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How cash-back strategy affect sale rate under refund and customers’ credit

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Abstract

This paper attempts to develop a dynamic model for a chain, including online shops and customers. In this chain, online shops can sell their goods directly and through a cash-back website. Cash-back market enables customers to receive part of a paid amount in cash or as a credit after the purchase by creating a link. The effects of this website and credit assignment are not clear to us yet, but cash-back helps those sensitive about the price and those who think markets are overselling to tend to shop more. Besides, we will investigate the refund-motivating policy in this paper, which enables customers to return the merchandise in case of dissatisfaction and receive the whole or part of the money they have paid. This paper's primary purpose is to focus on the effect of credit and motivating policies explained above on the online market's benefit. In this study, we are going to consider motivational policies, in both centralized and decentralized frameworks. Also, choosing a centralized and decentralized method by an online shop is often influenced by societal variables, as shown in this paper. The results suggest that cash-back results in more profit for the retailer; moreover, customer credit leads to more customer loyalty; however, cash-back websites might even result in loss of customers' trust in case of unwise use. On top of that, refund strategy works as a back-up for customers, so if the return rate of the market gets calculated wisely, carrying out this strategy might lead to gaining more customers as well.

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Notes

  1. https://www.digikala.com/.

  2. https://takhfifan.com/.

  3. https://snapp.ir/.

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Acknowledgements

The authors wish to thank the two anonymous reviewers for their critical and constructive comments.

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Correspondence to Ioannis Konstantaras.

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Appendices

Appendix 1

The Hessian matrix of Eq. (10) is defined as follows:

$$H = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\frac{{\partial^{2} \Pi_{ec} }}{{\partial s^{2} }}} & {\frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{cu} }}} & {\frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{r} }}} \\ \end{array} } \\ {\begin{array}{*{20}c} {\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu} \partial s}}} & {\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu}^{2} }}} & {\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu} \partial f_{r} }}} \\ \end{array} } \\ {\begin{array}{*{20}c} {\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r} \partial s}}} & {\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r} \partial f_{cu} }}} & {\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r}^{2} }}} \\ \end{array} } \\ \end{array} } \right],$$
(36)

where,

$$\frac{{\partial^{2} \Pi_{ec} }}{{\partial s^{2} }} = - 2\frac{D}{v\lambda },$$
(37)
$$\frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{cu} }} = \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu} \partial s}} = \frac{2D}{{\lambda v}},$$
(38)
$$\frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{r} }} = \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r} \partial s}} = \frac{D(1 + r)}{{v\lambda }},$$
(39)
$$\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu}^{2} }} = - 2\frac{D}{(1 - \lambda )\lambda v},$$
(40)
$$\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu} \partial f_{r} }} = \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r} \partial f_{cu} }} = - \frac{D}{\lambda v}(1 + r),$$
(41)
$$\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r}^{2} }} = - 2\frac{Dr}{{\lambda v}}.$$
(42)

When Eqs. (43) to (45) are established, the function is concave:

$$\left| H \right|_{1} < 0 \to \frac{{\partial^{2} \Pi_{ec} }}{{\partial s^{2} }} < 0$$
(43)
$$\left| H \right|_{2} > 0 \to \frac{{\partial^{2} \Pi_{ec} }}{{\partial s^{2} }}.\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu}^{2} }} - \frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{cu} }}.\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu} \partial s}} > 0$$
(44)
$$\left| H \right|_{3} < 0 \to \left[ \begin{gathered} \frac{{\partial^{2} \Pi_{ec} }}{{\partial s^{2} }}\left( {\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu}^{2} }}.\frac{{\partial^{2} \Pi_{rc} }}{{\partial f_{r}^{2} }} - \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu} \partial f_{r} }}.\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r} \partial f_{cu} }}} \right) \hfill \\ - \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu} \partial s}}\left( {\frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{cu} }}.\frac{{\partial^{2} \Pi_{rc} }}{{\partial f_{r}^{2} }} - \frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{r} }}.\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r} \partial f_{cu} }}} \right) \hfill \\ + \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r} \partial s}}\left( {\frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{cu} }}.\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu} \partial f_{r} }} - \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu}^{2} }}.\frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{r} }}} \right) \hfill \\ \end{gathered} \right] < 0$$
(45)

