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OPERATIONAL RESEARCH
TOPIC: BUSINESS APPLICATIONS OF ASSIGNMENT MODEL AND TRAVELLING
SALESMAN MODEL
SUBMITTED TO: DR.SUBHOJIT BHATTACHARYA
SUBMITTED BY :
AKHILESH KR MISHRA
PGDM MARKETING
XAVIER INSTITUTE OF SOCIAL SERVICE[XISS];
RANCHI
Introduction
 It involves assignment of people to projects, jobs to machines, workers to
jobs and teachers to classes etc., while minimizing the total assignment
costs.
 One of the important characteristics of assignment problem is that only
one job (or worker) is assigned to one machine (or project).
 An assignment problem is a special type of linear programming problem
where the objective is to minimize the cost or time of completing a
number of jobs by a number of persons.
Examples
 Aadhunik spices company has four men available for work on four
separate jobs. Only one man can work on any one job. The cost of
assigning each man to each job is given in the following table. The
objective is to assign men to jobs such that the total cost of
assignment is minimum.
 Step 1
 Identify the minimum element in each row and subtract it from every
element of that row.
 Step 2
 Identify the minimum element in each column and subtract it from
every element of that column.
 Make the assignment for the reduced matrix obtain from steps 1 and 2 in
the following way:
 Draw the minimum number of vertical and horizontal lines necessary
to cover all the zeros in the reduced matrix obtained from last step
Since the number of assignments is equal to the number
of rows (& columns), this is the optimal solution.
The total cost of assignment = A1 + B4 + C2 + D3
Substitute the values from original table: 20 + 17 + 24 + 17
= 78.
The Travelling Salesman Problem (TSP) is an NP-hard problem
in combinatorial optimization studied in operations
research and theoretical computer science. Given a list of cities and
their pairwise distances, the task is to find a shortest possible tour
that visits each city exactly once.
The problem was first formulated as a mathematical problem in
1930 and is one of the most intensively studied problems in
optimization. It is used as a benchmark for many optimization
methods. Even though the problem is computationally difficult, a
large number of heuristics and exact methods are known, so that
some instances with tens of thousands of cities can be solved.
THE TRAVELLING SALESMAN PROBLEM
The TSP has several applications even in its purest formulation, such
as planning,logistics, and the manufacture of microchips. Slightly
modified, it appears as a sub-problem in many areas, such as DNA
sequencing. In these applications, the conceptcity represents, for
example, customers, soldering paints, or DNA fragments, and the
concept distance represents travelling times or cost, or a similarity
measure between DNA fragments. In many applications, additional
constraints such as limited resources or time windows make the
problem considerably harder.
APPLICATIONS OF TSP
A travelling salesman of the above firm has to visit five cities. He wishes to
start from a particular city, visit one city once and then return to his starting
point. The travelling cost (in '000 Rupees) of each city from a particular city is
given below:
Emertxe IT Firm Co Ltd
The cities are encoded as A=Vellore,B=Chennai,C=Tirupati ,D=Pondicherry ,E=Bangalore
Now the salesman thinks what should be the sequence of his visit so that the cost is minimum?
Salesman solves the given travelling salesman problem as an assignment problem by Hungarian
method of assignment, an optimal solution is shown. However, this solution is not the solution to
the travelling salesman problem as it gives the sequence A — E — A. This violates the condition
that salesmen can visit each city only once.
The 'next best' solution to the problem which also satisfies this extra condition of
unbroken sequence to all cities, can he obtained by bringing the next (non-zero)
minimum element, i.e. 1 into the solution.
Case 1 Make the unit assignment in the cell (A, B) instead of zero
assignment in the cell (A, E) and delete row A and column so as to
eliminate the possibility of any other assignment in row A and column IN
make the assignments in the usual manner. The resulting assignments
are -:
The solution gives the sequence: A-> B,B->C,C->D, D->E, E->A.
