mixed integer linear programming
problem of assigning operations to machines and each operation require number of operators. there is a limited number of available resource (operators).
during the solution $S_i$ is determined.
my question is:
I don't know how to connect constraint 1 with 2
how to say when $S_i$ starting time of operation i starts at time t then $X_{ijt} = 1 ? $ for example, if $S_1 = 5 $ and assigned to machine 2 then $X_{125}=1$
Indices
t time index 1...T
j machine index 1...M
parameters
$res_i $ is resource requirement of job i
$p_ij$ is processing time of operation i on machine j
b is available resource
$A_j$ ready time of machine j
variables
$ S_i$ is an operation start time
decision variables
$ X_{ijt} = 1$ if operation i on machine j starts at time t and 0 otherwise
$ Z_{ij} = 1 $ if operation i is assigned to machine j, and 0 otherwise
$Si \geq 0 $ and integer i=1...N
$X_{ijt} ∈ {0, 1}, ∀i ∈ N, ∀ j ∈ M, t ∈ T$
$Si \le \sum_{j \in M} A_j Z_{ij} ...(1) $ determine starting time for each operation
$\sum_{i\in N}\sum_{j \in M}\sum_{s=max{0,t-p_{ij}}}^{t-1} res_i \times X_{ijs} \le b$ t =1...T (2) ensure that the total number of operators assigned to operations doesn't exceed the available operators at any time period.
I still don’t quite follow, but maybe you want this: $$S_i = \sum_{j,t} t\ X_{i,j,t}$$