Research ArticleInformed Principal Model and Contract in Supply Chain withDemand Disruption Asymmetric Information

Huan Zhang and Jianli Jiang

Department of Economic Management North China Electric Power University Baoding Hebei 071100 China

Correspondence should be addressed to Huan Zhang hd0086126com

Received 24 March 2016 Accepted 19 May 2016

Academic Editor Monica A Lopez-Campos

Copyright copy 2016 H Zhang and J Jiang This is an open access article distributed under the Creative Commons AttributionLicense which permits unrestricted use distribution and reproduction in any medium provided the original work is properlycited

Because of the frequency and disastrous influence the supply chain disruption has caused extensive concern both in the industryand in the academia In a supply chainwith onemanufacturer andone retailer the demandof the retailer is uncertain andmeanwhilemay suffer disruption with a probability Taking the demand disruption probability as the retailerrsquos asymmetric information aninformed principal model with the retailer as the principal is explored to make the contract The retailer can show its informationto the manufacturer through the contract It is found out that the high-risk retailer intends to pretend to be the low-risk one Sothe separating contract is given through the low-information-intensity allocation in which the order quantity and the transferringpayment for the low-risk retailer distort upwards but those of high-risk retailer do not distort In order to reduce the signaling costwhich the low-risk retailer pays the interim efficient model is introduced which ends up with the order quantity and transferringpayment distorting upwards again but less than before In the numerical examples with two different mutation probabilities theinformed principal contracts show the application of the informed principal model in the supply chain with demand disruption

1 Introduction

Supply chain risk management is becoming an increasinglyimportant area In the past several years there has been ashift of focus from creating efficient supply chains to reliableand efficient supply chains This shift is due to the large-scale negative impacts of supply chain disruptions in globalsupply chain networks For example in 2013 the ldquohorsemeatrdquoincident in Europe led to a plunge in demand of beef products[1] And the 2015 terrorist attack in Paris depressed thetourism in whole Europe and reduced the demand of aircraftindustry Walmart Home and some other large companieshave set up specialized disruption management departmentto deal with disruptions in supply chain [2] Because of thefrequency and disastrous influence the demand disruptionin supply chain has caused extensive concern both in theindustry and in the academia

Snyder et al figure out that contract designing is oneof the most important strategic measures in supply chaindisruption management [3] There is an abundant amountof research available on the topic of contract in supply chain

with demand disruption [4ndash7] Most research considers thedemand disruption as symmetric information [8ndash11] Thedemand disruption may be influenced by the nature politicseconomics society or even the finance management tech-nology human and so on so there is a different probabilityor damage degree to suffer disruption by different companies[3] Through the operation the company in supply chainknows the environment and its interior so it can know moreabout the disruption compared to the others [12] Hendrickset al find out that the managers choose not to report orunderstate the disruption which has a negative effect onthe finance or stock price [13 14] Bunkley reports that themotor components suppliers hide some important disruptioninformation after the Tohoku earthquake which results inthe second half to the American motor industry [15] Sothe disruption information is not symmetric between theanticipants of supply chain The experiment in Sarkar andKumar confirms that the asymmetry of demand disruptioninformation reduces the efficiency of supply chain [16]Asymmetric information leads to low supply chain efficiency[17] so lots of research focuses on coordinating the supply

Hindawi Publishing CorporationMathematical Problems in EngineeringVolume 2016 Article ID 2306583 12 pageshttpdxdoiorg10115520162306583

2 Mathematical Problems in Engineering

chains with asymmetric information [18ndash23] So it is impor-tant to consider the asymmetric disruption informationwhenwe design the supply chain contract [24] But very fewresearchers pay attention to the asymmetric disruptioninformation Gumus et al consider the supply disruptionprobability as the supplierrsquos asymmetric information andgive the optimal incentive contract [12 25] Lei et al derivethe optimal linear contract and wholesale contract with theimpact of asymmetric demanddisruption and cost disruptioninformation [24] Huang and Yang set up the principal agentmodel and make the incentive contract considering thedemand disruption information asymmetric [26]

In this paper we consider the demand disruption proba-bility as the retailerrsquos private information in order to investi-gate the contract between the supplier and the retailer In theavailable research about asymmetric information in supplychain researchers always try to reveal the retailerrsquos privateinformation by amenu of incentive contracts [27ndash29] or shar-ing the information [30 31] to improve the interests In all ofthe researches the principal agent theory is explored and theparty with no private information is seen as the principal AsMyerson et al point out there are limitations of the assump-tion of the principal with no private information In manycontracts for example the sales contract employment con-tract and regulation contract it is common for the party withprivate information to be the principal who makes the con-tract [32ndash35] So we consider the following situation in a sup-ply chain with one manufacturer and one retailer the retailermay suffer demand disruptionwith a certain probability Andthe retailerwith the asymmetric demanddisruption probabil-ity informationmakes the supply chain contract to express itsown information to the manufacturer Then the retailer candiffer from the other retailer by providing this contract

Differing from those of prior studies the major innova-tions of the research are as follows

(1)We consider demand uncertainty and demand disrup-tion simultaneously and the demanddisruption probability isconsidered as the retailerrsquos asymmetric information Demanduncertainty is commonly considered in the supply chainresearch and demand disruption happens on occasion Boththe demand uncertainty and demand disruption impactthe supply chain contract making And demand disruptionis totally different from demand uncertainty so demanduncertainty and demand disruption should be consideredtogether in the supply chain contract [3] Lots of researchpays more attention to the coordination contract in supplychain with demand disruption but most is studied withoutasymmetric disruption information [24 26]

(2) The informed principal model is explored for theseparating contract in the supply chain with asymmetricdemand disruption information In the research [24 26] thesupplier as the principal makes the contract to encouragethe retailer to show its own demand disruption informationBut we use the informed principal model and consider theretailer with the demand disruption information to be theprincipal The retailer can deliver its own information to themanufacturer and can be distinguished from the others fromthe separating contract [35ndash38]

The reminder of this paper is organized as followsIn Section 2 the concerned problem is defined and theassumptions and notations are given The full informationmodel and optimal contract are given in Section 3 Section 4goes into the informed principal model In Section 5 theinterim efficient allocation is given to reduce the signalingcost in the informed principal model Numerical examplesare presented in Section 6 and sensitivity analysis is carriedout with respect to some key model parameters The paperconcludes with Section 7 in which we provide a summary ofthe paper and future research directions

2 Model Description

We consider a supply chain consisting of one risk neutralmanufacturer and one risk neutral retailer When the sellingseason comes the retailer purchases a kind of product fromthe manufacturer and then sells it into marketThemarket ofthe retailer may suffer disruption And different retailers havedifferent probability to experience the disruption Supposingthe disruption probability as the retailerrsquos private informa-tion the informed principal model is set up to design thecontracts with which the retailer can demonstrate its trueinformation to the manufacturer

21 Assumptions Theinformedprincipalmodel of the supplychain with demand disruption meets the following assump-tions To make the presentation clear we also list the relevantvariables or notations (see VariableNotations section)

Assumption 1 Before the selling season comes the retailermakes order 119902 from the manufacturer and the manufacturersupplies the right quantity 119902 to the retailer with the unitman-ufacturing cost 119888 Then the retailer pays 119905 as the transferringpayment Andwhen the selling season comes the retailer sellsthe products to the demand market with the price 119901

Assumption 2 The retailer faces an unstable market whichmeans the demand in this market may suffer disruptionWithout disruption the demand 119910 belongs to [0 119860] with thedistribution function 119865(119910) and the density function 119891(119910)With the demand disruption the retailerrsquos demand 119909 belongsto [0 119863] with the distribution function 119866(119909) and densityfunction 119892(119909) The demand belongs to the same distributionstyle whenever with disruption or without disruption forexample uniform distribution and normal distribution andthe market scale without disruption is larger than that withdisruption 119860 gt 119863 And 119864(119909) = 1198602 119864(119910) = 1198632 Var(119909) gt

Var(119910) So for the same demand 1199090

= 1199100 the distribution

function meets 119866(1199090) ge 119865(119910

0) It means when the demand

disruption happens the demand which is less than the givendemand 119909

0 or 1199100 happens in a larger probability than that ofthe demand without disruption

Assumption 3 There are two kinds of retailers the low-riskone with demand disruption probability 120572

1and the high-risk

one with demand disruption probability 1205722 while 120572

1lt 1205722

The demand disruption probability is the retailerrsquos individualinformationThemanufacturer only knows there is a low-risk

Mathematical Problems in Engineering 3

Time

R gets typeinformation

R designsthe contracts

R choosesthe contract

and makes order

M suppliesproducts andgets payment

R sellssome

products

M gets theunsold

products

M acceptsor rejects

t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 t = 7

Figure 1 Informed principal game sequence in supply chain with demand disruption R means the retailer M means the manufacturer

retailer with the probability 120582 and a high-risk retailer with theprobability 1 minus 120582 Both the high-risk retailer and the low-riskretailer are risk neutral

Assumption 4 The retailer as the informed principal designsthe transferring contracts by which the retailer can demon-strate its true information to the manufacturer The transfer-ring contract for low-risk retailer is 119905

1 1199021 while for high-

risk one it is 1199052 1199022

Assumption 5 Every unsold product has the salvage value119904 (119904 lt 119888) And the manufacturer has the salvage value of theunsold products For example the unsold products can bereused for the manufacturer but the retailer has to payfor disposing them The manufacturer can get the unsoldproducts free from the retailer and reuse them In thiscase the manufacturer has the salvage value of the unsoldproducts

119878(120572119894 119902119894) (119894 = 1 2) denote the low-risk retailerrsquos and high-

risk retailerrsquos expected sales

119878 (120572119894 119902119894) = 120572119894(119902119894minus int

119902119894

0

119866 (119909) 119889119909)

+ (1 minus 120572119894) (119902119894minus int

119902119894

0

119865 (119910) 119889119910)

(1)

119868(120572119894 119902119894) (119894 = 1 2) denote the low-risk retailerrsquos and high-

risk retailerrsquos expected unsold quantity

119868 (120572119894 119902119894) = 120572119894int

119902119894

0

119866 (119909) 119889119909 + (1 minus 120572119894) int

119902119894

0

119865 (119910) 119889119910 (2)

119880119894(119894 = 1 2) denote the low-risk retailerrsquos and high-risk

retailerrsquos utility

119880119894= 119901119878 (120572

119894 119902119894) minus 119905119894 (3)

119881119894(119894 = 1 2) denote the manufacturerrsquos utility when it

cooperates with the low-risk retailer or high-risk retailer

119881119894= 119905119894minus 119888119902119894+ 119904119868 (120572

119894 119902119894) (4)

22 Game Sequence When the retailer faces the demanddisruption the sequence of the informed principal gamebetween themanufacturer and the retailer is shown as follows(Figure 1)

(1) The retailer knows its type information 120572119894

(2) The retailer as the informed principal designs thetransferring contract package (119905

1 1199021) (1199052 1199022)

(3) The manufacturer accepts or rejects the contract(4) Before the selling season the retailer chooses the

transferring contract according to its type and makesthe order 119902

119894

(5) The manufacturer supplies products and the retailerpays 119905

119894for them

(6) In the selling season the demand disruption occursin a certain probability and the retailer sells someproducts

(7) After the selling season the manufacturer gets theunsold products

3 Full Information Model

31 Modeling When the demand disruption probability ispublic information the different retailers should provide thesingle selected contracts (119905

1 1199021) or (119905

2 1199022) We can get this

pair of single selected contracts by solving the followingprograms 1198751

119865 and 119875

2

119865

1198751

119865

max1199051 1199021

1198801= 119901119878 (120572

1 1199021) minus 1199051 (5)

st 1198811= 1199051minus 1198881199021+ 119904119868 (120572

1 1199021) ge 0 (6)

1198752

119865

max1199052 1199022

1198802= 119901119878 (120572

2 1199022) minus 1199052 (7)

st 1198812= 1199052minus 1198881199022+ 119904119868 (120572

2 1199022) ge 0 (8)

The targets of these programs are the retailersrsquo revenuewhich are the principal As in the full information situationeach participant knows the true information so there is noincentive compatible constraint Whatever the manufacturershould get no less than the retrained revenue The incentiverationality constraint of the manufacturer facing differentretailers is useful here (see (6) and (8))

32 Results By solving programs 1198751119865 and 119875

2

119865 there comes

Proposition 6 (Proof of Proposition 6 is given inAppendixA)

Proposition 6 (1) Under full information the optimal con-tract in supply chain with demand disruption satisfies thatboth the optimal order and optimal transferring payment for

4 Mathematical Problems in Engineering

t

Alowast

Blowast

q

tlowast1

tlowast2

qlowast1qlowast2

Ulowast1

Ulowast2

Vlowast1

Vlowast2

Figure 2 Supply chain contract under full information

low-risk retailer are higher than those for high-risk retailer as119902lowast

1gt 119902lowast

2 119905lowast1gt 119905lowast

2

(2) Under full information the utility of the low-riskretailer is larger than that of high-risk retailer as 119880lowast

1gt 119880lowast

2

and the utility of the manufacturer cooperating with low-riskretailer and with high-risk retailer is equal to the retrainedrevenue as 119881lowast

1= 119881lowast

2= 0

From Proposition 6 it is obvious that under full infor-mation the low-risk retailer should make more orders thanthe high-risk retailer (119902lowast

1gt 119902lowast

2) and meanwhile pay more

(119905lowast1gt 119905lowast

2) Let us show the optimal contract in Figure 2 using

the retailer and themanufacturer indifferent utility curve Forthe low-risk retailer the optimal contract is shown at point119861lowast And for the high-risk retailer the optimal contract is

shown at point 119860lowast The retailerrsquos utility grows more whenits indifferent curve goes into southeast So when the low-risk retailer pretends to be high-risk one using allocation119860

lowastwhich is to say that the low-risk retailerrsquos indifferent utilitycurve goes northwest its utility decreases But if the high-risk retailer pretends to be low-risk one using allocation 119861

lowastits utility increases So as the self-interested participant thehigh-risk retailer tends to pretend to be low-risk one Thenwe can get Proposition 7

Proposition 7 Considering the demand disruption proba-bility as the retailerrsquos individual information the optimalallocation for the low-risk retailer is not separating equilibriumallocation

4 Informed Principal Model

In order to show its true type information the retailer can usethe informed principalmodel to offer an option contract [32]The term ldquooption contract (119905

1 1199021) (1199052 1199022)rdquo comes from the

fact that if the manufacturer accepts the contract the retailermust then exercise its built-in option and choose between(1199051 1199021) and (119905

2 1199022) The retailer will choose the term which

is fit to its own type by probability 100 as 1199011205721| (1199051 1199021) =

1199011205722

| (1199052 1199022) = 1 And the retailer will choose the term

which is not fit to its own probability by probability 0 as1199011205721

| (1199052 1199022) = 119901120572

2| (1199051 1199021) = 0 [33 34] Then the

informed principal model of the supply chain with demanddisruption is built up for the Perfect Bayesian Equilibrium

41 Modeling Firstly the incentive compatible constraint isgivenThe option contract is incentive compatible if the low-risk retailer prefers the contract item (119905

1 1199021) and the high-

risk retailer prefers the contract item (1199052 1199022) So the incentive

compatible constraint should keep the low-risk retailerrsquosutility no less than when it pretends to be a high-risk one

119901119878 (1205721 1199021) minus 1199051ge 119901119878 (120572

1 1199022) minus 1199052 (9)

And the high-risk retailerrsquos utility is no less than when itpretends to be a low-risk one

119901119878 (1205722 1199022) minus 1199052ge 119901119878 (120572

2 1199021) minus 1199051 (10)

Secondly the incentive rationality constraint is givenTheretailer with the individual information is the principal andthemanufacturerwithout individual information is the agentSo the agentrsquos incentive rationality is the same as that (as in (6)and (8)) in full information situation

Thirdly the target function is given The target of theinformed principal model is also to maximize the utility ofthe principal or to say the retailer

So the programs under informed principal of supplychain with demand disruption are 119875

1

119868 and 119875

2

119868 Consider

the following

1198751

119868

max1199051 1199021

1198801= 119901119878 (120572

1 1199021) minus 1199051

st (6) (8) (9) (10)

(11)

1198752

119868

max1199052 1199022

1198802= 119901119878 (120572

2 1199022) minus 1199052

st (6) (8) (9) (10)

(12)

Definition 8 Utility 1198801(1199050

1 1199020

1)1198802(1199050

2 1199020

2) for low-riskhigh-

risk retailer is the low-information-intensity optimum forthat type if (1199050

1 1199020

1)(1199050

2 1199020

2)maximizes low-risk retailerrsquos utility

in the set of incentive compatible constraint and incentiverationality constraint They are (part of) the solution to theprograms 1198751

119868 and 119875

2

119868

For solving the informedprincipalmodel the assumptionof weak monotonic profit is listed as follows

Assumption 9 It is weak monotonic profit when the manu-facturer makes a nonnegative profit if the contractual termsare those of low-risk retailer under symmetric informationand the retailer is a high-risk one 119881

1(119905lowast

2 119902lowast

2) ge 0

From Figure 2 if the contractual term for the low-riskretailer changes from119861

lowast to119860lowast the curve119881lowast1will go southeast

and themanufacturerrsquos profit increases So themanufacturerrsquosprofit is no less than 0 and Assumption 9 is satisfied

Mathematical Problems in Engineering 5

Definition 10 The separating allocation is the allocation (119905119904

1

119902119904

1) for the low-risk retailer and the symmetric information

contractual terms (119905lowast2 119902lowast

2) are for the high-risk retailer where

(119905119904

1 119902119904

1) maximizes the low-risk retailerrsquos utility subject to the

manufacturerrsquos breaking even for the low-risk retailer and tothe high-risk retailer not preferring (119905

119904

1 119902119904

1) to (119905

lowast

2 119902lowast

2) The

program is listed as 1198751119904

1198751

119904

max11990511199021

1198801= 119901119878 (120572

1 1199021) minus 1199051

st (6)

(13)

119901119878 (1205722 119902lowast

2) minus 119905lowast

2ge 119901119878 (120572

2 1199021) minus 1199051 (14)

42 Results

Proposition 11 Under the weak monotonic-profit assump-tion the separating allocation is the low-information-intensityoptimum

Proof (1) The high-risk retailer can get its asymmetric infor-mation utility even under asymmetric information Compar-ing programs 1198752

119865 and 119875

2

119868 it can be found that these two

programs have the same target function but 1198752

119865 has less

constraints so 119880(119905lowast

2 119902lowast

2) ge 119880(119905

0

2 1199020

2) According to Assump-

tion 9 the manufacturer at least breaks even regardless of theretailerrsquos type Hence (119905lowast

2 119902lowast

2) is the separating allocation item

for high-risk retailer(2) Because 119880(119905

lowast

2 119902lowast

2) ge 119880(119905

0

2 1199020

2) constraint (14) is more

restricted than (10) For the low-risk retailer the programs1198751

119868 and 119875

119904

1 have the same target functions but there are

more constraints in the former program so 119880(119905119904

1 119902119904

1) ge

119880(1199050

1 1199020

1) is satisfied And the low-risk retailer at least can

get 119880(119905119904

1 119902119904

1) from the option contract From the incentive

compatibility constraint (14) which can be rewritten as1198802(119905lowast

2 119902lowast

2) ge 1198802(119905119904

1 119902119904

1) it has been guaranteed that the high-

risk retailer will not choose 119905119904

1 119902119904

1 From the program 119875

119904

1

the low-risk retailer can get more utility when the contractualitem is 119905119904

1 119902119904

1 than that of 119905lowast

2 119902lowast

2 So we can get 1199050

1 1199020

1 =

119905119904

1 119902119904

1

Then we can get Proposition 12 (see details in Appen-dix B)

Proposition 12 As for the option contract by informed retailerin supply chain with demand disruption the low-information-intensity allocation satisfies the following

(1) Comparing to the full information situation the low-information-intensity allocation for high-risk retailerand its utility does not distort as 1199050

2= 119905lowast

2 11990202

= 119902lowast

2

1198802(1199050

2 1199020

2) = 119880lowast

2

(2) Comparing to the full information situation the low-information-intensity allocation for low-risk retailerhas upward distortion 119905

0

1gt 119905lowast

1 11990201gt 119902lowast

1 while its utility

has downward distortion 1198801(1199050

1 1199020

1) lt 119880lowast

1

(3) The order quantities in the low-information-intensityallocation for the retailers in different types satisfy

119901119878 (1205722 1199020

1) minus 119888119902

0

1+ 119904119868 (120572

1 1199020

1)

= 119901119878 (1205722 119902lowast

2) minus 119888119902

lowast

2+ 119904119868 (120572

2 119902lowast

2)

(15)

In the informed principal model in order to prevent thehigh-risk retailer from pretending to be a low-risk one thelow-information-intensity allocation items for the low-riskretailer distort The low-risk retailer should order more thanthe optimal order and paymore than the optimal transferringpayment But the utility of the low-risk retailer is less than theoptimal onewhichmeans the low-risk retailer pays some rentto separate from the high-risk retailer So the rent is calledsignaling cost

5 Interim Efficient Allocation

The retailer can deliver its own type information to themanufacturer by low-information-intensity allocation in theinformed principal model But the low-risk retailerrsquos orderand transferring payment distort relative to the full infor-mation situation and it has to pay the signaling cost Thesignaling cost is the part utility which the low-risk retailergets less than the optimal So maybe we can decrease thesignaling cost by increasing the high-risk retailerrsquos utilityand meanwhile increasing the low-risk retailerrsquos utility So inthis part we try to find a separating equilibrium with lesssignaling cost [32ndash34 38] by the interim efficient model

Let us consider the interim efficient model which candecrease the signaling cost by giving the high-risk retailermore than optimal utility Let be the utility which the high-risk retailer gets more than the optimal one And let 119871() betheminimal loss of themanufacturer when it cooperates withthe high-risk retailer119871() can be gotten by the program 119875

2

119898

1198752

119898

minus119871 () = max1199052 1199022

1199052minus 1198881199022+ 119904119868 (120572

2 1199022) (16)

st 119901119878 (1205722 1199022) minus 1199052

ge 119901119878 (1205722 119902lowast

2) minus 119905lowast

2+

(17)

It is easy to find out that when constraint (17) is binding1199022= 119902lowast

2and 119871() =

Definition 13 Utility 1198801(119905119898

1 119902119898

1)1198802(119905119898

2 119902119898

2) for low-risk

high-risk retailer is the interim efficient optimum for thattype if (119905119898

1 119902119898

1)(119905119898

2 119902119898

2) maximizes low-risk retailerrsquos utility

in the set of incentive compatible constraint and themanufac-turerrsquos expected profit And the contract (119905119898

1 119902119898

1) (119905119898

2 119902119898

2) is

the interim efficient allocationThey are (part of) the solutionto the programs 1198751

119898 and 119875

2

119898

6 Mathematical Problems in Engineering

Table 1 Full information contract versus low-information-intensity allocation

Type ofretailer Full information

Lowinformationintensity

Type ofretailer Full information

Lowinformationintensity

1205721= 02

119902lowast

1= 1001 119902

0

1= 1014

1205721= 03

119902lowast

1= 990 119902

0

1= 1002

119905lowast

1= 1862 119905

0

1= 1881 119905

lowast

1= 1840 119905

0

1= 1824

119880lowast

1= 5340 119880

0

1= 5338 119880

lowast

1= 5278 119880

0

1= 5192

1205722= 04

119902lowast

2= 978 119902

0

1= 978

1205722= 05

119902lowast

2= 967 119902

0

2= 967

119905lowast

2= 1819 119905

0

2= 1819 119905

lowast

2= 1799 119905

0

2= 1799

119880lowast

2= 5217 119880

0

2= 5217 119880

lowast

2= 5159 119880

0

2= 5159

1198751

119898

max11990511199021

1198801= 119901119878 (120572

1 1199021) minus 1199051 (18)

st 120582 (119905119898

1minus 119888119902119898

1+ 119904119868 (120572

1 119902119898

1)) minus (1 minus 120582) ge 0 (19)

119901119878 (1205722 119902lowast

2) minus 119905lowast

2+ ge 119901119878 (120572

2 119902119898

1) minus 119905119898

1 (20)

The high-risk retailer can get the rent above theoptimum so themanufacturer will lose some utilityThen theincentive rational constraint (see (19)) for themanufacturer isthe expected profit rather than the individual profit Equation(20) is the incentive compatible constraint for the high-riskretailer

From programs 1198751119898 and 119875

2

119898 we can get Propositions 14

and 15 (see Proof in Appendix C)

Proposition 14 The interim efficient allocation by informedprincipal for the supply chain with demand disruption satisfiesthe following

(1) Comparing to the full information situation the orderquantity for the high-risk retailer does not distort 119902119898

2=

119902lowast

2

(2) Comparing to the full information situation the orderquantity for the low-risk retailer satisfies (21) anddistorts upwards 119902119898

1gt 119902lowast

1 Consider

1199041198681015840(1205721 119902119898

1) + 119901119878

1015840(1205721 119902119898

1) minus 119888

=1 minus 120582

120582[1199011198781015840(1205722 119902119898

1) minus 119901119878

1015840(1205721 119902119898

1)]

(21)

(3) The transferring payment for the low-risk retailer meets(22) and the transferring payment for the high-riskretailer meets (23)

119905119898

1= (1 minus 120582)

sdot [119901119878 (1205722 119902119898

1) minus 119901119878 (120572

2 119902lowast

2) + 119888119902

lowast

2minus 119904119868 (120572

2 119902lowast

2)]

+ 120582 [119888119902119898

1minus 119904119868 (120572

1 119902119898

1)]

(22)

119905119898

2= (1 minus 120582) [119888119902

lowast

2minus 119904119868 (120572

2 119902lowast

2)] + 120582119901119878 (120572

2 119902lowast

2)

+ 120582 [119888119902119898

1minus 119904119868 (120572

1 119902119898

1) minus 119901119878 (120572

2 119902119898

1)]

(23)

Proposition 15 There is a threshold 1205820

= (1199011198781015840(1205722 1199020

1) minus

1199011198781015840(1205721 1199020

1))(119904119868

1015840(1205721 1199020

1) + 119901119878

1015840(1205722 1199020

1) minus 119888) for the low-risk

retailer probability If and only if the probability of low-riskretailer is no more than the threshold 120582 le 120582

0 the low-infor-mation-intensity allocation is also interim efficient and isthe unique Perfect Bayesian Equilibrium If the probabilityof low-risk retailer is more than the threshold 120582 gt 120582

0 thelow-information-intensity allocation is not the unique PerfectBayesian Equilibrium

If and only if the probability of low-risk retailer is smallenough (120582 le 120582

0) the low-information-intensity allocation isthe unique Perfect Bayesian EquilibriumAnd the retailer candeliver its own type information by the separating contractBut if the low-risk retailer is more than that (120582 gt 120582

0) the low-information-intensity allocation is not the unique PerfectBayesian Equilibrium and the interim efficient allocation canimprove the distortion situation to a certain degree So for thesupply chain with demand disruption the optimal allocationcannot be reached if the informed principal provides aseparating contract

6 Numerical Example

This section gives some numerical examples to inspect thesupply chain contract with demand disruption by informedprincipal We let the sales price 119901 = 10 the unit manu-facturing cost is 119888 = 2 and the salvage product value is119904 = 05 The demand distribution function before disruptionis 119865(119910) = 119910

2125 and the demand distribution function after

disruption is 119866(119909) = 1199092100

61 Main Results From the example listed above the fullinformation contracts and low-information-intensity alloca-tion are given in Table 1

Table 1 shows that for the full information contract theorder quantity for low-risk retailer is larger than that for thehigh-risk retailer 119902lowast

1gt 119902lowast

2 and also the transferring payment

119905lowast

1gt 119905lowast

2 which is performed in Proposition 6 In the informed

principal model the order quantity for the low-risk retailerdistorts in order to show its type (when 120572

1= 02 versus 120572

2=

04 119902lowast1

= 1001 lt 119902sb1

= 1014 and when 1205721= 03 versus

1205722= 05 119902lowast

1= 990 lt 119902

0

1= 1002) Meanwhile we notice that

the utility obtained by the low-risk retailer in the informed

Mathematical Problems in Engineering 7

0 10

5

10

15

q1

02 04 06 08

120582

Figure 3 The order quantity in interim efficient allocation for low-risk retailer changes by its probability

0

1

101112

1314

150

5

10

15

20

25

30

0204

0608

tm 1

q1 120582

Figure 4 The transferring payment in interim efficient allocationfor low-risk retailer changes by the order quantity and its probability

principal model is lower than that in full information model(when 120572

1= 02 versus 120572

2= 04 119880lowast

1= 5340 gt 119880

0

1= 5338

and when 1205721= 03 versus 120572

2= 05 119905lowast

1= 1840 gt 119905

0

1= 1824)

This implies that it costs the low-risk retailer something tocertify its typeThese results are the samewith Proposition 12

Considering the situation of 1205721= 02 versus 120572

2= 04 we

obtain 1205820= 019 The order quantity for the low-risk retailer

119902119898

1changes with the proportion it has 120582 in interim efficient

allocation As Figure 3 shows if and only if 120582 varies withincertain area (120582 le 120582

0) we obtain 119902119898

1gt 119902lowast

1 as in the results in

Proposition 14The transferring payment in the interim efficient allo-

cation is depicted in Figure 4 (for the low-risk retailer)and Figure 5 (for the high-risk retailer) The transferringpayments are changing by the order quantity for the low-risk retailer in the interim efficient allocation and the low-risk retailerrsquos probability From Figure 4 when the low-riskretailerrsquos probability 120582 is less enough and the order quantityfor the low-risk retailer 119902119898

1is high enough the transferring

0

05

1

10 11 12 13 14 15

20

25

30

35

40

45

tm 2

q1

120582

Figure 5 The transferring payment in interim efficient allocationfor high-risk retailer changes by the low-risk retailerrsquos order quantityand probability

payment 1199051198981increases by the low-risk retailerrsquos probability 120582

but decreases by the order quantity for the low-risk retailer119902119898

1(bottom left in Figure 4) But mostly the transferring

payment 1199051198981increases by the low-risk retailerrsquos probability 120582

but decreases by the order quantity for the low-risk retailer 1199021198981

(red and orange part in Figure 4) Normally the transferringpayment should increase by the order quantity which meansldquoget more pay morerdquo From this view the low-risk retailerrsquosprobability 120582 should not be very small and the order quantityfor the low-risk retailer 119902

119898

1should not be close to the

maximum value 15From Figure 5 it is obviously seen that the transferring

payment for high-risk retailer in the interim efficient alloca-tion 119905119898

2increases by the order quantity for the low-risk retailer

119902119898

1but decreases by the low-risk retailerrsquos probability 120582

62 Sensitivity Analyses The sensitivity analyses for the low-information-intensity allocation versus the unit manufactur-ing cost 119888 are shown in Table 2 It shows the outcomes whenthe unit manufacturing cost 119888 changes by +25 and minus25once at a time and keeping remaining parameters The orderquantities decrease by the unit manufacturing cost but theinfluence for the high-risk retailerrsquos order quantity is largerthan that for the low-risk retailer (minus069 versus minus317 and+217 versus +307) The transferring payments increaseby the unit manufacturing cost More interesting the utilitiesdecrease by the unit manufacturing cost and the low-riskretailer receives a greater impact (minus940 versus minus922 and+955 versus +953) So from this point of view the low-risk retailer should encourage themanufacturer to reduce thecost more

Figure 6 shows the sensitivity analyses for the order quan-tities versus the unit manufacturing cost in which the orderquantities include the order quantity for the low-risk retailerin low-information-intensity allocation 119902

0

1 the order quantity

8 Mathematical Problems in Engineering

Table 2 Sensitivity analyses for low-information-intensity allocation versus the unit manufacturing cost

Changes of 119888 The low-information allocation for low-riskretailer (120572

1= 02)

The low-information allocationfor high-risk retailer (120572

2= 04)

Value Change rate Value Change rate Value Change rate

119888 = 25 +251199020

1= 1007 minus069 119902

0

2= 947 minus317

1199050

1= 2375 +2626 119905

0

2= 2243 +2331

1198800

1= 4836 minus940 119880

0

2= 4736 minus922

119888 = 2 Baseline1199020

1= 1014 mdash 119902

sb2

= 978 mdash1199050

1= 1881 mdash 119905

sb2

= 1819 mdash1198800

1= 5338 mdash 119880

sb2

= 5217 mdash

119888 = 15 minus251199020

1= 1036 +217 119902

0

2= 1008 +307

1199050

1= 1398 minus2568 119905

0

2= 1362 minus2512

1198800

1= 5848 +955 119880

0

2= 5714 +953

108

106

104

102

10

98

96

94

92

C

14 16 18 2 22 24 26 28 3

qsb1

qlowast1q2

Figure 6 The order quantities change by the unit manufacturingcost

for the low-risk retailer in full information contract 119902lowast1 and

the order quantity for the high-risk retailer in full informationcontract which is equal to that in low-information-intensityallocation 119902

2 Both the order quantity for the low-risk retailer

in full information and the order quantity for the high-riskretailer decrease by the unit manufacturing cost and theychange in the same level basically But the change of the orderquantity for the low-risk retailer in low information intensitydepends on different values of the unit manufacturing costWhen the unit manufacturing cost 119888 = 275 the orderquantity is minimum and when 119888 = 175 or 119888 = 15 the orderquantity is the maximum

Figure 7 shows the sensitivity analyses for the transferringpayment versus the unit manufacturing cost in which thetransferring payments include the transferring payment forthe low-risk retailer in low-information-intensity allocation1199050

1 the transferring payment for the low-risk retailer in full

information contract 119905lowast

1 and the transferring payment for

26

24

22

20

18

16

14

C

14 16 18 2 22 24 26 28 3

tsb1

tlowast1t2

Figure 7 The transferring payments change by the unit manufac-turing cost

the high-risk retailer in full information contract which isequal to that in low-information-intensity allocation 119905

2 All

the transferring payments of the retailers increase by the unitmanufacturing cost The transferring payments for the low-risk retailer and the high-risk retailer in full informationcontract decrease in the same level

Figure 8 shows the sensitivity analyses for the retailersrsquoutilities versus the unit manufacturing cost in which theutilities include the utility for the low-risk retailer in low-information-intensity allocation 119880

0

1 the transferring pay-

ment for the low-risk retailer in full information contract119880lowast

1 and the transferring payment for the high-risk retailer

in full information contract which is equal to that in low-information-intensity allocation 119880

2 All the retailersrsquo utilities

decrease by the unit manufacturing costFigure 9 shows the sensitivity analyses for the low-risk

retailerrsquos signaling cost versus the unit manufacturing cost

Mathematical Problems in Engineering 9

44

46

48

50

52

54

56

58

60

C

14 16 18 2 22 24 26 28

Usb1

Ulowast1

U2

Figure 8 The retailerrsquos utilities change by the unit manufacturingcost

0

005

01

015

02

025

C

14 16 18 2 22 24 26 28

ΔU1

Figure 9The signaling cost changes by the saved unit manufactur-ing cost

The signaling cost of the low-risk retailer is the utility earnedin the low-information-intensity allocation which is sepa-rating contract less than that in full information contractWhen the unit manufacturing cost 119888 = 2 the signaling costis the least while when the saved unit manufacturing cost119888 = 175 the signaling cost is the most We do not find theinternal relationship between the signaling cost and the unitmanufacturing cost As the signaling cost is so important toput the contracts into practice investigation on the signalingcost is the research direction for us

7 Conclusion

In the recent years supply chain with demand disruptionhas become more and more attractive to both business and

academic research There is a lot of research available on thetopic of supply chain disruption contract In the real supplychain practice demand disruption is totally different fromthe stochastic demand So in our research both the demanduncertainty and the demand disruption are consideredTaking the demand disruption probability as the asymmetricinformation an informed principal model is explored tomake the contract for supply chain with demand disruptionAnd the conclusions we obtain are shown as follows

(1) The contracts for low-risk retailer and high-riskretailer in full information are provided and it isfound out that the high-risk retailer has the incentiveto pretend to be the low-risk retailer This result issimilar to the other informed principal model [3435] but totally different from the normal principalagent model [24 26] In the normal principal agentmodel the high efficient agent (low-risk retailer in oursetting) tries to pretend to be low efficient agent (high-risk retailer) to get more That is why the informedprincipal model is established

(2) The low-information-intensity allocation which is theresult of the informed principal model and also is theseparating contract shows that the order quantity andthe transferring payment for the low-risk retailer inlow-information-intensity allocation distort upwardsbut that of high-risk retailer does not distort Inorder to avoid imitation the imitatorsrsquo efficiencyshould be always distorted But in normal principalagent model the high-risk retailerrsquos quantity distortsdownwards [24 26] on the contrary the low-riskretailerrsquos quantity in informed principal model dis-torts upwards

(3) In order to reduce the signaling cost which the low-risk retailer pays the interim efficient model is intro-duced which ends up with the order quantity andtransferring payment distorted upward again but lessthan that of low information intensity Comparingto [38] focusing on analyzing the signaling costmore attention is paid to showing the interim efficientmodel to get a solution which can increase efficiency

From the managerial aspect retailers should try toincrease their antirisk capability to enhance competitivepower And when the retailer has lower demand disruptionprobability than the others it can show its style to the supplierthrough the separating contract For example the retailer cansay ldquobecause I am the low-risk one I can cooperate with thesupplier with this contract while the others who are high-riskones can only use the other contractrdquo But in order to be sep-arated from the others the low-risk retailer has to pay somesignaling cost and the quantity has to be distorted upwards

Above all there are some limitations of the research Wewill continue our research in the following directions (1)The signaling cost is a key factor impacting the separatingcontract which should be paid more attention (2) We onlyconsider the demand disruption but disruptions happen atnot only demand side but also supply side Sowewill continue

10 Mathematical Problems in Engineering

our research with demand disruption together with supplydisruption [39]

Appendix

A Proof of Proposition 6

For the two types of retailers we can obtain the optimalallocation if and only if the whole margin utility of the supplychain is the same with the margin cost So 119902

lowast

1and 119902

lowast

2satisfy

the following first-order conditions

1199011198781015840(1205721 1199021) + 1199041198681015840(1205721 1199021) = 119888 (A1)

1199011198781015840(1205722 1199022) + 1199041198681015840(1205722 1199022) = 119888 (A2)

And the optimal transferring payments are

119905lowast

1= 119888119902lowast

1minus 119904119868 (120572

1 119902lowast

1)

119905lowast

2= 119888119902lowast

2minus 119904119868 (120572

2 119902lowast

2)

(A3)

From (A1) and (A2) we obtain

1205791119866 (119902lowast

1) + (1 minus 120579

1) 119865 (119902lowast

1)

= 1205792119866 (119902lowast

2) + (1 minus 120579

2) 119865 (119902lowast

2) =

(119901 minus 119888)

(119901 minus 119904)

(A4)

In addition 119866(119902lowast

1) ge 119865(119902

lowast

1) 1205721

lt 1205722 so we know that

1205721119866(119902lowast

1) + (1minus120572

1)119865(119902lowast

1) lt 1205722119866(119902lowast

1) + (1minus120572

2)119865(119902lowast

1) and from

(A4) we obtain 1205722119866(119902lowast

2) + (1 minus 120572

2)119865(119902lowast

2) lt 1205722119866(119902lowast

1) + (1 minus

1205722)119865(119902lowast

1) and thus 119902lowast

2lt 119902lowast

1 From (A3) 119905lowast

1minus119905lowast

2= 119888(119902lowast

1minus119902lowast

2)minus

119904(119868(1205721 119902lowast

1) minus 119868(120572

2 119902lowast

2)) 119905lowast1minus 119905lowast

2=119902lowast

1=119902lowast

2

(1205721minus 1205722)119904(int119902

0119865(119910)119889119910 minus

int119902

0119866(119909)119889119909) gt 0 and 120597119905

lowast

1120597119902lowast

1gt 0 so 119905

lowast

1minus 119905lowast

2gt 0 and

thus we obtain 119905lowast

1gt 119905lowast

2 119881lowast

1= 119881lowast

2= 0 Furthermore we get

119880lowast

1gt 119880lowast

2from 120597119880120597120579 lt 0 and 120597119880120597119902 gt 0 Finally we have

Proposition 6

B Proof of Proposition 12

We denote 1205831and 120583

2as the Lagrange multipliers of (6) and

(14) and find the derivate of 1199051and 1199021 and thus we obtain

120597119880 (1205721 1199021)

1205971199051

= minus1 + 1205831+ 1205832= 0

120597119880 (1205721 1199021)

1205971199021

= 1199011198781015840(1205721 1199021) minus 1205831119888 + 12058311199041198681015840(1205721 1199021)

minus 12058321199011198781015840(1205722 1199021) = 0

(B1)

By rewriting (B1) we get the following formulas

1199041198681015840(1205721 119902119904

1) + 119901119878

1015840(1205721 119902119904

1) minus 119888

=(1 minus 120583

1)

1205831

[1199011198781015840(1205722 119902119904

1) minus 119901119878

1015840(1205721 119902119904

1)]

(B2)

1205831=

1199011198781015840(1205722 119902119904

1) minus 119901119878

1015840(1205721 119902119904

1)

1199041198681015840 (1205721 119902119904

1) + 1199011198781015840 (120572

2 119902119904

1) minus 119888

(B3)

1205832=

1199041198681015840(1205721 119902119904

1) + 119901119878

1015840(1205721 119902119904

1) minus 119888

1199041198681015840 (1205721 119902119904

1) + 1199011198781015840 (120572

2 119902119904

1) minus 119888

(B4)

Because the numerator of 1205831is negative so 120583

1is positive

and the denominator of 1205831is negative because 120583

2is positive

(when 1205832= 0 it cannot be separated from the former one)

the numerator of 1205832is negative nevertheless 1199011198781015840(120572

1 119902lowast

1) minus

119888 + 1199041198681015840(1205721 119902lowast

1) = 0 and thus 1199011198781015840(120572

1 119902119904

1) minus 119888 + 119904119868

1015840(1205721 119902119904

1) lt

1199011198781015840(1205721 119902lowast

1)minus 119888+ 119904119868

1015840(1205721 119902lowast

1) in addition 1205972119880120597119902

2lt 0 so 119902

119904

1gt

119902lowast

1 From 120583

1gt 0 and 120583

2gt 0 we deduce that (6) and (14)

are tight Furthermore the programs 1198751119865 and 119875

1

119898 have the

same objective function but the constraint of 1198751119898 is more

tight so there is distortion of the utility obtained from 1198751

119898 in

other words 1198801(119905119904

1 119902119904

1) lt 119880lowast

1 From Proposition 11 we obtain

1198802(1199050

2 1199020

2) = 119880lowast

2 So we have Proposition 12

C Proof of Propositions 14 and 15

We denote 120573 and 120574 as the Lagrange multipliers of (19) and(20) and find the derivate of 119905

1 1199021 and and thus we obtain

120597119871 (1199051 1199021 )

1205971199051

= minus1 + 120573120582 + 120574 = 0 (C1)

120597119871 (1199051 1199021 )

1205971199021

= 1199011198781015840(1205721 1199021) + 120573120582 [minus119888 + 119904119868

1015840(1205721 1199021)]

minus 1205741199011198781015840(1205721 1199021) = 0

(C2)

120597119871 (1199051 1199021 )

120597= minus120573 (1 minus 120582) + 120574 = 0 (C3)

To solve (C1)ndash(C3) we obtain 120573 = 1 120574 = 1 minus 120582 Andconsidering (C2) we get

1199041198681015840(1205721 119902119898

1) + 119901119878

1015840(1205721 119902119898

1) minus 119888

=1 minus 120582

120582[1199011198781015840(1205722 119902119898

1) minus 119901119878

1015840(1205721 119902119898

1)]

(C4)

Because (19) and (20) are tight we obtain the followingresults

119905119898

1= (1 minus 120582)

sdot [119901119878 (1205722 119902119898

1) minus 119901119878 (120572

2 119902lowast

2) + 119888119902

lowast

2minus 119904119868 (120572

2 119902lowast

2)]

+ 120582 [119888119902119898

1minus 119904119868 (120572

1 119902119898

1)]

119905119898

2= (1 minus 120582) [119888119902

lowast

2minus 119904119868 (120572

2 119902lowast

2)] + 120582119901119878 (120572

2 119902lowast

2)

+ 120582 [119888119902119898

1minus 119904119868 (120572

1 119902119898

1) minus 119901119878 (120572

2 119902119898

1)]

(C5)

Mathematical Problems in Engineering 11

To compare (A1) with (C4) we know 119902119898

1gt 119902lowast

1because

12059721198801205971199022

lt 0 and 1199011198781015840(1205722 119902) minus 119901119878

1015840(1205721 119902) lt 0 and the dis-

tortion level is changing with 120572 Finally we have Proposi-tion 14

To compare (C4) with (B2) and designate 1205820

= 120582 =

(1199011198781015840(1205722 1199020

1) minus 119901119878

1015840(1205721 1199020

1))(119904119868

1015840(1205721 1199020

1) + 119901119878

1015840(1205722 1199020

1) minus 119888) we

have the following conclusions

(1) When 120582 = 1205820 1199021198981

= 119902119904

1= 1199020

1 we obtain the same

contracts with the low-information-intensity situa-tion

(2) When 120582 lt 1205820 1199021198981

lt 119902119904

1= 1199020

1 we know there is more

distortion than the low-information-intensity alloca-tion and it improves anything