Corresponding to this feasible solution is Rs.15,000.
Case 2: If we make the assignment in the cell (D, C) instead of (D. E),
then no feasible solution is obtained in terms of zeros or which may give
cost less than Rs.15,000. Hence the best solution is: A->B,B-> C,C->D,D-
>E,E-> A, and the total cost associated with this solution is Rs.15,000
The Rollick’s Ice Cream Company has a distribution depot in
Greater Kailash Part I for distributing ice-cream in South
Delhi. There are four vendors located in different parts of
South Delhi (call them A. B, C and D) who have to be
supplied ice-cream every day. The following matrix displays
the distances (in kilometers) between the depot and the
four vendors:
How company van follow so that the total distance travelled
is minimized?
Company solves the given travelling salesman problem as an assignment
problem by using the Hungarian method of assignment
All four cases of possible solution to travelling salesman problem tried with
element 0.5 as well as zero element do not provide a desired solution. Thus,
we look for the 'next best' solution by bringing the next (non-zero) element 1
along, with 0.5 and zero elements into the solution. Make the assignment in
cell (Vendor C, Vendor B) and delete row 4 and column 3.In the remaining
matrix assignments are made using zeros and 0.5 in the cells.
The set of assignments given is a feasible solution to the travelling
salesman problem. The route for the salesman is: Vendor C->
Vendor B-> Depot ->Vendor D->Vendor A ->Vendor C. The total
distance (in km) to be covered in this sequence is 15 km
In this presentation, we presented the dynamic Hungarian algorithm for the
assignment problem
with changing costs and travelling salesman method for path optimization.
The goal of the algorithm is to efficiently repair an optimal
assignment when changes in the edge costs occur, as can happen in many
real-world
scenarios.
The dynamic Hungarian algorithm
is useful in any domain that requires the repeated solution of the
assignment problem
when costs may change dynamically. In future work, we will apply this
algorithm to
various transportation-related problems
CONCLUSION

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OPERATIONAL RESEARCH BUSINESS APPLICATIONS

  • 1. OPERATIONAL RESEARCH TOPIC: BUSINESS APPLICATIONS OF ASSIGNMENT MODEL AND TRAVELLING SALESMAN MODEL SUBMITTED TO: DR.SUBHOJIT BHATTACHARYA SUBMITTED BY : AKHILESH KR MISHRA PGDM MARKETING XAVIER INSTITUTE OF SOCIAL SERVICE[XISS]; RANCHI
  • 2. Introduction  It involves assignment of people to projects, jobs to machines, workers to jobs and teachers to classes etc., while minimizing the total assignment costs.  One of the important characteristics of assignment problem is that only one job (or worker) is assigned to one machine (or project).  An assignment problem is a special type of linear programming problem where the objective is to minimize the cost or time of completing a number of jobs by a number of persons.
  • 3. Examples  Aadhunik spices company has four men available for work on four separate jobs. Only one man can work on any one job. The cost of assigning each man to each job is given in the following table. The objective is to assign men to jobs such that the total cost of assignment is minimum.
  • 4.  Step 1  Identify the minimum element in each row and subtract it from every element of that row.
  • 5.  Step 2  Identify the minimum element in each column and subtract it from every element of that column.
  • 6.  Make the assignment for the reduced matrix obtain from steps 1 and 2 in the following way:
  • 7.  Draw the minimum number of vertical and horizontal lines necessary to cover all the zeros in the reduced matrix obtained from last step
  • 8. Since the number of assignments is equal to the number of rows (& columns), this is the optimal solution. The total cost of assignment = A1 + B4 + C2 + D3 Substitute the values from original table: 20 + 17 + 24 + 17 = 78.
