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Procedía CIRP 17 (2014) 639 - 644
Variety Management in Manufacturing. Proceedings of the 47th CIRP Conference on Manufacturing
Systems
A Supply Chain Planning Model with Supplier Selection under Uncertain Demands and Asymmetric Information
Sisi Yina, Tatsushi Nishia'*
aGraduate School of Engineering Science, Osaka University,1-3 Machikaneyama-cho,Toyona City, Japan * Corresponding author. Tel.:+81-6-6850-6351; fax:+81-6-6850-6351. E-mail address: nishi@sys.es.osaka-u.ac.jp .
Abstract
In this paper, a supply chain planning model including a manufacturer, a retailer and multiple suppliers under demand uncertainty with asymmetric information is considered. The manufacturer determines production, estimated quantity of defective components and the selection of suppliers. Quantities and quality of components are decided by the selected suppliers. The negotiation between the manufacturer and the retailer is based on buyback contracts. Due to asymmetric information, the quality information of components purchased from suppliers is unknown for the manufacturer. Thus, two scenarios are investigated for the manufacturer to estimate uncertainty of risk. The problem is analysed by a Stackelberg game where the manufacturer is a leader and the suppliers are followers. An optimization approach is proposed to solve the problem under demand uncertainty. A Stackelberg equilibrium is obtained by the proposed solution approach. Computational experiments are conducted to illustrate the features of the proposed models with different parameters. The results show the validity of the proposed model.
© 2014 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.Org/licenses/by-nc-nd/3.0/).
Selectionandpeer-reviewunderresponsibilityoftheInternational Scientific Committee of "The 47th CIRP Conference on Manufacturing Systems" in the person of the Conference Chair Professor Hoda ElMaraghy"
Keywords: global supply chain, supply chain planning; supplier selection, demand uncertainty; asymmetric information;; quality management; game theory
1. Introduction
Nowadays, the outsourcing from suppliers to countries such as China or India has become popular. Companies establish their factories globally to decrease costs. The increased outsourcing accelerates the competition on quality in the global market. In this paper, a game theoretic model for global supply chain planning with uncertain demands and the supplier selection is proposed in order to improve quality and obtain optimal profits.
In order to reduce costs and improve quality in global supply chain planning, the supplier selection becomes crucial. Many researchers consider a variety of supplier selection problems from different perspectives. Vanteddu et al. [1] presents a supplier selection problem as a supply chain configuration problem that competing suppliers at a stage
differ only in terms of costs and responsiveness. Xu and Nozick [2] formulate a multi-period single product supply chain as a two-stage mixed-integer stochastic program to optimize the supplier selection to hedge against the loss of the production capability at supplier sites. Mendoza and Ventura [3] propose inventory planning models to select the best set if suppliers determine the proper allocation of order quantities while minimizing the annual ordering, inventory holding, and purchasing costs under suppliers' capacity and quality constraints.
The quality problem associating with defects from suppliers or in the production process is widely investigated [4, 5, 6]. Quality issues have been studied intensively in supply chain planning. However, incomplete quality information has not been studied. From more practical perspective, it is important to assume asymmetric information for supply chain members due to different business strategies
2212-8271 © 2014 Published by Elsevier B.V. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/3.0/).
Selection and peer-review under responsibility of the International Scientific Committee of "The 47th CIRP Conference on Manufacturing Systems"
in the person of the Conference Chair Professor Hoda ElMaraghy"
doi:10.1016/j.procir.2014.01.109
in the global supply chain planning. Esmaeili and Zeephongsekul [7] introduce a seller-buyer supply chain model with an asymmetric information structure. They assume that only buyer knows the demand function and is aware of the seller's setup cost and purchasing cost. Lei et al. [8] investigates the impact of asymmetric information on disruption management when disruptions of demand and costs are private information. Most of works related to asymmetric information by applying game theory assume that demand information is asymmetric. There are rarely researches considering asymmetric quality information by applying game theoretic approaches. Tse and Tan [9] study the unclear information of quality risk and visibility in a multi-tier supply chain. They consider the situation of asymmetric information between a manufacturer and a supplier. They focus on the manufacturer's decision making to manage risk and visibility in supply chain planning. The coordination between the manufacturers and the supplier is not investigated. A game theoretic model for single manufacturer and suppliers has been presented by Yin et al. [10]. However, the supplier selection and the coordination with retailers have not been studied in the conventional works.
The objective of this paper is to study a three echelon supply chain model with the supplier selection and asymmetric quality information under demand uncertainty. The problem is solved by a game theoretical approach. It is assumed that the quality information between the manufacturer and suppliers is asymmetric. Thus, a worst case and an average case are analyzed to estimate uncertainty due to asymmetric information.
