FACETS FOR CONTINUOUS MULTI-MIXING SET AND ITS GENERALIZATIONS: STRONG CUTS FOR MULTI-MODULE CAPACITATED LOT-SIZING PROBLEM.

Transcription

1 FACETS FOR CONTINUOUS MULTI-MIXING SET AND ITS GENERALIZATIONS: STRONG CUTS FOR MULTI-MODULE CAPACITATED LOT-SIZING PROBLEM A Dssertaton by MANISH BANSAL Submtted to the Offce of Graduate and Professonal Studes of Texas A&M Unversty n partal fulfllment of the requrements for the degree of DOCTOR OF PHILOSOPHY Char of Commttee, Commttee Members, Head of Department, Kavash Kanfar Sergy Butenko Wlbert E. Wlhelm Anxao Jang César O. Malavé December 2014 Major Subject: Industral Engneerng Copyrght 2014

2 ABSTRACT The research objectve of ths dssertaton s to develop new facet-defnng vald nequaltes for several new mult-parameter mult-constrant mxed nteger sets. These vald nequaltes result n cuttng planes that sgnfcantly mprove the effcency of algorthms for solvng mxed nteger programmng (MIP) problems nvolvng multmodule capacty constrants. These MIPs arse n many classcal and modern applcatons rangng from producton plannng to cloud computng. The research n ths dssertaton generalzes cut-generatng methods such as mxed nteger roundng (MIR), mxed MIR, contnuous mxng, n-step MIR, mxed n-step MIR, mglng, and n-step mnglng, along wth varous well-known famles of cuts for problems such as mult-module capactated lot-szng (MMLS), mult-module capactated faclty locaton (MMFL), and mult-module capactated network desgn (MMND) problems. More specfcally, n the frst step, we ntroduce a new generalzaton of the contnuous mxng set, referred to as the contnuous mult-mxng set, where the coeffcents satsfy certan condtons. For each n {1,..., n}, we develop a class of vald nequaltes for ths set, referred to as the n -step cycle nequaltes, and present ther facet-defnng propertes. We also present a compact extended formulaton for ths set and an exact separaton algorthm to separate over the set of all n -step cycle nequaltes for a gven n {1,..., n}. In the next step, we extend the results of the frst step to the case where condtons on the coeffcents of the contnuous mult-mxng set are relaxed. Ths leads to an extended formulaton and a generalzaton of the n-step cycle nequaltes, n N, for the contnuous mult-mxng set wth general coeffcents. We also show that these nequaltes are facet-defnng n many cases.

3 In the thrd step, we further generalze the contnuous mult-mxng set (where no condtons are mposed on the coeffcents) by ncorporatng upper bounds on the nteger varables. We ntroduce a compact extended formulaton and new famles of mult-row cuts for ths set, referred to as the mngled n-step cycle nequaltes (n N), through a generalzaton of the n-step mnglng. We also provde an exact separaton algorthm to separate over a set of all these nequaltes. Furthermore, we present the condtons under whch a subset of the mngled n-step cycle nequaltes are facet-defnng for ths set. Fnally, n the fourth step, we utlze the results of frst step to ntroduce new famles of vald nequaltes for MMLS, MMFL, and MMND problems. Our computatonal results show that the developed cuts are very effectve n solvng the MMLS nstances wth two capacty modules, resultng n consderable reducton n the ntegralty gap, the number of nodes, and total soluton tme.

4 DEDICATION To My Teachers v

5 ACKNOWLEDGEMENTS The research n ths dssertaton was supported by the Natonal Scence Foundaton grant CMMI , whch s greatfully acknowledged. Ths research would not have been possble wthout the gudance, support, and sacrfce of varous people; and I would lke to thank all of them. A strong OR group and affordable tuton prompted me to jon MS program at Texas A&M Unversty where I began my research journey under the gudance of Dr. Kavash Kanfar. The frst lesson he taught me was how to fnd a research problem. I must admt, n the begnnng t was not easy to thoroughly understand the algorthms and mathematcal models proposed n varous papers. But Dr. Kanfar used to tell me that the joy of achevements and the enlghtenment that followed after crossng the barrer wll be worth the pan [Dr. Kanfar, I concur and wll never forget about t!]. Hs passon for research and teachng motvated me to apply for the Ph.D. program and Dr. Kanfar gladly accepted to be my advsor. He ntroduced me to the communty where the culture of exploraton and dscoverng the unknown are the essence of human lves. Most mportantly, hs joval, motvatng, and postve atttude n every meetng made me to enjoy the research. As a result, I always felt refreshed and charged at the end of each meetng. I aspre to be a teacher and researcher lke hm; and lead a cheerful work-lfe wth lessons learned from my experence wth hm. I wsh to thank my commttee members and teachers, Dr. Wlbert Wlhelm, Dr. Sergy Butenko, and Dr. Anxo Jang for ther valuable nputs n my dssertaton and teachng me (mxed nteger) lnear programmng, non-lnear programmng, and analyss of algorthms, respectvely. Dr. Wlhelm s devoton toward academc prov

6 fesson, Dr. Butenko s encouragement for theoretcal research, and Dr. Jang s art of teachng algorthms wll always reman my sources of nspraton. In addton, I would lke to thank Dr. Amarnath Banerjee for fnancally supportng me durng the frst year of my Ph.D. program and sharng hs valuable experence n academa. Also, I wsh to thank to my department faculty and staff, external faculty, campus employers, and my frends (Abhlasha, Anurag, Arupa, Avnash, Chtra, Dvya, Gokhan, Krshna, Mansh Jan, Mchelle, Samyukta, Saurav, Sujeev, Vatsal, and many more) for makng my tme at Texas A&M Unversty a great experence. Ths thess s dedcated to all my teachers, partcularly to my frst teacher, Mrs. Raksha Bansal (my mother). She helps me to mantan a constant focus for my educaton and career development. It s also dedcated to my father, Mr. Yashpal Bansal (an ndustralst), who teaches me the lessons of hard work and honesty. Snce my chldhood, my father has been sharng our famly s entrepreneural experences (both falures and successes) wth me. At a very early age, my parents ntroduced me to a work envronment where I saw people developng effcent methods to solve a wde varety of ndustral problems, and processes to acheve hgher productvty and hence attan a compettve advantage. I am ndebted and eternally grateful to my parents. And now, last but not the least, I wsh to acknowledge the moral support from my younger brother (Aayush Bansal) whose work-ethcs have always been my source of nspraton. I aspre to be dedcated, focused, and hard-workng lke hm. v

