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September  2022, 12(3): 537-550. doi: 10.3934/naco.2021020

## Convex optimization without convexity of constraints on non-necessarily convex sets and its applications in customer satisfaction in automotive industry

 Faculty of Mathematical Sciences, University of Guilan., Rasht, 41996-13776, Iran

* Corresponding author: Kamran Jalilian

Received  August 2020 Revised  March 2021 Published  September 2022 Early access  June 2021

In the present paper, some necessary and su?cient optimality conditions for a convex optimization problem over inequality constraints are presented which are not necessarily convex and are based on convex intersection of non-necessarily convex sets. The oriented distance function and a characterization of the normal cone of the feasible set are used to obtain the optimality conditions. In the second part of the paper, a non-linear smooth optimization model for customer satisfaction in automotive industry is introduced. The results of the first part are applied to solve this problem theoretically.

Citation: Kamran Jalilian, Kameleh Nasiri Pirbazari. Convex optimization without convexity of constraints on non-necessarily convex sets and its applications in customer satisfaction in automotive industry. Numerical Algebra, Control and Optimization, 2022, 12 (3) : 537-550. doi: 10.3934/naco.2021020
##### References:
 [1] E. Allevi, J. E. Martínez-Legaz and R. Riccardi, Optimality conditions for convex problems on intersections of non necessarily convex sets, Journal of Global Optimization, 77 (2020), 143-155.  doi: 10.1007/s10898-019-00849-z. [2] T. W. Andreassen and B. Lindestad, Customer loyalty and complex services, International Journal of Service Industry Management, 9 (1998), 7-23. [3] M. S. Bazaraa, H. D. Sherali and C. M. Shetty, Nonlinear Programming, Wiley, New Jersey, 2006. [4] Y. Bilan, Sustainable development of a company: Building of new level relationship with the consumers of XXI. Century, Amfiteatru Economic, 15 (2013), 687-701. [5] M. Bruhn and M. A. Grund, Development and implementation of national customer satisfaction indices: the Swiss Index of Customer Satisfaction (SWICS), Total Quality Management, 11 (2000), 1017-1028. [6] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, New York, 2004.  doi: 10.1017/CBO9780511804441. [7] N. H. Chieu, V. Jeyakumar, G. Li and H. Mohebi, Constraint qualifications for convex optimization without convexity of constraints: New connections and applications to best approximation, European Journal of Operational Research, 265 (2018), 19-25.  doi: 10.1016/j.ejor.2017.07.038. [8] F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer, New York, 1998. [9] J. Dutta and C. S. Lalitha, Optimality conditions in convex optimization revisited, Optimization Letters, 7 (2013), 221-229.  doi: 10.1007/s11590-011-0410-3. [10] Z. Ehsani and M. H. Ehsani, Effect of quality and price on customer satisfaction and commitment in Iran auto industry, International Journal of Service Science, Management and Engineering, 1 (2015), 52-59. [11] C. Fornell, A national customer satisfaction barometer: The Swedish experience, Journal of Marketing, 56 (1992), 6-21. [12] C. Fornell, M. D. Johnson, E. W. Anderson, J. Cha and B. E. Bryant, The American customer satisfaction index: nature, purpose, and findings, Journal of Marketing, 60 (1996), 7-18. [13] J. B. Hirriart-Urruty, New concepts in nondifferentiable programming, Bull. Soc. Math. France, 60 (1979), 57-85. [14] R. Hussain, A. Al Nasser and Y. K. Hussain, Service quality and customer satisfaction of a UAE-based airline: An empirical investigation, Journal of Air Transport Management, 42 (2015), 167-175. [15] A. A. Jahanshahi, M. A. H. Gashti, S. A. Mirdamadi, K. Nawaser and S. M. S. Khaksar, Study the effects of customer service