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ECE 307 - Techniques for Engineering Decisions Hungarian Method George Gross Department of Electrical and Computer Engineering University of Illinois at Urbana-Champaign ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved ASSIGNMENT PROBLEM ‰ We are given n machines M1 , M2 , … , Mn ↔ i n jobs J1 , J2 , … , Jn ↔ j cost of doing job j on machine i c ij = Q if job j cannot be done on machine i ‰ Each machine can only one job and each job requires one machine ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved ASSIGNMENT PROBLEM ‰ We wish to determine the optimal match, i.e., the assignment with the lowest total costs of doing the jobs on the n machines ‰ The brute force approach is simply enumeration: consider n = 10 and there are 3,628,800 possible choices! ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved SOLUTION APPROACH ‰ We can, however, introduce categorical decision variables x ij ⎧⎪ = ⎨ ⎪⎩0 job j is assigned to machine i otherwise ‰ And the constraints can be stated as n ∑ j =1 n ∑ i =1 x ij = ∀ i each machine does exactly job x ij = ∀ j each job is assigned to only machine ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved SOLUTION APPROACH ‰ The assignment problem, then, is formulated as n n Z = ∑ ∑ c ij x ij i =1 j =1 s.t n ∑ j =1 n ∑ i =1 x ij = ∀i x ij = ∀j x ij ∈ { , } ∀i, j ‰ Thus, the assignment problem can be viewed as a special case of transportation problem ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved COST MATRIX job j mach i M1 J1 J2 … x 11 x 12 … x 21 M2 x 22 c 22 c 21 … … x n1 x n2 c n1 demands ECE 307 x 1n c n2 1 c 1n … … … … x 2n c 2n … x nn … … Mn Jn c 12 c 11 supplies c nn © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved SIMPLIFIED COST MATRIX ‰ Since demands and supplies are for all assignment problems, we represent the assignment problem by the cost matrix below job j J1 J2 … Jn M1 c 11 c 12 … c 1n M2 c 21 c 22 … c 2n … … … … … Mn c n1 c n2 … c n1 mach i ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved HISTORY OF HUNGARIAN METHOD ‰ First published by Harold Kuhn in 1955 ‰ Based on earlier works of two Hungarian mathematicians, Dénes König and Jenő Egerváry ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved FACT n n n n Z = ∑∑ c ij x ij − k Z = ∑∑ c ij x ij i =1 j =1 i =1 j =1 s.t s.t n ∑ j =1 n ∑ i =1 x ij = ∀ i x ij = ∀ j x ij ∈ { , } ∀ i , j n (i) ∑ j =1 n ∑ i =1 x ij = ∀ i (ii) x ij = ∀ j x ij ∈ { , } ∀ i , j ‰ If x *ij ≤ for i , j ≤ n optimizes problem (i ), then x *ij ≤ for i , j ≤ n also optimizes problem (ii ) ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved BASIC IDEA ‰ The optimal assignment is not affected by a constant added or subtracted from any row of the original assignment cost matrix by the fact in the previous slide and Z = ∑(c n j =1 = n qj n ) − k x qj + n n c ij x ij ∑∑ i =1 j =1 i ≠q n c ij x ij − k ∑ xq j ∑∑ i =1 j =1 j =1 =Z −k ‰ A similar statement holds with respect to the column of the cost matrix ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 10 HUNGARIAN METHOD ‰ The rationale for this operation is that we subtract the smallest value from each element in a row including any element that is covered by a line; to compensate we also need to add an equal value to the element which is covered by the intersection of two lines and therefore the operation keeps the value of the elements not at an intersection unchanged ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 19 EXAMPLE job j J1 J2 J3 J4 M1 10 M2 8 M3 M4 mach i ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 20 EXAMPLE job j J1 J2 J3 J4 M1 10 M2 8 M3 M4 mach i ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 21 EXAMPLE job j J1 J2 J3 J4 M1 M2 3 M3 M4 mach i ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 22 EXAMPLE job j J1 J2 J3 J4 M1 M2 3 M3 M4 mach i ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 23 EXAMPLE job j J1 J2 J3 J4 M1 0 M2 2 M3 M4 2 mach i ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 24 EXAMPLE job j J1 J2 J3 J4 M1 0 M2 2 M3 M4 2 mach i ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 25 EXAMPLE job j J1 J2 J3 J4 M1 0 M2 1 M3 2 M4 0 1 mach i ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 26 EXAMPLE J1 J2 J3 J4 M1 0 M2 1 M3 2 M4 0 1 ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 27 PROBLEM 3-13 ‰ We cast the problem as an assignment with the days being the machines and the courses being the jobs ‰ In order for the assignment problem to be balanced, we introduce an additional course whose costs are zero for each day ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 28 PROBLEM 3-13 job j J1 J2 J3 J4 J5 M1 