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QM 670 Exam II

1. Kelvin Shoe Stores carries a basic black dress shoe for men that sells at a rate of 500 each

quarter. Their current policy is to order 500 per quarter, with a fixed cost of $30/order.

The annual holding cost is 20% of the cost of items held. The following cost structure is

applicable:

Order Quantity

Price/pair

0-125

$35

126-225

34

226-350

33

351+

32

For a price of $35, the optimal order quantity (ignoring the price breaks) is ___________.

(4)

2. Redo #1 if they allow backordered items with a shortage cost of $4/year.

Optimal order quantity = ___________. (4)

3. Kelvin Shoe Stores carries a basic black dress shoe for men that sells at a rate of 500 each

quarter. Their current policy is to order 500 per quarter, with a fixed cost of $30/order.

The annual holding cost is 20% of the cost of items held. The following cost structure is

applicable:

Order Quantity

Price/pair

0-125

$35

126-225

34

226-350

33

351+

32

The optimal order quantity is ______________. (5)

4. The Employee Credit Union at Directional State University is planning the allocation of

funds for the coming year. ECU makes four types of loans and has three additional

investment instruments. Each loan/investment has a corresponding risk and liquidity

factor (on a scale of 0-100, with 100 being the most risky/liquid). The various revenueproducing instruments are summarized in the table below:

Instrument

Automobile loans

Furniture loans

Other secured loans

Unsecured loans

Risk-free securities

Corporate stock fund

Corporate bond fund

Annual Rate of Return (%)

8

10

11

14

5

9

8

Risk Factor

50

60

70

80

0

60

50

Liquidity Factor

0

0

0

0

100

90

80

ECU has $2,000,000 available for investment during the coming year. However, state laws and

pesky stakeholders impose certain restrictions on choice of investment instruments. Risk-free

securities may not exceed 30% of total funds available for investment. Unsecured loans may not

exceed 10% of total funds invested in loans. The funds invested in automobile loans must not be

less than the total of funds invested in furniture and other secured loans. The average risk factor

may not exceed 60, and the average liquidity factor must be at least 40. (14)

Check/shade ALL that apply.

There are seven decisionvariables ____

There are sixconstraints (not counting non-negativity).____

We determine the average risk factor by summing risk values and dividing

by 7.____

Risk-free security totalinvestment may exceed$800,000.____

All $2,000,000 must be invested.____

This isa maximization problem.____

This problemcannot be run as anintegerprogram.____

5. You are asked to assign four patients to five nurses using the following travel distances.

We want to minimize total mileage. (10)

Nurse

Amy

Brenda

Connor

Danielle

Emily

Warren

20

15

32

41

24

Patient

Xavier

Yolanda

25

15

18

27

22

17

16

12

29

19

Zanthia

12

20

23

28

16

a. The optimal solution is:

b. The objective function value = _________.

c. If we formulated this as a linear program, how many decision variables would we have?

d. If we added a fifth patient, would she be assigned to Emily? Explain

e. This is aninteger program. (True / False)

6. Hungry Birds, Inc. manufactures birdseed. One variety consists of wheat. They are

trying to determine the optimal mix of buckwheat (X1), sunflower (X2), and poppy (X3)

(each in lbs.). Relevant information is provided in the following table. In addition, the

final mix is required to contain at least 500 lbs. of poppy. Also, the total weight of the

buckwheat may not exceed the total weight of the sunflower in the final mix. (15)

Nutritional Item

Fat

Protein

Roughage

Cost/lb.

Buckwheat

0.04

0.12

0.10

$0.18

Proportional Content

Sunflower

0.06

0.10

0.15

$0.10

Poppy

0.05

0.10

0.07

$0.11

The output of the linear program is givenbelow and on the following page.

LINEAR PROGRAMMING PROBLEM

MIN 0.18X1+0.1X2+0.11X3

S.T.

1)

2)

3)

4)

5)

