A Fuzzy Decision Making Inventory Model for Deteriorating Items Under Discounted Partial Advance-Partial Delayed Payment Strategy

Abstract

This article presents a buyer’s EOQ model under discounted partial advance payment and partial delayed payment strategies for a single supplier–single buyer supply chain. The durations of advance payment and delayed payment are set by the supplier. Supplier also offers cash discount to the buyer due to advance payment. The rate of cash discount depends on the amount of advance payment made at the time of placing the order, which seems to be realistic in the present business world. It is often observed in the market that a large pile of goods on the shelf has positive affect on the demand; this model considers demand depending on instantaneous stock level of the commodity. Further, to deal with the uncertainty, several inventory parameters are taken as fuzzy numbers (parabolic-flat fuzzy number). For defuzzification of the total average profit of the system, Graded Mean Integration Representation (GMIR) method is used. The aim of this paper is to determine optimal cycle time and optimal ordering quantity for the buyer under no shortages in order to maximize the profit of the system. The necessary theoretical results have been derived in connection with optimality of the total average profit function. To illustrate the model, numerical examples describing a hypothetical system are considered. The concavity of the profit function is visible from graph. Finally, in order to investigate the effects of changes of different system parameters, post optimality analysis is performed.

Appendices

Appendix 1

This appendix presents the explicit expressions of Eqs. (5), (8), (12) and (15).

The explicit expression for Eq. (5) is

$$\begin{aligned} & \frac{{{\text{dTPU}}_{1} }}{{{\text{d}}T}} = 0\;{\text{that is,}} \\ & 2A + aT^{2} \left[ \begin{array}{*{20}l} - h - 2xI_{c} - 2xI_{B} m_{1} + 2xI_{c} m_{1} + 2xd_{1} I_{B} m_{1}^{2} \hfill \\ - 2xI_{B} m_{2} + 2xI_{c} m_{2} + I_{e} p \, \hfill \\ + b ( p + x ( - 1 + d_{1} m_{1}^{2} - I_{c} ( - 1 + m_{1} + m_{2} )(M - 2T) ) \hfill \\ + xI_{B} ( Lm_{1} ( - 1 + d_{1} m_{1} ) + Mm_{2} - 2m_{1} T + 2d_{1} m_{1}^{2} T - 2m_{2} T) ) \hfill \\ - 2x\theta + x\eta \theta + xI_{c} M\theta - xI_{B} Lm_{1} \theta - xI_{c} Mm_{1} \theta \hfill \\ + xd_{1} m_{1}^{2} \theta + xd_{1} I_{B} Lm_{1}^{2} \theta + xI_{B} Mm_{2} \theta - xI_{c} Mm_{2} \theta - 2xI_{c} T\theta \hfill \\ - 2xI_{B} m_{1} T\theta + 2xI_{c} m_{1} T\theta + 2xd_{1} I_{B} m_{1}^{2} T\theta \hfill \\ - 2xI_{B} m_{2} T\theta + 2xI_{c} m_{2} T\theta \hfill \\ \end{array} \right] = 0 \\ \end{aligned}$$

(18)

