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Proactive and reactive purchasing planning under dependent demand, price, and yield risks

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Abstract

The trend of globalization and outsourcing makes supply unreliable and companies begin to have supplier diversity embedded into their procurement departments. Traditionally, contract suppliers are a major supply channel for many companies, while the effectiveness of reactive supply sources, such as spot markets, is often ignored. Spot markets have negligible lead times and higher average prices comparing with contract suppliers. In our research, procurement utilizes the combinatorial benefits of proactive supply (a contract supplier) and reactive supply (a spot market). The uncertainties of yields, spot prices, and demand, and the correlations among them are also taken into consideration when designing procurement plans. The objectives of this paper were to evaluate the effectiveness of dealing with uncertain supply using the spot market along with the contract supplier and to model the dependences among all the potential uncertainties. This research also seeks high expected profits without overlooking the associated variances. The analytical expression to determine the optimal order quantity is obtained under the most general situations where commodities can be both bought and sold via the spot market. Some properties are derived to provide useful managerial insights. In addition, reference scenarios, such as pure contract sourcing and the spot market restricted for buying or selling only, are included for comparison purposes.

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Correspondence to Xiaofeng Nie.

Appendices

Appendix 1: Derivation of the expected profit for the BS model

Under the assumption of buying and selling, the buyer’s profit can be expressed as follows:

$$\begin{aligned} \Pi _\mathrm{BS} =rd+s(Y-d)-wY. \end{aligned}$$

So, the expected value of the buyer’s profit can be computed as follows:

$$\begin{aligned} E( {\Pi _\mathrm{BS} })&= E\left[ {rd+s(Y-d)-wY} \right] \\&= E( {rd})+E( {sY})-E( {sd})-E( {wY}) \\&= r\mu _d +(\mu _s \mu _Y +\rho _{y,s} \sigma _s \sigma _Y )-(\mu _s \mu _d +\rho _{d,s} \sigma _s \sigma _d )-w\mu _Y \\&= ( {r-\mu _s })\mu _d -\rho _{d,s} \sigma _d \sigma _s +[(\mu _s -w)\mu _y +\rho _{y,s} \sigma _y \sigma _s ]Q. \end{aligned}$$

Appendix 2: Derivation of the profit variance for the BS model

The variance of the buyer’s profit can be computed as follows:

$$\begin{aligned} \mathrm{Var}( {\Pi _\mathrm{BS} })&= E[(\Pi _\mathrm{BS} )^2]-[E( {\Pi _\mathrm{BS} })]^2 \\&= E\{{[ {rd+s( {Y-d})-wY} ]^2} \}-[E( {\Pi _\mathrm{BS} })]^2 \\&= E[ {( {r-s})^2d^2} ]+2E [ {{d}Y( {r-s})( {s-w})} ]\\&+E[( {s-w})^2Y^2]-[E( {\Pi _\mathrm{BS} })]^2. \end{aligned}$$

As for \(E [{( {r-s})^2d^2}]\), we have that

$$\begin{aligned}&E[ {( {r-s})^2d^2}]=\mathrm{Var} [ {( {r-s})d}]+\{E[( {r-s})d]\}^2 \\&\quad =[E( {r-s})]^2\mathrm{Var}( d)+\mathrm{Var}( {r-s})[E( d)]^2+\mathrm{Var}( {r-s})\mathrm{Var}( d) \\&\qquad +2E( {s-r})E( d)\mathrm{Cov}( {s-r,d})+[\mathrm{Cov}( {s-r,d})]^2+\left\{ {E[( {r-s})d]} \right\} ^2 \\&\quad =(r-\mu _s )^2\sigma _d^2 +\sigma _s^2 \mu _d^2 +\sigma _s^2 \sigma _d^2 +2( {\mu _s -r})\mu _d \rho _{d,s} \sigma _s \sigma _d +\rho _{d,s}^2 \sigma _s^2 \sigma _d^2 \\&\qquad +[( {r-\mu _s })\mu _d -\rho _{d,s} \sigma _s \sigma _d ]^2 \\&\quad = [ {( {r-\mu _s })^2+\sigma _s^2 }](\mu _d^2 +\sigma _d^2 )+2\rho _{d,s}^2 \sigma _s^2 \sigma _d^2 -4\rho _{d,s} \sigma _s \sigma _d \mu _d (r-\mu _s ). \end{aligned}$$

