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Joint economic lot sizing model with stochastic demand and controllable lead-time by reducing ordering cost and setup cost

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Revista de la Real Academia de Ciencias Exactas, Físicas y Naturales. Serie A. Matemáticas Aims and scope Submit manuscript

Abstract

This paper derives a mathematical integrated production inventory model of single vendor and buyer to investigate the effects of capital investment in setup cost of the vendor and order processing cost of the buyer on the decision variables of the model. Main purpose of this study is to minimize the integrated total cost of the vendor and buyer by optimizing the delivery lot size, lead-time, setup and order processing cost and number of deliveries per order simultaneously while lead-time demand follows normal distribution and unknown distribution, i.e., distribution free case. This model is analyzed by calculus method and justified with appropriate numerical illustrations. Numerical results show that significant savings can be achieved at optimal investments in reducing lead time, ordering cost and setup cost. Finally, sensitivity analysis of the key parameters is carried out and some managerial implications are also included.

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References

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

    Article  Google Scholar 

  2. Ben-Daya, M., Raouf, A.: Inventory models involving lead-time as a decision variable. J. Oper. Res. Soc. 45, 579–582 (1994)

    Article  Google Scholar 

  3. Cárdenas-Barrón, L.E., Treviño-Garza, G., Wee, H.M.: A simple and better algorithm to solve the vendor managed inventory control system of multi-product multi-constraint economic order quantity model. Expert Syst. Appl. 29, 3888–3895 (2012)

    Article  Google Scholar 

  4. Chang, H.C., Ouyang, L.Y., Wu, K.S., Ho, C.H.: Integrated vendor–buyer cooperative inventory models with controllable lead-time and ordering cost reduction. Eur. J. Oper. Res. 170, 481–495 (2006)

    Article  MathSciNet  Google Scholar 

  5. Chu, P., Yang, K.L., Chen, P.S.: Improved inventory models with service level and lead time. Comput. Oper. Res. 32(2), 285–296 (2005)

    Article  MathSciNet  Google Scholar 

  6. Gallego, G., Moon, I.: The distribution free newsboy problem: review and extensions. J. Oper. Res. Soc. 44(8), 825–834 (1993)

    Article  Google Scholar 

  7. Glock, C.H.: Lead-time reduction strategies in a single vendor-single buyer integrated inventory model with lot size-dependent lead-time and stochastic demand. Int. J. Prod. Econ. 136, 37–44 (2012a)

    Article  Google Scholar 

  8. Glock, C.H.: The joint economic lot size problem: a review. Int. J. Prod. Econ. 135, 671–686 (2012b)

    Article  Google Scholar 

  9. Goyal, S.K.: An integrated inventory model for a single supplier-single customer problem. Int. J. Prod. Res. 15, 107–111 (1976)

    Article  Google Scholar 

  10. Goyal, S.K.: A joint economic-lot-size model for purchaser and vendor: a comment. Decis. Sci. 19, 236–241 (1988)

    Article  Google Scholar 

  11. Goyal, S.K.: A one-vendor multi-buyer integrated inventory model: a comment. Eur. J. Oper. Res. 82, 209–210 (1995)

    Article  Google Scholar 

  12. Goyal, S.K., Nebebe, F.: Determination of economic production-shipment policy for a single-vendor-single-buyer system. Eur. J. Oper. Res. 121, 175–178 (2000)

    Article  Google Scholar 

  13. Ha, D., Kim, S.L.: Implementation of JIT purchasing: an integrated approach. Prod. Plann. Control 8, 152–157 (1997)

    Article  Google Scholar 

  14. Hall, R.W.: Zero Inventories. Dow Jones-Irwin, Homewood (1983)

    Google Scholar 

  15. Hariga, M., Ben-Daya, M.: Some stochastic inventory models with deterministic variable lead-time. Eur. J. Oper. Res. 113, 42–51 (1999)

    Article  Google Scholar 

  16. Hill, R.M.: The single-vendor single-buyer integrated production-inventory model with a generalized policy. Eur. J. Oper. Res. 97, 493–499 (1997)

    Article  Google Scholar 

  17. Hill, R.M.: The optimal production and shipment policy for the single-vendor single-buyer integrated production inventory problem. Int. J. Prod. Res. 37, 2463–2475 (1999)

