# A Novel Constraint Programming Decomposition Approach for the Total Flow Time Fixed Group Shop Scheduling Problem

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## Abstract

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## 1. Introduction

## 2. Literature Review

## 3. The Fixed Group Shop Scheduling Problem

#### 3.1. Problem Definition

#### 3.2. An Illustrative Example

## 4. FGSSP Formulations and Lower Bounds

#### 4.1. MILP Formulation

- $nbJobs$: Number of jobs.
- $nbMchs$: Number of machines.
- $nbStgs$: Number of stages.
- j, k: Indices for jobs, $\{1,\dots ,nbJobs\}$.
- i, l: Indices for machines, $\{1,\dots ,nbMchs\}$.
- s: Index for the stages, $\{1,\dots ,nbStgs\}$.
- ${o}_{ji}$: Operation associated to job j at machine i.
- ${p}_{ji}$: Processing time of operation ${o}_{ji}$.
- ${S}_{jik}$: Setup time of job j if it is performed immediately after job k on machine i ($j\ne k$).
- ${B}_{is}$: 1 if machine i belongs to stage s; and 0 otherwise.
- ${A}_{li}$: 1 if machine l belongs to the stage immediately before the stage to which machine i belongs; and 0 otherwise ($i\ne l$).
- M: A sufficiently large number.

- ${C}_{j}$: Continuous variable that takes the value of the completion time of job j.
- ${C}_{ji}$: Continuous variable that takes the value of the completion time of job j at machine i.
- ${f}_{js}$: Continuous variable that takes the value of the completion time of job j at stage s.
- ${x}_{jil}$: Binary variable that takes value equal to 1 if operation ${o}_{ji}$ is performed after operation ${o}_{jl}$; or 0 in any other case.
- ${y}_{jik}$: Binary variable that takes value equal to 1 if operation ${o}_{ji}$ is performed after operation ${o}_{ki}$; or 0 in any other case.

#### 4.2. CP Formulation

- $nbJobs$: Number of jobs.
- $nbMchs$: Number of machines.
- j, k: Indices for the jobs, $\{1,\dots ,nbJobs\}$.
- i, l: Indices for the machines, $\{1,\dots ,nbMchs\}$.
- ${o}_{ji}$: Operation associated to job j at machine i.
- ${p}_{ji}$: Processing time of operation ${o}_{ji}$.
- ${A}_{li}$: 1 if machine l belongs to the stage immediately before the stage to which machine i belongs; and 0 otherwise ($i\ne l$).
- ${T}_{i}$: A transition matrix that reports the minimum delay required by any pair of jobs j, k, to perform in machine i. The transition matrix values equal ${S}_{jik}$.

- $itv{s}_{ji}$: Interval variables that define the start and the end of the operation of job j at machine i. The interval variable ensures that the difference between the start and the end value equals the processing time ${p}_{ji}$.
- $job{s}_{j}$: Sequence of interval variables $itv{s}_{ji}$ associated to the operations of job j.
- $mch{s}_{i}$: Sequence of interval variables $itv{s}_{ji}$ associated to operations performed in machine i.

