Skip to main content
SpringerLink
Log in

An augmented Lagrangian approach with general constraints to solve nonlinear models of the large-scale reliable inventory systems

  • Published:
Journal of Combinatorial Optimization Aims and scope Submit manuscript

Abstract

The Augmented Lagrangian method (ALM) is one of the algorithms in a class of methods for constrained optimization of nonlinear problems (NLP) that seeks a solution by replacing the original constrained problem using a sequence of unconstrained subproblems. Also known as the method of multipliers, the ALM approach introduces explicit Lagrangian multiplier estimates at each step. In this paper, an ALM is developed to solve the nonlinear models of the large-scale inventory systems. The proposed ALM is based on successive minimization of the augmented Lagrangian with respect to the possibly occurring between iterations. Our suggested approach is relatively easy to implement because the main computational operation at each iteration of NLP models is minimization of the smooth function to solve the bound-constrained subproblem. Accordingly, a large-scale NLP inventory system is designed and optimized using the ALM. The objectives are to simultaneously minimize the total inventory cost and maximize the total reliability in large-scale NLP inventory systems, while the constraints are satisfied. The results of numerical analyses, and performance comparison show that the proposed approach has satisfactory performance in terms of optimality criteria such as quality of solutions, complementarity, infeasibility, and optimality error.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2

Similar content being viewed by others

Data availability

The data that support the findings of this study is not taken from any resources. It means the data for numerical examples of this research is randomly generated by authors.

References

  • Ahmadi E, Jasemi M, Monplaisir L, Nabavi MA, Mahmoodi A, Jam PA (2018) New efficient hybrid candlestick technical analysis model for stock market timing on the basis of the Support Vector Machine and Heuristic Algorithms of Imperialist Competition and Genetic. Expert Syst Appl 94:21–31

    Google Scholar 

  • Ala A, Simic V, Pamucar D, Tirkolaee EB (2022) Appointment scheduling problem under fairness policy in healthcare services: fuzzy ant lion optimizer. Expert Syst Appl 207:117949

    Google Scholar 

  • Amirteimoori A, Mahdavi I, Solimanpur M, Ali SS, Tirkolaee EB (2022) A parallel hybrid PSO-GA algorithm for the flexible flow-shop scheduling with transportation. Comput Ind Eng 173:108672

    Google Scholar 

  • Amjadian A, Gharaei A (2021) An integrated reliable five-level closed-loop supply chain with multi-stage products under quality control and green policies: generalised outer approximation with exact penalty. Int J Syst Sci Oper Logist 9(3):429–449

    Google Scholar 

  • Askari, R., Sebt, M. V., & Amjadian, A. (2020). A multi-product EPQ model for defective production and ınspection with single machine, and operational constraints: stochastic programming approach. In International Conference on Logistics and Supply Chain Management. Springer, Cham, p 161–193

  • Babaeinesami A, Tohidi H, Ghasemi P, Goodarzian F, Tirkolaee EB (2022) A closed-loop supply chain configuration considering environmental impacts: a self-adaptive NSGA-II algorithm. Appl Intell 52(12):13478–13496

    Google Scholar 

  • Baradaran Rezaei H, Amjadian A, Sebt MV, Askari R, Gharaei A (2022) An ensemble method of the machine learning to prognosticate the gastric cancer. Annal Oper Res 17:1–42

    Google Scholar 

  • Ben-Daya M, As’ad R, Seliaman M (2013) An integrated production inventory model with raw material replenishment considerations in a three layer supply chain. Int J Prod Econ 143(1):53–61

    Google Scholar 

  • Christopher M, Peck H, Towill D (2006) A Taxonomy for selecting global supply chain strategies. Int J Logist Manag 17:277–287

    Google Scholar 

  • Diabat A, Al-Salem M (2015) An integrated supply chain problem with environmental considerations. Int J Prod Econ 164:330–338

    Google Scholar 

  • Diabat A, Deskoores R (2016) A hybrid genetic algorithm based heuristic for an integrated supply chain problem. J Manuf Syst 38:172–180

