Uncertain flexible flow shop scheduling problem subject to breakdowns

Transcription

1 Journal of Intelligent & Fuzzy Systems 32 (2017) DOI: /JIFS IOS Press 207 Uncertain flexible flow shop scheduling problem subject to breakdowns Jiayu Shen and Yuanguo Zhu School of Science, Nanjing University of Science and Technology, Nanjing, Jiangsu, P.R. China Abstract. Flexible flow shop scheduling problems become more complex when uncertain factors are taken into consideration. Most literature are under the assumption that machines are continuous available. But, a machine can be unavailable for many reasons, such as breakdown and planned preventive maintenance. Once a machine breaks down, then the original schedule can not be executed and we must make the corresponding adjustment according to the actual situation. This paper deals with a flexible flow shop scheduling problem with uncertain processing and repair time subject to breakdowns. Machines are noncontinuously available, i.e., they break down at arbitrary time instance not known in advance. The problem with breakdowns is modeled as a series of problems without breakdowns. To solve the problem, approaches including genetic algorithm and particle swarm optimization are used in this paper. A numerical example shows the effectiveness of the proposed approach. Keywords: Flexible flow shop scheduling, uncertain variable, machine breakdown, genetic algorithm, particle swarm optimization 1. Introduction Flexible flow shop scheduling problem is a generalization of the classical flow shop problem. There are k stages and some stages may have only one machine, but at least at one stage there are multiple machines. At each stage, a number of functionally identical machines operate in parallel. One machine can process at most one job at a time and a job can be processed by at most one machine at a time. The jobs have to visit the stages in the same order string from the stage one to the stage k. In addition, preemption of each job is not allowed. Our aim is to find an optimal allocation for each job and arrange the sequence of jobs on each machine. The flexible flow shop scheduling problem has been intensively studied for decades. Corresponding author. Yuanguo Zhu, School of Science, Nanjing University of Science and Technology, Nanjing , Jiangsu, P.R. China. Tel.: ; ygzhu@njust.edu.cn. However, most papers in literature assume that the scheduling environment is static and deterministic. While in real life, machine breakdowns is common when machine runs for long periods without maintenance. Such breakdowns make shop behavior hard to predict and reduce the efficiency of a manufacturing system. Some results about scheduling problem under machine breakdowns have been obtained. Li and Cao [16] considered single machine stochastic scheduling problems subject to random breakdowns. They assumed that processing time, repair time uptime and due date are random variables and obtained some optimal policies about these problems. Kasap et al. [14] investigated optimal sequencing policies for the expected makespan problem on a single machine subject to random breakdowns, where jobs have to be processed in their entirety if preemptions occur because of breakdowns. They showed that LPT is optimal under some specific conditions /17/$ IOS Press and the authors. All rights reserved

2 208 J. Shen and Y. Zhu / Uncertain flexible flow shop scheduling problem subject to breakdowns Flow shop problems with random breakdowns are studied by Allahverdi [2, 3] and Allahverdi and Mittenthal [1, 4]. In several different cases, they got a series of optimal policies about some two machine flow shop problems. Al-Hinai and ElMekkawy [8] studied the flexible job shop problems with random machine breakdowns using a two-stage hybrid genetic algorithm. The two-stage hybrid genetic algorithm outperformed three other methods from literature in terms of robustness and stability. Alcaide et al. [5] considered stochastic flow shop scheduling problems subject to breakdowns and proposed a procedure to convert these problems with breakdowns into a finite sequence of problems without breakdowns. Later, Alcaide et al. [6] applied the above conversion method to minimize the expected