#### 4.3. Lower Bounds

#### 4.3.1. Lower Bound $L{B}_{1}$

#### 4.3.2. Lower Bounds $L{B}_{2}$ and $L{B}_{3}$

## 5. Proposed Solution Method

Algorithm 1: Outline of the DEC procedure. |

#### 5.1. Initial Solution

#### 5.2. Neighborhood Exploration

#### 5.3. Shake Procedure

## 6. Computational Experiments

#### 6.1. Instance Generation

#### 6.2. Results for Small Size Instances

- If we consider the behavior of the exact methods, i.e., the CP variants as well as the MILP, the results show that each of these methods have difficulties even for moderately small instances with 10 jobs and 10 machines. In fact, we do not report the number of optimal solutions found by any of these methods because they fail to verify optimality even for instances with 7 jobs and 7 machines and up. Note that these methods solve all instances with 4 or 5 jobs and machines to optimality, but the combined effort of all the exact methods only verifies optimality for four additional instances.
- Among the different search strategies available in the CP solver, all methods perform similarly except for DF. If we consider this result together with the difficulty of each exact method to verify optimality, we are led to believe that a depth-first search approach as conducted by the DF strategy fails to backtrack to the initial stages of the problem, leading to suboptimal early decision never being reconsidered.
- When we compare the CP approaches and the MILP approach, the CP outperforms the MILP method in every instance group and metric (either number of best found or relative gap to best known). Moreover, the additional time allocated to the MILP does not result in better solutions and the CP approaches, except for the DF strategy, outperform the MILP. Specifically, for instances with 10 jobs and machines, the MILP fails to find solutions of the quality provided by the CP approaches. Consequently, we recommend the use of a CP strategy for the problem and avoid the use of the MILP approach in larger instances.
- The CP methods do not perform as well on instances with sequence-dependent setup times. Specifically, relative gaps increase and two search strategies, i.e., Auto and RS, tend to provide the best solutions among the five search methods. This result may be attributed to shortcomings of the CP approach that makes use of internal components within its search procedure that are more efficient in problems with fewer features to consider.
- The performance of the DEC approach using a CP solver to tackle the subproblems for small and medium instances is similar to the exact CP methods. The same does not hold true for the DEC method using the MILP solver, as their results are inferior to either the CP or the DEC method using CP.While the DEC method finds better solutions than the CP methods, specially on instances with fewer stages and the relative gaps are small, it does not outperform the exact methods for these instances. Please note that for small instances, the exact method benefits from considering the problem as a whole, unlike our method that tackles smaller parts of the complete problem. For small sized instances, dividing the problem into part leads to disadvantages in terms of the ability of the method to optimize all stages simultaneously.The similarity between the results of both methods was statistically checked using a paired t-test for statistical significance. The paired t-test compares the best solution found by any CP method with the best found among the ten replicates of the DEC method using the CP solver, as well as with each of its individual runs.The tests between the best solutions show that the results are not statistically different, with a p-value of 0.204 for the instances without sequence dependent setup times, and a p-value of 0.981 for the case with setup times. Note that Anderson–Darling tests show that the differences among values are not normally distributed, and thus we conduct Wilcoxon signed-rank non-parametric tests that confirm the results from the parametric tests. With regards to the statistical test between individual run of the DEC method when compared to the CP method, similar results are found. For the cases without sequence dependent setup times, six report statistical differences for the parametric test, but after a Bonferroni correction is run to account for multiple comparisons, none of the p-values suffice to point to statistically significant differences. For instances with setup times, none of the replicates report statistically significant differences to the best CP solutions.

#### 6.3. Results for Medium and Large Size Instances

#### 6.4. An Industrial Case Study

## 7. Conclusions and Future Work

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## References

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**Figure 1.**Graphical representations of a solution to the example instance provided in Table 1. (

**a**) Gantt chart representation of an arbitrary feasible solution for the FGSSP instance presented in Table 1 with ${\sum}_{j}{C}_{j}=60$. (

**b**) Disjunctive graph representation of an arbitrary feasible solution for the FGSSP instance presented in Table 1 with ${\sum}_{j}{C}_{j}=60$.

**Figure 2.**Graphical representation of the constructive heuristic solution of the DEC method for the example instance provided in Table 1. (

**a**) Gantt chart representation of the solution provided by the constructive heuristic for the FGSSP instance presented in Table 1. (

**b**) Disjunctive graph representation of the solution provided by the constructive heuristic for the FGSSP instance given in Table 1.

**Figure 3.**Graphical representation of the neighborhood exploration phase of the DEC method for the example instance provided in Table 1. (

**a**) Disjunctive graph representation of the problem associated to the stage 2. The arcs represent the fixed decisions (i.e., the decisions from stage 1 and 3). (

**b**) Gantt chart representation of the solution after solving the problem of stage 2. The solution improves the problem by rearranging the order of operations of the stage.

