A Stabilized Structured Dantzig-Wolfe Decomposition Method

We discuss an algorithmic scheme, which we call the stabilized structured Dantzig-Wolfe decomposition method, for solving large-scale structured linear programs. It can be applied when the subproblem of the standard Dantzig-Wolfe approach admits an alternative master model amenable to column generation, other than the standard one in which there is a variable for each of the extreme points and extreme rays of the corresponding polyhedron. Stabilization is achieved by the same techniques developed for the standard Dantzig-Wolfe approach and it is equally useful to improve the performance, as shown by computational results obtained on an application to the multicommodity capacitated network design problem

A note on "A LP-based heuristic for a time-constrained routing problem"

In their paper, Avella et al. (2006) investigate a time-constrained routing problem. The core of the proposed solution approach is a large-scale linear program that grows both row- and column-wise when new variables are introduced. Thus, a column-and-row generation algorithm is proposed to solve this linear program optimally, and an optimality condition is presented to terminate the column-and-row generation algorithm. We demonstrate by using Lagrangian duality that this optimality condition is incorrect and may lead to a suboptimal solution at termination

Coordination mechanisms with mathematical programming models for decentralized decision-making, a literature review

[EN] The increase in the complexity of supply chains requires greater efforts to align the activities of all its members in order to improve the creation of value of their products or services offered to customers. In general, the information is asymmetric; each member has its own objective and limitations that may be in conflict with other members. Operations managements face the challenge of coordinating activities in such a way that the supply chain as a whole remains competitive, while each member improves by cooperating. This document aims to offer a systematic review of the collaborative planning in the last decade on the mechanisms of coordination in mathematical programming models that allow us to position existing concepts and identify areas where more research is needed.Rius-Sorolla, G.; Maheut, J.; Estelles Miguel, S.; García Sabater, JP. (2020). Coordination mechanisms with mathematical programming models for decentralized decision-making, a literature review. Central European Journal of Operations Research. 28(1):61-104. https://doi.org/10.1007/s10100-018-0594-zS61104281Acar Y, Atadeniz SN (2015) Comparison of integrated and local planning approaches for the supply network of a globally-dispersed enterprise. Int J Prod Econ 167:204–219. https://doi.org/10.1016/j.ijpe.2015.05.028Agnetis A, Hall NG, Pacciarelli D (2006) Supply chain scheduling: sequence coordination. Discrete Appl Math 154(15):2044–2063. https://doi.org/10.1016/j.dam.2005.04.019Agnetis A, Aloulou MA, Fu LL (2016) Production and interplant batch delivery scheduling: Dominance and cooperation. Int J Prod Econ 182:38–49. https://doi.org/10.1016/j.ijpe.2016.08.007Albrecht M (2010) Supply chain coordination mechanisms Lecture notes in economics and mathematical systems, vol 628. Springer, Berlin. https://doi.org/10.1007/978-3-642-02833-5Albrecht M, Stadtler H (2015) Coordinating decentralized linear programs by exchange of primal information. Eur J Oper Res 247(3):788–796. https://doi.org/10.1016/j.ejor.2015.06.045Arkan A, Hejazi SR (2012) Coordinating orders in a two echelon supply chain with controllable lead time and ordering cost using the credit period. Comput Ind Eng 62(1):56–69. https://doi.org/10.1016/j.cie.2011.08.016Arshinder, Kanda A, Deshmukh SG (2008) Supply chain coordination: perspectives, empirical studies and research directions. Int J Prod Econ 115(2):316–335. https://doi.org/10.1016/j.ijpe.2008.05.011Attanasio