# Joint Pricing and Inventory Management of Interbasin Water Transfer Supply Chain

Authors: Xue Chen; Zhisong Chen
Journal: Complexity
 (2020)

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## Abstract

Four game-theoretical decision models without/with backlogging for the interbasin water transfer (IBWT) supply chain considering water delivery loss under joint pricing and inventory management (JPIM) are first developed, analyzed, and compared; then, the corresponding numerical and sensitivity analyses are conducted and compared; finally, the managerial insights and practical implementations are summarized in this paper. The research results indicate that (1) a revenue and cost sharing contract could effectively coordinate the IBWT supply chain and improve the operational performance of the IBWT supply chain under JPIM; (2) the partial backlogging strategy of water demand could effectively improve the operational performance of the IBWT supply chain under JPIM; (3) coordination strategy with partial backlogging is the best strategy for improving the operational performance of the IBWT supply chain under JPIM; (4) reducing water delivery loss rate and operational costs and increasing backlogging ratio are beneficial to improving the operational performance of the IBWT supply chain under JPIM.


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## Body

## 1. Introduction

To alleviate the shortage of water resources in arid and semiarid areas, various kinds of interbasin water transfer (IBWT) projects have been constructed and operated all over the world, such as the South-to-North Water Diversion (SNWD) Project in China, the California State Water Project, the Central Arizona Project and the Colorado River Aqueduct in the US, the Indira Gandhi Canal and the Telugu Ganga Project in India, the Snowy Mountains Scheme in Australia, the North Sinai Development Project in Egypt, and the National Water Carrier in Israel [1, 2]. In the practical operation management of the IBWT project, the existing rigid water price mechanism for the IBWT project is decoupled from the water supply-demand relationship and cannot effectively exert the regulatory role of the market mechanism and coordinate the interests of all parties involved. Thus, how to optimize pricing to achieve operational performance improvement under a flexible water price mechanism that is linked to the water supply-demand relationship is an urgent problem for the IBWT projects. Furthermore, due to the random water demand, the order quantity may mismatch with water demand. The water demand may be lower than the order quantity, and the holding cost of excess water inventory will thus be incurred; on the contrary, the water demand may be higher than the order quantity, and the shortage cost of excess water demand will thus be incurred. Hence, how to jointly make optimal pricing and inventory decisions to achieve operational performance improvement is also an important issue for the IBWT projects.From the perspective of supply chain management, the available research has explored the subsidy policy and the operational strategy of the IBWT green supply chain under social welfare maximization [3, 4], the impact of the supply capacity constraint and fairness concern on the operational decisions and outcomes of the IBWT supply chain under random precipitation [5], and the impact of fully/partial backlogging on the operational decisions and outcomes of IBWT green supply chain coordination considering water delivery loss under random precipitation [6]. However, the joint pricing-inventory management decisions and operational strategies for an IBWT supply chain considering water delivery loss and partial backlogging are rarely investigated in the current literatures and practices.Therefore, this paper will try to explore a novelty research issue regarding the operation management of the IBWT supply chain—the joint pricing-inventory management (JPIM) decisions and operational strategies for the IBWT supply chain considering water delivery loss and partial backlogging under random water demand.In the following sections, the related literatures are reviewed first in Section2; the theoretical modeling notations and assumptions for a generic IBWT supply chain are defined in Section 3; four game-theoretical decision models for the IBWT supply chain without/with backlogging under joint pricing-inventory management (JPIM) are developed, analyzed, and compared in Section 4; the corresponding numerical and sensitivity analyses for all models are implemented, and the corresponding results are compared in Section 5; the managerial insights and practical implementations are then summarized in Section 6; finally, the theoretical and practical contributions are summarized.


## 2. Literature Review

Currently, the interaction relationships among multiple stakeholders in the IBWT projects are investigated through game theory, such as the water conflict game-theoretical model of the SNWD project [7], game-theoretical model of the IBWT project considering both the water quantity and water quality [8], water allocation option contract for the IBWT projects [9], and incentive-compatible payment design for the SNWD project [10].Besides, cooperative game theory is applied to achieve Pareto improvement in the IBWT projects, such as the crisp and fuzzy Shapley game model for the IBWT water allocation [11], cooperative game model for the IBWT water allocation [12], IBWT water resource allocation using the least core game [13], and robust multiobjective bargaining game model for the IBWT water resource allocation [14].Currently, theories and techniques of supply chain management (SCM) have been applied in the IBWT projects to investigate the interactions among multiple stakeholders and develop equilibrium/coordination operational mechanisms, such as optimal pricing and coordination schemes for the SNWD supply chain [15], coordination mechanism based on revenue sharing contract for the SNWD supply chain with strategic customer [16], asymmetric Nash bargaining model for the SNWD supply chain [17], two-echelon water inventory model with inflow forecasting updates in an IBWT project [18], two-tier pricing and allocation schemes for the SNWD supply chain [19], competition intensity in the water supply chain under two contracts [20], power structures for the competitive water supply chains [21], optimal pricing and ordering strategies for dual competing water supply chains under three contracts [22], subsidy policies and operational strategies for the IBWT green supply chain under social welfare maximization [3, 4], impact of the supply capacity constraint and fairness concern on the operational decisions and outcomes of the IBWT supply chain under random precipitation [5], and impact of fully/partial backlogging on the IBWT green supply chain coordination considering water delivery loss under random precipitation [6].Nevertheless, these existing literatures regarding IBWT supply chain management neither explored the equilibrium/coordination strategies of the IBWT supply chain under JPIM, nor investigated the impact of the partial backlogging, the choice of operational strategies, and the water delivery loss on the operational performance of the IBWT supply chain. This paper tries to address the shortcomings in the available literatures and explore the operational strategies for an IBWT supply chain without/with partial backlogging under JPIM. An equilibrium decision model and a coordination decision model for the IBWT supply chain without backlogging/with partial backlogging under JPIM are developed, solved, and compared, respectively, to explore the optimal operational strategies and optimal joint pricing and inventory decisions for the IBWT supply chain.


## 3. Modeling Notations and Assumptions

An IBWT distribution system is a typical “embedded” supply chain structure. In this supply chain system, a horizontal water supply system embeds itself in a vertical water distribution system (see Figure1). In the horizontal water supply system, a local supplier and an external supplier work as a joint IBWT supplier via an efficient cooperation mechanism. Water resources are transferred and supplied by the local supplier from the water source to the external supplier within the trunk channel and then distributed to water resource distributors of all water intakes via river channels and artificial canals. Finally, the water resources are sold by each distributor to the water resource consumers in his/her service region. The water distributor and the corresponding water market in the ith water intake are indexed by i=1,2,…,n. It is assumed that there are m distributors supplied by the local supplier and n–m distributors supplied by the external supplier.Figure 1
A generic interbasin water transfer supply chain system.On this basis, the parameters used in the models are defined as follows:cdi = the water transfer cost from the ith water intake to the ith distributorck = the water transfer cost from the (k − 1)th water intake to the kth water intake within the horizontal supply chain, k=1,2,…,nδk = the water delivery loss rate from the (k − 1)th water intake to the kth water intake within the horizontal supply chain, and δk∈0,1, k=1,2,…,ncfi = the fixed cost for the ith water intake of the IBWT suppliercfl = the fixed cost for the local supplier, cfl=∑i=1mcficfe = the fixed cost for the external supplier, cfe=∑i=m+1ncficf = the fixed cost for the IBWT supplier, cf=cfl+cfe=∑i=1ncfiw = the wholesale price of water resources transferred from the local supplier to the external supplierwi = the wholesale price of water resources transferred from the IBWT supplier to the ith distributorpi = the retail price of water resources sold from the ith distributor to the consumers in his service regionκh = the holding cost coefficient, and 0<κh<1hi = the unit cost of holding water inventory for the ith distributor, and hi=κhpiκs = the shortage cost coefficient, and 0<κh<κs<1si = the shortage cost of unmet water demand for the ith distributor, and si=κspiQi = the original pumping quantity from the water source to the ith water intakeqi = the order quantity of the ith water intakeτ = the bargaining powers of the local supplier, and τ∈0,1λ = the bargaining powers of the ith water intake of the IBWT supplier, and λ∈0,1As mentioned above, the unmet water demand may be partially backlogged due to the capacity constraint of the IBWT project. The backlogging ratio of unmet water demand for theith distributor is φ, and φ∈0,1. The relationship between the water demand of the ith water intake qi and the original pumping quantity Qi is qi=Qi∏k=1i1−δk, and the total transfer cost of the original pumping quantity Qi is TCiQi=Qi∑k=1ick∏j=0k−11−δj; hereinto, δ0=0. Therefore, the total transfer cost of the water demand (order quantity) of the ith water intake is TCiqi=∑k=1ick∏j=0k−11−δj/∏k=1i1−δkqi. Defining Ci=∑k=1ick∏j=0k−11−δj/∏k=1i1−δk, then TCiqi=Ciqi. Following Howe and Linaweaver [23, 24], Petruzzi and Dada [25], and Wang et al. [26], the water demand for the ith distributor is dipi, and dipi=yipixi. Hereinto, yipi=aipi−b, where ai is the potential maximum water demand quantity and b is the price elasticity of the expected demand. xi is a random disturbance defined in the range A,B with B>A>0. The cumulative distribution function (CDF) and the probability density function (PDF) of xi are Fi⋅ and fi⋅, and the mean value and the standard deviation of xi are μi and σi. Following Petruzzi and Dada [25], Wang et al. [27], and Wang [28], zi=qi/yipi is defined as the “water stock factor” for the ith distributor; thus, the order quantity function of water resources for the ith water intake is qi=yipizi. The distribution of xi satisfies the IGFR (increasing generalized failure rate) condition: dgxi/dxi>0, where gixi≡xifixi/1−Fixi, and there exists a unique solution to the maximal expected problem. Many distributions, such as normal distribution and exponential distribution, satisfy the IGFR condition [27, 29, 30].Based on the foregoing modeling notations and assumptions, the profit functions of theith distributor, the ith water intake of the IBWT supplier, and the ith water intake of the IBWT supply chain with partial backlogging can be formulated as follows:(1)ΠDizi,pi=piyipiEminzi,xi−κhpiyipiEzi−xi++φpi−wi+cdi−κspiyipiExi−zi+−wi+cdiyipizi,ΠSiwi=wi−Ciyipizi+φwi−CiyipiExi−zi+−cfi,ΠSCizi,pi=piyipiEminzi,xi−κhpiyipiEzi−xi++φpi−Ci+cdi−κspiyipiExi−zi+−Ci+cdiyipizi−cfi.On this basis, the profit functions of the IBWT supplier, the local supplier, the external supplier, and the IBWT supply chain with partial backlogging can be formulated as follows:(2)ΠSwi=∑i=1nΠSiwi,ΠLSwi,w=∑i=1mΠSiwi+w∑i=m+1nyipizi,ΠESwi,w=∑i=m+1nΠSiwi−w∑i=m+1nyipizi,ΠSCzi,pi=∑i=1nΠSCizi,pi.


