An optimal freshness-keeping effort model for fresh produce with constraints of special funds

Bing Zhou; Yufeng Li; Xin Fang

doi:10.3934/jimo.2021215

Article Contents

2023, Volume 19, Issue 2: 984-1000. 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
^* Corresponding author: Yufeng Li

Received: July 31, 2021

Revised: October 31, 2021

Early access: December 2021

Published: February 2023

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

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Abstract

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.

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Mathematics Subject Classification: Primary: 91B06; Secondary: 65K05.

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Figure 1. The freshness-keeping sequence process of enterprise $ F $

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Figure 2. The change trend of the optimal value under different allocation ratios $ \lambda $ when initial self-owned funds $ B = 25 $

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Figure 3. The change trend of the optimal decision of the enterprise with initial self-owned funds $ B $, $ i = [1, 2, 3] $

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Table 1. 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 $

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Table 2. 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

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Table 3. 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

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Table 4. 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

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