A computational study of dicutreformulation for the Single SourceCapacitated Facility Location problem

Pasquale Avella, Maurizio Boccia * ,Saverio Salerno **

* DINGUniversità degli Studi del Sannio

e-mail: {avella, maurizio.boccia}@unisannio.it,** DIIMA

Università degli Studi di Salernoe-mail: salerno@unisa.it

Abstract. The Capacitated Facility Location Problem is to locate a set of facilities with capacityconstraints, with the aim of satisfying at the minimum cost the demands of a set of clients. TheSingle Source version (SingleCFLP ) of the problem has the additional requirement that eachclient must be assigned to a single facility.In this paper a reformulation of the problem based on Dicut Inequalities is proposed. Based onthe observation that Dicut Inequalities are that are knapsack inequalities, an exact knapsackseparation procedure is used to derive cutting planes from the inequalities which are valid forthe Dicut Polytope.The separation procedure for Dicut inequalities has been embedded into a Branch-and-Cutframework and a computational experience is reported on a wide set of benchmark instances.

Keywords: Facility Location ; Cutting Planes; Branch-and-Cut.

1. Introduction

The Capacitated Facility Location Problem is to locate a set of fa-cilities with capacity constraints, with the aim of satisfying at the min-

Studia Informatica Universalis.

22 Studia Informatica Universalis.

imum cost the demands of a set of clients. The Single Source version(SingleCFLP ) of the problem has the additional requirement that eachclient must be assigned to a single facility.

The Single Source Capacitated Facility Location is a very challeng-ing Integer Programming problem: several small size instances cannotbe solved to optimality by commercial MIP solvers. This can be partlyexplained by the large integrality gaps returned by the LP-relaxation.

Tightening MIP formulations by deriving Lifted Cover Inequalitiesfrom knapsack constraints is now a common technique, embedded intomost of the MIP solvers. In this paper we go a step further in thisdirection and present a cutting plane generation routine based on thefollowing idea:i) Reformulate the Capacitated Facility Location problem using a new class

of valid inequalites - the Dicut Inequalities - which are in fact knapsack in-equalities.ii) Let bTx ! b0 (b, b0 ! 0) be a Dicut Inequality and let F (b, b0) = {x "

{0, 1}n : bTx ! b0} denote the set of the feasible solutions satisfying the

Dicut inequality.iii) Run an exact knapsack separation routine to generate facet-inducing in-equalities of conv(F (b, b0)) violating a given fractional solution.

We will show how a fine-tuned implementation of this idea leads tobreakthrough computational results for this difficult problem.

The remainder of the paper is organized as follows. In Section 2 wesummarize the main solution approaches for SingleCFLP and otherrelated problems. In Section 3 we provide a formal statement of theproblem and describe its formulation and reformulation based on Di-cut Inequalities. In Section 4 we outline the exact knapsack separa-tion procedure, whose implementation issues are addressed in Section5. Finally in Section 6 we report on computational experience with aBranch-and-Cut algorithm, validating the effectiveness of the proposedapproach.

SSCFLP dicut reformulation 23

2. State of the art

SingleCFLP has received a considerable amount of attention in theliterature. Local search heuristics have been proposed by Filho and Gal-vao [FG98] and Delmaire et al. [DDFO99]. Ahuja et al. [AOP+04] in-vestigated a Very Large-Scale Neighborhood (VLSN) search algorithm.More recently Chen and Ting [CT08] combine Lagrangian heuristic andant colony system to find a high quality feasible solution for the prob-lem.

Lagrangean heuristics have been proposed by Klincewicz and Luss[KL86], Pirkul [Pir87], Barcelo and Casanovas [BC84], Guignard andOpaswongkarn [GO90], Hindi and Pienkosz [HP99], Sridharan [Sri93],Darby-Dowman and Lewis [DDL88]. Ronnqvist et al. [RTH99] pro-posed a repeated matching heuristic.

Exact algorithms have been proposed by Neebe and Rao [NR83] andBarcelo et al. [BFJ91]. Holmberg et al. [HRY99] presented a Branch-and-Bound algorithm where the lower bound is computed by solvinga Lagrangean relaxation providing a tighter lower bound than the LP-relaxation.

Diaz and Fernandez [DF02] proposed a two-stage algorithm. Theset of the candidate facilities is first reduced by preprocessing. Thenfor each possible subset of open facilities, a Generalized Assignmentproblem is solved by Branch-and-Price. The authors report on a com-putational experience on instances with 30 facilities and 90 clients.

2.1. Related problems

Let M be a set of facilities and let N be a set of clients. If weassume that all the facilities in M are open, SingleCFLP (M,N) re-duces to the Generalized Assignment Problem [PdAaU05]. In [ABV10]a Branch-and-Cut algorithm for theGeneralized Assignment Problem isproposed, based on the exact separation of the knapsack constraints. If,in addition, it is assumed that a feasible solution can admit unservedclients, the multiple knapsack problem is obtained [FMW96].

