BIRLA INSTITUTE OF TECHNOLOGY & SCIENCE PILANI
First Semester, 2022-23
MATH F212 (Optimization)
Quiz 1 (Regular, Closed Book)
Max. Time: 35 minutes
ID:
1
A
Date & Time: Saturday, October 01, 2022, 5:00-5:35 PM
Max. Marks: 20
Name:
Write the solution in the following table
6
2
7
3
8
4
9
5
10
Note: The notations have usual meaning as and when required. There are 10 questions and each question
carries 02 marks. There is no negative mark and no partial mark. Write the final answer in legitimate
manner in provided blank.
10
1
1
If there are three alternative basic optima of LP problem as (𝑥1 , 𝑥2 , 𝑥3 ) = (0,0, 3 ) , (0,5,0), and (1,4, 3),
then the associated non-basic alternative optima are given by ________________ .
2
For the LP problem with objective function max 𝑧 = 2𝑥1 + 𝑥2 and two constraints of ≤ type, the simplex
iteration tableau is as follows
𝐵𝑉 𝑥1 𝑥2 𝑠1 𝑠2 Solution
0 −3 2
0
20
𝑧
0
10
𝑥1 1 −1 1
2 −2 1
20
𝑠2 0
The solution is ________________ .
3
For the LP problem
max 𝑧 = 3𝑥1 + 2𝑥2 + 3𝑥3 subject to 2𝑥1 + 𝑥2 + 𝑥3 ≤ 2; 3𝑥1 + 4𝑥2 + 2𝑥3 ≥ 8; 𝑥1 , 𝑥2 , 𝑥3 ≥ 0
with 𝑠1 , 𝑠2 , 𝑅 as slack, surplus, and artificial variable respectively, the 𝑀 −method simplex iteration tableau
is as follows
𝐵𝑉
𝑥1
𝑥2
𝑥3
𝑠2
𝑠1
𝑅 Solution
4
𝑧 −1 + 5𝑀 0 −1 + 2𝑀 𝑀 2 + 4𝑀 0
2
1
1
0
1
0
2
𝑥2
0
1
0
𝑅
−5
−2
−1
−4
The solution is ________________ .
4
The optimal tableau for the LP problem
max 𝑧 = 2𝑥1 − 𝑥2 + 3𝑥3 subject to 𝑥1 − 𝑥2 + 5𝑥3 ≤ 10; 2𝑥1 − 𝑥2 + 3𝑥3 ≤ 40; 𝑥1 , 𝑥2 , 𝑥3 ≥ 0
with 𝑠1 , 𝑠2 as slack variables is
𝐵𝑉 𝑥1 𝑥2 𝑥3 𝑠1 𝑠2 Solution
0 0
0
0
1
40
𝑧
30
𝑥1 1 0 −2 −1 0
20
𝑥2 0 1 −7 −2 1
The type of solution(s) is (are) ________________ (Degenerate, Alternative optima, unbounded,
infeasible).
5
Which property (ies) is (are) followed by the LP models ________________ .
6
The system 𝑩𝑿 = 𝒃 has no solution if 𝑩 is __________ and 𝒃 is __________ of 𝑩.
7
For the LP problem, the optimal tableau is
𝐵𝑉 𝑥1 𝑥2
𝑠1
𝑠2
𝑠3
𝑠4 Solution
0 0
3/4
1/2
0
0
21
𝑧
1/4 −1/2 0
0
3
𝑥1 1 0
0
3/2
𝑥2 0 1 −1/8 3/4 0
3/8 −5/4 1
0
5/2
𝑠3 0 0
1/8 −3/4 0
1
1/2
𝑠4 0 0
The second best optimal value of 𝑧 is equal to ________________ .
8
John must work at least 20 hours a week to supplement his income while attending school. He has the
opportunity to work in two retail stores. In store 1, he can work between 5 and 12 hours a week, and in
store 2 he is allowed between 6 and 10 hours. Both stores pay the same hourly wage. In deciding how many
hours to work in each store, John wants to base his decision on work stress. Based on interview with present
employees, John estimates that, on an ascending scale of 1 to 10, the stress factors are 8 and 6 at stores 1
and 2, respectively. Because stress mounts by the hour, he assumes that the total stress for each store at the
end of the week is proportional to the number of hours he works in the store. The optimum stress index is
equal to ________________ .
