Robust solutions in multi-objective stochastic permutation flow shop problem

Computers & Industrial Engineering 137 (2019) 106026

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Robust solutions in multi-objective stochastic permutation flow shop problem

T

Eliana María González-Neiraa, , Ana María Urrego-Torresa, Ana María Cruz-Riverosa, Catalina Henao-Garcíaa, Jairo R. Montoya-Torresb, Lina Paola Molina-Sáncheza, Jose-Fernando Jiméneza ⁎

a b

Departamento de Ingeniería Industrial, Facultad de Ingeniería, Pontificia Universidad Javeriana, Carrera 7 No. 40-62, Bogotá, D.C., Colombia Grupo de investigación en Sistemas Logísticos, Facultad de Ingeniería, Universidad de La Sabana, Km 7, autopista norte de Bogotá, D.C., Chía, Cundinamarca, Colombia

ARTICLE INFO

ABSTRACT

Keywords: Stochastic permutation flow shop Tardiness Robustness Tabu search Stochastic processing times Multi-objective

The aim of this paper is to present a simheuristic approach that obtains robust schedules for a multi-objective permutation flow shop problem with stochastic processing times. In fact, this approach minimizes the expected tardiness and standard deviation of tardiness, as a efficiency and a robustness measure for the stated problem. The simheuristic algorithm hybridize the Tabu Search metaheuristic and the Pareto Archived Evolution Strategy algorithm with a Monte Carlo Simulation process. At first, this approach is tested in 540 benchmarked instances for the deterministic case. It uses a zero-standard-deviation strategy to show the competitiveness compared with other implemented tabu search algorithms. Afterwards, two experimental designs are carried out, with the same 540 instances, where two factors of interest are considered, such as the probability distribution and coefficient of variation of processing times. The probability distributions used were the lognormal and uniform distributions, and three coefficients of variation (0.3, 0.4, and 0.5). Results show that both probability distributions and coefficients of variation have a significant effect in the objective functions, showing the importance of an accurate fitting of probability distributions of the parameter under uncertainty. In addition, these results evidence that the usage of deterministic methods in presence of random events are not desirable or recommended. Finally, the simheuristic was implemented to solve the scheduling problem in an optical laboratory showing better results for expected tardiness and standard deviation of tardiness in comparison with company schedules.

1. Introduction For manufacturing and service enterprises, the assignment of activities or task to resources in predetermined time periods is key process for costs reduction and reaching of operational efficiency. This process is called scheduling (Pinedo, 2012). In general, the scheduling has many types of environments depending on the configuration of the manufacturing environment. However, among several configurations, this paper focuses on a flow shop problem (FSP) since it can be found in lots of application in chemical, pharmaceutical, steel, food and assembly industries (Sturrock, 2012). A FSP consists in determining the sequence of a set of N jobs that must be processed in a set of M machines in series (Pan & Ruiz, 2012). The following assumptions are commonly made when studying the basic FSP configuration (Baker, 1974; Ruiz & Maroto, 2005; Yenisey & Yagmahan, 2014): each job j can

only be processed in one machine at time, each machine i can only process one job j at time, preemption is not allowed, all jobs are independent and their release dates are zero, and there are no breakdowns. A kind of of a FSP is the permutation flow shop problem (PFSP), when the processing sequence of all jobs in all machines is the same. This paper focuses in this kind of problem as it is a common configuration in real-world industries, in which in-process storage of products is very limited (Ciavotta, Minella, & Ruiz, 2013). For several years, the study of deterministic FSP has received a lot of attention due to the ease that all parameters are certain and known in advance for the scheduling decision (Gourgand, Grangeon, & Norre, 2003). However, in the real industry, several of these parameters may be subject to uncertainty (Juan, Barrios, Vallada, Riera, & Jorba, 2014). According to Beyer and Sendhoff (2007), there are three possibilities to model uncertainties: deterministic (defines parameter domains in

Corresponding author. E-mail addresses: [email protected] (E.M. González-Neira), [email protected] (A.M. Urrego-Torres), [email protected] (A.M. Cruz-Riveros), [email protected] (C. Henao-García), [email protected] (J.R. Montoya-Torres), [email protected] (L.P. Molina-Sánchez), [email protected] (J.-F. Jiménez). ⁎

https://doi.org/10.1016/j.cie.2019.106026 Received 29 June 2018; Received in revised form 18 August 2019; Accepted 27 August 2019 Available online 28 August 2019 0360-8352/ © 2019 Elsevier Ltd. All rights reserved.