The calculation process of \(E[( {s-w})^2Y^2]\) is similar to that of \(E[ {( {r-s})^2d^2}]\), and the result is as follows:

$$\begin{aligned} E[ {( {s-w})^2Y^2}]&= [ {( {w-\mu _s })^2+\sigma _s^2 }](\mu _Y^2 +\sigma _Y^2 )+2\rho _{y,s}^2 \sigma _s^2 \sigma _Y^2 \\&-4\rho _{y,s} \sigma _s \sigma _Y \mu _Y ( {w-\mu _s }). \end{aligned}$$

As for \(E [{{d}Y( {r-s})( {s-w})}]\), defining \(u=s-r\) and \(v=s-w\), we have that \(E [{dY( {r-s})( {s-w})} ]=-E( {dYuv})\). Moreover, \(E( {dYuv})=E( {dY\cdot uv})=E( {dY})E( {uv})+\mathrm{Cov}(dY,uv)\), where \(E( {dY})=\mu _d \mu _Y +\rho _{d,y} \sigma _d \sigma _Y \) and \(E( {uv})=E( {s^2-sw-rs+rw})=( {\mu _s^2 +\sigma _s^2 })-(w+r)\mu _s +rw \quad =( {\mu _s -w})( {\mu _s -r})+\sigma _s^2 \).

Before we compute \(\mathrm{Cov}(dY,uv)\), let us introduce an important result from Bohrnstedt and Goldberger (1969). Let \(x_1 ,x_2 ,x_3,\) and \(x_4 \) be jointly distributed random variables. By definition, the covariance between products \(x_1 x_2 \) and \(x_3 x_4 \) is as follows:

$$\begin{aligned} C( {x_1 x_2 ,x_3 x_4 })=E \{{[{x_1 x_2 -E( {x_1 x_2 })} ][ {x_3 x_4 -E( {x_3 x_4 })}]}\}, \end{aligned}$$

where \(C\) is short for covariance. Let \(\Delta x_1 =x_1 -E(x_1 )\), \(\Delta x_2 =x_2 -E(x_2 )\), \(\Delta x_3 =x_3 -E( {x_3 }),\) and \(\Delta x_4 =x_4 -E(x_4 )\). We have that

$$\begin{aligned} C( {x_1 x_2 ,x_3 x_4 })&= E( {x_1 })E( {x_3 })C( {x_2 ,x_4 })+E( {x_1 })E( {x_4 })C( {x_2 ,x_3 })\\&+E( {x_2 })E( {x_3 })C( {x_1 ,x_4 })+E( {x_2 })E( {x_4 })C( {x_1 ,x_3 }) \\&+E(\Delta x_1 \Delta x_2 \Delta x_3 \Delta x_4 )+E( {x_1 })E(\Delta x_2 \Delta x_3 \Delta x_4 ) \\&+E( {x_2 })E(\Delta x_1 \Delta x_3 \Delta x_4 ) +E( {x_3 })E(\Delta x_1 \Delta x_2 \Delta x_4 )\\&+E( {x_4 })E(\Delta x_1 \Delta x_2 \Delta x_3 )-C( {x_1 ,x_2 })C( {x_3 ,x_4 }). \end{aligned}$$

Based on the result from Anderson (1958), under multivariate normality, all third moments vanish and

$$\begin{aligned} E(\Delta x_1 \Delta x_2 \Delta x_3 \Delta x_4 )&= C( {x_1 ,x_2 })C( {x_3 ,x_4 })+C( {x_1 ,x_3 })C( {x_2 ,x_4 })\\&+C( {x_1 ,x_4 })C( {x_2 ,x_3 }). \end{aligned}$$

Therefore,

$$\begin{aligned} C( {x_1 x_2 ,x_3 x_4 })&= E( {x_1 })E( {x_3 })C( {x_2 ,x_4 })+E( {x_1 })E( {x_4 })C( {x_2 ,x_3 }) \\&+E( {x_2 })E( {x_3 })C( {x_1 ,x_4 })+E( {x_2 })E( {x_4 })C( {x_1 ,x_3 })\\&+C( {x_1 ,x_3 })C( {x_2 ,x_4 })+C( {x_1 ,x_4 })C( {x_2 ,x_3 }). \end{aligned}$$

