PAUL JASON YU
MANAGEMENT SCIENCE ACC421-11343
MIDTERM PROBLEM SETS
June 28, 2021
Let’s Check
Weeks 4-5 ULOa
2. A minor league ballpark attracts 88 fans and draws in ₱553 in revenue from
ticket sales. A child's ticket costs ₱4 and an adult's ticket is ₱7. How many of each
type of ticket were sold?
Let
x = children’s ticket
y = adult’s ticket
x + y = 88
4x + 7y = 553
x + y = 88
y = 88-x
4x + 7y = 553
4x+7(88-x)=553
4x+616-7x=553
-3x=-63
x = 21
x + y = 88
21+y=88
y=67
Answer: 21 children’s tickets and 67 adult’s tickets were sold.
3. A woman owns 21 pets. Each of her pets is either a cat or a bird. If the pets
have a total of 76 legs, and assuming that none of the bird's legs are protruding
from any of the cats' jaws, how many cats and how many birds does the woman
own?
Let
x = number of cats
y = number of birds
x + y = 21
4x + 2y = 76
x + y = 21
y = 21 – x
4x + 2y = 76
4x+2(21-x)=76
4x+42-2x=76
2x=34
x=17
x + y = 21
17+y=21
y=4
Answer: There are 17 cats and 4 birds.
Let’s Analyze
Weeks 4-5 ULOa
9. A student has money in three accounts that pay 5%, 7%, and 8%, in annual
interest. She has three times as much invested at 8% as she does at 5%. If the
total amount she has invested is $1600 and her interest for the year comes to
$115, how much money does she have in each account?
Let
x = money with 5% interest
y = money with 7% interest
z = money with 8% interest
x + y + z = 1,600
.05x + .07y + .08z = 115
3x=z
x+y+z=1,600
x+y+3x=1,600
4x+y=1,600
y=1,600-4x
y=1,600-4(300)
y=1,600-1,200
y=400
.05x + .07y + .08z = 115
.05x + .07y + .08(3x) = 115
.05x + .07y + .24x = 115
.29x + .07y = 115
.29x + .07(1,600-4x) =115
.29x + 112 -.28x =115
.01x = 3
x = 300
z=3x
z=3(300)
z=900
Answer: The student has $300, $400 and $900 in her bank accounts with
5%, 7% and 8% interests, respectively.
7. How many gallons of 20% alcohol solution and 50% alcohol solution must be
mixed to get 9 gallons of 30% alcohol solution?
Let
x = gallons of 20% alcohol solution
y = gallons of 50% alcohol solution
.20x + .50y = 2.7
x+y=9
.20x + .50y = 2.7
-.20(x + y = 9)
-.20x - .20y = -1.8
.30y = .90
y=3
x+y=9
x+3=9
x=6
Answer: 6 gallons of 20% alcohol solution and 3 gallons of 50% alcohol
solution must be mixed in order to get 9 gallons of 30% alcohol solution.
Let’s Check
Weeks 4-5 ULOb
Let’s Analyze
Weeks 4-5 ULOb
1. A fruit juice company makes two special drinks by blending apple and pineapple
juices. The first drink uses 30% apple juice and 70% pineapple, while the
second drink uses 60% apple and 40% pineapple. There are 1000 liters of
apple and 1500 liters of pineapple juice available. If the profit for the first drink
is ₱0.60 per liter and that for the second drink is ₱0.50, use the simplex method
to find the number of liters of each drink that should be produced in order to
maximize the profit.
Let
X be the number of liters of drink 1
Y be the number of liters of drink 2
Therefore, we shall maximize:
Z=0.60X + 0.50Y
X,Y≥0
.30X + .60Y ≤ 1,000
.70X + .40Y ≤ 1,500
X
1
0
0
FINAL TABLEAU
Y
S1
S2
0
-1.33
2
1
2.33
-1
0
0.37
0.7
b
1,666.67
833.33
1,416.67
(X,Y,S1,S2) = (1,666.67,833.33,0,0)
Z=0.60X + 0.50Y
Z=0.60(1,666.67) + 0.50(833.33)
Z=1,000 + 416.67
Z=1,416.67
Answer: 1,667 liters of drink 1 and 833 liters of drink 2 should be
produced in order to generate the maximum revenue of ₱1,416.67
2. An accounting firm has 900 hours of staff time and 100 hours of reviewing time
available each week. The firm charges ₱2000 for an audit and ₱300 for a tax
return. Each audit requires 100 hours of staff time and 10 hours of review time,
and each tax return requires 12.5 hours of staff time and 2.5 hours of review
time. What number of audits and tax returns will bring in a maximum revenue?
