Coordinating Supply Chain with Demand and Price Factor Disruptions

LEI Dong', WENG Ming21 Economics and Management School, Wuhan University, P.R.China, 430072

2 Guangxi University of Finance and Economics, P.R.China, 530003

Abstract: This paper represents coordinationpolicies for a one-supplier-one-retailer supply chain thatexperiences demand and price factor disruptions at thesame time in the planning horizon. Up to now, in thecontext of supply chain the disruption managementmodels have only incorporated a single disruption factor.We analyze the disruption management decisions fromthe supplier's point of view when demand and pricefactor disrupt simultaneously. We provide thecoordination policies when the supplier and retailer takepart in a Stackelberg game and supplier is the leader. Theresults show that the original production plan of thesupplier has some robustness when the two disruptionshappen at the same time. But the retail price of theretailer should change with the disruptions. The all-unitquantity discount policy AQDP or the capacitated linearprice policy CLPP can be used to coordinate the supplychain in this case but the coordination policies shouldchange with the disruptions too.

Keywords: Supply chain coordination, Disruptionmanagement, Demand disruption, Price factor disruption

1 Introduction

In classical supply chain circumstance, productsflow in the system and change their forms frequently indifferent stages which include suppliers, manufacturers,distributors, retailers and customers. When design supplychain, managers often think that the supply chain can beoperated smoothly and the circumstance doesn't change.But changes of circumstance, such as machinebreak-down, worker strikes, natural disasters and othersudden events, often bring different kinds of disruptionsto supply chain and influence the performance of supplychain. Hendrick and SinghalI'l show that the disruptionsdramatically decrease the stock returns and increase therisk of equity of the firms on the supply chain.Disruption management of supply chain has been a maintopic of some researchers 2,3'4,5]. These papers representthe catalog of supply chain disruptions and give someconceptual frame-works of managing them. But in thepractice of supply chain management, it is also aproblem how to find the optimal plans to overcome thesedisruptions when the supply chain faces them.

The term of disruption management is firstlyproposed in OR/MS Today by Jens ClausenE6] in 2001and the research has developed quickly. In supply chainmanagement, in the case of convex production function,when production cost disrupts, Yang 7]et al. analyze howto overcome the original production plan in the use ofdisruption management methods so that the system canoperate smoothly and cause the least effects to the

production and inventory . Xia et al. [8,9] analyze thereal-time disruption management of the system ofproduction and inventory. QiE'01 analyze the supply chaincoordination to optimize the functions of the supplychain when demand disruption appears in aone-supplier-one-retailer supply chain. In the case ofnonlinear demand and convex production function, Xu etal. [11,12] study the disruption management and supplychain coordination with demand disruption in aone-supplier-one-retailer supply chain. Yu et al. [13] studycoordination policies when the disruption of price factorhappens. All of above research works are about a singlefactor disruption in the supply chain, and doesn't careabout two or more factors disrupt simultaneously. But inthe practice of management, there are many morecomplicated phenomena that several factors disruptsimultaneously, or one disruption doesn't end, anotherhas happened. In this paper, we analysis the supplier howto modify the production plan to make disruptionmanagement decisions with the demand and price factordisruptions in a one-supplier-one-retailer supply chain.Then we represent the coordination mechanism betweenthe supplier and retailer when they take part in aStackelberg game and supplier is the leader.

2 Demand and price factor disruptsimultaneously model

2.1 Certainty modelIn this paper, we study a one-supplier-one-retailer

supply chain, the relationship between the price of theproduction and real market demand is certain and known.The supplier manufacture the products and sale to theretailer and the retailer sale the products in the market.This state can be regarded as a Stackelberg game inwhich the supplier is the leader and the retailer is thefollower.

In this game, the supplier and the retailer are bothindependent decision makers, and their objective is tomaximize their profits. Let us denote the real marketdemand d and it is a linear function of retail price,moreover, d = D - kp, where D is the market scale(forexample, the largest market demand) and k > 0 is thecoefficient of price sensitivity. In the rest of this paper,we call k the price factor. Suppose the unit productioncost of the supplier is c, and the profit of the wholesupply chain, the supplier and the retailer is fSC fSand f' , then fSC (D - kp)(p - c) , fSC +frThrough simple calculation, we can obtain the profit ofthe supply chain is maximized at the retailprice p=(D + kc)/2k and the optimal order quantity of

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the retailer is Q =(D-kc) 2, then the highest profit ofthe supply chain is fsa, = (D - kc)2 1 4k . In all aboveproposes, QiE'01 et al. prove that the supplier cancoordinate the supply chain through an all-unit quantitydiscount policy AQDP ( w1, w2, qo) (w1 > 2 ) whichworks as follows: if the retailer orders Q < q0, the unitwholesale price is w1; if the retailer orders Q > q0, theunit wholesale price becomes w2. That is to say, thesupplier pronounces a wholesale price policy firstly, andthen the retailer decides the order quantity and retailprice. The supplier must provide the quantity of theproducts the retailer orders.

