Towards an Artificial Immune System for Scheduling Jobs and Preventive Maintenance Operations in

Flowshop Problems Fatima BENBOUZID-SI TAYEB, Wahiba BELKAALOUL

Laboratoire des Méthodes de Conception de Systèmes (LMCS) Ecole nationale Supérieure d’Informatique (ESI ex INI)

BP 68M, Oued Smar – Algiers, Algeria E-mail. {f_sitayeb; w_belkaaloul}@esi.dz

Abstract— This paper investigates permutation flowshop

problem with preventive maintenance (PM). The objective functions are to minimize the total completion time for the production part and the total earliness/tardiness for PM part. The resolution consists of two steps: the one consists on scheduling production jobs using an artificial immune algorithm (AIA); the second one consists on deploying PM operations, taking the production schedule as a mandatory constraint of resources unavailability in the resolution of the problem. Furthermore, we use the principles of vaccination and receptor editing in order to strengthen search ability. The efficiency of the proposed AIAs with respect to minimization of makespan for the production part and performance loss after PM insertion, is compared to some referred in the related scheduling literature metaheuristics. Simulation results on both standard PFSP problems and non- standard integrated PFSP with PM problems show the superiority of our proposed algorithms.

Keywords— Production; Preventive Maintenance; Scheduling; Artificial Immune System; Permutation Flowshop.

I. INTRODUCTION Production scheduling and preventive maintenance (PM)

planning are among the most common and significant problems faced in manufacturing industries. These activities directly operate on the same resources and equipment. Therefore, if the production scheduling does not observe the expected period of PM operations, the interruptions coming from the PM interference or machine breakdown may increase the probability of machine failure. Evading this conflict, it is essential that production scheduling and PM operations be carried out in an integrated way to hedge against these often avoidable failures and re-scheduling occurrences, allowing the decision maker to find trade-off solutions between both objectives of production and maintenance.

Nowadays, the issue of integrating production scheduling and PM planning, in failure-prone manufacturing systems, is becoming an active area of research due to its importance in the current highly competitive environment. Budai, Dekker and Nicolai [1] reviewed the majority of integrated maintenance and production models. In the production/maintenance optimization models, where our work is situated, many problems have been presented in the literature. Most of these models aim to optimize a combination of maintenance and/or production costs, production makespan or system availability

(or unavailability). A general result is that these scheduling problems, under different machine configurations and various objective functions, are strongly NP-hard, since for each separate criterion the problem is strongly NP-hard. Thus, the only feasible way to solve it is, therefore, the use of heuristics since exhaustive methods take a prohibitive execution time to find the best solution. Some works explicitly try to integrate flowshop scheduling problem (FSP) and maintenance decisions by optimizing them simultaneously [2, 3, 4, 5, 6, 7, 8].

During the last decade, inspired by biological immune system, artificial immune systems (AIS) has emerged a novel and power-full populations based metaheuristic and maturing computational paradigm. AIS have been applied successfully to a variety of optimization problems and studies have shown that it possesses several attractive immune properties, such as strong self-learning, long lasting memory, self-identity, fault tolerance and strong adaptability to the surroundings [9], that allow evolutionary algorithms to avoid premature convergence [10] and improve local search [11] due to the global search ability and quick convergence ability. However, it was pointed out that there are a few literatures considering the artificial immune systems for the solution of scheduling problems of any type.

