So what are the constraints that we're going to face? Well, for now we're going to assume that the failure rate of a product is 0.5 incidents per year. And we're going to assume that it comes from a Poisson distribution, we'll talk about that in a moment. And we're going to assume that there is a markup on repairs services. So let's assume that the retailer can get repairs done for 50% of what a third party company is going to charge consumers. But that's something that we're going to want to manipulate. Because what if I can find even a lower price? What if I can get repairs on for 25% of the cost? If I can find someone that does repairs for a lower price point, if I can manage to do that, I can keep some of those savings and pass some of those savings on to the consumer. All right, so what do we need to determine? What's the price for one year of warranty coverage? What's the additional cost for a second year of coverage? What's the deductible going to be, and that might be zero or some number greater than zero, and what's the quota going to be for the contract? All right, now if we take the most general case, warranty at a fixed price, optional second year of coverage, a deductible that might be zero and a quota that might be infinite. That's the most general scenario or special cases where there is no deductible and there is no quota or where there is a deductible but no quota are all nested within that. But we're going to work on building this up. So we're going to start by saying let's build out the case where I pay for the warranty as a consumer and there's no deductible, no quota. Then we'll say, let's bring in that deductible. Then on top of that, we'll say, okay, let's bring in this quota. So we want to build as general a decision support tool as we can. Now the piece that we don't know with certainty is how many incidents are we going to have to plan for for each customer. So depending on what I'm selling, let's say I'm selling flat screen TVs, how many incidents can I expect? Is it going to be zero incidents, one incident during the life of the contract, two incidents over the life of the contract? How many are we talking about? That's the unknown for us. So that's where we're going to use the Poisson distribution to characterize the number of failures during the course of the contract. Very similar to how we use the binomial distribution to characterize the number of no shows. All right, so just to give you a little bit of background on the Poisson distribution, and I've got a small Excel file that you can download where some of this is already set up for you. The Poisson distribution, very common for what's referred to as count data within marketing. So count data, so if we think about how many units somebody buys during a fixed period of time, that's the count that we're referring to. You go to the grocery store, how many bottles of soda did you buy? How many bags of bagels did you buy? How many oranges did you buy on this trip? Well that's a count that we're trying to model. That's where the Poisson distribution gets used. A feature of the Poisson distribution to bare in mind is the average and the variance of the Poisson distribution characterized by the same parameter. So when the mean and the variance are equal to each other, the Poisson distribution tends to fit the data very well. If this assumption doesn't hold, well, this isn't going to be the right distribution for you. For this particular exercise, let's assume that we know the Poisson distribution is appropriate and we know that the failure rate is going to be 0.5 incidents per year. One characteristic of the Poisson that's going to serve us well in this particular case, if I know that I'm dealing with a failure rate of 0.5 incidents per year, that's my expected number of failures in a given year, well what's my expected number of failures over two years? Well, let's say my X is the number of failures in the first year, my Y is going to be number of failures in the second year. X + Y is going to be my combined number of failures during that two year period. Well, I can just add up the two parameters, lambda 1 and lambda 2. And so in this case, if I have a 0.5 expected failures in each year, lambda 1 is 0.5, lambda two is 0.5, I add those up, the number of expected product failures during a two year period is going to be 1. All right, so let's take a look at this illustrative worksheet. What I've done here is calculated the probability of a particular event occurring given the value of lambda under the Poisson distribution. So for lambda of 0.5, again this is our one year, what can we expect as far as the number of failures that someone is going to experience. According to the Poisson distribution there is a 60% chance of 0 failures occurring, 30% of one failure occurring, 7.5% chance of two failures and so forth. Now you see the number of failures as illustrated in this plot looks nothing like a normal distribution. So again, this where it's important for us to match the probability distribution to the context that we're working with. All right, what about if we deal with over a two-year period? We said that lambda for year is 0.5, so over a two-year period we're expecting lambda of 1. Notice that our probability of not having any failures drops considerable. So over a two year period it goes from, we were at around 60% when lambda was 0.5, it goes from just over 60% to now around 37% for 0 failures, same percentage chance for only 1 failure. So we're expecting, actually, more failures over a two year period than over a one year period. So you can play around using that tool just to get a sense for how the failure rate or the rate at which service incidents occurs is going to affect our decision making.
