# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2021215
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## An optimal freshness-keeping effort model for fresh produce with constraints of special funds

 1 Research Center for Economy of Upper Reaches of the Yangtse River, Chongqing Technology and Business University, Chongqing 400067, China 2 School of Accounting, Chongqing Technology and Business University, Chongqing 400067, China 3 School of Management Science and Engineering, Chongqing Technology, and Business University, Chongqing 400067, China

* Corresponding author: Yufeng Li

Received  August 2021 Revised  November 2021 Early access December 2021

Fund Project: This research was funded by the Chongqing social science foundation, grant number 2019WT42 and the Chongqing major decision - making consulting research project, grant number 2019WT02

The quality deterioration in the post-production process of fresh products is very serious, and the life-cycle freshness-keeping technology investment is an effective way to reduce the deterioration. Because the investment cost is high in practice, enterprises need to allocate special funds for each stage to maximize their marginal revenue. In this paper, we use freshness to characterize the quality level of fresh products and investigate a maximize marginal revenue problem where a firm assigns special funds for the freshness-keeping effort with each post-production process. An optimal freshness-keeping model with the constraints of special funds is discussed. The investigation shows that both the optimal freshness-keeping effort and the closed-form optimal solutions of enterprises exist uniquely. A reasonable freshness-keeping investment in different post-production processes can improve the performance of enterprises with limited fund constraints. We then simulate the effect rules of funds constraint on these solutions based on numerical analysis and give some management insights.

Citation: Bing Zhou, Yufeng Li, Xin Fang. An optimal freshness-keeping effort model for fresh produce with constraints of special funds. Journal of Industrial and Management Optimization, doi: 10.3934/jimo.2021215
##### References:

show all references

##### References:
The freshness-keeping sequence process of enterprise $F$
The change trend of the optimal value under different allocation ratios $\lambda$ when initial self-owned funds $B = 25$
The change trend of the optimal decision of the enterprise with initial self-owned funds $B$, $i = [1, 2, 3]$
List of symbols
 Footmark $i$ $i \in\{S, R\}$, Where $i=S$ represents lead time, $i=R$ represents shelf-life Variables $Q$ market demand $e_i$ freshness-keeping input level Parameters $\lambda$ allocation ratio of lead time for initial funds $B$ $\beta$ sensitivity coefficient of freshness $\eta$ natural attenuation coefficient of fresh produce $\phi$ potential market size of fresh produce $t_S$ subscribe lead time $B$ initial funds of enterprise $F$ $c$ subscription cost of unit fresh produce $p$ market retail price of unit fresh produce $B_i$ freshness-keeping input cost at stage $i$ $k_i$ sensitivity coefficient of freshness-keeping input level to freshness $h_i$ sensitivity coefficient of freshness-keeping input level to freshness-keeping cost at stage $i$ $t_R$ shelf-life of fresh produce $\Pi_{F}$ profit function of enterprise $F$
 Footmark $i$ $i \in\{S, R\}$, Where $i=S$ represents lead time, $i=R$ represents shelf-life Variables $Q$ market demand $e_i$ freshness-keeping input level Parameters $\lambda$ allocation ratio of lead time for initial funds $B$ $\beta$ sensitivity coefficient of freshness $\eta$ natural attenuation coefficient of fresh produce $\phi$ potential market size of fresh produce $t_S$ subscribe lead time $B$ initial funds of enterprise $F$ $c$ subscription cost of unit fresh produce $p$ market retail price of unit fresh produce $B_i$ freshness-keeping input cost at stage $i$ $k_i$ sensitivity coefficient of freshness-keeping input level to freshness $h_i$ sensitivity coefficient of freshness-keeping input level to freshness-keeping cost at stage $i$ $t_R$ shelf-life of fresh produce $\Pi_{F}$ profit function of enterprise $F$
Optimal freshness-keeping input decision of the enterprise without funds constraints
 $e_S^*$ $e_R^*$ $B_S^*$ $B_R^*$ $Q^*$ $\Pi_{E}^{*}$ 0.6048 0.8384 9.1446 21.0897 78.7676 127.3009
 $e_S^*$ $e_R^*$ $B_S^*$ $B_R^*$ $Q^*$ $\Pi_{E}^{*}$ 0.6048 0.8384 9.1446 21.0897 78.7676 127.3009
The optimal value under different allocation ratios $\lambda$ when initial self-owned funds $B = 20$, $j = [1, 2, 3]$
 $\lambda$ $e_{Sj}^*$ $e_{Rj}^*$ $Q^*$ $\Pi_{F}^{*}$ 0.1 0.3162 0.8384 74.4044 125.2191 0.2 0.4472 0.8165 75.8328 126.6656 0.3 0.5477 0.7638 76.0261 127.0521 0.4 0.6048 0.7071 75.4640 126.7834 0.5 0.6048 0.6455 73.9143 126.1841 0.6 0.6048 0.5774 72.2002 125.2558 0.7 0.6048 0.5000 70.2546 123.8646 0.8 0.6048 0.4082 67.9467 121.7489 0.9 0.6048 0.2887 64.9391 118.2335
 $\lambda$ $e_{Sj}^*$ $e_{Rj}^*$ $Q^*$ $\Pi_{F}^{*}$ 0.1 0.3162 0.8384 74.4044 125.2191 0.2 0.4472 0.8165 75.8328 126.6656 0.3 0.5477 0.7638 76.0261 127.0521 0.4 0.6048 0.7071 75.4640 126.7834 0.5 0.6048 0.6455 73.9143 126.1841 0.6 0.6048 0.5774 72.2002 125.2558 0.7 0.6048 0.5000 70.2546 123.8646 0.8 0.6048 0.4082 67.9467 121.7489 0.9 0.6048 0.2887 64.9391 118.2335
The optimal decision of the enterprise when initial self-owned funds $B$ changes, $j = [1, 2, 3]$
 $B$ $e_{Sj}^*$ $e_{Rj}^*$ $Q^*$ $\Pi_{F}^{*}$ 5 0.2454 0.3413 60.8285 116.6570 10 0.3470 0.4827 65.9213 121.8426 15 0.4250 0.5912 69.8292 124.6583 20 0.4907 0.6826 73.1236 126.2473 25 0.5477 0.7638 76.0261 127.0521 30 0.6010 0.8361 78.6502 127.3004
 $B$ $e_{Sj}^*$ $e_{Rj}^*$ $Q^*$ $\Pi_{F}^{*}$ 5 0.2454 0.3413 60.8285 116.6570 10 0.3470 0.4827 65.9213 121.8426 15 0.4250 0.5912 69.8292 124.6583 20 0.4907 0.6826 73.1236 126.2473 25 0.5477 0.7638 76.0261 127.0521 30 0.6010 0.8361 78.6502 127.3004
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