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Based on the uncertainty theory, market demand information updating as the background, we study the coordination and optimization problem of three-stage supply chain in this paper. In half a asymmetric market information, participants are risk neutral; under the situation of the manufacturers and wholesalers having twice pre-season decision-making opportunity, wholesalers can be replenished in the season; manufacturers join the lowest supply contract of commitment: manufacturers for exchanging the information that they cannot get directly from the market will promise wholesalers to have a season lowest supply in pre-season. According to this contract, we establish optimization models of manufacturers and wholesalers respectively, and get the optimal strategy of supply chain members by analyzing the supply chain system. Finally, by giving a numerical example and comparing the results with that under random circumstances, the result is reasonable.

The uncertainty theory [

With further research, we got to know and researched Stackelberg game behavior in the supply chain and problems of supply chain coordination generated. Malone and Crowston [

Numerous studies illustrate that it is indeed effective strategy that using such Revenue Sharing Contract, Quantity Flexibility Contract, the Wholesale Price Flexibility Contract, the Commitment Contract and the Option Contract in order to achieve supply chain system coordinated, no matter analyzing from either the perspective of the manufacturer and wholesaler or the perspective of a multi-stage supply chain or multi-level one. With consumers’ “battle” intensified, relationship between supply chain members also began to develop from the original simple division of profit to mutual benefit and win-win results. The members can gain more profits only by improving the competitiveness of the entire supply chain, so it has become hot spots of researches that the relational contract is concerned about the relationship between supply chain members. Relational contract is a convention, more accurately, a trust that could not specified by stringent legal provisions. Therefore the commitment itself may be unlike the general traditional contract that can put into effect through a simple contract. Wang Yingjun [

The commitment contract is a typical relational contract. Helper [

Aiming to seasonal products with long production cycle, short sales period, unknown demand and large volatility, combining with the previous researches on the supply chain coordination with market demand information updates and based on the uncertainty theory, this paper further deepens and expands the application of commitment contract, which is a kind of relational contract, in the supply chain management. Based on the general newsboy model with uncertainty demand, this paper gives a two-stage supply chain model in case of market demand information updating and establishes the appropriate decision-making model from the perspective of the supply chain members (the manufacturer and the wholesaler) respectively, and then optimizes the corresponding supply chain contract with updating information from the two aspects.

Based on the lowest supply commitment contract, this paper establishes a two-stage three-stage supply chain system made up of a manufacturer and a wholesaler firstly: before the season beginning, the wholesaler and the manufacturer carry on the first time “blind” production and order. With the season coming, the wholesaler corrects the order quantity while he provides demand information with certain loyalty to the manufacturer. At the same time, the manufacturer of production revised inventory and promised the lowest supply. After the season beginning, the wholesaler can carry on replenishing once again according to the situation. Based on the supply chain above, this paper establishes the optimal decision model for the manufacturer and the wholesaler respectively and ultimately gets the wholesaler’s optimal order quantity, the value of the manufacturer’s optimal commitment, the optimal production quantity and other strategic decisions. Finally, it gives exponential analysis and obtains a more realistic conclusion.

Definition 1 [Liu1]: The uncertainty distribution Φ of an uncertain variable

Definition 2 [Liu1]: An uncertain variable

Definition 3 [Liu1]: Let

provided that at least one of the two integrals is finite.

Theorem 1[Liu1]: Let

Based on the newsboy model, a production enterprise with unit production costs C, unit retail price P(P > C), D is stochastic market demand that policy makers facing.

Calculating its derivative, we get

It means that

In the situation that information of market demand is uncertain and updating, as manufacturers can’t contact with the market, they get the updated information only through information sharing with wholesalers. Note

Phase 1: Manufacturers and wholesalers confirm uncertainty distribution of market demand with years of experience in sales and expert assessment, denoting as

Phase 2: With the sales season approaching, wholesalers get the market updates

Phase 3: With the sales season starting, we can get the market demand x, wholesalers selectively emit replenishment orders to manufacturers to resive the final order quantity according to the total order quantity d and demand x. As a manufacturer, they can only mechanically do their ability to distribute the surplus goods to wholesalers as complement goods. If the market needs are not met, give manufacturers and wholesalers s punishment considering loss from the customer, loss of reputation and so on. It is called shortage punishment.

Known from the above analysis, for a manufacturer, the quantity he committed must be greater than the wholesalers amount in the first two stages, that is,

that caused by understanding updates insufficiently. That caused by limited production is

Wholesalers take initiative relatively in the supply chain system and manufacturers make decisions due to the behavior of wholesalers. We assume that wholesalers know the manufacturers different responses to different

Known from a series of analysis above, we should set about from manufacturers. Obviously the order quantities were known in the two production decisions, so

It should be noted that manufacturers need not make any decisions in phase III, the producers do not need to make any decisions considering the collaboration process in the entire supply chain. Decomposing the expression by the stages of production decision and combining stage II and stage III, we can get the optimization model of manufacturer’s expected profits in this situation:

In which,

Analyzing the model above in the reverse order, we can analyze the decision problem in view of the stage II and solve

Theorem 2: In the decision problem

Proof: reducing decision problem:

In the decision problem

Discussing different circumstances:

When

(i) When

(ii) When

uncertain distribution function,

(iii) When

the second term in the expression of

way, we take the derivative of C. Then we get

That is,

When

In this case, since the total production has exceeded the amount of updating information

(i) When

(ii) When

To sum up, the manufacturers’ the most optimal minimum commitment supply quantity satisfies

Known from Theorem 2, the manufacturer will choose to avoid shortage punishment

Based on these, we continue to analyze the optimal production capacity q.

