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A new adaptive method to nonlinear semi-infinite programming
Optimization of a Multi-Item Inventory model for deteriorating items with capacity constraint using dynamic programming
Department of Industrial Engineering, Iran University of Science and Technology, Tehran, Iran |
In recent years, numerous studies have been conducted regarding inventory control of deteriorating items. However, due to the complexity of the solution methods, various real assumptions such as discrete variables and capacity constraints were neglected. In this study, we presented a multi-item inventory model for deteriorating items with limited carrier capacity. The proposed research considered the carrier, which transports the order has limited capacity and the quantity of orders cannot be infinite. Dynamic programming is used for problem optimization. The results show that the proposed solution method can solve the mixed-integer problem, and it can provide the global optimum solution.
References:
[1] |
R. Bellman,
Dynamic Programming, Science, 153 (1966), 34-37.
|
[2] |
D. Chakraborty, D. K. Jana and T. K. Roy,
Multi-warehouse partial backlogging inventory system with inflation for non-instantaneous deteriorating multi-item under imprecise environment, Soft Computing, 24 (2020), 14471-14490.
doi: 10.1007/s00500-020-04800-3. |
[3] |
C.-Y. Dye, L.-Y. Ouyang and T.-P. Hsieh,
Deterministic inventory model for deteriorating items with capacity constraint and time-proportional backlogging rate, European J. Oper. Res., 178 (2007), 789-807.
doi: 10.1016/j.ejor.2006.02.024. |
[4] |
S. K. Ghosh, T. Sarkar and K. Chaudhuri,
A multi-item inventory model for deteriorating items in limited storage space with stock-dependent demand, American Journal of Mathematical and Management Sciences, 34 (2015), 147-161.
doi: 10.1080/01966324.2014.980870. |
[5] |
S. K. Goyal and B. C. Giri,
Recent trends in modeling of deteriorating inventory, European J. Oper. Res., 134 (2001), 1-16.
doi: 10.1016/S0377-2217(00)00248-4. |
[6] |
M. Karimi, S. J. Sadjadi and A. G. Bijaghini,
An economic order quantity for deteriorating items with allowable rework of deteriorated products, J. Ind. Manag. Optim., 15 (2019), 1857-1879.
doi: 10.3934/jimo.2018126. |
[7] |
G. Li, X. He, J. Zhou and H. Wu,
Pricing, replenishment and preservation technology investment decisions for non-instantaneous deteriorating items, Omega, 84 (2019), 114-126.
doi: 10.1016/j.omega.2018.05.001. |
[8] |
J.-J. Liao, K.-N. Huang, K.-J. Chung, S.-D. Lin, S.-T. Chuang and H. M. Srivastava, Optimal ordering policy in an economic order quantity (EOQ) model for non-instantaneous deteriorating items with defective quality and permissible delay in payments, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 114 (2020), Paper No. 41, 26 pp.
doi: 10.1007/s13398-019-00777-3. |
[9] |
P. Mahata, G. C. Mahata and S. K. De,
An economic order quantity model under two-level partial trade credit for time varying deteriorating items, International Journal of Systems Science: Operations and Logistics, 7 (2020), 1-17.
doi: 10.1080/23302674.2018.1473526. |
[10] |
A. K. Malik and A. Sharma,
An inventory model for deteriorating items with multi-variate demand and partial backlogging under inflation, International Journal of Mathematical Sciences, 10 (2011), 315-321.
|
[11] |
M. Rezagholifam, S. J. Sadjadi, M. Heydari and M. Karimi, Optimal pricing and ordering strategy for non-instantaneous deteriorating items with price and stock sensitive demand and capacity constraint, International Journal of Systems Science: Operations and Logistics, (2020).
doi: 10.1080/23302674.2020.1833259. |
[12] |
G. P. Samanta and A. Roy,
A production inventory model with deteriorating items and shortages, Yugosl. J. Oper. Res., 14 (2004), 219-230.
doi: 10.2298/YJOR0402219S. |
[13] |
S. Sana, S. K. Goyal and K. S. Chaudhuri,
A production-inventory model for a deteriorating item with trended demand and shortages, European J. Oper. Res., 157 (2004), 357-371.