7 TABLE OF CONTENTS Page ABSTRACT DEDICATION v ACKNOWLEDGEMENTS v TABLE OF CONTENTS v LIST OF FIGURES x LIST OF TABLES x I. INTRODUCTION I.1 Contnuous Mult-Mxng Set I.2 Contnuous Mult-Mxng Set wth General Coeffcents I.3 Contnuous Mult-Mxng Set wth Bounded Integer Varables... 8 I.4 Cuts for MMLS, MMFL, and MMND Problems I.4.1 Computatonal Results I.5 Dssertaton Structure II. MIXED INTEGER PROGRAMMING, POLYHEDRAL THEORY, AND GENERALIZATIONS OF MIXED INTEGER ROUNDING II.1 Mxed Integer Programmng II.1.1 Some Defntons and Theoretcal Results n Polyhedral Theory 14 II.1.2 Algorthms for Solvng MIP Problems II.2 Generalzatons of Mxed Integer Roundng II.2.1 Mxed Integer Roundng (MIR) II.2.2 Contnuous Mxng II.2.3 n-step MIR Inequaltes II.2.4 n-step Mnglng Inequaltes II.2.5 Mxed n-step MIR Inequaltes v

8 III. CONTINUOUS MULTI-MIXING SET III.1 Vald Inequaltes and Extended Formulaton III.2 Facet-Defnng n-step Cycle Inequaltes III.3 Separaton Algorthm IV. CONTINUOUS MULTI-MIXING SET WITH GENERAL COEFFICIENTS IV.1 Vald Inequaltes and Extended Formulaton IV.2 Facet-Defnng n-step Cycle Inequaltes V. CONTINUOUS MULTI-MIXING SET WITH GENERAL COEFFICIENTS AND BOUNDED INTEGER VARIABLES V.1 Vald Inequaltes and Extended Formulaton V.2 Facet-Defnng Mngled n-step Cycle Inequaltes VI. CUTS FOR MMLS, MMFL, AND MMND PROBLEMS VI.1 Cuts for Mult-Module Capactated Lot-Szng Problem VI.2 Cuts for Mult-Module Capactated Faclty Locaton Problem VI.3 Cuts for Mult-Module Capactated Network Desgn Problem VI.4 Computatonal Results VII. CONCLUSION AND FUTURE RESEARCH VII.1 Concluson VII.2 Future Plans REFERENCES v

9 LIST OF FIGURES FIGURE Page 1 Generalzatons of Mxed Integer Roundng (MIR) Each cycle n graph G gves rse to a cycle nequalty x

10 LIST OF TABLES TABLE Page 1 Relaton between known nequaltes and procedures n lterature Results of computatonal experments on 2ML-WB nstances Results of computatonal experments on 2ML-B nstances x

11 CHAPTER I INTRODUCTION Mxed nteger programmng (MIP) s a major optmzaton technque to solve a wde varety of real-world problems nvolvng decsons of dscrete nature [81, 111]. In general, MIPs are NP-hard to solve [43]. The branch-and-cut algorthm [83] s among the most successful algorthms used to solve MIPs. Branch-and-cut s a branch-andbound algorthm [67, 81] n whch cuttng planes are used to tghten the formulatons of node problems and hence acheve better bounds (refer to Secton II.1.2 for detals). As a result, developng strong vald nequaltes as cuttng planes s crucal for effectveness of the branch-and-cut algorthm. To ths end, studyng the polyhedral structure of mxed nteger base sets whch consttute well-structured relaxatons of mportant MIP problems s a promsng approach. Ths s because oftentmes one can develop procedures n whch the vald nequaltes (or facets) developed for the base set are used to generate vald nequaltes (or facets) for the orgnal MIPs (see [6, 16, 15, 14, 36, 51, 62, 96, 111] for a few examples among many others). Mxed nteger roundng (MIR) [82, 111] s one of the most basc procedures for dervng cuts for MIPs whch utlzes the facet of a sngle-constrant two-varable mxed nteger base set. Several mportant generalzatons of MIR (shown n Fg. 1), ncludng mxed MIR [51], contnuous mxng [105], n-step MIR [62], mnglng [6], mxed n-step MIR [96], and n-step mnglng [7], are derved by studyng the polyhedral structure of more complex mxed nteger base sets (see Sectons I.1, I.2, and I.3 for detals). *Some parts of ths chapter are reprnted wth permsson from n-step cycle nequaltes: facets for contnuous n-mxng set and strong cuts for mult-module capactated lot-szng problem by Mansh Bansal and Kavash Kanfar, Integer Programmng and Combnatoral Optmzaton Conference, Lecture Notes n Computer Scence, 8494, , Copyrght 2014 by Sprnger. 1

12 Number of constrants n the base polyhedron ( ) Number of nteger varables n each constrant of the base polyhedron ( ) MIR [82, 111] (Nemhauser and Wolsey, 1990; Wolsey, 1998) Mnglng [6] (Atamtürk and Günlük, 2010) n-step MIR [62] (Kanfar and Fath, 2009) n-step mnglng [7] (Atamtürk and Kanfar, 2012) Mxed MIR [51] (Günlük and Pochet, 2001) Mxed n-step MIR [96] (Sanjeev and Kanfar, 2012) Contnuous mxng [105] (Van Vyve, 2005) Contnuous mult-mxng [14,15] Contnuous mult-mxng wth general coeffcents and bounded nteger varables [16] (Ths Dssertaton) Fgure 1: Generalzatons of Mxed Integer Roundng (MIR) Many well-known famles of vald nequaltes developed for MIP problems such as knapsack set, lot-szng (producton plannng), faclty locaton, and network desgn, are (or can be) derved usng MIR and ts aforementoned generalzatons (see Table 1 for detals). As shown n Fgure 1, n ths dssertaton, we generalze the aforementoned cut-generatng procedures by developng facet-defnng vald nequaltes for the followng generalzatons of the well-studed contnuous mxng set [105] (a sngleparameter mult-constrant mxed nteger set): (1) Contnuous mult-mxng set (a mult-parameter mult-constrant mxed nteger set) wth certan condtons on the coeffcents, (2) Contnuous mult-mxng set wth general coeffcents, and (3) Contnuous mult-mxng set wth general coeffcents and bounded nteger varables. We also present compact extended formulatons for these sets and an exact separaton algorthm to separate over each famly of vald nequaltes developed for these sets (see Sectons I.1, I.2, and I.3 for detals). These results provde a knowledge base for developng new famles of cuttng planes for MIP problems nvolvng multmodularty capacty constrants (MMCCs). Exstence of multple modulartes (module szes) of (producton/servce/process- 2

