# American Institute of Mathematical Sciences

doi: 10.3934/jimo.2020148

## Effect of reliability on varying demand and holding cost on inventory system incorporating probabilistic deterioration

 Department of Mathematics, National Institute of Technology Puducherry, Karaikal-609609, India

*Corresponding author: G. S. Mahapatra

Received  June 2020 Revised  August 2020 Published  September 2020

This paper presents a mathematical framework to derive an inventory model for time, reliability, and advertisement dependent demand. This paper considers the demand rate is high initially, and then the demand rate reduces later stage, which reflects the situation related to cash in hand. The uncertain deterioration of the product presents through Uniform, Triangular, and Double Triangular probability distributions. The holding cost of the proposed inventory system is dependent on the reliability of the item to make this study a more realistic one. This proposed inventory system allows the situation of shortage and partially backlogged at a fixed rate. Numerical examples, along with managerial implications and sensitivity analysis of the inventory parameters, discuss to examine the effect of changes on the optimal total inventory cost.

Citation: Sudip Adak, G. S. Mahapatra. Effect of reliability on varying demand and holding cost on inventory system incorporating probabilistic deterioration. Journal of Industrial & Management Optimization, doi: 10.3934/jimo.2020148
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##### References:
Proposed inventory model with inventory vs time
Total cost vs. $T_{1}$ vs. $T_{2}$ for Uniformly distributed deterioration
Total cost vs. $T_{1}$ vs. $T_{2}$ for Triangular distribution deterioration
Total cost vs. $T_{1}$ vs. $T_{2}$ for Triangular distribution deterioration
Percentage change of total profit vs change of parameter
Percentage change of total profit vs change of parameter
Percentage change of total profit vs change of parameter
Percentage change of total profit vs change of parameter
Contributions of the proposed model with compare to previous studies
 Author's Cash in hand Demand depend on Holding Cost depend on Deterioration Backlog Giri & Chaudhuri (1998) NA stock Non linear NA No Chang (2004) NA stock Non linear constant No Skouri et al. (2009) NA ramp type NA Weibull Yes Sana (2010) NA stock NA probabilistic No Sett et al. (2012) NA time demand NA NA No Sarkar & Sarkar (2013) NA time NA probabilistic No Chowdhury et al. (2014) NA time-quadratic NA time demand Yes Ghoreishi et al. (2014) NA price & time NA non-instantaneous Yes Ghosh et al. (2015) NA stock Constant constant No Wu & Zhao (2015) NA inventory & time NA constant No Bhunia et al. (2015) NA time, advertisement NA constant Yes Alfares & Ghaithan (2016) NA price time NA No Chanda & Kumar (2016) NA advertising & price NA NA No Sanni & Chukwu (2016) NA deterministic NA Weibull Yes Shah & Vaghela (2016) NA time & advertisement NA constant No Mahapatra et al. (2017) NA time & reliability NA constant Yes Mokhtari et al. (2017) NA stochastic NA constant Yes Pervin et al. (2018) NA time time Weibull Yes Lotfi et al. (2018) NA interdependent NA NA Yes Dey et al. (2019) NA selling price NA NA Yes Pervin et al. (2019) NA price and stock purchasing cost constant Yes Pervin et al. (2020) NA time & price NA constant Yes Roy et al. (2020) NA probabilistic Constant Weibull Yes This paper Consider time, reliability, advertisement reliability probabilistic Yes
