Two files of mathematics questions.

MidTerm exam

NICO MATH 484

Due day: 3-14-2017 Topics: Chapters 1 to 5

February 26, 2017

Exercise 1. Consider the following relations: g1 = x1 + x2 + 2x3, g2 = 2x1 + 0x2 + 3x3 , g3 = 2x1 + x2 + 3x3 and Z = 3x1 + 2x2 + 4x3. Solve the followings problems using the matrix representation equation (not table form).

1. max Z, s.t: g1 ≤ 4, 3g2 ≤ 5, g3 ≤ 7, x1, x2, x3 ≥ 0.

2. max Z + 4x4, s.t: 3g1 + x4 ≤ 7, 2g3 + 3x4 ≤ 5, x1, x2, x3, x4 ≥ 0.

Exercise 2. Prove that if C ⊆ Rn is a convex cone, and D is the set of all direction of C ( Cc and Dc are the complements set):

1. Prove that every x ∈ Dc also belong to x ∈ Cc

2. Construct and example to show that every element x ∈ C is also a direction of C.

Exercise 3.

Lemma 1. Suppose C is a convex set:

1. Prove that if x is NOT on the boundary of C, then x is NOT an extreme point of C

2. Construct and example to show the previous statement.

Exercise 4. Consider the polyhedral set defined by the system of inequalities:

ax1 + (b + 1)x2 ≤ 120 x1 + (a + b)x2 ≤ 160 (a− b)x1 + x2 ≤ 30

x1 ≥ 0 x2 ≥ 0

Find the necessary conditions for the numbers a and b to obtain (If it is possible):

1. Edges

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2. Extreme points No degenerate

3. Extreme points degenerate

4. Family of directions

5. Extreme directions.

6. No extreme directions.

7. No extreme points

8. Is it possible to find the sufficient conditions in order to characterize the extreme points (degenerate).

9. Find the necessary conditions for the numbers a, b to make the L.P problem have an optimal solution. (If it is possible)

Exercise 5. Construct two examples one in R2 and the other in R3 in order to have:

1. A convex cone.

2. A convex Functions

3. No convex cone.

Exercise 6. Prove that in R3, every cube has a No-degenerate extreme points. Exercise 7. A merchant plans to sell two models of home computers at costs of $250 and $400, respectively. The $250 model yields a profit of $45 and the $400 model yields a profit of $50. The merchant estimates that the total monthly demand will not exceed 250units. Find the number of units of each model that should be stocked in order to maximize profit. Assume that the merchant does not want to invest more than $70, 000 in computer inventory. Solve the problem using:

1. Graphical solution.

2. Use matrix representation equation (not table form)

3. Table form

4. Construct a computer algorithm to verify your solution.

5. the computer algorithm to get the solution for a minimization problem.

6. Use table form to solve the corresponding minimization problem and compare with the solution that you get with the computer software.

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Exercise 8. A local furniture shop makes chairs, tables, sofas and lamps The projected profits for the four products are, respectively, 20 per chair, 30 per table, 50 per sofa and 10 per lamp. The projected demand is 400 chairs, 100 tables, 50 sofas and 160 lamps. Each chair requires 2 cubic feet of type 1-wood, each sofa requires 4 cubic feet of type 1-wood, and each lamp requires 1 cubic feet of type 2- wood while, each table requires 5 cubic feet of type 2-wood. The shop has a total amount of 1,000 cubic feet of type-1 wood and 1500 cubic feet of type-2 wood in stock. How many chairs and tables should the shop make in order to maximize its profit?

1. Use table form to solve the problem

2. Construct a computer algorithm to verify your solution.

Conditions:

1. Total points 25

2.

 

1, 4, 7, 8− > 4pts, each 2, 3− > 3pts, each 5− > 1pts 6− > 2pts

3. March 14 is the last day, There are no extensions.

4. Only allowed, the use of white sheets

5. Writing is important. (if you want the maximum score)

6. Order is important. (if you want the maximum score)

7. Places allowed to drop off: Office or Mail box in McAllister.

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