Linear Programming - Model understaffing

Question

I am reading up a bit on Linear Programming and have taken a lot from "Scheduling Emergency Room Physicians" (by Michael W. Carter & Sophie D. LaPierre, Health Care Management Science 4, 347–360 (2001)) for this.

I have a question regarding constraint (3.2.2) from page 350. This constraint prevents understaffing and overstaffing. It goes as following: $\sum_{k=1}^{K}x_{ijk}=c_{ij}~\forall I,J$

It indicates wether worker $k$ works shift $i$ on day $j$.

Introducing overstaffing could easily be done by changing the equality to $\ge c_{ij}$

My question is, how do I change the constraint so that I can still allow for understaffing? Of course, the constraint should prevent that every shift is understaffed ($\sum_{k\in K}^{}x_{ijk}=0$), because this would logically minimize the costs.

I'm interested in how to introduce some understaffing, for example, that a physician can be admitted less than necessary, sort of as a lower minimum?

Answer 1

You can choose multiple varied ways but I'd go with something like the min of daily requirement (across all shifts) as the floor for total manpower needed for the day

$ \min \{c_{i,j}: \ \forall i \} \le \sum_k \sum_i x_{ijk} \quad \forall j$

Answer 2

You could add (positive) slack variables $o_{ij}$ for overstaffing and $u_{ij}$ for understaffing: $$ \sum_k x_{ijk}+ u_{ij}=c_{ij}+o_{ij} \quad \forall i,j $$ And add the cost of overstaffing/understaffing by weighting these variables in the objective function.