A, Ghiani G, Grandinetti L, Guerriero F (2006) Auction algorithms for decentralized parallel machine scheduling. Parallel Comput 32(9):701–709. https://doi.org/10.1016/j.parco.2006.03.002Badole CM, Jain R, Rathore APS, Nepal B (2012) Research and opportunities in supply chain modeling: a review. Int J Supply Chain Manag 1(3):63–86Bajgiran OS, Zanjani MK, Nourelfath M (2016) The value of integrated tactical planning optimization in the lumber supply chain. Int J Prod Econ 171:22–33. https://doi.org/10.1016/j.ijpe.2015.10.021Behnamian J (2014) Multi-cut Benders decomposition approach to collaborative scheduling. Int J Comput Integr Manuf 28(11):1–11. https://doi.org/10.1080/0951192X.2014.961963Ben-Daya M, Darwish M, Ertogral K (2008) The joint economic lot sizing problem: review and extensions. Eur J Oper Res 185(2):726–742. https://doi.org/10.1016/j.ejor.2006.12.026Benders JF (1962) Partitioning procedures for solving mixed-variables programming problems. Numer Math 4(1):238–252. https://doi.org/10.1007/BF01386316Bhatnagar R, Chandra P, Goyal SK (1993) Models for multi-plant coordination. Eur J Oper Res 67(2):141–160. https://doi.org/10.1016/0377-2217(93)90058-UBuer T, Homberger JJ, Gehring H (2013) A collaborative ant colony metaheuristic for distributed multi-level uncapacitated lot-sizing. Int J Prod Res 51(17):5253–5270. https://doi.org/10.1080/00207543.2013.802822Buer T, Ziebuhr M, Kopfer H (2015) A coordination mechanism for a collaborative lot-sizing problem with rivaling agents. In: Mattfeld D, Spengler T, Brinkmann J, Grunewald M (eds) Logistics management. Springer, Cham. https://doi.org/10.1007/978-3-319-13177-1_26Buxmann P, Ahsen A Von, Díaz LM (2008) Economic evaluation of cooperation scenarios in supply chains. J Enterp Inf Manag 21(3):247–262. https://doi.org/10.1108/17410390810866628Chakraborty A, Chatterjee AK (2015) A surcharge pricing scheme for supply chain coordination under JIT environment. Eur J Oper Res 253(1):14–24. https://doi.org/10.1016/j.ejor.2016.02.001Chen IJ, Paulraj A, Lado AA (2004) Strategic purchasing, supply management, and firm performance. J Oper Manag 22(5):505–523. https://doi.org/10.1016/j.jom.2004.06.002Cheng JH (2011) Inter-organizational relationships and information sharing in supply chains. Int J Inf Manag 31(4):374–384. https://doi.org/10.1016/j.ijinfomgt.2010.09.004Cheng R, Forbes JF, San Yip W, Fraser Forbes J, Yip WS (2008) Dantzig–Wolfe decomposition and plant-wide MPC coordination. Comput Chem Eng 32(7):1507–1522. https://doi.org/10.1016/j.compchemeng.2007.07.003Cooper MC, Lambert DM, Pagh JD (1997) Supply chain management: more than a new name for logistics. Int J Logist Manag 8(1):1–14. https://doi.org/10.1108/09574099710805556Dantzig GB, Wolfe P (1960) Decomposition principle for linear programs. Oper Res 8(1):101–111. https://doi.org/10.1287/opre.8.1.101Dash RK, Vytelingum P, Rogers A, David E, Jennings NR (2007) Market-based task allocation mechanisms for limited-capacity suppliers. IEEE Trans Syst Man Cybern Part A Syst Hum 37(3):391–405. https://doi.org/10.1109/TSMCA.2007.893474Dudek G, Stadtler H (2005) Negotiation-based collaborative planning between supply chains partners. Eur J Operat Res 163(3):668–687. https://doi.org/10.1016/j.ejor.2004.01.014Dudek G, Stadtler H (2007) Negotiation-based collaborative planning in divergent two-tier supply chains. Int J Prod Res 45(2):465–484Ertogral K, David Wu S (2000) Auction-theoretic coordination of production planning in the supply chain. IIE Trans 32:931–940. https://doi.org/10.1080/07408170008967451Eslikizi S, Ziebuhr M, Kopfer H, Buer T (2015) Shapley-based side payments and simulated annealing for distributed lot-sizing. IFAC-PapersOnLine 48(3):1592–1597. https://doi.org/10.1016/j.ifacol.2015.06.313Fan M, Stallaert J, Whinston AB (2003) Decentralized mechanism design for supply chain organizations using an auction market. Inf Syst Res 14(1):1–22. https://doi.org/10.1287/isre.14.1.1.14763Feng