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A cutting plane algorithm for the

Capacitated Facility Location Problem

Pasquale Avella Maurizio Boccia

March 15, 2005

RCOST Research Center on Software Technology, Universita del Sannio,

C.so Garibaldi 107, 82100 Benevento, Italy, [email protected] di Ingegneria, Universita del Sannio, Piazza Roma 21, palazzo ex INPS,

82100 Benevento, Italy, [email protected]


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Abstract

The Capacitated Facility Location Problem (CFLP) is to locate a set of facilities

with capacity constraints, to satisfy at the minimum cost the order-demands of a set of

clients.

In this paper we present a reformulation ofCFLPbased on Mixed Dicut Inequalities,

a family of knapsack inequalities of a mixed type, containing both binary and continuous

(flow) variables. By aggregating flow variables, any Mixed Dicut Inequality turns into a

binary knapsack inequality with a single continuous variable. We will refer to the convex

hull of the feasible solutions of this knapsack problem as the Mixed Dicut polytope.

We observe that the Mixed Dicut polytope is a rich source of valid inequalities for

CFLP: basic families of valid CFLP inequalities, like Variable Upper Bounds, Cover,

Flow Cover and Effective Capacity Inequalities, are valid for the Mixed Dicut polytope.

Furthermore we observe that new families of valid inequalities for CFLP can be derived

by the lifting procedures studied for the knapsack with a single continuous variable.

To deal with large-scale instances, we have developed a Branch-and-Price algorithm,

where the separation algorithm consists of the complete enumeration of the facets of the

Mixed Dicut polytope for a set of candidate Mixed Dicut Inequalities. We observe that

our procedure returns inequalities that dominate most of the known classes presented

in the literature. We report on computational experience with instances up to 1000

facilities and 1000 clients to validate the approach.

Keywords: Capacitated Facility Location Problem, Mixed Dicut Inequalities, Facet-

enumeration.


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and Flow Cover and Effective Capacity Inequalities, are valid for the Mixed Dicut

polytope. Furthermore new families of valid inequalities can be derived by using the

lifting procedures introduced by Marchand and Wolsey [17] for the knapsack with a

single continuous variable.

A Branch-and-Price algorithm is proposed to handle large scale instances. The

LP-relaxation is solved by delayed column generation. The cutting plane generation

consists of the complete enumeration of the facets of the Mixed Dicut polytope for a set

of candidate base Mixed Dicut Inequalities. The procedure returns valid inequalities

dominating most of the known classes presented in the literature.

The remainder of the paper is organized as follows. In Section 2 we describe the

standard formulation of CFLP. In Section 3 we review the literature concerning poly-

hedral properties of CFLP. In Section 4 we present the reformulation of CFLP as a

Fixed Charge Network Flow problem and introduce the family of the Mixed Dicut In-

equalities. In Section 5 we review all the known families of valid inequalities for CFLP

and prove that they are valid for the Mixed Dicut Polytope. In Section 6 we define

a new class of valid inequalities by extending to the Mixed Dicut Polytope the results

presented in [17] for the 0-1 Knapsack Problem with a Single Continuous Variable. In

Section 7 we describe the facet-enumeration and the selection of candidate Mixed Dicut

Inequalities. In Section 8 we describe the Branch-and-Cut-and-Price algorithm devel-

oped to deal with large-scale instances. Finally in Section 9 we report on computational

results validating the usefulness of the proposed approach.

2 Problem formulation

Let M = {1, . . . , m} be a set of facilities, N = {1, . . . , n} a set of clients and let

G(M N, A) be a complete bipartite graph. Let uk be the capacity of the facility k

and let dj be the demand of client j N. Let hk be the fixed cost of opening facility

k M. Let ckj be the cost of sending one unit of demand from the facility k M to

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the client j N through the arc kj A and let fkj be the flow of the demand from

facility k M to the client j N.

A subset K M of facilities is feasible if the client demands can be assigned to the

facilities in K while respecting the capacity bounds, i.e. if the following Transportation

Problem, denoted as Transp(K), admits a feasible solution:

min

kK

jN

ckjfkj

kK

fkj = dj, j N

jN

fkj uk, k K (1)

fkj 0, k M, j N (2)

Let c(K) be the value of the optimal solution ofTransp(K) and let h(K) =

kKhk

be the sum of the fixed costs of the open facilities. The Capacitated Facility Location

Problem (CFLP) is to find a feasible subset K M of open facilities, minimizing the

objective function z(K) = h(K) + c(H(K)).

Let yk be the binary variable associated with the facility k M, i.e. yk = 1 ifk is

open (i.e.k K), 0 otherwise. Formulation F1 of CFLP is:

min

kM

hkyk +

kM

jN

ckjfkj

kM

fkj = dj, j N (3)

jN

fkj ukyk, k M (4)

yk {0, 1}, k M

fkj 0, k M, j N

Constraints 3 require that the whole demand of each client be satisfied. Capacity

constraints 4 require that no client can be supplied from a closed facility and that the

total demand supplied from the facility k K does not exceed its capacity uk.

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3 Basic polyhedral results

Let X(G,d,u) be the set of the integer feasible solutions of formulation F1 and let

CAP(G,d,u) = conv(X(G,d,u)) be the Capacitated Facility Location Polytope. The

dimension of the CAP(G,d,u) is |M| + |N| + |M| |N| if G is a complete bipartite

graph [1]. In the following of this section we review the main classes of valid inequalities

introduced in the literature. For any S N, we will denote d(S) =jS

dj .

3.1 Variable Upper Bounds

Let k M, j N. The Variable Upper Bound

fkj ukyk (5)

is valid for CAP(G,d,u). The (5) can be strengthened to fkj djyk, if dj < uk.

Proposition 3.1 ([1]) Let k M, j N. The Variable Upper Bound fkj djyk is

facet-defining for CAP(G,d,u) ifhM

uh maxhM

uh > d(N) and dj < uk.

