Management Science Class Exercise in year 2024

Management Science Excercises

1. Management
Science
-Class Exercises

2. George is planning to sell cookies for tomorrow’s party. He will make
chocolate cookies and milk cookies. He can only make 100 cookies
total. And he will make chocolate cookies less or equal to milk
cookies due to recipe issues. If profit for chocolate cookies and milk
cookies is $6 and $4 each, respectively, construct a linear model to
maximize his profit.
Problem 1

3. Problem 1 Answer –Model
Let x = number of chocolate cookie produce
Let y = number of milk cookie produce
Maxz=6x+4y( Maximize profit)
s.tx+y≤100( He can only make 100 cookies )
x−y≤0( He will not make chocolate cookies more than
twice much as milk cookies )
x,y≧0( Non-negativity)

4. Problem 1 Answer –Excel
Let x = number of chocolate cookie produce
Let y = number of milk cookie produce

5. Problem 2
YonseiFarms uses at least 740kg of house meal daily. The house
meal is a mixture of corn and rice with the following:
If house meal should contain at least 70 kg of protein and at most
25 kg of fiber, construct a linear model to minimize the total cost
kg per kg of feedstuff
Cost ($/kg)
Protein Fiber
Corn 0.11 0.03 0.3
Rice 0.65 0.08 0.93

6. Problem 2 Answer –Model
Let C = amount of corn used
Let R = amount of rice used
minz=0.3C+0.93R ( Minimize cost)
s.t C+R≥740( Amount use for house meal daily )
0.11C+0.65R≥70( At least 70 kg of protein )
0.03C+0.08R≤25( At most 25 kg of fiber )
C,R≧0( Non-negativity)

7. Problem 2 Answer –Excel
Let C = amount of corn used
Let R = amount of rice used

8. Problem 3
Luke can either play game or read book for next 4 hours before
dinner time (assume that he does not do anything else but those
two options). He promises to his father that he will not spend twice
more time in game than reading. Luke personally thinks that stress
from reading book is 4 times more than from playing game. If we let
stress from game be 1, he does not want to get more than 12 total
stresses. What is the maximum time that Luke can play game?

9. Problem 3 Answer –Model
�
1=������������??????������
�
2=������������??????������
Maxz=�
1 ( Total time spend for playing game)
�.��
1+�
2=4 ( Time availability )
�
1−2�
2≤0 ( not spend twice more time in game than book )
�
1+4�
2≤12 ( no more than 12 total stresses )
�
??????≥0

10. Problem 3 Answer –Excel
�
1=������������??????������
�
2=������������??????������

11. Problem 4
David is trying to invest through domestic and foreign stocks.
Domestic stocks will return 11% per year and foreign stocks will return
17% per year. David has $12 million on his pocket. And by some
restriction, no more than $10 million of the fund should go into
domestic stocks and no more than $7 million into foreign stocks.
Moreover, at least half as much should be invested in foreign as
domestic, and at least half as much in domestic as foreign to
maintain some balance. Construct a linear model to maximize this
annual return.

12. Problem 4 Answer –Model
Let D = amount of domestic fund invest
Let F = amount of foreign fund invest
minz=0.11D+0.17F( Maximize annual return )
s.t D ≤10( No more than $10 million into domestic )
F≤7( No more than $7million into domestic )
0.5D−F≤0( At least half as much should be invested in
foreign as domestic )
−D+0.5F≤0( Vice versa )
D+F≤12( Money availability )
x,y≧0( Non-negativity)

13. Problem 3 Answer –Excel
Let D = amount of domestic fund invest
Let F = amount of foreign fund invest

14. Problem 4
max�=3�+2�
s.tx+y≦2
3x+y≧2
x,y≧0
Draw a graph and find the optimal solution using two different
approaches (Extreme point and isoprofit line).

15. Problem 4 Answer –Extreme Point
max�=3�+2�
(0,2) : z = 4
(2,0) : z = 6
(
2
3
,0) : z = 2
Thus, (2,0) is optimal
solution, where z = 6

16. Problem 4 Answer –IsoProfit Line
max�=3�+2�
3x + 2y can be
maximize until (2,0)
where z value is 6
Thus, (2,0) is optimal
solution, where z = 6

17. Problem 5
Katherine is CEO of an elite women’s clothing company. In this fall fashions, she considers three
different clothing item, cotton sweater, wool blazer, and tailored skirt. All of them are
manufactured in two different departments: cutting and dyeing. It follows:
Unit profit for cotton sweater, wool blazer, and tailored skirt is $30, $20, and $10 respectively. For
customers’ satisfaction, all of the item should be manufactured at least 10 percent of total
production. What is the optimal production plan to maximize profit?
Time per units (hr)
Dept Cotton Sweater Wool Blazer Tailored SkirtCapacity(hr)
Cutting .30 .25 .15 450
Dyeing .45 .40 .22 1200

18. Problem 5 Answer –Model
�
1=����������������������������
�
2=�������������������������
�
3=����������??????�������??????���������
Maxz=30�
1+20�
2+10�
3 ( Total profit )
�.�0.3�
1+0.25�
2+0.15�
3≤450 ( Cutting capacity )
0.45�
1+0.4�
2+0.22�
3≤1200 ( Dyeing capacity )
9�
1−�
2−�
3≥0 ( at least 10% of cotton sweater )
−�
1+9�
2−�
3≥0 ( at least 10% of wool blazer )
−�
1−�
2+9�
3≥0 ( at least 10% of tailored skirt )
�
??????≥0

19. Problem 5 Answer –Excel
�
1=����������������������������
�
2=�������������������������
�
3=����������??????�������??????���������

20. Problem 6
YonseiManufacturing Company has a contract to deliver 100, 250,
190, 140, 220, and 110 home windows over the next 6 months.
Production cost per window varies by period and is estimated to be
$50, $45, $55, $48, $52, and $50 over the next 6 months. To take
advantage of the fluctuations in manufacturing cost, Yonseican
produce more windows than needed in a given month and hold
the extra units for delivery in later months. This will incur a storage
cost at the rate of $8 per window per month, assessed on end-of-
month inventory. Develop a linear program to determine the
optimum production schedule.

21. Problem 6 Answer –Model
�
??????=??????���������??????����������??????�����ℎ??????
??????
??????=??????����������??????���������ℎ����������ℎ??????
Minz=50�
1+45�
2+55�
3+48�
4+52�
5+50�
6+8(??????
1+??????
2+??????
3+??????
4+??????
5)
s.t �
1−??????
1=100 ( Month 1 )
??????
1+�
2−??????
2=250 ( Month 2 )
??????
2+�
3−??????
3=190 ( Month 3 )
??????
3+�
4−??????
4=140 ( Month 4 )
??????
4+�
5−??????
5=220 ( Month 5 )
??????
5+�
6=110 ( Month 6 )
�
??????,??????
??????≥0

22. Problem 6 Answer –Excel
�
??????=??????���������??????����������??????�����ℎ??????
??????
??????=??????����������??????���������ℎ����������ℎ??????