Space structures12 designed for zero gravity have no structural weight to support and nothing to which a foundation can be attached.

The structure needs only to withstand vibrations in space. This is accomplished by replacing truss members with dampers at a given number of p sites among j = 1,

c, n candidate locations throughout the structure. Engineering analysis can identify the principal strain modes i = 1,

c, m, and estimate fractions di,j of total modal strain energy imparted in mode i to truss site j. The best design places dampers to absorb as much energy as possible. Specifically we want to maximize the minimum total of di,j reaching chosen sites over all modes i. Formulate an ILP model to choose an optimal design in terms of p, the di,j, and the decision variables 1j = 1,

c, n2 xj ! e 1 if a damper is placed at j 0 otherwise z ! smallest modal d@total

(Hint: Maximize z.)