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A new approach based on inventory control using interval differential equation with application to manufacturing system
1. | Department of Mathematics, The University of Burdwan, Burdwan-713104, India |
2. | Institute of IR 4.0, The National University of Malaysia, 43600 Bangi, Malaysia |
3. | Department of Mathematics, Near East University, Nicosia, TRNC, Mersin 10, Turkey |
4. | Faculty of Engineering, Bahcesehir Universiti, Istanbul, Turkey |
Now-a-days, uncertainty conditions play an important role in modelling of real-world problems. In this regard, the aim of this study is two folded. Firstly, the concept of system of interval differential equations and its solution procedure in the parametric approach have been proposed. To serve this purpose, using parametric representation of interval and its arithmetic, system of linear interval differential equations is converted to the system of differential equations in parametric form. Then, a mixing problem with three liquids is considered and the mixing process is governed by system of interval differential equations. Thereafter, the mixing liquid is used in the production process of a manufacturing firm. Secondly, using this concept, a production inventory model for single item has been developed by employing mixture of liquids and the proposed production system is formulated mathematically by using system of interval differential equations.The corresponding interval valued average profit of the proposed model has been obtained in parametric form and it is maximized by centre-radius optimization technique. Then to validate the proposed model, two numerical examples have been solved using MATHEMATICA software. In addition, we have shown the concavity of the objective function graphically using the code of 3D plot in MATHEMATICA. Finally, the post optimality analyses are carried out with respect to different system parameters.
References:
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F. J. Agocs, W. J. Handley, A. N. Lasenby and M. P. Hobson,
Efficient method for solving highly oscillatory ordinary differential equations with applications to physical systems, Physical Review Research, 2 (2020), 013030.
|
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N. Ahmady,
A numerical method for solving fuzzy differential equations with fractional order, International Journal of Industrial Mathematics, 11 (2019), 71-77.
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B. Bede, I. J. Rudas and A. L. Bencsik,
First order linear fuzzy differential equations under generalized differentiability, Information Sciences, 177 (2007), 1648-1662.
doi: 10.1016/j.ins.2006.08.021. |
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A. R. Bergstrom, Non recursive models as discrete approximations to systems of stochastic differential equations, Econometrica: Journal of the Econometric Society, (1966), 173–182.
doi: 10.2307/1909861. |
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A. K. Bhunia and S. S. Samanta,
A study of interval metric and its application in multi-objective optimization with interval objectives, Computers & Industrial Engineering, 74 (2014), 169-178.
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J. J. Buckley and T. Feuring,
Fuzzy differential equations, Fuzzy Sets and Systems, 110 (2000), 43-54.
doi: 10.1016/S0165-0114(98)00141-9. |
[8] |
C. Chicone, Ordinary Differential Equations with Applications, 34, Springer Science & Business Media, 2006. |
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D. P. Covei and T. A. Pirvu,
An elliptic partial differential equation and its application, Applied Mathematics Letters, 101 (2020), 106059.
doi: 10.1016/j.aml.2019.106059. |
[10] |
S. Cui and A. Friedman,
A free boundary problem for a singular system of differential equations: An application to a model of tumour growth, Transactions of the American Mathematical Society, 355 (2003), 3537-3590.
doi: 10.1090/S0002-9947-03-03137-4. |
[11] |
L. M. B. da Costa Campos, Non-linear differential equations and dynamical systems, CRC Press, 2019.
![]() |
[12] |
T. M. da Costa, Y. Chalco-Cano, W. A. Lodwick and G. N. Silva,
A new approach to linear interval differential equations as a first step toward solving fuzzy differential, Fuzzy Sets and Systems, 347 (2018), 129-141.
doi: 10.1016/j.fss.2017.10.008. |
[13] |
B. Das, N. K. Mahapatra and M. Maiti,
Initial-valued first order fuzzy differential equation in Bi-level inventory model with fuzzy demand, Mathematical Modelling and Analysis, 13 (2008), 493-512.
