A surcharge pricing policy for supply chain coordination under the just-in-time (JIT) environment in the presence of backordering

Abstract

In the just-in-time (JIT) manufacturing system, the vendors ship in small batch sizes. Hence the buyers are at an advantageous position as they can reduce their inventory holding costs. The practice of shipping in smaller batch sizes often results in the vendors incurring high costs as the number of production setups and the number of shipments increases for the vendors. The vendors could only switch to JIT mode when there is an assurance from the buyer of providing some compensation for the increased costs. This compensation from the buyer to the vendors is called ‘surcharge price’. In our work, we have studied the surcharge pricing mechanism in the presence of backorder when there is a powerful vendor and have shown that the it can coordinate a decentralized supply chain. Our study incorporates both the cases of the buyer holding full information about the vendors’ costs as well as the case of information asymmetry. In the case of full information, we have shown that the optimal order quantity in the presence of surcharge pricing is less than that of the vendor dominated scenario. At the same time, the optimal order quantity is higher than that in the buyer dominated scenario. Further, the total supply chain costs are also minimized when a surcharge pricing contract is offered in comparison to both the buyer and the vendor dominated scenarios. For the information asymmetry case, we have framed the optimal set of screening contracts and have shown that an optimal surcharge pricing contract to the vendor is free from the probability distribution of the vendor type.

This is a preview of subscription content, access via your institution.

Fig. 1
Fig. 2
Fig. 3
Fig. 4

Notes

  1. 1.

    Generalized Hazard or failure rate for a non-negative random variable X with probability density function \(\phi \) and cumulative distribution function \(\Phi \) is defined as \(g\left(\xi \right)=\xi \phi \left(\xi \right)/\left(1-\Phi \left(\xi \right)\right)\). The random variable X is said to have an increased generalized hazard rate and \(\Phi \) to have an Increased Generalized Failure Rate (IGFR) distribution if \(g\left(\xi \right)\) is weakly increasing for all \(\xi \) with \(\Phi \left(\xi \right)<1\) (Lariviere 2006).

References

  1. 1.

    Albrecht, M.: Supply chain coordination mechanisms: New approaches for collaborative planning. Springer, Berlin (2010)

    Google Scholar 

  2. 2.

    Aktas, E., Ulengin, F.: Penalty and reward contracts between a manufacturer and its logistics service provider. Logist. Res. 9(8), 1–14 (2016)

    Google Scholar 

  3. 3.

    Banerjee, A.: A joint economic lot-size model for purchaser and vendor. Decis. Sci. 17(3), 292–311 (1986)

    Article  Google Scholar 

  4. 4.

    Banerjee, A.: On “A quantity discount pricing model to increase vendor profits.” Manage. Sci. 32(11), 1513–1517 (1986)

    Article  Google Scholar 

  5. 5.

    Bazan, E., Jaber, M.Y., Zanoni, S.: Supply chain models with greenhouse gases emissions, energy usage and different coordination decisions. Appl. Math. Model. 39(17), 5131–5151 (2015)

    Article  Google Scholar 

  6. 6.

    Cachon, G.: Supply chain coordination with contracts. In: Graves, S., de Kok, A. (eds.) Handbook in Operations Research and Management Science: Supply Chain Management. North Holland, Amsterdam (2003)

    Google Scholar 

  7. 7.

    Cachon, G.P., Lariviere, M.A.: Supply chain coordination with revenue-sharing contracts: strengths and limitations. Manage. Sci. 51(1), 30–44 (2005)

    Article  Google Scholar 

  8. 8.

    Cai, J., Hu, X., Han, Y., Cheng, H., Huang, W.: Supply chain coordination with an option contract under vendor-managed inventory. Int. Trans. Oper. Res. 23, 1163–1183 (2016)

    Article  Google Scholar 

  9. 9.

    Chakraborty, A., Chatterjee, A.K.: A reverse discount model for dynamic demands. In: Presented at: XVI Latin-Ibero-American Conference on Operations Research, Rio de Janeiro (2012)

  10. 10.

    Chakraborty, A., Chatterjee, A.K., Mateen, A.: A vendor-managed inventory scheme as a supply chain coordination mechanism. Int. J. Prod. Res. 53(1), 13–24 (2015)

    Article  Google Scholar 

  11. 11.

    Chakraborty, A., Chatterjee, A.K.: A surcharge pricing scheme for supply chain coordination under JIT environment. Eur. J. Oper. Res. 253(1), 14–24 (2016)

    Article  Google Scholar 

  12. 12.

    Chan, C.K., Lee, Y.C.E., Goyal, S.K.: A delayed payment method in coordinating a single-vendor multi-buyer supply chain. Int. J. Prod. Econ. 127(1), 95–102 (2010)

    Article  Google Scholar 

  13. 13.

    Chatterjee, A.K., Ravi, R.: Joint economic lot-size model with delivery in sub-batches. Opsearch 28(2), 118–124 (1991)

    Google Scholar 

  14. 14.

    Chatterjee, A.K., Mateen, A., Chakraborty, A.: On the equivalence of some supply chain coordination models. Opsearch 52(2), 392–400 (2015)

    Article  Google Scholar 

  15. 15.

    Corbett, C.J., De Groote, X.: A supplier’s optimal quantity discount policy under asymmetric information. Manage. Sci. 46(3), 444–450 (2000)

    Article  Google Scholar 

  16. 16.

    Corbett, C.J., Zhou, D., Tang, C.S.: Designing supply contracts: Contract type and information asymmetry. Manage. Sci. 50(4), 550–559 (2004)

    Article  Google Scholar 

  17. 17.

    Darwish, M.A., Odah, O.M.: Vendor managed inventory model for single-vendor multi-retailer supply chains. Eur. J. Oper. Res. 204, 473–484 (2010)

    Article  Google Scholar 

  18. 18.

    Dong, Y., Carter, C.R., Dresner, M.E.: JIT purchasing and performance: An exploratory analysis of buyer and supplier perspectives. J. Oper. Manag. 19, 471–483 (2001)

    Article  Google Scholar 

  19. 19.

    Esmaeili, M., Aryanezhad, M.B., Zeephongsekul, P.: A game theory approach in seller–buyer supply chain. Eur. J. Oper. Res. 195(2), 442–448 (2009)

    Article  Google Scholar 

  20. 20.

