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This paper is devoted to the Truck-to-dock Door Assignment Problem. Two integer programming formulations introduced after 2009 are examined. Our review of the literature takes note of the criticisms and limitations addressed to the seminal work of 2009. Although the published adjustments that followed present strong argument and technical background, we have identified several errors, inaccuracies and incompleteness. This paper therefore brings a rigorous presentation and clarification of these two formulations. In particular, both are described, corrected if necessary, analyzed and discussed. From this study, a bi-objective variant derived from these two formulations is proposed, resulting in four formulations in total. They have been implemented and the codes are open-source and available online. Numerical experiments on instances from the literature have been conducted. Among the numerical results collected, we underline the advantage shown by the multi-objective formulations for the quality of the solutions produced.