An EPQ model for an imperfect production process with fuzzy cycle time and quality screening

Document Type : Research Paper


1 Suleyman Demirel University, Department of Business Administration, 32260, Isparta, Turkey

2 Suleyman Demirel University, Department of Mathematics, 32260, Isparta, Turkey

3 Suleyman Demirel University, Department of Computer Engineering, , 32260, Isparta, Turkey


This study has developed a production inventory model where the cycle time
is fuzzy, the existence of defective products is assumed in each batch and
product screening is performed both in-production and after-production.
Triangular fuzzy numbers serve to model uncertainties in the cycle time, and
a fuzzified total inventory profit function is created by the
defuzzification method known as the signed distance method. The classical
approach is used to determine the optimal policy, with the ideal cycle time
matched to the total profit. Although assuming asymmetric triangular fuzzy
numbers prevents the calculation of a clear analytical solution, the method
approaches as closely as possible to an analytical solution. A numerical
solution to only one equation is needed to obtain the optimal configuration.
Conversely, there is a positive trade-off, with an analytical solution to
the optimization problem if there is an assumption of symmetrical triangular
fuzzy numbers. The proposed model is illustrated by a numerical example. The
paper presents results and sensitivity analyses, in both tables and graphic
illustrations. The effects on total profit are discussed in relation to
various parameters. From the numerical studies, it is observed that the
level of fuzziness influences the cycle time and an approximately linear
relationship, in the opposite direction, was found between the total profit
and the level of fuzziness, when it was increased.


