Performance Comparison

We summarize the optimal decision variables and profit functions determined by each participant of the two closed-loop supply chain models for comparison purposes as shown in Table 1.

Table 1: Analytical Result of the Two Recycling Models

Current Practice Model Retailer Collection Model

Total Profits

Market Price 2

We evaluate the effect of recycling systems from different aspects of the return rates, the profits of the manufacturer, and the total profits. A rational player in recycling systems would like to increase its profits. The manufacturer who is the first mover in closed-loop supply chains has the most bargaining power and it is reasonable assuming that the manufacturer is seeking the maximum of its profits. However, in the perspective of the system designer, the return rate and the total profits are two important performance indicators in recycling systems. In this study, we are also interested in the trend of the return rate and total profits in the system designer’s perspective. Therefore, we compare these performance measures in terms of the return rates, the manufacturer’s profits, and the total profits between the current recycling system and the retailer collection model.

Observation 1 The optimal return rate in the retailer collection model, τ , is greater than ^* the optimal return rate in the current practice model, τ , whenever the condition ^C*

32C r A_L( + ) -16C b A_L( + ) 2(+ r A b A+ )( + )2 >0 holds.

+ + . Note that, in the retailer collection model, the retailer’s profits from collecting obsolete products are τ⋅ +(r A)(φ−p). In the current practice, the third-party player earns A⋅τ φ( −p)+ ⋅F τ from returned products. In the retailer collection model, the

retailer’s profits from collection work would be increased when the third-party player provides more payment to the retailer for those returned products. Therefore, the retailer would determine the return rate which is higher than the return rate in the current practices model when r is large enough. More specifically, ^{8 ( )} ₂

Observation 2 The optimal profits of the manufacturer in the retailer collection model are

high than the profits in the current practice, i.e., Π > Π_M ^C_M , whenever the condition 4(r A+ ) (> b A+ ) holds.

Proof. From Table 1, we observe that the profits of the manufacturer in the retailer

collection model increase with r. The manufacturer’s profits in the retailer collection model would be greater than the profits of the manufacturer in the current practice whenever

4 r b A+ A

> − . In the retailer collection model, when the retailer’s unit revenue from returned products, r, increases, it gives the retailer incentives to increase the quantity of recycled products. The quantity of recycled products, τ φ⋅( - )p , is positively related to the return rate and the market demands of new products. The profits of the manufacturer are also positively related to the market demands of new products. Then in the retailer collection model, the manufacturer can earn more profits with more recycled products as r increases. Therefore, when r is large enough, i.e.,

4

r>b A+ −A, the manufacturer’s

profits in the retailer collection model would be higher than the profits of manufacturer in the current practice model. ■

Finally, we compare the total profits between the retailer collection model and the current practice model. The unit payment between the retailer and the third-party player, r, is not a market parameter so it can be negotiated by the players in the retailer collection model.

We study how the change in the total profits as parameter, r, changes. From Assumption 2, we know that there exists an upper bound of r, 8C^L

condition, r A+ >0, we have the lower bound of r. Then we observe the total profit functions within the feasible region of r. In the current practice, the total profits, Π^C_T, are fixed with different r. Then we observe variation of the total profits in the retailer collection model, Π , and compare it with the total profits in the current practice model. _T However, from the second-order conditions, the concavity of Π is undetermined when _T parameter settings vary. In this study, we use two numerical examples to demonstrate different types of the total profit function, Π , which is affected by the unit payment from _T the third-party player to the retailer, r.

Example 1 The parameters, C , _L φ , c , b , and A, are given in Table 2.

Table 2: The Parameters in Example 1

CL φ c b A

Parameters 500 40 10 15 5

Under the parameter settings in Example 1, the feasible region of r is between -5 and 53. The graphs of total profits in each model are depicted in Figure 6.

Figure 6: Graphs of Total Profits under Example 1

When b is lower than the upper bound of r, i.e., 15 53< , the total profit function in the retailer collection model is not monotonic increasing within the feasible region of r. Then the total profits in the retailer collection model would not always be higher than that in the current practice model. In Example 1, the total profits in the retailer collection model would be higher than the total profits in the current practice model whenever r falls within

50

1.76 to 43.12. From Table 1, the term, (b r− ), of the total profits in the retailer collection model would be negative whenever r b> . Therefore, the total profits in the retailer collection model would be lower than the total profits in the current practice model when r still increases.

Example 2 The parameters, C , _L φ , c , b , and A, are given in Table 3.

Table 3: The Parameters in Example 2

CL φ c b A

Parameters 500 40 10 50 5

Under this parameter settings in Example 2, the feasible region of r is between -5 and 24. The graphs of total profits in each model are depicted in Figure 7.

Figure 7: Graphs of Total Profits under Example 2

When b exceeds the feasible region of r, i.e., 50 24> , the total profit function of the retailer collection model is an increasing function within the feasible region of r. When r is large, i.e., r>12.02 in this example, the total profits of the retailer collection model would be higher than that in the current practice model.