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Network equilibrium of production, transportation and pricing for multi-product multi-market

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Abstract

In this paper, we consider a production, transportation and pricing problem for multi-product multi-market (PTPMM) as a system, and develop a PTPMM network equilibrium model. After allocating each product’s production cost and revenue to each path, we establish a profit network graph. An equilibrium PTPMM matrix and a \(\lambda \)-combination equilibrium are proposed based on a generalization of the well-known Wardrop’s equilibrium principle. The necessary and sufficient conditions for the \(\lambda \)-combination equilibrium are proposed using a linear scalarized profit function. We prove that solving the PTPMM network equilibrium problem can be reduced to the solving of the weak vector variational inequality problem, which proposes an algorithm for the PTPMM problem. Finally, an illustrative example is given to demonstrate an application of these theoretical results.

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References

  • Aadachi T (2005) Third-degree price discrimination, consumption externalities and social welfare. Economica 72:171–178

    Article  Google Scholar 

  • Ahn H-S, Gumus M, Kaminsky P (2007) Pricing and manufacturing decisions when demand is a function of prices in multiple periods. Oper Res 55(6):1039–1057

    Article  MathSciNet  MATH  Google Scholar 

  • Boyd S, Vandenberghe L (2009) Convex optimization. Cambridge University Press, New York

    MATH  Google Scholar 

  • Chalam GA (1994) Fuzzy goal programming (FGP) approach to a stochastic transportation problem under budgetary constraint. Fuzzy Sets Syst 66:293–299

    Article  Google Scholar 

  • Chen GY (1992) Existence of solutions for a vector variational inequality: an extension of Hartmann-Stampacchia theorem. J Optim Theory Appl 74:445–456

    Article  MathSciNet  MATH  Google Scholar 

  • Chen GY, Goh CJ, Yang XQ (1999) Vector network equilibrium problems and nonlinear scalarization methods. Math Meth Oper Res 49:239–253

    MathSciNet  MATH  Google Scholar 

  • Cheng TCE, Wu YN (2006) A multiproduct, multicriterion supply-demand network equilibrium model. Oper Res 54(3):544–554

    Article  MathSciNet  MATH  Google Scholar 

  • Chen GY, Yang XQ (1990) The vector complementarity problem and its equivalences with the weak minimal element in ordered Banach spaces. J Math Anal Appl 153:136–158

    Article  MathSciNet  MATH  Google Scholar 

  • Dalal AJ, Alghalith M (2009) Production decisions under joint price and production uncertainty. Eur J Oper Res 197(1):84–92

    Article  MathSciNet  MATH  Google Scholar 

  • Dantzig GB, Thapa MN (1963) Linear programming 2: theory and extensions. Princeton University Press, New Jersey

    Book  MATH  Google Scholar 

  • Deng SM, Yano CA (2006) Joint production and pricing decisions with setup costs and capacity constraints. Manag Sci 52:741–756

    Article  MATH  Google Scholar 

  • Ebrahimnejad A (2014) A simplified new approach for solving fuzzy transportation problems with generalized trapezoidal fuzzy numbers. Appl Soft Comput 19:171–176

    Article  MathSciNet  Google Scholar 

  • Eksioglu SD, Romeijn HE, Pardalos PM (2006) Cross-facility management of production and transportation planning problem. Comput Oper Res 33:3231–3251

    Article  MATH  Google Scholar 

  • Fischer A (1992) A special Newton-type optimization method. Optimization 24:269–284

    Article  MathSciNet  MATH  Google Scholar 

  • Goh CJ, Yang XQ (2000) Scalarization methods for vector variational inequality. In: Giannessi F (ed) Vector variational inequalities and vector equilibria. Kluwer Academic Publishers, Dordrecht, pp 217–232

  • He Y, He J, Zhu DL, Zhou J (2010) Traffic network equilibrium with capacity constraints and generalized Wardrop equilibrium. Nonlinear Anal Real World Appl 11:4248–4253

    Article  MathSciNet  MATH  Google Scholar 

  • He YJ (2013) Sequential price and quantity decisions under supply and demand risks. Int J Prod Econ 141:541–551