    Article  Google Scholar 

  18. Hoque, M.A.: A vendor–buyer integrated production inventory model with normal distribution of lead-time. Int. J. Prod. Econ. 144, 409–417 (2013)

    Article  Google Scholar 

  19. Hou, K.L.: An EPQ model with setup cost and process quality as functions of capital expenditure. Appl. Math. Model. 31, 10–17 (2007)

    Article  Google Scholar 

  20. Jalbar, B.A., Gutierrez, J., Sicilia, J.: Integer-ratio policies for distribution/inventory systems. Int. J. Prod. Econ. 93–94, 407–415 (2005)

    Article  Google Scholar 

  21. 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 

  22. Keller, G., Noori, H.: Justifying new technology acquisition through its impact on the cost of running an inventory policy. IIE Trans. 20, 284–291 (1988)

    Article  Google Scholar 

  23. Kim, K.L., Hayya, J.C., Hong, J.D.: Setup reduction in economic production quantity model. Decis. Sci. 23, 500–508 (1992)

    Article  Google Scholar 

  24. Liao, C.J., Shyu, C.H.: An analytical determination of lead-time with normal demand. Int. J. Oper. Prod. Manag. 11, 72–78 (1991)

    Article  Google Scholar 

  25. Lu, L.: A one-vendor multi-buyer integrated inventory model. Eur. J. Oper. Res. 81, 312–323 (1995)

    Article  Google Scholar 

  26. Nasri, F., Affisco, J.F., Paknejad, M.J.: Setup cost reduction in an inventory model with finite-range stochastic lead-times. Int. J. Prod. Res. 28, 199–212 (1990)

    Article  Google Scholar 

  27. Ouyang, L.Y., Chang, H.C.: Lot size reorder point inventory model with controllable lead-time and set-up cost. Int. J. Syst. Sci. 33, 635–642 (2002)

    Article  MathSciNet  Google Scholar 

  28. Ouyang, L.Y., Chen, C.K., Chang, H.C.: Lead-time and ordering cost reductions in continuous review inventory systems with partial backorders. J. Oper. Res. Soc. 50, 1272–1279 (1999)

    Article  Google Scholar 

  29. Ouyang, L.Y., Wu, K.S., Ho, C.H.: Integrated vendor–buyer cooperative models with stochastic demand in controllable lead-time. Int. J. Prod. Econ. 92, 255–266 (2004)

    Article  Google Scholar 

  30. Ouyang, L.Y., Yeh, N.C., Wu, K.S.: Mixture inventory model with backorders and lost sales for variable lead-time. J. Oper. Res. Soc. 47, 829–832 (1996)

    Article  Google Scholar 

  31. Ouyang, L.Y., Wu, K.S.: Mixture inventory model involving variable lead time with a service level constraint. Comput. Oper. Res. 24(9), 875–882 (1997)

    Article  MathSciNet  Google Scholar 

  32. Paknejad, M.J., Nasri, F., Affisco, J.F.: Defective units in a continuous review (s, Q) system. Int. J. Prod. Res. 33, 2767–2777 (1995)

    Article  Google Scholar 

  33. Pan, J.C.H., Yang, J.S.: A study of an integrated inventory with controllable lead-time. Int. J. Prod. Res. 40, 1263–1273 (2002)

    Article  Google Scholar 

  34. Panda, S.: Coordination of a socially responsible supply chain using revenue sharing contract. Transp. Res. Part E Logist. Transp. Rev. 67, 92–104 (2014)

    Article  Google Scholar 

  35. Panda, S., Shah, S.: Optimal production rate and production stopping time for perishable seasonal products with ramp-type time-dependent demand. Int. J. Math. Oper. Res. 2, 657–673 (2010)

    Article  MathSciNet  Google Scholar 

  36. Panda, S., Modak, N.M., Basu, M.: Disposal cost sharing and bargaining for coordination and profit division in a three-echelon supply chain. Int. J. Manag. Sci. Eng. Manag. 9, 276–285 (2015)

    Google Scholar 

  37. Panda, S., Modak, N.M., Pradhan, D.: Corporate social responsibility, channel coordination and profit division in a two-echelon supply chain. Int. J. Manag. Sci. Eng. Manag. 11, 22–33 (2016)