(3) When 120582 gt 1205820 1199021198981

lt 119902119904

1= 1199020

1 the interim efficient

allocation can improve the low-information-intensityallocation So we have Proposition 15

VariablesNotations

119888 Unit manufacturing cost1205721 Disruption probability of low-risk retailer

119910 Market demand without disruption119865(sdot) Distribution function of demand without

disruption119904 Unit salvage value119860 Market scale without disruption119878(120572119894 119902119894) Expected sales

119868(120572119894 119902119894) Expected unsold quantity

119880119894 Utility of the retailer

120582 Probability of low-risk retailer1205722 Disruption probability of high-risk retailer

119909 Market demand with disruption119866(sdot) Distribution function of demand with

disruption119901 Retail price119863 Market scale with disruption119881119894 Utility of the manufacturer

Control Variables

119902119894 Order quantity

119905119894 Transferring payment

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported in part by a project supported bythe Humanities and Social Sciences Project of the EducationMinistry (no 14YJC630187) Hebei Social Sciences Project(no HB15GL058) and the Fundamental Research Funds forthe Central Universities (no 2016MS123)

References

[1] J Barnett F Begen S Howes et al ldquoConsumersrsquo confidencereflections and response strategies following the horsemeatincidentrdquo Food Control vol 59 pp 721ndash730 2016

[2] A J Schmitt and L V Snyder ldquoInfinite-horizon models forinventory control under yield uncertainty and disruptionsrdquoComputers amp Operations Research vol 39 no 4 pp 850ndash8622012

[3] L V Snyder Z Atan P Peng et al ldquoORMS models for supplychain disruptions a reviewrdquo IIE Transactions vol 48 no 2 pp89ndash109 2016

[4] X Qi J F Bard and G Yu ldquoSupply chain coordination withdemand disruptionsrdquo Omega vol 32 no 4 pp 301ndash312 2004

[5] T Xiao G Yu Z Sheng and Y Xia ldquoCoordination of a supplychain with one-manufacturer and two-retailers under demandpromotion and disruption management decisionsrdquo Annals ofOperations Research vol 135 pp 87ndash109 2005

[6] F Hu C-C Lim Z Lu and X Sun ldquoCoordination in a single-retailer two-supplier supply chain under random demand andrandom supply with disruptionrdquo Discrete Dynamics in Natureand Society vol 2013 Article ID 484062 12 pages 2013

[7] J Li X Liu J Wu and F Yang ldquoCoordination of supplychain with a dominant retailer under demand disruptionsrdquoMathematical Problems in Engineering vol 2014 Article ID854681 10 pages 2014

[8] K Chen and T Xiao ldquoDemand disruption and coordination ofthe supply chain with a dominant retailerrdquo European Journal ofOperational Research vol 197 no 1 pp 225ndash234 2009

[9] J Li and F T S Chan ldquoThe impact of collaborative transporta-tion management on demand disruption of manufacturingsupply chainsrdquo International Journal of Production Research vol50 no 19 pp 5635ndash5650 2012

[10] Y Zheng T Shu S Wang S Chen K K Lai and L GanldquoDemand disruption and coordination of supply chain via effortand revenue sharingrdquo Applied Economics vol 47 no 54 pp5886ndash5901 2015

[11] Q Pang Y Hou and Y Lv ldquoCoordinating three-level supplychain under disruptions using revenue-sharing contract witheffort dependent demandrdquoMathematical Problems in Engineer-ing vol 2016 Article ID 9167864 10 pages 2016

[12] M Gumus S Ray and H Gurnani ldquoSupply-side story risksguarantees competition and information asymmetryrdquo Man-agement Science vol 58 no 9 pp 1694ndash1714 2012

[13] K B Hendricks and V R Singhal ldquoAn empirical analysisof the effect of supply chain disruptions on long-run stockprice performance and equity risk of the firmrdquo Production ampOperations Management vol 14 no 1 pp 35ndash52 2005

[14] W Schimidt and A Raman ldquoWhen supply-chain disruptionsmatterrdquo Harvard Business School Working Paper 13-006 2012

[15] N Bunkley Piecing together a supply chain The New YorkTimes 2013 httpwwwnytimescom20110513businessglobal13autohtml r=0

[16] S Sarkar and S Kumar ldquoA behavioral experiment on inventorymanagement with supply chain disruptionrdquo International Jour-nal of Production Economics vol 169 pp 169ndash178 2015

[17] G P Cachon ldquoSupply chain coordination with contractsrdquoHandbooks in Operations Research amp Management Science vol11 pp 227ndash339 2003

[18] C J Corbett and X De Groote ldquoA supplierrsquos optimal quantitydiscount policy under asymmetric informationrdquo ManagementScience vol 46 no 3 pp 444ndash450 2000

12 Mathematical Problems in Engineering

[19] C J Corbett ldquoStochastic inventory systems in a supply chainwith asymmetric information cycle stocks safety stocks andconsignment stockrdquoOperations Research vol 49 no 4 pp 487ndash500 2001

[20] A Matopoulos M Vlachopoulou V Manthou and B ManosldquoA conceptual framework for supply chain collaboration empir-ical evidence from the agri-food industryrdquo Supply Chain Man-agement vol 12 no 3 pp 177ndash186 2007

[21] A Y Ha and S Tong ldquoContracting and information sharingunder supply chain competitionrdquoManagement Science vol 54no 4 pp 701ndash715 2008

[22] Y-W Zhou ldquoA comparison of different quantity discountpricing policies in a two-echelon channel with stochastic andasymmetric demand informationrdquo European Journal of Opera-tional Research vol 181 no 2 pp 686ndash703 2007

[23] XGan S P Sethi and J Zhou ldquoCommitment-penalty contractsin drop-shipping supply chains with asymmetric demand infor-mationrdquo European Journal of Operational Research vol 204 no3 pp 449ndash462 2010

[24] D Lei J Li and Z Liu ldquoSupply chain contracts under demandand cost disruptions with asymmetric informationrdquo Interna-tional Journal of Production Economics vol 139 no 1 pp 116ndash126 2012

[25] Z B Yang G Aydın V Babich and D R Beil ldquoSupplydisruptions asymmetric information and a backup productionoptionrdquoManagement Science vol 55 no 2 pp 192ndash209 2009

[26] S Huang and C Yang ldquoSupply chain revelation mechanismdesign under asymmetric demand disruption informationrdquoOperations Research andManagement Science vol 23 no 6 pp116ndash127 2014

[27] S Oh and O Ozer ldquoMechanism design for capacity planningunder dynamic evolutions of asymmetric demand forecastsrdquoManagement Science vol 59 no 4 pp 987ndash1007 2013

[28] Q Feng G Lai and L X Lu ldquoDynamic bargaining in a supplychain with asymmetric demand informationrdquo ManagementScience vol 61 no 2 pp 301ndash315 2015

[29] Q Li B Li P Chen and P Hou ldquoDual-channel supply chaindecisions under asymmetric information with a risk-averseretailerrdquo Annals of Operations Research 2015

[30] J Wei K Govindan Y Li and J Zhao ldquoPricing and collectingdecisions in a closed-loop supply chain with symmetric andasymmetric informationrdquo Computers and Operations Researchvol 54 pp 257ndash265 2015

[31] K Inderfurth A Sadrieh and G Voigt ldquoThe impact of infor-mation sharing on supply chain performance under asymmetricinformationrdquo Production amp Operations Management vol 22no 2 pp 410ndash425 2013

[32] R B Myerson ldquoMechanism design by an informed principalrdquoEconometrica vol 51 no 6 pp 1767ndash1797 1983

[33] E Maskin and J Tirole ldquoThe principal-agent relationship withan informed principal the case of private valuesrdquo Econometricavol 58 no 2 pp 379ndash409 1990

[34] E Maskin and J Tirole ldquoThe principal-agent relationship withan informed principal II common valuesrdquo Econometrica vol60 no 1 pp 1ndash42 1992

[35] T Mylovanov and T Troger ldquoInformed-principal problemsin environments with generalized private valuesrdquo TheoreticalEconomics vol 7 no 3 pp 465ndash488 2012

[36] TMylovanov andT Troger ldquoMechanismdesign by an informedprincipal private values with transferable utilityrdquo Review ofEconomic Studies vol 81 no 4 pp 1668ndash1707 2014

[37] C Wagner T Mylovanov and T Troger ldquoInformed-principalproblem with moral hazard risk neutrality and no limitedliabilityrdquo Journal of EconomicTheory vol 159 pp 280ndash289 2015

[38] S Galperti ldquoCommon agency with informed principals menusand signalsrdquo Journal of Economic Theory vol 157 pp 648ndash6672015

[39] H Zhang Y Liu and J Huang ldquoSupply chain coordina-tion contracts under double sided disruptions simultaneouslyrdquoMathematical Problems in Engineering vol 2015 Article ID812043 9 pages 2015

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2 Mathematical Problems in Engineering

chains with asymmetric information [18ndash23] So it is impor-tant to consider the asymmetric disruption informationwhenwe design the supply chain contract [24] But very fewresearchers pay attention to the asymmetric disruptioninformation Gumus et al consider the supply disruptionprobability as the supplierrsquos asymmetric information andgive the optimal incentive contract [12 25] Lei et al derivethe optimal linear contract and wholesale contract with theimpact of asymmetric demanddisruption and cost disruptioninformation [24] Huang and Yang set up the principal agentmodel and make the incentive contract considering thedemand disruption information asymmetric [26]

In this paper we consider the demand disruption proba-bility as the retailerrsquos private information in order to investi-gate the contract between the supplier and the retailer In theavailable research about asymmetric information in supplychain researchers always try to reveal the retailerrsquos privateinformation by amenu of incentive contracts [27ndash29] or shar-ing the information [30 31] to improve the interests In all ofthe researches the principal agent theory is explored and theparty with no private information is seen as the principal AsMyerson et al point out there are limitations of the assump-tion of the principal with no private information In manycontracts for example the sales contract employment con-tract and regulation contract it is common for the party withprivate information to be the principal who makes the con-tract [32ndash35] So we consider the following situation in a sup-ply chain with one manufacturer and one retailer the retailermay suffer demand disruptionwith a certain probability Andthe retailerwith the asymmetric demanddisruption probabil-ity informationmakes the supply chain contract to express itsown information to the manufacturer Then the retailer candiffer from the other retailer by providing this contract

Differing from those of prior studies the major innova-tions of the research are as follows

(1)We consider demand uncertainty and demand disrup-tion simultaneously and the demanddisruption probability isconsidered as the retailerrsquos asymmetric information Demanduncertainty is commonly considered in the supply chainresearch and demand disruption happens on occasion Boththe demand uncertainty and demand disruption impactthe supply chain contract making And demand disruptionis totally different from demand uncertainty so demanduncertainty and demand disruption should be consideredtogether in the supply chain contract [3] Lots of researchpays more attention to the coordination contract in supplychain with demand disruption but most is studied withoutasymmetric disruption information [24 26]

(2) The informed principal model is explored for theseparating contract in the supply chain with asymmetricdemand disruption information In the research [24 26] thesupplier as the principal makes the contract to encouragethe retailer to show its own demand disruption informationBut we use the informed principal model and consider theretailer with the demand disruption information to be theprincipal The retailer can deliver its own information to themanufacturer and can be distinguished from the others fromthe separating contract [35ndash38]

The reminder of this paper is organized as followsIn Section 2 the concerned problem is defined and theassumptions and notations are given The full informationmodel and optimal contract are given in Section 3 Section 4goes into the informed principal model In Section 5 theinterim efficient allocation is given to reduce the signalingcost in the informed principal model Numerical examplesare presented in Section 6 and sensitivity analysis is carriedout with respect to some key model parameters The paperconcludes with Section 7 in which we provide a summary ofthe paper and future research directions

2 Model Description

We consider a supply chain consisting of one risk neutralmanufacturer and one risk neutral retailer When the sellingseason comes the retailer purchases a kind of product fromthe manufacturer and then sells it into marketThemarket ofthe retailer may suffer disruption And different retailers havedifferent probability to experience the disruption Supposingthe disruption probability as the retailerrsquos private informa-tion the informed principal model is set up to design thecontracts with which the retailer can demonstrate its trueinformation to the manufacturer

21 Assumptions Theinformedprincipalmodel of the supplychain with demand disruption meets the following assump-tions To make the presentation clear we also list the relevantvariables or notations (see VariableNotations section)

Assumption 1 Before the selling season comes the retailermakes order 119902 from the manufacturer and the manufacturersupplies the right quantity 119902 to the retailer with the unitman-ufacturing cost 119888 Then the retailer pays 119905 as the transferringpayment Andwhen the selling season comes the retailer sellsthe products to the demand market with the price 119901

Assumption 2 The retailer faces an unstable market whichmeans the demand in this market may suffer disruptionWithout disruption the demand 119910 belongs to [0 119860] with thedistribution function 119865(119910) and the density function 119891(119910)With the demand disruption the retailerrsquos demand 119909 belongsto [0 119863] with the distribution function 119866(119909) and densityfunction 119892(119909) The demand belongs to the same distributionstyle whenever with disruption or without disruption forexample uniform distribution and normal distribution andthe market scale without disruption is larger than that withdisruption 119860 gt 119863 And 119864(119909) = 1198602 119864(119910) = 1198632 Var(119909) gt

Var(119910) So for the same demand 1199090

= 1199100 the distribution

function meets 119866(1199090) ge 119865(119910

0) It means when the demand

disruption happens the demand which is less than the givendemand 119909

0 or 1199100 happens in a larger probability than that ofthe demand without disruption

Assumption 3 There are two kinds of retailers the low-riskone with demand disruption probability 120572

1and the high-risk

one with demand disruption probability 1205722 while 120572

1lt 1205722

The demand disruption probability is the retailerrsquos individualinformationThemanufacturer only knows there is a low-risk

Mathematical Problems in Engineering 3

Time

R gets typeinformation

R designsthe contracts

R choosesthe contract

and makes order

M suppliesproducts andgets payment

R sellssome

products

M gets theunsold

products

M acceptsor rejects

t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 t = 7

Figure 1 Informed principal game sequence in supply chain with demand disruption R means the retailer M means the manufacturer

retailer with the probability 120582 and a high-risk retailer with theprobability 1 minus 120582 Both the high-risk retailer and the low-riskretailer are risk neutral

Assumption 4 The retailer as the informed principal designsthe transferring contracts by which the retailer can demon-strate its true information to the manufacturer The transfer-ring contract for low-risk retailer is 119905

1 1199021 while for high-

risk one it is 1199052 1199022

Assumption 5 Every unsold product has the salvage value119904 (119904 lt 119888) And the manufacturer has the salvage value of theunsold products For example the unsold products can bereused for the manufacturer but the retailer has to payfor disposing them The manufacturer can get the unsoldproducts free from the retailer and reuse them In thiscase the manufacturer has the salvage value of the unsoldproducts

119878(120572119894 119902119894) (119894 = 1 2) denote the low-risk retailerrsquos and high-

risk retailerrsquos expected sales

119878 (120572119894 119902119894) = 120572119894(119902119894minus int

119902119894

0

119866 (119909) 119889119909)

+ (1 minus 120572119894) (119902119894minus int

119902119894

0

119865 (119910) 119889119910)

(1)

119868(120572119894 119902119894) (119894 = 1 2) denote the low-risk retailerrsquos and high-

risk retailerrsquos expected unsold quantity

119868 (120572119894 119902119894) = 120572119894int

119902119894

0

119866 (119909) 119889119909 + (1 minus 120572119894) int

119902119894

0

119865 (119910) 119889119910 (2)

119880119894(119894 = 1 2) denote the low-risk retailerrsquos and high-risk

retailerrsquos utility

119880119894= 119901119878 (120572

119894 119902119894) minus 119905119894 (3)

119881119894(119894 = 1 2) denote the manufacturerrsquos utility when it

cooperates with the low-risk retailer or high-risk retailer

119881119894= 119905119894minus 119888119902119894+ 119904119868 (120572

119894 119902119894) (4)

22 Game Sequence When the retailer faces the demanddisruption the sequence of the informed principal gamebetween themanufacturer and the retailer is shown as follows(Figure 1)

(1) The retailer knows its type information 120572119894

(2) The retailer as the informed principal designs thetransferring contract package (119905

1 1199021) (1199052 1199022)

(3) The manufacturer accepts or rejects the contract(4) Before the selling season the retailer chooses the

transferring contract according to its type and makesthe order 119902

119894

(5) The manufacturer supplies products and the retailerpays 119905

119894for them

(6) In the selling season the demand disruption occursin a certain probability and the retailer sells someproducts

(7) After the selling season the manufacturer gets theunsold products

3 Full Information Model

31 Modeling When the demand disruption probability ispublic information the different retailers should provide thesingle selected contracts (119905

1 1199021) or (119905

2 1199022) We can get this

pair of single selected contracts by solving the followingprograms 1198751

119865 and 119875

2

119865

1198751

119865

max1199051 1199021

1198801= 119901119878 (120572

1 1199021) minus 1199051 (5)

st 1198811= 1199051minus 1198881199021+ 119904119868 (120572

1 1199021) ge 0 (6)

1198752

119865

max1199052 1199022

1198802= 119901119878 (120572

2 1199022) minus 1199052 (7)

st 1198812= 1199052minus 1198881199022+ 119904119868 (120572

2 1199022) ge 0 (8)

The targets of these programs are the retailersrsquo revenuewhich are the principal As in the full information situationeach participant knows the true information so there is noincentive compatible constraint Whatever the manufacturershould get no less than the retrained revenue The incentiverationality constraint of the manufacturer facing differentretailers is useful here (see (6) and (8))

32 Results By solving programs 1198751119865 and 119875

2

119865 there comes

Proposition 6 (Proof of Proposition 6 is given inAppendixA)

Proposition 6 (1) Under full information the optimal con-tract in supply chain with demand disruption satisfies thatboth the optimal order and optimal transferring payment for

4 Mathematical Problems in Engineering

t

Alowast

Blowast

q

tlowast1

tlowast2

qlowast1qlowast2

Ulowast1

Ulowast2

Vlowast1

Vlowast2

Figure 2 Supply chain contract under full information

low-risk retailer are higher than those for high-risk retailer as119902lowast

1gt 119902lowast

2 119905lowast1gt 119905lowast

2

(2) Under full information the utility of the low-riskretailer is larger than that of high-risk retailer as 119880lowast

1gt 119880lowast

2

and the utility of the manufacturer cooperating with low-riskretailer and with high-risk retailer is equal to the retrainedrevenue as 119881lowast

1= 119881lowast

2= 0

From Proposition 6 it is obvious that under full infor-mation the low-risk retailer should make more orders thanthe high-risk retailer (119902lowast

1gt 119902lowast

2) and meanwhile pay more

(119905lowast1gt 119905lowast

2) Let us show the optimal contract in Figure 2 using

the retailer and themanufacturer indifferent utility curve Forthe low-risk retailer the optimal contract is shown at point119861lowast And for the high-risk retailer the optimal contract is

shown at point 119860lowast The retailerrsquos utility grows more whenits indifferent curve goes into southeast So when the low-risk retailer pretends to be high-risk one using allocation119860

lowastwhich is to say that the low-risk retailerrsquos indifferent utilitycurve goes northwest its utility decreases But if the high-risk retailer pretends to be low-risk one using allocation 119861

lowastits utility increases So as the self-interested participant thehigh-risk retailer tends to pretend to be low-risk one Thenwe can get Proposition 7

Proposition 7 Considering the demand disruption proba-bility as the retailerrsquos individual information the optimalallocation for the low-risk retailer is not separating equilibriumallocation

4 Informed Principal Model

In order to show its true type information the retailer can usethe informed principalmodel to offer an option contract [32]The term ldquooption contract (119905

1 1199021) (1199052 1199022)rdquo comes from the

fact that if the manufacturer accepts the contract the retailermust then exercise its built-in option and choose between(1199051 1199021) and (119905

2 1199022) The retailer will choose the term which

is fit to its own type by probability 100 as 1199011205721| (1199051 1199021) =

1199011205722

| (1199052 1199022) = 1 And the retailer will choose the term

which is not fit to its own probability by probability 0 as1199011205721

| (1199052 1199022) = 119901120572

2| (1199051 1199021) = 0 [33 34] Then the

informed principal model of the supply chain with demanddisruption is built up for the Perfect Bayesian Equilibrium

41 Modeling Firstly the incentive compatible constraint isgivenThe option contract is incentive compatible if the low-risk retailer prefers the contract item (119905

1 1199021) and the high-

risk retailer prefers the contract item (1199052 1199022) So the incentive

compatible constraint should keep the low-risk retailerrsquosutility no less than when it pretends to be a high-risk one

119901119878 (1205721 1199021) minus 1199051ge 119901119878 (120572

1 1199022) minus 1199052 (9)

And the high-risk retailerrsquos utility is no less than when itpretends to be a low-risk one

119901119878 (1205722 1199022) minus 1199052ge 119901119878 (120572

2 1199021) minus 1199051 (10)

Secondly the incentive rationality constraint is givenTheretailer with the individual information is the principal andthemanufacturerwithout individual information is the agentSo the agentrsquos incentive rationality is the same as that (as in (6)and (8)) in full information situation

Thirdly the target function is given The target of theinformed principal model is also to maximize the utility ofthe principal or to say the retailer

So the programs under informed principal of supplychain with demand disruption are 119875

1

119868 and 119875

2

119868 Consider

the following

1198751

119868

max1199051 1199021

1198801= 119901119878 (120572

1 1199021) minus 1199051

st (6) (8) (9) (10)

(11)

1198752

119868

max1199052 1199022

1198802= 119901119878 (120572

2 1199022) minus 1199052

st (6) (8) (9) (10)

(12)

Definition 8 Utility 1198801(1199050

1 1199020

1)1198802(1199050

2 1199020

2) for low-riskhigh-

risk retailer is the low-information-intensity optimum forthat type if (1199050

1 1199020

1)(1199050

2 1199020

2)maximizes low-risk retailerrsquos utility

in the set of incentive compatible constraint and incentiverationality constraint They are (part of) the solution to theprograms 1198751

119868 and 119875

2

119868

For solving the informedprincipalmodel the assumptionof weak monotonic profit is listed as follows

Assumption 9 It is weak monotonic profit when the manu-facturer makes a nonnegative profit if the contractual termsare those of low-risk retailer under symmetric informationand the retailer is a high-risk one 119881

1(119905lowast

2 119902lowast

2) ge 0

From Figure 2 if the contractual term for the low-riskretailer changes from119861

lowast to119860lowast the curve119881lowast1will go southeast

and themanufacturerrsquos profit increases So themanufacturerrsquosprofit is no less than 0 and Assumption 9 is satisfied

Mathematical Problems in Engineering 5

Definition 10 The separating allocation is the allocation (119905119904

1

119902119904

1) for the low-risk retailer and the symmetric information

contractual terms (119905lowast2 119902lowast

2) are for the high-risk retailer where

(119905119904

1 119902119904

1) maximizes the low-risk retailerrsquos utility subject to the

manufacturerrsquos breaking even for the low-risk retailer and tothe high-risk retailer not preferring (119905

119904

1 119902119904

1) to (119905

lowast

2 119902lowast

2) The

program is listed as 1198751119904

1198751

119904

max11990511199021

1198801= 119901119878 (120572

1 1199021) minus 1199051

st (6)

(13)

119901119878 (1205722 119902lowast

2) minus 119905lowast

2ge 119901119878 (120572

2 1199021) minus 1199051 (14)

42 Results

Proposition 11 Under the weak monotonic-profit assump-tion the separating allocation is the low-information-intensityoptimum

Proof (1) The high-risk retailer can get its asymmetric infor-mation utility even under asymmetric information Compar-ing programs 1198752

119865 and 119875

2

119868 it can be found that these two

programs have the same target function but 1198752

119865 has less

constraints so 119880(119905lowast

2 119902lowast

2) ge 119880(119905

0

2 1199020

2) According to Assump-

tion 9 the manufacturer at least breaks even regardless of theretailerrsquos type Hence (119905lowast

2 119902lowast

2) is the separating allocation item

for high-risk retailer(2) Because 119880(119905

lowast

2 119902lowast

2) ge 119880(119905

0

2 1199020

2) constraint (14) is more

restricted than (10) For the low-risk retailer the programs1198751

119868 and 119875

119904

1 have the same target functions but there are

more constraints in the former program so 119880(119905119904

1 119902119904

1) ge

119880(1199050

1 1199020

1) is satisfied And the low-risk retailer at least can

get 119880(119905119904

1 119902119904

1) from the option contract From the incentive

compatibility constraint (14) which can be rewritten as1198802(119905lowast

2 119902lowast

2) ge 1198802(119905119904

1 119902119904

1) it has been guaranteed that the high-

risk retailer will not choose 119905119904

1 119902119904

1 From the program 119875

119904

1

the low-risk retailer can get more utility when the contractualitem is 119905119904

1 119902119904

1 than that of 119905lowast

2 119902lowast

2 So we can get 1199050

1 1199020

1 =

119905119904

1 119902119904

1

Then we can get Proposition 12 (see details in Appen-dix B)

Proposition 12 As for the option contract by informed retailerin supply chain with demand disruption the low-information-intensity allocation satisfies the following

(1) Comparing to the full information situation the low-information-intensity allocation for high-risk retailerand its utility does not distort as 1199050

2= 119905lowast

2 11990202

= 119902lowast

2

1198802(1199050

2 1199020

2) = 119880lowast

2

(2) Comparing to the full information situation the low-information-intensity allocation for low-risk retailerhas upward distortion 119905

0

1gt 119905lowast

1 11990201gt 119902lowast

1 while its utility

has downward distortion 1198801(1199050

1 1199020

1) lt 119880lowast

1

(3) The order quantities in the low-information-intensityallocation for the retailers in different types satisfy

119901119878 (1205722 1199020

1) minus 119888119902

0

1+ 119904119868 (120572

1 1199020

1)

= 119901119878 (1205722 119902lowast

2) minus 119888119902

lowast

2+ 119904119868 (120572

2 119902lowast

2)

(15)

In the informed principal model in order to prevent thehigh-risk retailer from pretending to be a low-risk one thelow-information-intensity allocation items for the low-riskretailer distort The low-risk retailer should order more thanthe optimal order and paymore than the optimal transferringpayment But the utility of the low-risk retailer is less than theoptimal onewhichmeans the low-risk retailer pays some rentto separate from the high-risk retailer So the rent is calledsignaling cost

5 Interim Efficient Allocation

The retailer can deliver its own type information to themanufacturer by low-information-intensity allocation in theinformed principal model But the low-risk retailerrsquos orderand transferring payment distort relative to the full infor-mation situation and it has to pay the signaling cost Thesignaling cost is the part utility which the low-risk retailergets less than the optimal So maybe we can decrease thesignaling cost by increasing the high-risk retailerrsquos utilityand meanwhile increasing the low-risk retailerrsquos utility So inthis part we try to find a separating equilibrium with lesssignaling cost [32ndash34 38] by the interim efficient model

Let us consider the interim efficient model which candecrease the signaling cost by giving the high-risk retailermore than optimal utility Let be the utility which the high-risk retailer gets more than the optimal one And let 119871() betheminimal loss of themanufacturer when it cooperates withthe high-risk retailer119871() can be gotten by the program 119875

2

119898

1198752

119898

minus119871 () = max1199052 1199022

1199052minus 1198881199022+ 119904119868 (120572

2 1199022) (16)

st 119901119878 (1205722 1199022) minus 1199052

ge 119901119878 (1205722 119902lowast

2) minus 119905lowast

2+

(17)

It is easy to find out that when constraint (17) is binding1199022= 119902lowast

2and 119871() =

Definition 13 Utility 1198801(119905119898

1 119902119898

1)1198802(119905119898

2 119902119898

2) for low-risk

high-risk retailer is the interim efficient optimum for thattype if (119905119898

1 119902119898

1)(119905119898

2 119902119898

2) maximizes low-risk retailerrsquos utility

in the set of incentive compatible constraint and themanufac-turerrsquos expected profit And the contract (119905119898

1 119902119898

1) (119905119898

2 119902119898

2) is

the interim efficient allocationThey are (part of) the solutionto the programs 1198751

119898 and 119875

2

119898

6 Mathematical Problems in Engineering

Table 1 Full information contract versus low-information-intensity allocation

Type ofretailer Full information

Lowinformationintensity

Type ofretailer Full information

Lowinformationintensity

1205721= 02

119902lowast

1= 1001 119902

0

1= 1014

1205721= 03

119902lowast

1= 990 119902

0

1= 1002

119905lowast

1= 1862 119905

0

1= 1881 119905

lowast

1= 1840 119905

0

1= 1824

119880lowast

1= 5340 119880

0

1= 5338 119880

lowast

1= 5278 119880

0

1= 5192

1205722= 04

119902lowast

2= 978 119902

0

1= 978

1205722= 05

119902lowast

2= 967 119902

0

2= 967

119905lowast

2= 1819 119905

0

2= 1819 119905

lowast

2= 1799 119905

0

2= 1799

119880lowast

2= 5217 119880

0

2= 5217 119880

lowast

2= 5159 119880

0

2= 5159

1198751

119898

max11990511199021

1198801= 119901119878 (120572

1 1199021) minus 1199051 (18)

st 120582 (119905119898

1minus 119888119902119898

1+ 119904119868 (120572

1 119902119898

1)) minus (1 minus 120582) ge 0 (19)

119901119878 (1205722 119902lowast

2) minus 119905lowast

2+ ge 119901119878 (120572

2 119902119898

1) minus 119905119898

1 (20)

The high-risk retailer can get the rent above theoptimum so themanufacturer will lose some utilityThen theincentive rational constraint (see (19)) for themanufacturer isthe expected profit rather than the individual profit Equation(20) is the incentive compatible constraint for the high-riskretailer

From programs 1198751119898 and 119875

2

119898 we can get Propositions 14

and 15 (see Proof in Appendix C)

Proposition 14 The interim efficient allocation by informedprincipal for the supply chain with demand disruption satisfiesthe following

(1) Comparing to the full information situation the orderquantity for the high-risk retailer does not distort 119902119898

2=

119902lowast

2

(2) Comparing to the full information situation the orderquantity for the low-risk retailer satisfies (21) anddistorts upwards 119902119898

1gt 119902lowast

1 Consider

1199041198681015840(1205721 119902119898

1) + 119901119878

1015840(1205721 119902119898

1) minus 119888

=1 minus 120582

120582[1199011198781015840(1205722 119902119898

1) minus 119901119878

1015840(1205721 119902119898

1)]

(21)

(3) The transferring payment for the low-risk retailer meets(22) and the transferring payment for the high-riskretailer meets (23)

119905119898

1= (1 minus 120582)

sdot [119901119878 (1205722 119902119898

1) minus 119901119878 (120572

2 119902lowast

2) + 119888119902

lowast

2minus 119904119868 (120572

2 119902lowast

2)]

+ 120582 [119888119902119898

1minus 119904119868 (120572

1 119902119898

1)]

(22)

119905119898

2= (1 minus 120582) [119888119902

lowast

2minus 119904119868 (120572

2 119902lowast

2)] + 120582119901119878 (120572

2 119902lowast

2)

+ 120582 [119888119902119898

1minus 119904119868 (120572

1 119902119898

1) minus 119901119878 (120572

2 119902119898

1)]

(23)

Proposition 15 There is a threshold 1205820

= (1199011198781015840(1205722 1199020

1) minus

1199011198781015840(1205721 1199020

1))(119904119868

1015840(1205721 1199020

1) + 119901119878

1015840(1205722 1199020

1) minus 119888) for the low-risk

retailer probability If and only if the probability of low-riskretailer is no more than the threshold 120582 le 120582

0 the low-infor-mation-intensity allocation is also interim efficient and isthe unique Perfect Bayesian Equilibrium If the probabilityof low-risk retailer is more than the threshold 120582 gt 120582

0 thelow-information-intensity allocation is not the unique PerfectBayesian Equilibrium

If and only if the probability of low-risk retailer is smallenough (120582 le 120582

0) the low-information-intensity allocation isthe unique Perfect Bayesian EquilibriumAnd the retailer candeliver its own type information by the separating contractBut if the low-risk retailer is more than that (120582 gt 120582

0) the low-information-intensity allocation is not the unique PerfectBayesian Equilibrium and the interim efficient allocation canimprove the distortion situation to a certain degree So for thesupply chain with demand disruption the optimal allocationcannot be reached if the informed principal provides aseparating contract

6 Numerical Example

This section gives some numerical examples to inspect thesupply chain contract with demand disruption by informedprincipal We let the sales price 119901 = 10 the unit manu-facturing cost is 119888 = 2 and the salvage product value is119904 = 05 The demand distribution function before disruptionis 119865(119910) = 119910

2125 and the demand distribution function after

disruption is 119866(119909) = 1199092100

61 Main Results From the example listed above the fullinformation contracts and low-information-intensity alloca-tion are given in Table 1

Table 1 shows that for the full information contract theorder quantity for low-risk retailer is larger than that for thehigh-risk retailer 119902lowast

1gt 119902lowast

2 and also the transferring payment

119905lowast

1gt 119905lowast

2 which is performed in Proposition 6 In the informed

principal model the order quantity for the low-risk retailerdistorts in order to show its type (when 120572

1= 02 versus 120572

2=

04 119902lowast1

= 1001 lt 119902sb1

= 1014 and when 1205721= 03 versus

1205722= 05 119902lowast

1= 990 lt 119902

0

1= 1002) Meanwhile we notice that

the utility obtained by the low-risk retailer in the informed

Mathematical Problems in Engineering 7

0 10

5

10

15

q1

02 04 06 08

120582

Figure 3 The order quantity in interim efficient allocation for low-risk retailer changes by its probability

0

1

101112

1314

150

5

10

15

20

25

30

0204

0608

tm 1

q1 120582

Figure 4 The transferring payment in interim efficient allocationfor low-risk retailer changes by the order quantity and its probability

principal model is lower than that in full information model(when 120572

1= 02 versus 120572

2= 04 119880lowast

1= 5340 gt 119880

0

1= 5338

and when 1205721= 03 versus 120572

2= 05 119905lowast

1= 1840 gt 119905

0

1= 1824)

This implies that it costs the low-risk retailer something tocertify its typeThese results are the samewith Proposition 12

Considering the situation of 1205721= 02 versus 120572

2= 04 we

obtain 1205820= 019 The order quantity for the low-risk retailer

119902119898

1changes with the proportion it has 120582 in interim efficient

allocation As Figure 3 shows if and only if 120582 varies withincertain area (120582 le 120582

0) we obtain 119902119898

1gt 119902lowast

1 as in the results in

Proposition 14The transferring payment in the interim efficient allo-

cation is depicted in Figure 4 (for the low-risk retailer)and Figure 5 (for the high-risk retailer) The transferringpayments are changing by the order quantity for the low-risk retailer in the interim efficient allocation and the low-risk retailerrsquos probability From Figure 4 when the low-riskretailerrsquos probability 120582 is less enough and the order quantityfor the low-risk retailer 119902119898

1is high enough the transferring

0

05

1

10 11 12 13 14 15

20

25

30

35

40

45

tm 2

q1

120582

Figure 5 The transferring payment in interim efficient allocationfor high-risk retailer changes by the low-risk retailerrsquos order quantityand probability

payment 1199051198981increases by the low-risk retailerrsquos probability 120582

but decreases by the order quantity for the low-risk retailer119902119898

1(bottom left in Figure 4) But mostly the transferring

payment 1199051198981increases by the low-risk retailerrsquos probability 120582

but decreases by the order quantity for the low-risk retailer 1199021198981

(red and orange part in Figure 4) Normally the transferringpayment should increase by the order quantity which meansldquoget more pay morerdquo From this view the low-risk retailerrsquosprobability 120582 should not be very small and the order quantityfor the low-risk retailer 119902

119898

1should not be close to the

maximum value 15From Figure 5 it is obviously seen that the transferring

payment for high-risk retailer in the interim efficient alloca-tion 119905119898

2increases by the order quantity for the low-risk retailer

119902119898

1but decreases by the low-risk retailerrsquos probability 120582

62 Sensitivity Analyses The sensitivity analyses for the low-information-intensity allocation versus the unit manufactur-ing cost 119888 are shown in Table 2 It shows the outcomes whenthe unit manufacturing cost 119888 changes by +25 and minus25once at a time and keeping remaining parameters The orderquantities decrease by the unit manufacturing cost but theinfluence for the high-risk retailerrsquos order quantity is largerthan that for the low-risk retailer (minus069 versus minus317 and+217 versus +307) The transferring payments increaseby the unit manufacturing cost More interesting the utilitiesdecrease by the unit manufacturing cost and the low-riskretailer receives a greater impact (minus940 versus minus922 and+955 versus +953) So from this point of view the low-risk retailer should encourage themanufacturer to reduce thecost more

Figure 6 shows the sensitivity analyses for the order quan-tities versus the unit manufacturing cost in which the orderquantities include the order quantity for the low-risk retailerin low-information-intensity allocation 119902

0

1 the order quantity

8 Mathematical Problems in Engineering

Table 2 Sensitivity analyses for low-information-intensity allocation versus the unit manufacturing cost

Changes of 119888 The low-information allocation for low-riskretailer (120572

1= 02)

The low-information allocationfor high-risk retailer (120572

2= 04)

Value Change rate Value Change rate Value Change rate

119888 = 25 +251199020

1= 1007 minus069 119902

0

2= 947 minus317

1199050

1= 2375 +2626 119905

0

2= 2243 +2331

1198800

1= 4836 minus940 119880

0

2= 4736 minus922

119888 = 2 Baseline1199020

1= 1014 mdash 119902

sb2

= 978 mdash1199050

1= 1881 mdash 119905

sb2

= 1819 mdash1198800

1= 5338 mdash 119880

sb2

= 5217 mdash

119888 = 15 minus251199020

1= 1036 +217 119902

0

2= 1008 +307

1199050

1= 1398 minus2568 119905

0

2= 1362 minus2512

1198800

1= 5848 +955 119880

0

2= 5714 +953

108

106

104

102

10

98

96

94

92

C

14 16 18 2 22 24 26 28 3

qsb1

qlowast1q2

Figure 6 The order quantities change by the unit manufacturingcost

for the low-risk retailer in full information contract 119902lowast1 and

the order quantity for the high-risk retailer in full informationcontract which is equal to that in low-information-intensityallocation 119902

2 Both the order quantity for the low-risk retailer

in full information and the order quantity for the high-riskretailer decrease by the unit manufacturing cost and theychange in the same level basically But the change of the orderquantity for the low-risk retailer in low information intensitydepends on different values of the unit manufacturing costWhen the unit manufacturing cost 119888 = 275 the orderquantity is minimum and when 119888 = 175 or 119888 = 15 the orderquantity is the maximum

Figure 7 shows the sensitivity analyses for the transferringpayment versus the unit manufacturing cost in which thetransferring payments include the transferring payment forthe low-risk retailer in low-information-intensity allocation1199050

1 the transferring payment for the low-risk retailer in full

information contract 119905lowast

1 and the transferring payment for

26

24

22

20

18

16

14

C

14 16 18 2 22 24 26 28 3

tsb1

tlowast1t2

Figure 7 The transferring payments change by the unit manufac-turing cost

the high-risk retailer in full information contract which isequal to that in low-information-intensity allocation 119905

2 All

the transferring payments of the retailers increase by the unitmanufacturing cost The transferring payments for the low-risk retailer and the high-risk retailer in full informationcontract decrease in the same level

Figure 8 shows the sensitivity analyses for the retailersrsquoutilities versus the unit manufacturing cost in which theutilities include the utility for the low-risk retailer in low-information-intensity allocation 119880

0

1 the transferring pay-

ment for the low-risk retailer in full information contract119880lowast

1 and the transferring payment for the high-risk retailer

in full information contract which is equal to that in low-information-intensity allocation 119880

2 All the retailersrsquo utilities

decrease by the unit manufacturing costFigure 9 shows the sensitivity analyses for the low-risk

retailerrsquos signaling cost versus the unit manufacturing cost

Mathematical Problems in Engineering 9

44

46

48

50

52

54

56

58

60

C

14 16 18 2 22 24 26 28

Usb1

Ulowast1

U2

Figure 8 The retailerrsquos utilities change by the unit manufacturingcost

0

005

01

015

02

025

C

14 16 18 2 22 24 26 28

ΔU1

Figure 9The signaling cost changes by the saved unit manufactur-ing cost

The signaling cost of the low-risk retailer is the utility earnedin the low-information-intensity allocation which is sepa-rating contract less than that in full information contractWhen the unit manufacturing cost 119888 = 2 the signaling costis the least while when the saved unit manufacturing cost119888 = 175 the signaling cost is the most We do not find theinternal relationship between the signaling cost and the unitmanufacturing cost As the signaling cost is so important toput the contracts into practice investigation on the signalingcost is the research direction for us

7 Conclusion

In the recent years supply chain with demand disruptionhas become more and more attractive to both business and

academic research There is a lot of research available on thetopic of supply chain disruption contract In the real supplychain practice demand disruption is totally different fromthe stochastic demand So in our research both the demanduncertainty and the demand disruption are consideredTaking the demand disruption probability as the asymmetricinformation an informed principal model is explored tomake the contract for supply chain with demand disruptionAnd the conclusions we obtain are shown as follows

(1) The contracts for low-risk retailer and high-riskretailer in full information are provided and it isfound out that the high-risk retailer has the incentiveto pretend to be the low-risk retailer This result issimilar to the other informed principal model [3435] but totally different from the normal principalagent model [24 26] In the normal principal agentmodel the high efficient agent (low-risk retailer in oursetting) tries to pretend to be low efficient agent (high-risk retailer) to get more That is why the informedprincipal model is established

(2) The low-information-intensity allocation which is theresult of the informed principal model and also is theseparating contract shows that the order quantity andthe transferring payment for the low-risk retailer inlow-information-intensity allocation distort upwardsbut that of high-risk retailer does not distort Inorder to avoid imitation the imitatorsrsquo efficiencyshould be always distorted But in normal principalagent model the high-risk retailerrsquos quantity distortsdownwards [24 26] on the contrary the low-riskretailerrsquos quantity in informed principal model dis-torts upwards

(3) In order to reduce the signaling cost which the low-risk retailer pays the interim efficient model is intro-duced which ends up with the order quantity andtransferring payment distorted upward again but lessthan that of low information intensity Comparingto [38] focusing on analyzing the signaling costmore attention is paid to showing the interim efficientmodel to get a solution which can increase efficiency

From the managerial aspect retailers should try toincrease their antirisk capability to enhance competitivepower And when the retailer has lower demand disruptionprobability than the others it can show its style to the supplierthrough the separating contract For example the retailer cansay ldquobecause I am the low-risk one I can cooperate with thesupplier with this contract while the others who are high-riskones can only use the other contractrdquo But in order to be sep-arated from the others the low-risk retailer has to pay somesignaling cost and the quantity has to be distorted upwards

Above all there are some limitations of the research Wewill continue our research in the following directions (1)The signaling cost is a key factor impacting the separatingcontract which should be paid more attention (2) We onlyconsider the demand disruption but disruptions happen atnot only demand side but also supply side Sowewill continue

10 Mathematical Problems in Engineering

our research with demand disruption together with supplydisruption [39]

Appendix

A Proof of Proposition 6

For the two types of retailers we can obtain the optimalallocation if and only if the whole margin utility of the supplychain is the same with the margin cost So 119902

lowast

1and 119902

lowast

2satisfy

the following first-order conditions

1199011198781015840(1205721 1199021) + 1199041198681015840(1205721 1199021) = 119888 (A1)

1199011198781015840(1205722 1199022) + 1199041198681015840(1205722 1199022) = 119888 (A2)

And the optimal transferring payments are

119905lowast

1= 119888119902lowast

1minus 119904119868 (120572

1 119902lowast

1)

119905lowast

2= 119888119902lowast

2minus 119904119868 (120572

2 119902lowast

2)

(A3)

From (A1) and (A2) we obtain

1205791119866 (119902lowast

1) + (1 minus 120579

1) 119865 (119902lowast

1)

= 1205792119866 (119902lowast

2) + (1 minus 120579

2) 119865 (119902lowast

2) =

(119901 minus 119888)

(119901 minus 119904)

(A4)