## 4. Game-Theoretical Decision Models

Based on the modeling notations and assumptions in Section3, two game-theoretical decision models without backlogging/with partial backlogging for the IBWT supply chain under JPIM considering water delivery loss, including the equilibrium and coordination decision models, are developed, analyzed, and compared in this section. In the models to follow, note that the super/subscript d represents an equilibrium decision; the super/subscript c represents a coordination decision; the super/subscript o represents the scenario without backlogging; the super/subscript b represents the scenario with partial backlogging.


### 4.1. Game-Theoretical Decision Models without Backlogging

Under the scenario without backlogging, the backlogging ratio of unmet water demand for theith distributor is φ=0. Two game-theoretical decision models of the IBWT supply chain without backlogging under JPIM considering water delivery loss, including the equilibrium and coordination decision models, are developed, analyzed, and compared in this section.


#### 4.1.1. Equilibrium Decision Model without Backlogging

Under this scenario, the detailed decision sequences are as follows: the local and external suppliers will first bargain over the wholesale price of the water resources within the IBWT horizontal supply chain via Nash bargaining theory [32–35] to achieve cooperative operations; then, the IBWT supplier decides the usage price of water resources for each water distributor in the IBWT vertical supply chain; finally, each water distributor independently and simultaneously decides the stock factor and the retail price of water resources for the consumer it serves.The two-stage Stackelberg and Nash bargaining game model for the IBWT supply chain without backlogging can be formulated as(3)maxwΩw=ΠLSodwiod,qiod,wτΠESodwiod,qiod,w1−τ,s.t.ΠLSodwiod,qiod,w+ΠESodwiod,qiod,w=ΠSodwiod,qiod,s.t.wiod,piod,ziodandqiodare derived from solving the following problem,maxwiΠSowi,piodwi,ziod,piodwiandziodare derived from solving the following problem,maxzi,piΠDiozi,pi.Solving this two-stage Stackelberg and Nash bargaining problem, we can get the equilibrium usage pricewiod in the ith water intake, the equilibrium retail price piod and the equilibrium stock factor ziod for the ith water distributor, the equilibrium ordering quantity qiod for the ith water distributor, and the bargaining wholesale price wdo. Furthermore, the profits of the local supplier, the external supplier, the IBWT supplier, the ith water distributor, and the IBWT supply chain can be calculated as ΠLSod, ΠESod, ΠSod, ΠDiod, and ΠSCod (see Table 1 for the detailed analytical results, and their derivations can be seen in sec10Supplementary Materials (available here)).Table 1
Analytical results of the IBWT supply chain without backlogging.
Scenario var.ResultEquilibrium decision (4.1.1)Coordination decision (4.1.2)wiowiod=bCi+cdi/b−1wioc=ϕcoCi−1−ϕcocdipiopiod=b/b−1piocpioc=b/b−1Ci+cdizioc/1+κszioc−ΛizioczioFiziod=FiziocFizioc=1+κs/b1+κh+κs+b−1Λizioc/b1+κh+κsziocqioqiod=b−1/bbqiocqioc=yipioczioc=aipioc−bziocwowdo=1/∑i=m+1nqiodτΠSod−∑i=1mΠSiodwco=1/∑i=m+1nqiocτΠSoc−∑i=1mΠSiocΠLSoΠLSod=τΠSod=τ∑i=1nb−1/bbΠSCioc+cfi−cfiΠLSoc=τΠSoc=τ∑i=1n1−ϕcoΠSCioc−ϕcocfiΠESoΠESod=1−τΠSod=1−τ∑i=1nb−1/bbΠSCioc+cfi−cfiΠESoc=1−τΠSoc=1−τ∑i=1n1−ϕcoΠSCioc−ϕcocfiΠSoΠSod=∑i=1nΠSiod=∑i=1nb−1/bbΠSCioc+cfi−cfiΠSoc=∑i=1nΠSioc=∑i=1n1−ϕcoΠSCioc−ϕcocfiΠDioΠDiod=b−1/bb−1ΠSCioc+cfiΠDioc=ϕcoΠSCioc+cfiΠSCoΠSCod=∑i=1nΠSCiod=∑i=1nb−1/bb+b−1/bb−1ΠSCioc+cfi−cfiΠSCoc=∑i=1nΠSCioc=∑i=1nCi+cdi/b−1qioc−cfiϕcoNAϕco=λb−1/bb−1+1−λ1−b−1/bbNoteΛizioc=1+κh+κs∫Azioczioc−xifixidxi+κs∫ABxifixidxi


#### 4.1.2. Coordination Decision Model without Backlogging

Under this scenario, the detailed decision sequences are as follows: the local and external suppliers will first bargain over the wholesale price of water resources within the IBWT horizontal supply chain to achieve cooperative operations; then, the IBWT supplier provides the distributors a revenue sharing contract in which the IBWT supplier charges a lower usage pricewi to the water distributors; if the distributors accept the contract, they will place orders with quantity qi to the IBWT supplier and decide the retail price of water resources pi and the stock factor of water resources zi; finally, they will share a proportion of their net revenues 1−ϕ to the IBWT supplier, where ϕ is the revenue keeping rate of the water distributors, 0≤ϕ≤1. The revenue shared by the ith distributor to the IBWT supplier is Ti=1−ϕpiyipiEminzi,xi−κhEzi−xi+−κsExi−zi+. Thus, the profit functions of the ith distributor and the IBWT supplier are as follows: ΠDioczi,pi=ΠDiozi,pi−Ti and ΠSocwi=∑i=1nΠSiocwi=∑i=1nΠSiowi+Ti.The two-stage coordination and Nash bargaining game model for the IBWT supply chain without backlogging can be formulated as(4)maxwΩw=ΠLSocwioc,qioc,wτΠESocwioc,qioc,w1−τ,s.t.ΠLSocwioc,qioc,w+ΠESocwioc,qioc,w=ΠSocwioc,qioc,wioc,qioc,ΠLSocwioc,qioc,w,ΠESocwioc,qioc,wandΠSocwioc,qiocare derived from solving the following problem,maxϕπioϕ=ΠSiocϕ−ΠSiodλΠDiocϕ−ΠDiod1−λ,s.t.ΠSiocϕ+ΠDiocϕ=ΠSCioc,wiocϕ,pioc,zioc,qioc,ΠSiocϕ,ΠDiocϕandΠSCiocare derived from solving the following problem,maxzi,piΠDioczi,pi,maxzi,piΠSCozi,pi.Solving this two-stage coordination and Nash bargaining problem, we can obtain the equilibrium usage pricewioc in the ith water intake, the equilibrium retail price pioc and the equilibrium stock factor zioc for the ith water distributor, the equilibrium ordering quantity qioc for the ith water distributor, and the bargaining wholesale price wcon. Furthermore, the profits of the local supplier, the external supplier, the IBWT supplier, the ith water distributor, and the IBWT supply chain can also be computed as ΠLSoc, ΠESoc, ΠSoc, ΠDioc, and ΠSCoc (see Table 1 for the detailed analytical results, and their derivations can be seen in Supplementary Materials).


### 4.2. Game-Theoretical Decision Models with Partial Backlogging

Under the scenario with partial backlogging, the backlogging ratio of unmet water demand for theith distributor is φ∈0,1. Two game-theoretical decision models of the IBWT supply chain with partial backlogging under JPIM considering water delivery loss, including the equilibrium and coordination decision models, are developed, analyzed, and compared in this section.