**Figure 4.**Graphical representation of the incumbent solution of the DEC method after the shake is performed for the example instance provided in Table 1. (

**a**) Disjunctive graph representation of the incumbent solution after the shake. (

**b**) Gantt chart representation of the incumbent solution after the shake.

**Table 1.**Processing times of the small-size instance. For every job, ${J}_{1}$, ${J}_{2}$ and ${J}_{3}$, and machine, ${M}_{1},\dots ,{M}_{7}$, the processing time is provided. Additionally, machines are grouped according to their stage, ${S}_{1}$, ${S}_{2}$ and ${S}_{3}$.

Stages | ${\mathit{S}}_{1}$ | ${\mathit{S}}_{2}$ | ${\mathit{S}}_{3}$ | ||||
---|---|---|---|---|---|---|---|

${\mathit{M}}_{\mathbf{1}}$ | ${\mathit{M}}_{\mathbf{2}}$ | ${\mathit{M}}_{\mathbf{3}}$ | ${\mathit{M}}_{\mathbf{4}}$ | ${\mathit{M}}_{\mathbf{5}}$ | ${\mathit{M}}_{\mathbf{6}}$ | ${\mathit{M}}_{\mathbf{7}}$ | |

${J}_{1}$ | 1 | 3 | 4 | 1 | 1 | 9 | 1 |

${J}_{2}$ | 2 | 1 | 3 | 2 | 2 | 1 | 1 |

${J}_{3}$ | 3 | 2 | 1 | 2 | 4 | 2 | 3 |

**Table 2.**Results for small instances without sequence-dependent setup times (problem $FGSS{P}_{s}\mid \mid {\sum}_{j}{C}_{j}$). For each instance size (represented by the number of jobs, column $nbJobs$, machines, column $nbMchs$, and stages, column $nbStgs$, we report the average gap to the best known solution and the number of best known solutions (in parentheses) provided by each CP search strategy (columns Auto, DF, RS, MP and ID), the best solution provided by all combined CP approaches (column CP), the results from the MILP approach (column MILP) and the best and the average found among 10 independent runs of the DEC approach (columns, (best) and (av.) respectively) using both the MILP and the CP solvers as their underlying methods to tackle the subproblems required by the approach (columns DEC MILP and DEC CP). The results of the best performing method for each group of instances are highlighted in boldface.

DEC MILP | DEC CP | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{nbJobs}$ | $\mathit{nbMchs}$ | $\mathit{nbStgs}$ | Auto | DF | RS | MP | ID | CP | MILP | (av.) | (Best) | (av.) | (Best) |

4 | 4 | 2 | 0.0 (10) | 0.0 (10) | 0.0 (10) | 0.0 (10) | 0.0 (10) | 0.0 (10) | 0.0
(10) | 0.0 (10) | 0.0 (10) | 0.0 (10) | 0.0 (10) |

5 | 5 | 2 | 0.0 (10) | 1.2 (0) | 0.0 (10) | 0.4 (4) | 0.4 (5) | 0.0 (10) | 0.0 (10) | 0.0 (10) | 0.0 (10) | 0.0 (10) | 0.0
(10) |

7 | 7 | 2 | 0.3 (4) | 10.3 (0) | 0.8 (2) | 2.4 (0) | 3.7 (0) | 0.1 (6) | 2.1 (0) | 2.7 (0) | 2.3 (0) | 0.4 (3) | 0.3 (4) |

3 | 0.1 (8) | 13.7 (0) | 0.3 (7) | 1.6 (0) | 2.5 (1) | 0.0 (10) | 0.5 (5) | 2.4 (0) | 1.9 (1) | 2.6 (0) | 2.5 (0) | ||

10 | 10 | 2 | 2.2 (1) | 14.7 (0) | 1.7 (0) | 3.7 (0) | 3.7 (0) | 1.3 (1) | 8.9 (0) | 11.1 (0) | 10.4 (0) | 0.5 (3) | 0.1 (9) |

3 | 1.0 (2) | 15.8 (0) | 1.2 (2) | 2.9 (1) | 3.1 (0) | 0.4
(5) | 6.2 (0) | 7.4 (0) | 7.4 (0) | 0.6 (1) | 0.5 (5) | ||