Y, D’Amours S, Beauregard R (2008) The value of sales and operations planning in oriented strand board industry with make-to-order manufacturing system: cross functional integration under deterministic demand and spot market recourse. Int J Prod Econ 115(1):189–209. https://doi.org/10.1016/j.ijpe.2008.06.002Fisher ML (1985) An applications oriented guide to Lagrangian relaxation. Interfaces 15(2):10–21. https://doi.org/10.1287/inte.15.2.10Fisher ML (2004) The Lagrangian relaxation method for solving integer programming problems. Manag Sci 50(12 Supplement):1861–1871. https://doi.org/10.1287/mnsc.1040.0263Frazzon E, Makuschewits T, Scholz-Reiter B, Novaes AGN (2010) Assessing the integrated scheduling of manufacturing and transportation systems along global supply chains. In: World conference on transport research, LisbonGaudreault J, Forget P, Frayret JMJ, Rousseau A, Lemieux S, D’Amours S (2010) Distributed operations planning in the softwood lumber supply chain: models and coordination. Int J Ind Eng Theory Appl Pract 17(3):168–189Gunnerud V, Foss B (2010) Oil production optimization—a piecewise linear model, solved with two decomposition strategies. Comput Chem Eng 34(11):1803–1812. https://doi.org/10.1016/j.compchemeng.2009.10.019Harb H, Paprott JN, Matthes P, Schütz T, Streblow R, Mueller D (2015) Decentralized scheduling strategy of heating systems for balancing the residual load. Build Environ 86:132–140. https://doi.org/10.1016/j.buildenv.2014.12.015Held M, Karp RM (1970) The traveling-salesman problem and minimum spanning trees. Oper Res 18(6):1138–1162. https://doi.org/10.1287/opre.18.6.1138Held M, Karp RM (1971) The traveling-salesman problem and minimum spanning trees: part II. Math Program 1(1):6–25. https://doi.org/10.1007/BF01584070Homberger J (2010) Decentralized multi-level uncapacitated lot-sizing by automated negotiation. 4OR 8(2):155–180. https://doi.org/10.1007/s10288-009-0104-1Homberger J (2011) A generic coordination mechanism for lot-sizing in supply chains. Electron Commer Res 11(2):123–149. https://doi.org/10.1007/s10660-010-9053-1Homberger J, Gehring H (2010) A pheromone-based negotiation mechanism for lot-sizing in supply chains. In: 2010 43rd Hawaii international conference on system sciences. IEEE, pp 1–10. https://doi.org/10.1109/hicss.2010.26Homberger J, Gehring H (2011) An ant colony optimization-based negotiation approach for lot-sizing in supply chains. Int J Inf Process Manag 2(3):86–99. https://doi.org/10.4156/ijipm.vol2.issue3.10Homberger J, Gehring H, Buer T (2015) Integrating side payments into collaborative planning for the distributed multi-level unconstrained lot sizing problem. In: Bui TX, Sprague RH (eds) 2015 48th Hawaii international conference on system sciences, vol 2015. IEEE, pp 1068–1077. https://doi.org/10.1109/hicss.2015.131Huang GQ, Lau JSK, Mak KL (2003) The impacts of sharing production information on supply chain dynamics: a review of the literature. Int J Prod Res 41(7):1483–1517. https://doi.org/10.1080/0020754031000069625Jeong I-J (2012) A centralized/decentralized design of a full return contract for a risk-free manufacturer and a risk-neutral retailer under partial information sharing. Int J Prod Econ 136(1):110–115. https://doi.org/10.1016/j.ijpe.2011.09.019Jeong IJ, Leon VJ (2002) Decision-making and cooperative interaction via coupling agents in organizationally distributed systems. IIE Trans (Inst Ind Eng) 34(9):789–802. https://doi.org/10.1023/A:1015548705266Jeong IJ, Yim SB (2009) A job shop distributed scheduling based on Lagrangian relaxation to minimise total completion time. Int J Prod Res 47(24):6783–6805. https://doi.org/10.1080/00207540701824217Jia ZZ, Deschamps JC, Dupas R (2016) A negotiation protocol to improve planning coordination in transport-driven supply chains. J Manuf Syst 38:13–26. https://doi.org/10.1016/j.jmsy.2015.10.003Jung H, Chen FF, Jeong B (2008) Decentralized supply chain planning framework