Research Rotation Presentation Bees Pollination Modeling

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Research Rotation PresentationBees Pollination
Modeling

• Xianyi Zeng
• Advisor
• Walter Murray
• Berry Brosi

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Outline

• ? A description of the problem
• ? Some basic assumptions and functions
• ? Discrete model
• ? Continuous model

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Bees Pollination Modeling

• Original problem
• ? Bees helps fertilizing crops.
• ? Wild bees are used to help with pollination.
  But they need habitats, which take up some
  fields.
• ? The more bees one unit of field have, the
  more crops it will produce. And there is a limit
  on how many crops the field may produce, no
  matter how many bees are there. So the marginal
  production of crops is a decreasing function of
  density of bees.
• ? On the other hand, costs of holding
  bee-habitats is supposed to be linear function of
  number of bees.
• ? There is a trade-off between the two, and
  the purpose is to find a optimized location of
  habitats, that will give the highest output.

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Bees Pollination Modeling

• Modeling
• ? Number of bee habitats
• The cost of placing the habitats is almost
  determined by the number of these habitats.
• ? Location of these habitats
• If the number of habitats is fixed, the
  cost is nearly fixed (although there may be small
  difference, depending on which model is used), so
  the problem is how to allocate these habitats.

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Bees Pollination Modeling

• Methods
• ? Discrete model
• The whole field is divided into cells,
  each habitat is considered to take up a whole
  cell, which means it will use a fixed percentage
  of fields.
• ? Continuous model
• Every habitat can be modeled as a point,
  or a region the position of the habitats center
  may change continuously.

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Basic Assumptions

• Bees distribution
• ? Bees are supposed to fly in some direction,
  from their home, and stop at every point, which
  is just the cells in the discrete model.
• ? Suppose the probability that a bee stop at
  its current point (cell) is qf. This probability
  may vary due to the diversity of different bees,
  so this qf is not fixed by nature.
• Production function
• ? Suppose d is the density (number) of bees at
  current point (cell), then the amount this unit
  of field produces is
• ?(d) b z
  ( 1 e dc )
• Where b is the baseline of production.
  i.e. if no bees exist, there will be still some
  output. z controls the maximum level of
  production, which is independent of d. And c is a
  constant controlling the slope of the curve.

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Discrete Model1-dimension

• Description
• ? In 1-dimension model, the field is
  considered as an array of cells, say 50 in
  number, and each cell is either used for crops,
  or for bees habitats
• The shadowed cells are where habitats
  located.
• In this case, each habitat uses up 1/50 of
  the land. Since there are 12 habitats located in
  the field, only 76 land are used for crops.
• ? The distribution of bees are simply
  calculated as
• Every single bee will choose one of two
  directions of chance by half, fly straight
  forward, and stop at every cell it arrives at the
  probability of qf. Thus,suppose there are M bees
  in the habitat located in the Rth cell, there
  will be bees of the number MF helping pollinating
  the ith cell. If the cell is also a habitat,
  these part of bees are wasted in the sense of
  helping pollination. And F is calculated as
• F0.5qf(1-qf)i - R

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Discrete Model1-dimension

• Modeling
• ? Actually, this is a discrete optimization
  problem with N binary variables. Where N is the
  number of cells.
• Procedure
• ? First, compute distribution of bees from
  every habitats.
• ? Second, calculate the total number bees in
  every cell that is used for crops.
• ? Third, calculate production of every such
  cell, take the opposite of the total production.
• ? Finally, try to find the minimum of this
  result.
• ? Note, it is supposed that in every habitat,
  there are M bees.

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Discrete Model1-dimension

• Original work
• ? Berry has used the GADS toolbox in MATLAB to
  solve this problem, with N equals 50.
• ? GADS uses genetic algorithm for this
  particular problem. Which randomly choose some
  sequences, calculate for some best performed
  ones. Then it discard the other, add some new
  sequences, which is either randomly chosen, or
  some offspring of the good sequences from the
  former iteration.
• ? One thing good about this algorithm is that
  it gives out the number of habitats and position
  of habitats simultaneously.
• ? However, this is achieved at the cost that
  the size of the problem is made much larger than
  the number of habitats.
• ? Furthermore, just like the case of gene
  development, the result of this algorithm is not
  unique, and even may be not a local minimizer.

