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    <p>One good way to approach LPs/IPs with a massive number of variables and constraints is to look for ways to group the decision variables in some logical way. Since you have only given a sketch of your problem, here's a solution idea.</p> <p><strong>Approach 1 : Group people into smaller batches</strong></p> <p>Instead of 1M people, think of them as 100 units of 10K people each. So now you only have 2400 (24 x 100) variables. This will get you part of the way there, and note that this won't be the optimal solution, but a good approximation. You can of course make 1000 batches of 1000 people and get a more fine-grained solution. You get the idea.</p> <p><strong>Approach 2: Grouping into cohorts, based on the Costs</strong></p> <p>Take a look at your R_ij's. Presumably you don't have a million different costs. There will typically be only a few unique cost values. The idea is to group many people with the same cost structure into one 'cohort'. Now you solve a much smaller problem - which cohorts go into which hour.</p> <p>Again, once you get the idea you can make it very tractable.</p> <p><strong>Update</strong> Based on OP's comment: By its very nature, making these groups is an <em>approximation technique</em>. There is no guarantee that the optimal solution will be obtained. However, the whole idea of careful grouping (by looking at cohorts with identical or very similar cost structures) is to get solutions as close to the optimal as possible, with far less computational effort.</p> <ul> <li>I should have also added that when scaling (grouping is just one way to scale-down the problem size), the other constants should also be scaled. That is, c_j should also be in the same units (10K). </li> <li>If persons A,B,C cannot be fit into time slot j, then the model will squeeze in as many of those as possible in the lowest cost time slot, and move the others to other slots where the cost is slightly higher, but they can be accommodated.</li> </ul> <p>Hope that helps you going in the right direction.</p>
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    1. COHow would I interpret the constraint that each hour has a finite amount of appointment slots? Lets say each hour has space for 100,000 people, and person A, B, and C's (belonging to batch X) true optimal time is at noon. Then, if we can't fit all of batch X into the noon appointments, A, B, and C will not be scheduled into their optimal time. Is that right?
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    2. CO@JCWong Yes, that is correct. Grouping is an *approximation* technique. I am updating my response to address what you bring up.
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