...com/locate/apthermeng Optimization of heat exchanger network Mofid Gorji-Bandpy, Hossein Yahyazadeh-Jelodar, Mohammadtaghi Khalili* Noshirvani University of Technology, P.O. Box 484, Babol, Iran a r t i c l e i n f o Article history: Received 6 September 2010 Accepted 26 October 2010 Available online 2 November 2010 Keywords: Heat exchanger network (HEN) Optimization Genetic algorithm Pinch Analysis Method Mathematical Optimization Method Sequential Quadratic Programming (SQP) a b s t r a c t In this paper, a new method is presented for optimization of heat exchanger networks making use of genetic algorithm and Sequential Quadratic Programming. The optimization problem is solved in the following two levels: 1- Structure of the optimized network is distinguished through genetic algorithm, and 2- The optimized thermal load of exchangers is determined through Sequential Quadratic Programming. Genetic algorithm uses these values for the determination of the fitness. For assuring the authenticity of the newly presented method, two standard heat exchanger networks are solved numerically. For representing the efficiency and applicability of this method for the industrial issues, an actual industrial optimization problem i.e. Aromatic Unit of Bandar Imam Petrochemistry in Iran is verified. The results indicate that the proposed multistage optimization algorithm of heat exchanger networks is better in all cases than those obtained using traditional optimization methods such as Pinch...
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...TABLES ...................................................................................................... vii LIST OF FIGURES ................................................................................................... viii 1. Introduction 1 1.1. Preventive Maintenance and Replacement Scheduling .................................. 1 1.2. Research Contributions .................................................................................... 2 1.3. Outline .............................................................................................................. 3 2. Literature Review 4 2.1. Introduction ...................................................................................................... 4 2.2. Optimization Models ........................................................................................ 4 2.2.1. Exact Algorithms...
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...An Iterated Dynasearch Algorithm for the Single-Machine Total Weighted Tardiness Scheduling Problem Faculty of Mathematical Studies, University of Southampton, Southampton, SO17 1BJ, UK Faculty of Mathematical Studies, University of Southampton, Southampton, SO17 1BJ, UK Department of Decision and Information Sciences, Rotterdam School of Management, Erasmus University, P.O. Box 1738, 3000 DR Rotterdam, The Netherlands Richard.Congram@paconsulting.com • C.N.Potts@maths.soton.ac.uk • S.Velde@fac.fbk.eur.nl Richard K. Congram • Chris N. Potts • Steef L. van de Velde T his paper introduces a new neighborhood search technique, called dynasearch, that uses dynamic programming to search an exponential size neighborhood in polynomial time. While traditional local search algorithms make a single move at each iteration, dynasearch allows a series of moves to be performed. The aim is for the lookahead capabilities of dynasearch to prevent the search from being attracted to poor local optima. We evaluate dynasearch by applying it to the problem of scheduling jobs on a single machine to minimize the total weighted tardiness of the jobs. Dynasearch is more effective than traditional first-improve or best-improve descent in our computational tests. Furthermore, this superiority is much greater for starting solutions close to previous local minima. Computational results also show that an iterated dynasearch algorithm in which descents are performed a few random moves away from previous...
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...Differential Evolution in Constrained Numerical Optimization. An Empirical Study Efr´n Mezura-Montesa,, Mariana Edith Miranda-Varelab , Rub´ del Carmen e ı c G´mez-Ram´n o o Laboratorio Nacional de Inform´tica Avanzada (LANIA A.C.) R´bsamen 80, Centro, a e Xalapa, Veracruz, 91000, MEXICO. b Universidad del Istmo, Campus Ixtepec. Ciudad Universitaria s/n, Cd. Ixtepec, Oaxaca, 70110, MEXICO c Universidad del Carmen. C. 56 #4, Ciudad del Carmen, Campeche, 24180, MEXICO a Abstract Motivated by the recent success of diverse approaches based on Differential Evolution (DE) to solve constrained numerical optimization problems, in this paper, the performance of this novel evolutionary algorithm is evaluated. Three experiments are designed to study the behavior of different DE variants on a set of benchmark problems by using different performance measures proposed in the specialized literature. The first experiment analyzes the behavior of four DE variants in 24 test functions considering dimensionality and the type of constraints of the problem. The second experiment presents a more in-depth analysis on two DE variants by varying two parameters (the scale factor F and the population size NP ), which control the convergence of the algorithm. From the results obtained, a simple but competitive combination of two DE variants is proposed and compared against state-of-the-art DE-based algorithms for constrained optimization in the third experiment. The study in this paper shows (1) important...
