K. Aardal, Y. Pochet, and L. A. Wolsey. Capacitated facility location: Valid inequalities and facets. Mathematics of Operations Research, 20:562--582, 1995.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....and Toth [15] for binpacking. We evaluate the binpacking inequality for this and J and add it to the formulation if it is violated. Otherwise we check whether the residual capacity inequality de ned by the same sets I and J is 22 violated. E ective Capacity Inequalities (Aardal et al. [1]) are also valid inequalities for the CFLP polytope. Let J I . For each j 2 J choose I j I and de ne I = j2J 0 I j . De ne also M j = minfM; d i g for each j 2 J = 0 M j D(I ) If 0 then the e ective capacity inequality d i x ij M j ) 1 x ....

Aardal, K., Pochet, Y., Wolsey, L.A. (1995): Capacitated Facility Location: Valid Inequalities and Facets. Mathematics of Operations Research 20, 562582

....complex flow models. In [103] a family of flow cover inequalities is described for a general single node flow model containing variable lower and upper bounds. Generalizations of flow cover inequalities to lot sizing and capacitated facility location problems can also be found respectively in [2] and [91] Flow cover inequalities have been used successfully in general purpose branch and cut algorithms to tighten formulations of mixed integer sets [104] Some examples are given in Section 4. Cover inequalities appear also in other contexts. In [29] cover inequalities are derived for the ....

....inequalities when Ct = C for t =1, n. First from (42) we obtain the MIR inequality s k 1 # j #S,k#j#t y j # rkt (kt # j#S,j#t x j ) where kt = #dkt C# and rkt = dkt (kt 1)C. N w if the rkt are placed in non decreasing order, and written r [1] # r [2] . # r [q] and [i] andX S [i] are the corresponding terms for and # j x j , the mixing procedure gives s k 1 # j #S,k#j#l y j # r [1] 1] X S [1] r [2] r [1] 2] X S [2] r [q] r [q 1] q] X S [q] An example of this inequality has been shown ....

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K. Aardal, Y. Pochet, and L.A. Wolsey, Capacitated facility location: valid inequalities and facets, Mathematics of Operations Research 20, 562 -- 582 (1995).

.... The single node fixed charge flow polyhedron, studied by Padberg et al. 10] and Van Roy and Wolsey [14] arises as an important relaxation of many 0 1 mixed integer programming problems with fixed charges, including lot sizing problems [4, 11] and capacitated facility location problems [1]. The valid inequalities derived for the single node fixed charge flow polyhedron have proven to be effective for solving these types of problems. Here we study a generalization of the single node fixed charge flow polyhedron that arises as a relaxation of network flow problems with additive ....

K. Aardal, Y. Pochet, and L. A. Wolsey. Capacitated facility location: Valid inequalities and facets. Mathematics of Operations Research, 20:562--582, 1995.

....[13] for a detailed discussion on using S and related structures as relaxations in mixed integer programming. The valid inequalities for S have also been instrumental in developing strong cutting planes for a variety of problems, including lotsizing problems [2, 10] and facility location problems [1]. Padberg et al. 9] were the first to study the polyhedral structure of the convex hull of S for the case with only incoming flow arcs, i.e. N Gamma = They introduced the flow cover inequalities, which were later generalized by Van Roy and Wolsey [12] Gu et al. 5] strengthened these ....

K. Aardal, Y. Pochet, and L. A. Wolsey. Capacitated facility location: Valid inequalities and facets. Mathematics of Operations Research, 20:562--582, 1995.

....# j#M S w j C # j#S y j # # k#K d k , for S # M = with S # # # k#K d k C#. Therefore, for i = 1, r, we can choose sets S i # M , K i # N and mix the corresponding base inequalities. The (k, l, S, I) based inequalities defined in Aardal, Pochet and Wolsey [1] can be obtained by mixing these base inequalities, when S i , K i form a nested family (i.e. K 1 # K 2 # . # K r and S 1 # S 2 # . # S r ) The corresponding mixed inequalities are typically strong (defining facets or high dimensional faces) We next give an ....

