Abstract
The present work investigates a manufacturing-inventory system with a single machine and multiple products, featuring returns on sales and backorders. In the proposed model, some imperfect items, including scrapped and defective items, are produced by the manufacturer. Such items can be classified, based on the severity of the failure, into several categories; as a result, the rework process is carried out at different rates. Moreover, the implementation of the quality control policy requires monitoring and checking the items during the production and reworking processes via an inspection process. The present study is aimed to calculate and obtain the optimal values of the cycle length and backorders quantity for every product in order to achieve the minimum total cost of system considering machine capacity, service level, warehouse space, and budget constraints. To solve the presented model, given as a Nonlinear Programming (NLP) problem, the GAMS software as well as four commonly used algorithms, which are categorized among the meta-heuristic algorithms, is used. These algorithms include the GA (Genetic Algorithm), IWO (Invasive Weed Optimization), GWO (Grey Wolf Optimizer) and HHO (Harris Hawks Optimization) algorithms. Along with these algorithms, the Response Surface Methodology (RSM) is applied to calibrate the parameters of the proposed algorithms. Finally, several numeric problems are solved, the results of which are then compared with each other. Moreover, an analytical hierarchy process (AHP) technique for order performance by similarity to ideal solution (TOPSIS), which is a hybrid method of decision making with multiple attributes, is used for ranking the algorithms.
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References
Ahmadianfar I, Bozorg-Haddad O, Chu X (2020) Gradient-based optimizer: a new Metaheuristic optimization algorithm. Inf Sci 540:131–159
Al-Salamah M (2019) Economic production quantity in an imperfect manufacturing process with synchronous and asynchronous flexible rework rates. Oper Res Persp. https://doi.org/10.1016/j.orp.2019.100103
Barzoki MR, Jahanbazi M, Bijari M (2011) Effects of imperfect products on lot sizing with work in process inventory. Appl Math Comput 217(21):8328–8336
Beheshti Fakher H, Nourelfath M, Gendreau M (2017) A cost minimisation model for joint production and maintenance planning under quality constraints. Int J Prod Res 55(8):2163–2176
Cheikhrouhou N, Sarkar B, Ganguly B, Malik AI, Batista R, Lee YH (2018) Optimization of sample size and order size in an inventory model with quality inspection and return of defective items. Ann Oper Res 271(2):445–467
Chiu SW, Chiu YSP, Yang JC (2012) Combining an alternative multi-delivery policy into economic production lot size problem with partial rework. Expert Syst Appl 39(3):2578–2583
Chiu SW, Wang SL, Chiu YSP (2007) Determining the optimal run time for EPQ model with scrap, rework, and stochastic breakdowns. Eur J Oper Res 180(2):664–676
Chiu YP (2003) Determining the optimal lot size for the finite production model with random defective rate, the rework process, and backlogging. Eng Optim 35(4):427–437
Chiu YSP, Chen KK, Cheng FT, Wu MF (2010) Optimization of the finite production rate model with scrap, rework and stochastic machine breakdown. Comput Math Appl 59(2):919–932
Fadlil IN, Novitasari R, Jauhari WA (2020) Sustainable economic production quantity model with rework and product return policy. In AIP Conference Proceedings (Vol. 2217, No. 1, p. 030078). AIP Publishing LLC.