In addition 119866(119902lowast

1) ge 119865(119902

lowast

1) 1205721

lt 1205722 so we know that

1205721119866(119902lowast

1) + (1minus120572

1)119865(119902lowast

1) lt 1205722119866(119902lowast

1) + (1minus120572

2)119865(119902lowast

1) and from

(A4) we obtain 1205722119866(119902lowast

2) + (1 minus 120572

2)119865(119902lowast

2) lt 1205722119866(119902lowast

1) + (1 minus

1205722)119865(119902lowast

1) and thus 119902lowast

2lt 119902lowast

1 From (A3) 119905lowast

1minus119905lowast

2= 119888(119902lowast

1minus119902lowast

2)minus

119904(119868(1205721 119902lowast

1) minus 119868(120572

2 119902lowast

2)) 119905lowast1minus 119905lowast

2=119902lowast

1=119902lowast

2

(1205721minus 1205722)119904(int119902

0119865(119910)119889119910 minus

int119902

0119866(119909)119889119909) gt 0 and 120597119905

lowast

1120597119902lowast

1gt 0 so 119905

lowast

1minus 119905lowast

2gt 0 and

thus we obtain 119905lowast

1gt 119905lowast

2 119881lowast

1= 119881lowast

2= 0 Furthermore we get

119880lowast

1gt 119880lowast

2from 120597119880120597120579 lt 0 and 120597119880120597119902 gt 0 Finally we have

Proposition 6

B Proof of Proposition 12

We denote 1205831and 120583

2as the Lagrange multipliers of (6) and

(14) and find the derivate of 1199051and 1199021 and thus we obtain

120597119880 (1205721 1199021)

1205971199051

= minus1 + 1205831+ 1205832= 0

120597119880 (1205721 1199021)

1205971199021

= 1199011198781015840(1205721 1199021) minus 1205831119888 + 12058311199041198681015840(1205721 1199021)

minus 12058321199011198781015840(1205722 1199021) = 0

(B1)

By rewriting (B1) we get the following formulas

1199041198681015840(1205721 119902119904

1) + 119901119878

1015840(1205721 119902119904

1) minus 119888

=(1 minus 120583

1)

1205831

[1199011198781015840(1205722 119902119904

1) minus 119901119878

1015840(1205721 119902119904

1)]

(B2)

1205831=

1199011198781015840(1205722 119902119904

1) minus 119901119878

1015840(1205721 119902119904

1)

1199041198681015840 (1205721 119902119904

1) + 1199011198781015840 (120572

2 119902119904

1) minus 119888

(B3)

1205832=

1199041198681015840(1205721 119902119904

1) + 119901119878

1015840(1205721 119902119904

1) minus 119888

1199041198681015840 (1205721 119902119904

1) + 1199011198781015840 (120572

2 119902119904

1) minus 119888

(B4)

Because the numerator of 1205831is negative so 120583

1is positive

and the denominator of 1205831is negative because 120583

2is positive

(when 1205832= 0 it cannot be separated from the former one)

the numerator of 1205832is negative nevertheless 1199011198781015840(120572

1 119902lowast

1) minus

119888 + 1199041198681015840(1205721 119902lowast

1) = 0 and thus 1199011198781015840(120572

1 119902119904

1) minus 119888 + 119904119868

1015840(1205721 119902119904

1) lt

1199011198781015840(1205721 119902lowast

1)minus 119888+ 119904119868

1015840(1205721 119902lowast

1) in addition 1205972119880120597119902

2lt 0 so 119902

119904

1gt

119902lowast

1 From 120583

1gt 0 and 120583

2gt 0 we deduce that (6) and (14)

are tight Furthermore the programs 1198751119865 and 119875

1

119898 have the

same objective function but the constraint of 1198751119898 is more

tight so there is distortion of the utility obtained from 1198751

119898 in

other words 1198801(119905119904

1 119902119904

1) lt 119880lowast

1 From Proposition 11 we obtain

1198802(1199050

2 1199020

2) = 119880lowast

2 So we have Proposition 12

C Proof of Propositions 14 and 15

We denote 120573 and 120574 as the Lagrange multipliers of (19) and(20) and find the derivate of 119905

1 1199021 and and thus we obtain

120597119871 (1199051 1199021 )

1205971199051

= minus1 + 120573120582 + 120574 = 0 (C1)

120597119871 (1199051 1199021 )

1205971199021

= 1199011198781015840(1205721 1199021) + 120573120582 [minus119888 + 119904119868

1015840(1205721 1199021)]

minus 1205741199011198781015840(1205721 1199021) = 0

(C2)

120597119871 (1199051 1199021 )

120597= minus120573 (1 minus 120582) + 120574 = 0 (C3)

To solve (C1)ndash(C3) we obtain 120573 = 1 120574 = 1 minus 120582 Andconsidering (C2) we get

1199041198681015840(1205721 119902119898

1) + 119901119878

1015840(1205721 119902119898

1) minus 119888

=1 minus 120582

120582[1199011198781015840(1205722 119902119898

1) minus 119901119878

1015840(1205721 119902119898

1)]

(C4)

Because (19) and (20) are tight we obtain the followingresults

119905119898

1= (1 minus 120582)

sdot [119901119878 (1205722 119902119898

1) minus 119901119878 (120572

2 119902lowast

2) + 119888119902

lowast

2minus 119904119868 (120572

2 119902lowast

2)]

+ 120582 [119888119902119898

1minus 119904119868 (120572

1 119902119898

1)]

119905119898

2= (1 minus 120582) [119888119902

lowast

2minus 119904119868 (120572

2 119902lowast

2)] + 120582119901119878 (120572

2 119902lowast

2)

+ 120582 [119888119902119898

1minus 119904119868 (120572

1 119902119898

1) minus 119901119878 (120572

2 119902119898

1)]

(C5)

Mathematical Problems in Engineering 11

To compare (A1) with (C4) we know 119902119898

1gt 119902lowast

1because

12059721198801205971199022

lt 0 and 1199011198781015840(1205722 119902) minus 119901119878

1015840(1205721 119902) lt 0 and the dis-

tortion level is changing with 120572 Finally we have Proposi-tion 14

To compare (C4) with (B2) and designate 1205820

= 120582 =

(1199011198781015840(1205722 1199020

1) minus 119901119878

1015840(1205721 1199020

1))(119904119868

1015840(1205721 1199020

1) + 119901119878

1015840(1205722 1199020

1) minus 119888) we

have the following conclusions

(1) When 120582 = 1205820 1199021198981

= 119902119904

1= 1199020

1 we obtain the same

contracts with the low-information-intensity situa-tion

(2) When 120582 lt 1205820 1199021198981

lt 119902119904

1= 1199020

1 we know there is more

distortion than the low-information-intensity alloca-tion and it improves anything

(3) When 120582 gt 1205820 1199021198981

lt 119902119904

1= 1199020

1 the interim efficient

allocation can improve the low-information-intensityallocation So we have Proposition 15

VariablesNotations

119888 Unit manufacturing cost1205721 Disruption probability of low-risk retailer

119910 Market demand without disruption119865(sdot) Distribution function of demand without

disruption119904 Unit salvage value119860 Market scale without disruption119878(120572119894 119902119894) Expected sales

119868(120572119894 119902119894) Expected unsold quantity

119880119894 Utility of the retailer

120582 Probability of low-risk retailer1205722 Disruption probability of high-risk retailer

119909 Market demand with disruption119866(sdot) Distribution function of demand with

disruption119901 Retail price119863 Market scale with disruption119881119894 Utility of the manufacturer

Control Variables

119902119894 Order quantity

119905119894 Transferring payment

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported in part by a project supported bythe Humanities and Social Sciences Project of the EducationMinistry (no 14YJC630187) Hebei Social Sciences Project(no HB15GL058) and the Fundamental Research Funds forthe Central Universities (no 2016MS123)

References

[1] J Barnett F Begen S Howes et al ldquoConsumersrsquo confidencereflections and response strategies following the horsemeatincidentrdquo Food Control vol 59 pp 721ndash730 2016

[2] A J Schmitt and L V Snyder ldquoInfinite-horizon models forinventory control under yield uncertainty and disruptionsrdquoComputers amp Operations Research vol 39 no 4 pp 850ndash8622012

[3] L V Snyder Z Atan P Peng et al ldquoORMS models for supplychain disruptions a reviewrdquo IIE Transactions vol 48 no 2 pp89ndash109 2016

[4] X Qi J F Bard and G Yu ldquoSupply chain coordination withdemand disruptionsrdquo Omega vol 32 no 4 pp 301ndash312 2004

[5] T Xiao G Yu Z Sheng and Y Xia ldquoCoordination of a supplychain with one-manufacturer and two-retailers under demandpromotion and disruption management decisionsrdquo Annals ofOperations Research vol 135 pp 87ndash109 2005

[6] F Hu C-C Lim Z Lu and X Sun ldquoCoordination in a single-retailer two-supplier supply chain under random demand andrandom supply with disruptionrdquo Discrete Dynamics in Natureand Society vol 2013 Article ID 484062 12 pages 2013

[7] J Li X Liu J Wu and F Yang ldquoCoordination of supplychain with a dominant retailer under demand disruptionsrdquoMathematical Problems in Engineering vol 2014 Article ID854681 10 pages 2014

[8] K Chen and T Xiao ldquoDemand disruption and coordination ofthe supply chain with a dominant retailerrdquo European Journal ofOperational Research vol 197 no 1 pp 225ndash234 2009

[9] J Li and F T S Chan ldquoThe impact of collaborative transporta-tion management on demand disruption of manufacturingsupply chainsrdquo International Journal of Production Research vol50 no 19 pp 5635ndash5650 2012

[10] Y Zheng T Shu S Wang S Chen K K Lai and L GanldquoDemand disruption and coordination of supply chain via effortand revenue sharingrdquo Applied Economics vol 47 no 54 pp5886ndash5901 2015

[11] Q Pang Y Hou and Y Lv ldquoCoordinating three-level supplychain under disruptions using revenue-sharing contract witheffort dependent demandrdquoMathematical Problems in Engineer-ing vol 2016 Article ID 9167864 10 pages 2016

[12] M Gumus S Ray and H Gurnani ldquoSupply-side story risksguarantees competition and information asymmetryrdquo Man-agement Science vol 58 no 9 pp 1694ndash1714 2012

[13] K B Hendricks and V R Singhal ldquoAn empirical analysisof the effect of supply chain disruptions on long-run stockprice performance and equity risk of the firmrdquo Production ampOperations Management vol 14 no 1 pp 35ndash52 2005

[14] W Schimidt and A Raman ldquoWhen supply-chain disruptionsmatterrdquo Harvard Business School Working Paper 13-006 2012

[15] N Bunkley Piecing together a supply chain The New YorkTimes 2013 httpwwwnytimescom20110513businessglobal13autohtml r=0

[16] S Sarkar and S Kumar ldquoA behavioral experiment on inventorymanagement with supply chain disruptionrdquo International Jour-nal of Production Economics vol 169 pp 169ndash178 2015

[17] G P Cachon ldquoSupply chain coordination with contractsrdquoHandbooks in Operations Research amp Management Science vol11 pp 227ndash339 2003

[18] C J Corbett and X De Groote ldquoA supplierrsquos optimal quantitydiscount policy under asymmetric informationrdquo ManagementScience vol 46 no 3 pp 444ndash450 2000

12 Mathematical Problems in Engineering

[19] C J Corbett ldquoStochastic inventory systems in a supply chainwith asymmetric information cycle stocks safety stocks andconsignment stockrdquoOperations Research vol 49 no 4 pp 487ndash500 2001

[20] A Matopoulos M Vlachopoulou V Manthou and B ManosldquoA conceptual framework for supply chain collaboration empir-ical evidence from the agri-food industryrdquo Supply Chain Man-agement vol 12 no 3 pp 177ndash186 2007

[21] A Y Ha and S Tong ldquoContracting and information sharingunder supply chain competitionrdquoManagement Science vol 54no 4 pp 701ndash715 2008

[22] Y-W Zhou ldquoA comparison of different quantity discountpricing policies in a two-echelon channel with stochastic andasymmetric demand informationrdquo European Journal of Opera-tional Research vol 181 no 2 pp 686ndash703 2007

[23] XGan S P Sethi and J Zhou ldquoCommitment-penalty contractsin drop-shipping supply chains with asymmetric demand infor-mationrdquo European Journal of Operational Research vol 204 no3 pp 449ndash462 2010

[24] D Lei J Li and Z Liu ldquoSupply chain contracts under demandand cost disruptions with asymmetric informationrdquo Interna-tional Journal of Production Economics vol 139 no 1 pp 116ndash126 2012

[25] Z B Yang G Aydın V Babich and D R Beil ldquoSupplydisruptions asymmetric information and a backup productionoptionrdquoManagement Science vol 55 no 2 pp 192ndash209 2009

[26] S Huang and C Yang ldquoSupply chain revelation mechanismdesign under asymmetric demand disruption informationrdquoOperations Research andManagement Science vol 23 no 6 pp116ndash127 2014

[27] S Oh and O Ozer ldquoMechanism design for capacity planningunder dynamic evolutions of asymmetric demand forecastsrdquoManagement Science vol 59 no 4 pp 987ndash1007 2013

[28] Q Feng G Lai and L X Lu ldquoDynamic bargaining in a supplychain with asymmetric demand informationrdquo ManagementScience vol 61 no 2 pp 301ndash315 2015

[29] Q Li B Li P Chen and P Hou ldquoDual-channel supply chaindecisions under asymmetric information with a risk-averseretailerrdquo Annals of Operations Research 2015

[30] J Wei K Govindan Y Li and J Zhao ldquoPricing and collectingdecisions in a closed-loop supply chain with symmetric andasymmetric informationrdquo Computers and Operations Researchvol 54 pp 257ndash265 2015

[31] K Inderfurth A Sadrieh and G Voigt ldquoThe impact of infor-mation sharing on supply chain performance under asymmetricinformationrdquo Production amp Operations Management vol 22no 2 pp 410ndash425 2013

[32] R B Myerson ldquoMechanism design by an informed principalrdquoEconometrica vol 51 no 6 pp 1767ndash1797 1983

[33] E Maskin and J Tirole ldquoThe principal-agent relationship withan informed principal the case of private valuesrdquo Econometricavol 58 no 2 pp 379ndash409 1990

[34] E Maskin and J Tirole ldquoThe principal-agent relationship withan informed principal II common valuesrdquo Econometrica vol60 no 1 pp 1ndash42 1992

[35] T Mylovanov and T Troger ldquoInformed-principal problemsin environments with generalized private valuesrdquo TheoreticalEconomics vol 7 no 3 pp 465ndash488 2012

[36] TMylovanov andT Troger ldquoMechanismdesign by an informedprincipal private values with transferable utilityrdquo Review ofEconomic Studies vol 81 no 4 pp 1668ndash1707 2014

[37] C Wagner T Mylovanov and T Troger ldquoInformed-principalproblem with moral hazard risk neutrality and no limitedliabilityrdquo Journal of EconomicTheory vol 159 pp 280ndash289 2015

[38] S Galperti ldquoCommon agency with informed principals menusand signalsrdquo Journal of Economic Theory vol 157 pp 648ndash6672015

[39] H Zhang Y Liu and J Huang ldquoSupply chain coordina-tion contracts under double sided disruptions simultaneouslyrdquoMathematical Problems in Engineering vol 2015 Article ID812043 9 pages 2015

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 3

Time

R gets typeinformation

R designsthe contracts

R choosesthe contract

and makes order

M suppliesproducts andgets payment

R sellssome

products

M gets theunsold

products

M acceptsor rejects

t = 1 t = 2 t = 3 t = 4 t = 5 t = 6 t = 7

Figure 1 Informed principal game sequence in supply chain with demand disruption R means the retailer M means the manufacturer

retailer with the probability 120582 and a high-risk retailer with theprobability 1 minus 120582 Both the high-risk retailer and the low-riskretailer are risk neutral

Assumption 4 The retailer as the informed principal designsthe transferring contracts by which the retailer can demon-strate its true information to the manufacturer The transfer-ring contract for low-risk retailer is 119905

1 1199021 while for high-

risk one it is 1199052 1199022

Assumption 5 Every unsold product has the salvage value119904 (119904 lt 119888) And the manufacturer has the salvage value of theunsold products For example the unsold products can bereused for the manufacturer but the retailer has to payfor disposing them The manufacturer can get the unsoldproducts free from the retailer and reuse them In thiscase the manufacturer has the salvage value of the unsoldproducts

119878(120572119894 119902119894) (119894 = 1 2) denote the low-risk retailerrsquos and high-

risk retailerrsquos expected sales

119878 (120572119894 119902119894) = 120572119894(119902119894minus int

119902119894

0

119866 (119909) 119889119909)

+ (1 minus 120572119894) (119902119894minus int

119902119894

0

119865 (119910) 119889119910)

(1)

119868(120572119894 119902119894) (119894 = 1 2) denote the low-risk retailerrsquos and high-

risk retailerrsquos expected unsold quantity

119868 (120572119894 119902119894) = 120572119894int

119902119894

0

119866 (119909) 119889119909 + (1 minus 120572119894) int

119902119894

0

119865 (119910) 119889119910 (2)

119880119894(119894 = 1 2) denote the low-risk retailerrsquos and high-risk

retailerrsquos utility

119880119894= 119901119878 (120572

119894 119902119894) minus 119905119894 (3)

119881119894(119894 = 1 2) denote the manufacturerrsquos utility when it

cooperates with the low-risk retailer or high-risk retailer

119881119894= 119905119894minus 119888119902119894+ 119904119868 (120572

119894 119902119894) (4)

22 Game Sequence When the retailer faces the demanddisruption the sequence of the informed principal gamebetween themanufacturer and the retailer is shown as follows(Figure 1)

(1) The retailer knows its type information 120572119894

(2) The retailer as the informed principal designs thetransferring contract package (119905

1 1199021) (1199052 1199022)

(3) The manufacturer accepts or rejects the contract(4) Before the selling season the retailer chooses the

transferring contract according to its type and makesthe order 119902

119894

(5) The manufacturer supplies products and the retailerpays 119905

119894for them

(6) In the selling season the demand disruption occursin a certain probability and the retailer sells someproducts

(7) After the selling season the manufacturer gets theunsold products

3 Full Information Model

31 Modeling When the demand disruption probability ispublic information the different retailers should provide thesingle selected contracts (119905

1 1199021) or (119905

2 1199022) We can get this

pair of single selected contracts by solving the followingprograms 1198751

119865 and 119875

2

119865

1198751

119865

max1199051 1199021

1198801= 119901119878 (120572

1 1199021) minus 1199051 (5)

st 1198811= 1199051minus 1198881199021+ 119904119868 (120572

1 1199021) ge 0 (6)

1198752

119865

max1199052 1199022

1198802= 119901119878 (120572

2 1199022) minus 1199052 (7)

st 1198812= 1199052minus 1198881199022+ 119904119868 (120572

2 1199022) ge 0 (8)

The targets of these programs are the retailersrsquo revenuewhich are the principal As in the full information situationeach participant knows the true information so there is noincentive compatible constraint Whatever the manufacturershould get no less than the retrained revenue The incentiverationality constraint of the manufacturer facing differentretailers is useful here (see (6) and (8))

32 Results By solving programs 1198751119865 and 119875

2

119865 there comes

Proposition 6 (Proof of Proposition 6 is given inAppendixA)

Proposition 6 (1) Under full information the optimal con-tract in supply chain with demand disruption satisfies thatboth the optimal order and optimal transferring payment for

4 Mathematical Problems in Engineering

t

Alowast

Blowast

q

tlowast1

tlowast2

qlowast1qlowast2

Ulowast1

Ulowast2

Vlowast1

Vlowast2

Figure 2 Supply chain contract under full information

low-risk retailer are higher than those for high-risk retailer as119902lowast

1gt 119902lowast

2 119905lowast1gt 119905lowast

2

(2) Under full information the utility of the low-riskretailer is larger than that of high-risk retailer as 119880lowast

1gt 119880lowast

2

and the utility of the manufacturer cooperating with low-riskretailer and with high-risk retailer is equal to the retrainedrevenue as 119881lowast

1= 119881lowast

2= 0

From Proposition 6 it is obvious that under full infor-mation the low-risk retailer should make more orders thanthe high-risk retailer (119902lowast

1gt 119902lowast

2) and meanwhile pay more

(119905lowast1gt 119905lowast

2) Let us show the optimal contract in Figure 2 using

the retailer and themanufacturer indifferent utility curve Forthe low-risk retailer the optimal contract is shown at point119861lowast And for the high-risk retailer the optimal contract is

shown at point 119860lowast The retailerrsquos utility grows more whenits indifferent curve goes into southeast So when the low-risk retailer pretends to be high-risk one using allocation119860

lowastwhich is to say that the low-risk retailerrsquos indifferent utilitycurve goes northwest its utility decreases But if the high-risk retailer pretends to be low-risk one using allocation 119861

lowastits utility increases So as the self-interested participant thehigh-risk retailer tends to pretend to be low-risk one Thenwe can get Proposition 7

Proposition 7 Considering the demand disruption proba-bility as the retailerrsquos individual information the optimalallocation for the low-risk retailer is not separating equilibriumallocation

4 Informed Principal Model

In order to show its true type information the retailer can usethe informed principalmodel to offer an option contract [32]The term ldquooption contract (119905

1 1199021) (1199052 1199022)rdquo comes from the

fact that if the manufacturer accepts the contract the retailermust then exercise its built-in option and choose between(1199051 1199021) and (119905

2 1199022) The retailer will choose the term which

is fit to its own type by probability 100 as 1199011205721| (1199051 1199021) =

1199011205722

| (1199052 1199022) = 1 And the retailer will choose the term

which is not fit to its own probability by probability 0 as1199011205721

| (1199052 1199022) = 119901120572

2| (1199051 1199021) = 0 [33 34] Then the

informed principal model of the supply chain with demanddisruption is built up for the Perfect Bayesian Equilibrium

41 Modeling Firstly the incentive compatible constraint isgivenThe option contract is incentive compatible if the low-risk retailer prefers the contract item (119905

1 1199021) and the high-

risk retailer prefers the contract item (1199052 1199022) So the incentive

compatible constraint should keep the low-risk retailerrsquosutility no less than when it pretends to be a high-risk one

119901119878 (1205721 1199021) minus 1199051ge 119901119878 (120572

1 1199022) minus 1199052 (9)

And the high-risk retailerrsquos utility is no less than when itpretends to be a low-risk one

119901119878 (1205722 1199022) minus 1199052ge 119901119878 (120572

2 1199021) minus 1199051 (10)

Secondly the incentive rationality constraint is givenTheretailer with the individual information is the principal andthemanufacturerwithout individual information is the agentSo the agentrsquos incentive rationality is the same as that (as in (6)and (8)) in full information situation

Thirdly the target function is given The target of theinformed principal model is also to maximize the utility ofthe principal or to say the retailer

So the programs under informed principal of supplychain with demand disruption are 119875

1

119868 and 119875

2

119868 Consider

the following

1198751

119868

max1199051 1199021

1198801= 119901119878 (120572

1 1199021) minus 1199051

st (6) (8) (9) (10)

(11)

1198752

119868

max1199052 1199022

1198802= 119901119878 (120572

2 1199022) minus 1199052

st (6) (8) (9) (10)

(12)

Definition 8 Utility 1198801(1199050

1 1199020

1)1198802(1199050

2 1199020

2) for low-riskhigh-

risk retailer is the low-information-intensity optimum forthat type if (1199050

1 1199020

1)(1199050

2 1199020

2)maximizes low-risk retailerrsquos utility

in the set of incentive compatible constraint and incentiverationality constraint They are (part of) the solution to theprograms 1198751

119868 and 119875

2

119868

For solving the informedprincipalmodel the assumptionof weak monotonic profit is listed as follows

Assumption 9 It is weak monotonic profit when the manu-facturer makes a nonnegative profit if the contractual termsare those of low-risk retailer under symmetric informationand the retailer is a high-risk one 119881

1(119905lowast

2 119902lowast

2) ge 0

From Figure 2 if the contractual term for the low-riskretailer changes from119861

lowast to119860lowast the curve119881lowast1will go southeast

and themanufacturerrsquos profit increases So themanufacturerrsquosprofit is no less than 0 and Assumption 9 is satisfied

Mathematical Problems in Engineering 5

Definition 10 The separating allocation is the allocation (119905119904

1

119902119904

1) for the low-risk retailer and the symmetric information

contractual terms (119905lowast2 119902lowast

2) are for the high-risk retailer where

(119905119904

1 119902119904

1) maximizes the low-risk retailerrsquos utility subject to the

manufacturerrsquos breaking even for the low-risk retailer and tothe high-risk retailer not preferring (119905

119904

1 119902119904

1) to (119905

lowast

2 119902lowast

2) The

program is listed as 1198751119904

1198751

119904

max11990511199021

1198801= 119901119878 (120572

1 1199021) minus 1199051

st (6)

(13)

119901119878 (1205722 119902lowast

2) minus 119905lowast

2ge 119901119878 (120572

2 1199021) minus 1199051 (14)

42 Results

Proposition 11 Under the weak monotonic-profit assump-tion the separating allocation is the low-information-intensityoptimum

Proof (1) The high-risk retailer can get its asymmetric infor-mation utility even under asymmetric information Compar-ing programs 1198752

119865 and 119875

2

119868 it can be found that these two

programs have the same target function but 1198752

119865 has less

constraints so 119880(119905lowast

2 119902lowast

2) ge 119880(119905

0

2 1199020

2) According to Assump-

tion 9 the manufacturer at least breaks even regardless of theretailerrsquos type Hence (119905lowast

2 119902lowast

2) is the separating allocation item

for high-risk retailer(2) Because 119880(119905

lowast

2 119902lowast

2) ge 119880(119905

0

2 1199020

2) constraint (14) is more

restricted than (10) For the low-risk retailer the programs1198751

119868 and 119875

119904

1 have the same target functions but there are

more constraints in the former program so 119880(119905119904

1 119902119904

1) ge

119880(1199050

1 1199020

1) is satisfied And the low-risk retailer at least can

get 119880(119905119904

1 119902119904

1) from the option contract From the incentive

compatibility constraint (14) which can be rewritten as1198802(119905lowast

2 119902lowast

2) ge 1198802(119905119904

1 119902119904

1) it has been guaranteed that the high-

risk retailer will not choose 119905119904

1 119902119904

1 From the program 119875

119904

1

the low-risk retailer can get more utility when the contractualitem is 119905119904

1 119902119904

1 than that of 119905lowast

2 119902lowast

2 So we can get 1199050

1 1199020

1 =

119905119904

1 119902119904

1

Then we can get Proposition 12 (see details in Appen-dix B)

Proposition 12 As for the option contract by informed retailerin supply chain with demand disruption the low-information-intensity allocation satisfies the following

(1) Comparing to the full information situation the low-information-intensity allocation for high-risk retailerand its utility does not distort as 1199050

2= 119905lowast

2 11990202

= 119902lowast

2

1198802(1199050

2 1199020

2) = 119880lowast

2

(2) Comparing to the full information situation the low-information-intensity allocation for low-risk retailerhas upward distortion 119905

0

1gt 119905lowast

1 11990201gt 119902lowast

1 while its utility

has downward distortion 1198801(1199050

1 1199020

1) lt 119880lowast

1

(3) The order quantities in the low-information-intensityallocation for the retailers in different types satisfy

119901119878 (1205722 1199020

1) minus 119888119902

0

1+ 119904119868 (120572

1 1199020

1)

= 119901119878 (1205722 119902lowast

2) minus 119888119902

lowast

2+ 119904119868 (120572

2 119902lowast

2)

(15)

In the informed principal model in order to prevent thehigh-risk retailer from pretending to be a low-risk one thelow-information-intensity allocation items for the low-riskretailer distort The low-risk retailer should order more thanthe optimal order and paymore than the optimal transferringpayment But the utility of the low-risk retailer is less than theoptimal onewhichmeans the low-risk retailer pays some rentto separate from the high-risk retailer So the rent is calledsignaling cost

5 Interim Efficient Allocation

The retailer can deliver its own type information to themanufacturer by low-information-intensity allocation in theinformed principal model But the low-risk retailerrsquos orderand transferring payment distort relative to the full infor-mation situation and it has to pay the signaling cost Thesignaling cost is the part utility which the low-risk retailergets less than the optimal So maybe we can decrease thesignaling cost by increasing the high-risk retailerrsquos utilityand meanwhile increasing the low-risk retailerrsquos utility So inthis part we try to find a separating equilibrium with lesssignaling cost [32ndash34 38] by the interim efficient model

Let us consider the interim efficient model which candecrease the signaling cost by giving the high-risk retailermore than optimal utility Let be the utility which the high-risk retailer gets more than the optimal one And let 119871() betheminimal loss of themanufacturer when it cooperates withthe high-risk retailer119871() can be gotten by the program 119875

2

119898

1198752

119898

minus119871 () = max1199052 1199022

1199052minus 1198881199022+ 119904119868 (120572

2 1199022) (16)

st 119901119878 (1205722 1199022) minus 1199052

ge 119901119878 (1205722 119902lowast

2) minus 119905lowast

2+

(17)

It is easy to find out that when constraint (17) is binding1199022= 119902lowast

2and 119871() =

Definition 13 Utility 1198801(119905119898

1 119902119898

1)1198802(119905119898

2 119902119898

2) for low-risk

high-risk retailer is the interim efficient optimum for thattype if (119905119898

1 119902119898

1)(119905119898

2 119902119898

2) maximizes low-risk retailerrsquos utility

in the set of incentive compatible constraint and themanufac-turerrsquos expected profit And the contract (119905119898

1 119902119898

1) (119905119898

2 119902119898

2) is

the interim efficient allocationThey are (part of) the solutionto the programs 1198751

119898 and 119875

2

119898

6 Mathematical Problems in Engineering

Table 1 Full information contract versus low-information-intensity allocation

Type ofretailer Full information

Lowinformationintensity

Type ofretailer Full information

Lowinformationintensity

1205721= 02

119902lowast

1= 1001 119902

0

1= 1014

1205721= 03

119902lowast

1= 990 119902

0

1= 1002

119905lowast

1= 1862 119905

0

1= 1881 119905

lowast

1= 1840 119905

0

1= 1824

119880lowast

1= 5340 119880

0

1= 5338 119880

lowast

1= 5278 119880

0

1= 5192

1205722= 04

119902lowast

2= 978 119902

0

1= 978

1205722= 05

119902lowast

2= 967 119902

0

2= 967

119905lowast

2= 1819 119905

0

2= 1819 119905

lowast

2= 1799 119905

0

2= 1799

119880lowast

2= 5217 119880

0

2= 5217 119880

lowast

2= 5159 119880

0

2= 5159

1198751

119898

max11990511199021

1198801= 119901119878 (120572

1 1199021) minus 1199051 (18)

st 120582 (119905119898

1minus 119888119902119898

1+ 119904119868 (120572

1 119902119898

1)) minus (1 minus 120582) ge 0 (19)

119901119878 (1205722 119902lowast

2) minus 119905lowast

2+ ge 119901119878 (120572

2 119902119898

1) minus 119905119898

1 (20)

The high-risk retailer can get the rent above theoptimum so themanufacturer will lose some utilityThen theincentive rational constraint (see (19)) for themanufacturer isthe expected profit rather than the individual profit Equation(20) is the incentive compatible constraint for the high-riskretailer

From programs 1198751119898 and 119875

2

119898 we can get Propositions 14

and 15 (see Proof in Appendix C)

Proposition 14 The interim efficient allocation by informedprincipal for the supply chain with demand disruption satisfiesthe following

(1) Comparing to the full information situation the orderquantity for the high-risk retailer does not distort 119902119898

2=

119902lowast

2

(2) Comparing to the full information situation the orderquantity for the low-risk retailer satisfies (21) anddistorts upwards 119902119898

1gt 119902lowast

1 Consider

1199041198681015840(1205721 119902119898

1) + 119901119878

1015840(1205721 119902119898

1) minus 119888

=1 minus 120582

120582[1199011198781015840(1205722 119902119898

1) minus 119901119878

1015840(1205721 119902119898

1)]

(21)

(3) The transferring payment for the low-risk retailer meets(22) and the transferring payment for the high-riskretailer meets (23)

119905119898

1= (1 minus 120582)

sdot [119901119878 (1205722 119902119898

1) minus 119901119878 (120572

2 119902lowast

2) + 119888119902

lowast

2minus 119904119868 (120572

2 119902lowast

2)]

+ 120582 [119888119902119898

1minus 119904119868 (120572

1 119902119898

1)]

(22)

119905119898

2= (1 minus 120582) [119888119902

lowast

2minus 119904119868 (120572

2 119902lowast

2)] + 120582119901119878 (120572

2 119902lowast

2)

+ 120582 [119888119902119898

1minus 119904119868 (120572

1 119902119898

1) minus 119901119878 (120572

2 119902119898

1)]

(23)

Proposition 15 There is a threshold 1205820

= (1199011198781015840(1205722 1199020

1) minus

1199011198781015840(1205721 1199020

1))(119904119868

1015840(1205721 1199020

1) + 119901119878

1015840(1205722 1199020

1) minus 119888) for the low-risk

retailer probability If and only if the probability of low-riskretailer is no more than the threshold 120582 le 120582

0 the low-infor-mation-intensity allocation is also interim efficient and isthe unique Perfect Bayesian Equilibrium If the probabilityof low-risk retailer is more than the threshold 120582 gt 120582

0 thelow-information-intensity allocation is not the unique PerfectBayesian Equilibrium

If and only if the probability of low-risk retailer is smallenough (120582 le 120582

0) the low-information-intensity allocation isthe unique Perfect Bayesian EquilibriumAnd the retailer candeliver its own type information by the separating contractBut if the low-risk retailer is more than that (120582 gt 120582

0) the low-information-intensity allocation is not the unique PerfectBayesian Equilibrium and the interim efficient allocation canimprove the distortion situation to a certain degree So for thesupply chain with demand disruption the optimal allocationcannot be reached if the informed principal provides aseparating contract

6 Numerical Example

This section gives some numerical examples to inspect thesupply chain contract with demand disruption by informedprincipal We let the sales price 119901 = 10 the unit manu-facturing cost is 119888 = 2 and the salvage product value is119904 = 05 The demand distribution function before disruptionis 119865(119910) = 119910

2125 and the demand distribution function after

disruption is 119866(119909) = 1199092100

61 Main Results From the example listed above the fullinformation contracts and low-information-intensity alloca-tion are given in Table 1

Table 1 shows that for the full information contract theorder quantity for low-risk retailer is larger than that for thehigh-risk retailer 119902lowast

1gt 119902lowast

2 and also the transferring payment

119905lowast

1gt 119905lowast

2 which is performed in Proposition 6 In the informed

principal model the order quantity for the low-risk retailerdistorts in order to show its type (when 120572

1= 02 versus 120572

2=

04 119902lowast1

= 1001 lt 119902sb1

= 1014 and when 1205721= 03 versus

1205722= 05 119902lowast

1= 990 lt 119902

0

1= 1002) Meanwhile we notice that

the utility obtained by the low-risk retailer in the informed

Mathematical Problems in Engineering 7

0 10

5

10

15

q1

02 04 06 08

120582

Figure 3 The order quantity in interim efficient allocation for low-risk retailer changes by its probability

0

1

101112

1314

150

5

10

15

20

25

30

0204

0608

tm 1

q1 120582

Figure 4 The transferring payment in interim efficient allocationfor low-risk retailer changes by the order quantity and its probability

principal model is lower than that in full information model(when 120572

1= 02 versus 120572

2= 04 119880lowast

1= 5340 gt 119880

0

1= 5338

and when 1205721= 03 versus 120572

2= 05 119905lowast

1= 1840 gt 119905

0

1= 1824)

This implies that it costs the low-risk retailer something tocertify its typeThese results are the samewith Proposition 12

Considering the situation of 1205721= 02 versus 120572

2= 04 we

obtain 1205820= 019 The order quantity for the low-risk retailer

119902119898

1changes with the proportion it has 120582 in interim efficient

allocation As Figure 3 shows if and only if 120582 varies withincertain area (120582 le 120582

0) we obtain 119902119898

1gt 119902lowast

1 as in the results in

Proposition 14The transferring payment in the interim efficient allo-

cation is depicted in Figure 4 (for the low-risk retailer)and Figure 5 (for the high-risk retailer) The transferringpayments are changing by the order quantity for the low-risk retailer in the interim efficient allocation and the low-risk retailerrsquos probability From Figure 4 when the low-riskretailerrsquos probability 120582 is less enough and the order quantityfor the low-risk retailer 119902119898

1is high enough the transferring

0

05

1

10 11 12 13 14 15

20

25

30

35

40

45

tm 2

q1

120582

Figure 5 The transferring payment in interim efficient allocationfor high-risk retailer changes by the low-risk retailerrsquos order quantityand probability

payment 1199051198981increases by the low-risk retailerrsquos probability 120582

but decreases by the order quantity for the low-risk retailer119902119898

1(bottom left in Figure 4) But mostly the transferring

payment 1199051198981increases by the low-risk retailerrsquos probability 120582

but decreases by the order quantity for the low-risk retailer 1199021198981

(red and orange part in Figure 4) Normally the transferringpayment should increase by the order quantity which meansldquoget more pay morerdquo From this view the low-risk retailerrsquosprobability 120582 should not be very small and the order quantityfor the low-risk retailer 119902

119898

1should not be close to the

maximum value 15From Figure 5 it is obviously seen that the transferring

payment for high-risk retailer in the interim efficient alloca-tion 119905119898

2increases by the order quantity for the low-risk retailer

119902119898

1but decreases by the low-risk retailerrsquos probability 120582

62 Sensitivity Analyses The sensitivity analyses for the low-information-intensity allocation versus the unit manufactur-ing cost 119888 are shown in Table 2 It shows the outcomes whenthe unit manufacturing cost 119888 changes by +25 and minus25once at a time and keeping remaining parameters The orderquantities decrease by the unit manufacturing cost but theinfluence for the high-risk retailerrsquos order quantity is largerthan that for the low-risk retailer (minus069 versus minus317 and+217 versus +307) The transferring payments increaseby the unit manufacturing cost More interesting the utilitiesdecrease by the unit manufacturing cost and the low-riskretailer receives a greater impact (minus940 versus minus922 and+955 versus +953) So from this point of view the low-risk retailer should encourage themanufacturer to reduce thecost more

Figure 6 shows the sensitivity analyses for the order quan-tities versus the unit manufacturing cost in which the orderquantities include the order quantity for the low-risk retailerin low-information-intensity allocation 119902

0

1 the order quantity

8 Mathematical Problems in Engineering

Table 2 Sensitivity analyses for low-information-intensity allocation versus the unit manufacturing cost

Changes of 119888 The low-information allocation for low-riskretailer (120572

1= 02)

The low-information allocationfor high-risk retailer (120572

2= 04)

Value Change rate Value Change rate Value Change rate

119888 = 25 +251199020

1= 1007 minus069 119902

0

2= 947 minus317

1199050

1= 2375 +2626 119905

0

2= 2243 +2331

1198800

1= 4836 minus940 119880

0

2= 4736 minus922

119888 = 2 Baseline1199020

1= 1014 mdash 119902

sb2

= 978 mdash1199050

1= 1881 mdash 119905

sb2

= 1819 mdash1198800

1= 5338 mdash 119880

sb2

= 5217 mdash

119888 = 15 minus251199020

1= 1036 +217 119902

0

2= 1008 +307

1199050

1= 1398 minus2568 119905

0

2= 1362 minus2512

1198800

1= 5848 +955 119880

0

2= 5714 +953

108

106

104

102

10

98

96

94

92

C

14 16 18 2 22 24 26 28 3

qsb1

qlowast1q2

Figure 6 The order quantities change by the unit manufacturingcost

for the low-risk retailer in full information contract 119902lowast1 and

the order quantity for the high-risk retailer in full informationcontract which is equal to that in low-information-intensityallocation 119902

2 Both the order quantity for the low-risk retailer

in full information and the order quantity for the high-riskretailer decrease by the unit manufacturing cost and theychange in the same level basically But the change of the orderquantity for the low-risk retailer in low information intensitydepends on different values of the unit manufacturing costWhen the unit manufacturing cost 119888 = 275 the orderquantity is minimum and when 119888 = 175 or 119888 = 15 the orderquantity is the maximum

Figure 7 shows the sensitivity analyses for the transferringpayment versus the unit manufacturing cost in which thetransferring payments include the transferring payment forthe low-risk retailer in low-information-intensity allocation1199050

1 the transferring payment for the low-risk retailer in full

information contract 119905lowast

1 and the transferring payment for

26

24

22

20

18

16

14

C

14 16 18 2 22 24 26 28 3

tsb1

tlowast1t2

Figure 7 The transferring payments change by the unit manufac-turing cost

the high-risk retailer in full information contract which isequal to that in low-information-intensity allocation 119905

2 All

the transferring payments of the retailers increase by the unitmanufacturing cost The transferring payments for the low-risk retailer and the high-risk retailer in full informationcontract decrease in the same level

Figure 8 shows the sensitivity analyses for the retailersrsquoutilities versus the unit manufacturing cost in which theutilities include the utility for the low-risk retailer in low-information-intensity allocation 119880

0

1 the transferring pay-

ment for the low-risk retailer in full information contract119880lowast

1 and the transferring payment for the high-risk retailer

in full information contract which is equal to that in low-information-intensity allocation 119880

2 All the retailersrsquo utilities

decrease by the unit manufacturing costFigure 9 shows the sensitivity analyses for the low-risk

retailerrsquos signaling cost versus the unit manufacturing cost

Mathematical Problems in Engineering 9

44

46

48

50

52

54

56

58

60

C

14 16 18 2 22 24 26 28

Usb1

Ulowast1

U2

Figure 8 The retailerrsquos utilities change by the unit manufacturingcost

0

005

01

015

02

025

C

14 16 18 2 22 24 26 28

ΔU1

Figure 9The signaling cost changes by the saved unit manufactur-ing cost

The signaling cost of the low-risk retailer is the utility earnedin the low-information-intensity allocation which is sepa-rating contract less than that in full information contractWhen the unit manufacturing cost 119888 = 2 the signaling costis the least while when the saved unit manufacturing cost119888 = 175 the signaling cost is the most We do not find theinternal relationship between the signaling cost and the unitmanufacturing cost As the signaling cost is so important toput the contracts into practice investigation on the signalingcost is the research direction for us

7 Conclusion

In the recent years supply chain with demand disruptionhas become more and more attractive to both business and

academic research There is a lot of research available on thetopic of supply chain disruption contract In the real supplychain practice demand disruption is totally different fromthe stochastic demand So in our research both the demanduncertainty and the demand disruption are consideredTaking the demand disruption probability as the asymmetricinformation an informed principal model is explored tomake the contract for supply chain with demand disruptionAnd the conclusions we obtain are shown as follows

(1) The contracts for low-risk retailer and high-riskretailer in full information are provided and it isfound out that the high-risk retailer has the incentiveto pretend to be the low-risk retailer This result issimilar to the other informed principal model [3435] but totally different from the normal principalagent model [24 26] In the normal principal agentmodel the high efficient agent (low-risk retailer in oursetting) tries to pretend to be low efficient agent (high-risk retailer) to get more That is why the informedprincipal model is established

(2) The low-information-intensity allocation which is theresult of the informed principal model and also is theseparating contract shows that the order quantity andthe transferring payment for the low-risk retailer inlow-information-intensity allocation distort upwardsbut that of high-risk retailer does not distort Inorder to avoid imitation the imitatorsrsquo efficiencyshould be always distorted But in normal principalagent model the high-risk retailerrsquos quantity distortsdownwards [24 26] on the contrary the low-riskretailerrsquos quantity in informed principal model dis-torts upwards

(3) In order to reduce the signaling cost which the low-risk retailer pays the interim efficient model is intro-duced which ends up with the order quantity andtransferring payment distorted upward again but lessthan that of low information intensity Comparingto [38] focusing on analyzing the signaling costmore attention is paid to showing the interim efficientmodel to get a solution which can increase efficiency

From the managerial aspect retailers should try toincrease their antirisk capability to enhance competitivepower And when the retailer has lower demand disruptionprobability than the others it can show its style to the supplierthrough the separating contract For example the retailer cansay ldquobecause I am the low-risk one I can cooperate with thesupplier with this contract while the others who are high-riskones can only use the other contractrdquo But in order to be sep-arated from the others the low-risk retailer has to pay somesignaling cost and the quantity has to be distorted upwards

Above all there are some limitations of the research Wewill continue our research in the following directions (1)The signaling cost is a key factor impacting the separatingcontract which should be paid more attention (2) We onlyconsider the demand disruption but disruptions happen atnot only demand side but also supply side Sowewill continue

10 Mathematical Problems in Engineering

our research with demand disruption together with supplydisruption [39]

Appendix

A Proof of Proposition 6

For the two types of retailers we can obtain the optimalallocation if and only if the whole margin utility of the supplychain is the same with the margin cost So 119902

lowast

1and 119902

lowast

2satisfy

the following first-order conditions

1199011198781015840(1205721 1199021) + 1199041198681015840(1205721 1199021) = 119888 (A1)

1199011198781015840(1205722 1199022) + 1199041198681015840(1205722 1199022) = 119888 (A2)