13 Problem type Inequaltes n lterature Are/can be developed by Contnuous cover [73] 2-step mnglng Knapsack Set Cover and pack [10, 11] 2-step mnglng n-step mnglng [6, 7] n-step mnglng (k, l, S, I) [87] Mxng Lot-Szng Mxed (k, l, S, I) [51] Mxed MIR Mult-module (k, l, S, I) [96] Mxed n-step MIR Flow cover [84] MIR Arc resdual [68] MIR Faclty Locaton (k, l, S, I) [2, 3, 1] Mxed MIR Mxed (k, l, S, I) [51] Mxed MIR Mult-module (k, l, S, I) Mxed n-step MIR (2-Modularty) cut-set [70] (2-step) MIR Flow cut-set [19] MIR Network Desgn Cut-set [9] MIR Mxed partton [52] Mxed MIR Partton [89] n-step MIR Table 1: Relaton between known nequaltes and procedures n lterature ng/transmsson/transportaton/storage/power generaton) capacty s nherent to many classcal and modern applcatons. One can easly fnd evdence of ths fact n the lterature of applcatons such as data centers [58, 97, 110, 114], cloud computng [27, 47, 55], (survvable fber-optc) communcaton networks [8, 17, 18, 19, 20, 32, 48, 49, 50, 52, 72, 115], batteres for electrc vehcles/wnd turbnes/solar panels [23, 38, 46, 64, 101], semconductor manufacturng [44, 53, 54, 60, 91], power/energy/smart grd systems [40, 57, 86, 104, 117], on-shore and off-shore constructon n ol ndustry [41, 80], offshore natural gas/ol ppelne systems [22, 69, 93, 94], pharmaceutcal manufacturng facltes [98, 102, 103], regonal wastewater treatment systems [56], chemcal processes [95], boreactors [109], transportaton systems [4, 42, 65, 66, 76, 85, 107, 108], and producton systems [90]. Nevertheless, the MIP cuttng plane lterature to date has almost entrely focused on problems wth sngle-modularty capacty 3

14 constrants. We ntroduce new classes of mult-row cuts for the MIP problems wth MMCCs, n partcular mult-module capactated lot-szng (MMLS), mult-module capactated faclty locaton (MMFL), and mult-module capactated network desgn (MMND). These nequaltes generalze varous well-known famles of cuts (mentoned n Table 1) for MMLS, MMFL, and MMND problems. Our computatonal results show that these cuttng planes sgnfcantly mprove the effcency of algorthms for solvng the MMLS problem wth(out) backloggng. See Secton I.4 for detals. In the followng sectons, we present bref summary of our research contrbuton. I.1 Contnuous Mult-Mxng Set A well-known mxed nteger base set s the contnuous mxng set Q := {(y, v, s) Z m R m+1 + : y + v + s β, = 1,..., m}, where β R, = 1,..., m [105]. Ths set s a generalzaton of the well-studed mxng set {(y, s) Z m R + : y + s β, = 1,..., m} [51], whch tself s a mult-constrant generalzaton of the base set {(y, s) Z R + : y + s β} that leads to the well-known mxed nteger roundng (MIR) nequalty (page 127 of [111]). In all these base sets each constrant has only one nteger varable. Fg. 1 presents a summary of the generalzaton relatonshp between these base sets and other base sets of nterest n ths dssertaton. The set Q arses as a substructure n relaxatons of problems such as lot-szng (producton plannng) wth backloggng [78], lot-szng wth stochastc demand [5], capactated faclty locaton [2], and capactated network desgn [50]. Mller and Wolsey [77] presented an extended formulaton for conv(q) wth O(m 2 ) varables and O(m 2 ) constrants. Later, Van Vyve [105] gave a compact and tght extended formulatons wth O(m) varables and O(m 2 ) constrants for 4

15 conv(q) and ts relaxaton to the case where s R. He also ntroduced the so-called cycle nequaltes (called 1-step cycle nequaltes n ths dssertaton) for these sets and showed that these nequaltes along wth bound constrants are suffcent to descrbe the convex hulls of these sets. The MIR nequaltes (called 1-step MIR nequaltes n ths dssertaton) of Nemhauser and Wolsey [82, 111] and the mxed (1-step) MIR nequaltes of Günlük and Pochet [51] are specal cases of the 1-step cycle nequaltes for Q (Fg. 1). It s mportant to note that the 1-step MIR cuts are equvalent to splt cuts of Cook et al. [31] and Gomory mxed nteger cuts [92], and are a specal case of the dsjunctve cuts [12, 13] (also see [21, 37]). Zhao and Faras [116] showed that the optmzaton over the relaxaton of Q n whch s R can be performed n O(m log m) tme. Furthermore, Confort et al. [30] studed two generalzatons of Q: frst, the ntersecton of several contnuous mxng sets wth dstnct s varables and common y and v varables, and second, the contnuous mxng set wth flows. They ntroduced two extended formulatons for the convex hull of each of these sets. In another drecton (Fg. 1), Kanfar and Fath [62] generalzed the 1-step MIR nequaltes [82] and developed the n-step MIR nequaltes for the mxed nteger knapsack set by studyng the base set { Q 1,n 0 = (y, s) Z Z n1 + R + : n t=1 } α t y t + s β, where α t R + \{0}, t = 1,..., n and β R. Note that ths base set has a sngle constrant and n nteger varables n ths constrant. The n-step MIR nequaltes are vald and facet-defnng for the base set Q 1,n 0 f α t s and β satsfy the so-called 5

16 n-step MIR condtons,.e. α t β (t1) /α t αt1, t = 2,..., n. (1) However, n-step MIR nequaltes can also be generated for a mxed nteger constrant wth no condtons mposed on the coeffcents. In that case, the external parameters used n generatng the nequalty are pcked such that they satsfy the n-step MIR condtons (see [62] for more detals). The n-step MIR nequaltes are facet-defnng for the mxed nteger knapsack set n many cases [7, 62]. The Gomory mxed nteger cut [92] and the 2-step MIR nequaltes [35, 36] are the specal cases of n-step MIR nequaltes, correspondng to n = 1, 2, respectvely. Kanfar and Fath [62, 63] showed that the n-step MIR nequaltes defne new famles of facets for the fnte and nfnte group problems. Recently, Sanjeev and Kanfar [96] showed that the procedure proposed by Günlük and Pochet [51] to mx 1-step MIR nequaltes can be generalzed and used to mx the n-step MIR nequaltes [62] (Fg. 1). As a result, they developed the mxed n-step MIR nequaltes for a generalzaton of the mxng set called the n- mxng set,.e. { Q m,n 0 = (y, s) (Z Z+ n1 ) m R + : n } α tyt + s β, = 1,..., m, t=1 where α t R + \{0}, t = 1,..., n, and β R, = 1,..., m, such that α t and β satsfy the n-step MIR condtons n each constrant. Note that ths s a mult-constrant base set wth n nteger varables n each constrant and a contnuous varable whch s common among all constrants. The mxed n-step MIR nequaltes are vald for Q m,n 0 and under certan condtons, these nequaltes are also facet defnng for the 6

