In the usual statistical inference problem, we estimate an unknown parameter of a statistical model using the information in the random sample. A priori information about the parameter is also known in several real-life situations. One such information is order restriction between the parameters. This prior formation improves the estimation quality. In this paper, we deal with the component-wise estimation of location parameters of two exponential distributions studied with ordered scale parameters under a bowl-shaped affine invariant loss function and generalized Pitman closeness criterion. We have shown that several benchmark estimators, such as maximum likelihood estimators (MLE), uniformly minimum variance unbiased estimators (UMVUE), and best affine equivariant estimators (BAEE), are inadmissible. We have given sufficient conditions under which the dominating estimators are derived. Under the generalized Pitman closeness criterion, a Stein-type improved estimator is proposed. As an application, we have considered special sampling schemes such as type-II censoring, progressive type-II censoring, and record values. Finally, we perform a simulation study to compare the risk performance of the improved estimators

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