In a clinic, the patients must be assigned to doctors. However, the clinic is struggling with the no-show patients problem, so they have developed a probability that the patient will effectively assist to the clinic. They need to maximize the assistance, as for each effective attended patient the clinic receives an income of 100 USD. Each doctor has a fixed cost of 300 USD and each one can handle a maximum of 5 patients in total.
We define variables as
$$X_{ij} = \begin{cases} 1 & \text{if patient } i \text{ is assigned to doctor } j\\ 0 & \text{otherwise}\end{cases}$$
$$D_j = \begin{cases} 1 & \text{if you will use the doctor } j\\ 0 & \text{otherwise}\end{cases}$$
The parameters are defined as
- $Y_i$ is the probability that the patient will effectively assist to his assignment.
The objective function to be maximized is income minus costs
$$Z = 100 \sum_i Y_i - 300 \sum_j D_j$$
How can I define the constraints of capacity with probabilities? Usually, if variable $Y_i$ is binary, I would do the following
$$\sum_i Y_i X_{ij} \le 5$$
for all $j=1,\dots,J$. In other words, each doctor has a maximum of $5$ patients assigned. The assignment depends on the patient show-up probability. But the problem I'm facing is that $Y_i$ is a probability, not a binary variable.