Continuous Decision Variables Research Articles

A mixed-integer non-linear problem formulation for miscible WAG injection

The paper presents a formulation to evaluate optimal injection scenarios or production strategies in an oil reservoir. The decisions variables involved in such strategy formulations are a mix of both continuous and integer nature variables. The time-line of the operation or the recovery strategy addressed in this paper is divided into three action periods: cyclic injection (a single or multiple/alternating cycles), continuous injection, and depletion. Two reservoir models description used in this study: a pattern sector model with a single producer and single injectors based on the SPE 5th comparative project miscible WAG (Killough and Kossack, 1987), and a multi-well case with 5 producers and 8 injectors with a more complex reservoir geology that is an extension of the SPE 5th comparative project. The study addresses two independent production modes, each with a different set of production constraints, artificial-lift from the start to the end of the timeline and natural-flow from the start to the end of the study period. For the two simplest injection strategies: continuous gas or continuous water injection, the only control variable is tubing-head pressure in the injection wells. For more complex injection strategies more decision variables are required including tubing-head pressures, injection volume, and time to change the injection strategy. The 8 studied strategies with their relevant decision variables are stated in a case matrix table in the paper. Each of the 8 strategies is studies with either natural flow mode or artificial lift mode. The objective function is the maximum of a Net Present Value (NPV) formula using revenue with a time escalation sales price, and a simple Operating Expenditures (OPEX) and Capital Expenditure (CAPEX). The optimization function is simple but captures the important trends in comparing oil recovery strategies. The optimization program is run for each strategy stated in the case matrix table, searching for the optimum by varying the continuous decision variables of that particular strategy. To ensure a proper location of the global optimum of each strategy, the optimization search employs several starting values of the continuous variables, with the number of starting values increasing with increased strategy complexity. The starting values are randomly generated. The overall optimum operation strategy is the one with the highest calculated NPV. The entire optimization study is conducted in a semi-automated manner. A proprietary program runs and manages the integration of the reservoir simulator, the NPV model and the optimizer. This is done automatically for each investigated strategy. The proposed methodology is applicable to any oil reservoir where both surface water and gas injection is available. This work contributes to the literature by establishing a general mixed-integer problem formulation for water and gas injection and providing an efficient heuristic method for solving the problem.

A Weighted Least Squares Algorithm for Optimal PMU Placement

This letter presents a new method for optimal placement of phasor measurement units (PMU) for complete power system observability. The optimal PMU placement (OPP) problem is formulated as a quadratic minimization problem with continuous decision variables subject to nonlinear observability constraints. The optimal solution is obtained by an unconstrained nonlinear weighted least squares (WLS) approach. Simulations are conducted on different IEEE systems (14-bus, 30-bus, 57-bus, and 118-bus) to prove the validity of the proposed algorithm.

An improved ant colony optimization method and its application for the thermodynamic modeling of phase equilibrium

Ant colony optimization (ACO) is a stochastic optimization method that emulates the indirect communication among the individuals of an ant colony. This multi-agent strategy has been used to solve optimization problems from engineering applications, especially for combinatorial problems. Although ACO seems to be effective for global optimization, only few ACO algorithms have been reported for solving engineering applications problems with continuous decision variables. In this study, a new continuous ACO algorithm, with feasible region selection, has been implemented and applied to perform thermodynamic calculations related to the modeling of phase equilibrium. In particular, the numerical capabilities of this ACO algorithm have been tested in the global optimization of thermodynamic functions from parameter estimation, phase equilibrium and phase stability problems. These thermodynamic calculations are relevant for chemical engineering process simulators. However, they are classed as challenging nonconvex optimization problems with continuous decision variables. The applicability and effectiveness of the proposed ACO algorithm, namely ACOFRS, have been studied using a challenging set of thermodynamic benchmark problems with both liquid–liquid and vapor–liquid equilibrium. Results show that the proposed ACOFRS is an alternative method for performing global optimization in phase equilibrium calculations of multicomponent systems. In particular, ACOFRS is more robust for solving VLE parameter estimation problems and it outperformed other stochastic optimization methods such as Particle Swarm Optimization, Differential Evolution and Genetic Algorithms.

