branch and bound - csuanderson/cs320/slides/branch and...branch and bound-algorithm design...
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Branch and Bound- algorithm design approach
- a tree search strategy to generateall possible solutions to a problem ,but with
- pruning rules to eliminate regionsof the tree that cannot lead to
solutions that are better than ones
found so far .
Three components to a branch and boundalgorithm :
- search strategy- branching strategy
- pruning rules
For some problems that cannot be solvedwith a greedy or dynamic programming
algorithm ,a branch and bound algorithm
may be a good ,and perhaps only , approach .
To design a branch and bound algorithm ,
specify- a search that will enumerate all possible
solutions
- pruning rules that eliminate a subtreeso it does not have to be searched
.
Pruning rules are based on upper and lower bounds
i I 23
o - I Knapsack,
W -
- 10,
w = 6 5 4V = 12 CO 7
-Ordered by decreasing %
Find upper
boundon value of packed knapsack
by considering it a fractional knapsack problemand finding greedy solution
.
Remaining Value inAdd
a
Capacityknapsactc( O O
all of item I 10 .
-6=4 Ot 12=12
W= 6,
ve 12
415 of item 24
-
485=012+41510=20
w=p,r= 10
Best value for fractional knapsack version
=2oSo
, we know a solution to the o - I knapsackwill have value I 20
ab= optimistic oriopperbound
i I 23
V is knapsa.deab
-
- 20 We 6 54V = 12 CO 7value V=o
fusing:} WHO addikml
ab -
- 17
&ipiem, \
ab = 20
✓ =o✓ = 12
W = tow -4
¥¥ skipperygdditem2"dominates
"
ab⇒atoo heavy
=
better v-42than W=4anythingYImssibhskipitmfadydi.tn
's
Bicycle Sharing SystemsStatic Rebalancing Problem :
- Vehicle moves bicycles from one stationto another
- Each station must be visited once andOnly once
.
Typical problem size : Panis 1,258stations13
,820 bicycles
Mathematical Formulations
For Station
io- Ei = number of bicycles•
Ri = minimum requiredRti
= maximum requireddij = travel time between
stations i and jWi = R
-
i- Ei or Ei - Rti
( - . . . - I -1 - --
Ei RI Rt .
' t
Ei
objective : Minimize § wi Cti - to )
ti - to is time that Station i hastoo few an too many bicycles
.
Ways to calculate -
lower bounds :
LB I - allow vehicle to visit a stationmore than once
LBZ,
LB 3 - other simplificationsupper bounds :
UP BI - genetic algorithm searchfor approximate solution
UPBZ - depth - first search of just thechild node with lowest lowerbound .Greedy
.
UP 133 - nearest - neighbor . Constructroute by picking next station
that minimizes travel time.
Experiments :
Tested network sizes to,
. . .
,100
.
< BI best overall
UBI, genetic algorithm best for small networks
,
UBZ,
depth first greedy best otherwise
Found optimal solutions for networks I 30.
in less than 134 hours.
Kadri,
et al .
,
"
A branch - and - boundalgorithm for solving The static rebalancing
problem in bicycle - sharing systems "
,
Computers & Industrial Engineering ( 95 ) 2016,41-52 .
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