Skip to main content
Springer Nature Link
Log in

A robust stochastic possibilistic programming model for dynamic supply chain network design with pricing and technology selection decisions

  • Application Article
  • Published:
OPSEARCH Aims and scope Submit manuscript

Abstract

This work considers a multi-period multi-echelon multi-product dynamic supply chain network design problem for both strategic and tactical decisions. Strategic decisions include the determination of the locations of the facilities, the capacities of the open facilities, and the capacities of the dedicated and the flexible technologies, and the tactical decisions include the determination of the prices of the products, the flows of materials and products among the locations, and the quantities of the products to produce in each plant in each period. The demand at each customer zone is modeled by a logit price-response function and is approximated by a piecewise linear function. A mixed-integer nonlinear programming model is developed to maximize the expected net present value while making these decisions. A robust possibilistic stochastic programming approach is used to deal with price-sensitive demands under hybrid, i.e., disruption and operational, uncertainties. The model considers the effect of robustness level on technology selection and price decisions, and enables tradeoffs between the robustness level and the expected net present value. The applicability of the model and the performance of the solution approach are examined through computational experiments. The results show that the optimal technology investment is a function of the types of uncertainties and the flexible-to-dedicated technology cost ratio. The results also show a significant advantage of the proposed robust possibilistic stochastic programming model over the other models in the simultaneous controllability of the possibilistic and scenario variabilities. The sensitivity of some key parameters in the model are analyzed in the computational experiments.

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

Access this article

Log in via an institution

Subscribe and save

Springer+ Basic
$34.99 /Month
  • Get 10 units per month
  • Download Article/Chapter or eBook
  • 1 Unit = 1 Article or 1 Chapter
  • Cancel anytime
Subscribe now

Buy Now

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6

Similar content being viewed by others

References

  1. Ivanov, D.: An adaptive framework for aligning (re)planning decisions on supply chain strategy, design, tactics, and operations. Int. J. Prod. Res. 48(13), 3999–4017 (2010). https://doi.org/10.1080/00207540902893417

    Article  Google Scholar 

  2. Melo, M.T., Nickel, S., Saldanha-Da-Gama, F.: A tabu search heuristic for redesigning a multi-echelon supply chain network over a planning horizon. Int. J. Prod. Econ. 136(1), 218–230 (2012). https://doi.org/10.1016/j.ijpe.2011.11.022

    Article  Google Scholar 

  3. Melo, M.T., Nickel, S., Saldanha-da-Gama, F.: Dynamic multi-commodity capacitated facility location: a mathematical modeling framework for strategic supply chain planning. Comput. Oper. Res. 33(1), 181–208 (2006)

    Article  Google Scholar 

  4. Correia, I., Melo, T.: A multi-period facility location problem with modular capacity adjustments and flexible demand fulfillment. Comput. Ind. Eng. 110, 307–321 (2017). https://doi.org/10.1016/j.cie.2017.06.003

    Article  Google Scholar 

  5. Correia, I., Melo, T., Saldanha-Da-Gama, F.: Comparing classical performance measures for a multi-period, two-echelon supply chain network design problem with sizing decisions. Comput. Ind. Eng. 64(1), 366–380 (2013). https://doi.org/10.1016/j.cie.2012.11.001

    Article  Google Scholar 

  6. Cortinhal, M.J., Lopes, M.J., Melo, M.T.: Dynamic design and re-design of multi-echelon, multi-product logistics networks with outsourcing opportunities: a computational study. Comput. Ind. Eng. 90, 118–131 (2015). https://doi.org/10.1016/j.cie.2015.08.019

    Article  Google Scholar 

  7. Guan, Z., Mou, Y., Sun, M.: Hybrid robust and stochastic optimization for a capital-constrained fresh product supply chain integrating risk-aversion behavior and financial strategies. Comput. Ind. Eng. (2022). https://doi.org/10.1016/j.cie.2022.108224

    Article  Google Scholar 

  8. Guan, Z., Tao, J., Sun, M.: Integrated optimization of resilient supply chain network design and operations under disruption risks. In: Khojasteh, J., Xu, H., Zolfaghari, S. (eds.) Supply Chain Risk Mitigation: Strategies, Methods and Applications, p. Forthcoming. Springer (2022)

  9. Melo, M.T., Nickel, S., Saldanha-da-Gama, F.: Facility location and supply chain management—a review. Eur. J. Oper. Res. 196(2), 401–412 (2009). https://doi.org/10.1016/j.ejor.2008.05.007

    Article  Google Scholar 

  10. Badri, H., Bashiri, M., Hejazi, T.H.: Integrated strategic and tactical planning in a supply chain network design with a heuristic solution method. Comput. Oper. Res. 40(4), 1143–1154 (2013). https://doi.org/10.1016/j.cor.2012.11.005

    Article  Google Scholar 

  11. Martínez-Costa, C., Mas-Machuca, M., Benedito, E., Corominas, A.: A review of mathematical programming models for strategic capacity planning in manufacturing. Int. J. Prod. Econ. 153, 66–85 (2014). https://doi.org/10.1016/j.ijpe.2014.03.011

