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Supply chain coordination by risk sharing contracts under random production yield and deterministic demand

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Abstract

From numerous contributions to literature we know that properly designed contracts can facilitate coordinated decision making of multiple actors in a supply chain (SC) so that efficiency losses for the whole SC can be avoided. In a newsvendor-type SC with stochastic demand it is well-known that the double marginalization effect hampers the simple wholesale price contract to achieve coordination. More complex contracts however can bring about coordination, especially those which enable appropriate sharing of risks between the actors. While the effectiveness of risk sharing contracts is well understood for SC situations with random demand and reliable supply, less is known about respective problems if demand is deterministic but supply is unreliable due to random production yield. This paper shows how in a buyer-supplier SC the distribution of risks affects the coordination of buyer’s ordering and supplier’s production decision in a basic random yield, deterministic demand setting. Both parties are exposed to risks of over-production or under-delivery, respectively, if a simple wholesale price contract is applied. The resulting risk distribution always impedes SC coordination. However, more sophisticated contract types which penalize or reward the supplier can change risk distribution so that SC coordination is possible under random yield. Additionally, it is proven that the wholesale price contract will guarantee SC coordination if the supplier has a second (emergency) procurement source that is more costly, but reliable. Moreover, it is shown that under wholesale price contracts it can be beneficial to utilize this emergency source even if it is unprofitable from a SC perspective. However, if such an emergency option is available to the buyer as opposed to the supplier, a wholesale price contract will not be able to coordinate the SC.

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Notes

  1. The profit function structure and the way of proving concavity and exploiting the FOC is identical for supplier’s profit functions under all following contracts. For that reason the respective derivations are not repeated hereafter.

  2. The profit function structure and the way of proving concavity and exploiting the FOC is identical for buyer’s profit functions under all following contracts. For that reason the respective derivations are not further repeated.

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Correspondence to Josephine Clemens.

Appendices

Appendix 1

Proof of concavity of \(\Pi _\mathrm{SC} (Q_\mathrm{SC} )\) in (1):Footnote 1

For the first-order and second-order derivative of \(\Pi _\mathrm{SC} (Q_\mathrm{SC})\) we get

$$\begin{aligned} \frac{\partial }{\partial Q_\mathrm{SC} }\Pi _\mathrm{SC} (Q_\mathrm{SC} )&= p\cdot \int \limits _0^{D/Q_\mathrm{SC}} {z\cdot \varphi (z)dz-c}\quad \text{ and}\\ \frac{\partial ^2}{\partial Q_\mathrm{SC}^2 }\Pi _\mathrm{SC} (Q_\mathrm{SC})&= -p\cdot \frac{D^2}{Q_\mathrm{SC}^3 }\cdot \varphi \left({\frac{D}{Q_\mathrm{SC} }}\right)<0. \end{aligned}$$

Thus, \(\Pi _\mathrm{SC}\) is concave in \(Q_\mathrm{SC}.\) Setting the first-order derivative equal to zero, i.e. \(\partial \Pi _\mathrm{SC} /\partial Q_\mathrm{SC} =0,\) equation (2) can immediately be derived.

Proof of concavity of \(\Pi _\mathrm{B}^\mathrm{WHP} (X\vert Q_\mathrm{S}^\mathrm{WHP} )\) in (11):Footnote 2

For the first-order and second-order derivative of \(\Pi _\mathrm{B}^\mathrm{WHP} (X\vert Q_\mathrm{S}^\mathrm{WHP} )\) we get \(\frac{\partial }{\partial X}\Pi _\mathrm{B}^\mathrm{WHP} (X\vert Q_\mathrm{S}^\mathrm{WHP} )\!=\!p\cdot K_\mathrm{S}^\mathrm{WHP} \cdot \int _0^{D/(K_\mathrm{S}^\mathrm{WHP} \cdot X)} z\cdot \varphi (z)dz\!-\!w\cdot \Big [K_\mathrm{S}^\mathrm{WHP} \cdot \int _0^{1/K_\mathrm{S}^\mathrm{WHP}} z\cdot \varphi (z)dz-\Phi (1/K_\mathrm{S}^\mathrm{WHP} )\!+\!1\Big ]\) and

\(\frac{\partial ^{2}}{\partial X^{2}}\Pi _{B}^\mathrm{WHP} (X\vert Q_\mathrm{S}^\mathrm{WHP} )=-p\cdot \frac{D^2}{(K_\mathrm{S}^\mathrm{WHP} )^{2} \cdot X^{3}}\cdot \varphi ({\frac{D}{K_\mathrm{S}^\mathrm{WHP} \cdot X}}) < 0.\)

Thus, \(\Pi _\mathrm{B}^\mathrm{WHP} (X\vert Q_\mathrm{S}^\mathrm{WHP})\)is concave in \(X.\) Setting the first-order derivative equal to zero, i.e. \(\frac{\partial }{\partial X}\Pi _\mathrm{B}^\mathrm{WHP} (X\vert Q_\mathrm{S}^\mathrm{WHP} )=0,\) and exploiting \(\int _0^{1/K_\mathrm{S}^\mathrm{WHP} } {z\cdot \varphi (z)dz=\frac{c}{w}}\) from equation (9) yields the respective optimality condition for \(X^\mathrm{WHP}\) in (12).