2.2 Demand and price factor disrupt simultaneouslymodel

When the market scale D and price factor kdisrupt simultaneously (or the sudden events cause thedisruption of k ), suppose the market scale change isAD and the price factor change is Ak. Obviously, onlywhen AD > -D , has the model the practice value.Therefore, the rest analysis of this paper bases on thiscondition. Suppose Q is the real demand, whendisruptions happen the real demandis Q = (D + AD) - (k +Ak)p . The change of the productionquantity is AQ = Q- Q . When AQ < 0, it is implied thatthere are excess raw materials and we suppose that theycan be saleable on the second market lower than the buyprice. When AQ > 0 , the supplier must increase theproduction to satisfy the larger order quantity. And theprofit of the whole supply chain is

D+AD k c) X(Q - Q ) - X2 (Q - Q) ()k+A\k

where (x)+ =max{0,x}. And X1 > 0 implies that the unitextra cost happens when the production quantity mustincrease in the case ofAQ > 0; X2 > 0 implies that theunit extra cost happens when the production quantitymust decrease in the case ofAQ < 0. Because the excessraw materials can be sold in the second market, wesuppose X2 < c Suppose demand is the decreasingfunction of price, therefore Ak > -k .

In order to analyze the effect of the disruptions ofdemand and price factor, we have the following lemma:

Lemma 1 If f(Q) is maximized at an optimalordering quantity Q*, when AD > Akc, we have Q* >Qwhen AD < Akc , we have Q* <Q.

Proof. Since Q maximizes fSC, for any Q > 0,D-Q D -QfSC(Q) = Q( -c) > Q( c) .

k kIfAD > Akc, but Q* <Q . Then

*(Q) = QD( +AkDQ_ c) X(Q _Q*~~ ~ k+A\k

k . D-Q* AD AkcAD Q*)=k Q ( c)+Q A 2(Q - Q)k+Ak k k+A

k D-Q AD -Akc< , Q ( k -c) + Q k = (Q)k+Ak k kA

This contradicts to the assumption that Q* maximizesf(Q)in (1). So we must have Q* Q whenAD >Akc.Similarly, Q* < Q when AD < Akc . E

From lemma 1, when AD > Akc, we can reducemaximization problem off(Q) to maximize the strictlyconcave function

D+AD-Q 1 Q Qfi() =Q(k+Ak c X,Q Q (2)

Subject to Q* >Q.From the first condition fl (Q) = 0 , f(Q) is

maximized at Q D k+ withoutl 2 2

constraint.In order to analyze Q1 in the constraint

ofAD > A/kc , we have two cases of the constraint:Case 1: AD >Akc + (k +Ak)XI. In this case Q, 2 Q is

true, because fl(Q) is a concave function, fl(Q) ismaximized at Q1 . The optimal production quantity is Q1 .

Case 2: Akc+(k+Ak)XI > AD >Akc In this case,

Ql < Q doesn't satisfy the constraint, so Q1 isn't theoptimal solution to (2). According to the property ofconcave function, f4(Q) is maximized at Q = (D - kc) / 2

in the area [Q,+oo]. The optimal production quantity is Q .From these two cases, we can see that if the change

of the market scale is larger than Akc + (k + Ak)XI thesupplier should increase his original production plan togain more profit. That is to say, compared with thechange of price factor, only when the change of themarket scale is large enough, will it be profitable toadjust production plan to overcome the disruptions. Butin these two cases, whether the retail price and the profitof the supply chain increase or not isn't clear.

For case 1, the optimal retail price isD+kc+AD+Akc X1

Pl 2(k +Ak) 2

where ax

-AD - x - Akpk++Ap+ k+A\kAD - Akc - (k +Ak)X1

2

(3)

And the profit of the supply chain isSC-c AD -Akp- cx

+((f.S= max + k/ (Q +)+(p-cXk+Ak ~+x+4 )For case 2, the optimal retail price is

D+kc AD AD - AkpP2 = ~ + = P +P22(k+Ak) k+Ak k+Ak

with supply chain profitQ(AD Akp _) f AD AkpQf2QC k+Ak fmax, k+Ak

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Now we deal with the case AD < Akc . In this casewe can reduce the objective function (1) to

D +AD-Q -

Subject to Q* < Q . From the fist condition, f2(Q) is.. ~D-kc AD-A\kc+(k+A\k)X2_maximized at Q2 = 2 + A (k2 without

constraint. Similarly, there are also two cases:Case3: A/kc - (k + Ak)X2 < AD < /kc . In this case,

Q2> Q doesn't satisfy the constraint, so Q2 isn't theoptimal solution to (4). According to the property ofconcave function, f2 (Q) is maximized at Q = (D - kc) 2

in the area [0,Q ]. The optimal production quantity is Q.Case 4: Akc-(k+Ak)X2 (D-kc) < AD <Akc-(k+Ak)X2.

