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

• 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.

 Citation: • • 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) Simultaneous Crisp (ordinary) Mathematical biology Das et al. (2008) Single Fuzzy Inventory Guchhait et al. (2013) Single Fuzzy Production inventory Jafari et al. (2016) Simultaneous Fuzzy Mathematical biology da Costa Campos (2019) Simultaneous Crisp (ordinary) Dynamical System Tsoularis (2019) Single Stochastic Inventory Kanekiyo and Agata (2019) Single Stochastic Inventory Overstall et al. (2020) Simultaneous Crisp (ordinary) Bio Science De et al. (2020) Single Fuzzy Production inventory Agocs et al. (2020) Simultaneous Crisp (ordinary) Dynamical system Rahman et al. (2020b) Single Interval Inventory Das et al. (2020) 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

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

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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Tables(4)

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