    Fazel, F.: A comparative analysis of inventory costs of JIT and EOQ purchasing. Int. J. Phys. Dist. Log. Manage. 27(8), 496–504 (1997)

    Article  Google Scholar 

  21. 21.

    Giri, B.C., Bardhan, S.: A vendor–buyer JELS model with stock-dependent demand and consigned inventory under buyer’s space constraint. Oper. Res. Int. J 15(1), 79–93 (2015)

    Article  Google Scholar 

  22. 22.

    Goyal, S.K.: An integrated inventory model for a single vendor-single customer problem. Int. J. Prod. Res. 15(1), 107–111 (1977)

    Article  Google Scholar 

  23. 23.

    Goyal, S.K.: “A joint economic lot-size model for purchaser and vendor”: A comment. Decision Sci. 19(1), 236–241 (1988)

    Article  Google Scholar 

  24. 24.

    Kelle, P., Al-khateeb, F., Miller, P.A.: Partnership and negotiation support by joint optimal ordering/setup policies for JIT. Int. J. Prod. Econ. 81–82, 431–441 (2003)

    Article  Google Scholar 

  25. 25.

    Laffont, J.J., Tirole, J.: A theory of incentives in procurement and regulation. MIT Press, Cambridge (1993)

    Google Scholar 

  26. 26.

    Lariviere, M.A.: A note on probability distributions with increasing generalized failure rates. Oper. Res. 54(3), 602–604 (2006)

    Article  Google Scholar 

  27. 27.

    Lee, J.Y., Johnson, G.: Contracting for vendor-managed inventory with a time-dependent stockout penalty. Int. Trans. Oper. Res. 27, 1573–1599 (2017)

    Article  Google Scholar 

  28. 28.

    Lee, H.L., Rosenblatt, M.J.: A generalized quantity discount pricing model to increase vendor’s profits. Manage. Sci. 32(9), 1177–1185 (1986)

    Article  Google Scholar 

  29. 29.

    Miller, P.A., Kelle, P.: Quantitative support for buyer-supplier negotiation in just-in-time purchasing. J. Supply Chain Manage. 34(1), 25–30 (1998)

    Google Scholar 

  30. 30.

    Monahan, J.P.: A quantity discount pricing model to increase vendor profits. Manage. Sci. 30(6), 720–726 (1984)

    Article  Google Scholar 

  31. 31.

    Mukhopadhyay, S.K., Zhu, X., Yue, X.: Optimal contract design for mixed channels under information asymmetry. Prod. Oper. Manage. 17(6), 641–650 (2008)

    Article  Google Scholar 

  32. 32.

    Myerson, R.B.: Optimal auction design. Math. Oper. Res. 6(1), 58–73 (1981)

    Article  Google Scholar 

  33. 33.

    Newman, G.: As just-in-time goes by. Across the Board 30(8), 7 (1993)

    Google Scholar 

  34. 34.

    Niu, R.H., Zhao, X., Castillo, I., Joro, T.: Pricing and inventory strategies for a two-stage dual-channel supply chain. Asia-Pacific J. Oper. Res. 29(01), 1240004 (2012)

    Article  Google Scholar 

  35. 35.

    Pasternack, B.A.: Optimal pricing and return policies for perishable commodities. Marketing Sci. 4(2), 166–176 (1985)

    Article  Google Scholar 

  36. 36.

    Sarmah, S.P., Acharya, D., Goyal, S.K.: Buyer vendor coordination models in supply chain management. Eur. J. Oper. Res. 175(1), 1–15 (2006)

    Article  Google Scholar 

  37. 37.

    Sucky, E.: Inventory management in supply chains: A bargaining problem. Int. J. Prod. Econ. 93–94, 253–262 (2005)

    Article  Google Scholar 

  38. 38.

    Sucky, E.: A bargaining model with asymmetric information for a single vendor-single buyer problem. Eur. J. Oper. Res. 171, 516–535 (2006)

    Article  Google Scholar 

  39. 39.

    Tersine, R. J., 1994. Principles of inventory and materials management. North Holland Publishing Company.

  40. 40.

    Verma, N.K., Chakraborty, A., Chatterjee, A.K.: Joint replenishment of multi retailer with variable replenishment cycle under VMI. Eur. J. Oper. Res. 233(3), 787–789 (2014)

    Article  Google Scholar 

  41. 41.

    Voigt, G., Inderfurth, K.: Supply chain coordination and setup cost reduction in case of asymmetric information. OR Spectrum 33, 99–122 (2011)

    Article  Google Scholar 

  42. 42.

    Yao, Y., Evers, P.T., Dresner, M.E.: Supply chain integration in vendor-managed inventory. Decis. Support Syst. 43(2), 663–674 (2007)

    Article  Google Scholar 

  43. 43.

    Zhang, Q.H., Luo, J.W.: Coordination of a buyer-vendor supply chain for a perishable product under symmetric and asymmetric information. Asia-Pacific J. Oper. Res. 28(05), 673–688 (2011)

    Article  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Abhishek Chakraborty.

Additional information

Publisher's Note

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

Appendices

Appendix A

Proof of Theorem 1

The optimization problem is \(Min \, TCB\left( {Q,Q_{1} } \right) = \frac{{DC_{O} }}{Q} + \frac{{h_{1} Q_{1}^{2} }}{2Q} + \frac{{h_{2} \left( {Q - Q_{1} } \right)^{2} }}{2Q} + \frac{D}{Q}C_{S} - C_{S}\).

