Bin Packing

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Bin Packing

• Chapter 1 Applied Mathematics of Logistics

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An instance of bin packing

• The maximum weight (capacity) of one box is 9 kg.
• Given the set of items whose weights are
  (6,6,5,5,5,4,4,4,4,2,2,2,2,3,3,7,7,5,5,8,8,4,4,5).
  
• How can we pack these items into the minimum
  number of boxes.

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Bin packing problem

• Given a set N of n items and an infinite number
  of bins each of which has a size (capacity) B.
• The size of item i 2 N is known and is denoted
  by wi .
• Find the minimum number of bins so that all the n
  items are packed and the total size of items
  packed into a bin does not exceed the bin size B.

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Big Oh(O)notation

• Function f(n) is O(g(n)) iff (if and only if)
  there exists a constant C such that

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Omega(O)notation

• Function f(n) is ?(g(n)) iff there exits a
  constant C such that

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Theta(T)notation

• f(n)O(g(n)) and f(n)? (g(n)) ? f(n)? (g(n))
• Function f(n) is ?(g(n)) iff there exists two
  constants c, C such that

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Small o and small omega notation

• Function f(n) is o(g(n)) (denoted by f(n)o(g(n))
  iff
• Function f(n) is ?(g(n)) (denoted by f(n)?(g(n))
  )iff

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Worst case analysis

• Optimal value of an instance I is denoted by
  OPT(I). Approximate solution value of an
  instance is denoted by A(I).The ratio r(I) is
  defined as
  
• If approximate algorithm A satisfies
• r(I) ?r
• for all instance I, A has a performance ratio
  r.

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Asymptotic performance ratio

• If approximate algorithm A satisfies
• . ,
• we call A has an asymptotic performance
  ratio r.

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Next fit heuristics

• Pack items in the order of their indices
  1,2,?,n.If the total size of items packed into
  the bin exceeds the size of bin, close the bin,
  prepare a new bin, and pack the item into the new
  bin.
• If we denote the number of bins obtained by next
  fit heuristics by Anf(I), we can prove
• Anf (I) ? 2 OPT (I)
• for all I.
• Next fit has a performance ration 2.
• The performance ratio 2 cannot be improved, i.e.,
  there exists an example whose performance ratio
  approaches to 2.

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Online and offline heuristics

• Online heuristics The items arrives in some
  order and they have to be assigned to a bin as
  soon as they arrive without knowledge of the
  remaining items.
• Offline heuristicsThe entire list of items and
  their sizes are known before the packing begins.
• Bounded space online heuristicsA fixed number K
  of partially filled bins are open. Once the bin
  is closed, it must remain so.
• Next fit heuristics is bounded space online with
  K1.

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First fit heuristics

• Pack items in the order of their indices
  1,2,?,n.If the item is packed into the bin that
  does not exceed its capacity and has the minimum
  index. If there does not exist any bin to be
  packed, close the bin, prepare a new bin, and
  pack the item into the new bin.
• First fit relaxes the bounded space condition K1
  of next fit.
• First fit has an asymptotic performance ratio
  17/10.
• The performance ratio 17/10 cannot be improved,
  i.e., there exists an example whose performance
  ratio approaches to 17/10.

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First fit decreasing heuristics

• Sort the items in the non-increasing order of
  their sizes. and then apply first fit heuristics.
  
• Offline heuristics
• First fit decreasing has an asymptotic
  performance ratio 11/9.
• The performance ratio 11/9 cannot be improved,
  i.e., there exists an example whose performance
  ratio approaches to 11/9.

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Approximation scheme

• Consider an approximate algorithm A that returns
  an approximate value A(I, ?) if we are given a
  problem instance I and an upper bound of
  relative error ? (gt0).
• A is called the approximation scheme if it
  satisfies
• for all instances I.
• Polynomial time approximate scheme an
  approximation scheme that runs in a polynomial
  time of the input size
• Fully polynomial time approximation scheme an
  approximation scheme that runs is a polynomial
  time both in the input size and 1/?

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Asymptotic approximation scheme

• A is called the asymptotic approximation scheme
  if it satisfies
• Polynomial time asymptotic approximate scheme an
  asymptotic approximation scheme that runs in a
  polynomial time of the input size
• Fully polynomial time asymptotic approximation
  scheme an asymptotic approximation scheme that
  runs is a polynomial time both in the input size
  and 1/?

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Approximation scheme for bin packing

• Assuming P ? NP , there does not exists a
  polynomial time approximate algorithm such that
  r(I)lt3/2.
• It can be proved by a polynomial time reduction
  from the number partitioning problem to the
  (approximate) bin packing problem.
• Thus we cannot expect a polynomial time
  approximation scheme.
• There exists an asymptotic approximation schme
  that returns A(I,?) ? (1?) OPT(I) 2/?2 1 .

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Probabilistic analysis

• Concern the average case behavior of heuristics
  under some assumptions of probabilistic
  distributions of problem instances.
• Let Ln(F) denote a random n-item list with item
  sizes chosen independently according to
  distribution F.
• Asymptotic performance ratio
• Expected waste

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Probabilistic analysis of next fit heuristics

• If item sizes are chosen independently according
  to uniform distribution U0,1 on an interval
  0,1 and the bin size B is 1, the asymptotic
  performance ratio of next fit is
• If F is uniform distribution U0,b(where 1/2 ? b
  ? 1) on an interval 0,b, the asymptotic
  performance ratio of next fit is

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Probabilistic analysis of first fit heuristics

• If F is U0,1 , the asymptotic performance ratio
  of first fit heuristics is
• The expected waste is

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Best fit heuristics

• Pack items in the order of their indices
  1,2,?,n.If the item is packed into the bin that
  does not exceed its capacity and has the maximum
  size (minimum remaining capacity) . If there does
  not exist any bin to be packed, close the bin,
  prepare a new bin, and pack the item into the new
  bin.
• If F is U0,1, the expected waste of best fit
  heuristics is

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Discrete distribution and perfect packable

• The bin size B is a positive integer.
• The item sizes are a sequence of positive
  integers s1 lts2 lt? ltsJ where sj (j1,?,J) is in
  1,B.
• The probability with which the item with size sj
  occurs is pj .
• The discrete distribution F is said to be
  perfectly packable if the expect waste of the
  optimal solution is o(n).

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Flow formulation and perfectly packablility
Variable xjh the ratio with which the item of
size sj is assigned to the bin withy height h.
The optimal value of the above linear
programming problem is 0.
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Sum-of-squares method

• Item sizes have a discrete distribution and the
  size of bin is a positive integer B.
• The height h of bin that is not empty nor full is
  an integer within an interval 1,B-1 .
• The number of bins with height h is denoted by
  M(h).
• In the sum-of-squares method, the arrived item is
  packed so that the sum of squares of M(h), SS
  below, is minimized.

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Evaluation of the difference in the
sum-of-squares and the expected waste

• The increase ? of SS when the item with size w
  is assigned to the bin with height h is
  If the distribution F is perfectly
  packable, the expected waste EWssn of the
  sum-of-squares method is