and product quality on customer satisfaction and loyalty, International Journal of Humanities and Social Science, 1 (2011), 253-260. [16] S. A. Jafari and A. M. Tehranchian, The effect of the optimal monetary and fiscal policies on major macroeconomic indexes in Iran: Av application of optimal control theory, Journal of Economic Research (Tahghighat- E-Eghtesadi), (in Persian), 39 (2004), 213-242. [17] M. D. Johnson and C. Fornell, A framework for comparing customer satisfaction across individuals and product categories, Journal of Economic Psychology, 12 (1991), 267-286. [18] A. Kabgani, M. Soleimani-Damaneh and M. Zamani, Optimality conditions in optimization problems with convex feasible set using convexificators, Mathematical Methods of Operations Research, 86 (2017), 103-121.  doi: 10.1007/s00186-017-0584-2. [19] D. E. Kirk, Optimal Control Theory: An Introduction, New York, Dover Publications Inc, 2012. [20] O. V. Krivobokova, Evaluating customer satisfaction as an aspect of quality management, World Academy of Science, Engineering and Technology, 53 (2009), 565-568. [21] P. Kotler and K. Keller, Dirección de Marketing (Decimocuarta ed), Naucalpan de Juárez, Pearson Education, 2012. [22] J. B. Lasserre, On representations of the feasible set in convex optimization, Optimization Letters, 4 (2010), 1-5.  doi: 10.1007/s11590-009-0153-6. [23] N. T. H. Linh and J. P. Penot, Optimality conditions for quasiconvex programs, SIAM Journal on Optimization, 17 (2006), 500-510.  doi: 10.1137/040621843. [24] J. E. Martínez-Legaz, Optimality conditions for pseudoconvex minimization over convex sets defined by tangentially convex constraints, Optimization Letters, 9 (2015), 1017-1023.  doi: 10.1007/s11590-014-0822-y. [25] S. Nair, Assessing customer satisfaction and brand awareness of branded bread, IOSR Journal of Business and Management, 12 (2013), 13-18. [26] V. M. Ngo and D. Pavelková, Moderating and mediating effects of switching costs on the relationship between service value, customer satisfaction and customer loyalty: investigation of retail banking in Vietnam, Journal of International Studies, 10 (2017), 9-33. [27] K. N. Pirbazari and K. Jalilian, Designing an optimal customer satisfaction model in automotive industry, Journal of Control, Automation and Electrical Systems, 31 (2020), 31-39. [28] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Translated by KN Trirogoff, New York, 1962. [29] M. Rod, N. J. Ashill, J. Shao and J. Carruthers, An examination of the relationship between service quality dimensions, overall internet banking service quality and customer satisfaction, Marketing Intelligence and Planning, 27 (2009), 103-126. [30] E. N. Saghier and D. Nathan, Service quality dimensions and customers' satisfactions of banks in Egypt, In Proceedings of 20th International Business Research Conference, 2013. [31] K. Srivastava and N. K. Sharma, Service quality, corporate brand image, and switching behavior: The mediating role of customer satisfaction and repurchase intention, Services Marketing Quarterly, 34 (2013), 274-291. [32] A. H. Susanto, The influence of customer purchase decision on customer satisfaction and it's impact to customer loyalty, Jurnal EMBA: Jurnal Riset Ekonomi, Manajemen, Bisnis dan Akuntansi, 1 (2013), 639-658. [33] D. Szwajca, Relationship between corporate image and corporate reputation in Polish banking sector, Oeconomia Copernicana, 9 (2018), 493-509. [34] G. T. Yeo, V. V. Thai and S. Y. Roh, An analysis of port service quality and customer satisfaction: The case of Korean container ports, The Asian Journal of Shipping and Logistics, 31 (2015), 437-447. [35] Y. Zhou and Z. Wang, A robust optimal trajectory tracking control for systems with an input delay, Journal of the Franklin Institute, 353 (2016), 2627-2649.  doi: 10.1016/j.jfranklin.2016.05.003.