    Google Scholar 

  38. Porteus, E.L.: Investing in reduced setups in the EOQ model. Manag. Sci. 31, 998–1010 (1985)

    Article  Google Scholar 

  39. Saha, S., Panda, S., Modak, N.M., Basu, M.: Mail-in-rebate coupled with revenue sharing downward direct discount for supply chain coordination. Int. J. Oper. Res. 23, 451–476 (2016)

    Article  MathSciNet  Google Scholar 

  40. Sarkar, B., Saren, S., Sarkar, M., Seo, Y.W.: A Stackelberg game approach in an integrated inventory model with carbon-emission and setup cost reduction. Sustainability 8, 1244 (2016). doi:10.3390/su8121244

    Article  Google Scholar 

  41. Sarkar, B., Saren, S., Sinha, D., Hur, S.: Effect of unequal lot sizes, variable setup cost, and carbon emission cost in a supply chain model. Math. Probl. Eng. 2015, 12 (2015). Article ID 469486

    Article  MathSciNet  Google Scholar 

  42. Tersine, R.J.: Principles of Inventory and Materials Management. Prentice Hall, Englewood Cliffs (1994)

    Google Scholar 

  43. Viswanathan, S.: Optimal strategy for the integrated vendor–buyer inventory model. Eur. J. Oper. Res. 105, 38–42 (1998)

    Article  Google Scholar 

  44. Wu, M.Y., Wee, H.M.: Buyer-seller joint cost for deteriorating items with multiple- lot-size deliveries. J. Chin. Inst. Ind. Eng. 18, 109–119 (2001)

    Google Scholar 

  45. Yang, P.C., Wee, H.M.: Economic order policy of deteriorated item for vendor and buyer: an integrated approach. Prod. Plan. Control 11, 474–480 (2000)

    Article  Google Scholar 

  46. Yang, P.C., Wee, H.M.: An arborescent inventory model in a supply chain system. Prod. Plan. Control 12, 728–735 (2001)

    Article  Google Scholar 

  47. Yang, P.C., Wee, H.M.: A single-vendor and multiple-buyers production- inventory policy for a deteriorating item. Eur. J. Oper. Res. 14, 570–581 (2002)

    Article  MathSciNet  Google Scholar 

  48. Yu, Y.G., Liang, L., Wang, C., Wang, Z.Q.: An integrated vendor- managed-inventory model for deteriorating item. Chin. J. Manag. Sci. 12, 32–37 (2004)

    Google Scholar 

  49. Zhang, T., Liang, L., Yu, Y., Yan, Yu.: An integrated vendor-managed inventory model for a two-echelon system with order cost reduction. Int. J. Prod. Econ. 109, 241–253 (2007)

    Article  Google Scholar 

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Author information

Authors and Affiliations

  1. Department of Operational Research, Faculty of Mathematical Sciences, New Academic Block, University of Delhi, Delhi, 110007, India

    Sunil Tiwari

  2. Department of Mathematics, Bhangar Mahavidyalaya, South 24 Parganas, Bhangar, 743502, India

    Shib Sankar Sana

  3. Department of Mathematics, Jadavpur University, Kolkata, India

    Sumon Sarkar

Authors
  1. Sunil Tiwari
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  2. Shib Sankar Sana
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  3. Sumon Sarkar
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Corresponding author

Correspondence to Shib Sankar Sana.

Appendix

Appendix

Now, we have to prove that the Hessian Matrix of \(JTC_{NvbAS} (Q,k,L,A,S,m)\) at point \((Q^{*},k^{*},A^{*},S^{*})\) for fixed m and \(L\in \left[ {L_i ,L_{i-1} } \right] \) is positive definite.

