Facility Location Models

Yuri Kochetov Sobolev Institute of Mathematics Novosibirsk Russia e-mail: [email protected]

Lecture 1. Facility Location Models Content 1. 2. 3. 4. 5. 6. 7.

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Uncapacitated facility location problem Theoretical and empirical results Polynomials in Boolean variables Equivalent reformulations and the “best” one Multi–stage facility location problems Uncapacitated facility location problems with user preferences Competitive facility location problems

Facility location problems. Discrete models and local search methods • Lecture 1

The Uncapacitated Facility Location Problem • Input: − − −

a set J of users; a set I of potential facilities; a fixed cost fi of opening facility i;

−

a production-transportation cost cij to service user j from facility i;

• Output: a set S ⊆ I of opening facilities;

• Goal: minimize the total cost to open facilities and service all users

F (S ) = ∑ fi + i∈S

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cij . ∑ min i∈S j∈ J


Example

I = {1,…, 8} is potential facility locations; J = {1,…, 15} is set of users Users are serviced from nearest facility

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Integer Programming Formulation Variables: 1 if facility i is open, xi = ⎧⎨ ⎩0, otherwise, 1 if user j is served by facility i, yij = ⎧⎨ ⎩0, otherwise, Mathematical model:

s.t.

⎧⎪ ⎫⎪ min ⎨∑ f i xi + ∑ ∑ cij yij ⎬ ⎪⎩i∈I ⎪⎭ i∈I j∈J ∑ yij = 1, j ∈ J ; i∈I

xi ≥ yij, i∈I, j∈J; xi, yij ∈ {0,1} i∈I, j∈J. - 5-


Societies and conferences on Location

Euro Working Group on Locational Analysis http://www.vub.ac.be/EWGLA/

International Symposium on Location Decisions http://www.aloj.us.es/isolde/

INFORMS Section on Location Analysis http://www.ent.ohiou.edu/~thale/sola/sola.html - 6-


Books and Journals H.A. Eiselt, C.-L. Sandblom. Decision Analysis, Location Models, and Scheduling Problems Springer. 2004.

Z. Drezner, H. Hamaher (Eds.) Facility Location. Applications and Theory Springer. 2004.

Z. Drezner (Ed.) Facility Location. A Survey of Applications and Methods Springer. 1995.

P.B. Mirchandani, R.L. Francis (Eds.) Discrete Location Theory. Wiley & Sons. 1990 Studies and Locational Analysis. ISENLORE. Editor-in-Chief F.Plastria - 7-


Theoretical results

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The UFL problem is strongly NP-hard even for metric case.

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An 1,463–factor approximation algorithm for the metric UFL problem would imply P = NP (Guna, Khuller 1999; Sviridenko).

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An 1,52–factor approximation algorithm for the metric UFL problem (Mahdian, Ye, and Zhang 2002).

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An ε–factor approximation algorithm for any ε > 0 in the special case when facilities and users are points in the plane and service costs are geometrical distances (Arora, Raghavan, Rao 1998; Kolliopoulos, Rao 1999).

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There is no constant–factor approximation algorithm for general UFL problem if P ≠ NP.


Empirical Results

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Efficient branch and bound methods based on the fast heuristics for dual problem (Lebedev, Kovalevskaya 1974; Trubin 1973; Beresnev 1974; Bilde, Krarup 1977; Erlenkotter 1978).

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Fast randomized heuristics for large scale metric instances (Chudak, Barahona 2000).

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Improved branch and bound method for large scale instances (Hansen, Mladenovic 2003).

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Effective and efficient local search methods for large scale instances (Resende, Werneck 2002, 2004; Hansen, Mladenović 1997)

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Benchmark library “Discrete Location Problems” (Kochetov, Kochetova, Paschenko, Alexseeva, Ivanenko 2004)


Reduction to Pseudo –Boolean Functions For a given vector gi, i∈I, with ranking

g i1 ≤ g i2 ≤ ... ≤ g im , m =| I | we introduce a vector ∆gi, i = 0,…, m

∆g 0 = g i1

∆g l = g il +1 − g il , 1 ≤ l ≤ m–1

∆g m = − g im . Lemma. For arbitrary vector zi∈{0,1}, i∈I, z ≠ (1,…,1) we have

min g i =

i| zi =0

m −1

∑ ∆gl zi1 ...zil .

