During the past few years, despite tremendous contribution on deterministic flow shop problem, there are only limited number of works dedicated on stochastic cases. This paper examines stochastic scheduling problems in two-machine flow shop environment for expected makespan minimization where processing times of jobs are normally distributed. Since jobs have stochastic processing times, to minimize the expected makespan, the expected sum of the second machine’s free times is minimized. In other words, by minimization waiting times for the second machine, it is possible to reach the minimum of the objective function. A mathematical method is proposed which utilizes the properties of the normal distributions. Furthermore, this method can be used as a heuristic method for other distributions, as long as the means and variances are available. The performance of the proposed method is explored using some numerical examples.

Heydari, M., Mazdeh, M & Bayat, M. (2013). Scheduling stochastic two-machine flow shop problems to minimize expected makespan.Decision Science Letters, 2(3), 163-174.

Adiri, I., & Frostig, E. (1984). A stochastic permutation-flowshop scheduling problem minimizing in distribution the schedule length. Operations Research Letters, 3(2), 101-103.

Allahverdi, A., & Mittenthal, J. (1995). Scheduling on a two-machine flowshop subject to random breakdowns with a makespan objective function. European Journal of Operational Research, 81(2), 376-387.

Allahverdi, A., & Fatih Tatari, M. (1997). Stochastic machine dominance in flowshops. Computers and Industrial Engineering, 32(4), 735-741.

Ancǎu, M. (2010). Weakness and strength of stochastic search in solving flowshop scheduling problem. Academic Journal of Manufacturing Engineering, 8(4), 6-10.

Araújo, D. C., & Nagano, M. S. (2010). A new effective heuristic method for the no-wait flowshop with sequence-dependent setup times problem. International Journal of Industrial Engineering Computations, 2, 155-166.

Baker, K.R., & Trietsch, D. (2011). Three heuristic procedures for the stochastic two-machine flow shop problem. Journal of Scheduling, 14(5), 445-454.

Braglia, M., Frosolini, M., Gabbrielli, R., & Zammori, F. (2011). CONWIP card setting in a flow-shop system with a batch production machine. International Journal of Industrial Engineering Computations, 2(1), 1-18.

Cunningham, A.A., & Dutta, S.K., (1973). Scheduling jobs with exponentially distributed processing times on two machines of a flow shop. Naval Research Logistics Quarterly, 16, 69–81.

Defersha, F. M. (2010). A comprehensive mathematical model for hybrid flexible flowshop lot streaming problem. International Journal of Industrial Engineering Computations, 2, 283-294.

Elmaghraby, S. E., & Thoney, K. A. (1999). The two-machine stochastic flowshop problem with arbitrary processing time distributions. IIE Transactions (Institute of Industrial Engineers), 31(5), 467-477.

Framinan, J.M., Gupta, J.N.D., & Leisten, R. (2004). A review and classification of heuristics for permutation flow-shop scheduling with makespan objective. Journal of the Operational Research Society, 55, 1243–1255.

Gourgand, M., Grangeon, N., & Norre, S. (2003). A contribution to the stochastic flow shop scheduling problem. European Journal of Operational Research, 151, 415–433.

Johnson, S.M. (1954). Optimal two- and three-stage production schedules with setup times included. Naval Research Logistics Quarterly, 1, 61–68.

Kalczynski, P.J., & Kamburowski, J. (2004). Generalization of Johnson’s and Talwar’s scheduling rules in two-machine stochastic flow shops. Journal of the Operational Research Society, 55, 1358 – 1362.

Kalczynski, P.J., & Kamburowski, J. (2006). A heuristic for minimizing the expected makespan in two-machine flow shops with consistent coefficients of variation. European Journal of Operational Research, 169, 742–750.

Kenneth R.B., & Dominik A. (2012). Heuristic solution methods for the stochastic flow shop problem. European Journal of Operational Research, 216, 172–17Ku, P.-S., & Niu, S.-C. (1986). On Johnson’s two-machine flow shop with random processing times. Operations Research, 34, 130–136.

Mahavi Mazdeh, M., Zaerpour, F., & Firouzi Jahantigh, F. (2010). A fuzzy modeling for single machine scheduling problem with deteriorating jobs. International Journal of Industrial Engineering Computations, 1(2), 147-156.

Mgwatu, M. I. (2011). Integration of part selection, machine loading and machining optimisation decisions for balanced workload in flexible manufacturing system. International Journal of Industrial Engineering Computations, 2(4), 913-930.

Naderi-Benia, M., Tavakkoli-Moghaddamb, R., Naderic, B., Ghobadiana, E., & Pourroustaa, A. (2012). A two-phase fuzzy programming model for a complex bi-objective no-wait flow shop scheduling. International Journal of Industrial Engineering, 3, 617-626.

Reisman, A., Kumar, A., & Motwani, J. (1997). Flowshop scheduling/sequencing research: A review of the literature, 1952–1994. IEEE Transactions on Engineering Management 44, 316–329.

Reza Hejazi, S., & Saghafian, S. (2005). Flowshop-scheduling problems with makespan criterion: a review. International Journal of Production Research, 43, 2895–2929.

Portougal, V., & Trietsch, D. (2006). Johnson’s problem with stochastic processing times and optimal service level. European Journal of Operational Research, 169, 751–760.

Talwar, P.P. (1967). A note on sequencing problems with uncertain job times. Journal of the Operations Research Society of Japan, 9, 93–97

Ruiz, R., & Maroto, C. (2005). A comprehensive review and evaluation of permutation flow shop heuristics. European Journal of Operational Research, 165, 479–494.

Ruiz, R., & Allahverdi, A. (2007). Some effective heuristics for no-wait flowshops with setup times to minimize total completion time. Annals of Operations Research, 156(1), 143-171.

Ruiz, R., & Stützle, T. (2008). An iterated greedy heuristic for the sequence dependent setup times flowshop problem with makespan and weighted tardiness objectives. European Journal of Operational Research, 187(3), 1143-1159.

Sayadi, M. K., Ramezanian, R., & Ghaffari-Nasab, N. (2010). A discrete firefly meta-heuristic with local search for makespan minimization in permutation flow shop scheduling problems. International Journal of Industrial Engineering Computations, 1(1), 1-10.

Sethi, S., Yan, H., Zhang, Q., & Zhou, X. Y. (1993). Feedback production planning in a stochastic two-machine flowshop: Asymptotic analysis and computational results. International Journal of Production Economics, 30-31(C), 79-93.

Soroush, H. M., & Allahverdi, A. (2005). Stochastic two-machine flowshop scheduling problem with total completion time criterion. International Journal of Industrial Engineering: Theory Applications and Practice, 12(2), 159-171.

Wang, X., & Tang, L. (2012). A discrete particle swarm optimization algorithm with self-adaptive diversity control for the permutation flowshop problem with blocking. Applied Soft Computing Journal, 12(2), 652-662.Zammori, F., Braglia, M., & Frosolini, M. (2011). CONWIP card setting in a flow-shop system with a batch processing machine. International Journal of Industrial Engineering Computations, 2, 593-616.