4 | 1.0 (2) | 15.7 (0) | 1.1 (1) | 1.6 (2) | 2.5 (1) | 0.4
(6) | 3.4 (1) | 4.9 (0) | 2.9 (2) | 1.9 (0) | 1.4 (4) |

**Table 3.**Results for small instances with sequence-dependent setup times (problem $FGSS{P}_{s}\mid {S}_{jik}\mid {\sum}_{j}{C}_{j}$). For each instance size (represented by the number of jobs, column $nbJobs$, machines, column $nbMchs$, and stages, column $nbStgs$, we report the average gap to the best-known solution (in parentheses) provided by each CP search strategy (columns Auto, DF, RS, MP and ID), the best solution provided by all combined CP approaches (column CP), the results from the MILP approach (column MILP) and the best and the average found among 10 independent runs of the DEC approach (columns, (best) and (av.) respectively) using both the MILP and the CP solvers as their underlying methods to tackle the subproblems required by the approach (columns DEC MILP and DEC CP). The results of the best performing method for each group of instances are highlighted in boldface.

DEC MILP | DEC CP | ||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|

$\mathit{nbJobs}$ | $\mathit{nbMchs}$ | $\mathit{nbStgs}$ | Auto | DF | RS | MP | ID | CP | MILP | (av.) | (Best) | (av.) | (Best) |

4 | 4 | 2 | 0.0 (10) | 0.0 (10) | 0.0 (10) | 0.0 (10) | 0.0 (10) | 0.0 (10) | 0.0
(10) | 0.0 (10) | 0.0 (10) | 0.0 (10) | 0.0 (10) |

5 | 5 | 2 | 0.0 (10) | 4.1 (0) | 0.0 (10) | 0.5 (4) | 2.2 (1) | 0.0 (10) | 0.0 (10) | 0.0 (10) | 0.0 (10) | 0.0 (10) | 0.0 (10) |

7 | 7 | 2 | 1.4 (1) | 15.7 (0) | 1.4 (1) | 2.3 (1) | 4.8 (0) | 0.7 (3) | 1.4 (2) | 2.6 (0) | 2.0 (0) | 0.3 (3) | 0.2 (5) |

3 | 0.7 (6) | 17.3 (0) | 0.8 (3) | 1.9 (2) | 4.2 (0) | 0.1
(8) | 0.6 (5) | 3.1 (0) | 2.3 (0) | 2.4 (1) | 2.3 (1) | ||

10 | 10 | 2 | 4.2 (0) | 22.0 (0) | 4.3 (0) | 5.7 (0) | 6.0 (0) | 2.8 (0) | 9.2 (0) | 13.1 (0) | 11.3 (0) | 0.5 (3) | 0.0 (10) |

3 | 2.2 (3) | 23.6 (0) | 1.4 (4) | 3.9 (1) | 5.5 (0) | 0.4
(8) | 5.4 (0) | 7.4 (0) | 6.1 (0) | 1.5 (0) | 1.2 (2) | ||

4 | 1.7 (3) | 22.3 (0) | 1.6 (3) | 3.9 (0) | 3.6 (1) | 0.3 (7) | 4.0 (1) | 6.1 (0) | 4.2 (0) | 1.8 (0) | 1.3 (2) |

**Table 4.**Results for the lower bounds for small-size instances without sequence-dependent setup times (problem $FGSS{P}_{s}\mid \mid {\sum}_{j}{C}_{j}$). For each combination of instance size (represented by the number of jobs, column $nbJobs$, machines, column $nbMchs$, and stages, column $nbStgs$, and solution method (columns $l{b}_{1}$, $l{b}_{2}$, $l{b}_{3}$, CP and MILP), we report the optimality gap and, in parentheses, the number of instances (out of 10) in which the method reported the best solution. The results of the best performing method for each group of instances are highlighted in boldface.