Appendix 2

First-order derivatives to following profit functions:

$$\begin{aligned} (15)\quad \, \frac{\partial }{\partial X}\Pi _\mathrm{B}^\mathrm{WHP} (X\vert Q_\mathrm{S}^\mathrm{WHP} )&= (p-w)\cdot \left[ {\frac{c}{w}\cdot K_\mathrm{S}^\mathrm{WHP} +1-\Phi (1/K_\mathrm{S}^\mathrm{WHP} )} \right]\\ (19)\,\,\,\;\;\quad \frac{\partial }{\partial Q_\mathrm{S} }\Pi _\mathrm{S}^\mathrm{ORS} (Q_\mathrm{S}\vert X)&= (w-w_{O})\cdot \int \limits _0^{X/Q_\mathrm{S}} {z\cdot \varphi (z)dz+w_O \cdot \mu _z -c}\\ (25)\quad \,\,\, \frac{\partial }{\partial X}\Pi _\mathrm{B}^\mathrm{ORS} (X\vert Q_\mathrm{S}^\mathrm{ORS} )&\! = p\cdot K_\mathrm{S}^\mathrm{ORS} \cdot \int \limits _0^{D/(K_\mathrm{S}^\mathrm{ORS} \cdot X)} {z\cdot \varphi (z)dz}\nonumber \\&\!-(w\!-\!w_O )\cdot \left[ {(1\!-\!\Phi (1/K_\mathrm{S}^\mathrm{ORS} ))\!+\!K_\mathrm{S}^\mathrm{ORS} \cdot \!\!\int \limits _0^{1/K_\mathrm{S}^\mathrm{ORS} }\!\! {z\cdot \varphi (z)dz} } \right]\nonumber \\&\!-w_O \cdot \mu _z \cdot K_\mathrm{S}^\mathrm{ORS}\\ (28)\,\,\,\quad \frac{\partial }{\partial X}\Pi _\mathrm{B}^\mathrm{ORS} (X\vert Q_\mathrm{S}^\mathrm{ORS} )&= \left[ {\frac{p\cdot (c-w_0 \cdot \mu _Z )}{w-w_0 }-c} \right]\cdot K_\mathrm{S}^\mathrm{ORS} \nonumber \\&+(p-w+w_O )\cdot \left[ {1-\Phi (1/K_\mathrm{S}^\mathrm{ORS} )} \right]\\ (33)\quad \,\,\,\,\,\, \frac{\partial }{\partial Q_\mathrm{S} }\Pi _\mathrm{S}^\mathrm{PEN} (Q_\mathrm{S} \vert X)&= (w+\pi )\cdot \int \limits _0^{X/Q_\mathrm{S}} {z\cdot \varphi (z)dz-c}\\ (39)\,\,\,\quad \frac{\partial }{\partial X}\Pi _\mathrm{B}^\mathrm{PEN} (X\vert Q_\mathrm{S}^\mathrm{PEN} )&= p\cdot K_\mathrm{S}^\mathrm{PEN} \cdot \int \limits _0^{D/(K_\mathrm{S}^\mathrm{PEN} \cdot X)} {z\cdot \varphi (z)dz}\\&-\left[ {w+c\cdot K_\mathrm{S}^\mathrm{PEN} -(w+\pi )\cdot \Phi (1/K_\mathrm{S}^\mathrm{PEN} )} \right]\\ (43)\,\,\,\quad \frac{\partial }{\partial X}\Pi _\mathrm{B}^\mathrm{PEN} (X\vert Q_\mathrm{S}^\mathrm{PEN} )&\!=&(p\!-w)\cdot \left[ {1\!-\!\Phi (1/K_\mathrm{S}^\mathrm{PEN} )} \right]\nonumber \\&\!+\!\left[ {\frac{(p\!-w\!-\pi )\cdot c}{w\!+\!\pi }} \right]\cdot K_\mathrm{S}^\mathrm{PEN} \!+\pi \cdot \Phi (1/K_\mathrm{S}^\mathrm{PEN} )\\ (53)\quad \frac{\partial }{\partial \hat{X}}\hat{\Pi }_\mathrm{B}^\mathrm{WHP} (\hat{X}\vert \hat{Q}_\mathrm{S}^\mathrm{WHP} )&\!=\!&-w\!\cdot \hat{K}_\mathrm{S}^\mathrm{WHP} \cdot \int \limits _0^{1/\hat{K}_\mathrm{S}^\mathrm{WHP} } {z\!\cdot \varphi (z)dz}\nonumber \\&\!-\!w\cdot \left[ {1\!-\!\Phi (1/\hat{K}_\mathrm{S}^\mathrm{WHP} )} \right]\!+\!c_E \cdot \hat{K}_\mathrm{S}^\mathrm{WHP} \cdot \!\int \limits _0^{D/(\hat{K}_\mathrm{S}^\mathrm{WHP}\! \cdot \hat{X})} {z\cdot \varphi (z)dz}\\ \end{aligned}$$

 

Appendix 3

The described potential for improving SC coordination under a WHP contact even for \(c_E >p\) will be illustrated by a numerical example. In order to exploit closed-form solutions we again assume a uniformly distributed yield rate in [0,1] with mean \(\mu _z =0.5\) like in Sect. 3.