doi: 10.3846/1392-6292.2008.13.493-512. |
[14] |
S. Das, M. A. A. Khan, E. E. Mahmoud, A. H. Abdel-Aty, K. M. Abualnaja and A. A. Shaikh,
A production inventory model with partial trade credit policy and reliability, Alexandria Engineering Journal, 60 (2021), 1325-1338.
|
[15] |
S. Das, A. K. Manna, E. E. Mahmoud, K. M. Abualnaja, A. H. Abdel-Aty and A. A. Shaikh, Product replacement policy in a production inventory model with replacement period-, stock-, and price-dependent demand, Journal of Mathematics, (2020).
doi: 10.1155/2020/6697279. |
[16] |
M. De, B. Das and M. Maiti,
EPL models with fuzzy imperfect production system including carbon emission: A fuzzy differential equation approach, Soft Computing, 24 (2020), 1293-1313.
|
[17] |
K. Engelborghs, V. Lemaire, J. Belair and D. Roose,
Numerical bifurcation analysis of delay differential equations arising from physiological modeling, Journal of Mathematical Biology, 42 (2001), 361-385.
doi: 10.1007/s002850000072. |
[18] |
N. A. Gasilov and S. E. Amrahov,
Solving a nonhomogeneous linear system of interval differential equations, Soft Computing, 22 (2018), 3817-3828.
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[19] |
N. A. Gasilov and S. E. Amrahov,
On differential equations with interval coefficients, Mathematical Methods in the Applied Sciences, 43 (2020), 1825-1837.
doi: 10.1002/mma.6006. |
[20] |
B. Ghanbari, H. Günerhan and H. M. Srivastava,
An application of the Atangana-Baleanu fractional derivative in mathematical biology: A three-species predator-prey model, Chaos, Solitons & Fractals, 138 (2020), 109910.
doi: 10.1016/j.chaos.2020.109910. |
[21] |
T. H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Annals of Mathematics, (1919), 292–296.
doi: 10.2307/1967124. |
[22] |
P. Guchhait, M. K. Maiti and M. Maiti,
A production inventory model with fuzzy production and demand using fuzzy differential equation: An interval compared genetic algorithm approach, Engineering Applications of Artificial Intelligence, 26 (2013), 766-778.
doi: 10.1142/S0218488514500457. |
[23] |
N. Ikeda and S. Watanabe,
A comparison theorem for solutions of stochastic differential equations and its applications, Osaka Journal of Mathematics, 14 (1977), 619-633.
|
[24] |
R. Jafari, W. Yu and X. Li,
Fuzzy differential equations for nonlinear system modelling with Bernstein neural networks, Ieee Access, 4 (2016), 9428-9436.
|
[25] |
H. T. Kanekiyo and S. Agata, Optimal control in an inventory management problem considering replenishment lead time based upon a non-diffusive stochastic differential equation, Journal of Advanced Mechanical Design, Systems, and Manufacturing, 13 (2019), JAMDSM0008-JAMDSM0008. |
[26] |
D. Kumar, A. R. Seadawy and A. K. Joardar,
Modified Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology, Chinese journal of physics, 56 (2018), 75-85.
|
[27] |
H. Liao and L. Li, Environmental sustainability EOQ model for closed-loop supply chain under market uncertainty: A case study of printer remanufacturing, Computers & Industrial Engineering, (2020), 106525. |
[28] |
W. Liu, M. Rockner, X. Sun and Y. Xie,
Averaging principle for slow-fast stochastic differential equations with time dependent locally Lipschitz coefficients, Journal of Differential Equations, 268 (2020), 2910-2948.
