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i am unsure if i have even identified the problem correctly, but reading up on Knapsack problem seems the closest to what i am trying to solve:

A cook has several ingredients of varying quantities. for example:

8 eggs 3 sausages 500 mL milk 12 strawberries

There is a finite list of recipes, each consisting varying ingredients of varying quantities. The universe of ingredients is finite, as is the quantity of each ingredient in all recipes.

Each recipe may or may not contain any of the ingredients the cook has.

The cook wants to use up all his ingredients as much as possible to minimize waste on 1 recipe.

There is a case where the cook wants to use all his ingredients on 2 or 3 different recipes, with minimal leftovers.

What is his optimized solution?

EDIT: My question is a more complex version of the following knapsack problem http://www.g12.cs.mu.oz.au/wiki/doku.php?id=simple_knapsack

bouncingHippo
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2 Answers2

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This doesn't sound like the knapsack problem if i'm not missing anything in your Q. The amount of each ingredient to go into each recipe is known so your slot size isn't a variable.

If i read your Q correctly, all you need to do is to run the ingredients thru each recipe, see whether the amounts are sufficient, and if so, calculate the value of the ingredients to be left out on that one. the recipe with the minimum of such positive values is your answer. takes \theta(m*n) time with direct access to the ingredients list-- m & n the number of ingredients and the recipes.

ashley
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  • yeah lets say i have 250,000+ recipes will that be `O (n)`? – bouncingHippo Nov 23 '12 at 03:11
  • Yes-- O(m*n). the **complexity** of the algorithm is a measure of how much faster the execution time grows by the size of your input. It doesn't depend in any way on the actual size of your input. The one i described grows linearly to the number of recipes-- at the same rate as the number of recipes. – ashley Nov 23 '12 at 03:53
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Let me rewrite your Q to see if i understand it correctly: You have a set of recipes, S, each containing a requirement list of ingredients. The cook has a set of ingredients, in which some amount of components may be even zero. You want to determine a subset of S such that each of the recipes in S can be satisfied by the ingredients the cook have, while minimizing the so called leftover.

say you have 1 egg, 5 sausages; two recipes: A. Egg with sausage-1E1S; B. Jumbo Sausage-0E5S. So the possible solutions are {A}.{B},{A,B}. with {A}, 4 sausage is left; with {B}, 1 egg is left, with {A,B}, it is actually illegal or infeasible solution.

My question is: how do you evaluate the leftover? if Avian influenza is on, the egg can be overpriced, {B} could be preferred. so maybe define some wighted sum function by pricing each ingredients, say f.

If you use wighted sum, this can be viewed as a multiple dimensional KP. The items are those receipts, with multiple "metrics" such as # of ingredients1(or say length), # of ingredients2 (or say width)... And they are to be packed into a bag happened to have multiple-dimension (length, width,...) so feasible packing must satisfy its restraint on each dimension, i.e, total usage of ingredients k must be less than the total ingredients k the cook has. The obj is to minimize the f(weight), or max f(price of the amount of ingredients used up)-total price of internets, or simply the first part.

As far as i know, this problem is solvable as long as the number of dimensions (in you case, the ingredients) is not high. This is some kinda like cutting stock.

jc W
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  • is there a way to determine what is the maximum number of ingredients such that the problem is still solvable? – bouncingHippo Apr 16 '13 at 19:50
  • When I say the number of ingredients is not high, I think under 100 is fine. I am referring the books named "knapsack problems" written by Hans Kellerer, Ulrich Pferschy, David Pisinger (ISBN-10: 3540402861 | ISBN-13: 978-3540402862). Practically, you can always treat this as a general MIP and use commercial solver to solve and usually get a promising solution with a relative small gap%. – jc W Apr 25 '13 at 12:31