American Institute of Mathematical Sciences

Advanced
March  2022, 18(2): 1145-1160. doi: 10.3934/jimo.2021013

Optimization of a Multi-Item Inventory model for deteriorating items with capacity constraint using dynamic programming

Mahdi Karimi , and  Seyed Jafar Sadjadi

Department of Industrial Engineering, Iran University of Science and Technology, Tehran, Iran

* Corresponding author

Received  May 2020 Revised  October 2020 Published  March 2022 Early access  December 2020

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.

Keywords: Inventory control, multi-item, deteriorating items, capacity constraint, knapsack problem, dynamic programming.
Mathematics Subject Classification: Primary: 90B05, 90C26; Secondary: 90C39.
Citation: Mahdi Karimi, Seyed Jafar Sadjadi. Optimization of a Multi-Item Inventory model for deteriorating items with capacity constraint using dynamic programming. Journal of Industrial and Management Optimization, 2022, 18 (2) : 1145-1160. doi: 10.3934/jimo.2021013
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.

Figure 1.  Inventory level of each item vs. time
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 2.  The flowchart of the proposed solution method
Figure Options

Download full-size image

Download as PowerPoint slide

Figure 3.  The inventory level of each item over time
Figure Options

Download full-size image

Download as PowerPoint slide

Table 1.  A. Review of previous works
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
Table Options

Download as excel

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
Table 2.  The required and remaining space for each action in the stage 1
$ k^{'}_{1} $ 0 1 2 3 4 5
$ k_{1} $ 0 110.7 221.7 330.7 435.4 533.5
$ k_{1}v_{1} $ 0 166.05 332.55 496.05 653.1 800.25
$ j_{1} $ 800 633.95 467.45 303.95 146.9 -0.25
(infeasible)
Table Options

Download as excel

$ k^{'}_{1} $ 0 1 2 3 4 5
$ k_{1} $ 0 110.7 221.7 330.7 435.4 533.5
$ k_{1}v_{1} $ 0 166.05 332.55 496.05 653.1 800.25
$ j_{1} $ 800 633.95 467.45 303.95 146.9 -0.25
(infeasible)
Table 3.  Different values of the state in the stage 1
$ i_{1} $ $ 0\leq i_{1} $ $ 166.05\leq i_{1} $ $ 332.555\leq i_{1} $ $ 496.05\leq i_{1} $ $ 653.1\leq i_{1} $
$<166.05 $ $<332.55 $ $<496.05 $ $<653.1 $ $ \leq800 $
$ i^{'}_{1} $ {0} {0, 1} {0, 1, 2} {0, 1, 2, 3} {0, 1, 2, 3, 4}
Table Options

Download as excel

$ i_{1} $ $ 0\leq i_{1} $ $ 166.05\leq i_{1} $ $ 332.555\leq i_{1} $ $ 496.05\leq i_{1} $ $ 653.1\leq i_{1} $
$<166.05 $ $<332.55 $ $<496.05 $ $<653.1 $ $ \leq800 $
$ i^{'}_{1} $ {0} {0, 1} {0, 1, 2} {0, 1, 2, 3} {0, 1, 2, 3, 4}
Table 4.  The required space for each action in the stage 2
$ k^{'}_{2} $ 0 1 2 3 4 5
$ k_{2} $ 0 36.9 77.5 127.1 200.4 338.3
$ k_{2}v_{1} $ 0 73.8 155 254.2 400.8 676.6
$ j_{2} $ 800 726.2 645 545.8 399.8 123.4
Table Options