We first obtain the Hessian Matrix H as follows:

$$\begin{aligned} H=\left[ {{\begin{array}{cccc} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial Q^{2}}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial Q\partial k}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial Q\partial A}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial Q\partial S}} \\ {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial Q}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k^{2}}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial A}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial S}} \\ {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A\partial Q}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A\partial k}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A^{2}}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A\partial S}} \\ {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S\partial Q}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S\partial k}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S\partial A}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S^{2}}} \\ \end{array} }} \right] \end{aligned}$$

where

$$\begin{aligned} \frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial Q^{2}}= & {} \frac{2D}{Q^{3}}\left( {A+\frac{S}{m}+\pi \sigma \sqrt{L}\psi (k)+C(L)} \right)>0\\ \frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k^{2}}= & {} \frac{D\pi \sigma \sqrt{L}\varphi (k)}{Q}>0\\ \frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A^{2}}= & {} \frac{\alpha a}{A^{2}}>0\\ \frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S^{2}}= & {} \frac{\alpha b}{S^{2}}>0\\ \frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial Q\partial k}= & {} \frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial Q}\\= & {} -\frac{D\pi \sigma \sqrt{L}\left( {\phi (k)-1} \right) }{Q^{2}}\\ \frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial Q\partial A}= & {} \frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A\partial Q}=-\frac{D}{Q^{2}}\\ \frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial Q\partial S}= & {} \frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S\partial Q}=-\frac{D}{Q^{2}m}\\ \frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial A}= & {} \frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A\partial k}=0\\ \frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial S}= & {} \frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S\partial k}=0\\ \frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A\partial S}= & {} \frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S\partial A}=0\\ \left| {H_{11} } \right|= & {} \left| {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial Q^{2}}} \right|>0 \nonumber \\= & {} \frac{2D}{Q^{3}}\left( {A+\frac{S}{m}+\pi \sigma \sqrt{L}\psi (k)+C(L)} \right) >0 \end{aligned}$$

Therefore, \(\left| {H_{11} } \right| >0\)

$$\begin{aligned} \left| {H_{22} } \right|= & {} \left| {{\begin{array}{cc} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial Q^{2}}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial Q\partial k}} \\ {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial Q}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k^{2}}} \\ \end{array} }} \right| \\= & {} \frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial Q^{2}}\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k^{2}}\\&-\;\left( {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial Q}} \right) ^{2}\\= & {} \frac{2D}{Q^{3}}\left( {A+\frac{S}{m}+\pi \sigma \sqrt{L}\psi (k)+C(L)} \right) \left( {\frac{D\pi \sigma \sqrt{L}\varphi (k)}{Q}} \right) \\&-\;\left( {-\frac{D\pi \sigma \sqrt{L}\left( {\phi (k)-1} \right) }{Q^{2}}} \right) ^{2}\\= & {} \left( {A+\frac{S}{m}{+}\pi \sigma \sqrt{L}\psi (k)+C(L)} \right) \left( {\frac{2D^{2}\pi \sigma \sqrt{L}\varphi (k)}{Q^{4}}} \right) {-}\frac{\left( {D\pi \sigma \sqrt{L}\left( {\phi (k)-1} \right) } \right) ^{2}}{Q^{4}}\\= & {} \frac{\left( {A+\frac{S}{m}+\pi \sigma \sqrt{L} \psi (k)+c(L)} \right) \left( {2D^{2}\pi \sigma \sqrt{L}\varphi (k)} \right) -\left( {D\pi \sigma \sqrt{L}\left( {\phi (k)-1} \right) } \right) ^{2}}{Q^{4}}\\= & {} \frac{2D^{2}}{Q^{4}}\pi \sigma \sqrt{L}\phi (k) \left( {A+\frac{S}{m}+\pi \sigma \sqrt{L}\psi (k)+C(L)} \right) \\&-\;\frac{D^{2}}{Q^{4}}\left( {\pi \sigma \sqrt{L}} \right) ^{2}\left( {2\varphi (k)\psi (k)- \left( {\phi (k)-1} \right) ^{2}} \right)>0\\= & {} \frac{2D^{2}}{Q^{4}}\pi \sigma \sqrt{L}\varphi \left( k \right) \left( {A+\frac{S}{m}+C\left( L \right) } \right) +\frac{D^{2}}{Q^{4}}\left( {\pi \sigma \sqrt{L}} \right) ^{2}\left( {\phi \left( k \right) -1} \right) ^{2}>0 \end{aligned}$$

Because, \(\phi (k)>0,\psi (k)>0\) and \(\left( {\phi (k)-1} \right) ^{2}>0\) for all \(k>0\) [28].

