Joint Pricing and Inventory Management of Interbasin Water

Research ArticleJoint Pricing and Inventory Management of Interbasin WaterTransfer Supply Chain

Xue Chen1 and Zhisong Chen 12

1Business School Nanjing Normal University Qixia District Nanjing 210023 China2Stern School of Business New York University 44 West Fourth Street New York 10012 NY USA

Correspondence should be addressed to Zhisong Chen zhisongchengmailcom

Received 19 June 2020 Revised 7 August 2020 Accepted 19 August 2020 Published 7 September 2020

Academic Editor Shouwei Li

Copyright copy 2020 Xue Chen and Zhisong Chen )is is an open access article distributed under the Creative CommonsAttribution License which permits unrestricted use distribution and reproduction in anymedium provided the original work isproperly cited

Four game-theoretical decision models withoutwith backlogging for the interbasin water transfer (IBWT) supply chain con-sidering water delivery loss under joint pricing and inventory management (JPIM) are first developed analyzed and comparedthen the corresponding numerical and sensitivity analyses are conducted and compared finally the managerial insights andpractical implementations are summarized in this paper )e research results indicate that (1) a revenue and cost sharing contractcould effectively coordinate the IBWT supply chain and improve the operational performance of the IBWT supply chain underJPIM (2) the partial backlogging strategy of water demand could effectively improve the operational performance of the IBWTsupply chain under JPIM (3) coordination strategy with partial backlogging is the best strategy for improving the operationalperformance of the IBWT supply chain under JPIM (4) reducing water delivery loss rate and operational costs and increasingbacklogging ratio are beneficial to improving the operational performance of the IBWT supply chain under JPIM

1 Introduction

To alleviate the shortage of water resources in arid andsemiarid areas various kinds of interbasin water transfer(IBWT) projects have been constructed and operated allover the world such as the South-to-North Water Diversion(SNWD) Project in China the California State WaterProject the Central Arizona Project and the Colorado RiverAqueduct in the US the Indira Gandhi Canal and the TeluguGanga Project in India the Snowy Mountains Scheme inAustralia the North Sinai Development Project in Egyptand the National Water Carrier in Israel [1 2] In thepractical operation management of the IBWT project theexisting rigid water price mechanism for the IBWTproject isdecoupled from the water supply-demand relationship andcannot effectively exert the regulatory role of the marketmechanism and coordinate the interests of all parties in-volved )us how to optimize pricing to achieve operationalperformance improvement under a flexible water pricemechanism that is linked to the water supply-demand

relationship is an urgent problem for the IBWT projectsFurthermore due to the random water demand the orderquantity may mismatch with water demand )e waterdemand may be lower than the order quantity and theholding cost of excess water inventory will thus be incurredon the contrary the water demand may be higher than theorder quantity and the shortage cost of excess water demandwill thus be incurred Hence how to jointly make optimalpricing and inventory decisions to achieve operationalperformance improvement is also an important issue for theIBWT projects

From the perspective of supply chain management theavailable research has explored the subsidy policy and theoperational strategy of the IBWT green supply chain undersocial welfare maximization [3 4] the impact of the supplycapacity constraint and fairness concern on the operationaldecisions and outcomes of the IBWT supply chain underrandom precipitation [5] and the impact of fullypartialbacklogging on the operational decisions and outcomes ofIBWT green supply chain coordination considering water

HindawiComplexityVolume 2020 Article ID 3954084 16 pageshttpsdoiorg10115520203954084

delivery loss under random precipitation [6] However thejoint pricing-inventory management decisions and opera-tional strategies for an IBWTsupply chain considering waterdelivery loss and partial backlogging are rarely investigatedin the current literatures and practices

)erefore this paper will try to explore a novelty re-search issue regarding the operation management of theIBWT supply chainmdashthe joint pricing-inventory manage-ment (JPIM) decisions and operational strategies for theIBWT supply chain considering water delivery loss andpartial backlogging under random water demand

In the following sections the related literatures arereviewed first in Section 2 the theoretical modeling nota-tions and assumptions for a generic IBWT supply chain aredefined in Section 3 four game-theoretical decision modelsfor the IBWT supply chain withoutwith backlogging underjoint pricing-inventory management (JPIM) are developedanalyzed and compared in Section 4 the correspondingnumerical and sensitivity analyses for all models areimplemented and the corresponding results are comparedin Section 5 the managerial insights and practical imple-mentations are then summarized in Section 6 finally thetheoretical and practical contributions are summarized

2 Literature Review

Currently the interaction relationships among multiplestakeholders in the IBWT projects are investigated throughgame theory such as the water conflict game-theoreticalmodel of the SNWD project [7] game-theoretical model ofthe IBWT project considering both the water quantity andwater quality [8] water allocation option contract for theIBWT projects [9] and incentive-compatible payment de-sign for the SNWD project [10]

Besides cooperative game theory is applied to achievePareto improvement in the IBWT projects such as thecrisp and fuzzy Shapley game model for the IBWT waterallocation [11] cooperative game model for the IBWTwater allocation [12] IBWT water resource allocationusing the least core game [13] and robust multiobjectivebargaining game model for the IBWT water resource al-location [14]

Currently theories and techniques of supply chainmanagement (SCM) have been applied in the IBWTprojectsto investigate the interactions among multiple stakeholdersand develop equilibriumcoordination operational mecha-nisms such as optimal pricing and coordination schemes forthe SNWD supply chain [15] coordination mechanismbased on revenue sharing contract for the SNWD supplychain with strategic customer [16] asymmetric Nash bar-gainingmodel for the SNWD supply chain [17] two-echelonwater inventory model with inflow forecasting updates in anIBWT project [18] two-tier pricing and allocation schemesfor the SNWD supply chain [19] competition intensity inthe water supply chain under two contracts [20] powerstructures for the competitive water supply chains [21]optimal pricing and ordering strategies for dual competingwater supply chains under three contracts [22] subsidypolicies and operational strategies for the IBWT green

supply chain under social welfare maximization [3 4]impact of the supply capacity constraint and fairness con-cern on the operational decisions and outcomes of the IBWTsupply chain under random precipitation [5] and impact offullypartial backlogging on the IBWT green supply chaincoordination considering water delivery loss under randomprecipitation [6]

Nevertheless these existing literatures regarding IBWTsupply chain management neither explored the equilibriumcoordination strategies of the IBWT supply chain underJPIM nor investigated the impact of the partial backloggingthe choice of operational strategies and the water deliveryloss on the operational performance of the IBWT supplychain )is paper tries to address the shortcomings in theavailable literatures and explore the operational strategies foran IBWT supply chain withoutwith partial backloggingunder JPIM An equilibrium decision model and a coor-dination decision model for the IBWTsupply chain withoutbackloggingwith partial backlogging under JPIM are de-veloped solved and compared respectively to explore theoptimal operational strategies and optimal joint pricing andinventory decisions for the IBWT supply chain

3 Modeling Notations and Assumptions

An IBWTdistribution system is a typical ldquoembeddedrdquo supplychain structure In this supply chain system a horizontalwater supply system embeds itself in a vertical water dis-tribution system (see Figure 1) In the horizontal watersupply system a local supplier and an external supplier workas a joint IBWT supplier via an efficient cooperationmechanismWater resources are transferred and supplied bythe local supplier from the water source to the externalsupplier within the trunk channel and then distributed towater resource distributors of all water intakes via riverchannels and artificial canals Finally the water resources aresold by each distributor to the water resource consumers inhisher service region )e water distributor and the cor-responding water market in the ith water intake are indexedby i 1 2 n It is assumed that there are m distributorssupplied by the local supplier and nndashm distributors suppliedby the external supplier

On this basis the parameters used in the models aredefined as follows

cdi the water transfer cost from the ith water intake tothe ith distributorck the water transfer cost from the (k minus 1)th waterintake to the kth water intake within the horizontalsupply chain k 1 2 n

δk the water delivery loss rate from the (k minus 1)th waterintake to the kth water intake within the horizontalsupply chain and δk isin (0 1) k 1 2 n

cfi the fixed cost for the ith water intake of the IBWTsuppliercfl the fixed cost for the local supplier cfl 1113936

mi1 cfi

cfe the fixed cost for the external suppliercfe 1113936

nim+1 cfi

2 Complexity

cf the fixed cost for the IBWT suppliercf cfl + cfe 1113936

ni1 cfi

w the wholesale price of water resources transferredfrom the local supplier to the external supplierwi the wholesale price of water resources transferredfrom the IBWT supplier to the ith distributorpi the retail price of water resources sold from the ithdistributor to the consumers in his service regionκh the holding cost coefficient and 0lt κh lt 1hi the unit cost of holding water inventory for the ithdistributor and hi κhpi

κs the shortage cost coefficient and 0lt κh lt κs lt 1si the shortage cost of unmet water demand for the ith

distributor and si κspi

Qi the original pumping quantity from the watersource to the ith water intakeqi the order quantity of the ith water intakeτ the bargaining powers of the local supplier andτ isin (0 1)

λ the bargaining powers of the ith water intake of theIBWT supplier and λ isin (0 1)

As mentioned above the unmet water demand may bepartially backlogged due to the capacity constraint of theIBWT project )e backlogging ratio of unmet water de-mand for the ith distributor is φ and φ isin [0 1] )e rela-tionship between the water demand of the ith water intake qi

and the original pumping quantity Qi isqi Qi 1113937

ik1 (1 minus δk) and the total transfer cost of the

original pumping quantity Qi is

TCi(Qi) Qi 1113936ik1[ck1113937

kminus 1j0(1 minus δj)] hereinto δ0 0

)erefore the total transfer cost of the water demand (orderquantity) of the ith water intake isTCi(qi) ((1113936

ik1[ck 1113937

kminus 1j0(1 minus δj)]1113937

ik1(1 minus δk))qi) De-

fining Ci (1113936ik1[ck 1113937


ik1(1 minus δk)) then

TCi(qi) Ciqi Following Howe and Linaweaver [23 24]Petruzzi and Dada [25] and Wang et al [26] the waterdemand for the ith distributor is di(pi) anddi(pi) yi(pi)xi Hereinto yi(pi) aip

minus bi where ai is the

potential maximum water demand quantity and b is theprice elasticity of the expected demand xi is a randomdisturbance defined in the range [A B] with BgtAgt 0 )ecumulative distribution function (CDF) and the probabilitydensity function (PDF) of xi are Fi(middot) and fi(middot) and themean value and the standard deviation of xi are μi and σiFollowing Petruzzi and Dada [25] Wang et al [27] andWang [28] zi (qiyi(pi)) is defined as the ldquowater stockfactorrdquo for the ith distributor thus the order quantityfunction of water resources for the ith water intake isqi yi(pi)zi )e distribution of xi satisfies the IGFR (in-creasing generalized failure rate) condition(dg(xi)dxi)gt 0 where gi(xi) equiv xi(fi(xi)[1 minus Fi(xi)])and there exists a unique solution to the maximal expectedproblem Many distributions such as normal distributionand exponential distribution satisfy the IGFR condition[27 29 30]

Based on the foregoing modeling notations and as-sumptions the profit functions of the ith distributor the ithwater intake of the IBWTsupplier and the ith water intake ofthe IBWT supply chain with partial backlogging can beformulated as follows

Horizontal supply chain IBWT water resource joint supplier

Local supplier

Reservoirs 1 Reservoirs 2 Reservoirs j Reservoirs J

Distributor1

Distributor2

Distributori

Distributorm

Distributorm + 1

(m + 1)th

market(m + 2)th

market(m + k)th

market

Distributorm + 2

Distributorm + k

Distributorn

1st

market2nd

marketith

marketmth

marketnth

market

Water-intake

Wat

er so

urce

Water supplier

helliphellip

hellip hellip

Vert

ical

supp

ly ch

ain

ibw

t wat

er d

istrib

utio

nPump PumpPumpPump

Figure 1 A generic interbasin water transfer supply chain system

Complexity 3

ΠDizi pi( 1113857 piyi pi( 1113857E min zi xi1113864 11138651113858 1113859 minus κhpiyi pi( 1113857E zi minus xi( 1113857

+1113858 1113859 + φ pi minus wi + cdi( 11138571113858 1113859 minus κspi1113864 1113865yi pi( 1113857E xi minus zi( 1113857

+1113858 1113859

minus wi + cdi( 1113857yi pi( 1113857zi

ΠSiwi( 1113857 wi minus Ci( 1113857yi pi( 1113857zi + φ wi minus Ci( 1113857yi pi( 1113857E xi minus zi( 1113857

+1113858 1113859 minus cfi

ΠSCizi pi( 1113857 piyi pi( 1113857E min zi xi1113864 11138651113858 1113859 minus κhpiyi pi( 1113857E zi minus xi( 1113857

+1113858 1113859 + φ pi minus Ci + cdi( 11138571113858 1113859 minus κspi1113864 1113865yi pi( 1113857E xi minus zi( 1113857

+1113858 1113859

minus Ci + cdi( 1113857yi pi( 1113857zi minus cfi

(1)

On this basis the profit functions of the IBWT supplierthe local supplier the external supplier and the IBWTsupply chain with partial backlogging can be formulated asfollows

ΠS wi( 1113857 1113944n

i1ΠSi

wi( 1113857

ΠLS wi w( 1113857 1113944m

i1ΠSi

wi( 1113857 + w 1113944n

im+1yi pi( 1113857zi

ΠES wi w( 1113857 1113944n

im+1ΠSi

wi( 1113857 minus w 1113944n


ΠSC zi pi( 1113857 1113944n

i1ΠSCi

zi pi( 1113857

(2)

4 Game-Theoretical Decision Models

Based on themodeling notations and assumptions in Section3 two game-theoretical decision models without back-loggingwith partial backlogging for the IBWT supply chainunder JPIM considering water delivery loss including theequilibrium and coordination decision models are devel-oped analyzed and compared in this section In the modelsto follow note that the supersubscript d represents anequilibrium decision the supersubscript c represents acoordination decision the supersubscript o represents the

scenario without backlogging the supersubscript b repre-sents the scenario with partial backlogging

41 Game-4eoretical Decision Models without BackloggingUnder the scenario without backlogging the backloggingratio of unmet water demand for the ith distributor is φ 0Two game-theoretical decision models of the IBWT supplychain without backlogging under JPIM considering waterdelivery loss including the equilibrium and coordinationdecision models are developed analyzed and compared inthis section

411 Equilibrium Decision Model without BackloggingUnder this scenario the detailed decision sequences are asfollows the local and external suppliers will first bargainover the wholesale price of the water resources within theIBWT horizontal supply chain via Nash bargaining theory[32ndash35] to achieve cooperative operations then the IBWTsupplier decides the usage price of water resources for eachwater distributor in the IBWT vertical supply chain finallyeach water distributor independently and simultaneouslydecides the stock factor and the retail price of water re-sources for the consumer it serves

)e two-stage Stackelberg and Nash bargaining gamemodel for the IBWT supply chain without backlogging canbe formulated as

maxwΩ(w) ΠodLS w

odi q

odi w1113872 11138731113960 1113961

τΠodES w

odi q

odi w1113872 11138731113960 1113961

1minus τ

st

ΠodLS wodi q

odi w1113872 1113873 +ΠodES w

odi q

odi w1113872 1113873 ΠodS w

odi q

odi1113872 1113873

st

wodi p

odi z

odi and q

odi are derived from solving the following problem

maxwi

ΠoS wi p

odi wi( 1113857 z

odi1113872 1113873

podi wi( 1113857 and z


maxzipi

ΠoDi

zi pi( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(3)

4 Complexity

Solving this two-stage Stackelberg and Nash bargainingproblem we can get the equilibrium usage price wod

i inthe ith water intake the equilibrium retail price pod

i and theequilibrium stock factor zod

i for the ith water distributor theequilibrium ordering quantity qodi for the ith water distrib-utor and the bargaining wholesale price wo

d Furthermorethe profits of the local supplier the external supplier theIBWTsupplier the ith water distributor and the IBWTsupplychain can be calculated as ΠodLS Π

odES Π

odS ΠodDi

and ΠodSC(see Table 1 for the detailed analytical results and theirderivations can be seen in sec10Supplementary Materials(available here))

412 Coordination Decision Model without BackloggingUnder this scenario the detailed decision sequences are asfollows the local and external suppliers will first bargainover the wholesale price of water resources within the IBWThorizontal supply chain to achieve cooperative operations

then the IBWT supplier provides the distributors a revenuesharing contract in which the IBWTsupplier charges a lowerusage price wi to the water distributors if the distributorsaccept the contract they will place orders with quantityqi to the IBWT supplier and decide the retail price of waterresources pi and the stock factor of water resources zifinally they will share a proportion of their net revenues(1 minus ϕ) to the IBWT supplier where ϕ is the revenuekeeping rate of the water distributors 0leϕle 1 )e revenueshared by the ith distributor to the IBWT supplier is Ti

(1 minus ϕ)piyi(pi) E[min zi xi1113864 1113865] minus κhE[(zi minus xi)+]minus1113864 κsE[(ximinus

zi)+] )us the profit functions of the ith distributor and the

IBWT supplier are as follows ΠocDi(zi pi) Πo

Di(zi pi) minus Ti

and ΠocS (wi) 1113936ni1Π

ocSi

(wi) 1113936ni1[Π

oSi

(wi) + Ti])e two-stage coordination and Nash bargaining game

model for the IBWT supply chain without backlogging canbe formulated as

maxwΩ(w) ΠocLS w

oci q

oci w( 11138571113858 1113859

τ ΠocES woci q

oci w( 11138571113858 1113859

1minus τ

st

ΠocLS woci q

oci w( 1113857 + ΠocES w

oci q

oci w( 1113857 ΠocS w

oci q

oci( 1113857

woci q

oci ΠocLS w

oci q

oci w( 1113857 ΠocES w

oci q

oci w( 1113857 andΠocS w

oci q

oci( 1113857

are derived from solving the following problem

maxϕ

πoi (ϕ) ΠocSi

(ϕ) minus ΠodSi1113960 1113961

λΠocDi

(ϕ) minus ΠodDi1113960 1113961

1minus λ

st

ΠocSi(ϕ) + ΠocDi

(ϕ) ΠocSCi

woci (ϕ) p

oci z

oci q

oci ΠocSi

(ϕ)ΠocDi(ϕ) andΠocSCi


maxzipi

ΠocDizi pi( 1113857

maxzipi

ΠoSC zi pi( 1113857

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(4)

