On-call scheduling optimization is a critical challenge in healthcare and IT industries. Balancing staff needs, patient care, and operational efficiency can be daunting. This article explores three powerful mathematical modeling techniques that revolutionize on-call scheduling, making it more efficient and fair.
Let’s go directly to the heart of the problem.
You want to use software to generate optimized on-call schedules. You have rules you want to enforce, but some flexibility is necessary. The challenge lies in making these complex requirements work within a mathematical model.
Don’t worry – we’ve got you covered. We’ll explore three approaches to on-call scheduling optimization, each suited for different scenarios.
1. Softening Equality Constraints in On-Call Scheduling
In on-call scheduling optimization, rigid constraints can lead to infeasible solutions. By introducing auxiliary variables, we can create flexible, “soft” constraints that balance rule adherence with practicality.
Implementing Soft Constraints in On-Call Schedules
If you declare a constraint in a mathematical model, it is held up very rigidly — the math solver will enforce the rule 100%, and if it’s not possible, it will tell you your problem is “infeasible”.
Therefore, we’d like to avoid this problem. When a soft constraint has to be violated, the solver should not return infeasible, but instead, it should be penalized by a weight.
This way, it will try to avoid breaking of the constraint, but if it has to break the constraint, it will tell you. The way we do this is by introducing an auxiliary variable z in a constraint.
Then, in the objective, we add a term to minimize the squared (or absolute) value of z.
If the constraint is followed, z = 0. The objective is not penalized.
If the constraint is not followed, z^2 and |z| ≥ 0, causing penalization to the objective.
You can see that, whenever the constraint is broken, z ≥ 0, which will penalize the objective. This makes the solver try to avoid breaking the constraint as much as possible.
Pros and Cons of Softening Equality Constraint
Advantages
- It is simple to set up.
- It also allows a proportionate penalization of breaking the constraint: the further you go, the greater the penalty to the objective.
Disadvantages
- It doesn’t offer integrality control, as in, you cannot formulate the model in such a way that, for every constraint broken, the objective is being penalized by exactly 1 (or any weight you’d like).
- Also, the objective is now non-linear if you don’t linearize it further.
When to Use this Approach in On-Call Scheduling
You can use it on equality constraints, in which you want to penalize the objective in a fashion that is dependent on how far the deviation away is from the constraint.
2. Softening Unitary Constraints in On-Call Scheduling
This approach works for some unitary constraints in on-call scheduling, which means constraints where the right hand side of the constraint is constrained to be either 1 or 0.
Implementing Unitary Constraints in On-Call Schedules
For example, if we want to state that, for each person, the total number of on-call shifts assigned per week is 1. Now, we know that the left hand side (aka the sum of shifts) must always be a non-negative integer, and for the right hand side, we have set it to 1.
If we want to detect whether the constraint has been broken, we can introduce an auxiliary variable z, and place it in the left side of the equation like below:
Because we know that the sum of left hand side is either 0 or 1, we can make the constraint this way.
If the constraint is followed, it means that the sum of shifts do equal 1, and therefore this must mean that z is 0.
On the contrary, if the constraint is violated, the value of z must be 1.
To complete the formulation, we penalize the objective by putting in z.
This way, the solver will try to minimize the breaking of such constraint.
Pros and Cons of Softening Unitary Constraints
Advantage
- Very simple to set up. You only need 1 auxiliary variable per constraint.
- It offers fine-grain integrality penalization to the objective when it is being violated.
- Objective is kept linear, which means faster solve times.
Disadvnatages
- This formulation can only be used in a very specific type of constraint: where the value of either side of the constraint is known to add up to either 1 or 0.
When to Use Unitary Constraints in On-Call Scheduling
This method is ideal for scenarios in on-call scheduling where you’re dealing with binary choices, such as assigning one shift per week to each staff member.
3. Softening Any Constraints in On-Call Scheduling
There are more forms of constraints to the above. This is why you might want a general-purpose soft constraint builder: for any constraint, you want to be able to transform them into an integer, where it is 1 when the constraint is violated, and 0 when the constraint is not.
Implementing General Soft Constraints in On-Call Scheduling
Naturally, the more general a formulation is, the more complicated it must be. Therefore we will be introducing many more variables in the constraint.
Before we start, I’d like to introduce 2 formulations to you. They look horrible at first, and it’s totally ok if you don’t understand it. But if you stare at it long enough the logic will become clear.
Checking if an Expression is non negative.
We want to set up an auxiliary variable b that returns to be 1 if the expression given equals 0, and 0 otherwise. The formulation is as such.
In this formulation, x+y represents the absolute value of the expression E.
If E is positive, it means z=1, y=0, and x=E.
Conversely if E is negative, it forces z=0, x=0, and y=E.
And the last constraint forces that, if |E|=0, then b is forced to be 1. Else it is forced to be 0.
Implications
If you want to check whether E is positive or negative, look at z.
If you want to check whether E is equal to 0, look at b.
Formulation for Equalities
Now we’re ready to formulate the functions.
If the constraint is an equality, it means the LHS should always equal the RHS. As such, we can check whether the expression LHS-RHS equals 0 using the formulation above.
Formulation for Inequalities
Below we assume LHS is always greater than RHS in the inequality formulation.
For inequalities, we will have to consider whether the inequality is of type “LHS>RHS”, or “LHS≥RHS”.
If the inequality demands LHS>RHS, we can take the value of LHS-RHS, and check if it is positive AND not 0.
If the inequality is LHS≥RHS, we can take the value of LHS-RHS, and check if it is positive AND if it is 0.
Pros and Cons of General Soft Constraints
Advantage
- This formulation applies to every single type of constraint.
- This formulation allows integrality control.
Disadvantage
- This formulation is more difficult and requires lots of auxiliary variables.
When to Use General Soft Constraints in On-Call Scheduling
This approach is best for complex on-call scheduling systems with diverse constraint types that cannot be easily handled by the previous two methods.
Comparison Table
Now, which type of soft constraint suits your use case the best?
| Comparison of Soft Constraints | Softening Equalities | Softening Integer Constraints | Softening All Constraints |
|---|---|---|---|
| Integrality Control | No | Yes | Yes |
| Applicability | Yes | No | Yes |
| Difficulty to Implement | Easy | Easy | Hard |
Conclusion
Mastering these mathematical modeling techniques for on-call scheduling optimization can significantly improve your organization’s efficiency and staff satisfaction. By choosing the right approach based on your specific needs, you can create fair, flexible, and optimized on-call schedules that benefit both your team and your patients or clients.
Remember, the key to successful on-call scheduling optimization lies in balancing mathematical precision with real-world flexibility. Start implementing these techniques today, and watch your on-call scheduling process transform from a headache into a strategic advantage.
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