First-order conditions for stationarity give

$$ \frac{{\partial TCB\left( {Q,Q_{1} } \right)}}{\partial Q} = - \frac{{DC_{O} }}{{Q^{2} }} - \frac{{h_{1} Q_{1}^{2} }}{{2Q^{2} }} + \frac{{4h_{2} \left( {Q - Q_{1} } \right)Q - 2h_{2} \left( {Q - Q_{1} } \right)^{2} }}{{4Q^{2} }} - \frac{D}{{Q^{2} }}C_{S} = 0 $$
$$ \frac{{\partial TCB\left( {Q,Q_{1} } \right)}}{{\partial Q_{1} }} = \frac{{2h_{1} Q_{1} }}{2Q} - \frac{{2h_{2} \left( {Q - Q_{1} } \right)}}{2Q} = 0 $$

Solving the above system of equations we get \(Q_{1} = \sqrt {\frac{{2Dh_{2} \left( {C_{O} + C_{S} } \right)}}{{h_{1} \left( {h_{1} + h_{2} } \right)}}}\) and \(Q = \sqrt {\frac{{2D\left( {C_{O} + C_{S} } \right)\left( {h_{1} + h_{2} } \right)}}{{h_{1} h_{2} }}}\). Thus, implying \(Q_{2} = \sqrt {\frac{{2D\left( {C_{O} + C_{S} } \right)h_{1} }}{{h_{2} \left( {h_{1} + h_{2} } \right)}}}\).\(\square \)

Proof of Lemma 1

The Hessian matrix for the objective function is given as:

$$ H = \left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} TC}}{{\partial^{2} Q_{1} }}} & {\frac{{\partial^{2} TC}}{{\partial Q\partial Q_{1} }}} \\ {\frac{{\partial^{2} TC}}{{\partial Q_{1} \partial Q}}} & {\frac{{\partial^{2} TC}}{{\partial^{2} Q}}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\frac{{h_{1} + h_{2} }}{Q}} & { - \frac{{Q_{1} \left( {h_{1} + h_{2} } \right)}}{{Q^{2} }}} \\ { - \frac{{Q_{1} \left( {h_{1} + h_{2} } \right)}}{{Q^{2} }}} & {\frac{{Q_{1}^{2} \left( {h_{1} + h_{2} } \right) + 2D\left( {C_{S} + C_{O} } \right)}}{{Q^{3} }}} \\ \end{array} } \right] $$
$$ \det H = \frac{{2D\left( {C_{S} + C_{O} } \right)\left( {h_{1} + h_{2} } \right)}}{{Q^{4} }} > 0 \;{\text{and}}\;\left. {\frac{{\partial^{2} TC}}{{\partial^{2} Q_{1} }}} \right|_{{Q = Q^{*} ,Q_{1} = Q_{1}^{*} }} = \sqrt {\frac{{\left( {h_{1} + h_{2} } \right)h_{1} h_{2} }}{{2D\left( {C_{O} + C_{S} } \right)}}} > 0 $$

This proves the convexity of the objective function.\(\square \)

Proof of Proposition 1

The optimal order quantity under surcharge pricing is \(Q_{SP}^{*} = \sqrt {\frac{{2D\left( {C_{O} + C_{S} } \right)\left( {h_{1} + h_{2} } \right)}}{{h_{1} h_{2} }}}\).

The optimal order quantity under Vendor dominated scenario is Q* = D.

The optimal order quantity under Buyer dominated scenario is \(Q_{BD}^{*} = \sqrt {\frac{{2DC_{O} \left( {h_{1} + h_{2} } \right)}}{{h_{1} h_{2} }}}\).

Clearly, \(Q_{SP}^{*} = \sqrt {\frac{{2D\left( {C_{O} + C_{S} } \right)\left( {h_{1} + h_{2} } \right)}}{{h_{1} h_{2} }}} > Q_{BD}^{*} = \sqrt {\frac{{2DC_{O} \left( {h_{1} + h_{2} } \right)}}{{h_{1} h_{2} }}}\).\(\square \)

Proof of Proposition 2

The total supply chain cost under vendor dominated scenario is TSSCISO = \(C_{S} + C_{O} + \frac{{Dh_{1} h_{2} }}{{2\left( {h_{1} + h_{2} } \right)}}\).

The total supply chain cost under surcharge pricing \(TSCC_{SP} = \sqrt {\frac{{2Dh_{1} h_{2} \left( {Co + C_{S} } \right)}}{{\left( {h_{1} + h_{2} } \right)}}}\)

$$ \left( {TSCC_{ISO} } \right)^{2} - \left( {TSCC_{SP} } \right)^{2} = \left( {C_{S} + C_{O} - \frac{{Dh_{1} h_{2} }}{{2\left( {h_{1} + h_{2} } \right)}}} \right)^{2} \ge 0 $$

Hence we can conclude that the total supply chain cost under surcharge pricing is less than the total supply chain cost under vendor dominated scenario.\(\square \)

Appendix B

Proof of Theorem 2

The optimization problem is

$$ Min \, TCB\left( {Q,Q_{1} } \right) = \frac{{DC_{O} }}{Q} + \frac{{h_{1} Q_{1}^{2} }}{2Q} + \frac{{h_{2} \left( {Q - Q_{1} } \right)_{{}}^{2} }}{2Q} + \frac{D}{{Q_{VEN} }}C_{S} + \frac{{Q_{VEN} }}{2}h_{S} - \frac{Q}{2}h_{S} - C_{S} $$

First-order conditions for stationarity give.

\(\frac{\partial TC}{{\partial Q}} = \frac{{\left( {Q - Q_{1} } \right)h_{2} }}{Q} - \frac{{\left( {Q - Q_{1} } \right)^{2} h_{2} }}{{2Q^{2} }} - \frac{{Q_{1}^{2} h_{1} }}{{2Q^{2} }} - \frac{{DC_{O} }}{{Q^{2} }} - \frac{{h_{s} }}{2} = 0\) and \(\frac{\partial TC}{{\partial Q_{1} }} = \frac{{Q_{1} h_{1} }}{Q} - \frac{{\left( {Q - Q_{1} } \right)h_{2} }}{Q} = 0\).

Solving them, we get \(Q = \sqrt {\frac{{2DC_{O} \left( {h_{1} + h_{2} } \right)}}{{h_{1} h_{2} - \left( {h_{1} + h_{2} } \right)h_{s} }}}\) and \(Q_{2}^{{}} = Q - Q_{1} = h_{1} \sqrt {\frac{{2DC_{O} }}{{\left( {h_{1} h_{2} - \left( {h_{1} + h_{2} } \right)h_{s} } \right)\left( {h_{1} + h_{2} } \right)}}}\).