In modern years, AIS algorithms are getting tremendous attention from researchers in the scheduling problem especially in flowshop scheduling. Alisantoso, Khoo and Jiang [12] considered a flexible PCB FSP by using an immune algorithm. Khoo and Situmdrang [13] dealt with the design of an assembly system for modular products by using an approach based on the principles of natural immune systems. Engin and Döyen [14] propose a computational method based on clonal selection principle and affinity maturation mechanism of the immune response to solve hybrid flowshop prolem. Kumar, Prakash, Shankar and Tiwari [15] used AIS to tackle a continuous flowshop problem with total flow times as the principal criterion. Chan, Swarnkar and Tiwari [16] and Zandieh, Fatemi Ghomi and Moattar Husseini [17] proposed approaches based on immune network interactions and genetic reproduction to address, respectively, flexible manufacturing system assignment and scheduling problems and hybrid flowshops scheduling with sequence dependant setup times. Tavakkoli-Moghaddam, Rahimi-Vahed and Mirzaei [18] considered a no-wait FSP with a new hybrid multi-objective

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algorithm based on a biological immune system. In 2007, they use AIS to address bi-criteria PFSP, in which the weighted mean completion time and the weighted mean tardiness are to be minimized simultaneously [19]. In [20] authors extend the artificial immune system approach by proposing a new methodology termed as immunoglobulin-based artificial immune system algorithm to solve the n-job, k-stage problem in a hybrid flowshop with the objective of minimizing makespan.

To the best of authors’ knowledge, there no previous study on the use of AIS for the integrated production and maintenance problem in operations research literature. In this way, the paper develops an AIA for scheduling jobs and PM operations in the flowshop sequencing problem which consists of two steps: the one consists on scheduling production jobs using an AIA; the second one consists on deploying PM operations, taking the production schedule as a mandatory constraint of resources unavailability in the resolution of the problem. The proposed AIA was, first tested with best found parameters on benchmark problems that were used by Taillard [21] for the production part. Then, the results were evaluated with respect to the minimization of performance loss after maintenance insertion with non-standard integrated PFSP with PM benchmark problems that were used by Benbouzid-Si Tayeb, Varnier and Zerhouni [22]. The main point is to show how the proposed s AIS approach is used to allow the decision maker to have compromise solutions meeting at best two criteria, one related to production and another to maintenance.

The remainder of this paper is organized as follows. In section 1, firstly we describe some approaches aiming at making production and maintenance cooperate, and then discuss some AIS relevant studies dedicated to flowshop scheduling problems. The problem description is provided in Section 2 and the proposed algorithms are described in Section 3; Section 4 is devoted to the computational experiments and in Section 5 some concluding remarks are offered.

II. PROBLEM DEFINITIONS This section describes the formulation of the production

scheduling problem and PM planning problem, then the objective functions to optimize. In this paper, we address the scheduling problem in the permutation flowshop (PFSP) with makespan minimization. For PM, among the various strategies, we focus on the systematic PM. The problem is to decide when to execute the PM operations on each machine in order to minimize the total earliness/tardiness of all PM operations.

A. Production scheduling problem The PFSP, one of the best known production scheduling

problems, can be viewed as a simplified version of the flowshop problem, and has been proved to be NP-hard [23].

In PFSP there are a set J of n different jobs J={1,…,n} to be scheduled on a set M of m machines M={1,…,m}, where the sequence of processing a job on all machines is identical. The processing time of job j on machine i Jij is pij and its completion time is cij. It is assumed that each job can be processed on at most one machine at a time and that each machine can process at most one job at a time. Furthermore, preemption is not allowed, each job is available and ready for

processing at time zero and the setup times are sequence independent. A schedule of this type, i.e., with the same job ordering on all machines, is called a permutation schedule and defined with a complete sequence of all jobs.

B. Preventive maintenance scheduling problem Systematic PM [24] at fixed predefined time intervals is

widely used in industry, and has been proved to be NP-hard [25].

The PM scheduling problem can be considered as a set NM of nm tasks NM={1,…,nm} that belong to a set M of m machines M={1,…,m}, periodically, at known intervals of time. PM operations are periodic interventions occurring every T periods and each occurrence of a PM operation depends on the ones preceding it on the same machine. The processing time of PM operation j on machine i Mij is p’ij and its completion time is c’ij. The processing times are fixed, nonnegative and evaluated with more or less certainty. Moreover, the periodicity T of these operations can vary in a tolerance interval noted [Tmin, Tmax]. This interval gives some flexibility to plan PM operations. Ideally a PM operation is planned in the interval [Tmin, Tmax]. However, it can be planed before Tmin and it is considered in advance (this Earliness is noted E’), or after Tmax and it is considered late (this Tardiness is noted T’) (Fig 1).

