An application of neutrosophic logic on an inventory model with two-level partial trade credit policy for time-dependent perishable products

Abstract

Fuzzy and neutrosophic sets are two effective tools to deal with the ambiguities and uncertainties present in real-world problems. To deal with the uncertainties of a real-world scenario, a neutrosophic set is better equipped than a fuzzy set. In this article, we develop a neutrosophic economic order quantity (EOQ) inventory model with the assumption that the market demand is sensitive to the retail price and promotional effort. The supplier and retailer both adopt a partial trade credit policy. We include preservation technology to restrict normal deterioration. We analyse the crisp model first, and then, neutrosophic logic is implemented in the proposed model. De-neutrosophication of total neutrosophic profit is performed based on the removal area approach. The present investigation yields that the de-neutrosophic value of the profit function is convex, that is, a unique solution exists. Mathematical outcomes are generated to efficiently determine the optimal inventory policy for the retailer. To demonstrate numerically and validate the model, a case study is carried out. The findings in this work generalize several existing results, and sensitivity analysis is also reported to support the model.

Appendices

A Proof of Theorem 3

$$\begin{aligned}{} & {} D(({\widetilde{\Pi }}_{Neu})_{_{_{11}}}(\digamma ,c_{_S}))\\{} & {} \quad =D((E_{11})_{11}\cdot ({\widetilde{X}}_{Neu}^{11})_{11} \ominus (E_{12})_{11}\cdot ({\widetilde{X}}_{Neu}^{12})_{11}\ominus (E_{13})_{11}\\{} & {} \quad \cdot ({\widetilde{X}}_{Neu}^{13})_{11}\ominus (E_{14})_{11}\cdot ({\widetilde{X}}_{Neu}^{14})_{11}\ominus (E_{15})_{11}\cdot ({\widetilde{X}}_{Neu}^{15})_{11}\ominus \\{} & {} \quad (E_{16})_{11}\cdot ({\widetilde{X}}_{Neu}^{16})_{11}\oplus (E_{17})_{11}\cdot ({\widetilde{X}}_{Neu}^{17})_{11}\ominus (E_{18})_{11}\cdot \\{} & {} \quad ({\widetilde{X}}_{Neu}^{18})_{11})\\{} & {} \quad =(E_{11})_{11}~~D(({\widetilde{X}}_{Neu}^{11})_{11})~~- ~~(E_{12})_{11}~~D~~(({\widetilde{X}}_{Neu}^{12})_{11})\\{} & {} \qquad -~~(E_{13})_{11}~D(({\widetilde{X}}_{Neu}^{13})_{11})~-~(E_{14})_{11}~~D(({\widetilde{X}}_{Neu}^{14})_{11})\\{} & {} \qquad -~(E_{15})_{11}~~D( ({\widetilde{X}}_{Neu}^{15})_{11})~-~(E_{16})_{11}~~D(({\widetilde{X}}_{Neu}^{16})_{11})\\{} & {} \qquad +~(E_{17})_{11}~~D(({\widetilde{X}}_{Neu}^{17})_{11})~-~(A_{18})_{11}~~D(({\widetilde{X}}_{Neu}^{18})_{11})\\{} & {} \quad =~~c_{_S}~~~e^VD(({\widetilde{X}}_{Neu}^{11})_{11})~~~-~~\frac{1}{\digamma }~~~D(({\widetilde{X}}_{Neu}^{12})_{11}) -\frac{1}{\digamma }\\{} & {} \quad D(({\widetilde{X}}_{Neu}^{13})_{11})-\frac{e^{V}}{\digamma }\left( \int \limits _0^{\digamma }e^{-\xi (t)}du\right) D(({\widetilde{X}}_{Neu}^{14})_{11})-\frac{e^V}{\digamma }\\{} & {} \quad \left( \int \limits _0^{\digamma }e^{-\xi (t)}(\int \limits _t^{\digamma }e^{\xi (u)}du)dt\right) D(({\widetilde{X}}_{Neu}^{15})-\frac{i_{_{IC}}e^V}{\digamma }\bigg \{(1-\lambda _1)\\{} & {} \quad (\digamma +V)\int \limits _0^{\digamma }e^{\xi (u)}du+\frac{1}{2}(1-\lambda _1)^2\digamma \int \limits _{0}^{\digamma }e^{\xi (u)}du+\frac{1}{2}\lambda _2\\{} & {} \quad (\digamma -U)^2+\frac{1}{2}(1-\lambda _2)(\digamma +V-U)^2\bigg \}D(({\widetilde{X}}_{Neu}^{16})\\{} & {} \quad +\frac{c_{_{S}}i_{_{IE}}e^V}{2\digamma }\bigg \{\lambda _2U^2+(1-\lambda _2)(U-V)^2\bigg \}\\{} & {} \quad D(({\widetilde{X}}_{Neu}^{17})-\frac{\eta }{\digamma }D(({\widetilde{X}}_{Neu}^{18})` \end{aligned}$$