makespan on an open shop problem subject to stochastic processing times and failures successfully. Hasan et al. [12] proposed an improved local search technique combining with genetic algorithm for job shop scheduling with machine breakdowns. Numerical experiments show that the algorithm outperforms the existing meta-heuristic algorithms. Holthaus [13] considered the simulation based analysis of dispatching rules for scheduling in job shops with machine breakdowns. Experimental results show that the relative performance of dispatching rules different from each other under different objective function. Albers and Schmidt [7] proposed an algorithm for an online scheduling problem with unexpected machine breakdowns. They obtained some properties about the remaining processing time. Allaoui and Artiba [9] studied a two stage hybrid flow shop scheduling problem with only one machine on the first stage and m machines on the second stage to minimize the makespan. They proposed a branch and bound algorithm for the small size problems and three algorithms for large size problems. At last, They proved that H heuristic is better than LPT heuristic in the worstcase performance. Shahul and Prabhaharana [21] proposed a greedy randomized adaptive search procedure algorithm for m-machine flow shop scheduling problem. They proved the proposed B-GRASP algorithm outperforms Chakravarthy and Rajendranąŕs heuristic in all cases. When no samples are available to estimate a probability distribution, we have to invite some domain experts to evaluate the belief degree that each event will occur. Perhaps some people think that the belief degree is subjective probability or fuzzy concept. However, it is usually inappropriate because both probability theory and fuzzy set theory may lead to counterintuitive results in this case [20]. In order to rationally deal with belief degrees, Liu [17] founded the uncertainty theory in 2007 and refined it [19] in 2010 as a branch of mathematics for modeling human uncertainty. For example, Ding and Zhu [10] proposed two empirical uncertain models for project scheduling problems. The uncertain pessimistic value model was solved by using inverse uncertainty distribution. The uncertain time range measure model was solved by genetic algorithm combining with uncertain simulation. Ke and Su [15] proposed an uncertain random multilevel programming and applied it to a production control problem. In order to solve the uncertain random model, a hybrid intelligent algorithm integrating the uncertain random simulations, neural networks and genetic algorithm was proposed. Yao [24] proposed a type of uncertain differential equation with jumps and provided sufficient conditions for having a unique solution and being stable. Sheng and Zhu [22] studied an optimistic value model for uncertain optimal control. They got the principle of optimality and equation of optimality. In addition, a portfolio selection problem was solved by the equation of optimality. Gao and Yao [11] investigated the analogous continuous dependence theorems in uncertain differential equation and proved two continuous dependence theorems. An uncertain bang-bang optimal control model for switched systems was introduced by Yan and Zhu [23]. They presented a two-stage algorithm and solved this problem successfully. In a manufacturing system such as steelmaking, continuous casting production process, electrical discharge machining process and mold production, the repair time may be uncertain because no one know the exact repair time and a repairman may estimate the repair time based on his prior experience. So, in contrast to previous papers, we consider the repair time as an uncertain variable. In addition, the processing time is usually recorded or collected under the influence of some unexpected factors such as improper manual operation, defects in job and machine themselves, and environmental impact. Thus the processing time may not be accurately known and may be considered as an uncertain variable. The rest of the paper is organized as follows. In Section 2, we briefly review some basic definitions and properties about uncertainty theory. In Section 3, we describe the problem formulation of a flexible flow shop scheduling problem. In Section 4, we introduce approaches for the problem. In Section 5, a numerical example is illustrated.

