5. (45 points) We are planning to run a specific projector for 99 hours in an exhibition. The lifetime of the light bulb used in this projector is a main concern because it is relatively short and has some uncertainty. There are four brands of bulbs for consideration: A, B, C, and D. Based on the specifications from their manufacturers, the lifetimes for a bulb for brands A, B, C, and D are EXP(0.04),EXP(0.1), Uniform [5,15], and Normal(25,52), respectively. The purchasing costs of one bulb for brands A, B, C, and D are $27,$11,$13, and $32, respectively. The constraint is that you have to purchase the bulbs before going to the exhibition and bring them with you. Our objective is to maximize the probability that the total lifetime of the bulbs you bring can sustain longer than the required 99 hours while the total cost of the bulbs is minimized. Assume that the lifetime of each bulb is independent. Conduct an extensive Monte Carlo simulation studies to decide what types of bulbs and how many you have to purchase. The following are some sample questions which may help you make a good decision. However, your study should not be limited by those questions. (a) What is the expected total lifetime if you decide to bring 5A bulbs? What is the probability that the total lifetime of 5 A bulbs exceed 99 hours? What is the total purchasing cost? (b) How will the answers in part (a) be changed if you bring 6 bulbs? Or if you bring B bulbs, C bulbs, or D bulbs? (c) How will the answers in part (b) be changed if you bring 5 A bulbs and 1B bulb? (d) How will the answers in part (c) be changed if you bring 3A bulbs, 1B bulb, 1C bulb and 1 D bulb? (e) An open question: what is your recommendation? You are encouraged to conduct more simulation to support your recommendation. For ease of analysis, you can assume that running 200 simulation replications for each alternative decision is sufficient to give you a good estimate