Assignment Problem

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Assignment Problem

• Assignment problem is also known as a special
  case of LP problem or transportation problem
  with which unit of demand and supply is 1
• Its LP formulation
• Our objective here is to determine its solution
  using heuristic algorithm similar to what we
  did in the transportation lecture.

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LP formulation
Total 1 1 1 1
Total 1 1
1 1
LP Min 210Xar 90Xaa 180Xad .. 120
Xdc s.t. XarXaaXadXac 1
XarXbrXcrXdr 1
XbrXbaXbdXbc 1 XaaXbaXcaXda
1 XcrXcaXcdXcc 1 XadXbdXcdXdd
1 XdrXdaXddXdc 1 XacXbcXccXdc 1
all Xij 0 or 1 for ia,b,c,d
jr,a,d,c
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Heuristic algorithm

• Its logical flow
• We make use of the opportunity cost concept
• It is defined as follows

How it works?
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Steps

• Step 1 For each column/row, find its minimum
  cost and subtract
• from its respective column/row
• Step 2 Determine its feasible solution by
  crossing
• rows/columns with most 0 values
• Step 3 Solution is obtained if
• total crossed lines total numbers of
  rows/column
• Otherwise,
• select min cost of uncrossed cells and
  subtracting it
• from all uncrossed and add it to
  double crossed cells
• Step 4 Repeat step 4 until solution is
  obtained.

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Example
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Example

• Consider the following example

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Step 1 For row, select its min and subtract
from them
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Step 1
Step 1 for column, select min cost and subtract
from them
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Step 2 Determine its feasible solution
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Step 2
Step 3 Only 3 lines. No good since we need four
lines Thus, we select the min cost for uncrossed
15 We subtract them from
uncrossed cells and add to it double
crossed Which resulting as .
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Steps 3 4
Step 4 We have four line above, Stop. Optimal
solution is obtained Solution is
or
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Important notes
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Important Note

• Note 1 It is a (nxn) matrix
• i.e. total supply total demand
• If not, we add row/column to them
• Note 2 We assign a big value M to
• a route that is not feasible one
• How computer package works?

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Tutorial
10
Tutorial

• Appendix B
• 37, 38, 40, 46

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