Solving a deterministic multi product single machine EPQ model withpartial backordering, scrapped products and rework

Document Type: Research Paper


1 Department of Industrial Engineering, Faculty of Engineering, Kharazmi University, Tehran, Iran

2 Department of Industrial Engineering, Faculty of Engineering, Kharazmi University, Karaj, Iran


In this paper, an economic production quantity (EPQ) inventory model with scrap and rework is developed. The inventory model is for multiple products and all products are manufactured in a single machine. Clearly, the existence of one machine results in limited production capacity and shortages. Therefore, shortages are permitted and partially backordered. We show that the model of the problem is a constrained non-linear program and use GAMS modelling language to solve it. Our objectiveis to minimize the joint total cost of the system and the supply cost of warehouse space, subject to capacity, service level, budget and warehouse space constraints. Subsequently, a nonlinear programming solver BARON is used to solve the model. At the end, numerical examples are provided to demonstrate the applicability of the model in real-world manufacturing problems.
To verify the solution obtained and to evaluate the performance of MCDM (Multi Criteria Decision Making) methods , TUKEY test is employed to compare the means of the primary objective values, the mean values of the second objective , and the mean needed CPU time of solving the problem using various methods of MCDM. Also, To compare the methods we used TOPSIS (Technique for Order Preference by Similarity to Ideal Solution). The results show that torabi-hasini method is the most efficient method to solve the model and the solution qualities of the methods differ significantly. Finally, some conclusions and future researches are included.


Main Subjects

Abdel-Aleem, A., El-Sharief, M.A., Hassan, M.A., El-Sebaie, M.G. (2017). A surface response optimization model for EPQ system with imperfect production process under rework and shortage. OPSEARCH , Vol. 54, pp. 735-751.

Al-Salamah, M. (2016). Economic production quantity in batch manufacturing with imperfect quality, imperfect inspection, and destructive and non-destructive acceptance sampling in a two-tier market Computers & Industrial Engineering,  Vol. 93, pp. 275-285.

AlDurgam, M., Adegbola, K., Glock, C.H. (2017) A single-vendor single-manufacturer integrated inventory model with stochastic demand and variable production rate. International Journal of Production Economics, Vol.  191, pp. 335-350.

Bhunia, A.K., Shaikh, A.A., Cárdenas-Barrón, L.E. (2017). A partially integrated production-inventory model with interval valued inventory costs, variable demand and flexible reliability. Applied Soft Computing, Vol. 55, pp. 491-502.

Biel, K., Glock, C.H. (2016). On the use of waste heat in a two-stage production system with controllable production rates. International Journal of Production Economics, Vol. 181, pp. 174-190.

Cárdenas-Barrón, L.E., Sarkar, B., Treviño-Garza, G. (2013). Easy and improved algorithms tojoint determination of the replenishment lot size and number of shipments for an EPQ modelwith rework. Mathematical and Computational Applications, Vol. 18, pp. 132–8.

Cárdenas-Barrón, L.E., Taleizadeh, A.A., Treviño-Garza, G. (2012). An improved solution to replenishment lot size problem with discontinuous issuing policy and rework, and the multi delivery policy into economic production lot size problem with partial rework. Expert Systems with Applications, Vol. 39, pp. 13540–6.

Cárdenas-Barrón, L.E. (2009). Economic production quantity with rework process at a single stage manufacturing system with planned backorders. Journal of Computers and Industrial Engineering, Vol.  57, pp. 1105–13.

Cárdenas-Barrón, L.E. (2007). On optimal manufacturing batch size with rework process atsingle-stage production system. Journal of Computers and Industrial Engineering, Vol. 53, pp.  196–8.

Cárdenas-Barrón, L.E. (2008). Optimal manufacturing batch size with rework in a single-stage production system – A simple derivation. Journal of Computers and Industrial Engineering, Vol. 55, pp. 758–65.

Chan, W.M., Ibrahim, R.N., Lochert, P.B. (2003). A new EPQ model: Integrating lower pricing, rework and reject situations. Production Planning and Control, Vol. 14, pp. 588–95.

Chan, C.K., Wong, W.H., Langevin, A., Lee, Y.C.E. (2017). An integrated production-inventory model for deteriorating items with consideration of optimal production rate and deterioration during delivery. International Journal of Production Economics, Vol. 189, pp. 1-13 .

Chiu, Y.S.P., Wu, M.F., Chiu, S.W., Chang, H.H. (2015). A simplified approach to the multi-item economic production quantity model with scrap, rework, and multi-delivery. Journal of Applied Research and Technology, Vol. 13, pp. 472-476.

