Tabu Search Based Hybrid Meta-Heuristic Approaches for Schedule-Based Production Cost Minimization PDF

INFORMATICA, 2022, Vol. 33, No. 3, 499–522 499 © 2022 Vilnius University DOI: https://doi.org/10.15388/21-INFOR471 Tabu Search Based Hybrid Meta-Heuristic Approaches for Schedule-Based Production Cost Minimization Problem for the Case of Cable Manufacturing Systems Fereshteh DANESHDOOST1 , Mostafa HAJIAGHAEI-KESHTELI2 , Ramazan SAHIN3 , Sadegh NIROOMAND4,∗ 1 Department of Industrial Engineering, University of Science and Technology of Mazandaran, Behshahr, Iran 2 Tecnologico de Monterrey, Escuela de Ingeniería y Ciencias, Puebla, Mexico 3 Department of Industrial Engineering, Faculty of Engineering, Gazi University, Ankara, Turkey 4 Department of Industrial Engineering, Firouzabad Institute of Higher Education, Firouzabad, Fars, Iran e-mail: [email protected], [email protected] Received: February 2021; accepted: December 2021 Abstract. This paper models and solves the scheduling problem of cable manufacturing industries that minimizes the total production cost, including processing, setup, and storing costs. Two hybrid meta-heuristics, which combine simulated annealing and variable neighbourhood search algorithms with tabu search algorithm, are proposed. Applying some case-based theorems and rules, a special initial solution with optimal setup cost is obtained for the algorithms. The computational experi- ments, including parameter tuning and ﬁnal experiments over the benchmarks obtained from a real cable manufacturing factory, show superiority of the combination of tabu search and simulated an- nealing comparing to the other proposed hybrid and classical meta-heuristics. Key words: scheduling theory, single machine scheduling, cable manufacturing, hybrid meta-heuristic, tabu search. 1. Introduction Organizations with manufacturing operations are sensitive to scheduling their tasks and operations to obtain better performance. The measures of performance in such organiza- tions can be on-time product delivery, less inventory cost, less tardiness, less earliness, etc. Therefore, applying scheduling policies, overall production cost may be decreased and less lead time for the products is obtained. To obtain a good schedule of tasks and operations in manufacturing organizations, the concepts of scheduling problems are used. In a scheduling problem, diﬀerent goals may be optimized under diﬀerent limitations and ∗ Corresponding author. 500 F. Daneshdoost et al. assumptions depending on the condition of production system under study. The goals such as earliness, tardiness, makespan, setup time, etc. (see Allahverdi et al., 2008; Santander- Mercado and Jubiz-Diaz, 2016; Nair et al., 2016) may be of interest for managers of man- ufacturing organizations. Such scheduling problems under the given goals are mathemat- ically modelled as a linear programming model (LP), or a mixed integer linear program- ming model (MILP), or even as a mixed integer nonlinear programming model (MINLP). For such models of scheduling problems the studies of Pinedo (2008), Vakhania et al. (2014), Hajiaghaei–Keshteli et al. (2014), Ku and Beck (2016), Ahmadi et al. (2016), He et al. (2016), Qu et al. (2016), Mahmoodirad et al. (2019), Niroomand et al. (2019), Mahmoodirad and Niroomand (2020), etc. can be referred. In addition to the goals that typically classify the scheduling problems, these problems may occur in diﬀerent production systems. The most famous production system is single machine system where the tasks should be scheduled to be produced on one machine. In this topic, the studies of Ozgur and Bai (2010), Ji et al. (2013), Wu et al. (2013), Fang and Lu (2016), Che et al. (2016), Ying et al. (2016), Hajiaghaei-Keshteli and Aminnayeri (2014), etc. can be exampled. The case of parallel machine scheduling problem is also interesting where two or more identical machines perform the same duties (see Lee et al., 2012; Kim and Lee, 2012; Zinder and Walker, 2015; Li et al., 2015). The task scheduling of job shop production systems considering various goals and limitations has been in a wide focus. The studies of Karimi-Nasab and Seyedhoseini (2013), Niu et al. (2013), Jamili (2016), Mirshekarian and Šormaz (2016), Koulamas and Panwalkar (2016), etc. work on diﬀerent versions of job shop scheduling problems. The scheduling problems of batch processing production systems is one of the most diﬃcult ,scheduling problems. In this type of problems, the jobs are to be performed in a limited numbers deﬁned as the batches. The batch processing concepts may also be combined with all the above- mentioned systems. In this topic the recent studies of Dong and Wang (2012), Bellanger et al. (2012), Mor and Mosheiov (2014), Chu et al. (2014), Zhou et al. (2016), etc. may be of interest. Notably, in all the above-mentioned scheduling problems, the tasks may have either sequence dependent setup times or independent setup time. In the case of sequence dependent setup time, the problem becomes more complex to be solved as any sequence of tasks may result in a diﬀerent total setup time. Moreover, the scheduling problems are even studied in certain and uncertain environments. In a certain environment all the data of the problem, e.g. task processing time, setup time, worker/machine cost, task delivery due date, etc., are of certain values. But these values in an uncertain environment cannot take a single value as those may be a fuzzy number, interval value or stochastically determined value. For the case of uncertain scheduling problems, the studies of Leite and Dimitrakopoulos (2014), Lu et al. (2014), Ebrahimi et al. (2014), Rahmani and Heydari (2014), Taassori et al. (2015), Hao et al. (2015), Tavana et al. (2018), Niroomand et al. (2018), etc. can be referred. The mathematical models of scheduling problems has been tackled by exact and meta- heuristic approaches in the literature. Generally, the introduced models of literature are of NP-hard problems. Therefore, they cannot be solved exactly with available general pur- pose solvers in medium and large sizes. For this reason the decomposition based algo- rithms like Lagrangian relaxation, Bender’s decomposition, etc. have been used to solve Tabu Search Based Hybrid Meta-Heuristic Approaches for Schedule-Based PCMP 501 some larger than small sized instances of such problems (see Chu and You, 2013; Paraste- gari et al., 2013; Xiao et al., 2015; Shi et al., 2015; Wolosewicz et al., 2015). On the other hand, to tackle the large and very large instances of these problems the researchers have applied meta-heuristic algorithms. These algorithms have been used either in their classic version or in a hybridized version (Zheng et al., 2020; Guo et al., 2020; Hosseini Shir- vani, 2020). The most frequently used algorithms in the literature are genetic algorithm, simulated annealing, variable neighbourhood search, tabu search, etc. (see Gomes et al., 2014; Reisi-Nafchi and Moslehi, 2015; Kurdi, 2015; Zhang and Wong, 2016; Martin et al., 2016; Akbari and Rashidi, 2016; Niroomand et al., 2016; Quintana et al., 2017; Hsieh, 2017; Hu et al., 2016; Ghadiri Nejad and Banar, 2018; Misevičius et al., 2018; Vizvári et al., 2018; Dugonik et al., 2019; Ullah et al., 2020; Hassanpour, 2020; Aliya et al., 2020; Fernández et al., 2020; Hussain and Khan, 2020). This paper contributes to solve an important scheduling problem in cable industries with a single machine as a case study which has been in focus of the literature previ- ously. As an important objective function, the total production cost, e.g. the sum of the costs related to inventory holding, setup, and processing, are to be minimized. Two novel hybrid meta-heuristic algorithms equipped with some theorems are introduced to tackle the problem eﬃciently. The proposed algorithms hybridize tabu search method with the classical algorithms such as simulated annealing algorithm and variable neighbourhood search separately. The obtained results prove the eﬀectiveness of the proposed approaches comparing to the approaches used in the previous studies. The next sections of the paper are addressed here. The next section explains the scheduling problem of cable industries. The problem is formulated in Section 3. The pro- posed hybrid meta-heuristic algorithms are detailed in Section 4. The proposed algorithms are experimentally examined and compared in Section 5. The paper ends with a conclusion in Section 6. 