The rest of paper is organized as follows. The problem description and modeling are described in Section 2. The solution approach is provided in Section 3. Numerical examples are shown in Section 4. Finally, the concluding remarks are stated in Section 5 with the future work.
2. Problem description
2.1. The three echelon supply chain
The outline of the supply chain model is shown in Fig. 1.
supplier selection asymmetric quality
buyback contract
component j = !,..,]
product k = 1,..,K
Fig. 1. The supply chain model.
In our model, a hierarchical supply chain consisting of one manufacturer which produces finished product k (k = 1,...,K) , suppliers j(j = !,■■,]) as followers producing component i (/ = 1,... I), and a retailer is addressed. In order to satisfy uncertain demands, the retailer orders finished
products from the manufacturer. The manufacturer and the retailer are coordinated by buyback contracts. The retailer decides order quantities of finished products under demand uncertainty. The asymmetric quality information is considered between one manufacturer and suppliers in the paper. The manufacturer decides production and the selection of suppliers. Quality of components and quantity of components are determined by suppliers. The problem is formulated by a Stackelberg game where the manufacturer is the leader and both of the suppliers and the retailer are the followers.
2.2. Quality
Suppliers pay higher cost for producing one component if the reliability of the component increases. Thus, the production cost for one unit component i for supplier j depends on the reliability x^ and the order quantity dy of component i for supplier j . The number of defective components is affected by reliability Xy. If the reliability Xy increases, the number of defective components decreases. The production cost is expressed by the cost function fry, such as h-ij = Aij + Bijdij + CijXij. Atj is the fixed cost of production for one unit component i paid by supplier j . By is the production cost responsiveness to the quantity of component i for supplier j. It indicates that if the order quantity dy is increased, production cost per one unit component increases. The increase of the order quantity of components causes the increase of production for the supplier. Thus, the supplier must pay more production costs such as machinery wearout costs or labor costs. Cy is the production cost responsiveness to the reliability of component i for supplier j. It indicates that higher reliability drives the increase of the production cost. It is assumed that the number of defective component i for supplier j is normally distributed where ¿¿y is mean value of number of defective component i for supplier, which is a constant known by both of the manufacturer and suppliers. However, the standard deviation of number of defective components 5y is a decision variable for suppliers which is unknown for the manufacturer. With the increase of the standard deviation 5y , the reliability of components will decrease. Therefore, there is an assumption in our paper such
that reliability Xy = e 2 . The higher 5y reduces the reliability of components seen in Fig. 2. The negative sign of this assumption is that it becomes difficult to evaluate the quality of components if 5y is large, because xtj tend to close to zero if Sij becomes large.
_i_1_1_1_1_1_1_1_1_1_1_1_1_1_i_
0 0.5 1 1.5 2 2.5 3 Sfj
Fig. 2. Reliability and deviation.
2.3. Formulation Indices
i : component for assembling finished products (i = 1,...,/) j : supplier (J = 1,...,/) k : product (k = 1, ...,K)
Decision variables
Retailer:
Dk: order quantity of finished product k from the retailer Manufacturer:
yk : production quantity of finished product k from the manufacturer for the retailer
Pij: estimated quantity of defective component i from supplier j
vtj : binary variable which takes 1 if the manufacturer buys component i from supplier j
Wj: binary variable to determine the selection of suppliers j for actual purchase which takes 1 if supplier j is contracted, and 0 otherwise.
