Combinations of more than two drugs are routinely used for the treatment of pathogens and tumors. High-order combinations may be chosen due to their non-overlapping resistance mechanisms or for favorable drug interactions. Synergistic/antagonistic interactions occur when the combination has a higher/lower effect than the sum of individual drug effects. The standard treatment of Mycobacterium tuberculosis (Mtb) is an additive cocktail of three drugs which have different targets. Herein, we experimentally measured all 190 pairwise interactions among 20 antibiotics against Mtb growth. We used the pairwise interaction data to rank all possible high-order combinations by strength of synergy/antagonism. We used drug interaction profile correlation as a proxy for drug similarity to establish exclusion criteria for ideal combination therapies. Using this ranking and exclusion design (R/ED) framework, we modeled ways to improve the standard 3-drug combination with the addition of new drugs. We applied this framework to find the best 4-drug combinations against drug-resistant Mtb by adding new exclusion criteria to R/ED. Finally, we modeled alternating 2-order combinations as a cycling treatment and found optimized regimens significantly reduced the overall effective dose. R/ED provides an adaptable framework for the design of high-order drug combinations against any pathogen or tumor.