2. Problem Deﬁnition and Case of the Study The case of this study is a problem which exists in a cable manufacturing system. This case study is exactly the same as the case that was focused in Niroomand and Vizvari (2015) with the same data set. The company produces various models of cables. They use metal (mainly copper) and plastic as the raw materials to produce the cables on a single machine. The cable types diﬀer in the diameter of copper and plastic cover colour. The copper is supplied to the company in large scales as “wire road”. As the wire road has just one diameter size, so while performing a “drawing” task it is changed to the wires with favourable diameters as demanded by the customers of the company. To ﬁnish the production of the cable, the raw wire which is just a copper string is covered by the plastic cover. The company also receives the plastic cover from the suppliers in various colours. Therefore, a type of cable can be deﬁned by a special diameter size and a special colour of plastic cover. In the company of this case, according to the demand of the customers, cables of diﬀerent sizea and diﬀerent coloura should be produced in a planning horizon. 502 F. Daneshdoost et al. The following real data can more describe the process of this company: Number of various sizes (diameters) of the copper (w) in the demand of one planning horizon; Number of various colours of the plastic cover (v) in the demand of one planning hori- zon; Demand values for all types of cable (vw types) of the planning horizon; All demands are responded on a predetermined single due date; Labour wage (cost) per unit of time (mainly seconds); The following data for setup times between two consecutive types of cable on produc- tion machine: – S1 setup: Setup time needed for two consecutive types of cable in production sched- ule, if only the colours are diﬀerent (the diameter sizes are the same) (this setup takes s1 seconds), – S2 setup: Setup time needed for two consecutive types of cable in production sched- ule, if only the diameter sizes are diﬀerent (the cover colours are the same) (this setup takes s2 seconds), – S3 setup: Setup time needed for two consecutive types of cable in production sched- ule, when their both diameter sizes and cover colours are diﬀerent (this setup takes s3 seconds), – This is known that s2 < s1 < s3 ; In each of the above-mentioned setups S1, S2 and S3, there is an amount of scrap in copper, plastic cover and both copper and plastic cover, respectively; Considering labour cost for setup times and scrap cost for the setup types, setup cost for each setup type is deﬁned. The setup costs have the following relation: – Setup cost (S2) < Setup cost (S1) < Setup cost (S3); Processing cost and time (machine cost per unit of time plus labour cost during machine process) for each cable type (for a bath of cable type based on metres) is known. The operating time increases when the cable diameter size is increased. There is no relation between the operating time and the colours of cables; The cost of holding the cables for unit of cable known for the produced cables. It does not depend on the cable type. The produced cables are held in the inventory until the due date. The company is interested to obtain such sequence of the cables to optimize the total production related cost which is the sum of setup (and scrap) cost, processing cost, and inventory cost. In continuation, a mathematical formulation and some hybrid meta-heuristic algo- rithms are suggested to optimize the total production cost of the described problem. 3. Mathematical Formulation The problem of the cable company is formulated as a mixed integer linear problem (MILP) in this section. This formulation also was suggested by Niroomand and Vizvari (2015). The notations used in the mathematical formulation are presented by Table 1. Tabu Search Based Hybrid Meta-Heuristic Approaches for Schedule-Based PCMP 503 Table 1 The notations used in the mathematical formulation. Notation Type Description n index the number of the cable types (n = vw) v index the number of diﬀerent colours of the cables w index the number of diﬀerent sizes of the cables i, j index the indexes of the cables k, p index the indexes of positions of the cables in their production schedule sij parameter the setup time of cable j if it is produced after cable i δij parameter the scrap cost if cable j is produced after cable i qi parameter the demand of cable i (in metres of cable) ti parameter the processing time of one unit of cable i φ parameter the processing cost for a unit of the cables per time unit γ parameter the labour cost per time unit μ parameter the inventory holding cost for a unit of the cables per time unit T parameter the single due date of the demands M parameter a large positive value T0 variable the time which the production of the ﬁrst cable is started (idle time of the production system) Xip variable a binary variable that takes 1 if cable i is assigned to position p in the production schedule Yijp variable a binary variable that takes 1 if cable j is produced after cable i at position p of the sequence Ci variable the completion time of cable i at its assigned position in the sequence of the cables The following assumptions are used to formulate the problem of the cable company: All types of cables are sequenced and produced to respond their demand; When the production of a type of cable starts, its whole demand is produced with no interruption; The inventory holding time (and its cost accordingly) for a cable type is the interval between the completion time and the single due date; The customers do not accept to receive the cables before the single due date. Based on the introduced assumptions, the mathematical formulation of the problem is written as follow: n n n n n min μqi (T − Ci ) + φqi ti + Yijp (γ sij + δij ) , (1) i=1 i=1 p=2 i=1 j =1 j =i subject to n Xip = 1, ∀i ∈ {1, 2,... , n}, (2) p=1 n Xip = 1, ∀p ∈ {1, 2,... , n}, (3) i=1 504 F. Daneshdoost et al. Ci T0 + qi ti Xi1 + M(1 − Xi1 ), ∀i ∈ {1, 2,... , n}, (4) Cj Ci + M(1 − Yijp ) + sij + qj tj , ∀i, j ∈ {1, 2,... , n}|i = j & p 2, (5) Xi(p−1) + Xjp Yijp + 1, ∀i, j, p ∈ {1, 2,... , n}|i = j & p 2, (6) Xi(p−1) + Xjp 2Yijp ∀i, j, p ∈ {1, 2,... , n}|i = j & p 2, (7) n n n n n sij Yijp + T0 + qi ti Xip T , (8) p=2 i=1 j =1 i=1 p=1 j =i Xip , Yijp ∈ {0, 1}, ∀i, j, p ∈ {1, 2,... , n}, (9) Ci 0, ∀i ∈ {1, 2,... , n}, (10) T0 0. (11) The objective function (1) minimizes the previous total production related cost of the company consisting of inventory holding cost, processing cost and sum of setup and scrap costs. These costs were explained in Section 2 and the above-mentioned