Supply chain performance for a risk inequity averse newsvendor

Abstract

This paper studies the impacts of risk inequity averse factor on supply chain performance in the newsvendor context. We apply a downside-risk approach to depict the risk caused by market demand uncertainty and develop the newsvendor model with a risk inequity averse agent. Contrary to the prior literature, we extend inequity averse profits to a broader range by including risk inequity aversion, and we find that risk inequity aversion has large impact on supply chain agents performance. Moreover, we attempt to consider the situation which a risk inequity averse retailer’s information is neglected by the supplier and performed a comparative analysis on different scenarios. If the supplier is only concerned with the risk inequity averse retailer while making decisions, the results range from worsening to improving. To mitigate the negative utilities caused by the factor of risk inequity aversion, we propose a mechanism that the risk-sharing parameter could achieve a better performance for a certain range. Several observations and managerial insights on risk inequity-averse newsvendor models are gathered.

Appendices

Appendices

Appendix A: The derivation process and proofs of the benchmark model

1.1 The derivation processes of expected profits are as follows:

Given retailer order quantity q and uncertainty demand x, the notation $min\left( q,x \right)$ denotes the minimum of the two. Therefore, the expected sales can be derived as follows:

$$\begin{aligned} {Emin\left( {q,x} \right) = \left\{ {\begin{array}{ll} {E\left( q \right) , \quad \hbox {if}\;q < x}\\ {E\left( x \right) , \quad \hbox {if}\;q > x} \end{array}} \right. } \end{aligned}$$

The derivation process is calculated separately are as follows.

$$\begin{aligned} \begin{array}{l} Emin\left( {q,x} \right) = \displaystyle \int _0^q {xf\left( x \right) dx} + \int _q^{ + \infty } {qf\left( x \right) dx} \\ \quad = xF\left( x \right) \left| {_q^{ + \infty }} \right. -\displaystyle \int _0^q {F\left( x \right) dx} + qF\left( x \right) \left| {_q^{ + \infty }} \right. \\ \quad = qF\left( q \right) - \displaystyle \int _0^q {F\left( x \right) dx} + q\left( {1 - F\left( q \right) } \right) \\ \quad = q -\displaystyle \int _0^q {F\left( x \right) dx} \end{array} \end{aligned}$$

The expected profit of the risk-neutral retailer and supply chain system can be derived as follows:

$$\begin{aligned} E({\pi _r})= & {} E\left[ {pmin\left( {q,x} \right) - wq} \right] = \left( {p - w} \right) q - p\int _0^q {F\left( x \right) dx} \\ E({\pi _s})= & {} \left( {w - c} \right) q \\ E(\varPi )= & {} E\left[ {pmin\left( {q,x} \right) - cq} \right] = \left( {p - c} \right) q - p\int _0^q {F\left( x \right) dx} \end{aligned}$$

Proof

We first discuss the benchmark model that the supply chain agents without risk inequity aversion.

1.2 Step 1: Solve the supply chain members profit maximization problem

In the decentralized system, the profit function of retailer and supplier are given in Eqs. (1) and (2). The first-order conditions of the utility function with respect to q and w respectively are as follows.

$$\begin{aligned} \left\{ {\begin{array}{ll} {\frac{{\partial E({\pi _r})}}{{\partial q}} = \left( {p - w} \right) - pF\left( q \right) }\\ {\frac{{\partial E({\pi _s})}}{{\partial w}} = \left( {w - c} \right) \cdot \frac{{\partial w}}{{\partial q}} + q } \end{array}} \right. \end{aligned}$$

Then we obtain the stationary point of the equations.

$$\begin{aligned} \left\{ {\begin{array}{ll} {\left( {p - w} \right) - pF\left( q \right) = 0}\\ {\left( {w - c} \right) + q \cdot \frac{{\partial q}}{{\partial w}} = 0} \end{array}} \right. \end{aligned}$$

Next, we take the second-order conditions of the utility function with respect to order quantity q:

$$\begin{aligned} \frac{{{\partial ^2}U\left( {{\pi _r}} \right) }}{{\partial {q^2}}} = - pf\left( q \right) < 0 \end{aligned}$$