The explicit expression for Eq. (8) is

$$\begin{aligned} & \frac{{{\text{d}}\,d_{F} \widetilde{{{\text{TPU}}}}_{1} }}{{{\text{d}}T}} = 0\;{\text{that is}}, \\ & \left[ \begin{array}{*{20}l} 7T\left( \begin{array}{*{20}l} - 2a_{2} h_{2} T + a_{3} b_{3} pT + 2a_{3} I_{e} pT + a_{3} p(2 + b_{3} T) \hfill \\ - 2a_{2} xT\theta_{2} + a_{2} x( - 1 + d_{1} m_{1}^{2} )T(b_{2} + \theta_{2} ) \hfill \\ + a_{2} x( - 1 + d_{1} m_{1}^{2} )(2 + b_{2} T + T\theta_{2} ) \hfill \\ - 2a_{2} x (I_{c} ( - 1 + m_{1} + m_{2} )(M - T) + I_{B} m_{1} (1 - d_{1} m_{1} )(L_{2} + T) \hfill \\ + I_{B} m_{2 } ( - M + T))(1 + T(b_{2} + \theta_{2} )) \hfill \\ - 2a_{2} x( - I_{c} ( - 1 + m1 + m2) + I_{B} (m_{1} - d_{1} m_{1}^{2} + m_{2} )) \hfill \\ \left( {T + \tfrac{1}{2}T^{2} (b_{2} + \theta_{2} )} \right) + 2a_{3 } x \eta_{3} T\eta_{3} \hfill \\ \end{array} \right) \hfill \\ - 7 \left( \begin{array}{*{20}l} - 2A_{2} - a_{2} h_{2} T^{2} + a_{3} I_{e} pT^{2} + a_{3} pT(2 + b_{3} T) - a_{2} xT\theta_{2} \hfill \\ + a_{2} x( - 1 + d_{1} m_{1}^{2} )T(2 + b_{2} T + T\theta_{2} ) \hfill \\ - 2a_{2} x(I_{c} ( - 1 + m_{1} + m_{2} )(M - T) + I_{B} m_{1} (1 - d_{1} m_{1} )(L_{2} + T) \hfill \\ + I_{B} m_{2} ( - M + T))\left( {T + \tfrac{1}{2}T^{2} (b_{2} + \theta_{2} )} \right) + a3x\eta_{3} T^{2} \theta_{3} \hfill \\ \end{array} \right) \hfill \\ + 7 T\left( \begin{array}{*{20}l} - 2a_{3} h_{3} T + a_{2} b_{2} pT + 2a_{2} I_{e} pT + a_{2} p(2 + b_{2} T) + 2a_{2} x\eta_{2} T\theta_{2} \hfill \\ - 2a_{3} xT\theta_{3} + a3x( - 1 + d_{1} m_{1}^{2} )T(b_{3} + \theta_{3} ) \hfill \\ + a_{3} x( - 1 + d_{1} m_{1}^{2} )(2 + b_{3} T + T\theta_{3} )\hfill \\ - 2a_{3} x(I_{c} ( - 1 + m_{1} + m_{2} )(M - T) + I_{B} m_{1} (1 - d_{1} m_{1} )(L_{3} + T) \hfill \\ + I_{B} m_{2} ( - M + T))(1 + T(b_{3} + \theta_{3} )) - 2a_{3} c( - I_{c} ( - 1 + m_{1} + m_{2} ) \hfill \\ + I_{B} (m_{1} - d_{1} m_{1}^{2} + m_{2} ))\left( {T + \tfrac{1}{2}T^{2} (b_{3} + \theta_{3} )} \right) \hfill \\ \end{array} \right) \hfill \\ - 7\left( \begin{array}{*{20}l} - 2A_{3} - a_{3} h_{3} T^{2} + a_{2} I_{e} pT^{2} + a_{2} pT(2 + b_{2} T) + a_{2} x\eta_{2} T^{2} \theta_{2} \hfill \\ - a_{3} xT^{2} \theta_{3} + a_{3} x( - 1 + d_{1} m_{1}^{2} )T(2 + b_{3} T + T\theta_{3} ) \hfill \\ - 2a_{3} x(I_{c} ( - 1 + m_{1} + m_{2} )(M - T) + I_{B} m_{1} (1 - d_{1} m_{1} )(L_{3} + T) \hfill \\ + I_{B} m_{2} ( - M + T))\left( {T + \tfrac{1}{2}T^{2} (b_{3} + \theta_{3} )} \right) \hfill \\ \end{array} \right) \hfill \\ + 8T\left( \begin{array}{*{20}l} - 2a_{1} h_{1} T + a_{4} b_{4} pT + 2a_{4} I_{e} pT + a_{4} p(2 + b_{4} T) - 2a_{1} xT\theta_{1} \hfill \\ + a_{1} x( - 1 + d_{1} m_{1}^{2} )T(b_{1} + \theta_{1} ) + a_{1} x( - 1 + d_{1} m_{1}^{2} )(2 + b_{1} T + T\theta_{1} ) \hfill \\ - 2a_{1} x(I_{c} ( - 1 + m_{1} + m_{2} )(M - T) + I_{B} m_{1} (1 - d_{1} m_{1} )(L_{1} + T) \hfill \\ + I_{B} m_{2} ( - M + T))(1 + T(b_{1} + \theta_{1} )) \hfill \\ - 2a_{1} x( - I_{c} ( - 1 + m_{1} + m_{2} ) + I_{B} (m_{1} - d_{1} m_{1}^{2} + m_{2} )) \hfill \\ \left( {T + \tfrac{1}{2}T^{2} (b_{1} + \theta_{1} )} \right) + 2a_{4} x\eta_{4} T\theta_{4} \hfill \\ \end{array} \right) \hfill \\ - 8 \left( \begin{array}{*{20}l} - 2A_{1} - a_{1} h_{1} T^{2} + a_{4} I_{e} pT^{2} + a_{4} pT(2 + b_{4} T) - a_{1} xT^{2} \theta_{1} \hfill \\ + a_{1} x( - 1 + d_{1} m_{1}^{2} )T(2 + b_{1} T + T\theta_{1} ) \hfill \\ - 2a_{1} x(I_{c} ( - 1 + m_{1} + m_{2} )(M - T) + I_{B} m_{1} (1 - d_{1} m_{1} )(L_{1} + T) \hfill \\ + I_{B} m_{2} ( - M + T))\left( {T + \tfrac{1}{2}T^{2} (b_{1} + \theta_{1} )} \right) + a_{4} x\eta_{4} T^{2} \theta_{4} \hfill \\ \end{array} \right) \hfill \\ + 8T\left( \begin{array}{*{20}l} - 2a_{4} h_{4} T + a_{1} b_{1} pT + 2a_{1} I_{e} pT + a_{1} p(2 + b_{1} T) + 2a_{1} x\eta_{1} T\theta_{1} \hfill \\ - 2a_{4} xT\theta_{4} + a_{4} x( - 1 + d_{1} m_{1}^{2} )T(b_{4} + \theta_{4} ) \hfill \\ + a_{4} x( - 1 + d_{1} m_{1}^{2} )(2 + b_{4} T + T\theta_{4} ) - 2a_{4} x(I_{c} ( - 1 + m_{1} + m_{2} ) \hfill \\ (M - T) + I_{B} m_{1} (1 - d_{1} m_{1} )(L_{4} + T) \hfill \\ + I_{B} m_{2} ( - M + T))(1 + T(b_{4} + \theta_{4} )) - 2a_{4} x( - I_{c} ( - 1 + m_{1} + m_{2} ) \hfill \\ + I_{B} (m_{1} - d_{1} m_{1}^{2} + m_{2} ))\left( {T + \tfrac{1}{2}T^{2} (b_{4} + \theta_{4} )} \right) \hfill \\ \end{array} \right) \hfill \\ - 8\left( \begin{array}{*{20}l} - 2A_{4} - a_{4} h_{4} T^{2} + a_{1} I_{e} pT^{2} + a_{1} pT(2 + b_{1} T) + a_{1} x\eta_{1} T^{2} \theta_{1} \hfill \\ - a_{4} xT^{2} \theta_{4} + a_{4} x( - 1 + d_{1} m_{1}^{2} )T(2 + b_{4} T + T\theta_{4} ) \hfill \\ - 2a_{4} x(I_{c} ( - 1 + m_{1} + m_{2} )(M - T) + I_{B} m_{1} (1 - d_{1} m_{1} )(L_{4} + T) \hfill \\ + I_{B} m_{2} ( - M + T))\left( {T + \tfrac{1}{2}T^{2} (b_{4} + \theta_{4} )} \right) \hfill \\ \end{array} \right) \hfill \\ \end{array} \right] = 0 \\ \end{aligned}$$