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...When LINGO finds a solution to a linear optimization model, it is the definitive best solution¾ we say it is the global optimum. A globally optimal solution is a feasible solution with an objective value that is as good or better than all other feasible solutions to the model. The ability to obtain a globally optimal solution is attributable to certain properties of linear models. This is not the case for nonlinear optimization. Nonlinear optimization models may have several solutions that are local optimums. All nonlinear solvers converge to a locally optimal point. That is, a solution for which no better feasible solutions can be found in the immediate neighborhood of the given solution. Although better solutions can't be found in the immediate neighborhood of the local optimum, additional local optimums may exist some distance away from the current solution. These additional locally optimal points may have objective values substantially better than the solver's current local optimum. Thus, when a nonlinear model is solved, we say the solution is merely a local optimum, and the user must be aware other local optimums may, or may not, exist with better objective values. Consider the following small nonlinear model involving the highly nonlinear cosine function: MIN = X * @COS( 3.1416 * X); X < 6; (Note, graph not shown. Go to Lingo and click Help Topics. Select the “Index” Window and open Local Optima topic) The following graph shows a plot of the...
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...Assignment 1–Advanced Operations Research - MATH 3010 Posted 23 August 2014 Due date: 19 September 2014, by 5pm In all the statements below, the notation, as well as references to page numbers, equations, etc, are as in the textbook Primal-dual interior-point methods, by Wright, Stephen, which is available online for UniSA staff and students. All relevant chapters of the textbook are also available in the webpage of the course. For solving this assignment, you need to read the handwritten Lecture Notes posted in the web and the material in the book up to Chapter 4, page 70. Question 1 (2+2+3+3+3+3=16 points) Fix A ∈ Rm×n , b ∈ Rm , and c ∈ Rn . (a) Write down the KKT conditions for the following problem, on the variable x ∈ Rn : min cT x Ax = b ; x ≥ 0. (1) (b) Write down the KKT conditions for the following problem, on the variable (λ, s) ∈ Rm+n AT λ max λT b + s = c; s ≥ 0, (2) Show that both the KKT conditions associated with both problems are identical. (c) Given x, s ∈ Rn , define the matrices X = diag(x1 , . . . , xn ), S = diag(s1 , . . . , sn ), and the vector e = (1, . . . , 1)T ∈ Rn . Let F : R2n+m → R2n+m be defined as T A λ+s−c . Ax − b F (x, λ, s) = XS e Show that a solution of F (x, λ, s) = 0 does not necessarily satisfy the KKT conditions of part (a) (or part (b)). Prove that, on the other hand, every vector (x, λ, s) that satisfies the KKT conditions must satisty F (x, λ, s) = 0. (d) Recall that the search direction (∆x, ∆λ, ∆s) generated by a Newton...
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...LINEAR PROGRAMMING II 1 Linear Programming II: Minimization © 2006 Samuel L. Baker Assignment 11 is on page 16. Introduction A minimization problem minimizes the value of the objective function rather than maximizing it. Minimization problems generally involve finding the least-cost way to meet a set of requirements. Classic example -- feeding farm animals. Animals need: 14 units of nutrient A, 12 units of nutrient B, and 18 units of nutrient C. Learning Objective 1: Recognize problems that linear programming can handle. Linear programming lets you optimize an objective function subject to some constraints. The objective function and constraints are all linear. Two feed grains are available, X and Y. A bag of X has 2 units of A, 1 unit of B, and 1 unit of C. A bag of Y has 1 unit of A, 1 unit of B, and 3 units of C. A bag of X costs $2. A bag of Y costs $4. Minimize the cost of meeting the nutrient requirements. To solve, express the problem in equation form: Cost = 2X + 4Y objective function to be minimized Constraints: 2X + 1Y $ 14 nutrient A requirement 1X + 1Y $ 12 nutrient B requirement 1X + 3Y $ 18 nutrient C requirement 8 8 Read vertically to see how much of each nutrient is in each grain. X $ 0, Y $ 0 non-negativity Learning objective 2: Know the elements of a linear programming problem -- what you need to calculate a solution. The elements are (1) an objective function that shows the cost or profit depending on what choices you make, (2) constraint inequalities...