Supplier i:
dtj : order quantity of component i offered by supplier j to the manufacturer
Sij: standard deviation of the reliability of production system
to produce component i for supplier j
xtj : reliability of component i for supplier j
Parameters
Retailer:
z^: credit cost per unit paid by the manufacturer to the retailer for returned product k
zk : goodwill penalty cost per one unit product k due to
stockout incurred by the retailer
pk: selling price of finished product k by the retailer
Dk : random variable of demand of finished products from
customers which follows a normal distribution such that
Dk~N(uk,Skz) where uk is the given mean value and 8'k is
the given standard deviation
fiP'k): probability density function followed by the demand
of product k for the retailer
Manufacturer:
Uij : upper bound of estimation of the worst case for component i for supplier j
rk: wholesale price of finished product k from the retailer
: inventory cost per one unit product k from the retailer paid by the manufacturer rk: wholesale price of finished product k ek: unit production cost for product k offering to retailer ak : opportunity loss cost for understocking of one unit of product k
bk : inventory holding cost for overstocking of one unit of product k
Cj: the capacity of supplier j
f(zk): probability density function followed by the demand of the product k
nij: the internal resource of component i for supplier j
Ty: penalty cost for extra defective component i for supplier j pij : cost of extra required component i purchased from the spot market
gik: number of units of component i required to produce one unit of product k
tk: resource required by the manufacturer to produce one unit of product k
Q : production capacity of the manufacturer Retj : random variable of realized quantity of defective component i for supplier j follows a normal distribution where is ^¡j is the mean value and Stj is the standard deviation. Here, is constant
zk : random variable of demand for product k from the
customers following by a normal distribution
Supplier:
hij : production cost for one unit component i paid by supplier j
qij: selling price of component i for supplier j
Yij : compensation cost of defective component i paid to the
manufacturer by supplier j
Atj : fixed production cost for one unit component i paid by supplier j
Bij : production cost responsiveness to order quantity of component i paid by supplier j
Cij : production cost responsiveness to risk degree of component i paid by supplier j
Due to asymmetric quality information, the manufacturer cannot observe full information of defective components during the negotiation process. Therefore, two scenarios (average case and worst case) for the manufacturer are investigated to estimate the number of defective components. For the average case, two situations are considered by the manufacturer. The number of defective components is estimated by minimizing the total expected penalty cost. For the worst case, the upper bound of defined situation is given.
2.4. The average case model
Due to incomplete information, the manufacturer faces amount of defective components after delivery from suppliers. Thus, the manufacturer should estimate the quantity of defective components beforehand. For the average case, once the estimated quantity of defective components is less than the realized quantity, the manufacturer has to order extra components from the outsourcing suppliers in order to achieve production. The manufacturer pays penalty costs when the estimated quantity is more than the realized quantity. However, the manufacturer is compensated by suppliers because of amount of defective components.
2.4.1 Manufacturer's model with supplier selection
The manufacturer decides production and supplier selection. Due to incomplete information, the manufacturer faces amount of defective components after delivery from suppliers. Thus, the manufacturer should estimate the quantity of defective components beforehand. Let Pi7- denote the estimated quantity of defective components and Retj be a random variable of the realized quantity of defective
components. It is assumed that the manufacturer knows partial quality information of components. Therefore, the realized quantity of defective component Retj follows a normal distribution (p-ij.Sfj) which is known by the manufacturer.
The manufacturer's objective function for the average case is the maximization of the total profit including the sale revenue of finished products, inventory cost, credit cost, production cost and purchasing cost of components. The last two terms represent the penalty cost caused by defective components and the compensation cost paid by suppliers. The formulation is as follows:
maxY,Kk=1[rkDk - z3k max(0,yk - Dk) - f0^k(Dk -
-Ei=1yfeefe -ZUZURIAJVij -
l'j=1mjwj-^i=1I,Jj=1{pljVljE[msK(p,Relj - PtJ)] + TijvijE[max(0,Pij - Rey)]} + E-=1Ej=1yyVy {E[max(0 ,Retj - Py)] + £[max(0,Py - Re y)]} (1) Subject to:
1) Production capacity
Ek=i tkyk<Q (2)
2) Required amount of components to assemble products
gikyk < - pij)vij ,vi (3)
3) Supplier's resource
ZIi=inijdij <CjWj,Vj (4)
4) Binary variable constraint
Wj>Vij,Vi,j (5)
where E is the expectation of the equation.
2.4.2 Retailer's model
The retailer determines order quantity Dk from the manufacturer with uncertain demands. The demand of products from customers is uncertain but it is assumed that probability density function f(Dk) is known. The retailer and the manufacturer are coordinated by the buyback contract. Thus, once the demand is realized, retailers could return unsold products to the manufacturer. Therefore, the retailer's decision model is formulated as the following unconstrained optimization problem:
max£*=1/^{[p^ + (Dk - D^f^dD^ +
IDklPkDk - M - Dk)zftf(.Di)dDi - Dkrk}
2.4.3 Suppliers' model
The supplier j's problem is formulated including sales revenue, the cost of components and the penalty cost for defective components.
