Easy and not so easy multifacility location problems... (In 20 minutes.)

Transcription

1 Easy and not so easy multifacility location problems... (In 20 minutes.) MINLP 2014 Pittsburgh, June 2014 Justo Puerto Institute of Mathematics (IMUS) Universidad de Sevilla

2 Outline 1 Introduction (In 3 minutes) 2 Discrete ordered median problem (in 7 minutes) 3 Continuous multifacility problems (10 minutes) 4 Dimensionality reduction of the SDP-rel for multifacility location problems (in no time) Mutifacility Location Problems MINLP

3 A few minutes of motivation Mutifacility Location Problems MINLP

4 A few minutes of motivation Logistic problem or in another jargon Clustering or Classification Problem Mutifacility Location Problems MINLP

5 A few minutes of motivation Logistic problem or in another jargon Clustering or Classification Problem Mutifacility Location Problems MINLP

6 Discrete ordered median problem: Modeling framework Mutifacility Location Problems MINLP

7 Discrete ordered median problem: Modeling framework r 1 r 2 r 3 r 4 r 5 λ 1r 1 + λ 2r 2 + λ 3r 3 + λ 4r 4 + λ 5r 5 Mutifacility Location Problems MINLP

8 Ordered Median Functions Definition Let x R n and λ R n. The ordered median function, f λ is defined as: f λ (x) = λ, sort(x) where sort(x) = (x (1),..., x (n) ) with x (i) {x 1,..., x n} and such that x (1) x (2) x (n). Mutifacility Location Problems MINLP

9 Special Cases Mean ( 1 n,, 1 n ) 1 n Minimum (1, 0,, 0) min i 1 i n Maximum (0, 0,, 1) max i 1 i n {}}{ k-centrum (0,, 0, 1,, 1) {}}{ anti-k-centrum ( 1,, 1, 0,, 0) {}}{{}}{ (k 1, k 2)-Trimmed mean ( 0,, 0, 1,, 1, 0,, 0) k 1 k k k 2 i=n k+1 k n k 2 i=k 1 +1 Range ( 1, 0,, 0, 1) max i min 1 i n x i x (i) x (i) x (i) 1 i n x i Mutifacility Location Problems MINLP

10 Our program starts about Mutifacility Location Problems MINLP

11 Previous Works Discrete Case: (Nickel & P., 2005; Boland, Marin, Nickel, P., 2006; Marín, Nickel, P., 2009). Networks: ( Nickel & P., 1999; Kalcsics, Nickel, P., 2002; Kalcsics, Nickel, P., Tamir, 2006; P. & Tamir, 2005); Continuous: Planar Weber Problem: (Weiszfeld, 1937; and extensions...) Planar Euclidean: BTST (Drezner, 2007); D.C. (Drezner& Nickel, 2009). Planar Convex OM with l p norms: (Espejo, et. al, 2009). Planar Euclidean k-centrum: (Rodríguez-Chía et. al, 2010). OM 1-Facility Location, any dimension, any l p or polyhedral norm: (Blanco, P. & ElHaj-BenAli, 2013, 2014). Mutifacility Location Problems MINLP

12 7 minutes for the DOMP: An INLP A = {a 1,..., a n} C = (c ij ) i,j=1,...,n X A with X = p p c i (X ) := min k X c ik σ X a permutation of {1,..., n} c σx (1)(X ) c σx (2)(X ) c σx (n)(x ) Discrete ordered median problem (DOpP) min X A, X =p λ i c σx (i)(x ). with λ = (λ 1,..., λ n) y λ i 0, i = 1,..., n. Mutifacility Location Problems MINLP

13 Sorting as an integer program (IP) INPUT: real numbers r 1,..., r n. OUTPUT: r σ(1) r σ(2)... r σ(n), σ Π({1,..., n}). Decision variables: s ki = { 1 if σ(k) = i 0 otherwise i, k = 1,..., n. Mutifacility Location Problems MINLP

