July  2017, 13(3): 1511-1535. doi: 10.3934/jimo.2017005

Stochastic machine breakdown and discrete delivery in an imperfect inventory-production system

1. 

School of Industrial Engineering, College of Engineering, University of Tehran, Tehran, 14155-6619, Iran

2. 

Department of Industrial Engineering, Iran University of Science and Technology, Tehran, 145888-9694, Iran

3. 

Department of Industrial & Management Engineering, Hanyang University, Ansan Gyeonggi-do, 15588, South Korea

* Corresponding author: bsbiswajitsarkar@gmail.com (Biswajit Sarkar), Phone Number-+82-10-7498-1981, Office Phone: +82-31-400-5259, Fax: +82-31-436-8146

Received  July 2015 Published  December 2016

In this paper, we develop an integrated inventory model to determine the optimal lot size and production uptime while considering stochastic machine breakdown and multiple shipments for a single-buyer and single-vendor. Machine breakdown cannot be controlled by the production house. Thus, we assume it as stochastic, not constant. Moreover, we assume that the manufacturing process produces defective items. When a breakdown takes place, the production system follows a no resumption policy. Some defective products cannot be reworked and are discarded from the system. To prevent shortages, we consider safety stock. The model assumes that both batch quantity and the distance between two shipments are identical and that the transportation cost is paid by the buyer. We prove the convexity of the total cost function and derive the closed-form solutions for decision variables analytically. To obtain the optimal production uptime, we determine both the lower and upper bounds for the optimal production uptime using a bisection searching algorithm. To illustrate the applicability of the proposed model, we provided a numerical example and sensitivity analysis.

Citation: Ata Allah Taleizadeh, Hadi Samimi, Biswajit Sarkar, Babak Mohammadi. Stochastic machine breakdown and discrete delivery in an imperfect inventory-production system. Journal of Industrial & Management Optimization, 2017, 13 (3) : 1511-1535. doi: 10.3934/jimo.2017005
References:
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L. E. Cárdenas-BarrónG. Trevińo-Garza and H. M. Wee, A simple and better algorithm to solve the vendor managed inventory control system of multi-product multi-constraint economic order quantity model, Expert Systems with Applications, 39 (2012), 3888-3895.   Google Scholar

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show all references

References:
[1]

N. E. AbboudM. Y. Jaber and N. A. Noueihed, Economic lot sizing with the consideration of random machine unavailability time, Computers & Operations Research, 27 (2000), 335-351.  doi: 10.1016/S0305-0548(99)00055-6.  Google Scholar

[2]

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[3]

L. E. Cárdenas-Barrón, The derivation of EOQ/EPQ inventory models with two backorders costs using analytic geometry and algebra, Applied Mathematical Modeling, 35 (2011), 2394-2407.  doi: 10.1016/j.apm.2010.11.053.  Google Scholar

[4]

L. E. Cárdenas-Barrón, A complement to A comprehensive note on: An economic order quantity with imperfect quality and quantity discounts, Applied Mathematical Modeling, 36 (2012), 6338-6340.  doi: 10.1016/j.apm.2012.02.021.  Google Scholar

[5]

L. E. Cárdenas-BarrónJ. T. TengG. Trevińo-GarzaH. M. Wee and K. R. Lou, An improved algorithm and solution on an integrated production-inventory model in a three-layer supply chain, International Journal of Production Economics, 136 (2015), 384-388.   Google Scholar

[6]

L. E. Cárdenas-BarrónG. Trevińo-Garza and H. M. Wee, A simple and better algorithm to solve the vendor managed inventory control system of multi-product multi-constraint economic order quantity model, Expert Systems with Applications, 39 (2012), 3888-3895.   Google Scholar

[7]

T. Chakraborty and B. C. Giri, Joint determination of optimal safety stocks and production policy for an imperfect production system, Applied Mathematical Modelling, 36 (2012), 712-722.  doi: 10.1016/j.apm.2011.07.012.  Google Scholar

[8]

T. ChakrabortyB. C. Giri and K. S. Chaudhuri, Production lot sizing with process deterioration and machine breakdown, European Journal of Operational Research, 185 (2008), 606-618.  doi: 10.1016/j.ejor.2007.01.011.  Google Scholar

