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Title: Water%20Resources%20Development%20and%20Management%20Optimization%20(Integer%20Programming)


1
Water Resources Development and
Management Optimization (Integer Programming)
CVEN 5393 Mar 11, 2013
2
  • Acknowledgements
  • Dr. Yicheng Wang (Visiting Researcher, CADSWES
    during Fall 2009 early Spring 2010) for slides
    from his Optimization course during Fall 2009
  • Introduction to Operations Research by Hillier
    and Lieberman, McGraw Hill

3
Todays Lecture
  • Integer Programming
  • Examples
  • R-resources / demonstration

4
Integer Programming
5
Integer Programming
In many practical problems, the decision
variables actually make sense only if they have
integer values. If some or all of the decision
variables in a linear programming formulation are
required to have integer values, then it is an
Integer Programming (IP) problem. The
mathematical model for integer programming is the
linear programming model with one additional
restriction that some or all of the decision
variables must have integer values. If only some
of the decision variables are required to have
integer values, then the model is called Mixed
Integer Programming (MIP) problem.
If all of the decision variables are required to
have integer values, then the model is called
Pure Integer Programming problem.
6
In some decision-making problems, the only two
possible choices for decisions are yes and no.
7
1. Examples of Integer Programming
(1) Example of BIP
8
All the decision variables have the binary form
x1 building a factory in Los Angeles? Yes (x11)
or No (x10)
x2 building a factory in San Francisco? Yes
(x21) or No (x20)
x3 building a warehouse in Los Angeles? Yes
(x31) or No (x30)
x4 building a warehouse in San Francisco? Yes
(x41) or No (x40)
Because the last two decisions represent mutually
exclusive alternatives (the company wants at most
one new warehouse), we need the constraint
9
Furthermore, decisions 3 and 4 are contingent
decisions, because they are contingent on
decisions 1 and 2, respectively (the company
would consider building a warehouse in a city
only if a new factory also were going there).
Thus, in the case of decision 3, we require that
x3 0 if x1 0. This restriction on x3 (when
x1 0) imposed by adding the constraint
Similarly, the requirement that x4 0 if x2 0
is imposed by adding the constraint
10
(2) Example of MIP
11
2. Some Perspectives on Solving Integer
Programming Problems
Question 1 Pure IP problems have a finite number
of feasible solutions. Is it possible to solve
pure IP problems by exhaustive enumeration ?
Consider the simple case of BIP problems.
With n variables, there are 2n solutions to be
considered. Each time n is increased by 1, the
number of solutions is doubled. n10, 1024
solutions n20, more than 1 million
solutions n30 more than 1 billion
solutions.
12
Question 2 Is IP easier to solve than LP because
IP tends to have much fewer feasible solutions
than LP ?
13
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14
3. The Branch-and-Bound Technique and Its
Application to BIP
The Branch-and-Bound Technique is the most
popular mode for IP algorithms. The basic
concept underlying the branch-and-bound technique
is to divide and conquer. The original problem
is divided into smaller and smaller subproblems
until these subproblems can be conquered. The
dividing (branching) is done by partitioning the
entire set of feasible solutions into smaller and
smaller subsets. The conquering (fathoming) is
done partially by bounding how good the best
solution in the subset can be and then discarding
the subset if its bound indicates that it cannot
possibly contain an optimal solution for the
original problem.
15
California Manufacturing Co. Example
Branching
16
Original problem
17
Bounding
LP relaxation of the whole problem
18
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19
Fathoming
Three cases where a subproblem is conquered
(fathomed).
(1) A subproblem is conquered if its LP
relaxation has an integer optimal solution
20
(2) A subproblem is conquered if it is inferior
to the current incumbent.
Since Z9, there is no reason to consider
further any subproblem whose bound 9, since
such a subproblem cannot have a feasible solution
