Water retailer managed inventory is a classical and inevitable inventory management mode in present economic society. Stochastic models can more clearly explain demand uncertainty and are closely related to water supply chains. Risk preferences are widely valued in behavioral operation management. Related to the risk preferences in inventory management, the research on risk aversion is dominant, while risk-seeking is insufficient. Based on the model assumptions, the risk-seeking retailer’s optimal decision-making inventory model with stochastic demand in a water supply chain is studied. The risk-seeking retailer’s optimal inventory quantity, optimal inventory cost, supplier profit, retailer profit, and the profit of the entire water supply chain are derived. The validity of the equations is proved. The sensitivity analysis of the risk-seeking retailer’s optimal inventory decision-making is carried out. The risk level effects on the five dimensions, the retail price, wholesale price, unit shortage cost, unit inventory cost, and unit residual value, are displayed through numerical simulation. The optimal inventory quantity and optimal inventory cost of the risk-seeking retailer are obtained.

With the global population growth and economic development, water resources are becoming increasingly scarce. Water is a crucial business resource, but subsistent evidence suggests that the management of water resources is often poorly operated [

In the traditional supply chain, retailers are at the end of the supply chain and directly face customers. On the one hand, the retailer keeps inventory and decides the inventory level to deal with demand uncertainty. On the other hand, the retailers also have to bear the shortage cost due to stockout. It is little motivated for upstream suppliers to reserve buffer stock to meet demand of end-users [

Schematic diagram of RMI mode.

Uncertainty is an essential feature of market. As the internal and external environments change rapidly, the uncertainty increases significantly. Several international organizations made predictions about world GDP and trade for this year and next year (Table

The recently updated forecasts for world Gross Domestic Product (GDP) and trade.

Real GDP (% change) | Trade volume (% change) | Elasticity (ratio) | ||||
---|---|---|---|---|---|---|

2020 | 2021 | 2020 | 2021 | 2020 | 2021 | |

Optimistic scenario | −2.5 | 7.4 | −12.9 | 21.3 | 5.3 | 2.9 |

Pessimistic scenario | −8.8 | 5.9 | −31.9 | 24.0 | 3.6 | 4.1 |

World Economic Prospects of the IMF (April 2020) | −3.0 | 5.8 | −11.0 | 8.4 | 3.6 | 1.4 |

Global Economic Outlook of the World Bank (May 2020) | −5.2 | 4.2 | −13.4 | 5.3 | 2.6 | 1.3 |

Single-hit scenario | −6.0 | 5.2 | −9.5 | 6.0 | 1.6 | 1.1 |

Double-hit scenario | −7.6 | 2.8 | −11.4 | 2.5 | 1.5 | 0.9 |

IMF GDP set at market exchange rate | −4.2 | 5.4 | −11.0 | 8.4 | 2.6 | 1.6 |

World Bank GDP set at purchasing power parity | −4.1 | 4.3 | −13.4 | 5.3 | 3.3 | 1.2 |

Source:

Owing to the background above, this paper focuses on the decision-making optimization of the risk-seeking retailer managed inventory mode under stochastic demand in a water supply chain, which consists of one supplier and one retailer. Based on model assumptions, the profit expressions of supplier, retailer, and the entire water supply chain are firstly provided. Then, sensitivity analyses of related factors are given through analytical equations combined with figures. The conclusions are drawn finally.

Based on the background discussed in the Introduction, our study involves three aspects of literature. The first stream is about the research on water supply chains, the second one is on the retailer managed inventory with stochastic demand, and the third one is on risk preferences in inventory management.

Regarding the first stream, the water footprint theory proposed by professor Hoekstra [

Water supply chain is characterized by high risk and multiple risks. It is obviously affected not only by climate [

Water resources can form a supply chain independently [

Because of the special form of water, it is transported in one direction through prebuilt pipes and is very dependent on the infrastructure.

Water supply chain is characterized by regionalization and this characteristic is also obvious for risks in water supply chains [

Inventory level is one of the important indicators reflecting economic benefits of enterprises. Many scholars have proved that RMI is an inevitable inventory management mode, and, in some cases, it has certain advantages. For example, Li et al. [

Because of uncertainty, the assumption of stochastic demand is more practical and representative for reality than that of deterministic demand [

“Rational man” or “economic man” was assumed in the classical decision theory and caused large differences between theoretical research results and real situations. The preference and expected utility theory proposed by Von Neumann and Morgenstern [

Nevertheless, the research on water supply chains from the perspective of inventory mode and risk-seeking preference is rarely carried out in the present literature and practices. This article happens to be a useful supplement to this field.

This paper studies the inventory decision-making of a two-echelon water supply chain under the RMI mode with stochastic demand. The basic assumptions are as follows:

The water supply chain is composed of one supplier and one retailer, and only the retailer is used as the main decision-making subject to study the effect of risk-seeking preference on its inventory decision. The supplier’s output and inventory decisions are not controlled. Therefore, it is assumed that the supplier’s ending profit is only the difference between sales revenue and production cost without supplier’s inventory cost.

The cost of water transportation is not considered.

The retailer has only one order opportunity during the sales cycle and cannot replenish water.

The wholesale price of water is determined by the supplier. The retail price of water is determined by the retailer without considering the functional relationship between price and demand.

It is assumed that the market is open and there is no upper limit to the market demand. Let demand

Based on the above assumptions, the RMI mode with stochastic demand in the water supply chain is described as follows.