3.2 Cover Inequalities

Let C M be a subset of facilities and let C = M \ C. The set C is said a cover if

kC

uk < d(N). Since the capacity of the facilities in C is not large enough to satisfy the

client demand, at least one of the facilities in C must be open and the Cover Inequality

kC

yk 1 (6)

is valid for CAP(G,d,u). The cover C is minimal if each subset S C is not a cover.

Proposition 3.2 (Aardal [1]) LetC M be a minimal cover and let umin = minkC

uk

be the minimum capacity of a facility in C. The Cover Inequality:

kC

yk 1

is facet-defining for conv(X(G,d,u) {yk = 1, k M\ C}) if

kM\C

uk + umin > d(N).

Sequential lifting can make the (6) facet-defining for CAP(G,d,u) [1].

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3.3 Flow Cover and Effective Capacity inequalties

Let J N and let C M be a flow-cover, i.e. a subset of facilities with the property

thatkC

uk d(J) = > 0.

Theorem 3.1 The Flow Cover Inequality

kC

jJ

fkj +

kC

(uk )+(1 yk) d(J) (7)

is valid for CAP(G,d,u).

Aardal [1] proved that Flow Cover inequalities are facet-defining if maxiI{ui} >

andkM

uk > d(J)+urr I. Flow Cover Inequalities generalize the Residual Capacity

Inequalities, introduced by Leung and Magnanti [16] for the special CFLP where all the

facilites have the same capacity u.

The same author introduced the family of the Effective Capacity Inequalities, that

dominate Flow Cover Inequalities. Suppose the bipartite graph G(M N, A) is not

complete and let Jk J be the subset of clients reachable by the facility k M.The effective capacity of the facility k is uk = min{uk, d(Jk)}. Let J N and let

C M be an effective flow-cover for J, i.e. a subset of facilities with the property thatkM

uk d(J) = > 0, maxkC

uk.

Theorem 3.2 The Effective Capacity Inequality

kC

jJk

fkj +

kC

(uk )+(1 yk) d(J) (8)


Aardal provides necessary and sufficient conditions guaranteeing that Effective Ca-

pacity Inequalities are facet-defining for CAP(G,d,u).

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4 Fixed Charge Network Flow reformulation

Let o be a supernode and let O = {ok : k M} be an additional set of arcs connecting

o to M. Let xkj be the variable representing the capacity installed on the arc kj . With

these additions, it is easy to see that CFLP is a Single Commodity Uncapacitated Fixed

Charge Network Flow problem on the graph Go(o M N, O A), where the amount

of flowjN

dj must be sent from the source node o to the sink nodes N through the

transhipment nodes M. A minor difference with the standard version is that capacities

xij are not forced to be integral.

min

kM

hkyk +

kM

jN

ckjxkj

kM

fok =

jN

dj (9)

jN

fkj fok = 0, k M (10)

kM

fkj = dj, j N (11)

fok ukyk, k M (12)

fkj xkj , k M (13)

yk {0, 1}, k M (14)

fkj 0, k M, j N (15)

xkj 0, k M, j N (16)

Let K M and let J N. The set of arcs (K, J) = {ok O : k K} {kj

A : k M \ K, j J} defines a dicut on Goseparating o and the node J. By Max

Flow-Min Cut theorem it is easy to show that CFLP can be reformulated using the

Mixed Dicut Inequalities introduced in Ortega and Wolsey [19].

Proposition 4.1 LetK M and let J N. The Mixed Dicut Inequality

kK

ukyk +

kM\K

jJ

xkj d(J) (17)

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Since the capacity xkj is fractional, in any optimal solution we will get xkj = fkj ,

for any value of fkj and it follows that we can project out variables xkj and the Mixed

Dicut Formulation F2 of CFLP is:

min

kM

hkyk +

kM

jN

ckjfkj

kKu

kyk +

kM\K

jJ f

kj jJ d

j, K M, J N (18)

yk {0, 1}, k M (19)

0 fkj dj, k M, j N (20)

5 CFLP inequalities and the Mixed Dicut Polytope

By letting s =

kM\K

jJ

fkj , any Mixed Dicut Inequality (17) turns into a knapsack

inequality with a single continuous variable. Let K M and J N and letkK

ukyk +

s d(J)} be a Mixed Dicut Inequality. Let Y(K, J) = {(y, s) Bn R+ :kK

ukyk +

s d(J)} be the set of incidence vectors of the feasible solutions and let CU T(K, J) =

conv(Y(K, J)) be the Mixed Dicut Polytope. It is easy to prove that CU T(K, J) is

full-dimensional. Studying CU T(K, J) can lead to tighter cutting planes families that

strengthen formulations F1 and F2. In the following we observe that the families of

valid inequalities described in Section 4 are valid for CU T(K, J) for some K M and

J N.

5.1 Variable Upper Bounds

Proposition 5.1 Letk M and letj N. The Variable Upper Bound (5) fkj ukyk

is valid for CU T({k}, {j}).

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Proof. By adding the equalitykM

fkj = dj to the Variable Upper Bound fkj ukyk,

we get the Mixed Dicut Inequality ukyk + s dj.

5.2 Cover Inequalities

Consider the dicut ({o} : M) = {ok : k M} formed by the arcs leaving the node o

and entering the set M.

Proposition 5.2 The Cover Inequality (6) is valid for CUT(M, N).

Proof. Trivial, since CU T(M, N) = conv{y {0, 1}|M| :kM

ukyk d(N)} is a

knapsack polytope.

5.3 Flow Cover Inequalities

In this subsection we show that the Flow Cover Inequality can be derived by sequential

lifting as a valid inequality for the Mixed Dicut Polytope. The following two lemmas

can be used to down lift a valid inequality for a Mixed Dicut Polytope. Theorem 5.1

shown that such an inequality is a Flow Cover Inequlitiy (7).

