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IERI Procedía 5 (2013) 118 - 125

2013 International Conference on Agricultural and Natural Resources Engineering

Coordination of Agricultural Products Supply Chain with Stochastic Yield by Price Compensation

Guoping Nonga, Sulin Pangb c*

a Management School, Jinan University,Guangzhou, 510632,China b School of Public Administration/School of Emergency Management, Jinan University, Guangzhou, 510632, China c Guangdong Emergency Technology Research Center of Risk Evaluation and Prewarning on Public Network Security, Guangzhou,510632, China

Abstract

In this paper, coordination problem of agricultural products supply chain with stochastic yield is studied based on prices compensation strategy. The agricultural producing is influenced by the natural conditions, and the yield is uncertain. While agricultural products is rigid demand goods, the fluctuations of yield cause greater volatility of prices. The two-echelon supply chain with one supplier and one retailor is studied, and the mathematical model is constructed. The model showed that prices compensation strategy is Pareto improvement for agricultural products supply chain with stochastic yield, and it also incentive agricultural products supplier to rise the production plan and balance the profit allocation of supply chain.

© 2013TheAuthors.Publishedby ElsevierB.V.

Selectionand peerreviewunder responsibility oflnformation EngineeringResearch Institute

Keywords: Agricultural products supply chain; Prices compensation strategy; Stochastic yield; Pareto improvement

Corresponding Author:Sulin Pang. Tel.: +(086)13610061308; fax: +(086)020-85226502. E-mail address: pangsulin@163.com.

2212-6678 © 2013 The Authors. Published by Elsevier B.V.

Selection and peer review under responsibility of Information Engineering Research Institute doi: 10.1016/j.ieri.2013.11.080

1. Introduction

Agricultural products are the necessities of human life and rigid consumer goods, and it is one of the most important social producing. Since the human society entered the modern time, agricultural production has become an industry with more detailed labor division, and most consumers do not need to produce agricultural products, which make production supply and consumption of agricultural products a supply chain. With the developing of supply chain management theory, many companies like IBM, P&G, DELL succeeded in applying supply chain management into bassness, and the agricultural product industry also have applied the

supply chain management theory to improve their own competitiveness. In developed countries, Agricultural product supply chain has been widely practice, and become a hot topic in academic research.

Kazaz, 2004 consider the random yield characteristics in the process of agricultural production, production planning and pricing decision problem under the background of random output. Deo, 2009 focuses on the influenza vaccine market of United States, studies the problem of stochastic output with Cournot competition model, and found that random yield incentive the industry tend to become gathering, and reduce the production plan, which damage consumer welfare.

Bohle, Maturana and Vera, 2010 focus on wine grape harvesting scheduling optimization problem subject to yield uncertainty. The research shows how effective robust optimization is solving this problem in practice and develop alternative robust models and show results for some test problems obtained from actual wine industry problems. Boyabatli, Kleindorfer and Koontz, 2011 analyzes the optimal procurement, processing, and production decisions of a meat-processing company in a beef supply chain subject to yield uncertainty, and the research show that higher variability of product increases the profits of the packer, but decreases the reliance on the contract market relative to the spot market.

He, 2010 study the optimization problems of production and procurement in a decentralized supply chain consisting of one supplier and one manufacturer, and the study shows that the supplier's profit function is piece-wise concave and explore structural properties for the manufacturer. He and Zhao, 2011 study the inventory, production, and contracting decisions of a multi-echelon supply chain with both demand and supply uncertainty, and find that the commonly used wholesale price contracts used by both up-stream and downstream supply-chain members cannot coordinate the system, and further provide contract terms that lead to win-win situation. The study also investigate the impact of the supplier's risk attitude on the decisions, as well as the impact of spot market price for raw material on the performance of the entire supply chain.

In this paper, we consider the coordination problem of agricultural products supply chain with stochastic output studies based on prices compensation strategy. Since the above study show the agricultural product supplier tend to reduce the production plan when it face the risk of yield uncertainty. We introduce the prices compensation strategy, which is that the retailer rise the procurement price higher than wholesale market price. We model the agricultural products supply chain with stochastic output, and prove prices compensation strategy is Pareto improvement.

2. Model Assumptions

In this paper, the agricultural product supply chain consist of one supplier and one retailer with two type of market, which are the wholesale market and the consumer market. The retailer could purchase agricultural products in wholesale market or make procurement contract with supplier directly. If the supplier has no contract with retailer, they would sale their agricultural products in wholesale market and the retailer buy the products in wholesale price. Since we focus on price compensation strategy, the contract of retailer and supplier with price compensation would be simplify as the compensation on wholesale price per product.