Solving this two-stage coordination and Nash bar-gaining problem we can obtain the equilibrium usageprice woc

i in the ith water intake the equilibrium retailprice poc

i and the equilibrium stock factor zoci for the ith

water distributor the equilibrium ordering quantity qoci

for the ith water distributor and the bargaining wholesale

price wonc Furthermore the profits of the local supplier

the external supplier the IBWT supplier the ith waterdistributor and the IBWT supply chain can also becomputed as ΠocLS Π

ocES Π

ocS ΠocDi

and ΠocSC (see Table 1 forthe detailed analytical results and their derivations can beseen in Supplementary Materials)

Complexity 5

Tabl

e1

Analytical

results

oftheIBWTsupp

lychainwith

outb

acklogging

Scenario

var

Result

Equilib

rium

decisio

n(411)

Coo

rdinationdecisio

n(412)

wo i

wod i

((

bC

i+

c di)(

bminus1)

)w

oc iϕo c

Ciminus

(1

minusϕo c

)cd

i

po i

pod i

(

b(

bminus1)

poc i

)poc i

(

b(

bminus1)

)((

Ci+

c di)

zoc i(1

+κ s

)zoc i

minusΛ

i(zoc i

))

zo i

Fi(

zod i

)

Fi(

zoc i

)F

i(zoc i

)

((1

+κ s

)b(1

+κ h

+κ s

))+

((b

minus1)Λ

i(zoc i

)b(1

+κ h

+κ s

)zoc i

)

qo i

qod i

((

bminus1)b

)bqoc i

qoc i

y

i(poc i

)zoc i

a

i(poc i

)minusbzoc i

wo

wo d

(1

1113936n i

m+1

qod i

)(τΠ

od Sminus

1113936m i1Π

od Si)

wo c

(1

1113936n i

m+1

qoc i

)(τΠ

oc Sminus

1113936m i1Π

oc Si)

Πo LS

Πod LS

τΠ

od Sτ1113936

n i1[

((b

minus1)b

)b(Π

oc SCi+

c fi)

minusc fi]

Πoc LS

τΠ

oc Sτ1113936

n i1[

(1

minusϕo c

)Πoc SC

iminusϕo c

c fi]

Πo ES

Πod ES

(1

minusτ)Π

od S

(1

minusτ)

1113936n i1[

((b

minus1)b

)b(Π

oc SCi+

c fi)

minusc fi]

Πoc ES

(1

minusτ)Π

oc S

(1

minusτ)

1113936n i1[

(1

minusϕo c

)Πoc SC

iminusϕo c

c fi]

Πo S

Πod S

1113936

n i1Π

od Si

1113936

n i1[

((b

minus1)b

)b(Π

oc SCi+

c fi)

minusc fi]

Πoc S

1113936

n i1Π

oc Si

1113936

n i1[

(1

minusϕo c

)Πoc SC

iminusϕo c

c fi]

Πo D

iΠ

od Di

((

bminus1)b

)bminus1 (Π

oc SCi+

c fi)

Πoc D

iϕo c

(Π

oc SCi+

c fi)

Πo SC

Πod SC

1113936

n i1Π

od SCi

1113936

n i1

[((

bminus1)b

)b+

((b

minus1)b

)bminus1 ](Π

oc SCi+

c fi)

minusc fi

11139661113967

Πoc SC

1113936

n i1Π

oc SCi

1113936

n i1(

(C

i+

c di)(

bminus1)

)qoc i

minusc fi

ϕo cNA

ϕo cλ(

(b

minus1)b

)bminus1

+(1

minusλ)

[1minus

((b

minus1)b

)b]

Note

Λi(

zoc i

)

(1

+κ h

+κ s

)1113938

zoc i

A(

zoc i

minusx

i)f

i(x

i)dx

i+κ s

1113938B A

xif

i(x

i)dx

i

6 Complexity

42 Game-4eoretical Decision Models with PartialBacklogging Under the scenario with partial backloggingthe backlogging ratio of unmet water demand for the ithdistributor is φ isin (0 1] Two game-theoretical decisionmodels of the IBWT supply chain with partial backloggingunder JPIM considering water delivery loss including theequilibrium and coordination decision models are devel-oped analyzed and compared in this section

421 Equilibrium Decision Model with Partial BackloggingUnder this scenario the detailed decision sequences are asfollows the local and external suppliers will first bargain

over the wholesale price of the water resources within theIBWT horizontal supply chain to achieve cooperative op-erations next the IBWT supplier decides the usage price ofwater resources for each water distributor in the IBWTvertical supply chain then each water distributor inde-pendently and simultaneously decides the stock factor andthe retail price of water resources for the consumer it servesfinally the unmet water demands of each market are par-tially backlogged and satisfied )e two-stage Stackelbergand Nash bargaining gamemodel for the IBWTsupply chainwith partial backlogging can be formulated as

maxwΩ(w) ΠbdLS w

bdi q

bdi w1113872 11138731113960 1113961

τΠbdES w

bdi q

bdi w1113872 11138731113960 1113961

1minus τ

st

ΠbdLS wbdi q

bdi w1113872 1113873 + ΠbdES w

bdi q

bdi w1113872 1113873 ΠbdS w

bdi q

bdi1113872 1113873

st

wbdi p

bdi z

bdi and q

bdi are derived from solving the following problem

maxwi

ΠbS wi p

bdi wi( 1113857 z

bdi1113872 1113873

pbdi wi( 1113857 and z


maxzipi

ΠbDi

zi pi( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)

Solving this two-stage Stackelberg and Nash bargainingproblem we can obtain the equilibrium usage price wbd

i inthe ith water intake the equilibrium retail price pbd

i and theequilibrium stock factor zbd

i for the ith water distributor theequilibrium ordering quantity qbdi for the ith water distrib-utor and the bargaining wholesale price wb

d Furthermorethe profits of the local supplier the external supplier theIBWT supplier the ith water distributor and the IBWTsupply chain can be calculated as ΠbdLS Π

bdES Π

bdS ΠbdDi

andΠbdSC (see Table 2 for the detailed analytical results and theirderivations can be seen in Supplementary Materials)

422 Coordination Decision Model with Partial BackloggingUnder this scenario the detailed decision sequences are asfollows the local and external suppliers will first bargainover the wholesale price of water resources within the IBWThorizontal supply chain to achieve cooperative operationsnext the IBWT supplier offers the distributors a revenue

sharing contract in which the IBWTsupplier charges a lowerusage price wi to the water distributors if the distributorsaccept the contract they will place orders with quantity qi tothe IBWT supplier and decide the retail price of water re-sources pi and the stock factor of water resources zi thenthe unmet water demands of each market are partiallybacklogged and satisfied finally all the distributors willshare a proportion of their net revenues (1 minus ϕ) to the IBWTsupplier where ϕ is the revenue keeping rate of the waterdistributors 0leϕle 1 )e revenue shared by the ith dis-tributor to the IBWT supplier is Ti (1 minus ϕ)piyi(pi)

E[min zi xi1113864 1113865] minus κhE[(zi minus xi)+] minus κsE[(xi minus zi)

+]1113864 1113865 )usthe profit functions of the ith distributor and the IBWTsupplier are as follows ΠbcDi

(zi pi) ΠbDi

(zi pi) minus Ti andΠbcS (wi) 1113936

ni1Π

bcSi

(wi) 1113936ni1[Π

bSi


model for the IBWT supply chain with partial backloggingcan be formulated as

Complexity 7

Tabl

e2

Analytical

results

oftheIBWTsupp

lychainwith

partialb

acklogging

Scenario

var

Result

Equilib

rium

decisio

n(421)

Coo

rdinationdecisio

n(422)

wb i

wbd i

((

bC

i+

c di)(

bminus1)

)w

bc iϕb c

Ciminus

(1

minusϕb c

)cd

i

pb i

pbd i

(

b(

bminus1)

)pbc i

pbc i

(

b(

Ci+

c di)(

bminus1)

)(((1

minusφ)

zbc i

+φΝ

i(zbc i

))(

(1

minusφ

+κ s

)zbc i

minusΜ

i(zbc i

)))

zb i

Fi(

zbd i

)

Fi(

zbc i

)F

i(zbc i

)

qb i

qbd i

((

bminus1)b

)bqbc i

qbc i

y

i(pbc i

)zbc i

a

i(pbc i

)minusbzbc i

wb

wb d

(1

1113936n i

m+1

qbd i

)(τΠ

bd Sminus

1113936m i1Π

bd Si)

wb c

(1

1113936n i

m+1

qbc i

)(τΠ

bc Sminus

1113936m i1Π

bc Si)

Πb LS

Πbd LS

τΠ

bd Sτ1113936

n i1[

((b

minus1)b

)b(Π

bc SCi+

c fi)

minusc f

i]Π

bc LSτΠ

bc Sτ1113936

n i1[

(1

minusϕb c

)Πbc SC

iminusϕb c

c fi]

Πb ES

Πbd ES

(1

minusτ)Π

bd S

(1

minusτ)

1113936n i1[

((b

minus1)b

)b(Π

bc SCi+

c fi)

minusc fi]

Πbc ES

(1

minusτ)Π

bc S

(1

minusτ)

1113936n i1[

(1

minusϕb c

)Πbc SC

iminusϕb c

c fi]

Πb S

Πbd S

1113936

n i1Π

bd Si

1113936

n i1[

((b

minus1)b

)b(Π

bc SCi+

c fi)

minusc fi]

Πbc S

1113936

n i1Π

bc Si

1113936

n i1[

(1

minusϕb c

)Πbc SC

iminusϕb c

c fi]

Πb D

iΠ

bd Di

((

bminus1)b

)bminus1 (Π

bc SCi+

c fi)

Πbc D

iϕb c

(Π

bc SCi+

c fi)

Πb SC

Πbd SC

1113936

n i1Π

bd SCi

1113936

n i1

[((

bminus1)b

)b+

((b

minus1)b

)bminus1 ](Π

bc SCi+

c fi)

minusc fi

11139661113967

Πbc SC

1113936

n i1Π

bc SCi

1113936

n i1

((C

i+

c di)(

bminus1)

)yi(

pbc i

)[(1

minusφ)

zbc i

+φΝ

i(zbc i

)]minus

c fi

11138641113865

ϕb cNA

ϕb cλ(

(b

minus1)b

)bminus1

+(1

minusλ)

[1minus

((b

minus1)b

)b]

Note

Fi(

zbc i

)

(((1

minusφ)

(1

minusφ

+κ s

)zbc i

+

bφ(

1minusφ

+κ s

)Ni(

zbc i

)+

(b

minus1)

(1

minusφ)

Mi(

zbc i

))(

[(b

minusφ)

(1

minusφ

+κ s

)+

b(1

minusφ)κ h

]zbc i

+bφ(

1minusφ

+κ s

+κ h

)Νi(

zbc i

)minus

(b

minus1)φΜ

i(zbc i

)))

Mi(

zbc i

)

(1

+κ h

+κ s

minus

φ)1113946

zbc i

A(

zbc i

minusx

i)f

i(x

i)dx

iminus

(φ

minusκ s

)1113946

B Ax

ifi(

xi)dx

iN

i(zbc i

)

1113946zbc i

A(

zbc i

minusx

i)f

i(x

i)dx

i+

1113946B A

xif

i(x

i)dx

i

8 Complexity

maxwΩ(w) ΠbcLS w

bci q

bci w1113872 11138731113960 1113961

τΠbcES w

bci q

bci w1113872 11138731113960 1113961

1minus τ

st

ΠbcLS wbci q

bci w1113872 1113873 +ΠbcES w

bci q

bci w1113872 1113873 ΠbcS w

bci q

bci1113872 1113873

wbci q

bci ΠbcLS w

bci q

bci w1113872 1113873ΠbcES w

bci q

bci w1113872 1113873 andΠbcS w

bci q

bci1113872 1113873


maxϕ

πbi (ϕ) ⊓bcSi

(ϕ) minus ΠbdSi1113960 1113961

λΠbcDi

(ϕ) minus ΠbdDi1113960 1113961

1minus λ

st

ΠbcSi(ϕ) + ΠbcDi

(ϕ) ΠbcSCi

wbci (ϕ) p

bci z

bci q

bci ΠbcSi

(ϕ)ΠbcDi(ϕ) andΠbcSCi


maxzipi

ΠbcDizi pi( 1113857

maxzipi

ΠbSC zi pi( 1113857

⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)

Solving this two-stage coordination andNash bargainingproblem we can obtain the equilibrium usage price wbc

i inthe ith water intake the equilibrium retail price pbc

i and theequilibrium stock factor zbc

i for the ith water distributor theequilibrium ordering quantity qbci for the ith water distrib-utor and the bargaining wholesale price wb

c Furthermorethe profits of the local supplier the external supplier theIBWT supplier the ith water distributor and the IBWTsupply chain can also be computed asΠbcLSΠ

bcESΠ

bcS ΠbcDi

andΠbcSC (see Table 1 for the detailed analytical results and theirderivations can be seen in Supplementary Materials)

5 Numerical and Sensitivity Analyses

Based on the game-theoretical decision modeling analysis areal-world case of the eastern route of the South-to-NorthWater Diversion (SNWD) project in China is selected toconduct numerical and sensitivity analyses

)e SNWD project is an important world-scale strategicwater resource engineering to solve the water shortageproblem in northern China )is project is divided into theeastern route western route and middle route )ese threeroutes of the SNWD project can transfer water resourcesseparately from the Yangtze River linking Yangtze RiverHuaihe River Yellow River and Haihe River to formulate anationwide water supply system with ldquoFour horizontal and)ree vertical South-North deployment East-West mutualsupportrdquo

)e eastern route of the SNWD project is constructedand extended based on the north water transfer project inJiangsu Province )rough the Jiangdu water control project

in Yangzhou City Jiangsu Province the water is extractedfrom the main stream of the lower reaches of the YangtzeRiver and transferred by the Beijing-Hangzhou Grand Canaland its parallel river channels to connect Hongze LakeLuoma Lake Nansi Lake and Dongping Lake After leavingDongping Lake there are two ways to deliver water one is tothe north passing through the Yellow River through atunnel near Weishan and then to Tianjin the other is to theeast Yantai and Weihai via the Jiaodong water transferbranch-line )ere are 13 pumping stations in this projectwith a total length of water transfer mainline at 14665kilometers a water raising capacity at 65 meters and a waterdiversion scale of 148 billion cubic meters)e eastern routeproject provides production and domestic water to the eastof Huang-Huai-Hai Plain Jiaodong area and Beijing-Tianjin-Hebei region In the water supply area there are 25cities at prefecture level or above from the Huaihe River theHaihe River and the Yellow River Basin

In the eastern route of the SNWD project there are sixsections for the mainline of the project Section 1 (JiangduStationsimSouth of Nansi Lake) Section 2 (Lower Cascade ofNansi Lake) Section 3 (Upper Cascade of Nansi Lake-simChanggou Pumping Station) Section 4 (ChanggouPumping StationsimDongping Lake including East ofDongping Lake) Section 5 (Dongping LakesimQiutun Sluicein Linqing City) and Section 6 (Qiutun Sluice in LinqingCitysimDatun Reservoir))ese six sections can be generalizedto six water intakes ie n 6 Hereinto Sections 1-2 aremanaged and operated by the local supplier (Jiangsu WaterSource Co Ltd) and Sections 3sim6 are managed and operatedby the external supplier (Shandong Mainline Co Ltd) ie

Complexity 9

m 2 and n minus m 4 )ere are two water distributors in theservice region of the local supplier and four water distrib-utors in the service region of the external supplier

Based on the real characteristics and managementpractices of the eastern route of the SNWD project in China[2] the numerical and sensitivity analyses are conducted inthe following sections and the corresponding values ofparameters relating to the IBWT supply chain are collectedand estimated from the publicly disclosed information of theeastern route of the SNWD project [34] and the watertransfer scheme for the eastern route of the SNWD project[36] )e parameters of water transfer costs and fixed costscan be estimated according to the cost accounting method)e parameters of the potential maximum water demandquantity and the price elasticity of the expected demand canbe estimated based on the historical operation data )erandom distribution and corresponding parameters of therandom factor can also be fitted based on the historicaloperation data )e parameter of the water delivery loss ratecan be estimated based on the historical operation data )eparameters of the holding cost coefficient and the shortagecost coefficient can be estimated via the cost accountingmethod or empirical parameters )e backlogging ratio ofunmet water demand can be calculated as total shortagequantity divided by extra supply capacity )e bargainingpowers of the local supplier and the ith water intake of theIBWT supplier can be calculated by the market powerevaluation On this basis the lower bound of the interval ofthe random factor (xi) is set at 000001 and the upperbounds of the interval of the random factor (xi) is set at 1ie A 1E minus 5 B 1 )e random factor xi obeys normaldistribution ie xi sim N(μi σ2i ) )e mean value of therandom factor μi is set at 010 and the standard deviation ofthe random factor σi is set at 001 )e corresponding pa-rameters of mainlinebranch-line water transfer costs andpotential maximum water demand quantities are collectedand estimated in Table 3 )e fixed cost of water delivery forthe ith water intake of the IBWT supplier cfi is 1000000According to the empirical parameters the water deliveryloss from the (i minus 1)th water intake to the ith water intakewithin the horizontal supply chain δi is about 15 )ebacklogging ratio of unmet water demand for the ith dis-tributor φ is 08 )e price elasticity of the expected demandb is 15 )e holding cost coefficient κh is 02 and theshortage cost coefficient κs is 05 Owing to the advantage ofthe local supplier the local supplierrsquos bargaining power τ is06 Likewise due to the advantage of the IBWTsupplier theith water intake of the IBWTsupplierrsquos bargaining power λ is06

51 Numerical Analysis )e joint pricing-inventory oper-ational decisions and performances for all the game-theo-retical decision models withoutwith backlogging arecompared and summarized in Tables 4ndash7 respectively Itshould be noted that the centralized decision neglects theroles of the membersrsquo decisions and therefore is inferior tocoordination decision regarding the derived solutions )usthe centralized decision model is not shown in the numerical

analysis separately )e main findings of the numericalanalysis are summarized below