Proof of Lemma 2

The Hessian matrix for the objective function is given as:

$$ H = \left[ {\begin{array}{*{20}c} {\frac{{\partial^{2} TC}}{{\partial^{2} Q_{1} }}} & {\frac{{\partial^{2} TC}}{{\partial Q\partial Q_{1} }}} \\ {\frac{{\partial^{2} TC}}{{\partial Q_{1} \partial Q}}} & {\frac{{\partial^{2} TC}}{{\partial^{2} Q}}} \\ \end{array} } \right] = \left[ {\begin{array}{*{20}c} {\frac{{h_{1} + h_{2} }}{Q}} & { - \frac{{Q_{1} \left( {h_{1} + h_{2} } \right)}}{{Q^{2} }}} \\ { - \frac{{Q_{1} \left( {h_{1} + h_{2} } \right)}}{{Q^{2} }}} & {\frac{{Q_{1}^{2} \left( {h_{1} + h_{2} } \right) + 2DC_{O} }}{{Q^{3} }}} \\ \end{array} } \right] $$
$$ \begin{aligned} \det H& = \frac{{2DC_{O} \left( {h_{1} + h_{2} } \right)^{2} }}{{Q^{4} }} > 0\,{\text{and}}\;\left. {\frac{{\partial^{2} TC}}{{\partial^{2} Q_{1} }}} \right|_{{Q = Q^{*} ,Q_{1} = Q_{1}^{*} }}\\ & = \sqrt {\frac{{\left( {h_{1} + h_{2} } \right)\left( {h_{1} h_{2} - \left( {h_{1} + h_{2} } \right)h_{s} } \right)}}{{2D\left( {C_{O} + C_{S} } \right)}}} > 0 \end{aligned} $$

This proves the convexity of the objective function.

Solution to the adverse selection problem

The Lagrangian function for the minimization problem can be written as:

\(\begin{gathered} L = \left[ {\frac{{DC_{O} }}{{Q^{H} }} + \frac{{h_{1} \left( {Q_{1}^{H} } \right)^{2} }}{{2Q^{H} }} + \frac{{h_{2} \left( {Q^{H} - Q_{1}^{H} } \right)_{{}}^{2} }}{{2Q^{H} }} + x^{H} D} \right]\lambda + \left[ {\frac{{DC_{O} }}{{Q^{L} }} + \frac{{h_{1} \left( {Q_{1}^{L} } \right)_{{}}^{2} }}{{2Q^{L} }} + \frac{{h_{2} \left( {Q^{L} - Q_{1}^{L} } \right)_{{}}^{2} }}{{2Q^{L} }} + x^{L} D} \right]\left( {1 - \lambda } \right) \hfill \\ + \mu_{1} \left( {\frac{D}{{Q^{H} }}C_{S}^{H} - C_{S}^{H} - x^{H} D} \right) + \mu_{2} \left( {\frac{D}{{Q^{L} }}C_{S}^{L} - C_{S}^{L} - x^{L} D} \right) + \mu_{3} \left[ {x^{H} D - \frac{D}{{Q^{H} }}C_{S}^{H} - x^{L} D + \frac{D}{{Q^{L} }}C_{S}^{H} } \right] \hfill \\ + \mu_{4} \left[ {x^{L} D - \frac{D}{{Q^{L} }}C_{S}^{L} - x^{H} D + \frac{D}{{Q^{H} }}C_{S}^{L} } \right] \hfill \\ \end{gathered}\) The Karush–Kuhn–Tucker (KKT) conditions can be written as:

Stationarity Conditions:

$$ \begin{aligned} \frac{\partial L}{{\partial Q^{H} }} =& \left( {\frac{{\left( {Q^{H} - Q_{1}^{H} } \right)h_{2} }}{{Q^{H} }} - \frac{{\left( {Q^{H} - Q_{1}^{H} } \right)^{2} h_{2} }}{{2\left( {Q^{H} } \right)^{2} }} - \frac{{\left( {Q_{1}^{H} } \right)^{2} h_{1} }}{{2\left( {Q^{H} } \right)^{2} }} - \frac{{DC_{O} }}{{\left( {Q^{H} } \right)^{2} }}} \right)\lambda\\ & - \frac{{C_{S}^{L} D\mu_{4} }}{{\left( {Q^{H} } \right)^{2} }} + \frac{{C_{S}^{H} D\mu_{3} }}{{\left( {Q^{H} } \right)^{2} }} - \frac{{C_{S}^{H} D\mu_{1} }}{{\left( {Q^{H} } \right)^{2} }} = 0 \end{aligned} $$

\(\frac{\partial L}{{\partial Q^{L} }} = \left( {\frac{{\left( {Q^{L} - Q_{1}^{L} } \right)h_{2} }}{{Q^{L} }} - \frac{{\left( {Q^{L} - Q_{1}^{L} } \right)^{2} h_{2} }}{{2\left( {Q^{L} } \right)^{2} }} - \frac{{\left( {Q_{1}^{L} } \right)^{2} h_{1} }}{{2\left( {Q^{L} } \right)^{2} }} - \frac{{C_{O} D}}{{\left( {Q^{L} } \right)^{2} }}} \right)\left( {1 - \lambda } \right) + \frac{{C_{S}^{L} D\mu_{4} }}{{\left( {Q^{L} } \right)^{2} }} - \frac{{C_{S}^{H} D\mu_{3} }}{{\left( {Q^{L} } \right)^{2} }} - \frac{{C_{S}^{L} D\mu_{2} }}{{\left( {Q^{L} } \right)^{2} }} = 0\)\(\frac{\partial L}{{\partial Q_{1}^{H} }} = \left( {\frac{{Q_{1}^{H} h_{1} }}{{Q^{H} }} - \frac{{\left( {Q^{H} - Q_{1}^{H} } \right)h_{2} }}{{Q^{H} }}} \right)\lambda = 0\),\(\frac{\partial L}{{\partial Q_{1}^{L} }} = \left( {\frac{{Q_{1}^{L} h_{1} }}{{Q^{L} }} - \frac{{\left( {Q^{L} - Q_{1}^{L} } \right)h_{2} }}{{Q^{L} }}} \right)\left( {1 - \lambda } \right) = 0\).

