There are so many different ways of reasoning this out that it would be impossible to have only one answer. Below, I have pointed out some of the possibilities.
The choices here could be different depending on which philosopher you choose and whether or not you are looking at this from an act or rule perspective. What we want to do here is look at this case study through the eyes of one of the philosophers in the textbook.
Mill is a rule utilitarian. Mill's approach would be to examine the higher and lower pleasures and he would have us test the utility of moral rules. This means we would not look at these cases individually but apply one, general moral rule that has already been decided on for its utility in these types of cases.
This would be an objective calculation based on statistics and would not be personal. A team of doctors making lung transplant choices would use this method. Most likely the mother. The child has an ongoing problem and could need another transplant or have other medical complications. The older man, while he did a lot of good, is too old. The smoker would not be chosen.
Bentham is an act utilitarian. Bentham's approach would be to determine what action to take based on all the pleasure and all the pain that results from the action. Therefore, you should list out all the pleasures and pains resulting from the action of saving each candidate and then discuss which candidate would result in the most pleasure for the most people. One possible problem here would be calculating all of the consequences from this choice. Who would Bentham choose? Possibly the oldest candidate because this would create the most amount of pleasure for the most amount of people.
Moore wants us to calculate the total good vs the total bad that results from an action. Here, we need to examine more than just pleasure but look at any consequence that counts as good and there are things other than just pleasure that may count as good. We can start to choose a candidate by considering flaws in the popular idea of good.
Here, we will evaluate the candidate by considering aesthetic characteristics and admirable mental qualities. In the end, we will rely on intuition. You would not want a doctor to rely on intuition to make this choice. Any candidate could be chosen depending on your intuition.
Hare is a preference utilitarian and suggests that we examine our total preferences. This is personal to our preferences but should also consider other people's preferences. This means we should ask ourselves what would my preferences be if I were in someone else's shoes.
In order to make sure this does not end in egoism instead of utilitarianism, we need to tally our personal preferences as well as those of others and weigh them against each other. Because this type of utilitarianism relies on our intuitions and gut instincts, it would not be a good one to use to calculate who should get a lung transplant. Any candidate would be chosen dependent on your preference and intuition.
In the end, lung transplant decisions are carefully calculated and use a variety of different statistical methods to calculate the best choice. In real life, this would be most closely related to Mill's utilitarian theory.