 in operations research, the glove problem (also known as the condom problemvardi, i. the condom problem. ch. 10 in computational recreations in mathematica. redwood city, ca: addison&ndash;wesley, pp.&nbsp;203&ndash;222, 1991. isbn 0-201-52989-0.) is an optimization problem used as an example that the cheapest capital cost often leads to dramatic increase in operational time, but that the shortest operational time need not be given by the most expensive capital cost. ==problem statement== m doctors are each to examine each of n patients, wearing gloves to avoid contamination. each glove can be used any number of times, but the same side of one glove cannot be exposed to more than one person. gloves can be re-used any number of times, and more than one can be used simultaneously. given m doctors and n patients, the minimum number of gloves g(m&nbsp;,n) required for all the doctors to examine all the patients is given by: * g(m,&nbsp;n) = m + n &minus; 2 if both m,&nbsp;n&nbsp;â¥&nbsp;2 * g(m,&nbsp;1) = m * g(1,&nbsp;n) = n * g(1,&nbsp;1) = 1 ==details== a naive approach would be to estimate the number of gloves as simply g(m,&nbsp;n) =&nbsp;mn. but this number can be significantly reduced by exploiting the fact that each glove has two sides, and it is not necessary to use both sides simultaneously. a better solution can be found by assigning each person his or her own glove, which is to be used for the entire operation. every pairwise encounter is then protected by a double layer. note that the outer surface of the doctors's gloves meets only the inner surface of the patients's gloves. this gives an answer of m&nbsp;+&nbsp;n gloves, which is significantly lower than&nbsp;mn. the makespan with this scheme is k&nbsp;Â·&nbsp;max(m,&nbsp;n), where k is the duration of one pairwise encounter. note that this is exactly the same makespan if mn gloves were used. clearly in this case, increasing capital cost has not produced a shorter operation time. the number g(m,&nbsp;n) may be refined further by allowing asymmetry in the initial distribution of gloves. the best scheme is given by: *doctor # 1 wears n gloves, layered one on top of another. he visits the n patients
Operations_research