One manufacturer outsources logistic transaction to n-3PL. But it is difficult for the manufacturer to maintain a good relationship with so many 3PLs in order to satisfy the high standard logistic service. On the contrary, too many 3PLs would greatly increase the production enterprise logistics subcontract and the difficulty of the management of logistics service quality. In practice, there are two main solutions. One is a self-built 4PL. Another is that the manufacturer delegates a 4PL to deal with the 3PL logistic service outsourcing transaction, manage the relationship between 4PL and 3PL, and control the quality risk.

The joint efforts among 4PL and 3PLs can increase the output of logistics services. Considering goods loss and damage, the logistic quality level is determined by the 3PLS’ efforts. Before the logistic operation process, the 4PL can’t monitor the 3PLS’ efforts. But when the logistics service finished, the 4PL can estimate 3PLS’ efforts with the good loss and damage level. The 3PLS should take the effort cost when exerting efforts. More efforts will generate fewer loss and damage level. In practice, the supply chain can’t get coordination because of 3PLS’ lower efforts than optimal level. The aim of this paper is to develop the 3PLS’ optimal effort with L-F game theory and Principal-agent theory.

Based on L-F game theory and principal-agent theory, the paper researches on examining the benefit relationship between the 4PL and 3PL.We assume that 4PL is a leader, playing a leading role in the process of the whole game, 3PLS are followers, always trying to decide whether or not to accept a contract parameters of the 4PL. Hypothesis 4PL and 3PLS are risk neutral and perfectly rational, namely, they will make decisions according to the expected profit maximization principle.

We set 2 continuous variables to measure the effort of 4PL and 3PLS. The

quality effort cost of them separates as, where k and k_i are cost parameters. The measure

is the service quality index, which stands for the logistics service quality influence of 3PLS’ effort; So is the goods loss and damage Ratio. is an random variable that affecting the logistics service quality which is exponential distribution. is the i^th parameter and is a constant. When the goods loss and damage happens, the 4PL will suffer punishment, where m is the punishment parameter. The output function is, where are the parameters of 4PL and 3PL and is an standard normality random variable. The 4PL offers and 2-part linear contact to 3PL in the form of where is the fixed shift and is a quality revenue parameter.

1) Virtual Centralized Decision

The aim of Supply Chain Coordination is to maximize the supply chain profit. In virtual centralize decision- making, the 4PL and 3PLs form an integrated company, which means 4PL choose 3PLs to execute logistics services. In order to get the best profit level, the decision maker determines the efforts of 4PL and 3PLs. The expected profit of the integrated company is as followed.

The function is jointly concave in each effort level of e and e_i, so we can get the optimal effort level e^* and with derivation.

Substituting the optimal solution (1) into the objective function, we can get the optimal profit.

2) Mixed Revenue Sharing Decision Model

In the Mixed Revenue Sharing subcontracting mechanism, 4PL offer every 3PL a fixed pay and based the 3PL_i’s profit. The expected profit of 4PL is as followed.

We introduce as the i^th logistics revenue, is his retained revenue. The 3PL make decision according to the aim to get the maximum revenue. Based on the principal-agent theory, the incentive compatibility constraints

(IC) and Participatory constraints (IR) are

(IC)

(IR)

As the leader of the game process, the 4PL choose his decision according to the L-F game theory. By solving IC, we get the optimal e_i

Considering IR, in order to get the maximum revenue, 4PL will always decrease to optimal

. Then substitute to the objective function, we get

The function is jointly concave, so we make derivation

So the i^th expected profit is and the 4PL’s expected profit is

As for, we compare (2) and (3) to get the coordination outcome. The equation is conflicted, so the supply chain cannot balance the coordination. And the 3PL’s effort level is lower than in the Virtual Centralized Model.

1) From Equation (4), We know that 3PLs’ effort level has an positive corelative with incentive parameter and side-payoff parameter, The larger value the 4PL offer to and, the higher effort level the 3PLs choose. 3PLs’ effort level has a negative correlative with the cost parameter k_i which stands for the operation capability of logistics service level. The stronger capability the 3PLs have, the lower goods loss and damage level is.