In this case, Q2 <Q is true, because f2(Q) is a concavefunction, f2(Q) is maximizes at Q2 . The optimalproduction quantity isQ2. Because Q2 is not positivewhen A/kc - (k + Ak)X2 - (D - kc) > AD , we obtain thecondition in case 4.

For case 3, the optimal retail price isD+kc AD AD - Akp

3 2(k+Ak) k+Ak k+Akwith the supply chain profit

fS l AD -A/p ) +AD -Akcp-f3sc = Q(p/+k)Ak _ mcax + k AkFor case 4, the optimal retail price is

D + kc AD +Akc - (k +Ak)X22(k +Ak) 2(k +Ak)_ AD - A/kp

k+Akwith supply chain profit

(5 )

Q + Ac ) + + p + 2)

ADfAk3A/pkc+Ak/cp 2

where P AD -Akc +(kc+Ak)X22

It is necessary to show that the optimal orderingquantity in case 4 and retail price in cases 2, 3 and 4 arepositive. Then we need the following lemma:

Lemma 2. The optimal ordering quantity (demand)in case 4 and retail price in case 2, 3, 4 are all positive.

Proof. From suppose AD > -D and X2 < c, wehave D+AD>Oandc-X2 >0.

In case 2, because AD > Akcc, the optimal retail priceis

D+kc ADP2 +kk2(kc+Ak) kc+Ak

D + AD +kc + AD2(k + Ak)

D + AD +kc+Akc+/c2(k + Ak)

In case 3, because Akc -((k + Ak)X2 <AD </ckc, theoptimal retail price is

D+kc AD D+AD+kc+ADA = + =2(k+Ak) k+Ak 2(k+Ak)

D+AD+kc+Akc-(k+Ak)X22(k + Ak)

D+AD+(k+Ak)(c -X2)> o

2(k + Ak)In case 4, the proof of p4 > 0 is similar as p3. And

because A/kc - (k + Ak)X2 - (D - kc) < AD < A/kc - (k + Ak)X2,the optimal ordering quantity is

D+AD -(k+Ak)(c-X2) >

2If AD < /kc - (k + A/k)X2 - (D - kc) , the optimal

ordering quantity will not be positive. This means thatthe relationship of supply chain will not exist.

Summarizing all above results, we haveTheorem 1. When the demand is a linear function

of price d = D - kp, if the disruptions of the market scaleand price factor are AD and Ak respectively, the supplychain profit is maximized for the following values ofproduction quantity and retail price

QAD -Akc + (k +A/k)X2Q + ~~2if Ak - (k +Ak)X2- (D - kc) < AD <A/kc - (k +Ak)X2;

Q*= Q, if Akc - (k +A/k)X2 < AD <Akc + (k +A/k)X;AD A/Scc- (kc+ A/c)X iAZD >A/\cc+(kc+A/ck)X

AD - Ac- kpk+A\k

Qf Akc -(kc+A/)X2 f(D-c)<AD </kc - (k +Ak)X2;AD --ACp

p k + Aki f AkAc- (k + /kc)X2 < AD < A/kc + (k+ /Ak)X;+AD a /p if AD > Akc + (k +Ak)XI.

where cc and f are defined as in (3) and (5).Theorem 1 says that although there are disruptions

from market scale and price factor, the originalproduction plan Q has some robustness. It is notnecessary to change the production plan when thedisruptions have slight effect on the supply chain. Theonly thing is to change the retail price to adjust to thedeviation cost derive from the disruptions. But if thechanges of the market scale and price factor exceed thethreshold in theorem 1, both the production plan and theretail price should be changed.

