ANALYSIS OF A LOCAL SEARCH HEURISTIC FOR FACILITY LOCATION - PowerPoint Presentation

Slide1

ANALYSIS OF A LOCAL SEARCH HEURISTIC FOR FACILITY LOCATION PROBLEM

Madhukar

R.

Korupolu

C. Greg

Plaxton

Rajmohan

Rajaraman

Proceedings

of the ninth annual ACM-SIAM

Symposium

on Discrete

Algorithms (SODA

), 1998Slide2

LOCAL SEARCH TECHNIQUE

Choose a feasible solution S arbitrarily.

Apply a local search operation on S to obtain a new solution with improved cost.

Repeat step 2 until no more operations can be performed that improves the existing cost.

Output the current solution S as local optimal solution.Slide3

LOCAL SEARCH OPERATIONS

Adding a facility

Dropping a facility

Swapping a facility

Assumption:- All the operations can be done in polynomial time.Slide4

Adding a facility

A new facility which does not belong to the current solution is added to it.

Reassignment of the clients is done.

If there is an improvement in the total cost, the facility is added, else rejected.Slide5

Dropping a facility

A facility is dropped from the current solution.

The clients of the dropped facility are reassigned to the other facilities in the solution.

If there is an improvement in the total cost, the facility is dropped, else not.Slide6

Swapping a facility

A facility of the current solution is swapped with a new facility which does not belong to the current solution.

Reassignment of the clients is done.

If there is an improvement in the total cost, the facility is swapped, else not.Slide7

IMPORTANT TERMS

Primary Facility

Secondary FacilitySlide8

PRIMARY FACILITY

For each facility

i

* in the optimal solution,

Neighborhood of

i

*, denoted as N(

i

*) is the set of

i

ɛ S such that

i* is closest to i in optimal.A facility i ɛ N(i*) is said to be a primary facility if it is closest to i*.

O

S

C

i

i

*

i

’

N(

i

*)

i

ndicates closest edge

Primary facilitySlide9

SECONDARY FACILITY

All

the facilities in the current solution which are not tagged as primary facilities are secondary facilities

.

O

S

C

For a secondary facility i’

ɛ

N(

i

*), i is said to be its

associated primary facility

if i is primary facility of

i

*.

Secondary Facility

Associated Primary Facility

N(

i

*)

i

*

i

’

iSlide10

ANALYSIS

S:- Local Optimal solution

O:- Global Optimal solution.

g

i

= total service cost paid by clients of facility

i

ε

S

g

i* = total service cost paid by clients of facility i ε OSp = set of all primary facilities in SSs = set of all secondary facilities in

SS ∩ O = φ

Since S is the local optimal solution, therefore, its cost cannot be improved further by performing any operation.Add operation is used to bound the service cost of S.

Drop and Swap operation are used to bound the facility opening cost of S.Slide11

+

+

+

+

+

+

+

+

+

+

+

+

C

S

O

-

-

-

-

-

-

-

-

-

-

-

-

BOUNDING SERVICE COST

Consider a facility o

ε

O added to S to give S’.

C(S’) – C(S) =

f

o

+

g

o

* – g

o

Since

S

is locally optimum,

f

o

+

g

o

* – g

o

≥

0 ==>

g

o

≤

f

o

+ g

o

*

After doing this for all the facilities of optimal and adding them, we get C

s

(

S

) ≤

C

f

(O) + C

s

(O).Slide12

BOUNDING FACILITY COST

OF SECONDARY FACILITIES

USING DROP OPERATION

Drop operation is performed on secondary facilities.

Consider a secondary facility i being dropped from S to give a new solution S’, i.e., S’= S - i.

All the clients of i are reassigned to its associated primary facility i’ in S.

Difference in total cost of S’ and S is, C(S’) - C(S).

Since S is locally optimum, => C(S’) – C(S) ≥ 0

Consider all the clients j of i, which are now reassigned to i’, therefore,

C(S’) – C(S) =

Σ

j (cji’ – cji) – fi ≥ 0 f

i ≤ Σj (

cji’ – c

ji) …..ISlide13

c

ji

’

≤

c

ji

+ c

ii*

+ c

i*i’

(by triangle inequality) ≤ cji + 2cii* (since i’ is primary facility of i*) ≤ cji + 2ci’’i (i*, and not i’’, is closest to i) cji’- cji ≤ 2(cji + cji’’) (by triangle inequality) cji’- cji ≤ 2(cji + c

ji*) (since

j is assigned to i’’)

Adding this for all the clients of i ε

Ss, we get

Σj(cji

’ - cji) ≤ 2(

gi + g

i*).Therefore eq I now becomes, f

i ≤ 2(gi

+ gi*).

Adding this for all secondary facilities i ε

Ss leads to,

Cf(

Ss) ≤ Σ

i 2(gi +

gi*)

i’’

i* O

j C

i

i

’

S

Let i

ε

N(i*). Then i’ also in N(i*)

-

+Slide14

BOUNDING FACILITY COST OF SECONDARY FACILITIES

USING

SWAP OPERATION

Swap operation is performed on primary facilities.

Consider a primary facility i being swapped with its closest facility i*

ε

O to give a new solution S’, i.e., S’= S – i + i*.

All the clients of i are reassigned to i*.

Difference in total cost of S’ and S is, C(S’) – C(S).

Since S is locally optimum, => C(S’) – C(S) ≥ 0

Consider all the clients j of i, which are now reassigned to i*, therefore,

C(S’) – C(S) = Σj (cji’ – cji) – fi + f

i* ≥ 0 f

i – fi*

≤ Σj (

cji’ – c

ji) …..ISlide15

c

ji

*

≤

c

ji

+ c

ii*

(by triangle inequality)

≤

cji + cii’’ (since i* is closest to i) cji* - cji ≤ (cji + cji’’) (by triangle inequality) cji* - cji ≤ cji + cji* (since j is assigned to i’’)Adding this for all the clients of i ε Sp, we get Σj

(cji* -

cji)≤ g

i+gi

*Therefore eq

I becomes, fi - fi* ≤ (

gi + gi

*).Adding this for all primary facilities

i ε S

p leads to,

Cf(Sp

)- Cf(O) ≤ Σ

i (gi

+ gi*)

i

’’

i* O

j C

i

S

+

-Slide16

BOUNDING TOTAL COST

Adding results

of

drop and swap, we

get

C

f

(S

)

≤

Cf(O) + 2Cs(S) + 2Cs(O)Using the following bound on service cost from add operation, Cs(S) ≤

Cf(O) + Cs(O

) we get, C(S

) ≤ 4 C

f(O) + 5

Cs(O)