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intervals or sets), probabilistic (using probability distributions) and possibilistic (through fuzzy sets). The authors mention that dealing with deterministic uncertainties is advisable to use robust regularization measures. Some examples of these methods are the conservative minmax family criteria (Kouvelis & Yu, 1997), the bw-robustness (Roy, 2010), the pw-robustness, the lexicographic α-robustness (Kalaï, Lamboray, & Vanderpooten, 2012), among many others. For the probabilistic case, the objective function is considered as a random variable (Beyer & Sendhoff, 2007), and some example are the expectancy measures (as expected value, variance and relative expected and variance values), the probabilistic measures (as the maximization of probability of obtaining a determined value) and the probabilistic or chance constraint to handle uncertain constraints. Finally, for the possibilistic uncertainties, fuzzy theory is used. In addition to the robust measures, Beyer and Sendhoff (2007) indicate that the approaches to deal with robust optimization can be classified in two main categories: (i) deterministic, which transform the robust optimization problem into an ordinary one in order to be solve it though optimization algorithms (it assumes the availability of first and second derivates of objective functions and (ii) simulation-optimization, that model uncertainties probabilistically and seeks the optimization of noisy functions and constraints and does not need of explicit first or second-order information. Although considerable research has been carried out on robust optimization, there has been little research in the use of probability distributions of uncertain parameters in robust optimization. One approach that has been proved to be successful for the robustness, that considers the probability distributions of uncertain parameters, in other optimization problems is the mean-variance (or mean - standard deviation) minimization (Jia & Ierapetritou, 2007; Nejlaoui, Houidi, Affi, & Romdhane, 2017). Particularly, considering that the makespan has been the most analyzed measure in literature and that there has been an increased interest in due-date based metrics (Fernandez-Viagas & Framinan, 2015), we aim to minimize the expected mean and standard deviation (measure of variability) of tardiness. In specific, according to well-known works from the scientific literature (Liu, Ullah, & Zhang, 2011), the variability of the tardiness as objective function is employed as a robust indicator. To deal with these two objectives, we propose the inclusion of a Pareto approach that allows obtaining the set of non-dominated solutions for the mean tardiness and the standard deviation of tardiness. Regarding the computational complexity, the deterministic version of the PFSP for tardiness minimization is classified as NP-hard problem (Garey, Johnson, & Sethi, 1976; Vallada, Ruiz, & Minella, 2008). Therefore, the stochastic permutation flow shop problem (SPFSP) is also NP-hard. In this sense, exact methods are not efficient for largesized instances and it opens the possibility of developing approximation methods for its resolution (Yenisey & Yagmahan, 2014). Hence, inspired by the work of Juan et al. (2014), this paper proposes a simheuristic solution procedure. For the optimization purpose, a Tabu Search (TS) has been selected. This is a metaheuristic that uses a memory list to avoid staying in local optima (Gao, Chen, & Deng, 2013). Although this metaheuristic plays an important role in scheduling problems, specifically in FS, little research have been reported in comparison with other methods like genetic algorithms (Armentano & Arroyo, 2004; Gao et al., 2013; Jones, Mirrazavi, & Tamiz, 2002). Moreover, Armentano and Arroyo (2004) mention that TS can be competitive in relation to GA at the moment of finding an approximation of Pareto frontier. Considering all aspects previously described, this research proposes a methodology that obtains robust solutions in a SPFSP, minimizing the expected tardiness and standard deviation of tardiness. The stochastic parameters are the processing times. According to notation presented by González-Neira, Montoya-Torres, and Barrera (2017), the problem is noted as

Fm |prmu, pij Lognormal (E [pij ], pij )| E [T ], Rob (T ) and F m |prmu, pij Uniform (E [pij ], pij )| E [T], Rob (T ) . To solve this problem, taking into account the presented elements, we propose a simheuristic that hybridizes TS, the Pareto Archive Evolution Strategy (PAES) and a Monte Carlo simulation. We attempt to obtain robust solutions considering the minimization of expected tardiness and its standard deviation. We expect that this approach obtain efficient results as other optimization problems have proved the effectiveness of this method (Nejlaoui et al., 2017). Finally, we evaluate the effect of two different types of probability distributions (lognormal and uniform) and three coefficients of variation (CVs) of the processing times in both measures. The remainder of the paper is organized as follows. Section 2 provides the literature review. Section 3 presents the mixed integer linear programming formulation. Section 4 describes the proposed solution approach. Section 4 exhibits the results of the computational experiments. Section 6 presents a case study in an optical laboratory. Finally, conclusions and opportunities for future work are presented in Section 7. 2. Literature review Considering that in this paper a multi-objective SPFSP with TS is solved, this section is divided in three main parts. The first part presents a review of current solution approaches that considers multi-objective mean-variance approaches for optimization problems under uncertainties, the second part resume the works on FSP under uncertainties and the third part the works related to the application of TS for scheduling problems. 2.1. Multi-objective mean-variance approaches for optimization problems Regarding robust multi-objective optimization and the usage of expectancy measures, various researches have been done using the objectives mean-variance or mean-deviation to obtain robust solutions. For example, Das (2000) used the Normal-Boundary Intersection optimization technique to find the Pareto front for optimality and robustness criteria that they mentioned respectively as expected measure and dispersion of the objective. Vin and Ierapetritou (2001) used and proposed standard deviation measures of makespan as robustness metrics for scheduling in a multi-period programming model under uncertain demand. They compared with the deterministic single-period scheduling problem, showing better results with the robustness metrics proposed. Jia and Ierapetritou (2007) dealt with a robust multi-objective scheduling problem in which the objectives were the minimization of expected makespan, expected unsatisfied demands, and the optimization of solution robustness with the minimization of positive deviations of makespan using the concept of upper partial mean. They used this concept instead of variance of makespan to avoid non-linearities in the formulation. Diaz, Handl, and Xu (2017) solved a production planning problem using minimization of expected and standard deviation of profit and demonstrated that the Pareto approach present better trade-offs between performance and robustness than solving the single objective problem. Nejlaoui et al. (2017) proposed a hybridization of a multi-objective Imperialist Competitive Algorithm with Monte Carlo Simulation for the robust design optimization of rail vehicle moving. Two objectives are analyzed, the minimization of expected and standard deviation of derailment angle to ensure respectively maximum safety levels and robustness on safety. Recently, Gotoh, Kim, and Lim (2018) demonstrated that most of robust empirical optimization problems are the same that solving a mean-variance optimization problem.