3 J. Shen and Y. Zhu / Uncertain flexible flow shop scheduling problem subject to breakdowns Preliminary In this section we will introduce some concepts of uncertainty theory. Let Ɣ be a nonempty set, and L a σ-algebra over Ɣ. Each element L is called an event. A set function M from L to [0, 1] is called an uncertain measure if it satisfies the following axioms [17]: M{Ɣ} =1 for the universal { set Ɣ; M{ }+M{ c }=1 for any event } ; M i M{ i } for every countable sequence of events 1, 2,. The triplet (Ɣ, L, M) is called an uncertainty space. To obtain an uncertain measure of a compound event, a product uncertain measure was defined by Liu [17] as: Let (Ɣ k, L k, M k ) be uncertainty spaces for k = 1, 2,. The product uncertain measure M { is an uncertain measure satisfying M } k = M k{ k }, where k are arbitrarily chosen events from L k for k = 1, 2,, respectively. An uncertain variable is a measurable function ξ from an uncertainty space (Ɣ, L, M) to the set R of real numbers, i.e., for any Borel set B of real numbers, the set {ξ B} ={γ ξ(γ) B} is an event. An uncertain vector ξ is (ξ 1,ξ 2,,ξ n ) where ξ 1,ξ 2,,ξ n are uncertain variables. The uncertain distribution of an uncertain variable ξ is defined by (x) = M{ξ x} for any real number x. Theorem 1. [19] Let ξ be an uncertain variable with continuous uncertainty distribution. Then for any real number x, we have M{ξ x} = (x), M{ξ x} =1 (x). Example 1. Linear uncertain variable ξ L(a, b) has an uncertainty distribution 0, if x a x a (x) = b a, if a x b 1, if x b where a and b are real numbers with a<b. Definition 1. [19] An uncertain distribution (x) is said to be regular if its inverse function 1 (x) exists and is unique for each α (0, 1). Then the inverse function 1 is called the inverse uncertainty distribution of ξ. Example 2. The inverse uncertainty distribution of linear uncertain variable L(a, b) is 1 (α) = (1 α)a + αb. Definition 2. [17] The uncertain variables ξ 1,ξ 2,,ξ m are said to be independent if M{ m (ξ i B i )}= min 1 i m M{ξ i B i } for any Borel sets B 1,B 2,,B m of real numbers. Theorem 2. [19] Let ξ 1,ξ 2,,ξ n be independent regular uncertain variables with uncertainty distributions 1, 2,, n, respectively. If function f (x 1,,x m,x m+1,,x n ) is strictly increasing with respect to x 1,x 2,,x m, and strictly decreasing with respect to x m+1,x m+2,,x n, then ξ = f (ξ 1,ξ 2,,ξ n ) is an uncertain variable with inverse uncertainty distribution 1 (α) = f ( 1 1 (α),, 1 m (α), (1 α),, 1(1 α)). 1 m+1 n Example 3. Let ξ 1,ξ 2,,ξ n be independent regular uncertain variables with uncertainty distributions 1, 2,, n, respectively. Since f (x 1,x 2,,x n ) = x 1 + x 2 + +x n is a strictly increasing function, the sum ξ = ξ 1 + ξ 2 + +ξ n is an uncertain variable with inverse uncertainty distribution 1 (α) = 1 1 (α) (α) n (α). Definition 3. [17] Let ξ be an uncertain variable, and α (0, 1]. Then ξ sup (α) = sup{r M{ξ r} α} is called the α-optimistic value to ξ, and ξ inf (α) = inf{r M{ξ r} α} is called the α-pessimistic value to ξ. Founded by Liu [18] in 2009 and refined by Liu [19] in 2010, uncertain programming is a type of mathematical programming involving uncertain variables. Assume that x is a decision vector, ξ is an uncertain vector, f (x, ξ) is an objective function, and g j (x, ξ) are constraint functions, j = 1, 2,,p.In order to obtain a decision with maximum expected objective value subject to a set of chance constraints,

4 210 J. Shen and Y. Zhu / Uncertain flexible flow shop scheduling problem subject to breakdowns Liu [18] proposed the following uncertain programming model, min min f x f subject to : M{f (x, ξ) f } β M{g j (x, ξ) 0} α j,j= 1, 2,,p, where α j and β are specified confidence levels for j = 1, 2,,p, and min f is the β-pessimistic value. If we want to minimize the pessimistic value of an objective (such as makespan or cost) subject to some chance constraints, we can use the above model. A key problem in uncertain programming is how to solve the above model. Fortunately, under the framework of uncertainty theory, an uncertain programming model may be transformed into a