    Article  Google Scholar 

  12. Jakubovskis, A.: Flexible production resources and capacity utilization rates: a robust optimization perspective. Int. J. Prod. Econ. 189, 77–85 (2017). https://doi.org/10.1016/j.ijpe.2017.03.011

    Article  Google Scholar 

  13. Verter, V., Dasci, A.: The plant location and fexible technology acquisition problem. Eur. J. Oper. Res. 136, 366–382 (2002). https://doi.org/10.1016/S0377-2217(01)00023-6

    Article  Google Scholar 

  14. Fattahi, M., Mahootchi, M., Govindan, K., Moattar Husseini, S.M.: Dynamic supply chain network design with capacity planning and multi-period pricing. Transp. Res. Part E: Log. Transp. Rev. 81, 169–202 (2015). https://doi.org/10.1016/j.tre.2015.06.007

    Article  Google Scholar 

  15. Bish, E.K., Wang, Q.: Optimal investment strategies for flexible resources, considering pricing and correlated demands. Oper. Res. 52(6), 954–964 (2004). https://doi.org/10.1287/opre.1040.0138

    Article  Google Scholar 

  16. Pishvaee, M.S., Razmi, J., Torabi, S.A.: Robust possibilistic programming for socially responsible supply chain network design: a new approach. Fuzzy Sets Syst. 206, 1–20 (2012). https://doi.org/10.1016/j.fss.2012.04.010

    Article  Google Scholar 

  17. Tang, C.S.: Perspectives in supply chain risk management. Int. J. Prod. Econ. 103(2), 451–488 (2006). https://doi.org/10.1016/j.ijpe.2005.12.006

    Article  Google Scholar 

  18. Li, X., Lu, S., Li, Z., Wang, Y., Zhu, L.: Modeling and optimization of bioethanol production planning under hybrid uncertainty: a heuristic multi-stage stochastic programming approach. Energy 245, 123285 (2022). https://doi.org/10.1016/j.energy.2022.123285

    Article  Google Scholar 

  19. Ahmadi, E., Masel, D., Hostetler, S.: A robust stochastic decision-making model for inventory allocation of surgical supplies to reduce logistics costs in hospitals: a case study. Oper. Res. Health Care 20, 33–44 (2019). https://doi.org/10.1016/J.ORHC.2018.09.001

    Article  Google Scholar 

  20. Ahmadi, E., Masel, D.T., Hostetler, S., Maihami, R., Ghalehkhondabi, I.: A centralized stochastic inventory control model for perishable products considering age-dependent purchase price and lead time. TOP 28(1), 231–269 (2020). https://doi.org/10.1007/s11750-019-00533-1

    Article  Google Scholar 

  21. Ahmadi, E., Mosadegh, H., Maihami, R., Ghalehkhondabi, I., Sun, M., Süer, G.A.: Intelligent inventory management approaches for perishable pharmaceutical products in a healthcare supply chain. Comput. Oper. Res. 147, 105968 (2022). https://doi.org/10.1016/j.cor.2022.105968

    Article  Google Scholar 

  22. Farrokh, M., Azar, A., Jandaghi, G., Ahmadi, E.: A novel robust fuzzy stochastic programming for closed loop supply chain network design under hybrid uncertainty. Fuzzy Sets Syst. 341, 69–91 (2018). https://doi.org/10.1016/j.fss.2017.03.019

    Article  Google Scholar 

  23. Guo, Y., Shi, Q., Guo, C., Li, J., You, Z., Wang, Y.: Designing a sustainable-remanufacturing closed-loop supply chain under hybrid uncertainty: cross-efficiency sorting multi-objective optimization. Comput. Ind. Eng. 172(PA), 108639 (2022). https://doi.org/10.1016/j.cie.2022.108639

    Article  Google Scholar 

  24. Keyvanshokooh, E., Ryan, S.M., Kabir, E.: Hybrid robust and stochastic optimization for closed-loop supply chain network design using accelerated Benders decomposition. Eur. J. Oper. Res. 249(1), 76–92 (2016). https://doi.org/10.1016/j.ejor.2015.08.028

    Article  Google Scholar 

  25. Bassamboo, A., Randhawa, R.S., Van Mieghem, J.A.: Optimal flexibility configurations in newsvendor networks: going beyond chaining and pairing. Manage. Sci. 56(8), 1285–1303 (2010)

    Article  Google Scholar 

  26. Bose, D., Chatterjee, A.K., Barman, S.: Towards dominant flexibility configurations in strategic capacity planning under demand uncertainty. Opsearch 53(3), 604–619 (2016)

    Article  Google Scholar 

  27. Eppen, G.D., Martin, R.K., Schrage, L.: OR practice—a scenario approach to capacity planning. Oper. Res. 37(4), 517–527 (1989)

