Designing structured supply contracts under demand and price uncertainty in an open supply chain

Abstract

This paper develops an integrated model for analyzing and designing structured supply contracts from the perspectives of the buyer, the supplier, and the entire supply chain in an open supply chain. We first present a flexible framework that encapsulates a wide range of contracting types that have been studied previously. We then introduce the concept of relative contract value vis-a-vis a reference alternative, which facilitates addressing explicitly both the demand uncertainty and the price uncertainty within which real supply relationships operate. To guide practitioners in designing optimal supply contracts, we derive closed-form expressions for optimal contract structure, quantity commitment and flexibility, pricing, and sharing policy as well as the conditions to maximize total supply chain profitability associated with a contract. Our research demonstrates that structured contracts consisting of several fixed and/or flexible components are capable of maximizing total supply chain profit and allocating profit between contracting parties arbitrarily.

Appendices

Appendix 1: Proof of Propositions

Proof of Proposition 2

Equation (10) gives the buyer’s maximum commitment

$$\begin{aligned} Q_{ti}^{U}(P_{ti})_{\max }=\Phi _{t}^{-1}\left( \frac{\theta E[\widetilde{S} _{t}]-P_{ti}}{\theta (1-h)E[\widetilde{S}_{t}]}\right) , \end{aligned}$$

(23)

which shows that the buyer will commit only $Q_{ti}^{U}(P_{ti})_{\max }=D_{t}^{a}$ if $\theta =P_{ti}/E[\widetilde{S}_{t}]$ and increase commitment to $Q_{ti}^{U}(P_{ti})_{\max }=D_{t}^{b}$ with the increase of $\theta$ to $P_{ti}/(hE[\widetilde{S}_{t}])$. Equations (23), (9), and (10) indicate that the buyer will commit the supply chain optimal quantity and the buyer’s contract value will be $E[V_{ti}^{B}]=\theta E[\pi _{ti}^{Fix}]$, if $\theta =P_{ti}/\beta E[ \widetilde{S}_{t}]$. $\square$

Proof of Proposition 3

The total commitment is dependent on $P_{tn}$, irrespective of the price of components 1 through $n-1$. By employing fixed components with decreasing component price $P_{ti}$ , $E[\widetilde{S}_{t}]\ge P_{t1}>\cdots >P_{tn}\ge \beta E[\widetilde{S} _{t}]$, $1\le i\le n$, the supplier can induce the buyer to commit up to $Q^{U}(P_{tn})_{\max }$ and gain from the higher price of components 1 through $n-1$. If $P_{tn}=\beta E[\widetilde{S}_{t}]$, the supply chain achieves its maximum joint value. As $n\rightarrow \infty$, if we keep setting $P_{t}=E[\widetilde{S}_{t}]-(1-h)E[\widetilde{S}_{t}]\Phi _{t}(Q_{t})$ for $Q_{t}$ over the entire range $[0,Q_{tn}^{U}]$, the buyer’s marginal value will tend to zero. The supplier will obtain the entire supply chain value, i.e.,

$$\begin{aligned} \sum E\left[ V_{ti}^{S}\right] =\int _{\beta E[\widetilde{S}_{t}]}^{E[\widetilde{S} _{t}]}\Phi _{t}^{-1}\left( \frac{E[\widetilde{S}_{t}]-P_{t}}{(1-h)E[\widetilde{S} _{t}]}\right) dP_{t}, \end{aligned}$$

(24)

which can be simplified to $E[\widetilde{S}_{t}]\int _{\beta }^{1} \Phi _{t}^{-1}((1-x)/(1-h))dx$. $\square$

