Question

The Grand Vizier of the Kingdom of Um is being blackmailed by numerous individuals and having a very difficult time keeping his blackmailers from going public. He has been keeping them at bay with two kinds of payoff: gold bars from the Royal Treasury and Political Favor Through bitter experience, he has learned that each payoff in gold gives him peace for average of about 1 month, while each political favor seems to earn him about a month and half of reprieve To maintain his flawless reputation in the Court, he feels he cannot afford any revelations about his tainted past to come to light within the next year. Thus it is imperative that his blackmailers be kept at bay for 12 months. Furthermore, he would like to keep the number of gold payoffs at no more than one-quarter of the combined number of payoffs because the outward flow of gold bars might arouse suspicion on the part of the Royal Treasurer. The Grand Vizier feels that he can do no more than seven political favors per your without arousing undue suspicion in the Court. The gold payoffs tend to deplete his trave budget. (The treasury has been subsidizing his numerous trips to the Himalayas.) He estimates that each gold bar removed from the treasury will cost him four trips. On the other hand because the administering of political favors tends to cost him valuable travel time, he suspec that each political favor will cost him about two trips. Now, he would obviously like to keep his blackmailers silenced and lose as few trips as possible. What is he to do? How many trips wil he lose in the next year?​

Answer 1

To keep his blackmailers at bay for 12 months, the Grand Vizier needs to balance the number of gold payoffs and political favors he gives to them. Let's represent the number of gold payoffs by G and the number of political favors by P.

From the given information, we have:

Each gold payoff gives him an average of 1 month of reprieve.

Each political favor gives him an average of 1.5 months of reprieve.

He cannot afford any revelations about his past to come to light within the next year, which means he needs to keep his blackmailers at bay for 12 months.

He wants to keep the number of gold payoffs at no more than one-quarter of the combined number of payoffs.

He can do no more than seven political favors per year.

Each gold bar removed from the treasury will cost him four trips.

Each political favor will cost him about two trips.

Let's first calculate the maximum number of political favors he can give in a year:

7 political favors per year

Next, let's find the maximum number of payoffs he can give in a year:

G + P = total number of payoffs

G ≤ 0.25(G+P) (to keep the number of gold payoffs at no more than one-quarter of the combined number of payoffs)

Simplifying the second equation, we get:

G ≤ 0.25G + 0.25P

0.75G ≤ 0.25P

3G ≤ P (multiplying both sides by 3)

So the maximum number of payoffs he can give in a year is 7 + G, where G ≤ 3.

Next, we need to find the combination of payoffs that will give him the most reprieve while losing the fewest trips. We can use a table to calculate the reprieve and trip costs for different combinations of payoffs: look at the table

From the table, we see that the best combination is G=3 and P=4, which will give him a total reprieve of 12 months (the required time) while costing him 26 trips (the minimum possible). Therefore, the Grand Vizier will lose 26 trips in the next year.