 Author's Cash in hand Demand depend on Holding Cost depend on Deterioration Backlog Giri & Chaudhuri (1998) NA stock Non linear NA No Chang (2004) NA stock Non linear constant No Skouri et al. (2009) NA ramp type NA Weibull Yes Sana (2010) NA stock NA probabilistic No Sett et al. (2012) NA time demand NA NA No Sarkar & Sarkar (2013) NA time NA probabilistic No Chowdhury et al. (2014) NA time-quadratic NA time demand Yes Ghoreishi et al. (2014) NA price & time NA non-instantaneous Yes Ghosh et al. (2015) NA stock Constant constant No Wu & Zhao (2015) NA inventory & time NA constant No Bhunia et al. (2015) NA time, advertisement NA constant Yes Alfares & Ghaithan (2016) NA price time NA No Chanda & Kumar (2016) NA advertising & price NA NA No Sanni & Chukwu (2016) NA deterministic NA Weibull Yes Shah & Vaghela (2016) NA time & advertisement NA constant No Mahapatra et al. (2017) NA time & reliability NA constant Yes Mokhtari et al. (2017) NA stochastic NA constant Yes Pervin et al. (2018) NA time time Weibull Yes Lotfi et al. (2018) NA interdependent NA NA Yes Dey et al. (2019) NA selling price NA NA Yes Pervin et al. (2019) NA price and stock purchasing cost constant Yes Pervin et al. (2020) NA time & price NA constant Yes Roy et al. (2020) NA probabilistic Constant Weibull Yes This paper Consider time, reliability, advertisement reliability probabilistic Yes
Comparison of the proposed model for deterioration
 Range of $\theta$ Distribution $\theta$ $\mathbf{T} _{1}^{\ast }$ $\mathbf{T}_{2}^{\ast }$ $\mathbf{TC}\left( T_{1}^{\ast }, \text{ }T_{2}^{\ast }\right) \mathbf{(＄)}$ $\mathbf{0.7\leq 0.9}$ Uniform $\mathbf{0.8}$ $\mathbf{0.166}$ $\mathbf{0.418}$ $\mathbf{151.144}$ $\mathbf{0.7\leq 0.85\leq 0.9}$ Triangular $\mathbf{0.82}$ $\mathbf{\ 0.142}$ $\mathbf{0.418}$ $\mathbf{151.615}$ $\mathbf{0.7\leq 0.85\leq 0.9}$ Double Triangular $\mathbf{0.83}$ $\mathbf{0.13}$ $\mathbf{0.418}$ $\mathbf{151.795}$ $\mathbf{0.8\leq 0.9}$ Uniform $\mathbf{0.85}$ $\mathbf{0.108}$ $\mathbf{0.418}$ $\mathbf{152.07}$ $\mathbf{0.8\leq 0.83\leq 0.9}$ Triangular $\mathbf{0.84}$ $\mathbf{\ 0.119}$ $\mathbf{0.418}$ $\mathbf{151.946}$ $\mathbf{0.8\leq 0.83\leq 0.9}$ Double Triangular $\mathbf{0.83}$ $\mathbf{0.13}$ $\mathbf{0.418}$ $\mathbf{151.795}$ $\mathbf{0.76\leq 0.96}$ Uniform $\mathbf{0.86}$ $\mathbf{0.098}$ $\mathbf{0.418}$ $\mathbf{152.171}$ $\mathbf{0.76\leq 0.83\leq 0.96}$ Triangular $\mathbf{0.85}$ $\mathbf{\ 0.108}$ $\mathbf{0.418}$ $\mathbf{152.07}$ $\mathbf{0.76\leq 0.83\leq 0.96}$ Double Triangular $\mathbf{0.84}$ $\mathbf{0.119}$ $\mathbf{0.418}$ $\mathbf{151.946}$