(1) Comparing the numerical analysis results betweenequilibrium decision (Table 4) and coordinationdecision (Table 5) without backlogging under JPIMthe findings are summarized as follows (i) )ewholesale prices of water resources under coordi-nation decision are lower than those under equi-librium decision (ii) )e retail prices of waterresources under coordination decision are lowerthan those under equilibrium decision (iii) )ewater stock factors under coordination decision areequal to those under equilibrium decision (iv) )eorder quantities of water resources under coordi-nation decision are higher than those under equi-librium decision (v) )e profits of the IBWT supplychain and its members under coordination decisionare higher than those under equilibrium decision

(2) Comparing the numerical analysis results betweenequilibrium decision (Table 6) and coordinationdecision (Table 7) with partial backlogging underJPIM the findings are summarized as follows (i))ewholesale prices of water resources under coordi-nation decision are lower than those under equi-librium decision (ii) )e retail prices of waterresources under coordination decision are lowerthan those under equilibrium decision (iii) )ewater stock factors under coordination decision areequal to those under equilibrium decision (iv) )eorder quantities of water resources under coordi-nation decision are higher than those under equi-librium decision (v) )e profits of the IBWT supplychain and its members under coordination decisionare higher than those under equilibrium decision

(3) Comparing the numerical analysis results betweenthe scenario without backlogging (Table 4) and thescenario with partial backlogging (Table 6) underequilibrium decision the findings are summarized asfollows (i) )e wholesale prices of water resourcesunder the scenario with partial backlogging are equalto those under the scenario without backlogging (ii))e retail prices of water resources under the sce-nario with partial backlogging are lower than thoseunder the scenario without backlogging (iii) )ewater stock factors under the scenario with partialbacklogging are lower than those under the scenariowithout backlogging (iv) )e order quantities ofwater resources under the scenario with partialbacklogging are lower than those under the scenariowithout backlogging (v) )e profits of the IBWTsupply chain and its members under the scenariowith partial backlogging are higher than those underthe scenario without backlogging

(4) Comparing the numerical analysis results betweenthe scenario without backlogging (Table 5) and thescenario with partial backlogging (Table 7) undercoordination decision the findings are summarized

10 Complexity

as follows (i))e wholesale prices of water resourcesunder the scenario with partial backlogging are equalto those under the scenario without backlogging (ii))e retail prices of water resources under the sce-nario with partial backlogging are lower than thoseunder the scenario without backlogging (iii) )ewater stock factors under the scenario with partialbacklogging are lower than those under the scenariowithout backlogging (iv) )e order quantities ofwater resources under the scenario with partialbacklogging are lower than those under the scenariowithout backlogging (v) )e profits of the IBWT

supply chain and its members under the scenariowith partial backlogging are higher than those underthe scenario without backlogging

52 Sensitivity Analysis Since in the previous analysiscoordination decision with partial backlogging is found to besuperior to the other decisions withoutwith partial back-logging the sensitivity analysis will focus on how thechanges in eight key parameters (including the mainlinetransfer cost the branch-line transfer cost the holding costcoefficient the shortage cost coefficient the backloggingratio the price elasticity of the expected water demand the

Table 3 Parameter setting

Section (water intake) i Mainline water transfer cost (ci) Branch-line water transfer cost (cdi) Potential maximum water demand (ai)

1 032 008 1796 +E92 024 010 1218 +E93 009 012 1169 +E94 014 076 1941 +E95 041 029 1117 +E96 081 040 1525 +E9

Table 4 Numerical analysis results of equilibrium decision without backlogging

i wodi pod

i qodi ΠodDiΠodS

1 129 452 19687772 53921334 481474442 238 817 5492511 27196481 ΠodLS3 312 1069 3525446 22826460 288884664 540 2033 2230602 27477799 ΠodES5 659 2272 1087114 14960682 192589786 1073 3674 721423 16059575 ΠodSCSum NA NA 32744868 162442332 210589776Note zodi 011 wo

d 051

Table 5 Numerical analysis results of coordination decision without backlogging

i woci poc

i qoci ΠocDiΠocS

1 023 151 102300667 62521086 870086022 045 272 28539924 31533966 ΠocLS3 060 356 18318752 26466983 522051614 061 678 11590549 31860151 ΠocES5 125 757 5648811 17346716 348034416 208 1225 3748622 18620868 ΠocSCSum NA NA 170147325 188349770 275358372Note zoc

i 011 woc 020 ϕo

c 067

Table 6 Numerical analysis results of equilibrium decision with partial backlogging

i wbdi pbd

i qbdi ΠbdDiΠbdS

1 129 442 19598802 55086318 493173122 238 799 5467690 27784068 ΠbdLS3 312 1045 3509514 23319631 295903874 540 1988 2220522 28071464 ΠbdES5 659 2221 1082201 15283911 197269256 1073 3593 718163 16406546 ΠbdSCSum NA NA 32596892 165951937 215269249Note zbdi 010 wb

d 053

Complexity 11

water delivery loss rate and the bargaining power of the ithwater intake of the IBWT supplier) impact the profits undercoordination decision with partial backlogging To capturethe impact of the change in key parameters we only selectthe parameters from the 1st distributor and the 1st waterintake to conduct sensitivity analysis including the mainlinetransfer cost and the branch-line transfer cost Figure 2(including eight subgraphs) shows the impact of eight keyparameters change on the profit of the IBWT supply chainwith partial backlogging )e findings from the sensitivityanalysis results are summarized as follows (1) the IBWTsupply chain profit decreases as the mainline transfer costincreases (2) the IBWT supply chain profit decreases as thebranch-line transfer cost increases (3) the IBWT supplychain profit decreases as the holding cost coefficient in-creases (4) the IBWT supply chain profit decreases as theshortage cost coefficient increases (5) the IBWT supplychain profit increases as the backlogging ratio increases (6)the IBWTsupply chain profit decreases as the price elasticityof the expected water demand increases (7) the IBWTsupply chain profit decreases as the water delivery loss in-creases (8) the IBWT supplierrsquos profit increases and thedistributorsrsquo profits decrease as the bargaining power of theith water intake of the IBWT supplier increases

6 Managerial Insights andPractical Implications

Based on the foregoing analysis this section will expound onthe managerial insights and practical implications

61Managerial Insights Based on the foregoing discussionsthe managerial insights can be derived and summarized asfollows

First be it under the scenario without backlogging orunder the scenario with partial backlogging the IBWTsupply chain and its members wouldmake lower retail pricesof water resources and order more water resources under thecoordination strategy than those under the equilibriumstrategy and could gain more profits under the coordinationstrategy than those under the equilibrium strategy Hencethe coordination strategy via a revenue and cost sharingcontract is beneficial to improving the operational perfor-mance for all the stakeholders and coordinate the IBWTsupply chain under JPIM

Second be it under equilibrium or coordination strat-egies the IBWT supply chain and its members would orderless water resources under the scenario with partial back-logging than those under the scenario without backloggingand could gain more profits under the scenario with partialbacklogging than those under the scenario without back-logging Hence partial backlogging of water demand isbeneficial to operational performance improvement for allthe stakeholders of the IBWT supply chain under JPIMFurthermore the higher the backlogging ratio is the moreprofits the IBWT supply chain and its members could gain)us increasing the backlogging ratio is beneficial to op-erational performance improvement for all the stakeholdersof the IBWT supply chain under JPIM

Finally be it under the scenario without backlogging orunder the scenario with partial backlogging reducing thewater delivery loss rate and operational costs (includingmainline transfer cost branch-line transfer cost holdingcost and shortage cost) is beneficial to operational per-formance improvement for all the stakeholders of the IBWTsupply chain under JPIM Furthermore a lower priceelasticity of the expected water demand is beneficial tooperational performance improvement for all the stake-holders of the IBWT supply chain under JPIM

In brief the coordination strategy using a revenue andcost sharing contract with partial backlogging outperformsall the other scenariosstrategies and is the best strategy forimproving the operational performance of all the stake-holders in the IBWT supply chain under JPIM

62 Practical Implications From the practical perspective ofthe IBWT project operational management the practicalimplications can be derived and summarized for the SNWDproject as follows

First a fixed water price mechanism is implemented inthe eastern and middle routes of the SNWD project cur-rently which cannot directly reflect the linkage effect be-tween water supply and demand and water price )is rigidexisting water price mechanism could not effectively exertthe regulatory role of the market mechanism and coordinatethe interests of all operating entities involved in the oper-ational management of the SNWD project )us it issuggested that some of the water pricing power may betransferred by the government to the main operating entitiesof the projects to give full play to the regulating role of the

Table 7 Numerical analysis results of coordination decision with partial backlogging

i wbci pbc

i qbci ΠbcDiΠbcS

1 023 147 101838362 63871869 890180752 045 266 28410950 32215265 ΠbcLS3 060 348 18235969 27038809 534108454 061 663 11538171 32548497 ΠbcES5 125 740 5623284 17721496 356072306 208 1198 3731682 19023177 ΠbcSCSum NA NA 169378417 192419112 281437186Note zbc

i 010 wbc 020 ϕb

c 067

12 Complexity

32

31

3

29

28

27

26

Prof

it

02 022 024 026 028 03 032 034 036 038 04Mainline transfer cost

Supply chain

times108

(a)

Prof

it

Supply chain

times108286

284

282

28

278

276

274005 006 007 008 009 01 011 012 013 014 015

Branchline transfer cost

(b)

Prof

it

284

283

282

281

28

279

278

277

27601 02 03 04 05

Holding cost coefficient

times108

Supply chain

(c)

Prof

it

284

285

286

287

288

283

282

281

2802 03 04 05 06

times108

Shortage cost coefficient

Supply chain

(d)

times108

Prof

it

284

283

282

281

28

279

278

277

276

2750 01 03 04 05 06 07 08 09 102

Backlogging ratio

Supply chain

(e)

times108

Prof

it

6

55

5

45

4

35

3

25

2

1511 12 13 14 15 16 17 18 19 2

Price-elasticity of the expected demand

Supply chain

(f )

Figure 2 Continued

Complexity 13

market mechanism In this situation the water prices foreach water intake and water market in the IBWTproject canbe adjusted freely according to the relationship between thewater supply and demand which is a flexible water pricemechanism On this basis a joint pricing and inventorymanagement (JPIM) mode could flexibly reflect the linkageeffect between water supply and demand and water price andregulate water supply and demand through the marketmechanism and thus is recommended to be considered andimplemented in the operational management of the SNWDproject

Second since the asset rights of the eastern route of theSNWD project belong to different entities each interestentity will tend to pursue its own interestsrsquo maximizationand will act in its own ways and lack of collaborative op-eration which will eventually lead to the reduction in op-eration efficiency Under the JPIM mode a coordinationstrategy via the revenue and cost sharing contract withpartial backlogging could effectively achieve Pareto im-provement of all the stakeholdersrsquo interests and thus isrecommended to be adopted to coordinate all the stake-holders in the eastern route of the SNWD project )edetailed action sequence of this coordination strategy is asfollows the local and the external suppliers first bargain overthe wholesale price of water resources within the IBWThorizontal supply chain to achieve cooperative operationsnext the IBWT supplier provides the distributors a revenuesharing contract in which the IBWTsupplier charges a lowerusage price to the water distributors if the distributorsaccept the contract they will place orders to the IBWTsupplier and decide the retail price of water resources andthe stock factor of water resources then the unmet water

demands of each market are partially backlogged and sat-isfied finally all the distributorsrsquo net revenues will be sharedto the IBWT supplier

Finally the order quantity of water resources mis-matching with the random water demand would induceunnecessary losses (including holding cost or shortage cost)in the operational management of the IBWTproject Underthe JPIM mode a partial backlogging strategy could effec-tively reduce these unnecessary losses and thus is recom-mended to be adopted in the operational management of theeastern route of the SNWD project )e decision makers ofthe project should make a lot of effort to enhance theirbacklogging abilities and increase the backlogging ratio toimprove the operational performance of all the stakeholdersFurthermore water delivery loss and water transfer costshave important impacts on the optimal operational deci-sionsperformance of the project the decision makers of theproject should make a lot of effort to reduce water deliveryloss and water transfer costs in the operational managementof the project

In sum a joint pricing and inventory management(JPIM) mode based on the flexible water price mechanism isrecommended to be implemented and a coordinationstrategy via the revenue and cost sharing contract withpartial backlogging is also recommended to be adopted toimprove operational efficiency for the eastern route of theSNWD project

7 Conclusions

From a perspective of supply chain management this papertries to explore the issues of joint pricing-inventory

33

32

31

3

29

28

27

26

times108

Prof

it

0 005 01 015 02Water delivery loss rate

(g)

01 03 04 05 06 07 08 0902Bargaining power

times107

Prof

it

11

10

9

8

7

6

5

4

3

2

1

+

Distributor 1Distributor 2Distributor 3Distributor 4

Distributor 5Distributor 6Distributor 7

(h)

Figure 2 )e impact of key parametersrsquo change on the profit of the IBWTsupply chain with partial backlogging (a) Mainline transfer cost(b) branch-line transfer cost (c) holding cost (d) shortage cost (e) backlogging ratio (f ) price elasticity (g) water delivery loss rate (h)bargaining power

14 Complexity

management decisions and operational strategies for theIBWT project considering water delivery loss and partialbacklogging )e equilibrium and coordination decisionmodels withoutwith partial backlogging for the IBWTsupply chain considering water delivery loss under jointpricing and inventory management (JPIM) are developedanalyzed and compared through a game-theoretical ap-proach and the corresponding numerical and sensitivityanalyses for all models are implemented and comparedfinally the managerial insights and practical implementa-tions are summarized in this paper )e research resultsshow that (1) a revenue and cost sharing contract couldeffectively coordinate the IBWT supply chain and improvethe operational performance of the IBWT supply chainunder JPIM (2) the partial backlogging strategy of waterdemand could effectively improve the operational perfor-mance of the IBWT supply chain under JPIM (3) coordi-nation strategy with partial backlogging is the best strategyfor improving the operational performance of the IBWTsupply chain under JPIM (4) reducing water delivery lossrate and operational costs (including mainline transfer costbranch-line transfer cost holding cost and shortage cost)are beneficial to improving operational performance of theIBWT supply chain under JPIM (5) increasing the back-logging ratio is beneficial to improving operational per-formance of the IBWTsupply chain under JPIM (6) a lowerprice elasticity of the expected water demand is beneficial toimproving operational performance of the IBWT supplychain under JPIM

In terms of theoretical contribution based on the game-theoretical approach the equilibrium and coordinationdecision models with partial backlogging under JPIMconsidering water delivery loss are developed analyzed andcompared for the IBWT supply chain respectively whichhave enriched the theories methodologies and applicationsin the optimal operation management of IBWT projectsWith regard to practical contribution the analytical andnumerical results provide the IBWT stakeholders a betterdecision support to make better operational decisions andstrategies

Data Availability

)e data used to support the findings of this study areavailable from the corresponding author upon request

Conflicts of Interest

)e authors declare that they have no conflicts of interest tothis work

Acknowledgments

)is work was supported by the National Natural ScienceFoundation of China (Grant no 71603125) China Schol-arship Council (Grant no 201706865020) National Plan-ning Office of Philosophy and Social Science (Grant no19BJY255) MOE (Ministry of Education in China) YouthFoundation Project of Humanities and Social Sciences

(Grant no 17YJC790002) China Postdoctoral ScienceFoundation (Grant no 2019M651833) Social ScienceFoundation of Jiangsu Province in China (Grant no19GLC003) National Key RampD Program of China (Grantno 2017YFC0404600) Natural Science Research Project ofColleges and Universities in Jiangsu Province (Grant no15KJB110012) and Young Leading Talent Program ofNanjing Normal University

Supplementary Materials

Proofs for analytical results of game-theoretical decisionmodels (Supplementary Materials)

References

[1] L Yang Foreign Projects of Water Diversion ChinaWater andPower Press Beijing China 2003

[2] G Wang Q Ouyang Y Zhang J Wei and Z Ren WorldrsquosWater Diversion Project Science Press Beijing China 2009

[3] Z Chen and L Pei ldquoInter-Basin water transfer green supplychain equilibrium and coordination under social welfaremaximizationrdquo Sustainability vol 10 no 4 p 1229 2018

[4] Z Chen S I Su and H Wang ldquoInter-basin water transfersupply chain equilibrium and coordination a social welfaremaximization perspectiverdquo Water Resources Managementvol 33 no 7 pp 2577ndash2598 2019

[5] Z Chen and H Wang ldquoInter-basin water transfer supplychain coordination with the fairness concern under capacityconstraint and random precipitationrdquoMarine Economics andManagement vol 2 no 1 pp 1ndash23 2019

[6] Z Chen and H Wang ldquoInter-basin water transfer greensupply chain coordination with partial backlogging underrandom precipitationrdquo Journal of Water and Climate Change2020

[7] S Wei H Yang K Abbaspour J Mousavi and A GnauckldquoGame theory based models to analyze water conflicts in themiddle route of the South-to-North water transfer project inChinardquo Water Research vol 44 no 8 pp 2499ndash2516 2010

[8] H D Manshadi M H Niksokhan and M Ardestani ldquoAquantity-quality model for inter-Basin water transfer systemusing game theoretic and virtual water approachesrdquo WaterResources Management vol 29 no 13 pp 4573ndash4588 2015

[9] D Rey A Garrido and J Calatrava ldquoAn innovative optioncontract for allocating water in Inter-Basin transfers the caseof the Tagus-Segura transfer in Spainrdquo Water ResourcesManagement vol 30 no 3 pp 1165ndash1182 2016

[10] J Sheng and M Webber ldquoIncentive-compatible payments forwatershed services along the eastern route of Chinarsquos South-North water transfer projectrdquo Ecosystem Services vol 25pp 213ndash226 2017

[11] M Sadegh N Mahjouri and R Kerachian ldquoOptimal inter-basin water allocation using crisp and fuzzy Shapley gamesrdquoWater Resources Management vol 24 no 10 pp 2291ndash23102010

[12] M R Nikoo R Kerachian and H Poorsepahy-Samian ldquoAninterval parameter model for cooperative inter-basin waterresources allocation considering the water quality issuesrdquoWater Resources Management vol 26 no 11 pp 3329ndash33432012

[13] K Jafarzadegan A Abed-Elmdoust and R Kerachian ldquoAfuzzy variable least core game for inter-basin water resources