Let $D(({\widetilde{\Pi }}_{Neu})_{_{_{11}}}(\digamma ))=\frac{f(\digamma )}{g(\digamma )}$, where

$$\begin{aligned} f(\digamma )&=c_{_S}e^V\digamma D(({\widetilde{X}}_{Neu}^{11})_{11})-D(({\widetilde{X}}_{Neu}^{12})_{11})\\&-D(({\widetilde{X}}_{Neu}^{13})_{11})-e^{V}\left( \int \limits _0^{\digamma }e^{-\xi (t)}du\right) \\&D(({\widetilde{X}}_{Neu}^{14})_{11})-e^V\left( \int \limits _0^{\digamma }e^{-\xi (t)}\left( \int \limits _t^{\digamma }e^{\xi (u)}du\right) dt\right) \\&D(({\widetilde{X}}_{Neu}^{15})-i_{_{IC}}e^V\bigg \{(1-\lambda _1)(\digamma +V)\int \limits _0^{\digamma }e^{\xi (u)}du\\&+\frac{1}{2}(1-\lambda _1)^2\digamma \int \limits _{0}^{\digamma }e^{\xi (u)}du+\frac{1}{2}\lambda _2(\digamma -U)^2+\\&\frac{1}{2}(1-\lambda _2)(\digamma +V-U)^2\bigg \}D(({\widetilde{X}}_{Neu}^{16})\\&+\frac{c_{_{S}}i_{_{IE}}e^V}{2}\bigg \{\lambda _2U^2+(1-\lambda _2)(U-V)^2\bigg \}\\&D(({\widetilde{X}}_{Neu}^{17})-\eta D(({\widetilde{X}}_{Neu}^{18})\\&\text{ and }~~~g(\digamma )=\digamma .\\ \qquad \qquad \dfrac{d f(\digamma )}{d\digamma }&=c_{_S}e^V D(({\widetilde{X}}_{Neu}^{11})_{11})-e^Ve^{\xi (\digamma )}D(({\widetilde{X}}_{Neu}^{14})_{11})\\&\quad -e^V\int \limits _0^{\digamma }e^{-\xi (t)+\xi (\digamma )}dt~ D(({\widetilde{X}}_{Neu}^{15})_{11})\\&\quad +i_{_{IC}}e^V D(({\widetilde{X}}_{Neu}^{16})_{11})\bigg \{e^{\xi (\digamma )}(V+x)(1-\lambda _1)\\&\quad +\frac{1}{2}x(1-\lambda _1)^2e^{\xi (\digamma )}+\frac{1}{2}(1-\lambda _1)(3-\lambda _1)\int \limits _0^{\digamma }e^{\xi (t)}dt\\&\quad +(\digamma -U+V)(1-\lambda _2)+(\digamma -U)\lambda _2\bigg \}\\ \dfrac{d^2f(\digamma )}{d \digamma ^2}&=\frac{1}{2}e^V\bigg [-2\bigg \{D(({\widetilde{X}}_{Neu}^{15})_{11})+i_{_{IC}}D(({\widetilde{X}}_{Neu}^{16})_{11})\\&+e^{\xi (\digamma )}D(({\widetilde{X}}_{Neu}^{16})_{11})(3-4\lambda _1+\lambda _1^2)\bigg \}\\&-2D(({\widetilde{X}}_{Neu}^{15})_{11})\int \limits _{0}^{\digamma }e^{-b\eta -\xi (t)+\xi (\digamma )}\zeta (\digamma )dt\\&-e^{-b\eta +\xi (\digamma )}\bigg \{2D(({\widetilde{X}}_{Neu}^{14})_{11})+D(({\widetilde{X}}_{Neu}^{15})_{11})\\&\left( -2V+\digamma (-3+\lambda _1)\right) (\lambda _1-1)\bigg \}\zeta (\digamma )\bigg ] \end{aligned}$$

Here, $\dfrac{d^2f(\digamma )}{d \digamma ^2}<0$. Hence, $f(\digamma )$ is a strictly concave function. Again $g(\digamma )=\digamma$ is positive, differentiable and convex function of $\digamma$ on $(0,~\infty )$. So, by the theorem 2, $D(({\widetilde{\Pi }}_{Neu})_{_{_{11}}}(\digamma ))$ is a strictly pseudo-concave function of $\digamma$ on $(0,~\infty )$.

