- The assignment problem
is one of the fundamental
combinatorial optimization problems
in the branch of optimization or
operations research
in mathematics.
- In an assignment problem, we must find a maximum matching that has the minimum weight in a weighted bipartite graph.
- Problem description:
- 3 men
apply for 3 jobs
Each applicant gets one job.
- The suitability of each candidate
for each job is represented by
a cost:
- The lower the cost, the more suitable the candidate is for that job
Example:
- Problem:
- Find a maximum matching (give jobs to as many men as possible) for which the sum of the cost of the edges is minimized
- 3 men
apply for 3 jobs
- In the previous lecture, we have learned:
- How to find a maximum matching
- We can solve the assignment problem
by:
- Find all maximum matchings
- Sum the cost of the edges
of each maximum matching
- Select the maximum matching with the lowest possible cost
- Find all maximum matchings
- Obviously, we want something better :-)
- The assignment problem
can be formulated as
a 0,1-integer
linear constrained optimization problem
(i.e.: IP)
- Example:
Cost Matrix: jobs -------------> 1 2 3 +- -+ 1 | 1 4 5 | C = 2 | 5 7 6 | 3 | 5 8 8 | +- -+Define:
- Cost matrix
C = [cij]
where
cij =
cost of
man i working on
job j
- Variable
xij = 0 or 1
- xij = 1 if man i is assigned to job j and
- xij = 0 otherwise
- Cost matrix
C = [cij]
where
cij =
cost of
man i working on
job j
- Integer program for the assignment problem:
min: 1x11 + 4x12 + 5x13 + 5x21 + 7x22 + 6x23 + 5x31 + 8x32 + 8x33 s.t.: x11 + x12 + x13 = 1 // Worker 1 gets 1 job x21 + x22 + x23 = 1 x31 + x32 + x33 = 1 x11 + x21 + x31 = 1 // Job 1 assigned to 1 worker x12 + x22 + x32 = 1 x13 + x23 + x33 = 1 xij = 0, or 1
- Example Program:
(Demo above code)
- IP Prog file: click here
Solution:
Value of objective function: 15.00 Actual values of the variables: x11 0 x12 1 x13 0 x21 0 x22 0 x23 1 x31 1 x32 0 x33 0
- Note:
- We can solve a Linear Program
instead of an integer program
--- this is due to a
special property
- The constraint matrix
of the assignment problem
is very special
It will only produce integer solutions
- These special constraint matrices are called uni-modular matrices (See: click here)
- We can solve a Linear Program
instead of an integer program
--- this is due to a
special property
- The Simplex Algorithm
is a general purpose
algorithm to find the
maximum/minimum
of a linear cost function
with
linear constraints
- Linear program that
are formulated from
network flow problems
often have a special structure
We can exploit the structure to improve the performance of the Simplex Algorithm for some special type of problem.
- Some specially adapted Simplex Algorithms:
- Network Simplex Algorithm
(See:
click here)
- The Network Simplex Method is
a (primal) Simplex Method adapted to solve the
maximum flow problem
in networks
(I did not discussed this for brevity)
- The Network Simplex Method is
a (primal) Simplex Method adapted to solve the
maximum flow problem
in networks
- The Hungarian Algorithm
(See:
click here)
- The Hungarian is
a (primal-dual) Simplex Method adapted to solve the
assignment problem
in bi-partite graphs
(I will discuss this algorithm next)
- The Hungarian is
a (primal-dual) Simplex Method adapted to solve the
assignment problem
in bi-partite graphs
- Network Simplex Algorithm
(See:
click here)