14 r σ(k) := s ki r i, k = 1,..., n Desired output obtained Sorting as an IP: (SORT) minimize 1 s.t. s ki = 1 k=1 s ki = 1 s ki r i i = 1,..., n k = 1,..., n s k+1,i r i k= 1,..., n 1 s ki {0, 1} i, k = 1,..., n. Mutifacility Location Problems MINLP

15 p-median formulation Decision variables: { 1 a new facility is built in aj y j = 0 otherwise { 1 site ai is served by facility a j and x ij = 0 otherwise minimize c ij x ij j=1 y j = p j=1 x ij = 1 j=1 y j x ij i = 1,..., n j, i = 1,..., n y j {0, 1}, x ij 0 j, i = 1,..., n Mutifacility Location Problems MINLP

16 Quadratic formulation for DOMP: Constraints to sort Setting r i = minimize c ij x ij, i = 1,..., n. (SORT+ p-median) (DOMP) j=1 n k=1 λ k s ki n s n ki j=1 c ijx ij s ki = 1 k=1 s ki = 1 c ij x ij j=1 s k+1,i i = 1,..., n k = 1,..., n c ij x ij k = 1,..., n 1 j=1 Mutifacility Location Problems MINLP

17 Quadratic formulation for DOMP: Constraints to sort Constraints corresponding to p-median problem y j = p j=1 x ij = 1 j=1 y j x ij x ij 0 s ki {0, 1} y j {0, 1} i = 1,..., n j, i = 1,..., n j, i = 1,..., n i, k = 1,..., n j = 1,..., n Mutifacility Location Problems MINLP

18 A first linearization: DOMP1 Constraints to sort. Adding artificial variables x k (L1) minimize n k=1 λ k n n j=1 c ij s ki x ij }{{} x k ij ij j=1 k=1 s ki }{{} j xk ij = 1 k = 1,..., n s ki = 1 k = 1,..., n }{{} j xk ij c ij s ki x ij k = 1,..., n 1 }{{} x k ij j=1 c ij s k+1,i x ij }{{} x k+1 ij Mutifacility Location Problems MINLP

19 Linearization Constraints corresponding to p-median problem y j = p j ( ) x ij = 1 i = 1,..., n }{{} j k xk ij y j x ij j, i = 1,..., n }{{} k xk ij xij k {0, 1} y j {0, 1} i, k, j = 1,..., n j = 1,..., n Mutifacility Location Problems MINLP

20 A new formulation for DOMP: DOMP3 Example j=1 c ij x k ij j=1 c ij x k+1 ij k = 1,..., n C = Mutifacility Location Problems MINLP

21 A new formulation for DOMP: DOMP3 Example j=1 c ij x k ij j=1 c ij x k+1 ij k = 1,..., n C = x k ij = y j = 1 if facility i allocated to server j and C ij is the k-th smallest cost 0 otherwise { 1 if j J 0 if j / J Mutifacility Location Problems MINLP

22 The rationale: Incompatibilities k k 1 Mutifacility Location Problems MINLP

23 The rationale: Incompatibilities k k 1 k k 1 Mutifacility Location Problems MINLP

24 DOMP 3 min s.t. îĵ ij j=1 k=1 j=1 k=1 k=1 x k 1 îĵ j=1 λ k C ij x k ij x k ij = 1 x k ij = 1 x k ij y j y j = p j=1 + îĵ ij x k ij {0, 1} x k îĵ 1 i = 1,..., n k = 2,..., n i, j = 1,..., n i, j = 1,, n k = 2,, n i, j, k = 1,..., n Mutifacility Location Problems MINLP