  • 9. The Travelling Salesman Problem (TSP) is an NP-hard problem in combinatorial optimization studied in operations research and theoretical computer science. Given a list of cities and their pairwise distances, the task is to find a shortest possible tour that visits each city exactly once. The problem was first formulated as a mathematical problem in 1930 and is one of the most intensively studied problems in optimization. It is used as a benchmark for many optimization methods. Even though the problem is computationally difficult, a large number of heuristics and exact methods are known, so that some instances with tens of thousands of cities can be solved. THE TRAVELLING SALESMAN PROBLEM
  • 10. The TSP has several applications even in its purest formulation, such as planning,logistics, and the manufacture of microchips. Slightly modified, it appears as a sub-problem in many areas, such as DNA sequencing. In these applications, the conceptcity represents, for example, customers, soldering paints, or DNA fragments, and the concept distance represents travelling times or cost, or a similarity measure between DNA fragments. In many applications, additional constraints such as limited resources or time windows make the problem considerably harder. APPLICATIONS OF TSP
  • 11. A travelling salesman of the above firm has to visit five cities. He wishes to start from a particular city, visit one city once and then return to his starting point. The travelling cost (in '000 Rupees) of each city from a particular city is given below: Emertxe IT Firm Co Ltd The cities are encoded as A=Vellore,B=Chennai,C=Tirupati ,D=Pondicherry ,E=Bangalore
  • 12. Now the salesman thinks what should be the sequence of his visit so that the cost is minimum? Salesman solves the given travelling salesman problem as an assignment problem by Hungarian method of assignment, an optimal solution is shown. However, this solution is not the solution to the travelling salesman problem as it gives the sequence A — E — A. This violates the condition that salesmen can visit each city only once. The 'next best' solution to the problem which also satisfies this extra condition of unbroken sequence to all cities, can he obtained by bringing the next (non-zero) minimum element, i.e. 1 into the solution.
  • 13. Case 1 Make the unit assignment in the cell (A, B) instead of zero assignment in the cell (A, E) and delete row A and column so as to eliminate the possibility of any other assignment in row A and column IN make the assignments in the usual manner. The resulting assignments are -: The solution gives the sequence: A-> B,B->C,C->D, D->E, E->A. Corresponding to this feasible solution is Rs.15,000. Case 2: If we make the assignment in the cell (D, C) instead of (D. E), then no feasible solution is obtained in terms of zeros or which may give cost less than Rs.15,000. Hence the best solution is: A->B,B-> C,C->D,D- >E,E-> A, and the total cost associated with this solution is Rs.15,000
  • 14. The Rollick’s Ice Cream Company has a distribution depot in Greater Kailash Part I for distributing ice-cream in South Delhi. There are four vendors located in different parts of South Delhi (call them A. B, C and D) who have to be supplied ice-cream every day. The following matrix displays the distances (in kilometers) between the depot and the four vendors: How company van follow so that the total distance travelled is minimized?
  • 15. Company solves the given travelling salesman problem as an assignment problem by using the Hungarian method of assignment
  • 16. All four cases of possible solution to travelling salesman problem tried with element 0.5 as well as zero element do not provide a desired solution. Thus, we look for the 'next best' solution by bringing the next (non-zero) element 1 along, with 0.5 and zero elements into the solution. Make the assignment in cell (Vendor C, Vendor B) and delete row 4 and column 3.In the remaining matrix assignments are made using zeros and 0.5 in the cells. The set of assignments given is a feasible solution to the travelling salesman problem. The route for the salesman is: Vendor C-> Vendor B-> Depot ->Vendor D->Vendor A ->Vendor C. The total distance (in km) to be covered in this sequence is 15 km
  • 17. In this presentation, we presented the dynamic Hungarian algorithm for the assignment problem with changing costs and travelling salesman method for path optimization. The goal of the algorithm is to efficiently repair an optimal assignment when changes in the edge costs occur, as can happen in many real-world scenarios. The dynamic Hungarian algorithm is useful in any domain that requires the repeated solution of the assignment problem when costs may change dynamically. In future work, we will apply this algorithm to various transportation-related problems CONCLUSION