Thus, there exists the critical point $q^*$ which is a local maximum, and this stationary point is the optimal order decision strategy. And the equilibrium solution $(q^*,w^*)$ can be obtained by solving the following equations.

$$\begin{aligned} \left\{ {\begin{array}{ll} {{q^*} = {F^{ - 1}}\left( {\frac{{p - {w^*}}}{p}} \right) }\\ {{w^*} = p{q^*}f\left( {{q^*}} \right) + c} \end{array}} \right. \end{aligned}$$

1.3 Step 2: Solve the supply chain systems profit maximization problem

In the supply chain system, we have derived the equation of the supply chain system’s expected profit function. Thus, the first-order condition of utility function with respect to supply chain order quantity q is as follows.

$$\begin{aligned} \frac{{\partial E\left( \varPi \right) }}{{\partial q}} = \left( {p - c} \right) - pF\left( q \right) \end{aligned}$$

The process of obtaining the stationary point and solving the optimal order quantity of centralized supply chain system is the same as previous. Thus, the optimal order decision strategy $q^o$ is expressed as follows.

$$\begin{aligned} {q^o} = {F^{ - 1}}\left( {\frac{{p - c}}{p}} \right) \end{aligned}$$

The assumption we proposed previously that $w>c$, and we assume that $F\left( \cdot \right)$ is strictly increasing and differentiable with the stochastic market demand is finite. Finally, the optimal order quantity of decentralized supply chain member is smaller than the centralized system: $q^* < q^o$.

Appendix B: The risk inequity averse newsvendor model, proof of Proposition 1

Proof

We discuss the model that considering the risk inequity aversion factor of the retailer under the wholesale price contract. The decentralized supply chain members play a Stackelberg game, the supplier is a leader while the retailer is a follower. $\square$

1.1 Step 1: Solve the supply chain members profit maximization problem

In the decentralized system, the profit function of retailer and supplier are given in Eqs. (15) and (16). To make the equation is tractable, we let $g(q) = \int _0^q {{\bar{F}}(x)} dx$, the first-order conditions of the utility function with respect to q and w respectively are as follows.

$$\begin{aligned} \left\{ \begin{aligned} \frac{{\partial E({\pi _{r,f}})}}{{\partial q}}&= \left( {p - w} \right) - pF\left( q \right) - \;\alpha pF\left( {g\left( q \right) } \right) {\bar{F}}\left( q \right) \\ \frac{{\partial E({\pi _{s,f}})}}{{\partial w}}&= \left( {w - c} \right) \cdot \frac{{\partial q}}{{\partial w}} + q \end{aligned} \right. \end{aligned}$$

Then we obtain the stationary point of the equations.

$$\begin{aligned} \left\{ \begin{aligned} \left( {p - w} \right)&- pF\left( q \right) - \alpha pF\left( {g\left( q \right) } \right) \cdot {\bar{F}}\left( q \right) = 0\\ \left( {w - c} \right)&+ q \cdot \frac{{\partial w}}{{\partial q}} = 0 \end{aligned} \right. \end{aligned}$$

Next, we take the second-order conditions of the utility function of risk-inequity averse retailer with respect to q:

$$\begin{aligned}\begin{array}{l} \frac{{{\partial ^2}U\left( {{\pi _{r,f}}} \right) }}{{\partial {q^2}}} = - pf\left( q \right) - \alpha p[ {f\left( {g\left( q \right) } \right) \cdot {{( {{\bar{F}}( q)} )}^2} - F\left( {g\left( q \right) } \right) \cdot f\left( q \right) } ]\\ \quad = - pf(q)[1 - \alpha F(g(q))] - \alpha pf\left( {g\left( q \right) } \right) \cdot {( {{\bar{F}}\left( q \right) })^2} \end{array} \end{aligned}$$

We can obtain the simplified equation $1 - \alpha F\left( {g\left( q \right) } \right) = \frac{w}{{p{\bar{F}}\left( q \right) }}$ and substitute into the equation of second-order conditions of the risk inequity averse retailer utility function with respect to q. Finally, the equation above can be simplified as: $\frac{{{\partial ^2}U\left( {{\pi _{r,f}}} \right) }}{{\partial {q^2}}} = - \frac{{wf\left( q \right) }}{{{\bar{F}}\left( q \right) }} - \alpha pf\left( {g\left( q \right) } \right) \cdot {( {{\bar{F}}\left( q \right) } )^2} < 0$.

