A multi-agent based negotiation for supply chain ... - Semantic Scholar

A multi-agent based negotiation for supply chain network using game theory Fang Yu, Toshiya Kaihara, and Nobutada Fujii Graduate School of System Information, Kobe University 1-1 Rokkodai,Nada, Kobe, Hyogo, 657-8501, Japan [email protected] [email protected] [email protected]

Abstract. This paper focuses on the single-issue negotiation between MA and MSA of the supply chain. When the MSA cannot finish the order independently, it will resort to find partners to cooperate. A two stage negotiation protocol is proposed in this paper. Cooperative game is combined with non-cooperative game to resolve the negotiation problem. The cooperative game is used to establish the coalitions. Then, the final determined coalition negotiates with MA to reach an agreement using the Stackelberg game. Numerical cases and comparisons are given to verify the effectiveness and superiority of the proposed protocol. Keywords: Multi-agent, supply chain, negotiation, game theory

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Introduction

Negotiation is the process of arriving at a state that is mutually agreeable to a set of agents, ranges from situations where resources must be allocated to agents to situations involving agent-to-agent bargaining. This paper focuses on the resources allocation of the negotiation using game theory. In SCN model there are many negotiations among MAs(Manufacture Agent), MSAs(Material Supplier Agent) and CAs(Customer Agent). Game theory has become a primary methodology used in SCN related problems. A prime methodological tool for dealing with these problems is non-cooperative and cooperative game theory that focuses on the simultaneous or sequential decision-making of multiple players under complete or incomplete information[2]. Game theoretical applications in supply chain management(SCM) were surveyed by [1] and [2]. They reviewed the application of game theory in SCM from the game theoretical techniques and SCM topics, respectively. In this paper only the negotiations between one MA and multiple MSAs are discussed using the game theory. It is assumed in this paper that MSAs will only accept the orders which are in their abilities. However, when the materials MA ordered are too large to provide by one MSA independently, thus they will be compelled to reject the order against their wills. In order to resolve this problem and maximize the total profit of the whole SCN, in general, MA will

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Fang Y., Toshiya K., Nobutada F.

decompose the order and then allocate to multiple MSAs. In this paper, we try to find another way to resolve this problem which can maintain the integrity of the order. We focus on the side of MSAs and devote our efforts to let the MSAs to find partners to build coalitions when the order is out of their abilities using game theory. A two stage negotiation protocol is proposed in this paper. In the first stage of negotiation, the coalition formation problem of the MSAs is modeled as a cooperative game. Theories of coalition formation were presented in [3]-[5]. In the second stage, the final determined coalition will negotiate with MA to reach an agreement on the price using non-cooperative game([6]-[7]). Firstly, MA announcing his strategy to the MSAs, then MSA choosing his best response to MA’s decision. Thus, the problem can be modeled as finding a Stackelberg equilibrium([8]-[9]). The main contributions of this paper is we present a twostage protocol which resolves the problem when the supplier cannot complete the order independently and at the same time maintain the integrity of the order and reduce the cost and workload of MA. This paper is organized as following. Section2 describes the SCN model used in this paper. Section3 gives a two-stage negotiation protocol and each stage of negotiations are discussed in section4 and section5 in detail. Experiments and comparisons are discussed in section6 to verify the effectiveness and feasibility of our proposed protocol. In the conclusion, we have commented on contributions and the direction of future work research

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Model descriptions

In this paper, all the organizations in the SCN are divided into three groups based on the multi-agent methodology(see Fig.1 (a)): MSA , MA and CA . We only discuss the negotiations between one MA and multiple MSAs

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Two-stage negotiation protocol

In this paper we mainly discuss the negotiations between one MA and multiple MSAs where the number of the order MA required is too large for each MSA to complete independently. When a MA wants some materials, it will broadcast an order (Mk , Qk , P Ak (0), T D), where Mk is the material MA wants to order, Qk is the quantity of Mk , P Ak (0) is the initial price of Mk of MA, T D is the due time. It wants to find the optimal MSA with the lowest price. We assume that the MSAs in our model will only accept the orders which are in their abilities. In real market, it is hackneyed that the order is out of the ability of MSAs, thus they will be compelled to reject the order against their wills. In order to resolve this problem, researchers tend to decompose the order and then allocate to multiple MSAs. In this paper, we try to find another way to resolve this problem which can maintain the integrity of the order. We devote our efforts to let the MSAs to combine together as a coalition and then to negotiate with MA to acquire the order. A two-stage negotiation protocol(see Figure.1 (b)) is proposed here as followings:

A Multi-agent based negotiation for SCN using game theory

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Fig. 1. (a)The model of SCN; (b)Flow chart of the two-stage negotiation protocol

Stage1: Negotiation among MSAs (see section4). MSAs evaluate the order and check whether it can be finished by themselves. If they can, then go to the second stage of negotiation directly; if they cannot, then they will negotiate with the other MSAs to build a coalition. A cooperative game is used for the coalition formation. Then, the final determined coalitions or MSA will enter into the second stage. Stage2: Negotiation between MA and final coalition (see section5). MA negotiate with the final coalition to find a Stackelberg equilibrium1 . Here the first stage is used for preparation since there exist MSAs which cannot complete the order by themselves, thus, they should to find partners to build a coalition. Then, the final negotiation about the price is started at the second stage.

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Negotiations among MSAs

When an order comes, if the MSA in the SCN cannot complete by itself, then it will start to negotiate with other MSAs in the SCN to establish coalitions. In the following subsections, we will discuss how to establish coalitions, how to determine the final coalition, and how to allocate the profit? The n-person Cooperative game is introduced to build the coalitions in the next section. 1

In our model the decision make of MA and the final coalition are not simultaneously. MA announces an offer firstly, and then MSAs give a counteroffer. Thus, the negotiation between MA and the final coalition can be regarded as a two-person Stackelberg game and the solution to this structure is to find the Stackelberg equilibrium.

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4.1


Coalition formation and determination

Coalition formation: A cooperative n-person game([4]) is a pair (N ,v) where N = 1, ..., n denotes the set of players(MSAs) and v is characteristic function and v(S) defined the amount that players in set S could win if they formed a coalition. Let S=(S1 ,...,SN ) denotes all partitions(coalition structure), Si =(si1 ,...,sim ) is the coalition structure of MSA i (m = 2n − 1), sij is one of the possible coalition of MSA i. Let Si∗ ={sij |v(sij ) = max v(sij ), i ∈ N } be the optimal coalition set sij ∈Si

of Pthe game (N ,v). A feasible payoff profile is defined as a vector ui such that ui = v(Si ). i∈Si

In this paper, we only focus on the single-issue (price) negotiation. Each player i ∈ N seeks to maximize its utility function ui , which is the payoff that it can obtain from belonging to a coalition. Therefore, the searching for optimal coalition can be converted to the calculating of the core of the game. In the following, we will apply the cooperative game into the SCN negotiation. Firstly, we give some useful definitions: P Rik = (1 + αi ) ∗ Cik .

(1)

IVik (T D) = IVik (T C) + γik ∗ (T D − T C).

(2)

P Cijk =

1 X P Rik ∗ (1 − σj ) | sij | i∈s

(3)

ij

and the payoff of the coalition Sj can be defined as: X u(sij ) = (P Cijk − Cik ) ∗ Qik .

(4)

i∈sij

where P Rik is the price of Mk of MSA i, αi is the percentage of profit of MSA i want to gain, Cik is the cost of MSA i produce Mk , IVik (T D) is the inventory level of Mk of i, γik is the productivity of Mk of i, T C is the current time, | sij | is the number of members in sij , P Cijk is the price of Mk of coalition sij , σj is the discount of coalition sij , Qk is the quantity of Mk MA ordered, Qik is the quantity of Mk of i acquired in coalition sij . Coalition determination: For each MSA i, it was expected to maximize its profit. Thus, the determination of the final coalition of each MSA i is transformed into the problem of finding the optimal coalition set S ∗ . We have discussed before that the searching for optimal coalition can be converted to the calculating of the core of the game. Therefore, the determination of the final coalition is equal to calculate the core of the game and it can be resolved by solving the following linear programming problem: X u∗i = max u(sij ) = max (P Cijk − Cik ) ∗ Qik (5) sij ∈Sj

sij ∈Sj

i∈sij


s.t.

