I am trying to solve a maximize minimum optimization. I have four different items that each of them has 10 values of Rates and for each value it has a corresponding weight. Then I have a free table with 20 column of time intervals and a capacity of 6 (i.e. 6 rows) for each time interval. I have to select the items for each time interval so that the overall minimum Rate is maximized and the capacity of each time interval is not exceeded.
First I tried to solve a multi knapsack problem (and it should be linear integer programming) but my result is the maximized rate instead of maximized minimum rate. In other words I would like to do a fairness in selection of items so that if two items with high rate is selected for the first time interval and capacity became full, for the second time interval I consider a priority for another two items that are not selected for the first time interval.
My evaluations follow: $$ \text{max} ~~ \epsilon $$
$$ \text{s.t.} ~~ \sum_{t=1}^{20} \sum_{i=1}^4 \sum_{j=1}^{10} \text{Rates}_{i,j} ~ x_{t,i,j} \geq \epsilon $$ $$ \sum_{i=1}^4 \sum_{j=1}^{10} \text{Weight}_{i,j} ~ x_{t,i,j} \leq 6 ~~ \text{for} ~~t \in \text{{1, ... ,20}} $$ $$ \sum_{j=1}^{10} x_{t,i,j} \leq 1 ~~\text{for} ~~ x \in \text{{1,0}} \text{,} ~ i \in \text{{1,2,3,4} and} ~t \in \text{{1, ... ,20}} $$
