This paper studies the optimal design of Type-I generalized progressive hybrid censoring schemes for life-testing experiments. The design problem involves simultaneously determining the inspection time, the guaranteed number of failures, and the progressive censoring scheme. First we develop a cost-constrained optimization framework for determining the optimal censoring scheme. Structural properties of the A-optimality criterion and the experimental cost with respect to the inspection time and the guaranteed number of failures are established. It reveals that they are conflicting behaviors which enables to develop an efficient search algorithm that substantially reduces the computational burden. Building on these theoretical results, a multi-objective optimization model is proposed to simultaneously minimize A-optimality criterion and the experimental cost. A Variable Neighborhood Search (VNS) algorithm is proposed to efficiently determine the optimal progressive removal vector by exploring the feasible design space while avoiding exhaustive enumeration. The resulting compromise designs simultaneously improve estimation precision and reduce experimental cost. In addition, the Shannon differential entropy of the observed lifetime distribution is derived and employed as a complementary information-theoretic measure for evaluating the selected censoring schemes. Numerical studies show that entropy-optimal designs generally differ from A-optimal designs, indicating that Shannon entropy characterizes uncertainty in the observed data rather than estimation precision. The proposed methodology provides an efficient computational framework for optimal life-test design and offers a foundation for future multi-objective optimization incorporating statistical efficiency, experimental cost, and information-theoretic uncertainty.

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