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Math 125 Study Guide

by: Lael Wynne

Math 125 Study Guide 35661

Lael Wynne

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About this Document

These notes includes subjects that are going to be on the exam, including example problems.There is also a practice homework that is very similar to the setup of the exam.
Elementary Linear Algebra
Roy Lowman
Study Guide
Math, linear elementary algebra, Mathematics, business, Algebra
50 ?




Popular in Elementary Linear Algebra

Popular in Mathematics (M)

This 4 page Study Guide was uploaded by Lael Wynne on Friday April 1, 2016. The Study Guide belongs to 35661 at University of Illinois at Chicago taught by Roy Lowman in Spring 2016. Since its upload, it has received 30 views. For similar materials see Elementary Linear Algebra in Mathematics (M) at University of Illinois at Chicago.


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Date Created: 04/01/16
Math 125 Exam 2 Subjects covered on the exam (Sections 3.2-5.3):  Transition matrix o Stable distribution probability  Solving Linear Programming o Use of the simplex method (Maximize & Minimize) o Graphing  Simplex Tableau o Pivoting o Slack variables o Objective value (min & max) o Sensitivity analysis  Linear programming o Duality o Pivoting  Trends o Percentage  Absorbing Stochastic Matrix o The corresponding column as a single 1 and the remaining entries are 0 o The single 1 must be located on the main diagonal of the matrix Example 1:  Absorbing matrix : 1 0 .3 0 I S 0 1 . 1 1 0 0 .5 0 0 R 0 0 .1 0 1) Identify S & R: S= .3 0 R= .5 0 .1 1 .1 0 2) Identify (I-R): 1 0 _ .5 0 = .5 0 0 1 .1 0 -.1 1 3) (I-R) (Can use calculator, matrix setting) .5 0 ^-1 = 2 0 -.1 0 .2 1 Example 2: A furniture manufacturer makes two types of furniture: chairs and sofas. The manufacture of a chair requires: 6 hours of carpentry, 1 hour of finishing, 2 hours of upholstery. Manufacture of a sofa requires: 3 hours of carpentry, 1 hour of finishing, 6 hours of upholstery. Each day the factory has available: 96 labor hours for carpentry, 18 labor-hours for finishing, 72 labor-hours for upholstery. The profit per chair is $80 and per sofa is $70. How many chairs and sofas should be produced each day to maximize the profit? Let x = number of chairs and y = number of sofas. X Y Resource Carpentry 6 3 96 Finishing 1 1 18 Upholstery 2 6 72 Profit per unit 80 70 Maximize M=80x+70y Carpentry: 6x + 3y ≤ 96 6x+ 3y+ u= 96 Finishing: x + y ≤ 18 x+ y + v= 18 Upholstery: 2x + 6y ≤ 72 2x+ 6y+ w =72 x ≥ 0 ; y ≥ 0 Initial Simplex Tableau X y u v w M R 6 3 1 0 0 0 96 1 1 0 1 0 0 18 2 6 0 0 1 0 72 -80 -70 0 0 0 1 0 Final Simplex Tableau X y u v w M R 1 0 1/3 -1 0 0 14 0 1 -1/3 2 0 0 4 0 0 4/3 -10 1 0 20 0 0 10/3 60 0 1 1400 This is a practice homework for these specific subjects: Setup and solve the LP problem using the Simplex method. Clearly label your answers and show all of your work. Maximize: z = 80x + 70y Subject To: 2x + 6y ≤ 72, constraint 1 6x + 3y ≤ 96, constraint 2 x + y ≤ 18, constraint 3 x ≥ 0 ; y ≥ 0 1. Give the final simplex tableau. 2. From the final simplex tableau give the solution to the problem and its dual. 3. Sensitivity analysis. o What is the range of feasibility for constraint 1? o If the resource for constraint 1 is increased by 1 what will be the new values of x and y? o If the resource for constraint 1 is increased by 1 what will be the new profit? o What is the shadow price corresponding to constraint 1? 4. Give the dual to the original problem. This is not a simplex tableau. Your answer should include Minimize . . . , subject to the constraints . . . . Your constraints should include the non-negativity constraints. 5. Solve the dual problem by converting it into a non-standard maximization problem. Do not solve by solving its dual. Are your answers the same as you found earlier? Explain why or why not. 6. Use the final simplex tableau to find the solution to the primal problem. Are your answers the same as you found earlier? Explain why or why not. 7. Given the following transition matrix A for a Markov process with states 1, 2, 3 and 4: 1 0 .6 0 0 1 0 .4 0 0 0 .6 0 0 .4 0 Find As the stead-state matrix for A. A. Is A an absorbing stochastic matrix? Explain why or why not. B. Find R and S C. Find (I − R)-1 D. -1 Find S(I − R) E. Use the previous results to find A S F. You can check your answers by raising A to a very high power.


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