In order to obtain the optimum values of the decision variables, we have:

$$\frac{{\partial \Pi_{ec} }}{\partial s} = D\left[ {\left( {1 - \frac{{f_{cu} }}{(1 - \lambda )v}} \right) - (s - f_{cu} )\frac{1}{\lambda v} + \left( {\frac{{f_{cu} }}{(1 - \lambda )v} - \frac{{s - f_{r} - f_{cu} }}{\lambda v}} \right) + \frac{{f_{r} r}}{\lambda v}} \right] = 0,$$
(46)
$$\frac{{\partial \Pi_{ec} }}{{\partial f_{cu} }} = D\left[ \begin{gathered} - (s - c)\frac{1}{(1 - \lambda )v} - 1\left( {\frac{{f_{cu} }}{(1 - \lambda )v} - \frac{{s - f_{r} - f_{cu} }}{\lambda v}} \right) \hfill \\ + \left( {\frac{1}{(1 - \lambda )v} + \frac{1}{\lambda v}} \right)(s - f_{cu} ) - f_{r} r\frac{1}{\lambda v} \hfill \\ \end{gathered} \right] = 0,$$
(47)
$$\frac{{\partial \Pi_{ec} }}{{\partial f_{r} }} = D\left[ {(s - f_{cu} )\frac{1}{\lambda v} - r\left( {\left[ {1 - \frac{{s - f_{r} - f_{cu} }}{\lambda v}} \right]} \right) - \frac{{f_{r} r}}{\lambda v}} \right] = 0.$$
(48)

By solving Eq. (46) to (48), the optimum values will be equal to:

$$s^{*} = \lambda v + f_{r} ,$$
(49)
$$f_{cu}^{*} = \frac{{\lambda v + f_{r} (1 - r)}}{2},$$
(50)
$$f_{cu}^{*} = \frac{{\lambda v + f_{r} (1 - r)}}{2},$$
$$f_{r}^{*} = \frac{2\lambda v(1 - r)}{{2r - r^{2} + 1}},$$
(51)

Appendix 2

The Hessian matrix of Eq. (14) is defined as follows:

$$H = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\frac{{\partial^{2} \Pi_{ec} }}{{\partial s^{2} }}} & {\frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{cu} }}} & {\frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{r} }}} \\ \end{array} } \\ {\begin{array}{*{20}c} {\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu} \partial s}}} & {\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu}^{2} }}} & {\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu} \partial f_{r} }}} \\ \end{array} } \\ {\begin{array}{*{20}c} {\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r} \partial s}}} & {\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r} \partial f_{cu} }}} & {\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r}^{2} }}} \\ \end{array} } \\ \end{array} } \right],$$
(52)

In other hand:

$$\frac{{\partial^{2} \Pi_{ec} }}{{\partial s^{2} }} = - 2\frac{D}{v\lambda },$$
(53)
$$\frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{cu} }} = \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu} \partial s}} = \frac{D}{\lambda v}(\theta + \psi ),$$
(54)
$$\frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{r} }} = \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r} \partial s}} = \frac{D(1 + r)}{{v\lambda }},$$
(55)
$$\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu}^{2} }} = - 2\frac{D\theta \psi }{{(1 - \lambda )\lambda v}},$$
(56)
$$\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu} \partial f_{r} }} = \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r} \partial f_{cu} }} = - \frac{D}{\lambda v}(1 + \psi r),$$
(57)
$$\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r}^{2} }} = - 2\frac{Dr}{{\lambda v}}.$$
(58)

With respect to Eq. (14), which is the condition for concavity of the function, when Eqs. (59) to (61) are established:

$$\left| H \right|_{1} < 0 \to \frac{{\partial^{2} \Pi_{ec} }}{{\partial s^{2} }} < 0,$$
(59)
$$\left| H \right|_{2} > 0 \to \frac{{\partial^{2} \Pi_{ec} }}{{\partial s^{2} }}\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu}^{2} }} - \frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{cu} }}\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu} \partial s}} > 0,$$
(60)
$$\left| H \right|_{3} < 0 \to \left[ \begin{gathered} \frac{{\partial^{2} \Pi_{ec} }}{{\partial s^{2} }}\left( {\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu}^{2} }}\frac{{\partial^{2} \Pi_{rc} }}{{\partial f_{r}^{2} }} - \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu} \partial f_{r} }}\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r} \partial f_{cu} }}} \right) \hfill \\ - \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu} \partial s}}\left( {\frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{cu} }}\frac{{\partial^{2} \Pi_{rc} }}{{\partial f_{r}^{2} }} - \frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{r} }}\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r} \partial f_{cu} }}} \right) \hfill \\ + \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{r} \partial s}}\left( {\frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{cu} }}\frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu} \partial f_{r} }} - \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu}^{2} }}\frac{{\partial^{2} \Pi_{ec} }}{{\partial s\partial f_{r} }}} \right) \hfill \\ \end{gathered} \right] < 0.$$
(61)