17 convex hull of Q m,n 0. In the frst step of ths dssertaton, we generalze the concepts of contnuous mxng [105] and mxed n-step MIR [96] by ntroducng a more general base set referred to as the contnuous mult-mxng set whch we defne as Q m,n := { (y, v, s) (Z Z n1 + ) m R m+1 + : n } α tyt + v + s β, = 1,..., m, t=1 where α t > 0, t = 1,..., n and β R, = 1,..., m such that α t and β satsfy the n- step MIR condtons (whch are automatcally satsfed f the parameters α 1,..., α n are dvsble) n each constrant (see Fg. 1). Note that ths set has multple (m) constrants wth multple (n) nteger varables n each constrant; but t s more general than the n-mxng set because n addton to the common contnuous varable s, each constrant has a contnuous varable v of ts own. The contnuous mxng set Q s the specal case of Q m,n, where n = 1 and α 1 = 1, and the n-mxng set of Sanjeev and Kanfar [96] s the projecton of Q m,n {v = 0} on (y, s). The contnuous mult-mxng set arses as a substructure n relaxatons of mult-module capactate lot-szng (MMLS) wth(out) backloggng, MMLS wth stochastc demand, mult-module capactated faclty locaton (MMFL), and mult-module capactated network desgn (MMND) problems (we wll descrbe these problems n Secton I.4). For each n {1,..., n}, we develop a class of vald nequaltes for Q m,n whch we refer to as n -step cycle nequaltes, and obtan condtons under whch these nequaltes are facet-defnng for conv(q m,n ). We dscuss how the n-step MIR nequaltes [62] and the mxed n-step MIR nequaltes [96] are specal cases of the n-step cycle nequaltes. We also ntroduce a compact extended formulaton for Q m,n and an effcent exact separaton algorthm to separate over the set of all n -step cycle nequaltes, n {1,..., n}, for set Q m,n. 7

18 I.2 Contnuous Mult-Mxng Set wth General Coeffcents In the next step, we relax the n-step MIR condtons on the coeffcents of Q m,n and consder the contnuous mult-mxng set wth general coeffcents, denoted by Y m := { (y, v, s) Z m N + R m + R + : N t=1 } a t yt + v + s b, = 1,..., m where a R mn and b R m. As mentoned before, Kanfar and Fath [62] showed that, for each n N, the n-step MIR facet of Q 1,n 0 can be used to generate a famly of vald nequaltes for the mxed nteger knapsack set whch s same as P roj y,s (Y 1 {v = 0}). Later Atamtürk and Kanfar [7] showed that these nequaltes defne facets for ths set under certan condtons. In ths dssertaton, we generalze the n-step cycle nequaltes to develop vald nequaltes for Y m and show that they are facet-defnng for conv(y m ) n many cases. I.3 Contnuous Mult-Mxng Set wth Bounded Integer Varables Despte the effectveness of MIR nequaltes to solve MIPs wth unbounded nteger varables, cuttng planes based on lftng technques appear to be more effectve for MIPs wth bounded nteger varables [6, 74]. Ths s because, unlke lftng technques, the MIR procedure does not explctly use bounds on nteger varables. To overcome ths drawback, Atamtürk and Günlük [6] ntroduced a smple procedure (called mnglng ) whch ncorporates the varable bound nformaton nto MIR and gves stronger vald nequaltes. They frst developed the so-called mnglng (and 2- step mnglng) nequaltes for the mxed nteger knapsack set and then showed that the facets of ths set derved earler by superaddtve lftng technques are specal cases of mnglng or 2-step mnglng nequaltes. In partcular, these nequaltes subsume the contnuous cover and reverse contnuous cover nequaltes of Marchand 8

19 and Wolsey [73] as well as the contnuous nteger knapsack cover and pack nequaltes of Atamtürk [10, 11]. Recently, Atamtürk and Kanfar [7] generalzed the mnglng procedure of Atamtürk and Günlük [6] and ntroduced a varant of the n-step MIR nequaltes [62] (whch they call n-step mnglng nequaltes) for the mxed-nteger knapsack set wth bounded nteger varables. Unlke n-step MIR nequaltes, the n-step mnglng nequaltes utlze the nformaton of bounds on nteger varables to gve stronger vald nequaltes, whch are facet-defnng n many cases [7]. In addton, they used n-step mnglng nequaltes to develop new vald nequaltes and facets based on covers and packs defned for mxed nteger knapsack sets. The thrd step of ths dssertaton s to unfy the concepts of contnuous multmxng and n-step mnglng by ncorporatng upper bounds on the nteger varables of the contnuous mult-mxng set (where no condtons are mposed on the coeffcents) and developng new famles of vald nequaltes for ths set (whch we refer to as the mngled n-step cycle nequaltes). We denote ths new generalzaton of contnuous mult-mxng set by { Z m := (y, v, s) Z m N + R m + R + : a t yt + } a k yk + v + s b, y u, = 1,..., m k K t T where (T, K) s a parttonng of {1,..., N} wth a t > 0 for t T, a k < 0 for k K, and u Z N + for {1,..., m}. We develop a compact extended formulaton for Z m and provde a separaton algorthm to separate over the set of all mngled n-step cycle nequaltes for a gven n N. Furthermore, we obtan the condtons under whch a specal case of mngled n-step cycle nequaltes (referred to as the mngled n-step mxng nequaltes) are facet-defnng for conv(z m ). 9

20 I.4 Cuts for MMLS, MMFL, and MMND Problems The objectve of ths step of dssertaton s to utlze the n-step cycle nequaltes to develop a new famly of vald nequaltes for MIPs nvolvng mult-modularty capacty constrants. In partcular, we focus on the mult-modularty generalzatons (where capacty can be composed of dscrete unts of multple dfferentally-szed modulartes) of three followng hgh-mpact classes of capactated MIPs: lot-szng (LS), faclty locaton (FL), and network desgn (ND) problems. Over the years a large volume of the MIP cuttng plane research has been dedcated to sngle modularty or constant-capacty versons of the LS [78, 87, 88, 90, 106, 112], FL [1, 2, 51], and ND [9, 26, 51, 70, 71] problems. Recently, Sanjeev and Kanfar [96] generalzed the lot-szng problem wth constant batches [87] (where the capacty n each perod can be some nteger multple of a sngle capacty module wth a gven sze) and ntroduced the mult-module capactated lot-szng (MMLS) problem. In ths problem, the total producton capacty n each perod can be the summaton of some nteger multples of several capacty modules of dfferent szes. They showed that the mxed n-step MIR nequaltes can be used to generate vald nequaltes for the MMLS problem wthout backloggng (whch we denote by MML-WB). They referred to these nequaltes as the mult-module (k, l, S, I) nequaltes. These nequaltes generalze the (k, l, S, I) nequaltes and mxed MIR nequaltes whch were ntroduced for the lot-szng problem wth constant batches by Pochet and Wolsey [87] and Günlük and Pochet [51], respectvely. Smlarly, they ntroduced mult-module capactated faclty locaton (MMFL) problem (a generalzaton of the capactated faclty locaton problem) and used mxed n-step MIR nequaltes to develop vald nequaltes for ths problem. These nequaltes generalze the mxed MIR [51] and (k, l, S, I) based [2, 3] nequaltes for 10

