A two-phase fuzzy programming model for a complex bi-objective no-wait flow shop scheduling

Article history: Received 25 January 2012 Accepted March, 5 2012 Available online 12 March 2012 In this paper, we study no-wait flow shop problem where setup times depend on sequence of operations. The proposed problem considers sequence-independent removal times, release date with an additional assumption that there are some preliminary setup times. There are two objectives of weighted mean tardiness and makespan associated with the proposed model of this paper. We formulate the resulted problem as a mixed integer programming, where a two-phase fuzzy programming is implemented to solve the model. To examine the performance of the proposed model, we generate several sample data, randomly and compare the results with other methods. The preliminary results indicate that the proposed two-phase model of this paper performed relatively better than Zimmerman's single-phase fuzzy method. © 2012 Growing Science Ltd. All rights reserved


Introduction
Flowshop is one of the well-known problems, which has attracted many researchers for a while.Manufacturing equipments, which produce one or two similar products using high-volume specialized equipments, are normally considered as an example of flowshop and as an example; we may consider an assembly line (Pinedo, 2008).No-wait flow shop problem (NWFP) is one of the primary restrictions in flowshop problem.According to this limitation, as soon as a job finishes its work on one machine, the operations of the next machine on this job must be initiated and no interruption is permitted.There are many industrial applications of NWFP such as steel industry, plastic injection, chemical and pharmaceutical industries (Hall & Sriskandarajah, 1996).There are literally various studies associated with NWFP problem.Liu et al. (2007), for instance presented particle swarm optimization to minimize makespan.Pen et al. (2008) presented a discrete PSO along with variable neighborhood descent to minimize makespan and total flowtime for NWFP problem.Tseng and Lin (2010) proposed a hybrid genetic algorithm for no-wait flowshop scheduling problem, which also uses a local search.Wang et al. (2010) used accelerated Tabu search for NWFP problem for maximum lateness ( ).They increased the basic operations of generating better candidate solutions using Tabu search and reduced the complexity of the operations of NEH.Note that Hall et al. (1996) performed a comprehensive survey of scheduling problems with blocking and no-wait in process.There are many industries where setup times as well as processing times are independent from the operational times.Setup times include the necessary times for adding or exchanging equipments on facilities (Eren, 2009).Setup times and other related times are divided into two groups of independent and dependent times.There are many industrial cases where setup times depend on the operations such producing different colors using some common machines and the problem is called sequence-dependent setup time flow shop problem (SDSTFP).
Based on the notations developed by Graham et al. (1979), Allahvedi et al. (2008), Eren (2007) et al. (1999, 2008).Franca et al. (2006)  In addition, they proposed some metaheuristics to solve the resulted problem, which seem to provide better quality solutions.They also used branch and bound method to provide a lower bound on this problem.Stafford and Tseng (2002) developed two models for family of flowshop sequencing problems of the form | , | .Gupta et al. (1997) provided a two-stage operations research models with setup and removal times separated.In their method, they considered setup times independent from operational times and proposed a method for minimizing makespan, which could be solved in polynomial time.Wang and Cheng (2006) proposed a heuristic approach for two-machine no-wait flowshop scheduling with due dates and class setups.They divided jobs to different groups with various setup times with an objective of minimizing the maximum lateness.The proposed model of this paper was developed for multiobjective problems.They proposed fuzzy goal programming technique to solve the resulted problem.Fuzzy programming has been considered in different ways to handle uncertainty associated with this type of problem.Ishibuchi et al. (1994) presented local search algorithms for flow shop scheduling with fuzzy due-dates where the primary objective was to maximize the minimum grade of satisfaction.Yao and Feng (2002) constructed a fuzzy flow-shop sequencing model based on statistical data.Wu (2010) presented the fuzzy earliness and tardiness in scheduling problems by using genetic algorithms.They presented a method to minimize the weighted lateness.The concept of fuzzy programming can be also used in different areas of scheduling (Gharegozli et al., 2009;Tavakkoli-Moghaddam et al., 2010;Khademi-Zare & Fakhrzad, 2011).
In this paper, we present a multi-objective NWFP problem where the first objective is the minimization of weighted average of lateness and the other objective is minimization of makespan.All setup times depend on the sequences of different jobs and machines.We also consider a preliminary setup time when the operations start, which is independent from the sequence of operations and only depends on the machines.The other assumption with the proposed model of this paper is that not all jobs are available at the beginning of the operations.In summary, the proposed model is denoted as , , , , | , .There are literally many real-world applications where we can use the proposed model of this paper (Ruiz et al., 2008).The proposed model of this paper considers two objectives and it is formulated in a form of mixed integer programming and it is solved using the method proposed by Li et al. (2006).The organization of this paper first presents the mathematical model in section 2. The proposed solution methodology is given in section 3 and the performance of the proposed model is examined using different test problems and the results are given in section 4. Finally, concluding remarks are given in the last to summarize the contribution of the paper.

The proposed model
The proposed model of this paper considers , , , , , where IST is the initial setup times for processing jobs located in the first stage of operations.

Assumptions
• All parameters are assumed deterministic.
• There must be no interruption when an operation starts on a particular machine.
• All jobs must precede the other operations on other machines as soon as they are finished with a particular machine unless the operations belong to the last machine.• Each machine can only process one job at the same time.
• Processing times do not depend on sequence and they are all deterministic.
• Not all jobs can be processed at the beginning of the operations.
• All machines are available in planning time horizon and there is no failure or interruption.
• Removal time is independent from the processing and order of sequence and it is predefined.