for third party logistics partnership. Comput Ind Eng 55(2):348–364. https://doi.org/10.1016/j.cie.2007.12.017Katok E, Pavlov V (2013) Fairness in supply chain contracts: a laboratory study. J Oper Manag 31(3):129–137. https://doi.org/10.1016/j.jom.2013.01.001Kelly JD, Zyngier D (2008) Hierarchical decomposition heuristic for scheduling: coordinated reasoning for decentralized and distributed decision-making problems. Comput Chem Eng 32(11):2684–2705. https://doi.org/10.1016/j.compchemeng.2007.08.007Kong J, Rönnqvist M (2014) Coordination between strategic forest management and tactical logistic and production planning in the forestry supply chain. Int Trans Oper Res 21(5):703–735. https://doi.org/10.1111/itor.12089Kovács A, Egri P, Kis T, Váncza J (2013) Inventory control in supply chains: alternative approaches to a two-stage lot-sizing problem. Int J Prod Econ 143(2):385–394. https://doi.org/10.1016/j.ijpe.2012.01.001Kumar BK, Nagaraju D, Narayanan S (2016) Supply chain coordination models: a literature review. Indian J Sci Technol. https://doi.org/10.17485/ijst/2016/v9i38/86938Kutanoglu E, David Wu S (1999) On combinatorial auction and Lagrangean relaxation for distributed resource scheduling. IIE Trans 31(9):813–826. https://doi.org/10.1080/07408179908969883Lau HC, Zhao ZJ, Ge SS, Lee TH (2011) Allocating resources in multiagent flowshops with adaptive auctions. IEEE Trans Autom Sci Eng 8(4):732–743. https://doi.org/10.1109/TASE.2011.2160536Lee DJ, Jeong IJ (2010) A distributed coordination for a single warehouse-multiple retailer problem under private information. Int J Prod Econ 125(1):190–199. https://doi.org/10.1016/j.ijpe.2010.02.001Lehoux N, D’Amours S, Frein Y, Langevin A, Penz B (2010a) Collaboration for a two-echelon supply chain in the pulp and paper industry: the use of incentives to increase profit. J Oper Res Soc 62(4):581–592. https://doi.org/10.1057/jors.2009.167Lehoux N, D’Amours S, Langevin A (2010b) A win–win collaboration approach for a two-echelon supply chain: a case study in the pulp and paper industry. Eur J Ind Eng 4(4):493. https://doi.org/10.1504/EJIE.2010.035656Lehoux N, D’Amours S, Langevin A (2014) Inter-firm collaborations and supply chain coordination: review of key elements and case study. Prod Plan Control 25(10):858–872. https://doi.org/10.1080/09537287.2013.771413Li X, Wang Q (2007) Coordination mechanisms of supply chain systems. Eur J Oper Res 179(1):1–16. https://doi.org/10.1016/j.ejor.2006.06.023Lu SYP, Lau HYK, Yiu CKF (2012) A hybrid solution to collaborative decision-making in a decentralized supply-chain. J Eng Technol Manag 29(1):95–111. https://doi.org/10.1016/j.jengtecman.2011.09.008Mahdiraji HA, Zavadskas EK, Hajiagha SHR (2015) Game theoretic approach for coordinating unlimited multi echelon supply chains. Transform Bus Econ 14(2):133–151Maheut J, Besga JM, Uribetxebarria J, Garcia-Sabater JP (2014a) A decision support system for modelling and implementing the supply network configuration and operations scheduling problem in the machine tool industry. Prod Plan Control 25(8):679–697. https://doi.org/10.1080/09537287.2013.798087Maheut J, Garcia-Sabater JP, Garcia-Sabater JJ, Marin-Garcia J (2014b) Coordination mechanism for MILP models to plan operations within an advanced planning and scheduling system in a motor company: a case study. In: Prado-Prado JC, García-Arca J (eds) Annals of industrial engineering 2012. Springer, London, pp 245–253. https://doi.org/10.1007/978-1-4471-5349-8_29Manrodt KB, Vitasek K (2004) Global process standardization: a case study. J Bus Logist 25(1):1–23. https://doi.org/10.1002/j.2158-1592.2004.tb00168.xMarin-Garcia JA, Ramirez Bayarri L, Atares Huerta L (2015) Protocol: comparing advantages and disadvantages of rating scales, behavior observation scales and paired comparison scales for behavior assessment of competencies in workers. A systematic literature review. Work Pap Oper Manag 