Are quantity flexibility contracts with discounts in the presence of spot market procurement relevant for the humanitarian supply chain? An exploration

Abstract

Procurement of aid material such as vaccines by a humanitarian agency (HA) is often fraught with uncertainties. For example, an epidemic outbreak can increase the demand for materials (such as vaccines) in a very short period. Most of the HAs depend on external donations (funding) to procure necessary vaccines to meet this demand. Hence, it is financially infeasible and operationally inefficient for the HA to procure large quantities of aid material in anticipation of a demand spike during an epidemic outbreak. Thus, the procurement processes for aid materials such as vaccines need to be flexible enough to meet these demand fluctuations. HAs can achieve this flexibility by employing a procurement mechanism portfolio that includes upfront buying, capacity reservation, spot market purchase, etc. However, the challenge lies in identifying the optimal combination of multiple procurement mechanisms and how they can be utilized to coordinate the humanitarian supply chain. In this study, we explore the feasibility of quantity flexibility contracts along with discount incentives combined with spot market procurement in humanitarian supply chains for aid material procurement. We also derive the conditions under which the contract can achieve systemic coordination between the supplier and HA. Furthermore, we also illustrate that under optimal conditions, the procurement of aid material using multiple procurement mechanisms by HA can also reduce the humanitarian supply chain’s total cost.

Appendices

Appendix

Profit function of HA

$$ \begin{aligned} \pi_{HA} = & \left( {S - P_{1} } \right)\left( {1 + \alpha } \right)q_{1} + \left( {S - P_{2} } \right)q_{2} \\ & - \left( {S - P_{1} } \right)\mathop \int \limits_{{\left( {1 - \omega } \right)q_{1} + q_{2} }}^{{\left( {1 + \alpha } \right)q_{1} + q_{2} }} F\left( X \right)dX - \left( {S - v} \right)\mathop \int \limits_{0}^{{\left( {1 - \omega } \right)q_{1} + q_{2} }} F\left( X \right)dX \\ & + \left( {S - P_{i} } \right)\mathop \int \limits_{{\left( {1 + \alpha } \right)q_{1} + q_{2} }}^{\infty } F\left( X \right)dX \\ \end{aligned} $$

(A1)

Taking the FOC,

$$ \frac{d}{{dq_{1} }}\left( {\pi_{HA} } \right) = \left( {P_{i} - P_{1} } \right)\left( {1 + \alpha } \right) - \left( {P_{i} - P_{1} } \right)\left( {1 + \alpha } \right)F\left( A \right) - \left( {P_{1} - v} \right)\left( {1 - \omega } \right)F\left( B \right) $$

(A2)

$$ \frac{d}{{dq_{2} }}\left( {\pi_{HA} } \right) = \left( {P_{i} - P_{2} } \right) - \left( {P_{i} - P_{1} } \right)F\left( A \right) - \left( {P_{1} - v} \right)F\left( B \right) $$

(A3)

where $A = \left( {1 + \alpha } \right)q_{1} + q_{2}$ and $B = \left( {1 - \omega } \right)q_{1} + q_{2}$.

Solving for F(A) by (A2)–(A3) $\left( {1 - \omega } \right)$

$$ \left( {P_{i} - P_{1} } \right)\left( {1 - \alpha } \right) - \left( {P_{i} - P_{2} } \right)\left( {1 - \omega } \right) - \left( {P_{i} - P_{1} } \right)F\left( A \right)\left( {\left( {1 - \alpha } \right) - \left( {1 - \omega } \right)} \right) = 0 $$

$$ F\left( A \right) = \frac{{\left( {P_{i} - P_{1} } \right)\left( {1 + \alpha } \right) - \left( {P_{i} - P_{2} } \right)\left( {1 - \omega } \right)}}{{\left( {P_{i} - P_{1} } \right)\left( {\alpha + \omega } \right)}} $$

$$ F\left( A \right) = \frac{{\left( {P_{i} - P_{1} } \right)\xi - \left( {P_{i} - P_{2} } \right)}}{{(P_{i} - P_{2} )\left( {\xi - 1} \right)}} $$

(A4)

Similarly, for F(B), $\left( {A2} \right) - \left( {A3} \right)\left( {1 - \alpha } \right)$

$$ \left( {P_{i} - P_{1} } \right)\left( {1 + \alpha } \right) - \left( {P_{i} - P_{2} } \right)\left( {1 + \alpha } \right) - \left( {P_{1} - v} \right)\left( {1 - \omega } \right)F\left( B \right) + \left( {P_{1} - v} \right)\left( {1 + \alpha } \right)F\left( B \right) = 0 $$

$$ F\left( B \right) = \frac{{\left( {P_{1} - P_{2} } \right)\xi }}{{\left( {P_{1} - v} \right)\left( {\xi - 1} \right)}} $$

