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1 - SENSATIVITY ANALYSIS / LINEAR PROGRAMMING MODELS
Two models of color TV sets designated Alpha and Beta are produced by Allison Company. The profit on Alpha is $300 and the profit on Beta is $250. There are 40 hours of labor each day in the production department and 45 hours of machine time available each day. The company can sell as many sets of model Beta as it can make, but it cannot sell more than 12 sets of Alpha. Each unit of the Alpha model requires two hours of labor and 1 hour of machine time. The Beta model requires 1 hour of labor and three hours of machine time.
FORMULATE AN LINEAR PROGRAM MODEL AND SOLVE THE PROBLEM.
Include the 4 steps of:
1. Defining the variables
2. State the objective function
3. State the content constraints
4. State the non negative constraints
1. slack variables,
2. create a tableau and
3. include range of optimality
2 - GOAL PROGRAMMING / LINEAR PROGRAMMING MODELS
Harrison Electric Company produces two products popular with home renovators: old-fashion chandeliers and ceiling fans. Both the chandeliers and fans require a two-step production process involving wiring and assembly. It takes about 2 hours to wire each chandelier, and 3 hours to wire a ceiling fan. Final assembly of the chandeliers and fans requires 6 and 5 hours respectively. The production capability is such that only 12 hours of wiring time and 30 hours of assembly time are available each day. If each chandelier nets the firm $7 and each fan $6, formulate a production mix decision LP.
In the above portion of this problem, you assumed that management had a single goal of maximizing profit. Now assume that the firm is moving to a new location during a particular production period and feels that maximizing profit is not a realistic goal. Management sets a profit level, which would be satisfactory during the adjustment period, of $30. Harrison has established the following goals in order of priority (i.e., the first one listed is the most important): Formulate a GP Model.
1. To produce as much profit above $30 as possible during the production period.
2. To fully utilize the available wiring department hours.
3. To avoid overtime in the assembly department.
4. To meet a contract requirement to produce at least seven ceiling fans.
Formulate and solve the GOAL PROGRAMMING Problem for parts 1 and 2
- NLP, convexity, Basic NLP algorithm,, Unconstrained optimization: , Newton's method,, line search and trust region methods to ensure convergence, Unconstrained optimization: , trust regions, steepest descent and its convergence , convergence rate, matrix condition number, quasi-Newton and conjugate gradient methods, Spreadsheet optimization, GAMS modeling language, Constrained problems-necessary and sufficient conditions, Saddle points, duality, Lagrangian relaxation, Reduced gradient methods, Penalty and Barrier methods, Successive quadratic programming , Successive linear programming (SLP), algorithm comparison.
- NLP applications including financial asset allocation using mean-variance models, Mixed integer nonlinear programming models and algorithms, Piece-by piece approach, global optimization, Global optimization, NLP applications, hand out take, Nature and Diversity of Nonlinear Programs . , Improving Search Paradigm, Improving and Feasible Directions . , Local and Global Optima, Initial Feasible Solutions . , Formulation on Unconstrained NLPs, Golden Section Search . , Bracketing and Quadratic Fit Search, First and Second Order Conditions for Local Optima . , Convex and Concave Functions, Gradient Search . , Newton's Method.
- Convergence of NLP Algorithms, Quasi-Newton Algorithms . , Conjugate Gradient Methods, Unconstrained Optimization without Derivatives . , Formulation of Constrained Nonlinear Programs, Convex, Separable, Quadratic and Geometric Programming , Lagrange Multiplier Techniques, Karush-Kuhn-Tucker Optimality Conditions . , Penalty and Barrier Methods . , Review of Simplex for LP, Reduced Gradient Methods . , Quadratic Programming, Sequential Quadratic Programming Methods . , Separable Programming, Posynomial Geometric Programming, Linear Programming Problems, Some Applications, The Basic Index Set, The Simplex Algorithm, Sensitivity.
- Cycling, Initialization, Convex Sets, Matrices, Calculation Schemes and the Revised Simplex Method, Sensitivity Revisited, Variables with Upper Bounds, The Dual Simplex, Complementary Slackness, The Dual Simplex Algorithm, Variables with Upper Bounds, The Transportation and Assignment Problem, The Transshipment Problem, The Dual Primal Algorithm, Assignment and Matching, Ellipsoid Method, Interior Point Methods (Karmarkar’s Algorithm), Networks and Maximal Flow and Minimal Cut Problems, Applications to Operations Research, Integer Programming, Quadratic Programming, Dantzig-Wolfe Algorithm, Linearization of Problems, Introduction to non Linear Programming, Unconstrained Miniminimization, Arithmetric-Geometric Inequatlity, Newton's Method
Topics for Linear and Nonlinear Programming
- Review of calculus , gradient, Hessian, Taylor expansion, Improving feasible directions, Optimality conditions, unconstrained problems , constrained problems, Finding initial feasible solutions , Iterative methods for optimization problems, Convexity, Modeling strategies, assumptions for linear programs, Graphical solutions to linear programs , Polytopes, Extreme points and extreme rays
- Simplex algorithm, Convergence of Simplex , Degeneracy, Primal and dual bounds for Linear Programming , Duality theory , Sensitivity analysis, post-optimal analysis, theory of linear programming, Simplex method, revised simplex method, duality, dual simplex method, Post-optimality analysis, Interior point methods, Decomposition methods, Network flow algorithms, Maximum flow
- shortest path, min cost flow problems, methods of non-linear optimization., Convex sets, convex and concave functions, Unconstrained and Constrained Optimization, Quadratic Programming, Optimality conditions and convergence results, Karush-Kuhn-Tucker conditions, penalty and barrier methods, Duality in nonlinear programming