And the optimal transferring payments are

119905lowast

1= 119888119902lowast

1minus 119904119868 (120572

1 119902lowast

1)

119905lowast

2= 119888119902lowast

2minus 119904119868 (120572

2 119902lowast

2)

(A3)

From (A1) and (A2) we obtain

1205791119866 (119902lowast

1) + (1 minus 120579

1) 119865 (119902lowast

1)

= 1205792119866 (119902lowast

2) + (1 minus 120579

2) 119865 (119902lowast

2) =

(119901 minus 119888)

(119901 minus 119904)

(A4)

In addition 119866(119902lowast

1) ge 119865(119902

lowast

1) 1205721

lt 1205722 so we know that

1205721119866(119902lowast

1) + (1minus120572

1)119865(119902lowast

1) lt 1205722119866(119902lowast

1) + (1minus120572

2)119865(119902lowast

1) and from

(A4) we obtain 1205722119866(119902lowast

2) + (1 minus 120572

2)119865(119902lowast

2) lt 1205722119866(119902lowast

1) + (1 minus

1205722)119865(119902lowast

1) and thus 119902lowast

2lt 119902lowast

1 From (A3) 119905lowast

1minus119905lowast

2= 119888(119902lowast

1minus119902lowast

2)minus

119904(119868(1205721 119902lowast

1) minus 119868(120572

2 119902lowast

2)) 119905lowast1minus 119905lowast

2=119902lowast

1=119902lowast

2

(1205721minus 1205722)119904(int119902

0119865(119910)119889119910 minus

int119902

0119866(119909)119889119909) gt 0 and 120597119905

lowast

1120597119902lowast

1gt 0 so 119905

lowast

1minus 119905lowast

2gt 0 and

thus we obtain 119905lowast

1gt 119905lowast

2 119881lowast

1= 119881lowast

2= 0 Furthermore we get

119880lowast

1gt 119880lowast

2from 120597119880120597120579 lt 0 and 120597119880120597119902 gt 0 Finally we have

Proposition 6

B Proof of Proposition 12

We denote 1205831and 120583

2as the Lagrange multipliers of (6) and

(14) and find the derivate of 1199051and 1199021 and thus we obtain

120597119880 (1205721 1199021)

1205971199051

= minus1 + 1205831+ 1205832= 0

120597119880 (1205721 1199021)

1205971199021

= 1199011198781015840(1205721 1199021) minus 1205831119888 + 12058311199041198681015840(1205721 1199021)

minus 12058321199011198781015840(1205722 1199021) = 0

(B1)

By rewriting (B1) we get the following formulas

1199041198681015840(1205721 119902119904

1) + 119901119878

1015840(1205721 119902119904

1) minus 119888

=(1 minus 120583

1)

1205831

[1199011198781015840(1205722 119902119904

1) minus 119901119878

1015840(1205721 119902119904

1)]

(B2)

1205831=

1199011198781015840(1205722 119902119904

1) minus 119901119878

1015840(1205721 119902119904

1)

1199041198681015840 (1205721 119902119904

1) + 1199011198781015840 (120572

2 119902119904

1) minus 119888

(B3)

1205832=

1199041198681015840(1205721 119902119904

1) + 119901119878

1015840(1205721 119902119904

1) minus 119888

1199041198681015840 (1205721 119902119904

1) + 1199011198781015840 (120572

2 119902119904

1) minus 119888

(B4)

Because the numerator of 1205831is negative so 120583

1is positive

and the denominator of 1205831is negative because 120583

2is positive

(when 1205832= 0 it cannot be separated from the former one)

the numerator of 1205832is negative nevertheless 1199011198781015840(120572

1 119902lowast

1) minus

119888 + 1199041198681015840(1205721 119902lowast

1) = 0 and thus 1199011198781015840(120572

1 119902119904

1) minus 119888 + 119904119868

1015840(1205721 119902119904

1) lt

1199011198781015840(1205721 119902lowast

1)minus 119888+ 119904119868

1015840(1205721 119902lowast

1) in addition 1205972119880120597119902

2lt 0 so 119902

119904

1gt

119902lowast

1 From 120583

1gt 0 and 120583

2gt 0 we deduce that (6) and (14)

are tight Furthermore the programs 1198751119865 and 119875

1

119898 have the

same objective function but the constraint of 1198751119898 is more

tight so there is distortion of the utility obtained from 1198751

119898 in

other words 1198801(119905119904

1 119902119904

1) lt 119880lowast

1 From Proposition 11 we obtain

1198802(1199050

2 1199020

2) = 119880lowast

2 So we have Proposition 12

C Proof of Propositions 14 and 15

We denote 120573 and 120574 as the Lagrange multipliers of (19) and(20) and find the derivate of 119905

1 1199021 and and thus we obtain

120597119871 (1199051 1199021 )

1205971199051

= minus1 + 120573120582 + 120574 = 0 (C1)

120597119871 (1199051 1199021 )

1205971199021

= 1199011198781015840(1205721 1199021) + 120573120582 [minus119888 + 119904119868

1015840(1205721 1199021)]

minus 1205741199011198781015840(1205721 1199021) = 0

(C2)

120597119871 (1199051 1199021 )

120597= minus120573 (1 minus 120582) + 120574 = 0 (C3)

To solve (C1)ndash(C3) we obtain 120573 = 1 120574 = 1 minus 120582 Andconsidering (C2) we get

1199041198681015840(1205721 119902119898

1) + 119901119878

1015840(1205721 119902119898

1) minus 119888

=1 minus 120582

120582[1199011198781015840(1205722 119902119898

1) minus 119901119878

1015840(1205721 119902119898

1)]

(C4)

Because (19) and (20) are tight we obtain the followingresults

119905119898

1= (1 minus 120582)

sdot [119901119878 (1205722 119902119898

1) minus 119901119878 (120572

2 119902lowast

2) + 119888119902

lowast

2minus 119904119868 (120572

2 119902lowast

2)]

+ 120582 [119888119902119898

1minus 119904119868 (120572

1 119902119898

1)]

119905119898

2= (1 minus 120582) [119888119902

lowast

2minus 119904119868 (120572

2 119902lowast

2)] + 120582119901119878 (120572

2 119902lowast

2)

+ 120582 [119888119902119898

1minus 119904119868 (120572

1 119902119898

1) minus 119901119878 (120572

2 119902119898

1)]

(C5)

Mathematical Problems in Engineering 11

To compare (A1) with (C4) we know 119902119898

1gt 119902lowast

1because

12059721198801205971199022

lt 0 and 1199011198781015840(1205722 119902) minus 119901119878

1015840(1205721 119902) lt 0 and the dis-

tortion level is changing with 120572 Finally we have Proposi-tion 14

To compare (C4) with (B2) and designate 1205820

= 120582 =

(1199011198781015840(1205722 1199020

1) minus 119901119878

1015840(1205721 1199020

1))(119904119868

1015840(1205721 1199020

1) + 119901119878

1015840(1205722 1199020

1) minus 119888) we

have the following conclusions

(1) When 120582 = 1205820 1199021198981

= 119902119904

1= 1199020

1 we obtain the same

contracts with the low-information-intensity situa-tion

(2) When 120582 lt 1205820 1199021198981

lt 119902119904

1= 1199020

1 we know there is more

distortion than the low-information-intensity alloca-tion and it improves anything

(3) When 120582 gt 1205820 1199021198981

lt 119902119904

1= 1199020

1 the interim efficient

allocation can improve the low-information-intensityallocation So we have Proposition 15

VariablesNotations

119888 Unit manufacturing cost1205721 Disruption probability of low-risk retailer

119910 Market demand without disruption119865(sdot) Distribution function of demand without

disruption119904 Unit salvage value119860 Market scale without disruption119878(120572119894 119902119894) Expected sales

119868(120572119894 119902119894) Expected unsold quantity

119880119894 Utility of the retailer

120582 Probability of low-risk retailer1205722 Disruption probability of high-risk retailer

119909 Market demand with disruption119866(sdot) Distribution function of demand with

disruption119901 Retail price119863 Market scale with disruption119881119894 Utility of the manufacturer

Control Variables

119902119894 Order quantity

119905119894 Transferring payment

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported in part by a project supported bythe Humanities and Social Sciences Project of the EducationMinistry (no 14YJC630187) Hebei Social Sciences Project(no HB15GL058) and the Fundamental Research Funds forthe Central Universities (no 2016MS123)

References

[1] J Barnett F Begen S Howes et al ldquoConsumersrsquo confidencereflections and response strategies following the horsemeatincidentrdquo Food Control vol 59 pp 721ndash730 2016

[2] A J Schmitt and L V Snyder ldquoInfinite-horizon models forinventory control under yield uncertainty and disruptionsrdquoComputers amp Operations Research vol 39 no 4 pp 850ndash8622012

[3] L V Snyder Z Atan P Peng et al ldquoORMS models for supplychain disruptions a reviewrdquo IIE Transactions vol 48 no 2 pp89ndash109 2016

[4] X Qi J F Bard and G Yu ldquoSupply chain coordination withdemand disruptionsrdquo Omega vol 32 no 4 pp 301ndash312 2004

[5] T Xiao G Yu Z Sheng and Y Xia ldquoCoordination of a supplychain with one-manufacturer and two-retailers under demandpromotion and disruption management decisionsrdquo Annals ofOperations Research vol 135 pp 87ndash109 2005

[6] F Hu C-C Lim Z Lu and X Sun ldquoCoordination in a single-retailer two-supplier supply chain under random demand andrandom supply with disruptionrdquo Discrete Dynamics in Natureand Society vol 2013 Article ID 484062 12 pages 2013

[7] J Li X Liu J Wu and F Yang ldquoCoordination of supplychain with a dominant retailer under demand disruptionsrdquoMathematical Problems in Engineering vol 2014 Article ID854681 10 pages 2014

[8] K Chen and T Xiao ldquoDemand disruption and coordination ofthe supply chain with a dominant retailerrdquo European Journal ofOperational Research vol 197 no 1 pp 225ndash234 2009

[9] J Li and F T S Chan ldquoThe impact of collaborative transporta-tion management on demand disruption of manufacturingsupply chainsrdquo International Journal of Production Research vol50 no 19 pp 5635ndash5650 2012

[10] Y Zheng T Shu S Wang S Chen K K Lai and L GanldquoDemand disruption and coordination of supply chain via effortand revenue sharingrdquo Applied Economics vol 47 no 54 pp5886ndash5901 2015

[11] Q Pang Y Hou and Y Lv ldquoCoordinating three-level supplychain under disruptions using revenue-sharing contract witheffort dependent demandrdquoMathematical Problems in Engineer-ing vol 2016 Article ID 9167864 10 pages 2016

[12] M Gumus S Ray and H Gurnani ldquoSupply-side story risksguarantees competition and information asymmetryrdquo Man-agement Science vol 58 no 9 pp 1694ndash1714 2012

[13] K B Hendricks and V R Singhal ldquoAn empirical analysisof the effect of supply chain disruptions on long-run stockprice performance and equity risk of the firmrdquo Production ampOperations Management vol 14 no 1 pp 35ndash52 2005

[14] W Schimidt and A Raman ldquoWhen supply-chain disruptionsmatterrdquo Harvard Business School Working Paper 13-006 2012

[15] N Bunkley Piecing together a supply chain The New YorkTimes 2013 httpwwwnytimescom20110513businessglobal13autohtml r=0

[16] S Sarkar and S Kumar ldquoA behavioral experiment on inventorymanagement with supply chain disruptionrdquo International Jour-nal of Production Economics vol 169 pp 169ndash178 2015

[17] G P Cachon ldquoSupply chain coordination with contractsrdquoHandbooks in Operations Research amp Management Science vol11 pp 227ndash339 2003

[18] C J Corbett and X De Groote ldquoA supplierrsquos optimal quantitydiscount policy under asymmetric informationrdquo ManagementScience vol 46 no 3 pp 444ndash450 2000

12 Mathematical Problems in Engineering

[19] C J Corbett ldquoStochastic inventory systems in a supply chainwith asymmetric information cycle stocks safety stocks andconsignment stockrdquoOperations Research vol 49 no 4 pp 487ndash500 2001

[20] A Matopoulos M Vlachopoulou V Manthou and B ManosldquoA conceptual framework for supply chain collaboration empir-ical evidence from the agri-food industryrdquo Supply Chain Man-agement vol 12 no 3 pp 177ndash186 2007

[21] A Y Ha and S Tong ldquoContracting and information sharingunder supply chain competitionrdquoManagement Science vol 54no 4 pp 701ndash715 2008

[22] Y-W Zhou ldquoA comparison of different quantity discountpricing policies in a two-echelon channel with stochastic andasymmetric demand informationrdquo European Journal of Opera-tional Research vol 181 no 2 pp 686ndash703 2007

[23] XGan S P Sethi and J Zhou ldquoCommitment-penalty contractsin drop-shipping supply chains with asymmetric demand infor-mationrdquo European Journal of Operational Research vol 204 no3 pp 449ndash462 2010

[24] D Lei J Li and Z Liu ldquoSupply chain contracts under demandand cost disruptions with asymmetric informationrdquo Interna-tional Journal of Production Economics vol 139 no 1 pp 116ndash126 2012

[25] Z B Yang G Aydın V Babich and D R Beil ldquoSupplydisruptions asymmetric information and a backup productionoptionrdquoManagement Science vol 55 no 2 pp 192ndash209 2009

[26] S Huang and C Yang ldquoSupply chain revelation mechanismdesign under asymmetric demand disruption informationrdquoOperations Research andManagement Science vol 23 no 6 pp116ndash127 2014

[27] S Oh and O Ozer ldquoMechanism design for capacity planningunder dynamic evolutions of asymmetric demand forecastsrdquoManagement Science vol 59 no 4 pp 987ndash1007 2013

[28] Q Feng G Lai and L X Lu ldquoDynamic bargaining in a supplychain with asymmetric demand informationrdquo ManagementScience vol 61 no 2 pp 301ndash315 2015

[29] Q Li B Li P Chen and P Hou ldquoDual-channel supply chaindecisions under asymmetric information with a risk-averseretailerrdquo Annals of Operations Research 2015

[30] J Wei K Govindan Y Li and J Zhao ldquoPricing and collectingdecisions in a closed-loop supply chain with symmetric andasymmetric informationrdquo Computers and Operations Researchvol 54 pp 257ndash265 2015

[31] K Inderfurth A Sadrieh and G Voigt ldquoThe impact of infor-mation sharing on supply chain performance under asymmetricinformationrdquo Production amp Operations Management vol 22no 2 pp 410ndash425 2013

[32] R B Myerson ldquoMechanism design by an informed principalrdquoEconometrica vol 51 no 6 pp 1767ndash1797 1983

[33] E Maskin and J Tirole ldquoThe principal-agent relationship withan informed principal the case of private valuesrdquo Econometricavol 58 no 2 pp 379ndash409 1990

[34] E Maskin and J Tirole ldquoThe principal-agent relationship withan informed principal II common valuesrdquo Econometrica vol60 no 1 pp 1ndash42 1992

[35] T Mylovanov and T Troger ldquoInformed-principal problemsin environments with generalized private valuesrdquo TheoreticalEconomics vol 7 no 3 pp 465ndash488 2012

[36] TMylovanov andT Troger ldquoMechanismdesign by an informedprincipal private values with transferable utilityrdquo Review ofEconomic Studies vol 81 no 4 pp 1668ndash1707 2014

[37] C Wagner T Mylovanov and T Troger ldquoInformed-principalproblem with moral hazard risk neutrality and no limitedliabilityrdquo Journal of EconomicTheory vol 159 pp 280ndash289 2015

[38] S Galperti ldquoCommon agency with informed principals menusand signalsrdquo Journal of Economic Theory vol 157 pp 648ndash6672015

[39] H Zhang Y Liu and J Huang ldquoSupply chain coordina-tion contracts under double sided disruptions simultaneouslyrdquoMathematical Problems in Engineering vol 2015 Article ID812043 9 pages 2015

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Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

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Mathematical Problems in Engineering

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Differential EquationsInternational Journal of

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Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

4 Mathematical Problems in Engineering

t

Alowast

Blowast

q

tlowast1

tlowast2

qlowast1qlowast2

Ulowast1

Ulowast2

Vlowast1

Vlowast2

Figure 2 Supply chain contract under full information

low-risk retailer are higher than those for high-risk retailer as119902lowast

1gt 119902lowast

2 119905lowast1gt 119905lowast

2

(2) Under full information the utility of the low-riskretailer is larger than that of high-risk retailer as 119880lowast

1gt 119880lowast

2

and the utility of the manufacturer cooperating with low-riskretailer and with high-risk retailer is equal to the retrainedrevenue as 119881lowast

1= 119881lowast

2= 0

From Proposition 6 it is obvious that under full infor-mation the low-risk retailer should make more orders thanthe high-risk retailer (119902lowast

1gt 119902lowast

2) and meanwhile pay more

(119905lowast1gt 119905lowast

2) Let us show the optimal contract in Figure 2 using

the retailer and themanufacturer indifferent utility curve Forthe low-risk retailer the optimal contract is shown at point119861lowast And for the high-risk retailer the optimal contract is

shown at point 119860lowast The retailerrsquos utility grows more whenits indifferent curve goes into southeast So when the low-risk retailer pretends to be high-risk one using allocation119860

lowastwhich is to say that the low-risk retailerrsquos indifferent utilitycurve goes northwest its utility decreases But if the high-risk retailer pretends to be low-risk one using allocation 119861

lowastits utility increases So as the self-interested participant thehigh-risk retailer tends to pretend to be low-risk one Thenwe can get Proposition 7

Proposition 7 Considering the demand disruption proba-bility as the retailerrsquos individual information the optimalallocation for the low-risk retailer is not separating equilibriumallocation

4 Informed Principal Model

In order to show its true type information the retailer can usethe informed principalmodel to offer an option contract [32]The term ldquooption contract (119905

1 1199021) (1199052 1199022)rdquo comes from the

fact that if the manufacturer accepts the contract the retailermust then exercise its built-in option and choose between(1199051 1199021) and (119905

2 1199022) The retailer will choose the term which

is fit to its own type by probability 100 as 1199011205721| (1199051 1199021) =

1199011205722

| (1199052 1199022) = 1 And the retailer will choose the term

which is not fit to its own probability by probability 0 as1199011205721

| (1199052 1199022) = 119901120572

2| (1199051 1199021) = 0 [33 34] Then the

informed principal model of the supply chain with demanddisruption is built up for the Perfect Bayesian Equilibrium

41 Modeling Firstly the incentive compatible constraint isgivenThe option contract is incentive compatible if the low-risk retailer prefers the contract item (119905

1 1199021) and the high-

risk retailer prefers the contract item (1199052 1199022) So the incentive

compatible constraint should keep the low-risk retailerrsquosutility no less than when it pretends to be a high-risk one

119901119878 (1205721 1199021) minus 1199051ge 119901119878 (120572

1 1199022) minus 1199052 (9)

And the high-risk retailerrsquos utility is no less than when itpretends to be a low-risk one

119901119878 (1205722 1199022) minus 1199052ge 119901119878 (120572

2 1199021) minus 1199051 (10)

Secondly the incentive rationality constraint is givenTheretailer with the individual information is the principal andthemanufacturerwithout individual information is the agentSo the agentrsquos incentive rationality is the same as that (as in (6)and (8)) in full information situation

Thirdly the target function is given The target of theinformed principal model is also to maximize the utility ofthe principal or to say the retailer

So the programs under informed principal of supplychain with demand disruption are 119875

1

119868 and 119875

2

119868 Consider

the following

1198751

119868

max1199051 1199021

1198801= 119901119878 (120572

1 1199021) minus 1199051

st (6) (8) (9) (10)

(11)

1198752

119868

max1199052 1199022

1198802= 119901119878 (120572

2 1199022) minus 1199052

st (6) (8) (9) (10)

(12)

Definition 8 Utility 1198801(1199050

1 1199020

1)1198802(1199050

2 1199020

2) for low-riskhigh-

risk retailer is the low-information-intensity optimum forthat type if (1199050

1 1199020

1)(1199050

2 1199020

2)maximizes low-risk retailerrsquos utility

in the set of incentive compatible constraint and incentiverationality constraint They are (part of) the solution to theprograms 1198751

119868 and 119875

2

119868

For solving the informedprincipalmodel the assumptionof weak monotonic profit is listed as follows

Assumption 9 It is weak monotonic profit when the manu-facturer makes a nonnegative profit if the contractual termsare those of low-risk retailer under symmetric informationand the retailer is a high-risk one 119881

1(119905lowast

2 119902lowast

2) ge 0

From Figure 2 if the contractual term for the low-riskretailer changes from119861

lowast to119860lowast the curve119881lowast1will go southeast

and themanufacturerrsquos profit increases So themanufacturerrsquosprofit is no less than 0 and Assumption 9 is satisfied

Mathematical Problems in Engineering 5

Definition 10 The separating allocation is the allocation (119905119904

1

119902119904

1) for the low-risk retailer and the symmetric information

contractual terms (119905lowast2 119902lowast

2) are for the high-risk retailer where

(119905119904

1 119902119904

1) maximizes the low-risk retailerrsquos utility subject to the

manufacturerrsquos breaking even for the low-risk retailer and tothe high-risk retailer not preferring (119905

119904

1 119902119904

1) to (119905

lowast

2 119902lowast

2) The

program is listed as 1198751119904

1198751

119904

max11990511199021

1198801= 119901119878 (120572

1 1199021) minus 1199051

st (6)

(13)

119901119878 (1205722 119902lowast

2) minus 119905lowast

2ge 119901119878 (120572

2 1199021) minus 1199051 (14)

42 Results

Proposition 11 Under the weak monotonic-profit assump-tion the separating allocation is the low-information-intensityoptimum

Proof (1) The high-risk retailer can get its asymmetric infor-mation utility even under asymmetric information Compar-ing programs 1198752

119865 and 119875

2

119868 it can be found that these two

programs have the same target function but 1198752

119865 has less

constraints so 119880(119905lowast

2 119902lowast

2) ge 119880(119905

0

2 1199020

2) According to Assump-

tion 9 the manufacturer at least breaks even regardless of theretailerrsquos type Hence (119905lowast

2 119902lowast

2) is the separating allocation item

for high-risk retailer(2) Because 119880(119905

lowast

2 119902lowast

2) ge 119880(119905

0

2 1199020

2) constraint (14) is more

restricted than (10) For the low-risk retailer the programs1198751

119868 and 119875

119904

1 have the same target functions but there are

more constraints in the former program so 119880(119905119904

1 119902119904

1) ge

119880(1199050

1 1199020

1) is satisfied And the low-risk retailer at least can

get 119880(119905119904

1 119902119904

1) from the option contract From the incentive

compatibility constraint (14) which can be rewritten as1198802(119905lowast

2 119902lowast

2) ge 1198802(119905119904

1 119902119904

1) it has been guaranteed that the high-

risk retailer will not choose 119905119904

1 119902119904

1 From the program 119875

119904

1

the low-risk retailer can get more utility when the contractualitem is 119905119904

1 119902119904

1 than that of 119905lowast

2 119902lowast

2 So we can get 1199050

1 1199020

1 =

119905119904

1 119902119904

1

Then we can get Proposition 12 (see details in Appen-dix B)

Proposition 12 As for the option contract by informed retailerin supply chain with demand disruption the low-information-intensity allocation satisfies the following

(1) Comparing to the full information situation the low-information-intensity allocation for high-risk retailerand its utility does not distort as 1199050

2= 119905lowast

2 11990202

= 119902lowast

2

1198802(1199050

2 1199020

2) = 119880lowast

2

(2) Comparing to the full information situation the low-information-intensity allocation for low-risk retailerhas upward distortion 119905

0

1gt 119905lowast

1 11990201gt 119902lowast

1 while its utility

has downward distortion 1198801(1199050

1 1199020

1) lt 119880lowast

1

(3) The order quantities in the low-information-intensityallocation for the retailers in different types satisfy

119901119878 (1205722 1199020

1) minus 119888119902

0

1+ 119904119868 (120572

1 1199020

1)

= 119901119878 (1205722 119902lowast

2) minus 119888119902

lowast

2+ 119904119868 (120572

2 119902lowast

2)

(15)

In the informed principal model in order to prevent thehigh-risk retailer from pretending to be a low-risk one thelow-information-intensity allocation items for the low-riskretailer distort The low-risk retailer should order more thanthe optimal order and paymore than the optimal transferringpayment But the utility of the low-risk retailer is less than theoptimal onewhichmeans the low-risk retailer pays some rentto separate from the high-risk retailer So the rent is calledsignaling cost

5 Interim Efficient Allocation

The retailer can deliver its own type information to themanufacturer by low-information-intensity allocation in theinformed principal model But the low-risk retailerrsquos orderand transferring payment distort relative to the full infor-mation situation and it has to pay the signaling cost Thesignaling cost is the part utility which the low-risk retailergets less than the optimal So maybe we can decrease thesignaling cost by increasing the high-risk retailerrsquos utilityand meanwhile increasing the low-risk retailerrsquos utility So inthis part we try to find a separating equilibrium with lesssignaling cost [32ndash34 38] by the interim efficient model

Let us consider the interim efficient model which candecrease the signaling cost by giving the high-risk retailermore than optimal utility Let be the utility which the high-risk retailer gets more than the optimal one And let 119871() betheminimal loss of themanufacturer when it cooperates withthe high-risk retailer119871() can be gotten by the program 119875

2

119898

1198752

119898

minus119871 () = max1199052 1199022

1199052minus 1198881199022+ 119904119868 (120572

2 1199022) (16)

st 119901119878 (1205722 1199022) minus 1199052

ge 119901119878 (1205722 119902lowast

2) minus 119905lowast

2+

(17)

It is easy to find out that when constraint (17) is binding1199022= 119902lowast

2and 119871() =

Definition 13 Utility 1198801(119905119898

1 119902119898

1)1198802(119905119898

2 119902119898

2) for low-risk

high-risk retailer is the interim efficient optimum for thattype if (119905119898

1 119902119898

1)(119905119898

2 119902119898

2) maximizes low-risk retailerrsquos utility

in the set of incentive compatible constraint and themanufac-turerrsquos expected profit And the contract (119905119898

1 119902119898

1) (119905119898

2 119902119898

2) is

the interim efficient allocationThey are (part of) the solutionto the programs 1198751

119898 and 119875

2

119898

6 Mathematical Problems in Engineering

Table 1 Full information contract versus low-information-intensity allocation

Type ofretailer Full information

Lowinformationintensity

Type ofretailer Full information

Lowinformationintensity

1205721= 02

119902lowast

1= 1001 119902

0

1= 1014

1205721= 03

119902lowast

1= 990 119902

0

1= 1002

119905lowast

1= 1862 119905

0

1= 1881 119905

lowast

1= 1840 119905

0

1= 1824

119880lowast

1= 5340 119880

0

1= 5338 119880

lowast

1= 5278 119880

0

1= 5192

1205722= 04

119902lowast

2= 978 119902

0

1= 978

1205722= 05

119902lowast

2= 967 119902

0

2= 967

119905lowast

2= 1819 119905

0

2= 1819 119905

lowast

2= 1799 119905

0

2= 1799

119880lowast

2= 5217 119880

0

2= 5217 119880

lowast

2= 5159 119880

0

2= 5159

1198751

119898

max11990511199021

1198801= 119901119878 (120572

1 1199021) minus 1199051 (18)

st 120582 (119905119898

1minus 119888119902119898

1+ 119904119868 (120572

1 119902119898

1)) minus (1 minus 120582) ge 0 (19)

119901119878 (1205722 119902lowast

2) minus 119905lowast

2+ ge 119901119878 (120572

2 119902119898

1) minus 119905119898

1 (20)

The high-risk retailer can get the rent above theoptimum so themanufacturer will lose some utilityThen theincentive rational constraint (see (19)) for themanufacturer isthe expected profit rather than the individual profit Equation(20) is the incentive compatible constraint for the high-riskretailer

From programs 1198751119898 and 119875

2

119898 we can get Propositions 14

and 15 (see Proof in Appendix C)

Proposition 14 The interim efficient allocation by informedprincipal for the supply chain with demand disruption satisfiesthe following

(1) Comparing to the full information situation the orderquantity for the high-risk retailer does not distort 119902119898

2=

119902lowast

2

(2) Comparing to the full information situation the orderquantity for the low-risk retailer satisfies (21) anddistorts upwards 119902119898

1gt 119902lowast

1 Consider

1199041198681015840(1205721 119902119898

1) + 119901119878

1015840(1205721 119902119898

1) minus 119888

=1 minus 120582

120582[1199011198781015840(1205722 119902119898

1) minus 119901119878

1015840(1205721 119902119898

1)]

(21)

(3) The transferring payment for the low-risk retailer meets(22) and the transferring payment for the high-riskretailer meets (23)

119905119898

1= (1 minus 120582)

sdot [119901119878 (1205722 119902119898

1) minus 119901119878 (120572

2 119902lowast

2) + 119888119902

lowast

2minus 119904119868 (120572

2 119902lowast

2)]

+ 120582 [119888119902119898

1minus 119904119868 (120572

1 119902119898

1)]

(22)

119905119898

2= (1 minus 120582) [119888119902

lowast

2minus 119904119868 (120572

2 119902lowast

2)] + 120582119901119878 (120572

2 119902lowast

2)

+ 120582 [119888119902119898

1minus 119904119868 (120572

1 119902119898

1) minus 119901119878 (120572

2 119902119898

1)]

(23)

Proposition 15 There is a threshold 1205820

= (1199011198781015840(1205722 1199020

1) minus

1199011198781015840(1205721 1199020

1))(119904119868

1015840(1205721 1199020

1) + 119901119878

1015840(1205722 1199020

1) minus 119888) for the low-risk

retailer probability If and only if the probability of low-riskretailer is no more than the threshold 120582 le 120582

0 the low-infor-mation-intensity allocation is also interim efficient and isthe unique Perfect Bayesian Equilibrium If the probabilityof low-risk retailer is more than the threshold 120582 gt 120582

0 thelow-information-intensity allocation is not the unique PerfectBayesian Equilibrium

If and only if the probability of low-risk retailer is smallenough (120582 le 120582

0) the low-information-intensity allocation isthe unique Perfect Bayesian EquilibriumAnd the retailer candeliver its own type information by the separating contractBut if the low-risk retailer is more than that (120582 gt 120582

0) the low-information-intensity allocation is not the unique PerfectBayesian Equilibrium and the interim efficient allocation canimprove the distortion situation to a certain degree So for thesupply chain with demand disruption the optimal allocationcannot be reached if the informed principal provides aseparating contract

6 Numerical Example

This section gives some numerical examples to inspect thesupply chain contract with demand disruption by informedprincipal We let the sales price 119901 = 10 the unit manu-facturing cost is 119888 = 2 and the salvage product value is119904 = 05 The demand distribution function before disruptionis 119865(119910) = 119910

2125 and the demand distribution function after

disruption is 119866(119909) = 1199092100

61 Main Results From the example listed above the fullinformation contracts and low-information-intensity alloca-tion are given in Table 1

Table 1 shows that for the full information contract theorder quantity for low-risk retailer is larger than that for thehigh-risk retailer 119902lowast

1gt 119902lowast

2 and also the transferring payment

119905lowast

1gt 119905lowast

2 which is performed in Proposition 6 In the informed

principal model the order quantity for the low-risk retailerdistorts in order to show its type (when 120572

1= 02 versus 120572

2=

04 119902lowast1

= 1001 lt 119902sb1

= 1014 and when 1205721= 03 versus

1205722= 05 119902lowast

1= 990 lt 119902

0

1= 1002) Meanwhile we notice that

the utility obtained by the low-risk retailer in the informed

Mathematical Problems in Engineering 7

0 10

5

10

15

q1

02 04 06 08

120582

Figure 3 The order quantity in interim efficient allocation for low-risk retailer changes by its probability

0

1

101112

1314

150

5

10

15

20

25

30

0204

0608

tm 1

q1 120582

Figure 4 The transferring payment in interim efficient allocationfor low-risk retailer changes by the order quantity and its probability

principal model is lower than that in full information model(when 120572

1= 02 versus 120572

2= 04 119880lowast

1= 5340 gt 119880

0

1= 5338

and when 1205721= 03 versus 120572

2= 05 119905lowast

1= 1840 gt 119905

0

1= 1824)

This implies that it costs the low-risk retailer something tocertify its typeThese results are the samewith Proposition 12

Considering the situation of 1205721= 02 versus 120572

2= 04 we

obtain 1205820= 019 The order quantity for the low-risk retailer

119902119898

1changes with the proportion it has 120582 in interim efficient

allocation As Figure 3 shows if and only if 120582 varies withincertain area (120582 le 120582

0) we obtain 119902119898

1gt 119902lowast

1 as in the results in

Proposition 14The transferring payment in the interim efficient allo-

cation is depicted in Figure 4 (for the low-risk retailer)and Figure 5 (for the high-risk retailer) The transferringpayments are changing by the order quantity for the low-risk retailer in the interim efficient allocation and the low-risk retailerrsquos probability From Figure 4 when the low-riskretailerrsquos probability 120582 is less enough and the order quantityfor the low-risk retailer 119902119898

1is high enough the transferring

0

05

1

10 11 12 13 14 15

20

25

30

35

40

45

tm 2

q1

120582

Figure 5 The transferring payment in interim efficient allocationfor high-risk retailer changes by the low-risk retailerrsquos order quantityand probability

payment 1199051198981increases by the low-risk retailerrsquos probability 120582

but decreases by the order quantity for the low-risk retailer119902119898

1(bottom left in Figure 4) But mostly the transferring

payment 1199051198981increases by the low-risk retailerrsquos probability 120582

but decreases by the order quantity for the low-risk retailer 1199021198981

(red and orange part in Figure 4) Normally the transferringpayment should increase by the order quantity which meansldquoget more pay morerdquo From this view the low-risk retailerrsquosprobability 120582 should not be very small and the order quantityfor the low-risk retailer 119902

119898

1should not be close to the

maximum value 15From Figure 5 it is obviously seen that the transferring

payment for high-risk retailer in the interim efficient alloca-tion 119905119898

2increases by the order quantity for the low-risk retailer

119902119898

1but decreases by the low-risk retailerrsquos probability 120582

62 Sensitivity Analyses The sensitivity analyses for the low-information-intensity allocation versus the unit manufactur-ing cost 119888 are shown in Table 2 It shows the outcomes whenthe unit manufacturing cost 119888 changes by +25 and minus25once at a time and keeping remaining parameters The orderquantities decrease by the unit manufacturing cost but theinfluence for the high-risk retailerrsquos order quantity is largerthan that for the low-risk retailer (minus069 versus minus317 and+217 versus +307) The transferring payments increaseby the unit manufacturing cost More interesting the utilitiesdecrease by the unit manufacturing cost and the low-riskretailer receives a greater impact (minus940 versus minus922 and+955 versus +953) So from this point of view the low-risk retailer should encourage themanufacturer to reduce thecost more

Figure 6 shows the sensitivity analyses for the order quan-tities versus the unit manufacturing cost in which the orderquantities include the order quantity for the low-risk retailerin low-information-intensity allocation 119902

0

1 the order quantity

8 Mathematical Problems in Engineering

Table 2 Sensitivity analyses for low-information-intensity allocation versus the unit manufacturing cost

Changes of 119888 The low-information allocation for low-riskretailer (120572

1= 02)

The low-information allocationfor high-risk retailer (120572

2= 04)

Value Change rate Value Change rate Value Change rate

119888 = 25 +251199020

1= 1007 minus069 119902

0

2= 947 minus317

1199050

1= 2375 +2626 119905

0

2= 2243 +2331

1198800

1= 4836 minus940 119880

0

2= 4736 minus922

119888 = 2 Baseline1199020

1= 1014 mdash 119902

sb2

= 978 mdash1199050

1= 1881 mdash 119905

sb2

= 1819 mdash1198800

1= 5338 mdash 119880

sb2

= 5217 mdash

119888 = 15 minus251199020

1= 1036 +217 119902

0

2= 1008 +307

1199050

1= 1398 minus2568 119905

0

2= 1362 minus2512

1198800

1= 5848 +955 119880

0

2= 5714 +953

108

106

104

102

10

98

96

94

92

C

14 16 18 2 22 24 26 28 3

qsb1

qlowast1q2

Figure 6 The order quantities change by the unit manufacturingcost

for the low-risk retailer in full information contract 119902lowast1 and

the order quantity for the high-risk retailer in full informationcontract which is equal to that in low-information-intensityallocation 119902

2 Both the order quantity for the low-risk retailer

in full information and the order quantity for the high-riskretailer decrease by the unit manufacturing cost and theychange in the same level basically But the change of the orderquantity for the low-risk retailer in low information intensitydepends on different values of the unit manufacturing costWhen the unit manufacturing cost 119888 = 275 the orderquantity is minimum and when 119888 = 175 or 119888 = 15 the orderquantity is the maximum

Figure 7 shows the sensitivity analyses for the transferringpayment versus the unit manufacturing cost in which thetransferring payments include the transferring payment forthe low-risk retailer in low-information-intensity allocation1199050

1 the transferring payment for the low-risk retailer in full

information contract 119905lowast

1 and the transferring payment for

26

24

22

20

18

16

14

C

14 16 18 2 22 24 26 28 3

tsb1

tlowast1t2

Figure 7 The transferring payments change by the unit manufac-turing cost

the high-risk retailer in full information contract which isequal to that in low-information-intensity allocation 119905

2 All

the transferring payments of the retailers increase by the unitmanufacturing cost The transferring payments for the low-risk retailer and the high-risk retailer in full informationcontract decrease in the same level

Figure 8 shows the sensitivity analyses for the retailersrsquoutilities versus the unit manufacturing cost in which theutilities include the utility for the low-risk retailer in low-information-intensity allocation 119880

0

1 the transferring pay-

ment for the low-risk retailer in full information contract119880lowast

1 and the transferring payment for the high-risk retailer

in full information contract which is equal to that in low-information-intensity allocation 119880

2 All the retailersrsquo utilities

decrease by the unit manufacturing costFigure 9 shows the sensitivity analyses for the low-risk

retailerrsquos signaling cost versus the unit manufacturing cost

Mathematical Problems in Engineering 9

44

46

48

50

52

54

56

58

60

C

14 16 18 2 22 24 26 28

Usb1

Ulowast1

U2

Figure 8 The retailerrsquos utilities change by the unit manufacturingcost

0

005

01

015

02

025

C

14 16 18 2 22 24 26 28

ΔU1

Figure 9The signaling cost changes by the saved unit manufactur-ing cost

The signaling cost of the low-risk retailer is the utility earnedin the low-information-intensity allocation which is sepa-rating contract less than that in full information contractWhen the unit manufacturing cost 119888 = 2 the signaling costis the least while when the saved unit manufacturing cost119888 = 175 the signaling cost is the most We do not find theinternal relationship between the signaling cost and the unitmanufacturing cost As the signaling cost is so important toput the contracts into practice investigation on the signalingcost is the research direction for us

7 Conclusion

In the recent years supply chain with demand disruptionhas become more and more attractive to both business and

academic research There is a lot of research available on thetopic of supply chain disruption contract In the real supplychain practice demand disruption is totally different fromthe stochastic demand So in our research both the demanduncertainty and the demand disruption are consideredTaking the demand disruption probability as the asymmetricinformation an informed principal model is explored tomake the contract for supply chain with demand disruptionAnd the conclusions we obtain are shown as follows

(1) The contracts for low-risk retailer and high-riskretailer in full information are provided and it isfound out that the high-risk retailer has the incentiveto pretend to be the low-risk retailer This result issimilar to the other informed principal model [3435] but totally different from the normal principalagent model [24 26] In the normal principal agentmodel the high efficient agent (low-risk retailer in oursetting) tries to pretend to be low efficient agent (high-risk retailer) to get more That is why the informedprincipal model is established

(2) The low-information-intensity allocation which is theresult of the informed principal model and also is theseparating contract shows that the order quantity andthe transferring payment for the low-risk retailer inlow-information-intensity allocation distort upwardsbut that of high-risk retailer does not distort Inorder to avoid imitation the imitatorsrsquo efficiencyshould be always distorted But in normal principalagent model the high-risk retailerrsquos quantity distortsdownwards [24 26] on the contrary the low-riskretailerrsquos quantity in informed principal model dis-torts upwards

(3) In order to reduce the signaling cost which the low-risk retailer pays the interim efficient model is intro-duced which ends up with the order quantity andtransferring payment distorted upward again but lessthan that of low information intensity Comparingto [38] focusing on analyzing the signaling costmore attention is paid to showing the interim efficientmodel to get a solution which can increase efficiency

From the managerial aspect retailers should try toincrease their antirisk capability to enhance competitivepower And when the retailer has lower demand disruptionprobability than the others it can show its style to the supplierthrough the separating contract For example the retailer cansay ldquobecause I am the low-risk one I can cooperate with thesupplier with this contract while the others who are high-riskones can only use the other contractrdquo But in order to be sep-arated from the others the low-risk retailer has to pay somesignaling cost and the quantity has to be distorted upwards

Above all there are some limitations of the research Wewill continue our research in the following directions (1)The signaling cost is a key factor impacting the separatingcontract which should be paid more attention (2) We onlyconsider the demand disruption but disruptions happen atnot only demand side but also supply side Sowewill continue

10 Mathematical Problems in Engineering

our research with demand disruption together with supplydisruption [39]

Appendix

A Proof of Proposition 6

For the two types of retailers we can obtain the optimalallocation if and only if the whole margin utility of the supplychain is the same with the margin cost So 119902

lowast

1and 119902

lowast

2satisfy

the following first-order conditions

1199011198781015840(1205721 1199021) + 1199041198681015840(1205721 1199021) = 119888 (A1)

1199011198781015840(1205722 1199022) + 1199041198681015840(1205722 1199022) = 119888 (A2)

And the optimal transferring payments are

119905lowast

1= 119888119902lowast

1minus 119904119868 (120572

1 119902lowast

1)

119905lowast

2= 119888119902lowast

2minus 119904119868 (120572

2 119902lowast

2)

(A3)

From (A1) and (A2) we obtain

1205791119866 (119902lowast

1) + (1 minus 120579

1) 119865 (119902lowast

1)

= 1205792119866 (119902lowast

2) + (1 minus 120579

2) 119865 (119902lowast

2) =

(119901 minus 119888)

(119901 minus 119904)

(A4)

In addition 119866(119902lowast

1) ge 119865(119902

lowast

1) 1205721

lt 1205722 so we know that

1205721119866(119902lowast

1) + (1minus120572

1)119865(119902lowast

1) lt 1205722119866(119902lowast

1) + (1minus120572

2)119865(119902lowast

1) and from

(A4) we obtain 1205722119866(119902lowast

2) + (1 minus 120572

2)119865(119902lowast

2) lt 1205722119866(119902lowast

1) + (1 minus

1205722)119865(119902lowast

1) and thus 119902lowast

2lt 119902lowast

1 From (A3) 119905lowast

1minus119905lowast

2= 119888(119902lowast

1minus119902lowast

2)minus

119904(119868(1205721 119902lowast

1) minus 119868(120572

2 119902lowast

2)) 119905lowast1minus 119905lowast

2=119902lowast

1=119902lowast

2

(1205721minus 1205722)119904(int119902

0119865(119910)119889119910 minus

int119902

0119866(119909)119889119909) gt 0 and 120597119905

lowast

1120597119902lowast

1gt 0 so 119905

lowast

1minus 119905lowast

2gt 0 and

thus we obtain 119905lowast

1gt 119905lowast

2 119881lowast

1= 119881lowast

2= 0 Furthermore we get

119880lowast

1gt 119880lowast

2from 120597119880120597120579 lt 0 and 120597119880120597119902 gt 0 Finally we have

Proposition 6

B Proof of Proposition 12

We denote 1205831and 120583

2as the Lagrange multipliers of (6) and

(14) and find the derivate of 1199051and 1199021 and thus we obtain

120597119880 (1205721 1199021)

1205971199051

= minus1 + 1205831+ 1205832= 0

120597119880 (1205721 1199021)

1205971199021

= 1199011198781015840(1205721 1199021) minus 1205831119888 + 12058311199041198681015840(1205721 1199021)

minus 12058321199011198781015840(1205722 1199021) = 0

(B1)

By rewriting (B1) we get the following formulas

1199041198681015840(1205721 119902119904

1) + 119901119878

1015840(1205721 119902119904

1) minus 119888

=(1 minus 120583

1)

1205831

[1199011198781015840(1205722 119902119904

1) minus 119901119878

1015840(1205721 119902119904

1)]

(B2)

1205831=

1199011198781015840(1205722 119902119904

1) minus 119901119878

1015840(1205721 119902119904

1)

1199041198681015840 (1205721 119902119904

1) + 1199011198781015840 (120572

2 119902119904

1) minus 119888

(B3)

1205832=

1199041198681015840(1205721 119902119904

1) + 119901119878

1015840(1205721 119902119904

1) minus 119888

1199041198681015840 (1205721 119902119904

1) + 1199011198781015840 (120572

2 119902119904

1) minus 119888

(B4)