doi: 10.1016/S0377-2217(03)00222-4. |
[14] |
N. H. Shah, U. Chaudhari and L. E. Cárdenas-Barrón,
Integrating credit and replenishment policies for deteriorating items under quadratic demand in a three echelon supply chain, International Journal of Systems Science: Operations and Logistics, 7 (2020), 34-45.
|
[15] |
N. H. Shah and M. K. Naik,
Inventory policies for deteriorating items with time-price backlog dependent demand, International Journal of Systems Science: Operations and Logistics, 7 (2020), 76-89.
doi: 10.1080/23302674.2018.1506062. |
[16] |
J.-T. Teng, L. E. Cárdenas-Barrón, H.-J. Chang, J. Wu and Y. Hu,
Inventory lot-size policies for deteriorating items with expiration dates and advance payments, Appl. Math. Model., 40 (2016), 8605-8616.
doi: 10.1016/j.apm.2016.05.022. |
[17] |
S. Tiwari, L. E. Cárdenas-Barrón, M. Goh and A. A. Shaikh,
Joint pricing and inventory model for deteriorating items with expiration dates and partial backlogging under two-level partial trade credits in supply chain, International Journal of Production Economics, 200 (2018), 16-36.
doi: 10.1016/j.ijpe.2018.03.006. |
[18] |
S. Tiwari, L. E. Cárdenas-Barrón, A. Khanna and C. K. Jaggi,
Impact of trade credit and inflation on retailer's ordering policies for non-instantaneous deteriorating items in a two-warehouse environment, International Journal of Production Economics, 176 (2016), 154-169.
doi: 10.1016/j.ijpe.2016.03.016. |
[19] |
Q. Wang, J. Wu, N. Zhao and Q. Zhu,
Inventory control and supply chain management: A green growth perspective, Resources, Conservation and Recycling, 145 (2019), 78-85.
doi: 10.1016/j.resconrec.2019.02.024. |
[20] |
J. Wu, F. B. Al-Khateeb, J.-T. Teng and L. E. Cárdenas-Barrón,
Inventory models for deteriorating items with maximum lifetime under downstream partial trade credits to credit-risk customers by discounted cash-flow analysis, International Journal of Production Economics, 171 (2016), 105-115.
doi: 10.1016/j.ijpe.2015.10.020. |
[21] |
J. Wu, L.-Y. Ouyang, L. E. Cárdenas-Barrón and S. K. Goyal,
Optimal credit period and lot size for deteriorating items with expiration dates under two-level trade credit financing, European Journal of Operational Research, 237 (2014), 898-908.
doi: 10.1016/j.ejor.2014.03.009. |
[22] |
J. Zhang, G. Liu, Q. Zhang and Z. Bai,
Coordinating a supply chain for deteriorating items with a revenue sharing and cooperative investment contract, Omega, 56 (2015), 37-49.
doi: 10.1016/j.omega.2015.03.004. |
show all references
References:
[1] |
R. Bellman,
Dynamic Programming, Science, 153 (1966), 34-37.
|
[2] |
D. Chakraborty, D. K. Jana and T. K. Roy,
Multi-warehouse partial backlogging inventory system with inflation for non-instantaneous deteriorating multi-item under imprecise environment, Soft Computing, 24 (2020), 14471-14490.
doi: 10.1007/s00500-020-04800-3. |
[3] |
C.-Y. Dye, L.-Y. Ouyang and T.-P. Hsieh,
Deterministic inventory model for deteriorating items with capacity constraint and time-proportional backlogging rate, European J. Oper. Res., 178 (2007), 789-807.
doi: 10.1016/j.ejor.2006.02.024. |
[4] |
S. K. Ghosh, T. Sarkar and K. Chaudhuri,
A multi-item inventory model for deteriorating items in limited storage space with stock-dependent demand, American Journal of Mathematical and Management Sciences, 34 (2015), 147-161.
doi: 10.1080/01966324.2014.980870. |
[5] |
S. K. Goyal and B. C. Giri,
Recent trends in modeling of deteriorating inventory, European J. Oper. Res., 134 (2001), 1-16.
doi: 10.1016/S0377-2217(00)00248-4. |
[6] |
M. Karimi, S. J. Sadjadi and A. G. Bijaghini,
An economic order quantity for deteriorating items with allowable rework of deteriorated products, J. Ind. Manag. Optim., 15 (2019), 1857-1879.
