Question

Leicester City Fanstore (LCF) will be selling the "new season jersey" for the 2018-2019 season. The regular price of the jersey is $80. Each jersey costs $40. Leftover jerseys will be sold at the end of the season (or later) at $30. Since jerseys are produced in China and lead time is long, Puma wants LCF to decide the quantity right now (December 2017).

a. After some analysis using historic data, LCF expects that the demand will follow a Normal distribution with a mean 40,000 and a standard deviation of 8,000 due to uncertainty in team performance. How many jerseys should LCF order?
b. If a customer cannot buy the jersey from LCF (in the case of a stock-out), they may leave the store disappointed and use other channels (such as Puma stores or puma.com) in future. LCF thinks that the lost customer goodwill is around $10. Should LCF change their decision in part (a)? If yes, please state the number of jerseys LCF should order.
c. Please state whether the following statement is always true, and give a brief explanation. If c=c, the newsvendor solution is the mean.

Answer 1

The cost of stock-outs is determined by multiplying the probability of stock-outs by the goodwill cost. LCF should not change their decision as the cost of stock-outs is lower than the difference in cost per jersey. The statement that the newsvendor solution is always the mean is not true when the cost per unit sold is different from the cost per unit not sold.

Step-by-step explanation:

The questions can be answered as -

a. To determine the quantity of jerseys LCF should order, we need to calculate the z-score for the desired service level using the formula:

z = (x - μ) / σ

where x is the desired service level, μ is the mean demand, and σ is the standard deviation. In this case, the desired service level is 40,000, the mean demand is 40,000, and the standard deviation is 8,000. Plugging these values into the formula gives:

z = (40,000 - 40,000) / 8,000 = 0

Looking up the corresponding value on the standard normal distribution table, we find that the probability of not stocking out is 0.5, which corresponds to a z-score of 0. Since this is exactly at the mean, LCF should order 40,000 jerseys.

b. To determine if LCF should change their decision based on the customer goodwill, we need to calculate the cost of stock-outs. The cost of stock-outs is calculated by multiplying the probability of a stock-out by the goodwill cost. In this case, the probability of a stock-out is the probability that demand exceeds the quantity ordered, which can be calculated by finding the area to the right of the z-score of 0 on the standard normal distribution table. This area is 0.5, so the probability of a stock-out is 0.5. Multiplying this by the goodwill cost of $10 gives a cost of stock-outs of $5 per jersey.

Comparing this cost to the difference in cost per jersey between the regular price and the price at the end of the season, we can determine if LCF should change their decision.

The difference in cost per jersey is $80 - $30 = $50. Since the cost of stock-outs per jersey is less than the difference in cost, LCF should not change their decision and should still order 40,000 jerseys.

c. The statement is not always true. The newsvendor solution is the mean only when the cost per unit sold is the same as the cost per unit not sold. In this case, the cost per unit sold is $80 and the cost per unit not sold is $30, so the statement does not hold.

Answer 2

Complete Question

Leicester City Fanstore (LCF) will be selling the "new season jersey" for the 2018-2019 season. The regular price of the jersey is $80. Each jersey costs $40. Leftover jerseys will be sold at the end of the season (or later) at $30. Since jerseys are produced in China and lead time is long, Puma wants LCF to decide the quantity right now (December 2017).

a. After some analysis using historic data, LCF expects that the demand will follow a Normal distribution with a mean 40,000 and a standard deviation of 8,000 due to uncertainty in team performance. How many jerseys should LCF order?

b. If a customer cannot buy the jersey from LCF (in the case of a stock-out), they may leave the store disappointed and use other channels (such as Puma stores or puma.com) in future. LCF thinks that the lost customer goodwill is around $10. Should LCF change their decision in part (a)? If yes, please state the number of jerseys LCF should order.

c. Please state whether the following statement is always true, and give a brief explanation. If
`C_o =C_i`, the news vendor solution is the mean.

Answer:

a

`N = 46728`

b

`n = 47728`

c

Yes it is always true

Step-by-step explanation:

From the question we are told that

The regular price of the jersey is
`P_r = \$ 80`

The cost of producing a jersey is
`C= \$ 40`

The left-over price of the jersey is
`P_o = $ 30`

The mean is
`\mu = 40000`

The standard deviation is
`\sigma = 8000`

The cost of lost customer goodwill is
`C_g = \$ 10`

Generally the fund that LCF will loss for one jersey if they order for too many jersey (i.e more than they need )is mathematically represented

`C_o = P_o - C`

=>
`C_o = 40 - 30`

=>
`C_o = \$ 10`

Generally the fund that LCF will loss for one jersey if they order lesser amount jersey (i.e less than they need )is mathematically represented

`C_i = P_r - C`

=>
`C_i = 80 - 40`

=>
`C_i = \$ 40`

Generally the critical ratio is mathematically represented as

`Z = (C_i )/( C_i + C_o)`

=>
`Z = (40)/( 40 + 10)`

=>
`Z = 0.8`

Generally the critical value of
`Z = 0.8` to the right of the normal curve is

`z = 0.841`

Generally the optimal quantity of jersey to order is mathematically represented as

`N = \mu * [z * \sigma]`

=>
`N = 40000 * [0.841 * 8000]`

=>
`N = 46728`

Considering question b

Generally considering the factor of customer goodwill the fund that LCF will loss for one jersey if they order lesser amount jersey (i.e less than they need )is mathematically represented as

`C_k = C_i + C_g`

=>
`C_k = 40 + 10`

=>
`C_k = \$ 50`

Now the critical ratio is mathematically represented as

`Z = (C_k )/( C_k + C_o)`

=>
`Z = (50)/( 50 + 10)`

=>
`Z = 0.833`

Generally the critical value of
`Z = 0.833` to the right of the normal curve is

`z = 0.966`

Generally the optimal quantity of jersey to order is mathematically represented as

`n = \mu * [z * \sigma]`

=>
`n = 40000 * [0.966 * 8000]`

=>
`n = 47728`

Considering question c

When
`C_o =C_i` then

The critical ratio is mathematically represented as

`Z = (C_k )/( C_k + C_k)`

=>
`Z = (1)/( 2)`

=>
`Z = 0.5`

Generally the critical value of
`Z = 0.5` to the right of the normal curve is

`z = 0`

So

The optimal quantity of jersey to order is mathematically represented as

`n = \mu * [z * \sigma]`

=>
`n = 40000 * [0* 8000]`

=>
`n = 40000 = \mu`

Hence the statement in c is true