A multiscale decision theory analysis for revenue sharing in three-stage supply chains

Abstract

Revenue sharing is an effective mechanism for coordinating decisions in a supply chain. For a three-stage supply chain, we explore how revenue-based incentives can be used by the stage 1 supply chain agent (retailer) to motivate cooperative behavior from its two upstream partners with conflicting interests. To illustrate our analysis, we provide a food supply chain example, with retailer, processor and farmer. Compared to the frequently studied two-stage problem, a three-stage supply chain leads to a more complex decision and incentive problem. To model and solve this more complex problem, we apply multiscale decision theory (MSDT), a novel approach for multi-level system analysis. MSDT enables us to account for uncertainties at all stages of the supply chain, not just at the final stage, and to derive analytic solutions. Results show and quantify the extent to which contracting and information sharing facilitate chain-wide cooperation. Further, it determines optimal decisions and incentives for agents at each stage. This paper is the first to apply MSDT to supply chains and contributes to its theory by advancing MSDT modeling and analysis capabilities. The modeling and solution approach can be applied to decision and inventive problems in other multi-level enterprise systems.

Appendix

Proof of Theorem 4

With $b_1^*>0$ according to Eq. (19), agent A2 takes the cooperative action. In addition, agent A2 is willing to pay an incentive to motivate agent A3’s cooperation if

$$\begin{aligned} E\left[ {r_{final}^{A2} \left| _{b_1^*,b_2}\right. a_1^{A1} ,a_1^{A2} ,a_1^{A3} } \right] \ge E\left[ {r_{final}^{A2} \left| _{b_1^*,b_2=0}\right. a_1^{A1} ,a_1^{A2} ,a_2^{A3} } \right] . \end{aligned}$$

(33)

Solving (33) for share coefficient $b_2$ yields the result in (25). For $b_1^*=0$, agent A2 will take its initially preferred non-cooperative action $a_2^{A2}$ as it is in its best interest. Consequentially, agent A2 prefers for agent A3 to choose its non-cooperative action as it too benefits agent A2. $\square$

Theorem 7

If agent A2’s influence on agent A1 is small, specifically if

$$\begin{aligned} c_1 <\frac{\left( {\rho _2^{A2} -\rho _1^{A2} } \right) }{2\left( {\rho _1^{A1} -\rho _2^{A1} } \right) }, \end{aligned}$$

(34)

no incentives will be shared and agents A2 and A3 will take their initially preferred non-cooperative actions (scenario 1).

Proof

Agent A1 compares its expected reward without incentives to its expected reward with a weak incentive $\hat{{b}}_1$. Agent A1 prefers not paying an incentive and no cooperative action from agent A2 if

$$\begin{aligned} E\left[ {r_{final}^{A1} \left| _{b_1=0}\right. a_1^{A1} ,a_2^{A2} ,a_2^{A3} } \right] \;>E\left[ {r_{final}^{A1}\left| _{\hat{{b}}_1}\right. a_1^{A1} ,a_1^{A2} ,a_2^{A3} } \right] . \end{aligned}$$

(35)

Solving the above inequality for influence level $c_1$ yields inequality (34) $\square$

If agent A2’s influence $c_1$ is larger than the threshold in (34), agent A1 must decide whether its sufficiently large to justify a strong incentive as opposed to a weak incentive contract.

Theorem 8

Agent A1 prefers scenario 3 (strong incentive) over scenario 2 (weak incentive), if

$$\begin{aligned}&\left[ {2c_1 \left( {\rho _1^{A1} -\rho _2^{A1} } \right) +\left( {\rho _2^{A2} -\rho _1^{A2} } \right) } \right] \left[ {\left( {\rho _1^{A1} -\rho _2^{A1} } \right) g_3 +\rho _2^{A1} } \right] >\nonumber \\&\qquad 2\left( {\rho _1^{A1} \!-\!\rho _2^{A1} } \right) c_1 \left[ {\left( {\rho _1^{A1} \!-\!\rho _2^{A1} } \right) g_4 \!+\!\rho _2^{A1} \left( {1\!+\!c_1} \right) \!+\!\left( {\rho _2^{A2} \!-\!\rho _1^{A2} } \right) \left( {\alpha _1^{A2} \!+\!\alpha _2^{A2} -1} \right) } \right] \nonumber \\ \end{aligned}$$

(36)

with

$$\begin{aligned} g_3&= \alpha _1^{A1} +c_1 \left( {2\alpha _1^{A2} -1} \right) +3c_1 c_2 \left( {2\alpha _1^{A3} -1} \right) \\ g_4&= -\alpha _1^{A1} -c_1 \left( {2\alpha _1^{A2} -1} \right) +3c_1 c_2 \left( {2\alpha _1^{A3} -1} \right) \end{aligned}$$