24 Studia Informatica Universalis.

On the other hand, if we assume that at most p facilities can beopened, we obtain the Capacitated p-Median problem [LS04]. ABranch-and-Cut algorithm based on the exact separation of the knap-sack constraints has been proposed in [BSSV08].

The Capacitated Concentrator Location problem consists of choos-ing a subset of a given terminal set to install concentrators and to assigneach remaining terminal node to a concentrator to minimize the costof installation and assignment. Concentrators can have capacity con-straints. Deng and Simchi-Levi [DSL92], Yaman [Yam05] and Yamanand Labbè [LYG05] study the polyhedral structure of this problem. In[LY06] Yaman and Labbè report on a computational experience witha Branch-and-Cut algorithm for a Concentrator Location problem withadditional quadratic constraints.

3. Problem formulation and reformulation

Let M = {1, . . . ,m} be a set of facilities and let N = {1, . . . , n}be a set of clients. Let uk be the capacity of the facility k and let dj bethe demand of client j " N . Let hk be the fixed cost of opening facilityk " M . Let ckj be the the transportation cost from the facility k " M

to the client j " N of all the client j demand dj .

We will denote by Q(M,N) the “standard" formulation ofSingleCFLP (M,N):

min!

k!M

hkyk +!

k!M

!

j!N

ckjdjxkj

!

k!M

xkj = 1, j " N (1)

!

j!N

djxkj # ukyk, k " M (2)

xkj " {0, 1}, k " M, j " N

where yk = 1 if the facility k " M is open, yk = 0 otherwise, and xkj =1 if the client j demand is assigned to facility k, xkj = 0 otherwise.

SSCFLP dicut reformulation 25

In Ardal [Aar98] Q(M,N) formulation is enforced by adding theTotal Capacity inequality

"

k!M ukyk !"

j!N dj . We will denote byQ

Cap(M,N) this formulation.

SingleCFLP (M,N) can be reformulated as in Avella and Boccia[AB09] by using the Dicut inequalities introduced in Ortega andWolsey[OW03]. Let Qdicut(M,N) denote the formulation based on Dicut in-equalities (3):

min!

k!M

hkyk +!

k!M

!

j!N

ckjdjxkj

!

k!K

ukyk +!

k!M\K

!

j!J

djxkj !!

j!J

dj, K $ M,J $ N (3)

yk " {0, 1}, k " M

xkj " {0, 1}, k " M, j " N

It is easy to prove that Q(M,N) and Qdicut(M,N) return the samelower bound.

Let K % M and J % N and let DICUT (K, J) = {yk "{0, 1}, xkj " {0, 1} :

"

k!K ukyk +"

k!M\K

"

j!J djxkj !"

j!J dj}denote the set of the solutions satisfying the Dicut inequality definedby K and J . Formulation Qdicut(M,N) can be tightened by replacingeach Dicut Inequality (3) by conv(DICUT (K, J)). Let Qknap(M,N)denote the new formulation:

min!

k!M

hkyk +!

k!M

!

j!N

ckjdjxkj

(yk, xkj) " conv(DICUT (K, J)), K $ M,J $ N (4)

yk " {0, 1}, k " M

xkj " {0, 1}, k " M, j " N

Formulation Qknap is not easy to handle. In the next section we willaddress the exact separation problem for conv(DICUT (K, J)).

26 Studia Informatica Universalis.

4. Exact knapsack separation

Dicut inequalities are in fact knapsack inequalities (complementingall the binary variables they become classical knapsack inequalities).The exact separation problem for the knapsack polytope was studiedby Boyd [Boy94], who developed a cutting plane algorithm for generalInteger Programming based on the so-called Fenchel duality. Given apolyhedron S, the basic idea of Boyd’s method is to prove that a certainpoint x belongs to conv(S) or to find a separating hyperplane, that is asfar as possible from x. Such a separating hyperplane is referred to asa Fenchel cut. To find a Fenchel cut a piecewise linear function has tobe maximized on a nonlinear domain. More recently, exact separationprocedures for knapsack constraints have been investigated in [KL10,ABV10, BSSV08]

Applegate at al. [ABCC03] describe a method to find local cuts forthe TSP that can be outlined in these terms: i) shrink different subsetsof nodes to map the original TSP problem and the current fractionalsolution onto a lower dimensional space and ii) look for linear inequal-ities that are satisfied by all the feasible solutions and violated by thefractional solution. To find such an inequality, they solve by columngeneration the dual of a Linear Programming problem with a variablefor each solution of the (graphical) TSP over the reduced problem.

Let F (b, b0) = {x " {0, 1} : bx ! b0} denote the set of the feasiblesolutions of a knapsack problem and let conv(F (b, b0)) be its convexhull. Facets of conv(F (b, b0)) can be efficiently generated using thefollowing procedure.