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Discrete Model2-dimension

• The original problem is actually a 2-dimensional
  problem. 1-dimensional case is meaningless in
  practical use, but just provide a instruction of
  how to deal with 2-D case.
• Discrete 2-D Case
• ? The problem is similar as the 1-D case. Just
  two thing is needed to be update the
  configuration of cells, and the distribution
  function.
• ? There may be at least 3 possible different
  configurations of cells
• figure 1 figure 2
  figure 3

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Discrete Model2-dimension

• ? Configuration 1
• Most common configuration, but the
  distance between two cells is not easy to define.
• ? Configuration 2
• The distance is easy to define, but every
  cell is small, resulting in more cells are
  needed, and the shape of each cell seems not
  practical.
• ? Configuration 3
• A combination of the former two, and it is
  good-looking
• Difficulty
• ? When generating the method to 2-D case, the
  size of the problem is squared, genetic algorithm
  performs not well.

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Continuous Model

• Description
• ? The field O is modeled as an interval 0,L
  in 1-D model, or a rectangle 0,L10, L2 in
  2-D model.
• ? Each habitat x is modeled as a point Xixi
  in 1-D model, or Xi(xi, yi) in 2-D model.
• ? Distribution of bees from a habitat centered
  at x is
• f(Y) f (dist(X,Y))
• i.e. the density at point Y from habitat X
  is only determined by the distance between X and
  Y.
• ? The total number of bees at point Y equals
  to
• D(Y)?f(dist(Xi,Y))M
• Where the summation take all habitats Xi,
  and M is number of bees in one habitat, which is
  supposed to be equal for all habitats.
• ? The production function of this allocation
  of habitats is
• ??(D(Y)) cost
• Where the integral is performed over all Y
  ?O. And cost is another function calculate the
  cost of such an allocation.

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Continuous Model

• Choice of f
• Based on different understanding of the
  parameter qf, there may be different function f
  to compute the bee density from one single
  habitat.
• For example, consider the function f is of
  the type f(l)ab-l, the condition that ?f(l)dl
  1 requires that alog(b)/2.
• Then suppose qf/(1- qf) is the proportion
  of bees go more than a distance l, to number of
  bees those arrived at a distance l (which is the
  same as in the discrete model). The function will
  be
• f(l)
  0.5qfexp(-qfl).
• A comparison between this and the one in
  discrete model is as follows

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Continuous Model

• figure 4

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Continuous Model

• Choices of Cost function
• It is mentioned in the beginning that in
  continuous model, the cost is highly related to
  the number of bee habitats.
• Actually, the most natural way to deal
  with this is to set the cost equals to a constant
  times the number of habitats.
• The constant may be chosen as bz
  multiplied by an area A. bz is the maximum level
  of production in the expression of function ?.
  And A is the area one habitat is supposed to use,
  for example Apr2, for some constant r.
• This is reasonable because near a habitat,
  according to figure 4, there will be quite a lot
  of bees there, and then sup? is nearly achieved
  in the neighbor of the center.

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Continuous Model

• Choices of Cost function
• Also, more complicated way to construct the
  cost function can be used.
• For example, the cost can be exactly
  computed as the integral of production in some
  set O , where O is defined as
• Y? O Y is near to some habitat Xi .
• Actually, in this case, every habitat is
  considered as some region around a center point
  Xi, like circle.
• The most significant difference between
  this cost function and the former one is that
  overlap of habitats is considered in this one.
• However, since it is not likely that an
  optimal solution will have 2 habitats quite near
  to each other, this difference may not be so
  significant in practical computation.

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Continuous Model

• Number of habitats
• Notice that in continuous model, number of
  variables is significantly reduced compared to
  the discrete model. i.e. n variables in 1-D case,
  and 2n variables in 2-D case are needed for the
  problem, where n is the number of habitats.
• Since the optimization need n as a given
  number, it can not compute out the optimal n as
  the discrete model do. So a little more work is
  needed to be done about the number of habitats.
• Suppose the optimal output can be
  calculate efficiently given the number n. As
  mentioned before, this output is the total
  production regardless of the habitats, minus the
  costs caused by the habitats. The former one
  should be an increasing function of n, with
  decreasing first derivative and the latter one,
  can be considered as a linear function of n,
  which is 0 as n0.
• The relationship is plotted in the
  following figure

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Continuous Model

• figure
  5
• The actual production is the difference
  between these two curves. Since both curve are
  quite well shaped, the problem is simply to find
  a n that the derivatives of this two curves are
  identical to each other.

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Thank you!

• ? Any questions?