B Proof of the Theorem 4

For any $\digamma$, differentiating

$D(({\widetilde{\Pi }}_{Neu})_{_{_{11}}}(c_{_S}))=$ $D(({\widetilde{\Pi }}_{Neu})_{_{_{11}}}(\digamma ,c_{_S}))$

with respect to $c_{_S}$, we get

$$\begin{aligned}&\dfrac{d D(({\widetilde{\Pi }}_{Neu})_{_{_{11}}}(\digamma ,c_{_S}))}{d c_{_S}}\\&=e^VD(({\widetilde{X}}_{Neu}^{11})_{11})+c_{_S}e^VD(({\widetilde{X}}_{Neu}^{11})_{11}\otimes (-{\widetilde{a}}_{Neu}))\\&\quad -\frac{e^{V}}{\digamma }\left( \int \limits _0^{\digamma }e^{-\xi (t)}du\right) D(({\widetilde{X}}_{Neu}^{14})_{11}\otimes (-{\widetilde{a}}_{Neu}))\\&\quad -\frac{e^V}{\digamma }\left( \int \limits _0^{\digamma }e^{-\xi (t)}\left( \int \limits _t^{\digamma }e^{\xi (u)}du\right) dt\right) D(({\widetilde{X}}_{Neu}^{15})_{11}\\&\otimes (-{\widetilde{a}}_{Neu})) -\frac{i_{_{IC}}e^V}{\digamma }\bigg \{(1-\lambda _1)(\digamma +V)\\&\int \limits _0^{\digamma }e^{\xi (u)}du+\frac{1}{2}(1-\lambda _1)^2\digamma \int \limits _{0}^{\digamma }e^{\xi (u)}du+\frac{1}{2}\\&\lambda _2(\digamma -U)^2+\frac{1}{2}(1-\lambda _2)(\digamma +V-U)^2\bigg \}\\&D(({\widetilde{X}}_{Neu}^{16})_{11}\otimes (-{\widetilde{a}}_{Neu}))+\frac{i_{_{IE}}e^V}{2\digamma }\bigg \{\lambda _2U^2\\&+(1-\lambda _2)(U-V)^2\bigg \}D(({\widetilde{X}}_{Neu}^{17}). \\&\dfrac{d^2 D(({\widetilde{\Pi }}_{Neu})_{_{_{11}}}(\digamma ,c_{_S}))}{d c_{_S}^2}\\&=2e^VD(({\widetilde{X}}_{Neu}^{11})_{11}\otimes (-{\widetilde{a}}_{Neu}))+c_{_S}e^V\\&D(({\widetilde{X}}_{Neu}^{11})_{11}\otimes (-{\widetilde{a}}_{Neu})\otimes (-{\widetilde{a}}_{Neu}))-\\&\frac{e^{V}}{\digamma }\left( \int \limits _0^{\digamma }e^{-\xi (t)}du\right) D(({\widetilde{X}}_{Neu}^{14})_{11}\otimes (-{\widetilde{a}}_{Neu})\\&\otimes (-{\widetilde{a}}_{Neu})) -\frac{e^V}{\digamma }\left( \int \limits _0^{\digamma }e^{-\xi (t)}\left( \int \limits _t^{\digamma }e^{\xi (u)}du\right) dt\right) \\&D(({\widetilde{X}}_{Neu}^{15})_{11}\otimes (-{\widetilde{a}}_{Neu})\otimes (-{\widetilde{a}}_{Neu})) -\frac{i_{_{IC}}e^V}{\digamma }\\&\bigg \{(1-\lambda _1)(\digamma +V)\int \limits _0^{\digamma }e^{\xi (u)}du+\frac{1}{2}(1-\lambda _1)^2\digamma \\&\int \limits _{0}^{\digamma }e^{\xi (u)}du+\frac{1}{2}\lambda _2(\digamma -U)^2+\frac{1}{2}(1-\lambda _2)(\digamma \\ {}&+V-U)^2\bigg \}D(({\widetilde{X}}_{Neu}^{16})_{11}\otimes (-{\widetilde{a}}_{Neu})\otimes (-{\widetilde{a}}_{Neu})). \end{aligned}$$