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...Goal Programming By Dr. Mojgan Afshari Goal Programming (GP) Goal programming involves solving problems containing not one specific objective function, but rather a collection of goals that we would like to achieve. Firms usually have more than one goal. For example, maximizing total profit, maximizing market share, maintaining full employment, providing quality ecological management It is not possible for LP to have multiple goals Goal Programming (GP) Most LP problems have hard constraints that cannot be violated... There are 1,566 labor hours available. There is $850,00 available for projects. In some cases, restrictive... hard constraints are too You have a maximum price in mind when buying a car (this is your “goal” or target price). If you can’t buy the car for this price you’ll likely find a way to spend more. We use soft constraints to represent such goals or targets we’d like to achieve. A Goal Programming Example: Myrtle Beach Hotel Expansion Davis McKeown wants to expand the convention center at his hotel in Myrtle Beach, SC. The types of conference rooms being considered are: Size (sq ft) Unit Cost Small Medium Large 400 750 1,050 $18,000 $33,000 $45,150 Davis would like to add 5 small, 10 medium and 15 large conference rooms. He also wants the total expansion to be 25,000 square feet and to limit the cost to $1,000,000. Defining the Decision Variables ...
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...used to get low variability, low cost, high quality, and high reliability in the process and product .The priority-based, mean squared–based, and Goal programming–based approaches in the RSM framework are attached to many drawbacks necessitating alternative methods to be used. The dual response optimization approach may be difficult for practitioners who do not have sufficient mathematical knowledge and are mostly found in journals that are difficult to find. RSM framework approaches have been proposed by Vining and Myers(1990), Copeland and Nelson(1996) and Lind and Tu(1995).Other data-driven approaches that are applicable include the ones by Ding et al.(2004), Jeong et al.(2005) and Shaibu and Cho(2009).Although the weighted square sum-based method allow overtaking of drawbacks of the priority based methods, they do not capture solutions. The proposed method uses less complicated techniques and it is assumed that an optimal solution for a dual response may require trade-off between process mean and standard deviation. Therefore, it minimizes both the mean and standard deviation departure from their target values. It is assumed that those involved in process and product optimization have the necessary background knowledge in allocating the appropriate values to targets and bounds for each response. This is necessary to...
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...planning and management decision making. Nowadays, with the development of technology, most of the real world Linear Programming problems are solved by computer programs. Excel Solver is a popular one. We work through different examples to demonstrate the applications of linear Programming model and the use of Excel Solver for various decision making in operation and supply chain management. Components of Linear Programming model To solve the linear programming problems, we first need to formulate the mathematical description called a mathematical model to represent the situation. Linear programming model usually consists of the following components * Decision variables: These represent the choices that the decision maker can control. For example, the number of items to produce, amounts of money to invest in and so on. The decisions variables are represented using symbols such as X1, X2, X3, ….Xn. * Objective function: The objective of the problem is expressed as a mathematical expression in decision variables. The objective may be maximizing the profit, minimizing...
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...in the product kind. Organizations have already adopted solutions with varying degrees of planning and scheduling capabilities. Yet, operations executive acknowledge that these same systems are becoming out dated, lacking the speed, flexibility and responsiveness to manage their increasing complex production environment. Optimization techniques are applied to find out whether resources available are effectively utilized in order to achieve optimum profit from the activities of the firm. There should be consistency in the use of various resources and the mix should be such that it brings down the cost for ensuring profit. Therefore, it is the duty of the management to exercise control over the resources and to see that the resources are effectively utilized. Similarly, organizations in general are involved in manufacturing a variety of products to cater the needs of the society and to maximize the profit. While doing so, they need to be familiar with different combinations of product mix which will maximize the profit. Or alternatively minimize the cost. The techniques such as ratio analysis, correlation and regression analyses, variance analysis, optimization and projection methods can be adopted for ascertaining the extend of resource utilization and selection of practically viable and profitable product mixes by taking in to account all possible constraints. Tulsan, P.C, and Vishal pandey, (2002: 231). The ratio analysis helps to evaluate the performance of the organization...
