A periodic review policy for a coordinated single vendor-multiple buyers supply chain with controllable lead time and distribution-free approach

Abstract

In this paper, we study a single vendor-multiple buyer integrated inventory model with controllable lead time and backorders-lost sales mixture. Each buyer adopts a periodic review policy in which the review period is an integer fraction of the production cycle time of the vendor. To reflect the practical circumstance characterized by the lack of complete information about the demand distribution, we assume that only the first two moments of the demand during the protection interval are known. The long-run expected total cost per time unit is derived, which includes stockout costs. The problem is to determine the length of the inventory cycle of the vendor, the produce-up-to level for the vendor, the replenishment policy of each buyer, and the length of lead times that minimize the cost function under the minimax distribution-free approach. Two alternative heuristics are proposed. Numerical experiments have been carried out to investigate the performance of the heuristics and to study the sensitivity of the model.

Appendices

Appendix 1

Point No. 1. If we take the second-order partial derivative of C_n with respect to L_n, with $L_{n} \in \left[ {L_{m,n} ,L_{m - 1,n} } \right]$, we have

$$ \frac{{\partial^{2} }}{{\partial L_{n}^{2} }}C_{n} \left( {T,z_{n} ,k_{n} ,L_{n} } \right) = - \frac{{\sigma_{n} }}{{4\left( {\frac{T}{{k_{n} }} + L_{n} } \right)^{{\frac{3}{2}}} }}\left\{ {h_{n} \left[ {z_{n} + \frac{{\beta_{n} }}{2}\left( {\sqrt {1 + z_{n}^{2} } - z_{n} } \right)} \right] + \frac{{\bar{\pi }_{n} k_{n} }}{2T}\left( {\sqrt {1 + z_{n}^{2} } - z_{n} } \right)} \right\} < 0, $$

which is valid for $m = 0,1, \ldots ,M_{n}$, for each n. This demonstrates that C is concave in L.

Point No. 2. We observe that Annadurai and Uthayakumar (2010) proved that a function structurally identical to C_n is convex in $\left( {T_{n} ,z_{n} } \right)$. Hence, since convexity is invariant under affine maps, we can deduce that C_n is convex in $\left( {T,z_{n} } \right)$, for fixed $\left( {k_{n} ,L_{n} } \right)$. Cleary, this is valid for each n. It can be easily noted that the same argument applies to $C_{V}$ permitting us to deduce that $C_{V}$ is convex in $\left( {T,z_{v} } \right)$. Since the sum of convex functions is itself a convex function, we can conclude that C is convex in $\left( {T,{\mathbf{z}}} \right)$, for fixed $\left( {{\mathbf{k}},{\mathbf{L}}} \right)$.

Finally, the limit property stated in the last point can readily be obtained. Hence, its proof is omitted. This ends the proof.

Appendix 2

$$ \begin{aligned} & \frac{\partial }{\partial T}C\left( {T,{\mathbf{z}},{\mathbf{k}},{\mathbf{L}}} \right) = - \frac{S}{{T^{2} }} + \frac{h}{2}\left[ {D\left( {1 - \frac{D}{P}} \right) + \mathop \sum \limits_{n = 1}^{N} \frac{{D_{n} }}{{k_{n} }}} \right] + \frac{{z_{v} }}{2T}\sqrt {T\mathop \sum \limits_{n = 1}^{N} \sigma_{n}^{2} } \\ & - \frac{{p_{v} }}{4T}\left( {\sqrt {1 + z_{v}^{2} } - z_{v} } \right)\sqrt {T\mathop \sum \limits_{n = 1}^{N} \sigma_{n}^{2} } + \mathop \sum \limits_{n = 1}^{N} \left\{ { - \left( {A_{n} + U_{n} \left( {L_{n} } \right)} \right)\frac{{k_{n} }}{{T^{2} }} + h_{n} } \right. \\ & \left[ {\frac{{D_{2} }}{{2k_{n} }} + \frac{{z_{n} \sigma_{n} }}{{2k_{n} \sqrt {\frac{T}{{k_{n} }} + L_{n} } }} + \frac{{\beta_{n} \sigma_{n} }}{{4k_{n} \sqrt {\frac{T}{{k_{n} }} + L_{n} } }}\left( {\sqrt {1 + z_{n}^{2} } - z_{n} } \right)} \right] \\ & \left. { + \frac{{k_{n} }}{2}\bar{\pi }_{n} \sigma_{n} \left( {\sqrt {1 + z_{n}^{2} } - z_{n} } \right)\left( {\frac{1}{{2k_{n} T\sqrt {\frac{T}{{k_{n} }} + L_{n} } }} - \frac{1}{{T^{2} }}\sqrt {\frac{T}{{k_{n} }} + L_{n} } } \right)} \right\} = 0, \\ \end{aligned} $$

