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Inventory Models with Lot-Size Dependent Discount for Deteriorating Items: Pricing and Inventory Policies

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Abstract

Two inventory models are studied for disintegrating products whose market demand be contingent upon the selling price. The retailer enjoys an opportunity on the per-unit purchasing charge of the products based on the order amount so that a lower purchase cost per unit is possible only for a sufficiently large enough order size. Under this lot-size related discount, the inventory models for without ending situation and a fully backlogged shortages situation are formulated mathematically. Deterioration starts instantly when the products are kept in the warehouse at a known certain rate. The ultimate intention is to find the best inventory strategy and price of the product to maximize both situations’ profit. The entire problem is formulated geometrically and solved by proposing an efficient algorithm. The optimum of the profit function is examined theoretically and also graphically by using MATLAB software. Finally, the proposed models are validated by using two numerical examples for solving each case. Sensitivity investigation is done by transforming each parameter in turn while keeping fixed the others.

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Appendices

Appendix 1

Differentiate (23) against p, one has

$$\frac{{\partial TP_i (p,Q)}}{{\partial p}} = \left\{ {\begin{array}{*{20}l} {m - 2np - \frac{{c_h nQ(m - pn)(Q - R)^2 }}{{2\{ Q(m - pn) + R\theta \} ^2 }} - \frac{{(c_i Q + A)nQ(m - pn)(m - pn + \theta )}}{{\{ Q(m - pn) + R\theta \} ^2 }}} \hfill \\ { - \frac{{c_b nQR^2 (m - pn + \theta )}}{{2\{ Q(m - pn) + R\theta \} ^2 }} + \frac{{(c_i Q + A)n(m - pn)}}{{Q(m - pn) + R\theta }}} \hfill \\ { + \frac{{c_h n(Q - R)^2 + c_b nR^2 }}{{2\{ Q(m - pn) + R\theta \} }} + \frac{{(c_i Q + A)n(m - pn + \theta )}}{{Q(m - pn) + R\theta }}} \hfill \\ \end{array} } \right\}$$
(A1)

Differentiate of the Eq. (23) with respect to Q, one finds

$$\frac{{\partial TP_i (p,Q)}}{{\partial Q}} = \left\{ {\begin{array}{*{20}l} {\frac{{c_h (m - pn)^2 (Q - R)^2 }}{{2\{ Q(m - pn) + R\theta \} ^2 }} + \frac{{(c_i Q + A)(m - pn)^2 (m - pn + \theta )}}{{\{ Q(m - pn) + R\theta \} ^2 }}} \hfill \\ { + \frac{{c_b (m - pn)R^2 (m - pn + \theta )}}{{2\{ Q(m - pn) + R\theta \} ^2 }} - \frac{{c_h (m - pn)(Q - R)}}{{Q(m - pn) + R\theta }}} \hfill \\ { - \frac{{c_i (m - pn)(m - pn + \theta )}}{{Q(m - pn) + R\theta }}} \hfill \\ \end{array} } \right\}$$
(A2)