Proof of Proposition 5

Proposition 4 demonstrates that the supply chain value is maximized if $P_{t}^{*}=F_{t}^{-1}(0)$. Substituting $P_{t}^{*}=F_{t}^{-1}(0)$ into Eq. (17), we obtain the optimal supply chain commitment $Q_{t}^{U^{*}}=\Phi _{t}^{-1}((1-\beta )/(1+e-\beta ))$, which indicates the corresponding $X_{t}^{*}=e(E[\widetilde{S}_{t}]-P_{t}^{*})/(1+e-\beta ))$ from Eq. (13). Equation (15) yields the second derivative with respect to the commitment $Q_{t}$ of the supplier’s value of a single flexible component:

$$\begin{aligned} d^{2}E\left[ V_{t}^{S}\right] \Big /dQ_{t}^{2}=-\Phi _{t}^{^{\prime }}(Q_{t})(P_{t}-(\beta -e)E[ \widetilde{S}_{t}]). \end{aligned}$$

(25)

$P_{t}^{*}=F_{t}^{-1}(0)$ indicates that $P_{t}^{*}\le S_{m}$. From Eq. (15), we know $dE[V_{t}^{S}]/dQ_{t}=0$ at $Q_{t}=Q_{t}^{U^{*}}.$ In some cases the supplier always suffers a loss by a supply chain optimal contract consisting of a single flexible component. For example, if $S_{m}<(\beta -e)E[\widetilde{S}_{t}]$, $d^{2}E[V_{t}^{S}]/dQ_{t}^{2}>0$, i.e., $E[V_{t}^{S}]$ is a convex function in $Q_{t}$, when $0<Q_{t}<Q_{t}^{U^{*}}$. Since $E[V_{t}^{S}]=0$ at $Q_{t}=0$ and $dE[V_{t}^{S}]/dQ_{t}=0$ at $Q_{t}=Q_{t}^{U^{*}}$, $E[V_{t}^{S}]<0$ at $Q_{t}=Q_{t}^{U^{*}}$. In other words, the supplier’s expected value from a single flexible component is negative. Therefore, the supply chain optimum cannot be achieved. $\square$

Proof of Proposition 6

Proposition 7 gives the upper and the lower bounds on $X_{ti}$ as in Eq. (20). If Eq. (20) holds for component n, the buyer will reserve the supply chain optimal quantity $\Phi _{t}^{-1}((1- \beta )/(1+e-\beta ))$. Adjusting $X_{ti}$ but keeping $P_{ti}=F_{t}^{-1}(0)$ for $1\le i\le n-1$ changes how the joint value will be split. As $n\rightarrow \infty$, at the lower bound, i.e., $X_{ti}=eE[\widetilde{S} _{t}]-(1-\Phi _{t}(Q_{ti}^{U}))(P_{ti}-(\beta -e)E[\widetilde{S}_{t}])$, the supplier’s marginal value $dE[V_{t}^{S}]/dQ_{t}^{U}$ is always zero and the buyer obtains the entirety of the supply chain value. Conversely, at the upper bound, i.e., $X_{ti}=(1-\Phi _{t}(Q_{ti}^{U}))(E[\widetilde{S} _{t}]-P_{ti})$, the supplier obtains the entire supply chain value. $\square$

Proof of Proposition 7

The marginal value for the buyer should be non-negative. Equation (15) indicates $X_{ti}\ge eE[ \widetilde{S}_{t}]-(1-\Phi _{t}(Q_{ti}^{U}))\int _{ti}^{\infty }[P_{ti}-( \beta -e)S_{t}]F_{t}^{^{\prime }}(S_{t})dS_{t}$. Similarly, Eq. (12 ) suggests $X_{ti}\le (1-\Phi _{t}(Q_{ti}^{U}))\int _{ti}^{ \infty }(S_{t}-P_{ti})F_{t}^{^{\prime }}(S_{t})dS_{t}$. The lower and the upper bounds on $X_{ti}$ intersect at

$$\begin{aligned} Q_{t}=\Phi _{t}^{-1}\left( 1-\frac{eE[\widetilde{S}_{t}]}{(1+e-\beta )(E[\widetilde{S }_{t}]-\int _{0}^{P_{ti}}S_{t}F_{t}^{^{\prime }}(S_{t})dS_{t})}\right) , \end{aligned}$$

(26)

which is exactly the supply chain optimal quantity $Q_{t}^{U^{*}}$ for a given $P_{ti}$, indicated by Eq. (17). $\square$