doi: 10.1016/j.jde.2019.09.047. |
[29] |
A. Mahata, S. P. Mondal, B. Roy and S. Alam, Study of two species prey-predator model in imprecise environment with MSY policy under different harvesting scenario, Environment, Development and Sustainability, (2021), 1–25. |
[30] |
X. Mao and C. Yuan, Stochastic differential equations with Markovian switching, Imperial college press, 2006.
doi: 10.1142/p473.![]() ![]() ![]() |
[31] |
W. Materi and D. S. Wishart,
Computational systems biology in drug discovery and development: Methods and applications, Drug Discovery Today, 12 (2007), 295-303.
|
[32] |
J. E. Moreno, M. A. Sanchez, O. Mendoza, A. Rodriguez-Diaz, O. Castillo, P. Melin and J. R. Castro,
Design of an interval type-2 fuzzy model with justifiable uncertainty, Information Sciences, 513 (2020), 206-221.
|
[33] |
A. M. Overstall, D. C. Woods and B. M. Parker, Bayesian optimal design for ordinary differential equation models with application in biological science, Journal of the American Statistical Association, (2020), 1–16.
doi: 10.1080/01621459.2019.1617154. |
[34] |
D. Pal, G. S. Mahapatra and G. P. Samanta,
New approach for stability and bifurcation analysis on predator-prey harvesting model for interval biological parameters with time delays, Computational and Applied Mathematics, 37 (2018), 3145-3171.
doi: 10.1007/s40314-017-0504-3. |
[35] |
P. Pandit and P. Singh, Fully Fuzzy Semi-linear Dynamical System Solved by Fuzzy Laplace Transform Under Modified Hukuhara Derivative, In Soft Computing for Problem Solving, Springer, Singapore, 2020,155–179. |
[36] |
M. S. Rahman, A. K. Manna, A. A. Shaikh and A. K. Bhunia,
An application of interval differential equation on a production inventory model with interval-valued demand via center-radius optimization technique and particle swarm optimization, International Journal of Intelligent Systems, 35 (2020), 1280-1326.
|
[37] |
M. S. Rahman, A. Duary, A. A. Shaikh and A. K. Bhunia, An application of parametric approach for interval differential equation in inventory model for deteriorating items with selling-price-dependent demand, Neural Computing and Applications, (2020), 1–17. |
[38] |
M. S. Rahman, A. A. Shaikh and A. K. Bhunia, On type-2 interval with interval mathematics and order relations: Its applications in inventory control, International Journal of Systems Science: Operations & Logistics, (2020), 1–13. |
[39] |
M. Ramezanzadeh, M. Heidari, O. Fard and A. Borzabadi, On the interval differential equation: Novel solution methodology, Advances in Difference Equations, (2015).
doi: 10.1186/s13662-015-0671-8. |
[40] |
C. Rout, D. Chakraborty and A. Goswami,
An EPQ model for deteriorating items with imperfect production, two types of inspection errors and rework under complete backordering, International Game Theory Review, 22 (2020), 2040011.
doi: 10.1142/S0219198920400113. |
[41] |
S. Salahshour, A. Ahmadian, S. Abbasbandy and D. Baleanu, M-fractional derivative under interval uncertainty: Theory, properties and applications, Chaos, Solitons and Fractals, (2018), 121–125.
doi: 10.1016/j.chaos.2018.10.002. |
[42] |
S. Salahshour, A. Ahmadian, M. Salimi, M. Ferarra and D. Baleanu,
Asymptotic solutions of fractional interval differential equations with nonsingular kernel derivative, Chaos: An Interdisciplinary Journal of Nonlinear Science, AIP, 29 (2019), 083110.
doi: 10.1063/1.5096022. |
[43] |
L. Stefanini and B. Bede,
Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), 1311-1328.
doi: 10.1016/j.na.2008.12.005. |
[44] |
M. Thongmoon and S. Pusjuso,
The numerical solutions of differential transform method and the Laplace transform method for a system of differential equations, Nonlinear Analysis: Hybrid Systems, 4 (2010), 425-431.
doi: 10.1016/j.nahs.2009.10.006. |
[45] |
A. Tsoularis,
A stochastic differential equation inventory model, International Journal of Applied and Computational Mathematics, 5 (2019), 8.
