1. 5.2 Mixed Integer Linear Programming
5.2.2 Implicit Enumeration

2. Assignment Problem

3. Assignment Problem
[Theorem] Any basic feasible solution of the assignment problem has every xij equal to either zero or one.
Implication: There are at most n variables that have the value 1.

4. Basic Concept
The basic idea of implicit enumeration is to explicitly enumerate a small subset of all possible solutions while concluding that it is not necessary to explicitly investigate the remaining solutions, because they are either infeasible or will result in an objective value that is inferior to the best integer solution already found.

5. Search Rules
A forward step is defined as the branching process of fixing a free variable to be 1.
The backtracking step is defined as the process to trace back to the origin until encounter the 1st node with only one descending branch.
The search process is continued until all pendant nodes are fathomed and each non-pendant node has exactly 2 branches.

7. Additional Terminologies
Completion: Given a node and a partial solution, a completion of the partial solution is a solution in which values are specified for all the remaining free variables.
Fathom: A partial solution is fathomed by either (1) demonstrating that there are no improving feasible completions or (2) finding the best feasible solution.

8. Standard Form

9. Zero Completion Test

10. Infeasibility Test
If no feasible completions (may or may not be zero completion), then the node should be fathomed.

11. Example

12. Subproblem P0 All variables are free.

13. Subproblem P1 (x1=1) 1

14. Subproblem P1 (x1=1)

16. Subproblem P2 (x1=1, x2=1)

18. Subproblem P3 (x1=1, x2=0)

19. 4

20. Subproblem P4 (x1=1, x2=0, x3=1)

22. Subproblem P5 (x1=1, x2=0, x3=0)

24. Subproblem P6 (x1=0)

26. Subproblem P7 (x1=0, x2=1)

28. Subproblem P8 (x1=0, x2=0)

29. Solution
The optimal integer solution is given by the incumbent solution found at node 7.
Note that the complete tree would have 5 levels and 2^5=32 pendant nodes.

30. Blending Products including Batch Sizes
In a plant we have 2 production units designated as number 1 and 2, making product 1 and 2, respectively, from the 3 feedstocks as shown. Unit 1 has a maximum capacity of 8000 lb/day, and unit 2 of lb/day.
To make 1 lb of product 1 requires 0.4 lb of A and 0.6 lb of B; to make 1 lb of product 2 requires 0.3 lb of B and 0.7 lb of C.
A maximum 6000 lb/day of B is available, but there are no limits on the available amounts of A and C.
Assume the net revenue after expenses from the manufacture of product 1 id $0.16/lb, and of product 2 is $0.20/lb.
How much of products 1 and 2 should be produced per day, assuming that each must be made in batches of 2000lb?