(A5)

The joint concavity in q₁ and q₂ follows directly from the objective function for HO in Eq. (1) since the second-order conditions on $\frac{{\partial^{2} E\left( {\pi_{HA} } \right)}}{{\partial q_{1}^{2} }} < 0$ and $\frac{{\partial^{2} E\left( {\pi_{HA} } \right)}}{{\partial q_{2}^{2} }} < 0$. Furthermore, the $Max\left( {X,0} \right)$ is an increasing function and linearity of $\left( {1 - \omega } \right)q_{1} + q_{2}$ and $\left( {1 + \alpha } \right)q_{1} + q_{2}$ preserves the concavity of the model.

Proof for Proposition 1

Since $0 \le F\left( B \right) \le F\left( A \right)$, we get,

$$ \frac{{\left( {P_{1} - P_{2} } \right)\xi }}{{\left( {P_{1} - v} \right)\left( {\xi - 1} \right)}} \le \frac{{\left( {P_{i} - P_{1} } \right)\xi - \left( {P_{i} - P_{2} } \right)}}{{\left( {P_{i} - P_{2} } \right)\left( {\xi - 1} \right)}} $$

$$ \frac{{P_{1} - v}}{{P_{1} - v}}\xi - \frac{{\left( {P_{2} - v} \right)}}{{P_{1} - v}}\xi = \xi - \frac{{P_{i} - P_{2} }}{{P_{i} - P_{1} }} $$

$$ \xi \ge \frac{{\left( {P_{i} - P_{2} } \right)\left( {P_{1} - v} \right)}}{{\left( {P_{i} - P_{1} } \right)\left( {P_{2} - v} \right)}} $$

(A6)

Proof for Proposition 2

The proof of proposition 2 has two parts. Firstly, the optimal values of $q_{1}^{*}$ and $q_{2}^{*}$ will be obtained and subsequently, the conditions in proposition 2 will be proved.

Let,

$$ \left( {1 + \alpha } \right)q_{1} + q_{2} = F^{ - 1} \left( H \right) $$

(A7)

$$ \left( {1 - \omega } \right)q_{1} + q_{2} = F^{ - 1} \left( K \right) $$

(A8)

Solving Eq. (A7 and A8)

$$ q_{1} = \frac{{F^{ - 1} \left( H \right) - F^{ - 1} \left( K \right)}}{{\left( {\xi - 1} \right)\left( {1 - \omega } \right)}} $$

(A9)

$$ q_{2} = \frac{{\xi F^{ - 1} \left( K \right) - F^{ - 1} \left( H \right)}}{\xi - 1} $$

(A10)

For $\xi \ge \frac{{\left( {P_{i} - P_{2} } \right)\left( {P_{1} - v} \right)}}{{\left( {P_{i} - P_{1} } \right)\left( {P_{2} - v} \right)}}$, HA will be incentivized to opt for QF contract. To show that $q_{1}^{*}$ is constrained by $q_{QF}$, let’s assume under QFDi, $q_{2}^{*} > 0$. Thus, the profit function of HA will be given by

$$ \begin{aligned} \pi_{HA} \left( {q_{1} ,q_{2}^{*} } \right) = & \left( {S - P_{1} } \right)\left( {1 + \alpha } \right)q_{1} + \left( {S - P_{2} } \right)q_{2}^{*} - \left( {S - P_{1} } \right)\mathop \int \limits_{{\left( {1 - \omega } \right)q_{1} + q_{2}^{*} }}^{{\left( {1 + \alpha } \right)q_{1} + q_{2}^{*} }} F\left( X \right)dX \\ & - \left( {S - v} \right)\mathop \int \limits_{0}^{{\left( {1 - \omega } \right)q_{1} + q_{2}^{*} }} F\left( X \right)dX + \left( {S - P_{i} } \right)\mathop \int \limits_{{\left( {1 + \alpha } \right)q_{1} + q_{2}^{*} }}^{\infty } F\left( X \right)dX \\ \end{aligned} $$

(A11)