21 constant capacty faclty locaton problem. In lterature, the cuttng planes have been derved for mult-module capactated network desgn (MMND) problem and ts specal cases [9, 19, 52, 61, 70, 72, 89]. Interestngly, the cuts developed n [19, 70, 72] for two-modularty ND wth dvsble capactes (2MND-DC) and n [9] for MMND can be derved just usng 1-step MIR procedure. The fact that the problem s mult-modularty, s not used n developng potentally many more classes of cuts. The same s true for the mxed partton nequaltes for 2MND-DC [52], whch can be derved just usng mxed MIR procedure. To our knowledge, the only classes of cuts derved by actually explotng the exstence of multple modulartes are the two-modularty cut-set nequaltes for 3MND-DC [70] (whch do not explot the thrd modularty) and the partton nequaltes for the sngle-arc MMND-DC [89]. The former can be derved usng the 2-step MIR [36, 62], and the n-step MIR not only generates the latter but also generalzes them to non-dvsble capactes [61]. In ths dssertaton, we ntroduce MMLS wth backloggng (MML-B) and use n- step cycle nequaltes to develop a new famly of cuttng planes for MML-(W)B, MMFL, and MMND problems whch subsume vald nequaltes ntroduced n [51, 87, 96] for LS problems, [2, 51, 96] for FL problems, and [9, 19, 51, 52, 61, 70, 72, 89] for ND problems, respectvely. We also computatonally evaluate the effectveness of the n-step cycle nequaltes for the MML-(W)B problem usng our separaton algorthm. I.4.1 Computatonal Results Our computatonal results on applyng 2-step cycle nequaltes usng our separaton algorthm show that our cuts are very effectve n solvng MML-WB and MML-B wth two capacty modules, resultng n consderable reducton n the ntegralty gap 11

22 (on average 85.90% for MML-WB and 86.32% for MML-B) and the number of nodes (on average 132 tmes for MML-WB and 31 tmes for MML-B). Also, the total tme taken to solve an nstance (whch also ncludes the cut generaton tme) s n average 58.3 tmes (for MML-WB) and 9.9 tmes (for MML-B) smaller than the tme taken by CPLEX wth default settngs (except for very easy nstances). More nterestngly, n these nstances addng cuts by applyng 2-step cycle nequaltes over 1-step cycle nequaltes has mproved the closed gap (on average 19.47% for MML-WB and 15.96% for MML-B), the number of nodes (on average 43 tmes for MML-WB and 14 tmes for MML-B), and the total soluton tme (on average 18 tmes for MML-WB and 4 tmes for MML-B). I.5 Dssertaton Structure The dssertaton s organzed as follows: In Chapter II, we present a bref ntroducton to mxed nteger programmng and revew some fundamental defntons, concepts, and theorems n MIP and polyhedra to the extent requred as background for the results n ths dssertaton. We present our research on contnuous multmxng set, contnuous mult-mxng set wth general coeffcents, contnuous multmxng set wth bounded nteger varables, and cuts for MMLS, MMFL, and MMND problems n Chapters III, IV, V, and VI, respectvely. We provde a concluson n Chapter VII along wth some future research plans. 12

23 CHAPTER II MIXED INTEGER PROGRAMMING, POLYHEDRAL THEORY, AND GENERALIZATIONS OF MIXED INTEGER ROUNDING Ths chapter presents an ntroducton to mxed nteger programmng and a theory of vald nequaltes for mxed nteger lnear sets to the extent requred as background for the results n ths dssertaton. In Secton II.1, we defne general (mxed) nteger program, brefly dscuss ther mportance and applcatons, and revew three algorthms used to solve them (.e. branch-and-bound, cuttng plane, and branchand-cut algorthms). We also reproduce the concept of extended formulaton along wth some fundamental defntons and theorems n polyhedral theory. In Secton II.2, we revew the MIR cut-generatng procedure [81, 111] and ts varous generalzatons (n partcular, contnuous mxng [105], n-step MIR [62], mxed n-step MIR [96], and n-step mnglng [6, 7]). II.1 Mxed Integer Programmng Mxed Integer Programmng s a powerful method to formulate and solve optmzaton problems contanng dscrete decson varables wth numerous applcatons n busness, scence, and engneerng. In general, MIPs are NP-hard problems. Therefore, t s challengng to mprove the exstng algorthms (or develop new effcent algorthms) for solvng MIP problems arsng n applcatons such as producton and dstrbuton plannng, faclty locaton, telecommuncaton, transportaton, arlne crew schedulng, electrcty generaton plannng, molecular bology, VLSI, and many more [81, 111]. 13

24 A mxed nteger program (MIP) can be wrtten as mn cv + hy Av + Gy b y Z n, v R p where A s an m by n matrx, G s an m by p matrx, c and h are row-vectors of dmensons n and p, respectvely, and v, y are the decson varables. In ths formulaton, f p = 0,.e. all varables are nteger, we get the pure nteger program mn{hy : Gy b, y Z n } and f all varables are bnary, we have the bnary nteger program mn{hy : Gy b, y {0, 1} n }. Furthermore, the lnear problem obtaned by droppng the ntegralty restrctons on decson varables of a MIP s called the lnear relaxaton of the MIP. II.1.1 Some Defntons and Theoretcal Results n Polyhedral Theory In ths secton, some defntons and fundamental theoretcal results n polyhedral theory are replcated from [81, 111] to the extent requred to present our research results. We also defne the concepts of extended formulaton and projecton (see [28, 29, 34, 113] for more detals). Defnton 1. The feasble regon of a MIP (denoted by P MIP Z n R p ) s the set 14

25 of ponts (y, v) Z n R p whch satsfy ts constrants: P MIP := {(y, v) Z n R p : Av + Gy b}. Defnton 2. A subset of R p descrbed by a fnte set of lnear constrants P = {v R p : Av b} s a polyhedron. Defnton 3. Gven a set X R n, the convex hull of X, denoted conv(x), s defned as: conv(x) = {x : x = t =1 λ x, t =1 λ = 1, λ 0 for = 1,..., t over all fnte subsets {x 1,..., x t } of X}. Theorem 1. conv(p MIP ) s a polyhedron, f the data A, G, b s ratonal. The proof of Theorem 1 s provded n [81]. Defnton 4. An nequalty πx π 0 s a vald nequalty for X R n f πx π 0 for all x X. Theorem 2. [81] If πx π 0 s vald for X R n, t s also vald for conv(x). Defnton 5. If πx π 0 and µx µ 0 are two vald nequaltes for P R n +, πx π 0 domnates µx µ 0 f there exsts u > 0 such that π uµ and π 0 uµ 0 and (π, π 0 ) (uµ, uµ 0 ). Observaton 1. If πx π 0 domnates µx µ 0, then {x R n + : πx π 0 } {x R n + : µx µ 0 }. Defnton 6. The ponts x 1,..., x k R n are affnely ndependent f the k 1 drectons x 2 x 1,..., x k x 1 are lnearly ndependent, or alternatvely the k vectors (x 1, 1),..., (x k, 1) R n+1 are lnearly ndependent. Defnton 7. The dmenson of P, denoted dm(p ), s one less than the maxmum number of affnely ndependent ponts n P. 15