J:
Set of jobs The completion time of job on machine if it is in position in terms of sequence C jip : The completion time of job on the last machine if it is in position in terms of sequence The earliest starting time of processing job on machine if it is in position ST jip : The earliest starting time of processing job on the first machine if it is in position ST j1p : The weighted mean lateness Eq. ( 1) and Eq. ( 2) are associated with minimization of weighted mean tardiness and makespan, respectively.Eq. (3) determines that in each position only one job can be located and Eq. ( 4) guarantees that each job can be only in one position.According to Eq. ( 5) and Eq. ( 6), is equal to one if both and are equal to one.Eq. ( 7) indicates that for each position 2 only one job can be started after another one.Eq. (8) indicates that the earliest time for processing job on machine cannot be less than its release date .Eq. ( 9) guarantees that earliest time to process job j cannot be sooner than its initial setup time if job is in the first position of sequence.Eq. ( 10) indicates that there is no interruption between jobs and machines.Eq. ( 11) computes the completion time of job on the first machine if job is located in position in terms of sequence.Eq. ( 12) computes the completion times of job j on the second machine and after and it guarantees that there is no interruption.Eq. ( 13) shows the relationship between completion times of two sequential jobs.According to Eq. ( 14) and Eq. ( 15), we can make sure that if job in not in position then the earliest starting and completion times are equal to zero, otherwise, 1. Eq. ( 16) shows that makespan cannot be less than completion time of the last machine.Eq. ( 17) is also associated with the tardiness of job .Finally, show variable types definitions.

The proposed model
The proposed model of this paper uses a two-stage fuzzy programming, which is based on the method developed by Li et al. (2006) and the first stage uses Zimmerman's method (Zimmerman, 1978).We first present the necessary notations associated with the proposed model.

Satisfaction degree :
Satisfaction degree after the first stage λ_ 1: Satisfaction degree after the second stage λ_ 2: Satisfaction degree of the th f objective function after the second stage 1,2 : weight of satisfaction degree of the th f objective function in the second stage 1,2 : The lower bound of the first objective function : The lower bound of the second objective function : The value of the first objective function when the second objective function has the minimum value z

:
The value of the second objective function when the first objective function has the minimum value z

:
The value of the first objective function after the first stage terminates : The value of the second objective function after the first stage terminates :

Phase one
Step 1.The proposed BMILP is solved based on and the optimal value is called .
Step 2. Based on the solution obtained from the first step, is computed.
Step 3. The proposed BMILP is solved based on and the optimal value is called .
Step 4. Based on the solution obtained from the third step, is computed.
Step 5. Solve the following linear programming problem.Step 6. Set λ_ 1 to computed λ in step 5.

Phase two
Step 7. Compute ‫و‬ based on the results of step 5 and store them in and , respectively.Solve the following linear programming problem, Step 8. _ 2 is equal to optimal value obtained from Eq. ( 26) achieved from step 7.
Step 9. Repeat computations to calculate and until the final solutions are achieved.

Numerical examples
In order to measure the relative efficiency of the proposed model of this paper, we have solved 102 randomly generated test problems.Except and , which are determined by decision maker, all other values are predefined.Table 1 shows details of our input parameters.

Discussion
In this section, we discuss the performance of the proposed model of this paper and compare the performance of two phase fuzzy method with Zimmerman's approach.We have solved all our test problems using LINGO-8 and the results are summarized in Table 2, Table 3 and Table 4.As we can observe from the results of Tables, the level of satisfaction degree of phase 2 is relatively better than the first level.In addition, Li et al. (2006) proved that the level of second level satisfaction will not be worse than the first level.Besides, 36 out of 45 sample test problems yields better results, which implies that the two-phase method performs better than the Zimmerman's method.Table 5 shows details of our results for different values of and .As we can observe the final solutions are sensitive to all these values.

Conclusions
In this paper, we have presented a no-wait flow shop problem where setup times depend on sequence of operations.The proposed problem considered sequence-independent removal times, release date with an additional assumption that there were some preliminary setup times.There were two objectives of weighted mean tardiness and makespan associated with the proposed model of this paper.The proposed model of this paper formulated the resulted problem as a mixed integer programming, where a two phase fuzzy programming was implemented to solve the model.To examine the performance of the proposed model, we have generated    , 187, 985-1032. Bianco, L., Dell'Olmo, P., & Giordani.S. (1999).Flowshop no-wait scheduling with sequence dependent setup times and release dates.INFOR, 37, 3-19.Eren, T. (2007) proposed a genetic algorithm for | , , | and compared the performance of their method with Bianco et al. (1999).Ruiz and Allahverdi (2007) investigated on SDSTFP problem in a form of | , | .They investigated different heuristics as well as metaheuristics to solve the resulted problem and presented a dominance ratio for three-machine problem.Aldowaisan and Allahverdi (1998) provided a solution methodology for | , | ∑ and presented one elimination criterion and developed optimal solution for two special cases.Aldowaisan (2001) considered | , | ∑ problem one more time and tried to find a local and global dominance relations.

Fig . 2 .Fig . 3 .
Fig .2.Comparison between satisfaction degrees of phase one and phase two for 3-machine problems

Fig. 4 .
Fig. 4. The impact of the changes in on satisfaction degree of phase two

Table 2
Results for 2-machine problems

Table 3
Results for 3-machine problems Comparison between satisfaction degrees of phase one and phase two for 2-machine problems several sample data, randomly and compared the results with other methods.The results indicated that the proposed two-phase model of this paper performed relatively better than Zimmerman's single phase fuzzy method.