doi: 10.1007/s40819-018-0594-7. |
show all references
References:
[1] |
F. J. Agocs, W. J. Handley, A. N. Lasenby and M. P. Hobson,
Efficient method for solving highly oscillatory ordinary differential equations with applications to physical systems, Physical Review Research, 2 (2020), 013030.
|
[2] |
N. Ahmady,
A numerical method for solving fuzzy differential equations with fractional order, International Journal of Industrial Mathematics, 11 (2019), 71-77.
|
[3] |
L. Arnold, Stochastic Differential Equations, New York, 1974. |
[4] |
B. Bede, I. J. Rudas and A. L. Bencsik,
First order linear fuzzy differential equations under generalized differentiability, Information Sciences, 177 (2007), 1648-1662.
doi: 10.1016/j.ins.2006.08.021. |
[5] |
A. R. Bergstrom, Non recursive models as discrete approximations to systems of stochastic differential equations, Econometrica: Journal of the Econometric Society, (1966), 173–182.
doi: 10.2307/1909861. |
[6] |
A. K. Bhunia and S. S. Samanta,
A study of interval metric and its application in multi-objective optimization with interval objectives, Computers & Industrial Engineering, 74 (2014), 169-178.
|
[7] |
J. J. Buckley and T. Feuring,
Fuzzy differential equations, Fuzzy Sets and Systems, 110 (2000), 43-54.
doi: 10.1016/S0165-0114(98)00141-9. |
[8] |
C. Chicone, Ordinary Differential Equations with Applications, 34, Springer Science & Business Media, 2006. |
[9] |
D. P. Covei and T. A. Pirvu,
An elliptic partial differential equation and its application, Applied Mathematics Letters, 101 (2020), 106059.
doi: 10.1016/j.aml.2019.106059. |
[10] |
S. Cui and A. Friedman,
A free boundary problem for a singular system of differential equations: An application to a model of tumour growth, Transactions of the American Mathematical Society, 355 (2003), 3537-3590.
doi: 10.1090/S0002-9947-03-03137-4. |
[11] |
L. M. B. da Costa Campos, Non-linear differential equations and dynamical systems, CRC Press, 2019.
![]() |
[12] |
T. M. da Costa, Y. Chalco-Cano, W. A. Lodwick and G. N. Silva,
A new approach to linear interval differential equations as a first step toward solving fuzzy differential, Fuzzy Sets and Systems, 347 (2018), 129-141.
doi: 10.1016/j.fss.2017.10.008. |
[13] |
B. Das, N. K. Mahapatra and M. Maiti,
Initial-valued first order fuzzy differential equation in Bi-level inventory model with fuzzy demand, Mathematical Modelling and Analysis, 13 (2008), 493-512.
doi: 10.3846/1392-6292.2008.13.493-512. |
[14] |
S. Das, M. A. A. Khan, E. E. Mahmoud, A. H. Abdel-Aty, K. M. Abualnaja and A. A. Shaikh,
A production inventory model with partial trade credit policy and reliability, Alexandria Engineering Journal, 60 (2021), 1325-1338.
|
[15] |
S. Das, A. K. Manna, E. E. Mahmoud, K. M. Abualnaja, A. H. Abdel-Aty and A. A. Shaikh, Product replacement policy in a production inventory model with replacement period-, stock-, and price-dependent demand, Journal of Mathematics, (2020).
doi: 10.1155/2020/6697279. |
[16] |
M. De, B. Das and M. Maiti,
EPL models with fuzzy imperfect production system including carbon emission: A fuzzy differential equation approach, Soft Computing, 24 (2020), 1293-1313.
|
[17] |
K. Engelborghs, V. Lemaire, J. Belair and D. Roose,
Numerical bifurcation analysis of delay differential equations arising from physiological modeling, Journal of Mathematical Biology, 42 (2001), 361-385.