(19)

The explicit expression for Eq. (12) is

$$\begin{aligned} & \frac{{{\text{dTPU}}_{2} (T)}}{{{\text{d}}T}} = 0\;{\text{that is}}, \\ & 2A + aT^{2} \left[ \begin{array}{*{20}l} - h - 2xI_{B} m_{1} + 2xd_{1} I_{B} m_{1}^{2} - I_{e} p + b \left\{ p ( (1 + I_{e} (M - 2T) ) \right. \hfill \\ \left. { + x( - 1 + d_{1} m_{1}^{2} + I_{B} m_{1} ( - 1 + d_{1} m_{1} )(L + 2T))} \right\} \hfill \\ - 2x\theta + x\eta \theta - xI_{B} Lm_{1} \theta + xd_{1} m_{1}^{2} \theta \hfill \\ + xd_{1} I_{B} Lm_{1}^{2} \theta - 2xI_{B} m_{1} T\theta + 2xd_{1} I_{B} m_{1}^{2} T\theta \hfill \\ \end{array} \right] = 0 \\ \end{aligned}$$

(20)

The explicit expression for Eq. (15) is

$$\begin{aligned} & \frac{{{\text{d}}d_{F} \widetilde{{{\text{TPU}}}}_{2} }}{{{\text{d}}T}} = 0\;{\text{that is}}, \\ & \left[ \begin{array}{*{20}l} 16A_{1} + 14A_{2} + 14A_{3} + 16A_{4} - 8a_{1} b_{1} xT^{2} - 7a_{2} b_{2} xT^{2} \hfill \\ - 7a_{3} b_{3} xT^{2} - 8a_{4} b_{4} xT^{2} - 8a_{1} h_{1} T^{2} - 7a_{2} h_{2} T^{2} \hfill \\ - 7a_{3} h{}_{3}T^{2} - 8a_{4} h_{4} T^{2} - 30a_{1} xI_{B} m_{1} T^{2} - 14a_{3} xI_{B} m_{1} T^{2} \hfill \\ - 16a_{4} xI_{B} m_{1} T^{2} - 8a_{1} b_{1} xI_{B} L_{1} m_{1} T^{2} \hfill \\ - 7a_{2} b_{2} xI_{B} L_{2} m_{1} T^{2} - 7a_{3} b_{3} xI_{B} L_{3} m_{1} T^{2} - 8a_{4} b_{4} xI_{B} L_{4} m_{1} T^{2} \hfill \\ + 8a_{1} b_{1} xd_{1} m_{1}^{2} T^{2} + 7a_{2} b_{2} xd_{1} m_{1}^{2} T^{2} \hfill \\ + 7a_{3} b_{3} xd_{1} m_{1}^{2} T^{2} + 8a_{4} b_{4} xd_{1} m_{1}^{2} T^{2} + 30a_{1} xd_{1} I_{B} m_{1}^{2} T^{2} \hfill \\ + 14a_{3} xd_{1} I_{B} m_{1}^{2} T^{2} + 16a_{4} xd_{1} I_{B} m_{1}^{2} T^{2} \hfill \\ + 8a_{1} b_{1} xd_{1} I_{B} L_{1} m_{1}^{2} T^{2} + 7a_{2} b_{2} xd_{1} I_{B} L_{2} m_{1}^{2} T^{2} \hfill \\ + 7a_{3} b_{3} xd_{1} I_{B} L_{3} m_{1}^{2} T^{2} + 8a_{4} b_{4} xd_{1} I_{B} L_{4} m_{1}^{2} T^{2} \hfill \\ + 8a_{1} b_{1} pT^{2} + 7a_{2} b_{2} pT^{2} + 7a_{3} b_{3} pT^{2} + 8a_{4} b_{4} pT^{2} \hfill \\ - 8a_{1} I_{e} \, pT^{2} - 7a_{2} I_{e} \, pT^{2} - 7a_{3} I_{e} pT^{2} \hfill \\ - 8a_{4} I_{e} pT^{2} + 8a_{1} b_{1} I_{e} MpT^{2} + 7a_{2} b_{2} I_{e} MpT^{2} + 7a_{3} b_{3} I_{e} MpT^{2} \hfill \\ + 8a_{4} b_{4} I_{e} MpT^{2} - 16a_{1} b_{1} xI_{B} m_{1} T^{3} \hfill \\ - 14a_{1} b_{2} xIBm_{1} T^{3} - 14a_{3} b_{3} xI_{B} m_{1} T^{3} - 16a_{4} b_{4} xI_{B} m_{1} T^{3} \hfill \\ + 16a_{1} b_{1} xd_{1} I_{B} m_{1}^{2} T^{3} + 14a_{1} b_{2} xd_{1} I_{B} m_{1}^{2} T^{3} \hfill \\ + 14a_{3} b_{3} xd_{1} I_{B} m_{1}^{2} T^{3} + 16a_{4} b_{4} xd_{1} I_{B} m_{1}^{2} T^{3} - 16a_{1} b_{1} I_{e} pT^{3} \hfill \\ - 14a_{2} b_{2} I_{e} pT^{3} - 14a_{3} b_{3} I_{e} pT^{3} \hfill \\ - 16a_{4} b_{4} I_{e} pT^{3} - 16a_{1} xT^{2} \theta_{1} + 8a_{1} x\eta_{1} T^{2} \theta_{1} - 8a_{1} xI_{B} L_{1} m_{1} T^{2} \theta_{1} \hfill \\ + 8a_{1} xd_{1} m_{1}^{2} T^{2} \theta_{1} + 8a_{1} xd_{1} I_{B} L_{1} m_{1}^{2} T^{2} \theta_{1} \hfill \\ - 16a_{1} xI_{B} m_{1} T^{3} \theta_{1} + 16a_{1} xd_{1} I_{B} m_{1}^{2} T^{3} \theta_{1} \hfill \\ - 14a_{2} xT^{2} \theta_{2} + 7a_{2} x\eta_{2} T^{2} \theta_{2} - 7a_{2} xI_{B} L_{2} m_{1} T^{2} \theta_{2} \hfill \\ + 7a_{2} xd_{1} m_{1}^{2} T^{2} \theta_{2} + 7a_{2} xd_{1} I_{B} L2m_{1}^{2} T^{2} \theta_{2} - 14a_{1} xI_{B} m_{1} T^{3} \theta_{2} \hfill \\ + 14a_{1} xd_{1} I_{B} m_{1}^{2} T^{3} \theta_{2} - 14a_{3} xT^{2} \theta_{3} \hfill \\ + 7a_{3} x\eta_{3} T^{2} \theta_{3} - 7a_{3} xI_{B} L_{3} m_{1} T^{2} \theta_{3} + 7a_{3} xd_{1} m_{1}^{2} T^{2} \theta_{3} \hfill \\ + 7a_{3} xd_{1} I_{B} L_{3} m_{1}^{2} \, T^{2} \theta_{3} - 14a_{3} xI_{B} m_{1} T^{3} \theta_{3} \hfill \\ + 14a_{3} xd_{1} I_{B} m_{1}^{2} T^{3} \theta_{3} - 16a_{4} xT^{2} \theta_{4} + 8a_{4} x\eta_{4} T^{2} \theta_{4} \hfill \\ - 8a_{4} xI_{B \, } L_{4} m_{1} T^{2} \theta_{4} + 8a_{4} xd_{1} m_{1}^{2} T^{2} \theta_{4} \hfill \\ + 8a_{4} xd_{1} I_{B} L_{4} m_{1}^{2} T^{2} \theta_{4} - 16a_{4} xI_{B} m_{1} T^{3} \theta_{4} + 16a_{4} xd_{1} I_{B} m_{1}^{2} T^{3} \theta_{4} \hfill \\ \end{array} \right] = 0 \\ \end{aligned}$$

(21)

Appendix 2

This appendix presents some results on optimality of the profit functions.

Lemma 1

If a function $G(y) = \frac{K(y)}{y}$, where $K(y)$ is twice differentiable function of y, then maximum value of $G(y)$ exists at $y = y^{*}$ obtained by solving $\frac{{{\text{d}}G}}{{{\text{d}}y}} = 0$ if $\frac{1}{y}\frac{{{\text{d}}^{2} K}}{{{\text{d}}y^{2} }} < 0$ at $y = y^{*}$.

Proof

We have $G(y) = \frac{K(y)}{y}$. For maximum or minimum, the necessary condition is $\frac{{{\text{d}}G}}{{{\text{d}}y}} = 0$.

