...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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...CHAPTER 1 INTRODUCTION Background Of the Study In mathematics, optimization problem is a problem where it consists of maximizing or minimizing a real function by systematically choose an input values within an allowed set and compute the value of the function. An additional, it also means solve the problem so that we can the goal as quickly as possible without wasting a lot of resources. Optimization also can be deviating from a target by the smallest possible margin. Generally, a large area of applied mathematics is comprised by the optimization theory and techniques to other formulations. In the simple case, optimization is like finding a good value or a best available value of some problems given a defined domain, including a many of different types of objectives functions and different types of domains. In vector calculus, the gradient of a scalar field is a vector field that points in the direction of the scalar field, and whose the magnitude is that rate of increase. The variation in space of any quantity can be represented by a slope in simple terms. The gradient is like represents the steepness and the direction of the slope. The gradient or gradient of a scalar function f〖:R〗^n→R^1 is denoted by ∇f or ∇ ⃗f where ∇ denotes the vector of the differential operator. Hessian matrix was developed in the 19th century by the German mathematician Ludwig Otto Hesse and this matrix is later named after him. Hessian matrix is the matrix of second derivatives...
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... It generalizes the existing literature by introducing combinatorial auctions of heterogeneous VMs, and models dynamic VM provisioning. Social welfare maximization under dynamic resource provisioning is proven NP-hard, and modeled with a linear integer program. An efficient α-approximation algorithm is designed, with α ∼ 2.72 in typical scenarios. We then employ this algorithm as a building block for designing a randomized combinatorial auction that is computationally efficient, truthful in expectation, and guarantees the same social welfare approximation factor α. A key technique in the design is to utilize a pair of tailored primal and dual LPs for exploiting the underlying packing structure of the social welfare maximization problem, to decompose its fractional solution into a convex combination of integral solutions. Empirical studies driven by Google Cluster traces verify the efficacy of the randomized auction. I. INTRODUCTION The cloud computing paradigm offers users rapid ondemand access to computing resources such as CPU, RAM and storage, with minimal management overhead. Recent commercial cloud platforms, exemplified by Amazon EC2 [1], Microsoft Azure and Linode [2], organize a shared resource pool for serving their users. Virtualization technologies help cloud providers pack their resources into different types of virtual machines (VMs), for allocation to cloud users. For example, Tab. I illustrates a number of VMs...
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...Solver: A solver is a generic term indicating a piece of mathematical software, possibly in the form of a stand-alone computer program or as a software library, that 'solves' a mathematical problem. A solver takes problem descriptions in some sort of generic form and calculates their solution. In a solver, the emphasis is on creating a program or library that can easily be applied to other problems of similar type. Types of problems with existing dedicated solvers include: * Linear and non-linear equations. In the case of a single equation, the "solver" is more appropriately called a root-finding algorithm. * Systems of linear equations. * Nonlinear systems. * Systems of polynomial equations, which are a special case of non linear systems, better solved by specific solvers. * Linear and non-linear optimisation problems * Systems of ordinary differential equations * Systems of differential algebraic equations * Logic/satisfiability problems * Constraint satisfaction problems * Shortest path problems * Minimum spanning tree problems * Search algorithms Excel Solver, introduced by Microsoft in 1991, is a powerful optimization application used in finance, production, distribution, purchasing and scheduling. It is part of the data analysis tools used for what-if analysis--a process of identifying changes in a cell by adjusting related cells. If your boss has asked you to find ways to increase company's profits by determining...
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...The manager of a shoe factory makes three varieties of shoes: football keds, jogging shoes and cricket boots. These shoes contribute Rs. 700, Rs. 300 and Rs. 900 per pair to the total profit. The shoes consume two types of raw material. They also consume time on a set of identical finishing machines. The table below has relevant figures. (10 marks) | Consumption per pair produced | | Material 1 (kg) | Material 2 (kg) | Machine Time (hours) | Football keds | 4 | 2 | 9 | Jogging shoes | 5 | 4 | 5 | Cricket boots | 6 | 6 | 6 | Daily availability | 360 | 300 | 600 | We solved the LP on Excel solver and our results are given in the Sensitivity Report on the next page. Based on this report, a. Write the optimal solution, and the optimal value of the objective function. (1) b. The manager wishes to expand the capacity of the finishing machines. For each unit expansion, he will spend Rs. 200. What is the maximum expansion advisable at this expense? Why? (1) c. Are any shoe types currently absent in the optimal solution? If yes, which ones? By changing their unit profit, can we bring them into the solution? If yes, by how much, respectively? (1) d. If management had a choice of obtaining a few units of either (but not both) Material 1 or Material 2 at the same cost as present, which one should be chosen? Why? (1) e. Management is considering a change in the profit that would result in increasing the profit on the...
