Optimal staffing of specialized programme trainees under uncertainty

Abstract

We propose a stochastic model and provide an easy-to-implement optimization tool for admission decisions to specialized training programmes designed for service industries. The model can be applied for staffing of trainees in medical residency programmes, vocational schools, management trainee programmes, and similar. Especially towards graduation, trainees in these programmes substantially contribute to workforce of their affiliated institutions, thus having a targeted number of advanced level students become a potential performance metric for administration. For uncertain attrition rates and study duration, we model and provide an iterative solution algorithm to find the optimal annual admission number for these programmes. Our numeric analysis results show that the solution is robust to changes in attrition and study duration probabilities; hence, our model is robust against specification errors for these parameters, which could be hard to estimate due to data unavailability and fluctuations in educational and economic conditions.

Appendix

Proof of Theorem 1:

• First we show that there is a unique solution to the admission decision model (AM) by using the convexity of the objective function. The objective function is a discrete function. A univariate function is discrete convex if it has non-decreasing first forward differences (please see Yüceer, 2002 and the references therein).

  The objective function in (1) can be written as: 𝔼[f(B(p₁, X)+B(p₂, Y))] where f:ℤ*↦ℝ is defined as f(k)=c₁max(k−N, 0)+c₂max(N−k, 0). Noting the similarity of f(k) with the absolute value function , we can conclude that f(k) has non-decreasing first forward differences and is discrete convex. Given that, we now show the convexity of our objective function in (1) using Lemma 1. A similar lemma for a different function and a single binomial random variable can be found in Aydın et al (2012, p 23). □

Lemma 1

• If the function f:ℤ*↦ℝ is discrete convex, then the function Y↦𝔼[f(B(p₁, X)+B(p₂, Y))] is also discrete convex.

Proof of Lemma 1:

• From hereon when mathematically more suitable, we represent binomial random variables with distribution-wise equivalent Bernouilli selection type random variables. Thus, here we use

  

  where U _k , k∈ℕ, is a sequence of i.i.d uniform random variables also independent of X; and we define B(p₂, Y) for any Y∈ℤ* similarly. We have (from independence of the random variables)

  
  
  

  Hence, for every possible value of B(p₁, X)+B(p₂, Y), the first forward difference of f function at that value will be multiplied with corresponding probabilities. Similarly we get

  
  

  Comparing (A.4) with (A.6), and from discrete convexity of f, we get that 𝔼[f(B(p₁, X)+B(p₂, Y))] has non-decreasing first forward differences. Thus, it is discrete convex.

  Given the convexity of the objective function, we can conclude that (AM) has a unique solution and this solution can be found via checking the first forward differences. From (A.4), we have

  
  
  
  
  

  which gives the formula of the first forward difference in (A.2). The solution space is non-negative integers and when the incrementation starts from (max(0, N−X)), the solution is reached at the smallest Y value with which incrementing by 1 gives a positive first forward difference for the objective function in (1). □

About Remark 1

Note that Median(B(p, X+Y)) is either ⌈(X+Y)p⌉ or ⌊(X+Y)p⌋ and it can be greater than or equal to N−1. This gives a total of four cases. The inequality ((N−2)/(p))−X<Y comes from the case N−1=Median (B(p, X+Y))=⌈(X+Y)p⌉. This case gives N−2<(X+Y)p⩽N−1 and hence a lower bound of ((N−2)/p)−X for Y. When compared with the bounds obtained from the other three cases in a similar way, it can be seen that it is the smallest lower bound for Y provided by these four cases. To ascertain that the algorithm does not miss the optimal value, the lowest lower bound provided by the four cases mentioned here should be taken as the starting point.