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

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

             $Z = \frac{C_i }{ C_i + C_o}$

=>           $Z = \frac{40}{ 40 + 10}$

=>           $Z = 0.8$

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

             $Z = \frac{C_k }{ C_k + C_o}$

=>           $Z = \frac{50}{ 50 + 10}$

=>           $Z = 0.833$

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

             $Z = \frac{C_k }{ C_k + C_k}$

=>           $Z = \frac{1}{ 2}$

=>           $Z = 0.5$          

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