2

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2.2. FSP under uncertainties

given solution space to avoid local optima. Specific applications of TS which authors demonstrated to be efficient and effective to solve FSP are reviewed next. Chen, Pan, and Wu (2008) analyzed a re-entrant PFSP with makespan objective. The TS algorithm was developed to improve the pure TS. Eren and Güner (2008) proposed a TS to minimize the total completion time and makespan in a two-machine flow shop with learning effects. Wang, Li, and Wang (2010) studied a no-wait FSP with maximum lateness criterion. Authors developed a TS in which they compared three different insertion and exchange procedures are compared in TS. Liao and Huang (2011) minimized the makespan in a 2-machine FPS with batch processing machines. A TS is proposed and compared with a MILP model in small problems, showing the efficiency of the algorithm. Mehravaran and Logendran (2012) studied a bi-objective FSP with sequence dependent setup times and two objectives, the minimization of work-in-process inventory for the producer and the maximization of customers’ service level. A MILP model was presented to solve small instances and a TS with embedded progressive perturbation to solve industry-size problems. Behnamian (2013) solved a biobjective FS problem considering sequence-dependent setup times in a just-in-time environment. The author developed two algorithms that decomposed population in parallel sub-populations. The min-max method is used for each sub-population to obtain an approximation of Pareto frontier. Then a variable neighborhood algorithm and a hybrid variable neighborhood search and TS are used to improve the Paretofront. Mehravaran and Logendran (2013) developed three meta-search algorithms based on TS to solve the PFSP and FSP and show that for small problems their TS present a gap of 2% respect to optimal values. Shen (2014) addressed a job shop problem with sequence dependent setup times are present. Authors designed a TS with neighborhood functions that use insertion techniques instead of easy swaps. TS results outperforms a previous published simulated annealing metaheuristic. Li and Gao (2016) hybridize a genetic algorithm with TS for solving a flexible job shop problem. Authors get new best solutions for several benchmark instances. Xu et al. (2019) solved the parallel machine scheduling problem to minimize the makespan. Authors proposed iterated local search (ILS) and a TS, showing best results with TS each of them showing good results than two existing algorithms. Bozejko, Hejducki, and Wodecki (2019) proposed a TS for a FSP application in a construction process (civil engineering) in presence of uncertainties modelled with fuzzy numbers and random variables.

Here we present a brief chronological review of the works published in the last 10 years. For more information the author is referred to Behnamian (2016), González-Neira et al. (2017), Gourgand, Grangeon, and Norre (2000). Allahverdi and Aydilek (2010) modeled the processing times with an interval approach to minimize maximum lateness. Authors proposed fourteen heuristics. Baker and Trietsch (2011) also minimized expected makespan with processing times that followed exponential, lognormal and uniform distributions. Authors designed thee different heuristic procedures. Liu et al. (2011) analyzed the robust makespan though a genetic algorithm that used a new generation scheme allowing the preservation of good characteristics of parents across generations. Azadeh, Jeihoonian, Shoja, and Seyedmahmoudi (2012) minimized weighted makespan plus weighted mean completion time in a SPFSP with two machines. Authors used simulation in a first stage which results are inputs of an artificial neural network metamodel. Baker and Altheimer (2012) presented a simulation-optimization approach for expected makespan minimization using three heuristics that worked well for deterministic problem. Liefooghe, Basseur, Humeau, Jourdan, and Talbi (2012) studied a bi-objective PFSP with stochastic processing times, which objectives were makespan and tardiness. To solve the problem the authors proposed an indicator-based evolutionary algorithm. Aydilek and Allahverdi (2013) developed a polynomial time heuristic algorithm to solve the problem with interval processing times. Johnson and Yoshida & Hitomi algorithms are special cases of the designed procedure. Elyasi and Salmasi (2013) developed a chance-constrained programming to minimize total tardy jobs considering stochastic processing times that followed normal and gamma distributions. Rahmani, Ramezanian, and Mehrabad (2014) applied Chance-constrained programming to transform a SFSP in a deterministic one. The stochastic parameters were processing and release times and the objectives were fuzzy weighted makespan, total flowtime and total tardiness solved. A genetic algorithm and goal programming was used to solve the deterministic and converted single objective problem. Mou, Li, Gao, and Yi (2015) solved a SPFS inverse scheduling problem to minimize hamming distance, the adjustment of total completion times and adjustments of processing times. Authors developed a hybrid multi-objective evolutionary algorithm with NEH-based insertion method. Fazayeli, Aleagha, Bashirzadeh, and Shafaei (2016) solved the robust makespan with the usage of -robustness criterion with genetic algorithm and simulated annealing metaheuristics. Gholami-Zanjani, Hakimifar, Nazemi, and Jolai (2017) proposed a robust and a fuzzy approach for makespan minimization in which the uncertain parameters were setup and processing times. Fu, Ding, Wang, and Wang (2017) designed a discrete fireworks algorithm to minimize expected makespan and expected tardiness with normal distributed processing times. Rahmani (2017) solved the robust mean completion time with stochastic processing times and machines breakdowns. Authors proposed a two phases approach which consisted in a proactive scheduling for the processing times and a reactive scheduling after machine failures. Cui, Lu, Li, and Han (2018) considered processing machine failures following a Weibull distribution to minimize expected makespan considering robust starting times. Authors proposed an algorithm with two loops: outer loop based on local search initialized with NEH and an inner loop based in a genetic algorithm. Framinan, Fernandez-Viagas, and Perez-Gonzalez (2019) presented different policies as a rescheduling approach after the occurrence of stochastic processing times. The policy based on critical path concept outperformed the other ones.

