Download GENERAL ASSIGNMENT PROBLEM via Branch and Price

GENERAL ASSIGNMENT PROBLEM
via Branch and Price
JOHN AND LEI
Outline
•  Review the column
generation in Generalized
Assignment Problem (GAP)
2
• 
GAP Examples in Branch
and Price
Assignment Problem
•  The assignment problem is to find a maximum profit assignment of n
tasks to n machines such that each task (i=1,2,…,n) is assigned to
exactly one machine (j=1,2,…n).
3
• 
To find a maximum weight
matching in a weighted bipartite
graph.
• 
One of the fundamental
combinatorial optimization
problem
Generalized Assignment Problem (GAP)
•  m tasks vs. n machines (m>=n)
• 
Each machine is allowed to be assigned to more than one task
• 
• 
• 
• 
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• 
z: total profit
pij: profit of task i on machine j
yij: association between task i and machine j
wij: resouce consumption of task i on machine j
dj: total available resource on machine j
Generalized Assignment Problem (GAP)
•  GAP:
5
• 
If there is only one machine
• 
Knapsack Problem (KP):
Generalized Assignment Problem (GAP)
•  GAP
KP1
KP2
.
.
.
KPn
6
• 
(NOT means the solution to the subproblems KPs can be combined to that of
the GAP)
• 
It motivates us to re-represent the set of points satisfying
constraint (13.10)
To use B & P
Review the B & P Approach
DANTZIG-WOLFE DECOMPOSITION
Conventional
Problem
reformulation
To choose the entering
basic variable:
Pricing Problems
(Sub-Problems)
7
Master
Problem
Column
Generation
Restricted Master
Problem
Master Problem Formulation
Consider the set of points satisfying:
S is just a finite set of points:
where
Conventional Problem
“Kj is the number of feasible solutions
for j” ---- textbook
Kj is the number of
feasible assignment
for the machine j
Z1j, Z2j, …….. ZKjj
are the feasible
solutions of KPj
8
Master Problem Formulation
Consider the set of points satisfying:
Every point y can be represented by
S is just a finite set of points:
where
Conventional Problem
“Kj is the number of feasible solutions
for j” ---- textbook
Kj is the number of
feasible assignment
for the machine j
Z1j, Z2j, …….. ZKjj
are the feasible
solutions of KPj
9
Master Problem Formulation
Every point y can be represented by
Master
Problem
Conventional Problem
Substitute y
The master problem of GAP is formulated in
terms of columns representing feasible
assignments of tasks to machines.
10
Structure of the formulation
Example: m=3, n=2
Master
Problem
Symbol 0/1 in the entries represents 0 or 1
value of z (assignment)
The task lines indicate the assignment
constraint
The machine lines indicate the
convexity constraint
11
The master problem of GAP is formulated in
terms of columns representing feasible
assignments of tasks to machines.
Issues with the formulation
Conventional Problem
• 
Fundamental difference between
these two:
replaced by
IP solution y in S
A finite set of points
Usually a very huge
number
(exponential)
• 
Issue 1: Too much columns in the
reformulation problem.
(So we choose to work with its restricted
version instead of solving directly)
• 
Issue 2: Choosing entering
variables
(Solve the sub-problem/pricing problem)
12
Reformulated Master Problem
Why bothering to re-formulate?
Conventional Problem
• 
A compact IP formulation (e.g.
conventional GAP) has a weak LP
relaxation.
IP solution y in S
• 
A finite set of points
The Master Problem formulation is
tighter than the conventional one.
Reformulated Master Problem
Intuitive explanation for GAP
Fractional solutions in LP that are not
convex combinations of the solution (z) to
the knapsack problems, are not feasible
to the master problem
• 
13
The decomposition formulation is not
to speed up but to tight the bound.
Sub-problem/Pricing Problem
• 
Issue 2: Choosing entering
variables
• 
(u,v) is the optimum dual solution
to the LP relaxation of the
restricted master problem
• 
KP has m+n columns, which is
usually much smaller than the
total number
• 
The relaxation of the restricted
master problem is solved by the
revised simplex method.
• 
The pivot column associated with
the entering variable is generated
by solving the pricing problem
(Solve the sub-problem/pricing problem)
sub-problem/pricing problem:
Then new decomposition
14
Sub-problem/Pricing Problem
sub-problem/pricing problem:
15
• 
If the optimum value of any pricing problem is positive, then the
column with positive reduced cost can be added to the restricted
master problem.
• 
If the maximal reduced costs of all the KPs are non-positive, then
the LP solution obtained is also maximal for the relaxation of the
UN-RESTRICTED master problem.
GAP Example
m=3, n=2
(d1,d2)=(11,18)
Conventional Formulation:
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GAP Example
Conventional Formulation:
17
GAP Example
Conventional Formulation:
Decomposition Re-formulation:
18
GAP Example
Initial feasible solution：
(Only a subset of all variables)
(13.18)
(13.19)
(13.20)
(13.21)
(13.22)
Two-phase
method
Decomposition Re-formulation:
19
GAP Example
Update the tableau
Passing the dual value u and v to the sub-problems, we have
Decomposition Re-formulation:
20
GAP Example
We do this until …
No negative objective value of sub-problems is found
Decomposition Re-formulation:
Z1=(1,0,0)
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Z2=(0,1,1)
Branch-and-Price Algorithm
Original Problem Formulation
Master Problem
Restricted Master Problem (RMP)
Solve LPR of RMP
Add such
columns to
RMP
Generate Duals from RMP
Solve Sub-problem by passing duals
Are there any columns generated with positive reduced cost?
N
Solution Integral ?
N
22
Branch
Y
Stop
Y
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