This problem considers a four-months inventory problem where a company must determine how many products should be produced at the beginning of each month. A setup cost of $6 is incurred during each month in which production takes place. The unit production cost of a product is $4, and you can assume that the company has infinite production capacity per month. For the first two months, the monthly demand is either 0 product with probability 0.25, 1 product with probability 0.25, or 2 products with probability 0.50. For the last two months, the monthly demand is either 0 product with probability 0.10, 1 product with probability 0.50, or 2 products with probability 0.40. All demand must be met on time either by producing or from inventory. Assume that the products produced in a month can be used to satisfy the demand of the same month. There is a $2 per-unit holding cost on each month's ending inventory. Assume that each month's ending inventory cannot exceed 3 products (warehouse capacity). a. Use dynamic programming to find a production policy minimizing the total expected cost incurred during the next four months. Assume that the initial inventory is 1 unit. Define the value function fr(i) explicitly, provide all details. b. Assume that the demands for the first three months are 1 unit. For this scenario, determine the optimal production amounts for all months by using the solution that you have found in (a). This problem considers a four-months inventory problem where a company must determine how many products should be produced at the beginning of each month. A setup cost of $6 is incurred during each month in which production takes place. The unit production cost of a product is $4, and you can assume that the company has infinite production capacity per month. For the first two months, the monthly demand is either 0 product with probability 0.25, 1 product with probability 0.25, or 2 products with probability 0.50. For the last two months, the monthly demand is either 0 product with probability 0.10, 1 product with probability 0.50, or 2 products with probability 0.40. All demand must be met on time either by producing or from inventory. Assume that the products produced in a month can be used to satisfy the demand of the same month. There is a $2 per-unit holding cost on each month's ending inventory. Assume that each month's ending inventory cannot exceed 3 products (warehouse capacity). a. Use dynamic programming to find a production policy minimizing the total expected cost incurred during the next four months. Assume that the initial inventory is 1 unit. Define the value function fr(i) explicitly, provide all details. b. Assume that the demands for the first three months are 1 unit. For this scenario, determine the optimal production amounts for all months by using the solution that you have found in (a)