crisp model if the related functions are increasing or decreasing. Thus, we can solve the equivalent crisp model of an uncertain programming model by an optimization method such as cutting plane method, tabu search, interior point method, and heuristic algorithms. Theorem 3. [17] Assume the constraint function g(x, ξ 1,ξ 2,,ξ n ) is strictly increasing with respect to ξ 1,ξ 2,,ξ k and strictly decreasing with respect to ξ k+1,ξ k+2,,ξ n.ifξ 1,ξ 2,,ξ n are independent uncertain variables with uncertainty distributions 1, 2,, n, respectively, then the chance constraint M {g(x, ξ 1,ξ 2,,ξ n ) 0)} α holds if and only if g(x, 1 1 (α),, 1 k (α), 1, 1 n (1 α)) Problem formulation k+1 (1 α), We consider a flexible flow shop machine problem with m stages in series. At stage s, s = 1, 2,,m, these are M s identical machines in parallel. There is unlimited intermediate storage between two successive stages. Assume: All jobs are independent and available for processing at the initial time; One machine can process only one job at a time and one job can be processed by only one machine at any time; Processing times of all jobs are uncertain variables; Travel time between consecutive stages is negligible; Preemption is not allowed; Repair times of machines are uncertain variables; The repaired machine will not fail again. We assume that at most one disruption take place at a time for all machines. The breakdown time is stochastic and the repair time is uncertain. The following notations are needed: ξ ij : processing time of job i at stage j (an uncertain variable); x il :1ifjobi is assigned to the lth position, and 0 otherwise; y ijk :1ifjobi is processed on the kth machine at stage j, and 0 otherwise; s ij : the starting time of job i at stage j; e ij : the completion time of job i at stage j; r ks : the repair time of machine k at stage s (an uncertain variable); L: a large enough number. Next we introduce the general procedure of converting flexible flow shop problem subject to breakdowns into a problem without breakdowns. At the first, we get the optimal scheduling without machine failure. Second, when one machine breaks down, the job which be processed on the failure machine and jobs whose starting time behind the failure time will be rearranged at machines without failure. However, those jobs whose starting time in front of failure instant on normal machines do not need to participate in the rearrangement. Third, when the failure machine is repaired, all the jobs whose starting time behind the repair instant must be rearranged at all normal machines. Fourth, at each stage, rearrange whenever every failure and repair occur. After the last rearrangement, we get the optimal schedule. Based on the aforementioned general procedure, we only need to set up an uncertain flexible flow shop model without breakdowns. Next we propose a models for the flexible flow shop problem with the objective of minimizing the makespan. The model is pessimistic value chance constrained model:

5 J. Shen and Y. Zhu / Uncertain flexible flow shop scheduling problem subject to breakdowns 211 min min f (1) x il,y ijk f subject to M{ max {e im} f } τ, (2),2,,n x il = 1,l= 1, 2,,n (3) x il = 1,i= 1, 2,,n (4) l=1 M s y ijk = 1,i= 1, 2,,n; j = 1, 2,,m (5) k=1 e ij = s ij + ξ ij,i= 1, 2,,n; j = 1, 2,,m (6) M{e ij s i(j+1) 0} α, i = 1, 2,,n; j = 1, 2,,m (7) { } M x il s ij x i(l+1) s ij 0 β, i, l = 1, 2,,n; j = 1, 2,,m (8) { M x il1 y ijk e ij x il2 y ijk s ij ( 1 ) } x il2 y ijk L 0 γ, j = 1, 2,,m; l 1,l 2 = 1,,n; l 1 l 2 ; k = 1,,M s (9) According Definition 3, the term (1) and Equation (2) indicate that our goal is to obtain τ-pessimistic value of completion time. Equation (3) ensures that every priority position belongs to only one job. Equation (4) ensures that each job is assigned to only one priority position. Equation (5) ensures that one job can be processed by only one machine at any time. Equation (6) determines the relationship of starting time and completion time of each job. Equation (7) ensures that each operation must satisfy the priority order on α confidence level. Equation (8) illustrates that each job must satisfy the priority order of itself on β confidence level. Equation (9) ensures that when two