Fig. 1. Tolerance interval of a PM operation.

The earliness and the tardiness of the kth occurrence of PM operation Mij is computed as follows:

- t’ijk : execution time of the kth occurrence of Mij. - E’ijk : earliness of the kth occurrence of Mij. E’ijk = max (0, t’ijk+p’ij+Tminij – t’ijk+1) - T’ijk : tardiness of the kth occurrence of Mij. T’ijk = max (0, t’ijk+1 – t’ijk – p’ij –Tmaxij).

C. Objectives functions In this paper, the production objective of minimizing the

maximum completion time i.e., the completion time of the last job on the last machine, or makespan, is considered. When all jobs are scheduled, the makespan Cmax is obtained by Cmax = cmn. One will note f1 the production objective function:

f1 = Cmax (1) The maintenance objective function respects the

maintenance periods which influence the constraints of the production system. One will note f2 the temporal deviation of the maintenance operations w.r.t. their ideal time allocations expressed as tolerance temporal intervals.

Machines

Tmaxj Tminj

Tj Time

p’j c’j t’j

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2

1 1 1( ' ' )

iji mmm

ijk ijki j k

f E T= = =

= +∑∑∑ 1 (2)

III. PROPOSED AIS APPROACH The proposed AIA is based on the clonal selection

algorithm (CSA) [26]. It is known that CSAs are more suitable to tackle optimization and scheduling problems [17,18]. Furthermore, our AIA uses the principles of vaccination and receptor editing in order to strengthen search ability.

In what follows, we will first describe the proposed AIA for the production scheduling with the objective of minimizing the makespan. Then we present the used heuristics to deploy PM operations in the resulting production sequence.

A. An artificial immune algorithm for the PFSP Assuming that the population is a set of antibodies, and

each of them represents a solution to the scheduling problem, we list the main procedures of the proposed AIA as follows:

Step 1 Initialization: (a) Parameters settings: set the number of initial population (Popsize), termination conditions (Max_gen, Improve_gen), the rate of cloning, the rate of mutation, the number nea of exchangeable antibodies and the frequency of elimination steps;

(b) Initial population: generate a population of Popsize antibodies (schedules); Step 2 Objective function evaluation: Evaluate the fitness function (makespan) for each antibody. Step 3 Affinity evaluation: Calculate affinity value of each antibody. Step 4 Clonal selection and expansion: (a) Select M antibodies from the population with the highest affinity. (b) Generate M clones (copies) from the selected antibodies in step 4(a) by using binary tournament rule (choose two antibodies from M antibodies randomly and select the antibody with higher affinity). (c) Select (nm) antibodies from M clones randomly and apply the mutations to make new antibodies. Select the best (nm) mutated antibodies with the highest affinity;

(d)Add (nm) new antibodies to the current generation; (e) Copy the best antibody to the next generation; Step 6 Vaccination: (a) Create a vaccine with the best antibodies of the current generation; (b) For each selected antibodies apply the vaccine to make new (vn) antibodies; (c) Copy the best antibody to the next generation. Step 7 Receptor editing: (a) Replace (nea) worst antibodies in the current population with the lowest affinity (nea is elimination ratio of antibodies) with nea new random antibodies; (b) Copy the best antibody to the next generation. Step 8 Producing next population: Select (Popsize-1) antibodies from current generation by a

1 mi : Number of PM operations on machine i mij : Occurrence number of the PM operation Mij.

suitable selection strategy and copy them into the next generation. Step 9 Check the stopping criterion. If it is met, return the best antibody; otherwise, go to step 2.