.04X1+.06X2+.05X3>480

.12X1+.1X2+.1X3>1200

.1X1+.15X2+.07X3<1500
1X3>500

1X1-1X2<0
OPTIMAL SOLUTION
Objective Function Value =
1237.500
Variable
-------------X1
X2
X3
Value
--------------0.000
8250.000
3750.000
Reduced Costs
-----------------0.050
0.000
0.000
Constraint
-------------1
2
3
4
5
Slack/Surplus
--------------202.500
0.000
0.000
3250.000
8250.000
Dual Prices
-----------------0.000
-1.188
0.125
0.000
0.000
Total
Requirement
480
1200
1500
OBJECTIVE COEFFICIENT RANGES
Variable
-----------X1
X2
X3
Lower Limit
--------------0.130
No Lower Limit
0.100
Current Value
--------------0.180
0.100
0.110
Upper Limit
--------------No Upper Limit
0.110
0.160
Current Value
--------------480.000
1200.000
1500.000
500.000
0.000
Upper Limit
--------------682.500
2142.857
1760.000
3750.000
No Upper Limit
RIGHT HAND SIDE RANGES
Constraint
-----------1
2
3
4
5
a.
Lower Limit
--------------No Lower Limit
1026.667
840.000
No Lower Limit
-8250.000
If this had been run as an integer program, we would have obtained a different
solution. (Check/shade if true.)
b. If we could reduce the fat requirement by 100 lbs., the optimal solution would not
change. (Check/shade if true.)
c. A new customer wants a mix with at least 20% buckwheat. Would this change the
optimal solution? If so, would it increase or decrease? Check/shade the following:
Change?
Decrease?
d.
Nora in Accounting noted a glitch in her software, and stated that the cost estimates
should be changed. She said the cost values should be $0.17 for buckwheat, $0.12 for
sunflower, and $0.12 for poppy. Would this be a cause for concern? If so, which
component(s) would be affected? Check/shade the following:
We should be concerned.
Buckwheat?
Sunflower?
Poppy?
e. If you could relax the requirement on one nutritional item, which would be the best
choice to achieve the lowest cost? Fill in the blank.
________________________
1. Kelvin Shoe Stores carries a basic black dress shoe for men that sells at a rate of 500 each
quarter. Their current policy is to order 500 per quarter, with a fixed cost of $30/order.
The annual holding cost is 20% of the cost of items held. The following cost structure is
applicable:
Order Quantity
Price/pair
0-125
$35
126-225
34
226-350
33
351+
32
For a price of $35, the optimal order quantity is ___________. (4)
Solution
EOQ = sqrt (2*(500*4)*30/(0.2*35)) = 130.93 = 131 units
As, for $35, only 0-125 can be ordered.
So, EOQ is infeasible at this cost structure
2. Redo #1 if they allow backordered items with a shortage cost of $4/quarter.
Optimal order quantity = ___________. (4)
Solution
If backordered items with shortage cost of $4/quarter was there, the decision will not change
regarding optimal quantity.
3. Kelvin Shoe Stores carries a basic black dress shoe for men that sells at a rate of 500 each
quarter. Their current policy is to order 500 per quarter, with a fixed cost of $30/order.
The annual holding cost is 20% of the cost of items held. The following cost structure is
applicable:
Order Quantity
Price/pair
0-125
$35
126-225
34
226-350
33
351+
32
The optimal order quantity is ______________. (5)
Solution
For $35
EOQ = sqrt (2*(500*4)*30/(0.2*35)) = 130.93 = 131 units
As, for $35, only 0-125 can be ordered, so this is not optimal
For $34
EOQ = sqrt (2*(500*4)*30/(0.2*34)) = 136.93 = 133 units
For $33
EOQ = sqrt (2*(500*4)*30/(0.2*33) = 134.83 = 135 units
As, for $33, only 226-350 can be ordered, so this is not optimal
For $32
EOQ = sqrt (2*(500*4)*30/(0.2*32)) = 136.93= 137units
As, for $32, only >351 can be ordered, so this is not optimal

So, optimal order quantity is 133 units

4. The Employee Credit Union at Directional State University is planning the allocation of

funds for the coming year. ECU makes four types of loans and has three additional

investment instruments. Each loan/investment has a corresponding risk and liquidity

factor (on a scale of 0-100, with 100 being the most risky/liquid). The various revenueproducing instruments are summarized in the table below:

Instrument

Automobile loans

Furniture loans

Other secured loans

Unsecured loans

Risk-free securities

Corporate stock fund

Corporate bond fund

Annual Rate of Return (%)

8

10

11

14

5

9

8

Risk Factor

50

60

70

80

0

60

50

Liquidity Factor

0

0

0

0

100

90

80

ECU has $2,000,000 available for investment during the coming year. However, state laws and

pesky stakeholders impose certain restrictions on choice of investment instruments. Risk-free

securities may not exceed 30% of total funds available for investment. Unsecured loans may not

exceed 10% of total funds invested in loans. The funds invested in automobile loans must not be

less than the total of funds invested in furniture and other secured loans. The average risk factor

may not exceed 60, and the average liquidity factor must be at least 40. Formulate a linear

program for ECU. DO NOT SOLVE THE LP. (10)

Solution

Let the variables be

X1 = Automobile loans

X2 = Furniture loans

X3 = Other secured loans

X4 = Unsecured loans

X5 = Risk-free securities

X6 = Corporate stock fund

X7 = Corporate bond fund

Objective function is to maximize return

Maximize Z = 0.08*X1 + 0.1*x2 +0.11*x3 +0.14*x4 +0.05*x5 +0.09*x6 +0.08*x7

Constraints:

Budget: X1 +x2 +x3+x4+x5+x6+x7 = 2,000,000

Risk Free: x5 = 0.3*(2,000,000) ? x5 = 600,000

Unsecured loans: x4 = 0.1*(x1 +x2 +x3 +x4) ? -0.1x 0.1×2 0.1×3 + 0.9×4 = 0

Automobile loans: x1 = (x2 + x3) ? x1 x2 x3 = 0

Risk: (50*X1 + 60*x2 +70*x3 +80*x4 +0*x5 +60*x6 +50*x7) / (X1 +x2 +x3+x4+x5+x6+x7) = 60

Liquidity: (0*X1 + 0*x2 +0*x3 +0*x4 +100*x5 +90*x6 +80*x7) / (X1 +x2 +x3+x4+x5+x6+x7) = 40

Non negative: X1 to X7 = 0

6. Hungry Birds, Inc. manufactures birdseed. One variety consists of wheat. They are

trying to determine the optimal mix of buckwheat (X1), sunflower (X2), and poppy (X3)

(each in lbs.). Relevant information is provided in the following table. In addition, the

final mix is required to contain at least 500 lbs. of poppy. Also, the total weight of the

buckwheat may not exceed the total weight of the sunflower in the final mix.

Nutritional Item

Fat

Protein

Roughage

Cost/lb.

Buckwheat

0.04

0.12

0.10

$0.18

Proportional Content

Sunflower

0.06

0.10

0.15

$0.10

Poppy

0.05

0.10

0.07

$0.11

The output of the linear program is given on the following page.

Total

Requirement

480

1200

1500

LINEAR PROGRAMMING PROBLEM

MIN 0.18X1+0.1X2+0.11X3

S.T.

1)

2)

3)

4)

5)

.04X1+.06X2+.05X3>480

.12X1+.1X2+.1X3>1200

.1X1+.15X2+.07X3<1500
1X3>500

1X1-1X2<0
OPTIMAL SOLUTION
Objective Function Value =
1237.500
Variable
-------------X1
X2
X3
Value
--------------0.000
8250.000
3750.000
Reduced Costs
-----------------0.050
0.000
0.000
Constraint
-------------1
2
3
4
5
Slack/Surplus
--------------202.500
0.000
0.000
3250.000
8250.000
Dual Prices
-----------------0.000
-1.188
0.125
0.000
0.000
OBJECTIVE COEFFICIENT RANGES
Variable
-----------X1
X2
X3
Lower Limit
--------------0.130
No Lower Limit
0.100
Current Value
--------------0.180
0.100
0.110
Upper Limit
--------------No Upper Limit
0.110
0.160
Current Value
--------------480.000
1200.000
1500.000
500.000
0.000
Upper Limit
--------------682.500
2142.857
1760.000
3750.000
No Upper Limit
RIGHT HAND SIDE RANGES
Constraint
-----------1
2
3
4
5
Lower Limit
--------------No Lower Limit
1026.667
840.000
No Lower Limit
-8250.000
(15)
a.
If this had been run as an integer program, we would obtain a different solution.
(Check/shade if true.)
b. If we could reduce the fat requirement by 100 lbs., the optimal solution would
change. (Check/shade if true.)
c. A new customer wants a mix with at least 20% buckwheat. Would this change the
optimal solution? If so, would it increase or decrease? Check/shade the following:
Change?
Increase?
d.
Nora in Accounting noted a glitch in her software, and stated that the cost estimates
should be changed. She said the cost values should be $0.17 for buckwheat, $0.12 for
sunflower, and $0.12 for poppy. Would this be a cause for concern? If so, which
component(s) would be affected? Check/shade the following:
We should be concerned.
Buckwheat?
Sunflower?
Poppy?
e. If you could relax the requirement on one nutritional item, which would be the best
choice to achieve the lowest cost? Fill in the blank.
Roughage requirement should be relaxed
5. You are asked to assign four patients to five nurses using the following travel distances.
We want to minimize total mileage. (10)
Nurse
Amy
Brenda
Connor
Danielle
Emily
Warren
20
15
32
41
24
Patient
Xavier
Yolanda
25
15
18
27
22
17
16
12
29
19
Zanthia
12
20
23
28
16
a. The optimal solution is:
Danielle is assigned to Yolanda.
b. The objective function value = ___60______.
c. If we formulated this as a linear program, how many decision variables would we have?
25
d. If we added a fifth patient, would she be assigned to Emily? Explain
Yes, she would be assigned to Emily.
e. This is aninteger program. (True / False)
FLASE. This is a transportation problem
...
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