A Multi-Item Replenishment Problem with Carbon Cap-and-Trade under Uncertainty
Abstract
:1. Introduction
2. Literature Review
3. Notation, Assumptions, and Problem Definitions
3.1. Notation
item, |
period, |
binary variable indicating the order during period |
inventory level of item at the end of period |
on-hand inventory of item at the end of period |
backorder level of item at the end of period |
order amount for item in period |
the order-up-to level of item in period |
if drops below , then , otherwise |
if drops below , then , otherwise |
if drops below and at least one item is ordered in period , then , otherwise |
if minor setup is done for item during period t, then , otherwise |
amount of buying carbon credit in period t |
amount of selling carbon credit in period t |
major ordering cost in period ($/order) |
minor ordering cost of item in period ($/order) |
per unit backorder cost of item in period ($/unit) |
per period holding cost of item in period ($/unit) |
carbon tax ($/ton) |
carbon cap for entire planning horizon |
amount of carbon emissions when a buyer holds inventory of item in period |
amount of carbon emissions when a buyer orders inventory of item in period |
demand for item during period |
volume of item |
storage capacity during period |
purchase price of item |
amount of budget during period |
can-order level of item |
reorder level of item |
big M, very big number |
3.2. Assumptions
- A single buyer orders multiple items from a single supplier and simultaneously considers carbon cap-and-trade under limited storage capacity and budget.
- The system considers a periodic review can-order policy to obtain the order-up to level. The supplier can utilize limited transportation, so the review period is dependent on the contract period between the buyer and the supplier.
- Both the buyer and the supplier share the demand information of the items in real time. Thus, the supplier can deliver multiple items with no lead time. Also, the demand for each item is known.
- The reorder level and the can-order level are assumed as constant.
- The storage capacity and budget are assumed as constant in the deterministic model and fuzzy numbers in the fuzzy model.
- The buyer’s carbon emissions occur throughout the ordering item and holding inventory. A buyer has a carbon cap and could buy or sell its own carbon credit to other company depending on the carbon emissions.
3.3. Problem Definition
4. Deterministic Model
5. Fuzzy Model
6. Numerical Experiments
6.1. Efficiency Test
6.2. Comparison Test between the Proposed Can-Order Policy and the Traditional can-order Policy
6.3. Fuzzy Model Test
7. Academic, Managerial, and Environmental Insights
7.1. Academic Insights
7.2. Managerial Insights
7.3. Environmental Insights
8. Conclusions
Author Contributions
Funding
Conflicts of Interest
References
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Item | 1 | 2 | 3 |
---|---|---|---|
Item | |||
---|---|---|---|
1 | 0 | 50 | 80 |
2 | 0 | 70 | 100 |
3 | 0 | 10 | 20 |
Traditional Can-Order Policy with Predetermined Values ($) | Proposed Model ($) |
---|---|
1,107,962.00 | 757,120.50 |
Test Number | Percent of the Highest Demand | Order-Up-to Level | Total Cost of The Traditional Can-Order Policy with Predetermined Values ($) | ||
---|---|---|---|---|---|
1 | 95% | 662 | 561 | 175 | 928,417.00 |
2 | 90% | 627 | 531 | 166 | 935,320.50 |
3 | 85% | 592 | 502 | 157 | 940,683.00 |
4 | 80% | 558 | 472 | 148 | 1,045,999.00 |
5 | 75% | 523 | 443 | 138 | 1,155,508.00 |
Item Number | ||||||
---|---|---|---|---|---|---|
1 | 1 | 13 | 1 | 0.1 | 1 | |
2 | 2 | 14 | 2 | 0.5 | 0.1 | |
3 | 1 | 10 | 1 | 1 | 0.2 | |
4 | 1 | 15 | 3 | 0.8 | 1 | |
5 | 2 | 8 | 1 | 1.1 | 2 | |
6 | 2 | 9 | 2 | 2 | 0.2 | |
7 | 1 | 6 | 1 | 1 | 0.1 | |
8 | 1 | 10 | 1 | 0.9 | 0.3 | |
9 | 2 | 11 | 2 | 1.5 | 2 | |
10 | 1 | 13 | 1 | 1.3 | 0.1 |
Number of Items | Periods | Total Cost ($) | |
---|---|---|---|
Traditional Can-Order Policy | Proposed Can-Order Policy | ||
6 | 5 | −5599.80 | −7264.20 |
10 | 8800.40 | 5471.60 | |
15 | 23,200.60 | 18,207.40 | |
20 | 37,600.80 | 30,943.20 | |
8 | 5 | −895.60 | −4021.20 |
10 | 18,208.80 | 11,957.60 | |
15 | 37,313.20 | 27,936.40 | |
20 | 56,417.60 | 43,915.20 | |
10 | 5 | 987.80 | −2448.60 |
10 | 21,975.60 | 15,102.80 | |
15 | 42,963.40 | 32,654.20 | |
20 | 63,951.20 | 50,205.60 |
Test Number | ||
---|---|---|
1 | 300 | 3000 |
2 | 300 | 4000 |
3 | 400 | 2000 |
4 | 400 | 4500 |
5 | 450 | 2500 |
Test Number | Fuzzy Total Cost ($) | |||||
---|---|---|---|---|---|---|
1 | 907,298.50 | 1,039,618.0 | 0.73 | 1081 | 8810 | 1,400,277.0 |
2 | 884,448.50 | 1,011,266.0 | 0.75 | 1075 | 9000 | |
3 | 930,148.50 | 1,078,992.0 | 0.68 | 1128 | 8640 | |
4 | 881,762.50 | 1,002,462.0 | 0.77 | 1092 | 9035 | |
5 | 918,723.50 | 1,062,622.0 | 0.72 | 1126 | 8700 |
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Noh, J.; Kim, J.S.; Hwang, S.-J. A Multi-Item Replenishment Problem with Carbon Cap-and-Trade under Uncertainty. Sustainability 2020, 12, 4877. https://doi.org/10.3390/su12124877
Noh J, Kim JS, Hwang S-J. A Multi-Item Replenishment Problem with Carbon Cap-and-Trade under Uncertainty. Sustainability. 2020; 12(12):4877. https://doi.org/10.3390/su12124877
Chicago/Turabian StyleNoh, Jiseong, Jong Soo Kim, and Seung-June Hwang. 2020. "A Multi-Item Replenishment Problem with Carbon Cap-and-Trade under Uncertainty" Sustainability 12, no. 12: 4877. https://doi.org/10.3390/su12124877