We consider in this paper a m-machine permutation flow shop scheduling problem and vehicle routing problem (VRP) integrated [1]. The manufacturing workshop is a flow shop and there is only one vehicle available. We are interested in the minimization of the total tardiness. related to the delivery completion times. Therefore, we are faced to three interdependent problems: scheduling the jobs in the flow shop environment, batching the completed jobs, and routing the batches. Then, we present the resolution method based on Tabu search and the results that have been obtained. Computational experiments are performed on random data sets and show the efficiency of the methods. Finally, a conclusion and some future research directions are proposed

Phuong Mai Pham

Ta, Quang Chieu

English

International Journal of Computer (IJC - Global Society of Scientific Research and Researchers, GSSRR)

   24 International Journal of Computer (IJC)  ISSN 2307-4523 (Print & Online)  http://ijcjournal.org/ Integrated Flow Shop and Vehicle Routing Problem Based on Tabu Search Algorithm Quang Chieu Taa*, Phuong Mai Phamb a,bDepartment of Artificial Intelligence, Thuyloi University, 175 Tay Son, Dong Da, Hanoi. Vietnam aEmail: quangchieu.ta@tlu.edu.vn, bEmail: maihoamieu@gmail.com    Abstract We consider in this paper a m-machine permutation flow shop scheduling problem and vehicle routing problem (VRP) integrated [1]. The manufacturing workshop is a flow shop and there is only one vehicle available. We are interested in the minimization of the total tardiness. related to the delivery completion times. Therefore, we are faced to three interdependent problems: scheduling the jobs in the flow shop environment, batching the completed jobs, and routing the batches. Then, we present the resolution method based on Tabu search and the results that have been obtained. Computational experiments are performed on random data sets and show the efficiency of the methods. Finally, a conclusion and some future research directions are proposed.  Keywords: flow shop problem; vehicle routing problem; metaheuristic algorithm; Tabu search. 1. Introduction and notations We consider that the jobs have to be delivered to the customers after their production by using a single vehicle. The processing time of each job on each machine and the due date of delivery for each job are known, and a matrix of travel times is given. The jobs have to be scheduled in a m-machine flow shop environment, then batches have to be defined (one batch corresponds to one trip of the vehicle) and a route has to be determined for each batch, so that the total tardiness of delivery is minimized. The vehicle routing problem consists in defining a route starting from the production site, visiting the customers associated to the jobs in the batch, and finishing at the production site. Each customer requiring goods is visited by the vehicle (see Figure 1).  The aim of this paper is to propose an algorithm for scheduling the jobs on the m-machines flow shop, for constituting batches of jobs and for determining the vehicle routing for each batch, so that the total tardiness of delivery is minimized. This problem is clearly an NP-hard problem [2]. ----------------------------------------------------------------------- * Corresponding author.  International Journal of Computer (IJC) (2022) Volume 43, No  1, pp 24-35 25  Figure 1: Illustration of the integrated production scheduling and vehicle routing problems. The specific notations that are used in this chapter are the following:  To each job 𝐽𝑗 is associated a customer location number j, the location of the production facility has an index 0.  