1. Unit Four: Assignment Model
Yalew Mamo
The Way Forward!

2. Introduction
• The basic objective of an assignment problem is to assign
n number of resources to n number of activities so as to
minimize the total cost or to maximize the total profit of
allocation in such a way that the measure of effectiveness
is optimized.
• The assignment model can be applied in many decision-
making processes like determining optimum processing
time in machine operators and jobs, effectiveness of
teachers and subjects, designing of good plant layout, etc.

3. Formulation of the Problem
• Let there are n jobs and n persons are available with
different skills. If the cost of doing jth work by ith person is
Cij. Then the cost matrix is given in the table 1 below:

4. Continued
• Now the problem is which work is to be assigned to whom so that the
cost of completion of work will be minimum. Mathematically, we can
express the problem as follows:
•
• To minimize z (cost) = 𝑖=1
𝑛
𝑗=1
𝑛
𝐶𝑖𝑗 𝑋𝑖𝑗; [𝑖 = 1,2, … 𝑛; 𝑗 = 1,2, … 𝑛
•
• Where xij =
1; 𝑖𝑓 𝑖𝑡ℎ 𝑝𝑟𝑠𝑜𝑛 𝑖𝑠 𝑎𝑠𝑠𝑖𝑔𝑛𝑒𝑑 𝑗𝑡ℎ 𝑤𝑜𝑟𝑘
0; 𝑖𝑓 𝑖𝑡ℎ 𝑝𝑒𝑟𝑠𝑜𝑛 𝑖𝑠 𝑛𝑜𝑡 𝑎𝑠𝑠𝑖𝑔𝑛𝑒𝑑 𝑡ℎ𝑒 𝑗𝑡ℎ𝑤𝑜𝑟𝑘
•
• with the restrictions
• 𝑖=1
𝑛
𝑋𝑖𝑗 = 1; 𝑗 = 1,2, . . 𝑛, 𝑖. 𝑒. , 𝑖𝑡ℎ 𝑝𝑒𝑟𝑠𝑜𝑛 𝑤𝑖𝑙𝑙 𝑑𝑜 𝑜𝑛𝑙𝑦 𝑜𝑛𝑒 𝑤𝑜𝑟𝑘
• 𝑗=1
𝑛
𝑋𝑖𝑗 = 1; 𝑖 = 1,2, . . 𝑛, 𝑖. 𝑒. , 𝑗𝑡ℎ 𝑤𝑜𝑟𝑘 𝑤𝑖𝑙𝑙 𝑏𝑒 𝑑𝑜𝑛𝑒 𝑜𝑛𝑙𝑦 𝑏𝑦 𝑜𝑛𝑒 𝑝𝑒𝑟𝑠𝑜𝑛

5. Comparison between Transportation
Problem and Assignment Problem
• Now let us see what are the similarities and differences
between Transportation problem and Assignment
Problem.
• Similarities
• 1. Both are special types of linear programming problems.
• 2. Both have objective function, structural constraints, and
non-negativity constraints. And the relationship between
variables and constraints are linear.
• 3. Both are basically minimization problems. For
converting them into maximization problem same
procedure is used.

6. Differences
Transportation Problem
• 1. The problem may have
rectangular matrix or square
matrix.
• 2. The rows and columns
may have any number of
allocations depending on the
rim conditions.
• 3. The basic feasible
solution is obtained by
northwest corner method or
matrix minimum method or
VAM
• 4. The optimality test is
given by stepping stone
method or by MODI method.
Assignment Problem
• 1. The matrix of the problem
must be a square matrix.
• 2. The rows and columns
must have one to one
allocation. Because of this
property, the matrix must be
a square matrix.
• 3. The basic feasible
solution is obtained by
Hungarian method or
Flood's technique or by
Assignment algorithm.
• 4. Optimality test is given by
drawing minimum number of
horizontal and vertical lines
to cover all the zeros in the
matrix.

7. Continued
Transportation Problem
• 5. The basic feasible
solution must have m + n
– 1 allocation.
• 6. The rim requirement
may have any numbers
(positive numbers).
• 7. In transportation
problem, the problem
deals with one commodity
being moved from various
origins to various
destinations.
Assignment Problem
• 5. Every column and row
must have at least one
zero. And one machine is
assigned to one job and
vice versa.
• 6. The rim requirements
are always 1 each for
every row and one each
for every column.
• 7. Here row represents
jobs or machines and
columns represent
machines or jobs.