Let

$$\begin{aligned} \Theta _{11}&=-2e^VD(({\widetilde{X}}_{Neu}^{11})_{11}\otimes (-{\widetilde{a}}_{Neu}))\\&+\frac{e^{V}}{\digamma }\left( \int \limits _0^{\digamma }e^{-\xi (t)}du\right) D(({\widetilde{X}}_{Neu}^{14})_{11}\otimes (-{\widetilde{a}}_{Neu})\otimes \\&(-{\widetilde{a}}_{Neu})) +\frac{e^V}{\digamma }\left( \int \limits _0^{\digamma }e^{-\xi (t)}\left( \int \limits _t^{\digamma }e^{\xi (u)}du\right) dt\right) D(({\widetilde{X}}_{Neu}^{15})_{11}\\ {}&\otimes (-{\widetilde{a}}_{Neu})\otimes (-{\widetilde{a}}_{Neu}))+\frac{i_{_{IC}}e^V}{\digamma }\bigg \{(1-\lambda _1)(\digamma +V)\\&\int \limits _0^{\digamma }e^{\xi (u)}du+\frac{1}{2}(1-\lambda _1)^2\digamma \int \limits _{0}^{\digamma }e^{\xi (u)}du+\frac{1}{2}\lambda _2(\digamma -U)^2\\&+\frac{1}{2}(1-\lambda _2)(\digamma +V-U)^2\bigg \} D(({\widetilde{X}}_{Neu}^{16})_{11}\otimes (-{\widetilde{a}}_{Neu})\\&~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~\otimes (-{\widetilde{a}}_{Neu})). \end{aligned}$$

So, $\dfrac{d^2 D(({\widetilde{\Pi }}_{Neu})_{_{_{11}}}(\digamma ,c_{_S}))}{d c_{_S}^2}<0$,

if $\Theta _{11}>c_{_S}e^V\dfrac{d^2 D(({\widetilde{X}}_{Neu}^{11})_{11})}{d c_{_S}^2}$.