[9]

T. ChakrabortyB. C. Giri and K. S. Chaudhuri, Production lot sizing with process deterioration and machine breakdown under inspection schedule, Omega, 37 (2009), 257-271.   Google Scholar

[10]

C. K. Chen and C. C. Lo, Optimal production run length for products sold with warranty in an imperfect production system with allowable shortage, Mathematical and Computer Modelling, 44 (2006), 319-331.  doi: 10.1016/j.mcm.2006.01.019.  Google Scholar

[11]

Y. S. P ChiuS. C. LiuC. L. Chiu and H. H. Chang, Mathematical modeling for determining the replenishment policy for EMQ model with rework and multiple shipments, Mathematical and Computer Modelling, 54 (2011), 2165-2174.  doi: 10.1016/j.mcm.2011.05.025.  Google Scholar

[12]

S. W. ChiuH. D. LinM. F. Wu and J. C. Yang, Determining replenishment lot size and shipment policy for an extended EPQ model with delivery and quality assurance issues, Scientia Iranica, 18 (2011), 1537-1544.  doi: 10.1016/j.scient.2011.09.008.  Google Scholar

[13]

S. W. ChiuS. L. Wang and Y. S. P. Chiu, Determining the optimal run time for EPQ model with scrap, rework, and stochastic breakdowns, European Journal of Operational Research, 180 (2007), 664-676.  doi: 10.1016/j.ejor.2006.05.005.  Google Scholar

[14]

S. W. Chiu, Robust planning in optimization for production system subject to random machine breakdown and failure in rework, Computers & Operations Research, 37 (2010), 899-908.  doi: 10.1016/j.cor.2009.03.016.  Google Scholar

[15]

Y. S. P. ChiuK. K. ChenF. T. Cheng and M. F. Wu, Optimization of the finite production rate model with scrap, rework and stochastic machine breakdown, Computers & Mathematics with Applications, 59 (2010), 919-932.  doi: 10.1016/j.camwa.2009.10.001.  Google Scholar

[16]

Y. S. P. ChiuH. D. Lin and H. H. Chang, Mathematical modeling for solving manufacturing run time problem with defective rate and random machine breakdown, Computers & Industrial Engineering, 60 (2011), 576-584.  doi: 10.1016/j.cie.2010.12.015.  Google Scholar

[17]

S. W. ChiuY. S. P. Chiu and J. C. Yang, Combining an alternative multi-delivery policy into economic production lot size problem with partial rework, Expert Systems with Applications, 39 (2012), 2578-2583.  doi: 10.1016/j.eswa.2011.08.112.  Google Scholar

[18]

Y. S. P. ChiuK. K. Chen and K. K. Ting, Replenishment run time problem with machine breakdown and failure in rework, Expert Systems with Applications, 39 (2012), 1291-1297.  doi: 10.1016/j.eswa.2011.08.005.  Google Scholar

[19]

K. -J. Chunga and K. -L. Hou, An optimal production runs time with imperfect production processes and allowable shortages, Computers & Operations Research 25 (2003), 483-490. Google Scholar

[20]

A. Diponegoro and B. R. Sarker, Determining manufacturing batch sizes for a lumpy delivery system with trend demand, International Journal of Production Economics, 77 (2002), 131-143.  doi: 10.1016/S0925-5273(02)00108-1.  Google Scholar

[21]

B. C. GiriW. Y. Yun and T. Dohi, Optimal design of unreliable production-inventory systems with variable production rate, European Journal of Operational Research, 162 (2005), 372-386.  doi: 10.1016/j.ejor.2003.10.015.  Google Scholar

[22]

G. C. Lin and D. C. Gong, On a production-inventory system of deteriorating items subject to random machine breakdowns with a fixed repair time, Mathematical and Computer Modelling, 43 (2006), 920-932.  doi: 10.1016/j.mcm.2005.12.013.  Google Scholar

[23]