3. Mixed integer linear programming (MILP) model In this section, a MILP model is proposed for the solution of deterministic counterpart of the problem. i.e. the minimization of total tardiness in a PFSP for minimizing total tardiness. The model is based on the one proposed by Molina-Sánchez and González-Neira (2016). Sets:

I: Machines {1, , m} J: Jobs {1, , n} K: Positions in the sequence {1,

, n}

Parameters:

dj : due date of job j, j J pij : processing time of job j in machine i , wj: weight of job j, j J M : Big possitve number

2.3. TS for scheduling problems

Decision Variables:

TS is a metaheuristic, originally proposed by Glover (1986), that uses the concept of a list of restricted moves to control the execution of the algorithm. In this sense, TS intensify or diversify its search in a

Xjk

i

I,

j

J

1, if job j is procesed in k position of the sequence,

3

J, k K 0, otherwise

j

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Table 1 Structure of proposed TS. Element

Description

Solution representation Initial solution Solution space Search space Tabu list Stopping criteria

Vector of n positions that storage the sequence of jobs Processing sequence obtained for the deterministic case when NEHEDD dispatching rule is applied All possible permutations of n jobs All possible permutations that can be obtained when the position of two jobs is interchanged given a specific solution Matrix of size [n × n] (TabuMatrix) which register the changes of positions previously selected and the remaining number of tabu iterations Tabu time = (n ∗ m) ∗ 0.01 s for deterministic case Tabu time = (n ∗ m) ∗ 0.1 s for bi-objective stochastic simheuristic

Cij: completion time of job j at machine i , Tj: tardiness of job j, j J

i

I,

j

1993) which has been shown very good results for tardiness problems (Fernandez-Viagas & Framinan, 2015).

J

The objective function is:

Minimize Z =

Tj

4.1.2. Neighborhood Considering that the solution of our problem is represented by a permutation of N jobs, we adopt as a neighborhood the swapping between two jobs in the permutation. In each iteration the best move that is not in tabu list is selected.

(1)

j J

Subject to:

Xjk = 1,

k

K

Xjk = 1,

j

J

(3)

k K

C1j

p1j ,

C(i + 1) j

Cij

p(i + 1) j

pij + (1 i

I,

Tj

Cij

Tj

0,

Xjk

j

J

{0, 1},

j

Xj (k + 1) ) M J, h

dj ,

i j

(

(4)

Cij,

h, j

4.1.3. Tabu list Once a move is accepted and performed, the tabu list is updated with the reverse move mentioned. The tabu list size is set in n n n 0.5, where n is the number of jobs of the instance, that 2 represents a 50% of the total interchanges executed to construct the neighborhood. It allows exploring multiple sections of sample space.

(2)

j J

J,

Cih j,

I,

I, i + 1

(1

Xhk ) M ,

L, k + 1

k

j

i

J

|K|

|I|

(5)

4.1.4. Stopping criteria The search process terminates under the given stop time (TabuTime) calculated as (n m) 0.01s , where m is the number of machines of the instance. This time is the same as the defined by Vallada et al. (2008) in order to make comparisons with the three TS algorithms reviewed by the authors.

(6) (7) (8)

J

j

J,

k

K

)

4.2. Framework of stochastic TS

(9)

The objective function (1) consist in minimizing the total tardiness. Constraint sets (2) and (3) guarantee that each job is sequenced in only one position and in every position of the sequence only one job is scheduled. Constraint set (4), (5) and (6) calculate the completion times of jobs at every machine, ensuring that the completion times on first machine are at least the value of processing times on that machine, the processing of a job in a machine only happens when that job have finished its processing in the previous machine and that in a in a specific machine there is no overlapping jobs. Constraint sets (7) and (8) calculate the tardiness of every job. Constraint set (9) defines the binary variables.

Considering that this research attempts to minimize the expected tardiness and the standard deviation of tardiness, this problem is solved though a multi-objective approach. In last years, different multi-objective techniques have been proposed using evolutionary algorithms. PAES algorithm has been one of the most important (Oltean, Grosan, Abraham, & Koppen, 2005). In its basic form PAES employs local search to generate new candida solutions but uses population information to help in the process of obtaining solutions. For this investigation the designed deterministic TS had to be adapted in three crucial aspects in order to solve the bi-objective SPFSP. Firstly, the processing times of each job at each machine correspond to random values that follows a determined PD. The mean of its distribution is the processing time given by the benchmark instances of Vallada et al. (2008). The standard deviation is calculated through a specific CV. Secondly, it is needed to integrate the deterministic TS with a Monte Carlo simulation to obtain expected tardiness and standard deviation of tardiness. For the stochastic TS the function that saved the total tardiness of each sequenced job in the deterministic TS is replaced by a simulation subroutine that obtains the mean and standard deviation of tardiness for each solution. Thirdly, in order to deal with both objectives simultaneously the PAES algorithm is adapted and hybridized with the proposed TS to construct Pareto frontier. PAES methodology implemented correspond to the variation (1 + λ), in which the λ generated solutions correspond to the neighborhood of the TS. The codification of the solutions for the stochastic TS is the same than for the deterministic case. Only the variables mean and standard deviation are included, and a new array that saves the Pareto solutions