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...corresponding elements, then adds the results. Example: [pic] If the row vector and the column vector are not of the same length, their product is not defined. Example: [pic] The Product of a Row Vector and Matrix When the number of elements in row vector is the same as the number of rows in the second matrix then this matrix multiplication can be performed. Example: [pic] If the number of elements in row vector is NOT the same as the number of rows in the second matrix then their product is not defined. Example: [pic] Linear programming (LP; also called linear optimization) is a method to achieve the best outcome (such as maximum profit or lowest cost) in a mathematical model whose requirements are represented by linear relationships. Linear programming is a special case of mathematical programming (mathematical optimization). More formally, linear programming is a technique for the optimization of a linear objective function, subject to linear equality and linear inequality constraints. Its feasible region is a convex polytope, which is a set defined as the intersection of finitely many half spaces, each of which is defined by a linear inequality. Its objective function is a real-valued affine function defined on this...
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...MATHEMATICAL PROGRAMMING - INDR. 363 (1) 2011 FALL Class Meeting Location ENG Z27 Class Meeting Times TH B3,TU B3 Instructor Office Hours Office Location Office Phone Email Web Address Number of Credits ETC Credit Prerequisites Language ONUR KAYA W 14:00-16:00 ENG 206 1583 okaya@ku.edu.tr 3 6 INDR. 262 English Assistant TA/RA/Lab Assistant Name AYLİN LELİZAR POLAT GÜLÇİN ERMİŞ Email aypolat@ku.edu.tr gulermis@ku.edu.tr Office Hours Office Location Course Description Introduction to modeling with integer variables and integer programming; network models, dynamic programming; convexity and nonlinear optimization; applications of various optimization methods in manufacturing, product design, communications networks, transportation, supply chain, and financial systems. Course Objectives The course is designed to teach the concepts of optimization models and solution methods that include integer variables and nonlinear constraints. Network models, integer, dynamic and nonlinear programming will be introduced to the students. Students will be exposed to applications of various optimization methods in manufacturing, product design, communications networks, transportation, supply chain, and financial systems. Several different types of algorithms will also be presented to solve these problems. The course also aims to teach how to use computer programs such as Matlab and GAMS to solve mathematical models. Learning Outcomes Students are expected to model...
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...Practice Problem Set 3 for Econ 4808: Optimization A. Unconstrained Optimization (Note: In each optimization problem, check the second-order condition.) 1. Consider the function: y = 2 x2. a. Determine the average rate of change of the function in the closed interval [1,4] of the argument. b. Determine whether the function is concave or convex, using geometric test and a specific value of ( = 0.3. c. Do the concavity/convexity test using the derivative conditions. 2. Consider the function: y = 2x. a. Determine the average rate of change of the function in the closed interval [1,4] of the argument. b. Determine whether the function is concave or convex, using geometric test and a specific value of ( = 0.3. c. Do the concavity/convexity test using the derivative conditions. 3. Use the derivative condition to test whether the function y = 8 + 10x - x2 is concave or convex over the domain [0,7]. 4. In a cross-country study of the relationship between income per-capita (Y) and pollution (S), Grossman and Krueger estimate a cubic relationship as S = 0.083Y3 - 2.2Y2 + 13.5Y + X, where X represents other factors not linked to income. For this exercise, consider X as a fixed quantity for a country. Identify and characterize the extreme values of this function. 5. In the model of perfect competition, all firms are price-takers since they treat price (P) as a market-determined constant. Assume that P = 12. A firm's total revenue (TR) function is TR(Q)...
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...these tasks may seem almost menial. “A robot would be able to recognize, from the activities its video cameras capture, what the people it’s observing are doing,” explains Dr. Raymond Perrault, director, Artificial Intelligence Center, SRI. “It can tell you that someone is walking through the door, or that two people have met and exchanged a package or that a person is digging a hole by the side of the road.” A robot like this with sensors could decide what information is pertinent and report the data to a group of warfighters. These intelligence systems can perceive their environment and adjust. “They manage to do [their mission] while the world changes around them,” Perrault says. To accomplish this task, they organize ideas utilizing mathematical logic. Using sensory data, the programs prove simple theorems by plugging the data into the algorithms, which results in a solution and consequent action.” Garegnani, J. (2010, December 15) Artificial Software is good for Aerospace because it gives us the capabilities to do more traveling into space than ever before. Intelligent System (IS) applications have gained popularity among aerospace professionals in the last decade due to the ease with which several of the IS tools can be...
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