show all references

##### References:
 [1] E. Allevi, J. E. Martínez-Legaz and R. Riccardi, Optimality conditions for convex problems on intersections of non necessarily convex sets, Journal of Global Optimization, 77 (2020), 143-155.  doi: 10.1007/s10898-019-00849-z. [2] T. W. Andreassen and B. Lindestad, Customer loyalty and complex services, International Journal of Service Industry Management, 9 (1998), 7-23. [3] M. S. Bazaraa, H. D. Sherali and C. M. Shetty, Nonlinear Programming, Wiley, New Jersey, 2006. [4] Y. Bilan, Sustainable development of a company: Building of new level relationship with the consumers of XXI. Century, Amfiteatru Economic, 15 (2013), 687-701. [5] M. Bruhn and M. A. Grund, Development and implementation of national customer satisfaction indices: the Swiss Index of Customer Satisfaction (SWICS), Total Quality Management, 11 (2000), 1017-1028. [6] S. Boyd and L. Vandenberghe, Convex Optimization, Cambridge University Press, New York, 2004.  doi: 10.1017/CBO9780511804441. [7] N. H. Chieu, V. Jeyakumar, G. Li and H. Mohebi, Constraint qualifications for convex optimization without convexity of constraints: New connections and applications to best approximation, European Journal of Operational Research, 265 (2018), 19-25.  doi: 10.1016/j.ejor.2017.07.038. [8] F. H. Clarke, Yu. S. Ledyaev, R. J. Stern and P. R. Wolenski, Nonsmooth Analysis and Control Theory, Springer, New York, 1998. [9] J. Dutta and C. S. Lalitha, Optimality conditions in convex optimization revisited, Optimization Letters, 7 (2013), 221-229.  doi: 10.1007/s11590-011-0410-3. [10] Z. Ehsani and M. H. Ehsani, Effect of quality and price on customer satisfaction and commitment in Iran auto industry, International Journal of Service Science, Management and Engineering, 1 (2015), 52-59. [11] C. Fornell, A national customer satisfaction barometer: The Swedish experience, Journal of Marketing, 56 (1992), 6-21. [12] C. Fornell, M. D. Johnson, E. W. Anderson, J. Cha and B. E. Bryant, The American customer satisfaction index: nature, purpose, and findings, Journal of Marketing, 60 (1996), 7-18. [13] J. B. Hirriart-Urruty, New concepts in nondifferentiable programming, Bull. Soc. Math. France, 60 (1979), 57-85. [14] R. Hussain, A. Al Nasser and Y. K. Hussain, Service quality and customer satisfaction of a UAE-based airline: An empirical investigation, Journal of Air Transport Management, 42 (2015), 167-175. [15] A. A. Jahanshahi, M. A. H. Gashti, S. A. Mirdamadi, K. Nawaser and S. M. S. Khaksar, Study the effects of customer service and product quality on customer satisfaction and loyalty, International Journal of Humanities and Social Science, 1 (2011), 253-260. [16] S. A. Jafari and A. M. Tehranchian, The effect of the optimal monetary and fiscal policies on major macroeconomic indexes in Iran: Av application of optimal control theory, Journal of Economic Research (Tahghighat- E-Eghtesadi), (in Persian), 39 (2004), 213-242. [17] M. D. Johnson and C. Fornell, A framework for comparing customer satisfaction across individuals and product categories, Journal of Economic Psychology, 12 (1991), 267-286. [18] A. Kabgani, M. Soleimani-Damaneh and M. Zamani, Optimality conditions in optimization problems with convex feasible set using convexificators, Mathematical Methods of Operations Research, 86 (2017), 103-121.  doi: 10.1007/s00186-017-0584-2. [19] D. E. Kirk, Optimal Control Theory: An Introduction, New York, Dover Publications Inc, 2012. [20] O. V. Krivobokova, Evaluating customer satisfaction as an aspect of quality management, World Academy of Science, Engineering and Technology, 53 (2009), 565-568. [21] P. Kotler and K. Keller, Dirección de Marketing (Decimocuarta ed), Naucalpan de Juárez, Pearson Education, 2012. [22] J. B. Lasserre, On representations of the feasible set in convex optimization, Optimization Letters, 4 (2010), 1-5.  doi: 10.1007/s11590-009-0153-6. [23] N. T. H. Linh and J. P. Penot, Optimality conditions for quasiconvex programs, SIAM Journal on Optimization, 17 (2006), 500-510.  