X

IVik (T D) ≥ Qk ,

i∈sij

X

Qik = Qk

5

(6)

i∈sij

0 < σj < αi < 1

(7)

Thus, we can get the optimal coalition Si∗ of each MSA i, but the coalition is determined if and only if the all the MSA i ∈ sij reach an agreement on the coalition. In other word, all the MSAs in the coalition must have the maximal profit for they are all selfish. After the agreement is reached then the Si∗ with the maximal value of u will be determined as the final coalition SFi of MSA i. Then, MA will select the final supplier SF ∗ = min{SFi } to reduce its cost and the negotiation enter into the second stage. 4.2

Profit allocation

In this section, we will discuss when a coalition acquires the order, how does it to allocate the profit among the members. It is easy to do when the order is just meet all the members’ demand. However, what should to do whenP the order is not IVik (T D) > enough to fulfill the demand of the coalition? In other words, i∈Sj

Qk . As we know, each player in the coalition is mainly interested in its individual benefit and tries to maximize its own profit. Thus, we should to assign the profit impartially. For this purpose, we present the following allocate rule: Rule: The MSA contributes more to the coalition will gains more. Therefore, we allocate the profit according to their contributions to the coalition. Let πi be the profit of player i, which is to be computed, so that the vector πsij =(π1 ,...,πm ) P denotes a profit allocation of the coalition sij . To be efficient, one must have: πi = u(sij ). i∈sij

According to the Rule we can get that πi = u(Sj ) ∗

Qik . Qk

(8)

We can see from the above equation that the profit allocation problem is transformed into the problem of how to allocate the order among the players in the coalition. To be fair, we allocate the order according to the ability(productivity): IVik (T D) ∗ Qk . Qik = P IVik (T D)

(9)

i∈Sj

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Negotiation between MA and the final coalition

After SF ∗ is determined, it will start to negotiate with MA to reach an agreement on the price of Mk . However, the target of SF ∗ is contrary to MA’s. Each individual wishes to maximize the utility to itself of the ultimate bargain. On one hand, SF ∗ aimed to maximize its payoff by increasing the price; on the

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other hand, the MA tried to maximize its profit by reducing the price to lower the product cost to minimize the total payment. Let P Fk (t) is the price of Mk of SF ∗ at t and P Fk (0)=P Cijk where Sj =SF ∗ , P F Ik is the minimal price of Mk of SF ∗ and P F Ik = {P Cijk |ai = amin } is the price of Mk of MA, P AAk is the maximal price of Mk of MA and P Ak (0) is given by the order, T N is the deadline of negotiation, T S is the negotiation step, we have: P Fk (t) = P Fk (t − 1) − P Ak (t) = P Ak (t − 1) +

5.1

P Fk (t − 1) − P F Ik (T N − t)/T S

(10)

P AAk − P Ak (t − 1) . (T N − t)/T S

(11)

Stackelberg equilibrium

Stackelberg equilibrium applies when one of the players can move before the other player and assumes the role of the leader[2]. In our model, the MA first announcing his strategy to the MSAs. Thus, the negotiation between MA and the coalition is transformed into finding a Stackelberg equilibrium of the game. Thus, the problem to determine the Stackelberg equilibrium, can be transformed into finding the optimal proposal so as to maximize the profit of MA(uM A ) and the profit of SF ∗ (uSF ∗ ). In other word, to maximize the total profit of the whole SCN(uSCN ). Thus, the problem of finding the Stackelberg equilibrium can be transformed into solving the following programming problem:

where uSF ∗ =

max P

uSCN = uSF ∗ + uM A

(12)

(P Fk (t) − Cik ) ∗ Qik , uM A = (P Sk − P Ak (t)) ∗ Qk . Here

i∈sij

P Sk is the sale price of Mk of MA. By resolving the programming problem (12) we can get the equilibrium point of the second stage of negotiation, which means that MA reaches an agreement with SF ∗ on the price of Mk . Thus, the negotiation terminate.