    Article  Google Scholar 

  28. Fine, C.H., Freund, R.M.: Optimal investment in product-flexible manufacturing capacity. Manage. Sci. 36(4), 449–466 (1990)

    Article  Google Scholar 

  29. Verter, V.: An integrated model for facility location and technology acquisition. Comput. Oper. Res. 29, 583–592 (2002). https://doi.org/10.1016/S0305-0548(00)00057-5

    Article  Google Scholar 

  30. Ahmed, S., Sahinidis, N.V.: Selection, acquisition, and allocation of manufacturing technology in a multi-period environment. Eur. J. Oper. Res. 189(3), 807–821 (2008). https://doi.org/10.1016/j.ejor.2006.11.046

    Article  Google Scholar 

  31. Chen, Z.-L., Li, S., Tirupati, D.: A scenario-based stochastic programming approach for technology and capacity planning. Comput. Oper. Res. 29(7), 781–806 (2002)

    Article  Google Scholar 

  32. Li, S., Tirupati, D.: Dynamic capacity expansion problem with multiple products: Technology selection and timing of capacity additions. Oper. Res. 42(5), 958–976 (1994)

    Article  Google Scholar 

  33. Lim, S., Kim, Y.: An integrated approach to dynamic plant location and capacity planning. J. Oper. Res. Soc. 50(12), 1205–1216 (2014). https://doi.org/10.1057/palgrave.jors.2600849

    Article  Google Scholar 

  34. Ji, S., Tang, J., Sun, M., Luo, R.: Multi-objective optimization for a combined location-routing-inventory system considering carbon-capped differences. J. Ind. Manag. Optim. 18(3), 1949–1977 (2022)

    Article  Google Scholar 

  35. Xin, C., Zhou, Y., Sun, M., Chen, X.: Strategic inventory and dynamic pricing for a two-echelon green product supply chain. J. Clean. Prod. 363, 132422 (2022)

    Article  Google Scholar 

  36. Govindan, K., Gholizadeh, H.: Robust network design for sustainable-resilient reverse logistics network using big data: a case study of end-of-life vehicles. Transp. Res. Part E: Log. Transp. Rev. 149, 102279 (2021). https://doi.org/10.1016/j.tre.2021.102279

    Article  Google Scholar 

  37. Papageorgiou, L.G.: Supply chain optimisation for the process industries: advances and opportunities. Comput. Chem. Eng. 33(12), 1931–1938 (2009). https://doi.org/10.1016/j.compchemeng.2009.06.014

    Article  Google Scholar 

  38. Özkir, V., Başligil, H.: Multi-objective optimization of closed-loop supply chains in uncertain environment. J. Clean. Prod. 41, 114–125 (2013). https://doi.org/10.1016/j.jclepro.2012.10.013

    Article  Google Scholar 

  39. Gholami, R.A., Sandal, L.K., Ubøe, J.: A solution algorithm for multi-period bi-level channel optimization with dynamic price-dependent stochastic demand. Omega 102, 102297 (2021)

    Article  Google Scholar 

  40. Raza, S.A., Abdullakutty, F.C., Rathinam, S., Govindaluri, S.M.: Multi-objective framework for process mean selection and price differentiation with leakage effects under price-dependent stochastic demand. Comput. Ind. Eng. 127, 698–708 (2019)

    Article  Google Scholar 

  41. Shah, N.H., Soni, H.: Continuous review inventory model for fuzzy price dependent demand. Int. J. Model. Oper. Manag. 1(3), 209–222 (2011)

    Google Scholar 

  42. Yu, Y., Zhu, J., Wang, C.: A newsvendor model with fuzzy price-dependent demand. Appl. Math. Model. 37(5), 2644–2661 (2013)

    Article  Google Scholar 

  43. Govindan, K., Darbari, J.D., Agarwal, V., Jha, P.C.: Fuzzy multi-objective approach for optimal selection of suppliers and transportation decisions in an eco-efficient closed loop supply chain network. J. Clean. Prod. 165, 1598–1619 (2017). https://doi.org/10.1016/j.jclepro.2017.06.180

    Article  Google Scholar 

  44. Govindan, K., Fattahi, M., Keyvanshokooh, E.: Supply chain network design under uncertainty: a comprehensive review and future research directions. Eur. J. Oper. Res. (2017). https://doi.org/10.1016/j.ejor.2017.04.009

    Article  Google Scholar 

  45. Mulvey, J.M., Vanderbei, M.J., Zenios, S.A.: Robust optimization of large scale systems. Oper. Res. 43(2), 264–281 (1995)

    Article  Google Scholar 

  46. Ghahremani-Nahr, J., Kian, R., Sabet, E.: A robust fuzzy mathematical programming model for the closed-loop supply chain network design and a whale optimization solution algorithm. Expert Syst. Appl. 116, 454–471 (2019). https://doi.org/10.1016/j.eswa.2018.09.027

