An economic order quantity for deteriorating items with allowable rework of deteriorated products

Mahdi Karimi; Seyed Jafar Sadjadi; Alireza Ghasemi Bijaghini

doi:10.3934/jimo.2018126

Article Contents

2019, Volume 15, Issue 4: 1857-1879. Doi: 10.3934/jimo.2018126

This issue Previous Article Capital-constrained supply chain with multiple decision attributes: Decision optimization and coordination analysis Next Article Sparse probabilistic Boolean network problems: A partial proximal-type operator splitting method

An economic order quantity for deteriorating items with allowable rework of deteriorated products

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

* Corresponding author
* Corresponding author

Received: December 2017

Revised: April 2018

Early access: July 2019

Published: July 2019

Abstract / Introduction Full Text(HTML) Figure(6) / Table(9) Related Papers Cited by

Abstract

This paper presents an inventory model for deteriorating items with variable demand when shortage is permitted and quantity discount in purchase cost, and rework on deteriorating products are also allowed. The main idea of this research is to study the effects of the discount and the rework on the inventory costs. In this paper, it is assumed that for a certain quantity of purchased items, the seller would offer a discount and the manager would have the choice to either accept the discount or dismiss. On the other hand, there is also a similar decision-making scenario, where the manager makes a decision to reduce the total costs by using the rework and reducing the shortage periods or reducing the total costs by ignoring the rework cost and increasing the shortage periods. The implementation of the mathematical model is illustrated with a numerical example and sensitivity analysis describes the effects of the parameters on the total costs. The results show that the rework will decrease the total costs of the inventory system, significantly.

Keywords:

Mathematics Subject Classification: Primary: 90B05, 90C30; Secondary: 90C25.

Citation:

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Figure 2. Inventory level $(I)$ vs. Time.

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Figure 1. Purchase cost vs. Ordering quantity$(Q_i)$

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Figure 3. Sensitivity analysis for backlogging rate

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Figure 4. Sensitivity analysis for deterioration rate

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Figure 5. Sensitivity analysis for rework percentage

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Figure 6. Sensitivity analysis for discount threshold

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Table 1. A. Review of previous works

Number		1	2	3	4	5	6	7	8	9	10
Paper		[49]	[29]	[12]	[46]	[48]	[5]	[9]	[7]	[37]	[54]
Model	Ordering	$\surd$	$\surd$	$\surd$	$\surd$			$\surd$	$\surd$	$\surd$
Type	Production					$\surd$	$\surd$				$\surd$
	Three									$\surd$
Levels	Two										$\surd$
	one	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$
Multi	No	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$
Product	Yes
Inflation	Discount							$\surd$		$\surd$
and	Inflation				$\surd$			$\surd$
Discount	Not Allowed	$\surd$	$\surd$	$\surd$		$\surd$	$\surd$		$\surd$		$\surd$
Time	Infinity			$\surd$						$\surd$
Horizon	Finitie	$\surd$	$\surd$		$\surd$	$\surd$	$\surd$	$\surd$	$\surd$		$\surd$
Ware	Two
-houses	One	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$
Suply	Open	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$
Chain	Cyclic
	Lost Sales
Shortage	Part. Backlog		$\surd$		$\surd$			$\surd$
	Com. Backlog	$\surd$		$\surd$		$\surd$
	Not Allowed						$\surd$		$\surd$	$\surd$	$\surd$
Lead	Psitive
Time	Zero	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$
Deter.	Variable	$\surd$	$\surd$	$\surd$						$\surd$
rate	Fixed				$\surd$	$\surd$	$\surd$	$\surd$	$\surd$		$\surd$
	Fuzzy
Demand	Prob.
	Det. Variable	$\surd$	$\surd$	$\surd$	$\surd$		$\surd$	$\surd$	$\surd$		$\surd$
	Det. Fixed					$\surd$				$\surd$
Number		11	12	13	14	15	16	17	18	19	20
Paper		[28]	[2]	[6]	[10]	[11]	[14]	[16]	[17]	[18]	[21]
Model	Ordering	$\surd$	$\surd$		$\surd$		$\surd$		$\surd$	$\surd$	$\surd$
Type	Production			$\surd$		$\surd$		$\surd$
	Three				$\surd$	$\surd$		$\surd$
Levels	Two			$\surd$
	one	$\surd$	$\surd$				$\surd$		$\surd$	$\surd$	$\surd$
Multi	No	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$		$\surd$
Product	Yes									$\surd$
Inflation	Discount				$\surd$
and	Inflation								$\surd$
Discount	Not Allowed	$\surd$	$\surd$	$\surd$		$\surd$	$\surd$	$\surd$		$\surd$	$\surd$
Time	Infinity	$\surd$	$\surd$	$\surd$						$\surd$	$\surd$
Horizon	Finitie				$\surd$	$\surd$	$\surd$	$\surd$	$\surd$
Ware	Two		$\surd$
-houses	One	$\surd$		$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$
Suply	Open	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$
Chain	Cyclic

