# 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:
 [1] M. Bakker, J. Riezebos and R. H. Teunter, Review of inventory systems with deterioration since 2001, European J. Oper. Res., 221 (2012), 275-284.  doi: 10.1016/j.ejor.2012.03.004. [2] S. Bardhan, H. Pal and B. C. Giri, Optimal replenishment policy and preservation technology investment for a non-instantaneous deteriorating item with stock-dependent demand, Operational Research, 19 (2019), 347-368. [3] Z. Bulut, Ü. Gürler and S. Alper, Bundle pricing of inventories with stochastic demand, J. Oper. Res., 197 (2009), 897-911.  doi: 10.1016/j.ejor.2006.09.106. [4] B. B. Cao, Z. P. Fan and T. H. You, Optimal pricing decision in finance-constrained supply chain considering deposits and loans of the bank based on the advance payment, Chinese Journal of Management Science, 28 (2020), 52-64. [5] J. Chen, M. Dong and L. Xu, A perishable product shipment consolidation model considering freshness-keeping effort, Transportation Research Part E: Logistics and Transportation Review, 115 (2018), 56-86.  doi: 10.1016/j.tre.2018.04.009. [6] J. Dong and D. D. Wu, Two-period pricing and quick response with strategic customers, International Journal of Production Economics, 215 (2019), 165-173.  doi: 10.1016/j.ijpe.2017.06.007. [7] T. J. Fan, C. Xu and F. Tao, Dynamic pricing and replenishment policy for fresh produce, Computers & Industrial Engineering, 139 (2020), 106127. [8] Y. Fang, Y. Jiang and X. Han, Bundle pricing decisions for fresh products with quality deterioration, Journal of Food Quality, 2018 (2018), 1595807.  doi: 10.1155/2018/1595807. [9] B. Gkgür and S. Karabat, Dynamic and targeted bundle pricing of two independently valued products, European J. Oper. Res., 279 (2019), 184-198.  doi: 10.1016/j.ejor.2019.05.022. [10] A. Herbon and E. Khmelnitsky, Optimal dynamic pricing and ordering of a perishable product under additive effects of price and time on demand, European J. Oper. Res., 260 (2017), 546-556.  doi: 10.1016/j.ejor.2016.12.033. [11] H. Huang and Z. P. Fan, The purchasing and financing strategies of capital-constrained retailer under different channel power structures, Chinese J. Management Science, 8 (2020), 79-88. [12] H. Huang, Y. He and D. Li, Coordination of pricing, inventory, and production reliability decisions in deteriorating product supply chains, International Journal of Production Research, 56 (2018), 6201-6224.  doi: 10.1080/00207543.2018.1480070. [13] M. Jian, X. Fang and L. Q. Jin, The impact of lead time compression on demand forecasting risk and production cost: A newsvendor model, Transportation Research Part E: Logistics and Transportation Review, 84 (2015), 61-72.  doi: 10.1016/j.tre.2015.10.006. [14] Y. Li, S. Zhang and J. Han, Dynamic pricing and periodic ordering for a stochastic inventory system with deteriorating items, Automatica, 76 (2017), 200-213.  doi: 10.1016/j.automatica.2016.11.003. [15] W. Li, D. M. Hardesty and A. W. Craig, The impact of dynamic bundling on price fairness perceptions, Journal of Retailing & Consumer Services, 40 (2018), 204-212. [16] Y. T. Li and Z. Qiao, Research on the collaborative decision-making method for preservation cost control in fruit and vegetable supply chain, Science & Technology and Economy, 27 (2014), 35-38. [17] C. Liu, W. Chen and Q. Zhou, Modelling dynamic freshness-keeping effort over a finite time horizon in a two-echelon online fresh product supply chain, European J. Oper. Res., 293 (2021), 511-528.  doi: 10.1016/j.ejor.2020.12.035. [18] L. Liu, L. Zhao and X. Ren, Optimal preservation technology investment and pricing policy for fresh food, Computers & Industrial Engineering, 135 (2019), 746-756.  doi: 10.1016/j.cie.2019.06.041. [19] M. L. Liu, B. Dan and S. G. Zhang, Information sharing in an e-tailing supply chain for fresh produce with freshness-keeping effort and value-added service, J. Oper. Res., 290 (2021), 572-584.  