Improved Harmony Search Algorithm for Solving Optimization Problem with Mixed Discrete Variables

The classical harmony search algorithm (HSA) can only be used to solve the unconstrained optimization problems with continuous decision variables. Therefore, the classical HSA is not suitable for solving an engineering optimization problem with mixed discrete variables. In order to improve the classical HSA, an engineering method for dealing with mixed discrete decision variables is introduced and an exact non-differentiable penalty function is used to transform the constrained optimization design model into an unconstrained mathematical model. Based on above improvements, a program of improved HSA is designed and it can be used for solving the constrained optimization design problems with continuous variables, integer variables and non-equidistant discrete variables. Finally, an optimization design example of single-stage cylindrical-gear reducer with mixed-discrete variables is given. The example shows that the designed program runs steadily and the proposed method is effective in engineering design.

Optimizing Bi-Objective Imperfect Preventive Maintenance Model for Multi-Component System Using a Proposed Improved PSO Algorithm

This study aims to establish a bi-objective imperfect preventive maintenance (BOIPM) model in which the total maintenance cost and the mean system reliability are optimized by determining the maintenance periods and maintenance activities simultaneously. To efficiently solve the established BOIPM model, this study proposes an improved particle swarm optimization (IPSO) algorithm. The IPSO extends the practicability of the conventional PSO originally designed to solve an optimization problem with continuous decision variables. Furthermore, time-varying mechanisms associated with search parameters of the PSO are utilized to enhance the particles search capability. An adjustment mechanism addressing the issue of particles falling into the infeasible area is constructed to enhance the exploring ability of the IPSO. A case verifies the effectiveness of the proposed approach.

Mixed Transportation Network Design under a Sustainable Development Perspective

A mixed transportation network design problem considering sustainable development was studied in this paper. Based on the discretization of continuous link-grade decision variables, a bilevel programming model was proposed to describe the problem, in which sustainability factors, including vehicle exhaust emissions, land-use scale, link load, and financial budget, are considered. The objective of the model is to minimize the total amount of resources exploited under the premise of meeting all the construction goals. A heuristic algorithm, which combined the simulated annealing and path-based gradient projection algorithm, was developed to solve the model. The numerical example shows that the transportation network optimized with the method above not only significantly alleviates the congestion on the link, but also reduces vehicle exhaust emissions within the network by up to 41.56%.

A Simulated Annealing Algorithm for Noisy Multiobjective Optimization

ABSTRACTThis paper presents a new simulated annealing (SA) algorithm for noisy multiobjective optimization with continuous decision variables. A novel feature of the algorithm in the context of SA is that the performance of a candidate solution is determined by estimating the probabilities that the candidate is dominated by the current non‐dominated solutions. The sum of these probabilities provides a scalar performance measure that is used to determine the acceptance of the candidate as the current solution and whether the candidate is inserted into the non‐dominated set. The second novel feature of the algorithm is the technique utilized for generating candidate solutions. Empirical probability distributions for sampling the new values of the decision variables are constructed on the basis of the values of the variables in the current non‐dominated set. Thus, the information contained by the non‐dominated set is utilized to improve the quality of the generated candidates, whereas this information is ignored in the existing multiobjective SA algorithms. The proposed algorithm is compared with a reference state‐of‐the‐art evolutionary algorithm as well as two other SA algorithms in numerical experiments involving 16 problems from commonly applied test suites. The proposed algorithm performs as good or better compared with the reference algorithms in majority of the experiments and therefore represents a promising solution method for noisy multiobjective optimization problems. Copyright © 2012 John Wiley & Sons, Ltd.

REINFORCEMENT LEARNING FOR DECISION-MAKING IN A BUSINESS SIMULATOR

Business simulators are powerful tools for both supporting the decision-making process of business managers as well as for business education. An example is SIMBA (SIMulator for Business Administration), a powerful simulator which is currently used as a web-based platform for business education in different institutions. In this paper, we propose the application of reinforcement learning (RL) for the creation of intelligent agents that can manage virtual companies in SIMBA. This application is not trivial, given the particular intrinsic characteristics of SIMBA: it is a generalized domain where hundreds of parameters modify the domain behavior; it is a multi-agent domain where both cooperation and competition among different agents can coexist; it is required to set dozens of continuous decision variables for a given business decision, which is made only after the study of hundreds of continuous variables. We will demonstrate empirically that all these challenges can be overcome through the use of RL, showing results for different learning scenarios.