Lemma 5.1 The optimal solution of the knapsack problem:

min z =

kS

(uk )+yk + s

s +

kS

ukyk d(J)

kC\(Sh)

uk uh (21)

s 0

yk {0, 1}, k S

is yk = 1 for k S, s = (uh )+. It follows that the optimalsolution value is

z =

kS(uk )+ + (uh )

+.

Proof. Since =kC

uk jJ

dj, we pose d(J)

kC\S

uk = +kS

uk, so we can

write:

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min z = +

kS

(uk )+yk

s +

kS

ukyk +

kS

uk uh

s 0

yk {0, 1}, k S

Suppose now that yk = 0 in any optimal solution. If we set yk = 1 for k = S, we

must add the yk coefficient (uk ), but at the same time s decreases by uk. The netamount (uk )

+ uk is negative, a contradiction.

Lemma 5.2 Let S C and let s +

kS(uk )+yk

kS(uk )

+ be a valid

inequality for CU T(C, J) {yk = 1, k C\ S}. Let h C\ S. The inequality

s +

kS

(uk )+yk + hyh

kS

(uk )+ +

is valid for CU T(C, J) {yk = 1, k C\ (S h)} if h = (uh )+.

Proof. Let z be the optimal solution of the knapsack problem (21). Using the standard

procedure for down-lifitng, the inequality s+kS

(uk)+yk+hyh

kS(uk)

++

is valid iff = z

kS(uk)+. By lemma 5.1 we have that z =

kS(uk)

+(uh

)+ and the proof is complete.

Now we are able to prove the following theorem.

Theorem 5.1 The Flow Cover Inequality (7) is valid for CUT(C, J)

Proof. The Inequality s 0 is valid for CU T(C, J) {yk = 1, k C}. Applying

iteratively lemma 5.2 to compute lifting coefficients, we have that the inequality

s +

kC

(uk )+yk

kC

(uk )+ (22)

is valid for CU T(C, J). By subtracting the equalitykM

jJ = d(J) to the (22), we

obtain the Flow Cover InequalitykC

jJ

fkj +kC

(uk )+(1 yk) d(J) .

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5.4 Effective Capacity Inequalities

Let K M be a subset of facilities and suppose that Go(M N, O A) is sparse. Let

Jk N be the subset of clients reachable by the facility k K and let J = kKJk be

the subset of clients reached by all the facilities in K. The set of arcs {ok O : k

C}{kj A : k C\M, j Jk} define a mixed dicut on Go. Let uk = min{uk, d(Jk)}.

As in [1], the Mixed Dicut Inequality defined by K and J can be strengthened as:

kK

ukyk + s

jJ

dj

that is valid for CAP(G,d,u). The Effective Mixed Dicut Polytope is:

ECUT(K, Jk) = conv{yk {0, 1}, k K, s R+ :

kK

ukyk + s d(J)

Studying this polytope we can define new cutting planes families valid for CAP(G,d,u).

Particularly, if C M is an effective flow-cover (section 3.3), using the lifting proce-

dure described in Section (5.3) for the Flow Cover Inequalities, we obtain the Effective

Capacity Inequality

kC

jJk

fkj +

kC

(uk )+(1 yk) d(J) (23)

defined in section (3.3).

6 More families of valid inequalities

Let Ys(K, J) = {(y, s) Y : s = s} be the subset of the incidence vectors of the

feasible solutions of a Mixed Dicut Inequality having the property that s = s. Let

CUTs(K, J) = conv(Y(K, J)).

Another way to generate facets of CU T(K, J) is to set s = s, consider facets of

the underlying knapsack polytope CU Ts(K, J) and reintroduce the continuous variable

s by lifting. This procedure leads to new families of valid inequalities for CUT(K, J)

and, consequently, for CAP(G,u,d). Marchand and Wolsey [17] introduced a lifting

procedure for the case wherekK

uk d(J) and s = 0.

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Theorem 6.1 (Marchand and Wolsey [17]) LetCU T0(K, J) = conv{(y, s) Y(K, J) :

s = 0} and letkK

kyk 0 be a valid inequality for CU T0(K, J). The inequality

kK

kyk 0 s

(24)

is valid for CU T(K, J) iff mins>0

s0(s)

, where

(s) = min{

kK

kyk :

kK

ukyk d(J) s, y B|K|}

Proof. We will give a geometric proof of the validity of the (24). (s) is a step function

(fig. 1). The inequalitykK

kyk (s) is valid for CU T(K, J) if (s) (s), for each

s R.

Let s be such that maxs>0

0(s)s

= 0(s)

s. Let (s) = 0

s

be the line of minimum

slope intersecting (s) and crossing the points (0, 0) and (s, (s)). It is easy to observe

that (s) (s) and the proof is complete.

Figure 1: The function (s)

The lifting procedure can be generalized by removing the restriction thatkK

uk

d(J).

Let smin = max(0, d(J) kK

uk) be the minimum value of s that guarantees that

the set of the feasible solutions Y(K, J) is nonempty.

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Theorem 6.2 Lets smin and letkK

kyk s be a valid inequality for the knapsack

polytope CU Ts(K, J).

The inequality

kK

kyk s s smin

(25)

is valid for CU T(K, J) if mins>smin

ssmins(s)

, where

(s) = min{

kK

kyk :

kK

ukyk d(J) s, y B|K|} (26)

Proof. The function (s) is defined for s smin. Let s be such that maxs>smins(s)ssmin =

s(s)

ssmin. Let s

ssmin

be the line of minimum slope intersecting (s) and crossing

the points (smin, (s)) and (s, (s)) (figure 2). It is immediate that (s) s

ssmin

and the proof is complete.

