- Knapsack problem(s):
- You are given a knapsack which
has a certain
capacity
(volume and/or weight):
- You have a number of items:
And so on.... Each item has:
- A value
(that gives a measure of benefit
if you pack one of these
items into the
knapsack)
- A weight (or volume) (that gives a measure of size reduction to the capacity of the knapsack if you pack one of these items into the knapsack)
- A value
(that gives a measure of benefit
if you pack one of these
items into the
knapsack)
- The problem:
- Pack as many items
as they will fit in the knapsack
such that:
- The total value packed in the knapsack is maximized
- Pack as many items
as they will fit in the knapsack
such that:
- You are given a knapsack which
has a certain
capacity
(volume and/or weight):
- Applications:
- The knapsack problem
is one of the most studied problems
in
combinatorial optimization,
with many real-life applications.
- Example: pack food in a knapsack for maximum nutritient value.
- The knapsack problem
is one of the most studied problems
in
combinatorial optimization,
with many real-life applications.
- Variations of
knapsack problems:
- Many variations of
knapsack problems exist:
- How many items
(of each kind) do you have
- How many knapsacks
do you have
- More types of
constraints
- And so on...
- How many items
(of each kind) do you have
- Many variations of
knapsack problems exist:
- Note:
- This course will not
study all kinds of
variations.
- We will study the following 2 variants:
- The 0,1 knapsack problem
- The unbounded knapsack problem (as a way to introduce the next topic)
- The 0,1 knapsack problem
- Here is a nice overview of the variants: click here
- This course will not
study all kinds of
variations.
- Fact:
- Knapsack problems
is a subset of a larger class
of problems called:
- constrainted optimization
- Knapsack problems
is a subset of a larger class
of problems called:
- Optimization problems:
- Optimization problem =
determine the
optimum value for
a certain system
Example:
max f( x1, x2, ..., xn );
(It the system can decribed as by a function f(x1, x2, ..., xn), the optimization problem is solved by finding the zero's of the derivitive function:
f'(x1, x2, ..., xn) = 0
- Optimization problem =
determine the
optimum value for
a certain system
- Types of
optimization problems
- Unconstrainted optimization
problems
- There are no constrainted on the range of values that x1, x2, ..., xn can take on
Example:
max ( -x12 + 3x1 - x24 + 5x2 + 4 )
- Constrainted optimization
problems
- The range of values that
x1, x2, ..., xn
can take on is limited
Example:
max ( -x12 + 3x1 - x24 + 5x2 + 4 ) subject to: x1 ≤ 9 (we only have 5 units of resource 1) x1 + x2 ≤ 5 (Resources 1 and 2 are related) - The range of values that
x1, x2, ..., xn
can take on is limited
- Unconstrainted optimization
problems
- The general form
of a constrained opimization problem:
max f( x1, x2, ..., xn ); subject to: g1( x1, x2, ..., xn ) = c1 // Equality constraints g2( x1, x2, ..., xn ) = c2 ... gu( x1, x2, ..., xn ) = cu h1( x1, x2, ..., xn ) ≤ d1 // Inequality constraints h2( x1, x2, ..., xn ) ≤ d2 ... hv( x1, x2, ..., xn ) ≤ dv
- Resource allocation problems:
- You have a limited amount of resources
- You have to maximize the profit using the limited (bounded) resources
- You have a limited amount of resources
- Decision problems
- You have a number of task to do
in a limited amount of time
- Select the set of task that will maximize some profit function
- You have a number of task to do
in a limited amount of time
- Remark:
- I will formulate
the knapsack problems as
constrained optimization problems
- There are many important problems in real life that are constrained optimization problems
- I will formulate
the knapsack problems as
constrained optimization problems
- Problem description:
- Given N types of items:
- The item type i has
a weight wi
- The item type i has a value vi
... 0,1 refers to:
- There is exactly one
item
of
each type of item available
(I.e.: you can pack an item (you will then one of that item) or you can do not pack that item (you will then zero of that item)
- The item type i has
a weight wi
- Given a knapsack with
capacity W:
- Problem:
- Pack as many items into
the knapsack
such that the
total value of the
items packed is
maximized
Note: you cannot exceed the capacity of the knapsack !
- Pack as many items into
the knapsack
such that the
total value of the
items packed is
maximized
- Given N types of items:
- Formulating the 0,1-knapsack problem as a
constained optimization problem:
- Define:
- x1 = # items of type 1 packed into the knapsack
- x2 = # items of type 2 packed into the knapsack
- ...
- xN = # items of type N packed into the knapsack
- Then:
- Total weight of item packed =
w1 × x1 +
w2 × x2 + ...
+ wN × xN
- wi = weight of an item i
- Total value of item packed =
v1 × x1 +
v2 × x2 + ...
+ vN × xN
- vi = value of an item i
- Total weight of item packed =
w1 × x1 +
w2 × x2 + ...
+ wN × xN
- The constrainted optimilization
formulation of the 0,1 knapsack problem:
max: v1×x1 + v2×x2 + ... + vN×xN s.t.: w1×x1 + w2×x2 + ... + wN×xN ≤ W x1 = 0, 1, 2, ..., c1 x2 = 0, 1, 2, ..., c2 ... xN = 0, 1, 2, ..., cN
- Define:
- Problem description:
- Given N types of items:
- The item type i has
a weight wi
- The item type i has a value vi
... Unbounded referes to:
- There are an unbounded
(= infinite) number
of
each type of item available
(I.e.: unbounded supply)
- The item type i has
a weight wi
- Given a knapsack with
capacity W:
- Problem:
- Pack as many items into
the knapsack
such that the
total value of the
items packed is
maximized
Note: you cannot exceed the capacity of the knapsack !
- Pack as many items into
the knapsack
such that the
total value of the
items packed is
maximized
- Given N types of items:
- The unbounded knapsack problem as a
constained optimization problem:
- Define:
- x1 = # items of type 1 packed into the knapsack
- x2 = # items of type 2 packed into the knapsack
- ...
- xN = # items of type N packed into the knapsack
- Then:
- Total weight of item packed =
w1 × x1 +
w2 × x2 + ...
+ wN × xN
- Total value of item packed = v1 × x1 + v2 × x2 + ... + vN × xN
- Total weight of item packed =
w1 × x1 +
w2 × x2 + ...
+ wN × xN
- The unconstrained knapsack problem:
max: v1×x1 + v2×x2 + ... + vN×xN s.t.: w1×x1 + w2×x2 + ... + wN×xN ≤ W
- Define:
- The bounded knapsack problem
- The bounded
knapsack problem:
- The bounded
knapsack problem (UKP)
places an general upper bound
on the number of copies
of each kind of item:
x1 ≤ C1 x2 ≤ C2 ... xN ≤ CN
- The bounded
knapsack problem (UKP)
places an general upper bound
on the number of copies
of each kind of item:
- The Mathematical formulation of
the bounded knapsack problem :
max: v1×x1 + v2×x2 + ... + vN×xN s.t.: w1×x1 + w2×x2 + ... + wN×xN ≤ W x1 = 0, 1, 2, ..., c1 x2 = 0, 1, 2, ..., c2 ... xN = 0, 1, 2, ..., cN
- The bounded
knapsack problem:
- The unbounded knapsack problem is
an example of:
- integer programming problem
that we will learn to solve later (when we talk about the branch and bound technique).