max£i=1{dy {qtj - htj) - YiJ{E[max(0,ReiJ - P;j)] + £[max(0,Pi7-flei7)]}} (7)
hij — Atj + Bijdij + CijXij
*ij = (9)
Eq. (8) represents the production cost which depends on the quantity of components and the quality of components. Eq. (9) is the reliability function with respect to the standard
deviation of the number of defective components. 2.5. The worst case model
The worst case is also considered in this paper so that the manufacturer could make pessimistic decisions according to his business strategies. Due to different business strategies, the definition of the worst case varies. In this paper, it is assumed that the worst case is that the actual quantity of defective components Retj is much larger than the estimated number of defective component Ptj. In other words, Retj > Pij. From the practical perspective, the manufacturers always suffer more loss once they have to purchase extra components from the spot markets. The manufacturer should determine an upper bound of this situation in order to design optimal production planning. Therefore, a chance constraint is utilized to estimate uncertainty of defective components. The chance constraint in this paper is to give an upper bound when the production cannot be completely achieved. The additional constraint for the manufacturer is given to estimate worst case which is expressed by
Pr[Reij > Pj] < Ui}
where Wy i s the upper bound on the probability Pr which is decided by decision makers. The chance-constraint is given to estimate the probability of worse situations that the realized quantity of defective components is greater than estimation, and the prob abi lity is b ounded by Uij.
In order to facilitate the calculation of chance-constraint, the equation is reformulated into a form introduced by Petkov and Maranas [11]. The chance-constraint is equivalently written as
0(Y„)>1 -Ut] (11)
where the left-hand side ^(Yjj) is a normal cumulative distribution function followed by . The cumulative
normal distribution is rewritten into the standardized normal form. Yij = Pl' where ^¡j is mean value of Retj and Stj is
deviation of Retj.
Thus, Yij > — Uij) which is equivalent to
Sij^-1(l-Uij)-Pij+^ij<0 (12)
The formulation for the worst case is considered by embedding the chance-constraint of Eq. (1) into the function for the average case.
3. Solution approach
The manufacturer and suppliers are analyzed by a Stackelberg game where the manufacturer is a leader. The optimal response functions should be derived firstly. Then, the manufacturer's decision is solved by substituting optimal resp onse function s as input parameters . The suppl ier j ' s objective function is rewritten as:
i1 = Y!i={dij ^qij - Atj - Buda - Cije~-T^ -Yij{E[max(0,Reij - Pij)] + £[max(0,Py - fley)]} (13)
where Ret]~NQit],ôlj)
The objective function can be reformulated by a normalization technique (Petkov and Maranas, 1997). The calculation procedure is as follows:
Let and YtJ=P-*p-
Oft O ¡j
Sij ~ «a J 1 Rejj-Hj > Pjj-Mj
¡■J
£[min(fley,Py)] = F(Yij)E[Rei]
F(Yij))E[Ptj = StJ + Slj{F(Ylj)E[Xlj\Xlj < YtJ F(YiJ))E[XiJ\XiJ < YtJ]
E[Ytj\Ytj — Xtj] = Yj
E[Xt]\Xt] — Ytj] =
' v äXjj f(Yij)
^S^e^SäXu ^y)
Thus, the resulting formulation is obtained as follows:
L1 = Y,'i=i{dij{qij-Aij - Bijdij - Ct]e 2 yijôij[2f(Yij) - Ytj + 2YtjFftj)]}
Where Ytj =
F(Ytj) is a cumulative distribution function, and /(Vy) is a probability density function.
The following optimal response functions are calculated by obtaining the first derivative of Eq. (14) with respect to Stj :
aSij ~
5?. y?. y?.
ZUidy Ctje-TStj - 2Ytj^=e-r - 2Ylj8ljj= e"^ *
(-Ytj) - 2rtj{Ptj-KjWtj) * ("^l =
ZUldtj CtJe-?StJ - 2rtjf(YtJ) - 2Yijf(Yij)Yi2j +
2Yijf(Yij)Yl2J] = E{=1[dy CtJe-?StJ - 2Yijf(Yij) ] (15)
By solving the equation —— = 0, the critical point of the
do ¿j
equation as an increasing and concave function of 5y is obtained. Because the following condition is satisfied:
32L1 _ 95?. "
Ei=i[- dijCije z5?--2y,
1 l> / \
Z'i=i[-dijCije~~tLSlj - -^f{Yij)Yjij\ < 0
because -
■ = -Zi=12Bt, <0.