Complexity 15

allocation under uncertaintyrdquo Water Resources Managementvol 27 no 9 pp 3247ndash3260 2013

[14] O Nasiri-Gheidari S Marofi and F Adabi ldquoA robust multi-objective bargaining methodology for inter-basin water re-source allocation a case studyrdquo Environmental Science andPollution Research vol 25 no 3 pp 2726ndash2737 2018

[15] H Wang Z Chen and S I Su ldquoOptimal pricing and co-ordination schemes for the eastern route of the South-to-North water diversion supply chain system in ChinardquoTransportation Journal vol 51 no 4 pp 487ndash505 2012

[16] Z Chen and H Wang ldquoOptimization and coordination ofSouth-to-North water diversion supply chain with strategiccustomer behaviorrdquo Water Science and Engineering vol 5no 4 pp 464ndash477 2012

[17] Z Chen and H Wang ldquoAsymmetric Nash bargaining modelfor the eastern route of South-to-North water diversionsupply chain cooperative operationsrdquo Journal of the ChineseInstitute of Industrial Engineers vol 29 no 6 pp 365ndash3742012

[18] Y Xu L Wang Z Chen S Shan and G Xia ldquoOptimizationand adjustment policy of two-echelon reservoir inventorymanagement with forecast updatesrdquo Computers amp IndustrialEngineering vol 63 no 4 pp 890ndash900 2012

[19] Z Chen H Wang and X Qi ldquoPricing and water resourceallocation scheme for the South-to-North water diversionproject in Chinardquo Water Resources Management vol 27no 5 pp 1457ndash1472 2013

[20] W Du Y Fan and X Tang ldquoTwo-part pricing contractsunder competition the South-to-North Water TransferProject supply chain system in Chinardquo International Journalof Water Resources Development vol 32 no 6 pp 895ndash9112016

[21] W Du Y Fan and L Yan ldquoPricing strategies for competitivewater supply chains under different power structures anapplication to the South-to-North water diversion project inChinardquo Sustainability vol 10 no 8 pp 1ndash13 2018

[22] W Du Y Fan X Liu S C Park and X Tang ldquoA game-basedproduction operation model for water resource managementan analysis of the South-to-North Water Transfer Project inChinardquo Journal of Cleaner Production vol 228 pp 1482ndash1493 2019

[23] C W Howe and F P Linaweaver Jr ldquo)e impact of price onresidential water demand and its relation to system design andprice structurerdquo Water Resources Research vol 3 no 1pp 13ndash32 1967

[24] K Schoengold D L Sunding andGMoreno ldquoPrice elasticityreconsidered panel estimation of an agricultural water de-mand functionrdquo Water Resources Research vol 42 no 9Article ID W09411 2006

[25] N C Petruzzi and M Dada ldquoPricing and the newsvendorproblem a review with extensionsrdquo Operations Researchvol 47 no 2 pp 183ndash194 1999

[26] L Wang L Fang and K W Hipel ldquoBasin-wide cooperativewater resources allocationrdquo European Journal of OperationalResearch vol 190 no 3 pp 798ndash817 2008

[27] Y Wang L Jiang and Z-J Shen ldquoChannel performanceunder consignment contract with revenue sharingrdquo Man-agement Science vol 50 no 1 pp 34ndash47 2004

[28] YWang ldquoJoint pricing-production decisions in supply chainsof complementary products with uncertain demandrdquo Oper-ations Research vol 54 no 6 pp 1110ndash1127 2006

[29] M A Lariviere and E L Porteus ldquoSelling to the newsvendoran analysis of price-only contractsrdquo Manufacturing amp ServiceOperations Management vol 3 no 4 pp 293ndash305 2001

[30] M A Lariviere ldquoA note on probability distributions withincreasing generalized failure ratesrdquo Operations Researchvol 54 no 3 pp 602ndash604 2006

[31] J F Nash ldquo)e bargaining problemrdquo Econometrica vol 18no 2 pp 155ndash162 1950

[32] E Kalai and M Smorodinsky ldquoOther solutions to Nashrsquosbargaining problemrdquo Econometrica vol 43 no 3pp 513ndash518 1975

[33] A Muthoo Bargaining 4eory with Applications CambridgeUniversity Press Cambridge MA USA 1999

[34] National Development and Reform Commission (NDRC)Notice on the Water Supply Price Policy for the First Phase ofthe East Line of the South-to-North Water Diversion Project(NDRC Price [2014] No 30) [EBOL] 2014 httpwwwnsbddxcomsingle_detail300html

[35] K Binmore A Rubinstein and A Wolinsky ldquo)e Nashbargaining solution in economic modellingrdquo 4e RANDJournal of Economics vol 17 no 2 pp 176ndash188 1986

[36] Ministry of Water Resources Notice on Printing and Dis-tributing the Water transfer Scheme (Trial) for the First Phaseof the East Route of the South-to-North Water Transfer Project(Water Resources [2013] No 466) [EBOL] 2013 httpwwwmwrgovcnzwgkgknr202008t20200803_1430819html

16 Complexity

delivery loss under random precipitation [6] However thejoint pricing-inventory management decisions and opera-tional strategies for an IBWTsupply chain considering waterdelivery loss and partial backlogging are rarely investigatedin the current literatures and practices

)erefore this paper will try to explore a novelty re-search issue regarding the operation management of theIBWT supply chainmdashthe joint pricing-inventory manage-ment (JPIM) decisions and operational strategies for theIBWT supply chain considering water delivery loss andpartial backlogging under random water demand

In the following sections the related literatures arereviewed first in Section 2 the theoretical modeling nota-tions and assumptions for a generic IBWT supply chain aredefined in Section 3 four game-theoretical decision modelsfor the IBWT supply chain withoutwith backlogging underjoint pricing-inventory management (JPIM) are developedanalyzed and compared in Section 4 the correspondingnumerical and sensitivity analyses for all models areimplemented and the corresponding results are comparedin Section 5 the managerial insights and practical imple-mentations are then summarized in Section 6 finally thetheoretical and practical contributions are summarized

2 Literature Review

Currently the interaction relationships among multiplestakeholders in the IBWT projects are investigated throughgame theory such as the water conflict game-theoreticalmodel of the SNWD project [7] game-theoretical model ofthe IBWT project considering both the water quantity andwater quality [8] water allocation option contract for theIBWT projects [9] and incentive-compatible payment de-sign for the SNWD project [10]

Besides cooperative game theory is applied to achievePareto improvement in the IBWT projects such as thecrisp and fuzzy Shapley game model for the IBWT waterallocation [11] cooperative game model for the IBWTwater allocation [12] IBWT water resource allocationusing the least core game [13] and robust multiobjectivebargaining game model for the IBWT water resource al-location [14]

Currently theories and techniques of supply chainmanagement (SCM) have been applied in the IBWTprojectsto investigate the interactions among multiple stakeholdersand develop equilibriumcoordination operational mecha-nisms such as optimal pricing and coordination schemes forthe SNWD supply chain [15] coordination mechanismbased on revenue sharing contract for the SNWD supplychain with strategic customer [16] asymmetric Nash bar-gainingmodel for the SNWD supply chain [17] two-echelonwater inventory model with inflow forecasting updates in anIBWT project [18] two-tier pricing and allocation schemesfor the SNWD supply chain [19] competition intensity inthe water supply chain under two contracts [20] powerstructures for the competitive water supply chains [21]optimal pricing and ordering strategies for dual competingwater supply chains under three contracts [22] subsidypolicies and operational strategies for the IBWT green

supply chain under social welfare maximization [3 4]impact of the supply capacity constraint and fairness con-cern on the operational decisions and outcomes of the IBWTsupply chain under random precipitation [5] and impact offullypartial backlogging on the IBWT green supply chaincoordination considering water delivery loss under randomprecipitation [6]

Nevertheless these existing literatures regarding IBWTsupply chain management neither explored the equilibriumcoordination strategies of the IBWT supply chain underJPIM nor investigated the impact of the partial backloggingthe choice of operational strategies and the water deliveryloss on the operational performance of the IBWT supplychain )is paper tries to address the shortcomings in theavailable literatures and explore the operational strategies foran IBWT supply chain withoutwith partial backloggingunder JPIM An equilibrium decision model and a coor-dination decision model for the IBWTsupply chain withoutbackloggingwith partial backlogging under JPIM are de-veloped solved and compared respectively to explore theoptimal operational strategies and optimal joint pricing andinventory decisions for the IBWT supply chain

3 Modeling Notations and Assumptions

An IBWTdistribution system is a typical ldquoembeddedrdquo supplychain structure In this supply chain system a horizontalwater supply system embeds itself in a vertical water dis-tribution system (see Figure 1) In the horizontal watersupply system a local supplier and an external supplier workas a joint IBWT supplier via an efficient cooperationmechanismWater resources are transferred and supplied bythe local supplier from the water source to the externalsupplier within the trunk channel and then distributed towater resource distributors of all water intakes via riverchannels and artificial canals Finally the water resources aresold by each distributor to the water resource consumers inhisher service region )e water distributor and the cor-responding water market in the ith water intake are indexedby i 1 2 n It is assumed that there are m distributorssupplied by the local supplier and nndashm distributors suppliedby the external supplier

On this basis the parameters used in the models aredefined as follows

cdi the water transfer cost from the ith water intake tothe ith distributorck the water transfer cost from the (k minus 1)th waterintake to the kth water intake within the horizontalsupply chain k 1 2 n

δk the water delivery loss rate from the (k minus 1)th waterintake to the kth water intake within the horizontalsupply chain and δk isin (0 1) k 1 2 n

cfi the fixed cost for the ith water intake of the IBWTsuppliercfl the fixed cost for the local supplier cfl 1113936

mi1 cfi

cfe the fixed cost for the external suppliercfe 1113936

nim+1 cfi

2 Complexity


ni1 cfi










TCi(Qi) Qi 1113936ik1[ck1113937



ik1[ck 1113937



fining Ci (1113936ik1[ck 1113937








Local supplier


Distributor1

Distributor2

Distributori

Distributorm

Distributorm + 1

(m + 1)th

market(m + 2)th

market(m + k)th

market

Distributorm + 2

Distributorm + k

Distributorn

1st

market2nd

marketith

marketmth

marketnth

market

Water-intake

Wat

er so

urce

Water supplier

helliphellip

hellip hellip

Vert

ical

supp

ly ch

ain

ibw

t wat

er d

istrib

utio

nPump PumpPumpPump


Complexity 3



+1113858 1113859



+1113858 1113859 minus cfi



+1113858 1113859


(1)


ΠS wi( 1113857 1113944n

i1ΠSi

wi( 1113857

ΠLS wi w( 1113857 1113944m

i1ΠSi

wi( 1113857 + w 1113944n


ΠES wi w( 1113857 1113944n

im+1ΠSi

wi( 1113857 minus w 1113944n


ΠSC zi pi( 1113857 1113944n

i1ΠSCi

zi pi( 1113857

(2)







maxwΩ(w) ΠodLS w

odi q

odi w1113872 11138731113960 1113961

τΠodES w

odi q

odi w1113872 11138731113960 1113961

1minus τ

st

ΠodLS wodi q

odi w1113872 1113873 +ΠodES w

odi q

odi w1113872 1113873 ΠodS w

odi q

odi1113872 1113873

st

wodi p

odi z

odi and q


maxwi

ΠoS wi p

odi wi( 1113857 z

odi1113872 1113873



maxzipi

ΠoDi

zi pi( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(3)

4 Complexity






odES Π

odS ΠodDi







Di(zi pi) minus Ti


ocSi

(wi) 1113936ni1[Π

oSi



maxwΩ(w) ΠocLS w

oci q

oci w( 11138571113858 1113859

τ ΠocES woci q

oci w( 11138571113858 1113859

1minus τ

st

ΠocLS woci q


oci q


oci q

oci( 1113857

woci q

oci ΠocLS w

oci q


oci q


oci q

oci( 1113857


maxϕ

πoi (ϕ) ΠocSi

(ϕ) minus ΠodSi1113960 1113961

λΠocDi

(ϕ) minus ΠodDi1113960 1113961

1minus λ

st

ΠocSi(ϕ) + ΠocDi

(ϕ) ΠocSCi

woci (ϕ) p

oci z

oci q

oci ΠocSi



maxzipi


maxzipi


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(4)








ocES Π

ocS ΠocDi


Complexity 5

Tabl

e1

Analytical

results

oftheIBWTsupp

lychainwith

outb

acklogging

Scenario

var

Result

Equilib

rium

decisio

n(411)

Coo

rdinationdecisio

n(412)

wo i

wod i

((

bC

i+

c di)(

bminus1)

)w

oc iϕo c

Ciminus

(1

minusϕo c

)cd

i

po i

pod i

(

b(

bminus1)

poc i

)poc i

(

b(

bminus1)

)((

Ci+

c di)

zoc i(1

+κ s

)zoc i

minusΛ

i(zoc i

))

zo i

Fi(

zod i

)

Fi(

zoc i

)F

i(zoc i

)

((1

+κ s

)b(1

+κ h

+κ s

))+

((b

minus1)Λ

i(zoc i

)b(1

+κ h

+κ s

)zoc i

)

qo i

qod i

((

bminus1)b

)bqoc i

qoc i

y

i(poc i

)zoc i

a

i(poc i

)minusbzoc i

wo

wo d

(1

1113936n i

m+1

qod i

)(τΠ

od Sminus

1113936m i1Π

od Si)

wo c

(1

1113936n i

m+1

qoc i

)(τΠ

oc Sminus

1113936m i1Π

oc Si)

Πo LS

Πod LS

τΠ

od Sτ1113936

n i1[

((b

minus1)b

)b(Π

oc SCi+

c fi)

minusc fi]

Πoc LS

τΠ

oc Sτ1113936

n i1[

(1

minusϕo c

)Πoc SC

iminusϕo c

c fi]

Πo ES

Πod ES

(1

minusτ)Π

od S

(1

minusτ)

1113936n i1[

((b

minus1)b

)b(Π

oc SCi+

c fi)

minusc fi]

Πoc ES

(1

minusτ)Π

oc S

(1

minusτ)

1113936n i1[

(1

minusϕo c

)Πoc SC

iminusϕo c

c fi]

Πo S

Πod S

1113936

n i1Π

od Si

1113936

n i1[

((b

minus1)b

)b(Π

oc SCi+

c fi)

minusc fi]

Πoc S

1113936

n i1Π

oc Si

1113936

n i1[

(1

minusϕo c

)Πoc SC

iminusϕo c

c fi]

Πo D

iΠ

od Di

((

bminus1)b

)bminus1 (Π

oc SCi+

c fi)

Πoc D

iϕo c

(Π

oc SCi+

c fi)

Πo SC

Πod SC

1113936

n i1Π

od SCi

1113936

n i1

[((

bminus1)b

)b+

((b

minus1)b

)bminus1 ](Π

oc SCi+

c fi)

minusc fi

11139661113967

Πoc SC

1113936

n i1Π

oc SCi

1113936

n i1(

(C

i+

c di)(

bminus1)

)qoc i

minusc fi

ϕo cNA

ϕo cλ(

(b

minus1)b

)bminus1

+(1

minusλ)

[1minus

((b

minus1)b

)b]

Note

Λi(

zoc i

)

(1

+κ h

+κ s

)1113938

zoc i

A(

zoc i

minusx

i)f

i(x

i)dx

i+κ s

1113938B A

xif

i(x

i)dx

i

6 Complexity




maxwΩ(w) ΠbdLS w

bdi q

bdi w1113872 11138731113960 1113961

τΠbdES w

bdi q

bdi w1113872 11138731113960 1113961

1minus τ

st

ΠbdLS wbdi q

bdi w1113872 1113873 + ΠbdES w

bdi q

bdi w1113872 1113873 ΠbdS w

bdi q

bdi1113872 1113873

st

wbdi p

bdi z

bdi and q


maxwi

ΠbS wi p

bdi wi( 1113857 z

bdi1113872 1113873



maxzipi

ΠbDi

zi pi( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)






bdES Π

bdS ΠbdDi






(zi pi) ΠbDi


ni1Π

bcSi

(wi) 1113936ni1[Π

bSi



Complexity 7

Tabl

e2

Analytical

results

oftheIBWTsupp

lychainwith

partialb

acklogging

Scenario

var

Result

Equilib

rium

decisio

n(421)

Coo

rdinationdecisio

n(422)

wb i

wbd i

((

bC

i+

c di)(

bminus1)

)w

bc iϕb c

Ciminus

(1

minusϕb c

)cd

i

pb i

pbd i

(

b(

bminus1)

)pbc i

pbc i

(

b(

Ci+

c di)(

bminus1)

)(((1

minusφ)

zbc i

+φΝ

i(zbc i

))(

(1

minusφ

+κ s

)zbc i

minusΜ

i(zbc i

)))

zb i

Fi(

zbd i

)

Fi(

zbc i

)F

i(zbc i

)

qb i

qbd i

((

bminus1)b

)bqbc i

qbc i

y

i(pbc i

)zbc i

a

i(pbc i

)minusbzbc i

wb

wb d

(1

1113936n i

m+1

qbd i

)(τΠ

bd Sminus

1113936m i1Π

bd Si)

wb c

(1

1113936n i

m+1

qbc i

)(τΠ

bc Sminus

1113936m i1Π

bc Si)

Πb LS

Πbd LS

τΠ

bd Sτ1113936

n i1[

((b

minus1)b

)b(Π

bc SCi+

c fi)

minusc f

i]Π

bc LSτΠ

bc Sτ1113936

n i1[

(1

minusϕb c

)Πbc SC

iminusϕb c

c fi]

Πb ES

Πbd ES

(1

minusτ)Π

bd S

(1

minusτ)

1113936n i1[

((b

minus1)b

)b(Π

bc SCi+

c fi)

minusc fi]

Πbc ES

(1

minusτ)Π

bc S

(1

minusτ)

1113936n i1[

(1

minusϕb c

)Πbc SC

iminusϕb c

c fi]

Πb S

Πbd S

1113936

n i1Π

bd Si

1113936

n i1[

((b

minus1)b

)b(Π

bc SCi+

c fi)

minusc fi]

Πbc S

1113936

n i1Π

bc Si

1113936

n i1[

(1

minusϕb c

)Πbc SC

iminusϕb c

c fi]

Πb D

iΠ

bd Di

((

bminus1)b

)bminus1 (Π

bc SCi+

c fi)

