Since March 2022, a new round of COVID-19 has broken out in China [1]. Consumption continues to be sluggish, and downward pressure on the economy is increasing. Provinces and cities in China have launched their own economic incentive policies in the hopes of accelerating the expansion of consumption and expanding domestic demand. Consumption coupons as an economic stimulus have been repeatedly reported in the press. For example, in April 2022, Shenzhen city issued 500 million yuan of consumption coupons to its consumers, and Ningbo issued 300 million yuan to consumers [2]. Despite different views [3], the effect in practice ^[4][5][4,5] has gradually gained widespread attention.

Consumers will be affected by various factors when consuming. The issuance of consumption coupons by the government to consumers does not mean that they will meet 100 percent of their consumption needs with those coupons [6]. Rational consumers will selectively use consumption coupons based on their personal preferences, resulting in a consumers’ probability consumption scenario. The probabilistic selling [7] strategies used by merchants also strengthen the trend of probabilistic consumer spending. This also urges the government to fully consider the probability consumption of consumers when using coupons to incentivize consumer demand.

In the era of commodity economy, the level of inventory management not only affects the cost of enterprises, but also plays a key role in stabilizing and coordinating consumer demand [8]. The dynamic changes in consumer demand preferences determine that enterprises must always control their inventory levels, neither out of stock, nor too much inventory, resulting in backlogs. Consumer behavior using coupons will inevitably challenge the traditional level of inventory management.

Therefore, it is of great significance to analyze the probabilistic consumption of consumers and the inventory optimization level of merchants in the process of using consumption coupons, which can provide some reference for the subsequent scientific development of incentive policies of consumption coupons.

2. Consumption Coupons, Consumption Probability and Inventory Optimization

In the face of the global recession and the impact of the COVID-19 pandemic on the economy, the economies of the world are facing enormous challenges. The economic contraction [9] has been triggered by the pandemic, and social and economic losses continue to increase in many countries [10]. The effect of investment pulling is not obvious in the short term. The pandemic has reduced investments and the fiscal revenues of governments, thus increasing uncertainty. A demand-side shock was triggered by global and national restrictions to limit the spread of COVID-19, such as lockdown measures and travel bans [11]. The revitalization of consumption has become an important tool to stabilize the economic development. For consumers, coupons mean the increase of their temporary incomes. Such fiscal policy would increase the demand for consumption and simulate aggregate demand ^[12][13][12,13]. Stimulating the economy by issuing consumption coupons and activating market demand can better alleviate market failures, promote economic development in the short term, and help economic recovery [14]. Consumption coupons are a temporary incentive for the government to stimulate consumption [15], and are also a special coupon, which can be used exclusively by dividing the consumption categories of designed products [16]. The government uses consumption coupons to boost consumer confidence [17], which, in turn, will increase consumers’ willingness to buy more goods and thus stimulate market dynamics, thus stimulating market vitality. Experts and scholars have various views on the role of consumption coupons, such as that they can effectively improve consumption, promote economic growth and increase employment [18]. Coupons can enhance the experience and outcomes for participants, benefiting more low-income households [19]. Consumption coupons are effective in stimulating short-term consumption levels. It will reduce the welfare level of society as a whole if the savings plan is not optimal [20]. The substitution effect of consumption coupons makes them difficult to pull effective demand for a long time [21]. The multiplier brought by consumption coupons in China during the 2020 epidemic is around 5–25, which has a significant effect on the GDP in the short term [22]. Consumption coupons are conducive to activating the frozen domestic demand [23] and market dynamics under the epidemic and make up for the short-term gap in overseas markets due to the global spread of the epidemic. The difference in consumer purchasing behavior mainly comes from the difference in shopping decisions, and there is uncertainty [24] due to the influence of various factors in the process of purchasing behaviors [25], which is generally expressed by probability. Uncertainties faced by consumers in their purchase decisions include preference uncertainty and product performance uncertainty [26]. People may be non-indifferent towards the timing of the resolution of consumption uncertainty [27]. Probability, as an expression of purchase intention [28], can produce more accurate consumption predictions and may elicit different responses than standard purchase intention scales. Businesses must pay attention to consumer dynamics [29] and make appropriate marketing decisions based on an accurate understanding of consumers. Consumer dynamics are defined as temporal changes in consumer attitudes and behaviors. Today, probabilistic selling ^[30][31][30,31] is increasingly becoming a sales strategy for businesses and growing in popularity. Probabilistic selling could generally increase a firm’s expected profit, and the firm will increase the price and the order quantity for the component product [32]. From a business perspective, adequate and controllable inventory can help companies better gain profits while reducing total costs [33]. Stock-outs will lead directly to lost sales, reduced profits, and the potential loss of customers [34]. Maintenance of inventories is a significant concern for any business organization due to decay or deterioration over time [35]. As one of the core problems of network optimization, the minimum-cost maximum-flow problem has a wide range of applications in reality, and the common special cases include the shortest path problem, the maximum-flow problem, and the assignment problem. A prognostic decision-making strategy is proposed to solve in real time the electric vehicle dynamic stochastic shortest path problem, aiming at the simultaneous utilization of historical and real-time traffic data [36]. A network is called uncertain if the arc capacities of the network are uncertain variables. Uncertainty theory is an efficient tool to deal with nondeterministic information, especially expert data and subjective estimation [37]. Scholars have conducted multidimensional studies based on different application scenarios ^[38][39][40][38,39,40], combined with the minimum-cost maximum-flow network model, and have achieved fruitful results.