B. Liu and J. Cao, Analysis of a production-inventory system with machine breakdowns and shutdowns, Computers & Operations Research, 26 (1999), 73-91.  doi: 10.1016/S0305-0548(98)00040-9.  Google Scholar

[24]

S. H. R. Pasandideh and S. T. A. Niaki, A genetic algorithm approach to optimize a multi-products EPQ model with discrete delivery orders and constrained space, Applied Mathematics and Computation, 195 (2008), 506-514.  doi: 10.1016/j.amc.2007.05.007.  Google Scholar

[25]

S. H. R. PasandidehS. T. A. Niaki and J. A. Yeganeh, A parameter-tuned genetic algorithm for multi-product economic production quantity model with space constraint, discrete delivery orders and shortages, Advances in Engineering Software, 41 (2010), 306-314.   Google Scholar

[26]

S. Sana, A production-inventory model of imperfect quality products in a three-layer supply chain, Decision Support Systems, 50 (2011), 539-547.  doi: 10.1016/j.dss.2010.11.012.  Google Scholar

[27]

S. SanaS. K. Goyal and K. S. Chaudhuri, An imperfect production process in a volume flexible inventory model, International Journal of Production Economics, 105 (2007), 548-559.  doi: 10.1016/j.ijpe.2006.05.005.  Google Scholar

[28]

B. Sarkar, Supply chain coordination with variable backorder, inspections, and discount policy for fixed lifetime products Mathematical Problems in Engineering 2016 (2016), Article ID 6318737, 14 pages. doi: 10.1155/2016/6318737.  Google Scholar

[29]

B. Sarkar, S. Saren, D. Sinha and S. Hur, Effect of unequal lot sizes, variable setup cost, and carbon emission cost in a supply chain model Mathematical Problems in Engineering 2015 (2015), Article ID 469486, 13 pages. doi: 10.1155/2015/469486.  Google Scholar

[30]

B. SarkarB. Mandal and S. Sarkar, Quality improvement and backorder price discount under controllable lead time in an inventory model, Journal of Manufacturing Systems, 35 (2015), 26-36.  doi: 10.1016/j.jmsy.2014.11.012.  Google Scholar

[31]

B. Sarkar and A. Mahapatra, Periodic review fuzzy inventory models with variable lead time and fuzzy demand International Transactions in Operational Research (2015), In press. doi: 10.1111/itor.12177.  Google Scholar

[32]

B. Sarkar and S. Saren, Product inspection policy for an imperfect production system with inspection errors and warranty cost, European Journal of Operational Research, 248 (2016), 263-271.  doi: 10.1016/j.ejor.2015.06.021.  Google Scholar

[33]

B. SarkarS. S. Sana and K. S. Chaudhuri, An imperfect production process for time varying demand with inflation and time value of money-An EMQ model, Expert Systems with Applications, 38 (2011), 13543-13548.  doi: 10.1016/j.eswa.2011.04.044.  Google Scholar

[34]

B. Sarkar and I. Moon, An EPQ model with inflation in an imperfect production system, App Math Comp, 217 (2011), 6159-6167.  doi: 10.1016/j.amc.2010.12.098.  Google Scholar

[35]

B. SarkarL. E. Cárdenas-BarrónM. Sarkar and M. L. Singgih, An economic production quantity model with random defective rate, rework process and backorders for a single stage production system, Journal of Manufacturing Systems, 33 (2014), 423-435.  doi: 10.1016/j.jmsy.2014.02.001.  Google Scholar

[36]

B. Sarkar and I. Moon, Improved quality, setup cost reduction, and variable backorder costs in an imperfect production process, International Journal of Production Economics, 155 (2014), 204-213.  doi: 10.1016/j.ijpe.2013.11.014.  Google Scholar

[37]

