Optimal Quantity Discount Strategy for an Inventory Model with Deteriorating Items

Abstract

This study discusses quantity discount pricing strategies in a channel of one seller (wholesaler) and one buyer (retailer). The seller purchases products from an upper-leveled supplier (manufacturer) and then sells them to the buyer who faces her/his customers’ demand. The seller attempts to increase her/his profit by controlling the buyer’s order quantity through a quantity discount strategy. The buyer tries to maximize her/his profit considering the seller’s proposal. We formulate the above problem for deteriorating items as a Stackelberg game between the seller and buyer to analyze the existence of the seller’s optimal quantity discount pricing policy, which maximizes her/his total profit per unit of time. Numerical examples are presented to illustrate the theoretical underpinnings of the proposed formulation.

Appendix A

In this appendix, we discuss the existence of the optimal quantity discount pricing policy, which attains ${\hat{P}_2}({N_2})$ in (31.19) when N ₂ is fixed to a suitable value.

• 1. ${N_2} = 1:$

  By differentiating P ₂(N ₂, T ₂) in (31.20) with respect to T ₂, we have

  $$ \frac{\partial }{{\partial {T_2}}}{P_2}({N_2},{T_2}) = - \frac{{[\rho {\theta_{\rm{b}}}\,{{\text{e}}^{{{\theta_{\rm{b}}}{T_2}}}} - Q({T_2})]({c_{\rm{s}}} + {h_{\rm{b}}}/{\theta_{\rm{b}}} - \alpha ) - ({a_{\rm{b}}} + {a_{\rm{s}}} + \beta )}}{{T_2^2}}. $$

  (31.31)

  It can easily be shown from (31.31) that the sign of $\partial {P_2}({N_2},{T_2})/\partial {T_2}$ is positive when (c _s + h _b/θ _b − α) = 0. In contrast, in the case of (c _s + h _b/θ _b − α) ${ > {<} \\}<!endgathered>$ 0, $\partial {P_2}({N_2},{T_2})/\partial {T_2} \geq 0$ agrees with

  $$ {\theta_{\rm{b}}}{T_2}\,{{{e}}^{{{\theta_{\rm{b}}}{T_2}}}} -({\theta_{\rm{b}}}\,{{{e}}^{{{\theta_{\rm{b}}}{T_2}}}} -1)\begin{array}{lllllllll} < \\= \\>\\\end{array} \frac{{{a_{\rm{b}}} + {a_{\rm{s}}} + \beta }}{{\rho({c_{\rm{s}}} + {h_{\rm{b}}}/{\theta_{\rm{b}}} - \alpha )}} .$$

  (31.32)

  Let L ₁(T ₂) express the left-hand side of Inequality (31.32), we have

  $$ {L^{\prime}_1}({T_2}) = \theta_{\rm{b}}^2{T_2}\,{{{e}}^{{{\theta_{\rm{b}}}{T_2}}}}\quad ( > 0), $$

  (31.33)
  $$ {L_1}(T_1^{*}) = \frac{{{a_{\rm{b}}}}}{{\rho ({p_{\rm{s}}} + {h_{\rm{b}}}/{\theta_{\rm{b}}})}}\quad ( > 0), $$

  (31.34)
  $$ \mathop{{\lim }}\limits_{{{T_2} \to + \infty }} {L_1}({T_2}) = + \infty . $$

  (31.35)

  From (31.33)–(31.35), the existence of an optimal quantity discount pricing policy can be discussed for the following two subcases:

  • $({c_{\rm{s}}} + {h_{\rm{b}}}/{\theta_{\rm{b}}} - \alpha ) > 0:$

    Equation (31.34) yields

    $$ {L_1}(T_1^{*}) < \frac{{{a_{\rm{b}}} + {a_{\rm{s}}} + \beta }}{{\rho ({c_{\rm{s}}} + {h_{\rm{b}}}/{\theta_{\rm{b}}} - \alpha )}}\quad ( > 0). $$

    (31.36)

    Equations (31.33), (31.35), and (31.36) indicate that the sign of $\partial {P_2}({N_2},{T_2})/\partial {T_2}$ changes from positive to negative only once. This signifies that P ₂(N ₂, T ₂) first increases and then decreases as T ₂ increases, and thus there exists a unique finite ${\tilde{T}_2}$ ($> T_1^{*}$), which maximizes P ₂(N ₂, T ₂) in (31.20). Hence, $(T_2^{*},{y^{*}})$is given by (31.24).

  • $({c_{\rm{s}}} + {h_{\rm{b}}}/{\theta_{\rm{b}}} - \alpha ) \leq 0:$

    In this subcase, we have

    $$ {L_1}(T_1^{*}) > \frac{{{a_{\rm{b}}} + {a_{\rm{s}}} + \beta }}{{\rho ({c_{\rm{s}}} + {h_{\rm{b}}}/{\theta_{\rm{b}}} - \alpha )}}\quad ( < 0). $$

    (31.37)

    Equations (31.33), (31.35), and (31.37) signify that the sign of $\partial {P_2}({N_2},{T_2})/\partial {T_2}$ is positive, and consequently the optimal policy can be expressed by (31.26).

