Assignment problems Operational Research Level 4

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Assignment problemsOperational Research- Level 4

• Prepared by T.M.J.A.Cooray
• Department of Marthematics

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Introduction

• This is a special type of transportation problem
  in which each source should have the capacity to
  fulfill the demand of any of the destinations.
• In other words any operator would be able perform
  any job regardless of his skills,although the
  cost( or the time taken) will be more if the job
  does not match with operators skill.

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Let m be the number of jobs as well as the
operators, and tij be the processing time of the
job i if it is assigned to the operator j. Here
the objective is to assign the jobs to the
operators such that the total processing time is
minimized.
General format of assignment problem
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• Examples of assignment problem

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Assignment problem as a zero-one ( Binary)
programming problem .

• Min Z c11x11cijXij.cmmXmm
• Subject to x11...x1m 1
• x21...x2m 1
• ..
• xm1...xmm 1
• x11...xm1 1
• x12...xm2 1
• ..
• x1m...xmm 1
• xij.0 or 1 for i1,2.m and j1,2..m.

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Types of assignment problems

• As in transportation problems assignment
  problems also can be balanced ( with equal
  number of rows and columns) or unbalanced.
• When it is unbalanced the necessary number of
  row/s or column/s are added to balance it. That
  is to make a square matrix.
• The values of the cell entries of the dummy rows
  or columns will be made equal to zero.

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Example Assign the 5 operators to the 5 jobs
such that the total processing time is minimized.
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Hungarian method

• Consists of two phases.
• First phase row reductions and column reductions
  are carried out.
• Second phase the solution is optimized in
  iterative basis.

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Phase 1 Row and column reductions

• Step 0 Consider the given cost matrix
• Step 1 Subtract the minimum value of each row
  from the entries of that row, to obtain the next
  matrix.
• Step 2 Subtract the minimum value of each column
  from the entries of that column , to obtain the
  next matrix.
• Treat the resulting matrix as the input for phase
  2.

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Phase 2 Optimization

• Step3 Draw a minimum number of lines to cover
  all the zeros of the matrix.
• Procedure for drawing the minimum number of
  lines
• 3.1 Row scanning
• 1 Starting from the first row ,if theres only
  one zero in a row mark a square round the zero
  entry and draw a vertical line passing through
  that zero. Otherwise skip the row.
• 2.After scanning the last row, check whether all
  the zeros are covered with lines. If yes go to
  step 4. Otherwise do column scanning. Ctd?

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• 3.2 Column scanning.
• 1. Starting from the first column if theres
  only one zero in a column mark a square round
  the zero entry and draw a horizontal line
  passing through that zero. otherwise skip the
  column.
• 2.After scanning the last column, check whether
  all the zeros are covered with lines. If yes go
  to step 4. Otherwise do row scanning. ctd ?

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• Step 4 check whether the number of squares
  marked is equal to the number of rows/columns of
  the matrix.
• If yes go to step 7. Otherwise go to step 5.
• Step 5 Identify the minimum value of the
  undeleted cell values ,say x. Obtain the next
  matrix by the following steps.
• 5.1 Copy the entries covered by the lines ,but
  not on the intersection points.
• 5.2 add x to the intersection points
• 5.3 subtract x from the undeleted cell values.
• Step 6 go to step 3.
• Step 7 optimal solution is obtained as marked
  by the squares

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Maximization problem

• If the problem is a maximization problem ,convert
  the problem into a minimization problem by
  multiplying by -1.
• Then apply the usual procedure of an assignment
  problem.

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Example Assign 4 sales persons to four
different sales regions such that the total
sales is maximized.
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Modified data , after multiplying the cell
entries by -1.
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After step 1
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After step 2
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Phase 2
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• Note that the number of squares is equal to the
  number of rows of the matrix. solution is
  feasible and optimal.
• Result

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Branch and Bound algorithm for the assignment
problem

• Terminology
• K-level number in the branching tree
• For root node k0
• ?-assignment made in the current node of a
  branching tree
• P?k assignment at level k of the branching tree
• A-set of assigned cells up to the node P?k from
  the root node
• V? - lower bound of partial assignment A up to
  P?k
• Such that V?

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• Cij is the cell entity of the cost matrix
• X rows which are not deleted up to node P?k from
  the root node in the branching tree.
• Y columns which are not deleted up to node P?k
• from the root node in the branching tree.
• ?

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Branching guidelines

• 1.At level k,the row marked as k of the
  assignment problem,will be assigned with the best
  column of the assignment problem.
• 2.if there is a lower bound ,then the terminal
  node at the lower most level is to be considered
  for further branching
• 3.stopping ruleif the minimum lower bound
  happens to be at any one of the terminal nodes at
  the (n-1)th level ,the optimality is reached.

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Example Assign the 5 operators to the 5 jobs
such that the total processing time is minimized.
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P?0
P121
P131
P151
P111
51
44
49
40
P141
44
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P?0
P151
P111
P131
P141
40
P121
49
44
49
44
P212
P232
P242
P222
43
49
47
50
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P?0
P151
P111
P131
P141
40
P121
49
44
49
44
P212
P232
P242
P222
43
49
47
50
P323
P333
43
P343
51
47
P424
P444
48
43
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• The optimum allocation will be
• Job operator time
• 1 5 8
• 2 1 7
• 3 3 7
• 4 2 10
• 5 4 11
• 43

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Example ROW SCANNING.
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Example Assign the 5 operators to the 5 jobs
such that the total processing time is minimized.
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Example Assign the 5 operators to the 5 jobs
such that the total processing time is minimized.
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Example Assign the 5 operators to the 5 jobs
such that the total processing time is minimized.
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Example Assign the 5 operators to the 5 jobs
such that the total processing time is minimized.