Assignment Problems

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Assignment Problems

• Hungarian method

Step 1 Balance the problem if it is
unbalanced Place an M as the cost element if some
assignment is prohibited Convert into equivalent
min problem if it is a max problem In a given
matrix subtract the smallest element in each row
from every element of that row and do the same in
the column.
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• Step2 In the reduced matrix obtain from step 1,
  subtract the smallest element in each column from
  every element of that column
• Step 3 Make the assignment for the reduced matrix
  obtained from step 1 and step 2
• (all the zeros in rows/columns are either
  marked (?) or (x) and there is exactly one
  assignment in each row and each column. In such a
  case optimum assignment policy for the given
  problem is obtained.
• If there is row or column with out an assignment
  go to the next step.

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• Step 4 Draw the minimum number of vertical and
  horizontal lines necessary to cover all the
  zeros in the reduced matrix obtained from step 3
  by adopting the following procedure.
• (i) mark(v) all rows that do not have assignments
• (ii) Mark (v) all columns (not already marked)
  which have zeros in the marked rows
• (iii) Mark (v) all rows (not already marked) that
  have assignments in marked columns
• (iv) Draw straight lines through all unmarked
  rows and marked columns

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• Step 5 If the number of lines drawn are equal to
• the number of rows or columns, then it is an
  optimum solution ,otherwise go to step 6
• Step 6 Select the smallest element among all the
  uncovered elements. Subtract this smallest
  element from all the uncovered elements an add it
  to the element which lies at the intersection of
  two line. Thus we obtain another reduced matrix
  for fresh assignments.
• Step 7 go the step 3 and repeat the procedure
  until the umber of assignment become equal to the
  number of rows or columns. In such a case, we
  shall observe that row/column has an assignment.
  Thus, the current solution is an optimum solution.

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• Problem 1
• A company centre has got four expert
  programmers. The centre needs four application
  programmes to be developed. The head of the
  computer centre, after studying carefully the
  programmes to be developed, estimate the
  computer time in minutes required by the
  respective experts to develop the application
  programmes as follows.

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• Problem 2Suggest optimum solution to the
  following assignment problem and also the minimum
  cost

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• Problem 3
• A company has taken the third floor of a
  multi-storied building for rent with a view to
  locate one of their zonal offices. There are
  five main rooms in this to be assigned to five
  managers. Each room has it sown advantages and
  disadvantages. Some have windows, some are closer
  to the wash rooms or to the canteen or
  secretarial pool. The rooms are of all different
  sizes and shapes. Each of the five managers were
  asked to rank their room preference amongst the
  rooms 301,302,303,304 and 305. Their preferences
  were recorded in a table as indicated below

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• Most of the managers did not list al the five
  rooms since they were not satisfied with some of
  these rooms and they have left these from the
  list. Assuming that their preferences can be
  quantified by numbers, find out as to which
  manager should be assigned to which room so that
  their total preference ranking is a minimum.

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• Problem 4 A company plans to assign 5 salesmen to
  5 districts in which it operates. Estimates of
  sales revenue in thousands of rupees for each
  salesman in different districts are given in the
  following table. In your opinion, what should be
  the placement of the salesmen if the objective is
  to maximize the expected sales revenue?.

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Expected sales data district wise
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• Problem5 A traveling salesman has to visit 5
  cities. He wishes to start from a particular
  city, visit each city once and then return to his
  starting point. The traveling cost for each city
  from a particular city is given below

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• What is the sequence of visit of the salesman, so
  that the cost is minimum.

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• Problem6. A solicitors firm employs typists on
  hourly piece rate basis for their work . There
  are five typists for service and their charges
  and speeds are different. According to an
  earlier understanding only one job is given to
  one typist and the typist is paid for full hour
  even if he works for a fraction of an hour. Find
  the least cost allocation for the following data.

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• Problem 7ABC airline, operating 7 days a week,
  has given the following time-table. The crews
  must have a minimum lay-over of 5 hours between
  flight. Obtain the pairing of flights that
  minimizes la-overtime away from home . For any
  given pairing the crew will be based at the city
  that results the smallest lay-over.

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• Problem 8.
• A company has four territories and four
  salesmen available for assignment. The
  territories are not equally rich in their sales
  potential. It is estimated that a typical
  salesman operating I each territory would bring
  the following weekly sales.

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• The four salesmen are also considered to differ
  in ability. It is estimated that working under
  the same conditions their yearly sales would be
  proportionately as follows

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• If the criteria is to maximize expected sales,
  the intuitive answer is to assign the best
  salesman to the richest territory, the next best
  salesman to the second richest and so on. Verify
  this answer by the assignment technique.

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