Under centralized decision making the SC optimal demand multiplier and profit are given as

$$\begin{aligned} \hat{K}_\mathrm{SC}^*=\sqrt{c_E /(2\cdot c)} \quad \text{ and} \quad \hat{\Pi }_\mathrm{SC}^*=\left[ {p-\sqrt{2\cdot c\cdot c_E } } \right]\cdot D, \text{ respectively}. \end{aligned}$$

When decision making is decentralized and the parties agree on the WHP contract, the supplier’s multiplier and profit are

$$\begin{aligned} \hat{K}_\mathrm{S}^\mathrm{WHP} =\sqrt{c_E /(2\cdot c)} =\hat{K}_\mathrm{SC}^*\quad \text{ and} \quad \hat{\Pi }_\mathrm{S}^\mathrm{WHP} =\left[ {w-\sqrt{2\cdot c\cdot c_E } } \right]\cdot D, \text{ respectively}.\nonumber \\ \end{aligned}$$
(55)

As coordination is achieved the buyer’s profit is simply

$$\begin{aligned} \hat{\Pi }_\mathrm{B}^\mathrm{WHP} =\hat{\Pi }_\mathrm{SC}^*-\hat{\Pi }_\mathrm{S}^\mathrm{WHP}. \end{aligned}$$

In Table 3, we provide a numerical example to illustrate the development of profits with increasing values for the emergency cost \(c_E.\) Assuming demand and price/cost data as for Table 2 in Sect. 3.3, we proceed from the parameter combination which yielded the largest profit deficit under the WHP contract without emergency option (i.e. \(w=6).\) From Table 2, recall the accompanying profits of \(\Pi _\mathrm{S}^\mathrm{WHP} =254 \text{ and} \,\Pi _\mathrm{B}^\mathrm{WHP} =569\) which result in a SC deficit of \(\Pi _\mathrm{SC}^*-(\Pi _\mathrm{S}^\mathrm{WHP} +\Pi _\mathrm{B}^\mathrm{WHP})=871-823=48.\)

Table 3 Impact of emergency cost values on profits

Table 3 now displays the results emerging from a supplier’s offer of a WHP contract under emergency procurement with different \(c_E\) cost levels where the buyer will be guaranteed to receive complete delivery of all ordered units so that his order equals external demand, i.e. \(\hat{X}^\mathrm{WHP}=D=100.\)

The results in Table 3 first illustrate that for an emergency cost \(c_E \) which is smaller than the retail price \(p=14,\) the SC profit is always higher than the sum of profits under the WHP contract without emergency production, which from Table 1 equals \(823 \text{ for} w=6.\) Second, it turns out that, even if the retail price is exceeded (i.e. \(c_E >14),\) utilizing the emergency option can be reasonable. For \(c_E =15\) and \(c_E =16\) the SC faces a reduction in the maximal profit \(\Pi _\mathrm{SC}^*=871\) without emergency procurement. This loss, however, is smaller than the coordination deficit of 48. Hence, from a SC perspective it is profitable to utilize emergency procurement even if the respective cost exceeds the price gained per unit. Furthermore, it can be shown that exercising the emergency option is not profitable for the supplier under all circumstances. From (55) it follows that the supplier’s profit will only be positive if \(c_E <w^2/(2\cdot c)\) holds. This means that, if \(c_E \ge 18\) in our numerical example, the supplier’s participation constraint no longer holds as \(\hat{\Pi }_\mathrm{S}^\mathrm{WHP} \le 0.\)

Nevertheless, the results also show that for some cases the supplier is worse off than under the WHP contract without emergency production while the buyer always benefits. In order to guarantee beneficial profit sharing for all parties, contract terms can be formulated appropriately. If the supplier is in the position to determine the wholesale price and has to guarantee the buyer a minimum profit of, e.g. 569, parameter combinations can be found which assure higher profits for all parties compared with the situation without emergency production. For \(p=14 \text{ and} c_E =15\) (i.e. \(c_E >p=14),\) Table 4 illustrates under which wholesale prices both buyer and supplier benefit from the emergency option and how the profit is split among the actors.

Table 4 Profit split for different WHP values under emergency option

Note that from Table 4 we find that the starting point for our consideration (based on \(w=6)\) is \(\Pi _\mathrm{S}^\mathrm{WHP} =254,\Pi _\mathrm{B}^\mathrm{WHP} =569\) and \(\Pi _\mathrm{SC}^\mathrm{WHP} =823.\) Thus, Table 4 describes contract terms that result in a win–win situation.

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Inderfurth, K., Clemens, J. Supply chain coordination by risk sharing contracts under random production yield and deterministic demand. OR Spectrum 36, 525–556 (2014). https://doi.org/10.1007/s00291-012-0314-3

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