Figure 2: The function (s) in the case

kKuk < d(J)

7 The separation routine

The above results suggest that the facial structure of CU T(K, J) can be exploited to

generate strong cutting planes. We propose a twp-step separation algorithm based

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on the enumeration of the facets of CU T(K, J). The first step consists of choosing a

Mixed Dicut InequalitykK

ukyk +

kM\K

jJ

xkj d(J), with K suitably small, as a

base inequality. Then for each base inequality we enumerate the feasible solutions of

Y(K, J) and run a facet-enumeration routine to list the facets ofCU T(K, J), returning

violated cutting planes.

An accurate selection of the base inequalities and a fast enumeration procedure of

the feasible solutions are crucial to the success of the separation algorithm.

7.1 Selection of base inequalities

Let (y, f) be the current fractional solution. The procedure to select candidate base

inequalities is inspired by the separation heuristic described in [2] for the Flow Cover

Inequalities and is based on the following observations.

Remark 7.1 A valid inequality for CU T(K, J) can be violated by the (y, f) only if

at least one of the variables yk, k K, has a fractional value.

Remark 7.2 The chances of finding violated a valid inequality for CU T(K, J) in-

crease if s is small (the smaller the s, the larger the r.h.s. of the knapsack inequalitykK

ukyk+ d(J) s).

Remark 7.3 The chances of finding a violated valid inequality for CU T(K, J) increase

if we choose K and J such that the facilities in K serve only the clients in J in the

current fractional solution.

The following heuristic attempts to generate base inequalities according to the re-marks (7.1), (7.2) and (7.3). The selection procedure is iterated for each triple of

facilities K = {k1, k2, k3} M such that there exists at least a k with 0 < ykj < 1.

The main steps of the procedure are summarized below. The facet-enumeration routine

is efficient only ifK is suitably small. Let maxK denote the maximum number of y-

variables in a base inequality used for the facet enumeration. Computational experience

showed us that a good choice is to set maxK = 7.

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Procedure 7.1 (Selection of the base inequalities)

Initialize K and J

0) LetK = {k1, k2, k3} be a set of facilities with the property that 0 < yk1 < 1.

1) LetJ be defined by all the clients j J such thatkK

fkj 0.01dj

2) Generate the base inequalitykK

ukyk + s d(J).

Enlarge K

3) Letk = arg maxkM\K

jJfkj. Add k to K.

4) Add to J all the clients j M \ J such thatkK

fkj 0.01dj.

5) If |K| maxK,kK

ukyk + s d(J) is a base inequality else STOP.

Adjust J

6) For each j J compute (j) =

kM\K

fkj .

7) Letjmin = arg maxjJ

(j).

8) IfkK

uk < d(J), remove jmin from J.

7) If |K| maxK generate the base inequalitykK

ukyk + s d(J) and go to

step 3), else STOP.

7.2 Enumeration of the feasible solutions satisying a base in-

equality

Let K M, J N and let Y(K, J) denote the set of the feasible solutions satisfying

the Mixed Dicut Inequality:

kK

ukyk + s d(J). (27)

For any partial solution y, let smin(y) = max(0, d(J) kK

ukyk) be the minimum

value of s that makes the solution (y, s) feasible for the (27). We refer to any solution

(y, smin(y)) as a minimal feasible solution of the (27) and we denote by Ysmin(K, J) =

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{(y, s) Y(K, J) : s = smin} the set defined by the minimal feasible solutions of the

(27). In this section we will prove that the facets of CU T(K, J) are in one-to-one

correspondence with a subset of the facets of conv(Ysmin(K, J)). We observe that the

polytope conv(Ysmin(K, J)) is full-dimensional.

Lemma 7.1 A inequalitykK

kyk + s is valid for CU T(K, J) only if 0.

Proof. LetkK

kyk + s be a valid inequality for CU T(K, J) and let (y, s)

Y(K, J) be a feasible solution satisfying the (27). If < 0, it is always possible to

set s large enough (and the solution remains feasible) such thatkK

kyk + s < , a

contradiction.

The following theorem proves that any facet ofCU T(K, J) is also a facet ofconv(Ysmin(K, J)).

Theorem 7.1 A inequality

kK

kyk + s (28)

is facet-defining for conv(Ysmin(K, J)) if it is facet-defining for CUT(K, J).

Proof. IfkK

kyk + s is valid for CU T(K, J) then > 0. IfkK

kyk + s is

facet-defining for CU T(K, J), there exist |K| +1 linearly independent feasible solutions

satisfying the (28) as an equality (roots).

Let (y, s) be a root of the (28) and suppose that s > smin. Since (y, s) is a root, we

havekK

kyk + s = . But (y, smin) is still a feasible solution and, since 0, it

violates the (28), i.e.kK

kyk + smin < , a contradiction.

It follows that all the |K| + 1 roots are of the form (y, sminy) and belong toYsmin(K, J), so we can conclude that the (28) is facet-defining for conv(Ysmin(K, J)).

Theorem 7.2 Let

kK

kyk + s (29)

be a facet of conv(Ysmin(K, J)). The (29) is facet-defining for CU T(K, J) if and only

if 0.

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promising initial set of columns. In our case we adopt the lagrangian core selection

strategy presented in [4]. This strategy use the lagrangean reduced costs in order to

select the variables of the core problem.

The branching strategy is to set node variables yu to 1 if facility u V is open, 0

otherwise. The node selection criterion is best-bound. The variable selection criterion

is to choose the most fractional node variable, i.e. the variable whose fractional value

is closest to 0.5.

Finally we use standard reduced cost fixing to reduce the size of the problem.

9 Computational results

In this section we report on the computational experience with the Branch-and-Cut-and-

Price algorithm. We ran the experiments on a Pentium IV 1.7 Ghz Personal Computer,

with 512 Mb RAM and we used Cplex 8.1 as LP solver and the routine Qhull [7],

developed by the Geometric Center of University of Minnesota, as facet-generation

routine. In the selection of base inequalities we found useful to consider only MixedDicut Inequaalites with |K| 9.