Download as excel

$ k^{'}_{2} $ 0 1 2 3 4 5
$ k_{2} $ 0 36.9 77.5 127.1 200.4 338.3
$ k_{2}v_{1} $ 0 73.8 155 254.2 400.8 676.6
$ j_{2} $ 800 726.2 645 545.8 399.8 123.4
Table 5.  Different values of the state in the stage n = 2
$ i_{2} $ $ 0\leq i_{2} $ $ 73.8\leq i_{2} $ $ 155\leq i_{2} $ $ 166.05\leq i_{2} $ $ 240.3\leq i_{2} $
$<73.8 $ $<155 $ $<166.05 $ $<240.3 $ $<254.2 $
$ i^{'}_2 $ {0} {0, 1} {0, 1, 2} {0, 1, 2} {0, 1, 2}
$ i_{2} $ $ 254.2\leq i_{2} $ $ 321.5\leq i_{2} $ $ 332.55\leq i_{2} $ $ 400.8\leq i_{2} $ $ 406.3\leq i_{2} $
$<321.5 $ $<332.55 $ $<400.8 $ $<406.3 $ $<420.7 $
$ i^{'}_2 $ {0, 1, 2, 3} {0, 1, 2, 3} {0, 1, 2, 3} {0, 1, 2, 3, 4} {0, 1, 2, 3, 4}
$ i_{2} $ $ 420.7\leq i_{2} $ $ 487.5\leq i_{2} $ $ 496.05\leq i_{2} $ $ 567.3\leq i_{2} $ $ 569.7\leq i_{2} $
$<487.5 $ $<496.05 $ $<567.3 $ $<569.7 $ $<586.7 $
$ i^{'}_2 $ {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}
$ i_{2} $ $ 586.7\leq i_{2} $ $ 650.9\leq i_{2} $ $ 653.1\leq i_{2} $ $ 676.6\leq i_{2} $ $ 726.9\leq i_{2} $
$<650.9 $ $<653.1 $ $<676.6 $ $<726.9 $ $<733.3 $
$ i^{'}_2 $ {0, 1, 2, 3, 4} {0, 1, 2, 3, 4} {0, 1, 2, 3, 4} {0, 1, ..., 5} {0, 1, ..., 5}
$ i_{2} $} $ 733.3\leq i_{2} $ $ 750.1\leq i_{2} $
$<750.1 $ $ \leq800 $
$ i^{'}_2 $ {0, 1, ..., 5} {0, 1, ..., 5}
Table Options

Download as excel

$ i_{2} $ $ 0\leq i_{2} $ $ 73.8\leq i_{2} $ $ 155\leq i_{2} $ $ 166.05\leq i_{2} $ $ 240.3\leq i_{2} $
$<73.8 $ $<155 $ $<166.05 $ $<240.3 $ $<254.2 $
$ i^{'}_2 $ {0} {0, 1} {0, 1, 2} {0, 1, 2} {0, 1, 2}
$ i_{2} $ $ 254.2\leq i_{2} $ $ 321.5\leq i_{2} $ $ 332.55\leq i_{2} $ $ 400.8\leq i_{2} $ $ 406.3\leq i_{2} $
$<321.5 $ $<332.55 $ $<400.8 $ $<406.3 $ $<420.7 $
$ i^{'}_2 $ {0, 1, 2, 3} {0, 1, 2, 3} {0, 1, 2, 3} {0, 1, 2, 3, 4} {0, 1, 2, 3, 4}
$ i_{2} $ $ 420.7\leq i_{2} $ $ 487.5\leq i_{2} $ $ 496.05\leq i_{2} $ $ 567.3\leq i_{2} $ $ 569.7\leq i_{2} $
$<487.5 $ $<496.05 $ $<567.3 $ $<569.7 $ $<586.7 $
$ i^{'}_2 $ {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}
$ i_{2} $ $ 586.7\leq i_{2} $ $ 650.9\leq i_{2} $ $ 653.1\leq i_{2} $ $ 676.6\leq i_{2} $ $ 726.9\leq i_{2} $
$<650.9 $ $<653.1 $ $<676.6 $ $<726.9 $ $<733.3 $
$ i^{'}_2 $ {0, 1, 2, 3, 4} {0, 1, 2, 3, 4} {0, 1, 2, 3, 4} {0, 1, ..., 5} {0, 1, ..., 5}
$ i_{2} $} $ 733.3\leq i_{2} $ $ 750.1\leq i_{2} $
$<750.1 $ $ \leq800 $
$ i^{'}_2 $ {0, 1, ..., 5} {0, 1, ..., 5}
Table 6.  The recursive function in the second stage
$ i_{2} $ $ 0\leq i_{2} $
$<73.8 $		 $ 73.8\leq i_{2} $
$<155 $		 $ 155\leq i_{2} $
$<166.05 $		 $ 166.05\leq i_{2} $
$<240.3 $		 $ 240.3\leq i_{2} $
$<254.2 $
$ f(2, i_{2}) $ 24404 24262 24132 23991 23849
$ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $
:0 :0 :0 :1 :0 :2 :1 :0 :1 :1
$ i_{2} $ $ 254.2\leq i_{2} $ $ 321.5\leq i_{2} $ $ 332.55\leq i_{2} $ $ 400.8\leq i_{2} $ $ 406.3\leq i_{2} $
$<321.5 $ $<332.55 $ $<400.8 $ $<406.3 $ $<420.7 $
$ f(2, i_{2}) $ 23849 23719 23651 23651 23509
$ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $
:1 :1 :1 :2 :2 :0 :2 :0 :2 :1
$ i_{2} $ $ 420.7\leq i_{2} $ $ 487.5\leq i_{2} $ $ 496.05\leq i_{2} $ $ 567.3\leq i_{2} $ $ 596.7\leq i_{2} $
$<487.5 $ $<496.05 $ $<567.3 $ $<569.7 $ $<586.7 $
$ f(2, i_{2}) $ 23509 23379 23379 23379 23258
$ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $
:2 :1 :2 :2 :2 :2 :2 :2 :3 :1
$ i_{2} $ $ 586.7\leq i_{2} $ $ 650.9\leq i_{2} $ $ 653.1\leq i_{2} $ $ 676.6\leq i_{2} $ $ 726.9\leq i_{2} $
$<650.9 $ $<653.1 $ $<676.6 $ $<726.9 $ $<733.3 $
$ f(2, i_{2}) $ 23256 23128 23128 23128 23127
$ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $
:2 :3 :3 :2 :3 :2 :3 :2 :4 :1
$ i_{2} $ $ 733.3\leq i_{2} $ $ 750.1\leq i_{2} $
$<750.1 $ $ \leq800 $
$ f(2, i_{2}) $ 23127 23005
$ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $
:4 :1 :3 :3
Table Options