For the demand distribution of $D^{\prime} = D - q_{2}^{*}$, the profit function becomes

$$ \begin{aligned} \pi_{HA} \left( {q_{1} ,q_{2}^{*} |D^{\prime}} \right) = & \left( {S - P_{1} } \right)\left( {1 + \alpha } \right)q_{1} + \left( {S - P_{2} } \right)q_{2}^{*} - \left( {S - P_{1} } \right)\mathop \int \limits_{{\left( {1 - \omega } \right)q_{1} }}^{{\left( {1 + \alpha } \right)q_{1} }} F\left( X \right)dX \\ & - \left( {S - v} \right)\mathop \int \limits_{0}^{{\left( {1 - \omega } \right)q_{1} }} F\left( X \right)dX + \left( {S - P_{i} } \right)\mathop \int \limits_{{\left( {1 + \alpha } \right)q_{1} }}^{\infty } F\left( X \right)dX \\ \end{aligned} $$

(A12)

Equation (A12) is similar to the profit function of a pure QF contract, but for the demand distribution function of $D^{\prime}$. The optimal solution of (A12) $q_{1}^{\prime* } q_{QF}$ since $D^{{\prime }} < D$ for $q_{2}^{*} > 0$. Thus, the limiting value for the $q_{1}^{*} = q_{QF}$ is obtained when $q_{2}^{*} = 0.$

For pure price discount contract, where the profit function for the HA is given by

$$ \pi_{HA} = \left( {S - P_{2} } \right)q_{2} - \left( {S - v} \right)\mathop \int \limits_{0}^{{q_{2} }} F\left( X \right)dX + \left( {S - P_{i} } \right)\mathop \int \limits_{{q_{2} }}^{\infty } F\left( X \right)dX $$

(A13)

FOC of Eq. (A13) gives,

$$ q_{2} = F^{ - 1} \left( {\frac{{P_{i} - P_{2} }}{{P_{i} - v}}} \right) $$

(A14)

Let $q_{1}^{*} > 0$, the optimal forecast under QFDi contact. Then, expected profit function of HA will be given by

$$ \begin{aligned} \pi_{HA} \left( {q_{1}^{*} ,q_{2} } \right) = & \left( {S - P_{1} } \right)\left( {1 + \alpha } \right)q_{1}^{*} + \left( {S - P_{2} } \right)q_{2} \\ & - \left( {S - P_{1} } \right)\mathop \int \limits_{{\left( {1 - \omega } \right)q_{1}^{*} + q_{2} }}^{{\left( {1 + \alpha } \right)q_{1}^{*} + q_{2} }} \left( {D - X} \right)dX \\ & - \left( {S - v} \right)\mathop \int \limits_{0}^{{\left( {1 - \omega } \right)q_{1}^{*} + q_{2} }} \left( {D - X} \right)dX \\ & + \left( {S - P_{i} } \right)\mathop \int \limits_{{\left( {1 + \alpha } \right)q_{1}^{*} + q_{2} }}^{\infty } \left( {X - D} \right)dX \\ \end{aligned} $$

(A15)

Let $D_{1} = D - \left( {1 + \alpha } \right)q_{1}^{*}$ and $D_{2} = D - \left( {1 + \omega } \right)q_{1}^{*}$. Thus, Eq. (A15) can be written as

$$ \begin{aligned} \pi_{HA} \left( {q_{1}^{*} ,q_{2} } \right) = & \left( {S - P_{1} } \right)\left( {1 + \alpha } \right)q_{1}^{*} + \left( {S - P_{2} } \right)q_{2} \\ & - \left( {S - P_{1} } \right)\left( {\mathop \int \limits_{0}^{{q_{2} }} \left( {D_{1} - X} \right)dX - \mathop \int \limits_{0}^{{q_{2} }} \left( {D_{2} - X} \right)dX} \right) \\ & - \left( {S - v} \right)\mathop \int \limits_{0}^{{q_{2} }} \left( {D_{2} - X} \right)dX + \left( {S - P_{i} } \right)\mathop \int \limits_{{q_{2} }}^{\infty } \left( {X - D_{1} } \right)dX \\ \end{aligned} $$

(A16)

For $\xi < \frac{{\left( {P_{i} - P_{2} } \right)\left( {P_{1} - v} \right)}}{{\left( {P_{i} - P_{1} } \right)\left( {P_{2} - v} \right)}}$, HA does not have any incentive to take part in QF contract over the quantity discount contracts since the flexibility does not outweigh cost of overstocking by HA. Thus, profit function for HA under this condition will be given by

$$ \pi_{HA}^{1} \left( {q_{2}^{*} } \right) = \left( {S - P_{2} } \right)q_{2} - \left( {S - v} \right)\mathop \int \limits_{0}^{{q_{2} }} \left( {D_{2} - X} \right)dX + \left( {S - P_{i} } \right)\mathop \int \limits_{{q_{2} }}^{\infty } \left( {X - D_{1} } \right)dX $$

(A17)

Equation (A17) is similar to pure quantity discount contract. The optimal value for

$$ q_{2}^{{\pi^{1} }} = F_{{\pi^{1} }}^{ - 1} \left( {\frac{{P_{i} - P_{2} }}{{P_{i} - v}}} \right) < F^{ - 1} \left( {\frac{{P_{i} - P_{2} }}{{P_{i} - v}}} \right) $$

since the demand distribution $D_{1} \& D_{2} < D$, indicating the limiting value for $q_{2}^{*}$ for any value of $q_{1} > 0$.