Let
X1 be the number of Audits
X2 the number of tax returns
Therefore, we shall maximize:
Z=2,000X1+300X2
X1, X2>=0
100X1+12.5X2≤900
10X1+2.5X2≤100
x1
1
0
0
FINAL TABLEAU
x2
s1
s2
0
0.02
-.1
1
-.08
0.8
0
16
40
b
8
8
18,400
(X1,X2,S1,S2) = (8,8,0,0)
Z=2,000 X1+300X2
Z=2,000(8)+300(8)
Z=16,000+2,400
Z=18,400
Answer: 8 audits and 8 tax returns will bring a maximum revenue of
₱18,400.
Let’s Check
Weeks 6-7 ULOa
3. What are the three information needs of the transportation model?
In order to use the transportation model, three information must first be
determined. First is the origin points and the capacity/supply per period at each.
Another is the destination points and the demand per period at each. Finally, the
cost of shipping one unit from each origin to each destination.
7. Explain what is meant by the term degeneracy within the context of
transportation modeling.
Degeneracy, in the context of transportation modeling, occurs when the
total number of used cells is not equal to the number of rows plus the number of
columns minus one. This may occur in an initial solution or in subsequent
solutions. A degenerate table cannot be subject to improvement due to lack of
entries.
Let’s Analyze
Weeks 6-7 ULOa
3. Engr. Santos has to assign 4 teams to work on 4 projects. The costs charged
by each team are as follows. Costs are in thousands of pesos. Determine the
least cost of assignment.
Projects (Initial Table)
Teams
1
2
3
4
A
15
18
20
16
B
18
17
14
20
C
21
20
13
15
D
17
15
18
19
1
0
4
8
2
Projects (Table 2)
2
3
3
5
3
0
7
0
0
3
4
1
6
2
4
Teams
A
B
C
D
1
0
4
8
2
Projects (Table 3)
2
3
3
5
3
0
7
0
0
3
4
0
5
1
3
Teams
A
B
C
D
1
0
3
7
2
Projects (Table 4)
2
3
3
6
2
0
6
0
0
4
4
0
4
0
3
Teams
A
B
C
D
Final Decision:
Team A to Project 1
Team B to Project 3
Team C to Project 4
Team D to Project 2
Minimum Cost
₱
₱
Cost
15,000
14,000
15,000
15,000
59,000
4. The purchasing agent for ROA Company wishes to purchase replacement tires
for the company’s service vehicles. He needs 400 units of type A, 400 units of
type B, and 400 units of type C. Three tire suppliers have bid at the prices
given below. Each of them says it will supply all or part of the order. The
purchasing agent would like to buy tires in such a way that the entire
requirement of each type is purchased from a single dealer. Use the
assignment model to determine the least cost of purchasing plan.
Company
Firestone
Goodrich
Goodyear
Tire Type (Initial Table)
A
B
C
850
1,000
880
860
880
890
800
890
890
Company
Firestone
Goodrich
Goodyear
Tire Type (Table 2)
A
B
C
0
150
30
0
20
30
0
90
90
Company
Firestone
Goodrich
Goodyear
Tire Type (Table 3)
A
B
C
0
130
0
0
0
0
0
70
60
Final Decision:
Firestone to Tire Type C
Goodrich to Tire Type B
Goodyear to Tire Type A
Minimum Cost
Price
₱ 880
880
800
₱2,560
Qty.
x 400
x 400
x 400
1,200
Total Cost
₱ 352,000
352,000
320,000
₱ 1,024,000