Now,

$$\frac{{{\text{d}}G}}{{{\text{d}}y}} = \frac{1}{{y^{2} }}\left[ {y\frac{{{\text{d}}K(y)}}{{{\text{d}}y}} - K(y)} \right],\frac{{{\text{d}}^{2} G}}{{{\text{d}}y^{2} }} = \frac{1}{{y^{3} }}\left[ {y^{2} \frac{{{\text{d}}^{2} K(y)}}{{{\text{d}}y^{2} }} - 2\left\{ {y\frac{{{\text{d}}K(y)}}{{{\text{d}}y}} - K(y)} \right\}} \right].$$

$$\frac{{{\text{d}}G}}{{{\text{d}}y}} = 0 \Rightarrow y\frac{{{\text{d}}K(y)}}{{{\text{d}}y}} - K(y) = 0$$

(22)

Let $y = y^{*}$ which satisfies Eq. (22). Therefore, at $y = y^{*}$ we have

$$\frac{{{\text{d}}^{2} G}}{{{\text{d}}y^{2} }} = \frac{1}{{y^{3} }}\left[ {y^{2} \frac{{{\text{d}}^{2} K(y)}}{{{\text{d}}y^{2} }} - 2\left\{ {y\frac{{{\text{d}}K(y)}}{{{\text{d}}y}} - K(y)} \right\}} \right] = \frac{1}{y}\frac{{{\text{d}}^{2} K(y)}}{{{\text{d}}y^{2} }}.$$

using Eq. (22).

Thus, $G(y)$ is maximum at $y = y^{*}$ if $\frac{{{\text{d}}^{2} G}}{{{\text{d}}y^{2} }} < 0$ at $y = y^{*}$ that is, if $\frac{1}{y}\frac{{{\text{d}}^{2} K(y)}}{{{\text{d}}y^{2} }} < 0$ at $y = y^{*}$. Hence the lemma.

Result 1

$${\text{TPU}}_{1} (T)$$

is maximum if

$$\begin{aligned} & a\left[ \begin{array}{*{20}l} - h - 2x\left( { - I_{c} \, ( - 1 + m_{1} + m_{2} ) + I_{B} (m1 - d_{1} m_{1}^{2} + m_{2} )} \right) \hfill \\ + bp + I_{e} p - x\theta + x\eta \theta \hfill \\ - xI_{B} Lm_{1} (1 - d_{1} m_{1} )(b + \theta ) + x( - 1 + \, d_{1} m_{1}^{2} )(b + \theta ) \hfill \\ + xI_{B} Mm_{2} (b + \theta ) - xI_{c} M( - 1 + m_{1} + m_{2} )(b + \theta ) \hfill \\ \end{array} \right] \\ & \quad + a\,T \, \left[ \begin{array}{*{20}l} - xI_{B} m_{1} (1 - d_{1} \, m_{1} )(b + \theta ) - xI_{B} m_{2} (b + \theta ) \hfill \\ + xI_{c} ( - 1 + m_{1} + m_{2} )(b + \theta ) \hfill \\ - 2x\left( { - I_{c} ( - 1 + m_{1} + m_{2} ) + I_{B} (m_{1} - d_{1} \, m_{1}^{2} + \, m_{2} )} \right)(b + \theta ) \hfill \\ \end{array} \right] < 0 \\ \end{aligned}$$

at $T = T^{*} > 0$ which is obtained by solving the Eq. (18).

Proof

Let ${\text{TPU}}_{1} (T) = \frac{K(T)}{T}$, where

$$K(T) = \frac{1}{2}\left[ \begin{array}{*{20}l} - 2A - ahT^{2} + aI_{e} pT^{2} + apT(2 + bT) - ax\theta T^{2} \hfill \\ + ax\eta \theta T^{2} + ax(d_{1} m_{1}^{2} - 1)(2 + bT + \theta T)T \hfill \\ - 2ax\left\{ {I_{c} (m_{1} + m_{2} - 1)(M - T) + I_{B} m_{1} (1 - d_{1} m_{1} )(L + T)} \right. \hfill \\ \left. {+ I_{B} m_{2} (T - M)} \right\}\left\{ {T + \tfrac{1}{2}T^{2} (b + \theta )} \right\} \hfill \\ \end{array} \right]$$

Let $T = T^{*} > 0$ be obtained by solving the Eq. (18). Since $T > 0$, by Lemma 1, it follows that average profit will be optimal if $\frac{{{\text{d}}^{2} K(T)}}{{{\text{d}}T^{2} }} < 0$ at $T = T^{*}$. That is if,