50 40 60 20 M2 40 30 40 30 M3 60 20 30 20 M4 30 30 20 30 M5 10 20 10 30 mach i ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 29 PROBLEM 3-13 job j J1 J2 J3 J4 J5 M1 40 20 50 0 M2 30 10 30 10 M3 50 20 0 M4 20 10 10 10 M5 0 10 mach i ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 30 PROBLEM 3-13 job j J1 J2 J3 J4 J5 M1 40 20 50 0 M2 30 10 30 10 M3 50 20 0 M4 20 10 10 10 M5 0 10 mach i ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 31 PROBLEM 3-13 J1 J2 J3 J4 J5 M1 40 20 50 10 M2 20 20 0 M3 50 20 10 M4 10 0 0 M5 0 10 10 ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 32 PROBLEM 3-13 J1 J2 J3 J4 J5 M1 40 20 50 10 M2 20 20 0 M3 50 20 10 M4 10 0 0 M5 0 10 10 ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 33 [...]... Reserved 12 EXAMPLE 1 job j J1 J2 J3 J4 M1 10 9 8 7 M2 3 4 5 6 M3 2 1 1 2 M4 4 3 5 6 mach i ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 13 EXAMPLE 1 job j J1 J2 J3 J4 M1 3 2 1 0 M2 3 4 5 6 M3 2 1 1 2 M4 4 3 5 6 mach i ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 14 EXAMPLE 1 job j J1 J2 J3 J4 M1 3 2 1. .. Urbana-Champaign, All Rights Reserved 30 PROBLEM 3 -13 job j J1 J2 J3 J4 J5 M1 40 20 50 0 0 M2 30 10 30 10 0 M3 50 0 20 0 0 M4 20 10 10 10 0 M5 0 0 0 10 0 mach i ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 31 PROBLEM 3 -13 J1 J2 J3 J4 J5 M1 40 20 50 0 10 M2 20 0 20 0 0 M3 50 0 20 0 10 M4 10 0 0 0 0 M5 0 0 0 10 10 ECE 307 © 2008-2009 George Gross, University... 3 2 0 1 M2 0 3 2 3 M3 1 0 2 1 M4 0 1 2 3 mach i ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 22 EXAMPLE 2 job j J1 J2 J3 J4 M1 3 2 0 1 M2 0 3 2 3 M3 1 0 2 1 M4 0 1 2 3 mach i ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 23 EXAMPLE 2 job j J1 J2 J3 J4 M1 3 2 0 0 M2 0 3 2 2 M3 1 0 2 0 M4 0 1 2 2... 2 job j J1 J2 J3 J4 M1 3 2 0 0 M2 0 3 2 2 M3 1 0 2 0 M4 0 1 2 2 mach i ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 25 EXAMPLE 2 job j J1 J2 J3 J4 M1 4 2 0 0 M2 0 2 1 1 M3 2 0 2 0 M4 0 0 1 1 mach i ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 26 EXAMPLE 2 J1 J2 J3 J4 M1 4 2 0 0 M2 0 2 1 1 M3 2 0... Urbana-Champaign, All Rights Reserved 28 PROBLEM 3 -13 job j J1 J2 J3 J4 J5 M1 50 40 60 20 0 M2 40 30 40 30 0 M3 60 20 30 20 0 M4 30 30 20 30 0 M5 10 20 10 30 0 mach i ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 29 PROBLEM 3 -13 job j J1 J2 J3 J4 J5 M1 40 20 50 0 0 M2 30 10 30 10 0 M3 50 0 20 0 0 M4 20 10 10 10 0 M5 0 0 0 10 0 mach i ECE 307 © 2008-2009 George... EXAMPLE 1 job j J1 J2 J3 J4 M1 3 2 1 0 M2 0 1 2 3 M3 1 0 0 1 M4 1 0 2 3 mach i ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 15 EXAMPLE 1 job j J1 J2 J3 J4 M1 3 2 1 0 M2 0 1 2 3 M3 1 0 0 1 M4 1 0 2 3 mach i ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 16 HUNGARIAN METHOD ‰ In general, feasible assignment... Reserved 19 EXAMPLE 2 job j J1 J2 J3 J4 M1 10 9 7 8 M2 5 8 7 8 M3 5 4 6 5 M4 2 3 4 5 mach i ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 20 EXAMPLE 2 job j J1 J2 J3 J4 M1 10 9 7 8 M2 5 8 7 8 M3 5 4 6 5 M4 2 3 4 5 mach i ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 21 EXAMPLE 2 job j J1 J2 J3 J4 M1 3... M3 50 0 20 0 10 M4 10 0 0 0 0 M5 0 0 0 10 10 ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 32 PROBLEM 3 -13 J1 J2 J3 J4 J5 M1 40 20 50 0 10 M2 20 0 20 0 0 M3 50 0 20 0 10 M4 10 0 0 0 0 M5 0 0 0 10 10 ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 33 ... optimal solution ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 11 THE HUNGARIAN METHOD ‰ For each i , we consider the elements i and compute ( c i = min c ij , 1 ≤ j ≤ n ) and subtract c i from each element in row i to get c ij = c ij − c i , 1 ≤ j ≤ n ‰ Then, we do the same procedure to each column ‰ We try to assign jobs only using the machines with... George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 26 EXAMPLE 2 J1 J2 J3 J4 M1 4 2 0 0 M2 0 2 1 1 M3 2 0 2 0 M4 0 0 1 1 ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 27 PROBLEM 3 -13 ‰ We cast the problem as an assignment with the days being the machines and the courses being the jobs ‰ In order for the assignment problem ... COST MATRIX job j mach i M1 J1 J2 … x 11 x 12 … x 21 M2 x 22 c 22 c 21 … … x n1 x n2 c n1 demands ECE 307 x 1n c n2 1 c 1n … … … … x 2n c 2n … x nn … … Mn Jn c 12 c 11 supplies c nn © 2008-2009... represent the assignment problem by the cost matrix below job j J1 J2 … Jn M1 c 11 c 12 … c 1n M2 c 21 c 22 … c 2n … … … … … Mn c n1 c n2 … c n1 mach i ECE 307 © 2008-2009 George Gross, University of... Reserved 12 EXAMPLE job j J1 J2 J3 J4 M1 10 M2 M3 1 M4 mach i ECE 307 © 2008-2009 George Gross, University of Illinois at Urbana-Champaign, All Rights Reserved 13 EXAMPLE job j J1 J2 J3 J4 M1 M2 M3 1