Πbc D

iϕb c

(Π

bc SCi+

c fi)

Πb SC

Πbd SC

1113936

n i1Π

bd SCi

1113936

n i1

[((

bminus1)b

)b+

((b

minus1)b

)bminus1 ](Π

bc SCi+

c fi)

minusc fi

11139661113967

Πbc SC

1113936

n i1Π

bc SCi

1113936

n i1

((C

i+

c di)(

bminus1)

)yi(

pbc i

)[(1

minusφ)

zbc i

+φΝ

i(zbc i

)]minus

c fi

11138641113865

ϕb cNA

ϕb cλ(

(b

minus1)b

)bminus1

+(1

minusλ)

[1minus

((b

minus1)b

)b]

Note

Fi(

zbc i

)

(((1

minusφ)

(1

minusφ

+κ s

)zbc i

+

bφ(

1minusφ

+κ s

)Ni(

zbc i

)+

(b

minus1)

(1

minusφ)

Mi(

zbc i

))(

[(b

minusφ)

(1

minusφ

+κ s

)+

b(1

minusφ)κ h

]zbc i

+bφ(

1minusφ

+κ s

+κ h

)Νi(

zbc i

)minus

(b

minus1)φΜ

i(zbc i

)))

Mi(

zbc i

)

(1

+κ h

+κ s

minus

φ)1113946

zbc i

A(

zbc i

minusx

i)f

i(x

i)dx

iminus

(φ

minusκ s

)1113946

B Ax

ifi(

xi)dx

iN

i(zbc i

)

1113946zbc i

A(

zbc i

minusx

i)f

i(x

i)dx

i+

1113946B A

xif

i(x

i)dx

i

8 Complexity

maxwΩ(w) ΠbcLS w

bci q

bci w1113872 11138731113960 1113961

τΠbcES w

bci q

bci w1113872 11138731113960 1113961

1minus τ

st

ΠbcLS wbci q

bci w1113872 1113873 +ΠbcES w

bci q

bci w1113872 1113873 ΠbcS w

bci q

bci1113872 1113873

wbci q

bci ΠbcLS w

bci q

bci w1113872 1113873ΠbcES w

bci q


bci q

bci1113872 1113873


maxϕ

πbi (ϕ) ⊓bcSi

(ϕ) minus ΠbdSi1113960 1113961

λΠbcDi

(ϕ) minus ΠbdDi1113960 1113961

1minus λ

st

ΠbcSi(ϕ) + ΠbcDi

(ϕ) ΠbcSCi

wbci (ϕ) p

bci z

bci q

bci ΠbcSi



maxzipi


maxzipi


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)






bcESΠ

bcS ΠbcDi








Complexity 9









10 Complexity






1 032 008 1796 +E92 024 010 1218 +E93 009 012 1169 +E94 014 076 1941 +E95 041 029 1117 +E96 081 040 1525 +E9


i wodi pod

i qodi ΠodDiΠodS


d 051


i woci poc

i qoci ΠocDiΠocS


i 011 woc 020 ϕo

c 067


i wbdi pbd

i qbdi ΠbdDiΠbdS


d 053

Complexity 11












i wbci pbc

i qbci ΠbcDiΠbcS


i 010 wbc 020 ϕb

c 067

12 Complexity

32

31

3

29

28

27

26

Prof

it


Supply chain

times108

(a)

Prof

it

Supply chain

times108286

284

282

28

278

276

274005 006 007 008 009 01 011 012 013 014 015


(b)

Prof

it

284

283

282

281

28

279

278

277

27601 02 03 04 05


times108

Supply chain

(c)

Prof

it

284

285

286

287

288

283

282

281

2802 03 04 05 06

times108


Supply chain

(d)

times108

Prof

it

284

283

282

281

28

279

278

277

276

2750 01 03 04 05 06 07 08 09 102

Backlogging ratio

Supply chain

(e)

times108

Prof

it

6

55

5

45

4

35

3

25

2

1511 12 13 14 15 16 17 18 19 2


Supply chain

(f )

Figure 2 Continued

Complexity 13






7 Conclusions


33

32

31

3

29

28

27

26

times108

Prof

it


(g)


times107

Prof

it

11

10

9

8

7

6

5

4

3

2

1

+



(h)


14 Complexity



Data Availability




Acknowledgments





References














Complexity 15

























16 Complexity


ni1 cfi










TCi(Qi) Qi 1113936ik1[ck1113937



ik1[ck 1113937



fining Ci (1113936ik1[ck 1113937








Local supplier


Distributor1

Distributor2

Distributori

Distributorm

Distributorm + 1

(m + 1)th

market(m + 2)th

market(m + k)th

market

Distributorm + 2

Distributorm + k

Distributorn

1st

market2nd

marketith

marketmth

marketnth

market

Water-intake

Wat

er so

urce

Water supplier

helliphellip

hellip hellip

Vert

ical

supp

ly ch

ain

ibw

t wat

er d

istrib

utio

nPump PumpPumpPump


Complexity 3



+1113858 1113859



+1113858 1113859 minus cfi



+1113858 1113859


(1)


ΠS wi( 1113857 1113944n

i1ΠSi

wi( 1113857

ΠLS wi w( 1113857 1113944m

i1ΠSi

wi( 1113857 + w 1113944n


ΠES wi w( 1113857 1113944n

im+1ΠSi

wi( 1113857 minus w 1113944n


ΠSC zi pi( 1113857 1113944n

i1ΠSCi

zi pi( 1113857

(2)







maxwΩ(w) ΠodLS w

odi q

odi w1113872 11138731113960 1113961

τΠodES w

odi q

odi w1113872 11138731113960 1113961

1minus τ

st

ΠodLS wodi q

odi w1113872 1113873 +ΠodES w

odi q

odi w1113872 1113873 ΠodS w

odi q

odi1113872 1113873

st

wodi p

odi z

odi and q


maxwi

ΠoS wi p

odi wi( 1113857 z

odi1113872 1113873



maxzipi

ΠoDi

zi pi( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(3)

4 Complexity






odES Π

odS ΠodDi







Di(zi pi) minus Ti


ocSi

(wi) 1113936ni1[Π

oSi



maxwΩ(w) ΠocLS w

oci q

oci w( 11138571113858 1113859

τ ΠocES woci q

oci w( 11138571113858 1113859

1minus τ

st

ΠocLS woci q


oci q


oci q

oci( 1113857

woci q

oci ΠocLS w

oci q


oci q


oci q

oci( 1113857


maxϕ

πoi (ϕ) ΠocSi

(ϕ) minus ΠodSi1113960 1113961

λΠocDi

(ϕ) minus ΠodDi1113960 1113961

1minus λ

st

ΠocSi(ϕ) + ΠocDi

(ϕ) ΠocSCi

woci (ϕ) p

oci z

oci q

oci ΠocSi



maxzipi


maxzipi


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(4)








ocES Π

ocS ΠocDi


Complexity 5

Tabl

e1

Analytical

results

oftheIBWTsupp

lychainwith

outb

acklogging

Scenario

var

Result

Equilib

rium

decisio

n(411)

Coo

rdinationdecisio

n(412)

wo i

wod i

((

bC

i+

c di)(

bminus1)

)w

oc iϕo c

Ciminus

(1

minusϕo c

)cd

i

po i

pod i

(

b(

bminus1)

poc i

)poc i

(

b(

bminus1)

)((

Ci+

c di)

zoc i(1

+κ s

)zoc i

minusΛ

i(zoc i

))

zo i

Fi(

zod i

)

Fi(

zoc i

)F

i(zoc i

)

((1

+κ s

)b(1

+κ h

+κ s

))+

((b

minus1)Λ

i(zoc i

)b(1

+κ h

+κ s

)zoc i

)

qo i

qod i

((

bminus1)b

)bqoc i

qoc i

y

i(poc i

)zoc i

a

i(poc i

)minusbzoc i

wo

wo d

(1

1113936n i

m+1

qod i

)(τΠ

od Sminus

1113936m i1Π

od Si)

wo c

(1

1113936n i

m+1

qoc i

)(τΠ

oc Sminus

1113936m i1Π

oc Si)

Πo LS

Πod LS

τΠ

od Sτ1113936

n i1[

((b

minus1)b

)b(Π

oc SCi+

c fi)

minusc fi]

Πoc LS

τΠ

oc Sτ1113936

n i1[

(1

minusϕo c

)Πoc SC

iminusϕo c

c fi]

Πo ES

Πod ES

(1

minusτ)Π

od S

(1

minusτ)

1113936n i1[

((b

minus1)b

)b(Π

oc SCi+

c fi)

minusc fi]

Πoc ES

(1

minusτ)Π

oc S

(1

minusτ)

1113936n i1[

(1

minusϕo c

)Πoc SC

iminusϕo c

c fi]

Πo S

Πod S

1113936

n i1Π

od Si

1113936

n i1[

((b

minus1)b

)b(Π

oc SCi+

c fi)

minusc fi]

Πoc S

1113936

n i1Π

oc Si

1113936

n i1[

(1

minusϕo c

)Πoc SC

iminusϕo c

c fi]

Πo D

iΠ

od Di

((

bminus1)b

)bminus1 (Π

oc SCi+

c fi)

Πoc D

iϕo c

(Π

oc SCi+

c fi)

Πo SC

Πod SC

1113936

n i1Π

od SCi

1113936

n i1

[((

bminus1)b

)b+

((b

minus1)b

)bminus1 ](Π

oc SCi+

c fi)

minusc fi

11139661113967

Πoc SC

1113936

n i1Π

oc SCi

1113936

n i1(

(C

i+

c di)(

bminus1)

)qoc i

minusc fi

ϕo cNA

ϕo cλ(

(b

minus1)b

)bminus1

+(1

minusλ)

[1minus

((b

minus1)b

)b]

Note

Λi(

zoc i

)

(1

+κ h

+κ s

)1113938

zoc i

A(

zoc i

minusx

i)f

i(x

i)dx

i+κ s

1113938B A

xif

i(x

i)dx

i

6 Complexity




maxwΩ(w) ΠbdLS w

bdi q

bdi w1113872 11138731113960 1113961

τΠbdES w

bdi q

bdi w1113872 11138731113960 1113961

1minus τ

st

ΠbdLS wbdi q

bdi w1113872 1113873 + ΠbdES w

bdi q

bdi w1113872 1113873 ΠbdS w

bdi q

bdi1113872 1113873

st

wbdi p

bdi z

bdi and q


maxwi

ΠbS wi p

bdi wi( 1113857 z

bdi1113872 1113873



maxzipi

ΠbDi

zi pi( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)






bdES Π

bdS ΠbdDi






(zi pi) ΠbDi


ni1Π

bcSi

(wi) 1113936ni1[Π

bSi



Complexity 7

Tabl

e2

Analytical

results

oftheIBWTsupp

lychainwith

partialb

acklogging

Scenario

var

Result

Equilib

rium

decisio

n(421)

Coo

rdinationdecisio

n(422)

wb i

wbd i

((

bC

i+

c di)(

bminus1)

)w

bc iϕb c

Ciminus

(1

minusϕb c

)cd

i

pb i

pbd i

(

b(

bminus1)

)pbc i

pbc i

(

b(

Ci+

c di)(

bminus1)

)(((1

minusφ)

zbc i

+φΝ

i(zbc i

))(

(1

minusφ

+κ s

)zbc i

minusΜ

i(zbc i

)))

zb i

Fi(

zbd i

)

Fi(

zbc i

)F

i(zbc i

)

qb i

qbd i

((

bminus1)b

)bqbc i

qbc i

y

i(pbc i

)zbc i

a

i(pbc i

)minusbzbc i

wb

wb d

(1

1113936n i

m+1

qbd i

)(τΠ

bd Sminus

1113936m i1Π

bd Si)

wb c

(1

1113936n i

m+1

qbc i

)(τΠ

bc Sminus

1113936m i1Π

bc Si)

Πb LS

Πbd LS

τΠ

bd Sτ1113936

n i1[

((b

minus1)b

)b(Π

bc SCi+

c fi)

minusc f

i]Π

bc LSτΠ

bc Sτ1113936

n i1[

(1

minusϕb c

)Πbc SC

iminusϕb c

c fi]

Πb ES

Πbd ES

(1

minusτ)Π

bd S

(1

minusτ)

1113936n i1[

((b

minus1)b

)b(Π

bc SCi+

c fi)

minusc fi]

Πbc ES

(1

minusτ)Π

bc S

(1

minusτ)

1113936n i1[

(1

minusϕb c

)Πbc SC

iminusϕb c

c fi]

Πb S

Πbd S

1113936

n i1Π

bd Si

1113936

n i1[

((b

minus1)b

)b(Π

bc SCi+

c fi)

minusc fi]

Πbc S

1113936

n i1Π

bc Si

1113936

n i1[

(1

minusϕb c

)Πbc SC

iminusϕb c

c fi]

Πb D

iΠ

bd Di

((

bminus1)b

)bminus1 (Π

bc SCi+

c fi)

Πbc D

iϕb c

(Π

bc SCi+

c fi)

Πb SC

Πbd SC

1113936

n i1Π

bd SCi

1113936

n i1

[((

bminus1)b

)b+

((b

minus1)b

)bminus1 ](Π

bc SCi+

c fi)

minusc fi

11139661113967

Πbc SC

1113936

n i1Π

bc SCi

1113936

n i1

((C

i+

c di)(

bminus1)

)yi(

pbc i

)[(1

minusφ)

zbc i

+φΝ

i(zbc i

)]minus

c fi

11138641113865

ϕb cNA

ϕb cλ(

(b

minus1)b

)bminus1

+(1

minusλ)

[1minus

((b

minus1)b

)b]

Note

Fi(

zbc i

)

(((1

minusφ)

(1

minusφ

+κ s

)zbc i

+

bφ(

1minusφ

+κ s

)Ni(

zbc i

)+

(b

minus1)

(1

minusφ)

Mi(

zbc i

))(

[(b

minusφ)

(1

minusφ

+κ s

)+

b(1

minusφ)κ h

]zbc i

+bφ(

1minusφ

+κ s

+κ h

)Νi(

zbc i

)minus

(b

minus1)φΜ

i(zbc i

)))

Mi(

zbc i

)

(1

+κ h

+κ s

minus

φ)1113946

zbc i

A(

zbc i

minusx

i)f

i(x

i)dx

iminus

(φ

minusκ s

)1113946

B Ax

ifi(

xi)dx

iN

i(zbc i

)

1113946zbc i

A(

zbc i

minusx

i)f

i(x

i)dx

i+

1113946B A

xif

i(x

i)dx

i

8 Complexity

maxwΩ(w) ΠbcLS w

bci q

bci w1113872 11138731113960 1113961

τΠbcES w

bci q

bci w1113872 11138731113960 1113961

1minus τ

st

ΠbcLS wbci q

bci w1113872 1113873 +ΠbcES w

bci q

bci w1113872 1113873 ΠbcS w

bci q

bci1113872 1113873

wbci q

bci ΠbcLS w

bci q

bci w1113872 1113873ΠbcES w

bci q


bci q

bci1113872 1113873


maxϕ

πbi (ϕ) ⊓bcSi

(ϕ) minus ΠbdSi1113960 1113961

λΠbcDi

(ϕ) minus ΠbdDi1113960 1113961

1minus λ

st

ΠbcSi(ϕ) + ΠbcDi

(ϕ) ΠbcSCi

wbci (ϕ) p

bci z

bci q

bci ΠbcSi



maxzipi


maxzipi


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)






bcESΠ

bcS ΠbcDi








Complexity 9









10 Complexity






1 032 008 1796 +E92 024 010 1218 +E93 009 012 1169 +E94 014 076 1941 +E95 041 029 1117 +E96 081 040 1525 +E9


i wodi pod

i qodi ΠodDiΠodS


d 051


i woci poc

i qoci ΠocDiΠocS


i 011 woc 020 ϕo

c 067


i wbdi pbd

i qbdi ΠbdDiΠbdS


d 053

Complexity 11












i wbci pbc

i qbci ΠbcDiΠbcS


i 010 wbc 020 ϕb

c 067

12 Complexity

32

31

3

29

28

27

26

Prof

it


Supply chain

times108

(a)

Prof

it

Supply chain

times108286

284

282

28

278

276

274005 006 007 008 009 01 011 012 013 014 015


(b)

Prof

it

284

283

282

281

28

279

278

277

27601 02 03 04 05


times108

Supply chain

(c)

Prof

it

284

285

286

287

288

283

282

281

2802 03 04 05 06

times108


Supply chain

(d)

times108

Prof

it

284

283

282

281

28

279

278

277

276

2750 01 03 04 05 06 07 08 09 102

Backlogging ratio

Supply chain

(e)

times108

Prof

it

6

55

5

45

4

35

3

25

2

1511 12 13 14 15 16 17 18 19 2


Supply chain

(f )

Figure 2 Continued

Complexity 13






7 Conclusions


33

32

31

3

29

28

27

26

times108

Prof

it


(g)


times107

Prof

it

11

10

9

8

7

6

5

4

3

2

1

+



(h)


14 Complexity



Data Availability




Acknowledgments





References














Complexity 15

























16 Complexity



+1113858 1113859



+1113858 1113859 minus cfi



+1113858 1113859


(1)


ΠS wi( 1113857 1113944n

i1ΠSi

wi( 1113857

ΠLS wi w( 1113857 1113944m

i1ΠSi

wi( 1113857 + w 1113944n


ΠES wi w( 1113857 1113944n

im+1ΠSi

wi( 1113857 minus w 1113944n


ΠSC zi pi( 1113857 1113944n

i1ΠSCi

zi pi( 1113857

(2)







maxwΩ(w) ΠodLS w

odi q

odi w1113872 11138731113960 1113961

τΠodES w

odi q

odi w1113872 11138731113960 1113961

1minus τ

st

ΠodLS wodi q

odi w1113872 1113873 +ΠodES w

odi q

odi w1113872 1113873 ΠodS w

odi q

odi1113872 1113873

st

wodi p

odi z

odi and q


maxwi

ΠoS wi p

odi wi( 1113857 z

odi1113872 1113873



maxzipi

ΠoDi

zi pi( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(3)

