A manufacturer can buy capacity from a trucking firm at a contract price of $150 per unit, and must decide how many units to contract for in the upcoming holiday season. The manufacturer will not know the actual capacity needed until after the season begins. If the contracted capacity turns out to be insufficient, then the manufacturer will have to buy additional capacity on the spot market for $175 per unit. On the other hand, if the contracted capacity turns out to be more than needed, then the manufacturer will get a partial refund of $100 per unused unit from the trucking firm. The manufacturer faces the following probability distribution of demand for trucking capacity. Demand (units) 100 200 300 400 Probability 0.2 0.2 0.3 0.3 a. Calculate the expected that is, average - demand. b. How many units should the manufacturer contract for, to minimize its expected trucking cost for the season? Validate your answer by calculating the manufacturer's expected trucking costs for the following contract quantities:0, 100, 200, 300, 400. (No other order quantity for example, 50 or 150 or 500 - can be optimal in this scenario.) Given this contract quantity, what is the expected spot market order size? c. Given the quantity in part b, calculate the manufacturer's (i) in-stock probability and (ii) fill rate. d. Given the quantity in part b, calculate the manufacturer's percentage excess cost, compared to the theoretical minimum cost, incurred because of demand-supply mismatch. A manufacturer can buy capacity from a trucking firm at a contract price of $150 per unit, and must decide how many units to contract for in the upcoming holiday season. The manufacturer will not know the actual capacity needed until after the season begins. If the contracted capacity turns out to be insufficient, then the manufacturer will have to buy additional capacity on the spot market for $175 per unit. On the other hand, if the contracted capacity turns out to be more than needed, then the manufacturer will get a partial refund of $100 per unused unit from the trucking firm. The manufacturer faces the following probability distribution of demand for trucking capacity. Demand (units) 100 200 300 400 Probability 0.2 0.2 0.3 0.3 a. Calculate the expected that is, average - demand. b. How many units should the manufacturer contract for, to minimize its expected trucking cost for the season? Validate your answer by calculating the manufacturer's expected trucking costs for the following contract quantities:0, 100, 200, 300, 400. (No other order quantity for example, 50 or 150 or 500 - can be optimal in this scenario.) Given this contract quantity, what is the expected spot market order size? c. Given the quantity in part b, calculate the manufacturer's (i) in-stock probability and (ii) fill rate. d. Given the quantity in part b, calculate the manufacturer's percentage excess cost, compared to the theoretical minimum cost, incurred because of demand-supply mismatch