Theorem 3: In the second stage of the decision problem

In which

Proof: Compute the expectation of

Here

Discussing different circumstances:

When

When

From the nature of condition uncertain distributed function, we know

When

As

So only let

When

When

When

To sun up, the decision problem in the second stage has 4 possible optimal strategy, noting as

Integrating above analysis, Theorem 2 and Theorem 3, at the beginning of stage II, the manufacturer obtain two decision programs for both strategies

Relative to the manufacturers and wholesalers have been in a dominant position throughout the entire supply chain operation process. They take the initiate to carry on two pre-season orders, obtain the market information directly, selectively share it with the manufacturer, and understand the manufacturer’s response to update information. That is, they now how the manufacturer correct the demand distribution function

Here

in which

Here

Theorem 4: In the second stage, the optimal pre-season order quantity that wholesalers selected is

Proof: Simplify the decision problem and get

It is necessary to pay attention to that

Discussing different circumstances:

When C = q,

(i) When q = d, we have

(1) if

Computing its derivation to d, we can get

In this case,

if

In this case, in order to maximize

(ii) When

Here the second term and the fourth one are independent of d.

If

And we can get further

In the similar way, we obtain the optimal decision is

If

At this moment, in order to maximize

b) When

(i) When

whose the last term is independent of d.

(1) if

So we have

(2) if

At this moment, in order to maximize

(ii) When

or

Instruction: For a wholesaler, when choosing the optimal pre-season order quantity d from

From the analysis in above part 2 and part 3, we get that in order to achieve coordination, supply chain should have the following conclusions, when the commitment amount C satisfy certain conditions.

Completion 1: Under the condition C = q, when

Proof: Under the condition C = q when

should have

When q = d and_{B} (

Completion 2: Under the condition

Proof: Under the condition

chain, we should have

only to adjust

Known from the above two completion and above analysis, when the manufacturer’s optimal production is larger than the wholesaler’s optimal order quantity, it is impossible to achieve optimal state of both of two members of the supply chain. Only one of them can be optimal, and the other will have some extra expense. In order to achieve the supply chain coordination, deduce more costs the other member paid from the more part of the profit the optimal member get than that without cooperation and the two members negotiate to allocate this part profits. Then the supply chain coordination can be achieved; when the manufacturer’s production is equal to the wholesaler’s order quantity, the supply chain coordination can be achieved. The two sides can reach optimal state under some conditions.

On the other hand, the total profits of the manufacturer is

Computing the derivative of the above formula, at this time it is necessary that

This completion is to reach the optimal supply chain coordination in the case of q = d. It is similar with the process of get the optimal order quantity

We assume that before market information updates, the manufacturer and wholesaler analyze and estimate that the market demand obey the normal uncertainty distribution

The wholesaler reflects to market information honestly. Here there is

As an manufacturer,^{*} = q_{A} = 64.61529. Corresponding optimal order quantity of the wholesaler is d_{A} = 62.89546, d_{B} = 61.48653, d_{C} = 61.7073 and d = 60.

The manufacturer gives a larger

As a manufacturer,

If the manufacturer does not consider new information the wholesaler provided, he still think

As a manufacturer,

Comparing the three cases above, we get the first one is the optimal and its cost is the lowest.

In addition, if complete the previous analyzed based on the probability theory, its result in the first case is as follows:

As a manufacturer,^{*} = q_{A} = 63.1682. Corresponding the wholesaler’s optimal order quantity (_{A} = 59.7678, d_{B} = 56.8318 and

Compared with results obtained in the previous uncertainty theory, the optimal production value of the manufacturer’s is less slightly and the optimal value of wholesalers is less than the previous value.

The calculation is similar in the other two cases, so here we do not compare them.

Based on the uncertainty theory, this paper studies seasonal merchandise with long production cycle, relatively short marketing period, uncertain and strong volatile market demand. Setting about the perspective of market demand information updated, it analyzes the contract coordination problem of a two-stage three-stage supply chain system with a risk-neutral manufacturer and a risk-neutral wholesaler. In the system of pure contract supply chain, it cannot achieve coordination. It introduces the pledge contract, in the case that producers have twice pre-season production and wholesalers have twice pre-season order and a replenishment opportunity in the season. It established contract model respectively, and according to these models, it carried on the optimization analysis. The optimal quantity, the output of eugenics and the optimal amount of commitment of the supply chain members are presented. In the case of the wholesalers predomination, it gives the information sharing strategies that regulate the supply chain. In this paper, it is just an attempt to apply the uncertainty theory in supply chain. There may also be many problems, but after all, it has been applied to the practical problem. I believe it will be applied to analyze the supply chain system by more researchers.

YuquanCui,XiaolinZhang,XiLu, (2015) Supply Chain Commitment Contract Model Based on Uncertainty Theory under Uncertain Market Information. Applied Mathematics,06,1727-1739. doi: 10.4236/am.2015.610153