Category Archives: Knapsack

Using Airline Miles to Buy Magazines: a Hidden Deeper Lesson

Last month, I received a letter from American Airlines’ Rewards Processing Center saying that I have 4735 miles that are about to expire and asking whether I’d be interested in redeeming them for some magazine subscriptions. These were the choices:

Magazine Issues Miles Needed
Afar 6 600
Architectural Digest 12 800
Barron’s 26 1700
Business Week 50 1600
Cat Fancy 12 500
Cigar Aficionado 6 400
Daily Variety 252 5500
Diabetes Forecast 12 500
ESPN the Magazine 26 500
Ebony and Jet 38 800
Elle 12 400
Entertainment Weekly 55 1300
Essence 12 500
Essence 2yrs 24 800
Fortune 25 1400
Golf Digest 12 700
Golf Magazine 12 700
Golfweek 45 1300
Martha Stewart Living 24 1400
Money 12 800
New York Magazine 46 700
Sports Illustrated 56 1400
SI and Golf Mag 68 1500
Sports Illustrated KIDS 12 1000
The Economist 51 3200
The Wall Street Journal 190 2800
US News & World Report 12 700
Variety 50 5500
Wine Spectator 15 900
Wired 12 400

Wow! 30 different magazines! How could I possibly decide…oh, wait! Maybe I can use some analytical techniques to help me make a better decision…what’s that thing called again…O.R.!!!

First, what do I want to accomplish?

a) Get as many issues of whatever magazine as possible: this is what a dentist’s office or any place with a waiting room might want to do. Note that the WSJ subscription provides 190 issues, but let’s say I only want to consider actual magazines. One possible answer is to subscribe to ESPN the Magazine, Ebony and Jet, Entertainment Weekly, New York Magazine, and Sports Illustrated. That will buy me 221 issues and use 4700 of my 4735 miles. Just out of curiosity, I could have gotten 286 issues if I hadn’t excluded the WSJ.

b) Get as many different subscriptions as possible: This one is easy. Just pick the magazines that require the least number of miles. One solution is Afar, Cat Fancy, Cigar Aficionado, Diabetes Forecast, ESPN the Magazine, Elle, Essence, US News & World Report, and Wired. That’s a total of 9 subscriptions, using 4500 of my 4735 miles (a waste of 235 miles).

c) Get the “best” 9 magazines you can afford: Since I know I can get 9 subscriptions, what are the 9 that make me use the most out of my 4735 miles? Maybe they’ll constitute a better set of 9 than those I found in item (b) above (positive correlation between quality and miles needed?). In this case the answer is to substitute New York Magazine for Essence in the set of 9 found in item (b). This alternative totals 144 issues and wastes only 35 miles.

d) Get the 9 magazines that provide the largest total number of issues: In that case I’d want to go with Cat Fancy, Cigar Aficionado, Diabetes Forecast, ESPN the Magazine, Ebony and Jet, Elle, Essence, New York Magazine and Wired. That’s a total of 176 issues and a waste of 35 miles as well.

For those of you who are thinking “who cares?”, don’t let appearances fool you. Say that I am United Way and 4735 is my budget (in thousands of US$). There are 30 charities (“magazines”) asking me for money (the “miles needed”) and telling me that they will help a certain number of people (the “issues”, in thousands). If I fund the charities in rows 9, 10, 12, 21 and 22 of the above table, I’ll spend US$ 4,700,000 and help 221 thousand people. That’s the best I can do! (if I am not allowed to fund the charity in row 26).

Lots of other problems can be cast into this framework. Another example: NASA is sending a spaceship to Mars. My available miles are the spaceship’s cargo capacity, the magazines are scientific experiments to be conducted in space (each requiring equipment to be loaded into the spaceship (the “miles needed”) and promising a certain (quantifiable) scientific benefit (the “issues”). This a generic resource allocation problem and the mathematical model we have used here is called The 0-1 Knapsack Problem.

Technical Details:

Want to solve your own budget allocation problem? Here’s how you can do it. We have n projects that require funding and a budget B . For each project i = 1,\ldots,n , let m_i be how much money it needs and let p_i be its payoff (e.g. lives saved). Let the binary variable x_i be equal to 1 if we decide to allocate money to project i , and equal to zero otherwise.

The objective is to maximize the payoff \sum_{i=1}^n p_i x_i .

The constraint is to respect the budget: \sum_{i=1}^n m_i x_i \leq B .

This model also allows you to include some useful side constraints. For example: “if I fund project i , then I must also fund project j ” would be written as x_i \leq x_j . And “if I fund project i , then I cannot fund project j “, would become x_i + x_j \leq 1 . I’ve talked about these kinds of logical conditions in another post.

Here’s the AMPL file I used to solve the magazine problem and its variations. Enjoy!

P.S.: Being an Economics major and an amazing cook, my wife requested The Economist and Martha Stewart Living; that’s what her objective function told her to do, I guess.

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Filed under Applications, Integer Programming, Knapsack, Modeling