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I assume for you have a binary matrix $S_{i,t}$ (tasks)$\times$(time steps) which tells you whether some task $i$ is active at time $t$. This approach introduces additional variables unless you have other parallelism constraints that can only be expressed in that format or of your objective depends on it like it does here. Define a new binary matrix of $G_{g,... 7 One simple approach is to impose the classical non-overlap constraints for each pair of tasks for which one task is in$T_1$and one task is in$T_2$, as shown here. 4 One way you can handle is to create combined tasks of (Batch #1 - Test 2) and (Batch #2 - Test 2) as a newer task (in your tasks set) and include this new task in your first set of covering/partitioning constraint. In more detail for 2 day case, Let say$t_{1,1}$represent (Batch #1 - Test 2) completing task-1 on day-1 and$t_{2,1}$represent (Batch #2 - ... 5 The general approach, whether you are using a mixed integer linear programming model or a constraint programming model, would be to have (nonnegative) variables representing the inventory of different materials at different times, plus flow constraints saying that the inventory at the end of each period is the starting inventory plus any production of the ... 4 What you are looking for sounds like, combining the job-shop scheduling problem with material requirement planning or Multi-Periods Job-Shop Scheduling Problem. I am not aware how you would like to use that in the paper or real situation, but in the second one, applying mixed-integer programming (without boosting from the special algorithm like column ... 1 Depending on what your decision variables are you are looking at an integer, mixed integer, or mixed integer linear programming problem. It seems to me that you want to solve it using a Python model. If that is the case, you can look at Pyomo or PuLP systems if you're looking for open-source solvers. 6 I would approach this as a mixed integer linear programming (MILP) problem. There are a number of MILP solvers, some open source, some commercial (with some of the commercial solvers providing free licenses for educational use). Many of them either have a Python API or can be used with PuLP (mentioned in comment to the selected answer). You might want to ... 5 I won't write a python solution as i am not familiar with any python modeling language but i can describe the approach i took in the past to solve problems like this. I would solve this problem using a technique of finite horizon optimal control where we have an$x_t$a state vector for each time point, a control signal$u_t$, a prediction$p_t\$. The core of ...