A Framework for Solving Hybrid Influence Diagrams Containing Deterministic Conditional Distributions

We describe a framework and an algorithm for approximately solving a class of hybrid influence diagrams (IDs) containing discrete and continuous chance variables, discrete and continuous decision variables, and deterministic conditional distributions for chance variables. A conditional distribution for a chance variable is said to be deterministic if its variances, for each state of its parents, are all zeroes. The solution algorithm is an extension of Shenoy's fusion algorithm for discrete influence diagrams. To mitigate the integration and optimization problems associated with solving hybrid IDs, we propose using mixture of polynomials approximations of conditional probability density and utility functions and piecewise linear approximations of nonlinear deterministic conditional distributions for continuous chance variables. The class of hybrid IDs that can be solved by our framework are those that do not involve divisions. The framework and algorithm are illustrated by solving two small examples of hybrid IDs.

Optimal composition of number and size of machines in a multi-stage make-to-order system with due dates

We develop an optimisation model to minimise costs for work-in-process (WIP), finished goods inventory (FGI), backorders, and capacity for a multi-stage production system applying a work-ahead window work release policy. Customers arrive to a make-to-order production system and have stochastic due dates. The parameters to optimise are the capacity invested at each stage, consisting of the number of machines and the processing rate (defined by a set of possible processing rates), as well as the work-ahead window. An optimality condition is provided showing when it is optimal to invest into a single machine at each stage if processing rates are continuous decision variables. The optimality of increasing capacity towards the customer end of the production line is proven under certain conditions. For a special case consisting of two stages with M/M/s queues with exponentially distributed customer required lead time, explicit expressions for WIP, FGI and backorders are developed. A set of numerical examples illustrates the influence of predefined processing rates on the optimal number and selection of machines. A simple solution heuristic for the problem with a predefined set of processing rates is proposed and its performance is shown in a numerical example. Additionally, the influence of uncertain input rates into the production system is discussed.

Reactive DWC leading the way to FAME and fortune

Reactive separation processes were recently proposed for the synthesis of fatty acid methyl esters (FAMEs), most of them making use of solid catalysts thus eliminating all conventional catalyst-related operations, improving process efficiency and reducing energy requirements. Such integrated systems require a stoichiometric reactants ratio in order to achieve complete conversion and high purity products. However, maintaining this ratio can be very difficult in practice, especially when the fatty acids feed composition is not constant in time.This study proposes a novel biodiesel process based on a reactive dividing-wall column (R-DWC) that allows the use of only ∼15% excess of methanol to completely convert the fatty acids feedstock. FAME are produced as pure bottom product, water as side stream, while the methanol excess is recovered as top distillate and recycled. The design is a challenging global optimization problem with discrete and continuous decision variables. The optimal setup was established by using simulated annealing as optimization method implemented in Matlab, and coupled with rigorous simulations carried out in Aspen Plus. Along with the FAME production, the novel design alternatives allow a fortune to be saved by reducing the energy requirements with over 25% and by using less equipment units than conventional processes.

Rebirthing genetic algorithm for storm sewer network design

Application of standard binary coded genetic algorithms for the solution of problems with continuous design variables requires discretization of the continuous decision variables. Coarse discretization of the design variables could adversely affect the final solution, while finer discretization would increasingly enlarge the scale of the problem, leading to higher computation cost. A rebirthing procedure is used in this paper as a remedy for the problem just outlined. The method is based on the idea of limiting the originally wide search space to a smaller one once a locally converged solution is obtained. The smaller search space is designed to contain the locally optimum solution at its center. The resulting search space is refined and a completely new search is conducted to find a better solution. The procedure is continued until no refinement is necessary or no improvement could be made by further refinement. The method is applied to a benchmark problem of a storm water network design, and the results are compared with those of the existing method. The method is shown to be very effective, efficient and insensitive to the population size of the genetic search and the search space size of the optimization problem.