$\mathit{nbJobs}$ | $\mathit{nbMchs}$ | $\mathit{nbStgs}$ | ${\mathit{lb}}_{1}$ | ${\mathit{lb}}_{2}$ | ${\mathit{lb}}_{3}$ | CP | MILP |
---|---|---|---|---|---|---|---|

4 | 4 | 2 | 21.5 (0) | 3.5 (0) | 5.0 (0) | 0.0 (10) | 0.0
(10) |

5 | 5 | 2 | 21.4 (0) | 3.6 (0) | 4.9 (0) | 0.0 (10) | 0.0
(10) |

7 | 7 | 2 | 20.7 (0) | 15.7 (2) | 12.7 (4) | 35.4 (0) | 12.7
(4) |

3 | 24.2 (0) | 16.7 (0) | 8.9 (0) | 21.1 (2) | 1.6 (10) | ||

10 | 10 | 2 | 22.1 (0) | 17.9 (10) | 19.5 (0) | 46.2 (0) | 40.4 (0) |

3 | 26.1 (0) | 19.3 (10) | 22.4 (0) | 35.5 (0) | 29.3 (0) | ||

4 | 28.9 (0) | 16.4 (6) | 19.7 (3) | 32.0 (0) | 22.6 (1) |

**Table 5.**Results for the lower bounds for small-size instances with sequence-dependent setup times (problem $FGSS{P}_{s}\mid {S}_{jik}\mid {\sum}_{j}{C}_{j}$). For each combination of instance size (represented by the number of jobs, column $nbJobs$, machines, column $nbMchs$, and stages, column $nbStgs$, and solution method (columns $l{b}_{1}$, $l{b}_{2}$, $l{b}_{3}$, CP and MILP), we report the optimality gap and, in parentheses, the number of instances (out of 10) in which the method reported the best solution. The results of the best performing method for each group of instances are highlighted in boldface.

$\mathit{nbJobs}$ | $\mathit{nbMchs}$ | $\mathit{nbStgs}$ | ${\mathit{lb}}_{1}$ | ${\mathit{lb}}_{2}$ | ${\mathit{lb}}_{3}$ | CP | MILP |
---|---|---|---|---|---|---|---|

4 | 4 | 2 | 28.6 (0) | 4.7 (0) | 7.2 (0) | 0.0 (10) | 0.0
(10) |

5 | 5 | 2 | 28.6 (0) | 5.1 (0) | 5.3 (0) | 0.0 (10) | 0.0
(10) |

7 | 7 | 2 | 28.2 (0) | 22.7 (0) | 20.5 (2) | 41.0 (0) | 13.9 (8) |

3 | 31.4 (0) | 24.7 (0) | 10.9 (0) | 34.2 (0) | 3.2 (10) | ||

10 | 10 | 2 | 30.6 (0) | 26.2 (10) | 27.6 (0) | 51.8 (0) | 44.4 (0) |

3 | 34.5 (0) | 28.3 (9) | 30.8 (0) | 42.7 (0) | 33.9 (1) | ||

4 | 36.9 (0) | 25.7 (8) | 30.8 (0) | 39.4 (0) | 27.6 (2) |

**Table 6.**Results for medium and large instances without sequence-dependent setup times (problem $FGSS{P}_{s}\mid \mid {\sum}_{j}{C}_{j}$). For each instance size (represented by the number of jobs, column $nbJobs$, machines, column $nbMchs$, and stages, column $nbStgs$, we report the average gap to the best solution and the number of instances where the best known solution was found (in parentheses) by the best CP search strategy (column Auto), the best solution provided among the CP approaches (column CP), the results from the MILP approach (column MILP) and the best and the average found among 10 independent runs of the DEC approach (columns, DEC (best) and DEC (av.) respectively). The results of the best performing method for each group of instances are highlighted in boldface.

$\mathit{nbJobs}$ | $\mathit{nbMchs}$ | $\mathit{nbStgs}$ | Auto | CP | DEC (av.) | DEC (Best) |
---|---|---|---|---|---|---|

