## Description

On this page, instances for the the Two-Stage selection problem under discrete uncertainty set could be found. In addition, information with regard to the size of instances provided as well as an overall description of the considered method of instance generation is available. For more general purposes, the instance generator software is also accessible through a link to our github repository. Finally, if more detail about theory or application of this method is desired, the main publication introducing this method could also be reached.

It must be noticed that in order to refer to the parameters of the robust selection problem, we use n for the number of items, p for the number of items we need to choose and N for the number of scenarios. Moreover, we use Ci to present the first-stage cost of item i and cij to show the second-stage cost of item i in scenario j.

Method Description: For all i ∈ [n], with 50% probability we choose Ci from {1, . . . , 100} and with 50% probability we set Ci = 50. Then, for all i ∈ [n] and j ∈ [N], if Ci = 50 with 50% probability we choose cij from {1, . . . , 10} and with 50%  probability we choose cij from {91, . . . , 100}; else, we choose cij from {Ci 5, . . . , C i + 5} and set negative values to be equal to zero.

## Instance Format

The instance set here consists of four datasets. In dataset1, we consider the two cases when N = n = 50, p = 25 and N = n = 100, p = 50. In dataset2, we fix n = N = 50 but change p ∈ {10, 20, 25, 30, 40}. In dataset3, we fix n = 50 and p = 25 but change N ∈ {10, 20, . . . , 60}. Finally, in dataset4, we consider problems with a large-scale number of scenarios N ∈ {100, 200, 500, 1000, 2000, 5000} with n = 50 and p = 25. Then, for each parameter choice, we generate 50 instances. Thus there are 100 instances in dataset1, 250 instances in dataset2, 300 instances in dataset3 and 300 instances in dataset4. These instances are named as “instance-n-p-N-0-21600-0-x” where x represents the number of instance (1 x 50). In addition, each instance file contains N+2 lines. The first line represents n, p, N, the second line represents the first stage scenario (including n item costs) and the remaining lines forms N given second stage scenarios.

## Generator Software

Although it is a good idea to have a library of instances for the robust optimization problems, it is not possible to upload all possible combination of problem parameters on a website. Alternatively, the generator software could be accessed so that any instance size could be generated. Therefore, it is possible to access a C++11 code which is used as the generator software.

## Reference

This page has been created based on the information provided in the following paper:

• Goerigk, M., & Khosravi, M. (2022). Benchmarking Problems for Robust Discrete Optimization. arXiv preprint arXiv:2201.04985.