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Experimental results

In this section, we suppose there are 5 MSAs and one MA distribute in the SCN (the information about the MSAs are shown as Table1, and the Price is calculated by (1)) and we assume the MA is price prior. We verify the feasibility of our protocol in three different cases compare with Protocol1: Protocol12 : In this approach, MA will select the MSA with the lowest price as the supplier, if the selected MSA cannot complete the order by itself, then MA will split the order and allocate the remains to the other MSAs. 2

Up to our knowledge, there are no references deal with such problems. In generally, MA will split the order, and allocate to several MSAs. It’s the most common solution. That’s why we compared with Protocol1.


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Table 1. Information and Results of the negotiation of our protocol Supplier Productivity (γik ) Cost (Cik ) Price of M1 (P Rik ) M SA1 M SA2 M SA3 M SA4 M SA5

150 150 200 150 100

7.91 8.20 7.80 7.67 7.82

10.283 10.660 10.140 9.971 10.166

Firstly, we define the parameters in our protocol as follows: αi =0.3, αmin =0.2, σj =0.2, T N =60s, T S=2s, P AAk =11, P Sk =15. Case1: MA submits an Order (M1 , 1000, 8.5, 10), which means that in this case all the MSAs can complete the order by themselves. Case2: MA submits an Order (M1 , 2000, 8.5, 10), we can see from Table1 that some MSAs cannot complete the order by themselves, thus, they need to find partners. Case3: MA submits an Order (M1 , 3000, 8.5, 10), we can see from Table1 that no MSA can complete the order by itself. 6.1

Results and Comparisons:

In Case1 all the MSAs in both protocols can complete the order by themselves, they will choose the same supplier with the lowest price. In Case2 and Case3 the order is out of the ability of some MSAs. Both Pr.1 and our protocol aimed to solve this problem. Pr.1 adopts the method to split the order and allocate to multiple MSAs which will increase the workload of MA. Our protocol solves this problem from the side of MSA, they try to build coalitions and what MA should to do is just announce the order and wait for the responses. Thus, our protocol is superior than Pr.1 from the view of providing quality service of SCN. Furthermore, from the standpoint of MA, it wants to reduce its payment. We can see from Table2 that our protocol reduce the payment of 320.1, 1376.3, 2000.0 in three cases respectively. It is obviously that our protocol is better than Pr.1 in all cases and can reduce more when the quantity increases. In summary, our protocol is a win-win protocol for both the MSA and MA. For MSA, it can get the profit from the order which must be rejected for the order is out of its ability. For MA, it not only reduce the cost but also maintain the integrity of the order, reduce the workload of MA to split the order. Table 2. Comparison results of different protocols under three cases Case1 Case2 Case3 Pr.1 Proposed Pr.1 Proposed Pr.1 Proposed Suppliers {4} {4} {4,3} {4,5} {4,3} {3,5} Payment 9971.0 9650.9 20039.5 18663.2 30166.5 28166.5

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Conclusion

In this paper, we have presented a multi-agent based negotiation protocol for supply chain network using game theory. A two stage negotiation protocol is proposed in this paper. A cooperative game is used in the first stage of negotiation for the coalition formation. The negotiation among the MSAs is transformed into the problem of calculating the core of the game. In the second stage, a twoperson non-cooperative game is adopted for the negotiation between MA and the final coalition. Thus, the negotiation problem can be dealt by finding the Stackelberg equilibrium of the non-cooperative game. The main contributions of our protocol are that our protocol resolves the problem when the supplier cannot complete the order independently and maintain the integrity of the order. Comparisons verified that our protocol had a good performance. It is a win-win solution for both side of MA and MSA. For MA it can find suppliers to get the material in time and at a lower price. On the other hand, MSAs can get the profitable order which is incapacitated for each of them. This paper only consider the single-attribute negotiation between one MA and multi-MSA, for a future work we will study the multi-attribute negotiation between one MA and multi-MSA. Furthermore, we will extend our negotiation protocol to the negotiations between multi-MA and multi-MSA which are more complex in the future. In this paper we assume that MA is price prior, but in real SCN, when the order is urgent the time must be taken into account, thus, we will discuss the price and time preference respectively in future.

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