    Article  Google Scholar 

  47. Ahmadi-Javid, A., Hoseinpour, P.: Incorporating location, inventory and price decisions into a supply chain distribution network design problem. Comput. Oper. Res. 56, 110–119 (2015). https://doi.org/10.1016/j.cor.2014.07.014

    Article  Google Scholar 

  48. Ghomi-Avili, M., Naeini, S.G.J., Tavakkoli-Moghaddam, R., Jabbarzadeh, A.: A fuzzy pricing model for a green competitive closed-loop supply chain network design in the presence of disruptions. J. Clean. Prod. 188, 425–442 (2018)

    Article  Google Scholar 

  49. Duc, T.T.H., Loi, N.T., Buddhakulsomsiri, J.: Buyback contract in a risk-averse supply chain with a return policy and price dependent demand. Int. J. Log. Syst. Manag. 30(3), 298–329 (2018)

    Google Scholar 

  50. Ullah, M., Khan, I., Sarkar, B.: Dynamic pricing in a multi-period newsvendor under stochastic price-dependent demand. Mathematics 7(6), 520 (2019)

    Article  Google Scholar 

  51. Ramezani, M., Kimiagari, A.M., Karimi, B., Hejazi, T.H.: Closed-loop supply chain network design under a fuzzy environment. Knowl.-Based Syst. 59, 108–120 (2014). https://doi.org/10.1016/j.knosys.2014.01.016

    Article  Google Scholar 

  52. Vijai, J.P.: Production network, technology choice, capacity investment and inventory sourcing decisions: operational hedging under demand uncertainty. Opsearch 58(4), 1164–1191 (2021). https://doi.org/10.1007/s12597-021-00511-x

    Article  Google Scholar 

  53. Boyabatlı, O., Toktay, L.B.: Stochastic capacity investment and flexible vs. dedicated technology choice in imperfect capital markets. Manag. Sci. 57(12), 2163–2179 (2011)

    Article  Google Scholar 

  54. Nagaraju, D., Kumar, B.K., Narayanan, S.: On the optimality of inventory and shipment policies in a two-level supply chain under quadratic price dependent demand. Int. J. Log. Syst. Manag. 35(4), 486–510 (2020)

    Google Scholar 

  55. Caliskan-Demirag, O., Chen, Y.F., Li, J.: Channel coordination under fairness concerns and nonlinear demand. Eur. J. Oper. Res. 207(3), 1321–1326 (2010)

    Article  Google Scholar 

  56. Xu, M., Qi, X., Yu, G., Zhang, H., Gao, C.: The demand disruption management problem for a supply chain system with nonlinear demand functions. J. Syst. Sci. Syst. Eng. 12(1), 82–97 (2003)

    Article  Google Scholar 

  57. Phillips, R.L.: Pricing and Revenue Optimization. Stanford University Press (2005)

    Book  Google Scholar 

  58. Ahmadi-Javid, A., Ghandali, R.: An efficient optimization procedure for designing a capacitated distribution network with price-sensitive demand. Optim. Eng. 15(3), 801–817 (2014). https://doi.org/10.1007/s11081-013-9245-3

    Article  Google Scholar 

  59. Talluri, K.T., van Ryzin, G.: The Theory and Practice of Revenue Management. Kluwer Academic Publishers, Boston (2004)

    Book  Google Scholar 

  60. Dubois, D., Prade, H.: Systems of linear fuzzy constraints. Fuzzy Sets Syst. 3(1), 37–48 (1980)

    Article  Google Scholar 

  61. Beale, E.M.L., Tomlin, J.A.: Special facilities in a general mathematical programming system for non-convex problems using ordered sets of variables. OR 69(447–754), 99 (1970)

    Google Scholar 

  62. Babazadeh, R., Razmi, J., Pishvaee, M.S., Rabbani, M.: A sustainable second-generation biodiesel supply chain network design problem under risk. Omega (UK) 66, 258–277 (2017). https://doi.org/10.1016/j.omega.2015.12.010

    Article  Google Scholar 

  63. Tomlin, J.A.: Special ordered sets and an application to gas supply operation planning. Math. Program. 42, 69–84 (1988)

    Article  Google Scholar 

  64. Tsai, W.-H., Chang, Y.-C., Lin, S.-J., Chen, H.-C., Chu, P.-Y.: A green approach to the weight reduction of aircraft cabins. J. Air Transp. Manag. 40, 65–77 (2014)

    Article  Google Scholar 

  65. Ben-Tal, A., El Ghaoui, L., Nemirovski, A.: Robust Optimization, vol. 28. Princeton University Press (2009)

    Book  Google Scholar 

  66. Liu, B., Iwamura, K.: Chance constrained programming with fuzzy parameters. Fuzzy Sets Syst. 94(2), 227–237 (1998). https://doi.org/10.1016/S0165-0114(96)00236-9

    Article  Google Scholar 

  67. Hasani, A., Khosrojerdi, A.: Robust global supply chain network design under disruption and uncertainty considering resilience strategies: a parallel memetic algorithm for a real-life case study. Transp. Res. Part E: Log. Transp. Rev. 87, 20–52 (2016)