$$\begin{aligned} & a\left[ \begin{array}{*{20}l} - h - 2x\left( { - I_{c} \, ( - 1 + m_{1} + m_{2} ) + I_{B} (m1 - d_{1} m_{1}^{2} + m_{2} )} \right) \hfill \\ + bp + I_{e} p - x\theta + x\eta \theta \hfill \\ - xI_{B} Lm_{1} (1 - d_{1} m_{1} )(b + \theta ) + x( - 1 + \, d_{1} m_{1}^{2} )(b + \theta ) \hfil \\ + xI_{B} Mm_{2} (b + \theta ) - xI_{c} M( - 1 + m_{1} + m_{2} )(b + \theta ) \hfill \\ \end{array} \right] \\ & \quad + a\,T \, \left[ \begin{array}{*{20}l} - xI_{B} m_{1} (1 - d_{1} \, m_{1} )(b + \theta ) - xI_{B} m_{2} (b + \theta ) \hfill \\ + xI_{c} \, ( - 1 + m_{1} + m_{2} )(b + \theta ) \hfill \\ - 2x\left( { - I_{c} ( - 1 + m_{1} + m_{2} ) + I_{B} (m_{1} - d_{1} \, m_{1}^{2} { + }m_{2} )} \right)(b + \theta ) \hfill \\ \end{array} \right] < 0 \\ \end{aligned}$$

Result 2

$d_{F} (\widetilde{{{\text{TPU}}}}_{1} )(T)$ is maximum if

$$\left[ \begin{array}{*{20}l} - 14a_{3} h_{3} - 16a_{4} h_{4} + 14a_{3} b_{3} p + 16a_{4} b_{4} p + 14a_{3} I_{e} p \hfill \\ + 16 a_{4} I_{e} p - 14a_{3} b_{3} x - 16a_{4} b_{4} x - 28a_{3} I_{C} x - 32a_{4} I_{C} x \hfill \\ + 14a_{3} b_{3} I_{C} Mx + 16a_{4} b_{4} I_{C} Mx - 2a_{3} I_{B} m_{1} x - 32a_{4} I_{B} m_{1} x \hfill \\ + 28a_{3} I_{C} m_{1} x + 32a_{4} I_{C} m_{1} x - 14a_{3} b_{3} I_{B} L_{3} m_{1} x \hfill \\ - 16a_{4} b_{4} I_{B} L_{4} m_{1} x - 14a_{3} b_{3} I_{C} Mm_{1} x - 16a_{4} b_{4} I_{C} Mm_{1} x \hfill \\ + 14a_{3} b_{3} d_{1} m_{1}^{2} x + 16a_{4} b_{4} d_{1} m_{1}^{2} x + 28a_{3} d_{1} m_{1}^{2} x \hfill \\ + 32a_{4} I_{B} d_{1} m_{1}^{2} x + 14a_{3} b_{3} I_{B} L_{3} d_{1} m_{1}^{2} x + 16a_{4} b_{4} I_{B} L_{4} d_{1} m_{1}^{2} x \hfill \\ - 28a_{3} I_{B} m_{2} x - 32a_{4} I_{B} m_{2} x + 28a_{3} I_{C} m_{2} x \hfill \\ + 32a_{4} I_{C} m_{2} x + 14a_{3} b_{3} I_{B} Mm_{2} x + 16a_{4} b_{4} I_{B} Mm_{2} x \hfill \\ - 14a_{3} b_{3} I_{C} Mm_{2} x - 16a_{4} b_{4} I_{C} Mm_{2} x - 42a_{3} b_{3} I_{C} Tx \hfill \\ - 48a_{4} b_{4} I_{C} Tx - 42a_{3} b_{3} I_{B} m_{1} Tx - 48a_{4} b_{4} I_{B} m_{1} Tx \hfill \\ + 42a_{3} b_{3} I_{C} m_{1} Tx + 48a_{4} b_{4} I_{C} m_{1} Tx + 42a_{3} b_{3} I_{B} d_{1} m_{{1}}^{2} Tx \hfill \\ + 48a_{4} b_{4} I_{B} d_{1} m_{1}^{2} Tx - 42a_{3} b_{3} I_{B} m_{2} Tx - 48a_{4} b_{4} I_{B} m_{2} Tx \hfill \\ + 42a_{3} b_{3} I_{C} m_{2} Tx + 48a_{4} b_{4} I_{C} m_{2} Tx - 28a_{3} \theta _{3} x \hfill \\ + 14a_{3} \eta _{3} \theta _{3} x + 14a_{3} I_{C} M\theta _{3} x - 14a_{3} I_{B} L_{3} m_{1} \theta _{3} x \hfill \\ - 14a_{3} I_{C} Mm_{1} \theta _{3} x + 14a_{3} d_{1} m_{1}^{2} \theta _{3} x + 14a_{3} I_{B} L_{3} d_{1} m_{{1}}^{2} \theta _{3} x \hfill \\ + 14a_{3} I_{B} Mm_{2} \theta _{3} x - 14a_{3} I_{C} Mm_{2} \theta _{3} x - 42a_{3} I_{C} T\theta _{3} x \hfill \\ - 42a_{3} I_{B} m_{1} T\theta _{3} x + 42a_{3} I_{C} m_{1} T\theta _{3} x + 42a_{3} I_{B} d_{1} m_{{1}}^{2} T\theta _{3} x \hfill \\ - 42a_{3} I_{B} m_{2} T\theta _{3} x + 42a_{3} I_{C} m_{2} T\theta _{3} x - 32a_{4} \theta _{4} x + 16a_{4} \eta _{4} \theta _{4} x \hfill \\ + 16a_{4} I_{C} M\theta _{4} x - 16a_{4} I_{B} L_{4} m_{1} \theta _{4} x - 16a_{4} I_{C} Mm_{1} \theta _{4} x \hfill \\ + 16a_{4} d_{1} m_{1}^{2} \theta _{4} x + 16a_{4} I_{B} L_{4} d_{1} m_{1}^{2} \theta _{4} x + 16a_{4} I_{B} Mm_{2} \theta _{4} x \hfill \\ - 16a_{4} I_{C} Mm_{2} \theta _{4} x - 48a_{4} I_{C} T\theta _{4} x - 48a_{4} I_{B} m_{1} T\theta _{4} x \hfill \\ + 48a_{4} I_{C} m_{1} T\theta _{4} x + 48a_{4} I_{B} d_{1} m_{{1}}^{2} T\theta _{4} x - 48a_{4} I_{B} m_{2} T\theta _{4} x \hfill \\ + 48a_{4} I_{C} m_{2} T\theta _{4} x + 8a_{1} ( - 2h_{1} + 2(b_{1} + I_{e} )p \hfill \\ + ( - 2I_{C} ( - 1 + m_{1} + m_{2} )( - 2 + b_{1} (M - 3T) + M\theta _{1} - 3T\theta _{1} ) \hfill \\ + 2( - b_{1} - 2\theta _{1} + \eta _{1} \theta _{1} + d_{1} m_{1}^{2} (b_{1} + \theta _{1} )) \hfill \\ + I_{B} (2m_{2} ( - 2 + b_{1} (M - 3T) + M\theta _{1} - 3T\theta _{1} ) \hfill \\ - 2m_{1} (2 + b_{1} (L_{1} + 3T) + L_{1} \theta _{1} + 3T\theta _{1} ) + 2d_{1} m_{1}^{2} (2 + b_{1} (L_{1} + 3T) \hfill \\ + L_{1} \theta _{1} + 3T\theta _{1} )))x) + 7a_{2} ( - 2h_{2} + 2(b_{2} + I_{e} )p \hfill \\ + ( - 2I_{C} ( - 1 + m_{1} + m_{2} )( - 2 + b_{2} (M - 3T) + M\theta _{2} - 3T\theta _{2} ) \hfill \\ + 2( - b_{2} - 2\theta _{2} + \eta _{2} \theta _{2} + d_{1} m_{1}^{2} (b_{2} + \theta _{2} )) \hfill \\ + I_{B} (2m_{2} ( - 2 + b_{2} (M - 3T) + M\theta _{2} - 3T\theta _{2} ) \hfill \\ - 2m_{1} (2 + b_{2} (L_{2} + 3T) + L_{2} \theta _{2} + 3T\theta _{2} ) \hfill \\ + 2d_{1} m_{1}^{2} (2 + b_{2} (L_{2} + 3T) + L_{2} \theta _{2} + 3T\theta _{2} )))x) \hfill \\ \end{array} \right] < 0$$

