Optimal lot-sizing policy for a failure prone production system with investment in process quality improvement and lead time variance reduction

Sumon Sarkar; Bibhas C. Giri

doi:10.3934/jimo.2021048

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2022, Volume 18, Issue 3: 1891-1913. Doi: 10.3934/jimo.2021048

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Optimal lot-sizing policy for a failure prone production system with investment in process quality improvement and lead time variance reduction

Sumon Sarkar^, and
Bibhas C. Giri

Department of Mathematics, Jadavpur University, Kolkata - 700032, India

^* Corresponding author: ss.sumonsarkar@gmail.com (Sumon Sarkar)
^* Corresponding author: ss.sumonsarkar@gmail.com (Sumon Sarkar)

Received: March 31, 2019

Revised: October 31, 2020

Published: May 2022

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Abstract

To survive in today's competitive market, it is not enough to produce low-cost products but also quality-related issues and lead time needs to be considered in the decision-making process. This paper extends the previous research by developing a stochastic economic manufacturing quantity (EMQ) model for a production system which is subject to process shifts from an in-control state to an out-of-control state at any random time. Moreover, we consider the option of investment to increase the process quality and decrease the lead-time variability. Closed-form solutions of the proposed models are obtained by applying the classical optimization technique. Some lemmas and theorems are developed to determine the optimal solution of the decision variables. Numerical results are obtained for each of these models and compared with those of the basic EMQ model without any investment. From the numerical analysis, it has been observed that our proposed model can significantly reduce the cost of the system compared to the basic model.

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Mathematics Subject Classification: Primary: 90B05, 90B30.

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Figure 1. Logistic diagram of the proposed model

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Table 1. Numerical results for different models

Model type		$ \; \; Q $	$ \; \; V $	$ \; \; \theta $	$ E[TC(\cdot)] $	reduction in $ \theta $ %	reduction in $ V $ %	reduction in $ E[TC(\cdot)] $ %
EMQ-SLT	Exact	$ \; \; 556.4 $	$ - $	$ - $	$ 3369.26 $	$ - $	$ - $	$ - $
	Approx.	$ \; \; 555.6 $	$ - $	$ - $	$ 3371.76 $	$ - $	$ - $	$ - $
SLT-QI	Exact	$ \; \; 602.3 $	$ - $	$ 0.02352 $	$ 3203.55 $	$ 84.32 $	$ - $	$ 4.92 $
	Approx.	$ \; \; 602.2 $	$ - $	$ 0.02331 $	$ 3204.00 $	$ 84.46 $	$ - $	$ 4.98 $
SLT-VR	Exact	$ \; \; 448.8 $	$ 0.0001660 $	$ - $	$ 3025.87 $	$ - $	$ 78.44 $	$ 10.19 $
	Approx	$ \; \; 448.2 $	$ 0.0001658 $	$ - $	$ 3027.5 $	$ - $	$ 78.47 $	$ 10.21 $
SLT-SI	Exact	$ \; \; 487.1 $	$ 0.0001802 $	$ 0.02903 $	$ 2908.06 $	$ 80.64 $	$ 76.60 $	13.69
	Approx.	$ \; \; 487.1 $	$ 0.0001801 $	$ 0.02882 $	$ 2908.43 $	$ 80.79 $	$ 76.61 $	13.68

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Table 2. Computational results for different lead-time interval