4. Proposed solution approach Previous to the design of simheuristic approach, a TS was designed for the deterministic PFSP that minimizes total tardiness in order to compare the procedure with other TS algorithms of literature. Then, supported in the works done by Juan et al. (2014) and Knowles and Corne (2000) we proposed a simheuristic that hybridizes the TS algorithm with Monte Carlo simulation, and the Pareto Archived Evolution Strategy (PAES) algorithm to address the stochastic and bi-objective natures of the problem, respectively. 4.1. Framework of deterministic TS 4.1.1. Initial solution With the purpose of increase the efficiency of the TS, we used an adaptation for the tardiness objective of the well-known NEH heuristic by Nawaz, Enscore, and Ham (1983) called NEHedd heuristic (Kim, 4

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Fig. 1. Multi-objective Simheuristic-TS proposed procedure.

is also incorporated. This array is updated each time the algorithm finds a solution than dominates the solutions already saved in the archive, or when finds a new solution than not dominates and is not dominated by the solutions of the archive. The stopping time is setting ten times greater than the one fixed for the deterministic TS due to the important quantity of short simulations that must be performed. Principal characteristics of the proposed TS for the deterministic case and the simheuristic are presented in Table 1 and the simheuristic

procedure in Fig. 1. The TS for the deterministic case is a simplification of the simheuristic in which Pareto archive is not constructed and no simulations are needed. 5. Computational results This section is divided in three parts. Firstly, we evaluate the results of the proposed TS for deterministic case in small instances by 5

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comparing TS results with the solutions of MILP model after 1000 s of running. Secondly, the results of TS for the deterministic version of the problem for the 540 benchmark instances proposed by Vallada et al. (2008) are compared with three TS algorithms previously presented in literature for the same problem. And thirdly, we present the results of the proposed simheuristics in the same 540 benchmark instances by using different probability distributions (PDs) of processing times (lognormal or uniform) and coefficients of variation (CVs) of processing times (0.3, 0.4 and 0.5). All tests were performed in a CPU Intel Core i74600 M @2.90 GHz8.00 GB in RAM.

Table 2 Performance of TS for deterministic PFSP in small instances. Instance

Proposed TS

MILP model until 3600 s of running

Tardiness factor

Due date range

Instance number

Tardiness

Time (seconds)

Tardiness

Time (seconds)

0.2

0.2

1 2 1 2 1 2

59 606 158 1938 396 38

0.5 0.5 0.5 0.5 0.5 0.5

59* 642 192 1932 536 38

87.00 3600.00 3600.00 3600.00 3600.00 3600.00

1 2 1 2 1 2

1906 1031 1698 0 118 0

0.5 0.5 0.5 0.5 0.5 0.5

1917 2397 1719 0* 152 0*

3600.00 3600.00 3600.00 1424.00 3600.00 8.60

1 2 1 2 1 2

1190 1314 913 1362 1245 621

0.5 0.5 0.5 0.5 0.5 0.5

1212 1337 958 1413 1274 2954

3600.00 3600.00 3600.00 3600.00 3600.00 3600.00

0.6 1.0 0.4

0.2 0.6 1.0

0.6

0.2 0.6 1.0

5.1. Performance of proposed TS for deterministic version of the problem in small instances To evaluate the performance of the designed TS for tardiness minimization in a deterministic PFSP in small instances we generated 18 different small instances of 10 jobs with 5 machines, two for each combination of these three due date ranges (0.2, 0.6 and 1.0) and three values of tardiness factor (0.2, 0.4 and 0.6). The processing times were generated through a uniform distribution between 1 and 99. This procedure was the same followed by Vallada et al. (2008) for generating the 540 benchmark instances. The 18 created small instances were solved with GLPK which was run for a maximum time of 1000 s for instance. During this time, we registered the best integer solution found until that moment. Table 2 presents the results obtained. As can be seen TS stopping time for this size of problem was 0.5 s, and in all cases in this time TS obtained better solutions than the MILP model after 3600 s of running. It demonstrates the effectiveness of our proposed TS for the deterministic case in small instances.

* This is the optimal solution found with GLPK before 3600 s of running. Table 3 Performance of TS for deterministic PFSP. Instance

Average RDI

Jobs

Machines

TS1

TS2

TS3

Proposed TS

50 50 50 150 150 150 250 250 250 350 350 350

10 30 50 10 30 50 10 30 50 10 30 50

23.50% 41.37% 42.39% 9.04% 10.91% 11.19% 9.29% 8.80% 9.97% 13.81% 13.30% 12.60%

15.98% 45.09% 51.32% 4.93% 10.58% 12.44% 5.64% 9.68% 10.77% 13.03% 15.20% 14.00%

18.76% 36.43% 38.99% 9.00% 9.74% 10.54% 9.34% 8.71% 9.74% 13.88% 13.22% 12.55%

13,02% 19,10% 20,23% 10,64% 14,67% 16,33% 9,86% 13,73% 16,90% 9,81% 14,22% 16,21%

17.18%

17.39%

15.91%

14.56%

To evaluate the performance of the designed TS for tardiness minimization in a deterministic PFSP it was computed a relative deviation index (10) presented by Vallada et al. (2008).

Relative deviation index (RDI ) =

Methodsol Bestsol × 100 Worstsol Bestsol

(10)

This allowed to compare the results of the proposed TS versus the following three different TS algorithms implemented in literature by Kim, Lim, and Park (1996) and reviewed by Vallada et al. (2008):

• TS1: movements are based on Jobs insertion and a complete evaluation of generated neighborhood. Tabu list has a size of 10, and it

Lognormal distribution

Uniform distribution 5000

Standard deviation of Tardiness

Standard deviation of Tardiness

Average

5.2. Performance of proposed TS for deterministic version of the problem in medium and large size instances

4500 4000 3500 3000 2500 2000 1500 1000 500 2500

7500

12500

17500

Expected Tardiness

0.3

0.4

5000 4500 4000 3500 3000 2500 2000 1500 1000 500 2500

7500

17500

12500

Expected Tardiness

0.5

0.3

Fig. 2. Pareto frontier for instance “I_0.2_1_50_10_5”.