1) Antibody encoding and decoding We encode each antibody by a permutation of jobs, which is

a natural and popular encoding mechanism for solving the PFSP. Given a permutation of jobs, we decode it by scheduling the jobs one by one from left to right onto the machines. Assuming that there are six jobs and four machines in the flowshop, a permutation π = [2,3,1,6,5,4] is an antibody that represents a schedule in which the sequence of jobs on each machine is J2, J3, J1, J6, J5, J4.

2) Fitness function The objective function is to minimize the maximum of

complete-time on all machines. Therefore, the reciprocal of the maximal makespan (mentioned in Eq.(1)) is selected as the fitness function, and the fitness of each antibody is calculated according to Eq.(3).

max

1( )( )

Affinity SC S

= (3)

3) Initial population As the size of the population has been fixed to popsize, we

use two mechanisms in order to generate the initial population: (1) to start the algorithm with a diversified population, we randomly generate Popsize-k antibodies (schedules), (2) to allow the algorithm starting the resolution with also good individuals, we generate an antibody using the best well-known heuristic for the FSP, NEH heuristic [27] and we apply k mutations to this individual to generate the rest of the population.

4) Mutation operators In this study, mutation operators are used to change the

assignment and sequence of the antibodies. The set of possible moves is defined by a neighbourhood of the current sequence. Most often for the FSP the following neighbourhood structures are used [28]: swap-moves, interchange-moves, insertion-moves

If the makespan value of one of the mutated sequences (after the mutations detailed above) is smaller than that of the original sequence (a generated clone from an antibody), then the mutated one is stored in the place of the original one. Otherwise, it stores the original sequence (generated clone).

5) Vaccination In this study, we apply vaccination as follows: • Let Pi the population of the ith generation

(1≤i≤Max_gen);

• Select the two best antibodies of Pi (with higher affinity);

• The vaccine represents the common sub-sequence of the two antibodies (schedules), i.e. same jobs, in the same position in the sequences.

• Applying the vaccine to the selected antibodies, means permuting each job of the vaccine (common sub-

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sequence) with the job, in the same position, of the antibody to vaccine.

The following example shows the vaccination principle: Let Si and Sj the two best antibodies of the current population

Pi and Sk the antibody to vaccine.

1 9 3 8 5 7 2 4 6 Si 7 9 8 3 5 4 2 1 6 Sj 1 3 6 7 2 9 5 4 8 Sk

The vaccine represents the sequence (9; 5; 6) at positions 2, 5 and 9. After the vaccination process, Sk will be the following sequence:

1 9 8 7 5 3 2 4 6 Sk The vaccination mechanism is activated, at each iteration, of

the algorithm, with the best solution produced by the search to permit the algorithm to start its next iteration with a good solution.

6) Receptor editing In some steps of the algorithm, after processes of cloning

and mutation, a number of antibodies (worst nea antibodies of the whole population) in the antibody population are eliminated and randomly created antibodies replace them. This mechanism allows us to find new schedules that correspond to new search regions in the entire search space.

7) Stopping criteria If a solution cannot be improved any more in consecutive

Improve_gen generations, the algorithm terminates or after a Max_gen generations.

B. Insertion of PM operations As we deal with PFSP, the insertion of PM operations will

be done first on one machine, then on all machines. For the single-machine problem two heuristics can be found in the literature: The Basic heuristic (BH) [29] and the In-depth research heuristic (IH) [8]. These heuristics will be the basis for the resolution of the multi-machines problem (PFSP with maintenance). Benbouzid-Si Tayeb, Guebli, Bessadi, Varnier and Zerhouni [8] propose three heuristics: Naïve (NH), Ascending (AH) and Descending (DH) heuristics. According to the results of [8], we used AH heuristic to insert PM operations in the production sequence. It inserts PM operations, on all machines, from the first until the last one. The insertion of PM operations on one machine is done according to IH heuristic.

