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doi: 10.3934/jimo.2022092
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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. BarryR. 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. EnkhbatN. TungalagJ. EnkhbayarO. 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. UlemjE. 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. BarryR. 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. EnkhbatN. TungalagJ. EnkhbayarO. 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. UlemjE. 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.

Table 1.  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} }$
Table 2.  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}}$
Table 3.  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
Table 4.  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 $
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