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With the development of technology and industry, new research issues keep emerging in the field of shop scheduling. Most of the existing research assumes that one job visits each machine only once or ignores the multiple resources in production activities, especially the operators with skill qualifications. In this paper, we consider a reentrant flow shop scheduling problem with multiresource considering qualification matching. The objective of the problem is to minimize the total number of tardy jobs. A mixed integer programming (MIP) model is formulated. Two heuristics, namely, the hill climbing algorithm and the adapted genetic algorithm (GA), are then developed to efficiently solve the problem. Numerical experiments on 30 randomly generated instances are conducted to evaluate the performance of proposed MIP formulation and heuristics.

Flow shop scheduling problem has been widely studied since it is first proposed ([

Reentrance.

Most existing works addressing flow shop scheduling problem with resource requirement only consider machines and raw materials. However, the impact of many other kinds of resources on the solution is not negligible, including the operators with different abilities. For example, doctors can be considered as expensive and rare surgical resources in the surgical scheduling problem, and drivers can be also regarded as resources in the vehicle scheduling problem, etc. In this paper, reentrance flow shop scheduling considering multiresource with personal qualification matching is investigated. The main contributions of this paper mainly include the following:

We consider a reentrant flow shop scheduling problem, taking personal qualification matching into consideration.

A new mixed integer programming (MIP) formulation is proposed.

Two heuristics are developed to efficiently solve the problem, i.e., the hill climbing algorithm and the adapted genetic algorithm (GA).

Numerical experiments are conducted to evaluate the performance of our developed heuristics. Computational results show that hill climbing algorithm is more time-saving, and GA performs better in terms of the solution quality.

The rest of this paper is structured as follows. Section

There are a significant amount of researches addressing the flow shop scheduling problem. Only few researches have been conducted to deal with the reentrance of jobs (e.g., [

Graves [

Chen (2006) addresses the reentrant permutation flow shop scheduling problem, where every job must be processed on machines in the same order,

Chu et al. (2008) investigate a reentrant shop problem, which can be considered as a special case of the problem studied by Wang et al. [

Desprez et al. [

Huang et al. [

Shop scheduling with multiresource has been studied in some research. However, researches focusing on shop scheduling problem with personal qualification matching are very rare.

Scheduling problem considering multiple resources has been investigated by many researchers. However, most existing works either ignore the personnel or assume they are identical and can be replaced with each other.

Dauz

Artigues et al. [

Wang and Wu [

Rajkumar et al. [

This paper extends the literature on reentrant flow shop scheduling by considering multiresource, which contains the operators with different skill qualification.

In this section, we first describe the problem and then propose a new mixed integer programming formulation.

In the deterministic problem, there are a set of jobs should be processed, i.e.,

The processing task is defined as one procedure of one job. We address the reentrance by considering that there should not be more than one processing task on one machine at a time. Multiple resources considered include raw materials or operators. Each processing task requires some raw materials, and the number of available raw materials is limited. Besides, some certain skill qualifications are also required by each processing task. Only operators processing the skill qualifications required can be assigned to a processing task, i.e., qualification matching. Moreover, one operator cannot assigned to two or more processing tasks at a time.

Each job should be processed after its own release time. It is assumed that the due dates of jobs are fixed and known in advance. The objective of the problem is to find a set of sequences and operators assignment of jobs in all procedures, in order to minimize the number of tardy jobs. To formally state the problem, a mixed integer programming (MIP) formulation is proposed in the following.

In the following, we give the definitions of parameters and decision variables. Then a new MIP model is formulated.

The objective function (

As we can see from the computational results reported in Section

In this section, two heuristics to solve the problem, i.e., a hill climbing algorithm and an adapted genetic algorithm (GA), are developed.

Solutions should be transformed into individuals that can be operated by the algorithms. For our problem, a solution is composed of three decision parts: (1) the first part is the sequence of jobs in each procedure, (2) the second part is the start time of each in each procedure, and (3) the third part is the operator assignment to each job in each procedure (see Figure

Coding.

Figure

For the generalization of an initial solution, first we generate a vector including random permutation of

Hill climbing algorithm is a local search approach, the basic idea of which is to find a string with better solution to replace the existing one. As shown in Algorithm

(

(

(

(

(

(

(

(

(

(

(

(

The procedure

Genetic algorithm (GA) is first introduced by Holland [

There are two common crossover operators in scheduling problem, namely, one-point crossover and multipoint crossover. In this paper, we adopt two-point crossover for our problem. For two-point crossover operator, two parent solutions are selected randomly. As shown in Figure

Crossover.