[1] A. Andriolo, D. Battini, R. W. Grubbstrom, A. Persona, F. Sgarbossa, A century of evolution from Harris's basic lot size
model: Survey and research agenda, International Journal of Production Economics, 155 (2014), 16-38.
[2] K. T. Atanassov, Intuitionistic fuzzy sets, Fuzzy Sets and Systems, 20 (1986), 87-96.
[3] A. Baykasoglu, T. Gocken, Solution of a fully fuzzy multi-item economic order quantity problem by using fuzzy ranking
functions, Engineering Optimization, 39 (2007), 919-39.
[4] K. M. Bjork, The economic production quantity problem with a fi nite production rate and fuzzy cycle time, Proceedings of the
41st Annual Hawaii International Conference on System Sciences, (2008), 68-77.
[5] K. M. Bjork, An analytical solution to a fuzzy economic order quantity problem, International Journal Approximate Reasoning,
50 (2009), 485-493.
[6] K. M. Bjork, A multi-item fuzzy economic production quantity problem with a finite production rate, International Journal of Production Economics, 135 (2012), 702-707.
[7] K. M. Bjork, C. Carlsson, The outcome of imprecise lead times on the distributors, Proceedings of the 38th Annual Hawaii
International Conference on System Sciences, (2005), 81-90.
[8] K. M. Bjork, C. Carlsson, The effect of flexible lead times on a paper producer, International Journal of Production Economics,
107 (2007), 139-150.
[9] M. E. Bredahl, J. R. Northen, A. Boecker, M. A. Normile, Consumer demand sparks the growth of quality assurance schemes
in the European food sector, Changing Structure of the Global Food Consumption and Trade: US Department of Agriculture,
Agriculture and Trade Report, (2001), 90-102.
[10] J. G. Brown, A note on fuzzy sets, Information and Control, 18 (1971), 32-39.
[11] C. Carlsson, R. Fuller, Soft computing and the bullwhip effect, Economics and Complexity, 2 (1999), 1-26.
[12] D. Chakraborty, D. K. Jana, T. K. Roy, Multi-item integrated supply chain model for deteriorating items with stock dependent
demand under fuzzy random and bifuzzy environments, Computers and Industrial Engineering, 88 (2015), 166-180.
[13] H. C. Chang, An application of fuzzy sets theory to the EOQ model with imperfect quality items, Computers and Operations
Research, 31 (2004), 2079-2092.
[14] T. H. Chen, Y. C. Tsao, Optimal lot-sizing integration policy under learning and rework effects in a manufacturer-retailer
chain, International Journal of Production Economics, 155 (2014), 239-248.
[15] S. H. Chen, C. C. Wang, R. Arthur, Backorder fuzzy inventory model under function principle, Information Sciences, 95
(1996), 71-79.
[16] S. W. Chiu, An optimization problem of manufacturing systems with stochastic machine breakdown and rework process,
Applied Stochastic Models in Business and Industry, 24 (2008), 203-219.
[17] B. Cui, W. Zeng, Approximate reasoning with interval-valued fuzzy sets, Fifth International Conference on Fuzzy Systems
and Knowledge Discovery, (2008), 60-64.
[18] A. De Luca, S. Termini, Algebraic properties of fuzzy sets, Journal of Mathematical Analysis and Applications, 40 (1972),
[19] P. Diamond, P. Kloeden, Metric spaces of fuzzy sets: Theory and applications, World scienti fic, 1994.
[20] D. Dubois, H. Prade, Operations on fuzzy numbers, International Journal of Systems Science, 9 (1978), 613-626.
[21] S. D. P. Flapper, J. C. Fransoo, R. A. C. M. Broekmeulen, K. Inderfurth, Planning and control of rework in the process
industries: A review, Production Planning and Control, 13 (2002), 26-34.
[22] S. B. Gershwin, How do quantity and quality really interact? Precise models instead of strong opinions, IFAC Proceedings
Volumes, 39 (2006), 33-39.
[23] C. H. Glock, M. Y. Jaber, A multi-stage production-inventory model with learning and forgetting effects, rework and scrap,
Computers and Industrial Engineering, 64 (2013), 708-720.
[24] J. A. Goguen, L-Fuzzy sets, Journal of Mathematical Analysis and Applications, 18 (1967), 145-174.
[25] M. B. Gorza lczany, A method of inference in approximate reasoning based on interval-valued fuzzy sets, Fuzzy Sets and
Systems, 21 (1987), 1-17.
[26] S. K. Goyal, A. Gunasekaran, T. Martikainen, P. Yli-Olli, Integrating production and quality control policies: A survey,
European Journal of Operational Research, 69 (1993), 1-13.
[27] R. W. Grubbstrom, B. G. Kingsman, Ordering and inventory policies for step changes in the unit item cost: A discounted
cash flow approach, Management Science, 50 (2004), 253-267.
[28] A. L. Guiffrida, Fuzzy inventory models, in Inventory Management Non-Classical Views, FL, Boca Raton: CRC Press,
(2009), 173-198.
[29] H. Gurnani, Z. Drezner, R. Akella, Capacity planning under different inspection strategies, European Journal of Operational
Research, 89 (1996), 2-12.
[30] P. A. Hayek, M. K. Salameh, Production lot sizing with the reworking of imperfect quality items produced, Production
Planning and Control, 12 (2001), 584-590.
[31] F. Herrera, E. Herrera-Viedma, Linguistic decision analysis: Steps for solving decision problems under linguistic information,
Fuzzy Sets and Systems, 115 (2000), 67-82.
[32] S. Islam, T. K. Roy, Fuzzy multi-item economic production quantity model under space constraint: A geometric programming
approach, Applied Mathematics and Computation, 184 (2007), 326-335.
[33] A. M. Jamal, B. R. Sarker, S. Mondal, Optimal manufacturing batch size with rework process at a single-stage production
system, Computer and Industrial Engineering, 47 (2004), 77-89.
[34] O. Kaleva, Fuzzy differential equations, Fuzzy Sets and Systems, 24 (1987), 301-317.