    Google Scholar 

  • Duan C, Deng C, Gharaei A, Wu J, Wang B (2018) Selective maintenance scheduling under stochastic maintenance quality with multiple maintenance actions. Int J Prod Res 56(23):7160–7178

    Google Scholar 

  • Ensafian H, Yaghoubi S, Yazdi MM (2017) Raising quality and safety of platelet transfusion services in a patient-based integrated supply chain under uncertainty. Comput Chem Eng 106:355–372

    Google Scholar 

  • Fabbe-Costes N, Jahre M (2008) Supply chain integration and performance: a review of the evidence. Int J Logist Manag 19:130–154

    Google Scholar 

  • Freund, R. M., & Mizuno, S. (2000). Interior point methods: current status and future directions. High Performance Optimization, 441–466

  • Frohlich MT, Westbrook R (2001) Arcs of integration: an international study of supply chain strategies. J Oper Manag 19:185–200

    Google Scholar 

  • Gharaei A, Almehdawe E (2020) Economic growing quantity. Int J Prod Econ 223:107517

    Google Scholar 

  • Gharaei A, Almehdawe E (2021) Optimal sustainable order quantities for growing items. J Clean Prod 307:127216

    Google Scholar 

  • Gharaei A, Pasandideh SHR (2017a) Four-echelon integrated supply chain model with stochastic constraints under shortage condition: Sequential quadratic programming. Ind Eng Manag Syst 16(3):316–329

    Google Scholar 

  • Gharaei A, Pasandideh SHR (2017b) Modeling and optimization of four-level integrated supply chain with the aim of determining the optimum stockpile and period length: Sequential quadratic programming. J Ind Prod Eng 34(7):529–541

    Google Scholar 

  • Gharaei A, Naderi B, Mohammadi M (2015) Optimization of rewards in single machine scheduling in the rewards-driven systems. Manag Sci Lett 5(6):629–638

    Google Scholar 

  • Gharaei A, Pasandideh SHR, Akhavan Niaki ST (2018) An optimal integrated lot sizing policy of inventory in a bi-objective multi-level supply chain with stochastic constraints and imperfect products. J Ind Prod Eng 35(1):6–20

    Google Scholar 

  • Gharaei A, Karimi M, Shekarabi SAH (2019b) An integrated multi-product, multi-buyer supply chain under penalty, green, and quality control polices and a vendor managed inventory with consignment stock agreement: the outer approximation with equality relaxation and augmented penalty algorithm. Appl Math Model 69:223–254

    MathSciNet  MATH  Google Scholar 

  • Gharaei A, Hoseini Shekarabi SA, Karimi M (2020a) Modelling and optimal lot-sizing of the replenishments in constrained, multi-product and bi-objective EPQ models with defective products: generalised cross decomposition. Int J Syst Sci Oper Logist 7(3):262–274

    Google Scholar 

  • Gharaei A, Karimi M, Hoseini Shekarabi SA (2020b) Joint economic lot-sizing in multi-product multi-level integrated supply chains: generalized benders decomposition. Int J Syst Sci Oper Logist 7(4):309–325

    Google Scholar 

  • Gharaei A, Diallo C, Venkatadri U (2022b) Optimal economic growing quantity for reproductive farmed animals under profitable by-products and carbon emission considerations. J Clean Prod 374:133849

    Google Scholar 

  • Gharaei A., & Pasandideh SHR (2016). Modeling and optimization the four-level integrated supply chain: sequential quadratic programming. Int J Comput Sci Inform Secur. 14(7)

  • Gharaei A, Pasandideh SHR, Khamseh AA (2017) Inventory model in a four-echelon integrated supply chain: modeling and optimization. J Modell Manag. 12(4):739–762

    Google Scholar 

  • Gharaei A, Hoseini Shekarabi SA, Karimi M, Pourjavad E, Amjadian A (2019a) An integrated stochastic EPQ model under quality and green policies: generalised cross decomposition under the separability approach. Int J Syst Sci Oper Logist 1–13:119–131

