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

Kamran Jalilian; Kameleh Nasiri Pirbazari

doi:10.3934/naco.2021020

Article Contents

2022, Volume 12, Issue 3: 537-550. 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

Kamran Jalilian^, and
Kameleh Nasiri Pirbazari

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

^* Corresponding author: Kamran Jalilian
^* Corresponding author: Kamran Jalilian

Received: July 31, 2020

Revised: February 28, 2021

Early access: May 2021

Published: September 2022

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Abstract

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.

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Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C35.

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Table 1. 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

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