Distributionally Robust Optimization for Scheduling Problem in Call Centers with Uncertain Forecasts

Abstract

This paper deals with the staffing and scheduling problem in call centers. We consider that the call arrival rates are subject to uncertainty and are following independent unknown continuous probability distributions. We assume that we only know the first and second moments of the distribution and thus propose to model this stochastic optimization problem as a distributionally robust program with joint chance constraints. Moreover, the risk level is dynamically shared throughout the entire scheduling horizon during the optimization process. We propose a deterministic equivalent of the problem and solve linear approximations of the Right-Hand Side of the program to provide upper and lower bounds of the optimal solution. We applied our approach on a real-life instance and give numerical results. Finally, we showed the practical interest of this approach compared to a stochastic approach in which the choice of the distribution is incorrect.

Appendix

Here we give the proof of the convexity of

$$\begin{aligned} f: ]0;1]\rightarrow & {} \qquad \qquad \qquad \qquad \,\mathbb {R}^+ \nonumber \\ y\mapsto & {} \qquad \qquad \qquad \sqrt{\frac{p^y}{1-p^y}} \end{aligned}$$

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with $p \in [0;1[$.

Function f is $C^\infty$, so we can compute the second derivative of function f. We have first:

$$\begin{aligned} \frac{df}{dy}=&\frac{\frac{\ln p}{2} p^{\frac{y}{2}} (1-p^y)^{\frac{1}{2}}+\frac{\ln p}{2} p^y (1-p^y)^{-\frac{1}{2}} p^{\frac{y}{2}}}{1-p^y} \\ =&\frac{\ln (p) (1-p^y)^{-\frac{1}{2}}(p^{\frac{y}{2}} (1-p^y) + p^{\frac{3}{2}y})}{2(1-p^y)}\\ =&f(y) \frac{\ln p}{2 (1-p^y)} \end{aligned}$$

Then,

$$\begin{aligned} \frac{d^2 f}{dy^2}=&\frac{\ln p}{2} \frac{f'(y)(1-p^y)+\ln (p) p^y f(y)}{(1-p^y)^2}\nonumber \\ =&\frac{\ln ^2(p) (1+2p^y)}{4 (1-p^y)^2} f(y)\nonumber \\ =&\frac{\ln ^2(p) (1+2p^y)}{4 (1-p^y)^2} \frac{p^{\frac{y}{2}}}{(1-p)^{\frac{y}{2}}} \end{aligned}$$

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Since every term of the second derivative is positive, we conclude that $\frac{d^2 f}{dy^2}$ is positive and then, f is convex. $\square$