1. Solving IPs – Implicit Enumeration Similar to Binary IP Branch and Bound General Idea: Fixed variables – those for which a value has been fixed. Free Variable – variables which whose values are unspecified. Completion – when all variables have been assigned a value. Upper Bound (minimization problem) – Best feasible solution found thus far. Lower Bound (minimization problem) – Optimal solution for a relaxed problem at a given node (e.g. some variables fixed, some free).

2. Solving IPs – Implicit Enumeration Example Sequencing Problem: Sequence a series of jobs to minimize the maximum lateness (L max ) for a set of jobs to be processed on a single machine. Each job belongs to a given part family. Jobs are denoted with an (i,j) subscript indicating the j th job from family i. Family (i)Job (j) Processing Time (p ij ) Due Date (d ij ) 1135 1227 2146 2239 3144 32210 L max = Max{L ij } L ij = C ij – d ij C ij is the completion time of job ij.

3. Solving IPs – Implicit Enumeration Example Sequencing Problem cont: Also, when starting the processing of a new family, a family setup time is incurred. All jobs are ready to be scheduled at time 0. Family (i) Setup TIme 12 22 32 Optimality Condition: All jobs within a family must be sequenced in earliest due date order (EDD).

4. Solving IPs – Implicit Enumeration Step 1 – Find an initial Upper bound What is a good upper bound? Step 2 – Perform implicit enumeration. Start building partial sequences and fathom nodes if lower bound for partial sequence exceeds upper bound. Update upper bound whenever a better value is found for a completion. What is a good lower bounding scheme?

5. Solving IPs – Implicit Enumeration Insert Hand Slides For Example Problem

6. Solving IPs – Using Lindo Statements – INT and GIN INT – forces binary solution (1 or 0) for decision variable. GIN – forces non-negative integer (0,1,2,3,4…) for decision variable. Knapsack problem: max 15xa + 20xb + 18xc + 13xd + 12xe st 18xa + 10xb + 21xc + 11xd + 11xe <= 50 end int xa int xb int xc int xd int xe