A production model and maintenance planning model for the process industry

zjt≤1-mj,t-1    j=1,...,M, t=1,...,T                           (2)

zjt≤mjt    j=1,...,M, t=1,...,T                           (3)

zjt≥(1-mj,t-1)+mjt-1    j=1,...,M, t=1,...,T                    (4)

The constraint sets (2), (3) and (4) also fix zjt at either 0 or 1.

To make sure that no jobs can be planned during a preventive maintenance (which requires RT
periods of time) we need the next constraints to make mjk one for every
k=t,...,t+RT-1:

zjt ≤ mjk    j= 1,...,M, t= 1,...,T, k = t,...,t+RT-1                  (5)

Besides constraints (1) through (5), some other key constraints are needed. A fundamental
material balance equation for each period is stated as:

Production + beginning inventory - ending inventory = demand

Or mathematically:

∑^M Rijδijt+(I⁺i,t-1-Ii^-,t-1)-(I⁺it-Ii^-t)=Dit   i=1,...,N t=1,...,T(6)

j=1

where inventory is defined as Iit⁺ - Iit^- , or positive net inventory minus backorders.

Rij is defined as the capacity of production line j divided by the processing time of product i on
production line j, or: Rij= Capj / pij.

Maintenance and production cannot be scheduled in the same period on a certain production line.
Constraint (7) prevents this from happening:

∑^N δijt+mjt≤1 j=1,...,M t = 0,...,T                      (7)

i=1

The variable yjmt must equal one when the last preventive maintenance job on production line j
occurred in period m and we are now in period t. To ensure this, we define yjmt as:

yjmt = (1-mjt)(1-mj,t-1)...(1-mj,m+1 )mjm

This is again nonlinear, so we use the same technique as applied in constraints (2) through (4):

yjmt≤1-mjk   j=1,..., M, t=1,...,T, k=m+1,...,t                 (8)

yjmt≤mjm    j=1,..., M, t=1,...,T, m=0,...,t-1                   (9)

t

y_jmt≥ ∑ (1-mjk)+mjm-(t-m) j=1,..., M, t=1,...,T, m=0,...,t-1      (10)

k=m+1

where yjmt is greater than or equal to 0.

To initialize some of the variables, we introduce a dummy period 0, where the previous
preventive maintenance was performed and where inventory was zero.

mj,0 = 1 j= 1,..., M

2

The general rule for linearizing expressions like x₁...x_k is as follows: replace x₁...x_k with y and add the next three constraints:

1) y<_ xj, 1_< j<_ k

2) y>_ Σ^kj=1 xj - (k-1)

3) y>_ 0

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