Let #x be a fractional point to separate, the knapsack separation prob-lem is:

!" = min #x

T"& "0

yT" ! "0 y " F (b, b0) (5)

1T" = 1 (6)

where the normalization constraint (6) prevents unboundedness.

SSCFLP dicut reformulation 27

Let ("","

"0) and !" be respectively the optimal solution of the sepa-

ration LP (5)-(6) and its value. If !" < 0 then #x /" conv(F (b, b0)) andthe valid inequality ""T

x ! a"0 is violated by #x.

The normalization constraint (6) corresponds to the #1 norm andgenerates a valid inequality maximing the normalized violation !0#!xT!

1T!

(it approximates the valid inequality with maximum distance from thepoint #x). It easy to observe that with this normalization the extremepoints of the LP (5)-(6) are in one-to-one correspondence with the facetsof conv(F (b, b0))) [Bon03].

Since the separation LP (5)-(6) contains a huge number of rows, itrequires a row generation approach, i.e. an iterative approach where,at each iteration, a partial separation problem - including only a sub-set of the constraints (5) - is considered. Let (","0) be the optimalsolution of the partial separation problem. If all the feasible solutionsof F (b, b0) satisfy the inequality ("y ! "0), then (","0) is the opti-mal solution of the complete separation problem. Otherwise the newinequality ("y ! "0) is added to the partial separation problem and theprocedure continues. The main steps of the row generation are summa-rized below.

Row generation procedureStep 1 Let S(b, b0) $ F (b, b0) = {x " {0, 1} : bTx ! b0} be a subset

of the feasible solutions of the knapsack problem (S(b, b0) can beinitialized to ').

Step 2 Solve the partial separation LP over S(b, b0):

! = min #xT"& "0

yT" ! "0 y " S(b, b0) (7)

1T" = 1

Let (","0) be its optimal solution.

28 Studia Informatica Universalis.

Step 3 Solve the following knapsack problem to look for a solutionw " F (b, b0) violating the inequality ("w ! "0) :

$ = min "Tw & "0 (8)

bTw ! b0

w " {0, 1}n

Let w and $ be respectively the optimal solution of the knapsackproblem (8) and its objective function value.

Step 4 If $ < 0 then set S = S ( {w} and goto Step 1.Step 5 (","0) is the optimal solution of the separation LP (5)-(6) and

the inequality "Tx ! "0 is valid for conv(F (b, b0)).

5. Implementation details

A careful implementation is crucial to make exact knapsack separa-tion effective. In this section we provide implementation details of theexact knapsack separation.

5.1. Selection of candidate Dicut inequalities

Dicut Inequalities are exponential in number and an accurate selec-tion of the “candidate for exact separation" inequalities is crucial to thesuccess of the algorithm. Our computational experience has shown thattwo special cases of Dicut Inequalities are very effective to reduce thegap, namely those with 1 # |K| # 2 and J = N and those with|M | & 1 # |K| # M and J = N . We observe that Dicut Inequalitieswith K = {k} and J = N correspond to the Capacity Inequalities (2)and that Dicut Inequalities with K = ' and J = N correspond to theTotal Capacity inequality [Aar98].

The selection strategy is:

i) We first consider the Dicut Inequalities withK = {k} and J = N ,corresponding to the Capacity Inequalities, for each k " M , andii) the Dicut Inequality with K = M and J = N , corresponding to

the Total Capacity Inequality.

SSCFLP dicut reformulation 29

iii) If no violated inequalities are found, we enumerate the pairs K ={k1, k2 " M : k1 )= k2} and consider for the exact separation theDicut Inequalities with K = {k1, k2 " M : k1 )= k2}, J = N , whoseslack with respect to the current fractional solution is less than a giventhreshold %. In our experiments we set % = 0.1 *

"

j!J dj .iv) Then we enumerate all the k " M and consider for the exact sep-aration the Dicut Inequaliites defined by K = M \ {k} and J = N ,whose slack with respect to the current fractional solution is less than agiven threshold %.

5.2. Solving the knapsack subproblems

Exact knapsack separation requires to solve a large number of knap-sack problems, so using efficient knapsack algorithms is a key issue.In our computational experience we used Pisinger’s MINKNAP algo-rithm [Pis95] that combines dynamic programming with bounding andreduction techniques to dynamically adjust the “core”. The boundingand reduction techniques allow to fix some variables to to 0 or 1 in theoptimal solution. The core is a subset of non-fixed variables. The algo-rithm starts defining a core with only one variable. At each iteration anew variable is added to the core and the optimal solution over the coreproblem is computed by Dynamic Programming. The algorithm stopswhen all the variables outside the core are fixed.