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Figure 1.  The vendors on-hand inventory of perfect-quality items in our EPQ model when machine breakdown does not occur
Figure 2.  The vendors on-hand inventory of defective items when machine breakdown does not occur
Figure 3.  The buyers inventory level when machine breakdown does not occur
Figure 4.  The vendors on-hand inventory of perfect-quality items in our EPQ model when machine breakdown occurs
Figure 5.  The vendors on-hand inventory of defective items when machine breakdown occurs
Figure 6.  The buyers inventory level when machine breakdown occurs
Figure 7.  The behavior of $E[TCU(t_1,q)]$ with respect to $t_1$
Figure 8.  The behavior of $E[TCU(t_1,q)]$ with respect to $\frac{1}{\beta}$
Figure 9.  The behavior of $E[TCU(t_1,q)]$ with respect to $q$
Figure 10.  Sensitivity analysis of $t^{*}_{1}$ for various parameter values
Figure 11.  Sensitivity analysis of $q^*$ for various parameter values
Decision variables
$t_1$ production uptime when a breakdown does not occur (year).
$q$ shipment quantity (units/delivery).
Parameters
$A$ setup cost of vendor ($/setup).
$C$ production cost of vendor ($/unit).
$C_s$ disposal cost of vendor ($/unit).
$h$ holding cost of vendor ($/unit/year).
$h_1$ holding cost for defective units of vendor ($/defective unit/year).
$C_t$ transportation cost of buyer ($/delivery).
$D$ demand rate of buyer (units/year).
$A_1$ ordering cost of buyer ($/order).
$h_2$ holding cost of buyer ($/unit/year).
$H_1$ maximum level of on-hand inventory when machine breakdown does not occur (units).
$H_2$ maximum level of on-hand inventory when machine breakdown occurs (units).
$M$ machine repair time (time unit).
$n$ number of shipments delivered during a cycle when machine breakdown does not occur.
$P$ production rate (units/year).
$T$ cycle length when breakdown does not occur (year).
$T'$ cycle length when breakdown occurs (year).
$T_U$ cycle length for integrated case (year).
$t$ production time before a random breakdown occurs (year).
$t_d$ time required to deplete all available perfect-quality items when machine breakdown does not occur (year).
$t'_{d}$ time required to deplete all available perfect-quality items when machine breakdown occurs (year).
$t_r$ machine repair time (year).
$TC(t,q)$ total inventory costs per cycle when machine breakdown occurs ($/cycle).
$TC(t_1,q)$ total inventory costs per cycle when machine breakdown does not occur ($/cycle).
$TCU(t_1,q)$ total inventory costs per unit time for integrated case ($/year).
Decision variables
$t_1$ production uptime when a breakdown does not occur (year).
$q$ shipment quantity (units/delivery).
Parameters
$A$ setup cost of vendor ($/setup).
$C$ production cost of vendor ($/unit).
$C_s$ disposal cost of vendor ($/unit).
$h$ holding cost of vendor ($/unit/year).
$h_1$ holding cost for defective units of vendor ($/defective unit/year).
$C_t$ transportation cost of buyer ($/delivery).
$D$ demand rate of buyer (units/year).
$A_1$ ordering cost of buyer ($/order).
$h_2$ holding cost of buyer ($/unit/year).
$H_1$ maximum level of on-hand inventory when machine breakdown does not occur (units).