4 Complexity






odES Π

odS ΠodDi







Di(zi pi) minus Ti


ocSi

(wi) 1113936ni1[Π

oSi



maxwΩ(w) ΠocLS w

oci q

oci w( 11138571113858 1113859

τ ΠocES woci q

oci w( 11138571113858 1113859

1minus τ

st

ΠocLS woci q


oci q


oci q

oci( 1113857

woci q

oci ΠocLS w

oci q


oci q


oci q

oci( 1113857


maxϕ

πoi (ϕ) ΠocSi

(ϕ) minus ΠodSi1113960 1113961

λΠocDi

(ϕ) minus ΠodDi1113960 1113961

1minus λ

st

ΠocSi(ϕ) + ΠocDi

(ϕ) ΠocSCi

woci (ϕ) p

oci z

oci q

oci ΠocSi



maxzipi


maxzipi


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(4)








ocES Π

ocS ΠocDi


Complexity 5

Tabl

e1

Analytical

results

oftheIBWTsupp

lychainwith

outb

acklogging

Scenario

var

Result

Equilib

rium

decisio

n(411)

Coo

rdinationdecisio

n(412)

wo i

wod i

((

bC

i+

c di)(

bminus1)

)w

oc iϕo c

Ciminus

(1

minusϕo c

)cd

i

po i

pod i

(

b(

bminus1)

poc i

)poc i

(

b(

bminus1)

)((

Ci+

c di)

zoc i(1

+κ s

)zoc i

minusΛ

i(zoc i

))

zo i

Fi(

zod i

)

Fi(

zoc i

)F

i(zoc i

)

((1

+κ s

)b(1

+κ h

+κ s

))+

((b

minus1)Λ

i(zoc i

)b(1

+κ h

+κ s

)zoc i

)

qo i

qod i

((

bminus1)b

)bqoc i

qoc i

y

i(poc i

)zoc i

a

i(poc i

)minusbzoc i

wo

wo d

(1

1113936n i

m+1

qod i

)(τΠ

od Sminus

1113936m i1Π

od Si)

wo c

(1

1113936n i

m+1

qoc i

)(τΠ

oc Sminus

1113936m i1Π

oc Si)

Πo LS

Πod LS

τΠ

od Sτ1113936

n i1[

((b

minus1)b

)b(Π

oc SCi+

c fi)

minusc fi]

Πoc LS

τΠ

oc Sτ1113936

n i1[

(1

minusϕo c

)Πoc SC

iminusϕo c

c fi]

Πo ES

Πod ES

(1

minusτ)Π

od S

(1

minusτ)

1113936n i1[

((b

minus1)b

)b(Π

oc SCi+

c fi)

minusc fi]

Πoc ES

(1

minusτ)Π

oc S

(1

minusτ)

1113936n i1[

(1

minusϕo c

)Πoc SC

iminusϕo c

c fi]

Πo S

Πod S

1113936

n i1Π

od Si

1113936

n i1[

((b

minus1)b

)b(Π

oc SCi+

c fi)

minusc fi]

Πoc S

1113936

n i1Π

oc Si

1113936

n i1[

(1

minusϕo c

)Πoc SC

iminusϕo c

c fi]

Πo D

iΠ

od Di

((

bminus1)b

)bminus1 (Π

oc SCi+

c fi)

Πoc D

iϕo c

(Π

oc SCi+

c fi)

Πo SC

Πod SC

1113936

n i1Π

od SCi

1113936

n i1

[((

bminus1)b

)b+

((b

minus1)b

)bminus1 ](Π

oc SCi+

c fi)

minusc fi

11139661113967

Πoc SC

1113936

n i1Π

oc SCi

1113936

n i1(

(C

i+

c di)(

bminus1)

)qoc i

minusc fi

ϕo cNA

ϕo cλ(

(b

minus1)b

)bminus1

+(1

minusλ)

[1minus

((b

minus1)b

)b]

Note

Λi(

zoc i

)

(1

+κ h

+κ s

)1113938

zoc i

A(

zoc i

minusx

i)f

i(x

i)dx

i+κ s

1113938B A

xif

i(x

i)dx

i

6 Complexity




maxwΩ(w) ΠbdLS w

bdi q

bdi w1113872 11138731113960 1113961

τΠbdES w

bdi q

bdi w1113872 11138731113960 1113961

1minus τ

st

ΠbdLS wbdi q

bdi w1113872 1113873 + ΠbdES w

bdi q

bdi w1113872 1113873 ΠbdS w

bdi q

bdi1113872 1113873

st

wbdi p

bdi z

bdi and q


maxwi

ΠbS wi p

bdi wi( 1113857 z

bdi1113872 1113873



maxzipi

ΠbDi

zi pi( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)






bdES Π

bdS ΠbdDi






(zi pi) ΠbDi


ni1Π

bcSi

(wi) 1113936ni1[Π

bSi



Complexity 7

Tabl

e2

Analytical

results

oftheIBWTsupp

lychainwith

partialb

acklogging

Scenario

var

Result

Equilib

rium

decisio

n(421)

Coo

rdinationdecisio

n(422)

wb i

wbd i

((

bC

i+

c di)(

bminus1)

)w

bc iϕb c

Ciminus

(1

minusϕb c

)cd

i

pb i

pbd i

(

b(

bminus1)

)pbc i

pbc i

(

b(

Ci+

c di)(

bminus1)

)(((1

minusφ)

zbc i

+φΝ

i(zbc i

))(

(1

minusφ

+κ s

)zbc i

minusΜ

i(zbc i

)))

zb i

Fi(

zbd i

)

Fi(

zbc i

)F

i(zbc i

)

qb i

qbd i

((

bminus1)b

)bqbc i

qbc i

y

i(pbc i

)zbc i

a

i(pbc i

)minusbzbc i

wb

wb d

(1

1113936n i

m+1

qbd i

)(τΠ

bd Sminus

1113936m i1Π

bd Si)

wb c

(1

1113936n i

m+1

qbc i

)(τΠ

bc Sminus

1113936m i1Π

bc Si)

Πb LS

Πbd LS

τΠ

bd Sτ1113936

n i1[

((b

minus1)b

)b(Π

bc SCi+

c fi)

minusc f

i]Π

bc LSτΠ

bc Sτ1113936

n i1[

(1

minusϕb c

)Πbc SC

iminusϕb c

c fi]

Πb ES

Πbd ES

(1

minusτ)Π

bd S

(1

minusτ)

1113936n i1[

((b

minus1)b

)b(Π

bc SCi+

c fi)

minusc fi]

Πbc ES

(1

minusτ)Π

bc S

(1

minusτ)

1113936n i1[

(1

minusϕb c

)Πbc SC

iminusϕb c

c fi]

Πb S

Πbd S

1113936

n i1Π

bd Si

1113936

n i1[

((b

minus1)b

)b(Π

bc SCi+

c fi)

minusc fi]

Πbc S

1113936

n i1Π

bc Si

1113936

n i1[

(1

minusϕb c

)Πbc SC

iminusϕb c

c fi]

Πb D

iΠ

bd Di

((

bminus1)b

)bminus1 (Π

bc SCi+

c fi)

Πbc D

iϕb c

(Π

bc SCi+

c fi)

Πb SC

Πbd SC

1113936

n i1Π

bd SCi

1113936

n i1

[((

bminus1)b

)b+

((b

minus1)b

)bminus1 ](Π

bc SCi+

c fi)

minusc fi

11139661113967

Πbc SC

1113936

n i1Π

bc SCi

1113936

n i1

((C

i+

c di)(

bminus1)

)yi(

pbc i

)[(1

minusφ)

zbc i

+φΝ

i(zbc i

)]minus

c fi

11138641113865

ϕb cNA

ϕb cλ(

(b

minus1)b

)bminus1

+(1

minusλ)

[1minus

((b

minus1)b

)b]

Note

Fi(

zbc i

)

(((1

minusφ)

(1

minusφ

+κ s

)zbc i

+

bφ(

1minusφ

+κ s

)Ni(

zbc i

)+

(b

minus1)

(1

minusφ)

Mi(

zbc i

))(

[(b

minusφ)

(1

minusφ

+κ s

)+

b(1

minusφ)κ h

]zbc i

+bφ(

1minusφ

+κ s

+κ h

)Νi(

zbc i

)minus

(b

minus1)φΜ

i(zbc i

)))

Mi(

zbc i

)

(1

+κ h

+κ s

minus

φ)1113946

zbc i

A(

zbc i

minusx

i)f

i(x

i)dx

iminus

(φ

minusκ s

)1113946

B Ax

ifi(

xi)dx

iN

i(zbc i

)

1113946zbc i

A(

zbc i

minusx

i)f

i(x

i)dx

i+

1113946B A

xif

i(x

i)dx

i

8 Complexity

maxwΩ(w) ΠbcLS w

bci q

bci w1113872 11138731113960 1113961

τΠbcES w

bci q

bci w1113872 11138731113960 1113961

1minus τ

st

ΠbcLS wbci q

bci w1113872 1113873 +ΠbcES w

bci q

bci w1113872 1113873 ΠbcS w

bci q

bci1113872 1113873

wbci q

bci ΠbcLS w

bci q

bci w1113872 1113873ΠbcES w

bci q


bci q

bci1113872 1113873


maxϕ

πbi (ϕ) ⊓bcSi

(ϕ) minus ΠbdSi1113960 1113961

λΠbcDi

(ϕ) minus ΠbdDi1113960 1113961

1minus λ

st

ΠbcSi(ϕ) + ΠbcDi

(ϕ) ΠbcSCi

wbci (ϕ) p

bci z

bci q

bci ΠbcSi



maxzipi


maxzipi


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)






bcESΠ

bcS ΠbcDi








Complexity 9









10 Complexity






1 032 008 1796 +E92 024 010 1218 +E93 009 012 1169 +E94 014 076 1941 +E95 041 029 1117 +E96 081 040 1525 +E9


i wodi pod

i qodi ΠodDiΠodS


d 051


i woci poc

i qoci ΠocDiΠocS


i 011 woc 020 ϕo

c 067


i wbdi pbd

i qbdi ΠbdDiΠbdS


d 053

Complexity 11












i wbci pbc

i qbci ΠbcDiΠbcS


i 010 wbc 020 ϕb

c 067

12 Complexity

32

31

3

29

28

27

26

Prof

it


Supply chain

times108

(a)

Prof

it

Supply chain

times108286

284

282

28

278

276

274005 006 007 008 009 01 011 012 013 014 015


(b)

Prof

it

284

283

282

281

28

279

278

277

27601 02 03 04 05


times108

Supply chain

(c)

Prof

it

284

285

286

287

288

283

282

281

2802 03 04 05 06

times108


Supply chain

(d)

times108

Prof

it

284

283

282

281

28

279

278

277

276

2750 01 03 04 05 06 07 08 09 102

Backlogging ratio

Supply chain

(e)

times108

Prof

it

6

55

5

45

4

35

3

25

2

1511 12 13 14 15 16 17 18 19 2


Supply chain

(f )

Figure 2 Continued

Complexity 13






7 Conclusions


33

32

31

3

29

28

27

26

times108

Prof

it


(g)


times107

Prof

it

11

10

9

8

7

6

5

4

3

2

1

+



(h)


14 Complexity



Data Availability




Acknowledgments





References














Complexity 15

























16 Complexity






odES Π

odS ΠodDi







Di(zi pi) minus Ti


ocSi

(wi) 1113936ni1[Π

oSi



maxwΩ(w) ΠocLS w

oci q

oci w( 11138571113858 1113859

τ ΠocES woci q

oci w( 11138571113858 1113859

1minus τ

st

ΠocLS woci q


oci q


oci q

oci( 1113857

woci q

oci ΠocLS w

oci q


oci q


oci q

oci( 1113857


maxϕ

πoi (ϕ) ΠocSi

(ϕ) minus ΠodSi1113960 1113961

λΠocDi

(ϕ) minus ΠodDi1113960 1113961

1minus λ

st

ΠocSi(ϕ) + ΠocDi

(ϕ) ΠocSCi

woci (ϕ) p

oci z

oci q

oci ΠocSi



maxzipi


maxzipi


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(4)








ocES Π

ocS ΠocDi


Complexity 5

Tabl

e1

Analytical

results

oftheIBWTsupp

lychainwith

outb

acklogging

Scenario

var

Result

Equilib

rium

decisio

n(411)

Coo

rdinationdecisio

n(412)

wo i

wod i

((

bC

i+

c di)(

bminus1)

)w

oc iϕo c

Ciminus

(1

minusϕo c

)cd

i

po i

pod i

(

b(

bminus1)

poc i

)poc i

(

b(

bminus1)

)((

Ci+

c di)

zoc i(1

+κ s

)zoc i

minusΛ

i(zoc i

))

zo i

Fi(

zod i

)

Fi(

zoc i

)F

i(zoc i

)

((1

+κ s

)b(1

+κ h

+κ s

))+

((b

minus1)Λ

i(zoc i

)b(1

+κ h

+κ s

)zoc i

)

qo i

qod i

((

bminus1)b

)bqoc i

qoc i

y

i(poc i

)zoc i

a

i(poc i

)minusbzoc i

wo

wo d

(1

1113936n i

m+1

qod i

)(τΠ

od Sminus

1113936m i1Π

od Si)

wo c

(1

1113936n i

m+1

qoc i

)(τΠ

oc Sminus

1113936m i1Π

oc Si)

Πo LS

Πod LS

τΠ

od Sτ1113936

n i1[

((b

minus1)b

)b(Π

oc SCi+

c fi)

minusc fi]

Πoc LS

τΠ

oc Sτ1113936

n i1[

(1

minusϕo c

)Πoc SC

iminusϕo c

c fi]

Πo ES

Πod ES

(1

minusτ)Π

od S

(1

minusτ)

1113936n i1[

((b

minus1)b

)b(Π

oc SCi+

c fi)

minusc fi]

Πoc ES

(1

minusτ)Π

oc S

(1

minusτ)

1113936n i1[

(1

minusϕo c

)Πoc SC

iminusϕo c

c fi]

Πo S

Πod S

1113936

n i1Π

od Si

1113936

n i1[

((b

minus1)b

)b(Π

oc SCi+

c fi)

minusc fi]

Πoc S

1113936

n i1Π

oc Si

1113936

n i1[

(1

minusϕo c

)Πoc SC

iminusϕo c

c fi]

Πo D

iΠ

od Di

((

bminus1)b

)bminus1 (Π

oc SCi+

c fi)

Πoc D

iϕo c

(Π

oc SCi+

c fi)

Πo SC

Πod SC

1113936

n i1Π

od SCi

1113936

n i1

[((

bminus1)b

)b+

((b

minus1)b

)bminus1 ](Π

oc SCi+

c fi)

minusc fi

11139661113967

Πoc SC

1113936

n i1Π

oc SCi

1113936

n i1(

(C

i+

c di)(

bminus1)

)qoc i

minusc fi

ϕo cNA

ϕo cλ(

(b

minus1)b

)bminus1

+(1

minusλ)

[1minus

((b

minus1)b

)b]

Note

Λi(

zoc i

)

(1

+κ h

+κ s

)1113938

zoc i

A(

zoc i

minusx

i)f

i(x

i)dx

i+κ s

1113938B A

xif

i(x

i)dx

i

6 Complexity




maxwΩ(w) ΠbdLS w

bdi q

bdi w1113872 11138731113960 1113961

τΠbdES w

bdi q

bdi w1113872 11138731113960 1113961

1minus τ

st

ΠbdLS wbdi q

bdi w1113872 1113873 + ΠbdES w

bdi q

bdi w1113872 1113873 ΠbdS w

bdi q

bdi1113872 1113873

st

wbdi p

bdi z

bdi and q


maxwi

ΠbS wi p

bdi wi( 1113857 z

bdi1113872 1113873



maxzipi

ΠbDi

zi pi( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)






bdES Π

bdS ΠbdDi






(zi pi) ΠbDi


ni1Π

bcSi

(wi) 1113936ni1[Π

bSi



Complexity 7

Tabl

e2

Analytical

results

oftheIBWTsupp

lychainwith

partialb

acklogging

Scenario

var

Result

Equilib

rium

decisio

n(421)

Coo

rdinationdecisio

n(422)

wb i

wbd i

((

bC

i+

c di)(

bminus1)

)w

bc iϕb c

Ciminus

(1

minusϕb c

)cd

i

pb i

pbd i

(

b(

bminus1)

)pbc i

pbc i

(

b(

Ci+

c di)(

bminus1)

)(((1

minusφ)

zbc i

+φΝ

i(zbc i

))(

(1

minusφ

+κ s

)zbc i

minusΜ

i(zbc i

)))

zb i

Fi(

zbd i

)

Fi(

zbc i

)F

i(zbc i

)

qb i

qbd i

((

bminus1)b

)bqbc i

qbc i

y

i(pbc i

)zbc i

a

i(pbc i

)minusbzbc i

wb

wb d

(1

1113936n i

m+1

qbd i

)(τΠ

bd Sminus

1113936m i1Π

bd Si)

wb c

(1

1113936n i

m+1

qbc i

)(τΠ

bc Sminus

1113936m i1Π

bc Si)

Πb LS

Πbd LS

τΠ

bd Sτ1113936

n i1[

((b

minus1)b

)b(Π

bc SCi+

c fi)

minusc f

i]Π

bc LSτΠ

bc Sτ1113936

n i1[

(1

minusϕb c

)Πbc SC

iminusϕb c

c fi]

Πb ES

Πbd ES

(1

minusτ)Π

bd S

(1

minusτ)

1113936n i1[

((b

minus1)b

)b(Π

bc SCi+

c fi)

minusc fi]

Πbc ES

(1

minusτ)Π

bc S

(1

minusτ)

1113936n i1[

(1

minusϕb c

)Πbc SC

iminusϕb c

c fi]

Πb S

Πbd S

1113936

n i1Π

bd Si

1113936

n i1[

((b

minus1)b

)b(Π

bc SCi+

c fi)

minusc fi]

Πbc S

1113936

n i1Π

bc Si

1113936

n i1[

(1

minusϕb c

)Πbc SC

iminusϕb c

c fi]

Πb D

iΠ

bd Di

((

bminus1)b

)bminus1 (Π

bc SCi+

c fi)

Πbc D

iϕb c

(Π

bc SCi+

c fi)