In a sales cycle, the retailer orders

Based on the above problem description, during the sales cycle, when the retailer’s order quantity is

Since demand

That is, the conditional value at risk

In the case of retailer managed inventory under stochastic demand, the objective function of the risk-seeking retailer is

We have defined the distribution function of random profit

When

Let

When

Let

Because of the target profit level

Now we present the relationship between the stochastic demand and profit in the case of retailer managed inventory under stochastic demand [

Schematic diagram of the relationship between stochastic demand and profit under the risk-seeking retailer managed inventory mode in the water supply chain. (a)

From Figure

When

Meanwhile, from

When

We get

In the water supply chain under RMI mode with stochastic demand, the risk-seeking retailer takes CVaR-profit maximization as its decision-making objective and its optimal inventory quantity

When

When

Based on the groundwork mentioned above, the solution process is divided into the two following situations:

When

Solving the first partial derivative of the above equation with respect to order quantity

We have

Therefore, the maximum exists. The optimal order quantity can be obtained by setting the first-order condition equal to 0:

When

Solving the first partial derivative of the above equation with respect to order quantity

The sign symbol of the above equation cannot be judged by the existing conditions, so we use the inverse method to solve it. Assuming that when

Let the first-order condition be 0; combining the equation

In conclusion, we have the following:

When

When

Thus, equations (

In the case of the risk-seeking retailer managed inventory under stochastic demand scenario in the water supply chain, the optimal inventory cost

In the case of the risk-seeking retailer managed inventory under stochastic demand scenario in the water supply chain, the supplier’s profit can be expressed by

In the case of the risk-seeking retailer managed inventory under stochastic demand scenario in the water supply chain, the retailer’s profit can be expressed by

In the case of the risk-seeking retailer managed inventory under stochastic demand scenario in the water supply chain, the profit of the entire supply chain is given by

Then, the sensitivity of the risk-seeking retailer’s optimal inventory strategy is analyzed under RMI mode with stochastic demand in the water supply chain. The effects of relevant factor changes on the retailer’s inventory strategy are discussed.

Since Proposition

In the case of the risk-seeking retailer managed inventory under stochastic demand scenario in the water supply chain, when

Solving the partial derivatives of the optimal inventory quantity

To sum up, the following conclusions are valid:

In other words, Corollary

Corollary

Effect of risk level

Effect of risk level

Effect of risk level

Effect of risk level

Effect of risk level

Effect of retail price

Effect of unit shortage cost

Effect of unit residual value

Effect of wholesale price

Effect of unit inventory cost

In the water supply chain, when

Because the proof process is similar to that of Corollary

Let

Thus,

Because it is not easy to judge the sign symbol on the right side of the equation, we discuss it in the two following cases:

When

When

That is, Corollary

Corollary

Effect of risk level

Effect of risk level

Effect of risk level

Effect of risk level

Effect of risk level

Effect of retail price

Effect of unit shortage cost

Effect of unit residual value

Effect of wholesale price

Effect of unit inventory cost

Water supply chain is characterized by high risk and multiple risks. The decision-maker’s attitude to risk is one of the important factors that affect the decision result. Uncertain demand is the most fundamental uncertainty in supply chains. So, stochastic demand is incorporated into our research hypothesis. Based on the model assumptions, we study the optimal decisions of the risk-seeking retailer under RMI mode considering stochastic demand in a water supply chain. Firstly, we discussed the relationship between stochastic demand and profit. Next, we proved the expressions of the risk-seeking retailer’s optimal inventory quantity, and then the expression of the risk-seeking retailer’s optimal inventory cost, the profit expressions of the supplier, retailer, and the entire supply chain are derived. Finally, through sensitivity analysis and numerical simulation, we explored the impact of risk level on the optimal inventory decisions. We get the following conclusions:

In the water supply chain, the risk-seeking level has no effect on the monotonicity of the optimal inventory quantity and optimal inventory cost in the five dimensions of the retail price, wholesale price, unit shortage cost, unit inventory cost, and unit residual value. But, under different risk levels, the five dimensions generate different effect strengths on the optimal inventory strategy.

With regard to the optimal inventory quantity and optimal inventory cost of the risk-seeking retailer, their monotonicities are positively associated with the retail price, unit shortage cost, and unit residual value, while they are negatively correlated with the wholesale price.

The unit inventory cost has different effects on the optimal inventory quantity and optimal inventory cost. It is negatively correlated with the optimal inventory quantity. Meanwhile, there is a certain threshold for unit inventory cost, which makes the monotonicity of the risk-seeking retailer’s optimal inventory cost change before and after this threshold. We can explain this counterintuitive phenomenon. Before the unit inventory cost is less than the certain threshold, the increase in the risk-seeking retailer’s inventory quantity can still increase the expected profit. At this time, the risk-seeking retailer still increases the inventory quantity, and the inventory cost increases accordingly. When the unit inventory cost exceeds the threshold, the increase in the risk-seeking retailer’s inventory quantity leads to a decrease in the expected profit. At this time, the risk-seeking retailer reduces the inventory quantity, and the corresponding inventory cost also decreases.

From the expressions of the supplier’s optimal profit and the profit of the entire supply chain, we can obtain that the supplier’s optimal profit and the profit of the entire supply chain are positively associated with the retailer’s order quantity. Therefore, the risk-seeking level of the retailer also affects the supplier’s profit and supply chain’s profit. It is important for the supplier to pay attention to the retailer’s risk preference.

We considered only demand as a random variable in our research. There are also price uncertainty and cost uncertainty in practice. It makes sense to study the association of multiple random variables. Some scholars have made attempt methods to describe the relationships between various random variables [

Water is regarded as a crucial business resource. This correlational research is carried out from the perspective of supply chain operations and focuses on the optimal inventory strategy of the risk-seeking retailer under RMI mode considering stochastic demand, which makes the research more practical.

The raw data supporting the conclusions of this article will be made available by the authors, without undue reservation, to any qualified researcher.

The authors declare that there are no conflicts of interest regarding the publication of this article.

This work was supported by the National Natural Social Science Foundation of China (Grant no. 20FJYB031).