assumptions were made for the model. The term T −Ci calculates the inventory holding time of cable i. Con- straints (2) and (3) determine the position of each type of cable in the production sequence of the cables. Constraint (4) determines the production starting time of the sequence of cables and the ﬁrst cable of the sequence together. Existence of M will help the constraint to be satisﬁed for the cables which are not assigned to the ﬁrst place of the sequence. Con- straint (5) calculates completion time of the cables which are placed at position p (p 2) of the sequence. Of course, this constraint is satisﬁed for those cables which are not as- signed to position p of the sequence using M value. The value of Cj is optimized in the optimal solution by help of minimization type objective function and the term T − Cj. Constraints (6) and (7) determine the value of variable Yijp according to its deﬁnition in Table 1. Constraint (8) guarantees that the last cable of the sequence is produced before the single due date of the demands. Constraints (9), (10) and (11) determine the type and sign of variables. According to Wan and Yuan (2013), most of the mathematical formulations used to op- timize the scheduling problems, are categorized as NP-complete and NP-hard type prob- lems. Therefore, as the model (1)–(11) is a typical form of single machine scheduling problems, it can be an NP-complete or NP-hard problem. Furthermore, complexity of the model (1)–(11) has been experimentally proved by Niroomand and Vizvari (2015). Therefore, the model should be tackled heuristically. For this aim Niroomand and Vizvari (2015) have introduced some meta-heuristic approaches to solve the model. In this study, also some new hybrid meta-heuristics are introduced to solve the model in order to obtain solutions better than what Niroomand and Vizvari (2015) obtained. 4. Meta-Heuristic Approaches The NP-hardness of the model (1)–(11) is a source of motivation for introducing meta- heuristic solution approaches. Earlier, Niroomand and Vizvari (2015) tackled this model Tabu Search Based Hybrid Meta-Heuristic Approaches for Schedule-Based PCMP 505 Fig. 1. A production schedule of the solution shown by vector (4, 2, 3, 1, 5). by applying some meta-heuristics. Before introducing any new approach, it is necessary to brieﬂy explain their methodology. In the study of Niroomand and Vizvari (2015), three approaches are used to obtain an initial solution, e.g. (1) the cables with the same colour are grouped and the total process- ing time of each group is calculated. Then, the groups are arranged in descending order of the total processing time, (2) the cables are grouped according to the size, and the total processing time of each group is calculated. Then, the groups are arranged in descending order of the total processing time, (3) no grouping policy is applied, and the cables are arranged in descending order of their processing time. In that study it is proved that the ﬁrst initial solution takes less total setup time which decreases the setup dependent labour cost which is a part of the objective function (1). Of course, all the three initial solutions try to have less inventory holding cost (because of descending order of processing times). Finally, they applied two meta-heuristic approaches, e.g. simulated annealing (SA) and variable neighbourhood search (VNS), using each of the obtained initial solutions sepa- rately. As a typical experimental result of their study, the initial solution (1) results in a much better ﬁnal solution than the others. In this study, we develop some hybrid meta-heuristics combining tabu search algorithm with SA and VNS separately. These hybrid approaches use the initial solution (1) proposed by Niroomand and Vizvari (2015) and try to improve it to obtain a better ﬁnal solution than what was obtained by Niroomand and Vizvari (2015). In the rest of this section, the initial solution and the proposed hybrid algorithms are explained. 4.1. Encoding-Decoding Method and Initial Solution Generation Any solution generated in the meta-heuristic approaches of this study is represented as a vector of numbers. Each number shows a cable type. So, a vector illustrates a sequence of cables to be produced according to their order in the vector. For instance, a solution represented by vector (4, 2, 3, 1, 5) is shown by Fig. 1. In the solution represented by Fig. 1, the completion time of cable 5 is considered as the given due date of the cables (T ). So, automatically a required idle time (T0 ) is applied here without needing to calculate it. Then, to calculate the objective function value of this solution, its inventory holding cost, processing cost, and setup cost, according to the concepts of the model (1)–(11), are calculated and summed up. Now, to generate a good initial solution, a sequence of cables should be minimized from three points of view, e.g. total setup cost, total inventory holding cost, and total processing cost. As the processing times are constant values, so the total processing cost is always ﬁxed and cannot be decreased. Therefore, in an initial solution we try to decrease the other cost types. 506 F. Daneshdoost et al. Fig. 2. A schematic representation of the example of Theorem 1. In order to generate an initial solution with lowest possible setup cost, the following theorem is introduced. Theorem 1. If in the vector of a solution, the cables with the same colour are produced consecutively in a way that the last cable of a colour has the same size with the ﬁrst cable of the next colour, the solution has the minimum possible setup cost. In this case, the cables with the same colour are called a set of cables. An example of this type of solutions for 3 colours (v = 3) and 3 sizes (w = 3) is illustrated by Fig. 2. Proof. According to the data given in Section 2, in general case when there exist n cables with v colours and w sizes, the number of S1 setup type is v−1 and the number of S2 setup type is equal to (vw − 1) − (v − 1), while there is no S3 setup type in this order of cables. Note that any change in the order of the cables in this type of solutions will increase the number of S1 and S3 setup types. As the relation Setup cost (S2) < Setup cost (S1) < Setup cost (S3) exists, the theorem is proved. For any solution which follows the concepts of Theorem 1, it is possible to decrease its inventory holding cost by applying the following rules. These rules may help to generate a more eﬃcient solution from inventory holding cost point of view. Rule 1. If the cables are sequenced based on Theorem 1, the sets of cables (as deﬁned in Theorem 1) are arranged in descending order of the below given value which is calculated for each set. This rule may cause less inventory holding time that results in less inventory holding cost. S + i∈I ti qi , ∀I ∈ {1, 2,... , v}, (12) i∈I qi where S is total setup time of each set of cables which is the same for all sets and is calculated as S = ws2 + s1. Note that s1 and s2 are the setup times of S1 and S2 setup types, respectively. An evidence for this rule is mentioned here. Consider two sets A and B with an extra assumption that the inventory holding time of all units of cables of each set starts together immediately after processing all cables of that set. If an initial order of these sets is A and then B, supposing that the interchanged order of these sets results in less total inventory holding time, therefore, Inventory holding time of sequence B, A Inventory holding time of sequence A, B. Tabu Search Based Hybrid Meta-Heuristic