Second-order partial derivatives against p and Q of \(TP_i (p,Q)\) are

$$\frac{{\partial ^2 TP_i (p,Q)}}{{\partial p^2 }} = \left\{ {\begin{array}{*{20}l} { - 2n - \frac{{c_h n^2 Q^2 (m - pn)(Q - R)^2 }}{{\{ Q(m - pn) + R\theta \} ^3 }}} \hfill \\ { - \frac{{2(c_i Q + A)n^2 Q^2 (m - pn)(m - pn + \theta )}}{{\{ Q(m - pn) + R\theta \} ^3 }}} \hfill \\ { - \frac{{c_b n^2 Q^2 R^2 (m - pn + \theta )}}{{\{ Q(m - pn) + R\theta \} ^3 }} + \frac{{2(c_i Q + A)n^2 Q(m - pn)}}{{\{ Q(m - pn) + R\theta \} ^2 }}} \hfill \\ { + \frac{{n^2 Q\{ c_h (Q - R)^2 + c_b R^2 \} }}{{\{ Q(m - pn) + R\theta \} ^2 }} + \frac{{2(c_i Q + A)n^2 Q(m - pn + \theta )}}{{\{ Q(m - pn) + R\theta \} ^2 }}} \hfill \\ { - \frac{{2(c_i Q + A)n^2 }}{{Q(m - pn) + R\theta }}} \hfill \\ \end{array} } \right\}$$
(A3)
$$\frac{{\partial ^2 TP_i (p,Q)}}{{\partial Q^2 }} = \left\{ {\begin{array}{*{20}l} { - \frac{{c_h (m - pn)^3 (Q - R)^2 }}{{\{ Q(m - pn) + R\theta \} ^3 }} - \frac{{2(c_i Q + A)(m - pn)^3 (m - pn + \theta )}}{{\{ Q(m - pn) + R\theta \} ^3 }}} \hfill \\ { - \frac{{c_b R^2 (m - pn)^2 (m - pn + \theta )}}{{\{ Q(m - pn) + R\theta \} ^3 }} + \frac{{2c_h (m - pn)^2 (Q - R)}}{{\{ Q(m - pn) + R\theta \} ^2 }}} \hfill \\ { + \frac{{2c_i (m - pn)^2 (m - pn + \theta )}}{{\{ Q(m - pn) + R\theta \} ^2 }} - \frac{{c_h (m - pn)}}{{Q(m - pn) + R\theta }}} \hfill \\ \end{array} } \right\}$$
(A4)
$$\frac{{\partial ^2 TP_i (p,Q)}}{{\partial p\partial Q}} = \left\{ {\begin{array}{*{20}l} {\frac{{c_h n(m - pn)^2 Q(Q - R)^2 }}{{\{ Q(m - pn) + R\theta \} ^3 }} + \frac{{2n(c_i Q + A)Q(m - pn)^2 (m - pn + \theta )}}{{\{ Q(m - pn) + R\theta \} ^3 }}} \hfill \\ { + \frac{{c_b nQR^2 (m - pn)(m - pn + \theta )}}{{\{ Q(m - pn) + R\theta \} ^3 }} - \frac{{(c_i Q + A)n(m - pn)^2 }}{{\{ Q(m - pn) + R\theta \} ^2 }}} \hfill \\ { - \frac{{c_h n(m - pn)(Q - R)(2Q - R)}}{{\{ Q(m - pn) + R\theta \} ^2 }} - \frac{{c_b nR^2 (2m - 2np + \theta )}}{{2\{ Q(m - pn) + R\theta \} ^2 }}} \hfill \\ { - \frac{{n(3c_i Q + 2A)Q(m - pn)(m - pn + \theta )}}{{\{ Q(m - pn) + R\theta \} ^2 }} + \frac{{c_i n(2m - 2np + \theta )}}{{Q(m - pn) + R\theta }}} \hfill \\ { + \frac{{c_h n(Q - R)}}{{Q(m - pn) + R\theta }}} \hfill \\ \end{array} } \right\}$$
(A5)

Appendix 2

For maximization of the objective function, one has

$$\begin{aligned} & \frac{{\partial ^2 TP_i (p,Q)}}{{\partial p^2 }} = \left\{ \begin{gathered} - 2n - \frac{{c_h n^2 Q^2 (m - pn)(Q - R)^2 }}{{\{ Q(m - pn) + R\theta \} ^3 }} \hfill \\ - \frac{{2(c_i Q + A)n^2 Q^2 (m - pn)(m - pn + \theta )}}{{\{ Q(m - pn) + R\theta \} ^3 }} \hfill \\ - \frac{{c_b n^2 Q^2 R^2 (m - pn + \theta )}}{{\{ Q(m - pn) + R\theta \} ^3 }} + \frac{{2(c_i Q + A)n^2 Q(m - pn)}}{{\{ Q(m - pn) + R\theta \} ^2 }} \hfill \\ + \frac{{n^2 Q\{ c_h (Q - R)^2 + c_b R^2 \} }}{{\{ Q(m - pn) + R\theta \} ^2 }} \hfill \\ + \frac{{2(c_i Q + A)n^2 Q(m - pn + \theta )}}{{\{ Q(m - pn) + R\theta \} ^2 }} - \frac{{2(c_i Q + A)b^2 }}{{Q(m - pn) + R\theta }} \hfill \\ \end{gathered} \right\} < 0 \\ & {\text{i}}{\rm{.e}}{\text{.}},\,\left\{ \begin{gathered} \frac{{2(c_i Q + A)n^2 Q(m - pn)}}{{\{ Q(m - pn) + R\theta \} ^2 }} + \frac{{n^2 Q\{ c_h (Q - R)^2 + c_b R^2 \} }}{{\{ Q(m - pn) + R\theta \} ^2 }} \hfill \\ + \frac{{2(c_i Q + A)n^2 Q(m - pn + \theta )}}{{\{ Q(m - pn) + R\theta \} ^2 }} \hfill \\ \end{gathered} \right\} \\ & \quad \quad > \left\{ \begin{gathered} 2n + \frac{{c_h n^2 Q^2 (m - pn)(Q - R)^2 }}{{\{ Q(m - pn) + R\theta \} ^3 }} + \frac{{2(c_i Q + A)n^2 Q^2 (m - pn)(m - pn + \theta )}}{{\{ Q(m - pn) + R\theta \} ^3 }} \hfill \\ + \frac{{c_b n^2 Q^2 R^2 (m - pn + \theta )}}{{\{ Q(m - pn) + R\theta \} ^3 }} + \frac{{2(c_i Q + A)n^2 }}{{Q(m - pn) + R\theta }} \hfill \\ \end{gathered} \right\} \\ \end{aligned}$$