Chiu, S.W., Chiu, Y.S.P., Yang, J.C. (2012). Combining an alternative multi-delivery policy intoeconomic production lot size problem with partial rework. Expert Systems with Applications, Vol. 39, pp. 2578–83.

Chiu, S.W., Chiu, Y.S.P. (2006). Mathematical modeling for production system withbacklogging and failure in repair. Journal of Scientific and Industrial Research, Vol. 65, pp.  499–506.

Chiu, S.W., Wang, S.L., Chiu, Y.S.P. (2007). Determining the optimal run time for EPQ modelwith scrap, rework, and stochastic breakdowns. European Journal of Operational Research, Vol. 180, pp. 664–76.

Chung, K.J., Her, C.C., Lin, S.D. (2009). A two-warehouse inventory model with imperfect quality production processes. Computers and Industrial Engineering, Vol.  53, pp. 193–7.

De, M., Das, B., Maiti, M. (2018). Green logistics under imperfect production system: A Rough age based Multi-Objective Genetic Algorithm approach. Computers & Industrial Engineering, Vol. 119, pp. 100-113.

Fakher, H.B., Nourelfath, M., Gendreau, M. (2018). Integrating production, maintenance and quality: A multi-period multi-product profit-maximization model. Reliability Engineering & System Safety, Vol. 170, pp. 191-201.

Haji, B., Haji, A., Haji, R. (2009). Optimal batch production with minimum rework cycles and constraint on accumulated defective units, Service Operations, Logistics and Informatics. IEEE/INFORMS International Conference pp. 633–8.

Haji, B., Haji, A., RahmatiTavakol, A. (2008). Scheduling accumulated rework in a normal cycle: Optimal batch production with minimum rework cycles. Journal of Industrial and Systems Engineering, Vol. 2, pp. 236–49.

Hou, K.L. (2007). An EPQ model with setup cost and process quality as functions of capital expenditure. Applied Mathematical Modelling, Vol.  31, pp. 10–17.

Hu, J.S., Zheng, H., Guo, C.Y., Ji, Y.P. (2010). Optimal production run length with imperfectproduction processes and backorder in fuzzy random environment. Computers and Industrial Engineering, Vol. 59, pp.  9–15.

Islam, S., Roy, T.K. (2006). A fuzzy EPQ model with flexibility and reliability consideration anddemand dependent unit production cost under a space constraint: A fuzzy geometric programming approach. Applied Mathematics and computation, Vol. 176, pp.  531–44.

Jamal, A.M.M., Sarker, B.R., Mondal, S. (2004). Optimal manufacturing batch size with reworkprocess at a single-stage production system. Computers and Industrial Engineering, Vol. 47, pp. 77–89.

Jiang, Y., Chen, M., Zhou, D. (2015). Joint optimization of preventive maintenance and inventory policies for multi-unit systems subject to deteriorating spare part inventory. Journal of Manufacturing Systems, Vol. 35, pp. 191-205.

Khan, M., Jaber, M.Y., Guiffrida, A.L., Zolfaghari, S. (2011). A review of the extensions of a modified EOQ model for imperfect quality items. International Journal of Production Economics, Vol. 132, pp. 1–12.

Liao, G.L., Chen, Y.H., Sheu, S.H. (2009). Optimal economic production quantity policy for imperfect process with imperfect repair and maintenance. European Journal of Operational Research, Vol. 195, pp. 348–57.

Liu, N., Kim,Y., Hwang, H. (2009). An optimal operating policy for the production system with rework. Computers and Industrial Engineering, Vol. 56, pp. 874–87.

Manna, A.K., Das, B., Dey, J.K., Mondal, S.K. (2016). An EPQ model with promotional demand in random planning horizon: population varying genetic algorithm approach. Journal of Intelligent Manufacturing, Vol. 1, pp. 1-17.

Manna, A.K., Das, B., Dey, J.K., Mondal, S.K. (2017a). Two layers green supply chain imperfect production inventory model under bi-level credit period. Tékhne, Vol. 15, pp. 124-142.

Manna, A.K., Dey, J.K., Mondal, S.K. (2017b). Imperfect production inventory model with production rate dependent defective rate and advertisement dependent demand. Computers & Industrial Engineering, Vol. 104, pp. 9-22.

Moussawi-Haidar, L., Salameh, M., Nasr, W. (2016). Production lot sizing with quality screening and rework. Applied Mathematical Modelling, Vol.  40, pp. 3242-3256.

Öztürk, H. (2017). A note on “Production lot sizing with quality screening and rework”. Applied Mathematical Modelling, Vol. 43, pp. 659-669.