\(\frac{\partial L}{{\partial x^{H} }} = D\left[ {\lambda - \mu_{4} + \mu_{3} - \mu_{1} } \right] = 0\), \(\frac{\partial L}{{\partial x^{L} }} = D\left[ {\left( {1 - \lambda } \right) + \mu_{4} - \mu_{3} - \mu_{2} } \right] = 0\).

Primal Feasibility Conditions: Q.H., Q.L., QH1, QL1, xH, xL ≥ 0.

Dual Feasibility Conditions: μ1, μ2, μ3, μ4 ≥ 0.

Complementary Slackness Conditions:

$$ \begin{aligned} \mu_{1} \left( {\frac{D}{{Q^{H} }}C_{S}^{H} - C_{S}^{H} - x^{H} D} \right) & = 0,\,\mu_{2} \left( {\frac{D}{{Q^{L} }}C_{S}^{L} - C_{S}^{L} - x^{L} D} \right) = 0 \\ \mu_{3} \left[ {x^{H} D - \frac{D}{{Q^{H} }}C_{S}^{H} - x^{L} D + \frac{D}{{Q^{L} }}C_{S}^{H} } \right] & = 0,\,\mu_{4} \left[ {x^{L} D - \frac{D}{{Q^{L} }}C_{S}^{L} - x^{H} D + \frac{D}{{Q^{H} }}C_{S}^{L} } \right] = 0 \\ \end{aligned} $$

The equations \(\lambda - \mu_{4} + \mu_{3} - \mu_{1} = 0\) and \(\left( {1 - \lambda } \right) + \mu_{4} - \mu_{3} - \mu_{2} = 0\) imply \(\mu_{1} + \mu_{2} = 1\).

Further, for λ ≠ 0 or 1 we also have

$$ \begin{aligned} & \frac{{Q_{1}^{H} h_{1} }}{{Q^{H} }} - \frac{{\left( {Q^{H} - Q_{1}^{H} } \right)h_{2} }}{{Q^{H} }} = 0 \;{\text{and}}\;\frac{{Q_{1}^{L} h_{1} }}{{Q^{L} }} - \frac{{\left( {Q^{L} - Q_{1}^{L} } \right)h_{2} }}{{Q^{L} }} = 0\, {\text{implying}}\;\\ &\frac{{Q_{1}^{H} }}{{Q^{H} }} = \frac{{h_{2} }}{{h_{1} + h_{2} }}\; {\text{and}}\;\frac{{Q_{1}^{L} }}{{Q^{L} }} = \frac{{h_{2} }}{{h_{1} + h_{2} }} \end{aligned} $$

The Lagrangian multipliers μi’s can either be equal to 0 or greater than 0 thus yielding 16 possible combinations which are given in the following table (Table 10):

Table 10 Various Combinations of Lagrangian Multipliers

Out of these 16 combinations, some combinations will be infeasible. For e.g., the relation \(\mu_{1} + \mu_{2} = 1\) would imply that any combination where μ1 = 0 = μ2 is present is ruled out. Similarly for Case I, i.e., when all μi > 0, the Complementary Slackness Conditions will imply \(\begin{aligned} & \frac{D}{{Q^{H} }}C_{S}^{H} - C_{S}^{H} - x^{H} D = 0 \Rightarrow x^{H} = C_{S}^{H} \left( {\frac{1}{{Q^{H} }} - \frac{1}{D}} \right) \\ & \frac{D}{{Q^{L} }}C_{S}^{L} - C_{S}^{L} - x^{L} D = 0 \Rightarrow x^{L} = C_{S}^{L} \left( {\frac{1}{{Q^{L} }} - \frac{1}{D}} \right) \\ & x^{H} D - \frac{D}{{Q^{H} }}C_{S}^{H} - x^{L} D + \frac{D}{{Q^{L} }}C_{S}^{H} = 0\,{\text{and}}\;x^{H} D - \frac{D}{{Q^{H} }}C_{S}^{H} - x^{L} D + \frac{D}{{Q^{L} }}C_{S}^{H} = 0 \\ \end{aligned}\).

Substituting the values for \(x^{H} = C_{S}^{H} \left( {\frac{1}{{Q^{H} }} - \frac{1}{D}} \right)\) and \(x^{L} = C_{S}^{L} \left( {\frac{1}{{Q^{L} }} - \frac{1}{D}} \right)\) above, we get

$$ C_{S}^{H} \left( {\frac{1}{{Q^{H} }} - \frac{1}{D}} \right)D - \frac{D}{{Q^{H} }}C_{S}^{H} - C_{S}^{L} \left( {\frac{1}{{Q^{L} }} - \frac{1}{D}} \right)D + \frac{D}{{Q^{L} }}C_{S}^{H} = 0 $$
$$ - C_{S}^{H} - C_{S}^{L} \frac{D}{{Q^{L} }} + C_{S}^{L} + \frac{D}{{Q^{L} }}C_{S}^{H} = 0 \Rightarrow \left( {\frac{D}{{Q^{L} }} - 1} \right)\left( {C_{S}^{H} - C_{S}^{L} } \right) = 0 \Rightarrow C_{S}^{H} = C_{S}^{L} {\text{ for }}D \ne Q^{L} $$

which violates the initial assumption made about the different states of the vendor’s setup cost. Hence this case becomes invalid. Under similar arguments, the Cases I, II, III, VII, VIII, IX, X, XI, XII, XIII, and XVI will become infeasible. Hence we are only left with the cases IV, V, VI, XIV, and XV to analyze further.

Case IV μ1 ≠ 0, μ2 ≠ 0, μ3 = 0 = μ4.

From the Complementary Slackness Conditions, we get.

\(\frac{D}{{Q^{H} }}C_{S}^{H} - C_{S}^{H} - x^{H} D = 0 \Rightarrow x^{H} = C_{S}^{H} \left( {\frac{1}{{Q^{H} }} - \frac{1}{D}} \right)\) and \(\frac{D}{{Q^{L} }}C_{S}^{L} - C_{S}^{L} - x^{L} D = 0 \Rightarrow x^{L} = C_{S}^{L} \left( {\frac{1}{{Q^{L} }} - \frac{1}{D}} \right)\).

Since μ3 = 0 = μ4, we must have μ1 = λ and μ2 = 1- λ.

