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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

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

* Corresponding author

Received  May 2020 Revised  October 2020 Published  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.

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
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##### References:
Inventory level of each item vs. time
The flowchart of the proposed solution method
The inventory level of each item over time
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
 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
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)
 $k^{'}_{1}$ 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)
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}
 $i_{1}$ $0\leq i_{1}$ $166.05\leq i_{1}$ $332.555\leq i_{1}$ $496.05\leq i_{1}$ $653.1\leq i_{1}$ $<166.05$ $<332.55$ $<496.05$ $<653.1$ $\leq800$ $i^{'}_{1}$ {0} {0, 1} {0, 1, 2} {0, 1, 2, 3} {0, 1, 2, 3, 4}
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
 $k^{'}_{2}$ 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
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}
 $i_{2}$ $0\leq i_{2}$ $73.8\leq i_{2}$ $155\leq i_{2}$ $166.05\leq i_{2}$ $240.3\leq i_{2}$ $<73.8$ $<155$ $<166.05$ $<240.3$ $<254.2$ $i^{'}_2$ {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}
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
 $i_{2}$ $0\leq i_{2}$ $<73.8$ $73.8\leq i_{2}$ $<155$ $155\leq i_{2}$ $<166.05$ $166.05\leq i_{2}$ $<240.3$ $240.3\leq i_{2}$ $<254.2$ $f(2, i_{2})$ 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
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)
 $k^{'}_{3}$ 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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