Interval goal programming for S-shaped penalty function

The approach of Jones and Tamiz (1995) [Jones, D.F., Tamiz, M., 1995. Expanding the flexibility of goal programming via preference modeling techniques. Omega 23, 41-48] has been accepted as the most efficient approach in the field of interval goal programming (IGP). Although several modifications to the original approach have been proposed recently [Vitoriano, B., Romero, C., 1999. Extended interval goal programming. Journal of the Operational Research Society 50, 1280-1283; Chang, C.-T., 2006. Mixed binary interval goal programming. Journal of the Operational Research Society 35, 389-396], all of them cannot formulate IGP with an S-shaped penalty function. In order to improve the utility of IGP, we extend the model of Chang (2006) [Chang, C.-T., 2006. Mixed binary interval goal programming. Journal of the Operational Research Society 35, 389-396] to be able to model an S-shaped penalty function. The newly formulated model is more concise and compact than the method of Li and Yu (2000) and it can easily be applied to a decision problem with the S-shaped penalty function. Finally, an illustrative example (i.e. how to build an appropriate E-learning system) is included for demonstrating the usefulness of the proposed model.