Approaches for Schedule-Based PCMP 507 So, T −S− ti q i qi + T − S − ti q i − S − ti q i qi i∈B i∈B i∈B i∈A i∈A T −S− ti q i qi + T − S − ti q i − S − ti q i qi , i∈A i∈A i∈A i∈B i∈B (13) which implies, S + i∈B ti qi S + i∈A ti qi . (14) i∈B qi i∈A qi Rule 2. In a solution satisfying Theorem 1 and Rule 1, in any set of cables (deﬁned in Theorem 1) all cables, except for the ﬁrst and the last cable of the set, are arranged in descending order of the below given value which is calculated for each of those cables. This rule also may result in less inventory holding time and, consequently, less inventory holding cost. s2 + tj , ∀j ∈ {2,... , w − 1} in each set. (15) qj An evidence for this rule is mentioned here. Consider two cables i and j from any set of cables with initial sequence of i and then j. Suppose that the interchanged order of these sets results in less total inventory holding time, therefore, Inventory holding time of sequence i, j Inventory holding time of sequence j , i. Then, T − (s2 + tj qj ) qj + T − (s2 + ti qi ) − (s2 + tj qj ) qi T − (s2 + ti qi ) qi + T − (s2 + tj qj ) − (s2 + ti qi ) qj , (16) which implies, s2 s2 + tj + ti. (17) qj qi The above-mentioned theorem and rules are used to generate a good initial solution for the meta-heuristic approaches that will be introduced in the rest of this section. The procedure of generating an initial solution is summarized in the pseudo code that is shown by Fig. 3. 4.2. Neighbourhood Search Operators As the initial solution obtained by Algorithm 1 will be used in some meta-heuristic ap- proaches which are presented in the following sections, three neighbourhood search oper- 508 F. Daneshdoost et al. Fig. 3. Pseudo code for generating an initial solution. Fig. 4. Pseudo code of neighbourhood search operator 1 (SO1 ). ators are introduced in this sub-section. These operators are designed in a way to respect the previously mentioned theorem and rules. 4.2.1. Neighbourhood Search Operator 1 (SO1 ) SO1 is designed in a way to maintain all characteristics of an initial solution which is obtained by Algorithm 1. In a neighbour obtained by SO1 , Theorem 1, Rule 1, and Rule 2 are respected. This operator is explained by the pseudo code of Fig. 4. SO1 may give a neighbour solution with better objective function value than the initial solution. The generated neighbour solution has the same setup, scrap, and processing cost comparing to the initial solution but its inventory holding cost may be decreased as the sequence of cables is changed. 4.2.2. Neighbourhood Search Operator 2 (SO2 ) Applying SO2 on a set of cables with colour i, two cables with diﬀerent sizes from this colour set are selected randomly by generating two random numbers between 2 and w − 1 (to respect Theorem 1, all sizes of the selected colour set can be selected, except for the ﬁrst and last cable of the set) and are interchanged. In the neighbour solution generated by this operator, Theorem 1 and Rule 1 are respected while Rule 2 is not considered in the set of cables with colour i. Tabu Search Based Hybrid Meta-Heuristic Approaches for Schedule-Based PCMP 509 4.2.3. Neighbourhood Search Operator 3 (SO3 ) Using SO3 , a set of cables with colour i is selected. Two random numbers between 2 and w−1 are generated. Then the sizes of cables which are associated to the generated random numbers in sets i and i + 1 are interchanged separately. It seems that SO2 is done on two consecutive sets with the same random numbers. 4.3. Tabu Search Hybridized by Simulated Annealing (TS-SA) Tabu Search (TS) (see Lamghari and Dimitrakopoulos, 2012; Li and Gao, 2016) and Sim- ulated Annealing (SA) (see Kirkpatrick et al., 1983; Niroomand et al., 2016) are two fa- mous single solution based meta-heuristic algorithms which have been widely applied to solve combinatorial optimization problems. TS starts with a current solution (initial solution). This current solution is also marked as the best solution. Then TS uses some directions of searching space and ﬁnds some neighbour solutions. The best neighbour solution of the discovered neighbours can be used for three purposes, (1) it is saved as the best solution if it is better than the current best solution, (2) the best neighbour and also its neighbourhood direction is marked as a tabu solution and direction and saved in a tabu list, and (3) it is saved as the current solution. Then the algorithm is repeated for a number of iterations. In order to avoid repeating some previously considered current solutions, in each iteration the neighbour solutions obtained by tabu searching directions are not considered as a new current solution unless they are better than the best obtained solution. This condition is named aspiration criteria. Considering tabu list and aspiration criteria together is one of the advantages of TS which avoid cycling in searching procedure. When last iteration is done, the best solution is introduced as the best obtained solution of TS. SA starts with an initial solution (called current solution and also best solution) in an initial state called initial temperature (current temperature). In the current temperature for a number of iterations, the current solution is tried to be improved by generating its neighbour solutions. Each generated neighbour solution could be used for the following purposes, (1) to be used instead of the best solution and the current solution if it has better objective function value than the best solution, (2) to be used instead of the current solution if it has better objective function value than just the current solution, and (3) to be used instead of the current solution if the condition − exp( f (neighbour)−f (current) currenttemperature ) r is held, where f is the objective function value and r is a fractional random number. Otherwise, the neighbour solution is ignored and the current and best solutions remain unchanged. Notably, after generating a neighbour solution the current and best solutions are updated (not necessarily) and the current solution is used for next neighbour generation. When the iterations of the current temperature are done, the current temperature is cooled down by a cooling ratio and the iterations are repeated in this current temperature. SA continues until the current temperature reaches a given ﬁnal temperature. At the end, the best solution is introduced as the best obtained solution of SA. In this study, TS is hybridized by SA. To combine these two algorithms and ﬁt them to the problem of Section 3, in the ﬁrst iteration of the TS the following is done: 510 F. Daneshdoost et al. The initial solution obtained by Section 4.1 is used as current (initial) solution. Each colour of the cables in the initial solution is considered as a searching direction (except for the last one, because neighbourhood search operator 1 cannot be applied in this case). In each searching direction of the initial solution (current solution), an SA algorithm using neighbourhood search operator 1, is applied as a local search to obtain the best possible neighbour solution in each searching direction separately. The best obtained neighbour among all searching directions is saved instead of the cur- rent solution of the TS and this searching direction is added to the list of tabu directions. The TS-SA algorithm continues to the consequent iterations considering the current solu- tion. Pseudo code of this hybrid algorithm is shown by Fig. 5, where the parameters used in the algorithm are noted in Table 2. 