#### 4.2.1. Equilibrium Decision Model with Partial Backlogging

Under this scenario, the detailed decision sequences are as follows: the local and external suppliers will first bargain over the wholesale price of the water resources within the IBWT horizontal supply chain to achieve cooperative operations; next, the IBWT supplier decides the usage price of water resources for each water distributor in the IBWT vertical supply chain; then, each water distributor independently and simultaneously decides the stock factor and the retail price of water resources for the consumer it serves; finally, the unmet water demands of each market are partially backlogged and satisfied. The two-stage Stackelberg and Nash bargaining game model for the IBWT supply chain with partial backlogging can be formulated as(5)maxwΩw=ΠLSbdwibd,qibd,wτΠESbdwibd,qibd,w1−τ,s.t.ΠLSbdwibd,qibd,w+ΠESbdwibd,qibd,w=ΠSbdwibd,qibd,s.t.wibd,pibd,zibdandqibdare derived from solving the following problem,maxwiΠSbwi,pibdwi,zibd,pibdwiandzibdare derived from solving the following problem,maxzi,piΠDibzi,pi.Solving this two-stage Stackelberg and Nash bargaining problem, we can obtain the equilibrium usage pricewibd in the ith water intake, the equilibrium retail price pibd and the equilibrium stock factor zibd for the ith water distributor, the equilibrium ordering quantity qibd for the ith water distributor, and the bargaining wholesale price wdb. Furthermore, the profits of the local supplier, the external supplier, the IBWT supplier, the ith water distributor, and the IBWT supply chain can be calculated as ΠLSbd, ΠESbd, ΠSbd, ΠDibd, and ΠSCbd (see Table 2 for the detailed analytical results, and their derivations can be seen in Supplementary Materials).Table 2
Analytical results of the IBWT supply chain with partial backlogging.
Scenario var.ResultEquilibrium decision (4.2.1)Coordination decision (4.2.2)wibwibd=bCi+cdi/b−1wibc=ϕcbCi−1−ϕcbcdipibpibd=b/b−1pibcpibc=bCi+cdi/b−11−φzibc+φΝizibc/1−φ+κszibc−ΜizibczibFizibd=FizibcFizibcqibqibd=b−1/bbqibcqibc=yipibczibc=aipibc−bzibcwbwdb=1/∑i=m+1nqibdτΠSbd−∑i=1mΠSibdwcb=1/∑i=m+1nqibcτΠSbc−∑i=1mΠSibcΠLSbΠLSbd=τΠSbd=τ∑i=1nb−1/bbΠSCibc+cfi−cfiΠLSbc=τΠSbc=τ∑i=1n1−ϕcbΠSCibc−ϕcbcfiΠESbΠESbd=1−τΠSbd=1−τ∑i=1nb−1/bbΠSCibc+cfi−cfiΠESbc=1−τΠSbc=1−τ∑i=1n1−ϕcbΠSCibc−ϕcbcfiΠSbΠSbd=∑i=1nΠSibd=∑i=1nb−1/bbΠSCibc+cfi−cfiΠSbc=∑i=1nΠSibc=∑i=1n1−ϕcbΠSCibc−ϕcbcfiΠDibΠDibd=b−1/bb−1ΠSCibc+cfiΠDibc=ϕcbΠSCibc+cfiΠSCbΠSCbd=∑i=1nΠSCibd=∑i=1nb−1/bb+b−1/bb−1ΠSCibc+cfi−cfiΠSCbc=∑i=1nΠSCibc=∑i=1nCi+cdi/b−1yipibc1−φzibc+φΝizibc−cfiϕcbNAϕcb=λb−1/bb−1+1−λ1−b−1/bbNoteFizibc=1−φ1−φ+κszibc+bφ1−φ+κsNizibc+b−11−φMizibc/b−φ1−φ+κs+b1−φκhzibc+bφ1−φ+κs+κhΝizibc−b−1φΜizibc,Mizibc=1+κh+κs−φ∫Azibczibc−xifixidxi−φ−κs∫ABxifixidxi,Nizibc=∫Azibczibc−xifixidxi+∫ABxifixidxi.


#### 4.2.2. Coordination Decision Model with Partial Backlogging

Under this scenario, the detailed decision sequences are as follows: the local and external suppliers will first bargain over the wholesale price of water resources within the IBWT horizontal supply chain to achieve cooperative operations; next, the IBWT supplier offers the distributors a revenue sharing contract in which the IBWT supplier charges a lower usage pricewi to the water distributors; if the distributors accept the contract, they will place orders with quantity qi to the IBWT supplier and decide the retail price of water resources pi and the stock factor of water resources zi; then, the unmet water demands of each market are partially backlogged and satisfied; finally, all the distributors will share a proportion of their net revenues 1−ϕ to the IBWT supplier, where ϕ is the revenue keeping rate of the water distributors, 0≤ϕ≤1. The revenue shared by the ith distributor to the IBWT supplier is Ti=1−ϕpiyipiEminzi,xi−κhEzi−xi+−κsExi−zi+. Thus, the profit functions of the ith distributor and the IBWT supplier are as follows: ΠDibczi,pi=ΠDibzi,pi−Ti and ΠSbcwi=∑i=1nΠSibcwi=∑i=1nΠSibwi+Ti.The two-stage coordination and Nash bargaining game model for the IBWT supply chain with partial backlogging can be formulated as(6)maxwΩw=ΠLSbcwibc,qibc,wτΠESbcwibc,qibc,w1−τ,s.t.ΠLSbcwibc,qibc,w+ΠESbcwibc,qibc,w=ΠSbcwibc,qibc,wibc,qibc,ΠLSbcwibc,qibc,w,ΠESbcwibc,qibc,wandΠSbcwibc,qibcare derived from solving the following problem,maxϕπibϕ=⊓Sibcϕ−ΠSibdλΠDibcϕ−ΠDibd1−λ,s.t.ΠSibcϕ+ΠDibcϕ=ΠSCibc,wibcϕ,pibc,zibc,qibc,ΠSibcϕ,ΠDibcϕandΠSCibc,are derived from solving the following problem,maxzi,piΠDibczi,pi,maxzi,piΠSCbzi,pi.Solving this two-stage coordination and Nash bargaining problem, we can obtain the equilibrium usage pricewibc in the ith water intake, the equilibrium retail price pibc and the equilibrium stock factor zibc for the ith water distributor, the equilibrium ordering quantity qibc for the ith water distributor, and the bargaining wholesale price wcb. Furthermore, the profits of the local supplier, the external supplier, the IBWT supplier, the ith water distributor, and the IBWT supply chain can also be computed as ΠLSbc, ΠESbc, ΠSbc, ΠDibc, and ΠSCbc (see Table 1 for the detailed analytical results, and their derivations can be seen in Supplementary Materials).


## 4.1. Game-Theoretical Decision Models without Backlogging

Under the scenario without backlogging, the backlogging ratio of unmet water demand for theith distributor is φ=0. Two game-theoretical decision models of the IBWT supply chain without backlogging under JPIM considering water delivery loss, including the equilibrium and coordination decision models, are developed, analyzed, and compared in this section.


### 4.1.1. Equilibrium Decision Model without Backlogging

Under this scenario, the detailed decision sequences are as follows: the local and external suppliers will first bargain over the wholesale price of the water resources within the IBWT horizontal supply chain via Nash bargaining theory [32–35] to achieve cooperative operations; then, the IBWT supplier decides the usage price of water resources for each water distributor in the IBWT vertical supply chain; finally, each water distributor independently and simultaneously decides the stock factor and the retail price of water resources for the consumer it serves.The two-stage Stackelberg and Nash bargaining game model for the IBWT supply chain without backlogging can be formulated as(3)maxwΩw=ΠLSodwiod,qiod,wτΠESodwiod,qiod,w1−τ,s.t.ΠLSodwiod,qiod,w+ΠESodwiod,qiod,w=ΠSodwiod,qiod,s.t.wiod,piod,ziodandqiodare derived from solving the following problem,maxwiΠSowi,piodwi,ziod,piodwiandziodare derived from solving the following problem,maxzi,piΠDiozi,pi.Solving this two-stage Stackelberg and Nash bargaining problem, we can get the equilibrium usage pricewiod in the ith water intake, the equilibrium retail price piod and the equilibrium stock factor ziod for the ith water distributor, the equilibrium ordering quantity qiod for the ith water distributor, and the bargaining wholesale price wdo. Furthermore, the profits of the local supplier, the external supplier, the IBWT supplier, the ith water distributor, and the IBWT supply chain can be calculated as ΠLSod, ΠESod, ΠSod, ΠDiod, and ΠSCod (see Table 1 for the detailed analytical results, and their derivations can be seen in sec10Supplementary Materials (available here)).Table 1
Analytical results of the IBWT supply chain without backlogging.
Scenario var.ResultEquilibrium decision (4.1.1)Coordination decision (4.1.2)wiowiod=bCi+cdi/b−1wioc=ϕcoCi−1−ϕcocdipiopiod=b/b−1piocpioc=b/b−1Ci+cdizioc/1+κszioc−ΛizioczioFiziod=FiziocFizioc=1+κs/b1+κh+κs+b−1Λizioc/b1+κh+κsziocqioqiod=b−1/bbqiocqioc=yipioczioc=aipioc−bziocwowdo=1/∑i=m+1nqiodτΠSod−∑i=1mΠSiodwco=1/∑i=m+1nqiocτΠSoc−∑i=1mΠSiocΠLSoΠLSod=τΠSod=τ∑i=1nb−1/bbΠSCioc+cfi−cfiΠLSoc=τΠSoc=τ∑i=1n1−ϕcoΠSCioc−ϕcocfiΠESoΠESod=1−τΠSod=1−τ∑i=1nb−1/bbΠSCioc+cfi−cfiΠESoc=1−τΠSoc=1−τ∑i=1n1−ϕcoΠSCioc−ϕcocfiΠSoΠSod=∑i=1nΠSiod=∑i=1nb−1/bbΠSCioc+cfi−cfiΠSoc=∑i=1nΠSioc=∑i=1n1−ϕcoΠSCioc−ϕcocfiΠDioΠDiod=b−1/bb−1ΠSCioc+cfiΠDioc=ϕcoΠSCioc+cfiΠSCoΠSCod=∑i=1nΠSCiod=∑i=1nb−1/bb+b−1/bb−1ΠSCioc+cfi−cfiΠSCoc=∑i=1nΠSCioc=∑i=1nCi+cdi/b−1qioc−cfiϕcoNAϕco=λb−1/bb−1+1−λ1−b−1/bbNoteΛizioc=1+κh+κs∫Azioczioc−xifixidxi+κs∫ABxifixidxi


### 4.1.2. Coordination Decision Model without Backlogging

Under this scenario, the detailed decision sequences are as follows: the local and external suppliers will first bargain over the wholesale price of water resources within the IBWT horizontal supply chain to achieve cooperative operations; then, the IBWT supplier provides the distributors a revenue sharing contract in which the IBWT supplier charges a lower usage pricewi to the water distributors; if the distributors accept the contract, they will place orders with quantity qi to the IBWT supplier and decide the retail price of water resources pi and the stock factor of water resources zi; finally, they will share a proportion of their net revenues 1−ϕ to the IBWT supplier, where ϕ is the revenue keeping rate of the water distributors, 0≤ϕ≤1. The revenue shared by the ith distributor to the IBWT supplier is Ti=1−ϕpiyipiEminzi,xi−κhEzi−xi+−κsExi−zi+. Thus, the profit functions of the ith distributor and the IBWT supplier are as follows: ΠDioczi,pi=ΠDiozi,pi−Ti and ΠSocwi=∑i=1nΠSiocwi=∑i=1nΠSiowi+Ti.The two-stage coordination and Nash bargaining game model for the IBWT supply chain without backlogging can be formulated as(4)maxwΩw=ΠLSocwioc,qioc,wτΠESocwioc,qioc,w1−τ,s.t.ΠLSocwioc,qioc,w+ΠESocwioc,qioc,w=ΠSocwioc,qioc,wioc,qioc,ΠLSocwioc,qioc,w,ΠESocwioc,qioc,wandΠSocwioc,qiocare derived from solving the following problem,maxϕπioϕ=ΠSiocϕ−ΠSiodλΠDiocϕ−ΠDiod1−λ,s.t.ΠSiocϕ+ΠDiocϕ=ΠSCioc,wiocϕ,pioc,zioc,qioc,ΠSiocϕ,ΠDiocϕandΠSCiocare derived from solving the following problem,maxzi,piΠDioczi,pi,maxzi,piΠSCozi,pi.Solving this two-stage coordination and Nash bargaining problem, we can obtain the equilibrium usage pricewioc in the ith water intake, the equilibrium retail price pioc and the equilibrium stock factor zioc for the ith water distributor, the equilibrium ordering quantity qioc for the ith water distributor, and the bargaining wholesale price wcon. Furthermore, the profits of the local supplier, the external supplier, the IBWT supplier, the ith water distributor, and the IBWT supply chain can also be computed as ΠLSoc, ΠESoc, ΠSoc, ΠDioc, and ΠSCoc (see Table 1 for the detailed analytical results, and their derivations can be seen in Supplementary Materials).


