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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 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 & Management Optimization, doi: 10.3934/jimo.2021013
References:
[1]

R. Bellman, Dynamic Programming, Science, 153 (1966), 34-37.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

show all references

References:
[1]

R. Bellman, Dynamic Programming, Science, 153 (1966), 34-37.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

Figure 1.  Inventory level of each item vs. time
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Figure 2.  The flowchart of the proposed solution method
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Figure 3.  The inventory level of each item over time
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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
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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)
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$ 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}
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$ 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
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$ 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}
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$ 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
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$ 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)
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$ 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)
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