doi: 10.1016/j.ejor.2020.08.026. [20] P. Liu and S. Wang, Logistics outsourcing of fresh enterprises considering fresh-keeping efforts based on evolutionary game analysis, IEEE Access, 9 (2021), 25659-25670.  doi: 10.1109/ACCESS.2021.3056699. [21] P. Ma, Y. M. Gong and M. Jin, Quality efforts in medical supply chains considering patient benefits, European J. Oper. Res., 279 (2019), 795-807.  doi: 10.1016/j.ejor.2019.06.030. [22] X. L. Ma, S. Y. Wang, S. M. N. Islam and X. B. Liu, Coordinating a three-echelon fresh agricultural products supply chain considering freshness-keeping effort with asymmetric information, Appl. Math. Model., 67 (2019), 337-356.  doi: 10.1016/j.apm.2018.10.028. [23] X. L. Ma, S. Y. Wang and H. Jin, Coordination and optimization of three-echelon agricultural product supply chain considering freshness-keeping effort and quantity/quality elasticity, Chinese Journal of Management Science, 26 (2018), 175-185. [24] R. Maihami and B. Karimi, Effect of two-echelon trade credit on pricing-inventory policy of non-instantaneous deteriorating products with probabilistic demand and deterioration functions, Ann. Oper. Res., 257 (2017), 237-273.  doi: 10.1007/s10479-016-2195-3. [25] H. Mohammadi, M. Ghazanfari, M. S. Pishvaee and E. Teimoury, Fresh-product supply chain coordination and waste reduction using a revenue-and-preservation-technology-investment-sharing contract: A real-life case study, J. Cleaner Production, 213 (2019), 262-282. [26] I. Moon, Y. J. Jeong and S. Saha, Investment and coordination decisions in a supply chain of fresh agricultural products, International Journal of Operational Research, 3 (2018), 1-25. [27] M. Pattnaik and P. Gahan, Preservation effort effects on retailers and manufacturers in integrated multi-deteriorating item discrete supply chain model, OPSEARCH, 58 (2021), 276-329.  doi: 10.1007/s12597-020-00477-2. [28] C. N. Rapolu and D. H. Kandpal, Joint pricing, advertisement, preservation technology investment and inventory policies for non-instantaneous deteriorating items under trade credit, OPSEARCH, 57 (2020), 274-300. [29] S. Saha, D. Chatterjee and B. Sarkar, The ramification of dynamic investment on the promotion and preservation technology for inventory management through a modified flower pollination algorithm, J. Retailing and Consumer Services, 58 (2021), 102326. [30] M. M. Siddh, G. Soni, R. Jain, M. K. Sharma, et al, Agri-fresh food supply chain quality (AFSCQ): A literature review, Industrial Management & Data Systems, (2017), 117. [31] Z. Song and S. He, Contract coordination of new fresh produce three-layer supply chain, Industrial Management & Data Systems, 119 (2019), 1-23.  doi: 10.1108/IMDS-12-2017-0559. [32] M. Thibaud, H. Chi, W. Zhou and S. Piramuthu, Internet of Things (IoT) in high-risk Environment, Health and Safety (EHS) industries: A comprehensive review, Decision Support Systems, 108 (2018), 79-95. [33] B. Wang and J. Wang, Price and service competition between new and remanufactured products, Math. Probl. Eng., 2015 (2015), 1-18.  doi: 10.1155/2015/325185. [34] G. L. Wang, P. Q. Ding and H. R. Chen, Green fresh product cost sharing contracts considering freshness-keeping effort, Soft Computing, 24 (2020), 2671-2691.  doi: 10.1007/s00500-019-03828-4. [35] L. Wang and B. Dan, Coordination of fresh agricultural supply chain considering retailer's freshness-keeping and consumer utility, Operations Research and Management Science, 24 (2015), 44-51. [36] M. Wang and L. Zhao, Cold chain investment and pricing decisions in a fresh food supply chain, Int. Trans. Oper. Res., 28 (2021), 1074-1097.  doi: 10.1111/itor.12564. [37] X. Z. Wang and G. Q. Wang, Integrating dynamic pricing and inventory control for fresh-agri product under consumer choice, Australian Economic Papers, 58 (2019), 96-111. [38] G. Xu, H. Wu and Y. Liu, A research on fresh-keeping strategies for fresh agricultural products from the perspective of green transportation, Discrete Dyn. Nat. Soc., 2020 (2020), 1307170.  