Managing water resources system in a mixed inexact environment using superiority and inferiority measures

A superiority–inferiority-based fuzzy-stochastic integer programming (SI-FSIP) method is developed for water resources management under uncertainty. In the SI-FSIP method, techniques of fuzzy mathematical programming with the superiority and inferiority measures and joint chance-constrained programming are integrated into an inexact mixed integer linear programming framework. The SI-FSIP improves upon conventional inexact fuzzy programming by directly reflecting the relationships among fuzzy coefficients in both the objective function and constraints with a high computational efficiency, and by comprehensively examining the risk of violating joint probabilistic constraints. The developed method is applied to a case study of water resources planning and flood control within a multi-stream and multi-reservoir context, where several studied cases (including policy scenarios) associated with different joint and individual probabilities are investigated. Reasonable solutions including binary and continuous decision variables are generated for identifying optimal strategies for water allocation, flood diversion and capacity expansion; the tradeoffs between total benefit and system-disruption risk are also analyzed. As the first attempt for planning such a water-resources system through the SI-FSIP method, it has potential to be applied to many other environmental management problems.

A Memetic Algorithm for Global Optimization in Chemical Process Synthesis Problems

Design optimization problems in chemical engineering and in many other engineering domains are characterized by the presence of a large number of discrete and continuous decision variables, complex nonlinear models that restrict the search space, nonlinear cost functions, and the presence of many local optima. The classical approach to such problems are mixed integer nonlinear program solvers that work on a superstructure formulation which explicitly represents all design alternatives. The structural decisions lead to a large number of discrete variables and an exponential increase in the computational effort. The mathematical programming (MP) methods which are usually employed to solve the continuous subproblems that arise by fixing the discrete variables provide only one local optimum which depends strongly on the initialization. Thus standard methods may not find the global optimum despite long computation times. In this contribution we introduce a memetic algorithm (MA) for the global optimization of a computational demanding real-world design problem from the chemical engineering domain. The MA overcomes the problem of getting stuck in local optima by the use of an evolution strategy (ES) which addresses the global optimization of the design decisions, whereas a robust MP solver is used to handle complex nonlinear constraints as well as to improve the individuals of the ES by performing a local search in continuous sub-spaces in an integrated fashion. The MA is discussed in detail, the novel decomposition of the problem class at hand is analyzed and the MA is tested for the example of the optimal design of a reactive distillation column with several thousand decision variables. The MA is the only algorithm that finds the global solution in reasonable computation times. The introduction of structural decisions and additional constraints and discontinuous penalty terms lead only to a moderate increase in the computational effort which demonstrates the potential of this class of memetic algorithms in real-world design optimization problems.

Management of Uncertain Information for Environmental Systems Using a Multistage Fuzzy-Stochastic Programming Model with Soft Constraints

In this study, a multistage fuzzy-stochastic programming model with soft constraints (FSPM-SC) is developed for managing environmental systems associated with uncertain information. The developed model can deal with uncertainties expressed as probability distributions and fuzzy sets; it can also reflect the dynamics in terms of decisions for waste-flow allocation and capacity expansion, through transactions at discrete points of a complete scenario set over a multistage context. The results indicate that solutions have been generated for binary and continuous decision variables under fuzzy and random information. They can be used for generating waste-flow-allocation pattern and facility-capacity-expansion scheme with a cost-effective manner. Sensitivity analyses are also conducted to demonstrate that the violation of waste-disposal-demand constraint has significant effect on reducing system cost.

Retrospective optimization of mixed-integer stochastic systems using dynamic simplex linear interpolation

We propose a family of retrospective optimization (RO) algorithms for optimizing stochastic systems with both integer and continuous decision variables. The algorithms are continuous search procedures embedded in a RO framework using dynamic simplex interpolation (RODSI). By decreasing dimensions (corresponding to the continuous variables) of simplex, the retrospective solutions become closer to an optimizer of the objective function. We present convergence results of RODSI algorithms for stochastic “convex” systems. Numerical results show that a simple implementation of RODSI algorithms significantly outperforms some random search algorithms such as Genetic Algorithm (GA) and Particle Swarm Optimization (PSO).

