Mysneak is a footwear distributor from New Zealand that carries most of the shoe brands sold in the country. The company owns several highly automated warehouses, where robots pick the right shoes and prepare orders to be sent to multiple retailers.

Joyce Roberts, director of Warehouse Operations, is responsible for managing the inventory of all the SKUs sold by Mysneak. The model U7 is a pair of winter boots. Joyce Roberts is looking forward to optimizing its inventory levels. The monthly demand for the U7 shoes follows a continuous uniform distribution with a minimum of a=105 and a maximum of b=204 units.

What is the probability that Mysneak will be able to fulfill the monthly demand for the U7 shoes if it has 190 units in stock?

Enter your answer as a decimal number with THREE decimal places. For example, if your answer is 76.06%, you should enter 0.761 in the answer box.

What is the probability of the monthly demand of the U7 shoes being between 160 and 190 units?

Enter your answer as a decimal number with THREE decimal places. For example, if your answer is 76.06%, you should enter 0.761 in the answer box.

Starting next month, Mysneak’s U7 shoes will be on sale, which Joyce Roberts thinks might impact product sales. For any fixed amount of inventory kept in stock for this SKU for the next month, which of the following options is true?

Select the correct answer.

If a increases by 20%, the probability of fulfilling the demand will also increase by 20%.

If b increases, the probability of fulfilling the demand could either increase or decrease.

If a and b both increase, the probability of fulfilling the demand will always decrease.

Question 2

The E6 model is Mysneak’s best selling product. For this SKU, Joyce found that monthly demand is normally distributed, with a mean of 548 and a standard deviation of 137. Taking this into consideration, answer the following questions:

How much stock of the E6 model should Mysneak have for one month of sales if the company wants to have a 98% probability of fulfilling the demand?

Round your answer to the nearest integer.

Joyce has just been notified that the automated warehouse has some capacity issues and it can only store 750 units of the E6 model every month. Which of the following options is true about this constraint?

Select the correct answer.

If Mysneak keeps 750 units of E6 in stock for the month, the probability of stocking out is only 1%.

This capacity limit will directly impact the probability of fulfilling the demand.

If the capacity limit doubles and Mysneak decides to keep 1500 units in stock, the probability of fulfilling the demand will also double.

Question 3

Besides inventory management, Joyce is facing another issue inherent in highly automated warehouses: the risk of the robots collapsing and stopping operations. To prevent this from happening, maintenance must be carried out timely. Given that the number of robots that require maintenance every hour follows a Poisson Distribution with lambda = 4.6, answer the following questions.

What is the probability of no robots needing maintenance in the next hour?

Enter your answer as a decimal number with THREE decimal places. For example, if your answer is 76.06%, you should enter 0.761 in the answer box.

Most recent reports showed that it takes 1 hour for a technician to fix a robot that collapses.

What is the probability that there will be a sufficient number of technicians to fix all robots that collapse in the next hour if Mysneak has 2 technicians available?

Enter your answer as a decimal number with THREE decimal places. For example, if your answer is 76.06%, you should enter 0.761 in the answer box.

Question 4

Finally, Joyce wants to analyze the demand data of a new model, the S9. This model was launched 15 weeks ago. The weekly demand data is still limited, which makes it challenging to forecast future sales. Joyce believes that the triangle distribution could be a good starting point to forecast the demand of S9. The available weekly demand data is presented in Table 1.

Table 1. Demand of S9 for the last 15 weeks

Week#             Demand (in units)

1                              11

2                            26

3                               60

4                              80

5                            60

6                           60

7                            41

8                             64

9                                37

10                                60

11                                  55

12                                  60

13                                   78

14                                  60

15                                    34

What would be the c parameter for this triangle distribution?

If Joyce decides to keep 70 units in stock for the next week, what is the probability of fulfilling the demand?

Enter your answer as a decimal number with THREE decimal places. For example, if your answer is 76.06%, you should enter 0.761 in the answer box.