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The expected order cost for the cycle is given by
$$E({\text{OC}}) = \mathop \sum \limits_{r = 1}^{n} C_{or}$$
(1)
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The expected varying holding cost for the cycle is given by
$$E\left( {{\text{HC}}\left( {N_{r} } \right)} \right) = \mathop \sum \limits_{r = 1}^{n} C_{hr} \left( {N_{r} } \right){ }\overline{I}_{r} = \mathop \sum \limits_{r = 1}^{n} C_{hr} { }N_{r}^{1 - \beta } \left( {Q_{mr} - \frac{{{\overline{D}}_{r} { }N_{r} }}{2} + \left( {1 - {\upgamma }_{r} } \right)\mathop \smallint \limits_{{Q_{mr} }}^{\infty } \left( {x_{r} - Q_{mr} } \right){ }f\left( {x_{r} } \right){\text{d}}x_{r} } \right)$$
(2)
where the expected average amount in inventory is given by
$$\overline{I}_{r} = N_{r} \left( {Q_{mr} - \frac{{{\overline{D}}_{r} { }N_{r} }}{2} + \left( {1 - {\upgamma }_{r} } \right)\mathop \smallint \limits_{{Q_{mr} }}^{\infty } \left( {x_{r} - Q_{mr} } \right){ }f\left( {x_{r} } \right)dx_{r} } \right).$$
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The expected varying backorder cost for the cycle is given by
$$E({\text{BC(}}N_{r} {)}) = \mathop \sum \limits_{r = 1}^{n} C_{br} \;\gamma_{r} \;N_{r}^{\beta } \overline{S}(Q_{mr} ) = \mathop \sum \limits_{r = 1}^{n} C_{br} \;\gamma_{r} \;N_{r}^{\beta } \mathop \smallint \limits_{{Q_{mr} }}^{\infty } (x_{r} - Q_{mr} )\;f(x_{r} )\;{\text{d}}x_{r}$$
(3)
where \(\overline{S}\left( {Q_{mr} } \right)\) represents the expected shortage quantity.
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The expected varying lost sales cost for the cycle is given by
$${ }E\left( {{\text{LC}}} \right) = \mathop \sum \limits_{r = 1}^{n} C_{Lr} \left( {1 - \gamma_{r} } \right) N_{r}^{\beta } \overline{S}\left( {Q_{mr} } \right) = \mathop \sum \limits_{r = 1}^{n} C_{Lr} \left( {1 - \gamma_{r} } \right)N_{r}^{\beta } \mathop \smallint \limits_{{Q_{mr} }}^{\infty } \left( {x_{r} - Q_{mr} } \right){ }f\left( {x_{r} } \right){\text{d}}x_{r} { }$$
(4)
And the expected varying refunding quantity cost for the cycle is given by
$$\begin{aligned} E\left( {{\text{RQC}}} \right) & = \mathop \sum \limits_{r = 1}^{n} C_{{{\text{RQ}}r}} \;N_{r}^{ - \beta } \mathop \smallint \limits_{0}^{{Q_{mr} }} x_{r} { }f\left( {x_{r} } \right)\;{\text{d}}x_{r} \\ & = \mathop \sum \limits_{r = 1}^{n} C_{hr} \;\rho_{r} \;N_{r}^{ - \beta } \mathop \smallint \limits_{0}^{{Q_{mr} }} x_{r} { }f\left( {x_{r} } \right)\;{\text{d}}x_{r} ,\quad 0 < \rho_{r} < 1 \\ \end{aligned}$$
(5)
The expected annual total cost will be the sum of the expected order cost, the expected varying holding cost, the expected varying backorder cost, the expected varying lost sales cost, and the expected varying refunding quantity cost
$$E\left( {{\text{TC}}\left( {Q_{mr} , N_{r} } \right)} \right) = \mathop \sum \limits_{r = 1}^{n} \left[ {E\left( {{\text{OC}}_{r} } \right) + E\left( {{\text{HC}}_{r} \left( {N_{r} } \right)} \right) + E\left( {{\text{BC}}_{r} \left( {N_{r} } \right)} \right) + E\left( {{\text{LC}}_{r} \left( {N_{r} } \right)} \right) + E\left( {{\text{RQC}}_{r} \left( {N_{r} } \right)} \right)} \right]$$
Then from Eqs. (1), (2), (3), (4) and (5), the expected annual total cost is given by
$$\begin{aligned} E\left( {{\text{TC}}\left( {Q_{mr} {,}N_{r} } \right)} \right) & = { }\mathop \sum \limits_{r = 1}^{n} \left[ {C_{or} + C_{hr} { }N_{r}^{1 - \beta } \left( {Q_{mr} - \frac{{{\overline{D}}_{r} { }N_{r} }}{2}} \right) + C_{hr} { }\rho_{r} { }N_{r}^{ - \beta } \mathop \smallint \limits_{0}^{{Q_{mr} }} x_{r} { }f\left( {x_{r} } \right){\text{d}}x_{r} } \right. \\ & \quad + \left( {C_{br} {\upgamma }_{r} { }N_{r}^{\beta } + \left( {1 - {\upgamma }_{r} } \right){ }\left( {C_{Lr} { }N_{r}^{\beta } + C_{hr} { }N_{r}^{1 - \beta } } \right)} \right)\mathop \smallint \limits_{{Q_{mr} }}^{\infty } \left( {x_{r} - Q_{mr} } \right){ }f\left( {x_{r} } \right){\text{d}}x_{r} \\ \end{aligned}$$
(6)
Note: Obviously, the expected order cost \(\left( {\sum\nolimits_{r = 1}^{n} {C_{or} } } \right)\) is fixed, so it can be temporarily neglected in calculating the minimum expected annual total cost and eventually added to it.
