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Question $1$

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Daily sales of an expensive item have been found to follow the distribution given below:

Sales

$0$

$1$

$2$

$3$

Probability

$0\mathrm{.}2$

$0\mathrm{.}1$

$0\mathrm{.}1$

$0\mathrm{.}6$

Which of the following is a valid sampling scheme for this distribution? Answer $1$ Question $1$

Simulate sales for the next three days, using the following stream of random numbers, starting at the leftmost number and using the correct sampling scheme above:

$784111627218$

What is your simulated total sales over the next three days? Answer $2$ Question $1$

Question $2$

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A spare part is demanded infrequently, with the average time between demands being $6$ weeks. Previous analysis has shown that the time between demands follows a negative exponential distribution.

Simulate the next inter$-$arrival times using the following stream of random numbers, starting at the leftmost number, and using four$-$digit random numbers:

$9042074505$

If a part has just been demanded, what is your simulated arrival time for the demand for the next part?

Multiple choice $1$ Question $2$

$14\mathrm{.}1$ weeks

$6\mathrm{.}0$ weeks

$0\mathrm{.}6$ weeks

$6\mathrm{.}1$ weeks

Question $3$

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A store places orders every week and receives the orders at the start of the next week. If demand is less than the stock available, then all demand is met and the remaining stock $($known as the $$closing stock$)$ is available for the next week. If demand exceeds supply, then any excess demand is not satisfied and the store has lost sales.

A simulation has been conducted for two weeks and is shown in the table below. Some elements of the simulation of the third week have also been completed.

Week

Opening Stock

Order

Arrival Week

Delivery

Demand

Closing Stock

Lost Sales

$1$

$30$

$20$

$2$

$0$

$25$

$5$

$0$

$2$

$5$

$30$

$3$

$20$

$20$

$5$

$0$

$3$

$30$

$4$

$40$

What is the simulated closing stock and lost sales at the end of the third week?

Multiple choice $1$ Question $3$

Closing stock of $0,$ Lost sales of $5$

Closing stock of $5,$ Lost sales of $15$

Closing stock of $0,$ Lost sales of $15$

Closing stock of $5,$ Lost sales of $0$