Complex Combinatorial Algorithms - algorithm

Question

So Wendy's advertises their sandwich as having 256 combinations - meaning there are 8 ingredients you can either have to not have (although I wonder why they would count the combination where you include nothing as valid, but I digress).
A generalized approach allows you to multiply the various states of each selection together, which allows more complex combinations. In this case Wendy's items can only be included or excluded. But some sandwiches might have the option of two kinds of mustard (but not both, to save costs).
These are fairly straightforward. You multiply the number of options together, so For Wendy's it's:
2*2*2*2*2*2*2*2 = 256
If they diversified their mustard selection as above it would be:
2*2*3*2*2*2*2*2 = 384
Going further appears to be harder.
If you make sesame seeds a separate item, then they require the bun item. You can have the sesame seed only if you include the bun, and you can have the bun without sesame seeds, but you cannot have sesame seeds without the bun. This can be simplified to a single bun item with three states (none, bun with seeds, bun without) but there are situations where that cannot be done.
Dell's computer configurator, for instance, disallows certain combinations (maybe the slots are all full, items are incompatible when put into same system, etc).
What are the appropriate combinatorial approaches when dealing with significantly more complex systems where items can conflict?
What are good, generalized, approaches to storing such information without having to code for each product/combination/item to catch conflicts?
Is there a simple way to say, "there are X ways to configure your system/sandwich" when the system has to deal with complex conflicting combinations?

Answer 1

HP's high-end server manufacturing facility in California used a custom rule-based system for many years to do just this.
The factory shopfloor build-cycle process included up-front checks to ensure the order was buildable prior to releasing it to the builders and testers.
One of these checks determined whether the order's bill of materials (BOM) conformed to a list of rules specified by the process engineers. For example, if the customer orders processors, ensure they have also ordered sufficient dc-converter parts; or, if they have ordered a certain quantity of memory DIMMs, ensure they have also ordered a daughter-board to accommodate the additional capacity.
A computer science student with a background in compilers would have recognized the code. The code parsed the BOM, internally generating a threaded tree of parts grouped by type. It then applied the rules to the internal tree to make the determination of whether the order conformed.
As a side-effect, the system also generated build documentation for each order which workers pulled up as they built each system. It also generated expected test results for the post-build burn-in process so the testing bays could reference them and determine whether everything was built correctly.

Answer 2

Adam Davis: If I understand correctly you intend to develop some sort of system that could in effect be used for shopping carts that assist users to purchase compatible parts.
Problem definition
Well this is a graph problem (aren't they all), you have items that are compatible with other items. For example, Pentium i3-2020 is compatible with any Socket 1155 Motherboard, The Asrock H61M-VS is a Socket 1155 Motherboard, which is compatible with 2xDDR3(speed = 1066), and requires a PCI-Express GPU, DDR3 PC RAM{Total(size) <= 16GB}, 4 pin ATX 12V power, etc.
You need to be able to (a) identify whether each item in the basket is satisfied by another item in the basket (i.e. RAM Card has a compatible Motherboard), (b) assign the most appropriate items (i.e. assign USB Hub to Motherboard USB port and Printer to USB Hub if motherboard runs out of USB ports, rather than do it the other way around and leave the hub dry), and (c) provide the user with a function to find a list of satisfying components. Perhaps USB Hubs can always take precedence as they are extensions (but do be aware of it).
Data structures you will need
You will need a simple classification system, i.e. H61M-VS is-a Motherboard, H61M-VS has-a DDR3 memory slot (with speed properties for each slot).
Second to classification and composition, you will need to identify requirements, which is quite simple. Now the simple classification can allow a simple SQL query to find all items that fit a classification.
Testing for a satisfying basket
To test the basket, a configuration need to be created, identifying which items are being matched with which (i.e. Motherboard's DDR3 slot matches with 4GB Ram module, SATA HDD cable connects to Motherboard SATA port and PSU's SATA power cable, while PSU's 4 pin ATX 12V power cable connects to the motherboard.
The simplest thing is just to check whether another satisfying item exists
Dell's Computer Configurator
You begin with one item, say a Processor. The processor requires a motherboard and a fan, so you can give them a choice of motherboard (adding the processor-fan to list_of_things_to_be_satisfied). This continues until there are no more items held in in list_of_things_to_be_satisfied. Of course this all depends on your exact-requirements and knowing what problem(s) you will solve for the user.