Proof of Proposition 8

Integration by parts transforms Eq. (13) into

$$\begin{aligned} Q_{t}^{U}(X_{ti},P_{ti})_{\max }=\Phi _{t}^{-1}\left( 1-\frac{X_{ti}}{E[\widetilde{S} _{t}]-P_{ti}+\int _{0}^{P_{ti}}F_{t}(S_{t})dS_{t}}\right) . \end{aligned}$$

(27)

To prove $\int _{0}^{P_{ti}}F_{t}(S_{t})dS_{t}$ increases in $\sigma _{t},$ let $S_{t}^{^{\prime }}=S_{t}-S_{m}$ and $P_{ti}^{^{\prime }}=P_{ti}-S_{m}$. Equation (1) gives

$$\begin{aligned} \frac{d\int _{0}^{P_{ti}}F_{t}(S_{t})dS_{t}}{d\sigma _{t}} =\int _{0}^{P_{ti}^{^{\prime }}}N_{0,1}^{^{\prime }}\left( \frac{\ln \left( S_{t}^{^{ \prime }}/S_{0}^{^{\prime }}\right) -\mu _{t}}{\sigma _{t}}\right) \left( 1-\frac{ \ln \left( S_{t}^{^{\prime }}/S_{0}^{^{\prime }}\right) -\mu _{t}}{\sigma _{t}^{2}} \right) dS_{t}^{^{\prime }}. \end{aligned}$$

(28)

Let $\mu _{t}=(a-\sigma ^{2}/2)t$, $\sigma _{t}^{2}=\sigma ^{2}t,$ and $x=(\ln (S_{t}^{^{\prime }}/S_{0}^{^{\prime }})-\mu _{t})/\sigma _{t},$ i.e., $S_{t}^{^{\prime }}=S_{0}^{^{\prime }}\exp (x\sigma _{t}+\mu _{t}).$ Then

$$\begin{aligned} \frac{d\int _{0}^{P_{ti}}F_{t}(S_{t})dS_{t}}{d\sigma _{t}}&=\frac{ S_{0}^{^{\prime }}}{\sqrt{2\pi }}\int _{-\infty }^{\frac{\ln (P_{ti}^{^{ \prime }}/S_{0}^{^{\prime }})-\mu _{t}}{\sigma _{t}}}\exp \left( -\frac{x^{2}}{2} \right) (-x+\sigma _{t})\exp (x\sigma _{t}+\mu _{t})dx \nonumber \\&=\frac{P_{ti}^{^{\prime }}}{\sqrt{2\pi }}\exp \left( -\frac{(\ln \left( P_{ti}^{^{ \prime }}/S_{0}^{^{\prime }}\right) -\mu _{t})^{2}}{2\sigma _{t}^{2}}\right) >0, \end{aligned}$$

(29)

which indicates that the buyer’s optimal reserved quantity is an increasing function of $\sigma _{t}$.

From Eq. (12), it is easy to prove that $dE[V_{t}^{B}]/dQ_{t}^{U}$ increases with $\sigma _{t}$. Whether the supply chain’s marginal value increases or decreases with $\sigma _{t}$ depends on $P_{ti}$. We rewrite Eq. (17) as:

$$\begin{aligned} dE\left[ \pi _{t}^{Flex}\right] \Big /dQ_{ti}^{U}= & {} \left( 1\!-\!\Phi _{t}\left( Q_{ti}^{U}\right) \right) (1\!+\!e\!-\!\beta )\left( E[ \widetilde{S}_{t}]\!-\!P_{ti}F_{t}(P_{ti})+ \int _{0}^{P_{ti}}F_{t}(S_{t})dS_{t}\right) \nonumber \\&-\,eE[\widetilde{S}_{t}]. \end{aligned}$$

(30)