Lead time interval	Model type	$ \; \; Q $	$ \; \; V $	$ \; \; \theta $	$ E[TC(\cdot)] $	reduction in $ \theta $ (%)	reduction in $ V $ (%)	reduction in $ E[TC(\cdot)] $ (%)
	EMQ-SLT	$ 420 $	$ - $	$ - $	$ 2552 $	$ - $	$ - $	$ - $
$ 1 $	SLT-QI	$ 453 $	$ - $	$ 0.0309 $	$ 2446 $	$ 79.4 $	$ - $	$ 4.15 $
week	SLT-VR	$ 420 $	$ - $	$ - $	$ 2552 $	$ - $	$ - $	$ - $
	SLT-SI	$ 453 $	$ - $	$ 0.0309 $	$ 2446 $	$ 79.4 $	$ - $	$ 4.15 $
	EMQ-SLT	$ \; \; 440 $	$ - $	$ - $	$ 2668 $	$ - $	$ - $	$ - $
$ 2 $	SLT-QI	$ \; \; 475 $	$ - $	$ 0.0296 $	$ 2554 $	$ 80.27 $	$ - $	$ 4.27 $
weeks	SLT-VR	$ \; \; 440 $	$ - $	$ - $	$ 2668 $	$ - $	$ - $	$ - $
	SLT-SI	$ \; \; 475 $	$ - $	$ 0.0296 $	$ 2554 $	$ 80.27 $	$ - $	$ 4.27 $
	EMQ-SLT	$ \; \; 470 $	$ - $	$ - $	$ 2852 $	$ - $	$ - $	$ - $
$ 3 $	SLT-QI	$ \; \; 508 $	$ - $	$ 0.0276 $	$ 2723 $	$ 81.60 $	$ - $	$ 4.52 $
weeks	SLT-VR	$ \; \; 448 $	$ 0.000166 $	$ - $	$ 2823 $	$ - $	$ 40.07 $	$ 1.02 $
	SLT-SI	$ \; \; 487 $	$ 0.000180 $	$ 0.0288 $	$ 2704 $	$ 76.62 $	35.02	5.19
	EMQ-SLT	$ \; \; 509 $	$ - $	$ - $	$ 3090 $	$ - $	$ - $	$ - $
$ 4 $	SLT-QI	$ \; \; 551 $	$ - $	$ 0.0255 $	$ 2944 $	$ 83.0 $	$ - $	$ 4.72 $
weeks	SLT-VR	$ \; \; 448 $	$ 0.000166 $	$ - $	$ 2938 $	$ - $	$ 66.33 $	$ 4.91 $
	SLT-SI	$ \; \; 487 $	$ 0.000180 $	$ 0.0288 $	$ 2819 $	$ 80.8 $	$ 63.49 $	8.77
	EMQ-SLT	$ \; \; 556 $	$ - $	$ - $	$ 3372 $	$ - $	$ - $	$ - $
$ 5 $	SLT-QI	$ \; \; 602 $	$ - $	$ 0.0233 $	$ 3204 $	$ 84.46 $	$ - $	$ 4.98 $
weeks	SLT-VR	$ \; \; 448 $	$ 0.000166 $	$ - $	$ 3028 $	$ - $	$ 78.47 $	$ 10.21 $
	SLT-SI	$ \; \; 487 $	$ 0.000180 $	$ 0.0288 $	$ 2908 $	$ 80.79 $	$ 76.61 $	13.68
	EMQ-SLT	$ \; \; 608 $	$ - $	$ - $	$ 3687 $	$ - $	$ - $	$ - $
$ 6 $	SLT-QI	$ \; \; 659 $	$ - $	$ 0.0213 $	$ 3495 $	$ 85.8 $	$ - $	$ 5.21 $
weeks	SLT-VR	$ \; \; 448 $	$ 0.000166 $	$ - $	$ 3100 $	$ - $	$ 85.03 $	$ 15.92 $
	SLT-SI	$ \; \; 487 $	$ 0.000180 $	$ 0.0288 $	$ 2981 $	$ 80.8 $	$ 83.77 $	19.15
	EMQ-SLT	$ \; \; 664 $	$ - $	$ - $	$ 4028 $	$ - $	$ - $	$ - $
$ 7 $	SLT-QI	$ \; \; 721 $	$ - $	$ 0.0194 $	$ 3808 $	$ 87.06 $	$ - $	$ 5.46 $
weeks	SLT-VR	$ \; \; 448 $	$ 0.000166 $	$ - $	$ 3162 $	$ - $	$ 89.01 $	$ 21.50 $
	SLT-SI	$ \; \; 487 $	$ 0.000180 $	$ 0.0288 $	$ 3043 $	$ 80.8 $	$ 88.08 $	24.45

