## March 2021

### Nurse scheduling problem

#### 29 March 2021 (165 words)

The Nurse Scheduling Problem (NSP) is a well-known optimization situation. The objective is to find an optimal way to assign nurses to shifts, where each nurse has their preferences for which shifts they would like to work. In addition, we need to meet the operational needs of the hospital in terms of nurse numbers and attributes for each shift.

Of course, a model of the NSP can be applied to constrained scheduling problems in many other situations.

A paper published in December 2018 describes a new formulation and solution for the Nurse Scheduling Problem, along with a real case study of applying the model in an Egyptian hospital.

The authors claim that the model improves the level of nurses’ satisfaction by considering their preferences, as well as decreasing the overall overtime cost by 36%.

A PDF of the paper is available at: A new formulation and solution for the nurse scheduling problem: A case study in Egypt.

### Chemical mix model reformulation

#### 24 March 2021 (151 words)

A recent project involved fixing a Solver model. The model was intended to be a straightforward Linear Program to find the optimal mix of three chemicals, subject to the proportions of the chemicals being within specific bounds.

That is, the model had constraints like: \({a \over a+b+c} \le p_a\)

where \(a\), \(b\), and \(c\) are variables for the quantity of each chemical, and \(p_a\) is a constant representing the upper bound on the proportion for chemical \(a\).

The problem was that this constraint is non-linear because a variable is divided by other variables. Solver's non-linear and evolutionary methods didn't always work well with this model.

Fortunately, we can rearrange the constraint to be in an equivalent linear form. That is: \({a \times (1 - p_a) \le p_a \times (b + c)}\)

With this reformulation, the model becomes linear while still achieving the desired intent. Consequently, the model now solves to optimality quickly and reliably using the Simplex method.

### On the shoulders of giants

#### 15 March 2021 (1,808 words)

Designing an optimization model can be difficult. In seeking guidance, we often look for inspiration from the published academic literature. Since the 1950s, there has been an enormous amount of research into how to design and solve optimization models for a wide range of situations, with models for just about anything we could want.

There is much that we can learn from the literature, if only we could understand it. Many academic papers include a statement of the model's design, known as the formulation, using mathematical notation that can look impenetrable. The notation is seldom explained, so translating a formulation into a working model can be a perplexing and frustrating process.

But with a little knowledge about what the notation means, we can gain a lot of insight into how to design or improve our models. The purpose of this article is to explain the meaning of common mathematical notation and illustrate how we can apply the knowledge contained in academic papers to help us build working models in Excel.

### Job sequencing to minimize completion time

#### 1 March 2021 (1,962 words)

"Job sequencing" is a common management problem, especially in manufacturing and production businesses.

That is, given a set of jobs, what is the job sequence that takes the minimum total time?

The benefits from improved sequencing can be substantial, while a poor sequencing solution can be expensive. To illustrate how an optimization model can help us make the decision, in this article we solve a simple job sequencing problem using Solver.