Faris H, Aljarah I, Al-Betar MA, Mirjalili S (2018) Grey wolf optimizer: a review of recent variants and applications. Neural Comput Appl 30(2):413–435
Gharaei A, Hoseini Shekarabi SA, Karimi M (2019) Modelling and optimal lot-sizing of the replenishments in constrained, multi-product and bi-objective EPQ models with defective products: Generalised cross decomposition. Int J Syst Sci Oper Log, 1–13
Haji A, Sikari SS, Shamsi R (2009) The effect of inspection errors on the optimal batch size in reworkable production systems with scraps. Int J Prod Dev 10(1–3):201–216
Hayek PA, Salameh MK (2001) Production lot sizing with the reworking of imperfect quality items produced. Prod Plan Cont 12(6):584–590
Heidari AA, Mirjalili S, Faris H, Aljarah I, Mafarja M, Chen H (2019) Harris hawks optimization: algorithm and applications. Futur Gener Comput Syst 97:849–872
Hou KL (2007) An EPQ model with setup cost and process quality as functions of capital expenditure. Appl Math Model 31(1):10–17
Hsu JT, Hsu LF (2013) Two EPQ models with imperfect production processes, inspection errors, planned backorders, and sales returns. Comput Ind Eng 64(1):389–402
Hwang CL, Yoon K (1981) Methods for multiple attribute decision making. In Multiple attribute decision making. Springer, Berlin
Jamal AMM, Sarker BR, Mondal S (2004) Optimal manufacturing batch size with rework process at a single-stage production system. Comput Ind Eng 47(1):77–89
Kang CW, Ramzan MB, Sarkar B, Imran M (2018) Effect of inspection performance in smart manufacturing system based on human quality control system. Int J Adv Manuf Technol 94(9–12):4351–4364
Kang CW, Ullah M, Sarkar B (2018) Optimum ordering policy for an imperfect single-stage manufacturing system with safety stock and planned backorder. Int J Adv Manuf Technol 95(1–4):109–120
Kang CW, Ullah M, Sarkar B, Hussain I, Akhtar R (2017) Impact of random defective rate on lot size focusing work-in-process inventory in manufacturing system. Int J Prod Res 55(6):1748–1766
Khalilpourazari S, Pasandideh SHR, Niaki STA (2016) Optimization of multi-product economic production quantity model with partial backordering and physical constraints: SQP, SFS, SA, and WCA. Appl Soft Comput 49:770–791
Korda N, Szorenyi B, Li S (2016) Distributed clustering of linear bandits in peer to peer networks. In International conference on machine learning (pp. 1301–1309). PMLR
Li S, Kar P (2015) Context-aware bandits. arXiv preprint arXiv:1510.03164
Li S, GentileC, Karatzoglou A (2016) Graph clustering bandits for recommendation. arXiv preprint arXiv:1605.00596
Li S, Karatzoglou A, Gentile C (2016) Collaborative filtering bandits. In Proceedings of the 39th International ACM SIGIR conference on Research and Development in Information Retrieval (pp. 539–548).
Liao GL, Chen YH, Sheu SH (2009) Optimal economic production quantity policy for imperfect process with imperfect repair and maintenance. Eur J Oper Res 195(2):348–357
Mahadik K, Wu Q, Li S, Sabne A (2020) Fast distributed bandits for online recommendation systems. In Proceedings of the 34th ACM International Conference on Supercomputing (pp. 1–13).
Mehrabian AR, Lucas C (2006) A novel numerical optimization algorithm inspired from weed colonization. Eco Inform 1(4):355–366
Mellouk L, Aaroud A, Boulmalf M, Zine-Dine K, Benhaddou D (2020) Development and performance validation of new parallel hybrid cuckoo search–genetic algorithm. Energy Syst 11(3):729–751
Mirjalili S, Mirjalili SM, Lewis A (2014) Grey wolf optimizer. Adv Eng Softw 69:46–61
Nobil AH, Afshar Sedigh AH, Cárdenas-Barrón LE (2018) Multi-machine economic production quantity for items with scrapped and rework with shortages and allocation decisions. Scientia Iranica 25(4):2331–2346
Nobil AH, Sedigh AHA, Cárdenas-Barrón LE (2016) A multi-machine multi-product EPQ problem for an imperfect manufacturing system considering utilization and allocation decisions. Expert Syst Appl 56:310–319
Nobil AH, Sedigh AHA, Cárdenas-Barrón LE (2020) A multiproduct single machine economic production quantity (EPQ) inventory model with discrete delivery order, joint production policy and budget constraints. Ann Oper Res 286(1):265–301
Ouyang LY, Chang CT (2013) Optimal production lot with imperfect production process under permissible delay in payments and complete backlogging. Int J Prod Econ 144(2):610–617
