# American Institute of Mathematical Sciences

• Previous Article
Non-zero-sum reinsurance and investment game with correlation between insurance market and financial market under CEV model
• JIMO Home
• This Issue
• Next Article
A hybrid metaheuristic algorithm for the multi-objective location-routing problem in the early post-disaster stage
doi: 10.3934/jimo.2022092
Online First

Online First articles are published articles within a journal that have not yet been assigned to a formal issue. This means they do not yet have a volume number, issue number, or page numbers assigned to them, however, they can still be found and cited using their DOI (Digital Object Identifier). Online First publication benefits the research community by making new scientific discoveries known as quickly as possible.

Readers can access Online First articles via the “Online First” tab for the selected journal.

## Application of sphere packing theory in financial management

 1 School of Business, National University of Mongolia, Ulaanbaatar, PC 14192, Mongolia 2 Institute of Mathematics and Digital Technology, Mongolian Academy of Sciences, Ulaanbaatar, PC 13330, Mongolia

* Corresponding author: Ankhbayar Chuluunbaatar

Received  November 2021 Revised  March 2022 Early access June 2022

The paper illustrates an application of sphere packing theory based on the capital structure of financial management. The entire process is modelled as a new optimization problem formulated as the sphere packing problem to define the capital structure. Numerical results are provided on an example of APU JSC of Mongolia.

Citation: Ankhbayar Chuluunbaatar, Burmaa Galaa, Enkhbat Rentsen. Application of sphere packing theory in financial management. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2022092
##### References:
 [1] P. J. Barry, R. W. Bierlen and N. L. Sotomayor, Financial structure of farm business under imperfect capital markets, American Journal of Agricultural Economics, 82 (2000), 920-933. [2] R. H. Bernhard, Mathematical programming models for capital budgeting: A survey, generalization, and critique, Journal of Financial and Quantitative Analysis, 4 (1969), 111-158. [3] R. Bierlen and A. M. Featherstone, Fundamental Q, cash flow and investment: Evidence from farm panel data, The Review of Economics and Statistics, 80 (1998), 427-435. [4] T. O. Boucher, A mixed-integer programming model for optimal investment and financing in segmented international markets, The Engineering Economist, 40 (1995), 145-170. [5] E. F. Brigham and M. C. Ehrhardt, Financial Management: Theory and Practice, 13$^{nd}$ edition, United States of America, 2011. [6] R. Enkhbat, Global optimization approach to Malfatti's problem, J. Global Optim., 65 (2016), 33-39.  doi: 10.1007/s10898-015-0372-6. [7] R. Enkhbat, Convex maximization formulation of general sphere packing problem, Izv. Irkutsk. Gos. Univ. Ser. Mat., 31 (2020), 142-149.  doi: 10.26516/1997-7670.2020.31.142. [8] R. Enkhbat, N. Tungalag, J. Enkhbayar, O. Battogtokh and L. Enkhtuvshin, Application of survival theory in mining industry, Numer. Algebra Control Optim., 11 (2021), 443-448.  doi: 10.3934/naco.2020036. [9] J. R. Freeland and M. J. Rosenblatt, An analysis of linear programming formulations for the capital rationing problem, The Engineering Economist, 23 (1978), 49-61. [10] C. A. Hawkins and R. A. Adams, A goal programming model for capital budgeting, Journal of Financial Management, 3 (1974), 52-57. [11] R. G. Hubbard and A. K. Kashyap, Internal net worth and the investment process: An application to U.S. agriculture, The Journal of Political Economy, 100 (1992), 506-534. [12] A. Kraus and R. H. Litzenberger, A state-preference model of optimal financial leverage, Journal of Finance, 28 (1973), 911-922. [13] R. R. Levary, A sequential solution procedure to stochastic capital budgeting models, Computers and Industrial Engineering, 14 (1988), 371-380. [14] J. H. Lorie and C. J. Savage, Three problems in rationing capital, Journal of Business, 28 (1955), 229-239. [15] G. L. Luenberger, Investment Science, Oxford University Press, 1998. [16] F. Modigliani and M. H. Miller, The cost of capital, corporation finance and the theory of investment, American Economic Review, 1 (1958), 261-297. [17] S. C. Myers, The capital structure puzzle, The Journal of Finance, 39 (1984), 575-592. [18] S. C. Myers, A note of linear programming and capital budgeting, The Journal of Finance, 27 (1972), 89-92. [19] S. A. Ross, The determination of financial structure: The incentive-signaling approach, the Bell Journal of Economics, 8 (1977), 23-40. [20] R. C. Salazar and K. S. Subrata, A simulation model of capital budgeting under uncertainty, Management Science, 15 (1968), 161-179. [21] B. Ulemj, E. Rentsen and B. Tsendpurev, Application of survival theory in taxation, J. Ind. Manag. Optim., 17 (2021), 2573-2578.  doi: 10.3934/jimo.2020083. [22] S. C. Vogt, The role of internal financial sources in firm financing and investment decision, Review of Financial Economics, 4 (1994), 1-24. [23] P. H. Zipkin, Simple ranking methods for allocation of one resource, Management Sci., 26 (1980), 34-43.  doi: 10.1287/mnsc.26.1.34.