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Table 1. B. Review of previous works

Number		11	12	13	14	15	16	17	18	19	20
Paper		[28]	[2]	[6]	[10]	[11]	[14]	[16]	[17]	[18]	[21]
	Lost Sales					$\surd$
Shortage	Part. Backlog		$\surd$					$\surd$
	Com. Backlog	$\surd$
	Not Allowed			$\surd$	$\surd$		$\surd$	$\surd$		$\surd$	$\surd$
Lead	Psitive		$\surd$
Time	Zero	$\surd$		$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$
Deter.	Variable	$\surd$			$\surd$
rate	Fixed		$\surd$	$\surd$		$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$
	Fuzzy	$\surd$
Demand	Prob.
	Det. Variable		$\surd$		$\surd$	$\surd$	$\surd$		$\surd$	$\surd$	$\surd$
	Det. Fixed			$\surd$				$\surd$
Number		21	22	23	24	25	26	27	28	29	30
Paper		[24]	[30]	[33]	[36]	[38]	[39]	[40]	[51]	[13]	[57]
Model	Ordering	$\surd$	$\surd$		$\surd$	$\surd$			$\surd$
Type	Production			$\surd$			$\surd$	$\surd$		$\surd$	$\surd$
	Three		$\surd$
Levels	Two			$\surd$					$\surd$		$\surd$
	one	$\surd$			$\surd$	$\surd$	$\surd$	$\surd$		$\surd$
Multi	No	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$
Product	Yes
Inflation	Discount								$\surd$
and	Inflation						$\surd$			$\surd$
Discount	Not Allowed	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$		$\surd$			$\surd$
Time	Infinity									$\surd$	$\surd$
Horizon	Finitie	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$
Ware	Two				$\surd$
-houses	One	$\surd$	$\surd$	$\surd$		$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$
Suply	Open	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$
Chain	Cyclic
	Lost Sales
Shortage	Part. Backlog	$\surd$	$\surd$		$\surd$		$\surd$
	Com. Backlog					$\surd$
	Not Allowed			$\surd$				$\surd$	$\surd$	$\surd$	$\surd$
Lead	Psitive									$\surd$
Time	Zero	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$		$\surd$
Deter.	Variable			$\surd$		$\surd$	$\surd$
rate	Fixed	$\surd$	$\surd$		$\surd$			$\surd$	$\surd$	$\surd$	$\surd$
	Fuzzy
Demand	Prob.			$\surd$
	Det. Variable	$\surd$	$\surd$		$\surd$	$\surd$	$\surd$	$\surd$	$\surd$		$\surd$
	Det. Fixed				$\surd$					$\surd$
Number		31	32	33	34	35	36	37	38	39	40
Paper		[58]	[60]	[65]	[66]	[59]	[27]	[68]	[3]	[20]	[23]
Model	Ordering	$\surd$	$\surd$	$\surd$	$\surd$		$\surd$		$\surd$	$\surd$	$\surd$
Type	Production					$\surd$		$\surd$
	Three
Levels	Two	$\surd$	$\surd$			$\surd$
	one			$\surd$	$\surd$		$\surd$	$\surd$	$\surd$	$\surd$	$\surd$

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Table 1. C. Review of previous works

Number		31	32	33	34	35	36	37	38	39	40
Paper		[58]	[60]	[65]	[66]	[59]	[27]	[68]	[3]	[20]	[23]
Multi	No	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$
Product	Yes
Inflation	Discount						$\surd$
and	Inflation		$\surd$	$\surd$	$\surd$					$\surd$	$\surd$
Discount	Not Allowed	$\surd$				$\surd$		$\surd$	$\surd$
Time	Infinity	$\surd$		$\surd$	$\surd$	$\surd$			$\surd$
Horizon	Finitie		$\surd$				$\surd$	$\surd$		$\surd$	$\surd$
Ware	Two			$\surd$	$\surd$		$\surd$	$\surd$	$\surd$
-houses	One	$\surd$	$\surd$			$\surd$				$\surd$	$\surd$
Suply	Open	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$
Chain	Cyclic
	Lost Sales
Shortage	Part. Backlog		$\surd$	$\surd$	$\surd$	$\surd$	$\surd$		$\surd$
	Com. Backlog	$\surd$								$\surd$	$\surd$
	Not Allowed							$\surd$
Lead	Psitive					$\surd$			$\surd$
Time	Zero	$\surd$	$\surd$	$\surd$	$\surd$		$\surd$	$\surd$		$\surd$	$\surd$
Deter.	Variable		$\surd$			$\surd$
rate	Fixed	$\surd$		$\surd$	$\surd$		$\surd$	$\surd$	$\surd$	$\surd$	$\surd$
	Fuzzy
Demand	Prob.
	Det. Variable				$\surd$	$\surd$	$\surd$		$\surd$	$\surd$	$\surd$
	Det. Fixed	$\surd$	$\surd$	$\surd$				$\surd$
Number		41	42	43	44	45	46	47	48	49	50
Paper		[32]	[34]	[35]	[47]	[56]	[63]	[69]	[55]	[31]	[8]
Model	Ordering		$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$
Type	Production	$\surd$
	Three
Levels	Two	$\surd$							$\surd$	$\surd$	$\surd$
	one		$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$
Multi	No	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$
Product	Yes
Inflation	Discount
and	Inflation					$\surd$		$\surd$
Discount	Not Allowed	$\surd$	$\surd$	$\surd$	$\surd$		$\surd$		$\surd$	$\surd$	$\surd$
Time	Infinity	$\surd$
Horizon	Finitie		$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$
Ware	Two							$\surd$
-houses	One	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$		$\surd$	$\surd$	$\surd$
Suply	Open	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$
Chain	Cyclic
	Lost Sales
Shortage	Part. Backlog		$\surd$				$\surd$	$\surd$
	Com. Backlog			$\surd$	$\surd$	$\surd$			$\surd$
	Not Allowed	$\surd$								$\surd$	$\surd$
Lead	Psitive
Time	Zero	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$