$$ \begin{aligned} & \frac{\partial }{{\partial z_{n} }}C\left( {T,{\mathbf{z}},{\mathbf{k}},{\mathbf{L}}} \right) = h_{n} \left[ {\sigma_{n} \sqrt {\frac{T}{{k_{n} }} + L_{n} } + \frac{1}{2}\beta_{n} \sigma_{n} \sqrt {\frac{T}{{k_{n} }} + L_{n} } \left( {\frac{{z_{n} }}{{\sqrt {1 + z_{n}^{2} } }} - 1} \right)} \right] \\ & \quad + \frac{1}{2}\bar{\pi }_{n} \sigma_{n} \sqrt {\frac{T}{{k_{n} }} + L_{n} } \left( {\frac{{z_{n} }}{{\sqrt {1 + z_{n}^{2} } }} - 1} \right)\frac{{k_{n} }}{T} = 0,\quad n = 1,2, \ldots ,N, \\ \end{aligned} $$

(25)

$$ \frac{\partial }{{\partial z_{v} }}C\left( {T,{\mathbf{z}},{\mathbf{k}},{\mathbf{L}}} \right) = h\sqrt {T\mathop \sum \limits_{n = 1}^{N} \sigma_{n}^{2} } + \frac{1}{2}\frac{{p_{v} }}{T}\sqrt {T\mathop \sum \limits_{n = 1}^{N} \sigma_{n}^{2} } \left( {\frac{{z_{v} }}{{\sqrt {1 + z_{v}^{2} } }} - 1} \right) = 0, $$

(26)

Appendix 3

From the first-order condition for optimality in z_n and z_v (see Eqs. 25, 26) we find

$$ \sqrt {1 + z_{n}^{2} } = z_{n} \frac{{h_{n} \frac{T}{{k_{n} }}\beta_{n} + \bar{\pi }_{n} }}{{h_{n} \frac{T}{{k_{n} }}\left( {\beta_{n} - 2} \right) + \bar{\pi }_{n} }},\quad n = 1,2, \ldots ,N, $$

(27)

and

$$ \sqrt {1 + z_{v}^{2} } = \frac{{p_{v} z_{v} }}{{p_{v} - 2hT}}, $$

(28)

respectively. If we now substitute Eq. (27) into Eq. (12) and Eq. (28) into Eq. (11), we obtain, after some algebraic manipulations,

$$ C_{n} \left( {T,k_{n} ,L_{n} } \right) = \left( {A_{n} + U_{n} \left( {L_{n} } \right)} \right)\frac{{k_{n} }}{T} + h_{n} \frac{{D_{n} }}{2}\frac{T}{{k_{n} }} + 2\bar{z}_{n} \left( {\frac{T}{{k_{n} }}} \right)h_{n} \sigma_{n} \sqrt {\frac{T}{{k_{n} }} + L_{n} } \frac{{h_{n} \frac{T}{{k_{n} }}\left( {\beta_{n} - 1} \right) + \bar{\pi }_{n} }}{{h_{n} \frac{T}{{k_{n} }}\left( {\beta_{n} - 2} \right) + \bar{\pi }_{n} }}, \quad n = 1,2, \ldots ,N, $$

(29)

and

$$ C_{V} \left( {T,{\mathbf{k}}} \right) = \frac{S}{T} + \frac{hT}{2}\left[ {D\left( {1 - \frac{D}{P}} \right) + \mathop \sum \limits_{n = 1}^{N} \frac{{D_{n} }}{{k_{n} }}} \right] + h\bar{z}_{v} \left( T \right)\sqrt {T\mathop \sum \limits_{n = 1}^{N} \sigma_{n}^{2} } \left( {1 + \frac{{p_{v} }}{{p_{v} - 2hT}}} \right), $$