Consider the term $-P_{ti}F_{t}(P_{ti})+\int _{0}^{P_{ti}}F_{t}(S_{t})dS_{t}.$ From Eqs. (1) and (29), we obtain

$$\begin{aligned} \frac{d(-P_{ti}F_{t}(P_{ti})+\int _{0}^{P_{ti}}F_{t}(S_{t})dS_{t})}{d\sigma _{t}}= & {} \frac{1}{\sqrt{2\pi }} \left[ P_{ti}^{^{\prime }}+\left( P _{ti}^{^{\prime }} {+S}_{m}\right) \left( \frac{\ln \left( P_{ti}^{^{\prime }}/S_{0}^{^{\prime }}\right) -at}{\sigma _{t}^{2}}{-}\frac{1 }{2}\right) \right] \nonumber \\&\exp \left( -\frac{\left( \ln \left( P_{ti}^{^{\prime }}/S_{0}^{^{\prime }}\right) -\mu _{t}\right) ^{2}}{2\sigma _{t}^{2}}\right) . \end{aligned}$$

(31)

Obviously, $P_{ti}^{^{\prime }}+(P_{ti}^{^{\prime }}+S_{m})\left( \frac{\ln (P_{ti}^{^{\prime }}/S_{0}^{^{\prime }})-at}{\sigma _{t}^{2}}-\frac{1}{2}\right)$ determines the sign of $\frac{d(dE[\pi _{t}^{Flex}]/dQ_{ti}^{U})}{d\sigma _{t}}$. We can see that there is a threshold $P_{ti}^{a}$ for $P_{ti}$ satisfying $\ln \left( \frac{P_{ti}^{^{a}}-S_{m}}{S_{0}-S_{m}}\right) =at+\frac{1}{2} \sigma _{t}^{2}-\frac{P_{ti}^{^{a}}-S_{m}}{P_{ti}^{^{a}}}\sigma _{t}^{2}$ such that $dE[\pi _{t}^{Flex}]/dQ_{ti}^{U}$ increases in $\sigma _{t}$ when $P_{ti}>P_{ti}^{a}$ and decreases in $\sigma _{t}$ when $P_{ti}<P_{ti}^{a}.$ Furthermore, we obtain $(S_{0}-S_{m})\exp (at-\frac{1}{2} \sigma _{t}^{2})+S_{m}\le P_{ti}^{a}<(S_{0}-S_{m})\exp (at+\frac{1}{2} \sigma _{t}^{2})+S_{m}$. $\square$

Proof of Proposition 9

From Eq. (8), we know the supply chain value of fixed components increases in $Q_{t}$ when $0\le Q_{t}\le$ $\Phi _{t}^{-1}((1-\beta )/(1-h))$. From Eq. (17), we know it is supply chain suboptimal to employ flexible components if $(1+e-\beta )\int _{0}^{P_{t}}S_{t}F_{t}^{^{\prime }}(S_{t})dS_{t}>(1-\beta )E[ \widetilde{S}_{t}]$, since the marginal supply chain value is negative and the increase in $Q_{t}$ only results in more loss.

Consider the range $(1+e-\beta )\int _{0}^{P_{t}}S_{t}F_{t}^{^{\prime }}(S_{t})dS_{t}\le (1-\beta )E[\widetilde{S}_{t}].$ Let $G(Q_{t})= \frac{dE[\pi _{t}^{Flex}]}{dQ_{t}}-\frac{dE[\pi _{t}^{Fix}]}{dQ_{t}}.$ Then

$$\begin{aligned} G(Q_{t})= & {} \Phi _{t}(Q_{t}) \left\{ (1+e-\beta ) \int _{0}^{P_{t}}{S}_{t}{F}_{t}^{^{\prime }} (S_{t} )dS_{t}-(e+h-\beta )E[\widetilde{S}_{t}]\right\} \nonumber \\&-\,(1+e-\beta )\int _{0}^{P_{t}}{S}_{t}{F}_{t}^{^{\prime }} (S_{t})dS_{t}. \end{aligned}$$

(32)