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Table 1. D. Review of previous works

Number		41	42	43	44	45	46	47	48	49	50
Paper		[32]	[34]	[35]	[47]	[56]	[63]	[69]	[55]	[31]	[8]
Deter.	Variable		$\surd$	$\surd$	$\surd$					$\surd$	$\surd$
rate	Fixed	$\surd$				$\surd$	$\surd$	$\surd$	$\surd$
	Fuzzy
Demand	Prob.
	Det. Variable		$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$		$\surd$	$\surd$
	Det. Fixed	$\surd$							$\surd$
Number		51	52	53	54	Percentage(rounded)				This Paper
Paper		[43]	[45]	[44]	[41]	$\%$				$^*$
Model	Ordering		$\surd$	$\surd$	$\surd$	$73\%$				$\surd$
Type	Production	$\surd$				$27\%$
	Three					$10\%$
Levels	Two		$\surd$			$26\%$
	one	$\surd$		$\surd$	$\surd$	$64\%$				$\surd$
Multi	No	$\surd$	$\surd$	$\surd$	$\surd$	$98\%$				$\surd$
Product	Yes					$2\%$
Inflation	Discount		$\surd$			$16\%$				$\surd$
and	Inflation		$\surd$			$24\%$
Discount	Not Allowed	$\surd$		$\surd$	$\surd$	$68\%$
Time	Infinity					$27\%$
Horizon	Finitie	$\surd$	$\surd$	$\surd$	$\surd$	$73\%$				$\surd$
Ware	Two					$15\%$
-houses	One	$\surd$	$\surd$	$\surd$	$\surd$	$85\%$				$\surd$
Suply	Open	$\surd$	$\surd$	$\surd$	$\surd$	$100\%$
Chain	Cyclic					$0\%$				$\surd$
	Lost Sales					$2\%$
Shortage	Part. Backlog		$\surd$	$\surd$	$\surd$	$39\%$				$\surd$
	Com. Backlog					$22\%$
	Not Allowed	$\surd$				$37\%$
Lead	Psitive					$8\%$
Time	Zero	$\surd$	$\surd$	$\surd$	$\surd$	$92\%$				$\surd$
Deter.	Variable			$\surd$		$32\%$
rate	Fixed	$\surd$	$\surd$		$\surd$	$68\%$				$\surd$
	Fuzzy					$2\%$
Demand	Prob.			$\surd$		$4\%$
	Det. Variable	$\surd$	$\surd$		$\surd$	$72\%$				$\surd$
	Det. Fixed					$22\%$

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Table 2. Optimal solution of Numerical Example

$n$	$TC$	$i$	$t_i$	$tw_i$	$s_i$
6	12785.79	1	0	0.0310	0.0310
7	12279.73	2	0.0310	0.0314	0.0410
8	7978.37	3	0.0410	0.0410	0.0520
9	14197.62	4	0.0521	0.0522	0.0814
10	14886.37	5	0.0816	0.0817	0.194
11	21056.36	6	0.1965	0.1972	0.2075
		7	0.2075	0.2086	0.2335
$n^*$=8		8	0.2345	0.2346	1.3667
$TC^*$=7978.37		9	-	1.3681	2

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Table 3. Sensitivity analysis for backlogging rate

$\beta$	0(Complete backlogging)	10	20	40	$\infty$(No shortage)
$TC$	7874.34	7894.87	7978.37	8078.46	8213.75

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Table 4. Sensitivity analysis for deterioration rate

$\rho$	0.01	0.016	0.02	0.024	0.03
$TC$	7975.35	7977.15	7978.37	7979.56	7981.37

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Table 5. Sensitivity analysis for rework percentage

$\eta$	0.1	0.16	0.2	0.24	0.3
$TC$	7978.63	7978.47	7978.37	7978.25	7978.09

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Table 6. Sensitivity analysis for discount threshold

$M$	50	200	400	600	$\infty$
$TC$	12917	7978.37	10546	13848	11180

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