Tuesday, 21 December 2010

Happy Holidays thanks to OR

At this time of the year many of my colleagues and fellow Dutchmen arrange to go ski in France or in Austria (Swiss is too expensive, you know the Dutch) or fly off to some sunny destination to enjoy the December holidays. Personally I like to stay at home and enjoy being with family and friends. It also allows me to write this blog and enter the December Blog Challenge of INFORMS.

How do Holidays and Operations Research relate? Well, a lot I would say. The obvious examples for this time of year are:

  • Given a (tight) budget which Xmas-presents to buy to maximise joy (Knapsack problem)
  • The shortest route along all the shops you want to go to for Xmas shopping or all the relatives you want to visit (Travelling salesman). Santa's route actually is that of a travelling salesman.
  • Getting all the presents you bought into your car (vehicle load optimisation)
  • Avoid gaining weight during the Xmas holidays, design a healthy Xmas menu (Diet problem)
  • And many more.
Those who don’t stay at home but book a flight, a train, a hotel room or any other holiday accommodation also come across Operations Research. It is used to determine the fee you have to pay for the flight or hotel room you book. It’s called revenue management or yield management. It explains why the person next to you paid a different fee than you and why prices for hotel rooms and flights fluctuated from day to day. The basic principle of revenue management is rather simple, try to maximise revenue given an uncertain demand. Remember selling lemonade as a child outside your house? You had to decide when to try and sell it, (it didn’t work on a rainy day) decide on a reasonable price and when/how to change the price as the day rolled on. Things are no different in selling flights or hotel rooms. There is however more to revenue management than just determine the right price.

One of my customers is in the travel industry. They offer flights, hotels and holiday accommodations or combinations thereof via the web. It is interesting to see how their planning cycle works and how Operations Research is used to support it. It all starts with having something to sell. Before you can sell, you need to have inventory (How many cans of lemonade to make?). Each year, ahead of the holiday season, it must be decided how many hotel rooms and flight seats are required to fulfil market demands or capture the market potential. This decision can have a lot of impact since, if a too small amount is bought, it is a missed opportunity for increased revenue. However if too much rooms or flights are bought, a lot of the capacity will by left unused leading to uncovered costs. Although deciding on the exact amounts is still a craftsman’s job, a lot can be learned from the past. This is where Operations Research offers a helping hand. Using advanced forecasting methods the capacity managers are able to make good esstimates for the required inventory that is than (together with strategic commercial objectives) input for the sales & booking process.

Given the above, you can imagine that the inventory risk is rather large. Huge amounts of flight seats and hotel rooms need to be bought in advance, which are perishable. If the flight seat, hotel room or accommodation isn’t sold, it expires. Also 95% of the inventory needs to be sold to have a healthy P&L, which leaves not much room to manoeuvre. This is already hard when you are selling just one product. Think about selling 18 million different products! Here is where the Operations Research comes in again. To deal with the complexity and size of the challenge, we developed a specific optimisation model that is incorporated into a revenue management system. The model is used to determine the optimal price for each of the 18 million products. On a daily basis 2-3 million product prices are updated, taking the actual bookings and updated forecasts for future demand into account.

At the introduction of the model, the mindset of the pricing managers was to set prices to sell 100% of the inventory. They achieved high occupation levels, but at the cost of many discounts. With the revenue management model this changed. First they used to model to maximise revenue by making timely price adjustments. Now they are even beyond that, using the model to find new growth opportunities. The revenue management model handles everything so they also can enjoy their holidays as well, thanks to OR. Happy Holidays to you all!

Wednesday, 8 December 2010

Statistical Reasoning for Dummies

"Statistical thinking and reasoning is necessary for efficient citizenship as the ability to read and write"

Is this statement to bold? I don’t think so. We are surrounded with statistics, uncertainties and probabilities and need to understand them, use them and make decisions with them. But, as it turns out, statistical reasoning is very difficult given the many mistakes that are made in newspapers, medical decision making, social science, gambling, politics. You name it, it’s everywhere and so are the mistakes. To give you an extreme example, in Innumeracy J.A. Paulos tells a story about a weather forecaster. The weather forecaster reports a 50% change of rain on Saturday, also a 50% chance of rain on Sunday. He concludes that it will rain the weekend for certain. More recently the publication of Stonewall stating that the average coming out age has been dropping was proven to be wrong by Ben Goldacre. The Stonewall survey is seriously flawed and proves the obvious point that people tend to get older when they get older, nothing more and nothing less. See Ben’s Bad Science weblog for more details. Yesterday a big news item on local television was that mother, son and grandson are born on the same date. Statistically it’s not that extraordinary, contrary what the journalist said (“It’s a miracle”). It’s easy to make a long list of these kinds of mistakes (the next Great Operations Research Blog Challenge theme?), but how to resolve this? Maybe some statistical reasoning for dummies could help? Let’s start with an introductory chapter, some basics.

As an Operations Researcher I am used to work with terms like probability, risk, variance, covariance, t-test, and many other statistical “Red” words as Sam Savage calls them is his book The Flaw of Averages. Many times these “Red” words are used to express a probability or risk, leading to many mistakes or confusion. Take for example the story from journalist David Duncan that was in Wired magazine a few years ago. David did a complete gene scan that checked for genetic decease markers in his DNA. Such tests will soon be part of everyday medical care (and insurance acceptance terms?). To his distress David receives the message that he has mutations in his DNA, raising his risk of having a heart attack. Such risks are expressed as the probability that you will have a heart attack is x%, a single event probability. It is similar to the statement that the probability that it will rain tomorrow is 30%. But what does it mean? Will it rain 30% of the time tomorrow, or in 30% of the country? Both inferences are wrong by the way. The problem with single event probabilities expressed in this way is that without a reference to the class of events the probability relates to, you are left in the dark as to how to interpret it. It causes Duncan to worry about having a heart attack, but should he have worries about it? A way around this confusion is to include the reference class to the probability. So, the weather forecaster should state something like that in 3 out of the 10 times he predicted rain for tomorrow, there was at least a trace of rain the next day. Much of the confusion of David could have been resolved if the doctor would have added a reference class, putting things in perspective.

Another classic misunderstanding is the interpretation of a conditional probability, like in interpreting diagnostics tests in medicine. See my earlier blog entry on that. The approach I used to explain the correct way to interpret the test results “translates” the probabilities (stated in percentages) into real numbers, making it easier to understand. Actually it does more or less the same as adding the reference class to the single event probability. It adds context. The last example of a much misunderstood statistic is relative risk. In the Netherlands there was much debate on whether girls should to be vaccinated against cervical cancer caused by the human Papilloma virus (HPV). To express the effectiveness of the vaccination, a relative risk reduction was used. Something like; “This vaccine will reduce the risk of getting cervical cancer from an HPV infection with x%”. This kind of statement is used regularly to express the effectiveness of preventive methods like screening, vaccines or other risk mitigation strategies. Using relative risk reduction as a measure can however be confusing. For example, if the number of women dying of cervical cancer reduces from 4 to 3 per 1000, the relative risk reduction is 25%. A massive risk reduction you would say. However, if you look at the actual reduction of women dying this is only 0,1% (=1/1000). The confusion, again, comes from not expressing the reference data causing many people to think that the relative risk reduction applies to those how take the vaccine, but it actually applies to those how don’t and die (25% less dead).

So first lesson in statistical reasoning is look for the reference and translate probabilities and risks into numbers. For us “professionals”, lest present our results in a smart and easy to understand way and skip using those Red words.

1) I used HG Wells’ statement on statistical reasoning from somewhere in the beginning of last century as a starting point.