doi: 10.1155/2020/1307170. [39] H. L. Yang, S. Y. Peng and J. J. Yuan, Ordering decisions in a dual-channel supply chain with retailer's hybrid financing, Computer Integrated Manufacturing Systems, (2021), 1–16. http://kns.cnki.net/kcms/detail/11.5946.TP.20200817.1019.016.html. [40] L. Yang and R. Tang, Comparisons of sales modes for a fresh product supply chain with freshness-keeping effort, Transportation Research Part E: Logistics and Transportation Review, 125 (2019), 425-448.  doi: 10.1016/j.tre.2019.03.020. [41] L. Yang, R. Tang and K. Chen, Call, put and bidirectional option contracts in agricultural supply chains with sales effort, Appl. Math. Model., 47 (2017), 1-16.  doi: 10.1016/j.apm.2017.03.002. [42] Y. Zhang and Z. Wang, A joint ordering, pricing, and freshness-keeping policy for perishable inventory systems with random demand over infinite horizon, IEEE Robotics and Automation Letters, 4 (2019), 2707-2713.  doi: 10.1109/LRA.2019.2916471. [43] N. Zhao, Q. Wang and J. Wu, Optimal pricing and ordering decisions with reference effect and quick replenishment policy, Int. Trans. Oper. Res., 29 (2022), 1188-1219.  doi: 10.1111/itor.12960. [44] Y. F. Zhou, T. G. Zou and C. S. Liu, Blood supply chain operation considering lifetime and transshipment under uncertain environment, Applied Soft Computing, 106 (2021), 107364.  doi: 10.1016/j.asoc.2021.107364. [45] Y. F. Zhou, B. Zheng, J. F Su, et al, The joint location-transportation model based on grey bi-level programming for early post-earthquake relief, Journal of Industrial and Management Optimization, 2020. doi: 10.3934/jimo.2020142. [46] F. E. Zulvia, R. J. Kuo and D. Y. Nugroho, A many-objective gradient evolution algorithm for solving a green vehicle routing problem with time windows and time dependency for perishable products, J. Cleaner Production, 242 (2019), 118428.

show all references

##### References:
 [1] M. Bakker, J. Riezebos and R. H. Teunter, Review of inventory systems with deterioration since 2001, European J. Oper. Res., 221 (2012), 275-284.  doi: 10.1016/j.ejor.2012.03.004. [2] S. Bardhan, H. Pal and B. C. Giri, Optimal replenishment policy and preservation technology investment for a non-instantaneous deteriorating item with stock-dependent demand, Operational Research, 19 (2019), 347-368. [3] Z. Bulut, Ü. Gürler and S. Alper, Bundle pricing of inventories with stochastic demand, J. Oper. Res., 197 (2009), 897-911.  doi: 10.1016/j.ejor.2006.09.106. [4] B. B. Cao, Z. P. Fan and T. H. You, Optimal pricing decision in finance-constrained supply chain considering deposits and loans of the bank based on the advance payment, Chinese Journal of Management Science, 28 (2020), 52-64. [5] J. Chen, M. Dong and L. Xu, A perishable product shipment consolidation model considering freshness-keeping effort, Transportation Research Part E: Logistics and Transportation Review, 115 (2018), 56-86.  doi: 10.1016/j.tre.2018.04.009. [6] J. Dong and D. D. Wu, Two-period pricing and quick response with strategic customers, International Journal of Production Economics, 215 (2019), 165-173.  doi: 10.1016/j.ijpe.2017.06.007. [7] T. J. Fan, C. Xu and F. Tao, Dynamic pricing and replenishment policy for fresh produce, Computers & Industrial Engineering, 139 (2020), 106127. [8] Y. Fang, Y. Jiang and X. Han, Bundle pricing decisions for fresh products with quality deterioration, Journal of Food Quality, 2018 (2018), 1595807.  doi: 10.1155/2018/1595807. [9] B. Gkgür and S. Karabat, Dynamic and targeted bundle pricing of two independently valued products, European J. Oper. Res., 279 (2019), 184-198.  doi: 10.1016/j.ejor.2019.05.022. [10] A. Herbon and E. Khmelnitsky, Optimal dynamic pricing and ordering of a perishable product under additive effects of price and time on demand, European J. Oper. Res., 260 (2017), 546-556.  