    Google Scholar 

  • Gharaei A, Amjadian A, Shavandi A (2021a) An integrated reliable four-level supply chain with multi-stage products under shortage and stochastic constraints. Int J Syst Sci Oper Logist 11:1–22

    Google Scholar 

  • Gharaei A, Hoseini Shekarabi SA, & Karimi M (2021b). Optimal lot-sizing of an integrated EPQ model with partial backorders and re-workable products: an outer approximation. Int J Syst Sci Oper Logist, 1–17

  • Gharaei A, Karimi M, & Hoseini Shekarabi SA. (2021c). Vendor-managed inventory for joint replenishment planning in the integrated qualitative supply chains: generalised benders decomposition under separability approach. Int J Syst Sci Oper Logist, 1–15

  • Gharaei A, Amjadian A, Amjadian A, Shavandi A, Hashemi A, Taher M, & Mohamadi N (2022a). An integrated lot-sizing policy for the inventory management of constrained multi-level supply chains: null-space method. Int J Syst Sci Oper Logist, 1–14

  • Goodarzian F, Ghasemi P, Gonzalez EDS, Tirkolaee EB (2023) A sustainable-circular citrus closed-loop supply chain configuration: pareto-based algorithms. J Environ Manage 328:116892

    Google Scholar 

  • Hashemi L, Mahmoodi A, Jasemi M, Millar RC, & Laliberté J (2021). Modeling a robust multi-objective locating-routing problem with bounded delivery time using meta-heuristic algorithms. Smart Resil Trans

  • Hoseini Shekarabi SA, Gharaei A, Karimi M (2019) Modelling and optimal lot-sizing of integrated multi-level multi-wholesaler supply chains under the shortage and limited warehouse space: generalised outer approximation. Int J Syst Sci Oper Logist 6(3):237–257

    Google Scholar 

  • Kahn KA (1998) Marketing’s integration with other departments. J Bus Res 42:53–62

    Google Scholar 

  • Kannan VR, Tan KC (2010) Supply chain integration: cluster analysis of the impact of span of integration. Supply Chain Manag Int J 15:207–215

    Google Scholar 

  • Karaoglan I, Altiparmak F, Kara I, Dengiz B (2012) The location-routing problem with simultaneous pickup and delivery: formulations and a heuristic approach. Omega 40(4):465–477

    MATH  Google Scholar 

  • Khan M, Jaber MY, Ahmad AR (2014) An integrated supply chain model with errors in quality inspection and learning in production. Omega 42:16–24

    Google Scholar 

  • Kim SW (2006) Effects of supply chain management practices, integration and competitive capability on performance. Supply Chain Manag Int J 11:241–248

    Google Scholar 

  • Lambert DM (2008) An executive summary of supply chain management: processes, partnerships, performance, Supply Chain Management Institute. Vol. 3

  • Mahmoudi A, Piri M (2013) An innovative methodology to make a questionnaire positive definite by the statistical software of SPSS. Middle-East J Sci Res 13(9):1267–1274

    Google Scholar 

  • Mahmoudi A, Hashemi L, Jasemi M, Pope J (2021) A comparison on particle swarm optimization and genetic algorithm performances in deriving the efficient frontier of stocks portfolios based on a mean-lower partial moment model. Int J Financ Econ 26(4):5659–5665

    Google Scholar 

  • Mehrjoo S, Jasemi M, Mahmoudi A (2014) A new methodology for deriving the efficient frontier of stocks portfolios: an advanced risk-return model. J AI Data Mining 2(2):113–123

    Google Scholar 

  • Ou CS, Liu FC, Hung YC, Yen DC (2010) A structural model of supply chain management on firm performance. Int J Oper Prod Manag 30:526–545

    Google Scholar 

  • Pagell M (2004) Understanding the factors that enable and inhibit the integration of operations, purchasing and logistics. J Oper Manag 22:459–487

    Google Scholar 

  • Pamela D, Pietro R (2011) Supply chain integration and efficiency performance: a study on the interactions between customer and supplier integration. Supply Chain Manag Int J 16:220–230

