In this paper, we study two-period subproblems proposed by Akartunali et al. (2015) for lot-sizing problems with big-bucket capacities and nonzero setup times, complementing our previous work investigating the special case of zero setup times. In particular, we study the polyhedral structure of the mixed integer sets related to various two-period relaxations. We derive several families of valid inequalities and investigate their facet-defining conditions. We also discuss the separation problems associated with these valid inequalities

Akartunali, Kerem

Doostmohammadi, Mahdi

Fragkos, Ioannis

University of Strathclyde Institutional Repository

Strathprints Institutional RepositoryAkartunali, Kerem and Doostmohammadi, Mahdi and Fragkos, Ioannis (2015) A theoretical study of two-period relaxations for lot-sizing problems with big-bucket capacities. In: International Workshop on Lot-Sizing, 2015-08-24 - 2015-08-26, University of Montreal, HEC. , This version is available at http://strathprints.strath.ac.uk/57489/Strathprints is  designed  to  allow  users  to  access  the  research  output  of  the  University  of Strathclyde. Unless otherwise explicitly stated on the manuscript, Copyright © and Moral Rights for the papers on this site are retained by the individual authors and/or other copyright owners. Please check the manuscript for details of any other licences that may have been applied. You may  not  engage  in  further  distribution  of  the  material  for  any  profitmaking  activities  or  any commercial gain. You may freely distribute both the url (http://strathprints.strath.ac.uk/) and the content of this paper for research or private study, educational, or not-for-profit purposes without prior permission or charge. Any  correspondence  concerning  this  service  should  be  sent  to  Strathprints  administrator: strathprints@strath.ac.ukA Theoretical Study of Two-Period Relaxations forLot-Sizing Problems with Big-Bucket CapacitiesKerem AkartunalıDept. of Management Science, University of Strathclyde, Glasgow, UKkerem.akartunali@strath.ac.ukMahdi DoostmohammadiDept. of Management Science, University of Strathclyde, Glasgow, UKmahdi.doostmohammadi@strath.ac.ukIoannis FragkosHEC Montre´al, Canadaioannis.fragkos@hec.caAbstractIn this paper, we study two-period subproblems proposed by [1] for lot-sizingproblems with big-bucket capacities and nonzero setup times, complementingour previous work [3] investigating the special case of zero setup times. Inparticular, we study the polyhedral structure of the mixed integer sets relatedto various two-period relaxations. We derive several families of valid inequalitiesand investigate their facet-defining conditions. We also discuss the separationproblems associated with these valid inequalities.1 IntroductionIn this study, we investigate multi-item production planning problems with big bucketcapacities, i.e., each resource is shared by multiple items, which can be producedin a specific time period. These real-world problems are very interesting, as theyremain challenging to solve to optimality and also to achieve strong bounds. Theuncapacitated and single-item relaxations of the problem have been previously studiedby [7]. The work of [6] introduced and studied the single-period relaxation with“preceding inventory”, where a number of cover and reverse cover inequalities aredefined for this relaxation. Finally, we also note the relevant study of [5], who studieda single-period relaxation and compared with other relaxations.We present a formulation for this problem following the notation of [2]. Let NT ,NI and NK indicate the number of periods, items, and machine types, respectively.We represent the production, setup, and inventory variables for item i in period t byxit, yit, and sit, respectively. We note that our model can be generalized to involvemultiple levels as in [1], however, we omit this for the sake of simplicity.1minNT∑t=1NI∑i=1f ityit +NT∑t=1NI∑i=1hitsit (1)s.t. xit + sit−1 − sit = dit t ∈ {1, . . . , NT}, i ∈ {1, . . . , NI} (2)NI∑i=1(aikxit + STikyit) ≤ Ckt t ∈ {1, . . . , NT}, k ∈ {1, . . . , NK} (3)xit ≤Mityit t ∈ {1, . . . , NT}, i ∈ {1, . . . , NI} (4)y ∈ {0, 1}NTxNI ; x, s ≥ 0 (5)The objective function (1) minimizes total cost, where f it and hit indicate the setupand inventory cost coefficients, respectively. The flow balance constraints (2) ensurethat the demand for each item i in period t, denoted by dit, is satisfied. The big bucketcapacity constraints (3) ensure that the capacity Ckt of machine k is not exceeded intime period t, where aik and STik represent the per unit production time and setuptime for item i, respectively. The constraints (4) guarantee that the setup variable isequal to 1 if production occurs, where M it represents the maximum number of itemi that can be produced in period t, based