doi: 10.1016/j.ejor.2016.12.033. [11] H. Huang and Z. P. Fan, The purchasing and financing strategies of capital-constrained retailer under different channel power structures, Chinese J. Management Science, 8 (2020), 79-88. [12] H. Huang, Y. He and D. Li, Coordination of pricing, inventory, and production reliability decisions in deteriorating product supply chains, International Journal of Production Research, 56 (2018), 6201-6224.  doi: 10.1080/00207543.2018.1480070. [13] M. Jian, X. Fang and L. Q. Jin, The impact of lead time compression on demand forecasting risk and production cost: A newsvendor model, Transportation Research Part E: Logistics and Transportation Review, 84 (2015), 61-72.  doi: 10.1016/j.tre.2015.10.006. [14] Y. Li, S. Zhang and J. Han, Dynamic pricing and periodic ordering for a stochastic inventory system with deteriorating items, Automatica, 76 (2017), 200-213.  doi: 10.1016/j.automatica.2016.11.003. [15] W. Li, D. M. Hardesty and A. W. Craig, The impact of dynamic bundling on price fairness perceptions, Journal of Retailing & Consumer Services, 40 (2018), 204-212. [16] Y. T. Li and Z. Qiao, Research on the collaborative decision-making method for preservation cost control in fruit and vegetable supply chain, Science & Technology and Economy, 27 (2014), 35-38. [17] C. Liu, W. Chen and Q. Zhou, Modelling dynamic freshness-keeping effort over a finite time horizon in a two-echelon online fresh product supply chain, European J. Oper. Res., 293 (2021), 511-528.  doi: 10.1016/j.ejor.2020.12.035. [18] L. Liu, L. Zhao and X. Ren, Optimal preservation technology investment and pricing policy for fresh food, Computers & Industrial Engineering, 135 (2019), 746-756.  doi: 10.1016/j.cie.2019.06.041. [19] M. L. Liu, B. Dan and S. G. Zhang, Information sharing in an e-tailing supply chain for fresh produce with freshness-keeping effort and value-added service, J. Oper. Res., 290 (2021), 572-584.  doi: 10.1016/j.ejor.2020.08.026. [20] P. Liu and S. Wang, Logistics outsourcing of fresh enterprises considering fresh-keeping efforts based on evolutionary game analysis, IEEE Access, 9 (2021), 25659-25670.  doi: 10.1109/ACCESS.2021.3056699. [21] P. Ma, Y. M. Gong and M. Jin, Quality efforts in medical supply chains considering patient benefits, European J. Oper. Res., 279 (2019), 795-807.  doi: 10.1016/j.ejor.2019.06.030. [22] X. L. Ma, S. Y. Wang, S. M. N. Islam and X. B. Liu, Coordinating a three-echelon fresh agricultural products supply chain considering freshness-keeping effort with asymmetric information, Appl. Math. Model., 67 (2019), 337-356.  doi: 10.1016/j.apm.2018.10.028. [23] X. L. Ma, S. Y. Wang and H. Jin, Coordination and optimization of three-echelon agricultural product supply chain considering freshness-keeping effort and quantity/quality elasticity, Chinese Journal of Management Science, 26 (2018), 175-185. [24] R. Maihami and B. Karimi, Effect of two-echelon trade credit on pricing-inventory policy of non-instantaneous deteriorating products with probabilistic demand and deterioration functions, Ann. Oper. Res., 257 (2017), 237-273.  doi: 10.1007/s10479-016-2195-3. [25] H. Mohammadi, M. Ghazanfari, M. S. Pishvaee and E. Teimoury, Fresh-product supply chain coordination and waste reduction using a revenue-and-preservation-technology-investment-sharing contract: A real-life case study, J. Cleaner Production, 213 (2019), 262-282. [26] I. Moon, Y. J. Jeong and S. Saha, Investment and coordination decisions in a supply chain of fresh agricultural products, International Journal of Operational Research, 3 (2018), 1-25. [27] M. Pattnaik and P. Gahan, Preservation effort effects on retailers and manufacturers in integrated multi-deteriorating item discrete supply chain model, OPSEARCH, 58 (2021), 276-329.  