4.4. Tabu Search Hybridized by Variable Neighbourhood Search (TS-VNS) Another hybridization of the TS is done by help of Variable Neighbourhood Search (VNS) algorithm. The structure of this hybrid algorithm (TS-VNS) is the same as the structure of TS-SA algorithm. The only diﬀerence is that instead of the SA algorithm, a VNS algo- rithm is used. The VNS is of single solution based meta-heuristics which starts with an initial (current) solution and uses a set of some diﬀerent neighbourhood search operators to explore neighbour solutions. Focusing on the initial (current) solution, ﬁrst search oper- ator is applied for a number of iterations. The best obtained neighbour solution is saved as the current solution and the algorithm is repeated in the same way for other neighbourhood search operators. At the end, current solution is introduced as the best obtained solution of the VNS. Pseudo code of this hybrid algorithm is shown by Fig. 6, where the parameters used in the algorithm are noted in Table 3. 4.5. On the Properties of the Proposed Hybrid Algorithms (TS-VNS and TS-SA) In this study, the hybrid meta-heuristic algorithms are proposed in order to improve the performance of the classical algorithms TS, VNS, and SA. Therefore, the proposed TS- VNS and TS-SA have the following advantages comparing to the classical algorithms: The classical TS has a simple neighbourhood search structure. The proposed TS-SA applies the SA as a local search mechanism in order to improve the search structure of TS. The proposed TS-VNS uses the VNS as a local search mechanism in order to improve the search structure of the TS. The proposed hybrid meta-heuristic algorithms applies more neighbourhood search methods comparing to the classical algorithms. Tabu Search Based Hybrid Meta-Heuristic Approaches for Schedule-Based PCMP 511 Fig. 5. Pseudo code of the TS-SA algorithm. 5. Computational Experiments The proposed algorithms of previous section are to be experimentally evaluated in this sec- tion. For thus aim, the proposed algorithms are coded in MATLAB and to perform all the 512 F. Daneshdoost et al. Table 2 Notations used in the TS-SA algorithm. Notation Description xc−T S Initial (current) solution of the TS obtained by Algorithm 1 xc−SA Initial (current) solution of the SA in each iteration of the TS (is the same as xc−T S ) xb−T S Best solution of the TS xb−SA Best solution of the SA xSA−i Neighbour solution obtained from i-th colour in the SA procedure xc−SA−i Current solution of the SA after searching in the neighbours of i-th colour Tin Initial (current) temperature of the SA Tf Final temperature of the SA R Number of iterations done in each temperature of the SA ε Cooling ratio of the SA r A random number (0 < r < 1) Q Number of iterations for the TS required experiments the codes are run on a computer with an Intel Core 2 Duo 2.53 GHz processor and 4.00 GB RAM. To evaluate and study the behaviour of the algorithms the following is needed: obtaining some benchmark problem from the factory under study, tuning the parameters of the algorithms, ﬁnal experiments and comparing the results with literature. The above-mentioned requirements are done in the sub-sections of this section. 5.1. Benchmark Problems To study the performance of the proposed algorithms, some benchmark problems are needed. The study of Niroomand and Vizvari (2015) contains just one benchmark (ob- tained from the case of the study) which is a large-sized benchmark containing 15 colours and 10 sizes. In this study, that benchmark is considered as the largest benchmark and for generating some other benchmarks, its sizes are decreased. Notably, the values of data (explained in Section 2) in the generated benchmarks remain unchanged, just the size of matrices and vectors which contain data are decreased by removing extra rows and columns. These benchmarks are detailed in Table 4. 5.2. Parameter Tuning Any meta-heuristic algorithm consists of some parameters which should be ﬁxed in ad- vance. As most of algorithms behave randomly, the values given for their parameters may aﬀect the quality of obtained solutions. Generally larger values for parameters result in larger CPU running time and maybe better result. This relation cannot be true always. Therefore, parameters of meta-heuristic algorithms should be tuned before performing ﬁ- nal experiments. The tuning procedure can be done to optimize one or more responses obtained from running an algorithm for a benchmark. In most of studies, the tuning pro- cedure is done to study the eﬀect of parameters’ levels for obtaining solutions with better Tabu Search Based Hybrid Meta-Heuristic Approaches for Schedule-Based PCMP 513 Fig. 6. Pseudo code of the TS-SA algorithm. Table 3 Notations used in the TS-VNS algorithm. Notation Description xc−TS Initial (current) solution of the TS obtained by Algorithm 1 xc−VNS Initial (current) solution of the VNS in each iteration of the TS (is the same as xc−TS ) xb−TS Best solution of the TS xb−VNS Best solution of the VNS xVNS−i Neighbour solution obtained from i-th colour in the VNS procedure xc−VNS−i Current solution of the VNS after searching in the neighbours of i-th colour n Number of neighbourhood search operators of the VNS R Number of iterations done for each neighbourhood search operator the VNS Q Number of iterations for the TS 514 F. Daneshdoost et al. Table 4 The benchmarks used for computational experiments. Benchmark Number of Number of Number of diﬀerent No. colours (v) sizes (w) cables (n) B1 6 4 24 B2 8 6 48 B3 10 8 80 B4 12 10 120 B5 15 10 150 objective function. As a more complete study on the behaviour of parameters, except qual- ity of solutions, CPU running time also can be considered as another response, meaning that a multi-response study can be done to tune the parameters of a meta-heuristic algo- rithm. To tune the parameters of the proposed algorithms of this study, two responses of objective function value (quality) and CPU running time are considered to be optimized. To optimize the parameters’ levels according to these two responses, a method based on regression analysis is done. The method previously was introduced by Pasandideh et al. (2015). The steps of this method are explained here: Step 1: Select an algorithm to tune its parameters. Select a proper interval for each pa- rameter of the selected algorithm. Step 2: Select a proper benchmark and randomly generate a proper number of test prob- lems for that benchmark. Notably, the values of parameters in each test problem are uni- formly generated within the intervals speciﬁed in Step 1. Step 3: Run each test problem by the selected algorithm and obtain the values of required responses. Step 4: Linearly normalize the values of parameters and responses of each test problem. Find quadratic regression function of each response separately. For example, a quadratic regression function for a response (say Ri ) dependent on three parameters of an algorithm (say P1 , P2 , P3 ) is as follows: E(Ri ) = β0 + β1 P1 + β2 P2 + β3 P3 + β11 P12 + β22 P22 + β33 P32 + β12 P1 P2 + β13 P1 P3 + β23 P2 P3 , (18) where βi , βii and βij are the linear, quadratic and interaction coeﬃcients, respectively. Step 5: Find the weighted sum of all E(Ri ) as follows: N E(R) = wi E(Ri ) , (19) i=1 where wi is the importance weight of response i, N is the number of responses and N i=1 wi = 1. Tabu Search Based Hybrid Meta-Heuristic Approaches for Schedule-Based PCMP 515 Table 5 The data and results of parameter tuning. Algorithm TS-SA TS-VNS Parameter Q Tin Tf R ε Q R Interval [50, 500] [50, 500] 5 [50, 300] [0.8, 0.99] [50, 500] [50, 300] Best level 55 252 5 50 0.85 65 85 Step 6: Solve the following non-linear mathematical model to obtain the most eﬀective levels of the parameters. max E(R) (20) subject to All interval of the parameters (21) To apply the above-mentioned steps to the proposed algorithms of this study, the proper interval of each parameter is selected (see Table 5). The benchmark number 4 is selected to tune the parameters. 