# Linear Programming - Model understaffing

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?

• Please show the constraint (3.2.2) as part of your question. May 28 at 22:43
• In addition to showing the constraint, please indicate what each index / parameter / variable represents.
– prubin
May 28 at 22:47
• Added both! Sorry for that May 29 at 4:51

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