Linear Programming Mathematics homework help

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Any one question from section B.

Due in 3 hours

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SECTION B: Candidates should answer one question from this section

3. Debbie Gibson is considering four investment options for a small inheritance that she has just received—crypto currency, foreign exchange market, stocks, and bonds. For Debbie, the crypto market has 3 options. Option 1 (BTC) is a well stablished crypto currency with a low deduction possibility but with also lower returns. Option 2 (SOL) is a rather established crypto currency and option 3 (Mina) is a newer crypto that has been recently added to the market. The returns and the probabilities for each of these options is shown in the following table:

Option

Strong

Average

Weak

Probability

Return

Probability

Return

Probability

Return

BTC

0.75

4%

0.15

1%

0.10

-2%

SOL

0.32

11%

0.43

0

0.25

-5%

Mina

0.14

38%

0.35

5%

0.51

-17%

Debbie considers two foreign currencies for her investment if she wants to invest in the foreign exchange market—US Dollars and Chinese Yuan. The position of these currencies in the market are not entirely predictable and following table gives returns and corresponding probabilities to it:

Option

Strong

Average

Weak

Probability

Return

Probability

Return

Probability

Return

US Dollar ($)

0.32

3%

0.32

1%

0.36

-2.5%

Chinese Yuan (¥)

0.21

18%

0.53

-1%

0.26

-6%

The returns from the stocks market and bonds are as follows: the probabilities of different cases are similar, e.g., the probabilities of a strong, average, or weak market are all equal to 1/3.

Investment

Strong

Average

Weak

Stocks

10%

4%

-9%

Bonds

4%

3%

2%

a) Construct a decision tree to help Debbie in her decision. Which option is the best for her? [30%]

b) What is the difference between decision making under risk and decision making under uncertainty? How do you categorize Debbie’s problem? Assume that Debbie does not have any information about the probabilities given above. How does this change the problem? What would be the best decisions for Debbie under different circumstances? [25%]

c) Assume that Debbie is planning to invest £10,000 of her inheritance. Debbie can pay for an online service that analyses different crypto currencies and identifies their future in the market. The outcome of this analysis can be either positive or negative for each of the crypto currencies. She must pay £100 for each individual crypto currency to have access to its analysis. While the return of each of the crypto currencies remain the same, from Debbie’s perspective, their corresponding probabilities will change once Debbie has access to their analysis.

In the case of positive or negative outcome, the probabilities for different markets are as following:

Positive Outcome

Negative Outcome

Option

Strong

Average

Weak

Strong

Average

Weak

BTC

0.95

0.04

0.01

0.10

0.72

0.18

SOL

0.80

0.15

0.05

0.06

0.42

0.52

Mina

0.79

0.18

0.03

0.02

0.23

0.75

Moreover, assume that the probability of a positive return for BTC is 70%, for SOL is 60% and for Mina is 50%. How does this change the structure of the decision tree? What is the best option for Debbie? [45%]

4. Company XYZ uses a production machine that has three gears. The machine can only operate if all these gears are working properly. Once one of the gears breaks, the machine stops until the gear is replaced with a new one. The engineers of company XYZ have studied the useful life of these gears and found that the life expectancy distribution of the gears is as follows:

Life of a gear (in hours)

Probability

1000

0.10

1100

0.13

1200

0.25

1300

0.13

1400

0.09

1500

0.12

1600

0.02

1700

0.06

1800

0.05

1900

0.05

Once a gear breaks, a mechanic replaces the gear but with a delay. The delay to bring in the mechanic has the following distribution:

Delay time (in minutes)

Probability

5

0.6

10

0.3

15

0.1

The mechanic’s working time to replacing one gear is random and follows the following probability distribution:

Time to replace a gear (in minutes)

Probability

15

0.3

20

0.4

25

0.3

The cost of a minute without the machine is estimated at £5 and the cost of the mechanic is £12 per hour. The cost of each gear is £16.

a) Set up a flow chart showing the logical sequence of events for simulating Company XYZ’s expected costs. Provide all the details of the formulas used for relevant calculations. [15%]

b) Assume that breakages of different gears do not occur at the same time. Use the random numbers (between 0 and 1) below (in the order as they appear) and simulate 20000 hours of this production machine. Organise all your calculation in a table. Calculate the costs and the number of times each of the gears had to be replaced. [35%]