26 Defnton 8. F defnes a face of the polyhedron P f F = {x P : πx = π 0 } for some vald nequalty πx π 0 of P. Defnton 9. F s a facet of P f F s a face of P and dm(f ) = dm(p ) 1. Defnton 10. If F s a face of P wth F = {x P : πx = π 0 }, the vald nequalty π x π 0 s sad to represent or defne the face. Defnton 11. Gven a polyhedron P (R n R p ), the projecton of P onto the space R n, denoted by Proj x (P ), s defned as Proj x (P ) := {x R n : (x, w) P for some w R p }. Defnton 12. Gven a set X R n and a polyhedron P := {(x, w) R n R p : Ax + Bw b} such that conv(x) Proj x (P ), the system Ax + Bw b provdes an extended formulaton for the set X. ) In case Proj x (P ) = conv(x), we call the extended formulaton s tght. ) An extended formulaton s compact f the addton of polynomal number of extra varables results n a formulaton wth a polynomal number of nequaltes. II.1.2 Algorthms for Solvng MIP Problems Branch-and-cut algorthm s among the most successful algorthms used to solve MIPs. Branch-and-cut s a branch-and-bound algorthm n whch cuttng planes are used to tghten the formulatons of node problems and hence acheve better bounds. Ths algorthm was frst ntroduced by Padberg and Rnald [83], and today most of the commercal and non-commercal MIP solvers use t. Ths s because t combnes the advantages of both branch-and-bound and cuttng plane algorthms, and hence overcomes the drawbacks assocated wth each of those algorthms. 16

27 Branch-and-bound (BB) was frst proposed by Land and Dog [67] for nteger programmng. The dea behnd the BB algorthm for a maxmzaton problem s as follows: The algorthm starts at the root node. The BB s done over a BB tree. Each node n the tree corresponds to a subset of the soluton space. At each node, the upper bound for the best soluton value obtanable n the soluton space correspondng to the node s calculated. Ths s done by solvng the lnear relaxaton (or any other easly solvable relaxaton) of the MIP. Based on the upper bound at the node and best known feasble soluton value (.e. best lower bound of the problem), the node s ether pruned or branched. A node can be pruned for two reasons: 1) f the upper bound value on that node s smaller than the best feasble soluton value found so far. In ths case there s no pont n searchng the node for optmal soluton anymore (ths s the man dea behnd BB). 2) f a soluton s found, the lower bound wll be updated f ths soluton has a larger objectve value. On the other hand, f a node cannot be pruned, the soluton space of the node s subdvded nto two or more subspaces (by generatng chld nodes). Ths acton s known as branchng. There are dfferent problem dependent strateges for choosng the branchng scheme n a node and also for choosng the next node n the tree. Whle solvng the MIP, one commonly used branchng strategy at a gven node s to create two chld nodes by addng the constrant (y y for frst node and y y for second node, where y s an nteger varable wth the fractonal LP soluton y ) to the lnear relaxaton at ths node. The problem s solved when all nodes are pruned and the best lower bound wll be the optmal value. The effcency of the method depends strongly on the branchng (node-splttng procedure) and on the upper and lower bound estmators. In order to solve mnmzaton problem usng BB, nterchange the lower bound and upper bound n the descrpton above. More detals and references can be found n [81, 111]. 17

28 Gomory [45, 92] presented the cuttng plane algorthm to solve (M)IPs. In [45], he showed how a modfed verson of the smplex algorthm provdes a fnte algorthm to solve pure nteger programs. Ths algorthm utlzes vald nequaltes (referred to as the cuts or cuttng planes) that are volated by the optmal soluton of the current lnear program, but satsfy all ntegral solutons. The algorthm n [92] s an extenson of the cuttng plane algorthm for pure nteger programs [45] to MIPs. The basc dea behnd ths algorthm s as follows: Gven a MIP, we solve ts LP relaxaton (LPR), generate a strong cut that s volated by the optmal soluton of LPR (n case t does not satsfy ntegralty constrants), and add the cut to the LPR whch tghten ts feasble regon wthout changng the feasble regon of MIP. Then we re-solve LPR and repeat the procedure untl all nteger constrants are satsfed. Note that a cuttng plane s called stronger than others f t cuts off bgger porton from the feasble regon of the LPR, n comparson to others. Therefore, facets of the convex hull of nteger solutons are the strongest possble cuts. The major advantage of ths algorthm s that t can solve a pure nteger program to optmalty n fnte number of steps. Despte that ths approach on ts own s not very effectve n practce because of the so-called talng-off phenomenon [24],.e. after some steps the porton cuts off from the feasble regon of the LPR by each cut becomes very small. In branch-and-cut algorthm, the cuttng planes are utlzed to provde a tghter formulaton of node problems and whenever the talng-off begns (due to the addton of cuttng planes) branchng s used to create new nodes (see [39, 59, 75, 79] for surveys on dfferent aspects of branch-and-cut algorthm). As a result, developng strong vald nequaltes as cuttng planes s crucal for effectveness of the branchand-cut algorthm. Ths fact s the major motvaton for the research n the area of cuttng planes. 18

29 II.2 Generalzatons of Mxed Integer Roundng Studyng the polyhedral structure of mxed nteger base sets whch consttute well-structured relaxatons of mportant MIP problems s a promsng approach. Ths s because oftentmes one can develop procedures n whch the vald nequaltes (or facets) developed for the base set are used to generate vald nequaltes (or facets) for the orgnal MIPs (see [6, 36, 51, 62, 96, 111] for a few examples among many others). In ths secton, we brefly revew the mxed nteger roundng (MIR) cut-generatng procedure [81, 111] and ts varous generalzatons (n partcular, contnuous mxng [105], n-step MIR [62], mxed n-step MIR [96], and n-step mnglng [6, 7]). II.2.1 Mxed Integer Roundng (MIR) One fundamental procedure to develop cuts for general MIPs s the MIR procedure [82, 111] whch utlzes the facet of a sngle-constrant mxed nteger base set, Q 1,1 0 := {(y, s) Z R + : α 1 y + s β} where α 1 > 0 and β R, referred to as the (1-step) MIR facet (page 127 of [111]). It s nterestng to note that all the facets of a general 0-1 MIP can be generated usng MIR [82] and for general MIP, MIR can be used to obtan strong vald nequaltes based on 1-row relaxatons [74]. Furthermore, the 1-step MIR cuts are equvalent to splt cuts of Cook et al. [31] and Gomory mxed nteger cuts [92], and are a specal case of the dsjunctve cuts [12, 13] (also see [21, 37]). Because of computatonal effectvenes, the MIR procedure s beng used n many MIP solvers today. Theorem 3. [111] The nequalty (1-step MIR facet) y 1 + v β β α 1 β/α 1, (2) α 1 19