Therefore, there exists a stationary point $q^*_{r,f}$ to be the local maximum point, and this is the optimal decision strategy of the retailer. And $(q^*_{r,f}, w^*_{r,f} )$ is the unique optimal decision strategy of the decentralized supply chain members by solving the set of equations as follows.

$$\begin{aligned} \left\{ \begin{aligned}&{\bar{F}}( {q_{r,f}^*} )[ {1 - \;\alpha F( {g( {q_{r,f}^*} )} )} ] = \frac{{w_{r,f}^*}}{p}\\&w_{r,f}^* = pq_{r,f}^*f( {q_{r,f}^*} )[ {1 - \;\alpha F( {g( {q_{r,f}^*} )} )} ] + \;\alpha pq_{r,f}^*f( {g( {q_{r,f}^*} )} ) \cdot {{( {{\bar{F}}( {q_{r,f}^*} )})}^2} + c \end{aligned} \right. \end{aligned}$$

Appendix C: Risk-sharing mechanism, Proof of Proposition 3

Under the risk-sharing mechanism, as we proved previously, the optimal order decision can be obtained by taking the first and second-order derivatives of the retailers utility [Eq. (24)] with respect to the order quantity $q^*_{r,RS}$. The equations are given as follows.

$$\begin{aligned}\begin{aligned} \frac{{\partial U({\pi _{r,RS}})}}{{\partial q}}&= \left( {p - w} \right) - pF\left( q \right) - (1 - \lambda )\alpha pF\left( {g\left( q \right) } \right) {\bar{F}}\left( q \right) = 0 \\ \frac{{{\partial ^2}U( {{\pi _{r,RS}}})}}{{\partial {q^2}}}&= - pf\left( q \right) - (1 - \lambda )\alpha p[ {f( {g( q)}) \cdot {{( {{\bar{F}}( q)} )}^2} - F( {g( q)} ) \cdot f( q )} ] \end{aligned} \end{aligned}$$

The equation can be derived by simplifying the equation of the first-order derivatives of the retailer’s utility.

$$\begin{aligned} p{\bar{F}}\left( q \right) \left[ {1 - \left( {1 - \lambda } \right) \alpha pF\left( {g\left( q \right) } \right) } \right] = w \end{aligned}$$

By substituting the above equation into the equation of the second-order derivatives of the retailer’s utility, then the second-order condition can be proved less than zero. Due to $0< \lambda <1$ and $0<(1-\lambda )<1$. The equation can be derived as follows.

$$\begin{aligned} \frac{{{\partial ^2}U( {{\pi _{r,RS}}})}}{{\partial {q^2}}} = - pf\left( q \right) \cdot \frac{w}{{p{\bar{F}}\left( q \right) }} - \left( {1 - \lambda } \right) \alpha pf\left( {g\left( q \right) } \right) \cdot {( {{\bar{F}}\left( q \right) } )^2} < 0 \end{aligned}$$

Then, we take the derivatives of suppliers utility [Eq. (24)] with respect to the wholesale price w. The equation can be given by:

$$\begin{aligned} \frac{{\partial E({\pi _{s,RS}})}}{{\partial w}} = \left( {w - c} \right) \cdot \frac{{\partial q}}{{\partial w}} + q - \;\alpha \cdot \frac{{\partial S\left( {\pi \left( q \right) } \right) }}{{\partial w}} = 0 \end{aligned}$$

This indicates that there is the stationary point $(q^*_{r,RS}, w^*_{r,RS})$ of the decentralized supply chain members can be calculated through simultaneous $p{\bar{F}}\left( q \right) \left[ {1 - \left( {1 - \lambda } \right) \alpha pF\left( {g\left( q \right) } \right) } \right] = w$ and $\frac{{\partial E({\pi _{s,RS}})}}{{\partial w}} = \left( {w - c} \right) \cdot \frac{{\partial q}}{{\partial w}} + q - \;\alpha \cdot \frac{{\partial S\left( {\pi \left( q \right) } \right) }}{{\partial w}} = 0$.

