Combinatorial optimization: Knapsack problem 2
The Knapsack-problem is an optimization problem, where a quantity of objects with corresponding weights and benefits is given. The aim is to choose a subset of these objects such that the total weight doesn't exceed a given threshold and the total benefit is maximal. This Problem is also called "backpack-problem" since it results from an example in which a hiker has to pack his backpack. He has to answer the question: how can I fill the 50 liter space of the backpack to get an optimal benefit.
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Definition and formulation
An instance of the Knapsack-problem consists of an index-set Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): I = \{ 1, \ldots, k \}
of objects, weights Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): a_i \in \mathbb{R}
and benefits Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): c_i \in \mathbb{R}
for each object Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): i \in I
and a maximum weight Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): b \in \mathbb{R}
. An optimal solution of that instance is an subset Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): I' \subseteq I
with Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \sum_{i \in I'} a_i \leq b
and Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \sum_{i \in I'} c_i
maximal.
The problem can be formulated as a Binary Linear Program as follows:
- Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): \begin{alignat}{3} \max & \sum_{i \in I} x_i \cdot c_i & \\ \mathrm{s.t.} & \sum_{i \in I} x_i \cdot a_i \leq b &\\ & x_i \in \{ 0, 1\} & \forall i \in I \end{alignat}
Replacing the binary conditions Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): x_i \in \{0,1\}
with the weaker conditions Fehler beim Parsen (http://mathoid.testme.wmflabs.org Serverantwort ist ungültiges JSON.): 0 \leq x_i \leq 1
, one gets the "relaxed" or "fractional knapsack problem".
Example
A woman wants to go on vacations. But she has to overcome a serious problem, because the airline prescribe a maximum weight of 25 kg () for her luggage (the worst that can happen to a woman). She has the choice between six different items which she can pack. Each object has a special benefit for the woman and a weight . What items should be selected by her to have the maximum benefit without exceeding the maximum weight of 25 kg?
Solution methods
For the knapsack problem, there are plenty of different solution methods. In this section, the most important solution methods are presented.
To be able to apply the first three of the presented methods a ranking must be made in advance.
For this purpose, a quotient is formed from the benefit and the weight . Afterwards, the objects are sorted according to these values.
Ordered list:
This list is the basis for the following three methods.
Kolesar
The goal is to obtain an integer solution. One begins with the object of rank 1, then rank 2 and so on and sets the variable equal 1 or 0. The items can be taken as long as the maximum weight of 25 kg is reached. This means that the last possible item mostly can't be taken completely. The solution with the highest objective function value is branched first. If the objective function value is less than the value that provides the best integer solution, this solution is not branched.
Note that the solution in the root node is an optimal solution to the fractional variant of the knapsack problem.
Greenberg & Hegerich
In this method, one starts with the non-integer variable and sets this variable equal 0 and 1 respectively to look for a better solution. This should be continued until an optimal solution is found. The procedure is similar to the first presented method.
Cascading Tree
Dynamic Programming
While the first three presented methods work in a Branch-and-Bound manner, a very popular method is based on Dynamic Programming.
