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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
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, Student
Female, Faculty
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,
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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