The Stationarity Conditions \(\frac{{Q_{1}^{H} }}{{Q^{H} }} = \frac{{h_{2} }}{{h_{1} + h_{2} }}\) and \(\frac{{Q_{1}^{L} }}{{Q^{L} }} = \frac{{h_{2} }}{{h_{1} + h_{2} }}\) together imply

$$ \begin{aligned} &\frac{{2\left( {Q^{H} - Q_{1}^{H} } \right)h_{2} }}{{Q^{H} }} - \frac{{\left( {Q^{H} - Q_{1}^{H} } \right)^{2} h_{2} }}{{\left( {Q^{H} } \right)^{2} }} - \frac{{\left( {Q_{1}^{H} } \right)^{2} h_{1} }}{{\left( {Q^{H} } \right)^{2} }} = \frac{{2DC_{O} }}{{\left( {Q^{H} } \right)^{2} }} + \frac{{2DC_{S}^{H} }}{{\left( {Q^{H} } \right)^{2} }} \\ &\quad\Rightarrow Q^{H} = \sqrt {\frac{{2D\left( {C_{O} + C_{S}^{H} } \right)\left( {h_{1} + h_{2} } \right)}}{{h_{1} h_{2} }}} \end{aligned} $$

Similarly, \(Q^{L} = \sqrt {\frac{{2D\left( {C_{O} + C_{S}^{L} } \right)\left( {h_{1} + h_{2} } \right)}}{{h_{1} h_{2} }}}\).

We can also find \(Q_{1}^{H} = \sqrt {\frac{{2D\left( {C_{O} + C_{S}^{H} } \right)h_{2} }}{{h_{1} \left( {h_{1} + h_{2} } \right)}}}\) and \(Q_{1}^{L} = \sqrt {\frac{{2D\left( {C_{O} + C_{S}^{L} } \right)h_{2} }}{{h_{1} \left( {h_{1} + h_{2} } \right)}}}\).

Case V μ1 ≠ 0, μ2 = 0, μ3 ≠ 0, μ4 ≠ 0.

From the Complementary Slackness Conditions, we get

$$ \frac{D}{{Q^{H} }}C_{S}^{H} - C_{S}^{H} - x^{H} D = 0 \Rightarrow x^{H} = C_{S}^{H} \left( {\frac{1}{{Q^{H} }} - \frac{1}{D}} \right) $$

\(x^{H} D - \frac{D}{{Q^{H} }}C_{S}^{H} - x^{L} D + \frac{D}{{Q^{L} }}C_{S}^{H} = 0\) and \(x^{L} D - \frac{D}{{Q^{L} }}C_{S}^{L} - x^{H} D + \frac{D}{{Q^{H} }}C_{S}^{L} = 0\).

Substituting the value of \(x^{H} = C_{S}^{H} \left( {\frac{1}{{Q^{H} }} - \frac{1}{D}} \right)\) in \(x^{H} D - \frac{D}{{Q^{H} }}C_{S}^{H} - x^{L} D + \frac{D}{{Q^{L} }}C_{S}^{H} = 0\) and \(x^{L} D - \frac{D}{{Q^{L} }}C_{S}^{L} - x^{H} D + \frac{D}{{Q^{H} }}C_{S}^{L} = 0\) respectively we get.

\(C_{S}^{H} \left( {\frac{D}{{Q^{H} }} - 1} \right) - \frac{D}{{Q^{H} }}C_{S}^{H} - x^{L} D + \frac{D}{{Q^{L} }}C_{S}^{H} = 0\) and \(x^{L} D - \frac{D}{{Q^{L} }}C_{S}^{L} - C_{S}^{H} \left( {\frac{D}{{Q^{H} }} - 1} \right) + \frac{D}{{Q^{H} }}C_{S}^{L} = 0\).

Adding these two equations we get \(\frac{1}{{Q^{H} }}C_{S}^{L} - \frac{1}{{Q^{H} }}C_{S}^{H} = \frac{1}{{Q^{L} }}C_{S}^{L} - \frac{1}{{Q^{L} }}C_{S}^{H} \Rightarrow Q^{H} = Q^{L} {\text{ for }}C_{S}^{L} \ne C_{S}^{H}\).

From the condition, \(x^{H} D - \frac{D}{{Q^{H} }}C_{S}^{H} - x^{L} D + \frac{D}{{Q^{L} }}C_{S}^{H} = 0\) we can also write \(x^{H} = x^{L}\). Also, we have \(Q_{1}^{H} = Q_{1}^{L}\).

Now, for μ2 = 0, we get μ1 = 1. Thus, from the Stationarity conditions we have

$$ \begin{aligned}& \left( {\frac{{\left( {Q^{H} - Q_{1}^{H} } \right)h_{2} }}{{Q^{H} }} - \frac{{\left( {Q^{H} - Q_{1}^{H} } \right)^{2} h_{2} }}{{2\left( {Q^{H} } \right)^{2} }} - \frac{{\left( {Q_{1}^{H} } \right)^{2} h_{1} }}{{2\left( {Q^{H} } \right)^{2} }} - \frac{{DC_{O} }}{{\left( {Q^{H} } \right)^{2} }}} \right)\lambda \\ &- \frac{{C_{S}^{L} D\mu_{4} }}{{\left( {Q^{H} } \right)^{2} }} + \frac{{C_{S}^{H} D\mu_{3} }}{{\left( {Q^{H} } \right)^{2} }} - \frac{{C_{S}^{H} D}}{{\left( {Q^{H} } \right)^{2} }} = 0 \end{aligned} $$
$$ \left( {\frac{{\left( {Q^{L} - Q_{1}^{L} } \right)h_{2} }}{{Q^{L} }} - \frac{{\left( {Q^{L} - Q_{1}^{L} } \right)^{2} h_{2} }}{{2\left( {Q^{L} } \right)^{2} }} - \frac{{\left( {Q_{1}^{L} } \right)^{2} h_{1} }}{{2\left( {Q^{L} } \right)^{2} }} - \frac{{C_{O} D}}{{\left( {Q^{L} } \right)^{2} }}} \right)\left( {1 - \lambda } \right) + \frac{{C_{S}^{L} D\mu_{4} }}{{\left( {Q^{L} } \right)^{2} }} - \frac{{C_{S}^{H} D\mu_{3} }}{{\left( {Q^{L} } \right)^{2} }} = 0 $$

Adding these two equations and substituting QL = QH and QL1 = QH1 we get \(Q^{H} = \sqrt {\frac{{2D\left( {C_{S}^{H} + C_{O} } \right)\left( {h_{1} + h_{2} } \right)}}{{h_{1} h_{2} }}} = Q^{L}\).