    Article  Google Scholar 

  • Hitchcock FL (1941) The distribution of a product from several sources to numerous localities. J Math Phys 20:224–230

    Article  MathSciNet  MATH  Google Scholar 

  • Hochbaum DS, Hong SP (1996) On the complexity of the production-transportation problem. SIAM J Optim 6(1):250–264

    Article  MathSciNet  MATH  Google Scholar 

  • Huang S, Yang C, Zhang X (2012) Pricing and production decisions in dual-channel supply chains with demand disruptions. Comput Ind Eng 62:70–83

    Article  Google Scholar 

  • Huang S, Yang C, Liu H (2013) Pricing and production decisions in a dual-channel supply chain when production costs are disrupted. Econ Model 30:521–538

    Article  Google Scholar 

  • Kaftal V, Pal D (2008) Third degree price discrimination in linear-demand markets: effects on number of markets served and social welfare. South Econ J 75:558–573

    Google Scholar 

  • Kaur A, Kumar A (2012) A new approach for solving fuzzy transportation problems using generalized trapezoidal fuzzy numbers. Appl Soft Comput 12:1201–1213

    Article  Google Scholar 

  • Kinderlehrer D, Stampacchia G (1980) An introduction to variational inequalities and their application. Academic Press, New York

    MATH  Google Scholar 

  • Koc U, Toptal A, Sabuncuoglu I (2013) A class of joint production and transportation planning problems under different delivery policies. Oper Res Lett 41:54–60

    Article  MathSciNet  MATH  Google Scholar 

  • Lee GM, Kim DS, Lee BS, Yen ND (1998) Vector variational inequality as a tool for studing vector optimization problems. Nonlinear Anal 34:745–765

    Article  MathSciNet  MATH  Google Scholar 

  • Lin Z (2010) On existence of vector equilibrium flows with capacity constraints of arcs. Nonlinear Anal 72:2076–2079

    Article  MathSciNet  MATH  Google Scholar 

  • Liu B, Zhang R, Xiao MD (2010) Joint decision on production and pricing for online dual channel supply chain system. Appl Math Model 34:4208–4218

    Article  MathSciNet  MATH  Google Scholar 

  • Liu SL, Kao C (2004) Solving fuzzy transportation problems based on extension principle. Eur J Oper Res 153:661–674

    Article  MathSciNet  MATH  Google Scholar 

  • Liu SS, Papageorgiou LG (2013) Multiobjective optimisation of production, distribution and capacity planning of global supply chains in the process industry. Omega 41:369–382

    Article  Google Scholar 

  • Li LY, Zhang TZ (2012) Study on the multi-products dynamic pricing model under uncertain demands. Energy Proced 16:1401–1407

    Article  Google Scholar 

  • Meisel F, Kirschstein T, Bierwirth C (2013) Integrated production and intermodal transportation planning in large scale production–distribution-networks. Transp Res Part E 60:62–78

    Article  Google Scholar 

  • Nagurney A (1999) Network economics: a variational inequality approach, Revised 2nd edn. Kluwer Academic Publishers, Berlin

    Book  Google Scholar 

  • Nagurney A (2000) A multiclass, multicriteria traffic network equilibrium model. Math Comput Model 32:393–411

    Article  MathSciNet  MATH  Google Scholar 

  • Nagurney A, Dong J (2002) A multiclass, multicriteria traffic network equilibrium model with elastic demand. Transp Res Part B 36:445–469

    Article  Google Scholar 

  • Ojha A, Das B, Mondal SK, Maiti M (2010) A stochastic discounted multi-objective solid transportation problem for breakable items using analytical hierarchy process. Appl Math Model 34:2256–2271

    Article  MathSciNet  MATH  Google Scholar 

  • Ojha A, Das B, Mondal SK, Maiti M (2014) A transportation problem with fuzzy-stochastic cost. Appl Math Model 38:1464–1481

    Article  MathSciNet  Google Scholar 

  • Pennings E (2001) Price or quantity setting under uncertain demand and capacity constraints: an Examination of the Profits. J Econ 2:157–171