2.1 Profit of the centralized HSC under QFDi

$$ E(\pi_{c} ) = \left( {S - c} \right)Q - \left( {S - v} \right)EMax\left( {Q - X,0} \right) + \left( {S - c_{i} } \right)EMax\left( {X - Q, 0} \right) $$

(A18)

thus, the optimal order quantity under centralized system is

$$ F^{ - 1} \left( {\frac{{c_{i} - c}}{{c_{i} - v}}} \right) = Q_{c}^{*} $$

(A19)

Proof for Proposition 3

The optimal production quantity should be equal to the centralized ordering quantity. Thus, for RHS

$$ \left( {1 + \alpha } \right)q_{1}^{*} + q_{2}^{*} = Q_{c}^{*} $$

(A20)

$$ \frac{{\left( {P_{i} - P_{1} } \right)\xi - \left( {P_{i} - P_{2} } \right)}}{{\left( {P_{i} - P_{2} } \right)\left( {\xi - 1} \right)}} = \frac{{c_{i} - c}}{{c_{i} - v}} $$

(A21)

Rearranging,

$$ P_{2} = P_{1} - \frac{{\left( {P_{i} - P_{1} } \right)\left( {\xi - 1} \right)\left( {c - v} \right)}}{{c_{i} - v}} $$

(A22)

Proof for Proposition 4

Since the spot market prices will always be more that regular prices, the $P_{1} < P_{i}$ is straightforward. For the RHS, proposition 1, we know that minimum flexibility required for QF part to be activated. Therefore, for $\xi < \frac{{\left( {P_{i} - P_{2} } \right)\left( {P_{1} - v} \right)}}{{\left( {P_{i} - P_{1} } \right)\left( {P_{2} - v} \right)}}$, HA will not have an incentive to order under QF. Thus,

$$ \xi < \frac{{\left( {P_{i} - P_{2} } \right)\left( {P_{1} - v} \right)}}{{\left( {P_{i} - P_{1} } \right)\left( {P_{2} - v} \right)}} $$

(A23)

rearranging,

$$ \frac{{\xi \left( {P_{2} - v} \right) - \left( {P_{1} - v} \right)}}{{P_{1} - v}} < \frac{{P_{1} - P_{2} }}{{P_{i} - P_{1} }} $$

(A24)

Combining both results,

$$ P_{1} < P_{i} < P_{1} + \frac{{\left( {P_{1} - P_{2} } \right)\left( {P_{1} - v} \right)}}{{\xi \left( {P_{2} - v} \right) - \left( {P_{1} - v} \right)}} $$

(A25)

Proof for Proposition 5

With an increase in flexibility, the PS will have to offer a lower price to incentivize HA to buy more under discounts. However, PS cannot reduce the price below $c_{i}$ which is the procurement price for PS under spot market. Hence, the limiting condition is given by $c_{i} = P_{2}$. Thus,

$$ c_{i} = P_{1} - \frac{{\left( {P_{i} - P_{1} } \right)\left( {\xi - 1} \right)\left( {c - v} \right)}}{{c_{i} - v}} $$

(A26)

Rearranging,

$$ \xi = 1 + \frac{{\left( {P_{1} - c_{i} } \right)\left( {c_{i} - v} \right)}}{{\left( {P_{i} - P_{1} } \right)\left( {c - v} \right)}} $$

(A27)

Proof for Proposition 6

Both PS and HA to benefit from the usage of quantity discounts at price $P_{2}$ only if net profit increases. Thus, $\frac{d}{{dq_{2} }}\left( {\pi_{HA} } \right) > 0$ and $\frac{d}{{dq_{2} }}\left( {\pi_{PS} } \right) > 0$ such that $q_{2} \in \left( {0, q_{2}^{*} } \right)$. The profit function for the HA becomes,

$$ \begin{aligned} \pi_{HA} = & \left( {S - P_{1} } \right)\left( {1 + \alpha } \right)q_{QF} + \left( {S - P_{2} } \right)q_{2} \\ & - \left( {S - P_{1} } \right)\mathop \int \limits_{{\left( {1 - \omega } \right)q_{QF} + q_{2} }}^{{\left( {1 + \alpha } \right)q_{QF} + q_{2} }} F\left( X \right)dX \\ & - \left( {S - v} \right)\mathop \int \limits_{0}^{{\left( {1 - \omega } \right)q_{QF} + q_{2} }} F\left( X \right)dX \\ & + \left( {S - P_{i} } \right)\mathop \int \limits_{{\left( {1 + \alpha } \right)q_{QF} + q_{2} }}^{\infty } F\left( X \right)dX \\ \end{aligned} $$