8. Types of Assignment Problem
• The assignment problems are of two types. It can be
either
• (i) Balanced or
• (ii) Unbalanced.
• If the number of rows is equal to the number of columns
or if the given problem is a square matrix, the problem is
termed as a balanced assignment problem. If the given
problem is not a square matrix, the problem is termed as
an unbalanced assignment problem.
• If the problem is an unbalanced one, add dummy rows
/dummy columns as required so that the matrix becomes
a square matrix or a balanced one. The cost or time
values for the dummy cells are assumed as zero.

9. Approach to Solution
• Let us consider a simple example and try to understand
the approach to solution and then discuss complicated
problems.
• 1. Solution by visual method
• In this method, first allocation is made to the cell having
lowest element. If there is more than one cell having
smallest element, tie exists and allocation may be made
to any one of them first and then second one is selected.
In such cases, there is a possibility of getting alternate
solution to the problem. This method is suitable for a
matrix of size 3 × 3 or 4 × 4. More than that, we may face
difficulty in allocating.

10. Continued
• There are 3 jobs A, B, and C and three machines X, Y,
and Z. All the jobs can be processed on all machines. The
time required for processing job on a machine is given
below in the form of matrix. Make allocation to minimize
the total processing time.
Machines (time in hours)
• Allocation: A to X, B to Y and C to Z and the total time =
11 + 13 +12 = 36 hours. (Since 11 is least, Allocate A to X,
12 is the next least, Allocate C to Z)
Jobs X Y Z
A 11 16 21
B 20 13 17
C 13 15 12

11. Continued
• 2. Solving the assignment problem by enumeration
• Let us take the same problem and workout the solution.
Machines (time in hours)
Jobs X Y Z
A 11 16 21
B 20 13 17
C 13 15 12
S.No Assignment Total cost in Birr.
1 AX BY CZ 11 + 13 + 12 = 36
2 AX BZ CY 11 + 17 + 15 = 43
3 AY BX CZ 16 + 20 + 12 = 48
4 AY BZ CX 16 + 17 + 13 = 46
5 AZ BY CX 21 + 13 + 13 = 47
6 AZ BX CY 21 + 20 + 15 = 56

12. Continued
• 3. Solution by Transportation method
• Let us take the same example and get the solution and
see the difference between transportation problem and
assignment problem. The rim requirements are 1 each
because of one to one allocation.
Machines (Time in hours)
Jobs X Y Z Available
A 11 16 21 1
B 20 13 17 1
C 13 15 12 1
Requirement 1 1 1 3
By using northwest corner method the assignments are:
Machines (Time in hours)
Jobs X Y Z Available
A 1 Ɛ 1
B 1 Ɛ 1
C 1 1
Requirement 1 1 1 3

13. Continued
• As the basic feasible solution must have m + n – 1
allocation, we have to add 2 epsilons. Next we have to
apply optimality test by MODI to get the optimal answer.
• This is a time consuming method. Hence it is better to go
for assignment algorithm to get the solution for an
assignment problem.

14. 4. Hungarian Method for Solving
Assignment Problem
Formulation and Solution of the Assignment problem
Assignment Problem
• A machine toll company decides to make four
subassemblies through four contractors. Each contractor
is to receive only one subassembly. The cost of each
subassembly is determined by the bids submitted by each
contractor and is shown in table below in hundreds of birr.
Subassemblies
Contractors
1 2 3 4
1 15 13 14 17
2 11 12 15 13
3 13 12 10 11
4 15 17 14 16

15. Continued
• (i) Formulate the mathematical model for the problem
• (ii) Show that the assignment model is a special case of
the transportation model
• (iii) Assign the difference subassemblies to contractors so
as to minimize the total cost.
• (i) Formulation of the Model
• Step I
• Key decision is what to whom i.e., which subassembly be
assigned to which contractor or what are the ‘n’ optimum
assignments on 1-1 basis.
• Step II
• Feasible alternatives are n! possible arrangement for n x
n assignment situation. In the given situation there are 4!

16. Continued
• Step III
• Objective is to minimize the total cost involved
• i.e., minimize
• Z= 𝑖=1
𝑛
𝑗=1
𝑛
𝐶𝑖𝑗 𝑋𝑖𝑗 = 𝑖=1
4
𝑗=1
4
=
15𝑋11 + 13𝑋12 + 14𝑋13 + 17𝑋14 +
11𝑋21 + 12𝑋22 + 15𝑋23 + 13𝑋24
+ 13𝑋31 + 12𝑋32 + 10𝑋33 + 11𝑋34 +
15𝑋41 + 17𝑋42 + 14𝑋43 + 16𝑋44
• Step IV
• Constraints:
• (a) Constraints on subassemblies are (b) Constraints on contractors are
• X11+ X12+ X13+X14 = 1 X11+ X21+ X31+X41 = 1
• X21+ X22+ X23+X24 = 1 X12+ X22+ X32+X42 = 1
• X31+ X32+ X33+X34 = 1 X13+ X23+ X33+X43 = 1
• X41+ X42+ X43+X44 = 1 X14+ X24+ X34+X44 = 1

