Deteriorating inventory with preservation technology under price- and stock-sensitive demand

Magfura Pervin; Sankar Kumar Roy; Gerhard Wilhelm Weber

doi:10.3934/jimo.2019019

Article Contents

2020, Volume 16, Issue 4: 1585-1612. Doi: 10.3934/jimo.2019019

This issue Previous Article Application of a modified CES production function model based on improved firefly algorithm Next Article A collaborative EPQ inventory model for a three-echelon supply chain with multiple products considering the effect of marketing effort on demand

Deteriorating inventory with preservation technology under price- and stock-sensitive demand

1.
Department of Applied Mathematics with Oceanology and Computer Programming, Vidyasagar University, Midnapore-721102, West Bengal, India
2.
Faculty of Engineering Management, Chair of Marketing and Economic Engineering, Poznan University of Technology, ul. Strzelecka 11, 60-965 Poznan, Poland

^* Corresponding Author: sankroy2006@gmail.com
^* Corresponding Author: sankroy2006@gmail.com

Received: July 2018

Revised: September 2018

Early access: March 2019

Published: May 2020

The author, Magfura Pervin is very much thankful to University Grants Commission (UGC) of India for providing financial support to continue this research work under [MANF(UGC)] scheme: Sanctioned letter number [F1-17.1/2012-13/MANF-2012-13-MUS-WES-19170 /(SA-Ⅲ/Website)] dated 28/02/2013.
The research of Sankar Kumar Roy is partially supported by the Portuguese Foundation for Science and Technology ("FCT-Fundação para a Ciência e a Tecnologia"), through the CIDMA - Center for Research and Development in Mathematics and Applications, University of Aveiro, Portugal, within project UID/MAT/04106/2019.
The research of Gerhard-Wilhelm Weber (Institute of Applied Mathematics, Middle East Technical University, 06800, Ankara, Turkey) is partially supported by the Portuguese Foundation for Science and Technology ("FCT-Fundação para a Ciência e a Tecnologia"), through the CIDMA - Center for Research and Development in Mathematics and Applications.

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Abstract

In this paper, we formulate and solve an Economic Production Quantity inventory model with deteriorating items. To reduce the rate of deterioration, we apply a preservation technology and calculate the amount for preservation technology investment. The demand function is dependent on stock-level and price. We assume that the production rate is linearly dependent on time, based on customer demand. Shortages are allowed in our consideration, and the shortages amount is partially backlogged for the interested customers for the next slot. The effect of inflation is incorporated, which indicates a critical factor in modern days. Our main objective is to find the optimal cycle length and the optimal amount of preservation technology investment by adjusting the inflation rate with maximizing the profit. A numerical example is provided to illustrate the features and advances of the model. A sensitivity analysis with respect to major parameters is performed in order to assess the stability of our model. The paper ends with a conclusion and an outlook at possible future directions.

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

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Figure 1. The inventory path-line

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Figure 2. Concavity of the profit function; including are $ x^* $, $ r_1^* $ and $ TP $, as x-axis, y-axis and z-axis, respectively

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Figure 3. Concavity of the profit function; including are $ r_1^* $, $ r_2^* $ and $ TP $, as x-axis, y-axis and z-axis, respectively

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Figure 4. Concavity of the profit function; including are $ x^* $, $ r_2^* $ and $ TP $, as x-axis, y-axis and z-axis, respectively

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Figure 5. Effect of $ \theta $ on total profit $ TP $

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Figure 6. Effect of $ \theta $ on selling price $ r_1 $

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Figure 7. Effect of demand parameter $ b_1 $ on preservation amount $ x $

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Table 1. Contributions of some authors related to inventory model

Authors	Deterio-rations	Preservation Technology	Stock-dependent	Price-dependent	Inflation	Partial backorder
Bhunia et al. (2009)	$\surd$
Dave & Patel (1981)	$\surd$
Papachristos & Skouri (2003)	$\surd$					$\surd$
Wee (1997)	$\surd$			$\surd$
Hsu et al. (2010)	$\surd$	$\surd$
Maihami and Abadi (2012)	$\surd$					$\surd$
Khedlekar et al. (2016)	$\surd$	$\surd$
Murr and Morris (1975)	$\surd$	$\surd$
Buzacott (1975)					$\surd$
Gupta and Vrat (1986)			$\surd$
Liao et al. (2000)	$\surd$				$\surd$	$\surd$
Ray and Chaudhuri (1997)			$\surd$		$\surd$
Pal et al. (2014)	$\surd$		$\surd$	$\surd$	$\surd$
Sarkar and Moon (2011)					$\surd$
Shah et al. (2017)	$\surd$	$\surd$
Teng and Chang (2005)	$\surd$		$\surd$	$\surd$
Widyadana and Wee (2010)	$\surd$			$\surd$
This work	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$	$\surd$

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Table 2. Sensitivity analysis on decision variables with respect to major parameters

Input Parameters	Output Variables and Parameters	$-50\%$	$-20\%$	$0\%$	$+20\%$	$+50\%$
	$T$	1.213	1.120	0.932	0.730	0.691
	$x$	4.5	4.0	3.5	2.7	2.1
$a_1$	$r_1$	12.7212	15.1139	16.2681	17.2527	18.1462
	$r_2$	11.58	12.86	13.42	14.23	15.70
	$TP$	1980.21	2378.00	2549.00	2645.72	2868.43
	$T$	0.647	0.720	0.839	0.906	0.987
	$x$	1.8	2.2	2.6	2.9	3.3
$c_1$	$r_1$	19.3510	18.7220	17.5701	16.0148	14.7370
	$r_2$	15.65	13.76	12.59	11.38	10.44
	$TP$	3376.30	3048.00	2849.21	2432.43	2135.67
	$T$	1.211	1.053	0.916	0.848	0.720
	$x$	3.4	3.0	2.9	2.6	2.1
$h_1$	$r_1$	15.5108	16.7233	17.3604	17.9451	18.4817
	$r_2$	11.55	11.89	12.10	12.93	13.41
	$TP$	3142.31	2991.00	2635.21	2307.10	2069.38
	$T$	1.120	0.984	0.764	0.793	0.807
	$x$	3.6	3.4	3.1	2.8	2.5
$b_1$	$r_1$	16.4707	16.8574	17.3612	18.7185	20.7354
	$r_2$	10.73	11.81	12.46	12.79	13.99
	$TP$	2376.00	2578.82	2854.06	3075.56	3243.61
	$T$	0.867	0.867	0.867	0.867	0.835
	$x$	2.7	2.8	2.9	3.0	3.2
$\delta$	$r_1$	19.4603	18.8746	18.2163	17.7563	17.3356
	$r_2$	11.77	12.13	12.85	13.23	13.79
	$TP$	2649.01	2546.24	2480.23	2465.35	2435.31
	$T$	0.979	0.964	0.932	0.881	0.831
	$x$	3.1	3.0	2.8	2.7	2.5
$c$	$r_1$	16.3745	16.8100	17.1344	17.6007	17.9103
	$r_2$	11.45	11.79	11.97	12.32	12.85
	$TP$	3036.54	2850.23	2662.04	2567.54	2387.37
	$T$	1.158	1.074	0.964	0.793	0.549
	$x$	3.45	3.16	2.90	2.75	2.53
$\theta$	$r_1$	14.0826	16.7284	18.2352	20.3577	23.1437
	$r_2$	9.76	10.45	13.12	15.67	17.88
	$TP$	2719.03	2680.25	2559.47	2478.53	2376.82

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