Pal, S., Mahapatra, G.S., Samanta, G.P. (2016). A three-layer supply chain EPQ model for price- and stock-dependent stochastic demand with imperfect item under rework. Journal of Uncertainty Analysis and Application, Vol.1, pp.  4-10.

Pasandideh, S.H.R., Niaki, S.T.A., Nobil, A.H., Cárdenas-Barrón, L.E. (2015). A multiproduct single machine economic production quantity model for an imperfect production system under warehouse construction cost. International Journal of Production Economics, Vol. 169, pp. 203-214.

Roy, A., Maity, K., Kar, S., Maiti, M. (2009). A production–inventory model withremanufacturing for defective and usable items in fuzzy-environment. Computers and Industrial Engineering, Vol. 56, pp. 87–96.

Sadeghi, J., Niaki, S.T.A., Malekian, M.R., Wang, Y. (2018). A lagrangian relaxation for a fuzzy random epq problem with shortages and redundancy allocation: two tuned meta-heuristics. International Journal of Fuzzy Systems,  Vol. 20, pp. 515-533.

Sadjadi, S.J., Yazdian, S.A., Shahanaghi, K. (2012). Optimal pricing, lot-sizing and marketingplanning in a capacitated imperfect production system. Computers and Industrial Engineering, Vol. 62, pp. 349–58.

Sarker, B.R., Jamal, A.M.M., Mondal, S. (2008). Optimal batch sizing in a process at multi-stageproduction system with network consideration. European Journal of Operational research, Vol.  184, pp. 915–29.

Shah, N.H., Patel, D.G., Shah, D.B. (2016). EPQ model for returned/reworked inventories during imperfect production process under price-sensitive stock-dependent demand. Operational Research, Vol. 1, pp. 1-17.

Shaikh, A.A., Cárdenas-Barrón, L.E., Tiwari, S. (2018). Closed-form solutions for the epq-based inventory model for exponentially deteriorating items under retailer partial trade credit policy in supply chain. International Journal of Applied and Computational Mathematics, Vol.  4, pp. 70 .

Taleizadeh, A.A., Akhavan, Niaki, S.T., HajiSeyedjavadi, S.M. (2012). Multi-product multi-chance-constraint stochastic inventory control problem with dynamic demand and partial back-ordering: a harmony search algorithm. Journal of Manufacturing Systems, Vol. 31(2), pp. 204–213.

Taleizadeh, A.A., Jalali-Naini, S.G., Wee, H.M., Kuo, T.C. (2013). An imperfect multi-product production system with rework. Journal of Scientia Iranica, Vol. 20, pp. 811–23.

Taleizadeh, A.A., Kalantari, S.S., Cárdenas-Barrón, L.E. (2016). Pricing and lot sizing for an EPQ inventory model with rework and multiple shipments. TOP, Vol.  24(1), pp. 143-155

Taleizadeh, A.A., Niaki, S.T.A., Najafi, A.A. (2010). Multi-product single-machine production system with stochastic scrapped production rate, partial backordering and service levelconstraint. Journal of Computational and Applied Mathematics, Vol. 233, pp. 1834–49.

Tiwari, R.N., Dharmar, S., Rao, J.R. (1987). Fuzzy goal programming – an additive method. Fuzzy Sets and Systems, Vol. 24, pp. 27–34.

Tsao, Y.C., Zhang, Q., Chang, F.C., Vu-Thuy, L. (2017). An imperfect production model under radio frequency identification adoption and trade credit, Applied Mathematical Modelling, Vol. 42, pp. 493-508.

Torabi, S.A., Hassini, E. (2008). An interactive possibilistic programming approach for multiple objective supply chain master planning. Fuzzy Sets and Systems, Vol. 159, pp. 193–214.

Viji, G., Karthikeyan, K. (2016). An economic production quantity model for three levels of production with Weibull distribution deterioration and shortage. Journal of Ain Shams Engineering, In press

Wang, X., Tang, W. (2009). Optimal production run length in deteriorating production processeswith fuzzy elapsed time. Computers and Industrial Engineering, Vol. 56, pp. 1627–32.

Wee, H.M., Wang, W.T., Cárdenas-Barrón, L.E. (2013). An alternative analysis and solutionprocedure for the EPQ model with rework process at a single-stage manufacturing system withplanned backorders. Journal of Computers and Industrial Engineering, Vol. 64, pp. 748–755.

White , A.S., Censlive, M. (2013). Using control theory to optimize profit in APVIOBPCS inventory systems. Journal of Manufacturing Systems, Vol. 4, pp. 680–688.