50 test problems were generated uniformly considering the speci- ﬁed intervals of the parameters. To overcome the randomness of the algorithms, each test problem was run for ﬁve times. The average response values (objective function value and CPU time) of the 5 runs of each test problem is considered in regression analysis. Finally, the best obtained levels of the parameters are shown by Table 5. Notably, when performing Step 6 of the tuning procedure, the wi values are assumed to be equal to 0.5. Although, in meta-heuristic algorithms, higher values of parameters may result in better solution, this is correct for the cases when the only response is objective function value. In the case of this study, as CPU time is also considered as a response, therefore, some parameters may not get their highest level as best level. The best level obtained for each parameter is applied to perform the ﬁnal experiments on the benchmarks. 5.3. Final Experiments on the Proposed Algorithms The best level of the parameters reported in Table 5 is applied in the proposed algorithms to perform the ﬁnal experiments on all the benchmarks of Table 4. To compare the results of the proposed algorithms with the methods of literature, the SA and VNS algorithms of Niroomand and Vizvari (2015) are simulated and applied. Notably, these algorithms are used with the best levels of their parameters reported in Table 5, except for the parameter Q (as these methods do not have such parameter). All the four algorithms for each bench- mark are allowed to run for 1000vw milliseconds. To obtain more reliable results, each benchmark with each algorithm is run for 10 times. For more comparison, the proposed formulation (1)–(11) is coded by the GAMS and the benchmarks of Table 4 are solved by the CPLEX solver of the GAMS as an exact method. This solver uses Branch and Bound method as a solution approach. For this aim, the running time of 3 hours (10800 seconds) 516 F. Daneshdoost et al. Table 6 The result obtained by the algorithms. Benchmarks Solution Algorithm B1 B2 B3 B4 B5 Minimum SA 4707.74 17560.54 51770.87 114117.25 187689.13 VNS 4702.78 17536.55 51773.56 114113.67 187685.51 TS-SA 4702.69 17537.93 51692.47 114059.51 187591.49 TS-VNS 4714.48 17559.89 51730.19 114102.09 187669.45 Average SA 4711.17 17566.58 51815.96 114216.07 187691.62 VNS 4707.07 17539.91 51804.06 114152.09 187685.76 TS-SA 4704.70 17537.93 51693.65 114063.17 187610.38 TS-VNS 4715.09 17562.51 51731.12 114119.07 187673.61 Maximum SA 4714.45 17573.87 51932.69 114287.21 187695.39 VNS 4711.21 17546.25 51837.64 114195.65 187686.79 TS-SA 4711.21 17537.93 51701.42 114071.62 187618.70 TS-VNS 4716.54 17563.17 51733.30 114126.82 187676.41 Branch and bound by GAMS 5626.30 32362.48 N.A. N.A. N.A. Fig. 7. Normalized plot of minimum obtained values for all the benchmarks by the algorithms. is considered for all benchmarks. The results obtained by the meta-heuristic algorithms and the exact method of GAMS are shown by Table 6. The results of Table 6 are normalized to be plotted for graphical illustrations. The graphs of normalized results for minimum, average and maximum values of Table 6 are shown by Figs. 7–9. As can be seen from the table and ﬁgures, in most of the benchmarks the TS-SA performs better than the other algorithms in the case of minimum, average, and maximum obtained solutions. There is just one exception that in the second benchmark (B2) the minimum obtained value for the VNS is better than the others. For the case of exact solution method, as mentioned above, 10800 seconds of running time is allowed to the GAMS for each of the benchmarks. The values of 5626.30 and 32362.48 obtained for the benchmarks B1 and B2 are not optimal and the optimality gap of 100% still remains Tabu Search Based Hybrid Meta-Heuristic Approaches for Schedule-Based PCMP 517 Fig. 8. Normalized plot of average of the obtained values for all the benchmarks by the algorithms. Fig. 9. Normalized plot of maximum obtained values for all the benchmarks by the algorithms. for each of them after 10800 seconds of running time. For the benchmarks B3, B4, and B5, after 10800 seconds the GAMS is unable to introduce any feasible solution. This fact shows the complexity of the problem. Concludingly, the proposed meta-heuristics perform better than the exact solution method of the GAMS. Also, the stability of the solutions obtained over 10 runs of each algorithm for all the benchmarks can be studied. For this aim, the diﬀerence of maximum and minimum values of each algorithm for each benchmark is calculated and plotted in Fig. 10. In this case, the TS-SA and TS-VNS algorithms perform better than the SA and VNS algorithms. 518 F. Daneshdoost et al. Fig. 10. The diﬀerence between maximum and minimum values obtained by each algorithm for each benchmark. 6. Concluding Remarks The study focused on solving an important scheduling problem of cable manufacturing industries. For the case that the cables vary in diameter size of cooper and plastic cover colour, the system was modelled as a single machine scheduling problem which minimizes the total production costs including processing cost, setup cost, and inventory holding cost. Two hybrid meta-heuristics which combine simulated annealing and variable neighbour- hood search algorithms with tabu search algorithm separately (say TS-SA and TS-VNS algorithms) were proposed. Applying some theorems and rules of the literature, a special initial solution with optimal setup cost was obtained for the algorithms. The computa- tional experiments over the benchmarks obtained from a real cable manufacturing factory show better performance of TS-SA comparing to TS-VNS and the methods of literature. Acknowledgements The authors are grateful to the editors and reviewers of the journal for their helpful and constructive comments that improved the quality of the paper. Funding This study was supported by Firouzabad Institute of Higher Education, Iran (research project no. 1399.001). The authors are grateful of the ﬁnancial support. References Ahmadi, E., Zandieh, M., Farrokh, M., Emami, S.M. (2016). A multi objective optimization approach for ﬂexible Tabu Search Based Hybrid Meta-Heuristic Approaches for Schedule-Based PCMP 519 job shop scheduling problem under random machine breakdown by evolutionary algorithms. Computers & Operations Research, 73, 56–66. Akbari, M., Rashidi, H. (2016). A multi-objectives scheduling algorithm based on cuckoo optimization for task allocation problem at compile time in heterogeneous systems. Expert Systems with Applications, 60, 234–248. Aliya, F., Fazli, A., Hussain, S.S.B. (2020). Geometric operators based on linguistic interval-valued intuitionistic neutrosophic fuzzy number and their application in decision making. Ann Optim Theory Practices, 3(1), 47–71. Allahverdi, A., Ng, C.T., Cheng, T.C.E., Kovalyov, M.Y. (2008). A survey of scheduling problems with setup times or costs. European Journal of Operational Research, 187, 985–1032. Bellanger, A., Janiak, A., Kovalyov, M.Y., Oulamara, A. (2012). Scheduling an unbounded batching machine with job processing time compatibilities. Discrete Applied Mathematics, 160(1–2), 