General resource-constrained assembly line balancing problem: conjunction normal form based constraint programming models

Abstract

In the literature, most of the researchers studying assembly line balancing have only considered task assignments. However, resources are needed to perform the tasks. Therefore, assigning resources related to tasks becomes more realistic when assigning tasks to stations. In the general case of the problem, the task is performed with a specified amount of resources. If resource types such as a, b, c are required to perform tasks in an assembly line, the combination of tasks required from these resources should also be assigned to the stations. This type of problem is defined as general resources-constrained assembly line balancing problem (GRCALBP). In this study, GRCALBP is addressed to minimize cycle time and resource usage for a given number of stations. New constraint programming (CP) models based on conjunction normal form are proposed. The CP models are tested with generated problem instances from the data set in the literature. The experimental results show that CP is an efficient and effective modeling technique to solve GRCALBP. Finally, suggestions are made regarding alternative objective functions.

Appendix 1

CNF-1 mathematical model of Corominas et al. (2011).

Additional decision variables and parameters are given in Table 9.

Table 9 Additional decision variables and parameters

$$ {\text{Min}}\;\sum\limits_{{j = m_{\min } + 1}}^{{m_{\max } }} {{\text{CE}}_{j} \cdot y_{j} } + \sum\limits_{j = 1}^{{m_{\max } }} {\sum\limits_{{\forall r|K_{rj} \ne \emptyset }} {{\text{CR}}_{r} \cdot s_{rj} } } $$

(23)

$$ \sum\limits_{{j = E_{i} }}^{{L_{i} }} {x_{ij} = 1} \quad i = 1, \ldots ,N $$

(24)

$$ \sum\limits_{{i \in W_{j} }} {t_{i} \cdot x_{ij} } \le {\text{CT}}\quad j = 1, \ldots ,m_{\max } $$

(25)

$$ \sum\limits_{{i \in W_{j} }} {x_{ij} } - {\text{MW}}_{j} \cdot y_{j} \le 0\quad j = m_{\min } + 1, \ldots ,m_{\max } $$

(26)

$$ \sum\limits_{{j = E_{i} }}^{{L_{i} }} {j \cdot x_{ij} } \le \sum\limits_{{j = E_{k} }}^{{L_{k} }} {j \cdot x_{kj} } \quad \forall (i,k) \in {\text{PR}} $$

(27)

$$ \sum\limits_{{\forall j|K_{rj} \ne \emptyset }} {s_{rj} } \le {\text{NM}}_{r} \quad r = 1, \ldots ,R $$

(28)

$$ \alpha_{rpi} \cdot x_{kj} \le s_{rj} + \alpha_{rpi} \cdot qc_{rpi} \quad p = 1, \ldots ,C_{i} ;\;\forall r|\alpha_{rpi} > 0;\;j \in [E_{i} , \ldots ,L_{i} ] $$

(29)

$$ \sum\limits_{{r|\alpha_{rpi} > 0}} {{\text{qc}}_{rpi} } \le \tau_{pi} - 1\quad p = 1, \ldots ,C_{i} $$

(30)

CNF-2 mathematical model of Corominas et al. (2011)

$$ {\text{Min}}\;\sum\limits_{{j = m_{\min } + 1}}^{{m_{\max } }} {{\text{CE}}_{j} .y_{j} } + \sum\limits_{j = 1}^{{m_{\max } }} {\sum\limits_{{\forall r|K_{rj} \ne \emptyset }} {{\text{CR}}_{r} \cdot \sum\limits_{q = 1}^{{{\text{MR}}_{rj} }} {v_{rqj} } } } $$

(31)

Equations 24, 25, 26 and 27

$$ \sum\limits_{{\forall j|K_{rj} \ne \emptyset }} {\sum\limits_{q = 1}^{{{\text{MR}}_{rj} }} {v_{rqj} } } \le {\text{NM}}_{r} \quad r = 1, \ldots ,R $$

(32)

$$ x_{ij} \le \sum\limits_{{r|\alpha_{rpi} > 0}} {v_{{r,\alpha_{rpi} ,j}} } \quad p = 1, \ldots ,C_{i} :j \in [E_{i} \ldots L_{i} ] $$

(33)

$$ v_{r,l,j} \le v_{r,l - 1,j} \quad l = 1, \ldots ,C_{i} :\;\forall r|\alpha_{rpi} > 1:\,2 \le q \le \alpha_{rpi} :j \in [E_{i} \ldots L_{i} ] $$

(34)