jobs are assigned to the same machine at a certain stage, jobs satisfy the priority order of each other on γ confidence level. This model can be transformed into a crisp model by Theorem 3. According to Definition 3, we note that term (1) and Equation (2) can be integrated together. So combining term (1) and Equation (2), we can get the objective function 1 max{e im } (τ), where max{e 1 im }(τ) is the inverse uncertainty distribution of im }. Then we have min x il,y max{e 1 im }(τ) (10) ijk subject to x il = 1,l= 1, 2,,n (11) x il = 1,i= 1, 2,,n (12) l=1 M s y ijk = 1,i= 1, 2,,n; j = 1, 2,,m (13) k=1 e ij = s ij +ξ ij,i= 1, 2,,n; j = 1, 2,,m(14) 1 e ij (α) 1 s i(j+1) (1 α),i= 1, 2,,n; j = 1, 2,,m (15) x il 1 s ij (β) x i(l+1) 1 s ij (1 β), i, l = 1, 2,,n; j = 1, 2,,m (16) x il1 y ijk 1 e ij (γ) x il2 y ijk 1 s ij (1 γ) ( + 1 ) x il2 y ijk L, j = 1, 2,,m; l 1,l 2 = 1,,n; l 1 l 2 ; k = 1,,M s. (17) 4. Heuristic approaches If a machine breaks down, the job currently being processed by the machine must be re-processed. Rescheduling is performed after the ongoing jobs at the disruption time are all completed. All jobs that start later than the breakdown time need to be rescheduled. The problem which is converted into scheduling problem without breakdowns may be solved by meta-heuristic algorithms. Genetic algorithm and particle swarm optimization will be adopted in our study. Genetic algorithm (i): Coding. Use matrix code as follows:

6 212 J. Shen and Y. Zhu / Uncertain flexible flow shop scheduling problem subject to breakdowns a 11 a a 1n a 21 a a 2n... A m n =...,... a m1 a m2... a mn where a ij is a real number in (1,M i + 1) (M i indicates the number of machines at stage i), and indicates the ith operation of job j processed on machine [a ij ]([ ] indicates rounding down operation). If [a ij ] = [a ik ], j/= k, this means multiple jobs processed on the same machine. If i = 1, we can determine the processing sequence according to the value of a ij.ifi>1, we follow the first input first output policy to determine the processing sequence. If the end time of previous operations are same, we determine the processing sequence according to the value of a ij. If the values of a ij are same, then we determine the processing sequence randomly. Then chromosome can be constructed based on the matrix code. It is a character string of length n m + m 1. The chromosome can be represented by [a 11,a 12,...,a 1n, 0,a 21,a 22,...,a 2n, 0,...,0, a m1,a m2,...,a mn ]. (ii): Initialization. Generate a set of matrices. The element a ij of a matrix is randomly generated in (1,M i + 1). Then we turn these matrices into chromosomes. (iii): Crossover. Exchange some genes of two chromosomes randomly. (iv): Mutation. Randomly generate K in the set { 1, 1}.IfK = 1, generate random number r = rand(0,m i + 1 a ij ); otherwise generate random number r = rand(0,a ij 1). Use a ij = a ij + K r to achieve mutation. Particle swarm optimization Coding method is the same as the aforementioned genetic algorithm. Then the update formula of particle velocity and position are as follows: V N S (t + 1) = w V N S (t) +c 1 rand 1 (0, 1) (P p j N S P N S(t)) +c 2 rand 2 (0, 1) (P g j N S P N S(t)) ( ) t +k 0.5, T N S(max) P N S (t + 1) = P N S (t) + V N S (t + 1) In the above formulas, w is the inertia weight, c 1 is the cognition coefficient, c 2 is the social coefficient, rand 1 (0, 1) and rand 2 (0, 1) are random numbers with uniform distribution U(0, 1), k is the adjustable parameter between 0 and 1, t is the current number of the iterations, and T N S(max) is the maximum number of iterations. V N S is the particle velocity, P p j N S is the local extremum, and Pg j N S is the global extremum. Do some revision after completion of each iteration to ensure particles legal right. If P ij < 1, then P ij = 1.1; if P ij >M j + 1, then P ij = M j A numerical example In this section we give an numerical example with pessimistic value chance constrained model. Assume that there are 3 stages and 7 jobs, each stage has 3 machines. Processing time ξ ij L(i, j), failure time subject to uniform distribution U(0, 10), repair time Fig. 1. Scheduling result of pessimistic value chance-constrained model (F indicates machine failure).