6

0.4

0.5

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Lognormal 0.3 0.4 1–540 PD CV I PD selected to generate the realizations of processing times in each run of the simulations CV used to obtain the standard deviation for generating the realizations of processing times in each run of the simulations Specific instance generated for each combination of n jobs and m machines and specific tardiness factor and due date range. This instances were taken from Vallada et al. (2008) Factors of interest

P-Values Source

Expected Tardiness

Standard Deviation of Tardiness

PD CV I PD * CV

0.000 0.000 0.000 0.015 99.13%

0.000 0.000 0.000 0.000 68.90%

2 R adj

• •

is constructed with the objective values obtained in previous interactions. TS2: similar to TS1. However, movements are based on interchanges of jobs. TS3: insertion based movements. Sequence is represented through a graphic which nodes represent Jobs and arcs the order of two Jobs (i, j) . these arcs exist only when job i precedes job j . Tabu list is defined by existing arcs, thence, job i do not sequenced before job j if the arc is tabu.

Results are presented in Table 3. The last column presents the results of the TS proposed in this paper. It is observed that proposed TS outperforms the other three presented in literature for instances of 50 jobs. In the case of 150 jobs the results are worse but not excessive far. 5.3. Pareto fronts for bi-objective stochastic problem A total of 3240 Pareto frontiers were obtained. They consist in one frontier for each combination of instance, PD (lognormal or uniform) and CV (0.3, 0.4 and 0.5) of processing times. As an example, the six Pareto frontiers for the instance “I_0.2_1_50_10_5” are presented in Fig. 2. This instance is an instance with due date range of 0.2, tardiness factor equal to 1, 50 jobs, 10 machines and is the fifth replication for this configuration. In this example it can be seen a contrary behavior of the two objectives evaluated. It can be appreciated that there is an increment in the positions of the frontiers as the CV increases. Both graphics are constructed in the same scale to compare the results of uniform distribution with lognormal distribution. For this specific instance it can be seen than the results for uniform distribution are lower than those of the lognormal distribution. These same behaviors can be observed in the Pareto frontiers of the other 539 instances. Nevertheless, it is important to analyze the results statistically to affirm these conclusions. The experiments carried out are explained in next subsection. 5.4. Experiments Two designs of experiments were executed to evaluate if there is an effect of two factors of interest in the results of expected tardiness and standard deviation of tardiness. The instance was used as a blocking factor. Both experiments analyzed the same factors (Table 4) implying a total of 6 treatments and 3240 observations each of them. It is important to note that the observations taken for the experiment were only the extremes of Pareto frontiers for each instance, that is the best results for each objective function. The p-values for each source of variation of the resulting ANOVAs and the corresponding adjusted determination coefficient are presented in Table 5. It can be noted that all main effects and the double interaction between the PD and CV are statistically significant for both expected tardiness and standard deviation of tardiness. It is very important because it demonstrates that the type of PD and CV of processing times affect the responses of the objectives (see Fig. 3). This implies that it is essential to have the correct adjustment and fitting of

Blocking factor

Probability Distribution Coefficient of variation Instance

Levels Notation Description Factor

Table 4 Factors and levels of experimental designs.

Uniform 0.5

Table 5 Friedman test for expected tardiness versus factors CV, Method and Distribution.

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Fig. 3. Main effects plots of experiments for Expected Tardiness and Standard Deviation of Tardiness.

– Spread of non-dominance solution (SNS): is an indicator of diversity of Pareto points as indicates Eq. (12).

Table 6 Friedman test for expected tardiness versus factors PD and CV. Factor CV PD

Levels of factor 0.3 0.4 0.5 Lognormal Uniform

N 1080 1080 1080 1620 1620

Expected median

Sum of Ranks

Overall median

DF

p-value

245258.0 287596.0 337442.0 288302.0 284215.0

1113.0 2150.0 3217.0 2818.0 2042.0

290098.0

2

0.000

286258.0

1

0.000

SNS =

Levels of factor

N

Expected median

Sum of Ranks

Overall median

DF

p-value

CV

0.3 0.4 0.5 Lognormal Uniform

1080 1080 1080 1620 1620

2257.0 5240.0 9453.0 10823.0 7964.0

1420.0 2157.0 2903.0 3045.0 1815.0

5650.0

2

0.000

9394.0

1

0.000

PD

ci

NPS

where ci =

,

E [Tardiness]i2

(11)

+

(12)

In this section, the proposed simheuristic is applied to a scheduling problem in an ophthalmic laboratory located in Colombia. In this environment, ophthalmic lenses are manufactured according to the patient prescription Rx and the required lenses material. The manufacturing process is divided into two big sub-processes: surfacing and edging. Surfacing consists in generating the patient Rx on two lens blanks. Edging consists in cutting the surfaced lenses with the form of the frame and mounting the cut lenses into the frame. Both processes consist in sequential operations that must follow the same order, which can be modeled as a flow shop environment. Surfacing has four operations namely: blocking, generating, fining and polishing. Edging consists in five operations: tracing, blocking, edging, mounting and quality control. These nine operations must be processed in the same order for all jobs. In this case, a job is an order of one patient conformed by two lenses and one frame. Fig. 4 shows the scheme of the process at the laboratory. Since patients generally have different lenses prescriptions, different required materials and different frames, the processing times at each operation are random variables that can be modeled through probability distributions. Probability distributions were fitted from data collected of each operation of about one-month historic processing times of jobs: Jobs were divided into nine types according to the combination of material of lenses (plastic, polycarbonate and hi-index) and type of vision (single vision, bifocal and progressive); then, a probability distribution was obtained for each type of job at each operation. Later, material and type of vision data of all jobs that were requested in other 10 different days were taken in order to run the simheuristic for those data (10 instances). Also, the real schedules proposed by company for all these 10 days were also simulated considering the probability distributions fitted, to test the effectiveness of our simheuristic. Table 9 shows the results of expected tardiness and standard deviation for the simulated company sequence and the two extremes of Pareto Frontier obtained with the proposed simheuristic. As can be seen, the simheuristic outperforms always the company

