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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

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

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

 Citation: • • Figure 1.  Logistic diagram of the proposed model

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

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

Table 3.  Critical points

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

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 $0.594$ $0.649$ $0.709$ $\Gamma$ $0.000445$ $0.000425$ $0.000398$ $0.000367$ $0.000337$ $0.000308$ $0.000282$

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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