    Article  Google Scholar 

  68. Ghavamifar, A., Makui, A., Taleizadeh, A.A.: Designing a resilient competitive supply chain network under disruption risks: a real-world application. Transp. Res. Part E: Log. Transp. Rev. 115, 87–109 (2018)

    Article  Google Scholar 

  69. Hanjoul, P., Hansen, P., Peeters, D., Thisse, J.-F.: Uncapacitated plant location under alternative spatial price policies. Manage. Sci. 36(1), 41–57 (1990)

    Article  Google Scholar 

  70. Hansen, P., Peeters, D., Thisse, J.: Facility location under zone pricing. J. Reg. Sci. 37(1), 1–22 (1997)

    Article  Google Scholar 

  71. Zeballos, L.J., M��ndez, C.A., Barbosa-Povoa, A.P., Novais, A.Q.: Multi-period design and planning of closed-loop supply chains with uncertain supply and demand. Comput. Chem. Eng. 66, 151–164 (2014). https://doi.org/10.1016/j.compchemeng.2014.02.027

    Article  Google Scholar 

Download references

Acknowledgements

The authors did not receive support from any organization for the submitted work.

Author information

Authors and Affiliations

  1. Department of Operations Management and Information Technology, Faculty of Management, Kharazmi University, Tehran, Iran

    Mojtaba Farrokh

  2. Stetson-Hatcher School of Business, Mercer University, Atlanta, GA, 30341, USA

    Ehsan Ahmadi

  3. Department of Management Science and Statistics, The University of Texas at San Antonio, San Antonio, TX, 78249, USA

    Minghe Sun

Authors
  1. Mojtaba Farrokh
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Ehsan Ahmadi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Minghe Sun
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Mojtaba Farrokh.

Ethics declarations

Conflict of interest

No conflict of interest exits in the submission of this manuscript, and the manuscript is approved by all authors for publication. The work described was original research that has not been published previously.

Ethical approval

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

Informed consent

Informed consent was obtained from all individual participants included in the study.

Additional information

Publisher's Note

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

Appendices

Appendices

Appendix A: The basic possibilistic stochastic programming (PSP) model

The basic PSP model for the considered SCND problem, formulated in subsection 0, is as follows.