– Number of Pareto solutions (NPS): the number of non-dominated points for each instance – Mean ideal distance (MID): it measures the closeness between Pareto solutions an ideal point (0,0). See Eq. (11). NPS i=1

NPS

ci ) 2

6. Case study

processing times distribution to obtain robust solutions more adjusted that if one only model the uncertain parameters with an interval. It is known that every experimental design must accomplish three assumptions that are normality, homoscedasticity and independence of residuals. Due to normality and homoscedasticity assumptions were not fulfilled we performed the non-parametric Friedman test for both objectives to validate the results of the ANOVAs. As can be seen in Tables 6 and 7 the null hypothesis is rejected for CV and PD indicating that there is a significant effect of these factors in the objective functions. Due to the best of our knowledge this is the only work that studies the minimization of expected tardiness simultaneously with minimization of standard deviation of tardiness in a SPFSP, we present the same three indicators used by Karimi, Zandieh, and Karamooz (2010) and Ebrahimi, Fatemi Ghomi, and Karimi (2014), which can be taken for future comparisons. For this work these indicators have been adjusted in order to consider the two analyzed objective functions. The measures are:

MID =

(MID

Table 8 shows the averages of NPS, MID and SNS for each combination of Jobs, Machines, PD and CV. As it was expected MID and SNS increase as the number of jobs and number of machines increases. Additionally, it can be observed that in a 75% of cases as CV increases MID augment.

Table 7 Friedman test for standard deviation of tardiness versus factors PD and CV. Factor

NPS i=1

SD [StDevTardiness]i2 8

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Table 8 Performance multi-objective measures for combinations of Jobs, Machines, PD and CV. Jobs

Machines

PD

CV 0.3

50

10 30 50

150

10 30 50

250

10 30 50

350

10 30 50

0.4

0.5

NPS

MID

SNS

NPS

MID

SNS

NPS

MID

SNS

lgn unf lgn unf lgn unf

29.37 137.01 138.93 406.04 284.68 607.50

34263.83 38120.67 83719.80 84458.74 130360.17 115603.85

663.31 1294.78 1130.46 2046.05 1759.33 2534.81

22.15 41.64 108.25 264.18 128.21 383.25

32117.84 33468.37 95757.55 92934.08 135328.33 124949.11

573.41 766.47 1010.89 1902.57 1171.32 2439.95

15.80 291.45 44.91 210.41 81.66 507.46

34156.71 46941.01 100070.03 97306.52 152840.51 136121.82

557.93 569.14 1116.16 1605.20 1355.14 2036.47

lgn unf lgn unf lgn unf

42.41 106.76 98.80 4146.13 118.79 176.76

149803.87 157305.16 374842.53 483488.50 541219.04 556576.14

1997.74 1801.88 3925.45 3593.07 3871.41 4319.47

26.49 103.79 56.27 121.69 52.64 1432.51

182855.03 237029.33 406267.96 428172.88 551840.62 596156.93

2023.55 2399.93 3117.98 3727.54 3591.68 3494.29

22.20 82.02 45.86 248.58 80.65 102.52

194933.83 220162.51 447453.88 506668.30 664467.59 606713.53

1868.68 2706.18 3478.09 3454.64 3603.08 3779.02

lgn unf lgn unf lgn unf

52.56 91.20 71.88 147.66 563.34 393.59

266498.37 307452.84 774808.56 855092.89 1359800.18 1205208.27

2846.39 2819.57 5588.19 7253.35 5534.23 5618.72

38.02 66.72 24.60 88.69 50.15 737.44

286236.95 288689.75 784203.41 951215.16 1164626.39 1342690.83

2867.79 2550.58 5332.49 5153.85 6693.12 5938.60

19.73 32.28 41.28 90.46 49.30 300.16

416327.15 390172.96 972269.17 1003900.12 1254820.19 1403800.94

3076.51 3038.56 5641.86 6081.29 7426.89 7887.04

lgn unf lgn unf lgn unf

41.27 58.48 57.23 553.76 1042.54 77.52

489487.15 609068.51 1419609.82 1644949.38 2236473.26 1726043.47

3150.15 4239.84 8606.03 8856.79 11946.49 11388.95

37.11 43.36 29.40 517.56 103.53 532.49

396782.97 511382.32 1230267.91 1683336.60 1938500.84 2301205.99

2420.25 2459.24 7633.74 7071.33 9011.24 11112.49

33.44 57.20 26.35 66.37 44.30 286.12

519935.47 648599.25 1336108.07 1508144.47 2292180.47 2125506.43

2149.15 3349.45 5859.81 10552.66 8861.83 11055.11

815.45

551945.84

4016.52

426.19

562080.53

3612.15

230.98

584616.27

3810.98

Total general

Fig. 4. Scheme process at ophthalmic laboratory. Table 9 Results of simheuristic for ophthalmic laboratory. Instance