show all references

##### References:
 [1] P. J. Barry, R. W. Bierlen and N. L. Sotomayor, Financial structure of farm business under imperfect capital markets, American Journal of Agricultural Economics, 82 (2000), 920-933. [2] R. H. Bernhard, Mathematical programming models for capital budgeting: A survey, generalization, and critique, Journal of Financial and Quantitative Analysis, 4 (1969), 111-158. [3] R. Bierlen and A. M. Featherstone, Fundamental Q, cash flow and investment: Evidence from farm panel data, The Review of Economics and Statistics, 80 (1998), 427-435. [4] T. O. Boucher, A mixed-integer programming model for optimal investment and financing in segmented international markets, The Engineering Economist, 40 (1995), 145-170. [5] E. F. Brigham and M. C. Ehrhardt, Financial Management: Theory and Practice, 13$^{nd}$ edition, United States of America, 2011. [6] R. Enkhbat, Global optimization approach to Malfatti's problem, J. Global Optim., 65 (2016), 33-39.  doi: 10.1007/s10898-015-0372-6. [7] R. Enkhbat, Convex maximization formulation of general sphere packing problem, Izv. Irkutsk. Gos. Univ. Ser. Mat., 31 (2020), 142-149.  doi: 10.26516/1997-7670.2020.31.142. [8] R. Enkhbat, N. Tungalag, J. Enkhbayar, O. Battogtokh and L. Enkhtuvshin, Application of survival theory in mining industry, Numer. Algebra Control Optim., 11 (2021), 443-448.  doi: 10.3934/naco.2020036. [9] J. R. Freeland and M. J. Rosenblatt, An analysis of linear programming formulations for the capital rationing problem, The Engineering Economist, 23 (1978), 49-61. [10] C. A. Hawkins and R. A. Adams, A goal programming model for capital budgeting, Journal of Financial Management, 3 (1974), 52-57. [11] R. G. Hubbard and A. K. Kashyap, Internal net worth and the investment process: An application to U.S. agriculture, The Journal of Political Economy, 100 (1992), 506-534. [12] A. Kraus and R. H. Litzenberger, A state-preference model of optimal financial leverage, Journal of Finance, 28 (1973), 911-922. [13] R. R. Levary, A sequential solution procedure to stochastic capital budgeting models, Computers and Industrial Engineering, 14 (1988), 371-380. [14] J. H. Lorie and C. J. Savage, Three problems in rationing capital, Journal of Business, 28 (1955), 229-239. [15] G. L. Luenberger, Investment Science, Oxford University Press, 1998. [16] F. Modigliani and M. H. Miller, The cost of capital, corporation finance and the theory of investment, American Economic Review, 1 (1958), 261-297. [17] S. C. Myers, The capital structure puzzle, The Journal of Finance, 39 (1984), 575-592. [18] S. C. Myers, A note of linear programming and capital budgeting, The Journal of Finance, 27 (1972), 89-92. [19] S. A. Ross, The determination of financial structure: The incentive-signaling approach, the Bell Journal of Economics, 8 (1977), 23-40. [20] R. C. Salazar and K. S. Subrata, A simulation model of capital budgeting under uncertainty, Management Science, 15 (1968), 161-179. [21] B. Ulemj, E. Rentsen and B. Tsendpurev, Application of survival theory in taxation, J. Ind. Manag. Optim., 17 (2021), 2573-2578.  doi: 10.3934/jimo.2020083. [22] S. C. Vogt, The role of internal financial sources in firm financing and investment decision, Review of Financial Economics, 4 (1994), 1-24. [23] P. H. Zipkin, Simple ranking methods for allocation of one resource, Management Sci., 26 (1980), 34-43.  doi: 10.1287/mnsc.26.1.34.