l =0 m −1

max g i = − ∑ ∆g m − l z m − l +1...zim

i | zi = 0 - 10-

l =0


Hence, we can get a pseudo–Boolean function for UFL problem: n −1

p ( z ) = ∑ f i (1 − zi ) + ∑∑ ∆clj z j ...z j , i∈I

j∈J l =0

i1

il

j

where the ranking i1 ,..., imj is generated by column j of the matrix (cij):

c j ≤ c j ≤ ... ≤ c j , j ∈ J i1

i2

im

and for optimal solutions we have

⎧ xi∗ = 1 − zi∗ , i ∈ I ⎪ ∗ ∗ = ∈ S i I x { | ⎨ i = 1} ⎪F ( S ∗ ) = p( z ∗ ) ⎩

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Example I = {1, 2, 3}, J = {1, 2, 3} and

⎛10 ⎞ ⎛ 0 3 10 ⎞ f i = ⎜10 ⎟ ; cij = ⎜ 5 0 0 ⎟ . ⎟ ⎜ ⎟ ⎜ 10 10 20 7 ⎠ ⎝ ⎠ ⎝ Pseudo–Boolean function: p(z) = 10(1 – z1) + 10(1 – z2) + 10(1 – z3) + (5z1 + 5z1z2) + (3z2 + 17z1z2) + + (7z2 + 3 z2z3) = 15 + 5(1 – z1) + 0(1 – z2) + 10(1 – z3) + 22z1z2 + 3 z2z3 . New UFL problem: I′ = I, J′ = {1, 2}

⎛5⎞ ⎛ 0 3⎞ f i′ = ⎜ 0 ⎟ ; cij′ = ⎜ 0 0 ⎟ . ⎜ ⎟ ⎜ ⎟ ⎝10 ⎠ ⎝ 22 0 ⎠ - 12-


From PB Function to UFL Problem PB Function with positive coefficients for nonlinear items:

p ( z ) = ∑ f i (1 − zi ) + ∑ al ∏ zi i∈I

where fi, al > 0, Il ⊂ I, l∈L.

l ∈L

i∈I l

Theorem. For given pseudo-Boolean function p(z) an equivalent instance of UFL problem with minimal number of users can be found in polynomial time. Proof. The family Il, l∈L with relation I l1 < I l 2 ⇔ I l1 ⊂ I l 2 is partially ordered set (poset). Chain in poset is a sequence I l1 < I l 2 ... < I l k . Each chain generates an element of the set J. We need to partition the family Il, l∈L into the minimal number of chains. It can be done in polynomial time (see Dilworth Theorem). g - 13-


Multi Stage Facility Location Problem Facilities

1 stage - 14-

2 stage

...

Users

k stage


• Input: − − − −

a set J of users; a set I of potential facilities; a set P of potential facility paths; a (0,1)-matrix (qpi) of inclusions facilities into paths;

− a fixed cost fi of opening facility i; − a production-transportation cost cpj to service user j from facility path p;

• Output: a set P′ ⊆ P of facility paths; • Goal: minimize the total cost to open facilities and service all users

⎧⎪ min ⎨∑ max f i q pi + P ′ ⊆ P ⎪i∈I p∈P ′ ⎩ - 15-

⎫⎪ c pj ⎬ . ∑ min ′ ⎪⎭ j∈J p∈P


Integer Programming Formulation Variables: 1 if facility i is open, ⎧ xi = ⎨ ⎩0, otherwise, 1 if user j is serviced by facility path p, y pj = ⎧⎨ ⎩0, otherwise. Mathematical model: s.t.

⎫⎪ ⎧⎪ min ⎨∑ f i xi + ∑ ∑ c pj y pj ⎬ ⎪⎭ ⎪⎩i∈I p∈P j∈J ∑ y pj = 1, j ∈ J ; p∈P

xi ≥

∑ q pi y pj ,

j ∈ J ,i ∈ I;

p∈P

xi, ypj ∈ {0,1}, i∈I, j∈J, p∈P. - 16-


UFL Problem with User Preferences (UFLPUP)

I = {1,…, 8} is potential facility locations; J = {1,…, 15} is set of users User is serviced by the most desirable facility. - 17-


• Input: − − −

a set J of users; a set I of potential facilities; a fixed cost fi of opening facility i;

−

a production-transportation cost cij to service user j from facility i;

−

a user preferences dij: facility i1 is more desirable then i2 for user j if d i1 j < d i2 j .