Πb SC

Πbd SC

1113936

n i1Π

bd SCi

1113936

n i1

[((

bminus1)b

)b+

((b

minus1)b

)bminus1 ](Π

bc SCi+

c fi)

minusc fi

11139661113967

Πbc SC

1113936

n i1Π

bc SCi

1113936

n i1

((C

i+

c di)(

bminus1)

)yi(

pbc i

)[(1

minusφ)

zbc i

+φΝ

i(zbc i

)]minus

c fi

11138641113865

ϕb cNA

ϕb cλ(

(b

minus1)b

)bminus1

+(1

minusλ)

[1minus

((b

minus1)b

)b]

Note

Fi(

zbc i

)

(((1

minusφ)

(1

minusφ

+κ s

)zbc i

+

bφ(

1minusφ

+κ s

)Ni(

zbc i

)+

(b

minus1)

(1

minusφ)

Mi(

zbc i

))(

[(b

minusφ)

(1

minusφ

+κ s

)+

b(1

minusφ)κ h

]zbc i

+bφ(

1minusφ

+κ s

+κ h

)Νi(

zbc i

)minus

(b

minus1)φΜ

i(zbc i

)))

Mi(

zbc i

)

(1

+κ h

+κ s

minus

φ)1113946

zbc i

A(

zbc i

minusx

i)f

i(x

i)dx

iminus

(φ

minusκ s

)1113946

B Ax

ifi(

xi)dx

iN

i(zbc i

)

1113946zbc i

A(

zbc i

minusx

i)f

i(x

i)dx

i+

1113946B A

xif

i(x

i)dx

i

8 Complexity

maxwΩ(w) ΠbcLS w

bci q

bci w1113872 11138731113960 1113961

τΠbcES w

bci q

bci w1113872 11138731113960 1113961

1minus τ

st

ΠbcLS wbci q

bci w1113872 1113873 +ΠbcES w

bci q

bci w1113872 1113873 ΠbcS w

bci q

bci1113872 1113873

wbci q

bci ΠbcLS w

bci q

bci w1113872 1113873ΠbcES w

bci q


bci q

bci1113872 1113873


maxϕ

πbi (ϕ) ⊓bcSi

(ϕ) minus ΠbdSi1113960 1113961

λΠbcDi

(ϕ) minus ΠbdDi1113960 1113961

1minus λ

st

ΠbcSi(ϕ) + ΠbcDi

(ϕ) ΠbcSCi

wbci (ϕ) p

bci z

bci q

bci ΠbcSi



maxzipi


maxzipi


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)






bcESΠ

bcS ΠbcDi








Complexity 9









10 Complexity






1 032 008 1796 +E92 024 010 1218 +E93 009 012 1169 +E94 014 076 1941 +E95 041 029 1117 +E96 081 040 1525 +E9


i wodi pod

i qodi ΠodDiΠodS


d 051


i woci poc

i qoci ΠocDiΠocS


i 011 woc 020 ϕo

c 067


i wbdi pbd

i qbdi ΠbdDiΠbdS


d 053

Complexity 11












i wbci pbc

i qbci ΠbcDiΠbcS


i 010 wbc 020 ϕb

c 067

12 Complexity

32

31

3

29

28

27

26

Prof

it


Supply chain

times108

(a)

Prof

it

Supply chain

times108286

284

282

28

278

276

274005 006 007 008 009 01 011 012 013 014 015


(b)

Prof

it

284

283

282

281

28

279

278

277

27601 02 03 04 05


times108

Supply chain

(c)

Prof

it

284

285

286

287

288

283

282

281

2802 03 04 05 06

times108


Supply chain

(d)

times108

Prof

it

284

283

282

281

28

279

278

277

276

2750 01 03 04 05 06 07 08 09 102

Backlogging ratio

Supply chain

(e)

times108

Prof

it

6

55

5

45

4

35

3

25

2

1511 12 13 14 15 16 17 18 19 2


Supply chain

(f )

Figure 2 Continued

Complexity 13






7 Conclusions


33

32

31

3

29

28

27

26

times108

Prof

it


(g)


times107

Prof

it

11

10

9

8

7

6

5

4

3

2

1

+



(h)


14 Complexity



Data Availability




Acknowledgments





References














Complexity 15

























16 Complexity

Tabl

e1

Analytical

results

oftheIBWTsupp

lychainwith

outb

acklogging

Scenario

var

Result

Equilib

rium

decisio

n(411)

Coo

rdinationdecisio

n(412)

wo i

wod i

((

bC

i+

c di)(

bminus1)

)w

oc iϕo c

Ciminus

(1

minusϕo c

)cd

i

po i

pod i

(

b(

bminus1)

poc i

)poc i

(

b(

bminus1)

)((

Ci+

c di)

zoc i(1

+κ s

)zoc i

minusΛ

i(zoc i

))

zo i

Fi(

zod i

)

Fi(

zoc i

)F

i(zoc i

)

((1

+κ s

)b(1

+κ h

+κ s

))+

((b

minus1)Λ

i(zoc i

)b(1

+κ h

+κ s

)zoc i

)

qo i

qod i

((

bminus1)b

)bqoc i

qoc i

y

i(poc i

)zoc i

a

i(poc i

)minusbzoc i

wo

wo d

(1

1113936n i

m+1

qod i

)(τΠ

od Sminus

1113936m i1Π

od Si)

wo c

(1

1113936n i

m+1

qoc i

)(τΠ

oc Sminus

1113936m i1Π

oc Si)

Πo LS

Πod LS

τΠ

od Sτ1113936

n i1[

((b

minus1)b

)b(Π

oc SCi+

c fi)

minusc fi]

Πoc LS

τΠ

oc Sτ1113936

n i1[

(1

minusϕo c

)Πoc SC

iminusϕo c

c fi]

Πo ES

Πod ES

(1

minusτ)Π

od S

(1

minusτ)

1113936n i1[

((b

minus1)b

)b(Π

oc SCi+

c fi)

minusc fi]

Πoc ES

(1

minusτ)Π

oc S

(1

minusτ)

1113936n i1[

(1

minusϕo c

)Πoc SC

iminusϕo c

c fi]

Πo S

Πod S

1113936

n i1Π

od Si

1113936

n i1[

((b

minus1)b

)b(Π

oc SCi+

c fi)

minusc fi]

Πoc S

1113936

n i1Π

oc Si

1113936

n i1[

(1

minusϕo c

)Πoc SC

iminusϕo c

c fi]

Πo D

iΠ

od Di

((

bminus1)b

)bminus1 (Π

oc SCi+

c fi)

Πoc D

iϕo c

(Π

oc SCi+

c fi)

Πo SC

Πod SC

1113936

n i1Π

od SCi

1113936

n i1

[((

bminus1)b

)b+

((b

minus1)b

)bminus1 ](Π

oc SCi+

c fi)

minusc fi

11139661113967

Πoc SC

1113936

n i1Π

oc SCi

1113936

n i1(

(C

i+

c di)(

bminus1)

)qoc i

minusc fi

ϕo cNA

ϕo cλ(

(b

minus1)b

)bminus1

+(1

minusλ)

[1minus

((b

minus1)b

)b]

Note

Λi(

zoc i

)

(1

+κ h

+κ s

)1113938

zoc i

A(

zoc i

minusx

i)f

i(x

i)dx

i+κ s

1113938B A

xif

i(x

i)dx

i

6 Complexity




maxwΩ(w) ΠbdLS w

bdi q

bdi w1113872 11138731113960 1113961

τΠbdES w

bdi q

bdi w1113872 11138731113960 1113961

1minus τ

st

ΠbdLS wbdi q

bdi w1113872 1113873 + ΠbdES w

bdi q

bdi w1113872 1113873 ΠbdS w

bdi q

bdi1113872 1113873

st

wbdi p

bdi z

bdi and q


maxwi

ΠbS wi p

bdi wi( 1113857 z

bdi1113872 1113873



maxzipi

ΠbDi

zi pi( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)






bdES Π

bdS ΠbdDi






(zi pi) ΠbDi


ni1Π

bcSi

(wi) 1113936ni1[Π

bSi



Complexity 7

Tabl

e2

Analytical

results

oftheIBWTsupp

lychainwith

partialb

acklogging

Scenario

var

Result

Equilib

rium

decisio

n(421)

Coo

rdinationdecisio

n(422)

wb i

wbd i

((

bC

i+

c di)(

bminus1)

)w

bc iϕb c

Ciminus

(1

minusϕb c

)cd

i

pb i

pbd i

(

b(

bminus1)

)pbc i

pbc i

(

b(

Ci+

c di)(

bminus1)

)(((1

minusφ)

zbc i

+φΝ

i(zbc i

))(

(1

minusφ

+κ s

)zbc i

minusΜ

i(zbc i

)))

zb i

Fi(

zbd i

)

Fi(

zbc i

)F

i(zbc i

)

qb i

qbd i

((

bminus1)b

)bqbc i

qbc i

y

i(pbc i

)zbc i

a

i(pbc i

)minusbzbc i

wb

wb d

(1

1113936n i

m+1

qbd i

)(τΠ

bd Sminus

1113936m i1Π

bd Si)

wb c

(1

1113936n i

m+1

qbc i

)(τΠ

bc Sminus

1113936m i1Π

bc Si)

Πb LS

Πbd LS

τΠ

bd Sτ1113936

n i1[

((b

minus1)b

)b(Π

bc SCi+

c fi)

minusc f

i]Π

bc LSτΠ

bc Sτ1113936

n i1[

(1

minusϕb c

)Πbc SC

iminusϕb c

c fi]

Πb ES

Πbd ES

(1

minusτ)Π

bd S

(1

minusτ)

1113936n i1[

((b

minus1)b

)b(Π

bc SCi+

c fi)

minusc fi]

Πbc ES

(1

minusτ)Π

bc S

(1

minusτ)

1113936n i1[

(1

minusϕb c

)Πbc SC

iminusϕb c

c fi]

Πb S

Πbd S

1113936

n i1Π

bd Si

1113936

n i1[

((b

minus1)b

)b(Π

bc SCi+

c fi)

minusc fi]

Πbc S

1113936

n i1Π

bc Si

1113936

n i1[

(1

minusϕb c

)Πbc SC

iminusϕb c

c fi]

Πb D

iΠ

bd Di

((

bminus1)b

)bminus1 (Π

bc SCi+

c fi)

Πbc D

iϕb c

(Π

bc SCi+

c fi)

Πb SC

Πbd SC

1113936

n i1Π

bd SCi

1113936

n i1

[((

bminus1)b

)b+

((b

minus1)b

)bminus1 ](Π

bc SCi+

c fi)

minusc fi

11139661113967

Πbc SC

1113936

n i1Π

bc SCi

1113936

n i1

((C

i+

c di)(

bminus1)

)yi(

pbc i

)[(1

minusφ)

zbc i

+φΝ

i(zbc i

)]minus

c fi

11138641113865

ϕb cNA

ϕb cλ(

(b

minus1)b

)bminus1

+(1

minusλ)

[1minus

((b

minus1)b

)b]

Note

Fi(

zbc i

)

(((1

minusφ)

(1

minusφ

+κ s

)zbc i

+

bφ(

1minusφ

+κ s

)Ni(

zbc i

)+

(b

minus1)

(1

minusφ)

Mi(

zbc i

))(

[(b

minusφ)

(1

minusφ

+κ s

)+

b(1

minusφ)κ h

]zbc i

+bφ(

1minusφ

+κ s

+κ h

)Νi(

zbc i

)minus

(b

minus1)φΜ

i(zbc i

)))

Mi(

zbc i

)

(1

+κ h

+κ s

minus

φ)1113946

zbc i

A(

zbc i

minusx

i)f

i(x

i)dx

iminus

(φ

minusκ s

)1113946

B Ax

ifi(

xi)dx

iN

i(zbc i

)

1113946zbc i

A(

zbc i

minusx

i)f

i(x

i)dx

i+

1113946B A

xif

i(x

i)dx

i

8 Complexity

maxwΩ(w) ΠbcLS w

bci q

bci w1113872 11138731113960 1113961

τΠbcES w

bci q

bci w1113872 11138731113960 1113961

1minus τ

st

ΠbcLS wbci q

bci w1113872 1113873 +ΠbcES w

bci q

bci w1113872 1113873 ΠbcS w

bci q

bci1113872 1113873

wbci q

bci ΠbcLS w

bci q

bci w1113872 1113873ΠbcES w

bci q


bci q

bci1113872 1113873


maxϕ

πbi (ϕ) ⊓bcSi

(ϕ) minus ΠbdSi1113960 1113961

λΠbcDi

(ϕ) minus ΠbdDi1113960 1113961

1minus λ

st

ΠbcSi(ϕ) + ΠbcDi

(ϕ) ΠbcSCi

wbci (ϕ) p

bci z

bci q

bci ΠbcSi



maxzipi


maxzipi


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)






bcESΠ

bcS ΠbcDi








Complexity 9









10 Complexity






1 032 008 1796 +E92 024 010 1218 +E93 009 012 1169 +E94 014 076 1941 +E95 041 029 1117 +E96 081 040 1525 +E9


i wodi pod

i qodi ΠodDiΠodS


d 051


i woci poc

i qoci ΠocDiΠocS


i 011 woc 020 ϕo

c 067


i wbdi pbd

i qbdi ΠbdDiΠbdS


d 053

Complexity 11












i wbci pbc

i qbci ΠbcDiΠbcS


i 010 wbc 020 ϕb

c 067

12 Complexity

32

31

3

29

28

27

26

Prof

it


Supply chain

times108

(a)

Prof

it

Supply chain

times108286

284

282

28

278

276

274005 006 007 008 009 01 011 012 013 014 015


(b)

Prof

it

284

283

282

281

28

279

278

277

27601 02 03 04 05


times108

Supply chain

(c)

Prof

it

284

285

286

287

288

283

282

281

2802 03 04 05 06

times108


Supply chain

(d)

times108

Prof

it

284

283

282

281

28

279

278

277

276

2750 01 03 04 05 06 07 08 09 102

Backlogging ratio

Supply chain

(e)

times108

Prof

it

6

55

5

45

4

35

3

25

2

1511 12 13 14 15 16 17 18 19 2


Supply chain

(f )

Figure 2 Continued

Complexity 13






7 Conclusions


33

32

31

3

29

28

27

26

times108

Prof

it


(g)


times107

Prof

it

11

10

9

8

7

6

5

4

3

2

1

+



(h)


14 Complexity



Data Availability




Acknowledgments





References














Complexity 15

























16 Complexity




maxwΩ(w) ΠbdLS w

bdi q

bdi w1113872 11138731113960 1113961

τΠbdES w

bdi q

bdi w1113872 11138731113960 1113961

1minus τ

st

ΠbdLS wbdi q

bdi w1113872 1113873 + ΠbdES w

bdi q

bdi w1113872 1113873 ΠbdS w

bdi q

bdi1113872 1113873

st

wbdi p

bdi z

bdi and q


maxwi

ΠbS wi p

bdi wi( 1113857 z

bdi1113872 1113873



maxzipi

ΠbDi

zi pi( 1113857

⎧⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(5)






bdES Π

bdS ΠbdDi






(zi pi) ΠbDi


ni1Π

bcSi

(wi) 1113936ni1[Π

bSi



Complexity 7

Tabl

e2

Analytical

results

oftheIBWTsupp

lychainwith

partialb

acklogging

Scenario

var

Result

Equilib

rium

decisio

n(421)

Coo

rdinationdecisio

n(422)

wb i

wbd i

((

bC

i+

c di)(

bminus1)

)w

bc iϕb c

Ciminus

(1

minusϕb c

)cd

i

pb i

pbd i

(

b(

bminus1)

)pbc i

pbc i

(

b(

Ci+

c di)(

bminus1)

)(((1

minusφ)

zbc i

+φΝ

i(zbc i

))(

(1

minusφ

+κ s

)zbc i

minusΜ

i(zbc i

)))

zb i

Fi(

zbd i

)

Fi(

zbc i

)F

i(zbc i

)

qb i

qbd i

((

bminus1)b

)bqbc i

qbc i

y

i(pbc i

)zbc i

a

i(pbc i

)minusbzbc i

wb

wb d

(1

1113936n i

m+1

qbd i

)(τΠ

bd Sminus

1113936m i1Π

bd Si)

wb c

(1

1113936n i

m+1

qbc i

)(τΠ

bc Sminus

1113936m i1Π

bc Si)

Πb LS

Πbd LS

τΠ

bd Sτ1113936

n i1[

((b

minus1)b

)b(Π

bc SCi+

c fi)

minusc f

i]Π

bc LSτΠ

bc Sτ1113936

n i1[

(1

minusϕb c

)Πbc SC

iminusϕb c

c fi]

Πb ES

Πbd ES

(1

minusτ)Π

bd S

(1

minusτ)

1113936n i1[

((b

minus1)b

)b(Π

bc SCi+

c fi)

minusc fi]

Πbc ES

(1

minusτ)Π

bc S

(1

minusτ)

1113936n i1[

(1

minusϕb c

)Πbc SC

iminusϕb c

c fi]

Πb S

Πbd S

1113936

n i1Π

bd Si

1113936

n i1[

((b

minus1)b

)b(Π

bc SCi+

c fi)

minusc fi]

Πbc S

1113936

n i1Π

bc Si

1113936

n i1[

(1

minusϕb c

)Πbc SC

iminusϕb c

c fi]

Πb D

iΠ

bd Di

((

bminus1)b

)bminus1 (Π

bc SCi+

c fi)

Πbc D

iϕb c

(Π

bc SCi+

c fi)

Πb SC

Πbd SC

1113936

n i1Π

bd SCi

1113936

n i1

[((

bminus1)b

)b+

((b

minus1)b

)bminus1 ](Π

bc SCi+

c fi)

minusc fi

11139661113967

Πbc SC

1113936

n i1Π

bc SCi

1113936

n i1

((C

i+

c di)(

bminus1)

)yi(

pbc i

)[(1

minusφ)

zbc i

+φΝ

i(zbc i

)]minus

c fi

11138641113865

ϕb cNA

ϕb cλ(

(b

minus1)b

)bminus1

+(1

minusλ)

[1minus

((b

minus1)b

)b]

Note

Fi(

zbc i

)

(((1

minusφ)

(1

minusφ

+κ s

)zbc i

+

bφ(

1minusφ

+κ s

)Ni(

zbc i

)+

(b

minus1)

(1

minusφ)