15–23. Che, A., Zeng, Y., Lyu, K. (2016). An eﬃcient greedy insertion heuristic for energy-conscious single machine scheduling problem under time-of-use electricity tariﬀs. Journal of Cleaner Production, 129, 565–577. Chu, Y., You, F. (2013). Integration of production scheduling and dynamic optimization for multi-product CSTRs: generalized benders decomposition coupled with global mixed-integer fractional programming. Computers & Chemical Engineering, 58, 315–333. Chu, Y., You, F., Wassick, J.M. (2014). Hybrid method integrating agent-based modeling and heuristic tree search for scheduling of complex batch processes. Computers & Chemical Engineering, 60, 277–296. Dong, M.G., Wang, N. (2012). A novel hybrid diﬀerential evolution approach to scheduling of large-scale zero- wait batch processes with setup times. Computers & Chemical Engineering, 45, 72–83. Dugonik, J., Bošković, B., Brest, J., Sepesy Maučec, M. (2019). Improving statistical machine translation quality using diﬀerential evolution. Informatica, 30(4), 629–645. Ebrahimi, M., Fatemi Ghomi, S.M.T., Karimi, B. (2014). Hybrid ﬂow shop scheduling with sequence dependent family setup time and uncertain due dates. Applied Mathematical Modelling, 38(9–10), 2490–2504. Fang, Y., Lu, X. (2016). Online parallel-batch scheduling to minimize total weighted completion time on single unbounded machine. Information Processing Letters, 116(8), 526–531. Fernández, P., Lančinskas, A., Pelegrín, B., Žilinskas, J. (2020). A discrete competitive facility location model with minimal market share constraints and equity-based ties breaking rule. Informatica, 31(2), 205–224. Ghadiri Nejad, M., Banar, M. (2018). Emergency response time minimization by incorporating ground and aerial transportation. Annals of Optimization Theory and Practice, 1(1), 43–57. Gomes, H.C., Neves, F.D.A.D., Souza, M.J.F. (2014). Multi-objective metaheuristic algorithms for the resource- constrained project scheduling problem with precedence relations. Computers & Operations Research, 44, 92–104. Guo, Y., Liu, X., Chen, L. (2020). Improved butterﬂy optimisation algorithm based on guiding weight and population restart. Journal of Experimental & Theoretical Artiﬁcial Intelligence, 33, 127–145. https://doi.org/ 10.1080/0952813X.2020.1725651. Hajiaghaei-Keshteli, M., Aminnayeri, M. (2014). Solving the integrated scheduling of production and rail trans- portation problem by Keshtel algorithm. Applied Soft Computing, 25, 184–203. Hajiaghaei–Keshteli, M., Aminnayeri, M., Fatemi Ghomi, S.M.T. (2014). Integrated scheduling of production and rail transportation. Computers and Industrial Engineering, 74, 240–256. Hao, J., Liu, M., Jiang, S., Wu, C. (2015). A soft-decision based two-layered scheduling approach for uncertain steelmaking-continuous casting process. European Journal of Operational Research, 244(3), 966–979. Hassanpour, M. (2020). Classiﬁcation of seven Iranian recycling industries using MCDM models. Annals of Optimization Theory and Practice, 3(4), 37–52. He, J., Li, Q., Xu, D. (2016). Scheduling two parallel machines with machine-dependent availabilities. Comput- ers & Operations Research, 72, 31–42. Hosseini Shirvani, M. (2020). Bi-objective web service composition problem in multi-cloud environment: a bi- objective time-varying particle swarm optimisation algorithm. Journal of Experimental & Theoretical Arti- ﬁcial Intelligence. https://doi.org/10.1080/0952813X.2020.1725652. Hsieh, F.S. (2017). A hybrid and scalable multi-agent approach for patient scheduling based on Petri net models. Applied Intelligence. 47, 1068–1086. https://doi.org/10.1007/s10489-017-0935-y. Hu, W., Wang, H., Yan, L., Du, B. (2016). A swarm intelligent method for traﬃc light scheduling: application to real urban traﬃc networks. Applied Intelligence, 44(1), 208–231. Hussain, A., Khan, M.S.A. (2020). Average operators based on spherical cubic fuzzy number and their applica- tion in multi-attribute decision making. Annals of Optimization Theory and Practice, 3(4), 83–111. 520 F. Daneshdoost et al. Jamili, A. (2016). Robust job shop scheduling problem: mathematical models, exact and heuristic algorithms. Expert Systems with Applications, 55, 341–350. Ji, M., Ge, J., Chen, K., Cheng, T.C.E. (2013). Single-machine due-window assignment and scheduling with resource allocation, aging eﬀect, and a deteriorating rate-modifying activity. Computers & Industrial Engi- neering, 66(4), 952–961. Karimi-Nasab, M., Seyedhoseini, S.M. (2013). Multi-level lot sizing and job shop scheduling with compressible process times: a cutting plane approach. European Journal of Operational Research, 231(3), 598–616. Kim, M.Y., Lee, Y.H. (2012). MIP models and hybrid algorithm for minimizing the makespan of parallel ma- chines scheduling problem with a single server. Computers & Operations Research, 39(11), 2457–2468. Kirkpatrick, S., Gelatt, C.D. Jr., Vecchi, M.P. (1983). Optimization by simulated annealing. Science, 220, 671–680. Koulamas, C., Panwalkar, S.S. (2016). The proportionate two-machine no-wait job shop scheduling problem. European Journal of Operational Research, 252(1), 131–135. Ku, W.Y., Beck, J.C. (2016). Mixed integer programming models for job shop scheduling: a computational analysis. Computers & Operations Research, 73, 165–173. Kurdi, M. (2015). A new hybrid island model genetic algorithm for job shop scheduling problem. Computers & Industrial Engineering, 88, 273–283. Lamghari, A., Dimitrakopoulos, R. (2012). A diversiﬁed Tabu search approach for the open-pit mine production scheduling problem with metal uncertainty. European Journal of Operational Research, 222(3), 642–652. Lee, W.C., Chuang, M.C., Yeh, W.C. (2012). Uniform parallel-machine scheduling to minimize makespan with position-based learning curves. Computers & Industrial Engineering, 63(4), 813–818. Leite, A., Dimitrakopoulos, R. (2014). Stochastic optimization of mine production scheduling with uncertain ore/metal/waste supply. International Journal of Mining Science and Technology, 24(6), 755–762. Li, X., Gao, L. (2016). An eﬀective hybrid genetic algorithm and tabu search for ﬂexible job shop scheduling problem. International Journal of Production Economics, 174, 93–110. Li, W., Li, J., Zhang, X., Chen, Z. (2015). Penalty cost constrained identical parallel machine scheduling problem. Theoretical Computer Science, 607(2), 181–192. Lu, C.C., Ying, K.C., Lin, S.W. (2014). Robust single machine scheduling for minimizing total ﬂow time in the presence of uncertain processing times. Computers & Industrial Engineering, 74, 102–110. Mahmoodirad, A., Niroomand, S. (2020). Uncertain location–allocation decisions for a bi-objective two-stage supply chain network design problem with environmental impacts. Expert Systems, e12558. Mahmoodirad, A., Heydari, A., Niroomand, S. (2019). An eﬀective hybrid fuzzy programming approach for an entropy-based multi-objective assembly line balancing problem. Informatica, 30(3), 503–528. Martin, S., Ouelhadj, D., Beullens, P., Ozcan, E., Juan, A.A., Burke, E.K. (2016). A multi-agent based cooper- ative approach to scheduling and routing. European Journal of Operational Research, 254(1), 169–178. Mirshekarian, S., Šormaz, D.N. (2016). Correlation of job-shop scheduling problem features with scheduling eﬃciency. Expert Systems with Applications, 62, 131–147. Misevičius, A., Kuznecovaitė, D., Platužienė, J. (2018). Some further experiments with crossover operators for genetic algorithms. Informatica, 29(3), 499–516. Mor, B., Mosheiov, G. (2014). Batch scheduling of identical jobs with controllable processing