The decision variables
 $Assets$ $Liabilities$ $A_1: {\rm{Cash\ and\ equivalents}}$ $L_1: {\rm{Accounts\ payable}}$ $A_2: {\rm{Accounts\ receivable} }$ $L_2: {\rm{Short-term\ debt}}$ $A_3: {\rm{Inventories} }$ $L_3: {\rm{Long-term\ debt} }$ $A_4: {\rm{Fixed\ assets} }$ $E: {\rm{Equity}}$ $A_5: {\rm{Other\ assets} }$
 $Assets$ $Liabilities$ $A_1: {\rm{Cash\ and\ equivalents}}$ $L_1: {\rm{Accounts\ payable}}$ $A_2: {\rm{Accounts\ receivable} }$ $L_2: {\rm{Short-term\ debt}}$ $A_3: {\rm{Inventories} }$ $L_3: {\rm{Long-term\ debt} }$ $A_4: {\rm{Fixed\ assets} }$ $E: {\rm{Equity}}$ $A_5: {\rm{Other\ assets} }$
The structure variables
 $Assets$ $Liabilities$ $x_1: {\rm{Cash\ and\ equivalents\ to\ total\ assets\ ratio}}$ $x_6: {\rm{Accounts\ payable\ to\ total\ assets\ ratio}}$ $x_2: {\rm{Accounts\ receivable\ to\ total\ assets\ ratio}}$ $x_7: {\rm{Short-term\ debt\ to\ total\ assets\ ratio}}$ $x_3: {\rm{Inventories\ to\ total\ assets\ ratio}}$ $x_8: {\rm{Long-term\ debt\ to\ total\ assets\ ratio}}$ $x_4: {\rm{Fixed\ assets\ to\ total\ assets\ ratio}}$ $x_9: {\rm{Equity\ to\ total\ assets\ ratio}}$ $x_5: {\rm{Other\ assets\ to\ total\ assets\ ratio}}$
 $Assets$ $Liabilities$ $x_1: {\rm{Cash\ and\ equivalents\ to\ total\ assets\ ratio}}$ $x_6: {\rm{Accounts\ payable\ to\ total\ assets\ ratio}}$ $x_2: {\rm{Accounts\ receivable\ to\ total\ assets\ ratio}}$ $x_7: {\rm{Short-term\ debt\ to\ total\ assets\ ratio}}$ $x_3: {\rm{Inventories\ to\ total\ assets\ ratio}}$ $x_8: {\rm{Long-term\ debt\ to\ total\ assets\ ratio}}$ $x_4: {\rm{Fixed\ assets\ to\ total\ assets\ ratio}}$ $x_9: {\rm{Equity\ to\ total\ assets\ ratio}}$ $x_5: {\rm{Other\ assets\ to\ total\ assets\ ratio}}$
Capital structure of APU JSC
 $Weights$ $2010$ $2011$ $2012$ $2013$ $2014$ $2015$ $2016$ $x_1$ $0.053$ $0.153$ $0.030$ $0.108$ $0.030$ $0.022$ $0.057$ $x_2$ $0.091$ $0.157$ $0.054$ $0.029$ $0.047$ $0.069$ $0.056$ $x_3$ $0.224$ $0.219$ $0.228$ $0.174$ $0.210$ $0.184$ $0.177$ $x_4$ $0.611$ $0.430$ $0.372$ $0.611$ $0.676$ $0.681$ $0.650$ $x_5$ $0.022$ $0.041$ $0.315$ $0.078$ $0.037$ $0.043$ $0.059$ $x_6$ $0.076$ $0.090$ $0.123$ $0.127$ $0.153$ $0.124$ $0.129$ $x_7$ $0.126$ $0.189$ $0.274$ $0.089$ $0.009$ $0.104$ $0.137$ $x_8$ $0.281$ $0.186$ $0.096$ $0.466$ $0.362$ $0.295$ $0.242$ $x_9$ $0.518$ $0.535$ $0.507$ $0.318$ $0.395$ $0.476$ $0.492$ continue $Weights$ $2017$ $2018$ $2019$ $2020$ $\mu$ $\sigma$ $x_1$ $0.080$ $0.115$ $0.093$ $0.115$ $0.078$ $0.043$ $x_2$ $0.070$ $0.067$ $0.066$ $0.070$ $0.071$ $0.033$ $x_3$ $0.110$ $0.147$ $0.167$ $0.150$ $0.181$ $0.037$ $x_4$ $0.373$ $0.467$ $0.478$ $0.472$ $0.529$ $0.119$ $x_5$ $0.367$ $0.203$ $0.195$ $0.192$ $0.141$ $0.121$ $x_6$ $0.087$ $0.127$ $0.052$ $0.118$ $0.110$ $0.030$ $x_7$ $0.062$ $0.000$ $0.000$ $0.000$ $0.097$ $0.085$ $x_8$ $0.050$ $0.035$ $0.052$ $0.051$ $0.192$ $0.148$ $x_9$ $0.801$ $0.837$ $0.897$ $0.831$ $0.601$ $0.202$ Source: mse.mn/en/company/90
