Multiobjective discrete optimization (MODO) techniques, including weight space decomposition, have received increasing attention in the last decade. The primary weight space decomposition technique in the literature is defined for the weighted sum utility function, through which sets of weights are assigned to a subset of the nondominated set. Recent work has begun to study the decomposition defined for weighted Tchebychev utility function, which provides the benefit of including all nondominated images but at the cost of “nice” convexity geometric properties. The current work presents the computational details for this weight space decomposition, which contributes an essential visualization technique for MODOs with three objectives that includes the entire nondominated set and overlapping weight set components. A thorough evaluation of the added value of the new weight set decomposition (in contrast to weighted sum) is con- ducted. Existing box-based MODO algorithms are shown to return insufficient information to compute the weight set decomposition, then the necessary modifications to the algorithm are proven. We further provide a computational study to illustrate the decomposition’s use in evaluating and improving design choices within box-based MODO algorithms.