A production model and maintenance planning model for the process industry

Ii⁺,0=Ii^-,0=0 i=1,...,N                                  (12)

Constraint (11), in combination with (7) initialized the δ_ij_t's to 0 in period 0.

As mentioned earlier, the objective function models the minimization of total costs, consisting of
preventive maintenance costs, expected breakdown costs, inventory and backorder costs, and
setup costs.

Preventive maintenance costs are one-time costs incurred when a preventive maintenance job is
started:

MT

∑∑zjtcpj                               (13)

j=1 t=1

Expected breakdown costs are incurred each period that a job is busy. The yjmt in the following
expression makes sure that the correct probability of a breakdown during that period is selected,
by only setting yj_mt to 1 if the previous maintenance was performed in period m :

N M T      t-1

∑∑∑δijt∑yjmtcjmt                          (14)

i=1 j=1 t=1     m=0

This expression is non-linear, but can be linearized by replacing δijtyjmt with the extra variable
α_ij_mt. Then (14) is transformed to:

N M T t-1
∑∑∑∑α ijmt cjmt

i=1 j=1 t=1 m=0

The linearization takes two extra constraints:

1

αij_mt ≤ -(δijt + y_tmt) i = 1,...,N, j = 1,...,M, t = 1,...,T, m = 0,...,t -1
2

αijmt≥δijt+yjmt-1 i=1,...,N, j=1,...,M, t=1,...,T, m=0,...,t-1

Again, the positive objective coefficients will make sure that αijmt is always restricted to either 0
or 1.

Inventory costs, back-order costs and setup costs are stated respectively as:

NT

∑∑I⁺ithi                                 (18)

i=1 t=1

NT

∑∑Ii^-t li                                     (19)

i=1 t=1

NMT

∑∑∑φijtsij                              (20)

i=1 j=1 t=1

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