doi: 10.1007/s002850000072. |
[18] |
N. A. Gasilov and S. E. Amrahov,
Solving a nonhomogeneous linear system of interval differential equations, Soft Computing, 22 (2018), 3817-3828.
|
[19] |
N. A. Gasilov and S. E. Amrahov,
On differential equations with interval coefficients, Mathematical Methods in the Applied Sciences, 43 (2020), 1825-1837.
doi: 10.1002/mma.6006. |
[20] |
B. Ghanbari, H. Günerhan and H. M. Srivastava,
An application of the Atangana-Baleanu fractional derivative in mathematical biology: A three-species predator-prey model, Chaos, Solitons & Fractals, 138 (2020), 109910.
doi: 10.1016/j.chaos.2020.109910. |
[21] |
T. H. Gronwall, Note on the derivatives with respect to a parameter of the solutions of a system of differential equations, Annals of Mathematics, (1919), 292–296.
doi: 10.2307/1967124. |
[22] |
P. Guchhait, M. K. Maiti and M. Maiti,
A production inventory model with fuzzy production and demand using fuzzy differential equation: An interval compared genetic algorithm approach, Engineering Applications of Artificial Intelligence, 26 (2013), 766-778.
doi: 10.1142/S0218488514500457. |
[23] |
N. Ikeda and S. Watanabe,
A comparison theorem for solutions of stochastic differential equations and its applications, Osaka Journal of Mathematics, 14 (1977), 619-633.
|
[24] |
R. Jafari, W. Yu and X. Li,
Fuzzy differential equations for nonlinear system modelling with Bernstein neural networks, Ieee Access, 4 (2016), 9428-9436.
|
[25] |
H. T. Kanekiyo and S. Agata, Optimal control in an inventory management problem considering replenishment lead time based upon a non-diffusive stochastic differential equation, Journal of Advanced Mechanical Design, Systems, and Manufacturing, 13 (2019), JAMDSM0008-JAMDSM0008. |
[26] |
D. Kumar, A. R. Seadawy and A. K. Joardar,
Modified Kudryashov method via new exact solutions for some conformable fractional differential equations arising in mathematical biology, Chinese journal of physics, 56 (2018), 75-85.
|
[27] |
H. Liao and L. Li, Environmental sustainability EOQ model for closed-loop supply chain under market uncertainty: A case study of printer remanufacturing, Computers & Industrial Engineering, (2020), 106525. |
[28] |
W. Liu, M. Rockner, X. Sun and Y. Xie,
Averaging principle for slow-fast stochastic differential equations with time dependent locally Lipschitz coefficients, Journal of Differential Equations, 268 (2020), 2910-2948.
doi: 10.1016/j.jde.2019.09.047. |
[29] |
A. Mahata, S. P. Mondal, B. Roy and S. Alam, Study of two species prey-predator model in imprecise environment with MSY policy under different harvesting scenario, Environment, Development and Sustainability, (2021), 1–25. |
[30] |
X. Mao and C. Yuan, Stochastic differential equations with Markovian switching, Imperial college press, 2006.
doi: 10.1142/p473.![]() ![]() ![]() |
[31] |
W. Materi and D. S. Wishart,
Computational systems biology in drug discovery and development: Methods and applications, Drug Discovery Today, 12 (2007), 295-303.
|
[32] |
J. E. Moreno, M. A. Sanchez, O. Mendoza, A. Rodriguez-Diaz, O. Castillo, P. Melin and J. R. Castro,
Design of an interval type-2 fuzzy model with justifiable uncertainty, Information Sciences, 513 (2020), 206-221.
|
[33] |
A. M. Overstall, D. C. Woods and B. M. Parker, Bayesian optimal design for ordinary differential equation models with application in biological science, Journal of the American Statistical Association, (2020), 1–16.
doi: 10.1080/01621459.2019.1617154. |
[34] |
D. Pal, G. S. Mahapatra and G. P. Samanta,
New approach for stability and bifurcation analysis on predator-prey harvesting model for interval biological parameters with time delays, Computational and Applied Mathematics, 37 (2018), 3145-3171.