• Output: a set S ⊆ I of opening facilities;

• Goal: minimize the total cost to open facilities and service all users

F ( S ) = ∑ f i + ∑ min ci ( s j ) j , where i ( s j ) = arg min d ij , j ∈ J . i∈S

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j∈J i∈S

i∈S


Integer Programming Formulation Variables: 1 if facility i is open, 1 if user j is served by facility i, xi = ⎧⎨ yij = ⎧⎨ ⎩0, otherwise, ⎩0, otherwise, Mathematical model: Company:

⎫⎪ ⎧⎪ ∗ min ⎨∑ f i xi + ∑ ∑ cij yij ( x)⎬ s.t. xi ∈ {0,1} i∈I; x ⎪i∈I ⎪⎭ i∈I j∈J ⎩

where yij∗ (x) is optimal solution of the user problem: Users:

min y

s.t.

∑∑ dij yij j∈J i∈I

∑ yij = 1,

j ∈ J;

i∈I

yij ≤ xi, xi, yij ∈ {0,1} i∈I, j∈J. - 19-


Reduction to Pseudo –Boolean Functions Let the ranking for user j∈J be

d i1 j < d i2 j < ... < dim j , m =| I | Sij = {k∈I | dkj < dij} and ∇Ci1 j = Ci1 j , ..., ∇Cil j = Cil j − Cil −1 j , 1 < l ≤ m; We get the equivalent minimization problem for Pseudo–Boolean function

P( z ) = ∑ f i (1 − zi ) + ∑ ∑ ∇Cij i∈I

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j∈J i∈I

∏ zkj

k∈Sij


From PB Function to UFPUP Problem We are given

P ( z ) = ∑ al ∏ z i l∈L

i∈Il

with arbitrary coefficients al, l∈L. Theorem 2. For PB Function P(z) the correspondence UFLP Problem with minimal number of us will minimal cardinality of the set J can be found in polynomial time. Proof. (similar to previous statement).

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Competitive Facility Location Problem • Input: − − − − −

a set J of users; a set I of potential facilities; a demand dj of user j; a distance function c : I×J → R; a number of facilities p0 to open by Leader;

−

a number of facilities p1 to open by Follower;

• Output: a set S ⊂ I , | S | = p0 of opening facilities by Leader;

• Goal: maximize the total number of users which are served by Leader.

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The Leader variables:

1 if facility i is open by Leader zi = ⎧⎨ ⎩0, otherwise; 1 if the demound of user j is satisfied by Leader y j = ⎧⎨ ⎩0, otherwise; The Follower variables:

1 if facility i is open by Follower ⎧ xi = ⎨ ⎩0, otherwise;

1 if the demound of user j is satisfied by Follower y j = ⎧⎨ ⎩0, otherwise; The set of appropriate point for Follower Ij(z) = {i∈I | cij < min (ckj | zk = 1)}

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Mathematical model

djyj ∑ y , z∈{0,1} max

Leader:

j∈J

s.t.

y j ≤ 1 − xi∗ , i ∈ I j ( z ), j ∈ J ;

∑ z i ≤ p0 ,

i∈I

where xi∗ is optimal solution of the problem:

djyj ∑ y , x∈{0,1} max

Follower:

j∈J

s.t.

yj ≤

∑ xi ,

j ∈ J;

i∈I j ( z )

∑ xi ≤ p1; i∈I

zi + xi ≤ 1, i∈I. - 24-


Mathematical model for K Followers Leader: max

d j y j; ∑ y , z∈{0,1} j∈ J

s.t.

y j ≤ 1 − xik∗ , i ∈ I j ( z ), j ∈ J , k ∈ K ; ∑ z i ≤ p0 , i∈I

∗

where xik is optimal solution of the problem:

Follower k:

d j y jk ; ∑ y , x∈{0,1} max

j∈ J

s.t.

y jk ≤

∑ xik ,

j ∈ J;

i∈I jk −1 ( z , x )

∑ xik ≤ pk ;

i∈I k −1 zi + xi∗k ′ k ′ =1

∑

+ xik ≤ 1, i ∈ I . k

∗ The set of appropriate point for Follower k: I jk ( z , x) = {i ∈ I | cij < min(clj | zl + ∑ xlk ′ > 0)} k ′=1

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