The test instances were randomly generated according to Cornuejols, Sridharan and

Thizy ([13]), Aardal ([1], [2]) and Barahona and Chudak ([6]). The generation procedure

is summarized below:

a) Points representing the clients and the facilities are uniformly randomly gener-

ated in a unit square. Transportation costs per unit flow are then computed by

multiplying the euclidean distances by 10.

b) Demands are drawn from U[5, 35] i.e. they are uniformly distributed in the inter-

val [5, 35].

c) Capacities uk are drawn from U[10, 160] and then are scaled by using the capacity

ratio r =

kMuk/d(J). In our experiments we set r = 5, 10, 15, 20.

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d) Fixed costs are generated according to the formula hk = U[0, 90]+ U[100, 110], to

take into account the economies of scale.

Moreover, we observe that in [13, 1, 2] the ratio r is set to 2, 3, 5, 10. We use larger

values of r because for sizes we deal with, setting r = 2 or r = 3 leads to instances

easily solvable by the lagrangean heuristic [4].

The test instances are organized into 4 groups, each including instances of the same

size. The sizes (|M| |N|) here considered are: 300 300, 500 500, 700 700 and

1000 1000. Each class of instances includes four different sets of problems according

to the ratio r. We used ratio indices r of 5, 10, 15 and 20 respectively.

The results are shown in tables 1-4, whose format is the following. Column Ratio is

the value ofr, UBHeur is the initial upper bound and column U BHeur Time is the time

spent to compute the initial upper bound. Columns LP and LPCuts report respectively

on the value of the LP relaxation and on the value of the LP relaxation after the addition

of cutting planes. Column %Gap shows the % of the gap closed by the cut generation

routine at the root node, i.e. % Gap =(LPCutsLP)/(BU BLP). Columns BLB abd

BU B report respectively on the best lower bound and on the best upper bound yielded

by Branch-and-Cut-and-Price within the time limit. Columns B&C&P and B&C&P

Time report respectively on the number of nodes and on the total computation time

for the Branch-and-Cut-and-Price-procedure.

The time limit for the Branch-and-Cut-and-Price was set to 100000 seconds. All

the instances in the test bed were solved to optimality, with the exceptions of I1000-8,

I1000-11 and I1000-12, where, however, the algorithm yielded a very small gap. We

observe that the cut generation strategy has led in most of the test to a significant

reduction of the gap.

We observe that the reduction of the gap is small for r = 5. This because the LP

lower bound is already very close to the optimal solution, so that it is more difficult to

find violated cutting planes by the LP solution. For these cases the pricing approach

assume a more significant role than cutting plane generation. On the other hand, for

18


21/27

higher values of r, also the initial gap increases and our cut generation strategy turned

out to be effective in solving the CFLP instances.

Tables 1-2 report also on lower bound at the root node computed by CPLEX 8.1

solver (CPLEXroot), and the total time necessary to CPLEX to solve the instances

(CPLEXTime). We remark that even if Cplex implements a flow cover separation

procedure, it is not able to appreciably raise the lower bound.

Acknowledgements

The authors wish to thank Antonio Sassano and Sara Mattia for the helpful suggestions.

References

[1] K. AARDAL 1992. On the solution of one and two-level capacitated facility location

problems by the cutting plane approach. Ph.D Thesis Universit Catholique de

Louvain, Lovain-la-Neuve, Belgium.

[2] K. AARDAL 1998. Capacitated Facility Location: Separation Algorithms and

Computational Experience, Mathematical Programming, 81, 149-175.

[3] K. AARDAL, Y. POCHET, L.A. WOLSEY 1995. Capacitated Facility Location:

Valid Inequalities and facets. Mathematics of Operations Research 20, 562-582.

[4] P. AVELLA, M. BOCCIA, A. SFORZA, I. VASILIEV 2004. An effective heuristic

for large-scale Capacitated Plant Location problems, Technical Report, available

on line at http://www.diima.unisa.it/boccia/.

[5] R. ANBIL, F. BARAHONA 2000. The volume algorithm: producing primal so-

lutions with a subgradient method. Mathematical Programming, published online

March 15.

19


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[6] F. BARAHONA, F. A. CHUDAK 1999. Near-optimal solution to large scale facility

location problems, internal report IBM research division T. J. Watson Research

Center , RC 21606.

[7] C.B. BARBER, D.P. DOBKIN, H.T. HUHDANPAA 1996. The Quickhull algo-

rithm for convex hulls. ACM Trans. on Mathematical Software, 22(4):469-483,

http://www.qhull.org.

[8] J. E. BEASLEY 1993, Lagrangean Heuristics for Location Problems, EJOR 65,

383-399.

[9] D. BERTSIMAS, J.N. TSITSIKLIS 1997. Introduction to Linear Optimization.

Athena Scientific, Belmont, Massachussetts.

[10] S. CERIA, C. CORDIER, H. MARCHAND, L.A. WOLSEY 2002. Cutting Planes

for Integer Programs with General Integer Variables. Discrete Applied Mathemat-

ics, 123, 397-446.

[11] T. CHRISTOF, G. REINELT 1997. Low-dimensional linear ordering polytopes.

Tech. Rep. Institut fur Angewandte Mathematik, Universitat Heidelberg.

[12] F. A. CHUDAK, D. P. WILLIAMSON 1999. Improved Approximation Algorithms

for Capacitated Facility Location Problems, 7th International IPCO Conference

Proceedings, 99-113.

[13] G. CORNUEJOLS, R. SRIDHARAN, J.M.THIZY 1991. A comparison of heuris-

tics and relaxations for the Capacitated Plant Location Problem. European Journal

of Operational Research, 50, 280-297.

[14] A. KLOSE 2001. A Column Generation Procedure for the Capacitated Facility

Location Problem. EURO XVIII conference on Operational Research, Erasmus

University Rotterdam, July 9-11.