25 DOMP 4 Formulation min j=1 k=1 λ k C ij x k ij s.t. j=1 k=1 x k ij = 1 i = 1,..., n j=1 x k ij = 1 k = 2,..., n x k ij y j k=1 y j = p j=1 i, j = 1,..., n i x k i j + j i =1 :i j j =1 ij i =1 :i j x k 1 j i j n2 =1 ij k = 2,, n x k ij {0, 1} i, j, k = 1,..., n Mutifacility Location Problems MINLP

26 Comparing formulations Notation We denote by P l the polytope defining the feasible set of the linear relaxation of formulation DOMP l, by z l ( ) the value of the objective function of DOMP l evaluated at the point ( ) and by Pl I the convex hull of the integer solutions within that polytope. Theorem 1 g(p 2) P 1. 2 f (P 3) P 2. 3 P 3 P 1. 4 z LP 3 z LP 2 z LP 1. Mutifacility Location Problems MINLP

27 Formulation GAP DOMP % DOMP % DOMP % DOMP % DOMP % Table: Average integrality gaps Mutifacility Location Problems MINLP

28 Formulation GAP DOMP % DOMP % DOMP % DOMP % DOMP % Table: Average integrality gaps Solution approach Time (s) #nodes DOMP 3(B&B) (8) DOMP 4 1(B&B) DOMP 4 1(B&C 3) DOMP 4(B&C 3) Table: CPU-Time and Number of nodes of the different formulations for n = 20, 30, 50. Mutifacility Location Problems MINLP

29 Continuous Multifacility Ordered Median Problems in 10 minutes Demand points : A = {a 1,..., a n } R d ( d ) 1 τ Norm : l τ ( x τ = x i τ, for all x R d ) Feasible Domain : R d Facilities : p facilities to be located. Goal : Find p points x1,..., xp R d minimizing some globalizing function of the distances to the set A. Mutifacility Location Problems MINLP

30 An easy model: Multiple allocation MF OMP Rodríguez-Chía, Nickel, P., and Fernández. (2000) Mutifacility Location Problems MINLP

31 Multiple allocation MF OMP where, µ jj 0. f NI λ (x 1, x 2,..., x p) = p j=1 p 1 λ ij d (i) (x j ) + X = {x 1,..., x p}, d i (x) = a i x τ, p j=1 j =j+1 d (1) (x) d (2) (x)... d (n) (x), λ 11 λ λ n1 0, λ 12 λ λ n2 0,... λ 1p λ 2p... λ np 0, µ jj x j x j τ, (1) Mutifacility Location Problems MINLP

32 Multiple allocation MF OMP where, µ jj 0. f NI λ (x 1, x 2,..., x p) = p j=1 p 1 λ ij d (i) (x j ) + X = {x 1,..., x p}, d i (x) = a i x τ, p j=1 j =j+1 d (1) (x) d (2) (x)... d (n) (x), λ 11 λ λ n1 0, λ 12 λ λ n2 0,... λ 1p λ 2p... λ np 0, µ jj x j x j τ, (1) ρ NI λ := min{fλ NI (x x 1,..., x p) : x j R d, j = 1,..., p}, (LOCOMF NI) Mutifacility Location Problems MINLP

33 Decision variables v ij and w lj continuous variables modelling the ordered weighted average in the o.f., x j = coordinates in R d of the j-th facility, y ijk = absolute value of the k-th coordinate of the vector x j a i, u ij = x j a i τ τ. z jj k = absolute value of the k-th coordinate of the vector x j x j, t jj = x j x j τ τ. Mutifacility Location Problems MINLP

