Bi-level programming problem in the supply chain and its solution algorithm

  • Haiyan LuoEmail author
  • Linzhong Liu
  • Xun Yang


Enterprise-wide supply chain planning problems naturally exhibit a multi-level decision network structure, where the upper level of a hierarchy may have his objective function and decision space partly determined by other levels. In addition, each planner’s control instruments may allow him to influence the policies at other levels and thereby to improve his own objective function. As a tool, bi-level programming is applied for modeling decentralized decisions in which two decision makers make decisions successively. In this paper, we specifically address bi-level decision-making problems with budget constraint as an attractive feature in the context of enterprise-wide supply chain. We first describe the typical bi-level linear programming problem (BLLPP) and its optimal solution to the penalty function problem, and then, a cooperative decision-making problem in supply chain is modeled as BLLPP. A particle swarm optimization-based computational algorithm is designed to solve the problem, and the numerical example is presented to illustrate the proposed framework.


Decentralized supply chain Bi-level linear programming Budget constraint Particle swarm optimization algorithm 



This study was funded by National Natural Science Foundation of China (Grant Numbers 71671079, 71361018) and Humanities and Social Science Foundation of Ministry of Education of China (Grant Number 15YJCZH107).

Compliance with ethical standards

Conflict of interest

All authors declare that they have no conflict of interest.

Ethical approval

This article does not contain any studies with human participants or animals performed by any of the authors.


