# Quantitative Method questions

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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 ... Purchase answer to see full attachment

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