Because the numerator of 1205831is negative so 120583

1is positive

and the denominator of 1205831is negative because 120583

2is positive

(when 1205832= 0 it cannot be separated from the former one)

the numerator of 1205832is negative nevertheless 1199011198781015840(120572

1 119902lowast

1) minus

119888 + 1199041198681015840(1205721 119902lowast

1) = 0 and thus 1199011198781015840(120572

1 119902119904

1) minus 119888 + 119904119868

1015840(1205721 119902119904

1) lt

1199011198781015840(1205721 119902lowast

1)minus 119888+ 119904119868

1015840(1205721 119902lowast

1) in addition 1205972119880120597119902

2lt 0 so 119902

119904

1gt

119902lowast

1 From 120583

1gt 0 and 120583

2gt 0 we deduce that (6) and (14)

are tight Furthermore the programs 1198751119865 and 119875

1

119898 have the

same objective function but the constraint of 1198751119898 is more

tight so there is distortion of the utility obtained from 1198751

119898 in

other words 1198801(119905119904

1 119902119904

1) lt 119880lowast

1 From Proposition 11 we obtain

1198802(1199050

2 1199020

2) = 119880lowast

2 So we have Proposition 12

C Proof of Propositions 14 and 15

We denote 120573 and 120574 as the Lagrange multipliers of (19) and(20) and find the derivate of 119905

1 1199021 and and thus we obtain

120597119871 (1199051 1199021 )

1205971199051

= minus1 + 120573120582 + 120574 = 0 (C1)

120597119871 (1199051 1199021 )

1205971199021

= 1199011198781015840(1205721 1199021) + 120573120582 [minus119888 + 119904119868

1015840(1205721 1199021)]

minus 1205741199011198781015840(1205721 1199021) = 0

(C2)

120597119871 (1199051 1199021 )

120597= minus120573 (1 minus 120582) + 120574 = 0 (C3)

To solve (C1)ndash(C3) we obtain 120573 = 1 120574 = 1 minus 120582 Andconsidering (C2) we get

1199041198681015840(1205721 119902119898

1) + 119901119878

1015840(1205721 119902119898

1) minus 119888

=1 minus 120582

120582[1199011198781015840(1205722 119902119898

1) minus 119901119878

1015840(1205721 119902119898

1)]

(C4)

Because (19) and (20) are tight we obtain the followingresults

119905119898

1= (1 minus 120582)

sdot [119901119878 (1205722 119902119898

1) minus 119901119878 (120572

2 119902lowast

2) + 119888119902

lowast

2minus 119904119868 (120572

2 119902lowast

2)]

+ 120582 [119888119902119898

1minus 119904119868 (120572

1 119902119898

1)]

119905119898

2= (1 minus 120582) [119888119902

lowast

2minus 119904119868 (120572

2 119902lowast

2)] + 120582119901119878 (120572

2 119902lowast

2)

+ 120582 [119888119902119898

1minus 119904119868 (120572

1 119902119898

1) minus 119901119878 (120572

2 119902119898

1)]

(C5)

Mathematical Problems in Engineering 11

To compare (A1) with (C4) we know 119902119898

1gt 119902lowast

1because

12059721198801205971199022

lt 0 and 1199011198781015840(1205722 119902) minus 119901119878

1015840(1205721 119902) lt 0 and the dis-

tortion level is changing with 120572 Finally we have Proposi-tion 14

To compare (C4) with (B2) and designate 1205820

= 120582 =

(1199011198781015840(1205722 1199020

1) minus 119901119878

1015840(1205721 1199020

1))(119904119868

1015840(1205721 1199020

1) + 119901119878

1015840(1205722 1199020

1) minus 119888) we

have the following conclusions

(1) When 120582 = 1205820 1199021198981

= 119902119904

1= 1199020

1 we obtain the same

contracts with the low-information-intensity situa-tion

(2) When 120582 lt 1205820 1199021198981

lt 119902119904

1= 1199020

1 we know there is more

distortion than the low-information-intensity alloca-tion and it improves anything

(3) When 120582 gt 1205820 1199021198981

lt 119902119904

1= 1199020

1 the interim efficient

allocation can improve the low-information-intensityallocation So we have Proposition 15

VariablesNotations

119888 Unit manufacturing cost1205721 Disruption probability of low-risk retailer

119910 Market demand without disruption119865(sdot) Distribution function of demand without

disruption119904 Unit salvage value119860 Market scale without disruption119878(120572119894 119902119894) Expected sales

119868(120572119894 119902119894) Expected unsold quantity

119880119894 Utility of the retailer

120582 Probability of low-risk retailer1205722 Disruption probability of high-risk retailer

119909 Market demand with disruption119866(sdot) Distribution function of demand with

disruption119901 Retail price119863 Market scale with disruption119881119894 Utility of the manufacturer

Control Variables

119902119894 Order quantity

119905119894 Transferring payment

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported in part by a project supported bythe Humanities and Social Sciences Project of the EducationMinistry (no 14YJC630187) Hebei Social Sciences Project(no HB15GL058) and the Fundamental Research Funds forthe Central Universities (no 2016MS123)

References

[1] J Barnett F Begen S Howes et al ldquoConsumersrsquo confidencereflections and response strategies following the horsemeatincidentrdquo Food Control vol 59 pp 721ndash730 2016

[2] A J Schmitt and L V Snyder ldquoInfinite-horizon models forinventory control under yield uncertainty and disruptionsrdquoComputers amp Operations Research vol 39 no 4 pp 850ndash8622012

[3] L V Snyder Z Atan P Peng et al ldquoORMS models for supplychain disruptions a reviewrdquo IIE Transactions vol 48 no 2 pp89ndash109 2016

[4] X Qi J F Bard and G Yu ldquoSupply chain coordination withdemand disruptionsrdquo Omega vol 32 no 4 pp 301ndash312 2004

[5] T Xiao G Yu Z Sheng and Y Xia ldquoCoordination of a supplychain with one-manufacturer and two-retailers under demandpromotion and disruption management decisionsrdquo Annals ofOperations Research vol 135 pp 87ndash109 2005

[6] F Hu C-C Lim Z Lu and X Sun ldquoCoordination in a single-retailer two-supplier supply chain under random demand andrandom supply with disruptionrdquo Discrete Dynamics in Natureand Society vol 2013 Article ID 484062 12 pages 2013

[7] J Li X Liu J Wu and F Yang ldquoCoordination of supplychain with a dominant retailer under demand disruptionsrdquoMathematical Problems in Engineering vol 2014 Article ID854681 10 pages 2014

[8] K Chen and T Xiao ldquoDemand disruption and coordination ofthe supply chain with a dominant retailerrdquo European Journal ofOperational Research vol 197 no 1 pp 225ndash234 2009

[9] J Li and F T S Chan ldquoThe impact of collaborative transporta-tion management on demand disruption of manufacturingsupply chainsrdquo International Journal of Production Research vol50 no 19 pp 5635ndash5650 2012

[10] Y Zheng T Shu S Wang S Chen K K Lai and L GanldquoDemand disruption and coordination of supply chain via effortand revenue sharingrdquo Applied Economics vol 47 no 54 pp5886ndash5901 2015

[11] Q Pang Y Hou and Y Lv ldquoCoordinating three-level supplychain under disruptions using revenue-sharing contract witheffort dependent demandrdquoMathematical Problems in Engineer-ing vol 2016 Article ID 9167864 10 pages 2016

[12] M Gumus S Ray and H Gurnani ldquoSupply-side story risksguarantees competition and information asymmetryrdquo Man-agement Science vol 58 no 9 pp 1694ndash1714 2012

[13] K B Hendricks and V R Singhal ldquoAn empirical analysisof the effect of supply chain disruptions on long-run stockprice performance and equity risk of the firmrdquo Production ampOperations Management vol 14 no 1 pp 35ndash52 2005

[14] W Schimidt and A Raman ldquoWhen supply-chain disruptionsmatterrdquo Harvard Business School Working Paper 13-006 2012

[15] N Bunkley Piecing together a supply chain The New YorkTimes 2013 httpwwwnytimescom20110513businessglobal13autohtml r=0

[16] S Sarkar and S Kumar ldquoA behavioral experiment on inventorymanagement with supply chain disruptionrdquo International Jour-nal of Production Economics vol 169 pp 169ndash178 2015

[17] G P Cachon ldquoSupply chain coordination with contractsrdquoHandbooks in Operations Research amp Management Science vol11 pp 227ndash339 2003

[18] C J Corbett and X De Groote ldquoA supplierrsquos optimal quantitydiscount policy under asymmetric informationrdquo ManagementScience vol 46 no 3 pp 444ndash450 2000

12 Mathematical Problems in Engineering

[19] C J Corbett ldquoStochastic inventory systems in a supply chainwith asymmetric information cycle stocks safety stocks andconsignment stockrdquoOperations Research vol 49 no 4 pp 487ndash500 2001

[20] A Matopoulos M Vlachopoulou V Manthou and B ManosldquoA conceptual framework for supply chain collaboration empir-ical evidence from the agri-food industryrdquo Supply Chain Man-agement vol 12 no 3 pp 177ndash186 2007

[21] A Y Ha and S Tong ldquoContracting and information sharingunder supply chain competitionrdquoManagement Science vol 54no 4 pp 701ndash715 2008

[22] Y-W Zhou ldquoA comparison of different quantity discountpricing policies in a two-echelon channel with stochastic andasymmetric demand informationrdquo European Journal of Opera-tional Research vol 181 no 2 pp 686ndash703 2007

[23] XGan S P Sethi and J Zhou ldquoCommitment-penalty contractsin drop-shipping supply chains with asymmetric demand infor-mationrdquo European Journal of Operational Research vol 204 no3 pp 449ndash462 2010

[24] D Lei J Li and Z Liu ldquoSupply chain contracts under demandand cost disruptions with asymmetric informationrdquo Interna-tional Journal of Production Economics vol 139 no 1 pp 116ndash126 2012

[25] Z B Yang G Aydın V Babich and D R Beil ldquoSupplydisruptions asymmetric information and a backup productionoptionrdquoManagement Science vol 55 no 2 pp 192ndash209 2009

[26] S Huang and C Yang ldquoSupply chain revelation mechanismdesign under asymmetric demand disruption informationrdquoOperations Research andManagement Science vol 23 no 6 pp116ndash127 2014

[27] S Oh and O Ozer ldquoMechanism design for capacity planningunder dynamic evolutions of asymmetric demand forecastsrdquoManagement Science vol 59 no 4 pp 987ndash1007 2013

[28] Q Feng G Lai and L X Lu ldquoDynamic bargaining in a supplychain with asymmetric demand informationrdquo ManagementScience vol 61 no 2 pp 301ndash315 2015

[29] Q Li B Li P Chen and P Hou ldquoDual-channel supply chaindecisions under asymmetric information with a risk-averseretailerrdquo Annals of Operations Research 2015

[30] J Wei K Govindan Y Li and J Zhao ldquoPricing and collectingdecisions in a closed-loop supply chain with symmetric andasymmetric informationrdquo Computers and Operations Researchvol 54 pp 257ndash265 2015

[31] K Inderfurth A Sadrieh and G Voigt ldquoThe impact of infor-mation sharing on supply chain performance under asymmetricinformationrdquo Production amp Operations Management vol 22no 2 pp 410ndash425 2013

[32] R B Myerson ldquoMechanism design by an informed principalrdquoEconometrica vol 51 no 6 pp 1767ndash1797 1983

[33] E Maskin and J Tirole ldquoThe principal-agent relationship withan informed principal the case of private valuesrdquo Econometricavol 58 no 2 pp 379ndash409 1990

[34] E Maskin and J Tirole ldquoThe principal-agent relationship withan informed principal II common valuesrdquo Econometrica vol60 no 1 pp 1ndash42 1992

[35] T Mylovanov and T Troger ldquoInformed-principal problemsin environments with generalized private valuesrdquo TheoreticalEconomics vol 7 no 3 pp 465ndash488 2012

[36] TMylovanov andT Troger ldquoMechanismdesign by an informedprincipal private values with transferable utilityrdquo Review ofEconomic Studies vol 81 no 4 pp 1668ndash1707 2014

[37] C Wagner T Mylovanov and T Troger ldquoInformed-principalproblem with moral hazard risk neutrality and no limitedliabilityrdquo Journal of EconomicTheory vol 159 pp 280ndash289 2015

[38] S Galperti ldquoCommon agency with informed principals menusand signalsrdquo Journal of Economic Theory vol 157 pp 648ndash6672015

[39] H Zhang Y Liu and J Huang ldquoSupply chain coordina-tion contracts under double sided disruptions simultaneouslyrdquoMathematical Problems in Engineering vol 2015 Article ID812043 9 pages 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 5

Definition 10 The separating allocation is the allocation (119905119904

1

119902119904

1) for the low-risk retailer and the symmetric information

contractual terms (119905lowast2 119902lowast

2) are for the high-risk retailer where

(119905119904

1 119902119904

1) maximizes the low-risk retailerrsquos utility subject to the

manufacturerrsquos breaking even for the low-risk retailer and tothe high-risk retailer not preferring (119905

119904

1 119902119904

1) to (119905

lowast

2 119902lowast

2) The

program is listed as 1198751119904

1198751

119904

max11990511199021

1198801= 119901119878 (120572

1 1199021) minus 1199051

st (6)

(13)

119901119878 (1205722 119902lowast

2) minus 119905lowast

2ge 119901119878 (120572

2 1199021) minus 1199051 (14)

42 Results

Proposition 11 Under the weak monotonic-profit assump-tion the separating allocation is the low-information-intensityoptimum

Proof (1) The high-risk retailer can get its asymmetric infor-mation utility even under asymmetric information Compar-ing programs 1198752

119865 and 119875

2

119868 it can be found that these two

programs have the same target function but 1198752

119865 has less

constraints so 119880(119905lowast

2 119902lowast

2) ge 119880(119905

0

2 1199020

2) According to Assump-

tion 9 the manufacturer at least breaks even regardless of theretailerrsquos type Hence (119905lowast

2 119902lowast

2) is the separating allocation item

for high-risk retailer(2) Because 119880(119905

lowast

2 119902lowast

2) ge 119880(119905

0

2 1199020

2) constraint (14) is more

restricted than (10) For the low-risk retailer the programs1198751

119868 and 119875

119904

1 have the same target functions but there are

more constraints in the former program so 119880(119905119904

1 119902119904

1) ge

119880(1199050

1 1199020

1) is satisfied And the low-risk retailer at least can

get 119880(119905119904

1 119902119904

1) from the option contract From the incentive

compatibility constraint (14) which can be rewritten as1198802(119905lowast

2 119902lowast

2) ge 1198802(119905119904

1 119902119904

1) it has been guaranteed that the high-

risk retailer will not choose 119905119904

1 119902119904

1 From the program 119875

119904

1

the low-risk retailer can get more utility when the contractualitem is 119905119904

1 119902119904

1 than that of 119905lowast

2 119902lowast

2 So we can get 1199050

1 1199020

1 =

119905119904

1 119902119904

1

Then we can get Proposition 12 (see details in Appen-dix B)

Proposition 12 As for the option contract by informed retailerin supply chain with demand disruption the low-information-intensity allocation satisfies the following

(1) Comparing to the full information situation the low-information-intensity allocation for high-risk retailerand its utility does not distort as 1199050

2= 119905lowast

2 11990202

= 119902lowast

2

1198802(1199050

2 1199020

2) = 119880lowast

2

(2) Comparing to the full information situation the low-information-intensity allocation for low-risk retailerhas upward distortion 119905

0

1gt 119905lowast

1 11990201gt 119902lowast

1 while its utility

has downward distortion 1198801(1199050

1 1199020

1) lt 119880lowast

1

(3) The order quantities in the low-information-intensityallocation for the retailers in different types satisfy

119901119878 (1205722 1199020

1) minus 119888119902

0

1+ 119904119868 (120572

1 1199020

1)

= 119901119878 (1205722 119902lowast

2) minus 119888119902

lowast

2+ 119904119868 (120572

2 119902lowast

2)

(15)

In the informed principal model in order to prevent thehigh-risk retailer from pretending to be a low-risk one thelow-information-intensity allocation items for the low-riskretailer distort The low-risk retailer should order more thanthe optimal order and paymore than the optimal transferringpayment But the utility of the low-risk retailer is less than theoptimal onewhichmeans the low-risk retailer pays some rentto separate from the high-risk retailer So the rent is calledsignaling cost

5 Interim Efficient Allocation

The retailer can deliver its own type information to themanufacturer by low-information-intensity allocation in theinformed principal model But the low-risk retailerrsquos orderand transferring payment distort relative to the full infor-mation situation and it has to pay the signaling cost Thesignaling cost is the part utility which the low-risk retailergets less than the optimal So maybe we can decrease thesignaling cost by increasing the high-risk retailerrsquos utilityand meanwhile increasing the low-risk retailerrsquos utility So inthis part we try to find a separating equilibrium with lesssignaling cost [32ndash34 38] by the interim efficient model

Let us consider the interim efficient model which candecrease the signaling cost by giving the high-risk retailermore than optimal utility Let be the utility which the high-risk retailer gets more than the optimal one And let 119871() betheminimal loss of themanufacturer when it cooperates withthe high-risk retailer119871() can be gotten by the program 119875

2

119898

1198752

119898

minus119871 () = max1199052 1199022

1199052minus 1198881199022+ 119904119868 (120572

2 1199022) (16)

st 119901119878 (1205722 1199022) minus 1199052

ge 119901119878 (1205722 119902lowast

2) minus 119905lowast

2+

(17)

It is easy to find out that when constraint (17) is binding1199022= 119902lowast

2and 119871() =

Definition 13 Utility 1198801(119905119898

1 119902119898

1)1198802(119905119898

2 119902119898

2) for low-risk

high-risk retailer is the interim efficient optimum for thattype if (119905119898

1 119902119898

1)(119905119898

2 119902119898

2) maximizes low-risk retailerrsquos utility

in the set of incentive compatible constraint and themanufac-turerrsquos expected profit And the contract (119905119898

1 119902119898

1) (119905119898

2 119902119898

2) is

the interim efficient allocationThey are (part of) the solutionto the programs 1198751

119898 and 119875

2

119898

6 Mathematical Problems in Engineering

Table 1 Full information contract versus low-information-intensity allocation

Type ofretailer Full information

Lowinformationintensity

Type ofretailer Full information

Lowinformationintensity

1205721= 02

119902lowast

1= 1001 119902

0

1= 1014

1205721= 03

119902lowast

1= 990 119902

0

1= 1002

119905lowast

1= 1862 119905

0

1= 1881 119905

lowast

1= 1840 119905

0

1= 1824

119880lowast

1= 5340 119880

0

1= 5338 119880

lowast

1= 5278 119880

0

1= 5192

1205722= 04

119902lowast

2= 978 119902

0

1= 978

1205722= 05

119902lowast

2= 967 119902

0

2= 967

119905lowast

2= 1819 119905

0

2= 1819 119905

lowast

2= 1799 119905

0

2= 1799

119880lowast

2= 5217 119880

0

2= 5217 119880

lowast

2= 5159 119880

0

2= 5159

1198751

119898

max11990511199021

1198801= 119901119878 (120572

1 1199021) minus 1199051 (18)

st 120582 (119905119898

1minus 119888119902119898

1+ 119904119868 (120572

1 119902119898

1)) minus (1 minus 120582) ge 0 (19)

119901119878 (1205722 119902lowast

2) minus 119905lowast

2+ ge 119901119878 (120572

2 119902119898

1) minus 119905119898

1 (20)

The high-risk retailer can get the rent above theoptimum so themanufacturer will lose some utilityThen theincentive rational constraint (see (19)) for themanufacturer isthe expected profit rather than the individual profit Equation(20) is the incentive compatible constraint for the high-riskretailer

From programs 1198751119898 and 119875

2

119898 we can get Propositions 14

and 15 (see Proof in Appendix C)

Proposition 14 The interim efficient allocation by informedprincipal for the supply chain with demand disruption satisfiesthe following

(1) Comparing to the full information situation the orderquantity for the high-risk retailer does not distort 119902119898

2=

119902lowast

2

(2) Comparing to the full information situation the orderquantity for the low-risk retailer satisfies (21) anddistorts upwards 119902119898

1gt 119902lowast

1 Consider

1199041198681015840(1205721 119902119898

1) + 119901119878

1015840(1205721 119902119898

1) minus 119888

=1 minus 120582

120582[1199011198781015840(1205722 119902119898

1) minus 119901119878

1015840(1205721 119902119898

1)]

(21)

(3) The transferring payment for the low-risk retailer meets(22) and the transferring payment for the high-riskretailer meets (23)

119905119898

1= (1 minus 120582)

sdot [119901119878 (1205722 119902119898

1) minus 119901119878 (120572

2 119902lowast

2) + 119888119902

lowast

2minus 119904119868 (120572

2 119902lowast

2)]

+ 120582 [119888119902119898

1minus 119904119868 (120572

1 119902119898

1)]

(22)

119905119898

2= (1 minus 120582) [119888119902

lowast

2minus 119904119868 (120572

2 119902lowast

2)] + 120582119901119878 (120572

2 119902lowast

2)

+ 120582 [119888119902119898

1minus 119904119868 (120572

1 119902119898

1) minus 119901119878 (120572

2 119902119898

1)]

(23)

Proposition 15 There is a threshold 1205820

= (1199011198781015840(1205722 1199020

1) minus

1199011198781015840(1205721 1199020

1))(119904119868

1015840(1205721 1199020

1) + 119901119878

1015840(1205722 1199020

1) minus 119888) for the low-risk

retailer probability If and only if the probability of low-riskretailer is no more than the threshold 120582 le 120582

0 the low-infor-mation-intensity allocation is also interim efficient and isthe unique Perfect Bayesian Equilibrium If the probabilityof low-risk retailer is more than the threshold 120582 gt 120582

0 thelow-information-intensity allocation is not the unique PerfectBayesian Equilibrium

If and only if the probability of low-risk retailer is smallenough (120582 le 120582

0) the low-information-intensity allocation isthe unique Perfect Bayesian EquilibriumAnd the retailer candeliver its own type information by the separating contractBut if the low-risk retailer is more than that (120582 gt 120582

0) the low-information-intensity allocation is not the unique PerfectBayesian Equilibrium and the interim efficient allocation canimprove the distortion situation to a certain degree So for thesupply chain with demand disruption the optimal allocationcannot be reached if the informed principal provides aseparating contract

6 Numerical Example

This section gives some numerical examples to inspect thesupply chain contract with demand disruption by informedprincipal We let the sales price 119901 = 10 the unit manu-facturing cost is 119888 = 2 and the salvage product value is119904 = 05 The demand distribution function before disruptionis 119865(119910) = 119910

2125 and the demand distribution function after

disruption is 119866(119909) = 1199092100

61 Main Results From the example listed above the fullinformation contracts and low-information-intensity alloca-tion are given in Table 1

Table 1 shows that for the full information contract theorder quantity for low-risk retailer is larger than that for thehigh-risk retailer 119902lowast

1gt 119902lowast

2 and also the transferring payment

119905lowast

1gt 119905lowast

2 which is performed in Proposition 6 In the informed

principal model the order quantity for the low-risk retailerdistorts in order to show its type (when 120572

1= 02 versus 120572

2=

04 119902lowast1

= 1001 lt 119902sb1

= 1014 and when 1205721= 03 versus

1205722= 05 119902lowast

1= 990 lt 119902

0

1= 1002) Meanwhile we notice that

the utility obtained by the low-risk retailer in the informed

Mathematical Problems in Engineering 7

0 10

5

10

15

q1

02 04 06 08

120582

Figure 3 The order quantity in interim efficient allocation for low-risk retailer changes by its probability

0

1

101112

1314

150

5

10

15

20

25

30

0204

0608

tm 1

q1 120582

Figure 4 The transferring payment in interim efficient allocationfor low-risk retailer changes by the order quantity and its probability

principal model is lower than that in full information model(when 120572

1= 02 versus 120572

2= 04 119880lowast

1= 5340 gt 119880

0

1= 5338

and when 1205721= 03 versus 120572

2= 05 119905lowast

1= 1840 gt 119905

0

1= 1824)

This implies that it costs the low-risk retailer something tocertify its typeThese results are the samewith Proposition 12

Considering the situation of 1205721= 02 versus 120572

2= 04 we

obtain 1205820= 019 The order quantity for the low-risk retailer

119902119898

1changes with the proportion it has 120582 in interim efficient

allocation As Figure 3 shows if and only if 120582 varies withincertain area (120582 le 120582

0) we obtain 119902119898

1gt 119902lowast

1 as in the results in

Proposition 14The transferring payment in the interim efficient allo-

cation is depicted in Figure 4 (for the low-risk retailer)and Figure 5 (for the high-risk retailer) The transferringpayments are changing by the order quantity for the low-risk retailer in the interim efficient allocation and the low-risk retailerrsquos probability From Figure 4 when the low-riskretailerrsquos probability 120582 is less enough and the order quantityfor the low-risk retailer 119902119898

1is high enough the transferring

0

05

1

10 11 12 13 14 15

20

25

30

35

40

45

tm 2

q1

120582

Figure 5 The transferring payment in interim efficient allocationfor high-risk retailer changes by the low-risk retailerrsquos order quantityand probability

payment 1199051198981increases by the low-risk retailerrsquos probability 120582

but decreases by the order quantity for the low-risk retailer119902119898

1(bottom left in Figure 4) But mostly the transferring

payment 1199051198981increases by the low-risk retailerrsquos probability 120582

but decreases by the order quantity for the low-risk retailer 1199021198981

(red and orange part in Figure 4) Normally the transferringpayment should increase by the order quantity which meansldquoget more pay morerdquo From this view the low-risk retailerrsquosprobability 120582 should not be very small and the order quantityfor the low-risk retailer 119902

119898

1should not be close to the

maximum value 15From Figure 5 it is obviously seen that the transferring

payment for high-risk retailer in the interim efficient alloca-tion 119905119898

2increases by the order quantity for the low-risk retailer

119902119898

1but decreases by the low-risk retailerrsquos probability 120582

62 Sensitivity Analyses The sensitivity analyses for the low-information-intensity allocation versus the unit manufactur-ing cost 119888 are shown in Table 2 It shows the outcomes whenthe unit manufacturing cost 119888 changes by +25 and minus25once at a time and keeping remaining parameters The orderquantities decrease by the unit manufacturing cost but theinfluence for the high-risk retailerrsquos order quantity is largerthan that for the low-risk retailer (minus069 versus minus317 and+217 versus +307) The transferring payments increaseby the unit manufacturing cost More interesting the utilitiesdecrease by the unit manufacturing cost and the low-riskretailer receives a greater impact (minus940 versus minus922 and+955 versus +953) So from this point of view the low-risk retailer should encourage themanufacturer to reduce thecost more

Figure 6 shows the sensitivity analyses for the order quan-tities versus the unit manufacturing cost in which the orderquantities include the order quantity for the low-risk retailerin low-information-intensity allocation 119902

0

1 the order quantity

8 Mathematical Problems in Engineering

Table 2 Sensitivity analyses for low-information-intensity allocation versus the unit manufacturing cost

Changes of 119888 The low-information allocation for low-riskretailer (120572

1= 02)

The low-information allocationfor high-risk retailer (120572

2= 04)

Value Change rate Value Change rate Value Change rate

119888 = 25 +251199020

1= 1007 minus069 119902

0

2= 947 minus317

1199050

1= 2375 +2626 119905

0

2= 2243 +2331

1198800

1= 4836 minus940 119880

0

2= 4736 minus922

119888 = 2 Baseline1199020

1= 1014 mdash 119902

sb2

= 978 mdash1199050

1= 1881 mdash 119905

sb2

= 1819 mdash1198800

1= 5338 mdash 119880

sb2

= 5217 mdash

119888 = 15 minus251199020

1= 1036 +217 119902

0

2= 1008 +307

1199050

1= 1398 minus2568 119905

0

2= 1362 minus2512

1198800

1= 5848 +955 119880

0

2= 5714 +953

108

106

104

102

10

98

96

94

92

C

14 16 18 2 22 24 26 28 3

qsb1

qlowast1q2

Figure 6 The order quantities change by the unit manufacturingcost

for the low-risk retailer in full information contract 119902lowast1 and

the order quantity for the high-risk retailer in full informationcontract which is equal to that in low-information-intensityallocation 119902

2 Both the order quantity for the low-risk retailer

in full information and the order quantity for the high-riskretailer decrease by the unit manufacturing cost and theychange in the same level basically But the change of the orderquantity for the low-risk retailer in low information intensitydepends on different values of the unit manufacturing costWhen the unit manufacturing cost 119888 = 275 the orderquantity is minimum and when 119888 = 175 or 119888 = 15 the orderquantity is the maximum

Figure 7 shows the sensitivity analyses for the transferringpayment versus the unit manufacturing cost in which thetransferring payments include the transferring payment forthe low-risk retailer in low-information-intensity allocation1199050

1 the transferring payment for the low-risk retailer in full

information contract 119905lowast

1 and the transferring payment for

26

24

22

20

18

16

14

C

14 16 18 2 22 24 26 28 3

tsb1

tlowast1t2

Figure 7 The transferring payments change by the unit manufac-turing cost

the high-risk retailer in full information contract which isequal to that in low-information-intensity allocation 119905

2 All

the transferring payments of the retailers increase by the unitmanufacturing cost The transferring payments for the low-risk retailer and the high-risk retailer in full informationcontract decrease in the same level

Figure 8 shows the sensitivity analyses for the retailersrsquoutilities versus the unit manufacturing cost in which theutilities include the utility for the low-risk retailer in low-information-intensity allocation 119880

0

1 the transferring pay-

ment for the low-risk retailer in full information contract119880lowast

1 and the transferring payment for the high-risk retailer

in full information contract which is equal to that in low-information-intensity allocation 119880

2 All the retailersrsquo utilities

decrease by the unit manufacturing costFigure 9 shows the sensitivity analyses for the low-risk

retailerrsquos signaling cost versus the unit manufacturing cost

Mathematical Problems in Engineering 9

44

46

48

50

52

54

56

58

60

C

14 16 18 2 22 24 26 28

Usb1

Ulowast1

U2

Figure 8 The retailerrsquos utilities change by the unit manufacturingcost

0

005

01

015

02

025

C

14 16 18 2 22 24 26 28

ΔU1

Figure 9The signaling cost changes by the saved unit manufactur-ing cost

The signaling cost of the low-risk retailer is the utility earnedin the low-information-intensity allocation which is sepa-rating contract less than that in full information contractWhen the unit manufacturing cost 119888 = 2 the signaling costis the least while when the saved unit manufacturing cost119888 = 175 the signaling cost is the most We do not find theinternal relationship between the signaling cost and the unitmanufacturing cost As the signaling cost is so important toput the contracts into practice investigation on the signalingcost is the research direction for us

7 Conclusion

In the recent years supply chain with demand disruptionhas become more and more attractive to both business and

academic research There is a lot of research available on thetopic of supply chain disruption contract In the real supplychain practice demand disruption is totally different fromthe stochastic demand So in our research both the demanduncertainty and the demand disruption are consideredTaking the demand disruption probability as the asymmetricinformation an informed principal model is explored tomake the contract for supply chain with demand disruptionAnd the conclusions we obtain are shown as follows

(1) The contracts for low-risk retailer and high-riskretailer in full information are provided and it isfound out that the high-risk retailer has the incentiveto pretend to be the low-risk retailer This result issimilar to the other informed principal model [3435] but totally different from the normal principalagent model [24 26] In the normal principal agentmodel the high efficient agent (low-risk retailer in oursetting) tries to pretend to be low efficient agent (high-risk retailer) to get more That is why the informedprincipal model is established

(2) The low-information-intensity allocation which is theresult of the informed principal model and also is theseparating contract shows that the order quantity andthe transferring payment for the low-risk retailer inlow-information-intensity allocation distort upwardsbut that of high-risk retailer does not distort Inorder to avoid imitation the imitatorsrsquo efficiencyshould be always distorted But in normal principalagent model the high-risk retailerrsquos quantity distortsdownwards [24 26] on the contrary the low-riskretailerrsquos quantity in informed principal model dis-torts upwards

(3) In order to reduce the signaling cost which the low-risk retailer pays the interim efficient model is intro-duced which ends up with the order quantity andtransferring payment distorted upward again but lessthan that of low information intensity Comparingto [38] focusing on analyzing the signaling costmore attention is paid to showing the interim efficientmodel to get a solution which can increase efficiency

From the managerial aspect retailers should try toincrease their antirisk capability to enhance competitivepower And when the retailer has lower demand disruptionprobability than the others it can show its style to the supplierthrough the separating contract For example the retailer cansay ldquobecause I am the low-risk one I can cooperate with thesupplier with this contract while the others who are high-riskones can only use the other contractrdquo But in order to be sep-arated from the others the low-risk retailer has to pay somesignaling cost and the quantity has to be distorted upwards

Above all there are some limitations of the research Wewill continue our research in the following directions (1)The signaling cost is a key factor impacting the separatingcontract which should be paid more attention (2) We onlyconsider the demand disruption but disruptions happen atnot only demand side but also supply side Sowewill continue

10 Mathematical Problems in Engineering

our research with demand disruption together with supplydisruption [39]

Appendix

A Proof of Proposition 6

For the two types of retailers we can obtain the optimalallocation if and only if the whole margin utility of the supplychain is the same with the margin cost So 119902

lowast

1and 119902

lowast

2satisfy

the following first-order conditions

1199011198781015840(1205721 1199021) + 1199041198681015840(1205721 1199021) = 119888 (A1)

1199011198781015840(1205722 1199022) + 1199041198681015840(1205722 1199022) = 119888 (A2)

And the optimal transferring payments are

119905lowast

1= 119888119902lowast

1minus 119904119868 (120572

1 119902lowast

1)

119905lowast

2= 119888119902lowast

2minus 119904119868 (120572

2 119902lowast

2)

(A3)

From (A1) and (A2) we obtain

1205791119866 (119902lowast

1) + (1 minus 120579

1) 119865 (119902lowast

1)

= 1205792119866 (119902lowast

2) + (1 minus 120579

2) 119865 (119902lowast

2) =

(119901 minus 119888)

(119901 minus 119904)

(A4)

In addition 119866(119902lowast

1) ge 119865(119902

lowast

1) 1205721

lt 1205722 so we know that

1205721119866(119902lowast

1) + (1minus120572

1)119865(119902lowast

1) lt 1205722119866(119902lowast

1) + (1minus120572

2)119865(119902lowast

1) and from

(A4) we obtain 1205722119866(119902lowast

2) + (1 minus 120572

2)119865(119902lowast

2) lt 1205722119866(119902lowast

1) + (1 minus

1205722)119865(119902lowast

1) and thus 119902lowast

2lt 119902lowast

1 From (A3) 119905lowast

1minus119905lowast

2= 119888(119902lowast

1minus119902lowast

2)minus

119904(119868(1205721 119902lowast

1) minus 119868(120572

2 119902lowast

2)) 119905lowast1minus 119905lowast

2=119902lowast

1=119902lowast

2

(1205721minus 1205722)119904(int119902

0119865(119910)119889119910 minus

int119902

0119866(119909)119889119909) gt 0 and 120597119905

lowast

1120597119902lowast

1gt 0 so 119905

lowast

1minus 119905lowast

2gt 0 and

thus we obtain 119905lowast

1gt 119905lowast

2 119881lowast

1= 119881lowast

2= 0 Furthermore we get

119880lowast

1gt 119880lowast

2from 120597119880120597120579 lt 0 and 120597119880120597119902 gt 0 Finally we have

Proposition 6

B Proof of Proposition 12

We denote 1205831and 120583

2as the Lagrange multipliers of (6) and

(14) and find the derivate of 1199051and 1199021 and thus we obtain

120597119880 (1205721 1199021)

1205971199051

= minus1 + 1205831+ 1205832= 0

120597119880 (1205721 1199021)

1205971199021

= 1199011198781015840(1205721 1199021) minus 1205831119888 + 12058311199041198681015840(1205721 1199021)

minus 12058321199011198781015840(1205722 1199021) = 0

(B1)

By rewriting (B1) we get the following formulas

1199041198681015840(1205721 119902119904

1) + 119901119878

1015840(1205721 119902119904

1) minus 119888

=(1 minus 120583

1)

1205831

[1199011198781015840(1205722 119902119904

1) minus 119901119878

1015840(1205721 119902119904

1)]

(B2)

1205831=

1199011198781015840(1205722 119902119904

1) minus 119901119878

1015840(1205721 119902119904

1)

1199041198681015840 (1205721 119902119904

1) + 1199011198781015840 (120572

2 119902119904

1) minus 119888

(B3)

1205832=

1199041198681015840(1205721 119902119904

1) + 119901119878

1015840(1205721 119902119904

1) minus 119888

1199041198681015840 (1205721 119902119904

1) + 1199011198781015840 (120572

2 119902119904

1) minus 119888

(B4)

Because the numerator of 1205831is negative so 120583

1is positive

and the denominator of 1205831is negative because 120583

2is positive

(when 1205832= 0 it cannot be separated from the former one)

the numerator of 1205832is negative nevertheless 1199011198781015840(120572

1 119902lowast

1) minus

119888 + 1199041198681015840(1205721 119902lowast

1) = 0 and thus 1199011198781015840(120572

1 119902119904

1) minus 119888 + 119904119868

1015840(1205721 119902119904

1) lt

1199011198781015840(1205721 119902lowast

1)minus 119888+ 119904119868

1015840(1205721 119902lowast

1) in addition 1205972119880120597119902

2lt 0 so 119902

119904

1gt

119902lowast

1 From 120583

1gt 0 and 120583

2gt 0 we deduce that (6) and (14)

are tight Furthermore the programs 1198751119865 and 119875

1

119898 have the

same objective function but the constraint of 1198751119898 is more

tight so there is distortion of the utility obtained from 1198751

119898 in

other words 1198801(119905119904

1 119902119904

1) lt 119880lowast

1 From Proposition 11 we obtain

1198802(1199050

2 1199020

2) = 119880lowast

2 So we have Proposition 12

C Proof of Propositions 14 and 15

We denote 120573 and 120574 as the Lagrange multipliers of (19) and(20) and find the derivate of 119905

1 1199021 and and thus we obtain

120597119871 (1199051 1199021 )

1205971199051

= minus1 + 120573120582 + 120574 = 0 (C1)

120597119871 (1199051 1199021 )

1205971199021

= 1199011198781015840(1205721 1199021) + 120573120582 [minus119888 + 119904119868

1015840(1205721 1199021)]

minus 1205741199011198781015840(1205721 1199021) = 0

(C2)

120597119871 (1199051 1199021 )

120597= minus120573 (1 minus 120582) + 120574 = 0 (C3)

To solve (C1)ndash(C3) we obtain 120573 = 1 120574 = 1 minus 120582 Andconsidering (C2) we get

1199041198681015840(1205721 119902119898

1) + 119901119878

1015840(1205721 119902119898

1) minus 119888

=1 minus 120582

120582[1199011198781015840(1205722 119902119898

1) minus 119901119878

1015840(1205721 119902119898

1)]

(C4)

Because (19) and (20) are tight we obtain the followingresults

119905119898

1= (1 minus 120582)

sdot [119901119878 (1205722 119902119898

1) minus 119901119878 (120572

2 119902lowast

2) + 119888119902

lowast

2minus 119904119868 (120572

2 119902lowast

2)]

+ 120582 [119888119902119898

1minus 119904119868 (120572

1 119902119898

1)]

119905119898

2= (1 minus 120582) [119888119902

lowast

2minus 119904119868 (120572

2 119902lowast

2)] + 120582119901119878 (120572

2 119902lowast

2)

+ 120582 [119888119902119898

1minus 119904119868 (120572

1 119902119898

1) minus 119901119878 (120572

2 119902119898

1)]

(C5)

Mathematical Problems in Engineering 11

To compare (A1) with (C4) we know 119902119898

1gt 119902lowast

1because

12059721198801205971199022

lt 0 and 1199011198781015840(1205722 119902) minus 119901119878

1015840(1205721 119902) lt 0 and the dis-

tortion level is changing with 120572 Finally we have Proposi-tion 14

To compare (C4) with (B2) and designate 1205820

= 120582 =

(1199011198781015840(1205722 1199020

1) minus 119901119878

1015840(1205721 1199020

1))(119904119868

1015840(1205721 1199020

1) + 119901119878

1015840(1205722 1199020

1) minus 119888) we

have the following conclusions

(1) When 120582 = 1205820 1199021198981

= 119902119904

1= 1199020

1 we obtain the same

contracts with the low-information-intensity situa-tion

(2) When 120582 lt 1205820 1199021198981

lt 119902119904

1= 1199020

1 we know there is more

distortion than the low-information-intensity alloca-tion and it improves anything

(3) When 120582 gt 1205820 1199021198981

lt 119902119904

1= 1199020

1 the interim efficient

allocation can improve the low-information-intensityallocation So we have Proposition 15

VariablesNotations

119888 Unit manufacturing cost1205721 Disruption probability of low-risk retailer

119910 Market demand without disruption119865(sdot) Distribution function of demand without

disruption119904 Unit salvage value119860 Market scale without disruption119878(120572119894 119902119894) Expected sales

119868(120572119894 119902119894) Expected unsold quantity

119880119894 Utility of the retailer

120582 Probability of low-risk retailer1205722 Disruption probability of high-risk retailer

119909 Market demand with disruption119866(sdot) Distribution function of demand with

disruption119901 Retail price119863 Market scale with disruption119881119894 Utility of the manufacturer

Control Variables

119902119894 Order quantity

119905119894 Transferring payment

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported in part by a project supported bythe Humanities and Social Sciences Project of the EducationMinistry (no 14YJC630187) Hebei Social Sciences Project(no HB15GL058) and the Fundamental Research Funds forthe Central Universities (no 2016MS123)

References

[1] J Barnett F Begen S Howes et al ldquoConsumersrsquo confidencereflections and response strategies following the horsemeatincidentrdquo Food Control vol 59 pp 721ndash730 2016

[2] A J Schmitt and L V Snyder ldquoInfinite-horizon models forinventory control under yield uncertainty and disruptionsrdquoComputers amp Operations Research vol 39 no 4 pp 850ndash8622012

[3] L V Snyder Z Atan P Peng et al ldquoORMS models for supplychain disruptions a reviewrdquo IIE Transactions vol 48 no 2 pp89ndash109 2016

[4] X Qi J F Bard and G Yu ldquoSupply chain coordination withdemand disruptionsrdquo Omega vol 32 no 4 pp 301ndash312 2004

[5] T Xiao G Yu Z Sheng and Y Xia ldquoCoordination of a supplychain with one-manufacturer and two-retailers under demandpromotion and disruption management decisionsrdquo Annals ofOperations Research vol 135 pp 87ndash109 2005

[6] F Hu C-C Lim Z Lu and X Sun ldquoCoordination in a single-retailer two-supplier supply chain under random demand andrandom supply with disruptionrdquo Discrete Dynamics in Natureand Society vol 2013 Article ID 484062 12 pages 2013

[7] J Li X Liu J Wu and F Yang ldquoCoordination of supplychain with a dominant retailer under demand disruptionsrdquoMathematical Problems in Engineering vol 2014 Article ID854681 10 pages 2014

[8] K Chen and T Xiao ldquoDemand disruption and coordination ofthe supply chain with a dominant retailerrdquo European Journal ofOperational Research vol 197 no 1 pp 225ndash234 2009

[9] J Li and F T S Chan ldquoThe impact of collaborative transporta-tion management on demand disruption of manufacturingsupply chainsrdquo International Journal of Production Research vol50 no 19 pp 5635ndash5650 2012

[10] Y Zheng T Shu S Wang S Chen K K Lai and L GanldquoDemand disruption and coordination of supply chain via effortand revenue sharingrdquo Applied Economics vol 47 no 54 pp5886ndash5901 2015

[11] Q Pang Y Hou and Y Lv ldquoCoordinating three-level supplychain under disruptions using revenue-sharing contract witheffort dependent demandrdquoMathematical Problems in Engineer-ing vol 2016 Article ID 9167864 10 pages 2016

[12] M Gumus S Ray and H Gurnani ldquoSupply-side story risksguarantees competition and information asymmetryrdquo Man-agement Science vol 58 no 9 pp 1694ndash1714 2012

[13] K B Hendricks and V R Singhal ldquoAn empirical analysisof the effect of supply chain disruptions on long-run stockprice performance and equity risk of the firmrdquo Production ampOperations Management vol 14 no 1 pp 35ndash52 2005

[14] W Schimidt and A Raman ldquoWhen supply-chain disruptionsmatterrdquo Harvard Business School Working Paper 13-006 2012

[15] N Bunkley Piecing together a supply chain The New YorkTimes 2013 httpwwwnytimescom20110513businessglobal13autohtml r=0

[16] S Sarkar and S Kumar ldquoA behavioral experiment on inventorymanagement with supply chain disruptionrdquo International Jour-nal of Production Economics vol 169 pp 169ndash178 2015