1 2 3 4 5 6 7 8 9 10

Quantity of jobs

200 193 208 187 219 197 222 207 192 185

Simulated company sequence

Proposed Simheuristic Extreme of pareto Frontier with best Expected Tardiness

Extreme of pareto Frontier with best Standard Deviation of Tardiness

Expected Tardiness

Standard Deviation of Tardiness

Expected Tardiness

Standard Deviation of Tardiness

Expected Tardiness

Standard Deviation of Tardiness

2006725.1 1859136.3 2186913.6 1758196.9 2410842.9 1948686.8 2498046.4 2138803.7 1860766.9 1715170.2

55474.1 52040.3 59679.9 52267.5 57568.8 56346.7 64508.5 54931.6 47317.1 43462.9

1874831.9 1741007.9 2039790.1 1666125.0 2239072.7 1837385.2 2356120.7 2011145.9 1741893.8 1623600.5

28557.7 47183.3 42788.3 41273.0 49343.0 34043.1 55761.0 24412.1 11189.6 35350.9

1998055.4 1822765.7 2131304.0 1712664.4 2322288.1 1887041.6 2400863.6 2096919.0 1779562.2 1660268.0

475.4 1740.5 1991.5 3615.8 571.7 384.4 238.7 383.5 826.3 7540.3

6.1%

32.5%

2.8%

96.4%

Improvement average of simheuristic in comparison with simulated company's sequence

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sequence. In fact, the company sequence is dominated by all the solutions in the Pareto Frontiers for each instance evaluated. Moreover, the best improvements are reached in the standard deviation of tardiness sequences, showing that, additional to the improvement of expected tardiness (between 2.8% and 6.1% in average), the robustness of schedules improve between 32.5% and 96.4% in comparison with company sequence results.

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7. Conclusions and future work This paper presented a simulation-optimization approach (simheuristic) that hybridizes a TS, a PAES algorithm and a Monte Carlo Simulation to obtain a Pareto frontier for expected tardiness and standard deviation of tardiness in a SPFSP. In a first instance an evaluation of performance of the simheuristic is done by testing deterministic instances with coefficient of variation of zero, to compare with other three benchmark TS algorithms proposed in literature. In second instance, two experimental designs were carried out to test the effects of different probability distributions (PDs) and coefficients of variation (CVs) in expected tardiness and standard deviation of tardiness. On the one hand, results of the performance of our simheuristic for the deterministic case showed that our algorithm presents competitive results. On the other hand, the experimental designs showed that both PDs and CVs have significant effects on the two objective functions (expected tardiness and standard deviation of tardiness). All of these indicate the importance of including uncertainty, modeled with accurate PDs, to obtain more adequate robust solutions than only analyzing an interval of occurrence. Additionally, the simheuristic was implemented in an optical laboratory showing better results in both expected and standard deviation of tardiness in comparison with the sequences implemented by the company. For future works we recommend including the analysis of qualitative decision criteria in the objective functions analyzed. Also, a robust Just in Time environment can be studied by adding earliness and deviation standard of earliness to the objective functions. Funding This research did not receive any specific grant from funding agencies in the public, commercial, or not-for-profit sectors. References Allahverdi, A., & Aydilek, H. (2010). Heuristics for the two-machine flowshop scheduling problem to minimize maximum lateness with bounded processing times. Computers & Mathematics with Applications, 60(5), 1374–1384. https://doi.org/10.1016/j.camwa. 2010.06.019. Armentano, V. A., & Arroyo, J. E. C. (2004). An application of a multi-objective tabu search algorithm to a bicriteria flowshop problem. Journal of Heuristics, 10(5), 463–481. https://doi.org/10.1023/B:HEUR.0000045320.79875.e3. Aydilek, H., & Allahverdi, A. (2013). A polynomial time heuristic for the two-machine flowshop scheduling problem with setup times and random processing times. Applied Mathematical Modelling, 37(12–13), 7164–7173. https://doi.org/10.1016/j.apm. 2013.02.003. Azadeh, A., Jeihoonian, M., Shoja, B. M., & Seyedmahmoudi, S. H. (2012). An integrated neural network–simulation algorithm for performance optimisation of the bi-criteria two-stage assembly flow-shop scheduling problem with stochastic activities. International Journal of Production Research, 50(24), 7271–7284. https://doi.org/10. 1080/00207543.2011.645511. Baker, K. R. (1974). Introduction to sequencing and scheduling (1st ed.). John Wiley & Sons. Baker, K. R., & Altheimer, D. (2012). Heuristic solution methods for the stochastic flow shop problem. European Journal of Operational Research, 216(1), 172–177. https:// doi.org/10.1016/j.ejor.2011.07.021. Baker, K. R., & Trietsch, D. (2011). Three heuristic procedures for the stochastic, twomachine flow shop problem. Journal of Scheduling, 14(5), 445–454. https://doi.org/ 10.1007/s10951-010-0219-4. Behnamian, J. (2013). A parallel competitive colonial algorithm for JIT flowshop scheduling. Journal of Computational Science, 5(5), 777–783. https://doi.org/10.1016/j. jocs.2013.11.002. Behnamian, J. (2016). Survey on fuzzy shop scheduling. Fuzzy Optimization and Decision Making, 15(3), 331–366. https://doi.org/10.1007/s10700-015-9225-5. Beyer, H. G., & Sendhoff, B. (2007). Robust optimization – A comprehensive survey.

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