3 Decentralized decision making afterdisruptions

When the disruptions happen, if the supplier and theretailer make decisions in a decentralized way, the

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quantity the retailer orders may not be the same as intheorem 1. Suppose that the supplier wants to gain theprofit f and the supplier sets the wholesale pricew.Then the retailer gets the rest profitf' of the supplychain. For production quantityQ, after the disruptions,the profit of the supplier is

JSd(w,Q) = Q(w c) kl(Q Q) X2(Q Q)And the profit of the retailer is

frd (w, Q) = Q(p-w) = Q(D+ADk Qw).k+AkBecause the discount policy is often used to

coordinate the supply chain,[14] as following, we showthat different all-unit quantity discount policies can alsobe used to coordinate the supply chain successfully afterdisruptions.3.1 Case 1: AD >Akc + (k +Ak)kl

In this case, the supply chain profit can be rewrittenas C (D + AD (k +Ak)(c + A1))2 X

4(k + Ak)Firstly, we consider the case f > kXQ . The profit

of the supplier can be written ass (D + AD - (k +Ak)(c + A1))2 X1Q (6)

4(k + Ak)where O< 1 < I .

Theorem 2.When AD > Akc + (k + Ak)kl and>2 kXQ, the supply chain can be coordinated by the

all-unit quantity discount policy AQDP (wv, w2, Q*)where

D+AD D+ADW1 k+Ak 1-AkH1)

D+ AD (k + Ak)(c + 1)w2c+X1+r~ 2(k+Ak)

where Q* is defined as in theorem 1.Proof. Suppose w2 is the wholesale price that the

retailer wants to take, the quantity he orders should notbe less than Q* and the profit of the retailer

is fjd (Q) = Q(D+ADQ - W2), Q > Q* , which isk+Ak

maximized at D + AD - (k + Ak)w2 According to theQ,= ~~2property of concave function, the retailer has to order thequantity of Q1 to maximize his profit because of Q1 < Q.Then we have

rl Q41 W2) (1 ) (D + AD (k +Ak)(c + ,1))24(k + Ak)

If the retailer orders less than Q*, he has to acceptthe wholesale price wI and his profit

iS fd (Q) = Q( kAQk w) , which is maximized

at " =D + AD - (k + Ak)wl . Hence the maximization ofaQ1 2

his profit is = (D+AD (k + Ak)w1)

D+AD D~4+ADBecause w1> k( - c - kl), we have

2 ' (D+AD (k+Ak)(c+X1))2~~~ ~4(k+Ak)

So in order to maximize his profit, the retailer has toorderQ*. The objective profit of the supplier and themaximal profit of the supply chain are both realized.

Because the all-unit quantity discount policy AQDP(wl, w2 , Q* ) makes sense only when wI > w2, now we

show that this condition holds in this case. A sufficientcondition of for w1 > w2 is

D+AD D+ADk1+Sk( k k-C-x1)k+Ak k+Ak

> C+ +qD+AD-(k+Ak)(c+X1)2(k + Ak)

and this equivalent to

(1 D)( AD D+AD c -1)k2 )(k+Ak 1k+Ak

or 1 11riS . Obviously, the inequality is true when2

° <q < 1. Therefore we have w1 > w2 DFIn fact, givenw2, we can always find w1 which is

large enough to guarantee w1 > w2 and the profit of theretailer is higher when he takes wholesale price w2 ratherthan w1 . In brief, in the following analysis we onlyindicate that w1 is large enough and don't give theaccurate value ofw1 .

The detailed proof of theorem 2 is given above,which illustrates the main idea of designing acoordination policy for a supply chain experiencingdisruptions. In the following theorems, we don't give theproofs because they are similar to the proof of theorem 2.

Now we consider the case f' < klQ . Suppose the

supplier wants to get profit f' =rqkQ where 0 < <1<.In this case the supply chain can not be coordinated by asimple AQDP policy.

Lemma 3. When AD > Akc + (k + Ak)kl and

f =rfklQ ( 0 <ri < 1 ), the all-unit quantity discountpolicy AQDP ( w1, w2, Q0 ) can not coordinate thesupply chain.

It is necessary to find a new coordinationmechanism for the supply chain in this case. Thecapacitated linear price policy introduced by Qi et alP01 ,denoted by CLPP ( w , q), can be used to coordinate thesupply chain. A CLPP (w , q) works as follows: thesupplier charges the retailer a constant unit wholesaleprice w, but the retailer cannot order than q .

Theorem 3. When AD > Akc + (k + Ak)kl and

f = lk,Q ( 0<-< I ), the supply chain can be

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coordinated by the capacitated linear price policy CLPP(w Q* ), where

w c +X1 +((1) X(D- kc)D+AD -(k+Ak)(c+XI)

Q* is defined as in theorem 1.

3.2 Case 2: Akc + (k + Ak)kl > AD > AkcIn this case, to maximize the profit of the supply

chain, theorem 1 implies that the retailer should orderQ* = Q = (D - kc) /2. The total profit of the supply chainisfSC Q(p+ AD - Akp (D kc)2 (D - kc)(AD - Akc)

k+Ak 4(k+Ak) 2(k+Ak)

Suppose that the profit the supplier wants to realizeis fS =f2SC (o<<1).