    Google Scholar 

  • Pasandideh SHR, Niaki STA, Gharaei A (2015) Optimization of a multiproduct economic production quantity problem with stochastic constraints using sequential quadratic programming. Knowl-Based Syst 84:98–107

    Google Scholar 

  • Ranjbarzadeh R, Dorosti S, Ghoushchi SJ, Caputo A, Tirkolaee EB, Ali SS, & Bendechache M (2022) Breast tumor localization and segmentation using machine learning techniques: Overview of datasets, findings, and methods. Comput Biol Med, 106443

  • Saffarian S, Mahmoudi A, Shafiee M, Jasemi M, Hashemi L (2020) Measuring the effectiveness of AHP and fuzzy AHP models in environmental risk assessment of a gas power plant. Hum Ecol Risk Assess Int J 27(5):1227–1241

    Google Scholar 

  • Samani MRG, Torabi SA, Hosseini-Motlagh SM (2018) Integrated blood supply chain planning for disaster relief. Int J Disaster Risk Reduct 27:168–188

    Google Scholar 

  • Singh PJ, Power D (2009) The nature and effectiveness of collaboration between firms, their customers and suppliers: a supply chain perspective. Supply Chain Manag Int J 14:189–200

    Google Scholar 

  • Souza RLC, Ghasemi A, Saif A, Gharaei A (2022) Robust job-shop scheduling under deterministic and stochastic unavailability constraints due to preventive and corrective maintenance. Comput Ind Eng 168:108130

    Google Scholar 

  • Stevens G (1989) Integrating the supply chain. Int J Phys Distrib Mater Manag 19:3–8

    Google Scholar 

  • Sun W, & Yuan YX (2006) Optimization theory and methods: nonlinear programming Vol 1 Springer Science & Business Media

  • Taleizadeh AA, Safaei AZ, Bhattacharya A, Amjadian A (2022) Online peer-to-peer lending platform and supply chain finance decisions and strategies. Annal Oper Res 315(1):397–427

    MathSciNet  MATH  Google Scholar 

  • Taleizadeh AA, Mahmoudzade Varzi A, Amjadian A, Noori-daryan M, & Konstantaras I (2023) How cash-back strategy affect sale rate under refund and customers credit. Oper Res, 1–31

  • Van der Vaart T, van Donk DP (2008) A critical review of survey-based research in supply chain integration. Int J Prod Econ 111:42–55

    Google Scholar 

  • Vickery SK, Jayaram J, Droge C, Calantone R (2003) The effects of an integrative supply chain strategy on customer service and financial performance: an analysis of direct versus indirect relationships. J Oper Manag, 21(5):523–539

    Google Scholar 

  • Wiengarten F, Humphreys P, Guangming C, Fynes B, McKittrick A (2010) Collaborative supply chain practices and performance: exploring the key role of information quality. Supply Chain Manag Int J 15:463–473

    Google Scholar 

Download references

Funding

The authors have not disclosed any funding.

Author information

Authors and Affiliations

  1. Department of Mechanical and Industrial Engineering, Faculty of Engineering, University of Toronto, Toronto, Canada

    Abolfazl Gharaei

  2. Department of Industrial Engineering, Faculty of Engineering, Kharazmi University, Tehran, Iran

    Alireza Amjadian

  3. Department of Industrial Engineering, Faculty of Engineering, Sharif University of Technology, Tehran, Iran

    Ali Shavandi

  4. Department of Industrial Engineering, Faculty of Engineering, Yazd University, Yazd, Iran

    Amir Amjadian

Authors
  1. Abolfazl Gharaei
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alireza Amjadian
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Ali Shavandi
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Amir Amjadian
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Alireza Amjadian.

Ethics declarations

Conflict of interest

The authors have not disclosed any competing interests.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Gharaei, A., Amjadian, A., Shavandi, A. et al. An augmented Lagrangian approach with general constraints to solve nonlinear models of the large-scale reliable inventory systems. J Comb Optim 45, 78 (2023). https://doi.org/10.1007/s10878-023-01002-z

Download citation

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10878-023-01002-z

Keywords

Search

Navigation