This paper develops algorithms to solve strong-substitutes product-mix auctions. That is, it finds competitive equilibrium prices and quantities for agents who use this auction's bidding language to truthfully express their strong-substitutes preferences over an arbitrary number of goods, each of which is available in multiple discrete units. (Strong substitutes preferences are also known, in other literatures, as $M^\natural$-concave, matroidal and well-layered maps, and valuated matroids). Our use of the bidding language, and the information it provides, contrasts with existing algorithms that rely on access to a valuation or demand oracle to find equilibrium. We compute market-clearing prices using algorithms that apply existing submodular minimisation methods. Allocating the supply among the bidders at these prices then requires solving a novel constrained matching problem. Our algorithm iteratively simplifies the allocation problem, perturbing bids and prices in a way that resolves tie-breaking choices created by bids that can be accepted on more than one good. We provide practical running time bounds on both price-finding and allocation, and illustrate experimentally that our allocation mechanism is practical.