# Consumption Coupons, Consumption Probability and Inventory Optimization: An Improved Minimum-Cost Maximum-Flow Approach

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## Abstract

**:**

## 1. Introduction

## 2. Literature Review

## 3. Materials and Methods

#### 3.1. The Improved Idea of a Minimum-Cost and Maximum-Flow Model, in Which the Capacity Value on Each Arc Would Vary with Its Own Probability Value

_{s}) and one single sink node (V

_{t}), each arc (v

_{i}, v

_{j}) belonged to the capacity set (C), and the cost per unit of flow on each arc b(v

_{i}, v

_{j}) was greater than or equal to 0. Under the premise of finding a maximum flow f in this network D and minimizing $b\left(f\right)=\sum _{\left({v}_{i},{v}_{j}\right)\u03f5A}{b}_{ij}{f}_{ij}$, the objective was to keep the sum of C over each arc in this directed network to be minimum.

_{1}was the initial probability corresponding to the capacity c

_{1}on an arc. As the probability value increased from P

_{1}to P

_{2}, …, P

_{n}, capacity value also synchronously increased from c

_{1}to c

_{2}, …, c

_{n}. P

_{n}was the maximum probability, and c

_{n}represented the maximum capacity value for the maximum probability. The capacity value under the initial probability P

_{1}was chosen to generate the directed network under this initial probability. Based on the cost unit of flow (b

_{ij}), a shortest path from V

_{s}to V

_{t}was calculated using Dijkstra’s method to find the corresponding augmented chain. Taking the initial flow f on this augmented chain as 0, the minimum value of the difference between the capacity value and the flow value of each arc on this augmented chain was used as the adjustment value to adjust the flow f. Next, the weighted digraph was constructed. We recalculated the shortest circuit, found out the corresponding augmented chain, and then adjusted the capacity value according to the probability value on each arc. We repeated the above steps until you could not find a shortest path from V

_{s}to V

_{t}, then ended the loop. At this point, the sum of capacity c on all arcs in the directed network was the minimum, and the maximum flow and minimum cost were satisfied.

#### 3.2. Calculation Steps to the Model

_{ij}were selected by the probability values, and the directed network was constructed by combining the cost per unit of flow on each arc b(v

_{i},v

_{j}).

_{s}to V

_{t}was calculated based on the cost per unit of flow, b(v

_{i},v

_{j}), on each arc, and the corresponding augmented chain was determined.

_{ij}on that augmented chain could be assumed to be 0. The minimum of all capacity values on that chain was used as the adjustment value θ for flow f

_{ij}, and then the flow value on each arc plus θ was used as the new flow value f

_{ij}. If there was already a flow value f

_{ij}on an arc on the augmented chain, the minimum value of the difference between the capacity value and the flow value on each arc in the incremental chain was used as the adjustment value θ to adjust the flow f

_{ij}when calculating the adjustment value, and the flow value on each arc plus θ was used as the new flow value f

_{ij}.

_{ij}by that adjacent probability value, replacing the capacity value on the existing arc.

_{i}, v

_{j}) and (v

_{j}, v

_{i}) that were inverse to each other, and the weights were recorded as ω

_{ij}and ω

_{ji}, respectively. The rules were as follows:

_{s}. The sum of the capacity values on all arcs was then calculated as the minimum total capacity on that network.

#### 3.3. Feasibility Analysis of an Improved Model for the Application of Consumption Coupons, Probabilistic Consumption and Inventory Optimization

_{s}could be viewed as the government, node V

_{c}as the consumer, and the intermediate nodes as the firms that satisfy the different needs of the consumer. It was assumed that the government-issued consumer coupons did not restrict consumers from purchasing any goods. It was only stipulated that only one consumption coupon could be used on one commodity.

_{s}to node v

_{t}. In the specific calculation, the connection between node v

_{t}and node v

_{c}could be deleted, and the improved minimum-cost maximum-flow model could be directly used for calculation. According to the above analysis, it could be seen that the improved minimum-cost maximum-flow model could better complete the research and had a strong practical guidance.

