Consider four production facilities numbered as 1, 2, 3 and 4. Each facility (plant) is dedicated to production of one product. Products produced at facilities 1 to 4 are labeled as A, B, C, and D, respectively. On average, the production time of each product takes 4 hours. It is estimated that the weekly demand for product A is equally likely to be 8, 10, or 12. That is, the probability is 1/3 for each of the possible outcomes, 8, 10, or 12 units. Each of the other three products has the same demand distribution. Furthermore, the demands are i.i.d. across time and product type. Each facility has a nominal work week of 40 hours at $5,500 per hour and the cost for 40 hours is incurred even if the facility works fewer than 40 hours in the week. But if a facility works more than 40 hours, then the overtime rate is 150% of the normal hour. The overtime rate applies only to the hours in excess of 40.

a) Calculate the expected yearly cost of the dedicated system (in which facilities 1, 2, 3 and 4 can only produce products A, B, C, and D, respectively).

b) The companys CEO decides to try a flexible system. Suppose first that a chaining structure is implemented such that plant 1 will also produce product B, plant 2 will produce product C, plant 3 will produce product D, and plant 4 will produce product A. Calculate the expected yearly cost of overtime for this system. Note: Given the assumptions made in this problem, for each realization of the random demands, it is possible to calculate the total overtime cost based on the total hours required without determining the assignments of demands to plants.

c) Suppose instead that the CEO chooses a full-flexibility system in which every plant is to produce all product types. Calculate the expected yearly cost of overtime for this system.

d)Determine the optimal assignment of demands to plants under both the chaining and full flexibility structures if the demands are 12, 8, 10, and 12 for products AD, respectively.

e) Suppose the cost of adding flexibility to produce one more product at a plant is $1,000,000. What is the expected number of years until the supply chain recoups the investment cost to convert the dedicated system to the partial flexibility system with the chaining structure? What about the full flexibility system?

With formula and explanations.