Optimal Financing Order Decisions of a Supply Chain under the Retailer's Delayed Payment

Table 1

Comparison of the optimal decisions under the delayed and up-front payment.

Condition	Delayed payment	Up-front payment
𝑞1,2	𝑞2=𝑦(1+𝑅𝑓)𝑤(1+𝑅𝑠)	𝑞1=𝑦(1+𝑅𝑓)𝑤1+𝑅𝑓+𝜆(𝑅𝑟−𝑅𝑓)
𝑞≤𝑞	𝑞^∗𝑠=𝐹^−1𝑤(1+𝑅𝑠)𝑝	𝑞^∗𝑏=𝐹^−1𝑤(1+𝑅𝑓+𝜆(𝑅𝑟−𝑅𝑓))𝑝
	𝑤𝑠^∗=𝑐+𝑝𝑞𝑓(𝑞)1+𝑅𝑓	𝑤𝑏^∗=𝑐+𝑝𝑞𝑓(𝑞)1+𝑅𝑓+𝜆(𝑅𝑟−𝑅𝑓)
𝑞>𝑞	𝑞^∗𝑠=𝐹^−1𝑤(1+𝑅𝑠)𝑝𝐹𝑥2	𝑞^∗𝑏=𝐹^−1𝑤(1+𝑅𝑓+𝜆(𝑅𝑟−𝑅𝑓))𝑝𝐹(𝑥1)
	𝑤𝑠^∗𝑥=𝑐+𝑝𝑉𝐹(2)1+𝑅𝑓+𝑝𝑞𝑓(𝑥2)𝑉^2−𝑓(𝑞)(1+𝑅𝑠)𝐹(𝑥2)−(1+𝑅𝑓)(1+𝑅𝑓)(1+𝑅𝑠)𝐹(𝑥2)−𝑉𝑞𝑓(𝑥2); where 𝑉=𝑤(1+𝑅𝑠)/𝑝	𝑤𝑏^∗=𝑐+𝑝𝑞𝑓(𝑞)−Ω^2𝑓(𝑥1)1+𝑅𝑓+𝜆(𝑅𝑟−𝑅𝑓)𝐹(𝑥1)−Ω𝑞𝑓(𝑥1); where Ω=𝑤1+𝑅𝑓+𝜆(𝑅𝑟−𝑅𝑓)/𝑝

Note: The results of the up-front payment, listed on column 2, source from literature [7]. The unsold products have no salvage value.