(A28)

Taking FOC,

$$ \frac{d}{{dq_{2} }}\left( {\pi_{HA} } \right) = \left( {P_{i} - P_{2} } \right) - \left( {P_{i} - P_{1} } \right)F\left( {\left( {1 + \alpha } \right)q_{QF} + q_{2} } \right) - \left( {P_{1} - v} \right)F\left( {\left( {1 - \omega } \right)q_{QF} + q_{2} } \right) > 0 $$

(A29)

Rearranging,

$$ P_{2} < P_{i} \left( {1 - F\left( {\left( {1 + \alpha } \right)q_{QF} + Q_{2} } \right)} \right) + P_{1} \left[ {F\left( {\left( {1 + \alpha } \right)q_{QF} + q_{2} } \right) - F\left( {\left( {1 - \omega } \right)q_{QF} + q_{2} } \right)} \right] + vF\left( {\left( {1 - \omega } \right)q_{QF} + q_{2} } \right) $$

(A30)

Similarly,

$$ \pi_{PS} = \left( {P_{1} - c} \right)\left( {1 + \alpha } \right)q_{QF} + \left( {P_{2} - c} \right)q_{2} - \left( {P_{1} - v} \right)\mathop \int \limits_{{\left( {1 - \omega } \right)q_{QF} + q_{2} }}^{{\left( {1 + \alpha } \right)q_{QF} + q_{2} }} F\left( X \right)dX + \left( {P_{i} - c_{i} } \right)\mathop \int \limits_{{\left( {1 + \alpha } \right)q_{QF} + q_{2} }}^{\infty } F\left( X \right)dX $$

(A31)

Taking FOC, and rearranging,

$$ \frac{d}{{dq_{2} }}\left( {\pi_{PS} } \right) = \left( {P_{2} - c - P_{i} + c_{i} } \right) + \left( {P_{i} - c_{i} - P_{1} + v} \right)F\left( {\left( {1 + \alpha } \right)q_{QF} - q_{2} } \right) + \left( {P_{1} - v} \right)F\left( {\left( {1 - \omega } \right)q_{QF} - q_{2} } \right) > 0 $$

(A32)

$$ P_{2} > c + \left( {P_{1} - v} \right)\left[ {F\left( {\left( {1 + \alpha } \right)q_{QF} + q_{2} } \right) - F\left( {\left( {1 - \omega } \right)q_{QF} + q_{2} } \right)} \right] + \left( {P_{i} - c_{i} } \right)\left( {1 - F\left( {\left( {1 + \alpha } \right)q_{QF} + q_{2} } \right)} \right) $$

(A33)

Now,

$$ \frac{d}{{dq_{2} }}\left( {\pi_{HA} } \right) - \frac{d}{{dq_{2} }}\left( {\pi_{PS} } \right) = vF\left( {\left( {1 + \alpha } \right)q_{QF} + q_{2} } \right) - c + c_{i} \left( {1 - F\left( {\left( {1 + \alpha } \right)q_{QF} + q_{2} } \right)} \right) $$

$$ \frac{d}{{dq_{2} }}\left( {\pi_{HA} } \right) - \frac{d}{{dq_{2} }}\left( {\pi_{PS} } \right) \ge \left( {c_{i} - c} \right) - \left( {c_{i} - v} \right)F\left( {\left( {1 + \alpha } \right)q_{QF} + q_{2} } \right) = 0 $$

(A34)

Combining Eqs. (A30) and (A33), we get the range between which $P_{2}$ can operate. The maximum amount that PS should offer under extended quantity discount contract is given by equilibrium condition for equation (A34). Thus,

$$ F\left( {\left( {1 + \alpha } \right)q_{QF} + q_{2} } \right) = \frac{{c_{i} - c}}{{c_{i} - v}} = Q_{c}^{*} $$

$$ q_{2}^{*} = F^{ - 1} \left( {\frac{{c_{i} - c}}{{c_{i} - v}}} \right) - \left( {1 + \alpha } \right)q_{QF} $$

(A35)