9
For the LP problem
min 𝑧 = 4𝑥1 + 𝑥2 subject to 3𝑥1 + 𝑥2 = 3; 4𝑥1 + 3𝑥2 − 𝑥3 = 6; 𝑥1 + 2𝑥2 + 𝑥4 = 4; 𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ≥ 0,
the updated 𝑧 −coefficient of non-basic variables 𝑥1 , 𝑥2 , 𝑥3 respectively, in starting 𝑀 −method simplex
iteration tableau are ________________ .
10 For the LP problem
max 𝑧 = 3𝑥1 + 2𝑥2 + 3𝑥3 subject to 2𝑥1 + 𝑥2 + 𝑥3 ≤ 2; 3𝑥1 + 4𝑥2 + 2𝑥3 ≥ 8; 𝑥1 , 𝑥2 , 𝑥3 ≥ 0,
the optimal simplex tableau at the end of phase I is given as
𝐵𝑉 𝑥1 𝑥2 𝑥3 𝑠1 𝑠2 𝑅 Solution
0
𝑟 −5 0 −2 −1 −4 0
2
1
1
0
1 0
2
𝑥2
0
1
0
𝑅 −5
−2 −1 −4
The optimal solution of LP problem is ________________ .
BIRLA INSTITUTE OF TECHNOLOGY & SCIENCE PILANI
First Semester, 2022-23
MATH F212 (Optimization)
Quiz 1 (Regular, Closed Book)
Max. Time: 35 minutes
ID:
1
Date & Time: Saturday, October 01, 2022, 5:00-5:35 PM
B
Max. Marks: 20
Name:
Write the solution in the following table
6
2
7
3
8
4
9
5
10
Note: The notations have usual meaning as and when required. There are 10 questions and each question
carries 02 marks. There is no negative mark and no partial mark. Write the final answer in legitimate
manner in provided blank.
1
The system 𝑩𝑿 = 𝒃 has no solution if 𝑩 is __________ and 𝒃 is __________ of 𝑩.
2
For the LP problem
min 𝑧 = 4𝑥1 + 𝑥2 subject to 3𝑥1 + 𝑥2 = 3; 4𝑥1 + 3𝑥2 − 𝑥3 = 6; 𝑥1 + 2𝑥2 + 𝑥4 = 4; 𝑥1 , 𝑥2 , 𝑥3 , 𝑥4 ≥ 0,
the updated 𝑧 −coefficient of non-basic variables 𝑥1 , 𝑥2 , 𝑥3 respectively, in starting 𝑀 −method simplex
iteration tableau are ________________ .
3
Which property (ies) is (are) followed by the LP models ________________ .
4
For the LP problem
max 𝑧 = 3𝑥1 + 2𝑥2 + 3𝑥3 subject to 2𝑥1 + 𝑥2 + 𝑥3 ≤ 2; 3𝑥1 + 4𝑥2 + 2𝑥3 ≥ 8; 𝑥1 , 𝑥2 , 𝑥3 ≥ 0,
the optimal simplex tableau at the end of phase I is given as
𝐵𝑉 𝑥1 𝑥2 𝑥3 𝑠1 𝑠2 𝑅 Solution
0
𝑟 −5 0 −2 −1 −4 0
2
1
1
0
1
0
2
𝑥2
0
𝑅 −5 0 −2 −1 −4 1
The optimal solution of LP problem is ________________ .
5
For the LP problem, the optimal tableau is
𝐵𝑉 𝑥1 𝑥2
𝑠1
𝑠2
𝑠3
𝑠4 Solution
0 0
3/4
1/2
0
0
21
𝑧
1/4 −1/2 0
0
3
𝑥1 1 0
0
3/2
𝑥2 0 1 −1/8 3/4 0
3/8 −5/4 1
0
5/2
𝑠3 0 0
0
0
1/8
0
1
1/2
𝑠4
−3/4
The second best optimal value of 𝑧 is equal to ________________ .