C Proof of Theorem 6

$$\begin{aligned}&D(({\widetilde{\Pi }}_{Neu})_{_{_{12}}}(\digamma ,c_{_S}))\\&=c_{_S}e^VD(({\widetilde{X}}_{Neu}^{11})_{12})-\frac{1}{\digamma }D(({\widetilde{X}}_{Neu}^{12})_{12})\\&\quad -\frac{1}{\digamma }D(({\widetilde{X}}_{Neu}^{13})_{12})-\frac{e^{V}}{\digamma }\left( \int \limits _0^{\digamma }e^{-\xi (t)}du\right) D(({\widetilde{X}}_{Neu}^{14})_{12})\\&\quad -\frac{e^V}{\digamma }\left( \int \limits _0^{\digamma }e^{-\xi (t)}\left( \int \limits _t^{\digamma }e^{\xi (u)}du\right) dt\right) D(({\widetilde{X}}_{Neu}^{15})_{12})-\frac{i_{_{IC}}e^V}{\digamma }\\&\bigg \{(1-\lambda _1) (\digamma +V)\int \limits _0^{\digamma }e^{\xi (u)}du+\frac{1}{2}(1-\lambda _1)^2\digamma \\&\int \limits _{0}^{\digamma }e^{\xi (u)}du +\frac{1}{2}(1-\lambda _2)(\digamma +V-U)^2\bigg \}D(({\widetilde{X}}_{Neu}^{16})_{12})\\&\quad +\frac{c_{_{S}}i_{_{IE}}e^V}{2\digamma }\bigg \{\lambda _2\digamma ^2+ 2\lambda _2 \digamma (U-\digamma )+(1-\lambda _2)(U-\\&V)^2\bigg \}D(({\widetilde{X}}_{Neu}^{17})_{12})-\frac{\eta }{\digamma }D(({\widetilde{X}}_{Neu}^{18})_{12}). \\&\dfrac{d D(({\widetilde{\Pi }}_{Neu})_{_{_{12}}}(\digamma ,c_{_S}))}{d c_{_S}}\\&=e^VD(({\widetilde{X}}_{Neu}^{11})_{12})+c_{_S}e^VD(({\widetilde{X}}_{Neu}^{11})_{12}\otimes (-{\widetilde{a}}_{Neu}))\\&\quad -\frac{e^{V}}{\digamma }\left( \int \limits _0^{\digamma }e^{-\xi (t)}du\right) D(({\widetilde{X}}_{Neu}^{14})_{12}\otimes (-{\widetilde{a}}_{Neu}))\\&\quad -\frac{e^V}{\digamma }\left( \int \limits _0^{\digamma }e^{-\xi (t)}\left( \int \limits _t^{\digamma }e^{\xi (u)}du\right) dt\right) D(({\widetilde{X}}_{Neu}^{15})_{12}\otimes (-{\widetilde{a}}_{Neu}))\\&\quad -\frac{i_{_{IC}}e^V}{\digamma }\bigg \{(1-\lambda _1)(\digamma +V)\int \limits _0^{\digamma }e^{\xi (u)}du+\frac{1}{2}(1-\lambda _1)^2\digamma \\&\int \limits _{0}^{\digamma }e^{\xi (u)}du+\frac{1}{2}(1-\lambda _2)(\digamma +V-U)^2\bigg \}\\&D(({\widetilde{X}}_{Neu}^{16})_{12}\otimes (-{\widetilde{a}}_{Neu}))+\frac{i_{_{IE}}e^V}{2\digamma }\bigg \{\lambda _2\digamma ^2+2\lambda _2\\&\digamma (U-\digamma )+(1-\lambda _2)(U-V)^2\bigg \}D(({\widetilde{X}}_{Neu}^{17})_{12}). \\&\dfrac{d^2 D(({\widetilde{\Pi }}_{Neu})_{_{_{12}}}(\digamma ,c_{_S}))}{d c_{_S}^2}\\&=2e^VD(({\widetilde{X}}_{Neu}^{11})_{12}\otimes (-{\widetilde{a}}_{Neu}))+c_{_S}e^VD(({\widetilde{X}}_{Neu}^{11})_{12}\\&\quad (-{\widetilde{a}}_{Neu})\otimes (-{\widetilde{a}}_{Neu}))\\&\quad -\frac{e^{V}}{\digamma }\left( \int \limits _0^{\digamma }e^{-\xi (t)}du\right) D(({\widetilde{X}}_{Neu}^{14})_{12}\\&\otimes (-{\widetilde{a}}_{Neu})\otimes (-{\widetilde{a}}_{Neu}))\\&\quad -\frac{e^V}{\digamma }\left( \int \limits _0^{\digamma }e^{-\xi (t)}\left( \int \limits _t^{\digamma }e^{\xi (u)}du\right) dt\right) \\&D(({\widetilde{X}}_{Neu}^{15})_{12}\otimes (-{\widetilde{a}}_{Neu})\otimes (-{\widetilde{a}}_{Neu}))\\&\quad -\frac{i_{_{IC}}e^V}{\digamma }\bigg \{(1-\lambda _1)\\&(\digamma +V)\int \limits _0^{\digamma }e^{\xi (u)}du+\frac{1}{2}(1-\lambda _1)^2\digamma \int \limits _{0}^{\digamma }e^{\xi (u)}du+\frac{1}{2}(1-\\&\lambda _2)(\digamma +V-U)^2\bigg \} D(({\widetilde{X}}_{Neu}^{16})_{12}\otimes (-{\widetilde{a}}_{Neu})\otimes (-{\widetilde{a}}_{Neu})). \end{aligned}$$

Let

$$\begin{aligned} \Theta _{12}&=-2e^VD(({\widetilde{X}}_{Neu}^{11})_{12}\otimes (-{\widetilde{a}}_{Neu}))\\&\quad +\frac{e^{V}}{\digamma }\left( \int \limits _0^{\digamma }e^{-\xi (t)}du\right) D(({\widetilde{X}}_{Neu}^{14})_{12}\otimes (-{\widetilde{a}}_{Neu})\\&\otimes (-{\widetilde{a}}_{Neu})) +\frac{e^V}{\digamma }\left( \int \limits _0^{\digamma }e^{-\xi (t)}\left( \int \limits _t^{\digamma }e^{\xi (u)}du\right) dt\right) \\&D(({\widetilde{X}}_{Neu}^{15})_{12}\otimes (-{\widetilde{a}}_{Neu})\otimes (-{\widetilde{a}}_{Neu})) +\frac{i_{_{IC}}e^V}{\digamma }\\&\bigg \{(1-\lambda _1)(\digamma +V)\int \limits _0^{\digamma }e^{\xi (u)}du+\frac{1}{2}(1-\lambda _1)^2\digamma \\&\int \limits _{0}^{\digamma }e^{\xi (u)}du+\frac{1}{2}(1-\lambda _2)(\digamma +V-U)^2\bigg \}\\&D(({\widetilde{X}}_{Neu}^{16})_{12}\otimes (-{\widetilde{a}}_{Neu})\otimes (-{\widetilde{a}}_{Neu})). \end{aligned}$$

So, $\dfrac{d^2 D(({\widetilde{\Pi }}_{Neu})_{_{_{12}}}(\digamma ,c_{_S}))}{d c_{_S}^2}<0$,

if $\Theta _{12}>c_{_S}e^V\dfrac{d^2 D(({\widetilde{X}}_{Neu}^{11})_{12})}{d c_{_S}^2}$.