 Range of $\theta$ Distribution $\theta$ $\mathbf{T} _{1}^{\ast }$ $\mathbf{T}_{2}^{\ast }$ $\mathbf{TC}\left( T_{1}^{\ast }, \text{ }T_{2}^{\ast }\right) \mathbf{(＄)}$ $\mathbf{0.7\leq 0.9}$ Uniform $\mathbf{0.8}$ $\mathbf{0.166}$ $\mathbf{0.418}$ $\mathbf{151.144}$ $\mathbf{0.7\leq 0.85\leq 0.9}$ Triangular $\mathbf{0.82}$ $\mathbf{\ 0.142}$ $\mathbf{0.418}$ $\mathbf{151.615}$ $\mathbf{0.7\leq 0.85\leq 0.9}$ Double Triangular $\mathbf{0.83}$ $\mathbf{0.13}$ $\mathbf{0.418}$ $\mathbf{151.795}$ $\mathbf{0.8\leq 0.9}$ Uniform $\mathbf{0.85}$ $\mathbf{0.108}$ $\mathbf{0.418}$ $\mathbf{152.07}$ $\mathbf{0.8\leq 0.83\leq 0.9}$ Triangular $\mathbf{0.84}$ $\mathbf{\ 0.119}$ $\mathbf{0.418}$ $\mathbf{151.946}$ $\mathbf{0.8\leq 0.83\leq 0.9}$ Double Triangular $\mathbf{0.83}$ $\mathbf{0.13}$ $\mathbf{0.418}$ $\mathbf{151.795}$ $\mathbf{0.76\leq 0.96}$ Uniform $\mathbf{0.86}$ $\mathbf{0.098}$ $\mathbf{0.418}$ $\mathbf{152.171}$ $\mathbf{0.76\leq 0.83\leq 0.96}$ Triangular $\mathbf{0.85}$ $\mathbf{\ 0.108}$ $\mathbf{0.418}$ $\mathbf{152.07}$ $\mathbf{0.76\leq 0.83\leq 0.96}$ Double Triangular $\mathbf{0.84}$ $\mathbf{0.119}$ $\mathbf{0.418}$ $\mathbf{151.946}$
Sensitivity analysis of the proposed inventory system for parameters
 Parameter Change(%) $T_{1}$ $T_{2}$ $TC$ Change $TC(\%)$ $-20$ $0.166$ $0.334$ $143.064$ $-5.346$ $T$ $-10$ $0.166$ $0.376$ $146.444$ $-3.110$ $10$ $0.166$ $0.459$ $156.796$ $3.739$ $20$ $0.166$ $0.501$ $163.153$ $7.946$ $-20$ $0.166$ $0.418$ $96.319$ $-36.274$ $A$ $-10$ $0.166$ $0.418$ $119.950$ $-20.639$ $10$ $0.166$ $0.418$ $191.193$ $26.498$ $20$ $0.166$ $0.418$ $241.458$ $59.754$ $-20$ $0.085$ $0.442$ $140.954$ $-6.742$ $C_{d}$ $-10$ $0.127$ $0.430$ $146.384$ $-3.149$ $10$ $0.201$ $0.406$ $155.207$ $2.688$ $20$ $0.235$ $0.396$ $158.566$ $4.911$ $-20$ $0.250$ $0.431$ $142.313$ $-5.843$ $C_{h}$ $-10$ $0.205$ $0.424$ $147.077$ $-2.691$ $10$ $0.131$ $0.411$ $154.715$ $2.363$ $20$ $0.099$ $0.405$ $157.929$ $4.489$ $-20$ $0.166$ $0.365$ $116.975$ $-22.607$ $C_{s}$ $-10$ $0.166$ $0.392$ $134.803$ $-10.812$ $10$ $0.166$ $0.441$ $166.178$ $9.947$ $20$ $0.166$ $0.462$ $180.055$ $19.128$ $-20$ $0.166$ $0.430$ $178.469$ $18.079$ $C_{l}$ $-10$ $0.166$ $0.424$ $164.852$ $9.069$ $10$ $0.166$ $0.411$ $137.350$ $-9.126$ $20$ $0.166$ $0.405$ $123.473$ $-18.308$ $-10$ $0.074$ $0.400$ $151.598$ $0.300$ $r$ $-5$ $0.121$ $0.409$ $151.493$ $0.231$ $5$ $0.210$ $0.425$ $150.459$ $-0.453$ $10$ $0.252$ $0.431$ $149.375$ $-1.171$ $-10$ $0.283$ $0.418$ $146.868$ $-2.829$ $\theta$ $-5$ $0.220$ $0.418$ $149.608$ $-1.016$ $5$ $0.119$ $0.418$ $151.946$ $0.530$ $10$ $0.079$ $0.418$ $152.317$ $0.776$
 Parameter Change(%) $T_{1}$ $T_{2}$ $TC$ Change $TC(\%)$ $-20$ $0.166$ $0.334$ $143.064$ $-5.346$ $T$ $-10$ $0.166$ $0.376$ $146.444$ $-3.110$ $10$ $0.166$ $0.459$ $156.796$ $3.739$ $20$ $0.166$ $0.501$ $163.153$ $7.946$ $-20$ $0.166$ $0.418$ $96.319$ $-36.274$ $A$ $-10$ $0.166$ $0.418$ $119.950$ $-20.639$ $10$ $0.166$ $0.418$ $191.193$ $26.498$ $20$ $0.166$ $0.418$ $241.458$ $59.754$ $-20$ $0.085$ $0.442$ $140.954$ $-6.742$ $C_{d}$ $-10$ $0.127$ $0.430$ $146.384$ $-3.149$ $10$ $0.201$ $0.406$ $155.207$ $2.688$ $20$ $0.235$ $0.396$ $158.566$ $4.911$ $-20$ $0.250$ $0.431$ $142.313$ $-5.843$ $C_{h}$ $-10$ $0.205$ $0.424$ $147.077$ $-2.691$ $10$ $0.131$ $0.411$ $154.715$ $2.363$ $20$ $0.099$ $0.405$ $157.929$ $4.489$ $-20$ $0.166$ $0.365$ $116.975$ $-22.607$ $C_{s}$ $-10$ $0.166$ $0.392$ $134.803$ $-10.812$ $10$ $0.166$ $0.441$ $166.178$ $9.947$ $20$ $0.166$ $0.462$ $180.055$ $19.128$ $-20$ $0.166$ $0.430$ $178.469$ $18.079$ $C_{l}$ $-10$ $0.166$ $0.424$ $164.852$ $9.069$ $10$ $0.166$ $0.411$ $137.350$ $-9.126$ $20$ $0.166$ $0.405$ $123.473$ $-18.308$ $-10$ $0.074$ $0.400$ $151.598$ $0.300$ $r$ $-5$ $0.121$ $0.409$ $151.493$ $0.231$ $5$ $0.210$ $0.425$ $150.459$ $-0.453$ $10$ $0.252$ $0.431$ $149.375$ $-1.171$ $-10$ $0.283$ $0.418$ $146.868$ $-2.829$ $\theta$ $-5$ $0.220$ $0.418$ $149.608$ $-1.016$ $5$ $0.119$ $0.418$ $151.946$ $0.530$ $10$ $0.079$ $0.418$ $152.317$ $0.776$
Sensitivity analysis of the proposed inventory model for parameters
 Parameter Change(%) $T_{1}$ $T_{2}$ $TC$ Change $TC(\%)$ $-20$ $0.306$ $0.418$ $145.956$ $-3.432$ $a$ $-10$ $0.227$ $0.418$ $149.419$ $-1.141$ $10$ $0.116$ $0.418$ $151.970$ $0.547$ $20$ $0.076$ $0.418$ $152.329$ $0.784$ $-10$ $0.047$ $0.436$ $160.574$ $6.239$ $b$ $-5$ $0.106$ $0.427$ $156.287$ $3.403$ $5$ $0.227$ $0.409$ $144.764$ $-4.221$ $10$ $0.289$ $0.400$ $136.737$ $-9.532$ $-20$ $0.058$ $0.418$ $131.929$ $-12.713$ $c$ $-10$ $0.111$ $0.418$ $141.825$ $-6.166$ $10$ $0.221$ $0.418$ $159.594$ $5.591$ $20$ $0.278$ $0.418$ $166.861$ $10.399$ $-20$ $0.166$ $0.418$ $82.784$ $-45.784$ $\nu$ $-10$ $0.166$ $0.418$ $107.584$ $-28.820$ $10$ $0.166$ $0.418$ $227.656$ $50.622$ $20$ $0.166$ $0.418$ $362.046$ $139.537$ $-20$ $0.166$ $0.418$ $155.908$ $3.152$ $k$ $-10$ $0.166$ $0.418$ $153.499$ $1.558$ $10$ $0.166$ $0.418$ $148.843$ $-1.523$ $20$ $0.166$ $0.418$ $146.594$ $-3.010$ $-20$ $0.166$ $0.418$ $151.242$ $0.065$ $\lambda$ $-10$ $0.166$ $0.418$ $151.193$ $0.032$ $10$ $0.166$ $0.417$ $151.095$ $-0.032$ $20$ $0.166$ $0.417$ $151.046$ $-0.065$ $-20$ $0.218$ $0.426$ $145.779$ $-3.550$ $u$ $-10$ $0.192$ $0.422$ $148.496$ $-1.752$ $10$ $0.140$ $0.413$ $153.738$ $1.716$ $20$ $0.115$ $0.408$ $156.298$ $3.410$
 Parameter Change(%) $T_{1}$ $T_{2}$ $TC$ Change $TC(\%)$ $-20$ $0.306$ $0.418$ $145.956$ $-3.432$ $a$ $-10$ $0.227$ $0.418$ $149.419$ $-1.141$ $10$ $0.116$ $0.418$ $151.970$ $0.547$ $20$ $0.076$ $0.418$ $152.329$ $0.784$ $-10$ $0.047$ $0.436$ $160.574$ $6.239$ $b$ $-5$ $0.106$ $0.427$ $156.287$ $3.403$ $5$ $0.227$ $0.409$ $144.764$ $-4.221$ $10$ $0.289$ $0.400$ $136.737$ $-9.532$ $-20$ $0.058$ $0.418$ $131.929$ $-12.713$ $c$ $-10$ $0.111$ $0.418$ $141.825$ $-6.166$ $10$ $0.221$ $0.418$ $159.594$ $5.591$ $20$ $0.278$ $0.418$ $166.861$ $10.399$ $-20$ $0.166$ $0.418$ $82.784$ $-45.784$ $\nu$ $-10$ $0.166$ $0.418$ $107.584$ $-28.820$ $10$ $0.166$ $0.418$ $227.656$ $50.622$ $20$ $0.166$ $0.418$ $362.046$ $139.537$ $-20$ $0.166$ $0.418$ $155.908$ $3.152$ $k$ $-10$ $0.166$ $0.418$ $153.499$ $1.558$ $10$ $0.166$ $0.418$ $148.843$ $-1.523$ $20$ $0.166$ $0.418$ $146.594$ $-3.010$ $-20$ $0.166$ $0.418$ $151.242$ $0.065$ $\lambda$ $-10$ $0.166$ $0.418$ $151.193$ $0.032$ $10$ $0.166$ $0.417$ $151.095$ $-0.032$ $20$ $0.166$ $0.417$ $151.046$ $-0.065$ $-20$ $0.218$ $0.426$ $145.779$ $-3.550$ $u$ $-10$ $0.192$ $0.422$ $148.496$ $-1.752$ $10$ $0.140$ $0.413$ $153.738$ $1.716$ $20$ $0.115$ $0.408$ $156.298$ $3.410$
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