IV. COMPUTATIONAL RESULTS In this chapter, we present the results of a series of

computational experiments, conducted to test the effectiveness of the newly proposed AIA for the integrated production and PM scheduling problem. We report on two sets of experiments. In the first set, we have undertaken a performance analysis of the proposed AIA for production scheduling. The second set analyses the effectiveness of the proposed AIA after PM insertion. Moreover, we compared the performance of our algorithms with those of other algorithms reported in the literature. Theses comparisons concerned the two steps of the resolution.

A. Test data Taillard’s benchmark problem datasets [21] were used to

evaluate the performance of the proposed AIA for the production part with objective of minimizing the makespan. There are 90 instances, 10 each of one particular size. Taillard’s datasets range from 20 to 100 jobs and 5 to 20 machines. For the second set of experiments, i.e. performance analysis of our AIA after PM operations insertion, as Taillard’s benchmark problems do not include PM data, we used Benbouzid-Si Tayeb, Varnier and Zerhouni [22] non-standard integrated benchmark problems. The used parameters are: the number of machines m and PM parameters (T, Tmin, Tmax and p’). We generated one PM operation type per machine. We consider two kinds of tolerance interval: Strict (STI) and Wide (WTI). For the strict intervals, the lower and upper bounds are respectively Tmin=Tmin – 5%T and Tmax=Tmax + 5%T. For the wide ones: Tmin = Tmin – 25%T and Tmax = Tmax + 25%T.

We average the results for all the 10 instances in a given combination. We run 10 independent replicates of each instance in order to have a better picture of the results.

B. Performance analysis of the proposed AIA for production scheduling

We compared the performance of the proposed AIA with those of five algorithms reported in the literature, namely Self-guided GA [30], SGA [30], PSOspv [31], DDE [32], and PACO [33], by running the proposed algorithm on the nine sets of benchmark problems. To address this issue, we present the experimental results of the average Error Ratio (ER) of the compared algorithms. In the literature, ER is often used to evaluate the performance of algorithms applied to deal with the PFSPs, whereby the ER of a solution Xi generated by an algorithm is calculated as follows:

max ( )i ii

i

C X UER

U−

=

(4)

where Ui is the makespan value of the best known or optimal solution provided by Taillard [21]. The parameters of our AIA are the following: (1) The number of antibody population is set to 50; (2) Number of clones M: We adopted values between 100 and 1000 depending on the complexity of the problem (these values were empirically found for each problem); (3) Receptor editing is applied every 20 generations; (4) the eliminating percentage B% of receptor editing to determine the nea antibodies to eliminate is set to 10% of the population size; (5) The maximum of iterative generations of AIA is set to 300 for all problems. However, the algorithm is stopped whenever there is no improvement after 20 successive generations which enables a reduction of running time.

Table 1 shows the statistics of the average ER values of all the algorithms on all the 90 test instances for the PFSP taken from Taillard [26].

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TABLE I. AVERAGE ER OF ALL THE ALGORITHMS ON TAILLARD’S INSTANCES

n m AIA Sel-guided GA SGA PSOspv DDE PACO

20

5 1.16 1.10 1.02 1.75 0.46 0.70 10 1.41 1.90 1.73 3.25 0.93 0.84 20 0.27 1.60 1.48 2.82 0.79 0.72

50 5 0.87 0.52 0.61 1.14 0.17 0.09

10 0.27 2.74 2.81 5.29 2.26 0.74 20 2.34 3.94 3.98 7.21 3.11 1.85

100 5 0.69 0.38 0.47 0.63 0.08 0.07

10 0.38 1.60 1.67 3.27 0.94 0.40 20 0.83 3.51 3.80 8.25 3.24 0.98

a. A larger value of an entry indicates an inferior performance, relative to the makespan value of the best known or optimal solution provided by [21].

