Consider the production of high$-$end electric bicycles $($e$-$bikes$).$

You are a procurement manager for a company that manufactures these e$-$bikes. The demand for the e$-$bikes in the summer months $($June$,$ July, August$)$ is normally distributed with a mean of $10,000$ units and a standard deviation of $2,000$ units.

Your supplier offers two types of contracts for the lithium$-$ion batteries needed in the e$-$bikes:

Forward Contract: You can buy batteries at a fixed price of $$100$ per battery.

Option Contract: You can pay an initial fee of $$10$ per battery for the right to buy batteries at a strike price of $$105$ per battery.

You also have the option to purchase on the spot market. The spot market price is forecasted to be normally distributed with a mean of $$105$ and a standard deviation of $$15.$

The question is: How many lithium$-$ion batteries should you purchase via forward contracts and option contracts?