The operations manager at MicroControl Limited believes that pessimistic demand has a probability of $20$percent$,$ expected demand has a probability of $50$percent$,$ and optimistic demand has a probability of $30$percent$.$ Currently, new machines must be purchased at a cost of $500,000$a piece, the price charged for each control unit is $$110$ and the variable cost of production is $$50$per unit. $[$Hint$]$ : since the price and variable cost for each control unit are the same, the profit maximizing product mix will be the same as the mix that maximizes the total number of units produced. a$.$Draw a decision tree for this problem. b$.$How many machines should the company purchase, and what is the expected payoff