34 Multiple allocation MF OMP Theorem Let τ = r s be such that r, s N \ {0}, r s and gcd(r, s) = 1. For any set of lambda weights satisfying λ 1j... λ nj 0 for all j = 1,..., p, Problem (LOCOMF NI) is equivalent to p p ρ NI λ = min p 1 p v ij + w lj + t jj (2) j=1 l=1 j=1 j=1 j =j+1 s.t v ij + w lj λ lj u ij, i, l = 1,..., n, j = 1,..., p y ijk x jk + a ik 0, i = 1,..., n, j = 1,..., p, k = 1,..., d y ijk + x jk a ik 0, i = 1,..., n, j = 1,..., p, k = 1,..., d, y r ijk ςs ijk ur s ij, i = 1,..., n, j = 1,..., p, k = 1,..., d, r ω s i d ς ijk u ij, k=1 i = 1,..., n, j = 1,..., p z jj k x jk + x j k 0, j, j = 1,..., p, k = 1,..., d, z jj k + x jk x j k 0, j, j = 1,..., p, k = 1,..., d, z r jj k ξs jj k tr s jj, j, j = 1,..., p, k = 1,..., d, r µ s jj d ξ jj k t jj, j, j = 1,..., p, k=1 ς ijk 0, i = 1,..., n, j = 1,..., p, k = 1,..., d, ξ jj k 0 j, j = 1,..., p, k = 1,..., d. Mutifacility Location Problems MINLP

35 Multiple allocation MF OMP Theorem Let τ = r s be such that r, s N \ {0}, r s and gcd(r, s) = 1. For any set of lambda weights satisfying λ 1j... λ nj 0 for all j = 1,..., p, Problem (LOCOMF NI) is equivalent to p p ρ NI λ = min p 1 p v ij + w lj + t jj (2) j=1 l=1 j=1 j=1 j =j+1 s.t v ij + w lj λ lj u ij, i, l = 1,..., n, j = 1,..., p y ijk x jk + a ik 0, i = 1,..., n, j = 1,..., p, k = 1,..., d y ijk + x jk a ik 0, i = 1,..., n, j = 1,..., p, k = 1,..., d, y r ijk ςs ijk ur s ij, i = 1,..., n, j = 1,..., p, k = 1,..., d, r ω s i d ς ijk u ij, k=1 i = 1,..., n, j = 1,..., p z jj k x jk + x j k 0, j, j = 1,..., p, k = 1,..., d, z jj k + x jk x j k 0, j, j = 1,..., p, k = 1,..., d, z r jj k ξs jj k tr s jj, j, j = 1,..., p, k = 1,..., d, r µ s jj d ξ jj k t jj, j, j = 1,..., p, k=1 ς ijk 0, i = 1,..., n, j = 1,..., p, k = 1,..., d, ξ jj k 0 j, j = 1,..., p, k = 1,..., d. Mutifacility Location Problems MINLP

36 Multiple allocation MF OMP Theorem Let τ = r s be such that r, s N \ {0}, r s and gcd(r, s) = 1. For any set of lambda weights satisfying λ 1j... λ nj 0 for all j = 1,..., p, Problem (LOCOMF NI) is equivalent to p p ρ NI λ = min p 1 p v ij + w lj + t jj (2) j=1 l=1 j=1 j=1 j =j+1 s.t v ij + w lj λ lj u ij, i, l = 1,..., n, j = 1,..., p y ijk x jk + a ik 0, i = 1,..., n, j = 1,..., p, k = 1,..., d y ijk + x jk a ik 0, i = 1,..., n, j = 1,..., p, k = 1,..., d, y r ijk ςs ijk ur s ij, i = 1,..., n, j = 1,..., p, k = 1,..., d, r ω s i d ς ijk u ij, k=1 i = 1,..., n, j = 1,..., p z jj k x jk + x j k 0, j, j = 1,..., p, k = 1,..., d, z jj k + x jk x j k 0, j, j = 1,..., p, k = 1,..., d, z r jj k ξs jj k tr s jj, j, j = 1,..., p, k = 1,..., d, r µ s jj d ξ jj k t jj, j, j = 1,..., p, k=1 ς ijk 0, i = 1,..., n, j = 1,..., p, k = 1,..., d, ξ jj k 0 j, j = 1,..., p, k = 1,..., d. Mutifacility Location Problems MINLP