Now, the main objective is to determine the optimal values \({Q}_{mr}^{*}\) and \({N}_{r}^{*}\) that minimize the expected annual total cost \(\mathrm{min }E\left(\mathrm{TC}\right)\). This paper puts a constraint on varying lost sales cost. The Karush–Kuhn–Tucker (KKT) conditions (Kuhn and Tucker [17]) are first-order necessary conditions for a solution of nonlinear programming to be optimal if some regularity conditions are satisfied. The Lagrange multiplier method is suitable to solve this constraint problem.
Consider a limitation on the expected varying lost sales cost, i.e.
$$\mathop \sum \limits_{r = 1}^{n} C_{Lr} { }\left( {1 - {\upgamma }_{r} } \right){ }N_{r}^{\beta } \mathop \smallint \limits_{{Q_{mr} }}^{\infty } \left( {x_{r} - Q_{mr} } \right){ }f\left( {x_{r} } \right)\;{\text{d}}x_{r} \le K_{Lr}$$
(7)
To solve this primal function which is a convex programming problem, Eqs. (6) and (7) can be written in the following form
$$\begin{aligned} & \min E\left( {{\text{TC}}\left( {Q_{mr} {, }N_{r} } \right)} \right) = \mathop \sum \limits_{r = 1}^{n} \left[ {C_{hr} { }N_{r}^{1 - \beta } \left( {Q_{mr} - \frac{{{\overline{D}}_{r} { }N_{r} }}{2}} \right) + C_{hr} { }\rho_{r} { }N_{r}^{ - \beta } \mathop \smallint \limits_{0}^{{Q_{mr} }} x_{r} { }f\left( {x_{r} } \right)\;{\text{d}}x_{r} } \right. \\ & \quad \left. { + \left( {C_{br} { }\gamma_{r} N_{r}^{\beta } + \left( {1 - \gamma_{r} } \right)\left( {C_{Lr} N_{r}^{\beta } + C_{hr} N_{r}^{1 - \beta } } \right)} \right)\mathop \smallint \limits_{{Q_{mr} }}^{\infty } \left( {x_{r} - Q_{mr} } \right){ }f\left( {x_{r} } \right)\;{\text{d}}x_{r} } \right] \\ \end{aligned}$$
(8)
\({\text{Subject to:}}\)
$$\mathop \sum \limits_{r = 1}^{n} C_{Lr} { }\left( {1 - \gamma_{r} } \right) N_{r}^{\beta } \mathop \smallint \limits_{{Q_{mr} }}^{\infty } \left( {x_{r} - Q_{mr} } \right){ }f\left( {x_{r} } \right)\;{\text{d}}x_{r} \le K_{Lr}$$
(9)
To find optimal values \(Q_{mr}^{*} { }\) and \({ }N_{r}^{*}\) which minimize Eq. (8) under the constraint (Eq. (9)), the Lagrange multipliers function with the Kuhn-Tucker conditions is given by
$$\begin{aligned} L\left( {Q_{mr} {, }N_{r} ,{ }\lambda_{Lr} } \right) & = \sum\limits_{r = 1}^{n} {\left[ {C_{hr} { }N_{r}^{1 - \beta } \left( {Q_{mr} - \frac{{{\overline{D}}_{r} { }N_{r} }}{2}} \right) + C_{hr} { }\rho_{r} { }N_{r}^{ - \beta } \mathop \smallint \limits_{0}^{{Q_{mr} }} x_{r} { }f\left( {x_{r} } \right)\;{\text{d}}x_{r} } \right.} \\ & \quad + \left( {C_{br} \gamma_{r} N_{r}^{\beta } + \left( {1 - \gamma_{r} } \right) \left( {C_{Lr} N_{r}^{\beta } + C_{hr } N_{r}^{1 - \beta } } \right)} \right)\mathop \smallint \limits_{{Q_{mr} }}^{\infty } \left( {x_{r} - Q_{mr} } \right) f\left( {x_{r} } \right)\;{\text{d}}x_{r} \\ & \quad \left. { + \lambda_{Lr} \left( {{ }C_{Lr} \left( {1 - \gamma_{r} } \right) N_{r}^{\beta } \mathop \smallint \limits_{{Q_{mr} }}^{\infty } \left( {x_{r} - Q_{mr} } \right){ }f\left( {x_{r} } \right)\;{\text{d}}x_{r} - K_{Lr} } \right)} \right] \\ \end{aligned}$$
(10)
where \({ }\lambda_{Lr}\) is a Lagrange multiplier.
