The attached file contains 5 questions that needs to be answered. It is preferred to be written in a pdf. All the questions must be answered in detail.1. (15 points) An automobile manufacturer has assembly plants located in the Northwest,

Midwest and Southeast. The cars are assembled and sent to major markets in the

Southwest, East, West and Northeast. The appropriate distance matrix, availabilities and

demands are given by the following table:

Northwest

Midwest

Southeast

Southwest

1200

400

800

East

8500

800

1200

West

1850

900

1000

Northeast

2250

1400

1100

demand

2,000,000

1,500,000

1,200,000

1,200,000

availability

2,500,000

1,800,000

1,600,000

a) (10 points) Assuming that the cost is proportional to distance. Formulate the problem to

minimize the total cost of transportation.

b) (5 points) Can the problem from part (a) be solved using the simplex method? Explain

your answer.

2. (15 points) The University is in the process of forming a committee to handle students’

grievances. The administration wants the committee to include at least one female, one male,

one student, one administrator, and one faculty member. Ten individuals (identified, for

simplicity, by the letter a to j) have been nominated. The mix of these individuals in the

different categories is given as follows:

Category

a

b

c

d

e

f

g

h

i

j

Individuals

Female, Student

Female, Administrator

Female, Student

Female, Faculty

Female, Administrator

Male, Administrator

Male, Student

Male, Faculty

Male, Faculty

Male, Student

The University wants to form the smallest committee with representation from each of the

five categories.

a) (10 points) Formulate the problem as an ILP.

b) (5 points) State which algorithm can be used to find the optimum solution.

3. (20 points) Consider the following function:

( ) = 3 12 + 9 1 + 2 2 − 1 2 + 4

(a) [12 points] Determine whether the function is convex, concave, or neither.

(b) [8 points] Suppose ∗ is a stationary point of the function . Is it a local max, local min,

global max, global min, or none of these? Explain your answer.

4. (30 points) Consider the following nonlinear program.

max ( − 2)

. . = 33 − 3

≥2

(a) [12 points] Derive the KKT necessary conditions for the nonlinear program.

(b) [18 points] Solve the nonlinear program using the KKT necessary conditions.

5. (20 points) A project is considered completed when activities A-F have all been completed.

The precedence relationships and duration of each activity are given in the table below:

Activity

Immediate

Predecessors

Duration (in

week)

A

2

–

B

3

–

C

1

B

D

5

A, C

E

7

B

F

5

C

G

2

D, F

H

6

E, F

I

4

G, H

a) (10 points) Draw the project network diagram.

b) (7 points) Determine the critical path and critical activities of this project.

c) (3 points) What is the earliest the project can be completed?

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