doi: 10.1137/040621843. [24] J. E. Martínez-Legaz, Optimality conditions for pseudoconvex minimization over convex sets defined by tangentially convex constraints, Optimization Letters, 9 (2015), 1017-1023.  doi: 10.1007/s11590-014-0822-y. [25] S. Nair, Assessing customer satisfaction and brand awareness of branded bread, IOSR Journal of Business and Management, 12 (2013), 13-18. [26] V. M. Ngo and D. Pavelková, Moderating and mediating effects of switching costs on the relationship between service value, customer satisfaction and customer loyalty: investigation of retail banking in Vietnam, Journal of International Studies, 10 (2017), 9-33. [27] K. N. Pirbazari and K. Jalilian, Designing an optimal customer satisfaction model in automotive industry, Journal of Control, Automation and Electrical Systems, 31 (2020), 31-39. [28] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Translated by KN Trirogoff, New York, 1962. [29] M. Rod, N. J. Ashill, J. Shao and J. Carruthers, An examination of the relationship between service quality dimensions, overall internet banking service quality and customer satisfaction, Marketing Intelligence and Planning, 27 (2009), 103-126. [30] E. N. Saghier and D. Nathan, Service quality dimensions and customers' satisfactions of banks in Egypt, In Proceedings of 20th International Business Research Conference, 2013. [31] K. Srivastava and N. K. Sharma, Service quality, corporate brand image, and switching behavior: The mediating role of customer satisfaction and repurchase intention, Services Marketing Quarterly, 34 (2013), 274-291. [32] A. H. Susanto, The influence of customer purchase decision on customer satisfaction and it's impact to customer loyalty, Jurnal EMBA: Jurnal Riset Ekonomi, Manajemen, Bisnis dan Akuntansi, 1 (2013), 639-658. [33] D. Szwajca, Relationship between corporate image and corporate reputation in Polish banking sector, Oeconomia Copernicana, 9 (2018), 493-509. [34] G. T. Yeo, V. V. Thai and S. Y. Roh, An analysis of port service quality and customer satisfaction: The case of Korean container ports, The Asian Journal of Shipping and Logistics, 31 (2015), 437-447. [35] Y. Zhou and Z. Wang, A robust optimal trajectory tracking control for systems with an input delay, Journal of the Franklin Institute, 353 (2016), 2627-2649.  doi: 10.1016/j.jfranklin.2016.05.003.
The parameters and variables
 Parameters and description $X_{10}$ customer satisfaction of after-sale services in the current year $X_{20}$ customer satisfaction of sale process in the current year $X_{30}$ customer satisfaction of IQS in the current year $X_{40}$ customer satisfaction of APEAL in the current year $\bar{X}_{1}>0$ at least customer satisfaction of after-sale services $\bar{X}_{2}>0$ at least customer satisfaction of sale process $\bar{X}_{3}>0$ at least customer satisfaction of IQS $\bar{X}_{4}>0$ at least customer satisfaction of APEAL $\mu, \gamma$ parameters in the cost of increasing satisfaction function $\mathcal{CO}(S_{0}, S_{1})$ Variables and description $X_{11}$ customer satisfaction of after-sale services in the next year $X_{21}$ customer satisfaction of sale process in the next year $X_{31}$ customer satisfaction of IQS in the next year $X_{41}$ customer satisfaction of APEAL in the next year
 Parameters and description $X_{10}$ customer satisfaction of after-sale services in the current year $X_{20}$ customer satisfaction of sale process in the current year $X_{30}$ customer satisfaction of IQS in the current year $X_{40}$ customer satisfaction of APEAL in the current year $\bar{X}_{1}>0$ at least customer satisfaction of after-sale services $\bar{X}_{2}>0$ at least customer satisfaction of sale process $\bar{X}_{3}>0$ at least customer satisfaction of IQS $\bar{X}_{4}>0$ at least customer satisfaction of APEAL $\mu, \gamma$ parameters in the cost of increasing satisfaction function $\mathcal{CO}(S_{0}, S_{1})$ Variables and description $X_{11}$ customer satisfaction of after-sale services in the next year $X_{21}$ customer satisfaction of sale process in the next year $X_{31}$ customer satisfaction of IQS in the next year $X_{41}$ customer satisfaction of APEAL in the next year
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