Similarly, we can find \(Q_{1}^{H} = \sqrt {\frac{{2D\left( {C_{O} + C_{S}^{H} } \right)h_{2} }}{{h_{1} \left( {h_{1} + h_{2} } \right)}}}\) and \(Q_{1}^{L} = \sqrt {\frac{{2D\left( {C_{O} + C_{S}^{H} } \right)h_{2} }}{{h_{1} \left( {h_{1} + h_{2} } \right)}}}\).

Case VI μ1 = 0, μ2 ≠ 0, μ3 ≠ 0, μ4 ≠ 0.

From the Complementary Slackness Conditions, we get

$$ \frac{D}{{Q^{L} }}C_{S}^{L} - C_{S}^{L} - x^{L} D = 0 \Rightarrow x^{L} = C_{S}^{L} \left( {\frac{1}{{Q^{L} }} - \frac{1}{D}} \right) $$

\(x^{H} D - \frac{D}{{Q^{H} }}C_{S}^{H} - x^{L} D + \frac{D}{{Q^{L} }}C_{S}^{H} = 0\) and \(x^{H} D - \frac{D}{{Q^{H} }}C_{S}^{H} - x^{L} D + \frac{D}{{Q^{L} }}C_{S}^{H} = 0\).

Substituting the value of \(x^{L} = C_{S}^{L} \left( {\frac{1}{{Q^{L} }} - \frac{1}{D}} \right)\) in \(x^{H} D - \frac{D}{{Q^{H} }}C_{S}^{H} - x^{L} D + \frac{D}{{Q^{L} }}C_{S}^{H} = 0\) and \(x^{L} D - \frac{D}{{Q^{L} }}C_{S}^{L} - x^{H} D + \frac{D}{{Q^{H} }}C_{S}^{L} = 0\) respectively we get.

\(x^{H} D - \frac{D}{{Q^{H} }}C_{S}^{H} - C_{S}^{L} \left( {\frac{D}{{Q^{L} }} - 1} \right) + \frac{D}{{Q^{L} }}C_{S}^{H} = 0\) and \(C_{S}^{L} \left( {\frac{D}{{Q^{L} }} - 1} \right) - \frac{D}{{Q^{L} }}C_{S}^{L} - x^{H} D + \frac{D}{{Q^{H} }}C_{S}^{L} = 0\).

Adding the above two equations we get

$$ - \frac{D}{{Q^{H} }}C_{S}^{H} + \frac{D}{{Q^{L} }}C_{S}^{H} - \frac{D}{{Q^{L} }}C_{S}^{L} + \frac{D}{{Q^{H} }}C_{S}^{L} = 0 \Rightarrow Q^{H} = Q^{L} {\text{ for }}C_{S}^{H} \ne C_{S}^{L} $$

From the condition, \(x^{H} D - \frac{D}{{Q^{H} }}C_{S}^{H} - x^{L} D + \frac{D}{{Q^{L} }}C_{S}^{H} = 0\) we can also write \(x^{H} = x^{L}\). Also, we have \(Q_{1}^{H} = Q_{1}^{L}\).

Since μ1 = 0 then μ2 = 1. Thus, from the Stationarity conditions we have

$$ \left( {\frac{{\left( {Q^{H} - Q_{1}^{H} } \right)h_{2} }}{{Q^{H} }} - \frac{{\left( {Q^{H} - Q_{1}^{H} } \right)^{2} h_{2} }}{{2\left( {Q^{H} } \right)^{2} }} - \frac{{\left( {Q_{1}^{H} } \right)^{2} h_{1} }}{{2\left( {Q^{H} } \right)^{2} }} - \frac{{DC_{O} }}{{\left( {Q^{H} } \right)^{2} }}} \right)\lambda - \frac{{C_{S}^{L} D\mu_{4} }}{{\left( {Q^{H} } \right)^{2} }} + \frac{{C_{S}^{H} D\mu_{3} }}{{\left( {Q^{H} } \right)^{2} }} = 0 $$
$$ \begin{aligned}& \left( {\frac{{\left( {Q^{L} - Q_{1}^{L} } \right)h_{2} }}{{Q^{L} }} - \frac{{\left( {Q^{L} - Q_{1}^{L} } \right)^{2} h_{2} }}{{2\left( {Q^{L} } \right)^{2} }} - \frac{{\left( {Q_{1}^{L} } \right)^{2} h_{1} }}{{2\left( {Q^{L} } \right)^{2} }} - \frac{{C_{O} D}}{{\left( {Q^{L} } \right)^{2} }}} \right)\left( {1 - \lambda } \right) \\ &\quad+ \frac{{C_{S}^{L} D\mu_{4} }}{{\left( {Q^{L} } \right)^{2} }} - \frac{{C_{S}^{H} D\mu_{3} }}{{\left( {Q^{L} } \right)^{2} }} - \frac{{C_{S}^{L} D}}{{\left( {Q^{L} } \right)^{2} }} = 0 \end{aligned} $$

Adding these two equations and substituting QL = QH and QL1 = QH1 we get \(\left( {\frac{{h_{1} h_{2} }}{{h_{1} + h_{2} }} - \frac{{2C_{O} D}}{{\left( {Q^{H} } \right)^{2} }}} \right) = \frac{{2C_{S}^{L} D}}{{\left( {Q^{H} } \right)^{2} }}\) implying \(Q^{H} = \sqrt {\frac{{2D\left( {C_{S}^{L} + C_{O} } \right)\left( {h_{1} + h_{2} } \right)}}{{h_{1} h_{2} }}} = Q^{L}\).

Similarly, we can find \(Q_{1}^{H} = \sqrt {\frac{{2D\left( {C_{O} + C_{S}^{L} } \right)h_{2} }}{{h_{1} \left( {h_{1} + h_{2} } \right)}}}\) and \(Q_{1}^{L} = \sqrt {\frac{{2D\left( {C_{O} + C_{S}^{L} } \right)h_{2} }}{{h_{1} \left( {h_{1} + h_{2} } \right)}}}\) respectively.