If $h>\beta -e,$ then $(e+h-\beta )E[\widetilde{S}_{t}]>0.$ Therefore,

$$\begin{aligned} {(1+e-\beta )}\int _{0}^{P_{t}}{S}_{t}{F}_{t}^{^{\prime }} {(S}_{t}{)dS}_{t}{-(e+h-\beta )E[}\widetilde{S}_{t} {]<(1+e-\beta )}\int _{0}^{P_{t}}{S}_{t}{F} _{t}^{^{\prime }}{(S}_{t}{)dS}_{t}{.} \end{aligned}$$

(33)

This indicates that $G(Q_{t})<0$, and in turn that fixed components are supply chain advantageous over flexible components. $h>\beta -e$ suggests that the buyer is better positioned than the supplier to process the excess inventory.

If $h<\beta -e$, then $(e+h-\beta )E[\widetilde{S}_{t}]<0$. The marginal supply chain value of optimal flexible components will be greater than that of optimal fixed components, i.e., $G(Q_{t})>0$, when $Q_{t}>\Phi _{t}^{-1}\left( \frac{{(1+e-\beta )}\int _{0}^{P_{t}}{S}_{t}{F} _{t}^{^{\prime }}{(S}_{t}{)dS}_{t}}{{(1+e-\beta )} \int _{0}^{P_{t}}{S}_{t}{F}_{t}^{^{\prime }}{(S}_{t} {)dS}_{t}-(e+h-\beta )E[\widetilde{S}_{t}]}\right)$. Otherwise, it will be less than or equal to that of optimal fixed components. $\square$

Appendix 2: Notation

The notation used in the analysis is summarized as follows.

a :

Expected appreciation rate of $S_{t}^{^{\prime }}$

b :

Return credit per unit in a return contract

${C}_{i}$ :

Component i of a contract

${D}_{t}$ :

Demand at period t

${D}_{t}^{a}$ :

Minimum possible demand at period t

${D}_{t}^{b}$ :

Maximum possible demand at period t

${eS}_{{t}}$ :

Unit loss at the supplier resulting from the quantity committed but not purchased by the buyer

${F}_{t}{(S}_{t}{)}$ :

Cumulative distribution function of $S_{t}$

${hS}_{t}$ :

Salvage value of unit excess inventory at the buyer after period t

${P}_{ti}$ :

Purchase price in contract component i at period t

${q}_{\max }$ :

Maximum quantity the buyer can purchase

${q}_{\min }$ :

Minimum quantity the buyer must purchase

${q}_{r}$ :

Reserved quantity in a back-up or return contract

${Q}_{ti}^{L}$ :

Lower purchase quantity breakpoint of contract component i at period t

${Q}_{ti}^{U}$ :

Upper purchase quantity breakpoint of contract component i at period t

${Q}_{t}^{U^{*}}$ :

Supply chain optimal purchase quantity at period t

${S}_{m}$ :

Minimum spot market price, beneath which suppliers will exit the industry

${S}_{t}$ :

Price of the reference alternative at period t

${V}_{ti}^{B}$ :

The buyer’s expected value from contract component i at period t

${V}_{ti}^{S}$ :

The supplier’s expected value from contract component i at period t

w :

Purchase price in a back-up or return contract

${w}_{p}$ :

Unit penalty cost in a back-up contract paid by the buyer to the supplier for each unit reserved but not purchased

${X}_{ti}$ :

Up-front unit cost incurred at the contracting time for contract component i at period t

${\beta S}_{{t}}$ :

Price at which the supplier can sell her products outside the supply chain

${\theta }$ :

Quota of the total revenue that the buyer can retain

${\mu }_{t}$ :

Mean of $\ln S_{t}^{^{\prime }}$ at period t

${\pi }_{ti}^{Fix}$ :

The supply chain’s expected value from fixed contract component i at period t

${\pi }_{ti}^{Flex}$ :

The supply chain’s expected value from flexible contract component i at period t

${\sigma }$ :

Volatility coefficient of $S_{t}^{^{\prime }}$

${\Phi }_{t}{(\cdot )}$ :

Truncated normal distribution function of $D_{t}$