doi: 10.3934/jimo.2018126. |
[7] |
G. Li, X. He, J. Zhou and H. Wu,
Pricing, replenishment and preservation technology investment decisions for non-instantaneous deteriorating items, Omega, 84 (2019), 114-126.
doi: 10.1016/j.omega.2018.05.001. |
[8] |
J.-J. Liao, K.-N. Huang, K.-J. Chung, S.-D. Lin, S.-T. Chuang and H. M. Srivastava, Optimal ordering policy in an economic order quantity (EOQ) model for non-instantaneous deteriorating items with defective quality and permissible delay in payments, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Mat. RACSAM, 114 (2020), Paper No. 41, 26 pp.
doi: 10.1007/s13398-019-00777-3. |
[9] |
P. Mahata, G. C. Mahata and S. K. De,
An economic order quantity model under two-level partial trade credit for time varying deteriorating items, International Journal of Systems Science: Operations and Logistics, 7 (2020), 1-17.
doi: 10.1080/23302674.2018.1473526. |
[10] |
A. K. Malik and A. Sharma,
An inventory model for deteriorating items with multi-variate demand and partial backlogging under inflation, International Journal of Mathematical Sciences, 10 (2011), 315-321.
|
[11] |
M. Rezagholifam, S. J. Sadjadi, M. Heydari and M. Karimi, Optimal pricing and ordering strategy for non-instantaneous deteriorating items with price and stock sensitive demand and capacity constraint, International Journal of Systems Science: Operations and Logistics, (2020).
doi: 10.1080/23302674.2020.1833259. |
[12] |
G. P. Samanta and A. Roy,
A production inventory model with deteriorating items and shortages, Yugosl. J. Oper. Res., 14 (2004), 219-230.
doi: 10.2298/YJOR0402219S. |
[13] |
S. Sana, S. K. Goyal and K. S. Chaudhuri,
A production-inventory model for a deteriorating item with trended demand and shortages, European J. Oper. Res., 157 (2004), 357-371.
doi: 10.1016/S0377-2217(03)00222-4. |
[14] |
N. H. Shah, U. Chaudhari and L. E. Cárdenas-Barrón,
Integrating credit and replenishment policies for deteriorating items under quadratic demand in a three echelon supply chain, International Journal of Systems Science: Operations and Logistics, 7 (2020), 34-45.
|
[15] |
N. H. Shah and M. K. Naik,
Inventory policies for deteriorating items with time-price backlog dependent demand, International Journal of Systems Science: Operations and Logistics, 7 (2020), 76-89.
doi: 10.1080/23302674.2018.1506062. |
[16] |
J.-T. Teng, L. E. Cárdenas-Barrón, H.-J. Chang, J. Wu and Y. Hu,
Inventory lot-size policies for deteriorating items with expiration dates and advance payments, Appl. Math. Model., 40 (2016), 8605-8616.
doi: 10.1016/j.apm.2016.05.022. |
[17] |
S. Tiwari, L. E. Cárdenas-Barrón, M. Goh and A. A. Shaikh,
Joint pricing and inventory model for deteriorating items with expiration dates and partial backlogging under two-level partial trade credits in supply chain, International Journal of Production Economics, 200 (2018), 16-36.
doi: 10.1016/j.ijpe.2018.03.006. |
[18] |
S. Tiwari, L. E. Cárdenas-Barrón, A. Khanna and C. K. Jaggi,
Impact of trade credit and inflation on retailer's ordering policies for non-instantaneous deteriorating items in a two-warehouse environment, International Journal of Production Economics, 176 (2016), 154-169.
doi: 10.1016/j.ijpe.2016.03.016. |
[19] |
Q. Wang, J. Wu, N. Zhao and Q. Zhu,
Inventory control and supply chain management: A green growth perspective, Resources, Conservation and Recycling, 145 (2019), 78-85.
doi: 10.1016/j.resconrec.2019.02.024. |
[20] |
J. Wu, F. B. Al-Khateeb, J.-T. Teng and L. E. Cárdenas-Barrón,
Inventory models for deteriorating items with maximum lifetime under downstream partial trade credits to credit-risk customers by discounted cash-flow analysis, International Journal of Production Economics, 171 (2016), 105-115.