Download as excel

$ i_{2} $ $ 0\leq i_{2} $
$<73.8 $		 $ 73.8\leq i_{2} $
$<155 $		 $ 155\leq i_{2} $
$<166.05 $		 $ 166.05\leq i_{2} $
$<240.3 $		 $ 240.3\leq i_{2} $
$<254.2 $
$ f(2, i_{2}) $ 24404 24262 24132 23991 23849
$ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $
:0 :0 :0 :1 :0 :2 :1 :0 :1 :1
$ i_{2} $ $ 254.2\leq i_{2} $ $ 321.5\leq i_{2} $ $ 332.55\leq i_{2} $ $ 400.8\leq i_{2} $ $ 406.3\leq i_{2} $
$<321.5 $ $<332.55 $ $<400.8 $ $<406.3 $ $<420.7 $
$ f(2, i_{2}) $ 23849 23719 23651 23651 23509
$ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $
:1 :1 :1 :2 :2 :0 :2 :0 :2 :1
$ i_{2} $ $ 420.7\leq i_{2} $ $ 487.5\leq i_{2} $ $ 496.05\leq i_{2} $ $ 567.3\leq i_{2} $ $ 596.7\leq i_{2} $
$<487.5 $ $<496.05 $ $<567.3 $ $<569.7 $ $<586.7 $
$ f(2, i_{2}) $ 23509 23379 23379 23379 23258
$ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $
:2 :1 :2 :2 :2 :2 :2 :2 :3 :1
$ i_{2} $ $ 586.7\leq i_{2} $ $ 650.9\leq i_{2} $ $ 653.1\leq i_{2} $ $ 676.6\leq i_{2} $ $ 726.9\leq i_{2} $
$<650.9 $ $<653.1 $ $<676.6 $ $<726.9 $ $<733.3 $
$ f(2, i_{2}) $ 23256 23128 23128 23128 23127
$ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $
:2 :3 :3 :2 :3 :2 :3 :2 :4 :1
$ i_{2} $ $ 733.3\leq i_{2} $ $ 750.1\leq i_{2} $
$<750.1 $ $ \leq800 $
$ f(2, i_{2}) $ 23127 23005
$ k^{'*}_1 $ $ k^{'*}_2 $ $ k^{'*}_1 $ $ k^{'*}_2 $
:4 :1 :3 :3
Table 7.  The required and remaining space for each action in stage n = 3
$ k^{'}_{3} $ 0 1 2 3 4 5
$ k_{3} $ 0 152.4 315.9 523.2 863.8 1517.5
$ k_{3}v_{3} $ 0 152.4 315.9 523.2 863.8 1517.5
$ j_{3} $ 800 647.6 484.1 276.8 -63.8 -717.5
(infeasible) (infeasible)
Table Options

Download as excel

$ k^{'}_{3} $ 0 1 2 3 4 5
$ k_{3} $ 0 152.4 315.9 523.2 863.8 1517.5
$ k_{3}v_{3} $ 0 152.4 315.9 523.2 863.8 1517.5
$ j_{3} $ 800 647.6 484.1 276.8 -63.8 -717.5
(infeasible) (infeasible)
[1]

Magfura Pervin, Sankar Kumar Roy, Gerhard Wilhelm Weber. Multi-item deteriorating two-echelon inventory model with price- and stock-dependent demand: A trade-credit policy. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1345-1373. doi: 10.3934/jimo.2018098
[2]

Yuli Zhang, Lin Han, Xiaotian Zhuang. Distributionally robust front distribution center inventory optimization with uncertain multi-item orders. Discrete and Continuous Dynamical Systems - S, 2022, 15 (7) : 1777-1795. doi: 10.3934/dcdss.2022006
[3]

Mohsen Lashgari, Ata Allah Taleizadeh, Shib Sankar Sana. An inventory control problem for deteriorating items with back-ordering and financial considerations under two levels of trade credit linked to order quantity. Journal of Industrial and Management Optimization, 2016, 12 (3) : 1091-1119. doi: 10.3934/jimo.2016.12.1091
[4]