D Proof of Theorem 8

For any given $c_{_S}$, to find the $\digamma ^*_{13}$, taking the first-order derivative of $D(({\widetilde{\Pi }}_{Neu})_{_{_{13}}}(\digamma ))=D(({\widetilde{\Pi }}_{Neu})_{_{_{13}}}(\digamma ,c_{_S}))$ with respect to $\digamma$ and equate it to zero, we get

$$\begin{aligned}&\Delta _{13}(\digamma )=\dfrac{d}{d\digamma }D(({\widetilde{\Pi }}_{Neu})_{_{_{13}}}(\digamma ,c_{_S}))\\&=\frac{1}{\digamma ^2}D(({\widetilde{X}}_{Neu}^{12})_{13})+\frac{1}{\digamma ^2}D(({\widetilde{X}}_{Neu}^{13})_{13}) +\frac{e^{V}}{\digamma ^2}\\&\bigg \{ \left( \int \limits _0^{\digamma }e^{-\xi (t)}du\right) -\digamma e^{\xi (\digamma )}\bigg \} D(({\widetilde{X}}_{Neu}^{14})_{13}) +\frac{e^V}{\digamma ^2}\\&\bigg \{\left( \int \limits _0^{\digamma }e^{-\xi (t)}\left( \int \limits _t^{\digamma }e^{\xi (u)}du\right) dt\right) - \digamma \left( \int \limits _0^{\digamma }e^{-\xi (t)+\xi (\digamma )}dt\right) \bigg \}\\&D(({\widetilde{X}}_{Neu}^{15})_{13}) +\frac{i_{_{IC}}e^V}{\digamma ^2}\bigg \{(1-\lambda _1)(\digamma +V)\int \limits _0^{\digamma }e^{\xi (u)}du\\&+\frac{1}{2}(1-\lambda _1)^2\digamma \int \limits _{0}^{\digamma }e^{\xi (u)}du\bigg \}D(({\widetilde{X}}_{Neu}^{16})_{13}) -\frac{e^Vi_{_{IC}}}{\digamma }\\&\bigg \{\big \{V+\digamma +\frac{1}{2}\digamma (1-\lambda _1)\big \}(1-\lambda _1)e^{\xi (\digamma )}+\frac{1}{2}(3-\lambda _1)\\&(1-\lambda _1)\int \limits _{0}^{\digamma }e^{\xi (t)}dt\bigg \}D(({\widetilde{X}}_{Neu}^{16})_{13})-\frac{c_{_{S}}i_{_{IE}}e^V}{2\digamma ^2}\bigg \{\digamma ^2+\\&2\lambda _2\digamma (U-\digamma )+2(1-\lambda _2)\digamma (U-\digamma -V)\bigg \}\\&D(({\widetilde{X}}_{Neu}^{17})_{13}) +\frac{e^V}{\digamma }7(U-\digamma -V(1-\lambda _2))\\&D(({\widetilde{X}}_{Neu}^{17})_{12})+\frac{\eta }{\digamma ^2}D(({\widetilde{X}}_{Neu}^{18})_{13}. \end{aligned}$$

So, $\dfrac{d^2 D(({\widetilde{\Pi }}_{Neu})_{_{_{13}}}(\digamma ,c_{_S}))}{d c_{_S}^2}<0$,

if $\Theta _{13}>c_{_S}e^V\dfrac{d^2 D(({\widetilde{X}}_{Neu}^{11})_{13})}{d c_{_S}^2}$.

E Proof of Theorem 9.

• 1.
  $$\begin{aligned} \lim \limits _{\digamma \rightarrow 0}\Delta (\digamma )=\infty \end{aligned}$$

  (29)

  If $\Delta _{13}(U-V)<0$, then by intermediate value property (IVP) and theorem 7, there exists a unique $\digamma ^*_{13}\in (0,~U-V)$ such that $\Delta (\digamma ^*_{13})=0$. Hence, $D(({\widetilde{\Pi }}_{Neu})_{_{_{13}}}$ $(\digamma ),c_{_S})$ is maximize at a unique $\digamma ^*_{13}$. Now

  $$\begin{aligned} \Delta _{13}(U-V)=\Delta _{12}(U-V). \end{aligned}$$

  So, if $\Delta _{13}(U-V)<0$, then $\Delta _{12}(U-V)<0$ and by the theorem 4, we get

  $$\begin{aligned} \dfrac{d}{d\digamma }D(({\widetilde{\Pi }}_{Neu})_{_{_{12}}}(\digamma ,c_{_S}))<0,~~\forall ~\digamma \in [U-V,~U]. \end{aligned}$$