K. Aardal, Y. Pochet and L.A. Wolsey, "Capacitated Facility Location : Valid Inequalities and Facets", Mathematics of Operations Research 20 (1995), 562-582.

....complex flow models. In [103] a family of flow cover inequalities is described for a general single node flow model containing variable lower and upper bounds. Generalizations of flow cover inequalities to lot sizing and capacitated facility location problems can also be found respectively in [2] and [91] Flow cover inequalities have been used successfully in general purpose branch and cut algorithms to tighten formulations of mixed integer sets [104] Some examples are given in Section 4. Cover inequalities appear also in other contexts. In [29] cover inequalities are derived for the ....

....inequalities when C t = C for t = 1, n. First from (42) we obtain the MIR inequality s k 1 # j #S,k#j#t y j # r kt ( kt # j#S,j#t x j ) where kt = #d kt C# and r kt = d kt ( kt 1)C. Now if the r kt are placed in non decreasing order, and written r [1] # r [2] . # r [q] and [i] and X S [i] are the corresponding terms for and # j x j , the mixing procedure gives s k 1 # j #S,k#j#l y j # r [1] 1] X S [1] r [2] r [1] 2] X S [2] r [q] r [q 1] q] X S [q] An example of this ....

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K. Aardal, Y. Pochet, and L.A. Wolsey, Capacitated facility location: valid inequalities and facets, Mathematics of Operations Research 20, 562 -- 582 (1995).

.... a dual feasible solution, given the same arbitrarily chosen primal objective function as above, and show that it has the same value as the optimal solution to the dual of the assignment problem AP, and hence to AP, which implies that it also has the same value as the optimal solution (x; y) Aardal et al. 1995) showed that the separation problem based on the family of extended flow cover inequalities can be solved in polynomial time if m j = m for all j 2 N . Van Roy and Wolsey (1986) also studied the single node flow model with both fixed charge inflow and outflow arcs. Separation heuristics for these ....

....CFL. generalizing the flow cover inequality is made by considering inequalities based on subsets K N of clients. One way of generalizing the flow cover inequalities further is by considering a subset of clients as well as subsets of arcs yielding the family of effective capacity inequalities (Aardal et al. 1995)) Let K j K for all j 2 M and let m j = minfm j ; d(K j )g. Let J define a flow cover with respect to K, i.e. P j2J m j = d(K) with 0. The effective capacity (EC) inequality X j2J X k2K j v jk d(K) Gamma X j2J ( m j Gamma ) 1 Gamma y j ) 59) is valid for ....

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K. Aardal, Y. Pochet and L.A. Wolsey (1995) "Capacitated facility location: valid inequalities and facets", Mathematics of Operations Research 20 562--582.

....on computational results from solving 60 small and medium size problems. Key words: Cutting planes; Facets; Location problems. We study the cutting plane approach to solving the capacitated facility location (CFL) problem. The polyhedral structure of CFL has been studied by Aardal (1992) and Aardal, Pochet and Wolsey (1993), by Leung and Magnanti (1989) for the equal capacity case, and by Deng and Simchi Levi (1993) for the case of equal capacities and unsplit demands. Cornu ejols, Sridharan and Thizy (1991) compared the strength of various Lagrangean relaxations of CFL, and relaxations obtained by completely ....

....is useful for computational purposes, even though it contains redundant information. 2 The Submodular Inequalities and Important Special Cases Submodular inequalities were first introduced by Wolsey (1989) for general fixed charge network flow problems, and adapted to CFL by Aardal (1992) and Aardal et al. 1993). Choose a subset K N of clients, and let J M be a subset of depots. For each depot j 2 J choose a subset K j K. Once the sets K; J and K j for j 2 J are known, we can define the effective capacity of depot j as m j = min(m j ; d(K j ) Definition 1 A set function f defined on N = ....

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K. Aardal, Y. Pochet and L.A. Wolsey (1993). "Capacitated facility location: valid inequalities and facets". CORE Discussion Paper 9323 (CORE, Louvain-la-Neuve) to appear in Mathematics of Operations Research.