doi: 10.1016/j.ijpe.2015.10.020. |
[21] |
J. Wu, L.-Y. Ouyang, L. E. Cárdenas-Barrón and S. K. Goyal,
Optimal credit period and lot size for deteriorating items with expiration dates under two-level trade credit financing, European Journal of Operational Research, 237 (2014), 898-908.
doi: 10.1016/j.ejor.2014.03.009. |
[22] |
J. Zhang, G. Liu, Q. Zhang and Z. Bai,
Coordinating a supply chain for deteriorating items with a revenue sharing and cooperative investment contract, Omega, 56 (2015), 37-49.
doi: 10.1016/j.omega.2015.03.004. |



Paper | Multi | Demand | Constraints | Variables | Solution | Shortages |
Item | Function | type | method | |||
[7] | No | Constant | Logical | Continuous | Soft | Allowed |
constraints | computing | |||||
[2] | Yes | Stock- | Capacity | Continuous | Soft | Allowed |
dependent | constraint | computing | ||||
[8] | No | Constant | No | Continuous | Mathematical | Not |
derivation | allowed | |||||
[15] | No | Time- | Logical | Continuous | Soft | Not |
dependent | constraints | computing | allowed | |||
[9] | No | Trade | No | Continuous | Soft | Not |
credit- | computing | allowed | ||||
dependent | ||||||
[14] | No | Time-price | No | Continuous | Mathematical | Allowed |
backlog | derivation | |||||
dependent | ||||||
[6] | No | Time- | No | Continuous | Mathematical | Allowed |
dependent | derivation | |||||
[11] | No | Stock and | Capacity | Continuous | Mathematical | Not |
price | constraint | derivation | allowed | |||
dependent | ||||||
This | Yes | Time | Capacity | Discrete and | Dynamic | Allowed |
Paper | -dependent | constraint | continuous | Programming |
Paper | Multi | Demand | Constraints | Variables | Solution | Shortages |
Item | Function | type | method | |||
[7] | No | Constant | Logical | Continuous | Soft | Allowed |
constraints | computing | |||||
[2] | Yes | Stock- | Capacity | Continuous | Soft | Allowed |
dependent | constraint | computing | ||||
[8] | No | Constant | No | Continuous | Mathematical | Not |
derivation | allowed | |||||
[15] | No | Time- | Logical | Continuous | Soft | Not |
dependent | constraints | computing | allowed | |||
[9] | No | Trade | No | Continuous | Soft | Not |
credit- | computing | allowed | ||||
dependent | ||||||
[14] | No | Time-price | No | Continuous | Mathematical | Allowed |
backlog | derivation | |||||
dependent | ||||||
[6] | No | Time- | No | Continuous | Mathematical | Allowed |
dependent | derivation | |||||
[11] | No | Stock and | Capacity | Continuous | Mathematical | Not |
price | constraint | derivation | allowed | |||
dependent | ||||||
This | Yes | Time | Capacity | Discrete and | Dynamic | Allowed |
Paper | -dependent | constraint | continuous | Programming |
0 | 1 | 2 | 3 | 4 | 5 | |
0 | 110.7 | 221.7 | 330.7 | 435.4 | 533.5 | |