  So, for all $\digamma \in [U-V,~U]$, $D(({\widetilde{\Pi }}_{Neu})_{_{_{12}}}(\digamma ,c_{_S}))$ is decreasing and maximize at $U-V$. Again,

  $$\begin{aligned} \Delta _{12}(U)=\Delta _{11}(U). \end{aligned}$$

  Since $\Delta _{12}(U)<0$, so $\Delta _{11}(U)<0$ and by the theorem 3, we have

  $$\begin{aligned} \dfrac{d}{d\digamma }D(({\widetilde{\Pi }}_{Neu})_{_{_{11}}}(\digamma ,c_{_S}))<0,~~\forall ~\digamma \in [U,~\infty ] . \end{aligned}$$

  So, for all $\digamma \in [U,~\infty ]$, $D(({\widetilde{\Pi }}_{Neu})_{_{_{11}}}(\digamma ,c_{_S}))$ is decreasing and maximize at U. So if $\Delta _{13}(U-V)<0$, then

  $$\begin{aligned} \begin{aligned} D(({\widetilde{\Pi }}_{Neu})_{_{_{13}}}(\digamma ^*_{13},c_{_S})&\ge D(({\widetilde{\Pi }}_{Neu})_{_{_{13}}}(U-V,c_{_S})\\&=D(({\widetilde{\Pi }}_{Neu})_{_{_{12}}}(U-V,c_{_S})\\&>D(({\widetilde{\Pi }}_{Neu})_{_{_{12}}}(\digamma ,c_{_S})\\&\ge D(({\widetilde{\Pi }}_{Neu})_{_{_{12}}}(U,c_{_S}),\\&~~~~~~~~~~~~~~~~~~\forall ~\digamma \in [U-V,U]. \end{aligned} \end{aligned}$$

  Again,

  $$\begin{aligned} \begin{aligned} D(({\widetilde{\Pi }}_{Neu})_{_{_{12}}}(U,c_{_S})&= D(({\widetilde{\Pi }}_{Neu})_{_{_{11}}}(U,c_{_S})\\&\ge D(({\widetilde{\Pi }}_{Neu})_{_{_{11}}}(\digamma ,c_{_S})~~\forall ~\digamma \in [U,\infty ]. \end{aligned} \end{aligned}$$

  Hence, the for $U\le V$ and for any $c_{_S}$, if $\Delta _{13}(U-V)<0$, then $D(({\widetilde{\Pi }}_{Neu})_{_{_{13}}}(\digamma ,c_{_S})$ is maximize at $\digamma ^*_{13}$.

• 2. If $\Delta _{13}(U-V)=0$, then from Eq. (29) and Theorem 7, we conclude that $U-V\in (0,U-V]$ is the unique point such that $\Delta _{13}(U-V)=0$. Hence, $D(({\widetilde{\Pi }}_{Neu})_{_{_{13}}}(\digamma ,c_{_S})$ is maximized at $U-V$. Again,

  $$\begin{aligned} \Delta _{13}(U-V)=\Delta _{12}(U-V)=0. \end{aligned}$$

  From Theorem5, we have $U-V\in [U-V,U]$ is the unique point such that

  $$\begin{aligned} \dfrac{d}{d\digamma }D(({\widetilde{\Pi }}_{Neu})_{_{_{12}}}(U-V,c_{_S}))=0. \end{aligned}$$

  Hence, $D(({\widetilde{\Pi }}_{Neu})_{_{_{12}}}(\digamma ,c_{_S})$ is maximized at $U-V$. Further, we can write

  $$\begin{aligned} \dfrac{d}{d\digamma }D(({\widetilde{\Pi }}_{Neu})_{_{_{12}}}(U-V,c_{_S}))<0,~~\forall ~\digamma \in (U-V,U]. \end{aligned}$$

  So,

  $$\begin{aligned} \dfrac{d}{d\digamma }D(({\widetilde{\Pi }}_{Neu})_{_{_{12}}}(U,c_{_S}))=\dfrac{d}{d\digamma }D(({\widetilde{\Pi }}_{Neu})_{_{_{11}}}(U,c_{_S}))<0. \end{aligned}$$