 $Weights$ $2010$ $2011$ $2012$ $2013$ $2014$ $2015$ $2016$ $x_1$ $0.053$ $0.153$ $0.030$ $0.108$ $0.030$ $0.022$ $0.057$ $x_2$ $0.091$ $0.157$ $0.054$ $0.029$ $0.047$ $0.069$ $0.056$ $x_3$ $0.224$ $0.219$ $0.228$ $0.174$ $0.210$ $0.184$ $0.177$ $x_4$ $0.611$ $0.430$ $0.372$ $0.611$ $0.676$ $0.681$ $0.650$ $x_5$ $0.022$ $0.041$ $0.315$ $0.078$ $0.037$ $0.043$ $0.059$ $x_6$ $0.076$ $0.090$ $0.123$ $0.127$ $0.153$ $0.124$ $0.129$ $x_7$ $0.126$ $0.189$ $0.274$ $0.089$ $0.009$ $0.104$ $0.137$ $x_8$ $0.281$ $0.186$ $0.096$ $0.466$ $0.362$ $0.295$ $0.242$ $x_9$ $0.518$ $0.535$ $0.507$ $0.318$ $0.395$ $0.476$ $0.492$ continue $Weights$ $2017$ $2018$ $2019$ $2020$ $\mu$ $\sigma$ $x_1$ $0.080$ $0.115$ $0.093$ $0.115$ $0.078$ $0.043$ $x_2$ $0.070$ $0.067$ $0.066$ $0.070$ $0.071$ $0.033$ $x_3$ $0.110$ $0.147$ $0.167$ $0.150$ $0.181$ $0.037$ $x_4$ $0.373$ $0.467$ $0.478$ $0.472$ $0.529$ $0.119$ $x_5$ $0.367$ $0.203$ $0.195$ $0.192$ $0.141$ $0.121$ $x_6$ $0.087$ $0.127$ $0.052$ $0.118$ $0.110$ $0.030$ $x_7$ $0.062$ $0.000$ $0.000$ $0.000$ $0.097$ $0.085$ $x_8$ $0.050$ $0.035$ $0.052$ $0.051$ $0.192$ $0.148$ $x_9$ $0.801$ $0.837$ $0.897$ $0.831$ $0.601$ $0.202$ Source: mse.mn/en/company/90
Algorithm ST
 $k$ $r$ $\gamma$ $\delta$ $Iterations$ $Solution (second)$ $0$ $0.0100$ $0.39900$ $-$ $-$ $-$ $1$ $0.0219$ $0.59880$ $0.1998$ $8$ $0.172$ $2$ $0.0219$ $0.61700$ $0.0182$ $2$ $0.063$ $3$ $0.0219$ $0.62606$ $0.0091$ $1$ $0.047$ $4*$ $0.0219$ $0.62607$ $0.0001$ $0$ $0.032$
 $k$ $r$ $\gamma$ $\delta$ $Iterations$ $Solution (second)$ $0$ $0.0100$ $0.39900$ $-$ $-$ $-$ $1$ $0.0219$ $0.59880$ $0.1998$ $8$ $0.172$ $2$ $0.0219$ $0.61700$ $0.0182$ $2$ $0.063$ $3$ $0.0219$ $0.62606$ $0.0091$ $1$ $0.047$ $4*$ $0.0219$ $0.62607$ $0.0001$ $0$ $0.032$
 [1] Hongwei Jiao, Junqiao Ma, Peiping Shen, Yongjian Qiu. Effective algorithm and computational complexity for solving sum of linear ratios problem. Journal of Industrial and Management Optimization, 2022  doi: 10.3934/jimo.2022135 [2] Alireza Bahiraie, A.K.M. Azhar, Noor Akma Ibrahim. A new dynamic geometric approach for empirical analysis of financial ratios and bankruptcy. Journal of Industrial and Management Optimization, 2011, 7 (4) : 947-965. doi: 10.3934/jimo.2011.7.947 [3] Qiong Liu, Ahmad Reza Rezaei, Kuan Yew Wong, Mohammad Mahdi Azami. Integrated modeling and optimization of material flow and financial flow of supply chain network considering financial ratios. Numerical Algebra, Control and Optimization, 2019, 9 (2) : 113-132. doi: 10.3934/naco.2019009 [4] Pablo Neme, Jorge Oviedo. A note on the lattice structure for matching markets via linear programming. Journal of Dynamics and Games, 2021, 8 (1) : 61-67. doi: 10.3934/jdg.2021001 [5] Qiang Du, M. D. Gunzburger, L. S. Hou, J. Lee. Analysis of a linear fluid-structure interaction problem. Discrete and Continuous Dynamical Systems, 2003, 9 (3) : 633-650. doi: 10.3934/dcds.2003.9.633 [6] Jinying Ma, Honglei Xu. Empirical analysis and optimization of capital structure adjustment. Journal of Industrial and Management Optimization, 2020, 16 (3) : 1037-1047. doi: 10.3934/jimo.2018191 [7] Hong Zhang, Fei Yang. Optimization of capital structure in real estate enterprises. Journal of Industrial and