The Joutec company sells many toys for children. Leaders know that the holiday season is the best time to introduce new toys, because many families use this time to look for new Christmas gift ideas. When the Joutec company discovers a new toy with high sales potential, it chooses to market it in October.

To have the toys on its shelves in October, Joutec places the order with its manufacturers in June or July each year. The demand for children's toys can be very volatile. If there is a craze for the new toy, a feeling of scarcity in the market often increases demand and significant profits can be made. However, the introduction of new toys can also result in failure, leaving the company with large stocks on their hands, which will have to be sold at a reduced price. The most important decision facing Joutec is to define the number of units that will be produced to meet potential demand. If too few toys are produced, the company loses sales (out of stock); if too many toys are produced, profits will be reduced because of the drop in price needed to run out of stocks.

For the coming season, Joutec plans to launch a new product called Mato l'ourson. This new version of a talking bear is made by a company in Taiwan. When a child presses the hand of the plush, the bear begins to speak. A barometer, placed inside the plush, selects one of five possible weather forecasts. Forecasts range from "It will be a great day. Take advantage! "To" I'm afraid it's raining today. Don't forget your umbrella! ". Tests have shown that, although not perfect, the sales forecasts were pretty good.

As with any product, Joutec must decide how many units to make. Different suggestions were made by the members of the management team: 15,000, 18,000, 24,000 or 28,000 units. The discrepancy between these proposals highlights the differences of opinion regarding the sales potential of this product. The managers call on you to analyze the probabilities that units remain unsold (on storage) in the different cases (15,000, 18,000, 24,000 or 28,000 units ordered) to estimate the potential profit and to make a recommendation as to the quantity to order. Joutec wants to sell Mato 24 dollars, knowing that the unit production cost is 16 dollars. If unsold inventory remains after the holidays, Joutec will sell each unit for $ 5. After reviewing the sales history for similar products, Joutec's chief sales forecaster forecasts demand for 20,000 units with a probability of 0.95 that demand will range from 10,000 to 30,000 units.

Report

Answer the following questions and recommend how much Mato to order.

1- Use sales forecasts to describe a normal probability distribution that can be used to estimate the distribution of demand. Represent the distribution and indicate its mean and its standard deviation.

2- Calculate the probability that there are unsold items for each of the order quantities suggested by the team of managers.

3- Calculate the expected profit for each of the quantities of orders suggested by the team of managers, under three alternative scenarios: the worst case with 10,000 units sold; the most likely case with 20,000 units sold; the most optimistic case with 30,000 units sold.

4- One of Joutec's managers thinks that the quantity ordered has a 0% chance of meeting demand and only a 30% chance of creating unsold inventory. In this context, how much should be ordered? What is the expected profit under the three sales scenarios?

5- Provide your own recommendation as to the quantity to order and give the expected profit for each of the three scenarios. Justify your recommendation.