Results

The best solution produced for the data set sched_3_12 is the following :

Cost: 92
1 -> 3:  2 - 15  [2, 32]
2 -> 3: 15 - 23  [4, 33]
3 -> 2: 15 - 32  [5, 36]
4 -> 1: 24 - 30  [7, 37]
5 -> 2: 32 - 38  [9, 39]
6 -> 2:  0 -  3  [0, 34]
7 -> 1:  3 - 13  [3, 30]
8 -> 1: 16 - 24  [6, 26]
9 -> 3: 23 - 36  [11, 36]
10 -> 1: 30 - 38  [2, 38]
11 -> 2:  3 - 15  [3, 31]
12 -> 1: 13 - 16  [4, 22]

A total of 1604 cuts are added to the MIP problem by 2691 CP model runs and the Branch-and-Bound search explores 12295 nodes. Optimality is proven within a few seconds on a Pentium IV PC.

It is possible to implement this problem entirely either with Xpress Optimizer or with Xpress Kalis. However, already for this three machines – 12 jobs instance the problem is extremely hard for either technique on its own. With CP it is difficult to prove optimality and with MIP the formulation of the disjunctions makes the definition of a large number of binary variables necessary (roughly in the order of number_of_machines · number_of_products²) which makes the problem impracticable to deal with.