(30)

respectively, where $\bar{z}_{n}$ is the value of z_n, which is a function of $\frac{T}{{k_{n} }}$, that solves Eq. (25), for $n = 1,2, \ldots ,N$, and $\bar{z}_{v}$ is the value of z_v, which is a function of T, that solves Eq. (26). In particular, it can be readily obtained

$$ \bar{z}_{n} \left( {\frac{T}{{k_{n} }}} \right) = \frac{{h_{n} \frac{T}{{k_{n} }}\left( {\beta_{n} - 2} \right) + \bar{\pi }_{n} }}{{2\sqrt {h_{n} \frac{T}{{k_{n} }}\left[ {\bar{\pi }_{n} - h_{n} \frac{T}{{k_{n} }}\left( {1 - \beta_{n} } \right)} \right]} }}, $$

(31)

$$ \bar{z}_{v} \left( T \right) = \frac{{p_{v} - 2hT}}{2}\sqrt {\frac{1}{{hT\left( {p_{v} - hT} \right)}}} . $$

(32)

Evidently, Eq. (31) is valid when $\bar{\pi }_{n} - h_{n} \frac{T}{{k_{n} }}\left( {1 - \beta_{n} } \right) > 0$, while Eq. (32) is valid when $p_{v} - hT > 0$. In practice, both conditions are often verified as the shortage cost is typically greater than the inventory holding cost over the inventory cycle time. If Eq. (32) is inserted into Eq. (30), C_n becomes

$$ C_{V} \left( {T,{\mathbf{k}}} \right) = \frac{S}{T} + \frac{hT}{2}\left[ {D\left( {1 - \frac{D}{P}} \right) + \mathop \sum \limits_{n = 1}^{N} \frac{{D_{n} }}{{k_{n} }}} \right] + \sqrt {h\left( {p_{v} - hT} \right)\mathop \sum \limits_{n = 1}^{N} \sigma_{n}^{2} } , $$

while, if we insert Eq. (31) into Eq. (29), we get

$$ C_{n} \left( {T,k_{n} ,L_{n} } \right) = \left( {A_{n} + U_{n} \left( {L_{n} } \right)} \right)\frac{{k_{n} }}{T} + h_{n} \frac{{D_{n} }}{2}\frac{T}{{k_{n} }} + \sigma_{n} \sqrt {\frac{{h_{n} \left( {\frac{T}{{k_{n} }} + L_{n} } \right)\left( {\bar{\pi }_{n} - h_{n} \frac{T}{{k_{n} }}\left( {1 - \beta_{n} } \right)} \right)}}{{\frac{T}{{k_{n} }}}}} , \quad n = 1,2, \ldots ,N, $$

which are Eqs. (16) and (17), respectively.

Appendix 4

With reference to a neighbourhood of $\hat{T}_{n}$, we can write

$$ \sqrt {\frac{{\left( {\frac{T}{{k_{n} }} + L_{n} } \right)\left( {\bar{\pi }_{n} - h_{n} \frac{T}{{k_{n} }}\left( {1 - \beta_{n} } \right)} \right)}}{{\frac{T}{{k_{n} }}}}} \approx d_{0,n} + d_{1,n} \left( {T_{n} - \hat{T}_{n} } \right) + \frac{1}{2}d_{2,n} \left( {T_{n} - \hat{T}_{n} } \right)^{2} , $$

(33)

where $T_{n} = \frac{T}{{k_{n} }}$ and

$$ d_{0,n} \equiv \sqrt {\frac{{\left( {\hat{T}_{n} + L_{n} } \right)\left( {\bar{\pi }_{n} - h_{n} \hat{T}_{n} \left( {1 - \beta_{n} } \right)} \right)}}{{\hat{T}_{n} }}} , $$

$$ d_{1,n} \equiv \frac{{h_{n} \left( {\beta_{n} - 1} \right)\hat{T}_{n}^{2} - \bar{\pi }_{n} L_{n} }}{{2\hat{T}_{n}^{{\frac{3}{2}}} \sqrt {\left( {\hat{T}_{n} + L_{n} } \right)\left( {\bar{\pi }_{n} - h_{n} \hat{T}_{n} \left( {1 - \beta_{n} } \right)} \right)} }}, $$

$$ d_{2,n} \equiv \frac{{L_{n} \bar{\pi }_{n}^{2} \left( {3L_{n} + 4\hat{T}_{n} } \right) + 2h_{n} \bar{\pi }_{n} L_{n} \hat{T}_{n} \left( {\beta_{n} - 1} \right)\left( {2L_{n} + 3\hat{T}_{n} } \right) - h_{n}^{2} \hat{T}_{n}^{4} \left( {\beta_{n} - 1} \right)^{2} }}{{4\hat{T}_{n}^{{\frac{5}{4}}} \left[ {\left( {\hat{T}_{n} + L_{n} } \right)\left( {\bar{\pi }_{n} - h_{n} \hat{T}_{n} \left( {1 - \beta_{n} } \right)} \right)} \right]^{{\frac{3}{2}}} }}. $$

According to Eq. (33), C_n can be approximated as follows:

$$ C_{n} \left( {T,k_{n} ,L_{n} } \right) \approx \hat{C}_{n} \left( {T,k_{n} ,L_{n} } \right) = \frac{{u_{n} k_{n} }}{T} + v_{n} \frac{T}{{k_{n} }} + w_{n} \left( {\frac{T}{{k_{n} }}} \right)^{2} + y_{n} , \quad n = 1,2, \ldots ,N, $$

(34)

where

$$ u_{n} \equiv A_{n} + U_{n} \left( {L_{n} } \right), $$

$$ v_{n} \equiv \frac{{h_{n} D_{n} }}{2} + \sigma_{n} \sqrt {h_{n} } \left( {d_{1,n} - d_{2,n} \hat{T}_{n} } \right), $$

$$ w_{n} \equiv \frac{1}{2}d_{2,n} \sigma_{n} \sqrt {h_{n} } , $$

$$ y_{n} \equiv \sigma_{n} \sqrt {h_{n} } \left( {d_{0,n} - d_{1,n} \hat{T}_{n} + \frac{1}{2}d_{2.n} \hat{T}_{n}^{2} } \right). $$

If we consider a neighbourhood of $\hat{T}$, we can write

$$ \sqrt {\left( {p_{v} - hT} \right)} \approx d_{0} + d_{1} \left( {T - \hat{T}} \right) + \frac{1}{2}d_{2} \left( {T - \hat{T}} \right)^{2} , $$

(35)

where

$$ d_{0} \equiv \sqrt {\left( {p_{v} - h\hat{T}} \right)} , $$

$$ d_{1} \equiv - \frac{h}{{2\sqrt {\left( {p_{v} - h\hat{T}} \right)} }}, $$

$$ d_{2} \equiv - \frac{{h^{2} }}{{4\left( {p_{v} - h\hat{T}} \right)^{{\frac{3}{2}}} }}, $$

Now, we can use Eq. (35) to approximate $C_{V}$ as follows:

$$ C_{V} \left( {T,{\mathbf{k}}} \right) \approx \hat{C}_{V} \left( {T,{\mathbf{k}}} \right) = \frac{{u_{0} }}{T} + \left( {v_{0} + \frac{h}{2}\mathop \sum \limits_{n = 1}^{N} \frac{{D_{n} }}{{k_{n} }}} \right)T + w_{0} T^{2} + y_{0} , $$

(36)

where

$$ u_{0} \equiv S, $$

$$ v_{0} \equiv \frac{Dh}{2}\left( {1 - \frac{D}{P}} \right) + \sqrt {h\mathop \sum \limits_{n = 1}^{N} \sigma_{n}^{2} } \left( {d_{1} - d_{2} \hat{T}} \right), $$

$$ w_{0} \equiv \frac{1}{2}d_{2} \sqrt {h\mathop \sum \limits_{n = 1}^{N} \sigma_{n}^{2} } , $$

$$ y_{0} \equiv \sqrt {h\mathop \sum \limits_{n = 1}^{N} \sigma_{n}^{2} } \left( {d_{0} - d_{1} \hat{T} + \frac{1}{2}d_{2} \hat{T}^{2} } \right). $$

Finally, Eqs. (34) and (36) permits us to consider the following approximation for cost function C:

$$ \begin{aligned} & \hat{C}\left( {T,{\mathbf{k}},{\mathbf{L}}} \right) = \hat{C}_{V} \left( {T,{\mathbf{k}}} \right) + \mathop \sum \limits_{n = 1}^{N} \hat{C}_{n} \left( {T,k_{n} ,L_{n} } \right) = \frac{{u_{0} }}{T} + \left( {v_{0} + \frac{h}{2}\mathop \sum \limits_{n = 1}^{N} \frac{{D_{n} }}{{k_{n} }}} \right)T \\ & \quad + w_{0} T^{2} + y_{0} + \mathop \sum \limits_{n = 1}^{N} \left[ {\frac{{u_{n} k_{n} }}{T} + v_{n} \frac{T}{{k_{n} }} + w_{n} \left( {\frac{T}{{k_{n} }}} \right)^{2} + y_{n} } \right]. \\ \end{aligned} $$