The optimal values \(Q_{mr}^{*}\) and \(N_{r}^{*}\) can be calculated by setting the corresponding first partial derivatives of Eq. (10) equal to zero. Then we obtain:
$$\frac{{\partial L\left( {Q_{mr} {, }N_{r} } \right)}}{{\partial Q_{mr} }} = 0{ },{ }\frac{{\partial L\left( {Q_{mr} {, }N_{r} } \right)}}{{\partial N_{r} }} = 0,{ }\frac{{\partial L\left( {Q_{mr} {, }N} \right)}}{{\partial \lambda_{Lr} }} = 0$$
$$\begin{aligned} & \mathop \smallint \limits_{{Q_{mr}^{*} }}^{\infty } { }f\left( {x_{r} } \right){\text{d}}x_{r} \\ & \quad = \frac{{C_{hr} N_{r}^{* - \beta } \left( {N_{r}^{*} + \rho_{r} Q_{mr}^{*} f\left( {Q_{mr}^{*} } \right)} \right)}}{{\left( {C_{br} \gamma_{r} N_{r}^{*\beta } + \left( {1 - \gamma_{r} } \right)\left( { C_{Lr} \left( {1 + \lambda_{Lr} } \right) N_{r}^{*\beta } + C_{hr} N_{r}^{*1 - \beta } } \right)} \right)}}, \\ \end{aligned}$$
(11)
$$\begin{aligned} & \mathop \smallint \limits_{{Q_{mr}^{*} }}^{\infty } \left( {x_{r} - Q_{mr}^{*} } \right){ }f\left( {x_{r} } \right){\text{d}}x_{r} \\ & \quad = \frac{{C_{hr} \beta \rho_{r} N_{r}^{* - (1 + \beta )} \mathop \smallint \nolimits_{0}^{{Q_{mr}^{*} }} x_{r} f\left( {x_{r} } \right){\text{d}}x_{r} + \frac{1}{2}C_{hr} {\overline{D}}_{r} N_{r}^{*1 - \beta } - C_{hr} \left( {1 - \beta } \right) N_{r}^{* - \beta } \left( {Q_{mr}^{*} - \frac{{{\overline{D}}_{r} N_{r}^{*} }}{2}} \right)}}{{C_{br} \gamma_{r} \beta N_{r}^{*\beta - 1} + C_{Lr} \beta \left( {1 - \gamma_{r} } \right) \left( {1 + \lambda_{Lr} } \right)N_{r}^{*\beta - 1} + C_{hr} \left( {1 - \gamma_{r} } \right)\left( {1 - \beta } \right)N_{r}^{* - \beta } }} \\ \end{aligned}$$
(12)
and
$$C_{Lr} (1 - \gamma_{r} ) N_{r}^{*\beta } \mathop \smallint \limits_{{Q_{mr}^{*} }}^{\infty } \left( {x_{r} - Q_{mr}^{*} } \right) f\left( {x_{r} } \right)\;{\text{d}}x_{r} = K_{Lr}$$
(13)
It can be determined the minimum expected annual total cost (min E(TC)), after finding the optimal values \(Q_{mr}^{*}\) and \({ }N_{r}^{*}\), substituting these in Eq. (8), then adding the fixed value \(\left( {\sum\nolimits_{r = 1}^{n} {C_{or} } } \right)\).