20


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[15] M. R. KORUPOLU, C. G. PLAXTON, R. RAJARAMAN 1998. Analysis of a

Local Search Heuristic for Facility Location Problems, DIMACS Technical Report

98-30.

[16] J.M.Y. LEUNG, T.L. MAGNANTI 1989. Valid Inequalities and Facets of the Ca-

pacitated Plant Location Problem. Mathematical Programming, 44, 271-291.

[17] H. MARCHAND, L.A. WOLSEY 1999. The 0-1 knapsack problem with a single

continuous variable. Mathematical Programming, 85, 15-33.

[18] T. L. MAGNANTI, P. MIRCHANDANI, R. VACHANI 1993. The convex hull of

two core capacitated network design problems. Mathematical Programming, 60,

233-250.

[19] F. ORTEGA, L. WOLSEY 2000. A Branch and Cut Algorithm for the Single

Commodity Uncapacitated Fixed Charge Network Flow Problem. Networks 41,

143-158.

21


24/27

Name

Ratio

UBHeur

UBHeur

LP

CPLEXroot

LPCuts

%

Gap

BLB

BUB

B&C&P

B&C&P

Tot.Time

CPLEXTime

Ti

me(secs.)

nodes

Time(secs.)

(secs.)

(secs.)

I300-1

5

16350,6

6

13,1

4

16292

16293,0

9

16296,4

9

7,6

5

16350,6

6

16350,6

6

374

432,4

3

445,5

5

1019.2

0

I300-2

5

15948,4

4

30,5

3

15862,2

1

15886.0

6

15894,3

9

37,3

2

15948,4

4

15948,4

4

904

873,7

1

904,4

5

1531,7

0

I300-3

5

15474,8

4

13,7

7

15414,9

5

15401,7

1

15438,7

3

39,7

1

15474,8

4

15474,8

4

248

307,3

6

321,1

3

1168,0

0

I300-4

5

17989,9

7

16,3

5

17926

17927,6

9

17938,3

9

19,3

7

17989,9

7

17989,9

7

534

650,8

6

667,2

1

1367,0

6

I300-5

5

18037,6

1

17,6

1

17962,8

17966,7

3

17982

25,6

7

18037,6

1

18037,6

1

570

850,1

3

867,7

4

1057,8

4

I300-6

10

11255,4

8

5,7

3

11177,5

4

11177,8

4

11209,5

4

43,4

5

11251,1

9

11251,1

9

310

624,0

5

629,7

8

1150,4

1

I300-7

10

11393,2

9

8,6

2

11307,6

1

11308,6

4

11343,4

4

42,2

11392,5

2

11392,5

2

464

657,5

2

660,1

4

1689,5

0

I300-8

10

11377,3

4

7,5

8

11331,0

9

11333,1

8

11347,3

1

35,0

7

11377,3

4

11377,3

4

46

120,5

6

128,1

4

559,3

9

I300-9

10

10980,0

5

8,2

9

10787,8

1

10790,3

5

10830,6

47,4

2

10878,0

5

10878,0

5

198

785,7

7

794,0

6

1437,8

9

I300-1

0

10

11265,1

3

7,4

2

11174,3

3

11183,1

8

11208,8

58,9

8

11232,7

7

11232,7

7

44

152,9

2

160,3

4

558,9

4

I300-1

1

15

10024,4

7

6,6

4

9977,1

4

9980,0

5

10003,4

56,1

1

10023,9

4

10023,9

4

26

90,7

2

97,3

6

268.7

2

I300-1

2

15

9398,1

9

6,6

1

9300,6

6

9304,2

1

9312,0

3

31,6

9336,6

4

9336,64

82

230,2

5

236,8

6

311.0

8

I300-1

3

15

10107,9

5

6,5

3

9980,7

3

9981,7

3

10016,5

0

45,5

4

10058,4

9

10058,4

9

242

157,0

3

202,5

7

699,2

0

I300-1

4

15

9768,7

2

7,5

5

9669,6

4

9670,3

2

9688,5

1

63,5

1

9699,3

5

9699,35

14

331,6

4

339,1

9

481.9

7

I300-1

5

15

9909,3

5,6

6

9788,4

9798,2

1

9823,1

2

64,5

7

9842,1

7

9842,17

26

131,0

6

136,7

2

253,8

8

I300-1

6

20

9331,7

7

4,7

9

9096,0

7

9102,6

5

9145,0

8

78,1

4

9158,7

9

9158,79

16

120,0

5

124,1

2

239,0

9

I300-1

7

20

9227,5

6

5,4

8

9133,8

7

9135,2

1

9148,7

5

39,3

7

9171,6

7

9171,67

52

313,5

2

319

382,8

9

I300-1

8

20

9584,0

7

5,2

5

9533,7

1

9535,8

0

9538,4

9

24,0

4

9553,5

9

9553,59

18

102,1

4

107,3

9

323.8

9

I300-1

9

20

9111,4

5,1

4

9013,6

9016,5

2

9041,7

9

66,7

4

9053,7

1

9053,71

28

221,5

3

226,6

7

302,2

1

I300-2

0

20

9068,4

2

4,9

2

9044,7

5

9044,8

3

9046,1

9

91,1

4

9046,3

3

9046,33

8

9,1

9

14,1

1

78.0

6

Table1:Computationalresultsfor

testinstanceswith300facilitiesan

d300clients

22


25/27

Name

Ratio

UBHeur

UBHeur

LP

CPLEXroot

LPCuts

%

Gap

BLB

BUB

B&C&P

B&C&P

Tot.Time

CPLEXTime

Ti

me(secs.)

nodes

Time(secs.)

(secs.)

(secs.)