37 Sorting with permutations: λ 1... λ n p j=1 λ ij u (i)j = max σ P n p λ ij u σ(i)j. The permutations in P n can be represented by the binary variables s ijk {0, 1} Next, combining the two sets of variables the objective function is: p p λ ij u (i)j = max λ ij u ij s ijk j=1 s.t j=1 k=1 j=1 s ijk = 1, j = 1,..., p, k = 1,..., n, s ijk = 1, i = 1,..., n, j = 1,..., p, k=1 s ijk {0, 1}. (3) Mutifacility Location Problems MINLP

38 Sorting with a linear program For fixed j {1,..., p} and for any vector u j R n, by using the dual of the assignment problem (7) we obtain the following expression λ ij u (i)j = min v ij + w lj (4) l=1 s.t v ij + w lj λ lj u ij, i, l = 1,..., n. Finally, we replace (4) for the objective function and we get min p v ij + j=1 l=1 p j=1 p 1 w lj + p j=1 j =j+1 s.t v ij + w lj λ lj u ij, i, l = 1,..., n, j = 1,..., p, u ij ω i x j a i τ, i = 1,..., n, j = 1,..., p, t jj µ jj x j x j τ, j, j = 1,..., p. t jj (5) Mutifacility Location Problems MINLP

39 Second Order Cone Constraints Lemma (Blanco, P., ElHaj-BenAli, 2014) Let τ = r s be such that r, s N \ {0}, r > s, gcd(r, s) = 1 and k = log 2 (r) 1. Let x, u and t be non negative and satisfying x r u s t r s. (6) Then, there exists w such that, (6) is equivalent to: w 2 1 A 1B 1,.. w 2 m A mb m, x 2 A m+1b m+1, (7) where A i, B i {w i 1, u, t, x} for i = 1,..., m and m = 1 + 2#{i : α i + β i + γ i 2, 1 i < k 1} + #{i : α i + β i + γ i 1, 1 i < k 1} O(log(r)) with: α i, β i, γ i {0, 1}. s = α k 1 2 k 1 + α k 2 2 k α α 02 0, r s = β k 1 2 k 1 + β k 2 2 k β β 02 0, 2 k r = γ k 1 2 k 1 + γ k 2 2 k γ γ 02 0, 2 k = (α k 1 + β k 1 + γ k 1 )2 k (α 0 + β 0 + γ 0)2 0, Mutifacility Location Problems MINLP

40 Corollary (sequel) Moreover, Problem (2) satisfies Slater condition and it can be represented as a second order cone (or semidefinite) program with (np + p 2 )(2d + 1) + p 2 linear inequalities and at most 4(p 2 d + npd) log r second order cone (or linear matrix) inequalities. Remark As consequence of the previous theorem, problem (2) is convex and can be solved, up to any given accuracy, in polynomial time for any dimension d. Mutifacility Location Problems MINLP

41 Experiments Coded in: Gurobi 5.6. Intel Core i7 processor; 2x 2.40 GHz; 4 GB of RAM Instance: 50-points data set in Eilon, Watson-Gandy & Christofides (1971) CPU-Times (secs.): τ p Mutifacility Location Problems MINLP

42 The program continues with some extensions... Mutifacility Location Problems MINLP

43 A hard MF problem: Single allocation MF OMP Miehle (1958), Cooper (1963) C. Pey-Chun, P. Hansen, B. Jaumard and T. Hoang (1998) Mutifacility Location Problems MINLP

44 Single allocation MF OMP Assume that: f i (x) : R pd R x = (x 1,..., x p) t i := min j a i τ }. j=1..p K R d. τ := r 1, r, s N with gcd(r, s) = 1. s λ l 0 for all l = 1,..., n. Consider the following problem ρ λ := min x { λ i f(i) (x) : x = (x 1,..., x p), x j K, j = 1,..., p}, (LOCOMF) Mutifacility Location Problems MINLP