Hence,

$$\begin{aligned} C( {dY,uv})&= E( d)E( u)C( {Y,v})+E( d)E( v)C( {Y,u})+E( Y)E( u)C( {d,v}) \\&+E( Y)E( v)C( {d,u})+C({d,u})C( {Y,v})+C( {d,v})C( {Y,u}). \end{aligned}$$

Because \(C( {Y,u})=C( {Y,s-r})=E \{ {[{Y-E( Y)}] [{( {s-r})-E( {s-r})}]}\}= E \{ {[{Y-E( Y)}] [{s-E( s)}]}\}=C(Y,s)\) and \(C( {Y,v})= C( {Y,s})\), we have that \(C( {Y,u})=C( {Y,v})=C( {Y,s})\). Thus,

$$\begin{aligned} C( {dY,uv})&= E( d)E( u)C( {Y,s})+E( d)E( v)C( {Y,s})+E( Y)E( u)C( {d,s})\\&+E( Y)E( v)C( {d,s}) +2C( {d,s})C( {Y,s}) \\&= \mu _d ( {\mu _s -r})\rho _{y,s} \sigma _Y \sigma _s +\mu _d ( {\mu _s -w})\rho _{y,s} \sigma _Y \sigma _s +\mu _Y ( {\mu _s -r})\rho _{d,s} \sigma _d \sigma _s \\&+\mu _Y ( {\mu _s -w})\rho _{d,s} \sigma _d \sigma _s +2\rho _{d,s} \sigma _d \sigma _s \rho _{y,s} \sigma _Y \sigma _s \\&= ( {2\mu _s -r-w})\sigma _s (\mu _d \rho _{y,s} \sigma _Y +\mu _Y \rho _{d,s} \sigma _d )+2\rho _{d,s} \sigma _d \sigma _s \rho _{y,s} \sigma _Y \sigma _s . \end{aligned}$$

Therefore, the profit variance is computed as follows:

$$\begin{aligned}&\mathrm{Var}( {\Pi _\mathrm{BS} })=E [{( {r-s})^2d^2}]+2E [{dY( {r-s})( {s-w})}]+E[( {s-w})^2Y^2] \\&\quad -[E( {\Pi _\mathrm{BS} })]^2=E [{( {r-s})^2d^2}]+E[( {s-w})^2Y^2]-2[E( {dY})E( {uv})+C(dY,uv)] \\&\quad -[E( {\Pi _\mathrm{BS} })]^2 =[{( {r-\mu _s })^2+\sigma _s^2 }](\mu _d^2 +\sigma _d^2 )+2\rho _{d,s}^2 \sigma _s^2 \sigma _d^2 -4\rho _{d,s} \sigma _s \sigma _d \mu _d ( {r-\mu _s }) \\&\quad +[ {( {w-\mu _s })^2+\sigma _s^2 } ]( {\mu _Y^2 +\sigma _Y^2 })+2\rho _{y,s}^2 \sigma _s^2 \sigma _Y^2 -4\rho _{y,s} \sigma _s \sigma _Y \mu _Y (w-\mu _s ) \\&\quad -2(\mu _d \mu _Y +\rho _{d,y} \sigma _d \sigma _Y ) [ {( {\mu _s -w})( {\mu _s -r})+\sigma _s^2 } ] \\&\quad -2( {2\mu _s -r-w})\sigma _s (\mu _d \rho _{y,s} \sigma _Y +\mu _Y \rho _{d,s} \sigma _d )-4\rho _{d,s} \sigma _d \sigma _s \rho _{y,s} \sigma _Y \sigma _s -[E( {\Pi _\mathrm{BS} })]^2. \end{aligned}$$

The square of the buyer’s expected profit is as follows:

$$\begin{aligned}&[E( {\Pi _\mathrm{BS} })]^2=[( {r-\mu _s })\mu _d +( {\mu _s -w})\mu _Y +(\rho _{y,s} \sigma _Y -\rho _{d,s} \sigma _d )\sigma _s ]^2 \\&\quad =( {r-\mu _s })^2\mu _d^2 +( {\mu _s -w})^2\mu _Y^2+ \rho _{y,s}^2 \sigma _s^2 \sigma _Y^2 +\rho _{d,s}^2 \sigma _s^2 \sigma _d^2 -2\rho _{d,s} \sigma _d \sigma _s \rho _{y,s} \sigma _Y \sigma _s \\&\qquad +2\mu _d \mu _Y ( {r-\mu _s })( {\mu _s -w})+2({r-\mu _s })\mu _d (\rho _{y,s} \sigma _Y -\rho _{d,s} \sigma _d )\sigma _s \\&\qquad + 2( {\mu _s -w})\mu _Y (\rho _{y,s} \sigma _Y -\rho _{d,s} \sigma _d )\sigma _s. \end{aligned}$$