Case XIV μ1 ≠ 0, μ2 = 0, μ3 ≠ 0, μ4 = 0.

From the Complementary Slackness Conditions, we get

$$ \frac{D}{{Q^{H} }}C_{S}^{H} - C_{S}^{H} - x^{H} D = 0 \Rightarrow x^{H} = C_{S}^{H} \left( {\frac{1}{{Q^{H} }} - \frac{1}{D}} \right) $$
$$ x^{H} D - \frac{D}{{Q^{H} }}C_{S}^{H} - x^{L} D + \frac{D}{{Q^{L} }}C_{S}^{H} = 0 $$

Since μ2 = 0, implies μ1 = 1. Also, μ4 = 0 implies μ3 = 1-λ.

Also from the Stationarity Conditions, we can write

$$ \left( {\frac{{\left( {Q^{H} - Q_{1}^{H} } \right)h_{2} }}{{Q^{H} }} - \frac{{\left( {Q^{H} - Q_{1}^{H} } \right)^{2} h_{2} }}{{2\left( {Q^{H} } \right)^{2} }} - \frac{{\left( {Q_{1}^{H} } \right)^{2} h_{1} }}{{2\left( {Q^{H} } \right)^{2} }} - \frac{{DC_{O} }}{{\left( {Q^{H} } \right)^{2} }}} \right)\lambda = \frac{{C_{S}^{H} D}}{{\left( {Q^{H} } \right)^{2} }} - \frac{{C_{S}^{H} D\left( {1 - \lambda } \right)}}{{\left( {Q^{H} } \right)^{2} }} $$
$$ Q^{H} = \sqrt {\frac{{2D\left( {h_{1} + h_{2} } \right)\left( {C_{S}^{H} + C_{O} } \right)}}{{h_{1} h_{2} }}} $$
$$ \begin{aligned}& \left( {\frac{{\left( {Q^{L} - Q_{1}^{L} } \right)h_{2} }}{{Q^{L} }} - \frac{{\left( {Q^{L} - Q_{1}^{L} } \right)^{2} h_{2} }}{{2\left( {Q^{L} } \right)^{2} }} - \frac{{\left( {Q_{1}^{L} } \right)^{2} h_{1} }}{{2\left( {Q^{L} } \right)^{2} }} - \frac{{C_{O} D}}{{\left( {Q^{L} } \right)^{2} }}} \right) = \frac{{C_{S}^{H} D}}{{\left( {Q^{L} } \right)^{2} }} \\ &\quad\Rightarrow Q^{L} = \sqrt {\frac{{2D\left( {C_{S}^{H} + C_{O} } \right)\left( {h_{1} + h_{2} } \right)}}{{h_{1} h_{2} }}} \end{aligned} $$

Case XV μ1 = 0, μ2 ≠ 0, μ3 = 0, μ4 ≠ 0.

From the Complementary Slackness Conditions, we get.

\(\frac{D}{{Q^{L} }}C_{S}^{L} - C_{S}^{L} - x^{L} D = 0 \Rightarrow x^{L} = C_{S}^{L} \left( {\frac{1}{{Q^{L} }} - \frac{1}{D}} \right)\) and \(x^{H} D - \frac{D}{{Q^{H} }}C_{S}^{H} - x^{L} D + \frac{D}{{Q^{L} }}C_{S}^{H} = 0\).

If μ1 = 0 then μ2 = 1. Also μ4 = λ since μ3 = 0.

Substituting the value of xL in the above equation we get

$$ x^{H} D - \frac{D}{{Q^{H} }}C_{S}^{H} - C_{S}^{L} \left( {\frac{D}{{Q^{L} }} - 1} \right) + \frac{D}{{Q^{L} }}C_{S}^{H} = 0 \Rightarrow $$
$$ \begin{aligned} & \left( {\frac{{\left( {Q^{H} - Q_{1}^{H} } \right)h_{2} }}{{Q^{H} }} - \frac{{\left( {Q^{H} - Q_{1}^{H} } \right)^{2} h_{2} }}{{2\left( {Q^{H} } \right)^{2} }} - \frac{{\left( {Q_{1}^{H} } \right)^{2} h_{1} }}{{2\left( {Q^{H} } \right)^{2} }} - \frac{{DC_{O} }}{{\left( {Q^{H} } \right)^{2} }}} \right)\lambda - \frac{{C_{S}^{L} D\lambda }}{{\left( {Q^{H} } \right)^{2} }} = 0\\ &\quad \Rightarrow Q^{H} = \sqrt {\frac{{2D\left( {C_{O} + C_{S}^{L} } \right)\left( {h_{1} + h_{2} } \right)}}{{h_{1} h_{2} }}} \end{aligned} $$
$$ \begin{gathered} \left( {\frac{{\left( {Q^{L} - Q_{1}^{L} } \right)h_{2} }}{{Q^{L} }} - \frac{{\left( {Q^{L} - Q_{1}^{L} } \right)^{2} h_{2} }}{{2\left( {Q^{L} } \right)^{2} }} - \frac{{\left( {Q_{1}^{L} } \right)^{2} h_{1} }}{{2\left( {Q^{L} } \right)^{2} }} - \frac{{DC_{O} }}{{\left( {Q^{L} } \right)^{2} }}} \right)\left( {1 - \lambda } \right) + \frac{{D\left( {\lambda C_{S}^{L} - C_{S}^{H} } \right)}}{{\left( {Q^{L} } \right)^{2} }} = 0 \hfill \\ \Rightarrow Q^{L} = \sqrt {\frac{{2D\left( {C_{O} + C_{S}^{L} } \right)\left( {h_{1} + h_{2} } \right)}}{{h_{1} h_{2} }}} \hfill \\ \end{gathered} $$

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Chakraborty, A. A surcharge pricing policy for supply chain coordination under the just-in-time (JIT) environment in the presence of backordering. OPSEARCH (2021). https://doi.org/10.1007/s12597-021-00509-5

Download citation

Keywords

  • Inventory theory
  • Supply chain coordination
  • Surcharge pricing
  • Information asymmetry
  • Non-linear programming