30 s vald and facet-defnng for conv(q 1,1 0 ). In a general settng, the 1-step MIR facet (2) for conv(q 1,1 0 ) can be used to generate strong vald nequaltes for a sngle-constrant mxed nteger knapsack set wth general coeffcents. We defne ths set as follows: Y 1 0 := {(y, s) Z N + R + : N a t y t + s b} t=1 where the coeffcents a t, t = 1,..., N and b are real numbers (no condtons mposed on them). Note that Y 1 0 = Proj y,s (Y 1 {v = 0}). By choosng a parameter α 1 > 0 such that b (1) = b α 1 b/α 1 > 0, the defnng nequalty of Y 1 0 can be relaxed to t J 0 α 1 at α 1 y t + ( ) aj + a (1) j y j + s b (3) α 1 t J 1 by parttonng {1,..., N} nto two dsjont subsets J 0, J 1, relaxng a t n the defnng nequalty of Y 1 0 to α 1 a t /α 1 ( a t ) for t J 0, and replacng a t n the defnng nequalty of Y0 1 by a j /α 1 + a (1) (= a t ) for t J 1. Ths s a relaxaton because j y t 0, t J 0. Observe that the terms n nequalty (3) can be rearranged to have a structure smlar to the defnng nequalty of Q 1,1 0,.e. nequalty (3) can be wrtten as Settng ( at α 1 y t + ) ( ) at y t + a (1) t y t + s b. (4) α 1 α 1 t J 0 t J 1 t J 1 y := at y t + at y t and s := a (1) t y t + s, (5) α 1 α 1 t J 0 t J 1 t J 1 nequalty (4) becomes of the same form as the defnng nequalty of Q 1,1 0 (notce 20

31 that s R + and y Z). Therefore the MIR nequalty for (4), gven by ( b (1) at y t + ) ( ) at y t + a (1) t y t + s b (1), (6) α 1 α 1 bα1 t J 0 t J 1 t J 1 s vald for Y 1 0. Interestngly, nequalty (6) becomes the Gomory Mxed Integer (GMI) cut [92] when α 1 = 1. In a compact form, the MIR nequalty (6) for Y 1 0 can be wrtten as follows: N µ 1 α 1,b(a t )y t + s µ 1 α 1,b(b), (7) t=1 where µ 1 α 1,b = b(1) t/α 1 + mn{b (1), t (1) } s referred to as the 1-step MIR functon. II.2.2 Contnuous Mxng Van Vyve [105] generated the cycle nequaltes for the contnuous mxng set Q as follows: Defne β 0 := 0, f := β β, {0,..., m} and wthout loss of generalty assume that f 1 f, = 1,..., m. Let G := (V, A) be a drected graph, where V := {0, 1,..., m} and A := {(, j) :, j V, f f j }. Note that G s a complete graph except for the arcs (, j) where f = f j. An arc (, j) A s called a forward arc f < j and a backward arc f > j. To each arc (, j) A, assocate a lnear functon ψ j (y, v, s) defned as s + v + (f f j + 1)(y β ) f j ψ j (y, v, s) := v + (f f j )(y β ) f (, j) s a forward arc, f (, j) s a backward arc, where v 0 = y 0 = 0. See Fg. 2. Theorem 4 ([105]). Gven an elementary cycle C = (V C, A C ) n the graph G, the 21

32 Background n-step Cycle Inequaltes Contnuous n-mxng set Varant and specal cases ψ j 0 j ψ j m Fgure 2: Each cycle n graph G gves rse to a cycle nequalty. Bansal and Kanfar - Texas A&M Unversty Contnuous n-mxng Set 17/27 nequalty referred to as the cycle nequalty, s vald for Q. (,j) A C ψ j (y, v, s) 0, (8) In [105], the valdty of the cycle nequalty (8) was proved ndrectly through the followng extended formulaton for Q: Q δ = { (y, v, s, δ) R m R m+1 + R m+1 : ψ j (y, v, s) δ δ j for all (, j) A, y + v + s β, = 1,..., m }. Note that the set of all orgnal nequaltes, all cycle nequaltes, along wth the bound constrants v, s 0, defne Proj y,v,s (Q δ ). Van Vyve [105] showed that for every extreme pont (or extreme ray) of Q, there exsts a pont (or a ray) n ts extended formulaton Q δ. Ths mples Q Proj y,v,s (Q δ ), and hence, the cycle nequaltes are vald for Q. Furthermore, t was shown n [105] that conv(q) = Proj y,v,s (Q δ ) and the separaton over conv(q) can be performed n O(m 3 ) tme by fndng a negatve weght cycle n G. Smlar results were presented for the relaxaton of Q to the case 22

33 where s R. II.2.3 n-step MIR Inequaltes In another drecton, Kanfar and Fath [62] developed the n-step MIR nequaltes (a generalzaton of MIR nequaltes [82, 111]) for the base set { Q 1,n 0 = (y, s) Z Z n1 + R + : n t=1 } α t y t + s β, where α t R + \{0}, t = 1,..., n, β R, and α t s and β satsfy the so-called n-step MIR condtons,.e. α t β (t1) /α t αt1, t = 2,..., n. (9) Note that Q 1,n 0 = Proj y,s ( Q 1,n {v = 0} ). The n-step MIR nequalty for ths set s ( n β s β (n) (l1) β (n) l=1 n n ) β (l1) y t, (10) t=1 l=t+1 where the recursve remanders β (t) are defned as β (t) := β (t1) α t β (t1) /α t, t = 1,..., n, (11) and β (0) := β (note that 0 β (t) < α t for t = 1,..., n). By defnton f a > b, then b a (.) = 0 and b a (.) = 1. For nequalty (10) to be non-trval, we assume that β (t1) /α t / Z, t = 1,..., n. Kanfar and Fath [62] showed that the n-step MIR nequalty (10) s vald and facet-defnng for the convex hull of Q 1,n 0. In a more general settng, Kanfar and Fath [62] used n-step MIR facets of Q 1,n 0 to generate n-step MIR nequaltes for Y 1 0, a sngle-constrant mxed nteger knapsack set wth 23

34 general coeffcents. Recall that Y 1 0 = Proj y,s (Y 1 {v = 0}). For each n N, by choosng a parameter vector α = (α 1,..., α n ) > 0 that satsfy the n-step MIR condtons, α t b (t1) /α t αt1, t = 2,..., n, (12) they ntroduced the so-called n-step MIR functon to generate an n-step MIR nequalty for Y 1 0. The n-step MIR functon s defned as follows: g µ n α,b (x) = n n q=1 l=q+1 n q=1 l=q+1 where for g = 0,..., n 1, x (q1) α q x (q1) α q b (n) + n l=g+2 b (n) + x (n) x (g) α g+1 b (n) f x Ig n, g = 0,..., n 1 f x I n n I n g := {x R : x (q) < b (q), q = 1,..., g, x (g+1) b (g+1) }; I n n := {x R : x (q) < b (q), q = 1,..., n}. The n-step MIR nequalty for Y 1 0 s then N µ n α,b(a t )y t + s µ n α,b(b). (13) t=1 Kanfar and Fath [62] proved that, for n N, nequalty (13) s vald for Y 1 0, and later, Atamtürk and Kanfar [7] showed that these nequaltes also have facetdefnng propertes n several cases. Please refer to [7, 62] for more detals. 24