Mi(

zbc i

))(

[(b

minusφ)

(1

minusφ

+κ s

)+

b(1

minusφ)κ h

]zbc i

+bφ(

1minusφ

+κ s

+κ h

)Νi(

zbc i

)minus

(b

minus1)φΜ

i(zbc i

)))

Mi(

zbc i

)

(1

+κ h

+κ s

minus

φ)1113946

zbc i

A(

zbc i

minusx

i)f

i(x

i)dx

iminus

(φ

minusκ s

)1113946

B Ax

ifi(

xi)dx

iN

i(zbc i

)

1113946zbc i

A(

zbc i

minusx

i)f

i(x

i)dx

i+

1113946B A

xif

i(x

i)dx

i

8 Complexity

maxwΩ(w) ΠbcLS w

bci q

bci w1113872 11138731113960 1113961

τΠbcES w

bci q

bci w1113872 11138731113960 1113961

1minus τ

st

ΠbcLS wbci q

bci w1113872 1113873 +ΠbcES w

bci q

bci w1113872 1113873 ΠbcS w

bci q

bci1113872 1113873

wbci q

bci ΠbcLS w

bci q

bci w1113872 1113873ΠbcES w

bci q


bci q

bci1113872 1113873


maxϕ

πbi (ϕ) ⊓bcSi

(ϕ) minus ΠbdSi1113960 1113961

λΠbcDi

(ϕ) minus ΠbdDi1113960 1113961

1minus λ

st

ΠbcSi(ϕ) + ΠbcDi

(ϕ) ΠbcSCi

wbci (ϕ) p

bci z

bci q

bci ΠbcSi



maxzipi


maxzipi


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)






bcESΠ

bcS ΠbcDi








Complexity 9









10 Complexity






1 032 008 1796 +E92 024 010 1218 +E93 009 012 1169 +E94 014 076 1941 +E95 041 029 1117 +E96 081 040 1525 +E9


i wodi pod

i qodi ΠodDiΠodS


d 051


i woci poc

i qoci ΠocDiΠocS


i 011 woc 020 ϕo

c 067


i wbdi pbd

i qbdi ΠbdDiΠbdS


d 053

Complexity 11












i wbci pbc

i qbci ΠbcDiΠbcS


i 010 wbc 020 ϕb

c 067

12 Complexity

32

31

3

29

28

27

26

Prof

it


Supply chain

times108

(a)

Prof

it

Supply chain

times108286

284

282

28

278

276

274005 006 007 008 009 01 011 012 013 014 015


(b)

Prof

it

284

283

282

281

28

279

278

277

27601 02 03 04 05


times108

Supply chain

(c)

Prof

it

284

285

286

287

288

283

282

281

2802 03 04 05 06

times108


Supply chain

(d)

times108

Prof

it

284

283

282

281

28

279

278

277

276

2750 01 03 04 05 06 07 08 09 102

Backlogging ratio

Supply chain

(e)

times108

Prof

it

6

55

5

45

4

35

3

25

2

1511 12 13 14 15 16 17 18 19 2


Supply chain

(f )

Figure 2 Continued

Complexity 13






7 Conclusions


33

32

31

3

29

28

27

26

times108

Prof

it


(g)


times107

Prof

it

11

10

9

8

7

6

5

4

3

2

1

+



(h)


14 Complexity



Data Availability




Acknowledgments





References














Complexity 15

























16 Complexity

Tabl

e2

Analytical

results

oftheIBWTsupp

lychainwith

partialb

acklogging

Scenario

var

Result

Equilib

rium

decisio

n(421)

Coo

rdinationdecisio

n(422)

wb i

wbd i

((

bC

i+

c di)(

bminus1)

)w

bc iϕb c

Ciminus

(1

minusϕb c

)cd

i

pb i

pbd i

(

b(

bminus1)

)pbc i

pbc i

(

b(

Ci+

c di)(

bminus1)

)(((1

minusφ)

zbc i

+φΝ

i(zbc i

))(

(1

minusφ

+κ s

)zbc i

minusΜ

i(zbc i

)))

zb i

Fi(

zbd i

)

Fi(

zbc i

)F

i(zbc i

)

qb i

qbd i

((

bminus1)b

)bqbc i

qbc i

y

i(pbc i

)zbc i

a

i(pbc i

)minusbzbc i

wb

wb d

(1

1113936n i

m+1

qbd i

)(τΠ

bd Sminus

1113936m i1Π

bd Si)

wb c

(1

1113936n i

m+1

qbc i

)(τΠ

bc Sminus

1113936m i1Π

bc Si)

Πb LS

Πbd LS

τΠ

bd Sτ1113936

n i1[

((b

minus1)b

)b(Π

bc SCi+

c fi)

minusc f

i]Π

bc LSτΠ

bc Sτ1113936

n i1[

(1

minusϕb c

)Πbc SC

iminusϕb c

c fi]

Πb ES

Πbd ES

(1

minusτ)Π

bd S

(1

minusτ)

1113936n i1[

((b

minus1)b

)b(Π

bc SCi+

c fi)

minusc fi]

Πbc ES

(1

minusτ)Π

bc S

(1

minusτ)

1113936n i1[

(1

minusϕb c

)Πbc SC

iminusϕb c

c fi]

Πb S

Πbd S

1113936

n i1Π

bd Si

1113936

n i1[

((b

minus1)b

)b(Π

bc SCi+

c fi)

minusc fi]

Πbc S

1113936

n i1Π

bc Si

1113936

n i1[

(1

minusϕb c

)Πbc SC

iminusϕb c

c fi]

Πb D

iΠ

bd Di

((

bminus1)b

)bminus1 (Π

bc SCi+

c fi)

Πbc D

iϕb c

(Π

bc SCi+

c fi)

Πb SC

Πbd SC

1113936

n i1Π

bd SCi

1113936

n i1

[((

bminus1)b

)b+

((b

minus1)b

)bminus1 ](Π

bc SCi+

c fi)

minusc fi

11139661113967

Πbc SC

1113936

n i1Π

bc SCi

1113936

n i1

((C

i+

c di)(

bminus1)

)yi(

pbc i

)[(1

minusφ)

zbc i

+φΝ

i(zbc i

)]minus

c fi

11138641113865

ϕb cNA

ϕb cλ(

(b

minus1)b

)bminus1

+(1

minusλ)

[1minus

((b

minus1)b

)b]

Note

Fi(

zbc i

)

(((1

minusφ)

(1

minusφ

+κ s

)zbc i

+

bφ(

1minusφ

+κ s

)Ni(

zbc i

)+

(b

minus1)

(1

minusφ)

Mi(

zbc i

))(

[(b

minusφ)

(1

minusφ

+κ s

)+

b(1

minusφ)κ h

]zbc i

+bφ(

1minusφ

+κ s

+κ h

)Νi(

zbc i

)minus

(b

minus1)φΜ

i(zbc i

)))

Mi(

zbc i

)

(1

+κ h

+κ s

minus

φ)1113946

zbc i

A(

zbc i

minusx

i)f

i(x

i)dx

iminus

(φ

minusκ s

)1113946

B Ax

ifi(

xi)dx

iN

i(zbc i

)

1113946zbc i

A(

zbc i

minusx

i)f

i(x

i)dx

i+

1113946B A

xif

i(x

i)dx

i

8 Complexity

maxwΩ(w) ΠbcLS w

bci q

bci w1113872 11138731113960 1113961

τΠbcES w

bci q

bci w1113872 11138731113960 1113961

1minus τ

st

ΠbcLS wbci q

bci w1113872 1113873 +ΠbcES w

bci q

bci w1113872 1113873 ΠbcS w

bci q

bci1113872 1113873

wbci q

bci ΠbcLS w

bci q

bci w1113872 1113873ΠbcES w

bci q


bci q

bci1113872 1113873


maxϕ

πbi (ϕ) ⊓bcSi

(ϕ) minus ΠbdSi1113960 1113961

λΠbcDi

(ϕ) minus ΠbdDi1113960 1113961

1minus λ

st

ΠbcSi(ϕ) + ΠbcDi

(ϕ) ΠbcSCi

wbci (ϕ) p

bci z

bci q

bci ΠbcSi



maxzipi


maxzipi


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)






bcESΠ

bcS ΠbcDi








Complexity 9









10 Complexity






1 032 008 1796 +E92 024 010 1218 +E93 009 012 1169 +E94 014 076 1941 +E95 041 029 1117 +E96 081 040 1525 +E9


i wodi pod

i qodi ΠodDiΠodS


d 051


i woci poc

i qoci ΠocDiΠocS


i 011 woc 020 ϕo

c 067


i wbdi pbd

i qbdi ΠbdDiΠbdS


d 053

Complexity 11












i wbci pbc

i qbci ΠbcDiΠbcS


i 010 wbc 020 ϕb

c 067

12 Complexity

32

31

3

29

28

27

26

Prof

it


Supply chain

times108

(a)

Prof

it

Supply chain

times108286

284

282

28

278

276

274005 006 007 008 009 01 011 012 013 014 015


(b)

Prof

it

284

283

282

281

28

279

278

277

27601 02 03 04 05


times108

Supply chain

(c)

Prof

it

284

285

286

287

288

283

282

281

2802 03 04 05 06

times108


Supply chain

(d)

times108

Prof

it

284

283

282

281

28

279

278

277

276

2750 01 03 04 05 06 07 08 09 102

Backlogging ratio

Supply chain

(e)

times108

Prof

it

6

55

5

45

4

35

3

25

2

1511 12 13 14 15 16 17 18 19 2


Supply chain

(f )

Figure 2 Continued

Complexity 13






7 Conclusions


33

32

31

3

29

28

27

26

times108

Prof

it


(g)


times107

Prof

it

11

10

9

8

7

6

5

4

3

2

1

+



(h)


14 Complexity



Data Availability




Acknowledgments





References














Complexity 15

























16 Complexity

maxwΩ(w) ΠbcLS w

bci q

bci w1113872 11138731113960 1113961

τΠbcES w

bci q

bci w1113872 11138731113960 1113961

1minus τ

st

ΠbcLS wbci q

bci w1113872 1113873 +ΠbcES w

bci q

bci w1113872 1113873 ΠbcS w

bci q

bci1113872 1113873

wbci q

bci ΠbcLS w

bci q

bci w1113872 1113873ΠbcES w

bci q


bci q

bci1113872 1113873


maxϕ

πbi (ϕ) ⊓bcSi

(ϕ) minus ΠbdSi1113960 1113961

λΠbcDi

(ϕ) minus ΠbdDi1113960 1113961

1minus λ

st

ΠbcSi(ϕ) + ΠbcDi

(ϕ) ΠbcSCi

wbci (ϕ) p

bci z

bci q

bci ΠbcSi



maxzipi


maxzipi


⎧⎪⎪⎪⎨

⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

⎧⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎨

⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎪⎩

(6)






bcESΠ

bcS ΠbcDi








Complexity 9









10 Complexity






1 032 008 1796 +E92 024 010 1218 +E93 009 012 1169 +E94 014 076 1941 +E95 041 029 1117 +E96 081 040 1525 +E9


i wodi pod

i qodi ΠodDiΠodS


d 051


i woci poc

i qoci ΠocDiΠocS


i 011 woc 020 ϕo

c 067


i wbdi pbd

i qbdi ΠbdDiΠbdS


d 053

Complexity 11












i wbci pbc

i qbci ΠbcDiΠbcS


i 010 wbc 020 ϕb

c 067

12 Complexity

32

31

3

29

28

27

26

Prof

it


Supply chain

times108

(a)

Prof

it

Supply chain

times108286

284

282

28

278

276

274005 006 007 008 009 01 011 012 013 014 015


(b)

Prof

it

284

283

282

281

28

279

278

277

27601 02 03 04 05


times108

Supply chain

(c)

Prof

it

284

285

286

287

288

283

282

281

2802 03 04 05 06

times108


Supply chain

(d)

times108

Prof

it

284

283

282

281

28

279

278

277

276

2750 01 03 04 05 06 07 08 09 102

Backlogging ratio

Supply chain

(e)

times108

Prof

it

6

55

5

45

4

35

3

25

2

1511 12 13 14 15 16 17 18 19 2


Supply chain

(f )

Figure 2 Continued

Complexity 13






7 Conclusions


33

32

31

3

29

28

27

26

times108

Prof

it


(g)


times107

Prof

it

11

10

9

8

7

6

5

4

3

2

1

+



(h)


14 Complexity



Data Availability




Acknowledgments





References














Complexity 15

























16 Complexity









10 Complexity






1 032 008 1796 +E92 024 010 1218 +E93 009 012 1169 +E94 014 076 1941 +E95 041 029 1117 +E96 081 040 1525 +E9


i wodi pod

i qodi ΠodDiΠodS


d 051


i woci poc

i qoci ΠocDiΠocS


i 011 woc 020 ϕo

c 067


i wbdi pbd

i qbdi ΠbdDiΠbdS


d 053

Complexity 11












i wbci pbc

i qbci ΠbcDiΠbcS


i 010 wbc 020 ϕb

c 067

12 Complexity

32

31

3

29

28

27

26

Prof

it


Supply chain

times108

(a)

Prof

it

Supply chain

times108286

284

282

28

278

276

274005 006 007 008 009 01 011 012 013 014 015


(b)

Prof

it

284

283

282

281

28

279

278

277

27601 02 03 04 05


times108

Supply chain

(c)

Prof

it

284

285

286

287

288

283

282

281

2802 03 04 05 06

times108


Supply chain

(d)

times108

Prof

it

284

283

282

281

28

279

278

277

276

2750 01 03 04 05 06 07 08 09 102

Backlogging ratio

Supply chain

(e)

times108

Prof

it

6

55

5

45

4

35

3

25

2

1511 12 13 14 15 16 17 18 19 2


Supply chain

(f )

Figure 2 Continued

Complexity 13






7 Conclusions


33

32

31

3

29

28

27

26

times108

Prof

it


(g)


times107

Prof

it

11

10

9

8

7

6

5

4

3

2

1

+



(h)


14 Complexity



Data Availability




Acknowledgments





References














Complexity 15

























16 Complexity






1 032 008 1796 +E92 024 010 1218 +E93 009 012 1169 +E94 014 076 1941 +E95 041 029 1117 +E96 081 040 1525 +E9


i wodi pod

i qodi ΠodDiΠodS


d 051


i woci poc

i qoci ΠocDiΠocS


i 011 woc 020 ϕo

c 067


i wbdi pbd

i qbdi ΠbdDiΠbdS


d 053

Complexity 11












i wbci pbc

i qbci ΠbcDiΠbcS


i 010 wbc 020 ϕb

c 067

12 Complexity

32

31

3

29

28

27

26

Prof

it


Supply chain

times108

(a)

Prof

it

Supply chain

times108286

284

282

28

278

276

274005 006 007 008 009 01 011 012 013 014 015


(b)

Prof

it

284

283

282

281

28

279

278

277

27601 02 03 04 05


times108

Supply chain

(c)

Prof

it

284

285

286

287

288

283

282

281

2802 03 04 05 06

times108


Supply chain

(d)

times108

Prof

it

284

283

282

281

28

279

278

277

276

2750 01 03 04 05 06 07 08 09 102

Backlogging ratio

Supply chain

(e)

times108

Prof

it

6

55

5

45

4

35

3

25

2

1511 12 13 14 15 16 17 18 19 2


Supply chain

(f )

Figure 2 Continued

Complexity 13






7 Conclusions


33

32

31

3

29

28

27

26

times108

Prof

it


(g)


times107

Prof

it

11

10

9

8

7

6

5

4

3

2

1

+



(h)


14 Complexity



Data Availability




Acknowledgments





References














Complexity 15

























16 Complexity












i wbci pbc

i qbci ΠbcDiΠbcS


i 010 wbc 020 ϕb

c 067

12 Complexity

32

31

3

29

28

27

26

Prof

it


Supply chain

times108

(a)

Prof

it

Supply chain

times108286

284

282

28

278

276

274005 006 007 008 009 01 011 012 013 014 015


(b)

Prof

it

284

283

282

281

28

279

278

277

27601 02 03 04 05


times108

Supply chain

(c)

Prof

it

284

285

286

287

288

283

282

281

2802 03 04 05 06

times108


Supply chain

(d)

times108

Prof

it

284

283

282

281

28

279

278

277

276

2750 01 03 04 05 06 07 08 09 102

Backlogging ratio

Supply chain

(e)

times108

Prof

it

6

55

5

45

4

35

3

25

2

1511 12 13 14 15 16 17 18 19 2


Supply chain

(f )

Figure 2 Continued

Complexity 13






7 Conclusions


33

32

31

3

29

28

27

26

times108

Prof

it


(g)


times107

Prof

it

11

10

9

8

7

6

5

4

3

2

1

+



(h)


14 Complexity



Data Availability




Acknowledgments





References














Complexity 15

























16 Complexity

32

31

3

29

28

27

26

Prof

it


Supply chain

times108

(a)

Prof

it

Supply chain

times108286

284

282

28

278

276

274005 006 007 008 009 01 011 012 013 014 015


(b)

Prof

it

284

283

282

281

28

279

278

277

27601 02 03 04 05


times108

Supply chain

(c)

Prof

it

284

285

286

287

288

283

282

281

2802 03 04 05 06

times108


Supply chain

(d)

times108

Prof

it

284

283

282

281

28

279

278

277

276

2750 01 03 04 05 06 07 08 09 102

Backlogging ratio

Supply chain

(e)

times108

Prof

it

6

55

5

45

4

35

3

25

2

1511 12 13 14 15 16 17 18 19 2


Supply chain

(f )

Figure 2 Continued

Complexity 13






7 Conclusions


33

32

31

3

29

28

27

26

times108

Prof

it


(g)


times107

Prof

it

11

10

9

8

7

6

5

4

3

2

1

+



(h)


14 Complexity



Data Availability




Acknowledgments





References














Complexity 15

























16 Complexity






7 Conclusions


33

32

31

3

29

28

27

26

times108

Prof

it


(g)


times107

Prof

it

11

10

9

8

7

6

5

4

3

2

1

+



(h)


14 Complexity



Data Availability




Acknowledgments





References














Complexity 15

























16 Complexity



Data Availability




Acknowledgments





References














Complexity 15

























16 Complexity

























16 Complexity

Documents

Joint Pricing and Inventory Management of Interbasin Water