doi: 10.1007/s12597-020-00477-2. [28] C. N. Rapolu and D. H. Kandpal, Joint pricing, advertisement, preservation technology investment and inventory policies for non-instantaneous deteriorating items under trade credit, OPSEARCH, 57 (2020), 274-300. [29] S. Saha, D. Chatterjee and B. Sarkar, The ramification of dynamic investment on the promotion and preservation technology for inventory management through a modified flower pollination algorithm, J. Retailing and Consumer Services, 58 (2021), 102326. [30] M. M. Siddh, G. Soni, R. Jain, M. K. Sharma, et al, Agri-fresh food supply chain quality (AFSCQ): A literature review, Industrial Management & Data Systems, (2017), 117. [31] Z. Song and S. He, Contract coordination of new fresh produce three-layer supply chain, Industrial Management & Data Systems, 119 (2019), 1-23.  doi: 10.1108/IMDS-12-2017-0559. [32] M. Thibaud, H. Chi, W. Zhou and S. Piramuthu, Internet of Things (IoT) in high-risk Environment, Health and Safety (EHS) industries: A comprehensive review, Decision Support Systems, 108 (2018), 79-95. [33] B. Wang and J. Wang, Price and service competition between new and remanufactured products, Math. Probl. Eng., 2015 (2015), 1-18.  doi: 10.1155/2015/325185. [34] G. L. Wang, P. Q. Ding and H. R. Chen, Green fresh product cost sharing contracts considering freshness-keeping effort, Soft Computing, 24 (2020), 2671-2691.  doi: 10.1007/s00500-019-03828-4. [35] L. Wang and B. Dan, Coordination of fresh agricultural supply chain considering retailer's freshness-keeping and consumer utility, Operations Research and Management Science, 24 (2015), 44-51. [36] M. Wang and L. Zhao, Cold chain investment and pricing decisions in a fresh food supply chain, Int. Trans. Oper. Res., 28 (2021), 1074-1097.  doi: 10.1111/itor.12564. [37] X. Z. Wang and G. Q. Wang, Integrating dynamic pricing and inventory control for fresh-agri product under consumer choice, Australian Economic Papers, 58 (2019), 96-111. [38] G. Xu, H. Wu and Y. Liu, A research on fresh-keeping strategies for fresh agricultural products from the perspective of green transportation, Discrete Dyn. Nat. Soc., 2020 (2020), 1307170.  doi: 10.1155/2020/1307170. [39] H. L. Yang, S. Y. Peng and J. J. Yuan, Ordering decisions in a dual-channel supply chain with retailer's hybrid financing, Computer Integrated Manufacturing Systems, (2021), 1–16. http://kns.cnki.net/kcms/detail/11.5946.TP.20200817.1019.016.html. [40] L. Yang and R. Tang, Comparisons of sales modes for a fresh product supply chain with freshness-keeping effort, Transportation Research Part E: Logistics and Transportation Review, 125 (2019), 425-448.  doi: 10.1016/j.tre.2019.03.020. [41] L. Yang, R. Tang and K. Chen, Call, put and bidirectional option contracts in agricultural supply chains with sales effort, Appl. Math. Model., 47 (2017), 1-16.  doi: 10.1016/j.apm.2017.03.002. [42] Y. Zhang and Z. Wang, A joint ordering, pricing, and freshness-keeping policy for perishable inventory systems with random demand over infinite horizon, IEEE Robotics and Automation Letters, 4 (2019), 2707-2713.  doi: 10.1109/LRA.2019.2916471. [43] N. Zhao, Q. Wang and J. Wu, Optimal pricing and ordering decisions with reference effect and quick replenishment policy, Int. Trans. Oper. Res., 29 (2022), 1188-1219.  doi: 10.1111/itor.12960. [44] Y. F. Zhou, T. G. Zou and C. S. Liu, Blood supply chain operation considering lifetime and transshipment under uncertain environment, Applied Soft Computing, 106 (2021), 107364.  doi: 10.1016/j.asoc.2021.107364. [45] Y. F. Zhou, B. Zheng, J. F Su, et al, The joint location-transportation model based on grey bi-level programming for early post-earthquake relief, Journal of Industrial and Management Optimization, 2020. doi: 10.3934/jimo.2020142. [46] F. E. Zulvia, R. J. Kuo and D. Y. Nugroho, A many-objective gradient evolution algorithm for solving a green vehicle routing problem with time windows and time dependency for perishable products, J. Cleaner Production, 242 (2019), 118428.
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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