6
John must work at least 20 hours a week to supplement his income while attending school. He has the
opportunity to work in two retail stores. In store 1, he can work between 5 and 12 hours a week, and in
store 2 he is allowed between 6 and 10 hours. Both stores pay the same hourly wage. In deciding how many
hours to work in each store, John wants to base his decision on work stress. Based on interview with present
employees, John estimates that, on an ascending scale of 1 to 10, the stress factors are 8 and 6 at stores 1
and 2, respectively. Because stress mounts by the hour, he assumes that the total stress for each store at the
end of the week is proportional to the number of hours he works in the store. The optimum stress index is
equal to ________________ .
7
For the LP problem
max 𝑧 = 3𝑥1 + 2𝑥2 + 3𝑥3 subject to 2𝑥1 + 𝑥2 + 𝑥3 ≤ 2; 3𝑥1 + 4𝑥2 + 2𝑥3 ≥ 8; 𝑥1 , 𝑥2 , 𝑥3 ≥ 0
with 𝑠1 , 𝑠2 , 𝑅 as slack, surplus, and artificial variable respectively, the 𝑀 −method simplex iteration tableau
is as follows
𝐵𝑉
𝑥1
𝑥2
𝑥3
𝑠2
𝑠1
𝑅 Solution
4
𝑧 −1 + 5𝑀 0 −1 + 2𝑀 𝑀 2 + 4𝑀 0
2
1
1
0
1
0
2
𝑥2
0
1
0
𝑅
−5
−2
−1
−4
The solution is ________________ .
10
1
If there are three alternative basic optima of LP problem as (𝑥1 , 𝑥2 , 𝑥3 ) = (0,0, 3 ) , (0,5,0), and (1,4, 3),
then the associated non-basic alternative optima are given by ________________ .
8
9
The optimal tableau for the LP problem
max 𝑧 = 2𝑥1 − 𝑥2 + 3𝑥3 subject to 𝑥1 − 𝑥2 + 5𝑥3 ≤ 10; 2𝑥1 − 𝑥2 + 3𝑥3 ≤ 40; 𝑥1 , 𝑥2 , 𝑥3 ≥ 0
with 𝑠1 , 𝑠2 as slack variables is
𝐵𝑉 𝑥1 𝑥2 𝑥3 𝑠1 𝑠2 Solution
0 0
0
0
1
40
𝑧
1
0
0
30
𝑥1
−2 −1
20
𝑥2 0 1 −7 −2 1
The type of solution(s) is (are) ________________ (Degenerate, Alternative optima, unbounded,
infeasible).
10 For the LP problem with objective function max 𝑧 = 2𝑥1 + 𝑥2 and two constraints of ≤ type, the simplex
iteration tableau is as follows
𝐵𝑉 𝑥1 𝑥2 𝑠1 𝑠2 Solution
0 −3 2
0
20
𝑧
1
1
0
10
𝑥1
−1
2 −2 1
20
𝑠2 0
The solution is ________________ .
Solution (Set A)
1
10
1
𝑥1 = 𝜆3 , 𝑥2 = 5𝜆2 + 4𝜆3 , 𝑥3 = 3 𝜆1 + 3 𝜆3 , 𝜆1 + 𝜆2 + 𝜆3 = 1, 0 ≤ 𝜆1 , 𝜆2 , 𝜆3
2 Unbounded
3 𝑧 = 4, 𝑥1 = 0, 𝑥2 = 2, 𝑥3 = 0
4 Alternative optima and unbounded
5 Proportionality, Additivity, Certainty
6 Singular, Independent
7 20
8 140
9 −4 + 7𝑀, −1 + 4𝑀, −𝑀
10 𝑧 = 4, 𝑥1 = 0, 𝑥2 = 2, 𝑥3 = 0
Solution (Set B)
Singular, Independent
−4 + 7𝑀, −1 + 4𝑀, −𝑀
Proportionality, Additivity, Certainty
𝑧 = 4, 𝑥1 = 0, 𝑥2 = 2, 𝑥3 = 0
20
140
𝑧 = 4, 𝑥1 = 0, 𝑥2 = 2, 𝑥3 = 0
10
1
𝑥1 = 𝜆3 , 𝑥2 = 5𝜆2 + 4𝜆3 , 𝑥3 = 3 𝜆1 + 3 𝜆3 , 𝜆1 + 𝜆2 + 𝜆3 = 1, 0 ≤ 𝜆1 , 𝜆2 , 𝜆3
9 Alternative optima and unbounded
10 Unbounded
1
2
3
4
5
6
7
8