The Correlated Knowledge Gradient for Simulation Optimization of Continuous Parameters using Gaussian Process Regression

We extend the concept of the correlated knowledge-gradient policy for the ranking and selection of a finite set of alternatives to the case of continuous decision variables. We propose an approximate knowledge gradient for problems with continuous decision variables in the context of a Gaussian process regression model in a Bayesian setting, along with an algorithm to maximize the approximate knowledge gradient. In the problem class considered, we use the knowledge gradient for continuous parameters to sequentially choose where to sample an expensive noisy function in order to find the maximum quickly. We show that the knowledge gradient for continuous decisions is a generalization of the efficient global optimization algorithm proposed in [D. R. Jones, M. Schonlau and W. J. Welch, J. Global Optim., 13 (1998), pp. 455–492].

Pure 0-1 programming approaches to wireless network design

This is a summary of the author’s Ph.D. Thesis supervised by Carlo Mannino and defended on January 18th, 2010 at Sapienza Universita di Roma. The thesis is written in English and is available from the author upon request to d.andreagiovanni@zib.de. This work deals with the Wireless Network Design Problem (WND), i.e. the problem of configuring a set of wireless transmitters in order to provide service coverage to the maximum number of receivers located in a target area. The thesis was awarded the 2010 INFORMS Doctoral Dissertation Award for Operations Research in Telecommunications, the most prestigious prize awarded in the USA to recognize outstanding scholarly achievements of young people in the field. A key task of the WND is establishing the power emitted by each transmitter. Classical optimization models represent power emissions as continuous decision variables. This modeling choice typically leads to Mixed-Integer Linear Programs with (very) ill-conditioned coefficient matrices and entails the introduction of the notorious big-M coefficients to represent disjunctive constraints that model coverage conditions. The resulting linear relaxations are very weak and the solutions returned by state-of-the-art MIP solvers are typically far from the optimum and may contain errors. The first aim of this thesis is to overcome these difficulties by introducing two innovative pure 0-1 Linear Programming formulations for the WND: the Power-Indexed Formulation (PI) and the Dyadic Formulation (DY). Both models are based on substituting the continuous power variables with the linear combination of a set of boolean variables multiplied by suitable power coefficients. Power discretization better fits a common approach to the representation of power that we have observed among practitioners. However, the only discretization is not sufficient to tackle all the numerical problems (see F. D’Andreagiovanni et al. “GUB Covers and Power-Indexed formulations for Wireless Network Design”, DIS Tech. Rep. (2) 14, Sapienza Universita di Roma, 2010): the resulting knapsack constraints still contain the coefficients that are sources of numerical issues and we thus replace them with families of (strong) valid inequalities. In particular, in the case of the PI, we define a family of lifted GUB cover inequalities, while in the case of the DY we define

Parameter Estimation of the Nonlinear Muskingum Model Using Parameter-Setting-Free Harmony Search

Although phenomenon-mimicking algorithms, such as genetic algorithms, particle swarm optimization, and harmony search, have overcome the disadvantages of mathematical algorithms, such as the nonlinear least-squares method, segmented least-squares method, Lagrange multiplier method, a hybrid of pattern search and local search, and the Broyden-Fletcher-Goldfarb-Shanno technique, the algorithms have an inherent shortcoming. They require a tedious and skillful parameter-setting process for the algorithm parameters, such as the crossover rate, mutation rate, acceleration coefficients, harmony memory considering rate, and pitch-adjusting rate. Thus, this study proposes a novel parameter-setting-free technique interfaced with a harmony search algorithm and applies it to the parameter estimation of the nonlinear Muskingum model, which is an optimization problem with continuous decision variables. Results show that the proposed technique found good model parameter values while outperforming a classical harmony sea...

A revised particle swarm optimization based discrete Lagrange multipliers method for nonlinear programming problems

In this paper, a new algorithm for solving constrained nonlinear programming problems is presented. The basis of our proposed algorithm is none other than the necessary and sufficient conditions that one deals within a discrete constrained local optimum in the context of the discrete Lagrange multipliers theory. We adopt a revised particle swarm optimization algorithm and extend it toward solving nonlinear programming problems with continuous decision variables. To measure the merits of our algorithm, we provide numerical experiments for several renowned benchmark problems and compare the outcome against the best results reported in the literature. The empirical assessments demonstrate that our algorithm is efficient and robust.