• 2. ${N_2} \geq 2:$

  By differentiating P ₂(N ₂, T ₂) in (31.20) with respect to T ₂, we have

  $$ \frac{\partial }{{\partial {T_2}}}{P_2}({N_2},{T_2}) = - \frac{{L({T_2}) - ({a_b} + \beta ){N_2} - {a_s}}}{{{N_2}T_2^2}}. $$

  (31.38)

  Then $\partial {P_2}({N_2},{T_2})/\partial {T_2} \geq 0$ agrees with

  $$ L({N_2},{T_2}) \leq ({a_{\rm{b}}} + \beta ){N_2} + {a_{\rm{s}}}. $$

  (31.39)

  If we assume that there exists a unique solution to (31.28), the optimal quantity discount pricing policy can be given by (31.24).

  In the special case where θ _s = θ _b = θ and $\alpha \leq ({h_{\rm{b}}} - {h_{\rm{s}}})/\theta$, by differentiating P ₂(N ₂, T ₂) in (31.20) with respect to T ₂, we have

  $$ \begin{array}{lllllllll} {\frac{\partial }{{\partial {T_2}}}{P_2}({N_2},{T_2}) = - \frac{1}{{{N_2}T_2^2}}\left\{ {C\rho \left[ {({N_2}\theta {T_2} - 1){{{e}}^{{{N_2}\theta {T_2}}}} + 1} \right]} \right. - ({a_b} + \beta ){N_2}} \\{\left. { - {a_{\rm{s}}} + H{N_2}\rho \left[ {(\theta {T_2} - 1)\,{{{e}}^{{\theta {T_2}}}} + 1} \right]} \right\}}. \\\end{array} $$

  (31.40)

  Then $\partial {P_2}({N_2},{T_2})/\partial {T_2} \geq 0$ agrees with

  $$ C\left[ {({N_2}\theta {T_2} - 1){{{e}}^{{{N_2}\theta {T_2}}}} + 1} \right] + H{N_2}\left[ {(\theta {T_2} - 1)\,{{{e}}^{{\theta {T_2}}}} + 1} \right] \leq \frac{{({a_{\rm{b}}} + \beta ){N_2} + {a_{\rm{s}}}}}{\rho }. $$

  (31.41)

  Let us denote, by L _a(T ₂), the left-hand side of Inequality (31.41), and we have

  $$ {L^{\prime}_{\rm{a}}}({T_2}) = {\theta^2}{N_2}{T_2}\left( {C{N_2}{{{e}}^{{{N_2}{\theta}{T_2}}}} + H{{{e}}^{{\theta {T_2}}}}} \right)\quad ( > 0), $$

  (31.42)
  $$ \begin{array}{lllllllll} {{L_{\rm{a}}}(T_1^{*}) = C\left[ {({N_2}\theta T_1^{*} - 1){{{e}}^{{{N_2}\theta T_1^{*}}}} + 1} \right]} \\{ + H{N_2}\left[ {(\theta T_1^{*} - 1){{{e}}^{{{\theta}T_1^{*}}}} + 1} \right]}, \\\end{array} $$

  (31.43)
  $$ \mathop{{\lim }}\limits_{{{T_2} \to + \infty }} {L_{\rm{a}}}({T_2}) = + \infty . $$

  (31.44)

  On the basis of the above results, we show below that an optimal quantity discount pricing strategy exits.

  • $C[({N_2}\theta T_1^{*} - 1){{{e}}^{{{N_2}\theta T_1^{*}}}} + 1] + H{N_2}[(\theta T_1^{*} - 1){{{e}}^{{\theta T_1^{*}}}} + 1] < [({a_{\rm{b}}} <$> <$>+ \beta ){N_2} + {a_{\rm{s}}}]/\rho :$

    In this subcase, the sign of $\partial {P_2}({N_2},{T_2})/\partial {T_2}$ varies from positive to negative only once, and consequently there exists a unique finite ${\tilde{T}_2}$ ($> T_1^{*}$), which maximizes P ₂(N ₂, T ₂) in (31.20). Hence, $(T_2^{*},{y^{*}})$is given by (31.24).

  • $C[({N_2}\theta T_1^{*} - 1){{{e}}^{{{N_2}\theta T_1^{*}}}} + 1] + H{N_2}[(\theta T_1^{*} - 1){{{e}}^{{\theta T_1^{*}}}} + 1] \geq [({a_{\rm{b}}} + \beta ){N_2} + {a_{\rm{s}}}]/\rho :$

    This subcase provides $\partial {P_2}({N_2},{T_2})/\partial {T_2} \leq 0$, and therefore the optimal policy can be expressed by (31.30).