## 4. Results and Discussion

#### 4.1. Simulation Problem and Data Descriptions

_{s}) in a region was ready to issue a number of consumption coupons to stimulate consumer demand, and consumers needed to satisfy four demands during that incentive period, which were borne by four different enterprises (four nodes, v

_{1}, v

_{2}, v

_{3}and v

_{t}, as shown in Figure 4). When purchasing goods, consumers could only use one consumption coupon per product, and the consumption coupon would not change the original price of the product. Enterprises needed to prepare the maximum inventory according to the maximum number of coupons issued to cope with the consumer’s purchase. There were various strategies for consumers to use consumer coupons to purchase goods;; for example, the arc (v

_{2}, v

_{1}) indicated that the consumer used the consumer coupon to purchase goods from the node enterprise v

_{1}after passing through node enterprise v

_{2}. The weight group value (2, 50) on the arc, where 2 indicated that the unit price of commodities purchased by consumers from node v

_{1}with coupons is 2; 50 represented the maximum amount of inventory prepared by node enterprise v

_{1}to cope with the consumption demand coming from node v

_{2}is 50.

#### 4.2. Simulation Scenario Calculations

#### 4.2.1. Scenario 1: Calculation Using the Traditional Minimum-Cost Maximum-Flow Model

_{s}to v

_{t}, and the augmented chain u = (v

_{s}, v

_{2}, v

_{1}, v

_{t}) was obtained. In the second step, the flow value on the augmented chain u was adjusted, and the adjusted flow figure I shown in Figure 5 was obtained.

#### 4.2.2. Scenario 2: Improved Model Calculation with Probability Change from 50% to 100%

_{s}to v

_{t}was solved to obtain the augmented chain u = (v

_{s}, v

_{2}, v

_{1}, v

_{t}).

_{2}, v

_{1}) were both equal to 25, the current probability value was 50% and the maximum 100% was not reached. We then went to step 6; otherwise, to step 7.

_{2}, v

_{1}) were equal, so from Table 1, it could be found that the next probability value was 80%, and the corresponding capacity value was 40. We replaced the existing capacity value with 40.

#### 4.2.3. Scenario 3: Improved Model Calculation with Probability Value on Part of the Arc Not Reaching 100%

_{2}, v

_{1}) and arc (v

_{1}, v

_{t}) were taken as 80%.

#### 4.3. Improved Simulation Results

## 5. Conclusions and Policy Recommendations

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

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arc | Probability Value | ||
---|---|---|---|

50% | 80% | 100% | |

(v_{s}, v_{1}) | 50 | 80 | 100 |

(v_{s}, v_{2}) | 40 | 64 | 80 |

(v_{2}, v_{1}) | 25 | 40 | 50 |

(v_{2}, v_{3}) | 50 | 80 | 100 |

(v_{1}, v_{t}) | 35 | 56 | 70 |

(v_{1}, v_{3}) | 10 | 16 | 20 |

(v_{3}, v_{t}) | 20 | 32 | 40 |

arc | Probability Value | ||
---|---|---|---|

50% | 80% | 100% | |

(v_{s}, v_{1}) | 50 | 80 | 100 |

(v_{s}, v_{2}) | 40 | 64 | 80 |

(v_{2}, v_{1}) | 25 | 40 | - |

(v_{2}, v_{3}) | 50 | 80 | 100 |

(v_{1}, v_{t}) | 35 | 56 | - |

(v_{1}, v_{3}) | 10 | 16 | 20 |

(v_{3}, v_{t}) | 20 | 32 | 40 |

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**MDPI and ACS Style**

Wang, S.; Chen, Y.
Consumption Coupons, Consumption Probability and Inventory Optimization: An Improved Minimum-Cost Maximum-Flow Approach. *Sustainability* **2022**, *14*, 7759.
https://doi.org/10.3390/su14137759

**AMA Style**

Wang S, Chen Y.
Consumption Coupons, Consumption Probability and Inventory Optimization: An Improved Minimum-Cost Maximum-Flow Approach. *Sustainability*. 2022; 14(13):7759.
https://doi.org/10.3390/su14137759

**Chicago/Turabian Style**

Wang, Shunlin, and Yifang Chen.
2022. "Consumption Coupons, Consumption Probability and Inventory Optimization: An Improved Minimum-Cost Maximum-Flow Approach" *Sustainability* 14, no. 13: 7759.
https://doi.org/10.3390/su14137759