Proof

Agent A1 prefers the strong incentive scenario 3 if its expected reward from that scenario is greater than its expected reward from the alternative weak incentive scenario 2. To make its decision, agent A1 evaluates

$$\begin{aligned} E\left[ {r_{final}^{A1} \left| _{b_1^*,\hat{{b}}_2}\right. a_1^{A1} ,a_1^{A2} ,a_1^{A3} } \right] >E\left[ {r_{final}^{A1} \left| _{\hat{{b}}_1 ,b_2 =0}\right. a_1^{A1} ,a_1^{A2} ,a_2^{A3} } \right] . \end{aligned}$$

(37)

Simplifying inequality (37) yields the inequality presented in (36). Solving for $c_1$ or $c_2$ is not possible. $\square$

Scenario 3 assumes agent A3’s cooperation. Therefore, in the incentive negotiation with agent A2, agent A1 will require that a strong incentive plan be accompanied by a clause that requires agent A2 to pay a cooperation inducing incentive to agent A3.

Agent A2, however, might prefer a weak incentive contract over a strong incentive contract. In this case, agent A2 would decline agent A1’s offer of a strong incentive contract, knowing that agent A1 would at least offer a weak incentive contract. The following theorem determines agent A2’s response to a strong incentive contract.

Theorem 9

Agent A2 prefers a strong incentive contract that includes the obligation to offer a weak incentive to agent A3 to a weak incentive contract from agent A1, if the following inequality is satisfied

$$\begin{aligned} f_1 f_2 >f_3 f_4 \end{aligned}$$

(38)

where,

$$\begin{aligned} f_1&= \left( {\rho _1^{A1} -\rho _2^{A1} } \right) \left( {\alpha _1^{A1} +c_1\left( {2\alpha _1^{A2} -1} \right) -3c_1 c_2 \left( {2\alpha _2^{A3} -1} \right) } \right) \\&\quad -\left( {\rho _2^{A2} -\rho _1^{A2} } \right) \left( {\alpha _1^{A2} +\alpha _2^{A2} -1} \right) +\rho _2^{A1}\\ f_2&= \left( {\rho _1^{A1} -\rho _2^{A1} } \right) \left( {\rho _2^{A2} -\rho _1^{A2} } \right) \left[ {\alpha _1^{A1} +c_1 c_2 \left( {2\alpha _1^{A3} -1} \right) } \right] \\&\quad +c_1 \left( {\rho _1^{A1} -\rho _2^{A1} } \right) \left( {\rho _2^{A2} +\rho _1^{A2} } \right) +\rho _2^{A1} \left( {\rho _2^{A2} -\rho _1^{A2} } \right) \\ f_3&= c_1 \left( {\rho _1^{A1} -\rho _2^{A1} } \right) \left[ {\left( {\rho _2^{A2} -\rho _1^{A2} } \right) \left( {-1+\alpha _2^{A2} +c_2 \left( {2\alpha _2^{A3} -1} \right) } \right) +\rho _2^{A2} } \right] \\ f_4&= \left( {\rho _1^{A1} -\rho _2^{A1} } \right) \left( {\alpha _1^{A1} +c_1 \left( {2\alpha _1^{A2} -1} \right) -3c_1 c_2 \left( {2\alpha _2^{A3} -1} \right) } \right) +\rho _2^{A1} \end{aligned}$$

Proof

Agent A2 prefers a strong incentive plan, with an agent A3 cooperation clause, if its expected reward is greater than the expected reward from the weak incentive contract, specifically if

$$\begin{aligned} E\left[ {r_{final}^{A2} \left| _{b_1^*,\hat{{b}}_2}\right. a_1^{A1} ,a_1^{A2} ,a_1^{A3} } \right] >E\left[ {r_{final}^{A2} \left| _{\hat{{b}}_1 ,b_2=0}\right. a_1^{A1} ,a_1^{A2} ,a_2^{A3} } \right] . \end{aligned}$$

(39)

Simplifying inequality (39) results in (38). $\square$

If inequalities (36) and (38) are satisfied, full chain cooperation is obtained $(b_1^{*} >0, \hat{{b}}_2 >0)$. If the inequality in (38) is not met, but $c_1$ is larger than threshold (34), agents A1 and A2 will agree on a weak incentive contract ($\hat{{b}}_1 >0,\;b_2 =0)$ leading to a partial chain cooperation, with agent A3 taking its initially preferred non-cooperative action. Only if $c_1$ is too small (34), will the result be no cooperation.