Under the consideration that the agricultural products are perishable goods, we assume the consumer market is market clear in order to ignore buy-back contract and simplify calculation. The notation of variables are defined as following:

x : Supplier Agricultural products plan;

PH : Wholesale prices;

PM : Consumer market price ;

Q: Price compensation;

RP : Coefficient of stochastic yield, which is random variables of mean 1 and variance a", and cr > 1;

The retail price and wholesale price of agricultural product are functions of the quantity of product, price function of demand in consumer markets is: PM (x) = aM—bMx, where aM is the passable highest consumer price, bM is the influence coefficient of quantity of supplier product; similarly, the price function of demand in wholesale markets is: PH (x) = aH—bHx ,where aH is the passable highest wholesale price, bH is the influence coefficient of quantity of supplier product. For the sake of simplicity, we assume the procurement price of retailer and the selling price of supplier are identical. Remark:

aM> aH, the retail price is larger than the wholesale price,;

aM—aH> 1, since the scale of measurement, difference of retail price and wholesale price could always set larger than 1;

(aM—aH ) > (aH — C), retail profit is greater than production profit;

bH> bM , due to the supplier is far away from the consumer market, quantity of supplier product is less influence on the retail price is less than the coefficient on consumer market price than wholesale price.

3. The optimal strategies of retailer and supplier

In this paper, we consider two kind structures of the supply chains: 1, suppliers sell product in the wholesale market, and retailer buy products from wholesale market and sell to consumer in consumer market price, there are no cooperation between supplier and retailer; 2, retailers and suppliers make a price compensation contract, the retailer compensate suppliers for every product; the supplier make producing plan according to price compensation, retailers would buy the product of supplier with price compensation, then sell to consumer in the consumer price.

3.1. Supply chain without cooperation

The supplier's production plan is set according to the price function of demand in the wholesale market, and its expectation profit is;

FS (x) = E(xRp (aH - bH xRp) - xC) = E(aH xRp) - E(bH x2RP) - Cx = (aH— C) x — bH (a +1) x2

The maximum expectation profit:

Max(FS (x)) = Max(E(xRp (aH- bH xRp) - xC) = Max((aH — C) x — bH (a +1) x2)

It is easy to see that FS ( x) is convex function, because of

d2 FS ( x) = -2bH (ct + 1)<0 Hence there is optimal solution of FS (x) , by the first order condition of optimization:

S _ aH ~C

2bH (^ + 1)

So the maximum expectation profit of supplier is :

FS _ (aH~C)2

Max 4bH (CT + 1)

Similarly, the maximum expectation profit of retailer could be obtained as following:

fr (x) = E (xRp (aM~ bM xRp-aH+bH xRp ) = (aM~aH ) x~ (bH~ bM )(° +1)x2

The maximum expectation profit of retailer:

Max( Fr ( x))

= Max((aM~aH ) x- (b^bM )(^ + 1) x2) It is easy to see that FR ( x) is convex function, because of

d2 Fr ( x ) = -2(bH-bM )(ct + 1)<0 Hence there is optimal solution of FR (x), by the first order condition of optimization:

xR _ aM~aH

2(bH-bM )(ct + 1) The maximum expectation profit of retailer:

R _ (aM aH)

4(bH-bM )(ct + 1)

Proposition 1: In the agricultural products supply chain without cooperation, the retailer's optimal order quantity is larger than the optimal production plan of supplier's.

Proof: by the assumptions of (aM—aH ) > (aH—C) and bH>bM , we get bH> (bH—bM ) > 0 . The numerator of optimal retailer order is larger than the supplier's, and the denominator is smaller, it is not difficult to see to x^>xsMax , QED.

Remark: Since the retailer's optimal order quantity is larger than the supplier's, the retailer profit is not maximized, consumer market is in unsaturated state. The raise of the purchase price of retailer would motives the product plan of supplier, then improve supply chain profit. Since the supplier's production plan is made before the selling season in supply chain without influence of retailer, the expect profits of retailer is shown as following:

FR (xsMx ) = (aM~aH) J'"" C - (b*-bu + 1Y°H~ C )2 2bH (a +1) 2bH (a +1)

3.2. Supply chain with cooperation

In order to encourage supplier to expand production plan, the retailer compensate on wholesale price of supplier. Then, the supplier make production plan according to the wholesale market wholesale price and compensation price, so the supplier's expected profit is present as following;

FSC (x, Q) = E (xRp (Gh- bH xRp+Q)- xC) = (aH +Q-C) x-bH (ct + 1) x2

The maximum expectation profit of supplier:

Max(FsC(x))= Max(E(xRp(aH-bHxRp+Q)- xRpC) = Max((aH - C + Q)x-bH(ct +1)x2)

It is easy to see that FS (x) is convex function, because of

d1FC (x) = -2bH (ct + 1)<0

Hence there is optimal solution of Ff (x) , by the first order condition of optimization:

xMl (Q)=-

CCS X (aH-C + Q)

2bH (ct + 1)

So the maximum expectation profit of supplier is :

FCS =■

_ ( aH — C + Q ) Max 4(b^bM )(a + 1)

From the above calculation, it is not difficult to see that the optimal production plan supplier is a function of price compensation, so retailer set price compensation to maximize their expected profit. The retailer expected profit is so a function of price compensation as shown following;

FRC ( x, Q ) = FRC ( xMx (Q ), Q )