Theorem 4. When Akc + (k + Ak)kl > AD > Akc and

f = 2SC (o < < 1), (1)if > D- +2(AD -Akc) the

~J)c(OrVzj)()if ~D -kc+2(AD -Akcc)supply chain can be coordinated by the policy AQDP(W1, W2, Q ) ; when r < Dkc+2(AD Akc) ,the

supply chain can be coordinated by the policy CLPP( w2, Q ) Where w1 is large enough,

D-kc+2(AD-Akc)anaW2 = C±+f-

2(k + Ak)

3.3 Case 3: Akc - (k + Ak)X2 < AD <AkcIn this case, the profit of the supply chain can be

rewritten as f3SC 4(D_ AC) + ( k)(4(k + Ak) 2(k + Ak)

if f3C > 0 is required, we have D - kc + 2(AD - Akc) > 0

or AD- Akc> -Q. Suppose that the profit the supplierwants to get is f5 =lf3)c (0 < y< 1) .

Theorem 5. When Akc - (k + A/k)X2 < AD <Akcand AD- Akc > -Q , the profit goal of the supplier is

s ~~~~~~~2(AD -A/cc) ,thf = f3sc (o< < 1) (1)if '>D> tc+2(D /c)'supply chain can be coordinated by the policy AQDP

k)+2(AD - Akc)

supply chain can be coordinated by the policy CLPP( w2, Q ), where w1 is large enough,

D - kc + 2(AD - Akc)ana W2 = C+11-

2(k + A/k)The supplier is the leader of the Stackelberg game,

so he can obtain the complete information of marketscale and price factor disruptions before he begins tomanufacture. When the total profit of the supply chain isnot positive, that is to say, AD- Akc < -Q, the retailerdoes not order any product . In order to lower the loss,

the supplier has to sale his products manufacturedaccording to the plan Q in the second market and bear

D -kc+2(AD -A/cc)-the loss _k + Ak) Q. Therefore, in this case,2(k + A/c)

the rational supplier doesn't manufacture any productand the coordination of the supply chain doesn't exist.

3.4 Case 4:Akc - (k + Ak)X2 - (D - kc) < AD < Akc - (k + Ak)X2In this case, the maximization of the total supply

chain profit isfSC (D+AD (k + Ak)(c - X2))X2 _

f4 ~~~4(kc+ A/c)We still consider the profit of the supply chain is

positive. From the retailer's point of view, suppose theobjective profit of the retailer is

(D + AD -(k + Ak)(c K)) -X2Q (6)4(kc+ A/c)

Then the profit of the supplier iSfS = ( -1)X2Q . Theprofits of them should be both positive and we have

<(D + AD - (k +Ak)(c - X2 ))2

4(k +Ak)X2QTheorem 6. When

Akc - (k + Ak)X2 - (D - kc) < AD < Akc - (k + Ak)X2 and

< <(D+AD-4(k/+Ak)(c-X2)) if the profit of the4(k + Ak)X I

retailer is defined as in (6 ), the supply chain can becoordinated by the policy AQDP ( w1, w2 , Q* , wherew1 is large enough,

w2 C X2+ gX2(D - kc)D+AD -(k+AAk)(c-X2)

Q* and g are defined as in theorem 1 and (6)respectively.

4 Conclusion

This paper investigates the production managementdecisions and the coordination policies in aone-supplier-one-retailer supply chain when itexperiences demand and price factor disruptionssimultaneously in which the supplier is the leader of aStackelberg game. Our investigation of disruptionmanagement in the supply chain finds that the originalproduction plan of the supplier has some robustness andthe production plan hasn't to be changed until thedisruptions between the demand and price factor satisfysome conditions. But the retail price has to change withthe disruptions to adjust to the deviation costs. And thecoordination policy has to change with the disruptionstoo. In addition, the changes of the production and thecoordination policy depend on both the demanddisruption and the price factor disruption.

The results derived in this paper shows that when

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the disruptions of the demand and the price factor happenat the same time, they have no effect on the productionplan in some condition, which is different from the casethat only one disruption happens. And if the policiesdon't change with the disruptions, although the totalsupply chain profit is positive, the supplier or the retailermay exit the supply chain because of no profit to gain.

For future research, it suggests that the nonlineardemand functions and the relationship between the unitproduction cost and the production quantity should beincorporated in the model. It would be worth-while toanalyze the influence of other disruptions besidesdemand and price factor on the management decisionsand coordination policies in a supply chain.

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