I. Barany, T. Van Roy, and L. A. Wolsey. Uncapacitated lot-sizing: The convex hull of solutions. Mathematical Programming Study, 22:32--43, 1984.  Home/Search   Document Not in Database   Summary   Related Articles   Check

....Belvaux and Wolsey [3,4] Pochet and Wolsey [17] and Wolsey [24] Therefore, a good understanding of the lot sizing polytope has immediate implications for many practical production problems. A complete linear description of the uncapacitated lot sizing polytope is given by Barany et al. [2]. The constant capacity lot sizing polytope is studied by Leung et al. 11] and Pochet and Wolsey [18] For the general case with A. Atamturk: Department of Industrial Engineering and Operations Research, 4135 Etcheverry Hall, University of California, Berkeley CA, 94720 1777 USA. Email: ....

.... i for all i [1, n] In this case, inequality (8) for S = 1, #] T [# 1, n] and # [0, n 1] reduces to y i (u # u # 1 ) 1 x # ) u # u # 1 )x i , 12) or y i u # 1 )x i , which is equivalent to the uncapacitated lot sizing inequality (Barany et al. [2]) d t# z t i # for T [1, #] # [1, n] There is an O(n log n) separation algorithm for these inequalities and it is su#cient to add them to the LP relaxation to obtain a complete description of the lot sizing polytope for the uncapacitated case [2] 3.2. Constant capacity case When ....

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I. Barany, T. J. Van Roy, and L. A. Wolsey. Uncapacitated lot sizing: The convex hull of solutions. Mathematical Programming Study, 22:32--43, 1984.

.... enumeration) when the objective function satisfies the Wagner Whitin assumption (see Pochet and Wolsey [19] The so called (l, S) inequalities known to describe the convex hull of feasible solutions of the uncapacitated (i.e. C = #) single item lot sizing problem (see Barany, Van Roy and Wolsey [5]) can also be obtained by the mixing procedure. 3.2 Capacitated Facility Location. The capacitated facility location problem consists of a set M = 1, m of potential depots with capacity C j , 1 # j # m, and a set N = 1, n of clients with demand d k , 1 # k # n. ....

I. Barany, T.J. Van Roy and L.A. Wolsey, "Uncapacitated Lot Sizing : the Convex Hull of Solutions", Mathematical Programming Study, 22 (1984), 32-43.

....facet inducing inequalities is given by Proposition 1. If c t i d i , the valid inequalities s i d i y i d i (7) induce facets of conv(X P I ) i = 1; P . These inequalities correspond to the (l; S) inequalities for the uncapacitated lot sizing problem, introduced by Barany, Van Roy, and Wolsey [1984]. Moreover, they imply the bounds s i 0; i = 1; P , which therefore never induce facets of conv(X P I ) We now characterize the extreme points and rays of conv(X P I ) Given an extreme point ( x; y; s) of conv(X P I ) let Q = fi 2 P : y i = 1g. Also, let Q u = fi 2 Q : ....

Barany, I., Roy, T. V., and Wolsey, L. (1984). Uncapacitated lot{sizing: the convex hull of solutions. Mathematical Programming Study, 22:32-43.

....to the Georgia Institute of Technology. 1 Introduction 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 ....

I. Barany, T. J. Van Roy, and L. A. Wolsey. Uncapacitated lot sizing: The convex hull of solutions. Mathematical Programming Study, 22:32--43, 1984.

....by lifting. We have also identified conditions under which each class of inequalities induces facets of conv(Y ) 2. 1 Continuous Cover and Continuous Reverse Cover Inequalities Theorem 1 (Marchand and Wolsey [1997] Given an (i; C; T ) cover pair for Y , order the elements of C such that a [1] : a [r C ] where r C is the number of elements of C with a j . Let A 0 = 0, A j = P j p=1 a [p] j = 1; r C , and define OE C (u) 8 : j Gamma 1) if A j Gamma1 u A j Gamma ; j = 1; r C ; j Gamma 1) u Gamma (A j Gamma ) if A j Gamma u A j ; j = ....

....all the variables in C n i, then all those in T n i. Because the lifting function for each of the two sets is superadditive, the lifting order within the two sets is immaterial. Theorem 2 (Marchand and Wolsey [1997] Given an (i; C; T ) cover pair for Y , order the elements of T such that a [1] : a [r T ] where r T is the number of elements of T with a j . Let A 0 = 0, A j = P j p=1 a [p] j = 1; r T , and define T (u) 8 : u Gamma j ; if A j u A j 1 Gamma ; j = 0; r T Gamma 1; A j Gamma j ; if A j Gamma u A j ; j = 1; r T Gamma 1; A ....

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I. Barany, T. van Roy, and L.A. Wolsey. Uncapacitated lot--sizing: the convex hull of solutions, Mathematical Programming Study 22 (1984), 32--43.

....optimization problems that are polynomially solvable (see Wagelmans et al. 1993) for the description of an O(T logT ) algorithm) For such problems we can expect to be able to give a compact characterization of the convex hull of feasible solutions, c.f. the matching problem (5) 7) Barany et al. 1984) showed that the constraints 0 y t 1 for all 1 t T , y 1 = 1, 33) 36) together with the exponential class of (l; S) inequalities (38) presented below, completely describe the convex hull of solutions. Take any 1 l T and S ae L = f1; lg. The (l; S) inequalities are written as X ....

...., is produced in a single period in S, which explains the coefficients of the y t variables. Although the class of (l; S) inequalities is exponential, we can still solve ELS efficiently by the polyhedral approach since the separation problem based on these inequalities is polynomially solvable (Barany et al. 1984)) We can generalize ELS by introducing startup costs, i.e. a payment for the first period in a set of consecutive periods in which production takes place. This new problem is referred to as ELSS. Below we demonstrate that the (l; S) inequalities can be generalized to incorporate the variables ....