Therefore, \(\left| {H_{22} } \right| >0\).

$$\begin{aligned} \left| {H_{33} } \right|= & {} \left| {{\begin{array}{ccc} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial Q^{2}}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial Q\partial k}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial Q\partial A}} \\ {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial Q}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k^{2}}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial A}} \\ {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A\partial Q}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A\partial k}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A^{2}}} \\ \end{array} }} \right| \\= & {} \frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial Q^{2}}\\&\times \,\left( {\begin{array}{c} \frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k^{2}}\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A^{2}} \\ -\left( {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial A}} \right) ^{2} \\ \end{array}} \right) \\&-\;\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial Q\partial k}\\&\times \,\left( {\begin{array}{c} \frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial Q}\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A^{2}} \\ -\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A\partial Q}\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial A} \\ \end{array}} \right) \\&+\;\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial Q\partial A}\\&\times \,\left( {\begin{array}{c} \frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial Q}\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A\partial k} \\ -\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A\partial Q}\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k^{2}} \\ \end{array}} \right) \\= & {} \frac{2D}{Q^{3}}\left( {A+\frac{S}{m}+\pi \sigma \sqrt{L}\psi (k)+C(L)} \right) \left[ {\left( {\frac{D\pi \sigma \sqrt{L}\phi (k)}{Q}} \right) \left( {\frac{\alpha a}{A^{2}}} \right) -0} \right] \\&+\left( {\frac{D\pi \sigma \sqrt{L}\left( {\phi (k)-1} \right) }{Q^{2}}} \right) \left( {-\frac{D\pi \sigma \sqrt{L}\left( {\phi (k)-1} \right) }{Q^{2}}\left( {\frac{\alpha a}{A^{2}}} \right) +\frac{D}{Q^{2}}.0} \right) \\&-\;\frac{D}{Q^{2}}\left( {-\frac{D\pi \sigma \sqrt{L}\left( {\phi (k)-1} \right) }{Q^{2}}.0+\frac{D}{Q^{2}}\frac{D\pi \sigma \sqrt{L}\phi (k)}{Q}} \right) \\= & {} \frac{2D}{Q^{3}}\left( {A+\frac{S}{m}+\pi \sigma \sqrt{L}\psi (k)+C(L)} \right) \left[ {\left( {\frac{D\pi \sigma \sqrt{L}\phi (k)}{Q}} \right) \left( {\frac{\alpha a}{A^{2}}} \right) } \right] \\&-\;\left( {\frac{D\pi \sigma \sqrt{L}\left( {\phi (k)-1} \right) }{Q^{2}}} \right) \left( {\frac{D\pi \sigma \sqrt{L}\left( {\phi (k)-1} \right) }{Q^{2}}\left( {\frac{\alpha a}{A^{2}}} \right) } \right) \\&+\;\frac{D}{Q^{2}}\left( {\frac{D}{Q^{2}}\frac{D\pi \sigma \sqrt{L}\phi (k)}{Q}} \right) \\ \end{aligned}$$

If

$$\begin{aligned}&\frac{2D}{Q^{3}}\left( {A+\frac{S}{m}+\pi \sigma \sqrt{L}\psi (k)+C(L)} \right) \left[ {\left( {\frac{D\pi \sigma \sqrt{L}\phi (k)}{Q}} \right) \left( {\frac{\alpha a}{A^{2}}} \right) } \right] \\&\qquad +\;\frac{D}{Q^{2}}\left( {\frac{D}{Q^{2}}\frac{D\pi \sigma \sqrt{L}\phi (k)}{Q}} \right) \\&\quad >\left( {\frac{D\pi \sigma \sqrt{L} \left( {\phi (k)-1} \right) }{Q^{2}}} \right) \left( {\frac{D\pi \sigma \sqrt{L}\left( {\phi (k)-1} \right) }{Q^{2}}\left( {\frac{\alpha a}{A^{2}}} \right) } \right) \end{aligned}$$

