NEW YORK CITY COLLEGE OF TECHNOLOGY The City University of New York

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1 NEW YORK CITY COLLEGE OF TECHNOLOGY The City University of New York DEPARTMENT: Mathematics COURSE: MAT 3770 TITLE: DESCRIPTION: TEXT: Mathematical Modeling I - Optimization The study of different types of optimization problems arising in different fields of business and industry. Examples are provided on sensitivity analysis of parameters of a model and calculating shadow prices. Mathematical Modeling M.M. Meerschaert 4 th edition San Diego, CA: Academic Press CREDITS: 3 (3 class hours) PREREQUISITES: MAT 2580 MAT 2675 CST 1101 A. Testing/Assessment Guidelines: The following exams should be scheduled: 1. A class exam at the end of the First Quarter. 2. A class exam at the end of the Second Quarter. 3. A final project and Final Examination. B. Computer Programming is required. Prepared by Professors Jonathan Natov and Urmi Ghosh-Dastidar (Spring 2008)

2 Learning Outcomes for MAT 3770 Mathematical Modeling I - Optimization 1. Students will be able to model real world problems (constrained and unconstrained, one variable problem, multivariable problem). 2. Students will be able to conduct parameter sensitivity analysis. 3. Students will be able to solve nonlinear optimization problems with equality constraints by using the Lagrange multiplier method. 4. Students will be able to calculate shadow prices. 5. Students will be able to interpret a linear program geometrically. 6. Students will be able to solve linear programming problems by using the simplex method. 7. Students will be able to apply integer programming to solve optimization problems arising in manufacturing, transportation, investment, and inventory. 8. Students will be able to use computer technology to assist in the above. New York City College of Technology Policy on Academic Integrity Students and all others who work with information, ideas, texts, images, music, inventions, and other intellectual property owe their audience and sources accuracy and honesty in using, crediting, and citing sources. As a community of intellectual and professional workers, the College recognizes its responsibility for providing instruction in information literacy and academic integrity, offering models of good practice, and responding vigilantly and appropriately to infractions of academic integrity. Accordingly, academic dishonesty is prohibited in The City University of New York and at New York City College of Technology and is punishable by penalties, including failing grades, suspension, and expulsion. The complete text of the College policy on Academic Integrity may be found in the catalog.

3 MAT 3770 Mathematical Modeling I Optimization Text: Mathematical Modeling by M. M. Meerschaert, 3 rd edition (2007) Session Mathematical Modeling I - Optimization Homework 1, Methods for Optimization Problems with One Variable: The Five-Step Method pp # 1, Sensitivity Analysis pp , Sensitivity Analysis pp pp # 4, 7, Sensitivity and Robustness pp , 6 Quiz, Project 7 First Examination 8, Unconstrained Multivariable Optimization Problems, Sensitivity Analysis, pp #1, 5 Robustness of the Model for Multivariable Problems pp , Lagrange Multipliers; Multivariable Optimization with One Constraint, Multivariable Optimization with Two Constraints. (Do Ex. 2.3, pp. 32, Ex. 2.4 before Ex 2.2) pp Supplemental problems: From Calculus III on Lagrange Multipliers 12, Application of Lagrange Multipliers (Ex. 2.2, pp. 29) pp pp # 6, Sensitivity Analysis and Shadow Prices with Lagrange Multipliers pp , Sensitivity Analysis and Shadow Prices with Lagrange Multipliers pp pp # 8, 9 16, 17 Quiz, Project 18 Second Examination

4 MAT 3770 Mathematical Modeling I Optimization Text: Mathematical modeling by M. M. Meerschaert, 3 rd edition (2007) 19, Graphical and Numerical Analysis (for one-variable optimization problems), Review of Newton s Method for One-variable Problem, pp Graphical and Numerical Analysis (for multivariable optimization problems), Review of Newton s Method for Multivariable Problems, pp , Graphical and Numerical Analysis (for multivariable optimization problems), Review of Newton s Method for Multivariable Problems, pp Quiz 23, Linear Programming for Multivariable Constrained Optimization Problems; Sensitivity Analysis and Shadow Prices, Introduction of Simplex Method, pp pp # 1, 6 pp # 10 pp # 16, 17 25, Discrete Optimization, pp pp # 23 27, 28 Project 29 Review 30 Final Examination

5 MAT 3770 Mathematical Modeling I Optimization Text: Mathematical modeling by M. M. Meerschaert, 3 rd edition (2007) Mathematical Modeling I - Optimization Homework 1.1 Methods for optimization problems with one variable: the five-step method pp # 1, Sensitivity Analysis pp Sensitivity and robustness pp pp # 4, 7, Unconstrained Multivariable Optimization Problems, sensitivity analysis, pp #1, 5 robustness of the model for multivariable problems pp Lagrange multipliers; Multivariable optimization with one constraint, Supplemental problems Multivariable optimization with two constraints Suggestion: Do Ex. 2.3, pp. 32, Ex. 2.4 before Ex 2.2, pp Application of Lagrange multipliers (Ex. 2.2, pp. 29) pp pp # Sensitivity Analysis and shadow prices with Lagrange multipliers pp Graphical and numerical analysis (for one-variable optimization problems), pp # 1, 6, 10 Review of Newton s method for one-variable problem, pp Graphical and numerical analysis (for multivariable optimization problems), Review of Newton s method for multivariable multi-variable problems, pp Linear programming for multivariable constrained optimization problems; pp # 16, 17 sensitivity analysis and shadow prices, introduction of simplex method, pp Discrete Optimization, pp pp. 108 # 23