PROBLEM 5 Consider a two-period production model where 1,000 kgs of a certain product are required in the first period (current period) and 1,000 kgs are required in the second period (future period). In this production system, the output quantity Pin kgs recovered from your production process is a fraction of the input quantity Q in kgs that you launch into your production process. Assume that a fraction of the input quantity launched into the process are lost due to factors beyond your control. Denote the output quantity by P = UQ where U is a random variable uniformly distributed between 0 and 1. Unfortunately, production cost depends on the input quantity, and is incurred at $10 per kg of input quantity. Furthermore, if you fail to deliver 1,000 kgs in each period, you incur a penalty of $20 per kg short. Determine the optimal production quantity in the first period. (20 pts)