Then,

$$\begin{aligned}&=\frac{2D}{Q^{3}}\left( {A+\frac{S}{m}+\pi \sigma \sqrt{L}\psi (k)+C(L)} \right) \left[ {\left( {\frac{D\pi \sigma \sqrt{L}\phi (k)}{Q}} \right) \left( {\frac{\alpha a}{A^{2}}} \right) } \right] \\&\quad -\;\left( {\frac{D\pi \sigma \sqrt{L}\left( {\phi (k)-1} \right) }{Q^{2}}}\right) \left( {\frac{D\pi \sigma \sqrt{L}\left( {\phi (k)-1} \right) }{Q^{2}}\left( {\frac{\alpha a}{A^{2}}} \right) } \right) +\frac{D}{Q^{2}}\left( {\frac{D}{Q^{2}}\frac{D\pi \sigma \sqrt{L}\phi (k)}{Q}} \right) \\&>0 \end{aligned}$$

Therefore, \(\left| {H_{33}}\right| >0\).

$$\begin{aligned} \left| {H_{44} } \right|= & {} \left| {{\begin{array}{cccc} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial Q^{2}}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial Q\partial k}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial Q\partial A}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial Q\partial S}} \\ \times \,{\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial Q}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k^{2}}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial A}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial S}} \\ {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A\partial Q}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A\partial k}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A^{2}}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A\partial S}} \\ \times \,{\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S\partial Q}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S\partial k}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S\partial A}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S^{2}}} \\ \end{array} }} \right| \\ \end{aligned}$$
$$\begin{aligned}= & {} \frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial Q^{2}}\\&\times \,\left| {{\begin{array}{ccc} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k^{2}}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial A}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial S}} \\ {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A\partial k}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A^{2}}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A\partial S}} \\ {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S\partial k}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S\partial A}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S^{2}}} \\ \end{array} }} \right| \\&-\;\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial Q\partial k}\\&\times \,\left| {{\begin{array}{ccc} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial Q}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial A}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial S}} \\ {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A\partial Q}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A^{2}}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A\partial S}} \\ {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S\partial Q}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S\partial A}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S^{2}}} \\ \end{array} }} \right| \\&+\;\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial Q\partial A}\\&\times \,\left| {{\begin{array}{ccc} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial Q}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k^{2}}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial S}} \\ {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A\partial Q}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A\partial k}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A\partial S}} \\ {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S\partial Q}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S\partial k}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S^{2}}} \\ \end{array} }} \right| \\&-\;\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial Q\partial S}\\&\times \,\left| {{\begin{array}{ccc} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial Q}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k^{2}}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial A}} \\ {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A\partial Q}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A\partial k}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A^{2}}} \\ {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S\partial Q}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S\partial k}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S\partial A}} \\ \end{array} }} \right| \end{aligned}$$