0 | 166.05 | 332.55 | 496.05 | 653.1 | 800.25 | |
800 | 633.95 | 467.45 | 303.95 | 146.9 | -0.25 | |
(infeasible) |
0 | 1 | 2 | 3 | 4 | 5 | |
0 | 110.7 | 221.7 | 330.7 | 435.4 | 533.5 | |
0 | 166.05 | 332.55 | 496.05 | 653.1 | 800.25 | |
800 | 633.95 | 467.45 | 303.95 | 146.9 | -0.25 | |
(infeasible) |
{0} | {0, 1} | {0, 1, 2} | {0, 1, 2, 3} | {0, 1, 2, 3, 4} |
{0} | {0, 1} | {0, 1, 2} | {0, 1, 2, 3} | {0, 1, 2, 3, 4} |
0 | 1 | 2 | 3 | 4 | 5 | |
0 | 36.9 | 77.5 | 127.1 | 200.4 | 338.3 | |
0 | 73.8 | 155 | 254.2 | 400.8 | 676.6 | |
800 | 726.2 | 645 | 545.8 | 399.8 | 123.4 |
0 | 1 | 2 | 3 | 4 | 5 | |
0 | 36.9 | 77.5 | 127.1 | 200.4 | 338.3 | |
0 | 73.8 | 155 | 254.2 | 400.8 | 676.6 | |
800 | 726.2 | 645 | 545.8 | 399.8 | 123.4 |
{0} | {0, 1} | {0, 1, 2} | {0, 1, 2} | {0, 1, 2} | |
{0, 1, 2, 3} | {0, 1, 2, 3} | {0, 1, 2, 3} | {0, 1, 2, 3, 4} | {0, 1, 2, 3, 4} | |
{0, 1, 2, 3, 4} | {0, 1, 2, 3, 4} | {0, 1, 2, 3, 4} | {0, 1, 2, 3, 4} | {0, 1, 2, 3, 4} | |
{0, 1, 2, 3, 4} | {0, 1, 2, 3, 4} | {0, 1, 2, 3, 4} | {0, 1, ..., 5} | {0, 1, ..., 5} | |
{0, 1, ..., 5} | {0, 1, ..., 5} |
{0} | {0, 1} | {0, 1, 2} | {0, 1, 2} | {0, 1, 2} | |
{0, 1, 2, 3} | {0, 1, 2, 3} | {0, 1, 2, 3} | {0, 1, 2, 3, 4} | {0, 1, 2, 3, 4} | |
{0, 1, 2, 3, 4} | {0, 1, 2, 3, 4} | {0, 1, 2, 3, 4} | {0, 1, 2, 3, 4} | {0, 1, 2, 3, 4} | |
{0, 1, 2, 3, 4} | {0, 1, 2, 3, 4} | {0, 1, 2, 3, 4} | {0, 1, ..., 5} | {0, 1, ..., 5} | |
{0, 1, ..., 5} | {0, 1, ..., 5} |
24404 | 24262 | 24132 | 23991 | 23849 | ||||||
:0 | :0 | :0 | :1 | :0 | :2 | :1 | :0 | :1 | :1 | |
23849 | 23719 | 23651 | 23651 | 23509 | ||||||
:1 | :1 | :1 | :2 | :2 | :0 | :2 | :0 | :2 | :1 | |
23509 | 23379 | 23379 | 23379 | 23258 | ||||||
:2 | :1 | :2 | :2 | :2 | :2 | :2 | :2 | :3 | :1 | |
23256 | 23128 | 23128 | 23128 | 23127 | ||||||
:2 | :3 | :3 | :2 | :3 | :2 | :3 | :2 | :4 | :1 | |
23127 | 23005 | |||||||||
:4 | :1 | :3 | :3 |
24404 | 24262 | 24132 | 23991 | 23849 | ||||||
:0 | :0 | :0 | :1 | :0 | :2 | :1 | :0 | :1 | :1 | |
23849 | 23719 | 23651 | 23651 | 23509 | ||||||
:1 | :1 | :1 | :2 | :2 | :0 | :2 | :0 | :2 | :1 | |
23509 | 23379 | 23379 | 23379 | 23258 | ||||||
:2 | :1 | :2 | :2 | :2 | :2 | :2 | :2 | :3 | :1 | |
23256 | 23128 | 23128 | 23128 | 23127 | ||||||
:2 | :3 | :3 | :2 | :3 | :2 | :3 | :2 | :4 | :1 | |
23127 | 23005 | |||||||||
:4 | :1 | :3 | :3 |
0 | 1 | 2 | 3 | 4 | 5 | |
0 | 152.4 | 315.9 | 523.2 | 863.8 | 1517.5 | |
0 | 152.4 | 315.9 | 523.2 | 863.8 | 1517.5 | |
800 | 647.6 | 484.1 | 276.8 | -63.8 | -717.5 | |
(infeasible) | (infeasible) |
0 | 1 | 2 | 3 | 4 | 5 | |
0 | 152.4 | 315.9 | 523.2 | 863.8 | 1517.5 | |
0 | 152.4 | 315.9 | 523.2 | 863.8 | 1517.5 | |
800 | 647.6 | 484.1 | 276.8 | -63.8 | -717.5 | |
(infeasible) | (infeasible) |
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