The delivery time between each pair of locations i and j is denoted by 𝑙𝑖.𝑗.  The number of vehicles is equal to 1. 2. Tabu search algorithm 2.1. Coding of a solution We use an array of 3n elements to represent a complete solution. The first n elements represent the sequence of jobs (i.e. the schedule), the next 2𝑛 elements give for each trip the number of jobs in the trip and the list of jobs (the routing of these jobs is implicitely the order of the jobs in this list). In the following, the first part of the coding of a solution 𝑆 (first 𝑛 elements) is denoted by 𝑆0 and the second part (2n elements) is denoted by 𝑆𝑇. The second part is decomposed into several trips. Each trip λ of 𝑆𝑇 is denoted by 𝑆𝜆𝑇 and is composed by the number of visits 𝑘𝜆 and the list of visits (𝑆𝜆[1]𝑇 ,…, 𝑆𝜆[𝑘𝜆]𝑇 ). The coding is illustrated in Figure 2.  Figure 2: Illustration of the coding of a solution and notations. International Journal of Computer (IJC) (2022) Volume 43, No  1, pp 24-35 26 Example: In the example presented in Figure 3, the schedule is {J1, J5, J2, J6, J4, J3, J7}, the number jobs in each trip is (2, 2, 3, 0, 0, 0, 0), i.e. two jobs in the first two trips and three jobs in the third trip, the remaining trips are empty.  Figure 3: Example of the coding of a solution. 2.2. Initial solution EDD algorithm is used for giving an initial solution for the scheduling problem. For the trips, each trip contains only one job. The customers are visited in the same order as in the sequence. Example: An example of an initial solution is given in Figure 4.  Figure 4: Example of initial solution. 2.3. Neighborhood definitions We assume that S = 𝑆0//𝑆𝑇 is the current complete solution with sequence 𝑆0 and trips 𝑆𝑇, where: International Journal of Computer (IJC) (2022) Volume 43, No  1, pp 24-35 27  𝑆0 =  𝑆10/𝑆[𝑖]0 /𝑆20/𝑆[𝑗]0 /𝑆30 with 𝑆[𝑖]0  and 𝑆[𝑗]0  the jobs in positions 𝑖 and 𝑗 (i < j) in 𝑆0 and 𝑆10, 𝑆20 and 𝑆30 three partial sequences.  𝑆𝑇  =  𝑘1, 𝑆1𝑇/ …/𝑘𝜆−1, 𝑆𝜆−1𝑇 /𝑘𝜆, 𝑆𝜆𝑇/𝑘𝜆 + 1, 𝑆𝜆+1𝑇 /𝑘𝑛 , 𝑆𝑛𝑇  with 𝑘𝜆  the number of jobs in trip λ. 𝑆𝜆𝑇  is the sequence of jobs in trip λ, ∀ λ ∈ {1, 2, …, n}. If 𝑘𝜆 = 0 then 𝑆𝜆𝑇 is empty. Notice also that ∑ 𝑘𝜆𝑛𝜆=1 = 𝑛. Neighborhood based on the sequence of jobs We use SWAP0 operator for creating the neighbors of sequence S0. This operator is the same as the one described [3].  SWAP0: A neighbor of S is created by interchanging the jobs in position i and j in sequence S0. The corresponding jobs are also swapped in ST. See Figure 5 for an illustration of this operator.  Figure 5: Illustration of SWAP0 operator. Neighborhood based on a single trip We use SWAPT, EBSRT, ESFRT and InversionT operators for creating the neighbors of a trip 𝑆λ𝑇. These operators are the same as those described [3,4]. Given a sequence of visits 𝑆λ𝑇 of trip λ, two random positions i and j in 𝑆λ𝑇 (𝑖 <  𝑗 and 𝑗 ≤ 𝑘𝜆): 𝑆λ𝑇 = 𝑆λ1𝑇 , 𝑆𝜆[𝑖]𝑇 , 𝑆λ2𝑇 , 𝑆𝜆[𝑗]𝑇 , 𝑆λ3𝑇 .  SWAPT: A neighbor of 𝑆λ𝑇 is created by interchanging the jobs in position i and j, leading to sequence 𝑆λ𝑇′ = 𝑆λ1𝑇 , 𝑆𝜆[𝑗]𝑇 , 𝑆λ2𝑇 , 𝑆𝜆[𝑖]𝑇 , 𝑆λ3𝑇 , See Figure 6 for example.  Figure 6: Illustration of SWAPT operator for a trip. International Journal of Computer (IJC) (2022) Volume 43, No  1, pp 24-35 28  EBSRT: A neighbor of 𝑆𝜆𝑇 is created by extracting the visit in position j and reinserting this visit backward just before the visit in position i, leading to sequence 𝑆𝜆𝑇′ = 𝑆𝜆1𝑇 , 𝑆𝜆[𝑗]𝑇 , 𝑆𝜆[𝑖]𝑇 ,  𝑆𝜆2𝑇 , 𝑆𝜆3𝑇 , See Figure 7 for example.  Figure 7: Illustration of EBSRT operator for a trip.  