After substituting \([E( {\Pi _\mathrm{BS} })]^2\) into \(\mathrm{Var}( {\Pi _\mathrm{BS} })\), we have that

$$\begin{aligned}&\mathrm{Var}( {\Pi _\mathrm{BS} })=( {r-\mu _s })^2\sigma _d^2 +( {\mu _s -w})^2\sigma _Y^2 +\sigma _s^2 (\mu _Y^2 +\sigma _Y^2 +\mu _d^2 +\sigma _d^2 +\rho _{d,s}^2 \sigma _d^2 \\&\quad +\rho _{y,s}^2 \sigma _Y^2-2\mu _Y \mu _d )-2\rho _{d,s} \sigma _d \sigma _s ( {r-\mu _s })( {\mu _d -\mu _Y })-2\rho _{y,s} \sigma _s \sigma _Y ( {\mu _s -w}) \\&\quad \times ( {\mu _d -\mu _Y })-2\rho _{d,s} \rho _{y,s} \sigma _d \sigma _Y \sigma _s^2 -2\rho _{d,y} \sigma _d \sigma _Y \big [ {( {\mu _s -r})( {\mu _s -w})+\sigma _s^2 } \big ]. \end{aligned}$$

Appendix 3: Proof of Proposition 1

Since \(\mu _Y =\mu _y Q\) and \(\sigma _Y =\sigma _y Q\), substituting them into \(\mathrm{Var}( {\Pi _\mathrm{BS} })\), we obtain that

$$\begin{aligned}&\mathrm{Var}( {\Pi _\mathrm{BS} })=( {r-\mu _s })^2\sigma _d^2 +( {\mu _s -w})^2\sigma _y^2 Q^2+\sigma _s^2 (\mu _y^2 Q^2+\sigma _y^2 Q^2+\mu _d^2 +\sigma _d^2 \\&\qquad +\rho _{d,s}^2 \sigma _d^2+\rho _{y,s}^2 \sigma _y^2 Q^2-2\mu _y Q\mu _d )-2\rho _{d,s} \sigma _d \sigma _s ( {r-\mu _s })(\mu _d -\mu _y Q) \\&\qquad -2\rho _{y,s} \sigma _s \sigma _y Q( {\mu _s -w})(\mu _d -\mu _y Q)-2\rho _{d,s} \rho _{y,s} \sigma _d \sigma _y Q\sigma _s^2 \\&\qquad -2\rho _{d,y} \sigma _d \sigma _y Q[( {\mu _s -r})( {\mu _s -w})+\sigma _s^2 ] \\&\quad =\big [ {( {\mu _s -w})^2\sigma _y^2 +( {\mu _y^2 +\sigma _y^2 })\sigma _s^2 +\rho _{y,s}^2 \sigma _y^2 \sigma _s^2 +2\rho _{y,s} \sigma _s \sigma _y \mu _y ( {\mu _s -w})} \big ]Q^2 \\&\qquad -2 \big \{\mu _y \mu _d \sigma _s^2 -\rho _{d,s} \sigma _d \sigma _s \mu _y ( {r-\mu _s })+\rho _{y,s} \sigma _s \sigma _y \mu _d ( {\mu _s -w})+\rho _{d,s} \rho _{y,s} \sigma _d \sigma _y \sigma _s^2 \\&\qquad +\rho _{d,y} \sigma _d \sigma _y \big [ {( {\mu _s -r})( {\mu _s -w})+\sigma _s^2 } \big ]\big \}Q+( {r-\mu _s })^2\sigma _d^2 +( \mu _d^2 \\&\qquad +\sigma _d^2 +\rho _{d,s}^2 \sigma _d^2 )\sigma _s^2-2\rho _{d,s} \sigma _d \sigma _s \mu _d ( {r-\mu _s }). \end{aligned}$$

