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

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  • 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.

    Mathematics Subject Classification: 65G40.


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  • Figure 1.  Representation of mixing procedure in production process

    Figure 2.  Pictorial representation of Production rate for different values of '$ \eta $' for Example 2

    Figure 3.  Pictorial representation of centre of interval-valued average profit for Example 2

    Figure 4.  Pictorial representation of average profit for different values of '$ \eta $' for Example 2

    Figure 5.  Lower and upper bounds of interval-valued average profit for Example 2

    Figure 6.  Pictorial representation of average profit in crisp environment for Example 3

    Figure 7.  Effect of $ [\underline{b}, \overline{b}] $ on optimal policy

    Figure 8.  Effect of $ [\underline{\theta}, \overline{\theta}] $ on optimal policy

    Figure 9.  Effect of $ [\underline{h}, \overline{h}] $ on optimal policy

    Figure 10.  Effect of $ [\underline{C}_o, \overline{C}_o] $ on optimal policy

    Figure 11.  Effect of $ [\underline{c}_p, \overline{c}_p] $ on optimal policy

    Figure 12.  Effect of $ [\underline{a}, \overline{a}] $ on optimal policy

    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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    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, $7397.71]/year
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    Table 3.  Optimal average profit for different values of '$ \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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    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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