Now,

$$\begin{aligned}&\left| {{\begin{array}{ccc} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k^{2}}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial A}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial S}} \\ {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A\partial k}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A^{2}}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A\partial S}} \\ {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S\partial k}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S\partial A}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S^{2}}} \\ \end{array} }} \right| \\&\quad =\left( {\frac{D\pi \sigma \sqrt{L}\phi (k)}{Q}} \right) \left( {\left( {\frac{\alpha a}{A^{2}}} \right) \left( {\frac{\alpha b}{S^{2}}-0} \right) } \right) -0.\left( {0.\frac{\alpha b}{S^{2}}-0.0} \right) +0.\left( {0.0-\frac{\alpha a}{A^{2}}.0} \right) \\&\quad =\left( {\frac{D\pi \sigma \sqrt{L}\phi (k)}{Q}} \right) \left( {\frac{\alpha a}{A^{2}}} \right) \left( {\frac{\alpha b}{S^{2}}} \right) \\&\qquad \times \; \left| {{\begin{array}{ccc} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial Q}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial A}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial S}} \\ {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A\partial Q}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A^{2}}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A\partial S}} \\ {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S\partial Q}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S\partial A}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S^{2}}} \\ \end{array} }} \right| \\&\quad =\left( {-\frac{D\pi \sigma \sqrt{L}\left( {\phi (k)-1} \right) }{Q^{2}}} \right) \left( {\left( {\frac{\alpha a}{A^{2}}.\frac{\alpha b}{S^{2}}-0} \right) } \right) -0.\left( {\left( {-\frac{D}{Q^{2}}} \right) .\frac{\alpha b}{S^{2}}-\frac{D}{Q^{2}m}.0} \right) \\&\qquad +\;0.\left( {\left( {-\frac{D}{Q^{2}}} \right) .0-\frac{\alpha a}{A^{2}}.\left( {-\frac{D}{Q^{2}m}} \right) } \right) \\&\quad =-\left( {\frac{D\pi \sigma \sqrt{L}\left( {\phi (k)-1} \right) }{Q}} \right) \left( {\frac{\alpha a}{A^{2}}} \right) \left( {\frac{\alpha b}{S^{2}}} \right) \\&\qquad \times \; \left| {{\begin{array}{ccc} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial Q}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k^{2}}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial S}} \\ {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A\partial Q}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A\partial k}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A\partial S}} \\ {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S\partial Q}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S\partial k}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S^{2}}} \\ \end{array} }} \right| \\&\quad =\left( {\frac{D\pi \sigma \sqrt{L}\left( {\phi (k)-1} \right) }{Q^{2}}} \right) \left( {\left( {0.\frac{\alpha b}{S^{2}}-0} \right) } \right) \\&\qquad -\;\frac{D\pi \sigma \sqrt{L}\phi (k)}{Q}.\left( {\left( {-\frac{D}{Q^{2}}} \right) .\frac{\alpha b}{S^{2}}-\frac{D}{Q^{2}m}.0} \right) +0.\left( {\left( {-\frac{D}{Q^{2}}} \right) .0-0.\left( {-\frac{D}{Q^{2}m}} \right) } \right) \\&\quad =-\left( {\frac{D^{2}\pi \sigma \sqrt{L}\phi (k)}{Q^{3}}} \right) \left( {\frac{\alpha b}{S^{2}}} \right) \\&\qquad \times \; \left| {{\begin{array}{ccc} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial Q}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k^{2}}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial A}} \\ {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A\partial Q}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A\partial k}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A^{2}}} \\ {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S\partial Q}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S\partial k}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S\partial A}} \\ \end{array} }} \right| \\&\quad =\left( {-\frac{D\pi \sigma \sqrt{L}\left( {\phi (k)-1} \right) }{Q^{2}}} \right) \left( {0.0-\frac{\alpha a}{A^{2}}.0} \right) \\&\qquad -\;\frac{D\pi \sigma \sqrt{L}\phi (k)}{Q}.\left( {\left( {-\frac{D}{Q^{2}}} \right) .0\left( {-\frac{D}{Q^{2}m}} \right) .\frac{\alpha a}{A^{2}}} \right) -\frac{D}{Q^{2}}.\left( {0.0-0.\left( {-\frac{D}{Q^{2}m}} \right) } \right) \\&\quad =-\left( {\frac{D^{2}\pi \sigma \sqrt{L}\phi (k)}{Q^{3}m}} \right) \left( {\frac{\alpha a}{A^{2}}} \right) \\ \end{aligned}$$
$$\begin{aligned} \left| {H_{44} } \right|= & {} \left| {{\begin{array}{cccc} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial Q^{2}}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial Q\partial k}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial Q\partial A}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial Q\partial S}} \\ {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial Q}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k^{2}}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial A}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial k\partial S}} \\ {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A\partial Q}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A\partial k}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A^{2}}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial A\partial S}} \\ {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S\partial Q}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S\partial k}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S\partial A}}&{} {\frac{\partial ^{2}JTC_{NvbAS} (Q,k,L,A,S,m)}{\partial S^{2}}} \\ \end{array} }} \right| \\ \end{aligned}$$
$$\begin{aligned}= & {} \left( {A+\frac{S}{m}+\pi \sigma \sqrt{L}\psi (k)+C(L)} \right) \left( {\frac{2D^{2}\pi \sigma \sqrt{L}\phi (k)}{Q^{4}}} \right) \left( {\frac{\alpha a}{A^{2}}} \right) \left( {\frac{\alpha b}{S^{2}}} \right) \\&-\;\left( {-\frac{D\pi \sigma \sqrt{L}\left( {\phi (k)-1} \right) }{Q^{2}}} \right) \left( {-\frac{D\pi \sigma \sqrt{L}\left( {\phi (k)-1} \right) }{Q^{2}}\left( {\frac{\alpha a}{A^{2}}} \right) \left( {\frac{\alpha b}{S^{2}}} \right) } \right) \\&+\;\left( {-\frac{D}{Q^{2}}} \right) \left( {\frac{D^{2}\pi \sigma \sqrt{L}\phi (k)}{Q^{3}}} \right) \left( {\frac{\alpha b}{S^{2}}} \right) -\left( {-\frac{D}{Q^{2}m}} \right) \left( {\frac{D^{2}\pi \sigma \sqrt{L}\phi (k)}{Q^{3}m}} \right) \left( {\frac{\alpha a}{A^{2}}} \right) \\= & {} \left( {A+\frac{S}{m}+\pi \sigma \sqrt{L}\psi (k)+C(L)} \right) \left( {\frac{2D^{2}\pi \sigma \sqrt{L}\phi (k)}{Q^{4}}} \right) \left( {\frac{\alpha a}{A^{2}}} \right) \left( {\frac{\alpha b}{S^{2}}} \right) \\&-\;\left( {\frac{D^{2}\pi \sigma \sqrt{L}\left( {\phi (k)-1} \right) }{Q^{2}}} \right) ^{2}\left( {\frac{\alpha ^{2}ab}{A^{2}S^{2}}} \right) -\left( {\frac{D^{3}\pi \sigma \sqrt{L}\phi (k)}{Q^{5}}} \right) \left( {\frac{\alpha b}{S^{2}}} \right) \\&-\;\left( {\frac{D^{3}\pi \sigma \sqrt{L}\phi (k)}{Q^{5}m^{2}}} \right) \left( {\frac{\alpha a}{A^{2}}} \right) \end{aligned}$$