Then, we find that \(\mathrm{Var}( {\Pi _\mathrm{BS} })\) is a quadratic function with the form of \(f( Q)=aQ^2+bQ+c\), where

$$\begin{aligned} a&= ( {\mu _s -w})^2\sigma _y^2 +( {\mu _y^2 +\sigma _y^2 })\sigma _s^2 +\rho _{y,s}^2 \sigma _y^2 \sigma _s^2 +2\rho _{y,s} \sigma _s \sigma _y \mu _y ( {\mu _s -w}) \\&= [( {\mu _s -w})\sigma _y +\rho _{y,s} \mu _y \sigma _s ]^2+( {\mu _y^2 +\sigma _y^2 })\sigma _s^2 -\rho _{y,s}^2 \sigma _s^2 \mu _y^2 +\rho _{y,s}^2 \sigma _y^2 \sigma _s^2 \\&= [( {\mu _s -w})\sigma _y +\rho _{y,s} \mu _y \sigma _s ]^2+( {1-\rho _{y,s}^2 })\sigma _s^2 \mu _y^2 +(1+\rho _{y,s}^2 )\sigma _s^2 \sigma _y^2 . \end{aligned}$$

Since \(0\le \rho _{y,s}^2 \le 1\), we have that \(a>0\). Therefore, the parabola opens upward.

Appendix 4: Derivation of the optimal order quantity \(Q^*\)

The objective function \(Z( Q)\) equals to \(E(\Pi _\mathrm{BS} )-k\mathrm{Var}(\Pi _\mathrm{BS} )\). Since \(Y=yQ\), \(\mu _Y =\mu _y Q\), and \(\sigma _Y =\sigma _y Q\), substituting them into \(Z(Q)\), we obtain that

$$\begin{aligned} Z( Q)&= \mu _d ( {r-\mu _s })+\mu _y Q( {\mu _s -w})+\sigma _s ( {\rho _{y,s} \sigma _y Q-\rho _{d,s} \sigma _d }) \\&-k\{( {r-\mu _s })^2\sigma _d^2 +( {\mu _s -w})^2\sigma _y^2 Q^2 \\&+\sigma _s^2 ( {\mu _y^2 Q^2+\sigma _y^2 Q^2+\mu _d^2 +\sigma _d^2 +\rho _{d,s}^2 \sigma _d^2 +\rho _{y,s}^2 \sigma _y^2 Q^2-2\mu _y Q\mu _d }) \\&-2\rho _{d,s} \sigma _d \sigma _s ( {r-\mu _s })( {\mu _d -\mu _y Q})-2\rho _{y,s} \sigma _s \sigma _y Q( {\mu _s -w})( {\mu _d -\mu _y Q}) \\&-2\rho _{d,s} \rho _{y,s} \sigma _d \sigma _y Q\sigma _s^2 -2\rho _{d,y} \sigma _d \sigma _y Q [{( {\mu _s -r})( {\mu _s -w})+\sigma _s^2 } ]\}. \end{aligned}$$

Thus,

$$\begin{aligned} \frac{d^2Z(Q)}{dQ^2}&= -k[2( {w-\mu _s })^2\sigma _y^2 +2\sigma _s^2 (\mu _y^2 +\sigma _y^2 )+2\rho _{y,s}^2 \sigma _s^2 \sigma _y^2 \\&+4( {\mu _s -w})\rho _{y,s} \mu _y \sigma _s \sigma _y ] \\&= -2k\big \{ {\left[ {( {\mu _s -w})\sigma _y +\rho _{y,s} \mu _y \sigma _s } \right] ^2+\sigma _s^2 [(1-\rho _{y,s}^2 )\mu _y^2 +(1+\rho _{y,s}^2 )\sigma _y^2 \big ]} \big \}. \end{aligned}$$

Since \(\rho _{y,s} \) is a correlation coefficient, we have that \(1-\rho _{y,s}^2 \ge 0\). Therefore, \(\frac{d^2Z(Q)}{dQ^2}\le 0\). So, \(Z( Q)\) is concave and the optimal order quantity \(Q^*\) can be obtained through \(\frac{dZ( Q)}{dQ}=0\). Based on the expression of \(Z( Q)\), we have that

$$\begin{aligned}&\frac{dZ( Q)}{dQ}=\mu _y ( {\mu _s -w})+\rho _{y,s} \sigma _y \sigma _s \\&\quad +2k \{\mu _y \mu _d \sigma _s^2 -\rho _{d,s} \mu _y \sigma _s \sigma _d ( {r-\mu _s })+\rho _{y,s} \mu _d \sigma _s \sigma _y ( {\mu _s -w})+\rho _{d,s} \rho _{y,s} \sigma _d \sigma _y \sigma _s^2 \\&\quad +\rho _{d,y} \sigma _d \sigma _y [( {\mu _s -r})( {\mu _s -w})+\sigma _s^2 ] \} \\&\quad -2kQ[( {w-\mu _s })^2\sigma _y^2 +(\mu _y^2 +\sigma _y^2 )\sigma _s^2 +\rho _{y,s}^2 \sigma _y^2 \sigma _s^2 +2\rho _{y,s} \mu _y \sigma _s \sigma _y ( {\mu _s -w})]. \end{aligned}$$

Let \(\frac{dZ( Q)}{dQ}=0\). Then, the optimal order quantity is as follows:

$$\begin{aligned} \frac{\begin{array}{ll} \frac{( {\mu _s -w})\mu _y +\rho _{y,s} \sigma _y \sigma _s }{2k}&{}+\mu _d \mu _y \sigma _s^2 -\rho _{d,s} \mu _y \sigma _s \sigma _d ( {r-\mu _s })+\rho _{y,s} \mu _d \sigma _s \sigma _y ( {\mu _s -w})\\ &{}\qquad +\rho _{d,s} \rho _{y,s} \sigma _d \sigma _y \sigma _s^2 +\rho _{d,y} \sigma _d \sigma _y [( {\mu _s -r})( {\mu _s -w})+\sigma _s^2 ]\end{array}}{( {w-\mu _s })^2\sigma _y^2 +( {\mu _y^2 +\sigma _y^2 })\sigma _s^2 +\rho _{y,s}^2 \sigma _y^2 \sigma _s^2 +2\rho _{y,s} \mu _y \sigma _s \sigma _y ( {\mu _s -w})}. \end{aligned}$$

Appendix 5: Proof of Proposition 2

Based on the analytical expression of the optimal order quantity \(Q^*\), \(\frac{dQ^*}{d\rho _{d,y} }\) is computed as follows:

$$\begin{aligned} \frac{dQ^*}{d\rho _{d,y} }=\frac{\sigma _d \sigma _y [{( {\mu _s -r})( {\mu _s -w})+\sigma _s^2 }]}{( {w-\mu _s })^2\sigma _y^2 +( {\mu _y^2 +\sigma _y^2 })\sigma _s^2 +\rho _{y,s}^2 \sigma _y^2 \sigma _s^2 +2\rho _{y,s} \mu _y \sigma _s \sigma _y ( {\mu _s -w})}. \end{aligned}$$

Since \(\rho _{d,y} <0\), in order to investigate the effect of \(\rho _{d,y} \) on the optimal order quantity \(Q^*\), we need to calculate \(\frac{dQ^*}{d(-\rho _{d,y} )}\), which is computed as follows:

$$\begin{aligned} \frac{dQ^*}{d(-\rho _{d,y} )}=\frac{-\sigma _d \sigma _y [ {( {\mu _s -r})( {\mu _s -w})+\sigma _s^2 } ]}{( {w-\mu _s })^2\sigma _y^2 +( {\mu _y^2 +\sigma _y^2 })\sigma _s^2 +\rho _{y,s}^2 \sigma _y^2 \sigma _s^2 +2\rho _{y,s} \mu _y \sigma _s \sigma _y ( {\mu _s -w})}. \end{aligned}$$

Let \(\frac{dQ^*}{d(-\rho _{d,y} )}>0\), we have that \(\frac{-\sigma _d \sigma _y [{( {\mu _s -r})( {\mu _s -w})+\sigma _s^2 }]}{( {w-\mu _s })^2\sigma _y^2 +( {\mu _y^2 +\sigma _y^2 })\sigma _s^2 +\rho _{y,s}^2 \sigma _y^2 \sigma _s^2 +2\rho _{y,s} \mu _y \sigma _s \sigma _y ( {\mu _s -w})}>0.\) According to a conclusion in Appendix 3, \(( {w-\mu _s })^2\sigma _y^2 +( {\mu _y^2 +\sigma _y^2 })\sigma _s^2 +\rho _{y,s}^2 \sigma _y^2 \sigma _s^2 +2\rho _{y,s} \mu _y \sigma _s \sigma _y ( {\mu _s -w})>0\). So, to guarantee that \(\frac{dQ^*}{d(-\rho _{d,y} )}>0\), we should have that \(-\sigma _d \sigma _y [{( {\mu _s -r})( {\mu _s -w})+\sigma _s^2 }]>0\).