  Hence, by Theorem 3, we have

  $$\begin{aligned} \dfrac{d}{d\digamma }D(({\widetilde{\Pi }}_{Neu})_{_{_{11}}}(\digamma ,c_{_S}))<0~\forall ~\digamma \in [U,\infty ). \end{aligned}$$

  Thus, $D(({\widetilde{\Pi }}_{Neu})_{_{_{11}}}(\digamma ,c_{_S})$ is strictly decreasing and maximize at U. Therefore,

  $$\begin{aligned} \begin{aligned} D(({\widetilde{\Pi }}_{Neu})_{_{_{13}}}(U-V,c_{_S})&= D(({\widetilde{\Pi }}_{Neu})_{_{_{12}}}(U-V,c_{_S})\\&\ge D(({\widetilde{\Pi }}_{Neu})_{_{_{12}}}(U,c_{_S})\\&=D(({\widetilde{\Pi }}_{Neu})_{_{_{11}}}(U,c_{_S}). \end{aligned} \end{aligned}$$

  Hence, if $\Delta _{13}(U-V)=0$, then for $U\le V$ De-neutrosophic value of retailer optimal profit is maximized at $U-V$.

• 3. If $\Delta _{13}(U-V)>0$, then by Theorem 7 and Eq. (29), we can write

  $$\begin{aligned} \dfrac{d}{d\digamma }D(({\widetilde{\Pi }}_{Neu})_{_{_{13}}}(\digamma ,c_{_S}))>0,~\forall ~\digamma \in (0,U-V]. \end{aligned}$$

  Hence, $D(({\widetilde{\Pi }}_{Neu})_{_{_{13}}}(\digamma ,c_{_S}))$ is monotonically strictly increasing function and is maximized at $U-V.$ Again,

  $$\begin{aligned} \dfrac{d}{d\digamma }D(({\widetilde{\Pi }}_{Neu})_{_{_{13}}}(U-V,c_{_S})) \end{aligned}$$
  $$\begin{aligned} =\dfrac{d}{d\digamma }D(({\widetilde{\Pi }}_{Neu})_{_{_{12}}}(U-V,c_{_S}))>0. \end{aligned}$$

  Now

  • (a) if $\Delta _{12}(U)<0$, then by Theorem 4, there exits a unique point $\digamma ^*_{12}\in (U-V,U)$ such that $\Delta _{12}(\digamma ^*_{12})=0$. Hence, $D(({\widetilde{\Pi }}_{Neu})_{_{_{12}}}(\digamma ,c_{_S}))$ is maximize at $\digamma ^*_{12}$. Also,

    $$\begin{aligned}{} & {} \dfrac{d}{d\digamma }D(({\widetilde{\Pi }}_{Neu})_{_{_{12}}}(U,c_{_S}))\\{} & {} \quad =\dfrac{d}{d\digamma }D(({\widetilde{\Pi }}_{Neu})_{_{_{11}}}(U,c_{_S}))<0. \end{aligned}$$

    By Theorem 3, we have

    $$\begin{aligned} \dfrac{d}{d\digamma }D(({\widetilde{\Pi }}_{Neu})_{_{_{11}}}(\digamma ,c_{_S}))<0,~\forall ~\digamma \in [U,\infty ]. \end{aligned}$$

    Hence, $D(({\widetilde{\Pi }}_{Neu})_{_{_{11}}}(\digamma ,c_{_S}))$ is strictly decreasing and is maximize at U. Now

    $$\begin{aligned} \begin{aligned} D(({\widetilde{\Pi }}_{Neu})_{_{_{12}}}(\digamma ^*_{12},c_{_S}))&\ge D(({\widetilde{\Pi }}_{Neu})_{_{_{12}}}(U-V,c_{_S}))\\&=D(({\widetilde{\Pi }}_{Neu})_{_{_{13}}}(U-V,c_{_S})) \end{aligned} \end{aligned}$$

    (30)

    and

    $$\begin{aligned} \begin{aligned} D(({\widetilde{\Pi }}_{Neu})_{_{_{12}}}(\digamma ^*_{12},c_{_S}))&\ge D(({\widetilde{\Pi }}_{Neu})_{_{_{12}}}(U,c_{_S}))\\&=D(({\widetilde{\Pi }}_{Neu})_{_{_{11}}}(U,c_{_S})). \end{aligned} \end{aligned}$$

    (31)

    Thus, from Eq.(30) and Eq.(31), we conclude that De-neutrosophic value of retailer optimal profit is maximized at $\digamma ^*_{12}$.

  • (b) The proof is similar to 3a.

  • (c) The proof is similar to 3a.