MINKNAP requires that all the coefficients of the knapsack prob-lem are integer, whereas in the exact knapsack separation the objectivefunction coefficients of the problem (8) are given by the optimal so-lution of the partial separation LP (5) and hence they can be fractional.Ceselli and Righini [CR05] modified MINKNAP algorithm to solve anykind of knapsack problems. They relaxed the bounding test so that theoptimal solution computed by the modified algorithm may differ fromoptimality by at most nk& where nk is the number of the variables out-side the core and & is a positive parameter. Using values of & in therange between 10#6 and 10#12 they observe that the solutions obtainedare very tight and computation time does not change significantly. Inour experiments we set & = 10#10.

30 Studia Informatica Universalis.

5.3. Sequential lifting

In the exact knapsack separation routine, the number of row gener-ation iterations grows exponentially with the size of the knapsack sub-problem, so to make separation efficient it is crucial to work on thefractional variables.

Let (y, x) be a fractional solution that is the optimal solution of theLP relaxation (1)-(2), let H = {i " 1, .., |M | : yi /" {0, 1} and H ={(k, j), k " 1, .., |M |, j " 1, .., |N | : xkj /" {0, 1} be the index sets ofthe fractional y and x variables, respectively. Let FDICUT (K, J) ={(yk, xkj) " DICUT (K, J) : yi = yi +i /" H, xhl = xhl +(h, l) /"Z} be the restriction of DICUT (X,N) to the fractional space, that isthe subset ofDICUT (X,N) obtained by setting to 0 or 1 the variablesthat in the fractional solution are respectively equal to 0 or 1.

After a valid inequality for conv(FDICUT (K, J)) has been found,we use standard sequential lifting [NW88] to make it valid forconv(DICUT (K, J)) too. Computing the lifting coefficients amountsto solving a knapsack problem. In our experiments we use the modifiedMINKNAP algorithm described in Section 5.2.

It is well-known that the resulting inequalities depend on the orderin which the variables are lifted, i.e. on the lifting sequence. For a givenvariable, a better lifting coefficient is obtained if the variable is liftedearlier in the sequence. In our experiments, down lifting is performedfirst. We lift the y variables first, then the x variables. To define thelifting sequence inside each group we consider the reduced costs in thecurrrent LP-relaxation. Variables with smaller reduction costs are liftedearlier.

5.4. Dealing with numerical errors

It is well-known that rounding errors affect the solution of linear sys-tems and hence of linear programming solvers. The influence of round-ing errors on the solution can be large if the problem is ill-conditioned.Ill-condition is likely to arise in LP subproblems created by Branch-and-Cut procedures at the nodes of the search tree for Integer Linear

SSCFLP dicut reformulation 31

programs, where rounding errors in the computation of the coefficientscan lead to a “non-valid" inequality, cutting-off a feasible solution.

Neumaier and Shcherbina [NS04] show a small Integer Program-ming problem (with 20 variables) that most of the commercial MIPsolvers erroneously recognize as “integer infeasible”. In [NS04] cheappre- and post-processing algorithms are defined to generate “safe" MIRand Gomory cuts.

In the exact knapsack separation, rounding errors can occur in thesolution of the partial separation LP (7) and of the row generation prob-lem (8). Such errors can impact on the sequential lifting, leading toweak cuts or, less frequently, to “non-valid" inequalities, i.e. inequali-ties violated by some feasible solutions.

To generate “safe" cutting planes, the inequalities generated by theexact knapsack separation on the fractional space are post-processed toverify its validity and to get equivalent cuts with integer coefficients. Let(""

,""0) be the optimal solution of the partial separation LP (5)-(6), we

post-process the inequality ""Tx ! "

"0 to get an equivalent inequality

with integer coefficients.

Let T be the index set of the n solutions corresponding to the activeconstraints in the optimal solution of the problem (5)-(6). The IntegerLinear Programming problem:

min '0

yTi ' = '0 i " T (9)

' " Zn

'0 " Z

returns the inequality 'Tx ! '0, which is equivalent to the original cut.

This problem has exactly n+1 variables and n constraints. If n is nottoo large (we work on the fractional variables, so that usually n # 25)this problem can be easily handled by a MIP solver.

To check the validity of the resulting inequality, a new iteration of therow generation procedure is performed. Since all inequality coefficientsare now integer, the checking problem can be solved by the MINKNAP

32 Studia Informatica Universalis.

algorithm so eliminating the possibility of rounding errors. If all thefeasible solutions of FDICUT (K, J) satisfy the inequality 'Tx ! '0,then it is facet-defining for conv(FDICUT (K, J)).

The integrality of the lifting coefficients ' determines that the liftingcoefficients are integer too, so the lifting procedure will not be affectedby rounding errors.

6. Computational results

The separation routine has been embedded into the Branch-and-Cutframework provided by ILOG CPLEX Callable Library 12.1. All theexperiments ran on a Computer Intel(R) Core(TM)2 CPU with 2.67GHz CPU and 4 GB of RAM.