## 4.1.1. Equilibrium Decision Model without Backlogging

Under this scenario, the detailed decision sequences are as follows: the local and external suppliers will first bargain over the wholesale price of the water resources within the IBWT horizontal supply chain via Nash bargaining theory [32–35] to achieve cooperative operations; then, the IBWT supplier decides the usage price of water resources for each water distributor in the IBWT vertical supply chain; finally, each water distributor independently and simultaneously decides the stock factor and the retail price of water resources for the consumer it serves.The two-stage Stackelberg and Nash bargaining game model for the IBWT supply chain without backlogging can be formulated as(3)maxwΩw=ΠLSodwiod,qiod,wτΠESodwiod,qiod,w1−τ,s.t.ΠLSodwiod,qiod,w+ΠESodwiod,qiod,w=ΠSodwiod,qiod,s.t.wiod,piod,ziodandqiodare derived from solving the following problem,maxwiΠSowi,piodwi,ziod,piodwiandziodare derived from solving the following problem,maxzi,piΠDiozi,pi.Solving this two-stage Stackelberg and Nash bargaining problem, we can get the equilibrium usage pricewiod in the ith water intake, the equilibrium retail price piod and the equilibrium stock factor ziod for the ith water distributor, the equilibrium ordering quantity qiod for the ith water distributor, and the bargaining wholesale price wdo. Furthermore, the profits of the local supplier, the external supplier, the IBWT supplier, the ith water distributor, and the IBWT supply chain can be calculated as ΠLSod, ΠESod, ΠSod, ΠDiod, and ΠSCod (see Table 1 for the detailed analytical results, and their derivations can be seen in sec10Supplementary Materials (available here)).Table 1
Analytical results of the IBWT supply chain without backlogging.
Scenario var.ResultEquilibrium decision (4.1.1)Coordination decision (4.1.2)wiowiod=bCi+cdi/b−1wioc=ϕcoCi−1−ϕcocdipiopiod=b/b−1piocpioc=b/b−1Ci+cdizioc/1+κszioc−ΛizioczioFiziod=FiziocFizioc=1+κs/b1+κh+κs+b−1Λizioc/b1+κh+κsziocqioqiod=b−1/bbqiocqioc=yipioczioc=aipioc−bziocwowdo=1/∑i=m+1nqiodτΠSod−∑i=1mΠSiodwco=1/∑i=m+1nqiocτΠSoc−∑i=1mΠSiocΠLSoΠLSod=τΠSod=τ∑i=1nb−1/bbΠSCioc+cfi−cfiΠLSoc=τΠSoc=τ∑i=1n1−ϕcoΠSCioc−ϕcocfiΠESoΠESod=1−τΠSod=1−τ∑i=1nb−1/bbΠSCioc+cfi−cfiΠESoc=1−τΠSoc=1−τ∑i=1n1−ϕcoΠSCioc−ϕcocfiΠSoΠSod=∑i=1nΠSiod=∑i=1nb−1/bbΠSCioc+cfi−cfiΠSoc=∑i=1nΠSioc=∑i=1n1−ϕcoΠSCioc−ϕcocfiΠDioΠDiod=b−1/bb−1ΠSCioc+cfiΠDioc=ϕcoΠSCioc+cfiΠSCoΠSCod=∑i=1nΠSCiod=∑i=1nb−1/bb+b−1/bb−1ΠSCioc+cfi−cfiΠSCoc=∑i=1nΠSCioc=∑i=1nCi+cdi/b−1qioc−cfiϕcoNAϕco=λb−1/bb−1+1−λ1−b−1/bbNoteΛizioc=1+κh+κs∫Azioczioc−xifixidxi+κs∫ABxifixidxi


## 4.1.2. Coordination Decision Model without Backlogging

Under this scenario, the detailed decision sequences are as follows: the local and external suppliers will first bargain over the wholesale price of water resources within the IBWT horizontal supply chain to achieve cooperative operations; then, the IBWT supplier provides the distributors a revenue sharing contract in which the IBWT supplier charges a lower usage pricewi to the water distributors; if the distributors accept the contract, they will place orders with quantity qi to the IBWT supplier and decide the retail price of water resources pi and the stock factor of water resources zi; finally, they will share a proportion of their net revenues 1−ϕ to the IBWT supplier, where ϕ is the revenue keeping rate of the water distributors, 0≤ϕ≤1. The revenue shared by the ith distributor to the IBWT supplier is Ti=1−ϕpiyipiEminzi,xi−κhEzi−xi+−κsExi−zi+. Thus, the profit functions of the ith distributor and the IBWT supplier are as follows: ΠDioczi,pi=ΠDiozi,pi−Ti and ΠSocwi=∑i=1nΠSiocwi=∑i=1nΠSiowi+Ti.The two-stage coordination and Nash bargaining game model for the IBWT supply chain without backlogging can be formulated as(4)maxwΩw=ΠLSocwioc,qioc,wτΠESocwioc,qioc,w1−τ,s.t.ΠLSocwioc,qioc,w+ΠESocwioc,qioc,w=ΠSocwioc,qioc,wioc,qioc,ΠLSocwioc,qioc,w,ΠESocwioc,qioc,wandΠSocwioc,qiocare derived from solving the following problem,maxϕπioϕ=ΠSiocϕ−ΠSiodλΠDiocϕ−ΠDiod1−λ,s.t.ΠSiocϕ+ΠDiocϕ=ΠSCioc,wiocϕ,pioc,zioc,qioc,ΠSiocϕ,ΠDiocϕandΠSCiocare derived from solving the following problem,maxzi,piΠDioczi,pi,maxzi,piΠSCozi,pi.Solving this two-stage coordination and Nash bargaining problem, we can obtain the equilibrium usage pricewioc in the ith water intake, the equilibrium retail price pioc and the equilibrium stock factor zioc for the ith water distributor, the equilibrium ordering quantity qioc for the ith water distributor, and the bargaining wholesale price wcon. Furthermore, the profits of the local supplier, the external supplier, the IBWT supplier, the ith water distributor, and the IBWT supply chain can also be computed as ΠLSoc, ΠESoc, ΠSoc, ΠDioc, and ΠSCoc (see Table 1 for the detailed analytical results, and their derivations can be seen in Supplementary Materials).


## 4.2. Game-Theoretical Decision Models with Partial Backlogging

Under the scenario with partial backlogging, the backlogging ratio of unmet water demand for theith distributor is φ∈0,1. Two game-theoretical decision models of the IBWT supply chain with partial backlogging under JPIM considering water delivery loss, including the equilibrium and coordination decision models, are developed, analyzed, and compared in this section.


### 4.2.1. Equilibrium Decision Model with Partial Backlogging

Under this scenario, the detailed decision sequences are as follows: the local and external suppliers will first bargain over the wholesale price of the water resources within the IBWT horizontal supply chain to achieve cooperative operations; next, the IBWT supplier decides the usage price of water resources for each water distributor in the IBWT vertical supply chain; then, each water distributor independently and simultaneously decides the stock factor and the retail price of water resources for the consumer it serves; finally, the unmet water demands of each market are partially backlogged and satisfied. The two-stage Stackelberg and Nash bargaining game model for the IBWT supply chain with partial backlogging can be formulated as(5)maxwΩw=ΠLSbdwibd,qibd,wτΠESbdwibd,qibd,w1−τ,s.t.ΠLSbdwibd,qibd,w+ΠESbdwibd,qibd,w=ΠSbdwibd,qibd,s.t.wibd,pibd,zibdandqibdare derived from solving the following problem,maxwiΠSbwi,pibdwi,zibd,pibdwiandzibdare derived from solving the following problem,maxzi,piΠDibzi,pi.Solving this two-stage Stackelberg and Nash bargaining problem, we can obtain the equilibrium usage pricewibd in the ith water intake, the equilibrium retail price pibd and the equilibrium stock factor zibd for the ith water distributor, the equilibrium ordering quantity qibd for the ith water distributor, and the bargaining wholesale price wdb. Furthermore, the profits of the local supplier, the external supplier, the IBWT supplier, the ith water distributor, and the IBWT supply chain can be calculated as ΠLSbd, ΠESbd, ΠSbd, ΠDibd, and ΠSCbd (see Table 2 for the detailed analytical results, and their derivations can be seen in Supplementary Materials).Table 2
Analytical results of the IBWT supply chain with partial backlogging.
Scenario var.ResultEquilibrium decision (4.2.1)Coordination decision (4.2.2)wibwibd=bCi+cdi/b−1wibc=ϕcbCi−1−ϕcbcdipibpibd=b/b−1pibcpibc=bCi+cdi/b−11−φzibc+φΝizibc/1−φ+κszibc−ΜizibczibFizibd=FizibcFizibcqibqibd=b−1/bbqibcqibc=yipibczibc=aipibc−bzibcwbwdb=1/∑i=m+1nqibdτΠSbd−∑i=1mΠSibdwcb=1/∑i=m+1nqibcτΠSbc−∑i=1mΠSibcΠLSbΠLSbd=τΠSbd=τ∑i=1nb−1/bbΠSCibc+cfi−cfiΠLSbc=τΠSbc=τ∑i=1n1−ϕcbΠSCibc−ϕcbcfiΠESbΠESbd=1−τΠSbd=1−τ∑i=1nb−1/bbΠSCibc+cfi−cfiΠESbc=1−τΠSbc=1−τ∑i=1n1−ϕcbΠSCibc−ϕcbcfiΠSbΠSbd=∑i=1nΠSibd=∑i=1nb−1/bbΠSCibc+cfi−cfiΠSbc=∑i=1nΠSibc=∑i=1n1−ϕcbΠSCibc−ϕcbcfiΠDibΠDibd=b−1/bb−1ΠSCibc+cfiΠDibc=ϕcbΠSCibc+cfiΠSCbΠSCbd=∑i=1nΠSCibd=∑i=1nb−1/bb+b−1/bb−1ΠSCibc+cfi−cfiΠSCbc=∑i=1nΠSCibc=∑i=1nCi+cdi/b−1yipibc1−φzibc+φΝizibc−cfiϕcbNAϕcb=λb−1/bb−1+1−λ1−b−1/bbNoteFizibc=1−φ1−φ+κszibc+bφ1−φ+κsNizibc+b−11−φMizibc/b−φ1−φ+κs+b1−φκhzibc+bφ1−φ+κs+κhΝizibc−b−1φΜizibc,Mizibc=1+κh+κs−φ∫Azibczibc−xifixidxi−φ−κs∫ABxifixidxi,Nizibc=∫Azibczibc−xifixidxi+∫ABxifixidxi.