better than the incumbent. Stated more
generally, a subproblem is fathomed whenever its
(3) If the simplex method finds that a
subproblems LP relaxation has no feasible
solutions, then the subproblem itself must have
no feasible solutions, so it can be dismissed
(fathomed).
21
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22
Using the BIP Branch-and Bround Algorithm to
Solve the California Manufacturing Co. Example
(1) Initiliazation
Set Z - 8. Solve the relaxation of the whole
problem by the simplex method. The optimal
solution of the relaxation is
Subproblem 1
1)The bound of the whole problem is less than
Z. 2)The relaxation of the whole problem has
feasible solution. 3)The optimal solution
includes a noninteger value of x1. So the whole
problem can not be fathomed and should be divided
(branched) into subproblems.
Subproblem 2
23
(3) Iteration 2
The only remaining subproblem corresponds to the
x11 node, so we shall branch from this node to
create the two new subproblems.
Subproblem 3 with x11, x20.
Subproblem 4 with x11, x21.
Subproblem 3
Subproblem 1
Subproblem 2
Subproblem 4
24
(4) Iteration 3
So far, the algorithm has created 4 subproblems.
Subproblem 1 has been fathomed, and subproblem 2
has been replaced by subproblems 3 and 4, but
these last two remain under consideration.
Because they were created simultaneously, but
subproblem 4 has the larger bound, the next
branching is done from subproblem 4, which
creates the following new subproblems
Subproblem 5 with x11, x21, x30.
Subproblem 1
Subproblem 6 with x11, x21, x31.
Subproblem 3
Subproblem 5
Subproblem 2
Subproblem 4
Subproblem 6
25
(5) Iteration 4
The subproblems 3 and 5 corresponding to nodes
(1,0) and (1,1,0) remain under consideration.
Since subproblem 5 was created most recently, so
it is selected for branching. Since x4 is the
last variable, fixing its value at either 0 or 1
actually creates a single solution rather than
subproblems. These single solutions are
(1,1,0,0) with Z14 is better than the incumbent
with Z9, so (1,1,0,0) with Z14 becomes the
new incumbent.
Subproblem 1
Subproblem 3
Subproblem 2
Subproblem 5
Subproblem 4
Subproblem 6
26
4. The Branch-and-Bound Algorithm for MIP
27
An MIP Example
28
Change 1The bounding step BIP algorithm With
integer coefficients in the objective function of
BIP, the value of Z for the optimal solution for
the subproblems LP relaxation is rounded down to
obtain the bound, because any feasible solution
for the subproblem must have an integer Z. MIP
algorithm With some of the variables not
integer-restricted, the bound is the value of Z
without rounding down.
Change 2 The fathoming test BIP algorithm With
a BIP problem, one of the fathoming tests is that
the optimal solution for the subproblems LP
relaxation is integer, since this ensures that
the solution is feasible, and therefore optimal,
for the subproblem.. MIP algorithm With a MIP
problem, the test requires only that the
integer-restricted variables be integer in the
optimal solution for the subproblems LP
relaxation, because this suffices to ensure that
the solution is feasible, and therefore optimal,
for the subproblem..
29
Subproblem 1
Subproblem 1
Subproblem 2
Subproblem 2
30
Change 3Choice of the branching variable BIP
algorithm The next variable in the natural
ordering x1, x2, , xn is chosen
automatically. MIP algorithm The only variables
considered are the integer-restricted variables
that have a noninteger value in the optimal
solution for the LP relaxation of the current
subproblem. The rule for choosing among these
variables is to select the first one in the
natural ordering.
Change 4The values assigned to the branching
variables BIP algorithm The binary variable is
fixed at 0 and 1, respectively, for the 2 new
subproblems. MIP algorithm The general
integer-restricted variable is given two ranges
for the 2 new subproblems.
31
Subproblem 3
Subproblem 1
Subproblem 2
Subproblem 4
32
Subproblem 3
Subproblem 5
Subproblem 6
33
Subproblem 5
Subproblem 3
Subproblem 1
Subproblem 6
Subproblem 4
Subproblem 2
34
Hughes-McMakee-notes\chapter-06.pdf
Integer / Mixed-integer programming (prof.
mcmakee notes) Introduction Examples
Hughes-McMakee-notes\chapter-07.pdf
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