$H_2$ maximum level of on-hand inventory when machine breakdown occurs (units).
$M$ machine repair time (time unit).
$n$ number of shipments delivered during a cycle when machine breakdown does not occur.
$P$ production rate (units/year).
$T$ cycle length when breakdown does not occur (year).
$T'$ cycle length when breakdown occurs (year).
$T_U$ cycle length for integrated case (year).
$t$ production time before a random breakdown occurs (year).
$t_d$ time required to deplete all available perfect-quality items when machine breakdown does not occur (year).
$t'_{d}$ time required to deplete all available perfect-quality items when machine breakdown occurs (year).
$t_r$ machine repair time (year).
$TC(t,q)$ total inventory costs per cycle when machine breakdown occurs ($/cycle).
$TC(t_1,q)$ total inventory costs per cycle when machine breakdown does not occur ($/cycle).
$TCU(t_1,q)$ total inventory costs per unit time for integrated case ($/year).
Table 1.  Variations of $\beta$ effects on $t^{*}_{1L}$, $w(t^{*}_{1L})$, $t^{*}_{1U}$, and $w(t^{*}_{1U})$
$\beta$ $\beta^{-1}$ $t^{*}_{1L}$ $w(t^{*}_{1L})$ $t^{*}_{1U}$ $w(t^{*}_{1U})$
0.1100.3986350.4065280060.4026280.410519
0.250.3985970.4142740.406620.422284
0.33.330.3985590.4219130.4106510.43396
0.42.50.3985210.4294430.414720.445535
0.520.3984830.4368680.4188260.456999
110.3982940.4724290.4399230.512281
1.10.9090.3982560.4792410.4442540.522854
1.20.8330.3982180.4859560.4486230.533239
1.30.7690.398180.4925770.4530280.543428
1.40.7140.3981420.4991060.457470.553411
1.50.6670.3981040.5055460.4619490.563177
20.50.3979150.53650.4848820.608482
30.3330.3975380.5933320.5333580.679759
40.250.3971610.6458990.5850750.727634
50.20.3967860.6968770.6397080.762221
60.1670.3964120.7482070.696910.794729
70.1420.3960380.8010530.7563380.831822
80.1250.3956660.8559610.8176680.875487
90.1110.3952950.9130660.8806060.925223
100.10.3949250.9722660.9448910.979797
$\beta$ $\beta^{-1}$ $t^{*}_{1L}$ $w(t^{*}_{1L})$ $t^{*}_{1U}$ $w(t^{*}_{1U})$
0.1100.3986350.4065280060.4026280.410519
0.250.3985970.4142740.406620.422284
0.33.330.3985590.4219130.4106510.43396
0.42.50.3985210.4294430.414720.445535
0.520.3984830.4368680.4188260.456999
110.3982940.4724290.4399230.512281
1.10.9090.3982560.4792410.4442540.522854
1.20.8330.3982180.4859560.4486230.533239
1.30.7690.398180.4925770.4530280.543428
1.40.7140.3981420.4991060.457470.553411
1.50.6670.3981040.5055460.4619490.563177
20.50.3979150.53650.4848820.608482
30.3330.3975380.5933320.5333580.679759
40.250.3971610.6458990.5850750.727634
50.20.3967860.6968770.6397080.762221
60.1670.3964120.7482070.696910.794729
70.1420.3960380.8010530.7563380.831822
80.1250.3956660.8559610.8176680.875487
90.1110.3952950.9130660.8806060.925223
100.10.3949250.9722660.9448910.979797
Table 2.  Sensitivity analysis of $t^{*}_{1}$, $q^{*}$, and $E[TCU(t^{*}_{1},q^{*})]$ for various parameter values
Rate of $q^{*}$ $t^{*}_{1}$$E[TCU(t^{*}_{1},q^{*})]$Rate of $q^{*}$ $t^{*}_{1}$$E[TCU(t^{*}_{1},q^{*})]$
A change h change
-0.3 730.29 0.3613 11485.71 -0.3 778.49 0.4845 11382.55
-0.2 730.29 0.3789 11544.89 -0.2 761.38 0.4564 11477.72
-0.1 730.29 0.3985 11601.68 -0.1 745.35 0.4326 11568.80