at $T = T^{*} > 0$ which is obtained by solving the Eq. (19).

Proof

Similar to Result 1.

Result 3

${\text{TPU}}_{2} (T)$ is maximum if

$$\begin{aligned} & aT\left[ {6bxI_{B} m_{1} ( - 1 + d_{1} m_{1} ) - 6bI_{e} p + 6xI_{B} m_{1} ( - 1 + d_{1} m_{1} )\theta } \right] \\ & \quad + a\left[ \begin{array}{*{20}l} - 2b - x - 2 \, h + 4xI_{B} m_{1} ( - 1 + d_{1} m_{1} ) \hfill \\ + 2bxI_{B} Lm_{1} ( - 1 + d_{1} m_{1} ) + 2bp - 2I_{e} p \hfill \\ + 2bI_{e} Mp - 4c\theta + 2x\eta \theta + 2xI_{B} Lm_{1} ( - 1 + d_{1} m_{1} )\theta \hfill \\ + 2xd_{1} m_{1}^{2} (b + \theta ) \hfill \\ \end{array} \right] < 0 \\ \end{aligned}$$

at $T = T^{*} > 0$ which is obtained by solving the Eq. (20).

Proof

${\text{TPU}}_{2} (T) = \frac{K(T)}{T}$, where

$$K(T) = \frac{1}{2}\left[ \begin{array}{*{20}l} - 2A - ahT^{2} - aI_{e} pT^{2} + apT(2 + bT) \hfill \\ + aI_{e} p(M - T)T(2 + bT) - axT^{2} \theta + ax\eta T^{2} \theta \hfill \\ + ax( - 1 + d_{1} m_{1}^{2} )T(2 + bT + T\theta ) \hfill \\ + axI_{B} m_{1} ( - 1 + d_{1} m_{1} )T(L + T)(2 + bT + T\theta ) \hfill \\ \end{array} \right]$$