.... a dual feasible solution, given the same arbitrarily chosen primal objective function as above, and show that it has the same value as the optimal solution to the dual of the assignment problem AP, and hence to AP, which implies that it also has the same value as the optimal solution (x; y) Aardal et al. 1995) showed that the separation problem based on the family of extended flow cover inequalities can be solved in polynomial time if m j = m for all j 2 N . This result can be obtained by considering the following extended formulation. Let n = jN j. P = f( x; y) 2 IR n Theta IR n Theta ZZ n ....

....considering the following extended formulation. Let n = jN j. P = f( x; y) 2 IR n Theta IR n Theta ZZ n : P j2N j b Gamma (m Gamma )l j x j Gamma (m Gamma )y j for all j 2 N P j2N x j = b 0 x j my j for all j 2 N 0 y j 1 for all j 2 N j 0 for all j 2 N Theorem 7 Aardal et al. 1995). The projection of P onto the subspace f n : 0g is equal to conv(X C FC ) Proof. By eliminating all j variables we obtain inequalities (33) and the defining constraints. From the separation point of view we note that the set S consists of all arcs j for which x j Gamma (m Gamma ....

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K. Aardal, Y. Pochet and L.A. Wolsey (1995) "Capacitated facility location: valid inequalities and facets", Mathematics of Operations Research 20 562--582.

.... have been used to solve problems such as the radio link frequency assignment problems by Aardal, Hipolito, Van Hoesel and Jansen (1995) and the uncapacitated facility location problem by Aardal and Van Hoesel (1995) Various more complex facility location problems have been studied by Aardal, see Aardal, Pochet and Wolsey (1993), Aardal (1994) and Aardal, Labb e, Leung and Queyranne (1994) Results on scheduling problems have been obtained by Nemhauser and Savelsbergh (1992) Crama and Spieksma (1995) Van den Akker, Van Hoesel and Savelsbergh (1993) and Van den Akker, Hurkens and Savelsbergh (1995) Savelsbergh has ....

....31 125 50332 1.2 691 1,560 58 54.3 51 450 50333 1.5 259 556 122 54.1 89 769 50334 0.7 239 493 42 76.6 23 213 50335 1.3 685 1,232 25 78.3 49 248 Table 3: Result of adding knapsack cover inequalities to CFL. The knapsack polytope is a quite drastic relaxation of CFL since it disregards all flows. Aardal et al. 1993) considered the general family of submodular inequalities X j2J X k2K j v jk f(J) Gamma X j2J (f(J) Gamma f(J n fjg) 1 Gamma y j ) 31) where K j K for all j 2 J , and where f(J) for J M is the maximum feasible flow from the facilities in J to the clients in K given the arc set ....

K. Aardal, Y. Pochet and L.A. Wolsey (1993) "Capacitated facility location: valid inequalities and facets", CORE Discussion Paper 9323, Louvain-la-Neuve, (to appear in Mathematics of Operations Research).

....base inequalities X j2MnS w j C X j2S y j X k2K d k ; for S M = with jSj d P k2K d k =Ce. Therefore, for i = 1; r, we can choose sets S i M , K i N and mix the corresponding base inequalities. The (k; l; S; I) based inequalities defined in Aardal, Pochet and Wolsey [1] can be obtained by mixing these base inequalities, when fS i ; K i g form a nested family (i.e. K 1 ae K 2 ae : ae K r and S 1 ae S 2 ae : ae S r ) The corresponding mixed inequalities are typically strong (defining facets or high dimensional faces) We next give an ....

K. Aardal, Y. Pochet and L.A. Wolsey, "Capacitated Facility Location : Valid Inequalities and Facets", Mathematics of Operations Research 20 (1995), 562-582.

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K. Aardal, Y. Pochet, and L. A. Wolsey. Capacitated facility location: Valid inequalities and facets. Mathematics of Operations Research, 20:562--582, 1995.

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Aardal, K., Y. Pochet and L.A. Wolsey, "Capacitated Facility Location: Valid Inequalities and Facets", Mathematics of Operations Research, (20), 562-582, 1995.

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