Robust Scheduling at a Large IT Services Delivery Center Elvin C ¸ oban ¨ Ozyeˇ gin University Istanbul, Turkey Email: elvin.coban@ozyegin.edu.tr Aliza Heching IBM Thomas J. Watson Research Center Yorktown Heights, USA Email: aheching@gmail.com J. N. Hooker and Alan Scheller-Wolf Carnegie Mellon University Pittsburgh, USA Email: jh38,awolf@andrew.cmu.edu Abstract—We study a project scheduling problem at a large IT services delivery center in which there are unpredictable delays. We apply robust optimization to minimize tardiness while informing the customer of a reasonable worst-case completion time. Due to the impracticality of quantifying joint probability distributions for delay times, we follow the recent practice of using empirically determined uncertainty sets, which to our knowledge have not been applied to service scheduling. To solve instances of a realistic size, we introduce a new solution method based on logic-based Benders decomposition. We show that when the uncertainty set is polyhedral, convexity properties of the problem allow us to simplify the decomposition substantially, leading to a model of tractable size. I. I NTRODUCTION We analyze a project scheduling problem at a large IT services delivery center in which there are uncertain processing times and unpredictable delays in start times. This study is motivated by a real problem at a global IT services delivery organization. To design a schedule that is not unduly disrupted by contingencies, we formulate a robust optimization problem. Our main objective is to minimize tardiness while informing the customer of a reasonable worst-case completion time in a complex scheduling enviroment. Due to the impracticality of quantifying joint probability distributions for delay times, we apply robust optimization with uncertainty sets rather than probabilistic information. An uncertainty set is an empirically determined space of possible outcomes for which one should realistically plan, without encompassing theoretically worst-case scenarios. To our knowledge, uncertainty sets have not prevously been applied to service scheduling. Optimization over uncertainty sets is a challenging two- stage optimization problem that can become intractable in large-scale instances using conventional techniques. We there- fore propose a new solution method based on logic-based Benders decomposition. We show that when the uncertainty set is polyhedral, the problem has convexity properties that result in a simplified decomposition that can be much easier to solve. Polyhedral uncertainty sets provide a great deal of flexibility for practical application. The size of the simplified model is proportional to the number of extreme points of the polyhedron. Thus if the polyhedron is a simplex, a natural option for the problem considered here, the model grows only linearly with the number of tasks to be scheduled. The remainder of the paper is organized as follows. After a review of previous work, we state the deterministic and then the robust scheduling problem. We describe logic-based Benders decomposition and show how it can be applied to robust scheduling. Due to the size of the problem, we apply a second level of Benders decomposition to the Benders master problem. This allows the subproblem to decouple and results in a three-stage model. We then focus on simplicial uncertainty sets, which are those described by a linear inequality and nonnegativity constraints. This allows us to formulate a much simpler two-stage model, based on convexity properties that are proved via a dynamic programming recursion. II. PREVIOUS WORK Robust optimization attempts to ensure that an optimal solution remains feasible and near optimal when uncertain parameters change somewhat [1]. Because the joint probability distribution of the parameters is often unavailable in practice, we focus on distribution-free robust optimization in which uncertain parameters belong to an uncertainty set [2]. This approach was first proposed in [3] for robust linear programming in which the columns of the coefficient matrix are uncertain and belong to a convex uncertainty set. Origi- nally, worst-case scenarios were included in the uncertainty set, but this is typically too conservative for practical im- plementations. To overcome this overconservatism, one can use ellipsoidal uncertainty sets that exclude unlikely outcomes of uncertain parameters, as in [4]–[8]. It can be shown that the robust counterpart of some important generic convex optimization problems under ellipsoidal uncertainty sets are exactly or approximately tractable problems that can be solved by interior point methods. However, the robust counterpart of a linear programming problem becomes a second-order conic problem. Thus, the complexity of the problem increases with the ellipsoidal uncertainty set. To avoid this increasing complexity, a polyhedral uncer- tainty set is proposed in [2], [9]. The robust counterpart of a linear programming problem remains a linear programming problem, and the method is more tractable, especially in large- scale settings. The level of conservatism is controlled via restricting the number of parameters that take their worst-case values [9]–[14]. Detailed reviews of robust optimization with uncertainty sets can be found in [1], [15], [16].