Computational experience has shown that cutting planes are moreeffective to reduce the gap in the upper level nodes of the search tree, sowe generate cutting planes at the nodes whose depth is less or equal than2. A violated cutting plane is added to the cut pool only if the amountof the violation is larger than a given threshold & (in our experiments weset & = 0.001).

Two different test-beds, denoted respectively as TBED1 andTBED2, were used to evaluate the algorithm.

TBED1 includes the nontrivial instances of Diaz and Fernandez[DF02]. It consists of 24 instances organized into 3 groups, namelyC1, C2 and C3, each containing 8 instances of the same size |M |,|N |.The sizes are 30,75 for C1, 30,60 for C2, 30,75 for C3 respectively.TBED1 instances are publically available at the URL http://www-eio.upc.es/-elena/sscplp/index.html.

In Table 1 we report on the experimental results for TBED1.Column LBLP reports on the lower bound provided by formulationQ(M,N). Column LBCplex reports on the lower bound provided byCplex 12.1 after the addition of built-in cutting planes to the formu-lation Q

Cap(M,N). Column LBDicut reports on the the lower boundyielded by the exact separation of the Dicut Inequalities described inSection 5.1.

SSCFLP dicut reformulation 33

Columns %GAPCplex and %GAPDicut show respectively the opti-mality gap for the lower bounds LBCplex and LBDicut, i.e. Gap =(Optimal V alue& Lower Bound)/Lower bound, 100. Finally col-umn OPT reports on the value of the optimal solution of the instances.

Computational results point out that for the TBED1 instancesLBDicut clearly outperforms the lower bound yielded by Cplex 12.1.Cutting planes can fill more than 50% and - in some cases - the 100%of the gap. Anyway these instances are easy to solve for Cplex 12.1 andthe addition of cutting planes turned out to be not usefull in terms ofcomputation times.

The instances of TBED2 were randomly generated according to theprocedure described in Cournuejols, Sridharan and Thizy [CST08] forthe Capacitated Facility Location Problem and have been used to gen-erate test instances in most the more recent papers on this problem. Thegeneration procedure is summarized below:

a) Points representing the facilities and the clients are uniformly ran-domly generated in a unit square. Transportation costs per unit flow arethen computed by multiplying the euclidean distances among them by10.b) Demands are generated from U [5, 35], i.e. they are uniformly dis-

tributed in the interval [5, 35].c) Let r =

"

k!M uk/"

j!N dj be the capacity ratio. Capacities uk

are generated from U [10, 160] and are then scaled by using the formula:uk = .uk/r/. In our tests r assumes the values: 2, 3 and 5.d) Fixed costs are generated by the formula hk = U [0, 90] +

U [100, 110]0uk to take into account the economies of scale.

e) The fixed costs hk are doubled when r = 2.

The instances of TBED2 are organized into 2 groups, each con-taining instances of the same size. The sizes here considered are:75 , 75 and 75 , 100. Each group includes three different sub-set of problems according to the ratio r (we set r = 2, 3, 5).Five instances were randomly generated for each value of r. Allthe the instances of this group are public available at the URLhttp://web.ing.unisannio.it/boccia/SSCFLP.htm.

34 Studia Informatica Universalis.

Tables 2 and 3 report on the experimental results for the TBED2instances. They present the same columns of Table 1 with the onlyaddition of column ratio, showing the capacity ratio r. We set a timelimit of 3000 seconds for the Branch-and-Cut algorithm.

All the instances were solved within the given time limit, except theinstances I75100& 3 and I75100& 7.

Furthermore we observe that exact separation for Dicut Inequalitiesis quite expensive in terms of computation time. Tables 4 and 5 focus oncomputation time. Column Cutgen time reports on time spent by theseparation routine to compute the lower bound. Column BandC time

reports on the total time spent by our algorithm to solve the problem.Column Cplex time reports on the total time spent by Cplex 12.1 tosolve the problem. The time limit for Cplex has been set to 15000 sec-onds.

Most of the TBED2, r = 5, instances are easy for Cplex and theaddition of cutting planes turned out to be not very useful. But forthe difficult instances, i.e. those where the Cplex computation time isgreater than 200 seconds, cutting planes become very effective. Manyinstances which are unsolvable for Cplex within the time limit of 15000seconds, are solved by our Branch-and-Cut algorithm in a reasonableamount of time. For the two instances that even our Branch-and-Cutcould not solve within the time limit of 3000 seconds, the best lowerand upper bound are much better than those provided by Cplex 12.1.

References

[Aar98] K. Aardal. Capacitated facility location: Separation algo-rithms and computational experience. Mathematical Pro-gramming, (81):149–175, 1998.

[AB09] P. Avella and M. Boccia. A cutting plane algorithm forthe capacitated facility location problem. ComputationalOptimization and Applications, 43:39–65, 2009.

[ABCC03] D. Applegate, R. Bixby, V. Chvátal, and W. Cook. Imple-menting the dantzig-fulkerson-johnson algorithm for large

SSCFLP dicut reformulation 35

traveling salesman problems.Mathematical Programming,(97):91–153, 2003.