To make the equation tractable, we multiply both sides of $\frac{{\partial E({\pi _{s,RS}})}}{{\partial w}} = \left( {w - c} \right) \cdot \frac{{\partial q}}{{\partial w}} + q - \;\alpha \cdot \frac{{\partial S\left( {\pi \left( q \right) } \right) }}{{\partial w}} = 0$ with $\frac{{\partial w}}{{\partial q}}$. The equation could be derived as follows.

$$\begin{aligned} \frac{{\partial E({\pi _{s,RS}})}}{{\partial w}} = \left( {w - c} \right) + q\cdot \frac{{\partial w}}{{\partial q}} - \alpha \cdot \frac{{\partial S\left( {\pi \left( q \right) } \right) }}{{\partial q}} = 0 \end{aligned}$$

The first order condition of w with respect to q and first derivation of $S\left( {\pi \left( q \right) } \right)$ regarding q respectively are:

$$\begin{aligned} \begin{aligned} \frac{{\partial w}}{{\partial q}}= - pf\left( q \right)&- \left( {1 - \lambda } \right) \alpha pF\left( {g\left( q \right) } \right) \cdot \left( { - f\left( q \right) } \right) - \left( {1 - \lambda } \right) \alpha pf\left( {g\left( q \right) } \right) \cdot {\left( {{\bar{F}}\left( q \right) } \right) ^2}\\ \frac{{\partial S\left( {\pi \left( q \right) } \right) }}{{\partial q}}&= pF\left( {g\left( q \right) } \right) \cdot {\bar{F}}\left( q \right) \end{aligned} \end{aligned}$$

The $\frac{{\partial E({\pi _{s,RS}})}}{{\partial w}} = \left( {w - c} \right) + q\cdot \frac{{\partial w}}{{\partial q}} - \alpha \cdot \frac{{\partial S\left( {\pi \left( q \right) } \right) }}{{\partial q}} = 0$ can be simplified by substituting the equations $\frac{{\partial w}}{{\partial q}}$ and $\frac{{\partial S\left( {\pi \left( q \right) } \right) }}{{\partial q}}$ we calculate above, respectively. And the optimal order quantity $q^*_{r,RS}$ of decentralized retailer is:

$$\begin{aligned} \begin{array}{l} p{\bar{F}}( {q_{r,RS}^*})[ {1 - \alpha pF( {g( {q_{r,RS}^*} )} )} ] - pq_{r,RS}^*f( {q_{r,RS}^*} ) + \alpha pq_{r,RS}^*f( {q_{r,RS}^*} ) \\ \quad \cdot ( {1 - \lambda } )F( {g( {q_{r,RS}^*})} ) - \alpha pq_{r,RS}^*( {1 - \lambda })( {g( {q_{r,RS}^*})} ) \cdot {( {{\bar{F}}( {q_{r,RS}^*})})^2} = c \end{array} \end{aligned}$$

As we proved previously, the optimal order quantity of the supply chain system is $q^*_{sc,RS}$, which satisfies the equation $\frac{{\partial U\left( {{\pi _{sc,ic}}} \right) }}{{\partial q}} = 0$. And the equation is given by:

$$\begin{aligned} q_{sc,RS}^* = {{{\bar{F}}}^{ - 1}}\left( {\frac{c}{{p[ {1 - \alpha F( {g( {q_{sc,RS}^*})} )} ]}}} \right) \end{aligned}$$

To achieve the channel coordination and performance improvement, let the optimal order quantity of decentralized supply chain members equals to that of the centralized system: $q^*_{r,RS}=q^*_{sc,RS}$, and we work out the shares of risk parameter $\lambda$ by simultaneously solving above two equations. The risk sharing parameter is obtained as follows.

$$\begin{aligned} \lambda = 1 - \frac{{q_{sc,RS}^*f(q_{sc,RS}^*)}}{{\alpha q_{sc,RS}^*[ {f( {g(q_{sc,RS}^*)} ){{( {{\bar{F}}(q_{sc,RS}^*)} )}^2} - F( {g(q_{sc,RS}^*)} )f(q_{sc,RS}^*)} ]}} \end{aligned}$$