  1. Anandalingam G, White D (1990) A solution method for the linear static Stackelberg problem using penalty functions. IEEE Trans Autom Control 35:1170–1173MathSciNetCrossRefzbMATHGoogle Scholar
  2. Bard J (1984) An investigation of the linear three level programming problem. IEEE Trans Syst Man Cybern 14:711–717MathSciNetCrossRefzbMATHGoogle Scholar
  3. Bard J (1998) Practical bi-level optimization: algorithms and applications. Kluwer Academic Publishers, BostonCrossRefzbMATHGoogle Scholar
  4. Bard J, Falk J (1982) An explicit solution to the multi-level programming problem. Comput Oper Res 9:77–100MathSciNetCrossRefGoogle Scholar
  5. Bard J, Moore J (1990) A branch and bound algorithm for the bi-level programming problem. SIAM J Sci Stat Comput 11:281–292CrossRefzbMATHGoogle Scholar
  6. Baumol W, Fabian T (1964) Decomposition, pricing for decentralization and external economies. Manag Sci 11:1–32CrossRefGoogle Scholar
  7. Ben-Ayed O (1993) Bi-level linear programming. Comput Oper Res 20:485–510MathSciNetCrossRefzbMATHGoogle Scholar
  8. Ben-Ayed O, Blair C (1990) Computational difficulties of bi-level linear programming. Oper Res 38:556–560MathSciNetCrossRefzbMATHGoogle Scholar
  9. Bernstein F, Federgruen A (2003) Pricing and replenishment strategies in a distribution system with competing retailers. Oper Res 51(3):409–426CrossRefzbMATHGoogle Scholar
  10. Bialas W, Karwan M (1984) Two-level linear programming. Manage Sci 30:1004–1020MathSciNetCrossRefzbMATHGoogle Scholar
  11. Boyaci Gallego (2002) Coordinating pricing and inventory replenishment policies for one wholesaler and one or more geographically dispersed retailers. Int J Prod Econ 77(2):95–111CrossRefGoogle Scholar
  12. Brandenburger AM, Stuart HW (1996) Value-based business strategy. J Econ Manag Strat 5(1):5–24CrossRefGoogle Scholar
  13. Calvete H, Gale C, Mateo P (2008) A new approach for solving linear bilevel problems using genetic algorithms. Eur J Oper Res 188:14–28MathSciNetCrossRefzbMATHGoogle Scholar
  14. Calvete H, Gale C, Dempe S, Lohse S (2012) Bi-level problems over polyhedra with extreme point optimal solutions. J Global Optim 53:573–586MathSciNetCrossRefzbMATHGoogle Scholar
  15. Carrasqueira P, Alves M, Antunes C (2017) Bi-level particle swarm optimization and evolutionary algorithm approaches for residential demand response with different user profiles. Inf Sci 418–419:405–420CrossRefGoogle Scholar
  16. Chirgui Z (2005) The economies of the smart card industry: towards coopetitive strategies. Econ Innov N Technol 14(6):455–477CrossRefGoogle Scholar
  17. Choi S (1991) Price competition in a channel structure with a common retailer. Market Sci 10(4):271–296MathSciNetCrossRefGoogle Scholar
  18. Clerc M, Kennedy J (2002) The particle swarm–explosion, stability, and convergence in a multidimensional complex space. IEEE Trans Evol Comput 6:58–73CrossRefGoogle Scholar
  19. Coello Coello C (2000) Use of a self-adaptive penalty approach for engineering optimization problems. Comput Ind 41:113–127CrossRefzbMATHGoogle Scholar
  20. Coello Coello C, Christiansen A (1997) A simple genetic algorithm for the design of reinforced concrete beams. Eng Comput 13:185–196CrossRefGoogle Scholar
  21. Cruz J (1978) Leader-follower strategies for multilevel systems. IEEE Trans Autom Control 23:244–255MathSciNetCrossRefzbMATHGoogle Scholar
  22. Dantzig G, Wolfe P (1960) Decomposition principle for linear programs. Oper Res 8:101–111CrossRefzbMATHGoogle Scholar
  23. Das T, Teng B (2000) Instabilities of strategic alliances: an internal tensions perspective. Organ Sci 11(1):77–101CrossRefGoogle Scholar
  24. Dasgupta D, Michalewicz Z (1997) Evolutionary algorithms in engineering applications, 1st edn. Springer, BerlinCrossRefzbMATHGoogle Scholar
  25. Davood M, Seyed H, Ashkan H (2016) A game theoretic analysis in capacity-constrained supplier-selection and cooperation by considering the total supply chain inventory costs. Int J Prod Econ 181:87–97CrossRefGoogle Scholar
  26. Deb K, Sinha A (2010) An efficient and accurate solution methodology for bi-level multi-objective programming problems using a hybrid evolutionary-local-search algorithm. Evol Comput 18:403–449CrossRefGoogle Scholar
  27. Dempe S, Zemkoho A (2012) On the Karush-Kuhn-Tucker reformulation of the bi-level optimization problem. Nonlinear Anal: Theory, Methods Appl 75:1202–1218CrossRefzbMATHGoogle Scholar
  28. Eberhart R, Kennedy J (1995) A new optimizer using particle swarm theory. In: Proceedings of the sixth international symposium on micromachine and human science, 1995 (MHS’95). IEEE Nagoya, Japan: IEEE, pp 39–43Google Scholar
  29. Esmaeili M, Aryanezhad MB, Zeephongsekul P (2009) A game theory approach in seller-buyer supply chain. Eur J Oper Res 195:442–448MathSciNetCrossRefzbMATHGoogle Scholar
  30. Gendreau M, Marcotte P, Savard G (1996) A hybrid tabu-ascent algorithm for the linear bi-level programming problem. J Global Optim 8:217–233MathSciNetCrossRefzbMATHGoogle Scholar
  31. Granot D, Sosic GA (2003) Three-stage model for a decentralized distribution system of retailers. Oper Res 51(5):771–784MathSciNetCrossRefzbMATHGoogle Scholar
  32. Hansen P, Jaumard B, Savard G (1992) New branch-and-bound rules for linear bi-level programming. SIAM J Sci Stat Comput 13:1194–1217CrossRefzbMATHGoogle Scholar
  33. He Q, Wang L (2007) An effective co-evolutionary particle swarm optimization for constrained engineering design problems. Eng Appl Artif Intell 20:89–99CrossRefGoogle Scholar
  34. Hejazi S, Memariani A, Jahanshahloo G, Sepehri M (2002) Linear bi-level programming solution by genetic algorithm. Comput Oper Res 29:1913–1925MathSciNetCrossRefzbMATHGoogle Scholar