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...Analysis The goal of "Bangalore Textile Company" is to make profit through production of the suit materials and sell in the market. In order to meet the demand from the market, in case of production shortage, the company may also has to purchase some materials from other production plant and then resell to the market as well. The information of production cost, selling price and purchase price from other plants are provided. The table below shows the profit for production profit and trading from purchasing and reselling profit. Table 1: Profit by production for each material Material | Selling price | Production cost | Production profit* | 1 | $18.50 | $14.10 | $4.40 | 2 | $20.50 | $16.30 | $4.20 | 3 | $22.60 | $17.90 | $4.70 | 4 | $27.90 | $22.40 | $5.50 | 5 | $29.85 | $24.15 | $5.70 | 6 | $32.75 | $26.70 | $6.05 | *Production profit equals selling price less production cost Table 2: Profit by reselling for each material Material | Selling price | Purchasing price | Resell profit* | 1 | $18.50 | $16.55 | $1.95 | 2 | $20.50 | $18.35 | $2.15 | 3 | $22.60 | $20.25 | $2.35 | 4 | $27.90 | $25.30 | $2.60 | 5 | $29.85 | $27.10 | $2.75 | 6 | $32.75 | $29.85 | $2.90 | *Resell profit equals selling price less purchasing price The two profit tables suggest that production profit are higher than resell profit for each material. The amount of materials to be produced by each type of machines and / or to be purchased from other plants are to be determined...
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...WAN optimization tools are used mainly in application delivery networks which consist of WAN Optimization Controllers (WOCs) and Application Delivery Controllers (ADCs). On the data center end Application Delivery Controllers are present to distribute the traffic among servers based on application specific criteria. In the branch office portion of Application Delivery Controllers, WDC are present that uses its advanced compression and flow optimization capabilities to provide application availability, security, visibility, and acceleration. WAN Optimization mitigates the latency and bandwidth limitations of the WAN to provide remote users with access to applications in the data center. IT departments have been trying to take initiatives to meet their business objectives to reduce costs and consolidate resources. WAN optimization...
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...deals with problems of maximizing or minimizing a linear function in the presence of linear equality and/or inequality constraints. In these problems, we find the optimal, or most efficient way of using limited resources to achieve the objective of the situation. Linear Programming enables users to model large and complex problems and solve in a short amount of time by the use of effective algorithm, hence it is a powerful and widely used tool in various fields such as science, industrial engineering, financial 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...
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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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...www.elsevier.com/locate/eswa A new multi-objective multi-mode model for solving preemptive time–cost–quality trade-off project scheduling problems Madjid Tavana a,b,⇑, Amir-Reza Abtahi c, Kaveh Khalili-Damghani d a Business Systems and Analytics Department, Lindback Distinguished Chair of Information Systems and Decision Sciences, La Salle University, Philadelphia, PA 19141, USA Business Information Systems Department, Faculty of Business Administration and Economics, University of Paderborn, D-33098 Paderborn, Germany c Department of Knowledge Engineering and Decision Sciences, University of Economic Sciences, Tehran, Iran d Department of Industrial Engineering, South-Tehran Branch, Islamic Azad University, Tehran, Iran b a r t i c l e i n f o a b s t r a c t Considering the trade-offs between conflicting objectives in project scheduling problems (PSPs) is a difficult task. We propose a new multi-objective multi-mode model for solving discrete time–cost–quality trade-off problems (DTCQTPs) with preemption and generalized precedence relations. The proposed model has three unique features: (1) preemption of activities (with some restrictions as a minimum time before the first interruption, a maximum number of interruptions for each activity, and a maximum time between interruption and restarting); (2) simultaneous optimization of conflicting objectives (i.e., time, cost, and quality); and (3) generalized precedence relations between activities. These assumptions are...