For the mutation operator, solutions are mutated in the same way as the

In this section, the performance of our proposed formulation and two heuristics are evaluated by 30 randomly generated instances. Formulation and proposed heuristics are coded in MATLAB_2014b. CPLEX 12.6 solver is called to solve the formulation. All numerical experiments are conducted on a personal computer with Core I5 and 3.30GHz processor and 8GB RAM under Windows 7 operating system. The computational times of formulation and two heuristics are limited to 3600 seconds.

A preliminary analysis is conducted to fine-tune the parameters of proposed two heuristics. For the hill climbing algorithm, the maximum number of iterations is set to be 500. For the genetic algorithm (GA), parameters are presented in Table

Parameters for GA.

Parameter | Value (GA) |
---|---|

Population size ( | 50 |

Generation number ( | 20 |

Crossover probability | 0.8 |

Mutation probability | 0.6 |

The tested data are generated following the way in Liu et al. (2015) and Chen (2009). The processing times of jobs are randomly generated from a discrete Uniform distribution over

Computational results on 30 randomly generated instances are shown in Table

In Table

Computational results.

Set | | CPLEX | HCA | GA | |||
---|---|---|---|---|---|---|---|

Obj | CT | Obj | CT | Obj | CT | ||

1 | | 1 | 142.5 | 1 | 0.9 | 1 | 1.6 |

2 | | 2 | 215.5 | 3 | 1.4 | 2 | 3.1 |

3 | | 2 | 628.7 | 3 | 2.3 | 2 | 6.9 |

4 | | 2 | 1230.4 | 5 | 3.8 | 3 | 8.6 |

5 | | 3 | 2352.1 | 8 | 5.7 | 4 | 13.2 |

6 | | 5 | 3325.3 | 8 | 74.8 | 7 | 105.8 |

7 | | 4 | 3600.0 | 6 | 105.7 | 6 | 197.2 |

8 | | 6 | 3600.0 | 9 | 126.9 | 7 | 256.6 |

9 | | 7 | 3600.0 | 10 | 189.8 | 9 | 385.1 |

10 | | 6 | 3600.0 | 10 | 251.7 | 8 | 493.3 |

11 | | 8 | 3600.0 | 15 | 386.1 | 12 | 621.9 |

12 | | - | - | 17 | 452.2 | 19 | 828.5 |

13 | | - | - | 18 | 514.3 | 15 | 956.4 |

14 | | - | - | 21 | 703.6 | 20 | 1129.3 |

15 | | - | - | 19 | 912.5 | 12 | 1408.2 |

16 | | - | - | 16 | 1030.4 | 15 | 1606.2 |

17 | | - | - | 24 | 1342.5 | 24 | 1946.8 |

18 | | - | - | 27 | 1608.0 | 22 | 2234.8 |

19 | | - | - | 29 | 1918.2 | 23 | 2415.3 |

20 | | - | - | 30 | 2213.4 | 24 | 2809.7 |

21 | | - | - | 37 | 2524.2 | 35 | 2998.7 |

22 | | - | - | 38 | 2896.1 | 33 | 3600.0 |

23 | | - | - | 31 | 3234.3 | 28 | 3600.0 |

24 | | - | - | 36 | 3577.1 | 30 | 3600.0 |

25 | | - | - | 38 | 3600.0 | 28 | 3600.0 |

26 | | - | - | 35 | 3600.0 | 30 | 3600.0 |

27 | | - | - | 42 | 3600.0 | 21 | 3600.0 |

28 | | - | - | 40 | 3600.0 | 24 | 3600.0 |

29 | | - | - | 45 | 3600.0 | 31 | 3600.0 |

30 | | - | - | 45 | 3600.0 | 30 | 3600.0 |

| |||||||

Average | - | - | 22.2 | 1522.5 | 17.5 | 1760.9 |

This work investigates the reentrant flow shop scheduling problem, considering multiresource qualification matching, in which operators processing required skill qualifications can be assigned to serve a job in a procedure. The objective of the problem is to schedule the jobs in each machine and assign operators to serve job processing. For the problem, a new mixed integer programming (MIP) formulation is proposed, and two heuristics are then developed, i.e., the hill climbing algorithm and the adapted genetic algorithm (GA). Numerical experiments on 30 randomly generated instances are conducted to evaluate the performance of our proposed MIP formulation and two heuristics. Computational results show that hill climbing is more time-saving, and GA performs better in terms of solution quality.

In the future, we should develop more effective algorithms to solve the problem and improve the quality of the solutions (on the basis of Yin et al. [

The data set in the numerical experiments are randomly generated, which can be accessed by the method stated in Section

The authors declare that they have no conflicts of interest.

This work was supported by the National Natural Science Foundation of China (NSFC) under Grants 71428002, 71531011, 71571134, and 71771048. This work was also supported by the Fundamental Research Funds for the Central Universities.