$$Max\sum\limits_{s \in S} {\pi^{s} \sum\limits_{g \in G} {\sum\limits_{t \in T} {\left( \begin{gathered} \sum\limits_{c \in C} {\sum\limits_{p \in P} {\sum\limits_{l \in L} {\rho_{pl}^{gt} \cdot \overline{M}(\tilde{D}m_{cpl}^{gts} ) \cdot \upsilon_{pl}^{gts} } } } - \sum\limits_{j \in J} {\overline{M}(\tilde{F}_{jg}^{P} )(X_{j}^{g + 1} - X_{j}^{g} )} - \hfill \\ \sum\limits_{k \in K} {\overline{M}(\tilde{F}_{kg}^{H} )(Y_{k}^{g + 1} - Y_{k}^{g} )} - \sum\limits_{j \in J} {\overline{M}(\tilde{g}_{j}^{g} ) \cdot X_{j}^{g} } - \sum\limits_{k \in K} {\overline{M}(\tilde{q}_{k}^{g} ) \cdot Y_{k}^{g} } - \hfill \\ \sum\limits_{j \in J} {\sum\limits_{d \in D} {\overline{M}(\tilde{\eta }_{dg}^{DM} ) \cdot A_{jdg}^{{}} } } - \sum\limits_{j \in J} {\sum\limits_{f \in F} {\overline{M}(\tilde{\eta }_{fg}^{FM} ) \cdot B_{jfg}^{{}} } } - \sum\limits_{j \in J} {\sum\limits_{d \in D} {\sum\limits_{{g \le g^{\prime}}} {\overline{M}(\tilde{\mu }_{dg}^{DM} ) \cdot A_{jdg}^{{}} } } } \hfill \\ - \sum\limits_{j \in J} {\sum\limits_{f \in F} {\sum\limits_{{g \le g^{\prime}}} {\overline{M}(\tilde{\mu }_{fg}^{FM} ) \cdot B_{jfg}^{{}} } } } - \sum\limits_{k \in K} {\sum\limits_{p \in P} {\overline{M}(\tilde{h}_{kp}^{gt} )\left( {H_{kp}^{gts} + \sum\limits_{j \in J} {\left( {\frac{1}{{2N_{jk} }}U_{jkp}^{gts} } \right)} } \right)} } - \hfill \\ \sum\limits_{j \in J} {\sum\limits_{p \in P} {\sum\limits_{o \in O} {\sum\limits_{d \in D} {\overline{M}(\tilde{\varepsilon }_{odgt}^{DM} ) \cdot Qd_{jpod}^{gts} } } } } - \sum\limits_{j \in J} {\sum\limits_{p \in P} {\sum\limits_{o \in O} {\sum\limits_{f \in F} {\overline{M}(\tilde{\varepsilon }_{ofgt}^{FM} )} } } } \cdot Qf_{jpof}^{gts} - \hfill \\ \sum\limits_{i \in I} {\sum\limits_{j \in J} {\sum\limits_{r \in R} {\overline{M}(\tilde{n}_{ijr}^{gt} + \tilde{\tau }_{ir}^{gt} ) \cdot M_{ijr}^{gts} } } } - \sum\limits_{j \in J} {\sum\limits_{k \in K} {\sum\limits_{p \in P} {\overline{M}(\tilde{u}_{jkp}^{gt} ) \cdot U_{jkp}^{gts} } } } - \sum\limits_{k \in K} {\sum\limits_{c \in C} {\sum\limits_{p \in P} {\overline{M}(\tilde{v}_{kcp}^{gt} ) \cdot R_{kcp}^{gts} } } } \hfill \\ \end{gathered} \right)} } }$$
(A.1)
$$s.t.$$
$$Cr\left\{ {\sum\limits_{k \in K} {R_{kcp}^{gts} } \ge \sum\limits_{l \in L} {\tilde{D}m_{cpl}^{gts} \cdot \upsilon_{pl}^{gts} } } \right\} \ge \alpha ,\,\,\,\,\forall c \in C,\quad p \in P,\quad t \in T,\quad g \in G,\quad s \in S$$
(A.2)
$$Cr\left\{ {\sum\limits_{j \in J} {M_{ijr}^{gts} } \le \tilde{C}_{irgt}^{S} \cdot Z_{ir}^{g} } \right\} \ge \alpha ,\,\,\,\,\forall i \in I,\quad r \in R,\quad t \in T,\quad g \in G,\quad s \in S$$
(A.3)
$$Cr\left\{ {\sum\limits_{p \in P} {\sum\limits_{o \in O} {\sum\limits_{d \in D} {w_{pod}^{DM} \cdot Qd_{jpod}^{gts} } } } \le \sum\limits_{d \in D} {\sum\limits_{{g \le g^{\prime}}} {\tilde{C}_{d}^{DM} \cdot A_{jdg}^{{}} } } } \right\} \ge \alpha ,\forall j \in J,g \in G,t \in T,s \in S$$
(A.4)
$$Cr\left\{ {\sum\limits_{p \in P} {\sum\limits_{o \in O} {\sum\limits_{f \in F} {w_{pof}^{FM} \cdot Qf_{jpof}^{gts} } } } \le \sum\limits_{f \in F} {\sum\limits_{{g \le g^{\prime}}} {\tilde{C}_{f}^{FM} \cdot B_{jfg}^{{}} } } } \right\} \ge \alpha ,\,\quad \forall j \in J,\quad g \in G,\quad t \in T,\quad s \in S$$
$$Cr\left\{ {\sum\limits_{p \in P} {\sum\limits_{o \in O} {\sum\limits_{d \in D} {\delta_{od}^{DM} \cdot Qd_{jpod}^{gts} } } } + \sum\limits_{p \in P} {\sum\limits_{o \in O} {\sum\limits_{f \in F} {\delta_{of}^{FM} \cdot Qf_{jpof}^{gts} } } } \le \tilde{C}_{j}^{P} \cdot X_{j}^{g} } \right\} \ge \alpha ,\quad \forall j \in J,\quad g \in G,\quad t \in T,\quad s \in S$$
(A.5)
$$Cr\left\{ {\sum\limits_{p \in P} {\delta_{p} \cdot H_{kp}^{gts} } + \sum\limits_{j \in J} {\sum\limits_{p \in P} {\delta_{p} \left( {\frac{1}{{N_{jk} }}U_{jkp}^{gts} } \right)} } \le \tilde{C}_{k}^{H} \cdot Y_{k}^{g} } \right\} \ge \alpha ,\quad \forall k \in K,\quad g \in G,\quad t \in T,\quad s \in S$$
(A.6)

Constraints (17)–(20), (22)–(25) and (28)–(33).

In constraints (A.2)–(A.6), \(\alpha\) indicates the minimum confidence level set by the decision makers for satisfying the possibilistic chance constraints under each scenario. For notational simplicity, the compact form of the basic PSP model stated as follows is used:

$$\begin{gathered} Max\,\,\,\,\sum\limits_{s \in S} {\pi^{s} \left( {\rho \overline{M}[{\tilde{\mathbf{d}}}^{s} ]{{\varvec{\upupsilon}}}^{s} - \overline{M}[{\tilde{\mathbf{f}}}]{\mathbf{x}} - \overline{M}[{\tilde{\mathbf{c}}}]{\mathbf{y}}^{s} } \right)} \hfill \\ s.t.\,\,\,\,\,\,\,Cr({\mathbf{y}}^{s} \ge {\tilde{\mathbf{d}}}^{s} {{\varvec{\upupsilon}}}^{s} ) \ge \alpha \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,{\mathbf{Ax}} = 0 \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,Cr({\mathbf{By}}^{s} \le {\tilde{\mathbf{D}}\mathbf{x}}) \ge \alpha \hfill \\ \,\,\,\,\,\,\,\,\,\,\,\,{\mathbf{x}} \in \{ 0,1\} ,\quad \,{{\varvec{\upupsilon}}}^{s} ,{\mathbf{y}}^{s} \ge 0. \hfill \\ \end{gathered}$$
(A.7)