The retailer's best response function is obtained analytically. The retailer's objective function is expressed by:
max L2 = ££=1 J°k{\pkDi + (Dk - +
iDk VkDk - {D'k - DJzlmDßdDi - Dkrk}
In order to determine the optimal order quantity Dk, the following equations are obtained by differentiating Eq. (17) with respect to the order quantity Dk and set this amount equal to 0 :
— = ^FCDp +Pk[l- F(Dk)] + z2[ 1 - F(Dk)] -rk = 0
F (D 'k) =
rk-Pk-Zk zk~Pk
This is a global maximum, since from Eq. (6)
4. Computational examples
In this section, the modeling approach is applied in an illustrative case study. There are assumed to be three outsourcing suppliers (/ = 3) which offers four electronic parts (I = 4). The manufacturer assembles two types of electronic devices (K = 2) to sell to the market. The demand of fini shed products from customers is as sumed to follow a normal distribution with mean value of 13 for each type of products and standard deviation of 19 and 18, respectively. Since profit monotonically increases with the decrease of Wy, the discussion on the selection of parameter a is neglected. A relatively much small value compared with the possibility when the actual quantity of defective components is large than the estimated defective components for the average case is chosen. Thus, the upper bound of worst case Wy is set to 0.54. The unit sales revenue for each product is 2000, and production cost for each product is 10. The opportunity loss cost for each product is 200, and the inventory cost for each product is 80 and 100, respectively. Other parameters are shown in Table 1.
Table 1. Parameters for computational experiments.
ci 250 % 105-130
Bik 1 Ty 10
3 Pll 5
1 Yu 4
Thus, the optimal solution 5y is obtained by solving = 0.
By taking the first derivative of L1 with respect to dy and setting the result to zero as follows:
= - At, - 2Blldl, - Cy-e"^) = 0 (16)
A PC with An Intel(R) Core TM i7-3770 3.4 GHz processor and 8GB memory is used for the computation. The pro gram is coded by GAMS and solved by BOMININ (Basic Open-source Nonlinear Mixed Integer programming). The optimal solutions for the manufacturer are shown in Table 1. It takes 0.031 seconds to solve the average case problem, and 0.016 seconds to solve the worst case problem. The absolute gap of this example is 7.275958e-012, and the relative gap is almost zero. A near-optimal solution can be derived by the proposed method. The results in Table 2 show that supplier 2 and supplier 3 are selected for both of the average case and worst case. The prices of components offered by supplier 1 are the most expensive. Moreover, the production capacity is
considered in the model. The purchasing quantity of components cannot exceed the capacity. Thus, the manufacturer will choose the cheaper one. Comparing with the results for the average case, the profit for the manufacturer decreases for the worst case. Meanwhile, the order quantity of components and estimated defective components increase for the worst case. The reason is that the manufacturer estimates that the quantity of defective components for the worst case is more than the quantity for the average case in order to avoid the huge loss. Thus, the required quantity of components also climbs up for the worst case. From the computational results, we conclude that it is necessary to optimize demand and quantity simultaneously for decision makers to obtain the maximum profits.
Table 2. Optimal solution.
Average case Product 1 Product 2
production 25.750 18.000 Compl Comp2 Comp3 Comp4
order quantity of supl: - - - -
components
sup 2: 68.395 72.558 76.722 76.722
sup3: 76.722 76.722 85.050 68.395
estimated defective supl: - - - -
components
sup 2: 3.000 3.000 3.000 3.000
sup3: 3.000 3.000 3.227 3.000
profit 53731.335
Worst case Product 1 Product 2
production 25.750 18.000 Comp1 Comp2 Comp3 Comp4
order quantity of supl: - - - -
components
sup2: 68.333 72.500 76.667 76.667
sup3: 76.722 76.722 85.050 68.395
estimated defective supl: - - - -
components
sup2: 3.367 3.356 3.346 3.346
sup3: 3.346 3.346 3.329 3.367
profit 51686.282
In this study, the demand of finished products is uncertain. In order to demonstrate the impact of demand uncertainty on profit for the average case and worst case, the comparative experiments are conducted by analyzing the deviation of demand of finished products. The results of sensitivity analysis are given in Table 3. The results show that the profits for both of the average case and the worst case always increase if the deviation is decreased. It indicates that if the demand from the market becomes stable, the manufacturer could obtain more profits.
Table 3. Sensitivity analysis of demand deviation.
Case 1 Deviation Product 1= 19 Product 2= 18
Average case (profit) 53731.335
Worst case (profit) 51686.282
Case 2 Deviation Product 1= 18 Product 2= 18
Average case (profit) 56190.443
Worst case (profit) 51739.674
Case 3 Deviation Product 1= 16 Product 2= 18
Average case (profit) 56824.598
Worst case (profit) 56797.638
5. Conclusion
A three-echelon supply chain optimization model under demand uncertainty with asymmetric quality information is addressed. The manufacturer faces supplier selection and uncertain quality information. The problem is approached by non-cooperative game. The quality information between the manufacturer and suppliers is asymmetric. Two scenarios ( average case and worst case ) for the manufacturer to estimate quality are studied. In future, more sophisticated estimation technique will be investigated.
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