I have a question regarding lot sizing model with sequence dependent setup times. The following model is by: "Single machine multi-product capacitated lot sizing with sequence-dependent setups" Bernardo Almada-lobo et. al.
$\sum_{t=1}^{T}\sum_{i=1}^{N}\sum_{j=1}^{N} C_{ij}T_{ijt} + \sum_{t=1}^{T}\sum_{i=1}^{N} h_{i}I_{it} \\$
s.t
$I_{it} = I_{i,t-1} + X_{it} - D_{it} $
$\sum_{i=1}^{N} X_{it}p_{i} + \sum_{i=1}^{N}\sum_{j=1}^{N}\sum_{t=1}^{T} T_{ijt}s_{ij} \leq 100 \hspace{1.2cm} \forall t$
$X_{it} \le M_{it}\bigg (\sum_{j=1}^N T_{jit}\alpha_{it} \bigg )$
$\sum_{i=1}^{N} \alpha_{it} = 1 \hspace{3.8cm}$
$\alpha_{it} + \sum_{j=1}^{N} T_{jit} = \alpha_{i,(t+1)} + \sum_{j=1}^{N} T_{ijt}$
$V_{ji} \ge V_{it}+NT_{ijt} - (N-1)-N\alpha_{i,t} \hspace{1.2cm} \forall i \neq j,t,$
$C_{ij}$ Cost of setup from product $i$ to product $j$
$D_{it}$ Demand of product $i$ in period $t$
$s_{ij}$ Time of setup from product $i$ to product $j$
$p_{i}$ Processing time for one unit of product $i$
$h_{i}$ Inventory holding cost for product $i$
$X_{it}$ Number of products $i$ produced in period $t$
$I_{it}$ Inventory levels for product $i$ in period $t$
$T_{ijt}$ If a setup from product $i$ to $j$ occurs in period $t$
$\alpha_{it}$ One if machine is setup for product $i$ in the beginning of period $t$ zero otherwise.
$V_{it}$ Auxiliary variable that assigns product $i$ in period $t$
$M_{it}$ Upper bound on the quantity of product $i$ in period $t$
I do not understand the need for the third sum in the capacity constraint. Why sum over t, when we have for all t? I get the correct answer (same as in the paper) if i remove the third sum, and I actually don't understand the need for it. Is it possible to remove the third sum and still have a valid model?
Yes, it does not make sense to have $\sum_t$ and $\forall t$ in the same constraint. Also, is the first sum instead supposed to be $\sum_i$?