T. W. Leung, Chi Kin Chan, Marvin D. Troutt. A mixed simulated annealing-genetic algorithm approach to the multi-buyer multi-item joint replenishment problem: advantages of meta-heuristics. Journal of Industrial and Management Optimization, 2008, 4 (1) : 53-66. doi: 10.3934/jimo.2008.4.53
[5]

Deepak Kumar Nayak, Sudhansu Sekhar Routray, Susanta Kumar Paikray, Hemen Dutta. A fuzzy inventory model for Weibull deteriorating items under completely backlogged shortages. Discrete and Continuous Dynamical Systems - S, 2021, 14 (7) : 2435-2453. doi: 10.3934/dcdss.2020401
[6]

Ying Kou, Zhong Wan. A new data-driven robust optimization approach to multi-item newsboy problems. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021180
[7]

Vincent Choudri, Mathiyazhgan Venkatachalam, Sethuraman Panayappan. Production inventory model with deteriorating items, two rates of production cost and taking account of time value of money. Journal of Industrial and Management Optimization, 2016, 12 (3) : 1153-1172. doi: 10.3934/jimo.2016.12.1153
[8]

Magfura Pervin, Sankar Kumar Roy, Gerhard Wilhelm Weber. A two-echelon inventory model with stock-dependent demand and variable holding cost for deteriorating items. Numerical Algebra, Control and Optimization, 2017, 7 (1) : 21-50. doi: 10.3934/naco.2017002
[9]

Prasenjit Pramanik, Sarama Malik Das, Manas Kumar Maiti. Note on : Supply chain inventory model for deteriorating items with maximum lifetime and partial trade credit to credit risk customers. Journal of Industrial and Management Optimization, 2019, 15 (3) : 1289-1315. doi: 10.3934/jimo.2018096
[10]

Yuzhong Zhang, Fan Zhang, Maocheng Cai. Some new results on multi-dimension Knapsack problem. Journal of Industrial and Management Optimization, 2005, 1 (3) : 315-321. doi: 10.3934/jimo.2005.1.315
[11]

Puspita Mahata, Gour Chandra Mahata. Two-echelon trade credit with default risk in an EOQ model for deteriorating items under dynamic demand. Journal of Industrial and Management Optimization, 2021, 17 (6) : 3659-3684. doi: 10.3934/jimo.2020138
[12]

Andrzej Nowakowski, Jan Sokolowski. On dual dynamic programming in shape control. Communications on Pure and Applied Analysis, 2012, 11 (6) : 2473-2485. doi: 10.3934/cpaa.2012.11.2473
[13]

Shuhua Zhang, Longzhou Cao, Zuliang Lu. An EOQ inventory model for deteriorating items with controllable deterioration rate under stock-dependent demand rate and non-linear holding cost. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021156
[14]

Rein Luus. Optimal control of oscillatory systems by iterative dynamic programming. Journal of Industrial and Management Optimization, 2008, 4 (1) : 1-15. doi: 10.3934/jimo.2008.4.1
[15]

Lianxia Zhao, Hui Qiao, Qi An. Optimal pre-sale policy for deteriorating items. Numerical Algebra, Control and Optimization, 2022, 12 (1) : 109-120. doi: 10.3934/naco.2021054
[16]

K. F. Cedric Yiu, S. Y. Wang, K. L. Mak. Optimal portfolios under a value-at-risk constraint with applications to inventory control in supply chains. Journal of Industrial and Management Optimization, 2008, 4 (1) : 81-94. doi: 10.3934/jimo.2008.4.81
[17]

Alexander Arguchintsev, Vasilisa Poplevko. An optimal control problem by parabolic equation with boundary smooth control and an integral constraint. Numerical Algebra, Control and Optimization, 2018, 8 (2) : 193-202. doi: 10.3934/naco.2018011
[18]

Behrouz Kheirfam. Multi-parametric sensitivity analysis of the constraint matrix in piecewise linear fractional programming. Journal of Industrial and Management Optimization, 2010, 6 (2) : 347-361. doi: 10.3934/jimo.2010.6.347
[19]

Konstantinos A. Draziotis, Anastasia Papadopoulou. Improved attacks on knapsack problem with their variants and a knapsack type ID-scheme. Advances in Mathematics of Communications, 2018, 12 (3) : 429-449. doi: 10.3934/amc.2018026
[20]

Majid Khalilzadeh, Hossein Neghabi, Ramin Ahadi. An application of approximate dynamic programming in multi-period multi-product advertising budgeting. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021202

2020 Impact Factor: 1.801

Tools

Metrics

  • PDF downloads (464)
  • HTML views (548)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy