Pre-positioning of relief inventories for non-profit organizations: a newsvendor approach

Abstract

Developing an efficient policy to determine the inventory a non-profit organization (NPO) should stockpile to respond to potential disasters plays a vital role in humanitarian relief. Incorporating social donation and emergency spot purchasing, this paper develops a (generalized) two-stage delivery process model to characterize relief materials’ delivery after a disaster. We propose an analytical model that aims to minimize the total costs when all demands incurred by an unexpected disaster need to be fully satisfied. Moreover, we determine that the optimal solution uniquely exists and characterize the effects of key parameters. Last, this paper also develops a risk-sharing scheme, in which the NPO and the supplier jointly stockpile relief materials. Under some mild conditions, we show that implementing the scheme could increase the storage quantity and decrease the total costs simultaneously. Several numerical examples are employed to validate the model, as well as the value of the proposed risk-sharing scheme.

Appendix 1: Mathematical proofs

Proof of Theorem 1

We first present the proof of the convexity of $TC\left( Q \right)$. Differentiating the expected total costs with respect to Q, we have the following

$$\begin{aligned} \frac{\partial TC(Q)}{\partial Q}= & {} c-v+\sum _{i=1}^n {[\pi _i (t_i +v-p_i )\Pr (\xi _i>Q)]} \nonumber \\&{-\sum _{e_i \in \mathbf{e}_{\mathbf{d}} } {[\pi _i (c^{d}+t_i^d )\Pr (\xi _i -Q<D_i ,\xi _i>Q)]} } \nonumber \\&{-\sum _{e_i \in \mathbf{e}_{\mathbf{d}} } {[\pi _i (c^{s}+t_i^s )\Pr (\xi _i -Q>D_i ,\xi _i>Q)]} } \nonumber \\&{-\sum _{e_i \in \mathbf{e}_{\bar{\mathbf{d}}} } {[\pi _i (c^{s}+t_i^s )\Pr (\xi _i >Q)]} }, \end{aligned}$$

(15)

$$\begin{aligned} \frac{\partial ^{2}TC(Q)}{\partial Q^{2}}= & {} \sum _{i=1}^n {[\pi _i (p_i -t_i -v)f_{\xi _i } (Q)]} +\sum _{e_i \in \mathbf{e}_{\bar{\mathbf{d}}} } {[\pi _i (c^{s}+t_i^s )f_{\xi _i } (Q)]} \nonumber \\&{+\sum _{e_i \in \mathbf{e}_{\mathbf{d}} } {[\pi _i (c^{s}+t_i^s -c^{d}-t_i^d )\Pr (\xi _i>Q)f_{\xi _i -D_i |\xi _i>Q} (Q)]} } \nonumber \\&{+\sum _{e_i \in \mathbf{e}_{\mathbf{d}} } {[\pi _i (c^{d}+t_i^d )\Pr (\xi _i -D_i <Q|\xi _i>Q)f_{\xi _i } (Q)]} } \nonumber \\&{+\sum _{e_i \in \mathbf{e}_{\mathbf{d}} } {[\pi _i (c^{s}+t_i^s )\Pr (\xi _i -D_i>Q|\xi _i >Q)f_{\xi _i } (Q)]} } \nonumber \\&-\, \sum _{e_i \in \mathbf{e}_{\mathbf{d}} } {[\pi _i (c^{d}+t_i^d )f_{\xi _i } (Q)]} . \end{aligned}$$

(16)

Rearranging condition (1) yields $\mathop \sum \nolimits _{i=1}^n \pi _i \left( {p_i -t_i -v} \right)>c-v>0$. Thus, the first term of the RHS of Eq. (16) is positive. Recall in the subset $e_{\mathbf{d}}$, the relation $c^{d}+t_i^d \le c^{s}+t_i^s$ holds. Together with the conditions showed in Eq. (2), we have the following:

$$\begin{aligned}&\sum _{e_i \in \mathbf{e}_{\mathbf{d}} } {[\pi _i (c^{d}+t_i^d )\Pr (\xi _i -D_i <Q|\xi _i>Q)f_{\xi _i } (Q)]} \nonumber \\&\quad {+\sum _{e_i \in \mathbf{e}_{\mathbf{d}} } {[\pi _i (c^{s}+t_i^s )\Pr (\xi _i -D_i>Q|\xi _i >Q)f_{\xi _i } (Q)]} } \nonumber \\&\quad -\sum _{e_i \in \mathbf{e}_{\mathbf{d}} } {[\pi _i (c^{d}+t_i^d )f_{\xi _i } (Q)]} \nonumber \\&\quad {\ge \sum _{e_i \in \mathbf{e}_{\mathbf{d}} } {[\pi _i (c^{d}+t_i^d )f_{\xi _i } (Q)]} -} \sum _{e_i \in \mathbf{e}_{\mathbf{d}} } {[\pi _i (c^{d}+t_i^d )f_{\xi _i } (Q)]} =0. \end{aligned}$$

(17)

Substituting Eqs. (17) into (16), we have the result $\partial ^{2}TC\left( Q \right) /\partial Q^{2}>0$, which means that $TC\left( Q \right)$ is strictly convex in Q. Thus, we can employ the first order condition (FOC) to determine the optimal pre-positioning inventory, as Eq. (3) shows. Moreover, from Eq. (15), we have $\mathop {\lim }\nolimits _{Q\rightarrow 0} \partial TC\left( Q \right) /\partial Q<0$ and $\mathop {\lim }\nolimits _{Q\rightarrow +\infty } \partial TC\left( Q \right) /\partial Q>0$. Together with the property that $\partial TC\left( Q \right) /\partial Q$ is increasing in Q, we determine that the optimal solution showed in Eq. (3) uniquely exist. $\square$

Proof of Proposition  1

Employing the implicit function theorem, we differentiate Eq. (5) w.r.t L and derive

$$\begin{aligned}&\sum _{i=1}^n {[\pi _i C_i f_{\xi _i } (Q^{*})]} \frac{\partial Q^{*}}{\partial L}-\sum _{e_i \in \mathbf{e}_{\mathbf{d}}} {[\pi _i (C_i^s -C_i^d )f_{\xi _i -D_i |\xi _i>Q^{*}} (Q^{*})]} \frac{\partial Q^{*}}{\partial L} \nonumber \\&\quad {-\sum _{e_i \in \mathbf{e}_{\mathbf{d}} } {[\pi _i C_i^d \Pr (\xi _i -Q^{*}<D_i |\xi _i>Q^{*})f_{\xi _i } (Q^{*})]} \frac{\partial Q^{*}}{\partial L}} \nonumber \\&\quad {-\sum _{e_i \in \mathbf{e}_{\mathbf{d}} } {[\pi _i C_i^s \Pr (\xi _i -Q^{*}>D_i |\xi _i >Q^{*})f_{\xi _i } (Q^{*})]} \frac{\partial Q^{*}}{\partial L}} \nonumber \\&\quad + \sum _{e_i \in \mathbf{e}_{\bar{\mathbf{d}}} } {[\pi _i C_i^d f_{\xi _i } (Q^{*})]} \frac{\partial Q^{*}}{\partial L}-\sum _{e_i \in \mathbf{e}_{\bar{\mathbf{d}}} } {[\pi _i C_i^s f_{\xi _i } (Q^{*})]} \frac{\partial Q^{*}}{\partial L}=1. \end{aligned}$$

(18)

Note that $\mathop \sum \nolimits _{i=1}^n \left[ {\pi _i C_i f_{\xi _i } \left( {Q^{*}} \right) } \right] =\mathop \sum \nolimits _{e_i \in \mathbf{e}_{\mathbf{d}} } \left[ {\pi _i C_i f_{\xi _i } \left( {Q^{*}} \right) } \right] +\mathop \sum \nolimits _{e_i \in \mathbf{e}_{\bar{\mathbf{d}}} } \left[ {\pi _i C_i f_{\xi _i } \left( {Q^{*}} \right) } \right]$. Using the relations established in Eqs. (3) and (17), we know that the coefficient of $\partial Q^{*}/\partial L$ on the LHS of Eq. (18) is negative, which means that $\partial Q^{*}/\partial L\le 0$ is established. Similarly, we can derive the results $\partial Q^{*}/\partial C_i \le 0$, $\partial Q^{*}/\partial C_i^d \ge 0$, and $\partial Q^{*}/\partial C_i^s \ge 0$. Moreover, for the effects of parameter L on the expected total cost, we using the following equation:

$$\begin{aligned} \frac{dTC(Q^{*})}{dL}=\frac{\partial TC(Q^{*})}{\partial L}+\frac{\partial TC(Q^{*})}{\partial Q^{*}}\frac{dQ^{*}}{dL}. \end{aligned}$$

(19)

Recognizing $\partial TC\left( {Q^{*}} \right) /\partial Q^{*}=0$, we have $\hbox {d}TC\left( {Q^{*}} \right) /\hbox {d}L\ge 0$. Using the similar methods, we have the other relations in Proposition 1. $\square$

Proof of Proposition 2

Consider the case of $\mathbf{e}_{\bar{\mathbf{d}}} =\emptyset$, the relation $C^{s}\ge C^{d}$ can be established. Hence, for a fixed denotation quantity D, we can reformulate Eq. (5) as

$$\begin{aligned}&-\pi C\bar{{F}}_{\xi _1 } (Q^{*})-(1-\pi -\bar{\pi })C\bar{{F}}_{\xi _2 } (Q^{*})+\pi C^{d}[F_{\xi _1 } (D+Q^{*})-F_{\xi _1 } (Q^{*})]\nonumber \\&\quad +\,{(1-\pi -\bar{\pi })C^{d}[F_{\xi _2 } (D+Q^{*})-F_{\xi _2 } (Q^{*})]} +\pi C^{s}\bar{{F}}_{\xi _1 } (D+Q^{*})\nonumber \\&\quad + \,(1-\pi -\bar{\pi })C^{s}\bar{{F}}_{\xi _2 } (D+Q^{*})=L. \end{aligned}$$

(20)

Differentiating Eq. (20) w.r.t $\pi$, we have

$$\begin{aligned} \frac{\partial Q^{*}}{\partial \pi }=\frac{\left( {C-C^{d}} \right) \left[ {\bar{{F}}_{\xi _2 } \left( {Q^{*}} \right) -\bar{{F}}_{\xi _1 } \left( {Q^{*}} \right) } \right] -\left( {C^{s}-C^{d}} \right) \left[ {\bar{{F}}_{\xi _2 } \left( {Q^{*}+D} \right) -\bar{{F}}_{\xi _1 } \left( {D+Q^{*}} \right) } \right] }{\left( {{\begin{array}{c} {\pi \left( {C^{d}-C} \right) f_{\xi _1 } \left( {Q^{*}} \right) +\left( {1-\pi -\bar{\pi }} \right) \left( {C^{d}-C} \right) f_{\xi _2 } \left( {Q^{*}} \right) } \\ {+\pi \left( {C^{s}-C^{d}} \right) f_{\xi _1 } \left( {D+Q^{*}} \right) +\left( {1-\pi -\bar{\pi }} \right) \left( {C^{s}-C^{d}} \right) f_{\xi _2 } \left( {D+Q^{*}} \right) } \\ \end{array} }} \right) }. \end{aligned}$$

Note that $\bar{{F}}_{\xi _2 } (\cdot )\ge \bar{{F}}_{\xi _1 } (\cdot )$ when $\xi _1 \le _{st} \xi _2$, $C\le C^{d}$, and $C^{s}\ge C^{d}$ for $\mathbf{e}_{\bar{\mathbf{d}}} =\emptyset$. Therefore, we have $\partial Q^{*}/\partial \pi \le 0$.