[17] G P Cachon ldquoSupply chain coordination with contractsrdquoHandbooks in Operations Research amp Management Science vol11 pp 227ndash339 2003

[18] C J Corbett and X De Groote ldquoA supplierrsquos optimal quantitydiscount policy under asymmetric informationrdquo ManagementScience vol 46 no 3 pp 444ndash450 2000

12 Mathematical Problems in Engineering

[19] C J Corbett ldquoStochastic inventory systems in a supply chainwith asymmetric information cycle stocks safety stocks andconsignment stockrdquoOperations Research vol 49 no 4 pp 487ndash500 2001

[20] A Matopoulos M Vlachopoulou V Manthou and B ManosldquoA conceptual framework for supply chain collaboration empir-ical evidence from the agri-food industryrdquo Supply Chain Man-agement vol 12 no 3 pp 177ndash186 2007

[21] A Y Ha and S Tong ldquoContracting and information sharingunder supply chain competitionrdquoManagement Science vol 54no 4 pp 701ndash715 2008

[22] Y-W Zhou ldquoA comparison of different quantity discountpricing policies in a two-echelon channel with stochastic andasymmetric demand informationrdquo European Journal of Opera-tional Research vol 181 no 2 pp 686ndash703 2007

[23] XGan S P Sethi and J Zhou ldquoCommitment-penalty contractsin drop-shipping supply chains with asymmetric demand infor-mationrdquo European Journal of Operational Research vol 204 no3 pp 449ndash462 2010

[24] D Lei J Li and Z Liu ldquoSupply chain contracts under demandand cost disruptions with asymmetric informationrdquo Interna-tional Journal of Production Economics vol 139 no 1 pp 116ndash126 2012

[25] Z B Yang G Aydın V Babich and D R Beil ldquoSupplydisruptions asymmetric information and a backup productionoptionrdquoManagement Science vol 55 no 2 pp 192ndash209 2009

[26] S Huang and C Yang ldquoSupply chain revelation mechanismdesign under asymmetric demand disruption informationrdquoOperations Research andManagement Science vol 23 no 6 pp116ndash127 2014

[27] S Oh and O Ozer ldquoMechanism design for capacity planningunder dynamic evolutions of asymmetric demand forecastsrdquoManagement Science vol 59 no 4 pp 987ndash1007 2013

[28] Q Feng G Lai and L X Lu ldquoDynamic bargaining in a supplychain with asymmetric demand informationrdquo ManagementScience vol 61 no 2 pp 301ndash315 2015

[29] Q Li B Li P Chen and P Hou ldquoDual-channel supply chaindecisions under asymmetric information with a risk-averseretailerrdquo Annals of Operations Research 2015

[30] J Wei K Govindan Y Li and J Zhao ldquoPricing and collectingdecisions in a closed-loop supply chain with symmetric andasymmetric informationrdquo Computers and Operations Researchvol 54 pp 257ndash265 2015

[31] K Inderfurth A Sadrieh and G Voigt ldquoThe impact of infor-mation sharing on supply chain performance under asymmetricinformationrdquo Production amp Operations Management vol 22no 2 pp 410ndash425 2013

[32] R B Myerson ldquoMechanism design by an informed principalrdquoEconometrica vol 51 no 6 pp 1767ndash1797 1983

[33] E Maskin and J Tirole ldquoThe principal-agent relationship withan informed principal the case of private valuesrdquo Econometricavol 58 no 2 pp 379ndash409 1990

[34] E Maskin and J Tirole ldquoThe principal-agent relationship withan informed principal II common valuesrdquo Econometrica vol60 no 1 pp 1ndash42 1992

[35] T Mylovanov and T Troger ldquoInformed-principal problemsin environments with generalized private valuesrdquo TheoreticalEconomics vol 7 no 3 pp 465ndash488 2012

[36] TMylovanov andT Troger ldquoMechanismdesign by an informedprincipal private values with transferable utilityrdquo Review ofEconomic Studies vol 81 no 4 pp 1668ndash1707 2014

[37] C Wagner T Mylovanov and T Troger ldquoInformed-principalproblem with moral hazard risk neutrality and no limitedliabilityrdquo Journal of EconomicTheory vol 159 pp 280ndash289 2015

[38] S Galperti ldquoCommon agency with informed principals menusand signalsrdquo Journal of Economic Theory vol 157 pp 648ndash6672015

[39] H Zhang Y Liu and J Huang ldquoSupply chain coordina-tion contracts under double sided disruptions simultaneouslyrdquoMathematical Problems in Engineering vol 2015 Article ID812043 9 pages 2015

Submit your manuscripts athttpwwwhindawicom

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Mathematical Problems in Engineering

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Stochastic AnalysisInternational Journal of

6 Mathematical Problems in Engineering

Table 1 Full information contract versus low-information-intensity allocation

Type ofretailer Full information

Lowinformationintensity

Type ofretailer Full information

Lowinformationintensity

1205721= 02

119902lowast

1= 1001 119902

0

1= 1014

1205721= 03

119902lowast

1= 990 119902

0

1= 1002

119905lowast

1= 1862 119905

0

1= 1881 119905

lowast

1= 1840 119905

0

1= 1824

119880lowast

1= 5340 119880

0

1= 5338 119880

lowast

1= 5278 119880

0

1= 5192

1205722= 04

119902lowast

2= 978 119902

0

1= 978

1205722= 05

119902lowast

2= 967 119902

0

2= 967

119905lowast

2= 1819 119905

0

2= 1819 119905

lowast

2= 1799 119905

0

2= 1799

119880lowast

2= 5217 119880

0

2= 5217 119880

lowast

2= 5159 119880

0

2= 5159

1198751

119898

max11990511199021

1198801= 119901119878 (120572

1 1199021) minus 1199051 (18)

st 120582 (119905119898

1minus 119888119902119898

1+ 119904119868 (120572

1 119902119898

1)) minus (1 minus 120582) ge 0 (19)

119901119878 (1205722 119902lowast

2) minus 119905lowast

2+ ge 119901119878 (120572

2 119902119898

1) minus 119905119898

1 (20)

The high-risk retailer can get the rent above theoptimum so themanufacturer will lose some utilityThen theincentive rational constraint (see (19)) for themanufacturer isthe expected profit rather than the individual profit Equation(20) is the incentive compatible constraint for the high-riskretailer

From programs 1198751119898 and 119875

2

119898 we can get Propositions 14

and 15 (see Proof in Appendix C)

Proposition 14 The interim efficient allocation by informedprincipal for the supply chain with demand disruption satisfiesthe following

(1) Comparing to the full information situation the orderquantity for the high-risk retailer does not distort 119902119898

2=

119902lowast

2

(2) Comparing to the full information situation the orderquantity for the low-risk retailer satisfies (21) anddistorts upwards 119902119898

1gt 119902lowast

1 Consider

1199041198681015840(1205721 119902119898

1) + 119901119878

1015840(1205721 119902119898

1) minus 119888

=1 minus 120582

120582[1199011198781015840(1205722 119902119898

1) minus 119901119878

1015840(1205721 119902119898

1)]

(21)

(3) The transferring payment for the low-risk retailer meets(22) and the transferring payment for the high-riskretailer meets (23)

119905119898

1= (1 minus 120582)

sdot [119901119878 (1205722 119902119898

1) minus 119901119878 (120572

2 119902lowast

2) + 119888119902

lowast

2minus 119904119868 (120572

2 119902lowast

2)]

+ 120582 [119888119902119898

1minus 119904119868 (120572

1 119902119898

1)]

(22)

119905119898

2= (1 minus 120582) [119888119902

lowast

2minus 119904119868 (120572

2 119902lowast

2)] + 120582119901119878 (120572

2 119902lowast

2)

+ 120582 [119888119902119898

1minus 119904119868 (120572

1 119902119898

1) minus 119901119878 (120572

2 119902119898

1)]

(23)

Proposition 15 There is a threshold 1205820

= (1199011198781015840(1205722 1199020

1) minus

1199011198781015840(1205721 1199020

1))(119904119868

1015840(1205721 1199020

1) + 119901119878

1015840(1205722 1199020

1) minus 119888) for the low-risk

retailer probability If and only if the probability of low-riskretailer is no more than the threshold 120582 le 120582

0 the low-infor-mation-intensity allocation is also interim efficient and isthe unique Perfect Bayesian Equilibrium If the probabilityof low-risk retailer is more than the threshold 120582 gt 120582

0 thelow-information-intensity allocation is not the unique PerfectBayesian Equilibrium

If and only if the probability of low-risk retailer is smallenough (120582 le 120582

0) the low-information-intensity allocation isthe unique Perfect Bayesian EquilibriumAnd the retailer candeliver its own type information by the separating contractBut if the low-risk retailer is more than that (120582 gt 120582

0) the low-information-intensity allocation is not the unique PerfectBayesian Equilibrium and the interim efficient allocation canimprove the distortion situation to a certain degree So for thesupply chain with demand disruption the optimal allocationcannot be reached if the informed principal provides aseparating contract

6 Numerical Example

This section gives some numerical examples to inspect thesupply chain contract with demand disruption by informedprincipal We let the sales price 119901 = 10 the unit manu-facturing cost is 119888 = 2 and the salvage product value is119904 = 05 The demand distribution function before disruptionis 119865(119910) = 119910

2125 and the demand distribution function after

disruption is 119866(119909) = 1199092100

61 Main Results From the example listed above the fullinformation contracts and low-information-intensity alloca-tion are given in Table 1

Table 1 shows that for the full information contract theorder quantity for low-risk retailer is larger than that for thehigh-risk retailer 119902lowast

1gt 119902lowast

2 and also the transferring payment

119905lowast

1gt 119905lowast

2 which is performed in Proposition 6 In the informed

principal model the order quantity for the low-risk retailerdistorts in order to show its type (when 120572

1= 02 versus 120572

2=

04 119902lowast1

= 1001 lt 119902sb1

= 1014 and when 1205721= 03 versus

1205722= 05 119902lowast

1= 990 lt 119902

0

1= 1002) Meanwhile we notice that

the utility obtained by the low-risk retailer in the informed

Mathematical Problems in Engineering 7

0 10

5

10

15

q1

02 04 06 08

120582

Figure 3 The order quantity in interim efficient allocation for low-risk retailer changes by its probability

0

1

101112

1314

150

5

10

15

20

25

30

0204

0608

tm 1

q1 120582

Figure 4 The transferring payment in interim efficient allocationfor low-risk retailer changes by the order quantity and its probability

principal model is lower than that in full information model(when 120572

1= 02 versus 120572

2= 04 119880lowast

1= 5340 gt 119880

0

1= 5338

and when 1205721= 03 versus 120572

2= 05 119905lowast

1= 1840 gt 119905

0

1= 1824)

This implies that it costs the low-risk retailer something tocertify its typeThese results are the samewith Proposition 12

Considering the situation of 1205721= 02 versus 120572

2= 04 we

obtain 1205820= 019 The order quantity for the low-risk retailer

119902119898

1changes with the proportion it has 120582 in interim efficient

allocation As Figure 3 shows if and only if 120582 varies withincertain area (120582 le 120582

0) we obtain 119902119898

1gt 119902lowast

1 as in the results in

Proposition 14The transferring payment in the interim efficient allo-

cation is depicted in Figure 4 (for the low-risk retailer)and Figure 5 (for the high-risk retailer) The transferringpayments are changing by the order quantity for the low-risk retailer in the interim efficient allocation and the low-risk retailerrsquos probability From Figure 4 when the low-riskretailerrsquos probability 120582 is less enough and the order quantityfor the low-risk retailer 119902119898

1is high enough the transferring

0

05

1

10 11 12 13 14 15

20

25

30

35

40

45

tm 2

q1

120582

Figure 5 The transferring payment in interim efficient allocationfor high-risk retailer changes by the low-risk retailerrsquos order quantityand probability

payment 1199051198981increases by the low-risk retailerrsquos probability 120582

but decreases by the order quantity for the low-risk retailer119902119898

1(bottom left in Figure 4) But mostly the transferring

payment 1199051198981increases by the low-risk retailerrsquos probability 120582

but decreases by the order quantity for the low-risk retailer 1199021198981

(red and orange part in Figure 4) Normally the transferringpayment should increase by the order quantity which meansldquoget more pay morerdquo From this view the low-risk retailerrsquosprobability 120582 should not be very small and the order quantityfor the low-risk retailer 119902

119898

1should not be close to the

maximum value 15From Figure 5 it is obviously seen that the transferring

payment for high-risk retailer in the interim efficient alloca-tion 119905119898

2increases by the order quantity for the low-risk retailer

119902119898

1but decreases by the low-risk retailerrsquos probability 120582

62 Sensitivity Analyses The sensitivity analyses for the low-information-intensity allocation versus the unit manufactur-ing cost 119888 are shown in Table 2 It shows the outcomes whenthe unit manufacturing cost 119888 changes by +25 and minus25once at a time and keeping remaining parameters The orderquantities decrease by the unit manufacturing cost but theinfluence for the high-risk retailerrsquos order quantity is largerthan that for the low-risk retailer (minus069 versus minus317 and+217 versus +307) The transferring payments increaseby the unit manufacturing cost More interesting the utilitiesdecrease by the unit manufacturing cost and the low-riskretailer receives a greater impact (minus940 versus minus922 and+955 versus +953) So from this point of view the low-risk retailer should encourage themanufacturer to reduce thecost more

Figure 6 shows the sensitivity analyses for the order quan-tities versus the unit manufacturing cost in which the orderquantities include the order quantity for the low-risk retailerin low-information-intensity allocation 119902

0

1 the order quantity

8 Mathematical Problems in Engineering

Table 2 Sensitivity analyses for low-information-intensity allocation versus the unit manufacturing cost

Changes of 119888 The low-information allocation for low-riskretailer (120572

1= 02)

The low-information allocationfor high-risk retailer (120572

2= 04)

Value Change rate Value Change rate Value Change rate

119888 = 25 +251199020

1= 1007 minus069 119902

0

2= 947 minus317

1199050

1= 2375 +2626 119905

0

2= 2243 +2331

1198800

1= 4836 minus940 119880

0

2= 4736 minus922

119888 = 2 Baseline1199020

1= 1014 mdash 119902

sb2

= 978 mdash1199050

1= 1881 mdash 119905

sb2

= 1819 mdash1198800

1= 5338 mdash 119880

sb2

= 5217 mdash

119888 = 15 minus251199020

1= 1036 +217 119902

0

2= 1008 +307

1199050

1= 1398 minus2568 119905

0

2= 1362 minus2512

1198800

1= 5848 +955 119880

0

2= 5714 +953

108

106

104

102

10

98

96

94

92

C

14 16 18 2 22 24 26 28 3

qsb1

qlowast1q2

Figure 6 The order quantities change by the unit manufacturingcost

for the low-risk retailer in full information contract 119902lowast1 and

the order quantity for the high-risk retailer in full informationcontract which is equal to that in low-information-intensityallocation 119902

2 Both the order quantity for the low-risk retailer

in full information and the order quantity for the high-riskretailer decrease by the unit manufacturing cost and theychange in the same level basically But the change of the orderquantity for the low-risk retailer in low information intensitydepends on different values of the unit manufacturing costWhen the unit manufacturing cost 119888 = 275 the orderquantity is minimum and when 119888 = 175 or 119888 = 15 the orderquantity is the maximum

Figure 7 shows the sensitivity analyses for the transferringpayment versus the unit manufacturing cost in which thetransferring payments include the transferring payment forthe low-risk retailer in low-information-intensity allocation1199050

1 the transferring payment for the low-risk retailer in full

information contract 119905lowast

1 and the transferring payment for

26

24

22

20

18

16

14

C

14 16 18 2 22 24 26 28 3

tsb1

tlowast1t2

Figure 7 The transferring payments change by the unit manufac-turing cost

the high-risk retailer in full information contract which isequal to that in low-information-intensity allocation 119905

2 All

the transferring payments of the retailers increase by the unitmanufacturing cost The transferring payments for the low-risk retailer and the high-risk retailer in full informationcontract decrease in the same level

Figure 8 shows the sensitivity analyses for the retailersrsquoutilities versus the unit manufacturing cost in which theutilities include the utility for the low-risk retailer in low-information-intensity allocation 119880

0

1 the transferring pay-

ment for the low-risk retailer in full information contract119880lowast

1 and the transferring payment for the high-risk retailer

in full information contract which is equal to that in low-information-intensity allocation 119880

2 All the retailersrsquo utilities

decrease by the unit manufacturing costFigure 9 shows the sensitivity analyses for the low-risk

retailerrsquos signaling cost versus the unit manufacturing cost

Mathematical Problems in Engineering 9

44

46

48

50

52

54

56

58

60

C

14 16 18 2 22 24 26 28

Usb1

Ulowast1

U2

Figure 8 The retailerrsquos utilities change by the unit manufacturingcost

0

005

01

015

02

025

C

14 16 18 2 22 24 26 28

ΔU1

Figure 9The signaling cost changes by the saved unit manufactur-ing cost

The signaling cost of the low-risk retailer is the utility earnedin the low-information-intensity allocation which is sepa-rating contract less than that in full information contractWhen the unit manufacturing cost 119888 = 2 the signaling costis the least while when the saved unit manufacturing cost119888 = 175 the signaling cost is the most We do not find theinternal relationship between the signaling cost and the unitmanufacturing cost As the signaling cost is so important toput the contracts into practice investigation on the signalingcost is the research direction for us

7 Conclusion

In the recent years supply chain with demand disruptionhas become more and more attractive to both business and

academic research There is a lot of research available on thetopic of supply chain disruption contract In the real supplychain practice demand disruption is totally different fromthe stochastic demand So in our research both the demanduncertainty and the demand disruption are consideredTaking the demand disruption probability as the asymmetricinformation an informed principal model is explored tomake the contract for supply chain with demand disruptionAnd the conclusions we obtain are shown as follows

(1) The contracts for low-risk retailer and high-riskretailer in full information are provided and it isfound out that the high-risk retailer has the incentiveto pretend to be the low-risk retailer This result issimilar to the other informed principal model [3435] but totally different from the normal principalagent model [24 26] In the normal principal agentmodel the high efficient agent (low-risk retailer in oursetting) tries to pretend to be low efficient agent (high-risk retailer) to get more That is why the informedprincipal model is established

(2) The low-information-intensity allocation which is theresult of the informed principal model and also is theseparating contract shows that the order quantity andthe transferring payment for the low-risk retailer inlow-information-intensity allocation distort upwardsbut that of high-risk retailer does not distort Inorder to avoid imitation the imitatorsrsquo efficiencyshould be always distorted But in normal principalagent model the high-risk retailerrsquos quantity distortsdownwards [24 26] on the contrary the low-riskretailerrsquos quantity in informed principal model dis-torts upwards

(3) In order to reduce the signaling cost which the low-risk retailer pays the interim efficient model is intro-duced which ends up with the order quantity andtransferring payment distorted upward again but lessthan that of low information intensity Comparingto [38] focusing on analyzing the signaling costmore attention is paid to showing the interim efficientmodel to get a solution which can increase efficiency

From the managerial aspect retailers should try toincrease their antirisk capability to enhance competitivepower And when the retailer has lower demand disruptionprobability than the others it can show its style to the supplierthrough the separating contract For example the retailer cansay ldquobecause I am the low-risk one I can cooperate with thesupplier with this contract while the others who are high-riskones can only use the other contractrdquo But in order to be sep-arated from the others the low-risk retailer has to pay somesignaling cost and the quantity has to be distorted upwards

Above all there are some limitations of the research Wewill continue our research in the following directions (1)The signaling cost is a key factor impacting the separatingcontract which should be paid more attention (2) We onlyconsider the demand disruption but disruptions happen atnot only demand side but also supply side Sowewill continue

10 Mathematical Problems in Engineering

our research with demand disruption together with supplydisruption [39]

Appendix

A Proof of Proposition 6

For the two types of retailers we can obtain the optimalallocation if and only if the whole margin utility of the supplychain is the same with the margin cost So 119902

lowast

1and 119902

lowast

2satisfy

the following first-order conditions

1199011198781015840(1205721 1199021) + 1199041198681015840(1205721 1199021) = 119888 (A1)

1199011198781015840(1205722 1199022) + 1199041198681015840(1205722 1199022) = 119888 (A2)

And the optimal transferring payments are

119905lowast

1= 119888119902lowast

1minus 119904119868 (120572

1 119902lowast

1)

119905lowast

2= 119888119902lowast

2minus 119904119868 (120572

2 119902lowast

2)

(A3)

From (A1) and (A2) we obtain

1205791119866 (119902lowast

1) + (1 minus 120579

1) 119865 (119902lowast

1)

= 1205792119866 (119902lowast

2) + (1 minus 120579

2) 119865 (119902lowast

2) =

(119901 minus 119888)

(119901 minus 119904)

(A4)

In addition 119866(119902lowast

1) ge 119865(119902

lowast

1) 1205721

lt 1205722 so we know that

1205721119866(119902lowast

1) + (1minus120572

1)119865(119902lowast

1) lt 1205722119866(119902lowast

1) + (1minus120572

2)119865(119902lowast

1) and from

(A4) we obtain 1205722119866(119902lowast

2) + (1 minus 120572

2)119865(119902lowast

2) lt 1205722119866(119902lowast

1) + (1 minus

1205722)119865(119902lowast

1) and thus 119902lowast

2lt 119902lowast

1 From (A3) 119905lowast

1minus119905lowast

2= 119888(119902lowast

1minus119902lowast

2)minus

119904(119868(1205721 119902lowast

1) minus 119868(120572

2 119902lowast

2)) 119905lowast1minus 119905lowast

2=119902lowast

1=119902lowast

2

(1205721minus 1205722)119904(int119902

0119865(119910)119889119910 minus

int119902

0119866(119909)119889119909) gt 0 and 120597119905

lowast

1120597119902lowast

1gt 0 so 119905

lowast

1minus 119905lowast

2gt 0 and

thus we obtain 119905lowast

1gt 119905lowast

2 119881lowast

1= 119881lowast

2= 0 Furthermore we get

119880lowast

1gt 119880lowast

2from 120597119880120597120579 lt 0 and 120597119880120597119902 gt 0 Finally we have

Proposition 6

B Proof of Proposition 12

We denote 1205831and 120583

2as the Lagrange multipliers of (6) and

(14) and find the derivate of 1199051and 1199021 and thus we obtain

120597119880 (1205721 1199021)

1205971199051

= minus1 + 1205831+ 1205832= 0

120597119880 (1205721 1199021)

1205971199021

= 1199011198781015840(1205721 1199021) minus 1205831119888 + 12058311199041198681015840(1205721 1199021)

minus 12058321199011198781015840(1205722 1199021) = 0

(B1)

By rewriting (B1) we get the following formulas

1199041198681015840(1205721 119902119904

1) + 119901119878

1015840(1205721 119902119904

1) minus 119888

=(1 minus 120583

1)

1205831

[1199011198781015840(1205722 119902119904

1) minus 119901119878

1015840(1205721 119902119904

1)]

(B2)

1205831=

1199011198781015840(1205722 119902119904

1) minus 119901119878

1015840(1205721 119902119904

1)

1199041198681015840 (1205721 119902119904

1) + 1199011198781015840 (120572

2 119902119904

1) minus 119888

(B3)

1205832=

1199041198681015840(1205721 119902119904

1) + 119901119878

1015840(1205721 119902119904

1) minus 119888

1199041198681015840 (1205721 119902119904

1) + 1199011198781015840 (120572

2 119902119904

1) minus 119888

(B4)

Because the numerator of 1205831is negative so 120583

1is positive

and the denominator of 1205831is negative because 120583

2is positive

(when 1205832= 0 it cannot be separated from the former one)

the numerator of 1205832is negative nevertheless 1199011198781015840(120572

1 119902lowast

1) minus

119888 + 1199041198681015840(1205721 119902lowast

1) = 0 and thus 1199011198781015840(120572

1 119902119904

1) minus 119888 + 119904119868

1015840(1205721 119902119904

1) lt

1199011198781015840(1205721 119902lowast

1)minus 119888+ 119904119868

1015840(1205721 119902lowast

1) in addition 1205972119880120597119902

2lt 0 so 119902

119904

1gt

119902lowast

1 From 120583

1gt 0 and 120583

2gt 0 we deduce that (6) and (14)

are tight Furthermore the programs 1198751119865 and 119875

1

119898 have the

same objective function but the constraint of 1198751119898 is more

tight so there is distortion of the utility obtained from 1198751

119898 in

other words 1198801(119905119904

1 119902119904

1) lt 119880lowast

1 From Proposition 11 we obtain

1198802(1199050

2 1199020

2) = 119880lowast

2 So we have Proposition 12

C Proof of Propositions 14 and 15

We denote 120573 and 120574 as the Lagrange multipliers of (19) and(20) and find the derivate of 119905

1 1199021 and and thus we obtain

120597119871 (1199051 1199021 )

1205971199051

= minus1 + 120573120582 + 120574 = 0 (C1)

120597119871 (1199051 1199021 )

1205971199021

= 1199011198781015840(1205721 1199021) + 120573120582 [minus119888 + 119904119868

1015840(1205721 1199021)]

minus 1205741199011198781015840(1205721 1199021) = 0

(C2)

120597119871 (1199051 1199021 )

120597= minus120573 (1 minus 120582) + 120574 = 0 (C3)

To solve (C1)ndash(C3) we obtain 120573 = 1 120574 = 1 minus 120582 Andconsidering (C2) we get

1199041198681015840(1205721 119902119898

1) + 119901119878

1015840(1205721 119902119898

1) minus 119888

=1 minus 120582

120582[1199011198781015840(1205722 119902119898

1) minus 119901119878

1015840(1205721 119902119898

1)]

(C4)

Because (19) and (20) are tight we obtain the followingresults

119905119898

1= (1 minus 120582)

sdot [119901119878 (1205722 119902119898

1) minus 119901119878 (120572

2 119902lowast

2) + 119888119902

lowast

2minus 119904119868 (120572

2 119902lowast

2)]

+ 120582 [119888119902119898

1minus 119904119868 (120572

1 119902119898

1)]

119905119898

2= (1 minus 120582) [119888119902

lowast

2minus 119904119868 (120572

2 119902lowast

2)] + 120582119901119878 (120572

2 119902lowast

2)

+ 120582 [119888119902119898

1minus 119904119868 (120572

1 119902119898

1) minus 119901119878 (120572

2 119902119898

1)]

(C5)

Mathematical Problems in Engineering 11

To compare (A1) with (C4) we know 119902119898

1gt 119902lowast

1because

12059721198801205971199022

lt 0 and 1199011198781015840(1205722 119902) minus 119901119878

1015840(1205721 119902) lt 0 and the dis-

tortion level is changing with 120572 Finally we have Proposi-tion 14

To compare (C4) with (B2) and designate 1205820

= 120582 =

(1199011198781015840(1205722 1199020

1) minus 119901119878

1015840(1205721 1199020

1))(119904119868

1015840(1205721 1199020

1) + 119901119878

1015840(1205722 1199020

1) minus 119888) we

have the following conclusions

(1) When 120582 = 1205820 1199021198981

= 119902119904

1= 1199020

1 we obtain the same

contracts with the low-information-intensity situa-tion

(2) When 120582 lt 1205820 1199021198981

lt 119902119904

1= 1199020

1 we know there is more

distortion than the low-information-intensity alloca-tion and it improves anything

(3) When 120582 gt 1205820 1199021198981

lt 119902119904

1= 1199020

1 the interim efficient

allocation can improve the low-information-intensityallocation So we have Proposition 15

VariablesNotations

119888 Unit manufacturing cost1205721 Disruption probability of low-risk retailer

119910 Market demand without disruption119865(sdot) Distribution function of demand without

disruption119904 Unit salvage value119860 Market scale without disruption119878(120572119894 119902119894) Expected sales

119868(120572119894 119902119894) Expected unsold quantity

119880119894 Utility of the retailer

120582 Probability of low-risk retailer1205722 Disruption probability of high-risk retailer

119909 Market demand with disruption119866(sdot) Distribution function of demand with

disruption119901 Retail price119863 Market scale with disruption119881119894 Utility of the manufacturer

Control Variables

119902119894 Order quantity

119905119894 Transferring payment

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported in part by a project supported bythe Humanities and Social Sciences Project of the EducationMinistry (no 14YJC630187) Hebei Social Sciences Project(no HB15GL058) and the Fundamental Research Funds forthe Central Universities (no 2016MS123)

References

[1] J Barnett F Begen S Howes et al ldquoConsumersrsquo confidencereflections and response strategies following the horsemeatincidentrdquo Food Control vol 59 pp 721ndash730 2016

[2] A J Schmitt and L V Snyder ldquoInfinite-horizon models forinventory control under yield uncertainty and disruptionsrdquoComputers amp Operations Research vol 39 no 4 pp 850ndash8622012

[3] L V Snyder Z Atan P Peng et al ldquoORMS models for supplychain disruptions a reviewrdquo IIE Transactions vol 48 no 2 pp89ndash109 2016

[4] X Qi J F Bard and G Yu ldquoSupply chain coordination withdemand disruptionsrdquo Omega vol 32 no 4 pp 301ndash312 2004

[5] T Xiao G Yu Z Sheng and Y Xia ldquoCoordination of a supplychain with one-manufacturer and two-retailers under demandpromotion and disruption management decisionsrdquo Annals ofOperations Research vol 135 pp 87ndash109 2005

[6] F Hu C-C Lim Z Lu and X Sun ldquoCoordination in a single-retailer two-supplier supply chain under random demand andrandom supply with disruptionrdquo Discrete Dynamics in Natureand Society vol 2013 Article ID 484062 12 pages 2013

[7] J Li X Liu J Wu and F Yang ldquoCoordination of supplychain with a dominant retailer under demand disruptionsrdquoMathematical Problems in Engineering vol 2014 Article ID854681 10 pages 2014

[8] K Chen and T Xiao ldquoDemand disruption and coordination ofthe supply chain with a dominant retailerrdquo European Journal ofOperational Research vol 197 no 1 pp 225ndash234 2009

[9] J Li and F T S Chan ldquoThe impact of collaborative transporta-tion management on demand disruption of manufacturingsupply chainsrdquo International Journal of Production Research vol50 no 19 pp 5635ndash5650 2012

[10] Y Zheng T Shu S Wang S Chen K K Lai and L GanldquoDemand disruption and coordination of supply chain via effortand revenue sharingrdquo Applied Economics vol 47 no 54 pp5886ndash5901 2015

[11] Q Pang Y Hou and Y Lv ldquoCoordinating three-level supplychain under disruptions using revenue-sharing contract witheffort dependent demandrdquoMathematical Problems in Engineer-ing vol 2016 Article ID 9167864 10 pages 2016

[12] M Gumus S Ray and H Gurnani ldquoSupply-side story risksguarantees competition and information asymmetryrdquo Man-agement Science vol 58 no 9 pp 1694ndash1714 2012

[13] K B Hendricks and V R Singhal ldquoAn empirical analysisof the effect of supply chain disruptions on long-run stockprice performance and equity risk of the firmrdquo Production ampOperations Management vol 14 no 1 pp 35ndash52 2005

[14] W Schimidt and A Raman ldquoWhen supply-chain disruptionsmatterrdquo Harvard Business School Working Paper 13-006 2012

[15] N Bunkley Piecing together a supply chain The New YorkTimes 2013 httpwwwnytimescom20110513businessglobal13autohtml r=0

[16] S Sarkar and S Kumar ldquoA behavioral experiment on inventorymanagement with supply chain disruptionrdquo International Jour-nal of Production Economics vol 169 pp 169ndash178 2015

[17] G P Cachon ldquoSupply chain coordination with contractsrdquoHandbooks in Operations Research amp Management Science vol11 pp 227ndash339 2003

[18] C J Corbett and X De Groote ldquoA supplierrsquos optimal quantitydiscount policy under asymmetric informationrdquo ManagementScience vol 46 no 3 pp 444ndash450 2000

12 Mathematical Problems in Engineering

[19] C J Corbett ldquoStochastic inventory systems in a supply chainwith asymmetric information cycle stocks safety stocks andconsignment stockrdquoOperations Research vol 49 no 4 pp 487ndash500 2001

[20] A Matopoulos M Vlachopoulou V Manthou and B ManosldquoA conceptual framework for supply chain collaboration empir-ical evidence from the agri-food industryrdquo Supply Chain Man-agement vol 12 no 3 pp 177ndash186 2007

[21] A Y Ha and S Tong ldquoContracting and information sharingunder supply chain competitionrdquoManagement Science vol 54no 4 pp 701ndash715 2008

[22] Y-W Zhou ldquoA comparison of different quantity discountpricing policies in a two-echelon channel with stochastic andasymmetric demand informationrdquo European Journal of Opera-tional Research vol 181 no 2 pp 686ndash703 2007

[23] XGan S P Sethi and J Zhou ldquoCommitment-penalty contractsin drop-shipping supply chains with asymmetric demand infor-mationrdquo European Journal of Operational Research vol 204 no3 pp 449ndash462 2010

[24] D Lei J Li and Z Liu ldquoSupply chain contracts under demandand cost disruptions with asymmetric informationrdquo Interna-tional Journal of Production Economics vol 139 no 1 pp 116ndash126 2012

[25] Z B Yang G Aydın V Babich and D R Beil ldquoSupplydisruptions asymmetric information and a backup productionoptionrdquoManagement Science vol 55 no 2 pp 192ndash209 2009

[26] S Huang and C Yang ldquoSupply chain revelation mechanismdesign under asymmetric demand disruption informationrdquoOperations Research andManagement Science vol 23 no 6 pp116ndash127 2014

[27] S Oh and O Ozer ldquoMechanism design for capacity planningunder dynamic evolutions of asymmetric demand forecastsrdquoManagement Science vol 59 no 4 pp 987ndash1007 2013

[28] Q Feng G Lai and L X Lu ldquoDynamic bargaining in a supplychain with asymmetric demand informationrdquo ManagementScience vol 61 no 2 pp 301ndash315 2015

[29] Q Li B Li P Chen and P Hou ldquoDual-channel supply chaindecisions under asymmetric information with a risk-averseretailerrdquo Annals of Operations Research 2015

[30] J Wei K Govindan Y Li and J Zhao ldquoPricing and collectingdecisions in a closed-loop supply chain with symmetric andasymmetric informationrdquo Computers and Operations Researchvol 54 pp 257ndash265 2015

[31] K Inderfurth A Sadrieh and G Voigt ldquoThe impact of infor-mation sharing on supply chain performance under asymmetricinformationrdquo Production amp Operations Management vol 22no 2 pp 410ndash425 2013

[32] R B Myerson ldquoMechanism design by an informed principalrdquoEconometrica vol 51 no 6 pp 1767ndash1797 1983

[33] E Maskin and J Tirole ldquoThe principal-agent relationship withan informed principal the case of private valuesrdquo Econometricavol 58 no 2 pp 379ndash409 1990

[34] E Maskin and J Tirole ldquoThe principal-agent relationship withan informed principal II common valuesrdquo Econometrica vol60 no 1 pp 1ndash42 1992

[35] T Mylovanov and T Troger ldquoInformed-principal problemsin environments with generalized private valuesrdquo TheoreticalEconomics vol 7 no 3 pp 465ndash488 2012

[36] TMylovanov andT Troger ldquoMechanismdesign by an informedprincipal private values with transferable utilityrdquo Review ofEconomic Studies vol 81 no 4 pp 1668ndash1707 2014

[37] C Wagner T Mylovanov and T Troger ldquoInformed-principalproblem with moral hazard risk neutrality and no limitedliabilityrdquo Journal of EconomicTheory vol 159 pp 280ndash289 2015

[38] S Galperti ldquoCommon agency with informed principals menusand signalsrdquo Journal of Economic Theory vol 157 pp 648ndash6672015

[39] H Zhang Y Liu and J Huang ldquoSupply chain coordina-tion contracts under double sided disruptions simultaneouslyrdquoMathematical Problems in Engineering vol 2015 Article ID812043 9 pages 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 7

0 10

5

10

15

q1

02 04 06 08

120582

Figure 3 The order quantity in interim efficient allocation for low-risk retailer changes by its probability

0

1

101112

1314

150

5

10

15

20

25

30

0204

0608

tm 1

q1 120582

Figure 4 The transferring payment in interim efficient allocationfor low-risk retailer changes by the order quantity and its probability

principal model is lower than that in full information model(when 120572

1= 02 versus 120572

2= 04 119880lowast

1= 5340 gt 119880

0

1= 5338

and when 1205721= 03 versus 120572

2= 05 119905lowast

1= 1840 gt 119905

0

1= 1824)

This implies that it costs the low-risk retailer something tocertify its typeThese results are the samewith Proposition 12

Considering the situation of 1205721= 02 versus 120572

2= 04 we

obtain 1205820= 019 The order quantity for the low-risk retailer

119902119898

1changes with the proportion it has 120582 in interim efficient

allocation As Figure 3 shows if and only if 120582 varies withincertain area (120582 le 120582

0) we obtain 119902119898

1gt 119902lowast

1 as in the results in

Proposition 14The transferring payment in the interim efficient allo-

cation is depicted in Figure 4 (for the low-risk retailer)and Figure 5 (for the high-risk retailer) The transferringpayments are changing by the order quantity for the low-risk retailer in the interim efficient allocation and the low-risk retailerrsquos probability From Figure 4 when the low-riskretailerrsquos probability 120582 is less enough and the order quantityfor the low-risk retailer 119902119898

1is high enough the transferring

0

05

1

10 11 12 13 14 15

20

25

30

35

40

45

tm 2

q1

120582

Figure 5 The transferring payment in interim efficient allocationfor high-risk retailer changes by the low-risk retailerrsquos order quantityand probability

payment 1199051198981increases by the low-risk retailerrsquos probability 120582

but decreases by the order quantity for the low-risk retailer119902119898

1(bottom left in Figure 4) But mostly the transferring

payment 1199051198981increases by the low-risk retailerrsquos probability 120582

but decreases by the order quantity for the low-risk retailer 1199021198981

(red and orange part in Figure 4) Normally the transferringpayment should increase by the order quantity which meansldquoget more pay morerdquo From this view the low-risk retailerrsquosprobability 120582 should not be very small and the order quantityfor the low-risk retailer 119902

119898

1should not be close to the

maximum value 15From Figure 5 it is obviously seen that the transferring

payment for high-risk retailer in the interim efficient alloca-tion 119905119898

2increases by the order quantity for the low-risk retailer

119902119898

1but decreases by the low-risk retailerrsquos probability 120582

62 Sensitivity Analyses The sensitivity analyses for the low-information-intensity allocation versus the unit manufactur-ing cost 119888 are shown in Table 2 It shows the outcomes whenthe unit manufacturing cost 119888 changes by +25 and minus25once at a time and keeping remaining parameters The orderquantities decrease by the unit manufacturing cost but theinfluence for the high-risk retailerrsquos order quantity is largerthan that for the low-risk retailer (minus069 versus minus317 and+217 versus +307) The transferring payments increaseby the unit manufacturing cost More interesting the utilitiesdecrease by the unit manufacturing cost and the low-riskretailer receives a greater impact (minus940 versus minus922 and+955 versus +953) So from this point of view the low-risk retailer should encourage themanufacturer to reduce thecost more

Figure 6 shows the sensitivity analyses for the order quan-tities versus the unit manufacturing cost in which the orderquantities include the order quantity for the low-risk retailerin low-information-intensity allocation 119902

0

1 the order quantity

8 Mathematical Problems in Engineering

Table 2 Sensitivity analyses for low-information-intensity allocation versus the unit manufacturing cost

Changes of 119888 The low-information allocation for low-riskretailer (120572

1= 02)

The low-information allocationfor high-risk retailer (120572

2= 04)

Value Change rate Value Change rate Value Change rate

119888 = 25 +251199020

1= 1007 minus069 119902

0

2= 947 minus317

1199050

1= 2375 +2626 119905

0

2= 2243 +2331

1198800

1= 4836 minus940 119880

0

2= 4736 minus922

119888 = 2 Baseline1199020

1= 1014 mdash 119902

sb2

= 978 mdash1199050

1= 1881 mdash 119905

sb2

= 1819 mdash1198800

1= 5338 mdash 119880

sb2

= 5217 mdash

119888 = 15 minus251199020

1= 1036 +217 119902

0

2= 1008 +307

1199050

1= 1398 minus2568 119905

0

2= 1362 minus2512

1198800

1= 5848 +955 119880

0

2= 5714 +953

108

106

104

102

10

98

96

94

92

C

14 16 18 2 22 24 26 28 3

qsb1

qlowast1q2

Figure 6 The order quantities change by the unit manufacturingcost

for the low-risk retailer in full information contract 119902lowast1 and

the order quantity for the high-risk retailer in full informationcontract which is equal to that in low-information-intensityallocation 119902

2 Both the order quantity for the low-risk retailer

in full information and the order quantity for the high-riskretailer decrease by the unit manufacturing cost and theychange in the same level basically But the change of the orderquantity for the low-risk retailer in low information intensitydepends on different values of the unit manufacturing costWhen the unit manufacturing cost 119888 = 275 the orderquantity is minimum and when 119888 = 175 or 119888 = 15 the orderquantity is the maximum

Figure 7 shows the sensitivity analyses for the transferringpayment versus the unit manufacturing cost in which thetransferring payments include the transferring payment forthe low-risk retailer in low-information-intensity allocation1199050

1 the transferring payment for the low-risk retailer in full

information contract 119905lowast

1 and the transferring payment for

26

24

22

20

18

16

14

C

14 16 18 2 22 24 26 28 3

tsb1

tlowast1t2

Figure 7 The transferring payments change by the unit manufac-turing cost

the high-risk retailer in full information contract which isequal to that in low-information-intensity allocation 119905

2 All

the transferring payments of the retailers increase by the unitmanufacturing cost The transferring payments for the low-risk retailer and the high-risk retailer in full informationcontract decrease in the same level

Figure 8 shows the sensitivity analyses for the retailersrsquoutilities versus the unit manufacturing cost in which theutilities include the utility for the low-risk retailer in low-information-intensity allocation 119880

0

1 the transferring pay-

ment for the low-risk retailer in full information contract119880lowast

1 and the transferring payment for the high-risk retailer

in full information contract which is equal to that in low-information-intensity allocation 119880

2 All the retailersrsquo utilities

decrease by the unit manufacturing costFigure 9 shows the sensitivity analyses for the low-risk

retailerrsquos signaling cost versus the unit manufacturing cost

Mathematical Problems in Engineering 9

44

46

48

50

52

54

56

58

60

C

14 16 18 2 22 24 26 28

Usb1

Ulowast1

U2

Figure 8 The retailerrsquos utilities change by the unit manufacturingcost

0

005

01

015

02

025

C

14 16 18 2 22 24 26 28

ΔU1

Figure 9The signaling cost changes by the saved unit manufactur-ing cost

The signaling cost of the low-risk retailer is the utility earnedin the low-information-intensity allocation which is sepa-rating contract less than that in full information contractWhen the unit manufacturing cost 119888 = 2 the signaling costis the least while when the saved unit manufacturing cost119888 = 175 the signaling cost is the most We do not find theinternal relationship between the signaling cost and the unitmanufacturing cost As the signaling cost is so important toput the contracts into practice investigation on the signalingcost is the research direction for us

7 Conclusion

In the recent years supply chain with demand disruptionhas become more and more attractive to both business and

academic research There is a lot of research available on thetopic of supply chain disruption contract In the real supplychain practice demand disruption is totally different fromthe stochastic demand So in our research both the demanduncertainty and the demand disruption are consideredTaking the demand disruption probability as the asymmetricinformation an informed principal model is explored tomake the contract for supply chain with demand disruptionAnd the conclusions we obtain are shown as follows

(1) The contracts for low-risk retailer and high-riskretailer in full information are provided and it isfound out that the high-risk retailer has the incentiveto pretend to be the low-risk retailer This result issimilar to the other informed principal model [3435] but totally different from the normal principalagent model [24 26] In the normal principal agentmodel the high efficient agent (low-risk retailer in oursetting) tries to pretend to be low efficient agent (high-risk retailer) to get more That is why the informedprincipal model is established

(2) The low-information-intensity allocation which is theresult of the informed principal model and also is theseparating contract shows that the order quantity andthe transferring payment for the low-risk retailer inlow-information-intensity allocation distort upwardsbut that of high-risk retailer does not distort Inorder to avoid imitation the imitatorsrsquo efficiencyshould be always distorted But in normal principalagent model the high-risk retailerrsquos quantity distortsdownwards [24 26] on the contrary the low-riskretailerrsquos quantity in informed principal model dis-torts upwards

(3) In order to reduce the signaling cost which the low-risk retailer pays the interim efficient model is intro-duced which ends up with the order quantity andtransferring payment distorted upward again but lessthan that of low information intensity Comparingto [38] focusing on analyzing the signaling costmore attention is paid to showing the interim efficientmodel to get a solution which can increase efficiency