2) The Equation (7) shows that 4PL’s incentive parameter and the side-payoff parameter has a negative correlative with the operation capability. 4PL will offer more shifts to 3PLs whose operation capability are stronger. The incentive effect is determined by the punishment parameter m, if 4PL gives more punishment, the 3PLs will give more effort to decrease the goods loss and damage ratio.

3) The Equation (6) shows that 4PL’s effort level has a positive correlative with its cost control capability of decrease loss and damage level. 4PL is regard as leader in the supply chain, forcing him to keep an optimal effort level to guarantee the coordination. If the yield parameter is higher, the 4PL will choose higher effort level to decrease the loss and damage.

4) 3PL s’ effort level under asymmetric information is not more than the effort level under symmetric information. The asymmetric information lowers the 3PL s’ effort level and prevents the coordination of the supply chain, but not the 4PL’s. When we just consider asymmetric information, the 3PL’s effort level has a positive correlative with the yield parameter which stands for the operation efficiency of the logistics service capability. The stronger operation efficiency the 3PLs have, the more effort level they will offer. Under asymmetric information, the 3PL s’ effort level has a negative correlative with the ratio of probability distribution function and probability density function. The larger the ratio is, the higher effort level distortion caused by asymmetric information is. So the 4PL must try its best to collect the true information to make the right decision.

5) From Equations (1) (3) (6) (13), we can get the conclusion that the 4PL’s quality effort level keeps constant in virtual centralized model, non-coordination, symmetric information and asymmetric information modes. The reason is that, the owner of 4PL entity is the manufacturer who has enough capital and supply 4PL logistics themselves. To guarantee high efficient logistics service, they always keep optimal effort and the main job of them is to guide 3PLs to offer enough quality to decrease loss goods and damage as well as timely delivery.

It is very important to identify the profit distribution between 3PL and 4PL. In virtual centralized decision, there is no profit distribution because 3PL and 4PL are regarded as an integrated entity. In decentralized decision making process with Mixed Revenue Sharing Decision Model, the 4PL can extract all system profits and leave 3PLs with just enough profit to ensure their participation, in other words, the 3PLs can only get their reservation profit R_i. When the cost information is asymmetric, the 4PL’s expected profit declines. This implies that the cost information is important to 4PL. Consequently, the 4PL should collect cost information of 3PLs to reduce the expected profit loss and improve logistics service quality. Certainly, there should be new cost in the process of collecting information.

3PLs’ quality effort level determines the goods loss and damage level. By comparison, we find that the 3PLs’ quality effort levels in mixed revenue-sharing decision and revenue-sharing with side-payoff decisions with symmetric cost information are the same, which means that there is no difference between the two kinds of decision model. The reason for this is that, though 4PL cannot monitor 3PLs’ quality effort, 4PL can design suitable contract parameters to drive 3PL exerting optimal effort level for maximizing his own profit. The 4PL is effectively the only decision maker. However, when the cost information is asymmetric, the 3PLs’ quality effort level is distorted by the asymmetric information.

We set a sample to identify the effort level and expected profit in different models. The value of the parameters is as follows.

Table 2Table 1

. Profit distribution in different model

Items	VC	M	S	A
				L-L	H-L
	0.8	0.8	0.8	0.8	0.8
	0.75	0.75	0.75	0.75	0.6
	0.6	0.6	0.6	0.6	0.4
		24.7	3	3	3
		24.8	5	5	5
		−33.4	8.1	8.1	7.885
Total profit	16.1	16.1	16.1	16.1	15.885

mode. Consequently, the 4PL should collect cost information of 3PLs to reduce the expected profit loss and improve logistics service quality. Certainly, there should be new cost in the process of collecting information. If the collecting cost is less than the expected profit loss 0.52, 4PL will collect information.