0 730.29 0.4121 11656.35 0 730.29 0.4121 11656.35
0.1 730.29 0.4278 11709.14 0.1 716.11 0.3943 11740.79
0.2 730.29 0.4431 11760.24 0.2 702.72 0.3786 11822.47
0.3 730.29 0.4578 11809.83 0.3 690.06 0.3646 11901.69
$h_2$ -0.3 806.47 0.4121 11552.87 $\beta$ -0.3 730.29 0.40802 11608.71
-0.2 778.49 0.4121 11588.52 -0.2 730.29 0.40939 11624.56
-0.1 753.24 0.4121 11622.97 -0.1 730.29 0.41077 11640.44
0 730.29 0.4121 11656.35 0 730.29 0.41216 11656.35
0.1 709.32 0.4121 11688.73 0.1 730.29 0.41356 11672.29
0.2 690.06 0.4121 11720.21 0.2 730.29 0.41497 11688.26
0.3 672.29 0.4121 11750.86 0.3 730.29 0.41639 11704.27
$C_t$ -0.3 730.29 0.4121 11477.42 $t_r$ -0.3 730.29 0.41222 11643.20
-0.2 730.29 0.4121 11540.70 -0.2 730.29 0.41220 11647.58
-0.1 730.29 0.4121 11600.13 -0.1 730.29 0.41218 11651.96
0 730.29 0.4121 11656.35 0 730.29 0.41216 11656.35
0.1 730.29 0.4121 11709.81 0.1 730.29 0.41214 11660.73
0.2 730.29 0.4121 11760.90 0.2 730.29 0.41212 11665.11
0.3 730.29 0.4121 11809.90 0.3 730.29 0.41210 11669.50
P -0.3 730.29 0.7005 11590.45 D -0.3 611.01 0.31465 8589.613
-0.2 730.29 0.5664 11537.73 -0.2 653.19 0.34625 9625.374
-0.1 730.29 0.4767 11628.08 -0.1 692.82 0.37862 10647.10
0 730.29 0.4121 11656.35 0 730.29 0.41216 11656.35
0.1 730.29 0.3633 11678.38 0.1 765.94 0.44731 12654.21
0.2 730.29 0.3249 11696.05 0.2 800.00 0.48453 13641.47
0.3 730.29 0.2940 11710.54 0.3 832.66 0.52439 14618.68
Rate of $q^{*}$ $t^{*}_{1}$$E[TCU(t^{*}_{1},q^{*})]$Rate of $q^{*}$ $t^{*}_{1}$$E[TCU(t^{*}_{1},q^{*})]$
A change h change
-0.3 730.29 0.3613 11485.71 -0.3 778.49 0.4845 11382.55
-0.2 730.29 0.3789 11544.89 -0.2 761.38 0.4564 11477.72
-0.1 730.29 0.3985 11601.68 -0.1 745.35 0.4326 11568.80
0 730.29 0.4121 11656.35 0 730.29 0.4121 11656.35
0.1 730.29 0.4278 11709.14 0.1 716.11 0.3943 11740.79
0.2 730.29 0.4431 11760.24 0.2 702.72 0.3786 11822.47
0.3 730.29 0.4578 11809.83 0.3 690.06 0.3646 11901.69
$h_2$ -0.3 806.47 0.4121 11552.87 $\beta$ -0.3 730.29 0.40802 11608.71
-0.2 778.49 0.4121 11588.52 -0.2 730.29 0.40939 11624.56
-0.1 753.24 0.4121 11622.97 -0.1 730.29 0.41077 11640.44
0 730.29 0.4121 11656.35 0 730.29 0.41216 11656.35
0.1 709.32 0.4121 11688.73 0.1 730.29 0.41356 11672.29
0.2 690.06 0.4121 11720.21 0.2 730.29 0.41497 11688.26
0.3 672.29 0.4121 11750.86 0.3 730.29 0.41639 11704.27
$C_t$ -0.3 730.29 0.4121 11477.42 $t_r$ -0.3 730.29 0.41222 11643.20
-0.2 730.29 0.4121 11540.70 -0.2 730.29 0.41220 11647.58
-0.1 730.29 0.4121 11600.13 -0.1 730.29 0.41218 11651.96
0 730.29 0.4121 11656.35 0 730.29 0.41216 11656.35
0.1 730.29 0.4121 11709.81 0.1 730.29 0.41214 11660.73
0.2 730.29 0.4121 11760.90 0.2 730.29 0.41212 11665.11
0.3 730.29 0.4121 11809.90 0.3 730.29 0.41210 11669.50
P -0.3 730.29 0.7005 11590.45 D -0.3 611.01 0.31465 8589.613
-0.2 730.29 0.5664 11537.73 -0.2 653.19 0.34625 9625.374
-0.1 730.29 0.4767 11628.08 -0.1 692.82 0.37862 10647.10
0 730.29 0.4121 11656.35 0 730.29 0.41216 11656.35
0.1 730.29 0.3633 11678.38 0.1 765.94 0.44731 12654.21
0.2 730.29 0.3249 11696.05 0.2 800.00 0.48453 13641.47
0.3 730.29 0.2940 11710.54 0.3 832.66 0.52439 14618.68
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