6(2):49. https://doi.org/10.4995/wpom.v6i2.4032Mason AN, Villalobos JR (2015) Coordination of perishable crop production using auction mechanisms. Agric Syst 138:18–30. https://doi.org/10.1016/j.agsy.2015.04.008McAfee RP, McMillan J (1987) Auctions and bidding. J Econ Lit 25(2):699–738Medina-Lopez C, Marin-Garcia JA, Alfalla-Luque R (2010) Una propuesta metodológica para la realización de búsquedas sistemáticas de bibliografía (A methodological proposal for the systematic literature review). Work Pap Oper Manag. https://doi.org/10.4995/wpom.v1i2.786Mouret S, Grossmann IE, Pestiaux P (2011) A new Lagrangian decomposition approach applied to the integration of refinery planning and crude-oil scheduling. Comput Chem Eng 35(12):2750–2766. https://doi.org/10.1016/j.compchemeng.2011.03.026Mula J, Peidro D, Díaz-Madroñero M, Vicens E (2010) Mathematical programming models for supply chain production and transport planning. Eur J Oper Res 204(3):377–390. https://doi.org/10.1016/j.ejor.2009.09.008Nie L, Xu X, Zhan D (2008) Collaborative planning in supply chains by lagrangian relaxation and genetic algorithms. Int J Inf Technol Decis Mak 7(1):183–197. https://doi.org/10.1142/s0219622008002879Nishi T, Shinozaki R, Konishi M (2008) An augmented Lagrangian approach for distributed supply chain planning for multiple companies. IEEE Trans Autom Sci Eng 5(2):259–274. https://doi.org/10.1109/TASE.2007.894727Ouelhadj D, Petrovic S (2009) A survey of dynamic scheduling in manufacturing systems. J Sched 12(4):417–431. https://doi.org/10.1007/s10951-008-0090-8Pibernik R, Sucky E (2007) An approach to inter-domain master planning in supply chains. Int J Prod Econ 108(1–2):200–212. https://doi.org/10.1016/j.ijpe.2006.12.010Pittman SD, Bare BB, Briggs DG (2007) Hierarchical production planning in forestry using price-directed decomposition. Can J For 37(10):2010–2021. https://doi.org/10.1139/X07-026Polyak BT (1969) Minimization of unsmooth functionals. USSR Comput Math Math Phys 9(3):14–29. https://doi.org/10.1016/0041-5553(69)90061-5Pukkala T, Heinonen T, Kurttila M (2009) An application of a reduced cost approach to spatial forest planning. For Sci 55(1):13–22Qu T, Nie DX, Chen X, Chen XD, Dai QY, Huang GQ (2015) Optimal configuration of cluster supply chains with augmented Lagrange coordination. Comput Ind Eng 84(SI):43–55. https://doi.org/10.1016/j.cie.2014.12.026Reiss F, Buer T (2014) A coordination mechanism for capacitated lot-sizing in non-hierarchical n-tier supply chains. In: 2014 IEEE symposium on computational intelligence in production and logistics systems (Cipls), pp 9–15. https://doi.org/10.1109/cipls.2014.7007155Rius-Sorolla G, Maheut J, Estelles-Miguel S, Garcia-Sabater JP (2017) Protocol: systematic literature review on coordination mechanisms for the mathematical programming models in production planning with decentralized decision making. Work Pap Oper Manag 8(2):22. https://doi.org/10.4995/wpom.v8i2.7858Sahin F, Robinson EPP (2002) Flow coordination and information sharing in supply chains: review, implications, and directions for future research. Decis Sci 33(4):505–535. https://doi.org/10.1111/j.1540-5915.2002.tb01654.xSilva CA, Sousa JMC, Runkler TA, Sá da Costa J (2009) Distributed supply chain management using ant colony optimization. Eur J Oper Res 199(2):349–358. https://doi.org/10.1016/j.ejor.2008.11.021Simatupang T, Sridharan R (2006) The collaboration index: a measure for supply chain collaboration. Int J Phys Distrib Logist Manag 35:44–62. https://doi.org/10.1108/09600030510577421Singh G, Ernst A (2011) Resource constraint scheduling with a fractional shared resource. Oper Res Lett 39(5):363–368. https://doi.org/10.1016/j.orl.2011.06.003Singh G, O’Keefe CM (2016) Decentralised scheduling with confidentiality protection. Oper Res Lett 44(4):514–519. https://doi.org/10.1016/j.orl.2016.05.004Sokoler LE, Standardi L, Edlund K, Poulsen NK, Madsen H, Jørgensen JB (2014) A Dantzig–Wolfe decomposition algorithm for