2.4.1 Model assumptions

The assumptions of the proposed model are presented below:


Restrictions for the formation of manufacturing cells:

A Methodology to Design and Balance Multiple Cell Manufacturing Systems

xik ¼ 1; ∀i ¼ 1, …, m (8)

xik ≤ Mmax � yk; ∀k ¼ 1, …, Kmax (9)

xik ≥ Mmin � yk; ∀k ¼ 1, …, Kmax (10)

ðuihGO þ uihGAÞ ¼ 1 ∀i ¼ 1, …, m (11)

tið Þ uihGO þ uihGA ≤C � rh ∀h ¼ 1, …, Smax (12)

ð Þ Smax � h þ 1 ð Þ uahGO � ubhGO ≥0 ∀ð Þ a, b ∈ GO (13)

ð Þ Smax � h þ 1 ð Þ ubhGA � uahGA ≥0 ∀ð Þ a, b ∈ GO (14)

gh � f kh ≤2 ∀k ¼ 1, …, Kmax (16)

f kh ≤ð Þ� θ � 1 gh þ 1 ∀h ¼ 1, …, Smax (17)

xik, yk, uihG,rh, f kh, gh ∈ f g 0, 1 ∀ h, i, j, k (18)

ðuihGO þ uihGAÞ � xik ≤ m � f kh ∀k ¼ 1, …, Kmax; h ¼ 1, …, Smax (15)

Linking restrictions between the formation and the balance of the cells:

The set of restrictions (8) restricts each machine to a single cell. The set of restrictions (9) restricts each created cell to a maximum of Mmax machines, while the set of restrictions (10) restricts them to a minimum of Mmin machines; the value of yk is equal to one for the first Kmin restrictions, since it is known that these cells are required. As to the balance of the lines, the objective is to minimize the cost per required work station in addition to the theoretical minimum, avoiding the need to have rh variables for stations 1 through ⌈Smin⌉. The set of restrictions (11) ensures that each machine is assigned to only one station, either in the original precedence graph or in the auxiliary one [18]. The set of restrictions (12) ensures that for every

station, the sum of the weighted average processing times of their assigned machines does not exceed the average cycle time; the values of rh are equal to one

K Xmax k¼1

Xm i¼1

DOI: http://dx.doi.org/10.5772/intechopen.89463

Xm i¼1

Restrictions for the balance of U-shaped cells:

X Smax

h¼1

X Smax

h¼1

K Xmax k¼1

Restrictions for defining binary variables:

Xm i¼1

X Smax

h¼1

X Smax

h¼1

Xm i¼1

49
