Possible combinations
This is a question with several answers. The easiest answer is the least useful. The number of possible build combinations, or unique configurations, is easily computed by multiplying the number of options for each feature. For example, if your product has feature A with 3 options, feature B with 2 options and feature C with 4 options, then there are 24 (3 x 2 x 4) possible build combinations.
These numbers grow very rapidly. If you have 5 features, each with 4 options, there are about 1,000 build combinations (exactly 1,024). With 10 such features, the number of combinations is about 1 million (1,048,576), and with 15 features it is over 1 billion (1,073,741,824).
Valid combinations
The number of possible combinations is not very useful because only a few products can be ordered in every possible way. A cup of coffee at Starbuck’s is one of these few. For most products, the number of valid build combinations is much smaller. Valid build combinations must satisfy compatibility rules. These rules, which may express engineering or marketing limitations, partition the set of possible combinations into valid and invalid. Counting the number of valid configurations is difficult, but it can be done. If the compatibility rules are expressed in a logical syntax, then automatic tree search methods can be used to count the exact number of valid combinations. These tree search methods are commonly used in operations research and artificial intelligence. For real products as complex as cars, the number of possible combinations is usually astronomical, while the number of valid configurations is only in the millions.
Historical combinations
A third answer is simply the number of unique configurations that have been ordered in the past. This number is typically in the thousands or tens of thousands. It is much smaller than the number of valid configurations, but of course it grows over time. Many build to order (BTO) manufacturing companies will compare each new customer order to the entire history of previous orders. If there is a match, then the order must be valid because it has been built before. If there is no match, then the order is sent to the engineering department to determine if it is valid. If it is valid, then it is built, added to the history and assigned a new SKU number. The great drawback with this system is that a new order may be almost the same as several earlier orders and differ only in options that the customer doesn’t really care about.
Next: the even smaller set of optimal combinations.
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