15 | 15 | 2 | 2.9 (0) | 2.3 (0) | 0.3 (0) | 0.0 (10) |

3 | 1.3 (2) | 1.0 (2) | 0.5 (3) | 0.2 (8) | ||

4 | 0.3 (5) | 0.3 (7) | 1.4 (0) | 1.1 (3) | ||

5 | 0.7 (4) | 0.4 (5) | 1.3 (1) | 1.0 (5) | ||

20 | 20 | 2 | 3.8 (0) | 3.1 (0) | 0.2 (3) | 0.0 (10) |

3 | 2.5 (0) | 1.9 (0) | 0.3 (0) | 0.0 (10) | ||

4 | 1.9 (2) | 1.7 (2) | 0.7 (0) | 0.1 (8) | ||

5 | 1.3 (2) | 1.1 (2) | 0.8 (0) | 0.2 (8) | ||

6 | 1.2 (3) | 0.8 (4) | 1.2 (0) | 0.6 (6) | ||

7 | 1.0 (2) | 0.4 (6) | 1.4 (0) | 1.0 (4) | ||

30 | 30 | 2 | 5.7 (0) | 4.5 (0) | 0.0 (5) | 0.0 (10) |

3 | 3.0 (0) | 2.9 (0) | 0.1 (5) | 0.0 (10) | ||

4 | 3.5 (0) | 2.5 (0) | 0.2 (3) | 0.0 (10) | ||

5 | 2.2 (0) | 1.9 (0) | 0.5 (2) | 0.0 (10) | ||

6 | 1.9 (1) | 1.6 (1) | 0.5 (1) | 0.1 (9) | ||

7 | 2.0 (1) | 1.7 (2) | 0.6 (0) | 0.0 (8) | ||

50 | 50 | 2 | 8.6 (0) | 6.6 (0) | 0.0 (2) | 0.0 (10) |

3 | 5.7 (0) | 5.4 (0) | 0.1 (1) | 0.0 (10) | ||

4 | 4.5 (0) | 4.3 (0) | 0.3 (0) | 0.0 (10) | ||

5 | 4.0 (0) | 3.7 (0) | 0.1 (0) | 0.0 (10) | ||

6 | 4.3 (0) | 4.0 (0) | 0.3 (0) | 0.0 (10) | ||

7 | 3.3 (0) | 3.2 (0) | 0.0 (0) | 0.0 (10) | ||

8 | 3.9 (0) | 3.7 (0) | 0.4 (0) | 0.0 (10) | ||

80 | 80 | 2 | 9.6 (0) | 9.6 (0) | 0.0 (7) | 0.0 (10) |

3 | 6.9 (0) | 6.6 (0) | 0.1 (1) | 0.0 (10) | ||

4 | 5.2 (0) | 5.0 (0) | 0.2 (1) | 0.0 (10) | ||

5 | 3.7 (0) | 3.6 (0) | 0.2 (0) | 0.0 (10) | ||

6 | 3.5 (0) | 3.3 (0) | 0.2 (0) | 0.0 (10) | ||

7 | 3.3 (0) | 3.1 (0) | 0.3 (0) | 0.0 (10) | ||

8 | 3.1 (0) | 3.1 (0) | 0.3 (0) | 0.0 (10) |

**Table 7.**Results for medium and large instances without sequence-dependent setup times (problem $FGSS{P}_{s}\mid {S}_{jik}\mid {\sum}_{j}{C}_{j}$). For each instance size (represented by the number of jobs, column $nbJobs$, machines, column $nbMchs$, and stages, column $nbStgs$, we provide the average gap to the best solution and (in parentheses) the number instances where the best CP strategy finds the best known solution (column Auto), the best solution provided among the CP approaches (column CP), the results from the MILP approach (column MILP) and the best and the average found among 10 independent runs of the DEC approach (columns, DEC (best) and DEC (av.) respectively). The results of the best performing method for each group of instances are highlighted in boldface.

$\mathit{nbJobs}$ | $\mathit{nbMchs}$ | $\mathit{nbStgs}$ | Auto | CP | DEC (av.) | DEC (Best) |
---|---|---|---|---|---|---|