Answer 3

There are many ways you can implement this in code, but here is in my humble opinion, the best way to go about solving the problem before programming anything:
Define Parts & Products (Pre-code)
When defining all the "parts" it will be paramount to identify hierarchy and categorization for the parts. This is true because some rules may be exclusive to a unique part (ex. "brown mustard only"), some categorical (ex. "all mustards"), some by type (ex. "all condiments"), etc.
Build Rule Sets (Pre-code)
Define the rule sets (prerequisites, exclusions, etc.) for each unique part, category, type, and finished product.
It may sound silly, but a lot of care must be taken to ensure the rules are defined with an appropriate scope. For example, if the finished product is a Burger:
Unique Item Rule - "Mushrooms only available with Blue Cheese selected" prerequisite
Categorical Rule - "Only 1 mustard may be selected" exclusive
Type Rule - "Pickles are incompatible with Peppers" exclusive
After having spent so much time on unique/category/type rules for "parts", many designers will overlook rules that apply only to the finished product even when the parts have no conflict.
Product Rule - "Maximum 5 condiments" condition
Product Rule - "Burger must have a bun" prerequisite
This graph of rule can quickly grow very complex.
Suggestions for Building Data Structures (Code)
Ensure your structures accommodate hierarchy and categorization. For example: "brown mustard" and "dijon mustard" are individual objects, and they are both mustards, and both condiments.
Carefully select the right combination of inheritance modeling (base classes) and object attributes (ex. Category property, or HasCondiments flag) to make this work.
Make a private field for RuleSets at each hierarchic object level.
Make public properties for a HasConflicts flag, and a RuleViolations collection.
When a part is added to a product, check against all levels of rules (its own, category, type, and product) -- do this via a public function that can be called from the product. Or for better internalization, you can make an event handler on the part itself.
Write your algorithms (Code)
This is where I suck, and good thing as it is sort of beyond the scope of your question.
The trick with this step will be how to implement in code the rules that travel up the tree/graph -- for example, when a specific part has issue with another part outside its scope, or how does it's validation get run when another part is added? My thought:
Use a public function methodology for each part. Pass it the product's CurrentParts collection.
On the Product object, have handlers defined to handle OnPartAdded and OnPartRemoved, and have them enumerate the CurrentParts collection and call each part's validation function.
Example Bare-bones Prototype
interface IProduct
{
void AddPart();
void OnAddPart();
}
// base class for products
public class Product() : IProduct
{
// private or no setter. write functions as you like to add/remove parts.
public ICollection<Part> CurrentParts { get; };
// Add part function adds to collection and triggers a handler.
public void AddPart(Part p)
{
CurrentParts.Add(p);
OnAddParts();
}
// handler for adding a part should trigger part validations
public void OnAddPart()
{
// validate part-scope rules, you'll want to return some message/exception
foreach(var part in CurrentParts) {
part.ValidateRules(CurrentParts);
}
ValidateRules(); // validate Product-scope rules.
}
}
interface IProduct
{
// "object" should be replaced with whatever way you implement your rules
void object RuleSet;
void ValidateRules(ICollection<Part> otherParts);
}
// base class for parts
public class Part : IPart
{
public object RuleSet; // see note in interface.
public ValidateRules(ICollection<Part> otherParts)
{
// insert your algorithms here for validating
// the product parts against this part's rule set.
}
}
Nice and clean.

Answer 4

As a programmer I would do the following (although I have never actually ever had to do this in real life):
Work out the total number of
combinations, usually a straight
multiplication of the options as
stated in your question will suffice.
There's no need to store all these
combinations.
Then divide your total by the
exceptions. The exceptions can be
stored as just a set of rules,
effectively saying which combinations
are not allowed.
To work out a total number of
combinations allowable, you will have
to run through the entire set of
exception rules.
If you think of all your combinations as a set, then the exceptions just remove members of that set. But you don't need to store the entire set, just the exceptions, since you can calculate the size of the set quite easily.

Answer 5

"Generating Functions" comes to mind as one construct that can be used when solving this type of problem. I'd note that there are several different generating functions depending on what you want.
In North America, car license plates can be an interesting combinatorial problem in counting all the permutations where there are 36 possible values for each place of the 6 or 7 that are the lengths of license plates depending on where one is getting a plate. However, some combinations are disqualified due to there being swear words or racist words in some of them that makes for a slightly harder problem. For example, there is an infamour N-word that has at least a couple of different spellings that wouldn't be allowed on license plates I'd think.
Another example would be determining all the different orders of words using a given alphabet that contains some items repeated multiple times. For example, if one wanted all the different ways to arrange the letters of say the word "letter" it isn't just 6! which would be the case of "abcdef" because there are 2 pairs of letters that make it slightly trickier to compute.
L33t can be another way to bring in more complexity in identifying inappropriate words as while a-s-s gets censored a$$ or #ss may not necessarily be treated the same way even though it is basically the same term expressed in different ways. I'm not sure if many special characters like $ or # would appear on license plates but one could think of parental controls on web content as having to have these kinds of algorithms to identify which terms to censor.

Answer 6

You'd probably want to create a data structure that represents an individual configuration uniquely. Then each compatibility rule should be defined in a way where it can generate a set containing all the individual configurations that fail that rule. Then you would take the union of all the sets generated by all the rules to get the set of all configurations that fail the rules. Then you count the size of that set and subtract it from the size of the set all possible configurations.
The hard part is defining the data structure in a way that can be generated by your rules and can have the set operations work on it! That's an exercise for the reader, AKA I've got nothing.

Answer 7

The only thing I can think of right now is building is if you can build a tree that defines the dependency between the parts you have a simple solution.
sandwitch
|
|__Bun(2)__sesame(1)
|
|__Mustard(3)
|
|__Mayo(2)
|
|__Ketchup(2)
|
|__Olives(3)
this simply says that you have 2 options for the Bun (bun or no bun) - 1 for the sesame (only if you have a bun - signifying the dependency - if you have a 7 here it means 7 types that can exist if you only have a bun)
3 for the mustard .. etc
then simply multiply the sum of all branches.

Answer 8

It is probably possible to formalize the problem as a k-sat problem. In some cases, the problem appear to be NP-complete and you will have to enumerate all the possibilities to check wether they satisfy or not all the conditions. In some other cases, the problem will be easily solvable (when few conditions are required for instance). This is an active field of research. You will find relevant references on google scholar.
In the case of the mustard, you would add a binary entry "mustard_type" for the mustard type and introduce the condition: not (not mustard and mustard_type) where mustard is the binary entry for mustard. It will impose the default choice mustard_type == 0 when you choose not mustard.
For the sesame choice, this is more explicit: not (sesame and not bun).
It thus seems that the cases you propose fall into the 2-sat family of problems.