[35] C. W. Kang, M. Ullah, B. Sarkar, Human errors incorporation in work-in-process group manufacturing system, Scientia
Iranica, 24 (2017), 2050-2061.
[36] A. Kaufman, M. M. Gupta, Introduction to fuzzy arithmetic, New York, USA: Van Nostrand Reinhold Company, 1991.
[37] N. Kazemi E. Ehsani, M. Y. Jaber, An inventory model with backorders with fuzzy parameters and decision variables,
International Journal of Approximate Reasoning, 51 (2010), 964-972.
[38] I. Konstantaras, K. Skouri, M. Y. Jaber, Inventory models for imperfect quality items with shortages and learning in inspection,
Applied Mathematical Modelling, 36 (2012), 5334-5343.
[39] H. M. Lee, J. S. Yao, Economic production quantity for fuzzy demand quantity, and fuzzy production quantity, European
Journal of Operational Research, 109 (1998), 203-211.
[40] F. T. Lin, Fuzzy job-shop scheduling based on ranking level (/spl lambda/, 1) interval-valued fuzzy numbers, IEEE Transactions
on Fuzzy Systems, 10 (2002), 510-522.
[41] W. N. Ma, D. C. Gong, G. C. Lin, An optimal common production cycle time for imperfect production processes with scrap,
Mathematical and Computer Modelling, 52 (2010), 724-737.
[42] J. Mezei, K. M. Bjork, An economic production quantity problem with backorders and fuzzy cycle times, Journal of Intelligent
and Fuzzy Systems, 28 (2015), 1861-1868.
[43] M. Mizumoto, K. Tanaka, Some properties of fuzzy sets of type 2, Information and Control, 31 (1976), 312-340.
[44] S. P. Mondal, Interval valued intuitionistic fuzzy number and its application in differential equation, Journal of Intelligent
and Fuzzy Systems, 34 (2018), 677-687.
[45] L. Moussawi-Haidar, M. Salameh, W. Nasr, Production lot sizing with quality screening and rework, Applied Mathematical
Modelling, 40 (2016), 3242-3256.
[46] H.  Ozturk, A note on production lot sizing with quality screening and rework, Applied Mathematical Modelling, 43 (2017),
[47] S. Papachristos, I. Konstantaras, Economic ordering quantity models for items with imperfect quality, International Journal
of Production Economics, 100 (2006), 148-154.
[48] K. S. Park, Fuzzy-set theoretic interpretation of economic order quantity, IEEE Transactions on Systems, Man and Cybernetics,
17 (1987), 1082-1084.
[49] S. Paul, D. Jana, S. P. Mondal, P. Bhattacharya, Optimal harvesting of two species mutualism model with interval parameters,
Journal of Intelligent and Fuzzy Systems, 33 (2017), 1991-2005.
[50] E. L. Porteus, Optimal lot sizing, process quality improvement and setup cost reduction, Operations Research, 34 (1986),
[51] M. J. Rosenblatt, H. L. Lee, Economic production cycles with imperfect production processes, IIE Transactions, 18 (1986),
[52] J. Sadeghi, S. T. A. Niaki, M. R. Malekian, S. Sadeghi, Optimising multi-item economic production quantity model with
trapezoidal fuzzy demand and backordering: Two tuned meta-heuristics, European Journal of Industrial Engineering, 10(2)
(2016), 170-195.
[53] M. K. Salameh, M. Y. Jaber, Economic production quantity model for items with imperfect quality, International Journal of
Production Economics, 64 (2000), 59-64.
[54] B. Sarkar, An inventory model with reliability in an imperfect production process, Applied Mathematics and Computation,
218 (2012), 4881-4891.
[55] B. K. Sett, S. Sarkar, B. Sarkar, Optimal bu er inventory and inspection errors in an imperfect production system with
preventive maintenance, The International Journal of Advanced Manufacturing Technology, 90 (2017), 545-560.
[56] C. Suntag, Inspection and inspection quality management, Milwaukee: ASQC, 1993.
[57] A. A. Taleizadeh, S. J. Sadjadi, S. T. A. Niaki, Multi product EPQ model with single machine, backordering and immediate
rework process, European Journal of Industrial Engineering, 5 (2011), 388-411.
[58] J. T. Teng, H. L. Yang, M. S. Chern, Economic order quantity models for deteriorating items and partial backlogging when
demand is quadratic in time, European Journal of Industrial Engineering, 5 (2011), 198-214.
[59] I. B. Turksen, Interval valued fuzzy sets based on normal forms, Fuzzy Sets and Systems, 20 (1986), 191-210.
[60] M. Ullah, C. W. Kang, Effect of rework, rejects and inspection on lot size with work-in-process inventory, International
Journal of Production Research, 52 (2014), 2448-2460.
[61] M. Vujosevic, D. Petrovic, R. Petrovic, EOQ formula when inventory cost is fuzzy, International Journal of Production
Economics, 45 (1996), 499-504.
62] H. M. Wee, W. T. Wang, P. C. Yang, A production quantity model for imperfect quality items with shortage and screening
constraint, International Journal Production Research, 51 (2013), 1869-1884.
[63] J. S. Yao, K. Wu, Ranking fuzzy numbers based on decomposition principle and signed distance, Fuzzy Sets and Systems,
116 (2000), 275-88.
[64] L. A. Zadeh, Fuzzy sets, Information and Control, 8 (1965), 338-353.
[65] L. A. Zadeh, Some reflections on soft computing, granular computing and their roles in the conception, design and utilization
of information/intelligent systems, Soft Computing, 2 (1998), 23-25.
[66] L. A. Zadeh, The concept of a linguistic variable and its application to approximate reasoning{II, Information Sciences, 8
(1975), 301-357.
[67] W. Zeng, H. Li, Relationship between similarity measure and entropy of interval valued fuzzy sets, Fuzzy Sets and Systems,
157(11) (2006), 1477-1484.
[68] X. Zhang, Y. Gerchak, Joint lot sizing and inspection policy in an EOQ model with random yield, IIE Transactions, 22
(1990), 41-4