The results show that the proposed AIA outperformed Sel-

guided GA, SGA, and PSOspv in terms of the ER value. The performance of our AIA, DDE, and PACO are quite similar. This is due to the fact that DDE employs the problem-specific heuristics NEH in permutation flowshop scheduling problems. The better performance of PACO is mainly due to the effectiveness of differential initial setting and differential updating of trail intensities, and those of our AIA for the diversified initial population, the vaccination process and the receptor editing which allow exploring new search regions helping the algorithm to escape from local optima.

C. Performance analysis of the proposed AIA after maintenance insertion

We compared the performance of our AIA with those of two algorithms reported in the literature, namely INTACO [34] and SEQAG [8]. Table 2 presents the results of the evaluation for 90 intergrated benchmarks taken from [22]. FS and Fw represent respectively the average of the function objective value of 10 instances of a given benchmark for the strict and wide tolerance interval. The mean percent relative increases of makespan yielded by the proposed algorithms, compared to the best makespan values reported by Taillard, are noted for every problem size. This deviation represents the mean percentage difference from Taillard’s UB, according to the performance loss after maintenance insertion, and is equal to (Cmax

2– UB)/UB. Thus, this loss is as an evaluation parameter for the proposed AIA after maintenance insertion. The best solution is the one that gave the least increase because the insertion of PM operations increases automatically the makespan values. A fair comparison of our results with the ones obtained with Taillard is difficult since it is dedicated to production scheduling only while ignoring maintenance.

2 This Cmax represents the completion time of the last job on the last

machine after maintenance insertion.

TABLE II. AVERAGE ER OF ALL THE ALGORITHMS ON TAILLARD’S INSTANCES AFTER MAINTENANCE INSERTION

n m Sequential AIA

SEQAG INTACO Fs Fw

20

5 20% 20% 24% 18% 10 35% 33% 38% 28% 20 31% 29% 35% 22%

50 5 21% 19% 24% 19%

10 30% 28% 28% 27% 20 54% 47% 62% 47%

100 5 37% 25% 39% 24%

10 34% 33% 39% 33% 20 38% 36% 43% 30%

Overall, it is observed from the results of table 2 that

INTACO performs better than, both SEQ and our AIA in its two versions. The main reason for such a superior performance is due to the fact that the optimisation in two phases of the sequential strategy (first production jobs are scheduled, which represents a strong constraint, then PM operations are inserted in the resulting schedule) decreases the quality of the obtained joint schedule. In the case of the integrated strategy, the strategy performed by INTACO, the simultaneous optimization of both production and maintenance criterion gives better results, according to a common representation of production jobs and PM operations.

However, the results of our sequential AIA with both strict and wide tolerance intervals are better than SEQAG. Moreover, the sequential AIA with wide tolerance interval perform better than the one with strict one. It is due to the fact that wide tolerance interval offers more possible PM operations insertion sites.

Furthermore, from table 2, one can notice that bigger the number of jobs and lower the number of machines are, worst are the results.

CONCLUSION AND FUTURE WORKS Considering maintenance planning and scheduling

relationship, and the multiple objectives requirement from the real-world production, the research has been conducted to develop artificial immune algorithm to facilitate the integrated production and PM scheduling problem. Experimental studies have been used to test the performance of the proposed approach. The results show that the proposed AIA indeed performs well in comparison with several others algorithms published in the literature. The contributions of this research include:

- the initial population was enhanced by using a problem-specific knowledge to diversify the population;

- the use of mutation operators for reproducing new individuals;

- the use of a vaccination process to enhance the performance of antibodies;

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- and finally, in some steps of the algorithm a model of receptor editing mechanism of the immune system was used.

Future topics along this line of research may include: - extending the idea to take into account more realistic

aspects of the problem such as sequence-dependent setup times (SDST flowshop), unrelated parallel machines at each stage (general hybrid flowshop) or the existence of due dates.;

- broadening the approach to deal with multi-objective optimization problems.

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