I500-1

5

26412,4

1

36,6

26374,6

1

26374,8

5

26382,5

2

20,9

26412,4

1

26412,4

1

458

1267,4

2

1304,0

2

7924.8

3

I500-2

5

28144,8

3

109,3

6

28082,0

1

28082,1

9

28106,7

7

51,0

1

28130,7

4

28130,7

4

2144

5106,7

1

5216,8

7

18621,5

5

I500-3

5

27915,1

2

80,1

6

27857,0

5

27857,0

5

27873,7

2

35,1

2

27904,5

1

27904,5

1

1114

2595,6

1

2675,7

7

5277,4

4

I500-4

5

28159,0

3

128,2

3

28093,0

8

28095,7

0

28118,1

2

37,3

8

28159,0

3

28159,0

3

1332

2548,8

9

2677,1

2

9524,7

2

I500-5

5

24702,7

7

136,1

2

24645,6

3

24645,8

6

24673,9

5

49,5

6

24702,7

7

24702,7

7

1872

4715,6

1

4851,7

3

15085,5

9

I500-6

10

15756,8

2

13,7

6

15672,3

6

15674,7

9

15726,2

5

63,8

1

15756,8

2

15756,8

2

128

844,2

7

858,0

3

6426,5

9

I500-7

10

16110,8

22,4

7

16029,0

1

16030,0

9

16076,9

4

59,7

1

16109,2

8

16109,2

8

436

2420,7

7

2443,2

4

8379.3

0

I500-8

10

16041,7

3

24,8

5

15969,9

4

15970,5

7

16011,2

2

57,5

16041,7

3

16041,7

3

458

3375,3

1

3300,1

6

9434,3

3

I500-9

10

16327,7

1

48,9

5

16237,4

7

16238,3

3

16265,1

30,6

1

16327,7

1

16327,7

1

2076

4277,3

4326,2

5

28226,2

8

I500-1

0

10

15815,1

3

19,3

7

15751,9

1

15752,0

2

15780,0

1

44,4

5

15815,1

3

15815,1

3

176

1226,7

8

1246,1

5

4709,1

9

I500-1

1

15

13442,9

1

17,7

6

13406,0

1

13406,2

5

13421,3

3

48,3

1

13437,7

2

13437,7

2

48

243,5

3

261,2

9

1292,3

7

I500-1

2

15

14698,0

3

33,7

1

14610,1

14610,5

2

14651,3

63,4

6

14675,0

2

14675,0

2

152

1018,9

7

1052,6

8

3401,3

9

I500-1

3

15

13667,2

3

12,7

1

13631,3

5

13631,3

8

13640,2

7

25,5

6

13666,2

5

13666,2

5

78

398,5

6

411,2

7

2732,6

4

I500-1

4

15

13595,1

21,7

4

13514,9

3

13516,2

5

13542,5

3

42,4

1

13580,0

1

13580,0

1

130

1486,0

6

1507,8

3909,2

3

I500-1

5

15

13930,6

8

35,6

6

13833,4

2

13834,1

9

13858,2

9

39,2

6

13896,7

6

13896,7

6

104

351,0

6

386,7

2

4954,7

2

I500-1

6

20

12786,2

9

8,5

9

12492,6

8

12493,6

4

12552,6

65,2

7

12584,4

9

12584,4

9

140

921,8

6

930,4

5

2503,0

6

I500-1

7

20

13347,4

17,3

8

13290,2

6

13291,3

1

13323,1

57,4

7

13347,4

13347,4

80

775,6

4

793,0

2

1972,2

3

I500-1

8

20

12831,1

2

14,3

8

12780,1

6

12780,7

5

12810,1

58,7

5

12831,1

2

12831,1

2

66

407,6

1

421,9

9

1241,2

2

I500-1

9

20

13489,6

1

14,4

1

13450,6

3

13451,7

2

13469,2

9

47,8

7

13489,6

1

13489,6

1

62

334,4

7

348,8

8

1635,1

1

I500-2

0

20

12342,2

6

14,7

5

12312,4

6

12311,5

1

12326,9

4

48,5

9

12342,2

6

12342,2

6

24

257,6

6

272,4

1

1267,2

7



d500clients

23


26/27

Name

Ratio

UBHeur

UBHeur

LP

LPCu

ts

%

Gap

BLB

BUB

B

&C&P

B&C&P

Tot.Time

Time(secs.)

nodes

Time(secs.)

(secs.)

