You've invented a new type of high-tech hair tie that comes equipped with Bluetooth technology. Demand...

 

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You've invented a new type of high-tech hair tie that comes equipped with Bluetooth technology. Demand for your new product is estimated to be a steady 1,200,000 units per year at a selling price of $3 per hair tic. You've signed a contract with a major manufacturer who has agreed to manufacture the hair ties for you. You will order hair ties from the manufacturer in batches and then you plan to store the hair ties in inventory as they await sale. When your inventory depletes to zero you will then order a new batch from the manufacturer. When you place an order with the manufacturer for a batch of hair ties, the manufacturer charges a flat fee of $5000 plus $1 per hair tie. So, for example if you ordered a batch of 1000 hair ties the manufacturer will charge you $5000 +$11000-$6000. You can order batches of any size up to a maximum of 1,200,000 hair ties, but you have to pay to store the resulting hair ties as they await sale. Due to the high-tech nature of your product, you need special climate-controlled storage facilities. The AAA Storage Company has the required facilities, and it has quoted you a price for storage of $0.12 per hair tie per year. Unfortunately, it only has enough space to store a maximum of 300,000 hair ties at any one time. You have to figure out the optimal batch size to order from the manufacturer that will maximize your total profit. a) Write down an equation for the total annual profit (i.e., Total annual revenue-total annual manufacturing cost- total annual storage cost) as a function of the batch size x (i.e., x is the number of hair ties you order in each batch). Make sure to state all constraints. b) Use calculus to determine the optimal batch size, and from this determine your anticipated total annual profit. c) If the storage facility had an unlimited capacity, how would your answers to part b change? You've invented a new type of high-tech hair tie that comes equipped with Bluetooth technology. Demand for your new product is estimated to be a steady 1,200,000 units per year at a selling price of $3 per hair tic. You've signed a contract with a major manufacturer who has agreed to manufacture the hair ties for you. You will order hair ties from the manufacturer in batches and then you plan to store the hair ties in inventory as they await sale. When your inventory depletes to zero you will then order a new batch from the manufacturer. When you place an order with the manufacturer for a batch of hair ties, the manufacturer charges a flat fee of $5000 plus $1 per hair tie. So, for example if you ordered a batch of 1000 hair ties the manufacturer will charge you $5000 +$11000-$6000. You can order batches of any size up to a maximum of 1,200,000 hair ties, but you have to pay to store the resulting hair ties as they await sale. Due to the high-tech nature of your product, you need special climate-controlled storage facilities. The AAA Storage Company has the required facilities, and it has quoted you a price for storage of $0.12 per hair tie per year. Unfortunately, it only has enough space to store a maximum of 300,000 hair ties at any one time. You have to figure out the optimal batch size to order from the manufacturer that will maximize your total profit. a) Write down an equation for the total annual profit (i.e., Total annual revenue-total annual manufacturing cost- total annual storage cost) as a function of the batch size x (i.e., x is the number of hair ties you order in each batch). Make sure to state all constraints. b) Use calculus to determine the optimal batch size, and from this determine your anticipated total annual profit. c) If the storage facility had an unlimited capacity, how would your answers to part b change?