Solution methodologies for the classical Grant van Dieman Friday 30 th November 2007 Supervisor: Prof. JH van Vuuren Co-supervisor: Mr JN Roux assignment problem

Slide 2 Overview  The classical assignment problem  Exact Solution methods  A maximum matching algorithm  Successive shortest path method  Hungarian method  Greedy heuristics  Comparison  Future work

Slide 3 The classical assignment problem  Votaw and Orden (1952)  Assumptions  x ij is 1 if assignee i is assigned to task j and 0 otherwise  The assignment problem is NP complete (Lloyd and Witzenhausen (1986))

Slide 4 The Weapon Target Assignment Problem  Flood (1957)  V j : priority of eliminating target j.  q ij : is the survival probability of target j if it is engaged by weapon i.  x ij =1 if weapon i engage target j and 0 otherwise

Slide 5 Overview  The classical assignment problem  Exact Solution methods  A maximum matching algorithm  Successive shortest path method  Hungarian method  Greedy heuristics  Comparison  Future work

Slide 6 A maximum matching algorithm for weighted bipartite graphs (MWM) q ij V 1 = {assignees} V 2 = {tasks} G :

Slide 7 A maximum matching algorithm for weighted bipartite graphs (MWM) V 1 = {assignees} V 2 = {tasks} q ij M :

Slide 8 Overview  The classical assignment problem  Exact Solution methods  A maximum matching algorithm  Successive shortest path method  Hungarian method  Greedy heuristics  Comparison  Future work

Slide 9 Successive shortest path algorithm (SSP)  Minimum cost flow algorithm  Why this algorithm can be used to solve the assignment problem  The value of x ij will be binary

Slide 10 Overview  The classical assignment problem  Exact Solution methods  Successive shortest path method  A maximum matching algorithm  Hungarian method  Greedy heuristics  Comparison  Future work

Slide 11 Hungarian Method  Kuhn(1955)  Special algorithm for the assignment problem  Construct reduced cost matrix

Slide 12 Overview  The classical assignment problem  Exact Solution methods  Successive shortest path method  A maximum matching algorithm  Hungarian method  Greedy heuristics  Comparison  Future work

Slide 13 Greedy Heuristics  Greedy RTB  Greedy RBT  Greedy RR  Greedy CLR  Greedy CRL  Greedy CR

Slide 14 Overview  The classical assignment problem  Exact Solution methods  Successive shortest path method  A maximum matching algorithm  Hungarian method  Greedy heuristics  Comparison  Future work

Slide 15 Comparisons  Benchmark set 1: JE Beasly (Randomly Generated)  3.4 Ghz, 1024 MB ram, Windows XP

Slide 16 Comparisons

Slide 17 Comparisons

Slide 18 Comparisons  Benchmarks set 2: Randomly Generated in Matlab

Slide 19 Comparisons

Slide 20 Comparisons

Slide 21 Future work  Advanced Heuristics and Meta-heuristics  More exact solution methods  Expand algorithms to solve variations of the assignment problem

Slide 22 References [1] [2] [3] [4] [5]