American Institute of Mathematical Sciences

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February  2022, 15(2): 457-480. doi: 10.3934/dcdss.2021117

A new approach based on inventory control using interval differential equation with application to manufacturing system

Md Sadikur Rahman 1, , Subhajit Das 1, , Amalesh Kumar Manna 1, , Ali Akbar Shaikh 1, , Asoke Kumar Bhunia 1, , Ali Ahmadian 2,3, and  Soheil Salahshour 4,,

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

*Corresponding author: Soheil Salahshour (soheil.salahshour@eng.bau.edu.tr)

Received  October 2020 Revised  June 2021 Published  February 2022 Early access  November 2021

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.

Keywords: Parametric approach, interval number, system of interval linear differential equation, mixing problem, production inventory.
Mathematics Subject Classification: 65G40.
Citation: Md Sadikur Rahman, Subhajit Das, Amalesh Kumar Manna, Ali Akbar Shaikh, Asoke Kumar Bhunia, Ali Ahmadian, Soheil Salahshour. A new approach based on inventory control using interval differential equation with application to manufacturing system. Discrete and Continuous Dynamical Systems - S, 2022, 15 (2) : 457-480. doi: 10.3934/dcdss.2021117
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.

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.

Figure 1.  Representation of mixing procedure in production process
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Figure 2.  Pictorial representation of Production rate for different values of '$ \eta $' for Example 2
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Figure 3.  Pictorial representation of centre of interval-valued average profit for Example 2
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Figure 4.  Pictorial representation of average profit for different values of '$ \eta $' for Example 2
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Figure 5.  Lower and upper bounds of interval-valued average profit for Example 2
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Figure 6.  Pictorial representation of average profit in crisp environment for Example 3
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Figure 7.  Effect of $ [\underline{b}, \overline{b}] $ on optimal policy
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Figure 8.  Effect of $ [\underline{\theta}, \overline{\theta}] $ on optimal policy
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Figure 9.  Effect of $ [\underline{h}, \overline{h}] $ on optimal policy
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Figure 10.  Effect of $ [\underline{C}_o, \overline{C}_o] $ on optimal policy
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Figure 11.  Effect of $ [\underline{c}_p, \overline{c}_p] $ on optimal policy
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Figure 12.  Effect of $ [\underline{a}, \overline{a}] $ on optimal policy
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Table 1.  Some previous works on applications of differential equations
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
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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
Table 2.  Optimal results of Example 2
Variable Optimal result
Production time ($ t_1 $) 1.743 year
Selling price ($ p $) ＄102.03/Lit.
Cycle length ($ T $) 1.865 year
Centre of the average profit ($ Z_c $) ＄7282.09/year
Interval valued average profit ($ [\underline{Z}, \overline{Z}] $) [＄7168.11, ＄7397.71]/year
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Variable Optimal result
Production time ($ t_1 $) 1.743 year
Selling price ($ p $) ＄102.03/Lit.
Cycle length ($ T $) 1.865 year
Centre of the average profit ($ Z_c $) ＄7282.09/year
Interval valued average profit ($ [\underline{Z}, \overline{Z}] $) [＄7168.11, ＄7397.71]/year
Table 3.  Optimal average profit for different values of '$ \eta $' of Example 2
$ \eta $ Average profit ($ Z(\eta) $)
0.0 ＄ 7397.71
0.2 ＄ 7351.26
0.4 ＄ 7305.08
0.5 ＄ 7282.09
0.6 ＄ 7259.17
0.8 ＄ 7213.52
1.0 ＄ 7168.11
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$ \eta $ Average profit ($ Z(\eta) $)
0.0 ＄ 7397.71
0.2 ＄ 7351.26
0.4 ＄ 7305.08
0.5 ＄ 7282.09
0.6 ＄ 7259.17
0.8 ＄ 7213.52
1.0 ＄ 7168.11
Table 4.  Optimal results of Example 3
Variable Optimal result
Production time ($ t_1 $) 1.746 year
Selling price ($ p $) ＄102.042/Lit.
Cycle length ($ T $) 1.868 year
Centre of the average profit ($ Z_c $) ＄ 7281.62/year
Interval valued average profit ($ [\underline{Z}, \overline{Z}] $) [＄ 7281.62, ＄ 7281.62]/year
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Variable Optimal result
Production time ($ t_1 $) 1.746 year
Selling price ($ p $) ＄102.042/Lit.
Cycle length ($ T $) 1.868 year
Centre of the average profit ($ Z_c $) ＄ 7281.62/year
Interval valued average profit ($ [\underline{Z}, \overline{Z}] $) [＄ 7281.62, ＄ 7281.62]/year
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