Pricing management for a closed-loop supply chain

Abstract

A closed-loop supply chain includes the forward supply chain and the reverse supply chain. In a reverse supply chain, the used products are collected from the end-customers. Hence, the return rate of used products is affected by the end-customer's willingness, and the end-customer's willingness is affected by the collecting price. In a forward supply chain, the wholesale price and the retail price will be affected by the collecting price. In this paper, we focus on the managements of the collecting price, the wholesale price and the retail price for the closed-loop supply chain. On the assumption that the return rate of the used products is an increasing function of the collecting price, we obtain the optimal collecting price, the optimal wholesale price and the optimal retail price based on the following models: Model CMRC (The manufacturer for collecting), Model CRMRC (The retailer for collecting) and Model CTMRC (The third party for collecting). By comparing the optimal pricing and the profits of the models, we find that the manufacturer for collecting is the best choice, and the retailer for collecting is another choice if the manufacturer has decided to transfer all its cost saving to the retailer. At the end of the paper, a numerical example is given to illustrate the optimal results.

Appendix

Proof of Proposition 5

• We divide the proof into two parts:

  (i) To prove that ω^*CMRC<ω^*CTMRC

  Note that,

  

  ω^*CTMRC is an increasing function of b when b>[k(c_m−c_rm)+c]/k+1, which is true by ∂ω^*CTMRC/∂b>0, ω^*CTMRC is an decreasing function of b when b<k(c_m−c_rm)+c/k+1, which is true by ∂ω^*CTMRC/∂b<0, and ω^*CTMRC takes its minimum value (we note that Minω^*CTMRC) at the point of b=[k(c_m−c_rm)+c]/(k+1) and Minω^*CTMRC=(φ+βc_m)/(2β)−[γk^k(c_m−c_rm−c)^k+1/(2(k+1)^k+1)]× (k/(k+1))^k.

  To prove that ω^*CMRC<ω^*CTMRC, we have to show

  

  which holds by 0<k<1. Hence, ω^*CMRC<ω^*CTMRC.

  (ii) To prove that ω^*CTMRC<ω^*CRMRC

  The proof of ω^*CTMRC<ω^*CRMRC can be easily observed from Table 1.

Proof of Proposition 6

• We divide the proof into two parts:

  (i) To prove that p^*CMRC⩽p^*CRMRC, we have to show that

  

  We note,

  

  In order to prove γk^k(c_m−c_rm−c)^k+1/(4(k+1)^k+1) ⩾[(c_m−c_rm−b)/4+(b−c)/(4(k+1))]×γk^k(b−c)^k/(k+1)^k, we have to show that Δ(b) is an increasing function of b and the max value of Δ(b) is γk^k(c_m−c_rm−c)^k+1/(4(k+1)^k+1). The former holds by ∂Δ(b)/∂b=[γk^k(b−c)^k+1/(4(k+1)^k+1)(k+1)(c_m−c_rm−b)>0, and the latter is true by 0<b<(c_m−c_rm) and Δ(c_m−c_rm)=γk^k(c_m−c_rm−c)^k+1/4(k+1)^k+1. Hence, p^*CMRC⩽p^*CRMRC.

  (ii) To prove that p^*CRMRC<p^*CTMRC

  The proof of p^*CRMRC<p^*CTMRC can be easily observed from Table 1.

  Because the ordering for the retail price p^* holds, the ordering for the demands of the channels follows trivially.

Proof of Proposition 7

• We divide the proof into three parts:

  (i) The proof of Π_M^*CMRC⩾Π_M^*CRMRC>Π_M^*CTMRC

  From the proof of p^*CMRC⩽p^*CRMRC<p^*CTMRC, we can obtain

  

  Note that β>0; we can show

  

  and

  

  Hence, Π_M^*CMRC⩾Π_M^*CRMRC>Π_M^*CTMRC.

  (ii) The proof of Π_R^*CMRC⩾Π_R^*CRMRC>Π_R^*CTMRC

  In a manner similar to that in the proof of Π_M^*CMRC⩾Π_M^*CRMRC>Π_M^*CTMRC, we can easily prove that Π_R^*CMRC⩾Π_R^*CRMRC>Π_R^*CTMRC.

  (iii) The proof of Π^*CMRC⩾Π^*CRMRC>Π^*CTMRC

  Obviously, Π^*CMRC⩾Π^*CRMRC holds by Π_M^*CMRC⩾Π_M^*CRMRC, Π_R^*CMRC⩾Π_R^*CRMRC, Π^*CMRC=Π_M^*CMRC+Π_R^*CMRC, and Π^*CRMRC=Π_M^*CRMRC+Π_R^*CRMRC.

  Easily, Π^*CRMRC>Π_R^*CTMRC is true by

  

  Hence, Π_M^*CMRC⩾Π_M^*CRMRC>Π_M^*CTMRC.