Appendix 6: Proof of Proposition 3

Based on the analytical expression of the optimal order quantity \(Q^*\), \(\frac{dQ^*}{d\rho _{d,s} }\) is computed as follows:

$$\begin{aligned} \frac{dQ^*}{d\rho _{d,s} }=\frac{-\sigma _d \sigma _s [{\mu _y ( {r-\mu _s })-\rho _{y,s} \sigma _s \sigma _y } ]}{( {w-\mu _s })^2\sigma _y^2 +( {\mu _y^2 +\sigma _y^2 })\sigma _s^2 +\rho _{y,s}^2 \sigma _y^2 \sigma _s^2 +2\rho _{y,s} \mu _y \sigma _s \sigma _y ( {\mu _s -w})}. \end{aligned}$$

Since \(( {w-\mu _s })^2\sigma _y^2 +( {\mu _y^2 +\sigma _y^2 })\sigma _s^2 +\rho _{y,s}^2 \sigma _y^2 \sigma _s^2 +2\rho _{y,s} \mu _y \sigma _s \sigma _y ( {\mu _s -w})>0\), to guarantee that \(\frac{dQ^*}{d\rho _{d,s} }>0,\,-\sigma _d \sigma _s [{\mu _y ( {r-\mu _s })-\rho _{y,s} \sigma _s \sigma _y }]\) must be greater than 0.

Appendix 7: Proof of Proposition 4

When all the risks are not correlated, we have that

$$\begin{aligned} Q^*=\frac{\frac{( {\mu _s -w})\mu _y }{2k}+\mu _d \mu _y \sigma _s^2 }{( {w-\mu _s })^2\sigma _y^2 +( {\mu _y^2 +\sigma _y^2 })\sigma _s^2 }. \end{aligned}$$

Thus, \(\frac{dQ^*}{d\mu _d }=\frac{\mu _y \sigma _s^2 }{( {w-\mu _s })^2\sigma _y^2 +( {\mu _y^2 +\sigma _y^2 })\sigma _s^2 }>0\).

Appendix 8: Proof of Proposition 5

When all the risks are not correlated, we have that

$$\begin{aligned} \frac{dQ^*}{d\mu _y }&= \frac{( {\frac{\mu _s -w}{2k}+\mu _d \sigma _s^2 }) [ {( {w-\mu _s })^2\sigma _y^2 +( {\mu _y^2 +\sigma _y^2 })\sigma _s^2 }]-2\mu _y \sigma _s^2 [ {\frac{( {\mu _s -w})\mu _y }{2k}+\mu _d \mu _y \sigma _s^2 }]}{[( {w-\mu _s })^2\sigma _y^2 +( {\mu _y^2 +\sigma _y^2 })\sigma _s^2 ]^2} \\&= \frac{( {\frac{\mu _s -w}{2k}+\mu _d \sigma _s^2 }) [ {( {w-\mu _s })^2\sigma _y^2 +( {\sigma _y^2 -\mu _y^2 })\sigma _s^2 } ]}{[( {w-\mu _s })^2\sigma _y^2 +( {\mu _y^2 +\sigma _y^2 })\sigma _s^2 ]^2}. \end{aligned}$$

To guarantee that \(\frac{dQ^*}{d\mu _y }>0\), we must have that \(( {\frac{\mu _s -w}{2k}+\mu _d \sigma _s^2 })[ ( {w-\mu _s })^2\sigma _y^2 +( \sigma _y^2-\mu _y^2 )\sigma _s^2 ]>0\).

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Hong, Z., Lee, C.K.M. & Nie, X. Proactive and reactive purchasing planning under dependent demand, price, and yield risks. OR Spectrum 36, 1055–1076 (2014). https://doi.org/10.1007/s00291-013-0344-5

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