Industrial optimization using three-factor Cobb-Douglas production function of non-linear programming with application

Shakoor Muhammad; Fazal Hanan; Sayyar Ali Shah; Aihua Yuan; Wahab Khan; Hua Sun; Shakoor Muhammad; Fazal Hanan; Sayyar Ali Shah; Aihua Yuan; Wahab Khan; Hua Sun

doi:10.3934/math.20231532

AIMS Mathematics

2023, Volume 8, Issue 12: 29956-29974. doi: 10.3934/math.20231532

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Research article

Industrial optimization using three-factor Cobb-Douglas production function of non-linear programming with application

1.
School of Environmental and Chemical Engineering Jiangsu University of Science and Technology Zhenjiang 212003, China
2.
Department of Mathematics, Abdul Wali Khan University Mardan, Khyber Pakhtunkhwa, Pakistan
3.
Department of Computer Science, University of Science and Technology, Bannu, KP, Pakistan

Received: 22 July 2023 Revised: 09 October 2023 Accepted: 16 October 2023 Published: 06 November 2023
MSC : 90C30, 90C90

This paper is about the effectiveness of the Cobb-Douglas (C-D) production function in industrial optimization, estimating the number of factors used in the production process of the water industry, for instance, capital and human labor. Moreover, we have modeled a nonlinear optimization problem for a local water industry using two and three factors of production. For this purpose, we have taken into account the Cobb-Douglas production function with different production factors using the Lagrange multiplier method with the ordinary least squares method. In the course of the solution, a linear function is used to calculate the cost function, and the C-D production function is used to calculate the production function. The Lagrange multiplier method with the ordinary least squares method is then used to solve the constrained optimization problem for the product of production. Furthermore, we compared the outcomes from both examples of two- and three-factor C-D production functions in order to validate the Lagrange multiplier method for the C-D production function. Moreover, the three-factor C-D production function is solved by the Lagrange multiplier method with the ordinary least squares method, which provides optimal results as compared to previous studies in literature. The validity of the proposed methodology is explained by using the products of a local production industry in Pakistan.
- Cobb-Douglas (C-D) production function,
- non-linear programming,
- optimization,
- cost function,
- human labors and capitals
Citation: Shakoor Muhammad, Fazal Hanan, Sayyar Ali Shah, Aihua Yuan, Wahab Khan, Hua Sun. Industrial optimization using three-factor Cobb-Douglas production function of non-linear programming with application[J]. AIMS Mathematics, 2023, 8(12): 29956-29974. doi: 10.3934/math.20231532

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Abstract

This paper is about the effectiveness of the Cobb-Douglas (C-D) production function in industrial optimization, estimating the number of factors used in the production process of the water industry, for instance, capital and human labor. Moreover, we have modeled a nonlinear optimization problem for a local water industry using two and three factors of production. For this purpose, we have taken into account the Cobb-Douglas production function with different production factors using the Lagrange multiplier method with the ordinary least squares method. In the course of the solution, a linear function is used to calculate the cost function, and the C-D production function is used to calculate the production function. The Lagrange multiplier method with the ordinary least squares method is then used to solve the constrained optimization problem for the product of production. Furthermore, we compared the outcomes from both examples of two- and three-factor C-D production functions in order to validate the Lagrange multiplier method for the C-D production function. Moreover, the three-factor C-D production function is solved by the Lagrange multiplier method with the ordinary least squares method, which provides optimal results as compared to previous studies in literature. The validity of the proposed methodology is explained by using the products of a local production industry in Pakistan.

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