[ABV10] P. Avella, M. Boccia, and I. Vasiliev. A computationalstudy of exact knapsack separation for the generalized as-signment problem. Computational Optimization and Ap-plications, 45:543–555, 2010.

[AOP+04] R.K. Ahuja, J.B. Orlin, S. Pallottino, M.P. Scaparra, andM.G. Scutella. A multi-exchange heuristic for the singlesource capacitated facility location. Management Science,(50):1749–760, 2004.

[BC84] J. Barceló and J. Casanovas. A heuristic lagrangean algo-rithm for the capacitated plant location problem. EuropeanJournal of Operational Research, 15:212–226, 1984.

[BFJ91] J. Barceló, E. Fernandez, and K. Jornsten. Computationalresults from a new relaxation algorithm for the capacitatedplant location problem. European Journal of OperationalResearch, 15:38–45, 1991.

[Bon03] P. Bonami. Etude et mise en oeuvre dŠapprochespolyt’edriques pour la rt’esolution de programmes en nom-bres entiers ou mixtes gt’ent’eraux. Ph.D. Thesis, Univer-sité de Paris, 2003.

[Boy94] E. A. Boyd. Fenchel cutting planes for integer programs.Operations Research, 42(1):53–64, 1994.

[BSSV08] M. Boccia, A. Sforza, C. Sterle, and I. Vasiliev. A cutand branch approach for the capacitated p-median problembased on fenchel cutting planes. Journal of MathematicalModelling and Algorithms, 7:43–58, 2008.

[CR05] A. Ceselli and G. Righini. A branch-and-price algorithmfor the capacitated p-median problem. Networks, 45:125–142, 2005.

[CST08] G. Cornuejols, R. Sridharan, and J.M. Thizy. A compari-son of heuristics and relaxations for the capacitated plantlocation problem. European Journal of Operational Re-search, 50:280–297, 2008.

36 Studia Informatica Universalis.

[CT08] CH. Chen and CJ. Ting. Combining lagrangian heuristicand ant colony system to solve the single source capaci-tated facility location problem. Transportation Researchpart E, 44:1099–1122, 2008.

[DDFO99] H. Delmaire, J. A. Diaz, E. Fernandez, and M. Ortega. Re-active grasp and tabu search based heuristics for the sin-gle source capacitated plant location problem. INFOR,Canadian Journal of Operational Research and Informa-tion Processing, 37(3):194U–225, 1999.

[DDL88] K. Darby-Dowman and H.S. Lewis. Lagrangean relaxationand the single source capacitated facility location problem.Journal of the Operational Research Society, 39:1035–1040, 1988.

[DF02] J.A. Diaz and E. Fernández. Branch and price algorithmfor the single source capacitated plant location problem.Journal of the Operational Research Society, 53(7):728–748, 2002.

[DSL92] Q. Deng and D. Simchi-Levi. Valid inequalities, facetsand computational results for the capacitated concentratorlocation problem. Department of Industrial Engineeringand Operations Research, Columbia University, New York,NY, 1992.

[FG98] V.J.M.F. Filho and R. D. Galvão. A tabu search heuristicfor the concentrator location problem. Location Science,6:189–209, 1998.

[FMW96] C.E. Ferreira, A. Martin, and R. Weismantel. Solving mul-tiple knapsack problems by cutting planes. SIAM Journalon Optimization, 6:858U–877, 1996.

[GO90] M. Guignard and K. Opaswongkarn. Lagrangean dual as-cent algorithms for computing bounds in capacitated plantlocation problems. European Journal of Operational Re-search, 46:73–83, 1990.

[HP99] K.S. Hindi and K. Pienkosz. Efficient solution of largescale single source capacitated plant location problems.Journal of the Operational Research Society, 50:268–274,1999.

SSCFLP dicut reformulation 37

[HRY99] K. Holmberg, M. Rönnqvist, and Di Yuan. An exact algo-rithm for the capacitated facility location problems withsingle sourcing. European Journal of Operational Re-search, 113:544–559, 1999.

[KL86] J.G. Klincewicz and H. Luss. A lagrangean relaxationheuristic for the capacitated facility location with single-source constraints. Journal of the Operational ResearchSociety, 37:495–500, 1986.

[KL10] K. Kaparis and A.N. Letchford. Separation algorithms for0-1 knapsack polytopes. Math. Programming, 124:69–91,2010.

[LS04] L.A.N. Lorena and E.L.F. Senne. A column generationapproach to capacitated p-median problems. Computersand Operations Research, 31:863–876, 2004.

[LY06] M. Labbè and H. Yaman. Polyhedral analysis for concen-trator location problems. Computational Optimization andApplications, 34:377–407, 2006.

[LYG05] M. Labbè, H. Yaman, and E. Gourdin. A branch and cut al-gorithm for hub location problems with single assignment.Math. Programming, 102:371–405, 2005.