### 4.2.2. Coordination Decision Model with Partial Backlogging

Under this scenario, the detailed decision sequences are as follows: the local and external suppliers will first bargain over the wholesale price of water resources within the IBWT horizontal supply chain to achieve cooperative operations; next, the IBWT supplier offers the distributors a revenue sharing contract in which the IBWT supplier charges a lower usage pricewi to the water distributors; if the distributors accept the contract, they will place orders with quantity qi to the IBWT supplier and decide the retail price of water resources pi and the stock factor of water resources zi; then, the unmet water demands of each market are partially backlogged and satisfied; finally, all the distributors will share a proportion of their net revenues 1−ϕ to the IBWT supplier, where ϕ is the revenue keeping rate of the water distributors, 0≤ϕ≤1. The revenue shared by the ith distributor to the IBWT supplier is Ti=1−ϕpiyipiEminzi,xi−κhEzi−xi+−κsExi−zi+. Thus, the profit functions of the ith distributor and the IBWT supplier are as follows: ΠDibczi,pi=ΠDibzi,pi−Ti and ΠSbcwi=∑i=1nΠSibcwi=∑i=1nΠSibwi+Ti.The two-stage coordination and Nash bargaining game model for the IBWT supply chain with partial backlogging can be formulated as(6)maxwΩw=ΠLSbcwibc,qibc,wτΠESbcwibc,qibc,w1−τ,s.t.ΠLSbcwibc,qibc,w+ΠESbcwibc,qibc,w=ΠSbcwibc,qibc,wibc,qibc,ΠLSbcwibc,qibc,w,ΠESbcwibc,qibc,wandΠSbcwibc,qibcare derived from solving the following problem,maxϕπibϕ=⊓Sibcϕ−ΠSibdλΠDibcϕ−ΠDibd1−λ,s.t.ΠSibcϕ+ΠDibcϕ=ΠSCibc,wibcϕ,pibc,zibc,qibc,ΠSibcϕ,ΠDibcϕandΠSCibc,are derived from solving the following problem,maxzi,piΠDibczi,pi,maxzi,piΠSCbzi,pi.Solving this two-stage coordination and Nash bargaining problem, we can obtain the equilibrium usage pricewibc in the ith water intake, the equilibrium retail price pibc and the equilibrium stock factor zibc for the ith water distributor, the equilibrium ordering quantity qibc for the ith water distributor, and the bargaining wholesale price wcb. Furthermore, the profits of the local supplier, the external supplier, the IBWT supplier, the ith water distributor, and the IBWT supply chain can also be computed as ΠLSbc, ΠESbc, ΠSbc, ΠDibc, and ΠSCbc (see Table 1 for the detailed analytical results, and their derivations can be seen in Supplementary Materials).


## 4.2.1. Equilibrium Decision Model with Partial Backlogging

Under this scenario, the detailed decision sequences are as follows: the local and external suppliers will first bargain over the wholesale price of the water resources within the IBWT horizontal supply chain to achieve cooperative operations; next, the IBWT supplier decides the usage price of water resources for each water distributor in the IBWT vertical supply chain; then, each water distributor independently and simultaneously decides the stock factor and the retail price of water resources for the consumer it serves; finally, the unmet water demands of each market are partially backlogged and satisfied. The two-stage Stackelberg and Nash bargaining game model for the IBWT supply chain with partial backlogging can be formulated as(5)maxwΩw=ΠLSbdwibd,qibd,wτΠESbdwibd,qibd,w1−τ,s.t.ΠLSbdwibd,qibd,w+ΠESbdwibd,qibd,w=ΠSbdwibd,qibd,s.t.wibd,pibd,zibdandqibdare derived from solving the following problem,maxwiΠSbwi,pibdwi,zibd,pibdwiandzibdare derived from solving the following problem,maxzi,piΠDibzi,pi.Solving this two-stage Stackelberg and Nash bargaining problem, we can obtain the equilibrium usage pricewibd in the ith water intake, the equilibrium retail price pibd and the equilibrium stock factor zibd for the ith water distributor, the equilibrium ordering quantity qibd for the ith water distributor, and the bargaining wholesale price wdb. Furthermore, the profits of the local supplier, the external supplier, the IBWT supplier, the ith water distributor, and the IBWT supply chain can be calculated as ΠLSbd, ΠESbd, ΠSbd, ΠDibd, and ΠSCbd (see Table 2 for the detailed analytical results, and their derivations can be seen in Supplementary Materials).Table 2
Analytical results of the IBWT supply chain with partial backlogging.
Scenario var.ResultEquilibrium decision (4.2.1)Coordination decision (4.2.2)wibwibd=bCi+cdi/b−1wibc=ϕcbCi−1−ϕcbcdipibpibd=b/b−1pibcpibc=bCi+cdi/b−11−φzibc+φΝizibc/1−φ+κszibc−ΜizibczibFizibd=FizibcFizibcqibqibd=b−1/bbqibcqibc=yipibczibc=aipibc−bzibcwbwdb=1/∑i=m+1nqibdτΠSbd−∑i=1mΠSibdwcb=1/∑i=m+1nqibcτΠSbc−∑i=1mΠSibcΠLSbΠLSbd=τΠSbd=τ∑i=1nb−1/bbΠSCibc+cfi−cfiΠLSbc=τΠSbc=τ∑i=1n1−ϕcbΠSCibc−ϕcbcfiΠESbΠESbd=1−τΠSbd=1−τ∑i=1nb−1/bbΠSCibc+cfi−cfiΠESbc=1−τΠSbc=1−τ∑i=1n1−ϕcbΠSCibc−ϕcbcfiΠSbΠSbd=∑i=1nΠSibd=∑i=1nb−1/bbΠSCibc+cfi−cfiΠSbc=∑i=1nΠSibc=∑i=1n1−ϕcbΠSCibc−ϕcbcfiΠDibΠDibd=b−1/bb−1ΠSCibc+cfiΠDibc=ϕcbΠSCibc+cfiΠSCbΠSCbd=∑i=1nΠSCibd=∑i=1nb−1/bb+b−1/bb−1ΠSCibc+cfi−cfiΠSCbc=∑i=1nΠSCibc=∑i=1nCi+cdi/b−1yipibc1−φzibc+φΝizibc−cfiϕcbNAϕcb=λb−1/bb−1+1−λ1−b−1/bbNoteFizibc=1−φ1−φ+κszibc+bφ1−φ+κsNizibc+b−11−φMizibc/b−φ1−φ+κs+b1−φκhzibc+bφ1−φ+κs+κhΝizibc−b−1φΜizibc,Mizibc=1+κh+κs−φ∫Azibczibc−xifixidxi−φ−κs∫ABxifixidxi,Nizibc=∫Azibczibc−xifixidxi+∫ABxifixidxi.


## 4.2.2. Coordination Decision Model with Partial Backlogging

Under this scenario, the detailed decision sequences are as follows: the local and external suppliers will first bargain over the wholesale price of water resources within the IBWT horizontal supply chain to achieve cooperative operations; next, the IBWT supplier offers the distributors a revenue sharing contract in which the IBWT supplier charges a lower usage pricewi to the water distributors; if the distributors accept the contract, they will place orders with quantity qi to the IBWT supplier and decide the retail price of water resources pi and the stock factor of water resources zi; then, the unmet water demands of each market are partially backlogged and satisfied; finally, all the distributors will share a proportion of their net revenues 1−ϕ to the IBWT supplier, where ϕ is the revenue keeping rate of the water distributors, 0≤ϕ≤1. The revenue shared by the ith distributor to the IBWT supplier is Ti=1−ϕpiyipiEminzi,xi−κhEzi−xi+−κsExi−zi+. Thus, the profit functions of the ith distributor and the IBWT supplier are as follows: ΠDibczi,pi=ΠDibzi,pi−Ti and ΠSbcwi=∑i=1nΠSibcwi=∑i=1nΠSibwi+Ti.The two-stage coordination and Nash bargaining game model for the IBWT supply chain with partial backlogging can be formulated as(6)maxwΩw=ΠLSbcwibc,qibc,wτΠESbcwibc,qibc,w1−τ,s.t.ΠLSbcwibc,qibc,w+ΠESbcwibc,qibc,w=ΠSbcwibc,qibc,wibc,qibc,ΠLSbcwibc,qibc,w,ΠESbcwibc,qibc,wandΠSbcwibc,qibcare derived from solving the following problem,maxϕπibϕ=⊓Sibcϕ−ΠSibdλΠDibcϕ−ΠDibd1−λ,s.t.ΠSibcϕ+ΠDibcϕ=ΠSCibc,wibcϕ,pibc,zibc,qibc,ΠSibcϕ,ΠDibcϕandΠSCibc,are derived from solving the following problem,maxzi,piΠDibczi,pi,maxzi,piΠSCbzi,pi.Solving this two-stage coordination and Nash bargaining problem, we can obtain the equilibrium usage pricewibc in the ith water intake, the equilibrium retail price pibc and the equilibrium stock factor zibc for the ith water distributor, the equilibrium ordering quantity qibc for the ith water distributor, and the bargaining wholesale price wcb. Furthermore, the profits of the local supplier, the external supplier, the IBWT supplier, the ith water distributor, and the IBWT supply chain can also be computed as ΠLSbc, ΠESbc, ΠSbc, ΠDibc, and ΠSCbc (see Table 1 for the detailed analytical results, and their derivations can be seen in Supplementary Materials).