doi: 10.1007/s40314-017-0504-3. |
[35] |
P. Pandit and P. Singh, Fully Fuzzy Semi-linear Dynamical System Solved by Fuzzy Laplace Transform Under Modified Hukuhara Derivative, In Soft Computing for Problem Solving, Springer, Singapore, 2020,155–179. |
[36] |
M. S. Rahman, A. K. Manna, A. A. Shaikh and A. K. Bhunia,
An application of interval differential equation on a production inventory model with interval-valued demand via center-radius optimization technique and particle swarm optimization, International Journal of Intelligent Systems, 35 (2020), 1280-1326.
|
[37] |
M. S. Rahman, A. Duary, A. A. Shaikh and A. K. Bhunia, An application of parametric approach for interval differential equation in inventory model for deteriorating items with selling-price-dependent demand, Neural Computing and Applications, (2020), 1–17. |
[38] |
M. S. Rahman, A. A. Shaikh and A. K. Bhunia, On type-2 interval with interval mathematics and order relations: Its applications in inventory control, International Journal of Systems Science: Operations & Logistics, (2020), 1–13. |
[39] |
M. Ramezanzadeh, M. Heidari, O. Fard and A. Borzabadi, On the interval differential equation: Novel solution methodology, Advances in Difference Equations, (2015).
doi: 10.1186/s13662-015-0671-8. |
[40] |
C. Rout, D. Chakraborty and A. Goswami,
An EPQ model for deteriorating items with imperfect production, two types of inspection errors and rework under complete backordering, International Game Theory Review, 22 (2020), 2040011.
doi: 10.1142/S0219198920400113. |
[41] |
S. Salahshour, A. Ahmadian, S. Abbasbandy and D. Baleanu, M-fractional derivative under interval uncertainty: Theory, properties and applications, Chaos, Solitons and Fractals, (2018), 121–125.
doi: 10.1016/j.chaos.2018.10.002. |
[42] |
S. Salahshour, A. Ahmadian, M. Salimi, M. Ferarra and D. Baleanu,
Asymptotic solutions of fractional interval differential equations with nonsingular kernel derivative, Chaos: An Interdisciplinary Journal of Nonlinear Science, AIP, 29 (2019), 083110.
doi: 10.1063/1.5096022. |
[43] |
L. Stefanini and B. Bede,
Generalized Hukuhara differentiability of interval-valued functions and interval differential equations, Nonlinear Analysis: Theory, Methods & Applications, 71 (2009), 1311-1328.
doi: 10.1016/j.na.2008.12.005. |
[44] |
M. Thongmoon and S. Pusjuso,
The numerical solutions of differential transform method and the Laplace transform method for a system of differential equations, Nonlinear Analysis: Hybrid Systems, 4 (2010), 425-431.
doi: 10.1016/j.nahs.2009.10.006. |
[45] |
A. Tsoularis,
A stochastic differential equation inventory model, International Journal of Applied and Computational Mathematics, 5 (2019), 8.