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Table 3. Critical points

Parameter	SLT-QI	SLT-VR	SLT-SI
$ \Gamma $	$>\frac{i}{R\zeta D\theta_{0}}\sqrt{\frac{4P(PZ+DR\theta_{0}\zeta)(P-D)}{D(2K(P-D)+D(B+H)PV_{0})}} $	$ - $	$>\frac{2(PZ+\theta_{0}R\zeta D)}{\theta_{0}R\zeta D\left(\frac{1}{\beta}+\frac{1}{i}\sqrt{2KD(PZ+\theta_{0}R\zeta D)/P+\frac{i^2}{\beta^2}}\right)} $
$ \Delta $	$ - $	$>\frac{4i(P-D)}{D^2V_{0}(B+H)}\sqrt{\frac{D(2K(P-D)+D(B+H)PV_{0})}{4P(PZ+DR\theta_{0}\zeta)(P-D)}} $	$>\frac{2(2K(1-D/P)+DV_{0}(B+H))}{DV_0(B+H)\left(\frac{1}{\delta}+\frac{1}{i}\sqrt{DZ(2K+V_{0}(B+H)D/(1-D/P))+\frac{i^2}{\delta^2}}\right)} $
$ i $	$<\frac{R\delta\zeta D\theta_{0}}{2}\sqrt{\frac{D(2K(P-D)+D(B+H)PV_{0})}{P(PZ+DR\theta_{0}\zeta)(P-D)}} $	$<\frac{D^2V_{0}(B+H)\beta}{2(P-D)}\sqrt{\frac{P(PZ+DR\theta_{0}\zeta)(P-D)}{D(2K(P-D)+D(B+H)PV_{0})}} $	$<\hbox{min}\; \Bigg\{\frac{R\zeta D\theta_{0}\delta}{2P}\sqrt{\frac{2KPD/\delta}{PZ/\delta+R\zeta D\theta_{0}(\frac{1}{\delta}-\frac{1}{\beta})}} $,
			$ \frac{D^2V_{0}Z}{2}\sqrt{\frac{R\zeta(B+H)P\beta}{(P-D)Z(2PK/\beta+DV_{0}(\frac{1}{\beta}-\frac{1}{\delta})R\zeta D)}}\Bigg\} $

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Table 4. Critical values for different lead-time intervals for SLT-QI model

Variables					Lead time interval
	1	2	3	4	5	6	7
$ \theta_{imp} $	$ 0.0309 $	$ 0.0296 $	$ 0.0276 $	$ 0.0255 $	$ 0.0233 $	$ 0.0213 $	$ 0.0194 $
$ i $	$ 0.450 $	$ 0.470 $	$ 0.502 $	0.544	$ 0.594 $	$ 0.649 $	$ 0.709 $
$ \Gamma $	$ 0.000445 $	$ 0.000425 $	$ 0.000398 $	$ 0.000367 $	$ 0.000337 $	$ 0.000308 $	$ 0.000282 $

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Table 5. Critical values for different lead-time intervals for SLT-VR model

Variables					Lead time interval
	1	2	3	4	5	6	7
$ V_{imp} $	$ - $	$ - $	$ 0.000166 $	$ 0.000166 $	$ 0.000166 $	$ 0.000166 $	$ 0.000166 $
$ i $	$ - $	$ - $	$ 0.160 $	0.262	$ 0.375 $	$ 0.494 $	$ 0.615 $
$ \Delta $	$ - $	$ - $	$ 0.000313 $	$ 0.000191 $	$ 0.000133 $	$ 0.000101 $	$ 0.0000813 $

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