...resource each produced item in each period is necessary. The "capacitated lot sizing problem with setup carry over" considers the possibility to conserve the setup state between two consecutive periods. Hence, there is no need for a setup if an item was produced last in the previous period and is produced first in the current one. In order to represent this aspect, further restrictions and an additional type of binary variables are introduced. Thus, the model size increases which means that heuristic solution methods are required for its solution. Sox and Gao (1999) propose a method which employs Lagrangian relaxation, subgradient optimization and an algorithm in order to solve the resulting subproblems optimally. However, not every aspect of the subproblems is considered which might cause suboptimal solutions of the subproblems.

In order to keep this paper self contained, we shortly introduce the model (GCLP1) and the notation and heuristic method of Sox and Gao (1999).

Problem (GCLP1):

min[N.summation over (i=1)] [T.summation over (t=1)] ([K.sub.i][z.sub.i,t] + [p.sub.i,t][x.sub.i,t] + [h.sub.i,t][I.sub.i,t]), (1)

subject to

[I.sub.i,t-1] + [x.sub.i,t] - [d.sub.i,t] = [I.sub.i,t] [for all]i, t, (2)

[N.summation over (i=1)] [a.sub.i][x.sub.i,t] [less than or equal to] [C.sub.t] [for all]t, (3)

[x.sub.i,t] [less than or equal to] [M.sub.i,t] x ([z.sub.i,t] + [[zeta].sub.i,t]) [for all]i, t, (4)

[N.summation over (i=1)] [[zeta].sub.i,t] = 1 [for all]t [greater than or equal to] 2, (5)

[[zeta].sub.i,t] - [z.sub.i,t-1] [less than or equal to] [for all]i, t [greater than or equal to] 2, (6)

[I.sub.i,t], [x.sub.i,t] [greater than or equal to] [for all]i, t, (7)

[z.sub.i,t], [[zeta].sub.i,t] [member of] {0, 1} [for all]i, t,...

NOTE: All illustrations and photos have been removed from this article.



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