## 3 The Proposed Algorithm
### The Number of Needed Operators  
The number of operators can be \(O=M\) (\(M\) obtained in the previous section) if a single operator is assigned to each machine. But, if a single operator can perform all the setups for all the \(M\) machines and meet the production plan, then \(O=1\). Therefore, the number of workers is in the range \(1\leq O\leq M\).  
Determining the minimum number of operators required to operate \(M\) machines, as established in the previous step (including the internal scheduling in each machine), can also be modeled as a special case of the bin packing problem. In this scenario, each machine functions as a different \"item\" to be packed, while each operator serves as a \"container\". Even though the machines are identical, since each machine has been assigned several different types of products, the machines can be considered as not identical \"items.\"  
For a single operator to effectively operate multiple machines, two critical conditions must be fulfilled:  
1. The cumulative setup time across all his/her assigned machines must be 1 or less than 1.
2. Under a single operator's responsibility, no two setup operations occur simultaneously on any machine.  
Consequently, this special case of the bin packing problem evolves into a conflict-based problem, necessitating a more complex approach than standard bin-packing algorithms. This variant accounts for both capacity constraints (total setup time) and scheduling conflicts (simultaneous setups), adding complexity to the optimization process.  
The problem where a single operator must operate several machines is the machine interference problem (see, for example, [1]). In our case, machine interference is forbidden. Here are some claims to demonstrate the problem of assigning several machines to a single operator without interference.  
**Claim 1**.: _If the cumulative setup time across all the assigned machines is 1 or less, and only one product is produced on each machine, then a single operator can operate up to \(\left\lfloor\frac{1}{Ts}\right\rfloor^{-}\) machines without overlap. Therefore, if there are \(M\) machines, each produces only one product, then a single operator will operate \(\min\left\{\left\lfloor\frac{1}{Ts}\right\rfloor^{-},M\right\}\) machines._  
Proof.: There is a simple way to schedule the M setups without overlap. Since the cycle time with the setup is 1 for all product types, and each cycle on each machine is represented by a circle with a perimeter of 1, the setup for the first machine can be completed, and then immediately the setup for the second machine can be performed, and so on, without  
Figure 3: A partition of three product types between two machines.  
any overlap. Since each machine requires only one setup and the total setup time across all machines does not exceed 1, the cycles can be scheduled across all machines without conflicts in setup timing. \(\square\)  
**Claim 2.**_Even in a situation with two machines, with two products on one machine and one product on the other, and the necessary condition is met, a single operator will not necessarily be able to operate both machines due to overlapping setup times._  
Proof.: Here is an example of such a situation. Three products must be produced on two machines, where each product needs \(T_{S}=0.33\) for setup. The total setup of the two machines is \(3T_{S}=3\times 0.33=0.99<1\), so the first condition that a single operator can operate these two machines is fulfilled. Moreover, the production time of the two products on the first machine are \(T_{\Phi,1}=T_{\Phi,2}=0.167\) and the production time of the product on the second machine is \(T_{\Phi,3}=0.670\). This example is illustrated in Figure 3, and it is easy to verify that there is no continuous range on machine 1 with a length of at least 0.330 that enables scheduling the setup of the second machine without setup overlapping. \(\square\)