## 5. Numerical and Sensitivity Analyses

Based on the game-theoretical decision modeling analysis, a real-world case of the eastern route of the South-to-North Water Diversion (SNWD) project in China is selected to conduct numerical and sensitivity analyses.The SNWD project is an important world-scale strategic water resource engineering to solve the water shortage problem in northern China. This project is divided into the eastern route, western route, and middle route. These three routes of the SNWD project can transfer water resources separately from the Yangtze River linking Yangtze River, Huaihe River, Yellow River, and Haihe River to formulate a nationwide water supply system with “Four horizontal and Three vertical, South-North deployment, East-West mutual support.”The eastern route of the SNWD project is constructed and extended based on the north water transfer project in Jiangsu Province. Through the Jiangdu water control project in Yangzhou City, Jiangsu Province, the water is extracted from the main stream of the lower reaches of the Yangtze River and transferred by the Beijing-Hangzhou Grand Canal and its parallel river channels to connect Hongze Lake, Luoma Lake, Nansi Lake, and Dongping Lake. After leaving Dongping Lake, there are two ways to deliver water: one is to the north, passing through the Yellow River through a tunnel near Weishan, and then to Tianjin; the other is to the east, Yantai, and Weihai, via the Jiaodong water transfer branch-line. There are 13 pumping stations in this project with a total length of water transfer mainline at 1466.5 kilometers, a water raising capacity at 65 meters, and a water diversion scale of 14.8 billion cubic meters. The eastern route project provides production and domestic water to the east of Huang-Huai-Hai Plain, Jiaodong area, and Beijing-Tianjin-Hebei region. In the water supply area, there are 25 cities at prefecture level or above from the Huaihe River, the Haihe River, and the Yellow River Basin.In the eastern route of the SNWD project, there are six sections for the mainline of the project: Section1 (Jiangdu Station∼South of Nansi Lake), Section 2 (Lower Cascade of Nansi Lake), Section 3 (Upper Cascade of Nansi Lake∼Changgou Pumping Station), Section 4 (Changgou Pumping Station∼Dongping Lake, including East of Dongping Lake), Section 5 (Dongping Lake∼Qiutun Sluice in Linqing City), and Section 6 (Qiutun Sluice in Linqing City∼Datun Reservoir). These six sections can be generalized to six water intakes, i.e., n = 6. Hereinto, Sections 1-2 are managed and operated by the local supplier (Jiangsu Water Source Co. Ltd), and Sections 3∼6 are managed and operated by the external supplier (Shandong Mainline Co. Ltd), i.e., m = 2 and n − m = 4. There are two water distributors in the service region of the local supplier and four water distributors in the service region of the external supplier.Based on the real characteristics and management practices of the eastern route of the SNWD project in China [2], the numerical and sensitivity analyses are conducted in the following sections, and the corresponding values of parameters relating to the IBWT supply chain are collected and estimated from the publicly disclosed information of the eastern route of the SNWD project [34] and the water transfer scheme for the eastern route of the SNWD project [36]. The parameters of water transfer costs and fixed costs can be estimated according to the cost accounting method. The parameters of the potential maximum water demand quantity and the price elasticity of the expected demand can be estimated based on the historical operation data. The random distribution and corresponding parameters of the random factor can also be fitted based on the historical operation data. The parameter of the water delivery loss rate can be estimated based on the historical operation data. The parameters of the holding cost coefficient and the shortage cost coefficient can be estimated via the cost accounting method or empirical parameters. The backlogging ratio of unmet water demand can be calculated as total shortage quantity divided by extra supply capacity. The bargaining powers of the local supplier and the ith water intake of the IBWT supplier can be calculated by the market power evaluation. On this basis, the lower bound of the interval of the random factor xi is set at 0.00001, and the upper bounds of the interval of the random factor xi is set at 1, i.e., A=1E−5,B=1. The random factor xi obeys normal distribution, i.e., xi∼Nμi,σi2. The mean value of the random factor μi is set at 0.10, and the standard deviation of the random factor σi is set at 0.01. The corresponding parameters of mainline/branch-line water transfer costs and potential maximum water demand quantities are collected and estimated in Table 3. The fixed cost of water delivery for the ith water intake of the IBWT supplier cfi is 1,000,000. According to the empirical parameters, the water delivery loss from the (i − 1)th water intake to the ith water intake within the horizontal supply chain δi is about 15%. The backlogging ratio of unmet water demand for the ith distributor φ is 0.8. The price elasticity of the expected demand b is 1.5. The holding cost coefficient κh is 0.2, and the shortage cost coefficient κs is 0.5. Owing to the advantage of the local supplier, the local supplier’s bargaining power τ is 0.6. Likewise, due to the advantage of the IBWT supplier, the ith water intake of the IBWT supplier’s bargaining power λ is 0.6.Table 3
Parameter setting.
Section (water intake)iMainline water transfer cost (ci)Branch-line water transfer cost (cdi)Potential maximum water demand (ai)10.320.081.796 +E920.240.101.218 +E930.090.121.169 +E940.140.761.941 +E950.410.291.117 +E960.810.401.525 +E9


### 5.1. Numerical Analysis

The joint pricing-inventory operational decisions and performances for all the game-theoretical decision models without/with backlogging are compared and summarized in Tables4–7, respectively. It should be noted that the centralized decision neglects the roles of the members’ decisions and therefore is inferior to coordination decision regarding the derived solutions. Thus, the centralized decision model is not shown in the numerical analysis separately. The main findings of the numerical analysis are summarized below:(1)
Comparing the numerical analysis results between equilibrium decision (Table4) and coordination decision (Table 5) without backlogging under JPIM, the findings are summarized as follows: (i) The wholesale prices of water resources under coordination decision are lower than those under equilibrium decision. (ii) The retail prices of water resources under coordination decision are lower than those under equilibrium decision. (iii) The water stock factors under coordination decision are equal to those under equilibrium decision. (iv) The order quantities of water resources under coordination decision are higher than those under equilibrium decision. (v) The profits of the IBWT supply chain and its members under coordination decision are higher than those under equilibrium decision.(2)
Comparing the numerical analysis results between equilibrium decision (Table6) and coordination decision (Table 7) with partial backlogging under JPIM, the findings are summarized as follows: (i) The wholesale prices of water resources under coordination decision are lower than those under equilibrium decision. (ii) The retail prices of water resources under coordination decision are lower than those under equilibrium decision. (iii) The water stock factors under coordination decision are equal to those under equilibrium decision. (iv) The order quantities of water resources under coordination decision are higher than those under equilibrium decision. (v) The profits of the IBWT supply chain and its members under coordination decision are higher than those under equilibrium decision.(3)
Comparing the numerical analysis results between the scenario without backlogging (Table4) and the scenario with partial backlogging (Table 6) under equilibrium decision, the findings are summarized as follows: (i) The wholesale prices of water resources under the scenario with partial backlogging are equal to those under the scenario without backlogging. (ii) The retail prices of water resources under the scenario with partial backlogging are lower than those under the scenario without backlogging. (iii) The water stock factors under the scenario with partial backlogging are lower than those under the scenario without backlogging. (iv) The order quantities of water resources under the scenario with partial backlogging are lower than those under the scenario without backlogging. (v) The profits of the IBWT supply chain and its members under the scenario with partial backlogging are higher than those under the scenario without backlogging.(4)
Comparing the numerical analysis results between the scenario without backlogging (Table5) and the scenario with partial backlogging (Table 7) under coordination decision, the findings are summarized as follows: (i) The wholesale prices of water resources under the scenario with partial backlogging are equal to those under the scenario without backlogging. (ii) The retail prices of water resources under the scenario with partial backlogging are lower than those under the scenario without backlogging. (iii) The water stock factors under the scenario with partial backlogging are lower than those under the scenario without backlogging. (iv) The order quantities of water resources under the scenario with partial backlogging are lower than those under the scenario without backlogging. (v) The profits of the IBWT supply chain and its members under the scenario with partial backlogging are higher than those under the scenario without backlogging.Table 4
Numerical analysis results of equilibrium decision without backlogging.
iwiodpiodqiodΠDiodΠSod11.294.5219,687,77253,921,33448,147,44422.388.175,492,51127,196,481ΠLSod33.1210.693,525,44622,826,46028,888,46645.4020.332,230,60227,477,799ΠESod56.5922.721,087,11414,960,68219,258,978610.7336.74721,42316,059,575ΠSCodSumNANA32,744,868162,442,332210,589,776Noteziod=0.11,wdo=0.51Table 5
Numerical analysis results of coordination decision without backlogging.
iwiocpiocqiocΠDiocΠSoc10.231.51102,300,66762,521,08687,008,60220.452.7228,539,92431,533,966ΠLSoc30.603.5618,318,75226,466,98352,205,16140.616.7811,590,54931,860,151ΠESoc51.257.575,648,81117,346,71634,803,44162.0812.253,748,62218,620,868ΠSCocSumNANA170,147,325188,349,770275,358,372Notezioc=0.11,wco=0.20,ϕco=0.67Table 6
Numerical analysis results of equilibrium decision with partial backlogging.
iwibdpibdqibdΠDibdΠSbd11.294.4219,598,80255,086,31849,317,31222.387.995,467,69027,784,068ΠLSbd33.1210.453,509,51423,319,63129,590,38745.4019.882,220,52228,071,464ΠESbd56.5922.211,082,20115,283,91119,726,925610.7335.93718,16316,406,546ΠSCbdSumNANA32,596,892165,951,937215,269,249Notezibd=0.10,wdb=0.53Table 7
Numerical analysis results of coordination decision with partial backlogging.
iwibcpibcqibcΠDibcΠSbc10.231.47101,838,36263,871,86989,018,07520.452.6628,410,95032,215,265ΠLSbc30.603.4818,235,96927,038,80953,410,84540.616.6311,538,17132,548,497ΠESbc51.257.405,623,28417,721,49635,607,23062.0811.983,731,68219,023,177ΠSCbcSumNANA169,378,417192,419,112281,437,186Notezibc=0.10,wcb=0.20,ϕcb=0.67