linear economic model predictive control of dynamically decoupled subsystems. J Process Control 24(8):1225–1236. https://doi.org/10.1016/j.jprocont.2014.05.013Sridharan R, Simatupang TM (2009) Managerial views of supply chain collaboration. Gadjah Mada Int J Bus 11(2):253–273Stadtler H (2007) A framework for collaborative planning and state-of-the-art. OR Spectr 31(1):5–30. https://doi.org/10.1007/s00291-007-0104-5Stadtler H, Kilger C (2008) Supply chain management and advanced planning. In: Stadtler H, Kilger C (eds) Supply chain management and advanced planning. Concepts, models, software, and case studies. Springer, BerlinStank TP, Goldsby TJ, Vickery SK (1999) Effect of service supplier performance on satisfaction and loyalty of store managers in the fast food industry. J Oper Manag 17(4):429–447. https://doi.org/10.1016/S0272-6963(98)00052-7Taghipour A, Frayret JM (2013) An algorithm to improve operations planning in decentralized supply chains. In: 2013 international conference on advanced logistics and transport, ICALT 2013, pp 100–103. https://doi.org/10.1109/icadlt.2013.6568442Tang SH, Rahimi I, Karimi H (2016a) Objectives, products and demand requirements in integrated supply chain network design: a review. Int J Ind Syst Eng 23(2):181. https://doi.org/10.1504/IJISE.2016.076399Tang J, Zeng C, Pan Z (2016b) Auction-based cooperation mechanism to parts scheduling for flexible job shop with inter-cells. Appl Soft Comput 49:590–602. https://doi.org/10.1016/j.asoc.2016.08.046Thomas A, Singh G, Krishnamoorthy M, Venkateswaran J (2013) Distributed optimisation method for multi-resource constrained scheduling in coal supply chains. Int J Prod Res 51(9):2740–2759. https://doi.org/10.1080/00207543.2012.737955Thomas A, Venkateswaran J, Singh G, Krishnamoorthy M (2014) A resource constrained scheduling problem with multiple independent producers and a single linking constraint: a coal supply chain example. Eur J Oper Res 236(3):946–956. https://doi.org/10.1016/j.ejor.2013.10.006Thomas A, Krishnamoorthy M, Singh G, Venkateswaran J (2015) Coordination in a multiple producers–distributor supply chain and the value of information. Int J Prod Econ 167:63–73. https://doi.org/10.1016/j.ijpe.2015.05.020VICS (2004) Collaborative planning, forecasting and replenishment. Retrieved January 21, 2017, from https://www.gs1us.org/Vitasek K (2016) Strategic sourcing business models. Strateg Outsour Int J 9(2):126–138. https://doi.org/10.1108/SO-02-2016-0003Walther G, Schmid E, Spengler TS (2008) Negotiation-based coordination in product recovery networks. Int J Prod Econ 111(2):334–350. https://doi.org/10.1016/j.ijpe.2006.12.069Wang L, Pfohl HC, Berbner U, Keck AK (2016) Supply chain collaboration or conflict? Information sharing and supply chain performance in the automotive industry. In: Clausen U, Friedrich H, Thaller C, Geiger C (eds) Commercial transport. Springer, Cham, pp 303–318. https://doi.org/10.1007/978-3-319-21266-1Wenzel S, Paulen R, Krämer S, Beisheim B, Engell S (2016a) Shared resource allocation in an integrated petrochemical site by price-based coordination using quadratic approximation. In: 2016 European control conference, ECC 2016, pp 1045–1050. https://doi.org/10.1109/ecc.2016.7810427Wenzel S, Paulen R, Stojanovski G, Kraemer S, Beisheim B, Engell S (2016b) Optimal resource allocation in industrial complexes by distributed optimization and dynamic pricing. At-Automatisierungstechnik 64(6):428–442. https://doi.org/10.1515/auto-2016-0003Whang S (1995) Coordination in operat

Stabilized Benders methods for large-scale combinatorial optimization, with appllication to data privacy