The notations in this compact form (A.7) are further explained in the following. The vectors \({{\varvec{\upupsilon}}}^{s}\) and \({\mathbf{y}}^{s}\) represent the continuous variables under each scenario and \({\mathbf{x}}\) represents the binary variables. A scenario is indexed by \(s \in S\) and is defined as a realization of all the fuzzy scenario-based variables. Each scenario \(s\) has an associated probability \(\pi^{s}\), with \(\sum\limits_{s \in S} {\pi^{s} } = 1\).

The coefficients of the constraints are represented by the matrices \({\mathbf{A}}\), \({\mathbf{B}}\) and \({\mathbf{D}}\), and the parameters are represented by the vectors \({\mathbf{c}}\), \({\mathbf{d}}\) and \({\mathbf{f}}\), respectively. The coefficients in \({\mathbf{A}}\) and \({\mathbf{B}}\) are known in advance. The price endpoints are represented by the vector \({\mathbf{Pr}}\). The elements of \({\mathbf{D}}\), \({\mathbf{f}}\) and \({\mathbf{c}}\) are the fuzzy variables describing the capacities, the fixed opening costs, and other costs, respectively, of each plant. Furthermore, the uncertain vector \({\mathbf{d}}\) represents the fuzzy scenario-based demands. Fuzzy coefficients corresponding to \({\mathbf{D}}\), \({\mathbf{f}}\) and \({\mathbf{c}}\) are represented by \({\tilde{\mathbf{D}}}\), \({\tilde{\mathbf{f}}}\) and \({\tilde{\mathbf{c}}}\), respectively, and the fuzzy scenario based variables corresponding to \({\mathbf{d}}\) are denoted by \({\tilde{\mathbf{d}}}^{s}\), where \({\mathbf{\tilde{f} = (f,\underline {f} ,\overline{f})}}_{{{\mathbf{LR}}}}\), \({\mathbf{\tilde{c} = (c,\underline {c} ,\overline{c})}}_{{{\mathbf{LR}}}}\) and \({\tilde{\mathbf{d}}}^{{\mathbf{s}}} {\mathbf{ = (d}}^{{\mathbf{s}}} {\mathbf{,\underline {d} }}^{{\mathbf{s}}} {\mathbf{,\overline{d}}}^{{\mathbf{s}}} {\mathbf{)}}_{{{\mathbf{LR}}}}\) for \(s \in S\) are described by LR trapezoidal fuzzy numbers.

Note that \(\tilde{F}^{s} ({{\varvec{\upupsilon}}}^{s} ,{\mathbf{x}}^{s} ,{\mathbf{y}}^{s} ) = \rho {\tilde{\mathbf{d}}}^{s} {{\varvec{\upupsilon}}}^{s} - {\tilde{\mathbf{f}}\mathbf{x}}^{s} - {\tilde{\mathbf{c}}\mathbf{y}}^{s}\) is a fuzzy profit of the solution \(({{\varvec{\upupsilon}}}^{s} ,{\mathbf{x}}^{s} ,{\mathbf{y}}^{s} )\), under scenario \(s\). The fuzzy profit under each scenario, i.e., \(\tilde{F}^{s}\), is stated in (A.8)

$$\tilde{F}^{s} = \left( {\rho {\mathbf{d}}^{s} {{\varvec{\upupsilon}}}^{s} - {\mathbf{fx}} - {\mathbf{c}}^{{\mathbf{s}}} {\mathbf{y}}^{s} ,\rho \underline{{\mathbf{d}}}^{s} {{\varvec{\upupsilon}}}^{s} - \underline{{\mathbf{f}}} {\mathbf{x}} - \underline{{\mathbf{c}}}^{s} {\mathbf{y}}^{s} ,\rho {\overline{\mathbf{d}}}^{s} {{\varvec{\upupsilon}}}^{s} - {\overline{\mathbf{f}}\mathbf{x}} - {\overline{\mathbf{c}}}^{s} {\mathbf{y}}^{s} } \right)_{LR} = \left( {F^{s} ,\underline{F}^{s} ,\overline{F}^{s} } \right)_{LR} .$$
(A.8)

The total fuzzy profit, i.e., \(\tilde{F}\), is stated in (A.9)

$$\tilde{F} = \left( {\sum\limits_{s \in S} {\pi^{s} F^{s} } ,\sum\limits_{s \in S} {\pi^{s} \underline{F}^{s} } ,\sum\limits_{s \in S} {\pi^{s} \overline{F}^{s} } } \right)_{LR} = (F,\underline{F} ,\overline{F})_{LR} .$$
(A.9)