### 5.2. Sensitivity Analysis

Since, in the previous analysis, coordination decision with partial backlogging is found to be superior to the other decisions without/with partial backlogging, the sensitivity analysis will focus on how the changes in eight key parameters (including the mainline transfer cost, the branch-line transfer cost, the holding cost coefficient, the shortage cost coefficient, the backlogging ratio, the price elasticity of the expected water demand, the water delivery loss rate, and the bargaining power of theith water intake of the IBWT supplier) impact the profits under coordination decision with partial backlogging. To capture the impact of the change in key parameters, we only select the parameters from the 1st distributor and the 1st water intake to conduct sensitivity analysis, including the mainline transfer cost and the branch-line transfer cost. Figure 2 (including eight subgraphs) shows the impact of eight key parameters change on the profit of the IBWT supply chain with partial backlogging. The findings from the sensitivity analysis results are summarized as follows: (1) the IBWT supply chain profit decreases as the mainline transfer cost increases; (2) the IBWT supply chain profit decreases as the branch-line transfer cost increases; (3) the IBWT supply chain profit decreases as the holding cost coefficient increases; (4) the IBWT supply chain profit decreases as the shortage cost coefficient increases; (5) the IBWT supply chain profit increases as the backlogging ratio increases; (6) the IBWT supply chain profit decreases as the price elasticity of the expected water demand increases; (7) the IBWT supply chain profit decreases as the water delivery loss increases; (8) the IBWT supplier’s profit increases and the distributors’ profits decrease as the bargaining power of the ith water intake of the IBWT supplier increases.Figure 2
The impact of key parameters’ change on the profit of the IBWT supply 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.
(a)(b)(c)(d)(e)(f)(g)(h)


## 5.1. Numerical Analysis

The joint pricing-inventory operational decisions and performances for all the game-theoretical decision models without/with backlogging are compared and summarized in Tables4–7, respectively. It should be noted that the centralized decision neglects the roles of the members’ decisions and therefore is inferior to coordination decision regarding the derived solutions. Thus, the centralized decision model is not shown in the numerical analysis separately. The main findings of the numerical analysis are summarized below:(1)
Comparing the numerical analysis results between equilibrium decision (Table4) and coordination decision (Table 5) without backlogging under JPIM, the findings are summarized as follows: (i) The wholesale prices of water resources under coordination decision are lower than those under equilibrium decision. (ii) The retail prices of water resources under coordination decision are lower than those under equilibrium decision. (iii) The water stock factors under coordination decision are equal to those under equilibrium decision. (iv) The order quantities of water resources under coordination decision are higher than those under equilibrium decision. (v) The profits of the IBWT supply chain and its members under coordination decision are higher than those under equilibrium decision.(2)
Comparing the numerical analysis results between equilibrium decision (Table6) and coordination decision (Table 7) with partial backlogging under JPIM, the findings are summarized as follows: (i) The wholesale prices of water resources under coordination decision are lower than those under equilibrium decision. (ii) The retail prices of water resources under coordination decision are lower than those under equilibrium decision. (iii) The water stock factors under coordination decision are equal to those under equilibrium decision. (iv) The order quantities of water resources under coordination decision are higher than those under equilibrium decision. (v) The profits of the IBWT supply chain and its members under coordination decision are higher than those under equilibrium decision.(3)
Comparing the numerical analysis results between the scenario without backlogging (Table4) and the scenario with partial backlogging (Table 6) under equilibrium decision, the findings are summarized as follows: (i) The wholesale prices of water resources under the scenario with partial backlogging are equal to those under the scenario without backlogging. (ii) The retail prices of water resources under the scenario with partial backlogging are lower than those under the scenario without backlogging. (iii) The water stock factors under the scenario with partial backlogging are lower than those under the scenario without backlogging. (iv) The order quantities of water resources under the scenario with partial backlogging are lower than those under the scenario without backlogging. (v) The profits of the IBWT supply chain and its members under the scenario with partial backlogging are higher than those under the scenario without backlogging.(4)
Comparing the numerical analysis results between the scenario without backlogging (Table5) and the scenario with partial backlogging (Table 7) under coordination decision, the findings are summarized as follows: (i) The wholesale prices of water resources under the scenario with partial backlogging are equal to those under the scenario without backlogging. (ii) The retail prices of water resources under the scenario with partial backlogging are lower than those under the scenario without backlogging. (iii) The water stock factors under the scenario with partial backlogging are lower than those under the scenario without backlogging. (iv) The order quantities of water resources under the scenario with partial backlogging are lower than those under the scenario without backlogging. (v) The profits of the IBWT supply chain and its members under the scenario with partial backlogging are higher than those under the scenario without backlogging.Table 4
Numerical analysis results of equilibrium decision without backlogging.
iwiodpiodqiodΠDiodΠSod11.294.5219,687,77253,921,33448,147,44422.388.175,492,51127,196,481ΠLSod33.1210.693,525,44622,826,46028,888,46645.4020.332,230,60227,477,799ΠESod56.5922.721,087,11414,960,68219,258,978610.7336.74721,42316,059,575ΠSCodSumNANA32,744,868162,442,332210,589,776Noteziod=0.11,wdo=0.51Table 5
Numerical analysis results of coordination decision without backlogging.
iwiocpiocqiocΠDiocΠSoc10.231.51102,300,66762,521,08687,008,60220.452.7228,539,92431,533,966ΠLSoc30.603.5618,318,75226,466,98352,205,16140.616.7811,590,54931,860,151ΠESoc51.257.575,648,81117,346,71634,803,44162.0812.253,748,62218,620,868ΠSCocSumNANA170,147,325188,349,770275,358,372Notezioc=0.11,wco=0.20,ϕco=0.67Table 6
Numerical analysis results of equilibrium decision with partial backlogging.
iwibdpibdqibdΠDibdΠSbd11.294.4219,598,80255,086,31849,317,31222.387.995,467,69027,784,068ΠLSbd33.1210.453,509,51423,319,63129,590,38745.4019.882,220,52228,071,464ΠESbd56.5922.211,082,20115,283,91119,726,925610.7335.93718,16316,406,546ΠSCbdSumNANA32,596,892165,951,937215,269,249Notezibd=0.10,wdb=0.53Table 7
Numerical analysis results of coordination decision with partial backlogging.
iwibcpibcqibcΠDibcΠSbc10.231.47101,838,36263,871,86989,018,07520.452.6628,410,95032,215,265ΠLSbc30.603.4818,235,96927,038,80953,410,84540.616.6311,538,17132,548,497ΠESbc51.257.405,623,28417,721,49635,607,23062.0811.983,731,68219,023,177ΠSCbcSumNANA169,378,417192,419,112281,437,186Notezibc=0.10,wcb=0.20,ϕcb=0.67


## 5.2. Sensitivity Analysis

Since, in the previous analysis, coordination decision with partial backlogging is found to be superior to the other decisions without/with partial backlogging, the sensitivity analysis will focus on how the changes in eight key parameters (including the mainline transfer cost, the branch-line transfer cost, the holding cost coefficient, the shortage cost coefficient, the backlogging ratio, the price elasticity of the expected water demand, the water delivery loss rate, and the bargaining power of theith water intake of the IBWT supplier) impact the profits under coordination decision with partial backlogging. To capture the impact of the change in key parameters, we only select the parameters from the 1st distributor and the 1st water intake to conduct sensitivity analysis, including the mainline transfer cost and the branch-line transfer cost. Figure 2 (including eight subgraphs) shows the impact of eight key parameters change on the profit of the IBWT supply chain with partial backlogging. The findings from the sensitivity analysis results are summarized as follows: (1) the IBWT supply chain profit decreases as the mainline transfer cost increases; (2) the IBWT supply chain profit decreases as the branch-line transfer cost increases; (3) the IBWT supply chain profit decreases as the holding cost coefficient increases; (4) the IBWT supply chain profit decreases as the shortage cost coefficient increases; (5) the IBWT supply chain profit increases as the backlogging ratio increases; (6) the IBWT supply chain profit decreases as the price elasticity of the expected water demand increases; (7) the IBWT supply chain profit decreases as the water delivery loss increases; (8) the IBWT supplier’s profit increases and the distributors’ profits decrease as the bargaining power of the ith water intake of the IBWT supplier increases.Figure 2
The impact of key parameters’ change on the profit of the IBWT supply 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.
(a)(b)(c)(d)(e)(f)(g)(h)


## 6. Managerial Insights and Practical Implications

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


### 6.1. Managerial Insights

Based on the foregoing discussions, the managerial insights can be derived and summarized as follows.First, be it under the scenario without backlogging or under the scenario with partial backlogging, the IBWT supply chain and its members would make lower retail prices of water resources and order more water resources under the coordination strategy than those under the equilibrium strategy and could gain more profits under the coordination strategy than those under the equilibrium strategy. Hence, the coordination strategy via a revenue and cost sharing contract is beneficial to improving the operational performance for all the stakeholders and coordinate the IBWT supply chain under JPIM.Second, be it under equilibrium or coordination strategies, the IBWT supply chain and its members would order less water resources under the scenario with partial backlogging than those under the scenario without backlogging and could gain more profits under the scenario with partial backlogging than those under the scenario without backlogging. Hence, partial backlogging of water demand is beneficial to operational performance improvement for all the stakeholders of the IBWT supply chain under JPIM. Furthermore, the higher the backlogging ratio is, the more profits the IBWT supply chain and its members could gain. Thus, increasing the backlogging ratio is beneficial to operational performance improvement for all the stakeholders of the IBWT supply chain under JPIM.Finally, be it under the scenario without backlogging or under the scenario with partial backlogging, reducing the water delivery loss rate and operational costs (including mainline transfer cost, branch-line transfer cost, holding cost, and shortage cost) is beneficial to operational performance improvement for all the stakeholders of the IBWT supply chain under JPIM. Furthermore, a lower price elasticity of the expected water demand is beneficial to operational performance improvement for all the stakeholders of the IBWT supply chain under JPIM.In brief, the coordination strategy using a revenue and cost sharing contract with partial backlogging outperforms all the other scenarios/strategies and is the best strategy for improving the operational performance of all the stakeholders in the IBWT supply chain under JPIM.