Employing Eq. (19) with respect to parameter $\pi$, we derive the partial derivative $dTC\left( {Q^{*}} \right) /d\pi$ for the case of $\mathbf{e}_{\bar{\mathbf{d}}} =\emptyset$:

$$\begin{aligned} \frac{dTC(Q^{*})}{d\pi }= & {} C\hbox {E}\min (\xi _1 ,Q)-C\hbox {E}\min (\xi _2 ,Q) \nonumber \\&{+\,C^{d}\hbox {E}\min (D,(\xi _1 -Q)^{+})-C^{d}\hbox {E}\min (D,(\xi _2 -Q)^{+})} \nonumber \\&{+\,C^{s}\hbox {E}((\xi _1 -Q)^{+}-D)^{+}-} C^{s}\hbox {E}((\xi _2 -Q)^{+}-D)^{+}. \end{aligned}$$

(21)

Note that if $\xi _1 \le _{st} \xi _2$, then the inequality $\hbox {E}f\left( {\xi _1 } \right) \le \hbox {E}f\left( {\xi _2 } \right)$ holds for all increasing functions f (Müller and Stoyan 2002). Since the functions $\hbox {min}\left( {\cdot ,Q} \right)$, $\hbox {min}\left( {D,\left( {\cdot -Q} \right) ^{+}} \right)$, and $\left( {\left( {\cdot -Q} \right) ^{+}-D} \right) ^{+}$ are increasing in variable, thus we have $\hbox {Emin}\left( {\xi _1 ,Q} \right) \le \hbox {Emin}\left( {\xi _2 ,Q} \right)$, $\hbox {Emin}\left( {D,\left( {\xi _1 -Q} \right) ^{+}} \right) \le \hbox {Emin}\left( {D,\left( {\xi _2 -Q} \right) ^{+}} \right)$, and $\hbox {E}\left( {\left( {\xi _1 -Q} \right) ^{+}-D} \right) ^{+}\le \hbox {E}\big ( \big ( {\xi _2 -Q} \big )^{+}-D \big )^{+}$. Therefore, we have $dTC\left( {Q^{*}} \right) /d\pi \le 0$. Moreover, one can use the similar approach to prove the case of $\mathbf{e}_{\mathbf{d}} =\emptyset$. $\square$

Proof of Theorem 2

We here only show the convexity of $TC\left( {\hat{Q}} \right)$. The other parts of Theorem  rm 2 can be proved by employing a similar approach of the proof of Theorem 1. Differentiating the expected total costs with respect to $\hat{Q}$ twice, we have the following:

$$\begin{aligned} \frac{\partial ^{2}TC(\hat{Q})}{\partial \hat{Q}^{2}}= & {} \sum _{i=1}^n {\lambda ^{2}\pi _i [(\hat{v}-v)-(\hat{{t}}_i -t)]f_{\xi _i } (\lambda \hat{Q})} \nonumber \\&{+\sum _{i=1}^n {\pi _i [p_i -\hat{v}-\hat{{t}}_i )]f_{\xi _i } (\hat{Q})} } +\sum _{e_i \in \mathbf{e}_{\bar{\mathbf{d}}} } {[\pi _i (c^{s}+t_i^s )f_{\xi _i } (\hat{Q})]} \nonumber \\&{+\sum _{e_i \in \mathbf{e}_{\mathbf{d}} } {[\pi _i (c^{s}+t_i^s -c^{d}-t_i^d )\Pr (\xi _i>\hat{Q})f_{\xi _i -D_i |\xi _i>\hat{Q}} (\hat{Q})]} } \nonumber \\&{+\sum _{e_i \in \mathbf{e}_{\mathbf{d}} } {[\pi _i (c^{d}+t_i^d )\Pr (\xi _i -D_i <\hat{Q}|\xi _i>\hat{Q})f_{\xi _i } (\hat{Q})]} } \nonumber \\&{+\sum _{e_i \in \mathbf{e}_{\mathbf{d}} } {[\pi _i (c^{s}+t_i^s )\Pr (\xi _i -D_i>\hat{Q}|\xi _i >\hat{Q})f_{\xi _i } (\hat{Q})]} } \nonumber \\&{-\sum _{e_i \in \mathbf{e}_{\mathbf{d}} } {[\pi _i (c^{d}+t_i^d )f_{\xi _i } (\hat{Q})]} }. \end{aligned}$$

(22)

From the relations $\hat{v}-v>\hat{{t}}_i -t_i$ and $p_i >\hat{v}+\hat{{t}}_i$, we can derive the result $\partial ^{2}TC\left( {\hat{Q}} \right) /\partial \hat{Q}^{2}>0$ based on Eq. (17). $\square$