times. Computers & Operations Research, 41, 115–124. Nair, A., Ataseven, C., Habermann, M., Dreyfus, D. (2016). Toward a continuum of measurement scales in Just- in-Time (JIT) research – an examination of the predictive validity of single-item and multiple-item measures. Operations Management Research, 9(1), 35–48. Niroomand, S., Vizvari, B. (2015). Exact mathematical formulations and metaheuristic algorithms for produc- tion cost minimization: a case study of the cable industry. International Transactions in Operational Research, 22, 519–544. Niroomand, S., Hadi-Vencheh, A., Mirzaei, N., Molla-Alizadeh-Zavardehi, S. (2016). Hybrid greedy algorithms for fuzzy tardiness/earliness minimization in a special single machine scheduling problem: case study and generalisation. International Journal of Computer Integrated Manufacturing, 29(8), 870–888. Niroomand, S., Bazyar, A., Alborzi, M., Mahmoodirad, A. (2018). A hybrid approach for multi-criteria emer- gency center location problem considering existing emergency centers with interval type data: a case study. Journal of Ambient Intelligence and Humanized Computing, 9(6), 1999–2008. Niroomand, S., Mahmoodirad, A., Mosallaeipour, S. (2019). A hybrid solution approach for fuzzy multiobjec- tive dual supplier and material selection problem of carton box production systems. Expert Systems, 36(1), e12341. Tabu Search Based Hybrid Meta-Heuristic Approaches for Schedule-Based PCMP 521 Niu, S.H., Ong, S.K., Nee, A.Y.C. (2013). An improved intelligent water drops algorithm for solving multi- objective job shop scheduling. Engineering Applications of Artiﬁcial Intelligence, 26(10), 2431–2442. Parastegari, M., Hooshmand, R.A., Khodabakhshian, A., Vatanpour, M. (2013). AC constrained hydro-thermal generation scheduling problem: Application of Benders decomposition method improved by BFPSO. Inter- national Journal of Electrical Power & Energy Systems, 49, 199–212. Pasandideh, S.H.R., Akhavan Niaki, S.T., Asadi, K. (2015). Bi-objective optimization of a multi-product multi- period three-echelon supply chain problem under uncertain environments: NSGA-II and NRGA. Information Sciences, 292, 57–74. Pinedo, M.L. (2008). Scheduling: Theory, Algorithms, and Systems, 3rd ed. Springer, New York. Qu, J., Zhong, X., Li, C.L. (2016). Faster algorithms for single machine scheduling with release dates and re- jection. Information Processing Letters, 116(8), 503–507. Quintana, D., Cervantes, A., Saez, Y., Isasi, P. (2017). Clustering technique for large-scale home care crew scheduling problems. Applied Intelligence, 47(2), 443–455. Ozgur, C.O., Bai, L. (2010). Hierarchical composition heuristic for asymmetric sequence dependent single ma- chine scheduling problems. Operations Management Research, 3(1), 98–106. Rahmani, D., Heydari, M. (2014). Robust and stable ﬂow shop scheduling with unexpected arrivals of new jobs and uncertain processing times. Journal of Manufacturing Systems, 33(1), 84–92. Reisi-Nafchi, M., Moslehi, G. (2015). A hybrid genetic and linear programming algorithm for two-agent order acceptance and scheduling problem. Applied Soft Computing, 33, 37–47. Santander-Mercado, A., Jubiz-Diaz, M. (2016). The economic lot scheduling problem: a survey. International Journal of Production Research, 54(16), 4973–4992. Shi, L., Jiang, Y., Wang, L., Huang, D. (2015). A novel two-stage Lagrangian decomposition approach for re- ﬁnery production scheduling with operational transitions in mode switching. Chinese Journal of Chemical Engineering, 23(11), 1793–1800. Taassori, M., Taassori, M., Niroomand, S., Vizvári, B., Uysal, S., Hadi-Vencheh, A. (2015). OPAIC: an opti- mization technique to improve energy consumption and performance in application speciﬁc network on chips. Measurement, 74, 208–220. Tavana, M., Santos-Arteaga, F.J., Mahmoodirad, A., Niroomand, S., Sanei, M. (2018). Multi-stage supply chain network solution methods: hybrid metaheuristics and performance measurement. International Journal of Systems Science: Operations & Logistics, 5(4), 356–373. Ullah, K., Mahmood, T., Jan, N., Ahmad, Z. (2020). Policy decision making based on some averaging aggrega- tion operators of t-spherical fuzzy sets; a multi-attribute decision making approach. Annals of Optimization Theory and Practice, 3(3), 69–92. Vakhania, N., Hernandez, J.A., Werner, F. (2014). Scheduling unrelated machines with two types of jobs. Inter- national Journal of Production Research, 52(13), 3793–3801. Vizvári, B., Guden, H., Nejad M, G. (2018). Local search based meta-heuristic algorithms for optimizing the cyclic ﬂexible manufacturing cell problem. Annals of Optimization Theory and Practice, 1(3), 15–32. Wan, L., Yuan, J. (2013). Single-machine scheduling to minimize the total earliness and tardiness is strongly NP-hard. Operations Research Letters, 41, 363–365. Wolosewicz, C., Dauzère-Pérès, S., Aggoune, R. (2015). A Lagrangian heuristic for an integrated lot-sizing and ﬁxed scheduling problem. European Journal of Operational Research, 244(1), 3–12. Wu, C.C., Lee, W.C., Liou, M.J. (2013). Single-machine scheduling with two competing agents and learning consideration. Information Sciences, 251, 136–149. Xiao, J., Yang, H., Zhang, C., Zheng, L., Gupta, J.N.D. (2015). A hybrid Lagrangian-simulated annealing-based heuristic for the parallel-machine capacitated lot-sizing and scheduling problem with sequence-dependent setup times. Computers & Operations Research, 63, 72–82. Ying, K.C., Lu, C.C., Chen, J.C. (2016). Exact algorithms for single-machine scheduling problems with a vari- able maintenance. Computers & Industrial Engineering, 98, 427–433. Zhang, L., Wong, T.N. (2016). Solving integrated process planning and scheduling problem with constructive meta-heuristics. Information Sciences, 340–341, 1–16. Zheng, Z.X., Li, J.Q., Han, Y.Y. (2020). An improved invasive weed optimization algorithm for solving dynamic economic dispatch problems with valve-point eﬀects. Journal of Experimental & Theoretical Artiﬁcial Intel- ligence, 32, 805–829. https://doi.org/10.1080/0952813X.2019.1673488. Zhou, S., Liu, M., Chen, H., Li, X. (2016). An eﬀective discrete diﬀerential evolution algorithm for scheduling uniform parallel batch processing machines with non-identical capacities and arbitrary job sizes. Interna- tional Journal of Production Economics, 179, 1–11. 522 F. Daneshdoost et al. Zinder, Y., Walker, S. (2015). Algorithms for scheduling with integer preemptions on parallel machines to min- imize the maximum lateness. Discrete Applied Mathematics, 196, 28–53. F. Daneshdoost is a lecturer with MSc degree of industrial engineering in Firouzabad Institute of Higher Education, Iran. M. Hajiaghaei-Keshteli is an assistant professor of industrial engineering in University of Science and Technology of Mazandaran, Behshahr, Iran. His research interests are op- erations research and exact and meta-heuristic solution approaches. R. Sahin in an associate professor of industrial engineering in Gazi University in Turkey. He received his PhD degree in industrial engineering from Gazi University. His research interests are operations research, fuzzy theory, and exact and meta-heuristic solution ap- proaches. S. Niroomand is an associate professor of industrial engineering in Firouzabad Institute of Higher Education in Iran. He received his PhD degree in industrial engineering from Eastern Mediterranean University (in Turkey), in 2013. His research interests are opera- tions research, fuzzy theory, and exact and meta-heuristic solution approaches. 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