### 6.2. Practical Implications

From the practical perspective of the IBWT project operational management, the practical implications can be derived and summarized for the SNWD project as follows.First, a fixed water price mechanism is implemented in the eastern and middle routes of the SNWD project currently, which cannot directly reflect the linkage effect between water supply and demand and water price. This rigid existing water price mechanism could not effectively exert the regulatory role of the market mechanism and coordinate the interests of all operating entities involved in the operational management of the SNWD project. Thus, it is suggested that some of the water pricing power may be transferred by the government to the main operating entities of the projects to give full play to the regulating role of the market mechanism. In this situation, the water prices for each water intake and water market in the IBWT project can be adjusted freely according to the relationship between the water supply and demand, which is a flexible water price mechanism. On this basis, a joint pricing and inventory management (JPIM) mode could flexibly reflect the linkage effect between water supply and demand and water price and regulate water supply and demand through the market mechanism and thus is recommended to be considered and implemented in the operational management of the SNWD project.Second, since the asset rights of the eastern route of the SNWD project belong to different entities, each interest entity will tend to pursue its own interests’ maximization and will act in its own ways and lack of collaborative operation, which will eventually lead to the reduction in operation efficiency. Under the JPIM mode, a coordination strategy via the revenue and cost sharing contract with partial backlogging could effectively achieve Pareto improvement of all the stakeholders’ interests and thus is recommended to be adopted to coordinate all the stakeholders in the eastern route of the SNWD project. The detailed action sequence of this coordination strategy is as follows: the local and the external suppliers first bargain over the wholesale price of water resources within the IBWT horizontal supply chain to achieve cooperative operations; next, the IBWT supplier provides the distributors a revenue sharing contract in which the IBWT supplier charges a lower usage price to the water distributors; if the distributors accept the contract, they will place orders to the IBWT supplier and decide the retail price of water resources and the stock factor of water resources; then, the unmet water demands of each market are partially backlogged and satisfied; finally, all the distributors’ net revenues will be shared to the IBWT supplier.Finally, the order quantity of water resources mismatching with the random water demand would induce unnecessary losses (including holding cost or shortage cost) in the operational management of the IBWT project. Under the JPIM mode, a partial backlogging strategy could effectively reduce these unnecessary losses and thus is recommended to be adopted in the operational management of the eastern route of the SNWD project. The decision makers of the project should make a lot of effort to enhance their backlogging abilities and increase the backlogging ratio to improve the operational performance of all the stakeholders. Furthermore, water delivery loss and water transfer costs have important impacts on the optimal operational decisions/performance of the project; the decision makers of the project should make a lot of effort to reduce water delivery loss and water transfer costs in the operational management of the project.In sum, a joint pricing and inventory management (JPIM) mode based on the flexible water price mechanism is recommended to be implemented, and a coordination strategy via the revenue and cost sharing contract with partial backlogging is also recommended to be adopted to improve operational efficiency for the eastern route of the SNWD project.


## 6.1. Managerial Insights

Based on the foregoing discussions, the managerial insights can be derived and summarized as follows.First, be it under the scenario without backlogging or under the scenario with partial backlogging, the IBWT supply chain and its members would make lower retail prices of water resources and order more water resources under the coordination strategy than those under the equilibrium strategy and could gain more profits under the coordination strategy than those under the equilibrium strategy. Hence, the coordination strategy via a revenue and cost sharing contract is beneficial to improving the operational performance for all the stakeholders and coordinate the IBWT supply chain under JPIM.Second, be it under equilibrium or coordination strategies, the IBWT supply chain and its members would order less water resources under the scenario with partial backlogging than those under the scenario without backlogging and could gain more profits under the scenario with partial backlogging than those under the scenario without backlogging. Hence, partial backlogging of water demand is beneficial to operational performance improvement for all the stakeholders of the IBWT supply chain under JPIM. Furthermore, the higher the backlogging ratio is, the more profits the IBWT supply chain and its members could gain. Thus, increasing the backlogging ratio is beneficial to operational performance improvement for all the stakeholders of the IBWT supply chain under JPIM.Finally, be it under the scenario without backlogging or under the scenario with partial backlogging, reducing the water delivery loss rate and operational costs (including mainline transfer cost, branch-line transfer cost, holding cost, and shortage cost) is beneficial to operational performance improvement for all the stakeholders of the IBWT supply chain under JPIM. Furthermore, a lower price elasticity of the expected water demand is beneficial to operational performance improvement for all the stakeholders of the IBWT supply chain under JPIM.In brief, the coordination strategy using a revenue and cost sharing contract with partial backlogging outperforms all the other scenarios/strategies and is the best strategy for improving the operational performance of all the stakeholders in the IBWT supply chain under JPIM.


## 6.2. Practical Implications

From the practical perspective of the IBWT project operational management, the practical implications can be derived and summarized for the SNWD project as follows.First, a fixed water price mechanism is implemented in the eastern and middle routes of the SNWD project currently, which cannot directly reflect the linkage effect between water supply and demand and water price. This rigid existing water price mechanism could not effectively exert the regulatory role of the market mechanism and coordinate the interests of all operating entities involved in the operational management of the SNWD project. Thus, it is suggested that some of the water pricing power may be transferred by the government to the main operating entities of the projects to give full play to the regulating role of the market mechanism. In this situation, the water prices for each water intake and water market in the IBWT project can be adjusted freely according to the relationship between the water supply and demand, which is a flexible water price mechanism. On this basis, a joint pricing and inventory management (JPIM) mode could flexibly reflect the linkage effect between water supply and demand and water price and regulate water supply and demand through the market mechanism and thus is recommended to be considered and implemented in the operational management of the SNWD project.Second, since the asset rights of the eastern route of the SNWD project belong to different entities, each interest entity will tend to pursue its own interests’ maximization and will act in its own ways and lack of collaborative operation, which will eventually lead to the reduction in operation efficiency. Under the JPIM mode, a coordination strategy via the revenue and cost sharing contract with partial backlogging could effectively achieve Pareto improvement of all the stakeholders’ interests and thus is recommended to be adopted to coordinate all the stakeholders in the eastern route of the SNWD project. The detailed action sequence of this coordination strategy is as follows: the local and the external suppliers first bargain over the wholesale price of water resources within the IBWT horizontal supply chain to achieve cooperative operations; next, the IBWT supplier provides the distributors a revenue sharing contract in which the IBWT supplier charges a lower usage price to the water distributors; if the distributors accept the contract, they will place orders to the IBWT supplier and decide the retail price of water resources and the stock factor of water resources; then, the unmet water demands of each market are partially backlogged and satisfied; finally, all the distributors’ net revenues will be shared to the IBWT supplier.Finally, the order quantity of water resources mismatching with the random water demand would induce unnecessary losses (including holding cost or shortage cost) in the operational management of the IBWT project. Under the JPIM mode, a partial backlogging strategy could effectively reduce these unnecessary losses and thus is recommended to be adopted in the operational management of the eastern route of the SNWD project. The decision makers of the project should make a lot of effort to enhance their backlogging abilities and increase the backlogging ratio to improve the operational performance of all the stakeholders. Furthermore, water delivery loss and water transfer costs have important impacts on the optimal operational decisions/performance of the project; the decision makers of the project should make a lot of effort to reduce water delivery loss and water transfer costs in the operational management of the project.In sum, a joint pricing and inventory management (JPIM) mode based on the flexible water price mechanism is recommended to be implemented, and a coordination strategy via the revenue and cost sharing contract with partial backlogging is also recommended to be adopted to improve operational efficiency for the eastern route of the SNWD project.


## 7. Conclusions

From a perspective of supply chain management, this paper tries to explore the issues of joint pricing-inventory management decisions and operational strategies for the IBWT project considering water delivery loss and partial backlogging. The equilibrium and coordination decision models without/with partial backlogging for the IBWT supply chain considering water delivery loss under joint pricing and inventory management (JPIM) are developed, analyzed, and compared through a game-theoretical approach, and the corresponding numerical and sensitivity analyses for all models are implemented and compared; finally, the managerial insights and practical implementations are summarized in this paper. The research results show that (1) a revenue and cost sharing contract could effectively coordinate the IBWT supply chain and improve the operational performance of the IBWT supply chain under JPIM; (2) the partial backlogging strategy of water demand could effectively improve the operational performance of the IBWT supply chain under JPIM; (3) coordination strategy with partial backlogging is the best strategy for improving the operational performance of the IBWT supply chain under JPIM; (4) reducing water delivery loss rate and operational costs (including mainline transfer cost, branch-line transfer cost, holding cost, and shortage cost) are beneficial to improving operational performance of the IBWT supply chain under JPIM; (5) increasing the backlogging ratio is beneficial to improving operational performance of the IBWT supply chain under JPIM; (6) a lower price elasticity of the expected water demand is beneficial to improving operational performance of the IBWT supply chain under JPIM.In terms of theoretical contribution, based on the game-theoretical approach, the equilibrium and coordination decision models with partial backlogging under JPIM considering water delivery loss are developed, analyzed, and compared for the IBWT supply chain, respectively, which have enriched the theories, methodologies, and applications in the optimal operation management of IBWT projects. With regard to practical contribution, the analytical and numerical results provide the IBWT stakeholders a better decision support to make better operational decisions and strategies.



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*Source: 3954084-2020-09-07.xml*