EXAMPLE 4.2 OPTIMIZATION INVOLVING AN INTEGERVALUED VARIABLE Consider a catalytic regeneration cycle in which there is a simple trade-off between costs incurred during regeneration and the increased revenues due to the regenerated catalyst. Let x1 be the number of days during which the catalyst is used in the reactor and x2 be the number of days for regeneration. The reactor start-up crew is only available in the morning shift, so x1+x2 must be an integer. We assume that the reactor feed flow rate q=(kg/day) is constant as is the cost of the feed C1(kg5), the value of the product C2(kg5), and the regeneration cost C3(5/ regeneration cycle ). We further assume that the catalyst deteriorates gradually according to the linear relation: d=1.0kx1anddavg=1.0(k2x1) where 1.0 represents the weight fraction conversion of feed at the start of the operating cycle, and k is the deterioration factor in units of weight fraction per day. Define an objective function and find the optimal value of x1