0.76 0.92 0.87 0.65 0.23 0.78 0.61 0.73 0.24 0.96 0.11 0.33 0.65 0.29 0.28 0.56

0.55 0.37 0.11 0.01 0.91 0.86 0.48 0.63 0.57 0.59 0.05 0.49 0.39 0.07 0.36 0.83

0.77 0.44 0.07 0.91 0.94 0.41 0.28 0.63 0.68 0.36 0.79 0.71 0.23 0.01 0.23 0.19

0.12 0.86 0.74 0.23 0.93 0.42 0.35 0.83 0.65 0.63 0.33 0.93 0.21 0.19 0.91 0.81

0.51 0.71 0.29 0.49 0.41 0.38 0.12 0.33 0.41 0.48 0.42 0.01 0.09 0.62 0.46 0.44

0.88 0.24 0.46 0.25 0.97 0.4 0.28 0.65 0.5 0.56 0.82 0.04 0.34 0.17 0.66 0.60

0.24 0.56 0.45 0.22 0.48 0.02 0.40 0.67 0.59 0.94 0.56 0.42 0.61 0.24 0.37 0.91

0.78 0.11 0.91 0.27 0.14 0.29 0.80 0.39 1.00 0.13 0.81 0.97 0.20 0.24 0.08 0.44

0.65 0.31 0.75 0.09 0.93 0.45 0.65 0.51 0.27 0.86 0.18 0.05 0.96 0.71 0.25 0.38

0.87 0.00 0.07 0.11 0.99 0.16 0.75 0.18 0.66 0.42 0.61 0.09 0.63 0.77 0.42 0.08

0.85 0.80 0.43 0.22 0.06 0.09 0.23 0.17 0.32 0.92 0.46 0.55 0.40 0.43 0.15 0.23

0.13 0.48 0.90 0.16 0.28 0.96 0.21 0.66 0.15 0.60 0.10 0.78 0.32 0.60 0.32 0.11

0.48 0.44 0.78 0.35 0.65 0.49 0.82 0.37 0.76 0.21 0.23 0.74 0.79 0.57 0.37 0.58

c) The engineers suggested that each time a gear breaks, instead of changing that gear, they replace all three gears. Changing all three gears at the same time takes a random amount of time for the mechanic:

Time to replace three gears (in minutes)

Probability

30

0.1

40

0.45

50

0.3

60

0.15

Using the following random numbers, simulate 20000 hours of this production machine with this new assumption. Calculate the costs and comment on your observations. [30%]

0.18 0.05 0.96 0.71 0.25 0.38 0.87 0.00 0.07 0.11 0.99 0.16 0.75 0.18 0.66 0.42

0.61 0.09 0.63 0.77 0.42 0.08 0.85 0.80 0.43 0.22 0.06 0.09 0.23 0.17 0.32 0.92

0.46 0.55 0.40 0.43 0.15 0.23 0.13 0.48 0.90 0.16 0.28 0.96 0.21 0.66 0.15 0.60

0.10 0.78 0.32 0.60 0.32 0.11 0.48 0.44 0.78 0.35 0.65 0.49 0.82 0.37 0.76 0.21

0.23 0.74 0.79 0.57 0.37 0.58 0.07 0.72 0.96 0.38 0.50 0.59 0.57 0.38 0.03 0.21

0.81 0.03 0.58 0.47 0.10 0.68 0.73 0.46 0.90 0.44 0.25 0.89 0.86 0.76 0.92 0.87

0.55 0.37 0.11 0.01 0.91 0.86 0.48 0.63 0.57 0.59 0.05 0.49 0.39 0.07 0.36 0.83

0.77 0.44 0.07 0.91 0.94 0.41 0.28 0.63 0.68 0.36 0.79 0.71 0.23 0.01 0.23 0.19

0.12 0.86 0.74 0.23 0.93 0.42 0.35 0.83 0.65 0.63 0.33 0.93 0.21 0.19 0.91 0.81

0.51 0.71 0.29 0.49 0.41 0.38 0.12 0.33 0.41 0.48 0.42 0.01 0.09 0.62 0.46 0.44

0.88 0.24 0.46 0.25 0.97 0.4 0.28 0.65 0.5 0.56 0.82 0.04 0.34 0.17 0.66 0.60

0.24 0.56 0.45 0.22 0.48 0.02 0.40 0.67 0.59 0.94 0.56 0.42 0.61 0.24 0.37 0.91

0.78 0.11 0.91 0.27 0.14 0.29 0.80 0.39 1.00 0.13 0.81 0.97 0.20 0.24 0.08 0.44

0.65 0.31 0.75 0.09 0.93 0.45 0.65 0.51 0.27 0.86 0.18 0.05 0.96 0.71 0.25 0.38

d) Based on your calculations in part c, if all the other parameters remain unchanged, what is the maximum value of a gear that justifies simultaneous replacement of gears even if only one of them is broken? [10%]

e) What is the minimum number of machines that justifies hiring a mechanic with a yearly salary of £18000? You may assume that machines run 24 hours a day during 365 days of the year and with the hired mechanic, the company will not face any delays in repair times. To justify your answer, you can make more assumptions. State them clearly. [10%]

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