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doi: 10.3934/naco.2021020
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## 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 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 & Optimization, doi: 10.3934/naco.2021020
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##### References:
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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