I700-1

5

36908,5

8

162,2

2

36839,9

6

36859,4

9

28,4

6

36905,9

3

36905,9

3

1600

9874,7

4

10036,1

9

I700-2

5

34311,7

1

188,4

4

34250,1

6

34274,8

7

40,1

5

34311,7

1

34311,7

1

3820

14823,7

8

15012,2

2

I700-3

5

34294,6

3

195,1

4

34253,7

3

34267,2

6

33,0

8

34294,6

3

34294,6

3

1036

2748,2

2

2943,3

6

I700-4

5

380

90,9

263,2

3

38029,8

7

38046,5

7

27,3

6

38090,9

38090,9

2400

8703,5

6

8966,7

9

I700-5

5

378

02,1

309,9

9

37741,1

2

37759,6

5

30,3

9

37802,1

37802,1

3112

15369,7

3

15679,7

2

I700-6

10

19927,3

4

50,7

9

19821,9

7

19854,3

8

36,5

4

19910,6

7

19910,6

7

3368

23088,9

4

23139,7

3

I700-7

10

212

97,3

49,7

8

21195,9

3

21233,1

5

36,7

2

21297,3

21297,3

1808

17096,0

6

17145,8

4

I700-8

10

20678,8

7

44,0

9

20592,1

8

20621,6

5

43,4

8

20659,9

6

20659,9

6

410

5495,9

2

5540,0

1

I700-9

10

20990,2

7

86,3

20894,1

9

20920,6

9

30,9

3

20979,8

8

20979,8

8

1638

18050,7

5

18137,0

5

I700-1

0

10

22

060

49,5

1

21960,8

8

22015,4

8

57,7

6

22055,4

1

22055,4

1

1872

27235,6

1

27285,1

2

I700-1

1

15

17164,3

1

22,9

9

17005,2

4

17056,5

6

44,6

6

17120,1

5

17120,1

5

5022

64296,5

3

64319,5

2

I700-1

2

15

18130,4

5

33,0

1

18048,2

18080,8

6

39,7

2

18130,4

2

18130,4

2

262

4459,7

4492,7

1

I700-1

3

15

17239,9

6

53,4

4

17163,1

5

17201,3

2

49,6

9

17239,9

6

17239,9

6

394

3068,0

8

3121,5

2

I700-1

4

15

17337,6

3

32,4

7

17247,1

6

17279,1

4

35,3

5

17337,6

3

17337,6

3

578

10716,7

5

10749,2

2

I700-1

5

15

18145,4

9

44,6

9

18069,7

9

18097,2

7

36,3

18145,4

9

18145,4

9

192

4885,2

5

4929,9

4

I700-1

6

20

16000,0

3

16,0

8

15945,2

8

15988,4

2

78,7

9

16000,0

3

16000,0

3

24

802,9

7

819,0

5

I700-1

7

20

16172,7

4

15,9

1

16132,8

3

16145,9

9

33,8

8

16171,6

7

16171,6

7

54

1292,1

9

1308,1

I700-1

8

20

164

14,8

14,4

16350,5

16384,2

6

52,5

16414,8

16414,8

66

1673,5

9

1687,9

9

I700-1

9

20

16366,7

8

20,3

4

16321,9

1

16346,3

8

54,5

4

16366,7

8

16366,7

8

94

1483,7

4

1504,0

8

I700-2

0

20

15434,2

1

19,1

4

15369,6

6

15404,4

2

53,8

5

15434,2

1

15434,2

1

64

2007,5

2026,6

4



d700clients

24


27/27

Name

Ratio

UB

Heur

UBHeur

LP

LPCuts

%

Gap

BLB

BUB

B

&C&P

B&C&P

Tot.Time

Time(secs.)

nodes

Time(secs.)

(secs.)

I1000-1

5

49537,2

780,2

9

49459,6

7

49476,1

9

32,9

5

49509,8

1

49509,8

1

7608

60122,6

9

60902,9

8

I1000-2

5

506

92,4

8

1042,1

1

50626,8

8

50641,7

4

24,0

9

50688,5

7

50688,5

7

8042

71738,0

6

72780,1

7

I1000-3

5

472

04,2

4

429,1

9

47153,1

5

47166,6

1

27,2

47202,6

4

47202,6

4

3454

29870,6

4

30299,8

3

I1000-4

5

488

71,0

6

1359,4

2

48807,6

8

48822,7

2

24,7

1

48868,5

4

48868,5

4

5250

42581,9

6

43941,3

8

I1000-5

5

50753,6

594,0

3

50704,3

8

50719,7

6

38,5

6

50743,5

4

50743,5

4

782

7430,6

5

8024,6

8

I1000-6

10

278

43,6

4

403,7

3

27736,3

2777

2,7

41,5

8

27823,8

4

27823,8

4

3214

70700,4

4

71104,1

7

I1000-7

10

272

64,7

7

436,9

1

27162,1

8

27189,6

7

30,5

27252,3

2

27252,3

2

4172

71356,3

8

71793,2

9

I1000-8

10

273

75,3

7

285,4

4

27253,5

3

27289,8

2

29,7

8

27369,0

4

27375,3

7

6386

100000

100285,4

I1000-9

10

268

88,1

2

381,5

7

26755,2

2

26802,0

6

45,9

8

26857,0

9

26857,0

9

4978

71681,7

2

72063,2

9

I1000-1

0

10

272

41,2

6

155

27106,9

5

27140,0

8

41,3

9

27186,9

9

27186,9

9

1564

40634,6

3

40789,6

3

I1000-1

1

15

221

80,3

3

445,8

2

22070,4

22103,3

4

29,9

6

22158,5

2

22180,3

3

4928

100000

100445,8

I1000-1

2

15

221

60,3

9

405,2

6

22054,7

6

22089,2

1

32,6

1

22145,9

6

22160,3

9

5270

100000

100405,3

I1000-1

3

15

226

57,0

9

228,8

6

22552,1

3

22590,9

1

36,9

5

22657,0

9

22657,0

9

1962

80693,0

3

80921,8

9

I1000-1

4

15

223

27,6

2

242,5

8

22241,7

6

22260,6

2

26,8

5

22312,0

1

22312,0

1

2086

51850,2

5

52092,8

3

I1000-1

5

15

22648,5

2

171,6

4

2523,3

6

22558,3

4

32,9

8

22629,4

4

22629,4

4

2914

85618,5

2

85790,1

6

I1000-1

6

20

213

31,8

1

119,0

4

21258,7

7

21279,0

2

27,7

2

21331,8

1

21331,8

1

1000

30657,8

30776,8

4

I1000-1

7

20

211

88,8

9

79,0

9

21153,1

2

21176,0

4

64,0

8

21188,8

9

21188,8

9

72

3403,0

8

3482,1

7

I1000-1

8

20

207

13,4

4

48,1

2

20658,4

2068

2,9

44,5

2

20713,4

3

20713,4

3

106

3222,2

4

3270,3

6

I1000-1

9

20

205

37,4

5

44,1

1

20470,1

1

20504,4

2

50,9

5

20537,4

5

20537,4

5

300

10417,0

9

10461,2

I1000-2

0

20

215

60,8

6

102,8

4

21496,8

9

21512

23,6

2

21560,8

6

21560,8

6

784

18619,6

9

18722,5

3

Table4:Computationalresultsfortestinstanceswith1000facilitiesan

d1000clients

Avella 05 - Bcp Cflp

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