Management Optimization, 2015, 11 (3) : 969-983. doi: 10.3934/jimo.2015.11.969 [8] Yue Zheng, Zhongping Wan, Shihui Jia, Guangmin Wang. A new method for strong-weak linear bilevel programming problem. Journal of Industrial and Management Optimization, 2015, 11 (2) : 529-547. doi: 10.3934/jimo.2015.11.529 [9] Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. A stability estimate for fluid structure interaction problem with non-linear beam. Conference Publications, 2009, 2009 (Special) : 424-432. doi: 10.3934/proc.2009.2009.424 [10] Emine Kaya, Eugenio Aulisa, Akif Ibragimov, Padmanabhan Seshaiyer. FLUID STRUCTURE INTERACTION PROBLEM WITH CHANGING THICKNESS NON-LINEAR BEAM Fluid structure interaction problem with changing thickness non-linear beam. Conference Publications, 2011, 2011 (Special) : 813-823. doi: 10.3934/proc.2011.2011.813 [11] Patrick Beißner, Emanuela Rosazza Gianin. The term structure of sharpe ratios and arbitrage-free asset pricing in continuous time. Probability, Uncertainty and Quantitative Risk, 2021, 6 (1) : 23-52. doi: 10.3934/puqr.2021002 [12] Charles Fefferman. Interpolation by linear programming I. Discrete and Continuous Dynamical Systems, 2011, 30 (2) : 477-492. doi: 10.3934/dcds.2011.30.477 [13] B. D. Craven, Sardar M. N. Islam. An optimal financing model: Implications for existence of optimal capital structure. Journal of Industrial and Management Optimization, 2013, 9 (2) : 431-436. doi: 10.3934/jimo.2013.9.431 [14] Vladimir Gaitsgory, Alex Parkinson, Ilya Shvartsman. Linear programming based optimality conditions and approximate solution of a deterministic infinite horizon discounted optimal control problem in discrete time. Discrete and Continuous Dynamical Systems - B, 2019, 24 (4) : 1743-1767. doi: 10.3934/dcdsb.2018235 [15] Yibing Lv, Tiesong Hu, Jianlin Jiang. Penalty method-based equilibrium point approach for solving the linear bilevel multiobjective programming problem. Discrete and Continuous Dynamical Systems - S, 2020, 13 (6) : 1743-1755. doi: 10.3934/dcdss.2020102 [16] Qingsong Duan, Mengwei Xu, Liwei Zhang, Sainan Zhang. Hadamard directional differentiability of the optimal value of a linear second-order conic programming problem. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3085-3098. doi: 10.3934/jimo.2020108 [17] Ali Mahmoodirad, Harish Garg, Sadegh Niroomand. Solving fuzzy linear fractional set covering problem by a goal programming based solution approach. Journal of Industrial and Management Optimization, 2022, 18 (1) : 439-456. doi: 10.3934/jimo.2020162 [18] Jean Creignou, Hervé Diet. Linear programming bounds for unitary codes. Advances in Mathematics of Communications, 2010, 4 (3) : 323-344. doi: 10.3934/amc.2010.4.323 [19] Yi Xu, Wenyu Sun. A filter successive linear programming method for nonlinear semidefinite programming problems. Numerical Algebra, Control and Optimization, 2012, 2 (1) : 193-206. doi: 10.3934/naco.2012.2.193 [20] Dan Xue, Wenyu Sun, Hongjin He. A structured trust region method for nonconvex programming with separable structure. Numerical Algebra, Control and Optimization, 2013, 3 (2) : 283-293. doi: 10.3934/naco.2013.3.283

2021 Impact Factor: 1.411