35 II.2.4 n-step Mnglng Inequaltes Atamtürk and Günlük [6] and Atamtürk and Kanfar [7] consdered the mxednteger knapsack set wth bounded nteger varables Z 1 0 := { (y, s) Z N + R + : a t y t + } a k y k + s b, y u, t T k K where (T, K) s a parttonng of {1,..., N} wth a t > 0 for t T, a k < 0 for k K, and u Z N +. Atamtürk and Günlük [6] ntroduced (1-step) mnglng and 2-step mnglng nequaltes for Z 1 0 whch are generalzed by Atamtürk and Kanfar [7] to n-step mnglng nequaltes, n N, for Z 1 0. Unlke n-step MIR nequalty (13), the n-step mnglng nequalty utlzes the nformaton about the bounds and s derved as follows [6, 7]. Assumng b 0, let T + := {1,..., n + } {t T : a t > b} and K := {k K : a k + t T + a t u t < 0}. We ndex T + n non-ncreasng order of a t s. For k K \ K, we defne a set T k, an nteger l k, and the numbers ū tk such that u tk u t for t T k as follows: T k l k { := {1,..., q(k)}, where q(k) := mn q T + : a k + { q(k)1 } := mn l Z + : a k + a t u t + a q(k) l 0 ; and u t, ū tk := l k, f t < q(k), f t = q(k). t=1 q t=1 } a t u t 0 ; Now for k K, let T k := T +, q(k) := n +, l k := u n +, and ū tk := u t for t T k. We also defne K t := {k K : k T k }; as a result, for t T \ T +, K t =. Also for k K, let τ k := mn {b, a k + } a t Tk t ū tk, and therefore, 0 τ k b for k K \ K and τ k < 0 for k K. Usng the (n 1)-step MIR functon, they then proved that 25

36 for n N, the n-step mnglng nequalty t T + µ n1 + k K α,b (b) [ y t k K t ū tk y k ] + t T \T + µ n1 α,b (a t)y t µ n1 α,b (τ k)y k + s µ n1 α,b (b) (14) s vald for Z 1 0 for a parameter vector α = (α 1,..., α n1 ) > 0 that satsfy the (n 1)- step MIR condtons (12). Note that for n = 1, we defne µ n1 α,b (x) = x. These nequaltes are used when nteger varables are bounded from both sdes. The n- step mnglng utlzes the bounds on nteger varables to gve stronger nequaltes, whch are facet-defnng n many cases [7]. Atamtürk and Günlük [6] proved that the 1-step mnglng nequaltes are facet-defnng for conv(z 1 0) f b mn{τ k : k K} max{a : a > b, T \T + }. For n 2, Atamtürk and Kanfar [7] proved that the n-step mnglng nequaltes are facet-defnng for conv(z 1 0) f the followng condtons are satsfed (Theorem 2 n [7]): ) b (n1) > 0 and α d = a d where d T \T + for k = 1,..., n 1; ) T + = { I : a α 1 b/α 1 } and α d1 α d b (d1) /α d for d = 2,..., n 1; b ) u t1 α 1 mn{τk :k K} α 1 and u td b (d1) α d for d = 2,..., n 1. It s mportant to note that for T + =, the 1-step mnglng nequalty reduces to the base nequalty and for n 2, the n-step mnglng nequalty reduces to the (n 1)-step MIR nequalty (13). Also, for n > 1, the n-step mnglng nequalty (14) domnates the nequalty obtaned by applyng the (n 1)-step MIR procedure on 1-step mnglng nequalty [7]. Moreover, the facet-defnng contnuous nteger cover nequalty [10] (obtaned by superaddtve lftng) for Z 1 0 s a specal case of 26

37 nequalty (14) for n = 2, b > 0, K =, T + = {t T : a t α 1 b/α 1 }, and α 1 = α d for some d T. Please refer to [6, 7] for more detals. II.2.5 Mxed n-step MIR Inequaltes As mentoned n Chapter I, Sanjeev and Kanfar [96] generalzed the MIR mxng procedure of Günlük and Pochet [51] to the case of n-step MIR and developed the mxed n-step MIR nequaltes for the n-mxng set Q m,n 0. Note that Q m,n 0 = Proj y,s ( Q m,n {v = 0} ). These nequaltes are generated as follows: Wthout loss of generalty, we assume β (n) 1 β(n), = 2,..., m. Let ˆK := { 1,..., K }, where 1 < 2 < < ˆK, be a non-empty subset of {1,..., m}. If the n-step MIR condtons (9) hold for each constrant ˆK,.e. α t β (t1) /α t α t1, t = 2,..., n, then the nequaltes ˆK ( ) s β (n) p β (n) p1 φ n p (y p ) (15) p=1 ˆK ( ) ( ) (φ s β (n) p β (n) p1 φ n p (y p ) + α n β (n) n 1 (y 1 ) 1 ), (16) ˆK p=1 are vald for Q m,n 0, where β (n) 0 = 0 and φ n (y ) := n β (l1) l=1 n n t=1 l=t+1 β (l1) yt (17) for ˆK. Inequaltes (15) and (16) are referred to as the type I and type II mxed n-step MIR nequaltes, respectvely. Inequalty (15) s shown to be facet-defnng for Q m,n 0. Inequalty (16) also defnes a facet for Q m,n 0 f some addtonal condtons are satsfed (see [96] for detals). Note that the functon φ n (y ) has the same form as the multple of β (n) n the rght-hand sde of the n-step MIR nequalty (10). Ths 27

38 functon can alternatvely be wrtten as follows (see proof of Lemma 10 n [96]): φ n (y ) := 1 + n n t=1 l=t+1 ( ) β (l1) β (t1) yt. (18) α t 28

39 CHAPTER III CONTINUOUS MULTI-MIXING SET In ths chapter, we ntroduce a mult-parameter mult-constrant mxed nteger base set referred to as the contnuous mult-mxng set whch we defne as Q m,n := { (y, v, s) (Z Z n1 + ) m R m+1 + : n t=1 } α t yt + v + s β, = 1,..., m, where α t > 0, t = 1,..., n and β R, = 1,..., m such that the n-step MIR condtons for {1,..., m} hold,.e. α t β (t1) /α t α t1, t = 2,..., n, {1,..., m}. (19) These n-step MIR condtons are automatcally satsfed f the parameters α 1,..., α n are dvsble. The polyhedral study of ths set generalzes the concepts of MIR [81, 111], mxed MIR [51], contnuous mxng [105], n-step MIR [62], and mxed n-step MIR [96] (see Fg. 1). Note that ths set has multple (m) constrants wth multple (n) nteger varables n each constrant; but t s more general than the n-mxng set (dscussed n Chapter II) because n addton to the common contnuous varable s, each constrant has a contnuous varable v of ts own. The contnuous mxng set Q s the specal case of Q m,n, where n = 1 and α 1 = 1, and the n-mxng set of Sanjeev and Kanfar [96] s the projecton of Q m,n {v = 0} on (y, s). The contnuous mult-mxng set arses as a substructure n relaxatons of MML-WB, MML wth *Some parts of ths chapter are reprnted wth permsson from n-step cycle nequaltes: facets for contnuous n-mxng set and strong cuts for mult-module capactated lot-szng problem by Mansh Bansal and Kavash Kanfar, Integer Programmng and Combnatoral Optmzaton Conference, Lecture Notes n Computer Scence, 8494, , Copyrght 2014 by Sprnger. 29