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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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...your progress if you ask for it. 3. Flexible situation: Read/understand the case, have some ideas about what method you will use to solve it and how you will set up the Excel model, and have a feasible plan that will help you finish the case by the final deadline. In this case, you can submit a draft or partially-complete Excel worksheet and I will provide feedback based on your submission. Case: Location, Distribution, and Capacity Expansion at a Beer Manufacturer A Case Study of Location, Distribution, and Capacity Expansion Decisions at Anadolu Efes Case Description and Objectives This case study involves the application of linear programming and integer linear programming methods in solving a distribution and capacity planning problem at Anadolu Efes Beverage Group. Faced with intense competition and anticipated increase in beer demand that will surpass the current production capacity, the company wants to optimize its malt and beer...
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...Deterministic Optimization Homework 1 Deadline : 27 Sept 2015 Problem 1 (40 Points). A wine Company produces two kinds of wine Nectar and Red. The wines are produced from 64 tons of grapes the company has acquired this season. A 1,000-gallon batch of Nectar requires 4 tons of grapes, and a batch of Red requires 8 tons. However, production is limited by the availability of only 50 cubic yards of storage space for aging and 120 hours of processing time. A batch of each type of wine requires 5 cubic yards of storage space. The processing time for a batch of Nectar is 15 hours, and the processing time for a batch of Red is 8 hours. Demand for each type of wine is limited to seven batches. The profit for a batch of Nectar is $9,000, and the profit for a batch of Red is $12,000. The company wants to determine the number of 1,000-gallon batches of Nectar and Red to produce in order to maximize profit. a Formulate a linear programming model for this problem. b Solve this model by using graphical analysis. c How much processing time will be left unused at the optimal solution? d What would be the effect on the optimal solution of increasing the available storage space from 50 to 60 cubic yards? Problem 2 (60 Points). A manufacturing firm into two products. Each product may undergo three processes (assembly, finishing and packing). The firm has 2400 hours available for assembly and 800 hours for finishing,and 1200 hours for packing. Each unit of product 1 has a profit of $5...
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...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 real life problems using mathematical models including integer variables and nonlinear equations. Students will be able to apply mathematical modeling techniques such as dynamic, integer and nonlinear programming to different types of problems. They will also be able to model and solve transportation and network problems such as shortest path, maximum flow and minimum cost network flow...
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...Natural Computing Series Series Editors: G. Rozenberg Th. Bäck A.E. Eiben J.N. Kok H.P. Spaink Leiden Center for Natural Computing Advisory Board: S. Amari G. Brassard K.A. De Jong C.C.A.M. Gielen T. Head L. Kari L. Landweber T. Martinetz Z. Michalewicz M.C. Mozer E. Oja G. P˘ un J. Reif H. Rubin A. Salomaa M. Schoenauer H.-P. Schwefel C. Torras a D. Whitley E. Winfree J.M. Zurada For further volumes: www.springer.com/series/4190 Franz Rothlauf Design of Modern Heuristics Principles and Application Prof. Dr. Franz Rothlauf Chair of Information Systems and Business Administration Johannes Gutenberg Universität Mainz Gutenberg School of Management and Economics Jakob-Welder-Weg 9 55099 Mainz Germany rothlauf@uni-mainz.de Series Editors G. Rozenberg (Managing Editor) rozenber@liacs.nl Th. Bäck, J.N. Kok, H.P. Spaink Leiden Center for Natural Computing Leiden University Niels Bohrweg 1 2333 CA Leiden, The Netherlands A.E. Eiben Vrije Universiteit Amsterdam The Netherlands ISSN 1619-7127 Natural Computing Series ISBN 978-3-540-72961-7 e-ISBN 978-3-540-72962-4 DOI 10.1007/978-3-540-72962-4 Springer Heidelberg Dordrecht London New York Library of Congress Control Number: 2011934137 ACM Computing Classification (1998): I.2.8, G.1.6, H.4.2 © Springer-Verlag Berlin Heidelberg 2011 This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations...
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