Retailer Collection Model

In order to make performances of recycling operations more efficient, we develop a model in this section. According to Savaskan et al. (2004), a closed-loop supply chain with the retailer engaging in collection effort is the most efficient in terms of the profits of manufacturer, the total profits, and the return rate, but recycling models with products remanufacturing do not suit for IT industry. In this section, we develop a model where the retailer collects used products with charging a service fee and the manufacturer cooperates with a third-party player. However, sometimes the retailer does not have enough ability to handle obsolete products. Then those returned products are sold to a third-party firm for

further appropriate recycled processes. The third-party player makes some profits by handling those used products collected by retailers. Figure 4 depicts this closed-loop model.

Figure 4: The Retailer Collection Model

We let r denote the revenue for the retailer from selling a unit of those used products to the third-party player. In other words, r is the unit cost of the third-party player for buying those used products from retailers. We assume that the retailer’s profits from collecting used products are positive so the condition, r A+ > , holds. The profit function 0 for member j is denoted by Π_j, where subscript j takes value M, R, or 3P, which denotes the manufacturer, the retailer, or the third-party player, respectively. Then the profit functions of each participant are

( )( )

A (collection service fee)

r (transfer price)

(profits of handling) (contract relationship)

The sequence of decision-making in this supply chain channel is shown in Figure 5.

After observing the unit manufacturing cost, the manufacturer determines the wholesale price, w , and the contract variable, F. Then the retailer decides the retail price, p, and the third-party player determines the return rate, τ, simultaneously based on the wholesale price and the contract information revealed by the manufacturer.

Figure 5: The Timeline of the Retailer Collection Model

The manufacturer, acting as a leader, considers the retailer’s and the third-party player’s best responses when making decisions. The followers, the retailer and the third-party player, make decisions after observing the manufacturer’s decision. We apply the backward induction to study this sequential two-stage model moving from the retailer’s and the third-party player’s decision problems in the second stage to the manufacturer’s decision problem in the first stage.

Step 1. The retailer’s decision in the second stage:

The retailer maximizes its profit function, Π , which is the profits from selling new _R products and recycling units minus the collection effort as shown in (4.21).

2

variable, F, and wholesale price, w.

Manufacturer observes unit cost, c.

Contract variable and wholesale price are revealed to retailer and

third-party player.

Market clears.

Retailer determines retail price, p, and return rate, τ. Third-party player decides whether to accept contract.

*

Solving the two equations for two unknown variables, we obtain the optimal market price p and return rate ^* τ , which satisfy the first-order conditions, as follows: ^*

*

Intuitively, the return rate, τ , must be lower than one. To ensure this, we assume that

retailer’s profit function is down-sloping at τ = , i.e., 1

1 return rate which makes the retailer’s profit function be maximized is lower than one. From this condition follows Assumption 2.

Assumption 2 Parameter C defined in the collection effort is assumed to be sufficiently _L

large such thatτ^* <1, i.e., which means

profit function is concave in p and τ , then (4.21) is maximized when the first-order conditions hold.

Step 2. The third-party player’s decision in the second stage:

In this step, we investigate the third-party player’s decision. The profit function of the third-party player, which includes the profits from those obsolete products plus the revenue from the contract, is shown below.

* * *

3_P τ ( - )( -b r φ p ) Fτ

Π = + (4.26)

Under our modeling setting, all of these notations in the profit function are given parameters for the third-party player. In other words, there is no decision variable in this step. Therefore, instead of maximizing its profits, the third-party player would make the decision about whether to accept the contract provided by the manufacturer or not.

The third-party player would accept the contract when its profits are positive, i.e.,

3_P 0

Π ≥ . Then we have a constraint about the contract variable F for any value of pto ensure that the third-party player is with a non-negative profit.

-( - )( *)

F ≥ b r φ−p (4.27)

In stage 2, the third-party player and the retailer make decisions simultaneously. Then we substitute (4.24) into (4.27) and get the constraint of the contract variable,

2

Constraint (4.28) shows that there exists a lower bound of F, which is the decision variable of the manufacturer. It implies that the contract must be attractive enough to the third-party player so that the third-party player has incentives to accept the contract. From (4.28), we know that the lower bound of F is positive whenever r b> . It means that if the unit cost of those obsolete products is higher than the unit revenue, the manufacturer would pay the third-party player to help it take the responsibility of the recycling processes.

On the contrary, if the third-party player can receive positive profits from those used products, it has incentives to join this closed-loop supply chain collection program without any payment from the manufacturer.

Step 3. The manufacturer’s decision in the first stage:

The manufacturer decides the wholesale price, w , and the contract, F, to maximize its profits from selling the new products minus the cost of contract as below.

* *

Max ( -, _M )( )

-w F Π = φ p w c Fτ (4.29)

When making the decision, the manufacturer would consider the retailer’s and the third-party player’s best responses. Substituting (4.24) and (4.25) into this profit function, we have

Lemma 1 The profit of the manufacturer is maximized when F reaches the lower bound.

Proof. We let F denote the lower bound of F. Assuming that there exists a F^'= + F ε

+ . Assumptions 2 and condition, ( - ) 0φ w > , contract this inequality. Therefore, the profits of the manufacturer is maximized while the contract variable, F, reaches its lower bound. ■

Lemma 1 simplifies the manufacturer’s problem into a single-variable problem.

Substituting the lower bound of the contact variable, F , into (4.30), then we have

2

From the first-order conditions, the optimal wholesale price is

2

+ + . Substituting (4.32) into the constraint from Assumption 2, ( )2 ( )( - )

2

2^M 0

d dw

Π < , hold. Then, the profit function of the manufacturer is concave in w so (4.31)

is maximized when the first-order conditions hold. Finally, the manufacturer’s profit is given by

The optimal return rate can be found by substitution of w^*, then we have

* ( )( - ) ( - )

From (4.33) and (4.34), we find that the optimal return rate, τ^*, is positively related to r and the profits of the manufacturer, Π , are also positively related to _M r. Therefore, the

profits of the manufacturer, Π , are positively related to the return rate, _M τ . Then the manufacturer would like to see a high products return rate.

Proposition 1 The manufacturer would like to provide the third-party player with an

appropriate contract, which induce the retailer to increase the return rate.

Proof. From Lemma 1, the optimal contract variable, F, which reaches its lower bound,

( )( ) indirectly by offering the third-party player an appropriate contract so that it can affect the retailer’s decision, the return rate, τ. Furthermore, the profits of the manufacturer, Π , _M are positively related to the return rate, τ . In order to maximize its profits, the manufacturer would determine an appropriate contract to induce the retailer to increase the return rate, τ . ■

In the next section, we summarize all decision variables and profit functions of the two proposed models, the retailer collection model and the current practice model. We compare

the return rates, the profits of manufacturer, and the total profits between these two models.

Then we investigate the performance comparison when the third-party player is a non-profit organization. We conduct sensitivity analysis to study some interesting issues about the collection effort parameter, C , and some interactions between parameters, _L C , _L b, r, and

A.