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QM 670

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 decision variables ____

There are six constraints (not counting non-negativity).____

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

by 7.____

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

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

This is a maximization problem.____

This problem cannot be run as an integer program.____

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 an integer 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

Total

Requirement

480

1200

1500

The output of the linear program is given below 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
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.
________________________
...
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