ESFRT: A neighbor of 𝑆𝜆𝑇 is created by extracting the visit in position i and reinserting it forward immediately after the visit in position j, leading to a sequence 𝑆𝜆𝑇′ = 𝑆𝜆1𝑇 , 𝑆𝜆2𝑇 , 𝑆𝜆[𝑗]𝑇 , 𝑆𝜆[𝑖]𝑇 , 𝑆𝜆3𝑇 . See Figure 8 for example.  Figure 8: Illustration of ESFRT operator for a trip.  InversionT: A neighbor of 𝑆𝜆𝑇 is created by inserting 𝑆𝜆[𝑖]𝑇 , 𝑆𝜆2𝑇 , 𝑆𝜆[𝑗]𝑇  in the inverse order between 𝑆𝜆1𝑇  and 𝑆𝜆3𝑇 . See Figure 9 for example.  Figure 9: Illustration of InversionT operator for a trip. Moves and selection of the best neighbor The aim of the problem is to minimize the total tardiness. The best neighbor in the candidate item that is non-tabu and with the smallest total tardiness. For managing the tabu list, we use the first-in-first-out (FIFO) International Journal of Computer (IJC) (2022) Volume 43, No  1, pp 24-35 29 strategy. Old attributes are deleted as new attributes are inserted. Tabu list We define three tabu lists and the size of the tabu lists is fixed. An element of the tabu list is defined by (λ, µ, Ji, Jj), λ, µ are the indexes of two trips, Ji and Jj are two jobs:  For a swap of two jobs Ji and Jj in the sequence, we insert in the Tabu list the element (0, 0, Ji, Jj). For example, see Error! Reference source not found., the element inserted in the Tabu list is (0, 0, J5, J2). In this case λ = µ = 0.  For a move in two trips λ and µ, we insert in the Tabu list the element (λ, µ, Ji, Jj) where i and j are the positions that are concerned in trips λ and µ respectively. For a move in a single trip, we have λ = µ. For example, see Error! Reference source not found., where the tabu list is (1, 1, J1, J5). Stopping condition The algorithm is stopped when the time limit has been reached. This time limit is denoted by TimeLimitTS = n(m/2)t ms [5], where t = 90. Detailed algorithm The detailed TS algorithm is given in Algorithm 1  FlagSwapo allow to make a selection of the neighbor operator of sequence.   LimitSwapo allow to limit the size of the neighborhood in sequence.   FlagOperaT allow to make a selection of the neighbor operators in sequence of a trip (FlagOperaT ∈ {FlagSwapT, FlagEBSRT, FlagEF SRT, FlagInversionT}).  LimitOperaT allow to limit the size of the neighborhood in a trip (LimitOperaT ∈ {LimitSwapT, LimitEBSRT, LimitEFSRT, LimitInversionT}).   FlagOperaT2 allow to make a selection of the neighbor operators in sequence of two different trips (FlagOperaT 2 ∈ {FlagSwapT2, FlagEBSRT2, FlagEFSRT2}).   Del(T) deletes the upper element of TS   Add(T,(λ, µ, Ji, Jj)) adds element (T,(λ, µ, Ji, Jj)) to T (tabu list), ∀λ, µ ∈ {0, 1, 2, ...}. In Algo. 15, the Test(SWAPo), Test(OperaT), Test(OperaT2) are described in Algorithm 2, 3, 4: 1: Initialization 2: S0 = initial solution, S = current solution 3: S’ = S0 // best solution of N(S) 4: S* = S0 // best solution of N(S) and non-tabu International Journal of Computer (IJC) (2022) Volume 43, No  1, pp 24-35 30 5: f* = f(S0) // f∗ value of S* and f(S0) value of S0 6: T = ∅ // T is the tabu list 7: while (CPU ≤ TimeLimitTS) do 8:  f(S0) = ∞, 9:  //Selecting neighbor in sequence 10:  for i = 0 to n − 1 do 11:   for j = i + 1 to n do 12:    Test(SWAPo) 13:   end for 14:  end for 15:  //Selecting neighbor of a trip 16:  for λ = 0 to n − 1 do 17:   for i = 0 to nbjoboftrip[λ] − 2 do 18:   //nbjobof trip[λ] is the number jobs of trip λ 19:    for j = i + 1 to nbjoboftrip[λ] − 1 do 20:     Test(OperaT) 21:    end for 22:   end for 23:  end for 24:  //Selecting neighbor in two trips 25:  for λ = 0 to n − 2 do 26:   for µ = 0 to n − 1 do International Journal of Computer (IJC) (2022) Volume 43, No  1, pp 24-35 31 27:    for i = 0 to nbjoboftrip[λ] − 1 do 28:    //nbjobof trip[λ], nbjoboftrip[µ] are the number