Given a scenario \(s\), the possibilistic mean of the profit, denoted by \(F^{s}\), can be computed using (A.10)

$$\begin{gathered} F^{s} = \overline{M}(\tilde{F}^{s} ) = \rho \overline{M}[{\tilde{\mathbf{d}}}^{s} ]{{\varvec{\upupsilon}}}^{s} - \overline{M}[{\tilde{\mathbf{f}}}]{\mathbf{x}} - \overline{M}[{\tilde{\mathbf{c}}}]{\mathbf{y}}^{s} \\ = \left( {\rho \left({\mathbf{d}}^{s} + \frac{{{\overline{\mathbf{d}}}^{s} - \underline{{\mathbf{d}}}^{s} }}{6}\right){{\varvec{\upupsilon}}}^{s} - \left({\mathbf{f}} + \frac{{{\overline{\mathbf{f}}} - \underline{{\mathbf{f}}} }}{6}\right){\mathbf{x}} - \left({\mathbf{c}} + \frac{{{\overline{\mathbf{c}}} - \underline{{\mathbf{c}}} }}{6}\right){\mathbf{y}}^{s} } \right). \\ \end{gathered}$$
(A.10)

Moreover, the weighted possibilistic mean of the objective function over all scenarios, denoted by \(F\), can be calculated using (A.11)

$$F = \overline{M}(\tilde{F}) = \sum\limits_{s \in S} {\pi^{s} F^{s} } .$$
(A.11)

Hence, by assuming \({\tilde{\mathbf{D}}} = ({\mathbf{D}},{\mathbf{\underline {D} }},{\overline{\mathbf{D}}})_{LR}\), the basic crisp non-robust PSP model is constructed as follows:

$$Max\,\,\,\,F$$
(A.12)
$$s.t.\,\,\,\,\,\,\,{\mathbf{y}}^{s} \ge \left[ {(2 - 2\alpha ){\mathbf{d}}^{s} + (2\alpha - 1)({\mathbf{d}}^{s} + {\overline{\mathbf{d}}}^{s} )} \right]{{\varvec{\upupsilon}}}^{s} ,\,\,\,\forall s$$
(A.13)
$$\,\,\,\,\,\,\,\,\,\,\,A{\mathbf{x}} = 0\,\,$$
(A.14)
$$\,\,\,\,\,\,\,\,\,\,\,{\mathbf{By}}^{s} \le \left[ {(2\alpha - 1)({\mathbf{D}} - \underline{{\mathbf{D}}} ) + (2 - 2\alpha ){\mathbf{D}}} \right]{\mathbf{x}},\,\,\,\forall s$$
(A.15)
$$\,\,\,\,\,\,\,\,\,\,\,{\mathbf{x}} \in \{ 0,1\} ,\,{{\varvec{\upupsilon}}}^{s} ,{\mathbf{y}}^{s} \ge 0,\,\,\,\,\,\,\,\forall s \in S.$$
(A.16)

Appendix B: The RSP and RPP models

The robust stochastic programming (RSP) model, according to Mulvey et al. [45], is stated as follows for the SCND problem:

$$Max\,F - \gamma \cdot \sum\limits_{s \in S} {\pi^{s} \theta^{s} } - \omega \cdot \sum\limits_{l \in L} {\pi^{s} \left[ {(Dm_{cpl}^{gts} + \overline{D}m_{cpl}^{gts} ) - } \right.} (2 - 2\alpha ) \cdot Dm_{cpl}^{gts} \left. { - (2\alpha - 1)(Dm_{cpl}^{gts} + \overline{D}m_{cpl}^{gts} )} \right]\upsilon_{pl}^{gts}$$
(B.1)

s.t. Constraints (17)–(20), (22)–(25), (28)–(33), (35)–(40), and (43)–(45).

The robust possibilistic programming (RPP) model, according to Pishvaee et al. [16], is stated as follows:

$$Max\,\,F - \lambda \cdot v(\tilde{F}) - \omega \cdot \sum\limits_{l \in L} {\pi^{s} \left[ {(Dm_{cpl}^{gts} + \overline{D}m_{cpl}^{gts} ) - } \right.} (2 - 2\alpha ) \cdot Dm_{cpl}^{gts} \left. { - (2\alpha - 1)(Dm_{cpl}^{gts} + \overline{D}m_{cpl}^{gts} )} \right]\upsilon_{pl}^{gts}$$
(B.2)

s.t.Constraints (17)–(20), (22)–(25), (28)–(33), and (35)–(40).

The RSP and the RPP models can control the scenario and possibilistic variabilities in the objective function, respectively.

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

Farrokh, M., Ahmadi, E. & Sun, M. A robust stochastic possibilistic programming model for dynamic supply chain network design with pricing and technology selection decisions. OPSEARCH 60, 1082–1120 (2023). https://doi.org/10.1007/s12597-023-00643-2

Download citation

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12597-023-00643-2

Keywords

Search

Navigation