and

$$\begin{aligned} & \frac{{\partial ^2 TP_i (p,Q)}}{{\partial Q^2 }} = \left\{ {\begin{array}{*{20}l} { - \frac{{c_h (m - pn)^3 (Q - R)^2 }}{{\{ Q(m - pn) + R\theta \} ^3 }} - \frac{{2(c_i Q + A)(m - pn)^3 (m - pn + \theta )}}{{\{ Q(m - pn) + R\theta \} ^3 }}} \hfill \\ { - \frac{{c_b R^2 (m - pn)^2 (m - pn + \theta )}}{{\{ Q(m - pn) + R\theta \} ^3 }} + \frac{{2c_h (m - pn)^2 (Q - R)}}{{\{ Q(m - pn) + R\theta \} ^2 }}} \hfill \\ { + \frac{{2c_i (m - pn)^2 (m - pn + \theta )}}{{\{ Q(m - pn) + R\theta \} ^2 }} - \frac{{c_h (m - pn)}}{{Q(m - pn) + R\theta }}} \hfill \\ \end{array} } \right\} < 0 \\ & {\text{i}}{\rm{.e}}{\text{.}},\,\left\{ {\frac{{2c_h (m - pn)^2 (Q - R)}}{{\{ Q(m - pn) + R\theta \} ^2 }} + \frac{{2c_i (m - pn)^2 (m - pn + \theta )}}{{\{ Q(m - pn) + R\theta \} ^2 }}} \right\} \\ & \quad \quad > \left\{ \begin{gathered} \frac{{c_h (m - pn)^3 (Q - R)^2 }}{{\{ Q(m - pn) + R\theta \} ^3 }} + \frac{{2(c_i Q + A)(m - pn)^3 (m - pn + \theta )}}{{\{ Q(m - pn) + R\theta \} ^3 }} \hfill \\ + \frac{{c_b R^2 (m - pn)^2 (m - pn + \theta )}}{{\{ Q(m - pn) + R\theta \} ^3 }} + \frac{{c_h (m - pn)}}{{Q(m - pn) + R\theta }} \hfill \\ \end{gathered} \right\} \\ \end{aligned}$$

Appendix 3

According to Lemma 1

\(\frac{\partial^2 TP_i (p,Q)}{{\partial p^2 }} < 0\,{\text{and}}\,\frac{\partial^2 TP_i (p,Q)}{{\partial Q^2 }} < 0\,\) then obviously the condition \(\left( {\frac{\partial^2 TP_i }{{\partial p^2 }}} \right)\left( {\frac{\partial^2 TP_i }{{\partial Q^2 }}} \right) - \left( {\frac{\partial^2 TP_i }{{\partial p\partial Q}}} \right)^2 \ge 0\).

and the objective function gives maximum value due to the optimal value of p and Q.

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Halim, M.A. (2022). Inventory Models with Lot-Size Dependent Discount for Deteriorating Items: Pricing and Inventory Policies. In: Ali, I., Chatterjee, P., Shaikh, A.A., Gupta, N., AlArjani, A. (eds) Computational Modelling in Industry 4.0. Springer, Singapore. https://doi.org/10.1007/978-981-16-7723-6_10

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