The Cell Suppression Problem (CSP) is a challenging Mixed-Integer Linear Problem arising in statistical tabular data protection. Medium sized instances of CSP involve thousands of binary variables and million of continuous variables and constraints. However, CSP has the typical structure that allows application of the renowned Benders’ decomposition method: once the “complicating” binary variables are fixed, the problem decomposes into a large set of linear subproblems on the “easy” continuous ones. This allows to project away the easy variables, reducing to a master problem in the complicating ones where the value functions of the subproblems are approximated with the standard cutting-plane approach. Hence, Benders’ decomposition suffers from the same drawbacks of the cutting-plane method, i.e., oscillation and slow convergence, compounded with the fact that the master problem is combinatorial. To overcome this drawback we present a stabilized Benders decomposition whose master is restricted to a neighborhood of successful candidates by local branching constraints, which are dynamically adjusted, and even dropped, during the iterations. Our experiments with randomly generated and real-world CSP instances with up to 3600 binary variables, 90M continuous variables and 15M inequality constraints show that our approach is competitive with both the current state-of-the-art (cutting-plane-based) code for cell suppression, and the Benders implementation in CPLEX 12.7. In some instances, stabilized Benders is able to quickly provide a very good solution in less than one minute, while the other approaches were not able to find any feasible solution in one hour.Peer ReviewedPreprin

Standard Bundle Methods: Untrusted Models and Duality

We review the basic ideas underlying the vast family of algorithms for nonsmooth convex optimization known as "bundle methods|. In a nutshell, these approaches are based on constructing models of the function, but lack of continuity of first-order information implies that these models cannot be trusted, not even close to an optimum. Therefore, many different forms of stabilization have been proposed to try to avoid being led to areas where the model is so inaccurate as to result in almost useless steps. In the development of these methods, duality arguments are useful, if not outright necessary, to better analyze the behaviour of the algorithms. Also, in many relevant applications the function at hand is itself a dual one, so that duality allows to map back algorithmic concepts and results into a "primal space" where they can be exploited; in turn, structure in that space can be exploited to improve the algorithms' behaviour, e.g. by developing better models. We present an updated picture of the many developments around the basic idea along at least three different axes: form of the stabilization, form of the model, and approximate evaluation of the function

Simultaneous column-and-row generation for large-scale linear programs with column-dependent-rows

In this paper, we develop a simultaneous column-and-row generation algorithm that could be applied to a general class of large-scale linear programming problems. These problems typically arise in the context of linear programming formulations with exponentially many variables. The defining property for these formulations is a set of linking constraints, which are either too many to be included in the formulation directly, or the full set of linking constraints can only be identified, if all variables are generated explicitly. Due to this dependence between columns and rows, we refer to this class of linear programs as problems with column-dependent-rows. To solve these problems, we need to be able to generate both columns and rows on-the-fly within an efficient solution approach. We emphasize that the generated rows are structural constraints and distinguish our work from the branch-and-cut-and-price framework. We first characterize the underlying assumptions for the proposed column-and-row generation algorithm. These assumptions are general enough and cover all problems with column-dependent-rows studied in the literature up until now to the best of our knowledge. We then introduce in detail a set of pricing subproblems, which are used within the proposed column-and-row generation algorithm. This is followed by a formal discussion on the optimality of the algorithm. To illustrate the proposed approach, the paper is concluded by applying the proposed framework to the multi-stage cutting stock and the quadratic set covering problems