Let $T = T^{*} > 0$ be obtained by solving the Eq. (20). Since $T > 0$, by Lemma 1, it follows that average profit will be optimal if $\frac{{{\text{d}}^{2} K(T)}}{{{\text{d}}T^{2} }} < 0$ at $T = T^{*}$. That is if,

$$\begin{aligned} & aT\left[ {6bxI_{B} m_{1} ( - 1 + d_{1} m_{1} ) - 6bI_{e} p + 6xI_{B} m_{1} ( - 1 + d_{1} m_{1} )\theta } \right] \, \\ & + a\left[ \begin{array}{*{20}l} - 2b - x - 2 \, h + 4xI_{B} m_{1} ( - 1 + d_{1} m_{1} ) \hfill \\ + 2bxI_{B} Lm_{1} ( - 1 + d_{1} m_{1} ) + 2bp - 2I_{e} p \hfill \\ + 2bI_{e} Mp - 4c\theta + 2x\eta \theta + 2xI_{B} Lm_{1} ( - 1 + d_{1} m_{1} )\theta \hfill \\ + 2xd_{1} m_{1}^{2} (b + \theta ) \hfill \\ \end{array} \right] < 0 \\ \end{aligned}$$

Result 4

$d_{F} (\widetilde{{{\text{TPU}}}}_{2} )(T)$ is maximum if

$$\left[ \begin{array}{*{20}l} - 14a_{3} h_{3} - 16a_{4} h_{4} + 14a_{3} b_{3} p + 16a_{4} b_{4} p - 14a_{3} I_{e} p - 16a_{4} I_{e} p \hfill \\ + 14a_{3} b_{3} I_{e} Mp + 16a_{4} b_{4} I_{e} Mp - 42a_{3} b_{3} I_{e} pT - 48a_{4} b_{4} I_{e} pT \hfill \\ - 14a_{3} b_{3} x - 16a_{4} b_{4} x - 28a_{3} I_{B} m_{1} x - 32a_{4} I_{B} m_{1} x - 14a_{3} b_{3} I_{B} L_{3} m_{1} x \hfill \\ - 16a_{4} b_{4} I_{B} L_{4} m_{1} x + 14a_{3} b_{3} d_{1} m_{1}^{2} x + 16a_{4} b_{4} d_{1} m_{1}^{2} x + 28a_{3} I_{B} d_{1} m_{1}^{2} x \hfill \\ + 32a_{4} I_{B} d_{1} m_{1}^{2} x + 14a_{3} b_{3} I_{B} L3d_{1} m_{1}^{2} x + 16a_{4} b_{4} I_{B} L_{4} d_{1} m_{1}^{2} x \hfill \\ - 42a_{3} b_{3} I_{B} m_{1} Tx - 48a_{4} b_{4} I_{B} m_{1} Tx + 42a_{3} b_{3} I_{B} d_{1} m_{1}^{2} Tx \hfill \\ + 48a_{4} b_{4} I_{B} d_{1} m_{1}^{2} Tx - 28a_{3} \theta _{3} x + 14a_{3} \eta _{3} \theta _{3} x - 14a_{3} I_{B} L_{3} m_{1} \theta _{3} x \hfill \\ + 14a_{3} d_{1} m_{1}^{2} \theta _{3} x + 14a_{3} I_{B} L_{3} d_{1} m_{1}^{2} \theta _{3} x - 42a_{3} I_{B} m_{1} T\theta _{3} x \hfill \\ + 42a_{3} I_{B} d_{1} m_{1}^{2} T\theta _{3} x - 32a_{4} \theta _{4} x + 16a_{4} \eta _{4} \theta _{4} x - 16a_{4} I_{B} L_{4} m_{1} \theta _{4} x \hfill \\ + 16a_{4} d_{1} m_{1}^{2} \theta _{4} x + 16a_{4} I_{B} L_{4} d_{1} m_{1}^{2} \theta _{4} x - 48a_{4} I_{B} m_{1} T\theta _{4} x \hfill \\ + 48a_{4} I_{B} d_{1} m_{1}^{2} T\theta _{4} x + 14a_{2} ( - h_{2} + p(b_{2} - I_{e} + b_{2} I_{e} M - 3b_{2} I_{e} T) \hfill \\ + ( - b_{2} - 2\theta _{2} + \eta _{2} \theta _{2} d_{1} m_{1}^{2} (b_{2} + \theta _{2} ) + I_{B} L_{2} m_{1} ( - 1 + d_{1} m_{1} )(b_{2} + \theta _{2} ))x) \hfill \\ + a_{1} ( - 16h_{1} + 16p(b_{1} - I_{e} + b_{1} I_{e} M - 3b_{1} I_{e} T) + (16( - b_{1} - 2\theta _{1} \hfill \\ + \eta _{1} \theta _{1} + d_{1} m_{1}^{2} (b_{1} + \theta _{1} )) + 2I_{B} m_{1} ( - 1 + d_{1} m_{1} )(30 + 21b_{2} T \hfill \\ + 8b_{1} (L_{1} + 3T) + 8L_{1} \theta _{1} + 24T\theta _{1} + 21T\theta _{2} ))x) \hfill \\ \end{array} \right] < 0$$

at $T = T^{*} > 0$ which is obtained by solving the Eq. (21).

Proof

Similar to Result 3.