Pasandideh SHR, Niaki STA, Gharaei A (2015) Optimization of a multiproduct economic production quantity problem with stochastic constraints using sequential quadratic programming. Knowl-Based Syst 84:98–107
Pasandideh SHR, Niaki STA, Nobil AH, Cárdenas-Barrón LE (2015) A multiproduct single machine economic production quantity model for an imperfect production system under warehouse construction cost. Int J Prod Econ 169:203–214
Pirayesh M, Poormoaied S (2015) GPSO-LS algorithm for a multi-item EPQ model with production capacity restriction. Appl Math Model 39(17):5011–5032
Porteus EL (1986) Optimal lot sizing, process quality improvement and setup cost reduction. Oper Res 34(1):137–144
Sarkar B, Saren S (2016) Product inspection policy for an imperfect production system with inspection errors and warranty cost. Eur J Oper Res 248(1):263–271
Sarkar B, Cárdenas-Barrón LE, Sarkar M, Singgih ML (2014) An economic production quantity model with random defective rate, rework process and backorders for a single stage production system. J Manuf Syst 33(3):423–435
Sarkar B, Chaudhuri K, Moon I (2015) Manufacturing setup cost reduction and quality improvement for the distribution free continuous-review inventory model with a service level constraint. J Manuf Syst 34:74–82
Sarkar B, Sett BK, Sarkar S (2018) Optimal production run time and inspection errors in an imperfect production system with warranty. J Indus Manag Optim 14(1):267–282
Taheri-Tolgari J, Mirzazadeh A, Jolai F (2012) An inventory model for imperfect items under inflationary conditions with considering inspection errors. Comput Math Appl 63(6):1007–1019
Taleizadeh AA, Kalantari SS, Cárdenas-Barrón LE (2015) Determining optimal price, replenishment lot size and number of shipments for an EPQ model with rework and multiple shipments. J Indus Manag Optim 11(4):1059–1071
Taleizadeh AA, Niaki STA, Najafi AA (2010) Multiproduct single-machine production system with stochastic scrapped production rate, partial backordering and service level constraint. J Comput Appl Math 233(8):1834–1849
Taleizadeh AA, Sadjadi SJ, Niaki STA (2011) Multiproduct EPQ model with single machine, backordering and immediate rework process. Eur J Indus Eng 5(4):388–411
Taleizadeh AA, Sarkar B, Hasani M (2019) Delayed payment policy in multi-product single-machine economic production quantity model with repair failure and partial backordering. J Indus Manag Optim 13(5):1
Taleizadeh AA, Wee HM, Jalali-Naini SG (2013) Economic production quantity model with repair failure and limited capacity. Appl Math Model 37(5):2765–2774
Taleizadeh AA, Yadegari M, Sana SS (2019) Production models of multiple products using a single machine under quality screening and reworking policies. J Modell Manag. https://doi.org/10.1108/JM2-06-2018-0086
Taleizadeh AA, Cárdenas-Barrón LE, Biabani J, Nikousokhan R (2012) Multi products single machine EPQ model with immediate rework process. Int J Ind Eng Comput 3(2):93–102
Taleizadeh AA, Moghadasi H, Niaki STA, Eftekhari AK (2009) An EOQ-joint replenishment policy to supply expensive imported raw materials with payment in advance. J Appl Sci 8(23):4263–4273
Taleizadeh A.A., Najafi A.A., Niaki S.T.A., (2010). “Multi Product EPQ Model with Scraped Items and limited Production Capacity,” International Journal of Science and Technology (Scientia Iranica) Transaction E, 17 (1): 58–69.
Taleizadeh AA, Kalantary SS, Cárdenas-Barrón LE (2016) Pricing and lot sizing for an EPQ inventory model with rework and multiple shipments. TOP 24:143–155
Tayyab M, Sarkar B (2016) Optimal batch quantity in a cleaner multi-stage lean production system with random defective rate. J Clean Prod 139:922–934
Wee HM, Widyadana GA (2012) Economic production quantity models for deteriorating items with rework and stochastic preventive maintenance time. Int J Prod Res 50(11):2940–2952
Yang XS (2010) Nature-inspired metaheuristic algorithms. Luniver press, UK
Yeniay Ö (2005) Penalty function methods for constrained optimization with genetic algorithms. Math Comput Appl 10(1):45–56
Yoo SH, Kim D, Park MS (2009) Economic production quantity model with imperfect-quality items, two-way imperfect inspection and sales return. Int J Prod Econ 121(1):255–265
Yoo SH, Kim D, Park MS (2012) Inventory models for imperfect production and inspection processes with various inspection options under one-time and continuous improvement investment. Comput Oper Res 39(9):2001–2015