Designing the Liver Allocation Hierarchy: Incorporating Equity and Uncertainty

Liver transplantation is the only available therapy for any acute or chronic condition resulting in irreversible liver dysfunction. The liver allocation system in the U.S. is administered by the United Network for Organ Sharing (UNOS), a scientific and educational nonprofit organization. The main components of the organ procurement and transplant network are Organ Procurement Organizations (OPOs), which are collections of transplant centers responsible for maintaining local waiting lists, harvesting donated organs and carrying out transplants. Currently in the U.S., OPOs are grouped into 11 regions to facilitate organ allocation, and a three-tier mechanism is utilized that aims to reduce organ preservation time and transport distance to maintain organ quality, while giving sicker patients higher priority. Livers are scarce and perishable resources that rapidly lose viability, which makes their transport distance a crucial factor in transplant outcomes. When a liver becomes available, it is matched with patients on the waiting list according to a complex mechanism that gives priority to patients within the harvesting OPO and region. Transplants at the regional level accounted for more than 50% of all transplants since 2000.This dissertation focuses on the design of regions for liver allocation hierarchy, and includes optimization models that incorporate geographic equity as well as uncertainty throughout the analysis. We employ multi-objective optimization algorithms that involve solving parametric integer programs to balance two possibly conflicting objectives in the system: maximizing efficiency, as measured by the number of viability adjusted transplants, and maximizing geographic equity, as measured by the minimum rate of organ flow into individual OPOs from outside of their own local area. Our results show that efficiency improvements of up to 6% or equity gains of about 70% can be achieved when compared to the current performance of the system by redesigning the regional configuration for the national liver allocation hierarchy.We also introduce a stochastic programming framework to capture the uncertainty of the system by considering scenarios that correspond to different snapshots of the national waiting list and maximize the expected benefit from liver transplants under this stochastic view of the system. We explore many algorithmic and computational strategies including sampling methods, column generation strategies, branching and integer-solution generation procedures, to aid the solution process of the resulting large-scale integer programs. We also explore an OPO-based extension to our two-stage stochastic programming framework that lends itself to more extensive computational testing. The regional configurations obtained using these models are estimated to increase expected life-time gained per transplant operation by up to 7% when compared to the current system.This dissertation also focuses on the general question of designing efficient algorithms that combine column and cut generation to solve large-scale two-stage stochastic linear programs. We introduce a flexible method to combine column generation and the L-shaped method for two-stage stochastic linear programming. We explore the performance of various algorithm designs that employ stabilization subroutines for strengthening both column and cut generation to effectively avoid degeneracy. We study two-stage stochastic versions of the cutting stock and multi-commodity network flow problems to analyze the performances of algorithms in this context