doi: 10.1007/s40819-018-0594-7. |












Reported Works | Simultaneous /Single differential equations | Nature of equations (Crisp/Fuzzy/Stochastic/Interval) | Area of applications |
Cui and Friedman (2003)[10] | Simultaneous | Crisp (ordinary) | Mathematical biology |
Das et al. (2008)[13] | Single | Fuzzy | Inventory |
Guchhait et al. (2013)[22] | Single | Fuzzy | Production inventory |
Jafari et al. (2016)[24] | Simultaneous | Fuzzy | Mathematical biology |
da Costa Campos (2019)[11] | Simultaneous | Crisp (ordinary) | Dynamical System |
Tsoularis (2019)[45] | Single | Stochastic | Inventory |
Kanekiyo and Agata (2019)[25] | Single | Stochastic | Inventory |
Overstall et al. (2020)[33] | Simultaneous | Crisp (ordinary) | Bio Science |
De et al. (2020)[16] | Single | Fuzzy | Production inventory |
Agocs et al. (2020)[1] | Simultaneous | Crisp (ordinary) | Dynamical system |
Rahman et al. (2020b)[37] | Single | Interval | Inventory |
Das et al. (2020)[15] | Single | Crisp (ordinary) | Production inventory |
This work | Simultaneous | Interval | Production inventory |
Reported Works | Simultaneous /Single differential equations | Nature of equations (Crisp/Fuzzy/Stochastic/Interval) | Area of applications |
Cui and Friedman (2003)[10] | Simultaneous | Crisp (ordinary) | Mathematical biology |
Das et al. (2008)[13] | Single | Fuzzy | Inventory |
Guchhait et al. (2013)[22] | Single | Fuzzy | Production inventory |
Jafari et al. (2016)[24] | Simultaneous | Fuzzy | Mathematical biology |
da Costa Campos (2019)[11] | Simultaneous | Crisp (ordinary) | Dynamical System |
Tsoularis (2019)[45] | Single | Stochastic | Inventory |
Kanekiyo and Agata (2019)[25] | Single | Stochastic | Inventory |
Overstall et al. (2020)[33] | Simultaneous | Crisp (ordinary) | Bio Science |
De et al. (2020)[16] | Single | Fuzzy | Production inventory |
Agocs et al. (2020)[1] | Simultaneous | Crisp (ordinary) | Dynamical system |
Rahman et al. (2020b)[37] | Single | Interval | Inventory |
Das et al. (2020)[15] | Single | Crisp (ordinary) | Production inventory |
This work | Simultaneous | Interval | Production inventory |
Variable | Optimal result |
Production time ( |
1.743 year |
Selling price ( |
$102.03/Lit. |
Cycle length ( |
1.865 year |
Centre of the average profit ( |
$7282.09/year |
Interval valued average profit ( |
[$7168.11, $7397.71]/year |
Variable | Optimal result |
Production time ( |
1.743 year |
Selling price ( |
$102.03/Lit. |
Cycle length ( |
1.865 year |
Centre of the average profit ( |
$7282.09/year |
Interval valued average profit ( |
[$7168.11, $7397.71]/year |
|
Average profit ( |
0.0 | $ 7397.71 |
0.2 | $ 7351.26 |
0.4 | $ 7305.08 |
0.5 | $ 7282.09 |
0.6 | $ 7259.17 |
0.8 | $ 7213.52 |
1.0 | $ 7168.11 |
|
Average profit ( |
0.0 | $ 7397.71 |
0.2 | $ 7351.26 |
0.4 | $ 7305.08 |
0.5 | $ 7282.09 |
0.6 | $ 7259.17 |
0.8 | $ 7213.52 |
1.0 | $ 7168.11 |
Variable | Optimal result |
Production time ( |
1.746 year |
Selling price ( |
$102.042/Lit. |
Cycle length ( |
1.868 year |
Centre of the average profit ( |
$ 7281.62/year |
Interval valued average profit ( |
[$ 7281.62, $ 7281.62]/year |
Variable | Optimal result |
Production time ( |
1.746 year |
Selling price ( |
$102.042/Lit. |
Cycle length ( |
1.868 year |
Centre of the average profit ( |
$ 7281.62/year |
Interval valued average profit ( |
[$ 7281.62, $ 7281.62]/year |
[1] |
Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete and Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203 |
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Jahnabi Chakravarty, Ashiho Athikho, Manideepa Saha. Convergence of interval AOR method for linear interval equations. Numerical Algebra, Control and Optimization, 2022, 12 (2) : 293-308. doi: 10.3934/naco.2021006 |
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Hadda Hmili. Non topologically weakly mixing interval exchanges. Discrete and Continuous Dynamical Systems, 2010, 27 (3) : 1079-1091. doi: 10.3934/dcds.2010.27.1079 |
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