17. Continued
• (ii) Comparing this model with the transportation
model, we find that ai = 1, i = 1,2,3,4, and bj = 1, j =
1,2,3,4. Thus, the assignment model can be represented
as in table below. Therefore, the assignment model is a
special case of the transportation model in which
• (a) All right- hand side constraints in the constraints are
unity i.e., ai =1, bj =1.
Subassemblies
(jobs, tasks or
requirements)
Contractors
1 2 3 4 Supply ai
1
1
1
1
1 15 13 14 17
2 11 12 15 13
3 13 12 10 11
4 15 17 14 16
Demand bj 1 1 1 1
(b) All coefficients of Xij in the constraints are unity.
(c) m = n

18. Continued
• (iii) Solution of the Model
• We shall apply the food’s technique for solving the assignment
problems. This techniques also known as the Hungarian
Method or Reducing Matrix Method consists the following
steps.
• Step I
• Prepare a square Matrix: since the situation involves a square
matrix, this step is not necessary.
• Step II
• Deduce the Matrix: this involves the following sub steps:
• Sub step 1: In the effectiveness matrix, subtract the minimum
element of each raw from all the elements of that row. The
resulting reduced matrix will have at least one zero element in
each row. Check if there is at least one zero element in each
column also. If so, stop here. If not proceed to subset 2.

19. Continued
• Sub step 2: Mark the columns that do not have zero
element. Now subtract the minimum element of each such
column from all the elements of that column.

20. Continued
• Step III
• Check if the Optimal assignment can be made in the
current solution or not
• Basis for making this check is that if the minimum number
of lines crossing all zeros in low than n (in our example n
=4), then an optional assignment cannot be made in the
current solution. If it is equal to n (=4), then optimal
assignment can be made in the current solution.
Subassemblies
Contractors
1 2 3 4
1 2 1 3
2 1 4 1
3 3 2 0
4 1 3 1
0
0
0
0

21. Continued
• The optimal assignment can be made in the current
solution. Thus minimum total cost is
• = birr (13x1 + 11x1 + 11x1 +14x1) x 100 = 4,900 birr.
• Subassembly 1 - Contractor 2,
• Subassembly 2 - Contractor 1,
• Subassembly 1 - Contractor 4,
• Subassembly 1 - Contractor 3
• The minimal cost of 4900 birr can be determined by
summing up all elements that were subtracted during the
solution procedure i.e., [(13+11+10+14) +1] x 100= 4900.

22. Example
• Four different jobs can be done on four different
machines. The set-up and take-down time costs are
assumed to be prohibitively high for changeovers. The
matrix below gives the costs in birrs of producing job i on
machine j.
• (i) How should the jobs be assigned to the various
machines so that the total cost is minimized?
Machines
Jobs
M1 M2 M3 M4
J1 5 7 11 6
J2 8 5 9 6
J3 4 7 10 7
J4 10 4 8 3

23. Continued
• Reduce the Matrix: this involves the following sub steps:
• Step III
• Check if the Optimal assignment can be made in the
current solution or not
M1 M2 M3 M4
J1 0 2 6 1
J2 3 0 4 1
J3 0 3 6 3
J4 7 1 5 0
Matrix after sub step 1
(contains no zero in column 3
M1 M2 M3 M4
First
Feasible
Solution
J1 0 2 2 1
J2 3 0 0 1
J3 0 3 2 3
J4 7 1 1 0
Matrix after sub step 2
M1 M2 M3 M4
J1 2 2 1
J2 3 0 1
J3 0 3 2 3
J4 7 1 1
0
0
0

24. Continued
• Sub step 4 Mark (✓) the rows for which assignment has not
been made. In our problem it is the third row.
• Sub step 5: Mark (✓) columns (not already marked) which
have zeros in marked rows. Thus column 1 is marked (✓)
• Sub step 6: Mark (✓) rows (not already marked) which have
zeros in the marked columns. Thus row 1 is marked (✓)
• Sub step 8: draw liens through all unmarked rows and
through all marked columns.

25. Continued
• Step IV
• Iterate Towards Optimality

26. Continued
• Step VI
• Iterate Towards Optimality

27. Continued
• As there is assignment in each row and in each column,
optimal assignment can be made in the current solution.
Hence optimal assignment policy is
• Job J1 should be assigned to machine M1,
• Job J2 should be assigned to machine M2,
• Job J3 should be assigned to machine M3,
• Job J4 should be assigned to machine M4,
• And optimal cost = (5+5+10+3) = 23 birr.