15 | 15 | 2 | 4.6 (0) | 3.2 (0) | 0.7 (0) | 0.0 (10) |

3 | 1.6 (1) | 1.3 (2) | 0.6 (0) | 0.1 (8) | ||

4 | 1.2 (5) | 0.6 (7) | 2.5 (0) | 1.6 (3) | ||

5 | 2.7 (2) | 0.4 (6) | 1.8 (0) | 1.3 (4) | ||

20 | 20 | 2 | 5.0 (0) | 4.1 (0) | 0.5 (0) | 0.0 (10) |

3 | 4.2 (0) | 3.1 (2) | 1.1 (0) | 0.3 (8) | ||

4 | 2.1 (4) | 1.1 (4) | 2.8 (2) | 2.0 (6) | ||

5 | 2.7 (1) | 0.8 (6) | 3.0 (2) | 2.5 (4) | ||

6 | 2.4 (3) | 0.8 (5) | 1.2 (0) | 0.4 (5) | ||

7 | 2.0 (0) | 0.5 (4) | 1.3 (0) | 0.7 (6) | ||

30 | 30 | 2 | 9.6 (0) | 7.7 (0) | 0.5 (4) | 0.0 (10) |

3 | 5.4 (0) | 4.2 (0) | 0.5 (0) | 0.0 (10) | ||

4 | 6.3 (0) | 5.3 (0) | 1.8 (0) | 0.0 (10) | ||

5 | 3.4 (1) | 3.3 (1) | 1.5 (0) | 0.2 (9) | ||

6 | 3.3 (1) | 2.8 (2) | 1.2 (0) | 0.1 (8) | ||

7 | 3.4 (1) | 2.5 (1) | 1.0 (0) | 0.0 (9) | ||

50 | 50 | 2 | 12.6 (0) | 8.4 (0) | 0.1 (8) | 0.0 (10) |

3 | 7.1 (0) | 5.4 (0) | 1.0 (0) | 0.0 (10) | ||

4 | 9.9 (0) | 9.1 (0) | 1.9 (0) | 0.0 (10) | ||

5 | 4.8 (0) | 4.4 (1) | 1.4 (0) | 0.1 (9) | ||

6 | 6.2 (0) | 6.0 (0) | 1.1 (0) | 0.0 (10) | ||

7 | 4.1 (0) | 4.0 (0) | 0.0 (0) | 0.0 (10) | ||

8 | 4.7 (1) | 4.3 (1) | 1.5 (0) | 0.4 (9) | ||

80 | 80 | 2 | 14.4 (1) | 12.2 (2) | 2.0 (2) | 1.4 (8) |

3 | 24.8 (0) | 23.5 (0) | 1.9 (0) | 0.0 (10) | ||

4 | 23.2 (0) | 21.9 (0) | 3.4 (0) | 0.0 (10) | ||

5 | 21.1 (0) | 19.9 (0) | 4.1 (0) | 0.0 (10) | ||

6 | 16.0 (0) | 14.0 (1) | 3.0 (0) | 0.3 (9) | ||

7 | 12.4 (0) | 11.7 (1) | 4.9 (0) | 0.1 (9) | ||

8 | 14.2 (0) | 13.1 (0) | 1.9 (0) | 0.0 (10) |

Algorithm | Objective Function Value (Minutes) |
---|---|

CP AUTO | 263,366 |

CP DF | 293,522 |

CP RS | 274,561 |

CP MP | 293,442 |

CP ID | 264,043 |

CP DEC (av.) | 261,532 |

CP DEC (best) | 260,349 |

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**MDPI and ACS Style**

Yuraszeck, F.; Mejía, G.; Pereira, J.; Vilà, M. A Novel Constraint Programming Decomposition Approach for the Total Flow Time Fixed Group Shop Scheduling Problem. *Mathematics* **2022**, *10*, 329.
https://doi.org/10.3390/math10030329

**AMA Style**

Yuraszeck F, Mejía G, Pereira J, Vilà M. A Novel Constraint Programming Decomposition Approach for the Total Flow Time Fixed Group Shop Scheduling Problem. *Mathematics*. 2022; 10(3):329.
https://doi.org/10.3390/math10030329

**Chicago/Turabian Style**

Yuraszeck, Francisco, Gonzalo Mejía, Jordi Pereira, and Mariona Vilà. 2022. "A Novel Constraint Programming Decomposition Approach for the Total Flow Time Fixed Group Shop Scheduling Problem" *Mathematics* 10, no. 3: 329.
https://doi.org/10.3390/math10030329