[NR83] A.W. Neebe and M.R. Rao. An algorithm for the fixed-charge assigning users to sources problem. Journal of theOperational Research Society, 34:1107–1113, 1983.

[NS04] A. Neumaier and O. Shcherbina. Safe bounds in linearand mixed-integer programming. Math. Programming A,99:283–296, 2004.

[NW88] G.L. Nemhauser and L.A. Wolsey. Integer and combinato-rial optimization. eds. Wiley-Interscience, New York, NY,USA, 1988.

[OW03] F. Ortega and L. Wolsey. A branch-and-cut algorithmfor the single-commodity, uncapacitated, fixed-charge net-work flow problem. Networks, 41(3):143–158, 2003.

[PdAaU05] A. Pigatti, M. Poggi de Aragão, and E. Uchoa. Stabi-lized branch-and-cut-and-price for the generalized assign-

38 Studia Informatica Universalis.

ment problem. Electronic Notes in Discrete Mathematics,19:389–395, 2005.

[Pir87] H. Pirkul. Efficient algorithms for the capacitated concen-trator location problem. Computers and Operations Re-search, 14:197–208, 1987.

[Pis95] D. Pisinger. A minimal algorithm for the 0-1 knapsackproblem. Operat. Res., 46:758–767, 1995.

[RTH99] M. Ronnqvist, S. Tragantalerngsak, and J. Holt. A repeatedmatching heuristic for the single-source capacitated facil-ity location problem. European Journal of Operational Re-search, 116:51–U68, 1999.

[Sri93] R. Sridharan. A lagrangean heuristic for the capacitatedplant problem with single source constraints. EuropeanJournal of Operational Research, 66:305–312, 1993.

[Yam05] H. Yaman. Concentrator location in telecommunicationsnetworks. Springer, Combinatorial Optimization Series,16, 2005.

SSCFLP dicut reformulation 39Name

|M|

|N|

LB

LP

LB

CPLEX

%GAPCplex

LB

Dicut

%GAPD

icut

OPT

P34

30

60

4449.56

4479.27

4.95

4669.56

0.67

4701.00

P35

30

60

5352.32

5437.09

0.35

5456.00

0.00

5456.00

P36

30

60

15872.35

16641.69

0.83

16774.01

0.04

16781.00

P37

30

60

14325.77

14402.13

1.85

14614.15

0.37

14668.00

P38

30

60

46120.35

47016.42

0.49

47246.36

0.01

47249.00

P39

30

60

38971.46

40633.71

0.92

41001.05

0.01

41007.00

P40

30

60

60336.90

60806.73

1.36

61619.69

0.02

61633.00

P41

30

60

15906.91

16684.52

3.36

17246.00

0.00

17246.00

P42

30

75

7397.59

7697.22

2.46

7886.00

0.00

7887.00

P43

30

75

4937.20

4992.31

2.43

5109.49

0.09

5114.00

P44

30

75

35718.46

35940.89

0.23

36016.14

0.02

36022.00

P45

30

75

16265.22

16349.86

1.99

16676.00

0.00

16676.00

P46

30

75

46708.09

47279.63

3.01

48698.98

0.00

48701.00

P47

30

75

65071.63

65726.61

0.76

66211.47

0.03

66230.00

P48

30

75

54269.79

56467.55

4.43

58955.10

0.02

58967.00

P49

30

75

78022.72

79145.78

0.59

79601.49

0.02

79614.00

P50

30

90

5667.24

5681.06

4.50

5908.13

0.49

5937.00

P51

30

90

8787.54

9013.81

0.51

9046.25

0.15

9060.00

P52

30

90

33384.33

34010.13

1.89

34644.56

0.02

34652.00

P53

30

90

29302.78

29419.19

2.10

30035.07

0.01

30038.00

P54

30

90

42198.79

43581.70

0.62

43853.00

0.00

43853.00

P55

30

90

68005.12

69532.97

0.11

69607.79

0.00

69610.00

P56

30

90

63911.60

63974.88

0.78

64474.00

0.00

64474.00

P57

30

90

47720.44

49791.00

0.00

49791.00

0.00

49791.00

Table1:

ComputationalresultsforT

BED1

40 Studia Informatica Universalis.

Name

|M|

|N|

rLB

LP

LB

CPLEX

%GAPCplex

LB

Dicut

%GAPD

icut

OPT

I7575-1

75

75

221937.98

22307.30

1.44

22602.28

0.11

22627.60

I7575-2

75

75

225074.98

25448.20

0.89

25643.36

0.12

25674.53

I7575-3

75

75

225044.85

25427.66

1.24

25702.32

0.16

25742.51

I7575-4

75

75

228733.17

29085.66

1.05

29349.58

0.14

29392.02

I7575-5

75

75

224627.96

25006.76

0.81

25180.42

0.11

25208.68

I7575-6

75

75

38305.93

8461.90

1.67

8565.26

0.44

8603.42

I7575-7

75

75

37841.38

8040.63

1.46

8123.16

0.43

8158.02

I7575-8

75

75

38251.13

8455.28

1.31

8553.32

0.15

8566.09

I7575-9

75

75

39141.82

9338.70

1.60

9448.49

0.42

9487.77

I7575-10

75

75

39190.00

9279.73

1.38

9391.66

0.18

9408.22

I7575-11

75

75

55752.51

5867.58

1.26

5910.82

0.52

5941.66

I7575-12

75

75

55099.87

5180.19

1.56

5230.48

0.59

5261.24

I7575-13

75

75

55585.38

5653.57

2.83

5736.73

1.34

5813.71

I7575-14

75

75

56027.85

6076.93

1.15

6136.35

0.17

6146.99

I7575-15

75

75

55192.05

5245.47

1.89

5295.19

0.93

5344.42

Table2:

ComputationalresultsforT

BED2class1instances

Name

|M|

|N|

rLB

LP

LB

CPLEX

%GAPCplex

LB

Dicut

%GAPD

icut

OPT

I75100-1

75

100

232449.76

32585.51

0.64

32767.28

0.08

32793.50

I75100-2

75

100

233321.31

33524.25

0.92

33753.05

0.24

33833.06

I75100-3

75

100

224161.73

24347.73

1.57

24604.40

0.51

$24730.12

I75100-4

75

100

229077.43

29302.25

1.28

29599.58

0.26

29675.97

I75100-5

75

100

225564.17

25770.26

1.95

25967.80

1.17

26271.59

I75100-6

75

100

39208.12

9319.16

1.51

9423.37

0.39

9459.94

I75100-7

75

100

310553.91

10659.46

1.15

10716.21

0.62

$10782.40

I75100-8

75

100

310765.18

10885.68

1.46

10987.52

0.52

11044.53

I75100-9

75

100

38993.01

9070.21

1.81

9179.05

0.60

9234.43

I75100-10

75

100

310782.02

10879.28

1.78

11013.96

0.54

11073.44

I75100-11

75

100

57394.84

7417.98

1.42

7488.38

0.46

7523.16

I75100-12

75

100

56032.73

6088.77

0.80

6127.86

0.16

6137.51

I75100-13

75

100

55821.03

5872.14

1.43

5913.01

0.73

5956.12

I75100-14

75

100

56098.99

6134.87

1.44

6194.09

0.47

6223.16

I75100-15

75

100

55720.14

5813.15

1.48

5865.51

0.58

5899.28

Table3:

ComputationalresultsforT

BED2class2instances

SSCFLP dicut reformulation 41

Name |M | |N | r Cutgen BandC CPLEXtime (secs.) time (secs.) time (secs.)

I7575-1 75 75 2 520.27 14.34 % 15000

I7575-2 75 75 2 232.81 67.72 % 15000

I7575-3 75 75 2 205.03 79, 92 % 15000

I7575-4 75 75 2 345, 56 224, 63 % 15000

I7575-5 75 75 2 349.72 119.13 % 15000

I7575-6 75 75 3 360.66 90, 36 1281, 77I7575-7 75 75 3 109.63 48.07 586.58I7575-8 75 75 3 80.09 13.56 1545.86I7575-9 75 75 3 291.30 54.11 2354.44I7575-10 75 75 3 126.19 17.18 731.94

I7575-11 75 75 5 25.94 46.23 261.91I7575-12 75 75 5 105.50 9.11 95.06I7575-13 75 75 5 247.19 77.91 500.59I7575-14 75 75 5 124.75 11.88 38.17I7575-15 75 75 5 93.77 13.41 107.38

Table 4: Computation times for TBED2 class 1 instances

Name |M | |N | r Cutgen BandC CPLEXtime (secs.) time (secs.) time (secs.)

I75100-1 75 100 2 356.38 299.33 % 15000

I75100-2 75 100 2 758.56 2215.49 % 15000

I75100-3 75 100 2 553.00 % 3000 % 15000

I75100-4 75 100 2 859.36 1676.78 % 15000

I75100-5 75 100 2 944.77 % 3000 % 15000

I75100-6 75 100 3 321.86 85.72 1072.45I75100-7 75 100 3 478.83 % 3000 % 15000

I75100-8 75 100 3 266.01 696.97 4071.53I75100-9 75 100 3 477.83 118.80 2134.77I75100-10 75 100 3 513.41 1142.03 % 15000

I75100-11 75 100 5 218.03 120.36 379.25I75100-12 75 100 5 19.81 5.13 55.64I75100-13 75 100 5 145.59 44.95 130.97I75100-14 75 100 5 733.39 271.06 753.94I75100-15 75 100 5 137.83 27.45 104.23

Table 5: Computation times for TBED2 class 2 instances

42 Studia Informatica Universalis.