I. Barany, T.J. Van Roy and L.A. Wolsey (1984) "Uncapacitated lot-sizing: the convex hull of solutions", Mathematical Programming Study 22 32-43.

....S among others. We refer the reader to [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 ....

I. Barany, T. J. Van Roy, and L. A. Wolsey. Uncapacitated lot sizing: The convex hull of solutions. Mathematical Programming Study, 22:32--43, 1984.

.... enumeration) when the objective function satisfies the Wagner Whitin assumption (see Pochet and Wolsey [19] The socalled (l; S) inequalities known to describe the convex hull of feasible solutions of the uncapacitated (i.e. C = 1) single item lot sizing problem (see Barany, Van Roy and Wolsey [5]) can also be obtained by the mixing procedure. 3.2 Capacitated Facility Location. The capacitated facility location problem consists of a set M = f1; mg of potential depots with capacity C j , 1 j m, and a set N = f1; ng of clients with demand d k , 1 k n. The objective is ....

I. Barany, T.J. Van Roy and L.A. Wolsey, "Uncapacitated Lot Sizing : the Convex Hull of Solutions", Mathematical Programming Study, 22 (1984), 32-43.

....50 and the initial stock cost h 0 at 60, while for sets 4, 5 and 6 the ratio parameter is 30 and h 0 is 40. All instances cover 30 time periods. The stock costs are h t = 10 for t = 1; n. The unit production cost p t varies uniformly in the range [0 20] and the demands d t in the range [1 19]. The Separation Heuristic. Given a fractional point (x ) the problem is to nd an inequality of the form (4) cutting o the point. The rst step is to nd an inequality of the form (2) The most important decision is the choice of the lifting set T , and a secondary question of interest is ....

I. Barany, T.J. Van Roy and L.A. Wolsey, Uncapacitated lot sizing : the convex hull of solutions, Mathematical Programming Study 22, 32-43 (1984).

....at 50 and the initial stock cost h0 at 60, while for sets 4, 5 and 6 the ratio parameter is 30 and h0 is 40. All instances cover 30 time periods. The stock costs are ht = 10 for t = 1, n. The unit production cost Pt varies uniformly in the range [0 20] and the demands dt in the range [1 19]. The Separation Heuristic. Given a fractional point (x ,y , s ) the problem is to find an inequality of the form (4) cutting off the point. The first step is to find an inequality of the form (2) The most important decision is the choice of the lifting set T, and a secondary question of ....

I. Barany, T.J. Van Roy and L.A. Wolsey, Uncapacitated lot sizing: the convex hull of solutions, Mathematical Programming Study 22, 32-43 (1984).

....and certain properties of optimal solutions were presented for WW costs. Almost thirty years later Wagelmans et al. 8] see also [1, 4] showed that the running time of the dynamic programming algorithm could be reduced to O(n log n) in the general case, and to O(n) with WW costs. Barany et al. [3] presented a description of the convex hull of solutions in the general case, and Pochet and Wolsey [5] showed how an alternative formulation leads to a simpler polyhedral description of the convex hull in the presence of WW costs. Aghezzaf and Wolsey [2] showed that the number of set ups in two ....

I. Barany, T.J. Van Roy and L.A. Wolsey, Uncapacitated lot sizing : the convex hull of solutions, Mathematical Programming Study 22, 32-43 (1984).

....algorithm and primal heuristics, and very recently Cruz et al. 12] report results obtained with a branch and bound algorithm also based on a Lagrangian relaxation. The original goal of this study was to test whether the e#ectiveness of cutting planes in solving uncapacitated lot sizing problems [4], 40] extended to the more general and more di#cult UFC. In particular we were interested in how the prototype branch and cut system bc opt [11] which among others generates path inequalities for fixed charge path networks generalizing the lot sizing inequalities) behaved on such problems. ....

Barany I., van Roy T.J., and Wolsey L.A. Uncapacitated lot-sizing: The convex hull of solutions. Math. Programming Stud., 22:32--43, 19984.

.... in this section, consider the small example shown in Figure 5, where d = 3; 2; 4; 1) and u = 1; 1; 1; 1) x 1 x 2 x 3 x 4 1 v 1 2 3 4 v 2 v 3 v 4 3 2 4 1 1 2 3 4 Figure 5: Small example Examining periods 3 and 4, the in ow out ow inequalities from [12] or the (l; S) inequality of [1] with l = 4, S = f3; 4g give the valid inequality x 3 x 4 5y 3 1y 4 v 3 v 4 4 where the coecient (d 3 d 4 ) of y 3 is the amount of in ow in x 3 that could escape through the demand nodes d 3 , d 4 , and not through the arcs v 3 , v 4 or 4 . However the above inequality does ....

I. Barany, T.J. Van Roy and L.A. Wolsey, Uncapacitated lot sizing: the convex hull of solutions, Mathematical Programming Study 22, 3243 (1984)

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I. Barany, T. Van Roy, and L. A. Wolsey. Uncapacitated lot-sizing: The convex hull of solutions. Mathematical Programming Study, 22:32--43, 1984.

No context found.

I. Barany, T. J. Van Roy, and L. A. Wolsey. Uncapacitated lot sizing: The convex hull of solutions. Mathematical Programming Study, 22:32--43, 1984.

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Combinatorica 6, 221-233. Barany, I., T. J. Van Roy, and L. A. Wolsey. 1984. Uncapacitated lot-sizing: the convex hull of solutions. Mathematical Programming Study 22, 32-43.

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Combinatorica 6, 221-233. Barany, I., T. J. Van Roy, and L. A. Wolsey. 1984. Uncapacitated lot-sizing: the convex hull of solutions. Mathematical Programming Study 22, 32-43.

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