    Article  MATH  Google Scholar 

  • Petruzzi N, Dada M (1999) Pricing and the newsvendor problem: a review with extensions. Oper Res 47(2):183–194

    Article  MATH  Google Scholar 

  • Pirkul H, Jayaraman V (1996) Production, transportation, and distribution planning in a multi-commodity tri-echelon system. Transp Sci 30(4):291–302

    Article  MATH  Google Scholar 

  • Sakawa M, Nishizaki I, Uemura Y (2001) Fuzzy programming and profit and cost allocation for a production and transportation problm. Eur J Oper Res 131:1–15

    Article  MathSciNet  MATH  Google Scholar 

  • Sharkey TC (2011) Network flow problems with pricing decisions. Optim Lett 5:71–83

    Article  MathSciNet  MATH  Google Scholar 

  • Sibdari S, Pyke DF (2014) Dynamic pricing with uncertain production cost: an alternating-move approach. Eur J Oper Res 236:218–228

    Article  MathSciNet  MATH  Google Scholar 

  • Subramanian PK (1993) Gauss–Newton methods for the complementarity problem. J Optim Theory Appl 77:467–482

    Article  MathSciNet  MATH  Google Scholar 

  • Thomas LJ (1970) Price-production decisions with deterministic demand. Manag Sci 16:747–750

    Article  MATH  Google Scholar 

  • Thomas LJ (1974) Price and production decisions with stochastic demand. Oper Res 22:513–518

    Article  MATH  Google Scholar 

  • Wang Y (2006) Joint pricing-production decisions in supply chains of complementary products with uncertain demand. Oper Res 54(6):1110–1127

    Article  MATH  Google Scholar 

  • Wei J, Zhao J (2011) Pricing decisions with retail competition in a fuzzy closed-loop supply chain. Expert Syst Appl 38:11209–11216

    Article  Google Scholar 

  • Yang XQ, Yu H (2005) Vector variational inequalities and dynamic traffic equilibria. In: Giannessi F, Maugeri A (eds) Variational analysis and applications. Springer, New York

    Google Scholar 

  • Yao DQ, Yue XH, John L (2008) Vertical cost information sharing in a supply chain with value-adding retailers. Omega 36:838–851

    Article  Google Scholar 

  • Zhao J, Tang WS, Zhao R, Wei J (2012) Pricing decisions for substitutable products with a common retailer in fuzzy environments. Eur J Oper Res 216:409–419

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The authors are indebted to the anonymous referees for their careful reading of the manuscript and for their useful comments and suggestions which improved the presentation of this work. This research was supported by National Natural Science Foundation of China (Grant No. 71301110) and the Humanities and Social Sciences Foundation of the Ministry of Education (Grant No. 13XJC630015), and also supported by Research Fund for the Doctoral Program of Higher Education of China (Grant No. 20130181120059) and Project of Education Department of Sichuan Province (Grant No. 14ZB0173).

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Authors and Affiliations

  1. Business School, Sichuan University, Chengdu, 610064, China

    Jiuping Xu & Guomin Fang

  2. College of Applied Mathematics, Chengdu University of Information Technology, Chengdu, 610225, China

    Guomin Fang & Zezhong Wu

Authors
  1. Jiuping Xu
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  2. Guomin Fang
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  3. Zezhong Wu
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Correspondence to Jiuping Xu.

Appendix

Appendix

In this section, we give some solutions with letting several special values of parameter \(\lambda \). It is worth pointing out that the solution of \((VI)_\lambda \) should be a set. The following, we only give one of solutions for each special parameter \(\lambda \). The main information is shown in Tables 5, 6 and 7.

Table 5 Some solutions and transportation flows
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Table 6 Production quantity and total profit of products
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Table 7 Demand and price on sub-market
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Xu, J., Fang, G. & Wu, Z. Network equilibrium of production, transportation and pricing for multi-product multi-market. Math Meth Oper Res 84, 567–595 (2016). https://doi.org/10.1007/s00186-016-0557-x

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