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

Mahdi Karimi; Seyed Jafar Sadjadi

doi:10.3934/jimo.2021013

Article Contents

2022, Volume 18, Issue 2: 1145-1160. Doi: 10.3934/jimo.2021013

This issue Previous Article A new adaptive method to nonlinear semi-infinite programming Next Article Decision-making in a retailer-led closed-loop supply chain involving a third-party logistics provider

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
^* Corresponding author

Received: May 2020

Revised: October 2020

Early access: December 2020

Published: March 2022

Abstract Full Text(HTML) Figure(3) / Table(7) Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 90B05, 90C26; Secondary: 90C39.

Citation:

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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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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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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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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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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} $
$ 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} $
$ 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} $
$ 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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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} $
$ 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} $
$ 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} $
$ 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} $
$ 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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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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