In this case, to solve the second part of Eq. (14), the first derivative is used. First derivative conditions are:

$$\frac{{\partial \Pi_{ec} }}{\partial s} = D\left[ {\left( {1 - \frac{{(\psi f_{cu} }}{(1 - \lambda )v}} \right) - (s - \theta f_{cu} )\frac{1}{\lambda v} + \left( {\frac{{\psi f_{cu} }}{(1 - \lambda )v} - \frac{{s - f_{r} - \psi f_{cu} }}{\lambda v}} \right) + \frac{{f_{r} r}}{\lambda v}} \right] = 0,$$
(62)
$$\frac{{\partial \Pi_{ec} }}{{\partial f_{cu} }} = D\left[ { - \frac{(s - c)\psi }{{(1 - \lambda )v}} - \theta \left( {\frac{{\psi f_{cu} }}{(1 - \lambda )v} - \frac{{s - f_{r} - \psi f_{cu} }}{\lambda v}} \right) + \left( {\frac{\psi }{(1 - \lambda )v} + \frac{\psi }{\lambda v}} \right)(s - \theta f_{cu} ) - f_{r} r\frac{\psi }{\lambda v}} \right] = 0,$$
(63)
$$\frac{{\partial \Pi_{ec} }}{{\partial f_{r} }} = D\left[ {(s - f_{cu} )\frac{1}{\lambda v} - r\left( {\left[ {1 - \frac{{s - f_{r} - \psi f_{cu} }}{\lambda v}} \right]} \right) - \frac{{f_{r} r}}{\lambda v}} \right] = 0.$$
(64)

By solving Eqs. (62) to (64), the optimum answers will be equal to:

$$s^{*} = \lambda v + f_{r} ,$$
(65)
$$f_{cu}^{*} = \frac{{\lambda v + f_{r} (1 - r)}}{2 + \psi },$$
(66)
$$f_{r}^{*} = \frac{{v\left( {(2 + \psi )\lambda - \frac{2\theta \lambda \psi }{{(1 - \lambda )(2 + \psi )}}} \right) - c\frac{\psi \lambda }{{1 - \lambda }}}}{{1 + \frac{2\theta \psi }{{(2 + \psi )(1 - \lambda )}} + r\psi - (2 + \psi )}}.$$
(67)

Appendix 3

The Hessian matrix of Eq. (20) is defined as follows:

$$\left| H \right| < 0 \to \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu}^{2} }} < 0 \to \left| H \right| = - 2\frac{D}{v\lambda (1 - \lambda )} < 0,$$
(68)

and

$$\frac{{\partial \Pi_{cw} }}{{\partial f_{cu} }} = D\left[ { - \left( {\frac{{f_{cu} }}{v(1 - \lambda )} - \frac{{s - f_{r} - f_{cu} }}{v\lambda }} \right) + \left( {f_{cw} - f_{cu} } \right)\left( {\frac{1}{v(1 - \lambda )} + \frac{1}{v\lambda }} \right)} \right] = 0.$$
(69)

According to Eq. (69), we have:

$$f_{cu}^{*} = \frac{{(s - f_{r} )(1 - \lambda ) + f_{cw} }}{2}$$
(70)

Appendix 4

Proof of the concavity of Eq. (23), the Hessian matrix is defined as follows:

$$H = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\frac{{\partial^{2} \Pi_{es} }}{{\partial s^{2} }}} & {\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{cw} }}} & {\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{r} }}} \\ \end{array} } \\ {\begin{array}{*{20}c} {\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial s}}} & {\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw}^{2} }}} & {\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial f_{r} }}} \\ \end{array} } \\ {\begin{array}{*{20}c} {\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{r} \partial s}}} & {\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{r} \partial f_{cw} }}} & {\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{r}^{2} }}} \\ \end{array} } \\ \end{array} } \right],$$
(71)

In other hand:

$$\frac{{\partial^{2} \Pi_{es} }}{{\partial s^{2} }} = - \frac{D}{v}\left( {\frac{1 + \lambda }{\lambda }} \right),$$
(72)
$$\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{cw} }} = \frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial s}} = \frac{D}{\lambda v},$$
(73)
$$\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{r} }} = \frac{{\partial^{2} \Pi_{es} }}{{\partial f_{r} \partial s}} = \frac{D}{v}\left( {\frac{1 + \lambda (1 + r)}{{2\lambda }}} \right),$$
(74)
$$\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw}^{2} }} = - \frac{D}{(1 - \lambda )\lambda v},$$
(75)
$$\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial f_{r} }} = \frac{{\partial^{2} \Pi_{es} }}{{\partial f_{r} \partial f_{cw} }} = - \frac{D}{2\lambda v}(1 + r),$$
(76)
$$\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{r}^{2} }} = - \frac{Dr}{v}.$$
(77)

When Eqs. (78) to (80) is established, the function is concave:

$$\left| H \right|_{1} < 0 \to \frac{{\partial^{2} \Pi_{es} }}{{\partial s^{2} }} = - \frac{D}{v}\left( {\frac{1 + \lambda }{\lambda }} \right) < 0,$$
(78)
$$\left| H \right|_{2} > 0 \to \frac{{\partial^{2} \Pi_{es} }}{{\partial s^{2} }}.\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw}^{2} }} - \frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{cw} }}.\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial s}} = \frac{{2D^{2} }}{{v^{2} \lambda }} > 0,$$
(79)
$$\begin{gathered} \left| H \right|_{3} < 0 \to \left[ \begin{gathered} \frac{{\partial^{2} \Pi_{es} }}{{\partial f_{r}^{2} }}\left( {\frac{{\partial^{2} \Pi_{es} }}{{\partial s^{2} }}.\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw}^{2} }} - \frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial s}}.\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{cw} }}} \right) \hfill \\ - \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cw} \partial f_{r} }}\left( {\frac{{\partial^{2} \Pi_{es} }}{{\partial s^{2} }}.\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial f_{r} }} - \frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{r} }}.\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{cw} }}} \right) \hfill \\ + \frac{{\partial^{2} \Pi_{es} }}{{\partial f_{r} \partial s}}\left( {\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{cw} }}.\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial f_{r} }} - \frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw}^{2} }}.\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{r} }}} \right) \hfill \\ \end{gathered} \right] < 0. \hfill \\ \to \frac{{D^{3} }}{{\lambda^{2} v^{3} }}\left( { - (1 + \lambda )\left( {\frac{r}{1 - \lambda } - \frac{{(1 + r)^{2} }}{4\lambda }} \right) + \frac{(1 + r)(1 + \lambda (1 + r))}{{4\lambda }} - r + \frac{(1 + \lambda (1 + r))}{{2\lambda (1 - \lambda )}} - \frac{1 + r}{{2\lambda }}} \right) < 0. \hfill \\ \end{gathered}$$
(80)

In order to obtain the optimum values of the decision variables, we have:

$$\frac{{\partial \Pi_{ec} }}{\partial s} = D\left[ {\left( {1 - \frac{{s - f_{r} }}{2v}} \right) - \left( {\frac{{f_{cw} }}{2v(1 - \lambda )}} \right) - \frac{(s - c)}{{2v}} + \left( {\frac{{f_{cw} }}{2(1 - \lambda )\lambda v} - \frac{{s - f_{r} }}{2\lambda v}} \right) - \frac{{(s - f_{cw} )}}{2v\lambda } + \frac{{f_{r} r}}{2\lambda }} \right] = 0,$$
(81)
$$\frac{{\partial \Pi_{ec} }}{{\partial f_{cw} }} = D\left[ { - \frac{(s - c)}{{2(1 - \lambda )v}} - \left( {\frac{{f_{cw} }}{2\lambda (1 - \lambda )v} - \frac{{s - f_{r} }}{2\lambda v}} \right) + \left( {\frac{{(s - f_{cw} )}}{2\lambda (1 - \lambda )v}} \right) - \frac{{f_{r} r}}{2\lambda v}} \right] = 0,$$
(82)
$$\frac{{\partial \Pi_{ec} }}{{\partial f_{r} }} = D\left[ {\frac{(s - c)}{{2v}} + (s - f_{cw} )\frac{1}{2\lambda v} - \frac{{f_{r} r}}{2v} - r\left( {1 - \frac{{(s - f_{r} )r}}{2v} - \frac{{rf_{cw} }}{2\lambda v}} \right)} \right] = 0.$$
(83)

By solving Eqs. (81) to (83), the optimum answers will be equal to:

$$s^{*} = \left( {\frac{v\lambda }{{1 + \lambda }}\left[ {\frac{\lambda (\xi - 2r)}{{\lambda \xi^{2} }} + 1} \right] + \frac{c\lambda }{{1 + \lambda }}\left[ {\frac{(1 + \lambda (1 + r))\lambda r}{{\xi^{2} }} + 1} \right] - f_{cw} \left[ {2\theta \left( {1 - r} \right)} \right]} \right),$$
(84)
$$f_{cw}^{*} = \frac{{\left( {v\left( {\left[ {\frac{2(\xi - 2r)}{{\lambda \xi^{2} (1 + \lambda )}} - \frac{1}{1 + \lambda }} \right] - \left[ {\frac{{2\left( {1 + r} \right)(2r - \xi )}}{{\xi^{2} }}} \right]} \right) + c\left( \begin{gathered} \left[ {\frac{2(1 + \lambda (1 + r))\lambda r}{{(1 - \lambda )\xi^{2} }} - \frac{1}{1 - \lambda }} \right] \hfill \\ + \left[ {\frac{{\left( {1 + r} \right)(2r - \xi )}}{{\xi^{2} }}} \right] + \left[ {\frac{1}{1 - \lambda }} \right] \hfill \\ \end{gathered} \right)} \right)}}{{\left( {2\frac{1}{1 - \lambda } - \left( {1 + r} \right)\left( {\frac{2r}{{\xi^{2} }}} \right) + \frac{2}{2\lambda (1 + \lambda )}\left( {2 + \frac{(1 + \lambda (1 + r))}{{\xi^{2} }}\left( {2r} \right)} \right)} \right)}},$$
(85)
$$f^{*}_{r} = \left( {2v\lambda \left[ {\frac{\xi - 2r}{{\xi^{2} }}} \right] + \frac{{rc\lambda^{2} }}{{\xi^{2} }} + f_{cw} \frac{{\left[ {\xi 2r} \right]}}{{\xi^{2} }}} \right)$$
(86)

Appendix 5

The Hessian matrix of Eq. (29) is defined as follows:

$$\left| H \right| < 0 \to \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cu}^{2} }} < 0 \to \left| H \right| = - 2\frac{D\theta \psi }{{v\lambda (1 - \lambda )}} < 0.$$
(87)

According to Eq. (87), Eq. (29) is concave and for obtaining the optimum values of the decision variables, we have:

$$\frac{{\partial \Pi_{cw} }}{{\partial f_{cu} }} = D\left[ { - \theta \left( {\frac{{\psi f_{cu} }}{v(1 - \lambda )} - \frac{{s - f_{r} - \psi f_{cu} }}{v\lambda }} \right) + \left( {f_{cw} - \theta f_{cu} } \right)\left( {\frac{\psi }{v(1 - \lambda )} + \frac{\psi }{v\lambda }} \right)} \right] = 0.$$
(88)

By solving Eq. (88), we have:

$$f_{cu}^{*} = \frac{{\theta (s - f_{r} )(1 - \lambda ) + \psi f_{cw} }}{2\theta \psi }.$$
(89)

Appendix 6

With regard to Eq. (89) in Appendix 5, we can say:

$$H = \left[ {\begin{array}{*{20}c} {\begin{array}{*{20}c} {\frac{{\partial^{2} \Pi_{es} }}{{\partial s^{2} }}} & {\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{cw} }}} & {\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{r} }}} \\ \end{array} } \\ {\begin{array}{*{20}c} {\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial s}}} & {\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw}^{2} }}} & {\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial f_{r} }}} \\ \end{array} } \\ {\begin{array}{*{20}c} {\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{r} \partial s}}} & {\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{r} \partial f_{cw} }}} & {\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{r}^{2} }}} \\ \end{array} } \\ \end{array} } \right],$$
(90)

where

$$\frac{{\partial^{2} \Pi_{es} }}{{\partial s^{2} }} = - \frac{D}{v}\left( {\frac{1 + \lambda }{\lambda }} \right),$$
(91)
$$\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{cw} }} = \frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial s}} = \frac{D}{2\lambda \theta v}(\theta + \psi ),$$
(92)
$$\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{r} }} = \frac{{\partial^{2} \Pi_{es} }}{{\partial f_{r} \partial s}} = \frac{D}{v}\left( {\frac{1 + \lambda (1 + r)}{{2\lambda }}} \right),$$
(93)
$$\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw}^{2} }} = - \frac{D\psi }{{\theta (1 - \lambda )\lambda v}},$$
(94)
$$\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial f_{r} }} = \frac{{\partial^{2} \Pi_{es} }}{{\partial f_{r} \partial f_{cw} }} = - \frac{D}{2\lambda \theta v}(\theta + \psi r),$$
(95)
$$\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{r}^{2} }} = - \frac{Dr}{v}.$$
(96)

When Eqs. (97) to (99) exist, the function is concave:

$$\left| H \right|_{1} < 0 \to \frac{{\partial^{2} \Pi_{es} }}{{\partial s^{2} }} = - \frac{D}{v}\left( {\frac{1 + \lambda }{\lambda }} \right) < 0,$$
(97)
$$\left| H \right|_{2} > 0 \to \frac{{\partial^{2} \Pi_{es} }}{{\partial s^{2} }}\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw}^{2} }} - \frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{cw} }}\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial s}} = \left( {\frac{D}{\lambda v}} \right)^{2} \left( {\frac{\psi (1 + \lambda )}{{\theta (1 - \lambda )}} - \left( {\frac{\theta + \psi }{{2\theta }}} \right)} \right) > 0,$$
(98)
$$\begin{gathered} \left| H \right|_{3} < 0 \to \left[ \begin{gathered} \frac{{\partial^{2} \Pi_{es} }}{{\partial f_{r}^{2} }}\left( {\frac{{\partial^{2} \Pi_{es} }}{{\partial s^{2} }}\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw}^{2} }} - \frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial s}}\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{cw} }}} \right) \hfill \\ - \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cw} \partial f_{r} }}\left( {\frac{{\partial^{2} \Pi_{es} }}{{\partial s^{2} }}\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial f_{r} }} - \frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{r} }}\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{cw} }}} \right) \hfill \\ + \frac{{\partial^{2} \Pi_{es} }}{{\partial f_{r} \partial s}}\left( {\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{cw} }}\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial f_{r} }} - \frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw}^{2} }}\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{r} }}} \right) \hfill \\ \end{gathered} \right] < 0. \hfill \\ \to \frac{{D^{3} }}{{v^{3} \lambda^{2} }}\left[ \begin{gathered} - (1 + \lambda )\left( {\frac{r\psi }{{\theta (1 - \lambda )}}} \right) - \frac{{(\theta + r\psi )^{2} }}{{4\theta^{2} \lambda }} \hfill \\ - \frac{{\partial^{2} \Pi_{ec} }}{{\partial f_{cw} \partial f_{r} }}\left( {\frac{{\partial^{2} \Pi_{es} }}{{\partial s^{2} }}\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial f_{r} }} - \frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{r} }}\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{cw} }}} \right) \hfill \\ + \frac{{\partial^{2} \Pi_{es} }}{{\partial f_{r} \partial s}}\left( {\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{cw} }}\frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw} \partial f_{r} }} - \frac{{\partial^{2} \Pi_{es} }}{{\partial f_{cw}^{2} }}\frac{{\partial^{2} \Pi_{es} }}{{\partial s\partial f_{r} }}} \right) \hfill \\ \end{gathered} \right] \hfill \\ \end{gathered}$$
(99)

With respect to Eq. (100) to (102), the function is indefinite, and the maximum point, in this case obtained by:

$$\frac{{\partial \Pi_{ec} }}{\partial s} = D\left[ {\left( {1 - \frac{{s - f_{r} }}{2v}} \right) + \left( { - \frac{{\psi f_{cw} }}{2v\theta (1 - \lambda )}} \right) - \frac{(s - c)}{{2v}} + \frac{{\psi f_{cw} }}{2\theta (1 - \lambda )\lambda v} - \frac{{(2s - f_{r} - f_{cw} )}}{2\lambda v} + \frac{{f_{r} r}}{2\lambda }} \right] = 0,$$
(100)
$$\frac{{\partial \Pi_{ec} }}{{\partial f_{cw} }} = D\left[ { - \frac{{(s - c)f_{cw} }}{2\theta (1 - \lambda )v} - \left( {\frac{{f_{cw} f_{cw} }}{2\theta \lambda (1 - \lambda )v} - \frac{{s - f_{r} }}{2\lambda v}} \right) + \left( {\frac{{f_{cw} }}{2\theta \lambda (1 - \lambda )v}} \right)(s - f_{cw} ) - \frac{{f_{r} rf_{cw} }}{2\theta \lambda v}} \right] = 0,$$
(101)
$$\frac{{\partial \Pi_{ec} }}{{\partial f_{r} }} = D\left[ {\frac{(s - c)}{{2v}} + (s - f_{cw} )\frac{1}{2\lambda v} - \frac{{f_{r} r}}{2v} - r\left( {1 - \frac{{(s - f_{r} )r}}{2v} - \frac{{\psi rf_{cw} }}{2\theta \lambda v}} \right)} \right] = 0.$$
(102)

By solving Eq. (100) to (102), we have:

$$s^{*} = \left( \begin{gathered} \frac{v\lambda }{{1 + \lambda }}\left[ {\frac{\lambda (\xi - 2r)}{{\lambda \xi^{2} }} + 1} \right] + \frac{c\lambda }{{1 + \lambda }}\left[ {\frac{(1 + \lambda (1 + r))\lambda r}{{\xi^{2} }} + 1} \right] \hfill \\ - f_{cw} \left[ {\theta \left( {2 - \frac{(1 + \lambda (1 + r))}{{\xi^{2} }}\left( {\xi (1 - \theta )\varepsilon - 2\psi r} \right)} \right) - (1 - \theta )\varepsilon } \right] \hfill \\ \end{gathered} \right),$$
(103)
$$f_{cw}^{*} = \frac{{\left( \begin{gathered} v\left( {\left[ {\frac{(2 + \psi )(\xi - 2r)}{{\lambda \xi^{2} (1 + \lambda )}} - \frac{1}{1 + \lambda }} \right] - \left[ {\frac{{2\left( {\theta + r\psi } \right)(2r - \xi )}}{{\xi^{2} }}} \right]} \right) \hfill \\ + c\left( {\left[ {\frac{(2 + \psi )(1 + \lambda (1 + r))\lambda r}{{(1 - \lambda )\xi^{2} }} - \frac{1}{1 - \lambda }} \right] + \left[ {\frac{{\left( {\theta + r\psi } \right)(2r - \xi )}}{{\xi^{2} }}} \right] + \left[ {\frac{\psi }{1 - \lambda }} \right]} \right) \hfill \\ \end{gathered} \right)}}{{\left( {\frac{2\psi }{{1 - \lambda }} + \left( {\theta + r\psi } \right)\left( {\frac{\xi (1 - \theta )\varepsilon - 2r\psi }{{\xi^{2} }}} \right) + \frac{2 + \psi }{{2\lambda (1 + \lambda )\theta }}\left( {2 - \frac{(1 + \lambda (1 + r))}{{\xi^{2} }}\left( {\xi (1 - \theta )\varepsilon - 2\psi r} \right)} \right)} \right)}},$$
(104)
$$f_{r}^{*} = \left( {2v\lambda \left[ {\frac{\xi - 2r}{{\xi^{2} }}} \right] + \frac{{rc\lambda^{2} }}{{\xi^{2} }} - f_{cw} \frac{{\left[ {\xi (1 - \theta )\varepsilon - 2r\psi } \right]}}{{\xi^{2} }}} \right).$$
(105)

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Taleizadeh, A.A., Varzi, A.M., Amjadian, A. et al. How cash-back strategy affect sale rate under refund and customers’ credit. Oper Res Int J 23, 19 (2023). https://doi.org/10.1007/s12351-023-00752-2

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