  35. Heppner F, Grenander U (1990) A stochastic non-linear model for bird flocking. In: Krasner S (ed) The Ubiquity of Chaos Washington, D.C.: American Association for the Advancement of Science, 1st edn. American Association for the Advancement of Science, Washington, D.C., pp 233–238Google Scholar
  36. Kennedy J, Eberhart R (1995) Particle swarm optimization. In: Proceedings of the 1995 IEEE international conference on neural networks. Perth, Australia: Piscataway, NJ, USA: IEEE, pp 1942–1948Google Scholar
  37. Kim K, Hwang H (1989) Simultaneous improvement of supplier’s profit and buyer’s cost by utilizing quantity discounts. J Oper Res Soc 40:255–256CrossRefzbMATHGoogle Scholar
  38. Kohli R, Park H (1989) A cooperative game theory model of quantity discounts. Manag Sci 35:693–707CrossRefzbMATHGoogle Scholar
  39. Kuo R, Han Y (2011) A hybrid of genetic algorithm and particle swarm optimization for solving bi-level linear programming problem—a case study on supply chain model. Appl Math Model 35:3905–3917MathSciNetCrossRefzbMATHGoogle Scholar
  40. Kuo R, Huang C (2009) Application of particle swarm optimization algorithm for solving bi-level linear programming problem. Comput Math Appl 58:678–685MathSciNetCrossRefzbMATHGoogle Scholar
  41. Kuo R, Lee Y, Zulvia F, Tien F (2015) Solving bi-level linear programming problem through hybrid of immune genetic algorithm and particle swarm optimization algorithm. Appl Math Comput 266:1013–1026MathSciNetzbMATHGoogle Scholar
  42. Li S, Huang Z, Ashley A (1996) Improving buyer seller system cooperation through inventory control. Int J Prod Econ 43(1):312–323CrossRefGoogle Scholar
  43. Liu Q, Wei W, Yuan H, Zhan ZH, Li Y (2016) Topology selection for particle swarm optimization. Inf Sci 363:154–173CrossRefGoogle Scholar
  44. Liu L, Luo H, Mu H, yang J, Li X (2018) A self-adaptive hybrid particle swarm optimization algorithm. Inf Sci, (submitted)Google Scholar
  45. Luo Y (2007) Coopetition perspective of global competition. J World Bus 42(1):129–144CrossRefGoogle Scholar
  46. Lynn N, Ali M, Suganthan P (2018) Population topologies for particle swarm optimization and differential evolution. Swarm Evol Comput 39:24–35CrossRefGoogle Scholar
  47. Marinakis Y, Marinaki M (2013) Particle swarm optimization with expanding neighborhood topology for the permutation flowshop scheduling problem. Soft Comput 17:1159–1173CrossRefzbMATHGoogle Scholar
  48. Mathieu R, Pittard L, Anandalingam G (1994) Genetic algorithm based approach to bi-level linear programming. Oper Res 28:1–21MathSciNetCrossRefzbMATHGoogle Scholar
  49. Maurice C (2006) Stagnation analysis in particle swarm optimization or what happens when nothing happens. Technical report. Accessed 9 Dec 2018
  50. Mendes R, Kennedy J, Neves J (2004) The fully informed particle swarm: simpler, maybe better. IEEE Trans Evol Comput 8:204–210CrossRefGoogle Scholar
  51. Metropolis N, Rosenbluth A, Rosenbluth M, Teller A, Teller E (1953) Equation of state calculations by fast computing machines. J Chem Phys 2:1087–1092CrossRefGoogle Scholar
  52. Mladenovic N, Hansen P (1997) Variable neighborhood search. Computers. Oper Res 24:1097–1100MathSciNetzbMATHGoogle Scholar
  53. Poli R, Kennedy J, Blackwell T (2007) Particle swarm optimization—an overview. Swarm Intell 1:33–57CrossRefGoogle Scholar
  54. Roghanian E, Sadiadi S, Aryanezhad M (2007) A probabilistic bi-level linear multi-objective programming problem to supply chain planning. Appl Math Comput 188:786–800MathSciNetzbMATHGoogle Scholar
  55. Ryu J, Dua V, Efstratios N (2004) A bi-level programming framework for enterprise-wide process networks under uncertainty. Comput Chem Eng 28:1121–1129CrossRefGoogle Scholar
  56. Samma H, Lim C, Saleh J (2016) A new reinforcement learning-based metric particle swarm optimizer. Appl Soft Comput 43:276–297CrossRefGoogle Scholar
  57. Shi Y, Eberhart R (1998) A modified particle swarm optimizer. In: 1998 IEEE international conference on evolutionary computation proceedings. IEEE World congress on computational intelligence. Anchorage, AK, USA: Piscataway, NJ, USA: IEEE, pp 69–73Google Scholar
  58. Wan Z, Wang G, Sun B (2013) A hybrid intelligent algorithm by combining particle swarm optimization with chaos searching technique for solving nonlinear bi-level programming problems. Swarm Evol Comput 8:26–32CrossRefGoogle Scholar
  59. Wang Q (2004) Coordinating independent buyers with integer-ratio time coordination and quantity discounts. Naval Res Log 51(3):316–331MathSciNetCrossRefzbMATHGoogle Scholar
  60. Wang L, Yang B, Orchard J (2016) Particle swarm optimization using dynamic tournament topology. Appl Soft Comput 48:584–596CrossRefGoogle Scholar
  61. White D, Anandalingam G (1993) A penalty function approach for solving bi-level linear programs. J Global Optim 3:397–419MathSciNetCrossRefzbMATHGoogle Scholar
  62. Wu T, Shi L, Geunes J et al (2011) An optimization framework for solving capacitated multi-level lot-sizing problems with backlogging. Eur J Oper Res 214:428–441MathSciNetCrossRefzbMATHGoogle Scholar
  63. Yeh K, Whittaker C, Realff M, Lee J (2015) Two stage stochastic bi-level programming model of a pre-established timberlands supply chain with biorefinery investment interests. Comput Chem Eng 73:141–153CrossRefGoogle Scholar
  64. Yin Y (2000) Genetic algorithm based approach for bi-level programming models. J Transp Eng 126:115–120CrossRefGoogle Scholar
  65. Yu G, Liang H, George Q (2006) Leader-follower game in vender-managed inventory system with limited production capacity considering wholesale and retail prices. Int J Log: Res Appl 9:335–350CrossRefGoogle Scholar

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.School of Traffic & TransportationLanzhou Jiaotong UniversityLanzhouChina
  2. 2.School of Economics & ManagementLanzhou Jiaotong UniversityLanzhouChina

Personalised recommendations