If

$$\begin{aligned}&\left( {A+\frac{S}{m}+\pi \sigma \sqrt{L}\psi (k)+C(L)} \right) \left( {\frac{2D^{2}\pi \sigma \sqrt{L}\phi (k)}{Q^{4}}} \right) \left( {\frac{\alpha a}{A^{2}}} \right) \left( {\frac{\alpha b}{S^{2}}} \right) \\&\quad >\left( {\frac{D^{2}\pi \sigma \sqrt{L}\left( {\phi (k)-1} \right) }{Q^{2}}} \right) ^{2}\left( {\frac{\alpha ^{2}ab}{A^{2}S^{2}}} \right) +\left( {\frac{D^{3}\pi \sigma \sqrt{L}\phi (k)}{Q^{5}}} \right) \left( {\frac{\alpha b}{S^{2}}} \right) \\&\qquad +\;\left( {\frac{D^{3}\pi \sigma \sqrt{L}\phi (k)}{Q^{5}m^{2}}} \right) \left( {\frac{\alpha a}{A^{2}}} \right) , \end{aligned}$$

then

$$\begin{aligned}&\left( {A+\frac{S}{m}+\pi \sigma \sqrt{L}\psi (k)+C(L)} \right) \left( {\frac{2D^{2}\pi \sigma \sqrt{L}\phi (k)}{Q^{4}}} \right) \left( {\frac{\alpha a}{A^{2}}} \right) \left( {\frac{\alpha b}{S^{2}}} \right) \\&\quad -\;\left( {\frac{D^{2}\pi \sigma \sqrt{L}\left( {\phi (k)-1} \right) }{Q^{2}}} \right) ^{2}\left( {\frac{\alpha ^{2}ab}{A^{2}S^{2}}} \right) -\left( {\frac{D^{3}\pi \sigma \sqrt{L}\phi (k)}{Q^{5}}} \right) \left( {\frac{\alpha b}{S^{2}}} \right) \\&\quad -\;\left( {\frac{D^{3}\pi \sigma \sqrt{L}\phi (k)}{Q^{5}m^{2}}} \right) \left( {\frac{\alpha a}{A^{2}}} \right) >0 \end{aligned}$$

Therefore, \(\left| {H_{44}}\right| >0\).

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Tiwari, S., Sana, S.S. & Sarkar, S. Joint economic lot sizing model with stochastic demand and controllable lead-time by reducing ordering cost and setup cost. RACSAM 112, 1075–1099 (2018). https://doi.org/10.1007/s13398-017-0410-y

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