jobs of trip λ, µ 29:     for j = 0 to nbjoboftrip[µ] − 1 do 30:      Test(OperaT2) 31:     end for 32:    end for 33:   end for 34:  end for 35:  if (f(S0) < f*) then S* = S0, f* = f(S), end if 36:  if (SizeTabu ≥ TabuMax) then Del(T) end if 37:  Add(T,(λ, µ, i, j)) 38: end while Algorithm 1. Tabu search algorithm for scheduling and vehicle routing 1: if (FlagSwapo = 1) and (j − i ≤ LimitSwapo) then  2:  S = S0, f(S) = f(S0), SWAPo (S,(0, 0, i, j)),  3:  if ((0, 0, i, j)  T) then  4:   Calculate(f(S)),  5:   if (f(S) < f(S0)) then S0 = S, f(S0) = f(S), move = (0, 0, i, j), end if  6:  end if  7: end if Algorithm 2. Test(SWAPo) 1: if (FlagOperaT = 1) and (j − i ≤ LimitOperaT) then International Journal of Computer (IJC) (2022) Volume 43, No  1, pp 24-35 32 2:  S = S0, f(S) = f(S0), OperaT(S,(λ, λ, i, j)), 3:  if ((λ, λ, i, j)  T) then 4:   Calculate(f(S)), 5:   if (f(S) < f(S0)) then S0 = S, f(S0) = f(S), move = (λ, λ, i, j), end if 6:  end if 7: end if Algorithm 3. Test(OperaT) 1: if (FlagOperaT2 = 1) then 2:  S = S0, f(S) = f(S0), OperaT2(S,(λ, µ, i, j)), 3:  if ((λ, µ, i, j)  T) then 4:   Calculate(f(S)), 5:   if (f(S) < f(S0)) then S0 = S, f(S0) = f(S), move = (λ, µ, i, j)), end if 6:  end if 7: end if Algorithm 4. Test(OperaT2) 3. Computational experiments We present in this section the generation of data and we discuss the results. Generation data We have tested the algorithms on a PC Intel coreTMi5 CPU 2.4GHz. Data sets have been randomly generated (notice that there is no benchmark instance for the m-machine flow shop and vehicle routing problem integrated). The processing times pi,j have been generated in [1,100], the due dates dj have been generated in [50, 50n], the position of “custom” j is given by its coordinates (xj, yj) generated in [1, 70] (see Figure 10). The delivery time li,j is the classical Euclidian distance 𝑙𝑖,𝑗 = √(𝑥𝑗 − 𝑥𝑖)2 + (𝑦𝑗 − 𝑦𝑖)2. Ten instances are used for each combination of n and m, with n  {20, 30, 50, 70, 100, 150, 200} and m  {2, 4}. For the TS algorithm, some preliminary experiments have conducted to the following parameters settings: TimeLimitTS = 10 seconds. International Journal of Computer (IJC) (2022) Volume 43, No  1, pp 24-35 33 Tabu list = {10, 40, 60,120} elements.  Figure 10: Illustration of calculation of li,j. 4. Results In Error! Reference source not found., column ‘Best’ for TSX (X  {10, 40, 60, 120}) indicates the number of times this method strictly outperforms other method. Column ‘∆TSX’ indicates the average deviation between TSX and the best method between TS10, TS40, TS60 and TS120. ∆𝑇𝑆𝑋 =𝑇𝑆𝑋 − min (𝑇𝑆10, 𝑇𝑆40, 𝑇𝑆60, 𝑇𝑆120)𝑇𝑆𝑋 Table 1: Tabu search results. n x m TS10 TS40 TS60 TS120 Best ∆TS10 Best ∆TS40 Best ∆TS60 Best ∆TS120 20 × 2  30 × 2  50 × 2  70 × 2  100 × 2  150 × 2  200 × 2  20 × 4  30 × 4  50 × 4  70 × 4  100 × 4  150 × 4  200 × 4  3  1  0  0  1  0  0  0  0  0  1  0  0  0 17.58% 11.62% 17.40% 16.12% 19.24% 23.42% 26.35% 9.51% 9.28% 8.67% 7.34% 17.60% 19.08% 18.52% 7  7  1  0  1   0  4  9  10  5  2  3  2  3 4.16% 13.34% 21.81% 21.88% 25.61% 15.27% 5.60% 0.00% 0.00% 3.19% 9.54% 9.84% 2.84% 2.60% 3  3  2  2  0  4  3  1  0  2  3  2  4  5 17.58% 10.97% 12.49% 10.99% 14.93% 5.29% 5.67% 9.51% 9.28% 3.27% 5.38% 9.23% 3.08% 4.16% 0  0  7  8  8  6  3  0  0  1  5  5  4  2 24.81% 30.19% 3.23% 2.50% 1.65% 5.95% 8.19% 12.04% 17.63% 4.60% 4.47% 4.75% 4.10% 10.73%  6 14.98% 54 10.46% 34 8.49% 49 9.94% We can see that the TS40 (with tabu list is 40) leads to the best results with a number of best solutions equal to 54. On average, the deviation between the solutions returned by this method and the best solutions is 10.46%. This value is around and 14.98% for TS10, 10.46% for TS40, 8.49% for TS60 and 9.94% for TS120. In Table 2, column ‘Best TS < EDD’ indicates the number of times the method TS outperforms method EDD, International Journal of Computer (IJC) (2022) Volume 43, No  1, pp 24-35 34 column Cpu(s) indicates the average computation time of TS per 10 instances. Column ‘∆’ indicates the average deviation between EDD and TS.  ∆=𝐸𝐷𝐷 − 𝑇𝑆𝐸𝐷𝐷 Table 2 : Comparison of the TS40 and EDD algorithm. n × m Best TS<EDD Cpu(s) ∆ 20 × 2 30 × 2 50 × 2 70 × 2 100 × 2 150 × 2 200 × 2 20 × 4 30 × 2 50 × 4 70 × 4 100 × 4 150 × 4 200 × 4 10 10 10 10 10 10 10 10 10 10 10 10 10 10 10.00 10.00 10.00 10.01 10.01 10.03 10.06 10.00 10.00 10.00 10.01 10.01 10.03 10.03 72.0% 79.7% 88.3% 85.2% 91.4% 90.0% 81.3% 55.9% 69.2% 75.9% 71.4% 83.7% 77.7% 70.4% As we can see in Table 2, TS improves significantly the initial solution given by EDD, with 78,0% of improvement in average. 5. Conclusions and discussions We approach a problem where a m-machine permutation flow shop scheduling problem and a vehicle routing problem are integrated to minimize the total tardiness. To our knowledge, this is the first time that this problem is approached in the literature. We present a direct coding for a complete solution and a neighborhood method for finding a sequence and trips. We propose a tabu search algorithm for this problem, the first results show that the TS greatly improves the initial solution given by EDD and where each trip serves only one job at a time.  In the future, the first research directions are about the evaluation of the method. We will develop genetic algorithms and other methods in order to see the real quality of the tabu search. Then, we will combine mathematical programming and local search (matheuristic), in order to see if matheuristic are performing methods for this problem. Acknowledgements The authors would like to thank to Professor Jean-Charles Billaut of University of Tours, France who provided insight and expertise that greatly assisted the research.  International Journal of Computer (IJC) (2022) Volume 43, No  1, pp 24-35 35 References  [1] Q. C. Ta, J. C. Billaut, J. L. Bouquard and P-E. Morin. “Minimisation de la somme des retards pour un problème d’ordonnancement de type flow-shop à deux machines et un problème de livraison intégrés.” 15ème congrès de la Société de la Société Française de Recherche Opérationnelle et d'Aide à la Décision Roadef’, 2014. Bordeaux, France. [2] J. K. Lenstra and A. H. G. Rinnooy Kan. “Complexity of vehicle routing and scheduling problems”. Networks, vol 11, pp. 221-227, 1981. [3] Q. C. Ta, J. C. Billaut and J. L. Bouquard. “An hybrid metaheuristic. a hybrid lower bound and a tabu search for the two-machine flowshop total tardiness problem.” Proceedings of the 10th IEEE RIVF International Conference on Computing and Communication Technologies (RIVF’13), 2013, Hanoi, Vietnam. [4] Q. C. Ta, J. C. Billaut and J. L. Bouquard. “Minimizing total tardiness in the m-machine flow-shop by genetic and matheuristic algorithms”. Journal of Intelligent Manufacturing, vol 29, pp. 617- 628, 2015. [5] E. Vallada, R. Ruiz and G. Minella. “Minimising total tardiness in the m- machine flowshop problem: A review and evaluation of heuristics and metaheuristics”. Computers & Operations Research, vol 35, pp. 1350-1373, 2008.   

Integrated Flow Shop and Vehicle Routing Problem Based on Tabu Search Algorithm

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Integrated Flow Shop and Vehicle Routing Problem Based on Tabu Search Algorithm

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