5.2 Mixed Integer Linear Programming 5.2.2 Implicit Enumeration

Assignment Problem – LP Model

Assignment LP Problem  

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

Forward steps: o-1-2-3 Backtracking step: 3-2-4 Pendant nodes: 3,4,6,7,10,11,13,14

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. At this node, the second descending branch is searched by changing the corresponding variable from 1 to 0. The search process is continued until all pendant nodes are fathomed and each non-pendant node has exactly 2 descending branches.

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.

Standard Form

Zero Completion Test

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

Example

Subproblem P0 All variables are free.

  1

 

 

 

Backtracking!

 

4

 

 

 

 

 

Solution  

In a plant we have 2 production units designated as number 1 and 2, making product 1 and 2, respectively, from the 3 feed stocks (A, B and C). Unit 1 has a maximum capacity of 8000 lb/day, and unit 2 of 10000 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.

Blending Products including Batch Sizes Assume the net revenue after expenses from the manufacture of product 1 is $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?