45 Single allocation MF OMP ˆρ λ = min λ l θ l l=1 (MFOMP λ ) s.t. h il 1 := t i θ l + UB i (1 w il ), i = 1,..., n, l = 1,..., n, h l 2 := θ l θ l+1, l = 1,..., n 1, h 3 ij := u ij t i + UB i (1 z ij ), i = 1,..., n, j = 1,..., p, hijk 4 := v ijk x jk + a ik 0, i = 1,..., n, j = 1,..., p, k = 1,..., d, h ijk 5 := v ijk + x jk a ik 0, i = 1,..., n, j = 1,..., p, k = 1,.., d, h ijk 6 := vr ijk ζs ijk ur s, i = 1,..., n, j = 1,..., p, k = 1,..., d, ij hij 7 d := ζ ijk u ij, i = 1,..., n, j = 1,..., p, k=1 hi 8 p := z ij = 1, i = 1,..., n, j=1 hl 9 := w il = 1, l = 1,..., n, h 10 i := w il = 1, i = 1,..., n, l=1 w il {0, 1}, i, l = 1,..., n, z ij {0, 1}, i = 1,..., n, j = 1,..., p, θ l, t i, v ijk, ζ ijk, u ij R +, i, l = 1,..., n, j = 1,..., p, k = 1,..., d, x j K, j = 1,..., p. Mutifacility Location Problems MINLP

46 Theorem Let x be a feasible solution of LOCOMF then there exists a solution (x, z, u, v, ζ, w, t, θ) for MFOMP λ such that their objective values are equal. Conversely, if (x, z, u, v, ζ, w, t, θ) is a feasible solution for MFOMP λ then x is a feasible solution for LOCOMF. Furthermore, if K satisfies Slater condition then the feasible region of the continuous relaxation of MFOMP λ also satisfies Slater condition and ρ λ = ˆρ λ. Mutifacility Location Problems MINLP

47 Experiments Problem p τ CPU Time f Problem p τ CPU Time f Problem p τ CPU Time f p-median p-center p-25-centrum Table: Computational results for p-median, p-center and p-25-centrum problems for the 50-points in Eilon, Watson-Gandy & Christofides data set. Mutifacility Location Problems MINLP

48 Constrained MF Ordered Median Problem A SDP relaxation of the LOCOMF Let h 0(θ) := m λ l θ l, and denote ξ j := (deg g j )/2 and ν j := (deg h j )/2, where l=1 {g 1,..., g nk }, and {h 1,..., h nc1} are, respectively, the polynomial constraints that define K and ˆK \ K in MFOMP λ. For r r 0 := max{ max ξ k, max ν j}, introduce the k=1,...,n K j=0,...,nc1 hierarchy of semidefinite programs: min y L y(p λ ) s.t. M r (y) 0, M r ξk (g k, y) 0, k = 1,..., n K, M r νj (h j, y) 0, j = 1,..., nc1, y 0 = 1, (Q1 r ) Mutifacility Location Problems MINLP

49 Theorem Let ˆK R nv 1 (compact) be the feasible domain of Problem MFOMP λ. Let inf Q1 r be the optimal value of the semidefinite program Q1 r. Then, with the notation above: (a) inf Q1 r ρ λ as r. (b) Let y r be an optimal solution of the SDP relaxation Q1 r. If rank M r (y r ) = rank M r r0 (y r ) = ϕ then min Q1 r = ρ λ and one may extract ϕ points (x 1 (k),..., x p (k), z (k), u (k), v (k), ζ (k), w (k), t (k), θ (k)) ϕ k=1 ˆK, all global minimizers of the MFOMP λ problem. Mutifacility Location Problems MINLP

50 The size of SDP relaxations grows rapidly with the original problem size. In view of the present status of SDP-solvers, only small to medium size problems can be solved. We found in our model patterns to exploit: Sparsity. Symmetry. Mutifacility Location Problems MINLP

51 Thank you! Mutifacility Location Problems MINLP