When I first entered BNL summer research program, I was told the cafeteria offers 20% discount to research students.
After some investigation, it is in fact, not correct. I feel like BNL lied to me, cheated on me. I don't feel safe at BNL anymore.
Suppose function f(x) = amount I pay for food that normally cost $x. The definition of f(x) is:
Graph of f(x) compare to x and 0.8x
Graph of d(x) compare to 0.8
Graph of f(x) using d(x) with some transformation as gradient. darker = less percent discount.
Function s(x) is the average discount from 0 to x. Defined as
s(15) = 0.85
s(x) is a monotonic increasing function for x>15.
So on average, if one spend $15 or more, one has less than 15% discount!
After analyze the entire system, how can we take advantage of it and get as much discount as possible?
You can break the order into pieces so it fit into some range, or combine order with someone else to get maximum profit.
For example, a $7(with tax) order composed of 2 $3.5 lasagna. Normally you pay $6, but split it into two orders, you pay$5.5. Doing it everyday can save up to $15 when the research program ends!
I don't have enough skills on problems like this, my guess is there is no simple analytical formula to find how exactly buy stuff to get the best result, it's best to use computer to find a numerical answer.
Without out a computer, you have to do a little calculations, bring a table of values would be nice, here is one from $3.5 to $15, every 10 cents gap. With amount of money before savings and discount expected. Orange = 20+% discount, Yellow = 18.75+% discount.(remember to add tax while looking at the menu)
| $ |
% |
$ |
% |
$ |
% |
$ |
% |
| 3.5 |
0.2143 |
6.4 |
0.1562 |
9.3 |
0.1613 |
12.2 |
0.1844 |
| 3.6 |
0.2083 |
6.5 |
0.1538 |
9.4 |
0.1596 |
12.3 |
0.1829 |
| 3.7 |
0.2027 |
6.6 |
0.1515 |
9.5 |
0.1579 |
12.4 |
0.1815 |
| 3.8 |
0.1974 |
6.7 |
0.1493 |
9.6 |
0.1562 |
12.5 |
0.1800 |
| 3.9 |
0.1923 |
6.8 |
0.1471 |
9.7 |
0.1546 |
12.6 |
0.1786 |
| 4.0 |
0.1875 |
6.9 |
0.1449 |
9.8 |
0.1531 |
12.7 |
0.1772 |
| 4.1 |
0.1829 |
7.0 |
0.1429 |
9.9 |
0.1515 |
12.8 |
0.1758 |
| 4.2 |
0.1786 |
7.1 |
0.1408 |
10.0 |
0.1500 |
12.9 |
0.1744 |
| 4.3 |
0.1744 |
7.2 |
0.1389 |
10.1 |
0.1485 |
13.0 |
0.1731 |
| 4.4 |
0.1705 |
7.3 |
0.1370 |
10.2 |
0.1471 |
13.1 |
0.1718 |
| 4.5 |
0.1667 |
7.4 |
0.1351 |
10.3 |
0.1456 |
13.2 |
0.1705 |
| 4.6 |
0.1630 |
7.5 |
0.1333 |
10.4 |
0.1442 |
13.3 |
0.1692 |
| 4.7 |
0.1596 |
7.6 |
0.1316 |
10.5 |
0.1429 |
13.4 |
0.1679 |
| 4.8 |
0.1562 |
7.7 |
0.1299 |
10.6 |
0.1415 |
13.5 |
0.1667 |
| 4.9 |
0.1531 |
7.8 |
0.1282 |
10.7 |
0.1402 |
13.6 |
0.1654 |
| 5.0 |
0.2000 |
7.9 |
0.1266 |
10.8 |
0.1389 |
13.7 |
0.1642 |
| 5.1 |
0.1961 |
8.0 |
0.1875 |
10.9 |
0.1376 |
13.8 |
0.1630 |
| 5.2 |
0.1923 |
8.1 |
0.1852 |
11.0 |
0.1364 |
13.9 |
0.1619 |
| 5.3 |
0.1887 |
8.2 |
0.1829 |
11.1 |
0.1351 |
14.0 |
0.1607 |
| 5.4 |
0.1852 |
8.3 |
0.1807 |
11.2 |
0.1339 |
14.1 |
0.1596 |
| 5.5 |
0.1818 |
8.4 |
0.1786 |
11.3 |
0.1327 |
14.2 |
0.1585 |
| 5.6 |
0.1786 |
8.5 |
0.1765 |
11.4 |
0.1316 |
14.3 |
0.1573 |
| 5.7 |
0.1754 |
8.6 |
0.1744 |
11.5 |
0.1304 |
14.4 |
0.1562 |
| 5.8 |
0.1724 |
8.7 |
0.1724 |
11.6 |
0.1293 |
14.5 |
0.1552 |
| 5.9 |
0.1695 |
8.8 |
0.1705 |
11.7 |
0.1282 |
14.6 |
0.1541 |
| 6.0 |
0.1667 |
8.9 |
0.1685 |
11.8 |
0.1271 |
14.7 |
0.1531 |
| 6.1 |
0.1639 |
9.0 |
0.1667 |
11.9 |
0.1261 |
14.8 |
0.1520 |
| 6.2 |
0.1613 |
9.1 |
0.1648 |
12.0 |
0.1875 |
14.9 |
0.1510 |
| 6.3 |
0.1587 |
9.2 |
0.1630 |
12.1 |
0.1860 |
15.0 |
0.2000 |
Someone can design a program so it's possible to calculate the most optimal solution if at most 
order splits is possible for 
individual food with each of the food's price 
.
P.S. The summer science program(Research Program:Science Program::Gentoo:Ubuntu) started today. So many Asian girls!(at least 8) it's unbelievable! It's like MIT! Haha, I was wondering where's the Asian when I entered the research program(only me + 1 more person).
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