# QM 670 Exam II

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

0-99

$36

100-199

32

200-299

30

300+

28

For a price of $36, the optimal order quantity is ___________. (4)

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

Optimal order quantity = ___________. (4)

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

0-99

$36

100-199

32

200-299

30

300+

28

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 revenue-producing instruments are summarized in the table below:

Instrument

Annual Rate of Return (%)

Risk Factor

Liquidity Factor

Automobile loans

8

50

0

Furniture loans

10

60

0

Other secured loans

11

70

0

Unsecured loans

14

80

0

Risk-free securities

5

0

100

Corporate stock fund

9

60

90

Corporate bond fund

8

50

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)

5. Powers Tire Company (which specializes in tires for vehicles ca. 1965) needs your expertise to help them schedule over the next three months. They can make tires using regular labor, overtime labor, or subcontracting. Their capacities (in tires) are as follows: 700/month for regular labor, 50/mo. for overtime labor, and 150/mo. for subcontracted labor in March and April, and 130/mo. for subcontracted labor for May. It costs $40/tire to produce with regular labor, $50/tire with overtime, and $70/tire with subcontracting. It costs $2/tire/month to carry a tire in inventory. Tires can be produced only for the current or future periods, e.g., tires cannot be produced in May to fill demand in April. Demands for the next three periods are 800, 1000, and 750, for March, April, and May, respectively, and all demand must be met. Check (or shade) all that apply. (12)

This is an integer program.

This is a transportation problem.

The cost of using regular labor in April to fill March demand is $40/tire.

The cost of using overtime labor in April to fill May demand is $52/tire.

We would need at least one dummy row.

We would need at least one dummy column.

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

Proportional Content

Total Requirement

Buckwheat

Sunflower

Poppy

Fat

0.04

0.06

0.05

480

Protein

0.12

0.10

0.10

1200

Roughage

0.10

0.15

0.07

1500

Cost/lb.

$0.18

$0.10

$0.11

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

LINEAR PROGRAMMING PROBLEM

MIN 0.18X1+0.1X2+0.11X3

S.T.

1) .04X1+.06X2+.05X3>480

2) .12X1+.1X2+.1X3>1200

3) .1X1+.15X2+.07X3<1500 4) 1X3>500

5) 1X1-1X2<0 OPTIMAL SOLUTION Objective Function Value = 1237.500 Variable Value Reduced Costs -------------- --------------- ------------------ X1 0.000 0.050 X2 8250.000 0.000 X3 3750.000 0.000 Constraint Slack/Surplus Dual Prices -------------- --------------- ------------------ 1 202.500 0.000 2 0.000 -1.188 3 0.000 0.125 4 3250.000 0.000 5 8250.000 0.000 OBJECTIVE COEFFICIENT RANGES Variable Lower Limit Current Value Upper Limit ------------ --------------- --------------- --------------- X1 0.130 0.180 No Upper Limit X2 No Lower Limit 0.100 0.110 X3 0.100 0.110 0.160 RIGHT HAND SIDE RANGES Constraint Lower Limit Current Value Upper Limit ------------ --------------- --------------- --------------- 1 No Lower Limit 480.000 682.500 2 1026.667 1200.000 2142.857 3 840.000 1500.000 1760.000 4 No Lower Limit 500.000 3750.000 5 -8250.000 0.000 No Upper Limit (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. ________________________

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