From the managerial aspect retailers should try toincrease their antirisk capability to enhance competitivepower And when the retailer has lower demand disruptionprobability than the others it can show its style to the supplierthrough the separating contract For example the retailer cansay ldquobecause I am the low-risk one I can cooperate with thesupplier with this contract while the others who are high-riskones can only use the other contractrdquo But in order to be sep-arated from the others the low-risk retailer has to pay somesignaling cost and the quantity has to be distorted upwards

Above all there are some limitations of the research Wewill continue our research in the following directions (1)The signaling cost is a key factor impacting the separatingcontract which should be paid more attention (2) We onlyconsider the demand disruption but disruptions happen atnot only demand side but also supply side Sowewill continue

10 Mathematical Problems in Engineering

our research with demand disruption together with supplydisruption [39]

Appendix

A Proof of Proposition 6

For the two types of retailers we can obtain the optimalallocation if and only if the whole margin utility of the supplychain is the same with the margin cost So 119902

lowast

1and 119902

lowast

2satisfy

the following first-order conditions

1199011198781015840(1205721 1199021) + 1199041198681015840(1205721 1199021) = 119888 (A1)

1199011198781015840(1205722 1199022) + 1199041198681015840(1205722 1199022) = 119888 (A2)

And the optimal transferring payments are

119905lowast

1= 119888119902lowast

1minus 119904119868 (120572

1 119902lowast

1)

119905lowast

2= 119888119902lowast

2minus 119904119868 (120572

2 119902lowast

2)

(A3)

From (A1) and (A2) we obtain

1205791119866 (119902lowast

1) + (1 minus 120579

1) 119865 (119902lowast

1)

= 1205792119866 (119902lowast

2) + (1 minus 120579

2) 119865 (119902lowast

2) =

(119901 minus 119888)

(119901 minus 119904)

(A4)

In addition 119866(119902lowast

1) ge 119865(119902

lowast

1) 1205721

lt 1205722 so we know that

1205721119866(119902lowast

1) + (1minus120572

1)119865(119902lowast

1) lt 1205722119866(119902lowast

1) + (1minus120572

2)119865(119902lowast

1) and from

(A4) we obtain 1205722119866(119902lowast

2) + (1 minus 120572

2)119865(119902lowast

2) lt 1205722119866(119902lowast

1) + (1 minus

1205722)119865(119902lowast

1) and thus 119902lowast

2lt 119902lowast

1 From (A3) 119905lowast

1minus119905lowast

2= 119888(119902lowast

1minus119902lowast

2)minus

119904(119868(1205721 119902lowast

1) minus 119868(120572

2 119902lowast

2)) 119905lowast1minus 119905lowast

2=119902lowast

1=119902lowast

2

(1205721minus 1205722)119904(int119902

0119865(119910)119889119910 minus

int119902

0119866(119909)119889119909) gt 0 and 120597119905

lowast

1120597119902lowast

1gt 0 so 119905

lowast

1minus 119905lowast

2gt 0 and

thus we obtain 119905lowast

1gt 119905lowast

2 119881lowast

1= 119881lowast

2= 0 Furthermore we get

119880lowast

1gt 119880lowast

2from 120597119880120597120579 lt 0 and 120597119880120597119902 gt 0 Finally we have

Proposition 6

B Proof of Proposition 12

We denote 1205831and 120583

2as the Lagrange multipliers of (6) and

(14) and find the derivate of 1199051and 1199021 and thus we obtain

120597119880 (1205721 1199021)

1205971199051

= minus1 + 1205831+ 1205832= 0

120597119880 (1205721 1199021)

1205971199021

= 1199011198781015840(1205721 1199021) minus 1205831119888 + 12058311199041198681015840(1205721 1199021)

minus 12058321199011198781015840(1205722 1199021) = 0

(B1)

By rewriting (B1) we get the following formulas

1199041198681015840(1205721 119902119904

1) + 119901119878

1015840(1205721 119902119904

1) minus 119888

=(1 minus 120583

1)

1205831

[1199011198781015840(1205722 119902119904

1) minus 119901119878

1015840(1205721 119902119904

1)]

(B2)

1205831=

1199011198781015840(1205722 119902119904

1) minus 119901119878

1015840(1205721 119902119904

1)

1199041198681015840 (1205721 119902119904

1) + 1199011198781015840 (120572

2 119902119904

1) minus 119888

(B3)

1205832=

1199041198681015840(1205721 119902119904

1) + 119901119878

1015840(1205721 119902119904

1) minus 119888

1199041198681015840 (1205721 119902119904

1) + 1199011198781015840 (120572

2 119902119904

1) minus 119888

(B4)

Because the numerator of 1205831is negative so 120583

1is positive

and the denominator of 1205831is negative because 120583

2is positive

(when 1205832= 0 it cannot be separated from the former one)

the numerator of 1205832is negative nevertheless 1199011198781015840(120572

1 119902lowast

1) minus

119888 + 1199041198681015840(1205721 119902lowast

1) = 0 and thus 1199011198781015840(120572

1 119902119904

1) minus 119888 + 119904119868

1015840(1205721 119902119904

1) lt

1199011198781015840(1205721 119902lowast

1)minus 119888+ 119904119868

1015840(1205721 119902lowast

1) in addition 1205972119880120597119902

2lt 0 so 119902

119904

1gt

119902lowast

1 From 120583

1gt 0 and 120583

2gt 0 we deduce that (6) and (14)

are tight Furthermore the programs 1198751119865 and 119875

1

119898 have the

same objective function but the constraint of 1198751119898 is more

tight so there is distortion of the utility obtained from 1198751

119898 in

other words 1198801(119905119904

1 119902119904

1) lt 119880lowast

1 From Proposition 11 we obtain

1198802(1199050

2 1199020

2) = 119880lowast

2 So we have Proposition 12

C Proof of Propositions 14 and 15

We denote 120573 and 120574 as the Lagrange multipliers of (19) and(20) and find the derivate of 119905

1 1199021 and and thus we obtain

120597119871 (1199051 1199021 )

1205971199051

= minus1 + 120573120582 + 120574 = 0 (C1)

120597119871 (1199051 1199021 )

1205971199021

= 1199011198781015840(1205721 1199021) + 120573120582 [minus119888 + 119904119868

1015840(1205721 1199021)]

minus 1205741199011198781015840(1205721 1199021) = 0

(C2)

120597119871 (1199051 1199021 )

120597= minus120573 (1 minus 120582) + 120574 = 0 (C3)

To solve (C1)ndash(C3) we obtain 120573 = 1 120574 = 1 minus 120582 Andconsidering (C2) we get

1199041198681015840(1205721 119902119898

1) + 119901119878

1015840(1205721 119902119898

1) minus 119888

=1 minus 120582

120582[1199011198781015840(1205722 119902119898

1) minus 119901119878

1015840(1205721 119902119898

1)]

(C4)

Because (19) and (20) are tight we obtain the followingresults

119905119898

1= (1 minus 120582)

sdot [119901119878 (1205722 119902119898

1) minus 119901119878 (120572

2 119902lowast

2) + 119888119902

lowast

2minus 119904119868 (120572

2 119902lowast

2)]

+ 120582 [119888119902119898

1minus 119904119868 (120572

1 119902119898

1)]

119905119898

2= (1 minus 120582) [119888119902

lowast

2minus 119904119868 (120572

2 119902lowast

2)] + 120582119901119878 (120572

2 119902lowast

2)

+ 120582 [119888119902119898

1minus 119904119868 (120572

1 119902119898

1) minus 119901119878 (120572

2 119902119898

1)]

(C5)

Mathematical Problems in Engineering 11

To compare (A1) with (C4) we know 119902119898

1gt 119902lowast

1because

12059721198801205971199022

lt 0 and 1199011198781015840(1205722 119902) minus 119901119878

1015840(1205721 119902) lt 0 and the dis-

tortion level is changing with 120572 Finally we have Proposi-tion 14

To compare (C4) with (B2) and designate 1205820

= 120582 =

(1199011198781015840(1205722 1199020

1) minus 119901119878

1015840(1205721 1199020

1))(119904119868

1015840(1205721 1199020

1) + 119901119878

1015840(1205722 1199020

1) minus 119888) we

have the following conclusions

(1) When 120582 = 1205820 1199021198981

= 119902119904

1= 1199020

1 we obtain the same

contracts with the low-information-intensity situa-tion

(2) When 120582 lt 1205820 1199021198981

lt 119902119904

1= 1199020

1 we know there is more

distortion than the low-information-intensity alloca-tion and it improves anything

(3) When 120582 gt 1205820 1199021198981

lt 119902119904

1= 1199020

1 the interim efficient

allocation can improve the low-information-intensityallocation So we have Proposition 15

VariablesNotations

119888 Unit manufacturing cost1205721 Disruption probability of low-risk retailer

119910 Market demand without disruption119865(sdot) Distribution function of demand without

disruption119904 Unit salvage value119860 Market scale without disruption119878(120572119894 119902119894) Expected sales

119868(120572119894 119902119894) Expected unsold quantity

119880119894 Utility of the retailer

120582 Probability of low-risk retailer1205722 Disruption probability of high-risk retailer

119909 Market demand with disruption119866(sdot) Distribution function of demand with

disruption119901 Retail price119863 Market scale with disruption119881119894 Utility of the manufacturer

Control Variables

119902119894 Order quantity

119905119894 Transferring payment

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported in part by a project supported bythe Humanities and Social Sciences Project of the EducationMinistry (no 14YJC630187) Hebei Social Sciences Project(no HB15GL058) and the Fundamental Research Funds forthe Central Universities (no 2016MS123)

References

[1] J Barnett F Begen S Howes et al ldquoConsumersrsquo confidencereflections and response strategies following the horsemeatincidentrdquo Food Control vol 59 pp 721ndash730 2016

[2] A J Schmitt and L V Snyder ldquoInfinite-horizon models forinventory control under yield uncertainty and disruptionsrdquoComputers amp Operations Research vol 39 no 4 pp 850ndash8622012

[3] L V Snyder Z Atan P Peng et al ldquoORMS models for supplychain disruptions a reviewrdquo IIE Transactions vol 48 no 2 pp89ndash109 2016

[4] X Qi J F Bard and G Yu ldquoSupply chain coordination withdemand disruptionsrdquo Omega vol 32 no 4 pp 301ndash312 2004

[5] T Xiao G Yu Z Sheng and Y Xia ldquoCoordination of a supplychain with one-manufacturer and two-retailers under demandpromotion and disruption management decisionsrdquo Annals ofOperations Research vol 135 pp 87ndash109 2005

[6] F Hu C-C Lim Z Lu and X Sun ldquoCoordination in a single-retailer two-supplier supply chain under random demand andrandom supply with disruptionrdquo Discrete Dynamics in Natureand Society vol 2013 Article ID 484062 12 pages 2013

[7] J Li X Liu J Wu and F Yang ldquoCoordination of supplychain with a dominant retailer under demand disruptionsrdquoMathematical Problems in Engineering vol 2014 Article ID854681 10 pages 2014

[8] K Chen and T Xiao ldquoDemand disruption and coordination ofthe supply chain with a dominant retailerrdquo European Journal ofOperational Research vol 197 no 1 pp 225ndash234 2009

[9] J Li and F T S Chan ldquoThe impact of collaborative transporta-tion management on demand disruption of manufacturingsupply chainsrdquo International Journal of Production Research vol50 no 19 pp 5635ndash5650 2012

[10] Y Zheng T Shu S Wang S Chen K K Lai and L GanldquoDemand disruption and coordination of supply chain via effortand revenue sharingrdquo Applied Economics vol 47 no 54 pp5886ndash5901 2015

[11] Q Pang Y Hou and Y Lv ldquoCoordinating three-level supplychain under disruptions using revenue-sharing contract witheffort dependent demandrdquoMathematical Problems in Engineer-ing vol 2016 Article ID 9167864 10 pages 2016

[12] M Gumus S Ray and H Gurnani ldquoSupply-side story risksguarantees competition and information asymmetryrdquo Man-agement Science vol 58 no 9 pp 1694ndash1714 2012

[13] K B Hendricks and V R Singhal ldquoAn empirical analysisof the effect of supply chain disruptions on long-run stockprice performance and equity risk of the firmrdquo Production ampOperations Management vol 14 no 1 pp 35ndash52 2005

[14] W Schimidt and A Raman ldquoWhen supply-chain disruptionsmatterrdquo Harvard Business School Working Paper 13-006 2012

[15] N Bunkley Piecing together a supply chain The New YorkTimes 2013 httpwwwnytimescom20110513businessglobal13autohtml r=0

[16] S Sarkar and S Kumar ldquoA behavioral experiment on inventorymanagement with supply chain disruptionrdquo International Jour-nal of Production Economics vol 169 pp 169ndash178 2015

[17] G P Cachon ldquoSupply chain coordination with contractsrdquoHandbooks in Operations Research amp Management Science vol11 pp 227ndash339 2003

[18] C J Corbett and X De Groote ldquoA supplierrsquos optimal quantitydiscount policy under asymmetric informationrdquo ManagementScience vol 46 no 3 pp 444ndash450 2000

12 Mathematical Problems in Engineering

[19] C J Corbett ldquoStochastic inventory systems in a supply chainwith asymmetric information cycle stocks safety stocks andconsignment stockrdquoOperations Research vol 49 no 4 pp 487ndash500 2001

[20] A Matopoulos M Vlachopoulou V Manthou and B ManosldquoA conceptual framework for supply chain collaboration empir-ical evidence from the agri-food industryrdquo Supply Chain Man-agement vol 12 no 3 pp 177ndash186 2007

[21] A Y Ha and S Tong ldquoContracting and information sharingunder supply chain competitionrdquoManagement Science vol 54no 4 pp 701ndash715 2008

[22] Y-W Zhou ldquoA comparison of different quantity discountpricing policies in a two-echelon channel with stochastic andasymmetric demand informationrdquo European Journal of Opera-tional Research vol 181 no 2 pp 686ndash703 2007

[23] XGan S P Sethi and J Zhou ldquoCommitment-penalty contractsin drop-shipping supply chains with asymmetric demand infor-mationrdquo European Journal of Operational Research vol 204 no3 pp 449ndash462 2010

[24] D Lei J Li and Z Liu ldquoSupply chain contracts under demandand cost disruptions with asymmetric informationrdquo Interna-tional Journal of Production Economics vol 139 no 1 pp 116ndash126 2012

[25] Z B Yang G Aydın V Babich and D R Beil ldquoSupplydisruptions asymmetric information and a backup productionoptionrdquoManagement Science vol 55 no 2 pp 192ndash209 2009

[26] S Huang and C Yang ldquoSupply chain revelation mechanismdesign under asymmetric demand disruption informationrdquoOperations Research andManagement Science vol 23 no 6 pp116ndash127 2014

[27] S Oh and O Ozer ldquoMechanism design for capacity planningunder dynamic evolutions of asymmetric demand forecastsrdquoManagement Science vol 59 no 4 pp 987ndash1007 2013

[28] Q Feng G Lai and L X Lu ldquoDynamic bargaining in a supplychain with asymmetric demand informationrdquo ManagementScience vol 61 no 2 pp 301ndash315 2015

[29] Q Li B Li P Chen and P Hou ldquoDual-channel supply chaindecisions under asymmetric information with a risk-averseretailerrdquo Annals of Operations Research 2015

[30] J Wei K Govindan Y Li and J Zhao ldquoPricing and collectingdecisions in a closed-loop supply chain with symmetric andasymmetric informationrdquo Computers and Operations Researchvol 54 pp 257ndash265 2015

[31] K Inderfurth A Sadrieh and G Voigt ldquoThe impact of infor-mation sharing on supply chain performance under asymmetricinformationrdquo Production amp Operations Management vol 22no 2 pp 410ndash425 2013

[32] R B Myerson ldquoMechanism design by an informed principalrdquoEconometrica vol 51 no 6 pp 1767ndash1797 1983

[33] E Maskin and J Tirole ldquoThe principal-agent relationship withan informed principal the case of private valuesrdquo Econometricavol 58 no 2 pp 379ndash409 1990

[34] E Maskin and J Tirole ldquoThe principal-agent relationship withan informed principal II common valuesrdquo Econometrica vol60 no 1 pp 1ndash42 1992

[35] T Mylovanov and T Troger ldquoInformed-principal problemsin environments with generalized private valuesrdquo TheoreticalEconomics vol 7 no 3 pp 465ndash488 2012

[36] TMylovanov andT Troger ldquoMechanismdesign by an informedprincipal private values with transferable utilityrdquo Review ofEconomic Studies vol 81 no 4 pp 1668ndash1707 2014

[37] C Wagner T Mylovanov and T Troger ldquoInformed-principalproblem with moral hazard risk neutrality and no limitedliabilityrdquo Journal of EconomicTheory vol 159 pp 280ndash289 2015

[38] S Galperti ldquoCommon agency with informed principals menusand signalsrdquo Journal of Economic Theory vol 157 pp 648ndash6672015

[39] H Zhang Y Liu and J Huang ldquoSupply chain coordina-tion contracts under double sided disruptions simultaneouslyrdquoMathematical Problems in Engineering vol 2015 Article ID812043 9 pages 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

8 Mathematical Problems in Engineering

Table 2 Sensitivity analyses for low-information-intensity allocation versus the unit manufacturing cost

Changes of 119888 The low-information allocation for low-riskretailer (120572

1= 02)

The low-information allocationfor high-risk retailer (120572

2= 04)

Value Change rate Value Change rate Value Change rate

119888 = 25 +251199020

1= 1007 minus069 119902

0

2= 947 minus317

1199050

1= 2375 +2626 119905

0

2= 2243 +2331

1198800

1= 4836 minus940 119880

0

2= 4736 minus922

119888 = 2 Baseline1199020

1= 1014 mdash 119902

sb2

= 978 mdash1199050

1= 1881 mdash 119905

sb2

= 1819 mdash1198800

1= 5338 mdash 119880

sb2

= 5217 mdash

119888 = 15 minus251199020

1= 1036 +217 119902

0

2= 1008 +307

1199050

1= 1398 minus2568 119905

0

2= 1362 minus2512

1198800

1= 5848 +955 119880

0

2= 5714 +953

108

106

104

102

10

98

96

94

92

C

14 16 18 2 22 24 26 28 3

qsb1

qlowast1q2

Figure 6 The order quantities change by the unit manufacturingcost

for the low-risk retailer in full information contract 119902lowast1 and

the order quantity for the high-risk retailer in full informationcontract which is equal to that in low-information-intensityallocation 119902

2 Both the order quantity for the low-risk retailer

in full information and the order quantity for the high-riskretailer decrease by the unit manufacturing cost and theychange in the same level basically But the change of the orderquantity for the low-risk retailer in low information intensitydepends on different values of the unit manufacturing costWhen the unit manufacturing cost 119888 = 275 the orderquantity is minimum and when 119888 = 175 or 119888 = 15 the orderquantity is the maximum

Figure 7 shows the sensitivity analyses for the transferringpayment versus the unit manufacturing cost in which thetransferring payments include the transferring payment forthe low-risk retailer in low-information-intensity allocation1199050

1 the transferring payment for the low-risk retailer in full

information contract 119905lowast

1 and the transferring payment for

26

24

22

20

18

16

14

C

14 16 18 2 22 24 26 28 3

tsb1

tlowast1t2

Figure 7 The transferring payments change by the unit manufac-turing cost

the high-risk retailer in full information contract which isequal to that in low-information-intensity allocation 119905

2 All

the transferring payments of the retailers increase by the unitmanufacturing cost The transferring payments for the low-risk retailer and the high-risk retailer in full informationcontract decrease in the same level

Figure 8 shows the sensitivity analyses for the retailersrsquoutilities versus the unit manufacturing cost in which theutilities include the utility for the low-risk retailer in low-information-intensity allocation 119880

0

1 the transferring pay-

ment for the low-risk retailer in full information contract119880lowast

1 and the transferring payment for the high-risk retailer

in full information contract which is equal to that in low-information-intensity allocation 119880

2 All the retailersrsquo utilities

decrease by the unit manufacturing costFigure 9 shows the sensitivity analyses for the low-risk

retailerrsquos signaling cost versus the unit manufacturing cost

Mathematical Problems in Engineering 9

44

46

48

50

52

54

56

58

60

C

14 16 18 2 22 24 26 28

Usb1

Ulowast1

U2

Figure 8 The retailerrsquos utilities change by the unit manufacturingcost

0

005

01

015

02

025

C

14 16 18 2 22 24 26 28

ΔU1

Figure 9The signaling cost changes by the saved unit manufactur-ing cost

The signaling cost of the low-risk retailer is the utility earnedin the low-information-intensity allocation which is sepa-rating contract less than that in full information contractWhen the unit manufacturing cost 119888 = 2 the signaling costis the least while when the saved unit manufacturing cost119888 = 175 the signaling cost is the most We do not find theinternal relationship between the signaling cost and the unitmanufacturing cost As the signaling cost is so important toput the contracts into practice investigation on the signalingcost is the research direction for us

7 Conclusion

In the recent years supply chain with demand disruptionhas become more and more attractive to both business and

academic research There is a lot of research available on thetopic of supply chain disruption contract In the real supplychain practice demand disruption is totally different fromthe stochastic demand So in our research both the demanduncertainty and the demand disruption are consideredTaking the demand disruption probability as the asymmetricinformation an informed principal model is explored tomake the contract for supply chain with demand disruptionAnd the conclusions we obtain are shown as follows

(1) The contracts for low-risk retailer and high-riskretailer in full information are provided and it isfound out that the high-risk retailer has the incentiveto pretend to be the low-risk retailer This result issimilar to the other informed principal model [3435] but totally different from the normal principalagent model [24 26] In the normal principal agentmodel the high efficient agent (low-risk retailer in oursetting) tries to pretend to be low efficient agent (high-risk retailer) to get more That is why the informedprincipal model is established

(2) The low-information-intensity allocation which is theresult of the informed principal model and also is theseparating contract shows that the order quantity andthe transferring payment for the low-risk retailer inlow-information-intensity allocation distort upwardsbut that of high-risk retailer does not distort Inorder to avoid imitation the imitatorsrsquo efficiencyshould be always distorted But in normal principalagent model the high-risk retailerrsquos quantity distortsdownwards [24 26] on the contrary the low-riskretailerrsquos quantity in informed principal model dis-torts upwards

(3) In order to reduce the signaling cost which the low-risk retailer pays the interim efficient model is intro-duced which ends up with the order quantity andtransferring payment distorted upward again but lessthan that of low information intensity Comparingto [38] focusing on analyzing the signaling costmore attention is paid to showing the interim efficientmodel to get a solution which can increase efficiency

From the managerial aspect retailers should try toincrease their antirisk capability to enhance competitivepower And when the retailer has lower demand disruptionprobability than the others it can show its style to the supplierthrough the separating contract For example the retailer cansay ldquobecause I am the low-risk one I can cooperate with thesupplier with this contract while the others who are high-riskones can only use the other contractrdquo But in order to be sep-arated from the others the low-risk retailer has to pay somesignaling cost and the quantity has to be distorted upwards

Above all there are some limitations of the research Wewill continue our research in the following directions (1)The signaling cost is a key factor impacting the separatingcontract which should be paid more attention (2) We onlyconsider the demand disruption but disruptions happen atnot only demand side but also supply side Sowewill continue

10 Mathematical Problems in Engineering

our research with demand disruption together with supplydisruption [39]

Appendix

A Proof of Proposition 6

For the two types of retailers we can obtain the optimalallocation if and only if the whole margin utility of the supplychain is the same with the margin cost So 119902

lowast

1and 119902

lowast

2satisfy

the following first-order conditions

1199011198781015840(1205721 1199021) + 1199041198681015840(1205721 1199021) = 119888 (A1)

1199011198781015840(1205722 1199022) + 1199041198681015840(1205722 1199022) = 119888 (A2)

And the optimal transferring payments are

119905lowast

1= 119888119902lowast

1minus 119904119868 (120572

1 119902lowast

1)

119905lowast

2= 119888119902lowast

2minus 119904119868 (120572

2 119902lowast

2)

(A3)

From (A1) and (A2) we obtain

1205791119866 (119902lowast

1) + (1 minus 120579

1) 119865 (119902lowast

1)

= 1205792119866 (119902lowast

2) + (1 minus 120579

2) 119865 (119902lowast

2) =

(119901 minus 119888)

(119901 minus 119904)

(A4)

In addition 119866(119902lowast

1) ge 119865(119902

lowast

1) 1205721

lt 1205722 so we know that

1205721119866(119902lowast

1) + (1minus120572

1)119865(119902lowast

1) lt 1205722119866(119902lowast

1) + (1minus120572

2)119865(119902lowast

1) and from

(A4) we obtain 1205722119866(119902lowast

2) + (1 minus 120572

2)119865(119902lowast

2) lt 1205722119866(119902lowast

1) + (1 minus

1205722)119865(119902lowast

1) and thus 119902lowast

2lt 119902lowast

1 From (A3) 119905lowast

1minus119905lowast

2= 119888(119902lowast

1minus119902lowast

2)minus

119904(119868(1205721 119902lowast

1) minus 119868(120572

2 119902lowast

2)) 119905lowast1minus 119905lowast

2=119902lowast

1=119902lowast

2

(1205721minus 1205722)119904(int119902

0119865(119910)119889119910 minus

int119902

0119866(119909)119889119909) gt 0 and 120597119905

lowast

1120597119902lowast

1gt 0 so 119905

lowast

1minus 119905lowast

2gt 0 and

thus we obtain 119905lowast

1gt 119905lowast

2 119881lowast

1= 119881lowast

2= 0 Furthermore we get

119880lowast

1gt 119880lowast

2from 120597119880120597120579 lt 0 and 120597119880120597119902 gt 0 Finally we have

Proposition 6

B Proof of Proposition 12

We denote 1205831and 120583

2as the Lagrange multipliers of (6) and

(14) and find the derivate of 1199051and 1199021 and thus we obtain

120597119880 (1205721 1199021)

1205971199051

= minus1 + 1205831+ 1205832= 0

120597119880 (1205721 1199021)

1205971199021

= 1199011198781015840(1205721 1199021) minus 1205831119888 + 12058311199041198681015840(1205721 1199021)

minus 12058321199011198781015840(1205722 1199021) = 0

(B1)

By rewriting (B1) we get the following formulas

1199041198681015840(1205721 119902119904

1) + 119901119878

1015840(1205721 119902119904

1) minus 119888

=(1 minus 120583

1)

1205831

[1199011198781015840(1205722 119902119904

1) minus 119901119878

1015840(1205721 119902119904

1)]

(B2)

1205831=

1199011198781015840(1205722 119902119904

1) minus 119901119878

1015840(1205721 119902119904

1)

1199041198681015840 (1205721 119902119904

1) + 1199011198781015840 (120572

2 119902119904

1) minus 119888

(B3)

1205832=

1199041198681015840(1205721 119902119904

1) + 119901119878

1015840(1205721 119902119904

1) minus 119888

1199041198681015840 (1205721 119902119904

1) + 1199011198781015840 (120572

2 119902119904

1) minus 119888

(B4)

Because the numerator of 1205831is negative so 120583

1is positive

and the denominator of 1205831is negative because 120583

2is positive

(when 1205832= 0 it cannot be separated from the former one)

the numerator of 1205832is negative nevertheless 1199011198781015840(120572

1 119902lowast

1) minus

119888 + 1199041198681015840(1205721 119902lowast

1) = 0 and thus 1199011198781015840(120572

1 119902119904

1) minus 119888 + 119904119868

1015840(1205721 119902119904

1) lt

1199011198781015840(1205721 119902lowast

1)minus 119888+ 119904119868

1015840(1205721 119902lowast

1) in addition 1205972119880120597119902

2lt 0 so 119902

119904

1gt

119902lowast

1 From 120583

1gt 0 and 120583

2gt 0 we deduce that (6) and (14)

are tight Furthermore the programs 1198751119865 and 119875

1

119898 have the

same objective function but the constraint of 1198751119898 is more

tight so there is distortion of the utility obtained from 1198751

119898 in

other words 1198801(119905119904

1 119902119904

1) lt 119880lowast

1 From Proposition 11 we obtain

1198802(1199050

2 1199020

2) = 119880lowast

2 So we have Proposition 12

C Proof of Propositions 14 and 15

We denote 120573 and 120574 as the Lagrange multipliers of (19) and(20) and find the derivate of 119905

1 1199021 and and thus we obtain

120597119871 (1199051 1199021 )

1205971199051

= minus1 + 120573120582 + 120574 = 0 (C1)

120597119871 (1199051 1199021 )

1205971199021

= 1199011198781015840(1205721 1199021) + 120573120582 [minus119888 + 119904119868

1015840(1205721 1199021)]

minus 1205741199011198781015840(1205721 1199021) = 0

(C2)

120597119871 (1199051 1199021 )

120597= minus120573 (1 minus 120582) + 120574 = 0 (C3)

To solve (C1)ndash(C3) we obtain 120573 = 1 120574 = 1 minus 120582 Andconsidering (C2) we get

1199041198681015840(1205721 119902119898

1) + 119901119878

1015840(1205721 119902119898

1) minus 119888

=1 minus 120582

120582[1199011198781015840(1205722 119902119898

1) minus 119901119878

1015840(1205721 119902119898

1)]

(C4)

Because (19) and (20) are tight we obtain the followingresults

119905119898

1= (1 minus 120582)

sdot [119901119878 (1205722 119902119898

1) minus 119901119878 (120572

2 119902lowast

2) + 119888119902

lowast

2minus 119904119868 (120572

2 119902lowast

2)]

+ 120582 [119888119902119898

1minus 119904119868 (120572

1 119902119898

1)]

119905119898

2= (1 minus 120582) [119888119902

lowast

2minus 119904119868 (120572

2 119902lowast

2)] + 120582119901119878 (120572

2 119902lowast

2)

+ 120582 [119888119902119898

1minus 119904119868 (120572

1 119902119898

1) minus 119901119878 (120572

2 119902119898

1)]

(C5)

Mathematical Problems in Engineering 11

To compare (A1) with (C4) we know 119902119898

1gt 119902lowast

1because

12059721198801205971199022

lt 0 and 1199011198781015840(1205722 119902) minus 119901119878

1015840(1205721 119902) lt 0 and the dis-

tortion level is changing with 120572 Finally we have Proposi-tion 14

To compare (C4) with (B2) and designate 1205820

= 120582 =

(1199011198781015840(1205722 1199020

1) minus 119901119878

1015840(1205721 1199020

1))(119904119868

1015840(1205721 1199020

1) + 119901119878

1015840(1205722 1199020

1) minus 119888) we

have the following conclusions

(1) When 120582 = 1205820 1199021198981

= 119902119904

1= 1199020

1 we obtain the same

contracts with the low-information-intensity situa-tion

(2) When 120582 lt 1205820 1199021198981

lt 119902119904

1= 1199020

1 we know there is more

distortion than the low-information-intensity alloca-tion and it improves anything

(3) When 120582 gt 1205820 1199021198981

lt 119902119904

1= 1199020

1 the interim efficient

allocation can improve the low-information-intensityallocation So we have Proposition 15

VariablesNotations

119888 Unit manufacturing cost1205721 Disruption probability of low-risk retailer

119910 Market demand without disruption119865(sdot) Distribution function of demand without

disruption119904 Unit salvage value119860 Market scale without disruption119878(120572119894 119902119894) Expected sales

119868(120572119894 119902119894) Expected unsold quantity

119880119894 Utility of the retailer

120582 Probability of low-risk retailer1205722 Disruption probability of high-risk retailer

119909 Market demand with disruption119866(sdot) Distribution function of demand with

disruption119901 Retail price119863 Market scale with disruption119881119894 Utility of the manufacturer

Control Variables

119902119894 Order quantity

119905119894 Transferring payment

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported in part by a project supported bythe Humanities and Social Sciences Project of the EducationMinistry (no 14YJC630187) Hebei Social Sciences Project(no HB15GL058) and the Fundamental Research Funds forthe Central Universities (no 2016MS123)

References

[1] J Barnett F Begen S Howes et al ldquoConsumersrsquo confidencereflections and response strategies following the horsemeatincidentrdquo Food Control vol 59 pp 721ndash730 2016

[2] A J Schmitt and L V Snyder ldquoInfinite-horizon models forinventory control under yield uncertainty and disruptionsrdquoComputers amp Operations Research vol 39 no 4 pp 850ndash8622012

[3] L V Snyder Z Atan P Peng et al ldquoORMS models for supplychain disruptions a reviewrdquo IIE Transactions vol 48 no 2 pp89ndash109 2016

[4] X Qi J F Bard and G Yu ldquoSupply chain coordination withdemand disruptionsrdquo Omega vol 32 no 4 pp 301ndash312 2004

[5] T Xiao G Yu Z Sheng and Y Xia ldquoCoordination of a supplychain with one-manufacturer and two-retailers under demandpromotion and disruption management decisionsrdquo Annals ofOperations Research vol 135 pp 87ndash109 2005

[6] F Hu C-C Lim Z Lu and X Sun ldquoCoordination in a single-retailer two-supplier supply chain under random demand andrandom supply with disruptionrdquo Discrete Dynamics in Natureand Society vol 2013 Article ID 484062 12 pages 2013

[7] J Li X Liu J Wu and F Yang ldquoCoordination of supplychain with a dominant retailer under demand disruptionsrdquoMathematical Problems in Engineering vol 2014 Article ID854681 10 pages 2014

[8] K Chen and T Xiao ldquoDemand disruption and coordination ofthe supply chain with a dominant retailerrdquo European Journal ofOperational Research vol 197 no 1 pp 225ndash234 2009

[9] J Li and F T S Chan ldquoThe impact of collaborative transporta-tion management on demand disruption of manufacturingsupply chainsrdquo International Journal of Production Research vol50 no 19 pp 5635ndash5650 2012

[10] Y Zheng T Shu S Wang S Chen K K Lai and L GanldquoDemand disruption and coordination of supply chain via effortand revenue sharingrdquo Applied Economics vol 47 no 54 pp5886ndash5901 2015

[11] Q Pang Y Hou and Y Lv ldquoCoordinating three-level supplychain under disruptions using revenue-sharing contract witheffort dependent demandrdquoMathematical Problems in Engineer-ing vol 2016 Article ID 9167864 10 pages 2016

[12] M Gumus S Ray and H Gurnani ldquoSupply-side story risksguarantees competition and information asymmetryrdquo Man-agement Science vol 58 no 9 pp 1694ndash1714 2012

[13] K B Hendricks and V R Singhal ldquoAn empirical analysisof the effect of supply chain disruptions on long-run stockprice performance and equity risk of the firmrdquo Production ampOperations Management vol 14 no 1 pp 35ndash52 2005

[14] W Schimidt and A Raman ldquoWhen supply-chain disruptionsmatterrdquo Harvard Business School Working Paper 13-006 2012

[15] N Bunkley Piecing together a supply chain The New YorkTimes 2013 httpwwwnytimescom20110513businessglobal13autohtml r=0

[16] S Sarkar and S Kumar ldquoA behavioral experiment on inventorymanagement with supply chain disruptionrdquo International Jour-nal of Production Economics vol 169 pp 169ndash178 2015

[17] G P Cachon ldquoSupply chain coordination with contractsrdquoHandbooks in Operations Research amp Management Science vol11 pp 227ndash339 2003

[18] C J Corbett and X De Groote ldquoA supplierrsquos optimal quantitydiscount policy under asymmetric informationrdquo ManagementScience vol 46 no 3 pp 444ndash450 2000

12 Mathematical Problems in Engineering

[19] C J Corbett ldquoStochastic inventory systems in a supply chainwith asymmetric information cycle stocks safety stocks andconsignment stockrdquoOperations Research vol 49 no 4 pp 487ndash500 2001

[20] A Matopoulos M Vlachopoulou V Manthou and B ManosldquoA conceptual framework for supply chain collaboration empir-ical evidence from the agri-food industryrdquo Supply Chain Man-agement vol 12 no 3 pp 177ndash186 2007

[21] A Y Ha and S Tong ldquoContracting and information sharingunder supply chain competitionrdquoManagement Science vol 54no 4 pp 701ndash715 2008

[22] Y-W Zhou ldquoA comparison of different quantity discountpricing policies in a two-echelon channel with stochastic andasymmetric demand informationrdquo European Journal of Opera-tional Research vol 181 no 2 pp 686ndash703 2007

[23] XGan S P Sethi and J Zhou ldquoCommitment-penalty contractsin drop-shipping supply chains with asymmetric demand infor-mationrdquo European Journal of Operational Research vol 204 no3 pp 449ndash462 2010

[24] D Lei J Li and Z Liu ldquoSupply chain contracts under demandand cost disruptions with asymmetric informationrdquo Interna-tional Journal of Production Economics vol 139 no 1 pp 116ndash126 2012

[25] Z B Yang G Aydın V Babich and D R Beil ldquoSupplydisruptions asymmetric information and a backup productionoptionrdquoManagement Science vol 55 no 2 pp 192ndash209 2009

[26] S Huang and C Yang ldquoSupply chain revelation mechanismdesign under asymmetric demand disruption informationrdquoOperations Research andManagement Science vol 23 no 6 pp116ndash127 2014

[27] S Oh and O Ozer ldquoMechanism design for capacity planningunder dynamic evolutions of asymmetric demand forecastsrdquoManagement Science vol 59 no 4 pp 987ndash1007 2013

[28] Q Feng G Lai and L X Lu ldquoDynamic bargaining in a supplychain with asymmetric demand informationrdquo ManagementScience vol 61 no 2 pp 301ndash315 2015

[29] Q Li B Li P Chen and P Hou ldquoDual-channel supply chaindecisions under asymmetric information with a risk-averseretailerrdquo Annals of Operations Research 2015

[30] J Wei K Govindan Y Li and J Zhao ldquoPricing and collectingdecisions in a closed-loop supply chain with symmetric andasymmetric informationrdquo Computers and Operations Researchvol 54 pp 257ndash265 2015

[31] K Inderfurth A Sadrieh and G Voigt ldquoThe impact of infor-mation sharing on supply chain performance under asymmetricinformationrdquo Production amp Operations Management vol 22no 2 pp 410ndash425 2013

[32] R B Myerson ldquoMechanism design by an informed principalrdquoEconometrica vol 51 no 6 pp 1767ndash1797 1983

[33] E Maskin and J Tirole ldquoThe principal-agent relationship withan informed principal the case of private valuesrdquo Econometricavol 58 no 2 pp 379ndash409 1990

[34] E Maskin and J Tirole ldquoThe principal-agent relationship withan informed principal II common valuesrdquo Econometrica vol60 no 1 pp 1ndash42 1992

[35] T Mylovanov and T Troger ldquoInformed-principal problemsin environments with generalized private valuesrdquo TheoreticalEconomics vol 7 no 3 pp 465ndash488 2012

[36] TMylovanov andT Troger ldquoMechanismdesign by an informedprincipal private values with transferable utilityrdquo Review ofEconomic Studies vol 81 no 4 pp 1668ndash1707 2014

[37] C Wagner T Mylovanov and T Troger ldquoInformed-principalproblem with moral hazard risk neutrality and no limitedliabilityrdquo Journal of EconomicTheory vol 159 pp 280ndash289 2015

[38] S Galperti ldquoCommon agency with informed principals menusand signalsrdquo Journal of Economic Theory vol 157 pp 648ndash6672015

[39] H Zhang Y Liu and J Huang ldquoSupply chain coordina-tion contracts under double sided disruptions simultaneouslyrdquoMathematical Problems in Engineering vol 2015 Article ID812043 9 pages 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 9

44

46

48

50

52

54

56

58

60

C

14 16 18 2 22 24 26 28

Usb1

Ulowast1

U2

Figure 8 The retailerrsquos utilities change by the unit manufacturingcost

0

005

01

015

02

025

C

14 16 18 2 22 24 26 28

ΔU1

Figure 9The signaling cost changes by the saved unit manufactur-ing cost

The signaling cost of the low-risk retailer is the utility earnedin the low-information-intensity allocation which is sepa-rating contract less than that in full information contractWhen the unit manufacturing cost 119888 = 2 the signaling costis the least while when the saved unit manufacturing cost119888 = 175 the signaling cost is the most We do not find theinternal relationship between the signaling cost and the unitmanufacturing cost As the signaling cost is so important toput the contracts into practice investigation on the signalingcost is the research direction for us

7 Conclusion

In the recent years supply chain with demand disruptionhas become more and more attractive to both business and

academic research There is a lot of research available on thetopic of supply chain disruption contract In the real supplychain practice demand disruption is totally different fromthe stochastic demand So in our research both the demanduncertainty and the demand disruption are consideredTaking the demand disruption probability as the asymmetricinformation an informed principal model is explored tomake the contract for supply chain with demand disruptionAnd the conclusions we obtain are shown as follows

(1) The contracts for low-risk retailer and high-riskretailer in full information are provided and it isfound out that the high-risk retailer has the incentiveto pretend to be the low-risk retailer This result issimilar to the other informed principal model [3435] but totally different from the normal principalagent model [24 26] In the normal principal agentmodel the high efficient agent (low-risk retailer in oursetting) tries to pretend to be low efficient agent (high-risk retailer) to get more That is why the informedprincipal model is established

(2) The low-information-intensity allocation which is theresult of the informed principal model and also is theseparating contract shows that the order quantity andthe transferring payment for the low-risk retailer inlow-information-intensity allocation distort upwardsbut that of high-risk retailer does not distort Inorder to avoid imitation the imitatorsrsquo efficiencyshould be always distorted But in normal principalagent model the high-risk retailerrsquos quantity distortsdownwards [24 26] on the contrary the low-riskretailerrsquos quantity in informed principal model dis-torts upwards

(3) In order to reduce the signaling cost which the low-risk retailer pays the interim efficient model is intro-duced which ends up with the order quantity andtransferring payment distorted upward again but lessthan that of low information intensity Comparingto [38] focusing on analyzing the signaling costmore attention is paid to showing the interim efficientmodel to get a solution which can increase efficiency

From the managerial aspect retailers should try toincrease their antirisk capability to enhance competitivepower And when the retailer has lower demand disruptionprobability than the others it can show its style to the supplierthrough the separating contract For example the retailer cansay ldquobecause I am the low-risk one I can cooperate with thesupplier with this contract while the others who are high-riskones can only use the other contractrdquo But in order to be sep-arated from the others the low-risk retailer has to pay somesignaling cost and the quantity has to be distorted upwards

Above all there are some limitations of the research Wewill continue our research in the following directions (1)The signaling cost is a key factor impacting the separatingcontract which should be paid more attention (2) We onlyconsider the demand disruption but disruptions happen atnot only demand side but also supply side Sowewill continue

10 Mathematical Problems in Engineering

our research with demand disruption together with supplydisruption [39]

Appendix

A Proof of Proposition 6

For the two types of retailers we can obtain the optimalallocation if and only if the whole margin utility of the supplychain is the same with the margin cost So 119902

lowast

1and 119902

lowast

2satisfy

the following first-order conditions

1199011198781015840(1205721 1199021) + 1199041198681015840(1205721 1199021) = 119888 (A1)

1199011198781015840(1205722 1199022) + 1199041198681015840(1205722 1199022) = 119888 (A2)

And the optimal transferring payments are

119905lowast

1= 119888119902lowast

1minus 119904119868 (120572

1 119902lowast

1)

119905lowast

2= 119888119902lowast

2minus 119904119868 (120572

2 119902lowast

2)

(A3)

From (A1) and (A2) we obtain

1205791119866 (119902lowast

1) + (1 minus 120579

1) 119865 (119902lowast

1)

= 1205792119866 (119902lowast

2) + (1 minus 120579

2) 119865 (119902lowast

2) =

(119901 minus 119888)

(119901 minus 119904)

(A4)

In addition 119866(119902lowast

1) ge 119865(119902

lowast

1) 1205721

lt 1205722 so we know that

1205721119866(119902lowast

1) + (1minus120572

1)119865(119902lowast

1) lt 1205722119866(119902lowast

1) + (1minus120572

2)119865(119902lowast

1) and from

(A4) we obtain 1205722119866(119902lowast

2) + (1 minus 120572

2)119865(119902lowast

2) lt 1205722119866(119902lowast

1) + (1 minus

1205722)119865(119902lowast

1) and thus 119902lowast

2lt 119902lowast

1 From (A3) 119905lowast

1minus119905lowast

2= 119888(119902lowast

1minus119902lowast

2)minus

119904(119868(1205721 119902lowast

1) minus 119868(120572

2 119902lowast

2)) 119905lowast1minus 119905lowast

2=119902lowast

1=119902lowast

2

(1205721minus 1205722)119904(int119902

0119865(119910)119889119910 minus

int119902

0119866(119909)119889119909) gt 0 and 120597119905

lowast

1120597119902lowast

1gt 0 so 119905

lowast

1minus 119905lowast

2gt 0 and

thus we obtain 119905lowast

1gt 119905lowast

2 119881lowast

1= 119881lowast

2= 0 Furthermore we get

119880lowast

1gt 119880lowast

2from 120597119880120597120579 lt 0 and 120597119880120597119902 gt 0 Finally we have

