## Complexity, Nonlinear Evolution, Computational Experiments, Agent-Based Modeling and Big Data Modeling for Complex Social Systems

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Xue Chen, Zhisong Chen, "Joint Pricing and Inventory Management of Interbasin Water Transfer Supply Chain", *Complexity*, vol. 2020, Article ID 3954084, 16 pages, 2020. https://doi.org/10.1155/2020/3954084

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

**Academic Editor:**Shouwei Li

#### 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.

#### 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 Section 2; 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 Figure 1). 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 *i*^{th} water intake are indexed by . It is assumed that there are distributors supplied by the local supplier and distributors supplied by the external supplier.

On this basis, the parameters used in the models are defined as follows: = the water transfer cost from the *i*^{th} water intake to the *i*^{th} distributor = the water transfer cost from the (*k* − 1)^{th} water intake to the *k*^{th} water intake within the horizontal supply chain, = the water delivery loss rate from the (*k* − 1)^{th} water intake to the *k*^{th} water intake within the horizontal supply chain, and , = the fixed cost for the *i*^{th} water intake of the IBWT supplier = the fixed cost for the local supplier, = the fixed cost for the external supplier, = the fixed cost for the IBWT supplier, = the wholesale price of water resources transferred from the local supplier to the external supplier = the wholesale price of water resources transferred from the IBWT supplier to the *i*^{th} distributor = the retail price of water resources sold from the *i*^{th} distributor to the consumers in his service region = the holding cost coefficient, and = the unit cost of holding water inventory for the *i*^{th} distributor, and = the shortage cost coefficient, and = the shortage cost of unmet water demand for the *i*^{th} distributor, and = the original pumping quantity from the water source to the *i*^{th} water intake = the order quantity of the *i*^{th} water intake = the bargaining powers of the local supplier, and = the bargaining powers of the *i*^{th} water intake of the IBWT supplier, and

As 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 the *i*^{th} distributor is , and . The relationship between the water demand of the *i*^{th} water intake and the original pumping quantity is , and the total transfer cost of the original pumping quantity is ; hereinto, . Therefore, the total transfer cost of the water demand (order quantity) of the *i*^{th} water intake is . Defining , then . Following Howe and Linaweaver [23, 24], Petruzzi and Dada [25], and Wang et al. [26], the water demand for the *i*^{th} distributor is , and . Hereinto, , where is the potential maximum water demand quantity and is the price elasticity of the expected demand. is a random disturbance defined in the range with . The cumulative distribution function (CDF) and the probability density function (PDF) of are and , and the mean value and the standard deviation of are and . Following Petruzzi and Dada [25], Wang et al. [27], and Wang [28], is defined as the “water stock factor” for the *i*^{th} distributor; thus, the order quantity function of water resources for the *i*^{th} water intake is . The distribution of satisfies the IGFR (increasing generalized failure rate) condition: , where , 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 the *i*^{th} distributor, the *i*^{th} water intake of the IBWT supplier, and the *i*^{th} water intake of the IBWT supply chain with partial backlogging can be formulated as follows:

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:

#### 4. Game-Theoretical Decision Models

Based on the modeling notations and assumptions in Section 3, 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 the *i*^{th} distributor is . 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

Solving this two-stage Stackelberg and Nash bargaining problem, we can get the equilibrium usage price in the *i*^{th} water intake, the equilibrium retail price and the equilibrium stock factor for the *i*^{th} water distributor, the equilibrium ordering quantity for the *i*^{th} water distributor, and the bargaining wholesale price . Furthermore, the profits of the local supplier, the external supplier, the IBWT supplier, the *i*^{th} water distributor, and the IBWT supply chain can be calculated as , , , , and (see Table 1 for the detailed analytical results, and their derivations can be seen in sec10Supplementary Materials (available here)).

###### 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 price to the water distributors; if the distributors accept the contract, they will place orders with quantity to the IBWT supplier and decide the retail price of water resources and the stock factor of water resources ; finally, they will share a proportion of their net revenues to the IBWT supplier, where is the revenue keeping rate of the water distributors, . The revenue shared by the *i*^{th} distributor to the IBWT supplier is . Thus, the profit functions of the *i*^{th} distributor and the IBWT supplier are as follows: and .

The two-stage coordination and Nash bargaining game model for the IBWT supply chain without backlogging can be formulated as

Solving this two-stage coordination and Nash bargaining problem, we can obtain the equilibrium usage price in the *i*^{th} water intake, the equilibrium retail price and the equilibrium stock factor for the *i*^{th} water distributor, the equilibrium ordering quantity for the *i*^{th} water distributor, and the bargaining wholesale price . Furthermore, the profits of the local supplier, the external supplier, the IBWT supplier, the *i*^{th} water distributor, and the IBWT supply chain can also be computed as , , , , and (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 the *i*^{th} distributor is . 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

Solving this two-stage Stackelberg and Nash bargaining problem, we can obtain the equilibrium usage price in the *i*^{th} water intake, the equilibrium retail price and the equilibrium stock factor for the *i*^{th} water distributor, the equilibrium ordering quantity for the *i*^{th} water distributor, and the bargaining wholesale price . Furthermore, the profits of the local supplier, the external supplier, the IBWT supplier, the *i*^{th} water distributor, and the IBWT supply chain can be calculated as , , , , and (see Table 2 for the detailed analytical results, and their derivations can be seen in Supplementary Materials).