level method with low computational effort is introduced, which is the main contribution of this work. In contrast to the reference binary formulation, the number of integer and continuous variables and constraints for the integer formulation does not scale with the number of aggregated units. Thus, depending on the UC problem, the computational effort can be drastically decreased, which is shown in test cases.