Proposition 6

B Proof of Proposition 12

We denote 1205831and 120583

2as the Lagrange multipliers of (6) and

(14) and find the derivate of 1199051and 1199021 and thus we obtain

120597119880 (1205721 1199021)

1205971199051

= minus1 + 1205831+ 1205832= 0

120597119880 (1205721 1199021)

1205971199021

= 1199011198781015840(1205721 1199021) minus 1205831119888 + 12058311199041198681015840(1205721 1199021)

minus 12058321199011198781015840(1205722 1199021) = 0

(B1)

By rewriting (B1) we get the following formulas

1199041198681015840(1205721 119902119904

1) + 119901119878

1015840(1205721 119902119904

1) minus 119888

=(1 minus 120583

1)

1205831

[1199011198781015840(1205722 119902119904

1) minus 119901119878

1015840(1205721 119902119904

1)]

(B2)

1205831=

1199011198781015840(1205722 119902119904

1) minus 119901119878

1015840(1205721 119902119904

1)

1199041198681015840 (1205721 119902119904

1) + 1199011198781015840 (120572

2 119902119904

1) minus 119888

(B3)

1205832=

1199041198681015840(1205721 119902119904

1) + 119901119878

1015840(1205721 119902119904

1) minus 119888

1199041198681015840 (1205721 119902119904

1) + 1199011198781015840 (120572

2 119902119904

1) minus 119888

(B4)

Because the numerator of 1205831is negative so 120583

1is positive

and the denominator of 1205831is negative because 120583

2is positive

(when 1205832= 0 it cannot be separated from the former one)

the numerator of 1205832is negative nevertheless 1199011198781015840(120572

1 119902lowast

1) minus

119888 + 1199041198681015840(1205721 119902lowast

1) = 0 and thus 1199011198781015840(120572

1 119902119904

1) minus 119888 + 119904119868

1015840(1205721 119902119904

1) lt

1199011198781015840(1205721 119902lowast

1)minus 119888+ 119904119868

1015840(1205721 119902lowast

1) in addition 1205972119880120597119902

2lt 0 so 119902

119904

1gt

119902lowast

1 From 120583

1gt 0 and 120583

2gt 0 we deduce that (6) and (14)

are tight Furthermore the programs 1198751119865 and 119875

1

119898 have the

same objective function but the constraint of 1198751119898 is more

tight so there is distortion of the utility obtained from 1198751

119898 in

other words 1198801(119905119904

1 119902119904

1) lt 119880lowast

1 From Proposition 11 we obtain

1198802(1199050

2 1199020

2) = 119880lowast

2 So we have Proposition 12

C Proof of Propositions 14 and 15

We denote 120573 and 120574 as the Lagrange multipliers of (19) and(20) and find the derivate of 119905

1 1199021 and and thus we obtain

120597119871 (1199051 1199021 )

1205971199051

= minus1 + 120573120582 + 120574 = 0 (C1)

120597119871 (1199051 1199021 )

1205971199021

= 1199011198781015840(1205721 1199021) + 120573120582 [minus119888 + 119904119868

1015840(1205721 1199021)]

minus 1205741199011198781015840(1205721 1199021) = 0

(C2)

120597119871 (1199051 1199021 )

120597= minus120573 (1 minus 120582) + 120574 = 0 (C3)

To solve (C1)ndash(C3) we obtain 120573 = 1 120574 = 1 minus 120582 Andconsidering (C2) we get

1199041198681015840(1205721 119902119898

1) + 119901119878

1015840(1205721 119902119898

1) minus 119888

=1 minus 120582

120582[1199011198781015840(1205722 119902119898

1) minus 119901119878

1015840(1205721 119902119898

1)]

(C4)

Because (19) and (20) are tight we obtain the followingresults

119905119898

1= (1 minus 120582)

sdot [119901119878 (1205722 119902119898

1) minus 119901119878 (120572

2 119902lowast

2) + 119888119902

lowast

2minus 119904119868 (120572

2 119902lowast

2)]

+ 120582 [119888119902119898

1minus 119904119868 (120572

1 119902119898

1)]

119905119898

2= (1 minus 120582) [119888119902

lowast

2minus 119904119868 (120572

2 119902lowast

2)] + 120582119901119878 (120572

2 119902lowast

2)

+ 120582 [119888119902119898

1minus 119904119868 (120572

1 119902119898

1) minus 119901119878 (120572

2 119902119898

1)]

(C5)

Mathematical Problems in Engineering 11

To compare (A1) with (C4) we know 119902119898

1gt 119902lowast

1because

12059721198801205971199022

lt 0 and 1199011198781015840(1205722 119902) minus 119901119878

1015840(1205721 119902) lt 0 and the dis-

tortion level is changing with 120572 Finally we have Proposi-tion 14

To compare (C4) with (B2) and designate 1205820

= 120582 =

(1199011198781015840(1205722 1199020

1) minus 119901119878

1015840(1205721 1199020

1))(119904119868

1015840(1205721 1199020

1) + 119901119878

1015840(1205722 1199020

1) minus 119888) we

have the following conclusions

(1) When 120582 = 1205820 1199021198981

= 119902119904

1= 1199020

1 we obtain the same

contracts with the low-information-intensity situa-tion

(2) When 120582 lt 1205820 1199021198981

lt 119902119904

1= 1199020

1 we know there is more

distortion than the low-information-intensity alloca-tion and it improves anything

(3) When 120582 gt 1205820 1199021198981

lt 119902119904

1= 1199020

1 the interim efficient

allocation can improve the low-information-intensityallocation So we have Proposition 15

VariablesNotations

119888 Unit manufacturing cost1205721 Disruption probability of low-risk retailer

119910 Market demand without disruption119865(sdot) Distribution function of demand without

disruption119904 Unit salvage value119860 Market scale without disruption119878(120572119894 119902119894) Expected sales

119868(120572119894 119902119894) Expected unsold quantity

119880119894 Utility of the retailer

120582 Probability of low-risk retailer1205722 Disruption probability of high-risk retailer

119909 Market demand with disruption119866(sdot) Distribution function of demand with

disruption119901 Retail price119863 Market scale with disruption119881119894 Utility of the manufacturer

Control Variables

119902119894 Order quantity

119905119894 Transferring payment

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported in part by a project supported bythe Humanities and Social Sciences Project of the EducationMinistry (no 14YJC630187) Hebei Social Sciences Project(no HB15GL058) and the Fundamental Research Funds forthe Central Universities (no 2016MS123)

References

[1] J Barnett F Begen S Howes et al ldquoConsumersrsquo confidencereflections and response strategies following the horsemeatincidentrdquo Food Control vol 59 pp 721ndash730 2016

[2] A J Schmitt and L V Snyder ldquoInfinite-horizon models forinventory control under yield uncertainty and disruptionsrdquoComputers amp Operations Research vol 39 no 4 pp 850ndash8622012

[3] L V Snyder Z Atan P Peng et al ldquoORMS models for supplychain disruptions a reviewrdquo IIE Transactions vol 48 no 2 pp89ndash109 2016

[4] X Qi J F Bard and G Yu ldquoSupply chain coordination withdemand disruptionsrdquo Omega vol 32 no 4 pp 301ndash312 2004

[5] T Xiao G Yu Z Sheng and Y Xia ldquoCoordination of a supplychain with one-manufacturer and two-retailers under demandpromotion and disruption management decisionsrdquo Annals ofOperations Research vol 135 pp 87ndash109 2005

[6] F Hu C-C Lim Z Lu and X Sun ldquoCoordination in a single-retailer two-supplier supply chain under random demand andrandom supply with disruptionrdquo Discrete Dynamics in Natureand Society vol 2013 Article ID 484062 12 pages 2013

[7] J Li X Liu J Wu and F Yang ldquoCoordination of supplychain with a dominant retailer under demand disruptionsrdquoMathematical Problems in Engineering vol 2014 Article ID854681 10 pages 2014

[8] K Chen and T Xiao ldquoDemand disruption and coordination ofthe supply chain with a dominant retailerrdquo European Journal ofOperational Research vol 197 no 1 pp 225ndash234 2009

[9] J Li and F T S Chan ldquoThe impact of collaborative transporta-tion management on demand disruption of manufacturingsupply chainsrdquo International Journal of Production Research vol50 no 19 pp 5635ndash5650 2012

[10] Y Zheng T Shu S Wang S Chen K K Lai and L GanldquoDemand disruption and coordination of supply chain via effortand revenue sharingrdquo Applied Economics vol 47 no 54 pp5886ndash5901 2015

[11] Q Pang Y Hou and Y Lv ldquoCoordinating three-level supplychain under disruptions using revenue-sharing contract witheffort dependent demandrdquoMathematical Problems in Engineer-ing vol 2016 Article ID 9167864 10 pages 2016

[12] M Gumus S Ray and H Gurnani ldquoSupply-side story risksguarantees competition and information asymmetryrdquo Man-agement Science vol 58 no 9 pp 1694ndash1714 2012

[13] K B Hendricks and V R Singhal ldquoAn empirical analysisof the effect of supply chain disruptions on long-run stockprice performance and equity risk of the firmrdquo Production ampOperations Management vol 14 no 1 pp 35ndash52 2005

[14] W Schimidt and A Raman ldquoWhen supply-chain disruptionsmatterrdquo Harvard Business School Working Paper 13-006 2012

[15] N Bunkley Piecing together a supply chain The New YorkTimes 2013 httpwwwnytimescom20110513businessglobal13autohtml r=0

[16] S Sarkar and S Kumar ldquoA behavioral experiment on inventorymanagement with supply chain disruptionrdquo International Jour-nal of Production Economics vol 169 pp 169ndash178 2015

[17] G P Cachon ldquoSupply chain coordination with contractsrdquoHandbooks in Operations Research amp Management Science vol11 pp 227ndash339 2003

[18] C J Corbett and X De Groote ldquoA supplierrsquos optimal quantitydiscount policy under asymmetric informationrdquo ManagementScience vol 46 no 3 pp 444ndash450 2000

12 Mathematical Problems in Engineering

[19] C J Corbett ldquoStochastic inventory systems in a supply chainwith asymmetric information cycle stocks safety stocks andconsignment stockrdquoOperations Research vol 49 no 4 pp 487ndash500 2001

[20] A Matopoulos M Vlachopoulou V Manthou and B ManosldquoA conceptual framework for supply chain collaboration empir-ical evidence from the agri-food industryrdquo Supply Chain Man-agement vol 12 no 3 pp 177ndash186 2007

[21] A Y Ha and S Tong ldquoContracting and information sharingunder supply chain competitionrdquoManagement Science vol 54no 4 pp 701ndash715 2008

[22] Y-W Zhou ldquoA comparison of different quantity discountpricing policies in a two-echelon channel with stochastic andasymmetric demand informationrdquo European Journal of Opera-tional Research vol 181 no 2 pp 686ndash703 2007

[23] XGan S P Sethi and J Zhou ldquoCommitment-penalty contractsin drop-shipping supply chains with asymmetric demand infor-mationrdquo European Journal of Operational Research vol 204 no3 pp 449ndash462 2010

[24] D Lei J Li and Z Liu ldquoSupply chain contracts under demandand cost disruptions with asymmetric informationrdquo Interna-tional Journal of Production Economics vol 139 no 1 pp 116ndash126 2012

[25] Z B Yang G Aydın V Babich and D R Beil ldquoSupplydisruptions asymmetric information and a backup productionoptionrdquoManagement Science vol 55 no 2 pp 192ndash209 2009

[26] S Huang and C Yang ldquoSupply chain revelation mechanismdesign under asymmetric demand disruption informationrdquoOperations Research andManagement Science vol 23 no 6 pp116ndash127 2014

[27] S Oh and O Ozer ldquoMechanism design for capacity planningunder dynamic evolutions of asymmetric demand forecastsrdquoManagement Science vol 59 no 4 pp 987ndash1007 2013

[28] Q Feng G Lai and L X Lu ldquoDynamic bargaining in a supplychain with asymmetric demand informationrdquo ManagementScience vol 61 no 2 pp 301ndash315 2015

[29] Q Li B Li P Chen and P Hou ldquoDual-channel supply chaindecisions under asymmetric information with a risk-averseretailerrdquo Annals of Operations Research 2015

[30] J Wei K Govindan Y Li and J Zhao ldquoPricing and collectingdecisions in a closed-loop supply chain with symmetric andasymmetric informationrdquo Computers and Operations Researchvol 54 pp 257ndash265 2015

[31] K Inderfurth A Sadrieh and G Voigt ldquoThe impact of infor-mation sharing on supply chain performance under asymmetricinformationrdquo Production amp Operations Management vol 22no 2 pp 410ndash425 2013

[32] R B Myerson ldquoMechanism design by an informed principalrdquoEconometrica vol 51 no 6 pp 1767ndash1797 1983

[33] E Maskin and J Tirole ldquoThe principal-agent relationship withan informed principal the case of private valuesrdquo Econometricavol 58 no 2 pp 379ndash409 1990

[34] E Maskin and J Tirole ldquoThe principal-agent relationship withan informed principal II common valuesrdquo Econometrica vol60 no 1 pp 1ndash42 1992

[35] T Mylovanov and T Troger ldquoInformed-principal problemsin environments with generalized private valuesrdquo TheoreticalEconomics vol 7 no 3 pp 465ndash488 2012

[36] TMylovanov andT Troger ldquoMechanismdesign by an informedprincipal private values with transferable utilityrdquo Review ofEconomic Studies vol 81 no 4 pp 1668ndash1707 2014

[37] C Wagner T Mylovanov and T Troger ldquoInformed-principalproblem with moral hazard risk neutrality and no limitedliabilityrdquo Journal of EconomicTheory vol 159 pp 280ndash289 2015

[38] S Galperti ldquoCommon agency with informed principals menusand signalsrdquo Journal of Economic Theory vol 157 pp 648ndash6672015

[39] H Zhang Y Liu and J Huang ldquoSupply chain coordina-tion contracts under double sided disruptions simultaneouslyrdquoMathematical Problems in Engineering vol 2015 Article ID812043 9 pages 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

10 Mathematical Problems in Engineering

our research with demand disruption together with supplydisruption [39]

Appendix

A Proof of Proposition 6

For the two types of retailers we can obtain the optimalallocation if and only if the whole margin utility of the supplychain is the same with the margin cost So 119902

lowast

1and 119902

lowast

2satisfy

the following first-order conditions

1199011198781015840(1205721 1199021) + 1199041198681015840(1205721 1199021) = 119888 (A1)

1199011198781015840(1205722 1199022) + 1199041198681015840(1205722 1199022) = 119888 (A2)

And the optimal transferring payments are

119905lowast

1= 119888119902lowast

1minus 119904119868 (120572

1 119902lowast

1)

119905lowast

2= 119888119902lowast

2minus 119904119868 (120572

2 119902lowast

2)

(A3)

From (A1) and (A2) we obtain

1205791119866 (119902lowast

1) + (1 minus 120579

1) 119865 (119902lowast

1)

= 1205792119866 (119902lowast

2) + (1 minus 120579

2) 119865 (119902lowast

2) =

(119901 minus 119888)

(119901 minus 119904)

(A4)

In addition 119866(119902lowast

1) ge 119865(119902

lowast

1) 1205721

lt 1205722 so we know that

1205721119866(119902lowast

1) + (1minus120572

1)119865(119902lowast

1) lt 1205722119866(119902lowast

1) + (1minus120572

2)119865(119902lowast

1) and from

(A4) we obtain 1205722119866(119902lowast

2) + (1 minus 120572

2)119865(119902lowast

2) lt 1205722119866(119902lowast

1) + (1 minus

1205722)119865(119902lowast

1) and thus 119902lowast

2lt 119902lowast

1 From (A3) 119905lowast

1minus119905lowast

2= 119888(119902lowast

1minus119902lowast

2)minus

119904(119868(1205721 119902lowast

1) minus 119868(120572

2 119902lowast

2)) 119905lowast1minus 119905lowast

2=119902lowast

1=119902lowast

2

(1205721minus 1205722)119904(int119902

0119865(119910)119889119910 minus

int119902

0119866(119909)119889119909) gt 0 and 120597119905

lowast

1120597119902lowast

1gt 0 so 119905

lowast

1minus 119905lowast

2gt 0 and

thus we obtain 119905lowast

1gt 119905lowast

2 119881lowast

1= 119881lowast

2= 0 Furthermore we get

119880lowast

1gt 119880lowast

2from 120597119880120597120579 lt 0 and 120597119880120597119902 gt 0 Finally we have

Proposition 6

B Proof of Proposition 12

We denote 1205831and 120583

2as the Lagrange multipliers of (6) and

(14) and find the derivate of 1199051and 1199021 and thus we obtain

120597119880 (1205721 1199021)

1205971199051

= minus1 + 1205831+ 1205832= 0

120597119880 (1205721 1199021)

1205971199021

= 1199011198781015840(1205721 1199021) minus 1205831119888 + 12058311199041198681015840(1205721 1199021)

minus 12058321199011198781015840(1205722 1199021) = 0

(B1)

By rewriting (B1) we get the following formulas

1199041198681015840(1205721 119902119904

1) + 119901119878

1015840(1205721 119902119904

1) minus 119888

=(1 minus 120583

1)

1205831

[1199011198781015840(1205722 119902119904

1) minus 119901119878

1015840(1205721 119902119904

1)]

(B2)

1205831=

1199011198781015840(1205722 119902119904

1) minus 119901119878

1015840(1205721 119902119904

1)

1199041198681015840 (1205721 119902119904

1) + 1199011198781015840 (120572

2 119902119904

1) minus 119888

(B3)

1205832=

1199041198681015840(1205721 119902119904

1) + 119901119878

1015840(1205721 119902119904

1) minus 119888

1199041198681015840 (1205721 119902119904

1) + 1199011198781015840 (120572

2 119902119904

1) minus 119888

(B4)

Because the numerator of 1205831is negative so 120583

1is positive

and the denominator of 1205831is negative because 120583

2is positive

(when 1205832= 0 it cannot be separated from the former one)

the numerator of 1205832is negative nevertheless 1199011198781015840(120572

1 119902lowast

1) minus

119888 + 1199041198681015840(1205721 119902lowast

1) = 0 and thus 1199011198781015840(120572

1 119902119904

1) minus 119888 + 119904119868

1015840(1205721 119902119904

1) lt

1199011198781015840(1205721 119902lowast

1)minus 119888+ 119904119868

1015840(1205721 119902lowast

1) in addition 1205972119880120597119902

2lt 0 so 119902

119904

1gt

119902lowast

1 From 120583

1gt 0 and 120583

2gt 0 we deduce that (6) and (14)

are tight Furthermore the programs 1198751119865 and 119875

1

119898 have the

same objective function but the constraint of 1198751119898 is more

tight so there is distortion of the utility obtained from 1198751

119898 in

other words 1198801(119905119904

1 119902119904

1) lt 119880lowast

1 From Proposition 11 we obtain

1198802(1199050

2 1199020

2) = 119880lowast

2 So we have Proposition 12

C Proof of Propositions 14 and 15

We denote 120573 and 120574 as the Lagrange multipliers of (19) and(20) and find the derivate of 119905

1 1199021 and and thus we obtain

120597119871 (1199051 1199021 )

1205971199051

= minus1 + 120573120582 + 120574 = 0 (C1)

120597119871 (1199051 1199021 )

1205971199021

= 1199011198781015840(1205721 1199021) + 120573120582 [minus119888 + 119904119868

1015840(1205721 1199021)]

minus 1205741199011198781015840(1205721 1199021) = 0

(C2)

120597119871 (1199051 1199021 )

120597= minus120573 (1 minus 120582) + 120574 = 0 (C3)

To solve (C1)ndash(C3) we obtain 120573 = 1 120574 = 1 minus 120582 Andconsidering (C2) we get

1199041198681015840(1205721 119902119898

1) + 119901119878

1015840(1205721 119902119898

1) minus 119888

=1 minus 120582

120582[1199011198781015840(1205722 119902119898

1) minus 119901119878

1015840(1205721 119902119898

1)]

(C4)

Because (19) and (20) are tight we obtain the followingresults

119905119898

1= (1 minus 120582)

sdot [119901119878 (1205722 119902119898

1) minus 119901119878 (120572

2 119902lowast

2) + 119888119902

lowast

2minus 119904119868 (120572

2 119902lowast

2)]

+ 120582 [119888119902119898

1minus 119904119868 (120572

1 119902119898

1)]

119905119898

2= (1 minus 120582) [119888119902

lowast

2minus 119904119868 (120572

2 119902lowast

2)] + 120582119901119878 (120572

2 119902lowast

2)

+ 120582 [119888119902119898

1minus 119904119868 (120572

1 119902119898

1) minus 119901119878 (120572

2 119902119898

1)]

(C5)

Mathematical Problems in Engineering 11

To compare (A1) with (C4) we know 119902119898

1gt 119902lowast

1because

12059721198801205971199022

lt 0 and 1199011198781015840(1205722 119902) minus 119901119878

1015840(1205721 119902) lt 0 and the dis-

tortion level is changing with 120572 Finally we have Proposi-tion 14

To compare (C4) with (B2) and designate 1205820

= 120582 =

(1199011198781015840(1205722 1199020

1) minus 119901119878

1015840(1205721 1199020

1))(119904119868

1015840(1205721 1199020

1) + 119901119878

1015840(1205722 1199020

1) minus 119888) we

have the following conclusions

(1) When 120582 = 1205820 1199021198981

= 119902119904

1= 1199020

1 we obtain the same

contracts with the low-information-intensity situa-tion

(2) When 120582 lt 1205820 1199021198981

lt 119902119904

1= 1199020

1 we know there is more

distortion than the low-information-intensity alloca-tion and it improves anything

(3) When 120582 gt 1205820 1199021198981

lt 119902119904

1= 1199020

1 the interim efficient

allocation can improve the low-information-intensityallocation So we have Proposition 15

VariablesNotations

119888 Unit manufacturing cost1205721 Disruption probability of low-risk retailer

119910 Market demand without disruption119865(sdot) Distribution function of demand without

disruption119904 Unit salvage value119860 Market scale without disruption119878(120572119894 119902119894) Expected sales

119868(120572119894 119902119894) Expected unsold quantity

119880119894 Utility of the retailer

120582 Probability of low-risk retailer1205722 Disruption probability of high-risk retailer

119909 Market demand with disruption119866(sdot) Distribution function of demand with

disruption119901 Retail price119863 Market scale with disruption119881119894 Utility of the manufacturer

Control Variables

119902119894 Order quantity

119905119894 Transferring payment

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported in part by a project supported bythe Humanities and Social Sciences Project of the EducationMinistry (no 14YJC630187) Hebei Social Sciences Project(no HB15GL058) and the Fundamental Research Funds forthe Central Universities (no 2016MS123)

References

[1] J Barnett F Begen S Howes et al ldquoConsumersrsquo confidencereflections and response strategies following the horsemeatincidentrdquo Food Control vol 59 pp 721ndash730 2016

[2] A J Schmitt and L V Snyder ldquoInfinite-horizon models forinventory control under yield uncertainty and disruptionsrdquoComputers amp Operations Research vol 39 no 4 pp 850ndash8622012

[3] L V Snyder Z Atan P Peng et al ldquoORMS models for supplychain disruptions a reviewrdquo IIE Transactions vol 48 no 2 pp89ndash109 2016

[4] X Qi J F Bard and G Yu ldquoSupply chain coordination withdemand disruptionsrdquo Omega vol 32 no 4 pp 301ndash312 2004

[5] T Xiao G Yu Z Sheng and Y Xia ldquoCoordination of a supplychain with one-manufacturer and two-retailers under demandpromotion and disruption management decisionsrdquo Annals ofOperations Research vol 135 pp 87ndash109 2005

[6] F Hu C-C Lim Z Lu and X Sun ldquoCoordination in a single-retailer two-supplier supply chain under random demand andrandom supply with disruptionrdquo Discrete Dynamics in Natureand Society vol 2013 Article ID 484062 12 pages 2013

[7] J Li X Liu J Wu and F Yang ldquoCoordination of supplychain with a dominant retailer under demand disruptionsrdquoMathematical Problems in Engineering vol 2014 Article ID854681 10 pages 2014

[8] K Chen and T Xiao ldquoDemand disruption and coordination ofthe supply chain with a dominant retailerrdquo European Journal ofOperational Research vol 197 no 1 pp 225ndash234 2009

[9] J Li and F T S Chan ldquoThe impact of collaborative transporta-tion management on demand disruption of manufacturingsupply chainsrdquo International Journal of Production Research vol50 no 19 pp 5635ndash5650 2012

[10] Y Zheng T Shu S Wang S Chen K K Lai and L GanldquoDemand disruption and coordination of supply chain via effortand revenue sharingrdquo Applied Economics vol 47 no 54 pp5886ndash5901 2015

[11] Q Pang Y Hou and Y Lv ldquoCoordinating three-level supplychain under disruptions using revenue-sharing contract witheffort dependent demandrdquoMathematical Problems in Engineer-ing vol 2016 Article ID 9167864 10 pages 2016

[12] M Gumus S Ray and H Gurnani ldquoSupply-side story risksguarantees competition and information asymmetryrdquo Man-agement Science vol 58 no 9 pp 1694ndash1714 2012

[13] K B Hendricks and V R Singhal ldquoAn empirical analysisof the effect of supply chain disruptions on long-run stockprice performance and equity risk of the firmrdquo Production ampOperations Management vol 14 no 1 pp 35ndash52 2005

[14] W Schimidt and A Raman ldquoWhen supply-chain disruptionsmatterrdquo Harvard Business School Working Paper 13-006 2012

[15] N Bunkley Piecing together a supply chain The New YorkTimes 2013 httpwwwnytimescom20110513businessglobal13autohtml r=0

[16] S Sarkar and S Kumar ldquoA behavioral experiment on inventorymanagement with supply chain disruptionrdquo International Jour-nal of Production Economics vol 169 pp 169ndash178 2015

[17] G P Cachon ldquoSupply chain coordination with contractsrdquoHandbooks in Operations Research amp Management Science vol11 pp 227ndash339 2003

[18] C J Corbett and X De Groote ldquoA supplierrsquos optimal quantitydiscount policy under asymmetric informationrdquo ManagementScience vol 46 no 3 pp 444ndash450 2000

12 Mathematical Problems in Engineering

[19] C J Corbett ldquoStochastic inventory systems in a supply chainwith asymmetric information cycle stocks safety stocks andconsignment stockrdquoOperations Research vol 49 no 4 pp 487ndash500 2001

[20] A Matopoulos M Vlachopoulou V Manthou and B ManosldquoA conceptual framework for supply chain collaboration empir-ical evidence from the agri-food industryrdquo Supply Chain Man-agement vol 12 no 3 pp 177ndash186 2007

[21] A Y Ha and S Tong ldquoContracting and information sharingunder supply chain competitionrdquoManagement Science vol 54no 4 pp 701ndash715 2008

[22] Y-W Zhou ldquoA comparison of different quantity discountpricing policies in a two-echelon channel with stochastic andasymmetric demand informationrdquo European Journal of Opera-tional Research vol 181 no 2 pp 686ndash703 2007

[23] XGan S P Sethi and J Zhou ldquoCommitment-penalty contractsin drop-shipping supply chains with asymmetric demand infor-mationrdquo European Journal of Operational Research vol 204 no3 pp 449ndash462 2010

[24] D Lei J Li and Z Liu ldquoSupply chain contracts under demandand cost disruptions with asymmetric informationrdquo Interna-tional Journal of Production Economics vol 139 no 1 pp 116ndash126 2012

[25] Z B Yang G Aydın V Babich and D R Beil ldquoSupplydisruptions asymmetric information and a backup productionoptionrdquoManagement Science vol 55 no 2 pp 192ndash209 2009

[26] S Huang and C Yang ldquoSupply chain revelation mechanismdesign under asymmetric demand disruption informationrdquoOperations Research andManagement Science vol 23 no 6 pp116ndash127 2014

[27] S Oh and O Ozer ldquoMechanism design for capacity planningunder dynamic evolutions of asymmetric demand forecastsrdquoManagement Science vol 59 no 4 pp 987ndash1007 2013

[28] Q Feng G Lai and L X Lu ldquoDynamic bargaining in a supplychain with asymmetric demand informationrdquo ManagementScience vol 61 no 2 pp 301ndash315 2015

[29] Q Li B Li P Chen and P Hou ldquoDual-channel supply chaindecisions under asymmetric information with a risk-averseretailerrdquo Annals of Operations Research 2015

[30] J Wei K Govindan Y Li and J Zhao ldquoPricing and collectingdecisions in a closed-loop supply chain with symmetric andasymmetric informationrdquo Computers and Operations Researchvol 54 pp 257ndash265 2015

[31] K Inderfurth A Sadrieh and G Voigt ldquoThe impact of infor-mation sharing on supply chain performance under asymmetricinformationrdquo Production amp Operations Management vol 22no 2 pp 410ndash425 2013

[32] R B Myerson ldquoMechanism design by an informed principalrdquoEconometrica vol 51 no 6 pp 1767ndash1797 1983

[33] E Maskin and J Tirole ldquoThe principal-agent relationship withan informed principal the case of private valuesrdquo Econometricavol 58 no 2 pp 379ndash409 1990

[34] E Maskin and J Tirole ldquoThe principal-agent relationship withan informed principal II common valuesrdquo Econometrica vol60 no 1 pp 1ndash42 1992

[35] T Mylovanov and T Troger ldquoInformed-principal problemsin environments with generalized private valuesrdquo TheoreticalEconomics vol 7 no 3 pp 465ndash488 2012

[36] TMylovanov andT Troger ldquoMechanismdesign by an informedprincipal private values with transferable utilityrdquo Review ofEconomic Studies vol 81 no 4 pp 1668ndash1707 2014

[37] C Wagner T Mylovanov and T Troger ldquoInformed-principalproblem with moral hazard risk neutrality and no limitedliabilityrdquo Journal of EconomicTheory vol 159 pp 280ndash289 2015

[38] S Galperti ldquoCommon agency with informed principals menusand signalsrdquo Journal of Economic Theory vol 157 pp 648ndash6672015

[39] H Zhang Y Liu and J Huang ldquoSupply chain coordina-tion contracts under double sided disruptions simultaneouslyrdquoMathematical Problems in Engineering vol 2015 Article ID812043 9 pages 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Mathematical Problems in Engineering 11

To compare (A1) with (C4) we know 119902119898

1gt 119902lowast

1because

12059721198801205971199022

lt 0 and 1199011198781015840(1205722 119902) minus 119901119878

1015840(1205721 119902) lt 0 and the dis-

tortion level is changing with 120572 Finally we have Proposi-tion 14

To compare (C4) with (B2) and designate 1205820

= 120582 =

(1199011198781015840(1205722 1199020

1) minus 119901119878

1015840(1205721 1199020

1))(119904119868

1015840(1205721 1199020

1) + 119901119878

1015840(1205722 1199020

1) minus 119888) we

have the following conclusions

(1) When 120582 = 1205820 1199021198981

= 119902119904

1= 1199020

1 we obtain the same

contracts with the low-information-intensity situa-tion

(2) When 120582 lt 1205820 1199021198981

lt 119902119904

1= 1199020

1 we know there is more

distortion than the low-information-intensity alloca-tion and it improves anything

(3) When 120582 gt 1205820 1199021198981

lt 119902119904

1= 1199020

1 the interim efficient

allocation can improve the low-information-intensityallocation So we have Proposition 15

VariablesNotations

119888 Unit manufacturing cost1205721 Disruption probability of low-risk retailer

119910 Market demand without disruption119865(sdot) Distribution function of demand without

disruption119904 Unit salvage value119860 Market scale without disruption119878(120572119894 119902119894) Expected sales

119868(120572119894 119902119894) Expected unsold quantity

119880119894 Utility of the retailer

120582 Probability of low-risk retailer1205722 Disruption probability of high-risk retailer

119909 Market demand with disruption119866(sdot) Distribution function of demand with

disruption119901 Retail price119863 Market scale with disruption119881119894 Utility of the manufacturer

Control Variables

119902119894 Order quantity

119905119894 Transferring payment

Competing Interests

The authors declare that there is no conflict of interestsregarding the publication of this paper

Acknowledgments

This research was supported in part by a project supported bythe Humanities and Social Sciences Project of the EducationMinistry (no 14YJC630187) Hebei Social Sciences Project(no HB15GL058) and the Fundamental Research Funds forthe Central Universities (no 2016MS123)

References

[1] J Barnett F Begen S Howes et al ldquoConsumersrsquo confidencereflections and response strategies following the horsemeatincidentrdquo Food Control vol 59 pp 721ndash730 2016

[2] A J Schmitt and L V Snyder ldquoInfinite-horizon models forinventory control under yield uncertainty and disruptionsrdquoComputers amp Operations Research vol 39 no 4 pp 850ndash8622012

[3] L V Snyder Z Atan P Peng et al ldquoORMS models for supplychain disruptions a reviewrdquo IIE Transactions vol 48 no 2 pp89ndash109 2016

[4] X Qi J F Bard and G Yu ldquoSupply chain coordination withdemand disruptionsrdquo Omega vol 32 no 4 pp 301ndash312 2004

[5] T Xiao G Yu Z Sheng and Y Xia ldquoCoordination of a supplychain with one-manufacturer and two-retailers under demandpromotion and disruption management decisionsrdquo Annals ofOperations Research vol 135 pp 87ndash109 2005

[6] F Hu C-C Lim Z Lu and X Sun ldquoCoordination in a single-retailer two-supplier supply chain under random demand andrandom supply with disruptionrdquo Discrete Dynamics in Natureand Society vol 2013 Article ID 484062 12 pages 2013

[7] J Li X Liu J Wu and F Yang ldquoCoordination of supplychain with a dominant retailer under demand disruptionsrdquoMathematical Problems in Engineering vol 2014 Article ID854681 10 pages 2014

[8] K Chen and T Xiao ldquoDemand disruption and coordination ofthe supply chain with a dominant retailerrdquo European Journal ofOperational Research vol 197 no 1 pp 225ndash234 2009

[9] J Li and F T S Chan ldquoThe impact of collaborative transporta-tion management on demand disruption of manufacturingsupply chainsrdquo International Journal of Production Research vol50 no 19 pp 5635ndash5650 2012

[10] Y Zheng T Shu S Wang S Chen K K Lai and L GanldquoDemand disruption and coordination of supply chain via effortand revenue sharingrdquo Applied Economics vol 47 no 54 pp5886ndash5901 2015

[11] Q Pang Y Hou and Y Lv ldquoCoordinating three-level supplychain under disruptions using revenue-sharing contract witheffort dependent demandrdquoMathematical Problems in Engineer-ing vol 2016 Article ID 9167864 10 pages 2016

[12] M Gumus S Ray and H Gurnani ldquoSupply-side story risksguarantees competition and information asymmetryrdquo Man-agement Science vol 58 no 9 pp 1694ndash1714 2012

[13] K B Hendricks and V R Singhal ldquoAn empirical analysisof the effect of supply chain disruptions on long-run stockprice performance and equity risk of the firmrdquo Production ampOperations Management vol 14 no 1 pp 35ndash52 2005

[14] W Schimidt and A Raman ldquoWhen supply-chain disruptionsmatterrdquo Harvard Business School Working Paper 13-006 2012

[15] N Bunkley Piecing together a supply chain The New YorkTimes 2013 httpwwwnytimescom20110513businessglobal13autohtml r=0

[16] S Sarkar and S Kumar ldquoA behavioral experiment on inventorymanagement with supply chain disruptionrdquo International Jour-nal of Production Economics vol 169 pp 169ndash178 2015

[17] G P Cachon ldquoSupply chain coordination with contractsrdquoHandbooks in Operations Research amp Management Science vol11 pp 227ndash339 2003

[18] C J Corbett and X De Groote ldquoA supplierrsquos optimal quantitydiscount policy under asymmetric informationrdquo ManagementScience vol 46 no 3 pp 444ndash450 2000

12 Mathematical Problems in Engineering

[19] C J Corbett ldquoStochastic inventory systems in a supply chainwith asymmetric information cycle stocks safety stocks andconsignment stockrdquoOperations Research vol 49 no 4 pp 487ndash500 2001

[20] A Matopoulos M Vlachopoulou V Manthou and B ManosldquoA conceptual framework for supply chain collaboration empir-ical evidence from the agri-food industryrdquo Supply Chain Man-agement vol 12 no 3 pp 177ndash186 2007

[21] A Y Ha and S Tong ldquoContracting and information sharingunder supply chain competitionrdquoManagement Science vol 54no 4 pp 701ndash715 2008

[22] Y-W Zhou ldquoA comparison of different quantity discountpricing policies in a two-echelon channel with stochastic andasymmetric demand informationrdquo European Journal of Opera-tional Research vol 181 no 2 pp 686ndash703 2007

[23] XGan S P Sethi and J Zhou ldquoCommitment-penalty contractsin drop-shipping supply chains with asymmetric demand infor-mationrdquo European Journal of Operational Research vol 204 no3 pp 449ndash462 2010

[24] D Lei J Li and Z Liu ldquoSupply chain contracts under demandand cost disruptions with asymmetric informationrdquo Interna-tional Journal of Production Economics vol 139 no 1 pp 116ndash126 2012

[25] Z B Yang G Aydın V Babich and D R Beil ldquoSupplydisruptions asymmetric information and a backup productionoptionrdquoManagement Science vol 55 no 2 pp 192ndash209 2009

[26] S Huang and C Yang ldquoSupply chain revelation mechanismdesign under asymmetric demand disruption informationrdquoOperations Research andManagement Science vol 23 no 6 pp116ndash127 2014

[27] S Oh and O Ozer ldquoMechanism design for capacity planningunder dynamic evolutions of asymmetric demand forecastsrdquoManagement Science vol 59 no 4 pp 987ndash1007 2013

[28] Q Feng G Lai and L X Lu ldquoDynamic bargaining in a supplychain with asymmetric demand informationrdquo ManagementScience vol 61 no 2 pp 301ndash315 2015

[29] Q Li B Li P Chen and P Hou ldquoDual-channel supply chaindecisions under asymmetric information with a risk-averseretailerrdquo Annals of Operations Research 2015

[30] J Wei K Govindan Y Li and J Zhao ldquoPricing and collectingdecisions in a closed-loop supply chain with symmetric andasymmetric informationrdquo Computers and Operations Researchvol 54 pp 257ndash265 2015

[31] K Inderfurth A Sadrieh and G Voigt ldquoThe impact of infor-mation sharing on supply chain performance under asymmetricinformationrdquo Production amp Operations Management vol 22no 2 pp 410ndash425 2013

[32] R B Myerson ldquoMechanism design by an informed principalrdquoEconometrica vol 51 no 6 pp 1767ndash1797 1983

[33] E Maskin and J Tirole ldquoThe principal-agent relationship withan informed principal the case of private valuesrdquo Econometricavol 58 no 2 pp 379ndash409 1990

[34] E Maskin and J Tirole ldquoThe principal-agent relationship withan informed principal II common valuesrdquo Econometrica vol60 no 1 pp 1ndash42 1992

[35] T Mylovanov and T Troger ldquoInformed-principal problemsin environments with generalized private valuesrdquo TheoreticalEconomics vol 7 no 3 pp 465ndash488 2012

[36] TMylovanov andT Troger ldquoMechanismdesign by an informedprincipal private values with transferable utilityrdquo Review ofEconomic Studies vol 81 no 4 pp 1668ndash1707 2014

[37] C Wagner T Mylovanov and T Troger ldquoInformed-principalproblem with moral hazard risk neutrality and no limitedliabilityrdquo Journal of EconomicTheory vol 159 pp 280ndash289 2015

[38] S Galperti ldquoCommon agency with informed principals menusand signalsrdquo Journal of Economic Theory vol 157 pp 648ndash6672015

[39] H Zhang Y Liu and J Huang ldquoSupply chain coordina-tion contracts under double sided disruptions simultaneouslyrdquoMathematical Problems in Engineering vol 2015 Article ID812043 9 pages 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

12 Mathematical Problems in Engineering

[19] C J Corbett ldquoStochastic inventory systems in a supply chainwith asymmetric information cycle stocks safety stocks andconsignment stockrdquoOperations Research vol 49 no 4 pp 487ndash500 2001

[20] A Matopoulos M Vlachopoulou V Manthou and B ManosldquoA conceptual framework for supply chain collaboration empir-ical evidence from the agri-food industryrdquo Supply Chain Man-agement vol 12 no 3 pp 177ndash186 2007

[21] A Y Ha and S Tong ldquoContracting and information sharingunder supply chain competitionrdquoManagement Science vol 54no 4 pp 701ndash715 2008

[22] Y-W Zhou ldquoA comparison of different quantity discountpricing policies in a two-echelon channel with stochastic andasymmetric demand informationrdquo European Journal of Opera-tional Research vol 181 no 2 pp 686ndash703 2007

[23] XGan S P Sethi and J Zhou ldquoCommitment-penalty contractsin drop-shipping supply chains with asymmetric demand infor-mationrdquo European Journal of Operational Research vol 204 no3 pp 449ndash462 2010

[24] D Lei J Li and Z Liu ldquoSupply chain contracts under demandand cost disruptions with asymmetric informationrdquo Interna-tional Journal of Production Economics vol 139 no 1 pp 116ndash126 2012

[25] Z B Yang G Aydın V Babich and D R Beil ldquoSupplydisruptions asymmetric information and a backup productionoptionrdquoManagement Science vol 55 no 2 pp 192ndash209 2009

[26] S Huang and C Yang ldquoSupply chain revelation mechanismdesign under asymmetric demand disruption informationrdquoOperations Research andManagement Science vol 23 no 6 pp116ndash127 2014

[27] S Oh and O Ozer ldquoMechanism design for capacity planningunder dynamic evolutions of asymmetric demand forecastsrdquoManagement Science vol 59 no 4 pp 987ndash1007 2013

[28] Q Feng G Lai and L X Lu ldquoDynamic bargaining in a supplychain with asymmetric demand informationrdquo ManagementScience vol 61 no 2 pp 301ndash315 2015

[29] Q Li B Li P Chen and P Hou ldquoDual-channel supply chaindecisions under asymmetric information with a risk-averseretailerrdquo Annals of Operations Research 2015

[30] J Wei K Govindan Y Li and J Zhao ldquoPricing and collectingdecisions in a closed-loop supply chain with symmetric andasymmetric informationrdquo Computers and Operations Researchvol 54 pp 257ndash265 2015

[31] K Inderfurth A Sadrieh and G Voigt ldquoThe impact of infor-mation sharing on supply chain performance under asymmetricinformationrdquo Production amp Operations Management vol 22no 2 pp 410ndash425 2013

[32] R B Myerson ldquoMechanism design by an informed principalrdquoEconometrica vol 51 no 6 pp 1767ndash1797 1983

[33] E Maskin and J Tirole ldquoThe principal-agent relationship withan informed principal the case of private valuesrdquo Econometricavol 58 no 2 pp 379ndash409 1990

[34] E Maskin and J Tirole ldquoThe principal-agent relationship withan informed principal II common valuesrdquo Econometrica vol60 no 1 pp 1ndash42 1992

[35] T Mylovanov and T Troger ldquoInformed-principal problemsin environments with generalized private valuesrdquo TheoreticalEconomics vol 7 no 3 pp 465ndash488 2012

[36] TMylovanov andT Troger ldquoMechanismdesign by an informedprincipal private values with transferable utilityrdquo Review ofEconomic Studies vol 81 no 4 pp 1668ndash1707 2014

[37] C Wagner T Mylovanov and T Troger ldquoInformed-principalproblem with moral hazard risk neutrality and no limitedliabilityrdquo Journal of EconomicTheory vol 159 pp 280ndash289 2015

[38] S Galperti ldquoCommon agency with informed principals menusand signalsrdquo Journal of Economic Theory vol 157 pp 648ndash6672015

[39] H Zhang Y Liu and J Huang ldquoSupply chain coordina-tion contracts under double sided disruptions simultaneouslyrdquoMathematical Problems in Engineering vol 2015 Article ID812043 9 pages 2015

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of

Submit your manuscripts athttpwwwhindawicom

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical Problems in Engineering

Hindawi Publishing Corporationhttpwwwhindawicom

Differential EquationsInternational Journal of

Volume 2014

Applied MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Probability and StatisticsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Mathematical PhysicsAdvances in

Complex AnalysisJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

OptimizationJournal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

CombinatoricsHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Operations ResearchAdvances in

Journal of

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Function Spaces

Abstract and Applied AnalysisHindawi Publishing Corporationhttpwwwhindawicom Volume 2014

International Journal of Mathematics and Mathematical Sciences

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

The Scientific World JournalHindawi Publishing Corporation httpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Algebra

Discrete Dynamics in Nature and Society

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Decision SciencesAdvances in

Discrete MathematicsJournal of

Hindawi Publishing Corporationhttpwwwhindawicom

Volume 2014 Hindawi Publishing Corporationhttpwwwhindawicom Volume 2014

Stochastic AnalysisInternational Journal of