7 J. Shen and Y. Zhu / Uncertain flexible flow shop scheduling problem subject to breakdowns 213 r ks L(k, s). α = β = γ = 0.9, τ = Adopt the following rank-based evaluation function Eval(V t ) = a(1 a) t 1, t = 1, 2,,pop size, where parameter a (0, 1). Crossover probability P c = 0.75, mutation probability P m = 0.25, the number of population size is 20, generation is 100 in genetic algorithm. The parameters in particle swarm optimization are set as follows: particle population size is 20, the maximum number of iterations is 100, k = 0.3, c 1 = c 2 = 2, w = 0.8. For our model, we run two algorithms five times respectively. The mean value of the objective is 20.4 and minimum value is 19.5 by genetic algorithm. The mean value of the objective is 22.6 and minimum value is 21.8 by particle swarm optimization. The result by genetic algorithm is better than that by particle swarm optimization. We use Fig. 1 to represent the optimal scheduling result of pessimistic value chance constrained model by genetic algorithm. 6. Conclusions The main work of this paper is that we adopted uncertainty theory to establish a chance constrained framework model for flexible flow shop programming problem with uncertain factors, which may be applied much widely in real industrial production. We considered flexible flow shop scheduling problems subject to random breakdowns under an uncertain environment. A pessimistic value chance constrained model for uncertain flexible flow shop scheduling problem is proposed. The genetic algorithm and particle swarm optimization algorithm are suggested for solving the problem. An example shows the effectiveness of the established model and these two heuristic algorithms. Acknowledgments This work is supported by the National Natural Science Foundation of China (No ). References [1] A. Allahverdi and J. Mittenthal, Two-machine ordered flowshop scheduling under breakdown, Mathematical and Computer Modelling 20(2) (1994), [2] A. Allahverdi, Two-stage production scheduling with separated setup times and stochastic breakdowns, The Journal of the Operational Research Society 46(7) (1995), [3] A. Allahverdi, Two-machine proportionate flowshop scheduling with breakdowns to minimize maximum lateness, Computers & Operations Research 23(10) (1996), [4] A. Allahverdi and J. Mittenthal, Dual criteria scheduling on a two-machine flowshop subject to random breakdowns, International Transactions in Operational Research 5(4) (1998), [5] D. Alcaide, A. Rodriguez-Gonzalez and J. Sicilia, An approach to solve the minimum expected makespan flowshop problem subject to breakdowns, European Journal of Operational Research 140(2) (2002), [6] D. Alcaide, A. Rodriguez-Gonzalez and J. Sicilia, A heuristic approach to minimize expected makespan in open shops subject to stochastic processing times and failures, International Journal of Flexible Manufacturing Systems 17(3) (2005), [7] S. Albers and G. Schmidt, Scheduling with unexpected machine breakdowns, Discrete Applied Mathematics 110(2) (2001), [8] N. Al-Hinai and T. ElMekkawy, Robust and stable flexible job shop scheduling with random machine breakdowns using a hybrid genetic algorithm, International Journal of Production Economics 132(2) (2011), [9] H. Allaoui and A. Artiba, Scheduling two-stage hybrid flow shop with availability constraints, Computers & Operations Research 33(5) (2006), [10] C. Ding and Y. Zhu, Two empirical uncertain models for project scheduling problem, Journal of the Operational Research Society 66(9) (2015), [11] Y. Gao and K. Yao, Continuous dependence theorems on solutions of uncertain differential equations, Applied Mathematical Modelling 38 (2014), [12] K. Hasan, R. Sarker and D. Essam, Genetic algorithm for job-shop scheduling with machine unavailability and breakdowns, International Journal of Production Research 49(16) (2011), [13] O. Holthaus, Scheduling in job shops with machine breakdowns: An experimental study, Computers & Industrial Engineering 36(1) (1999), [14] N. Kasap, H. Aytug and A. Paul, Minimizing makespan on a single machine subject to random breakdowns, Operations Research Letters 34(1) (2006), [15] H. Ke and T. Su, Uncertain random multilevel programming with application to production control problem, Soft Computing 19(6) (2015), [16] W. Li and J. Cao, Stochastic scheduling on a single machine subject to multiple breakdown according to different probabilities, Operations Research Letters 18(2) (1995), [17] B. Liu, Uncertainty Theory, 2nd Edition, Springer-Verlag, Berlin, [18] B. Liu, Theory and Practice of Uncertain Programming, 2nd Edition, Springer-Verlag, Berlin, [19] B. Liu, Uncertainty Theory: A Branch of Mathematics for Modeling Human Uncertainty, Springer-Verlag, Berlin, [20] B. Liu, Why is there a need for uncertainty theroy? Journal of Uncertain Systems 6(1) (2012), [21] B. Shahul and G. Prabhaharan, A grasp algorithm for m- machine flowshop scheduling problem with bicriteria of makespan and maximum tardiness, International Journal of Computer Mathematics 84(12) (2007),

8 214 J. Shen and Y. Zhu / Uncertain flexible flow shop scheduling problem subject to breakdowns [22] L. Sheng and Y. Zhu, Optimistic value model of uncertain optimal control, International Journal of Uncertainty, Fuzziness and Knowledge-Based Systems 21(7) (2013), [23] H. Yan and Y. Zhu, Bang-bang control model for uncertain switched systems, Applied Mathematical Modelling 39(10-11) (2014), [24] K. Yao, Uncertain differential equation with jumps, Soft Computing 19(7) (2015),