on the minimum of remaining cumulativedemand and capacity available. Finally, the integrality and non-negativity constraintsare given by (5).2 Two-Period RelaxationLet I = {1, . . . , NI}. We present the feasible region of a two-period, single-machinerelaxation of the multi-item production planning problem, denoted by X2PL (see [1]for details).xit′ ≤ M˜it′yit′ i ∈ I, t′ = 1, 2 (6)xit′ ≤ d˜it′yit′ + si i ∈ I, t′ = 1, 2 (7)xi1 + xi2 ≤ d˜i1yi1 + d˜i2yi2 + si i ∈ I (8)xi1 + xi2 ≤ d˜i1 + si i ∈ I (9)∑i∈I(aixit′ + STiyit′) ≤ C˜t′ t′ = 1, 2 (10)x, s ≥ 0, y ∈ {0, 1}2×NI (11)Since we consider a single machine, we dropped the k index from this formulation,however, all parameters are defined in the same lines as before. The obvious choice2for the horizon would be t+1, in which case the definition of the parameter M˜ it′ is thesame as of the basic definition of M it+t′−1, for all i and t′ = 1, 2. Similarly, capacityparameter C˜t′ is the same as Ct+t′−1, for all t′ = 1, 2. Cumulative demand parameterd˜it′ represents simply dit+t′−1, t+1, for all i and t′ = 1, 2, i.e., d˜i1 = di1,2 and d˜i2 = di2. Wenote the following polyhedral result for X2PL from [1].Proposition 2.1 Assume that M˜ it > 0, ∀t ∈ {1, . . . , NT}, ∀i ∈ {1, . . . , NI} andST i < C˜t, ∀t ∈ {1, . . . , NT}, ∀i ∈ {1, . . . , NI}. Then conv(X2PL) is full-dimensional.For the sake of simplicity, we will drop subscript t and symbol ˜ in the followingnotations. In this paper, we investigate the case of ai = 1, ∀i ∈ {1, . . . , NI} withnonzero setups. We establish two relaxations of X2PL and study their polyhedralstructures. For a given t, we define the first relaxation of X2PL, denoted by LR1, asset of (x, y) ∈ RNI × ZNI satisfyingxi ≤M iyi, i ∈ INI∑i=1(xi + ST iyi) ≤ Cxi ≥ 0, yi ∈ {0, 1}, i ∈ INext, we present a result from the literature [4] concerning this relaxation.Definition 2.1 Let S1 ⊆ I and S2 ⊆ I such that S1 ∩ S2 = ∅. We say that (S1, S2)is a generalized cover of I if∑i∈S1(M i + ST i) +∑i∈S2ST i − C = δ > 0.Proposition 2.2 (see [4]) Let (S1, S2) be a generalized cover of I, and let L1 ⊆I\(S1 ∪ S2) and L2 ⊆ I\(S1 ∪ S2) such that L1 ∩ L2 = ∅. Then,∑i∈S1∪L1xi +∑i∈S1∪S2∪L1∪L2ST iyi −∑i∈S1(M i + ST i − δ)+yi −∑i∈S2(ST i − δ)+yi−∑i∈L1(qi − δ)yi −∑i∈L2(STi− δ)yi ≤ C −∑i∈S1(M i + ST i − δ)+ −∑i∈S2(ST i − δ)+is valid for LR1, where A ≥ max(maxi∈S1(Mi+ST i),maxi∈S2 STi, δ), qi = max(A,M i+ST i), and STi= max(A, ST i).For a given t, second relaxation of X2PL, denoted by LR2, can be defined as theset of (x, y, s) ∈ RNI × ZNI × RNI satisfyingxi ≤M iyi, i ∈ Ixi ≤ diyi + si, i ∈ INI∑i=1(xi + ST iyi) ≤ Cxi ≥ 0, yi ∈ {0, 1}, si ≥ 0, i ∈ I3In this talk, we will present the trivial facet-defining inequalities for LR2, andthen derive several classes of valid inequalities such as cover and partition inequali-ties. We will also present item- and period-extended versions of some of these familiesof inequalities, and we will establish facet-defining conditions for all families of in-equalities. We will also discuss the separation problems associated with these validinequalities.References[1] K. Akartunalı, I. Fragkos, A.J. Miller and T. Wu, Local cuts and two-periodcovex hull closures for big bucket lot-sizing problems, Technical Report, Dept.of Management Science, University of Strathclyde, Available at Optimiza-tion Online, http://www.optimization-online.org/DB_HTML/2014/07/4423.html (2014).[2] K. Akartunalı and A.J. Miller, A computational analysis of lower bounds for bigbucket production planning problems, Computational Optimization and Appli-cations, 53, 729-753 (2012).[3] M. Doostmohammadi and K. Akartunalı, A Polyhedral Study of Two-PeriodRelaxations for Big-Bucket Lot-Sizing Problems: Zero Setup Case, TechnicalReport, Dept. of Management Science, University of Strathclyde, Available atOptimization Online, http://www.optimization-online.org/DB_HTML/2015/02/4767.html (2015).[4] M. Goemans, Valid ineqialites and separation for mixed 0-1 constraints withvariable upper bounds, Operations Research Letters, 8, 315-322 (1989).[5] R. Jans and Z. Degraeve, Improved lower bounds for the capacitated lot sizingproblem with setup times, Operations Research Letters, 32, 185-195 (2004).[6] A.J. Miller, G.L. Nemhauser and M.W.P. Savelsbergh, Solving multi-item capac-itated lot-sizing problems with setup times by branch-and-cut, Technical reportCORE DP 2000/39, Universite´ Catholique de Louvain, Louvain-la-Neuve (2000).[7] Y. Pochet, Valid inequalities and separation for capacitated economic lot-sizing,Operations Research Letters, 7, 109-116 (1988).4

A theoretical study of two-period relaxations for lot-sizing problems with big-bucket capacities

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A theoretical study of two-period relaxations for lot-sizing problems with big-bucket capacities

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