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Appendices
Appendices
Appendix A Determining the length of the cycle
In accordance with Eq. (8), we will have:
Appendix B Calculating the cost of holding
Considering Eq. (23), we have:
According to Eq. (24), \({CH}_{a}\) is:
and
Also
Therefore
Then, based on Eqs. (2–6), we have:\({CH}_{a}=\sum_{i=1}^{n}\left(\left(\frac{{{h}_{i}a}_{i}}{2}{\left(\frac{{D}_{i}}{{ P}_{i}\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)}\right)}^{2}\right)\left(T\right)+\left({{h}_{i}a}_{i}{\left(\frac{{D}_{i}}{{ P}_{i}\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)}\right)}^{2}\sum_{j=1}^{m}\left(\frac{\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]}{{V}_{i}^{j}}\right)\right)\left(T\right)+\left(\frac{{h}_{i}}{2}{\left(\frac{{D}_{i}}{{ P}_{i}\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)}\right)}^{2}\sum_{i=1}^{m}\left({\left(\frac{\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]}{{V}_{i}^{j}}\right)}^{2}{y}_{i}^{j}\right)\right)\left(T\right)+ \left({h}_{i}{\left(\frac{{D}_{i}}{{ P}_{i}\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)}\right)}^{2}\sum_{j=1}^{m}\left(\frac{\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]}{{V}_{i}^{j}}+ \sum_{k=1}^{j-1}\left(\frac{\left[{\alpha }_{i}^{k}+\left(1-{\sigma }_{i}\right){e1}_{i}^{k}\right]}{{V}_{i}^{k}}{y}_{i}^{k}\right)\right)\right)\left(T\right) + \left(\frac{{h}_{i}{D}_{i}}{2}{\left(\frac{{D}_{i}}{{ P}_{i}\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)}\right)}^{2}\right)\left(T\right)+ \left(\frac{{a}_{i}{h}_{i}{D}_{i}}{{\left({ P}_{i}\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)\right)}^{2}}\sum_{j=1}^{m}\left(\frac{\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]{y}_{i}^{j}}{{V}_{i}^{j}}\right)\right)\left(T\right)+ \left(\frac{{h}_{i}{D}_{i}}{2}{\left(\frac{1}{{ P}_{i}\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)}\right)}^{2}\sum_{j=1}^{m}{\left(\frac{\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]}{{V}_{i}^{j}}\right)}^{2}\right)\left(T\right) + \left(\frac{{h}_{i}{D}_{i}}{{\left({ P}_{i}\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)\right)}^{2}}{\sum_{j=1}^{m}\left(\frac{\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]}{{V}_{i}^{j}}+ \sum_{k=1}^{j-1}\left(\frac{\left[{\alpha }_{i}^{k}+\left(1-{\sigma }_{i}\right){e1}_{i}^{k}\right]}{{V}_{i}^{k}}{y}_{i}^{k}\right)\right)}^{2}\right)\left(T\right)+ \frac{{h}_{i}}{2{D}_{i}}\left(\frac{{\left({B}_{i}\right)}^{2}}{T}\right) + \frac{{h}_{i}}{2{a}_{i}}\left(\frac{{\left({B}_{i}\right)}^{2}}{T}\right)- \left(\frac{{h}_{i}{a}_{i}}{{ P}_{i}\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)}\right)\left({B}_{i}\right) - \left(\frac{{h}_{i}}{{ P}_{i} (1-{\theta }_{i})}\sum_{j=1}^{m}\left(\frac{\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]{y}_{i}^{j}}{{V}_{i}^{j}}\right)\right)\left({B}_{i}\right)- \left(\frac{{h}_{i}{D}_{i}}{{ P}_{i}\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)}\right)\left({B}_{i}\right)- \left(\left(\frac{{h}_{i}{D}_{i}}{{ P}_{i}\left(\left[\left(1-{\sigma }_{i}\right)\left(1-{e1}_{i}\right)+{\alpha }_{i}{e2}_{i}^{j}\right] + \sum_{j=1}^{m}{\gamma }_{i}^{j}\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]\right)}\right)\sum_{j=1}^{{\varvec{m}}}{\left(\frac{\left[{\alpha }_{i}^{j}+\left(1-{\sigma }_{i}\right){e1}_{i}^{j}\right]{y}_{i}^{j}}{{V}_{i}^{j}}\right)}^{2}\right)\left({B}_{i}\right)\right)\)
According to Eq. (25), \({CH}_{b}\) is:
Using Eq. (14), we have:
According to Eq. (26), \({CH}_{c}\) is:
Using Eq. (14), we have:
According to Eq. (27), \({CH}_{d}\) is:
According to Eqs. (3–5), we have:
Then, using Eq. (14), we have:
Therefore,
Appendix C: Determining the machine capacity constraint
Based on Eqs. (1)–(5), we have:
By Inserting Eq. (14), the Machine capacity constraint is determined as follow.
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Taleizadeh, A.A., Askari, R. & Konstantaras, I. An optimization model for a manufacturing-inventory system with rework process based on failure severity under multiple constraints. Neural Comput & Applic 34, 4221–4264 (2022). https://doi.org/10.1007/s00521-021-06513-6
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DOI: https://doi.org/10.1007/s00521-021-06513-6