-in a ) aH~C + Q (b b )(_ , 1)( aH~C + Q )2 aH-C + Q Q

2bH (ct +1) 2bH (ct +1) 2bH (a +1)

The maximum expectation profit of retailer: Max( FRC (Q))

-Max((a a ) aH~C + Q (b b . 1)( aH~C + Q )2 aH~C+Q Q

-Max((aM~aH ) H ~(bM~bH )^ + 1)L w ^ n ) - H (

2bH (a +1) 2bH +1) 2bH (a +1)

It is easy to see that FR ( x) is convex function, because of the second order condition is:

d2FRC (Q) = (-2bH +1) - 2(bM - bH +1))( 1 )2< 0

2bH ^ + 1)

Hence there is optimal solution of FR (x) , by the first order condition of optimization:

QR =■

¿-■■Max

(aM~ aH )bH~ (bM~ 2bH )(aH~ C)

The maximum expectation profit of retailer is shown as following:

CR _ (aM aH )

4(bH-bM )^-1)

Proposition 2: In the agricultural products supply chain with cooperation, retailer incentive the product plan of supplier by price compensation, the expected profit of supplier and retailer are increase, thus the price compensation strategy is a Pareto improvement in supply chain.

Proof: Supplier's expected revenue, through subsidies, molecular becomes larger, the denominator becomes small, apparently, The numerator of optimal expected profit of supplier in cooperation supply chain is larger than the one without cooperation, and the denominator is smaller, it is apparent that FMisax> FIMfax . Similarly, the numerator of optimal expected profit of supplier in cooperation supply chain is larger than the one without cooperation, and the denominator is smaller, it is apparent that F^ > FMRax . So the price compensation strategy is a Pareto improvement in supply chain, QED.

Remark: By applying the price compensation strategy, the retailer share profit of the consumer market, which is bigger than the wholesale market to supplier and incentives the products plan, then chang the production supply, and improve the expected profits both of supplier and retailer.

4. Numerical Experiment

In this paper, numerical experiments apply the data based on banana market of Hainan province of China based. By the aid of Matlab7, optimal compensation price and optimal expected profit of retailer and supplier in both non-cooperation and cooperation supply chain are calculated, and the sensitive of coefficient of stochastic yield in supply chain is examined. The data used are shown as following: aM = 2000 : the possible highest price of banana consumer market, RMB per ton; bM = 0.01: coefficient of banana yield influence on the consumer market; aH = 1100: the possible highest price of banana wholesale market,RMB per ton; bH = 0.015: coefficient of banana yield influence on the wholesale market; <7 = 1.5: variance of coefficient of stochastic yield; C = 500 : production cost of banana, RMB per ton;

Table 1. supply chains performance under difference variances of coefficient of stochastic yield

variance of stochastic yield coefficient 1.2 1.3 1.4 1.8 2.5

non-cooperation supplier produce plan (ton) 9090 8695 8333 7142 5143

non-cooperation supplier expected profit (RMB) 2727300 2608700 2500000 2142900 1714300

non-cooperation retailer expected profit (RMB) 8181800 7826100 7500000 6428600 5142900

Price compensation (RMB per ton) 136.36 130.43 125.00 107.14 85.71

cooperation supplier produce plan (ton) 24793 23629 22569 22569 22569

cooperation supplier expected profit (RMB) 12323000 11599000 10951000 8929500 6717200

cooperation retailer expected profit (RMB) 18409000 17609000 16875000 14464000 11571000

From table 1, we can see:

1, under any variances condition, cooperative supply chain perform better than non-cooperative supply chain, both retailer and supplier improve the expected profit; the price compensation strategy is Pareto improvement strategy;

2, with the increase of variance of coefficient of stochastic yield, both non-cooperation and cooperation supply chain production plan decrease, so does the expected profit. It testifies the theory of Deo 2009, the producer tends to reduce the produce plan, under the high risk of stochastic yield;

3, with the increase of variance of coefficient of stochastic yield, the price compensation of retailer is reduced, which explains economic principle of the risk reduce the price;

4, in the non-cooperation supply chain, suppliers and expected profit of retailers is the 3-4 times larger than supplier; by the application of price compensation strategy, the profit allocation is significantly balance and expected profit difference between retailer and supplier reduce to about 2 times.

5. Summary

The coordination problems of agricultural product supply chain with stochastic yield risk is studied. The agricultural producing is high risk by the suffering natural disaster, and the low profit reduce the produce plan. The market is shrink and the welfare of consumer is reduce. By introducing the price compensation strategy in supply chain, the supplier is encouraged to raise the production plan, and the expected profit of whole supply chain is raise. Mathematical models is established to prove the price compensation strategy of agricultural products supply chain is Pareto improvement, increased both the the supplier and the retailer's expected profit

and balance the profit allocation in supply chain. The study provides constructive suggestion to coordination in the supply chain of agricultural products.

Acknowledgements

This paper is financially supported by the National Natural Science Foundation of China (71173089), Guangdong Province Science and technology plan project (2010A032000002) and The Project of High Level Talents Guangdong Province Funding.

References

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