Assignment Problem
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The document discusses the Hungarian method for solving assignment problems. It begins by defining an assignment problem and providing examples. It then explains the steps of the Hungarian method, which involves reducing the cost matrix to find the optimal assignment that minimizes total cost. Three example problems are provided and solved using the Hungarian method. The key steps are row reduction, column reduction, and eliminating zeros with lines to reach the optimal solution.
Assignment Problem
- 1. PGDM(2nd Trimester) Compiled By : Prof Amit Kumar
- 2. An Assignment problem is a particular case of a transportation problem where the sources are assignees and the destination are tasks. Each assignee is to be assigned exactly one task Such kind of problem of assignment arises because the resources that are such as men, machine, etc, have varying degree of efficiency for performing different activities(these have to be judges on the basis of time, cost or profit which also different for different task) The problem is : How should the assignments be made so as to optimize the given objective? Some of the assignment problem which we are going to face off : Workers to Machine, Job Seeker to Job Providers, Salesman to Different sales area,Clerks to various check counters, classes to rooms, vehicle to routes, Investment to returns, contract to bidders, Flight Crews Scheduling etc. Compiled By : Prof Amit Kumar
- 3. a) Enumeration Method : In this method , a list of all possible assignment among the given resources (men, machine etc.) and activities(jobs, sales area, worker etc) is prepared. If there are nth worker/jobs , then there are n! possible assignment b) Simplex Method : In this technique, assignment can be formulated as 0 and 1 Integer Programming Problem, by simplex table.( A complex one solve manually) Compiled By : Prof Amit Kumar
- 4. Transportation Problem : An assignment problem can be form into transportation problem. But here only n assignment possible, which make the assignment problem degenerate.(so it’s not a good technique at all for assignment cases) Hungarian Method* : It is developed by Hungarian Mathematician D.Konig ; its provide us with an efficient method of finding the optimal solution , without having direct comparison. It works on the principle of reducing the given cost matrix(opportunity loss) to a matrix of opportunity cost. Compiled By : Prof Amit Kumar
- 5. The project work consists of four major jobs for which an equal number of contractors have submitted tenders. The tenders amount quoted (in lakh of rupees) is given in the matrix. Find the assignment which minimizes the total cost of the project when each contractor has to be assigned at least one job by hungarian method. Compiled By : Prof Amit Kumar Jobs A B C d Contrac tors C1 10 24 30 15 C2 16 22 28 12 C3 12 20 32 10 C4 9 26 34 16
- 6. Step 1 : No. of Assignee (row)= No. of Task (Column) Step 2 : Check whether the case minimize or maximize** Step 3 : Row Reduction Step 4 : Column Reduction-After we get Opportunity Cost table Step 5 : Now eliminate(cut) Zero by horizontal or vertical lines Compiled By : Prof Amit Kumar
- 7. Step 1 : No. of row = No. of column ; if not we have to introduce dummy make it equal(all values of dummy is Zero) Step 2 : Case of Minimization can be directly start using step 3 ; But in case of maximization cases one more step involved that is step2A in which highest value from the whole table is get subtracted so the opportunity loss table made Compiled By : Prof Amit Kumar
- 8. Step 3 : Row Reduction (choosing minimum from each row and then subtracted from their respective row) Step 4 : Column Reduction (after Row reduction from the reduced table , choosing minimum from each column and then subtracted from their respective column) Step 5 : Now zero elimination by using vertical or horizontal lines (maximum lines upto = max. Row = max. Column)(Try to use less lines to cut) Step 6** : Now a moment when all lines(max lines) were used , we called the moment of Optimal solution/assignment, assign square boxes to each zero so that assignment process can start. Compiled By : Prof Amit Kumar
- 9. Problem 2 : A construction company has requested bids for subcontracts on five different projects. Five companies have responded and their Bids are represented below : Determine the minimum cost assignment of subcontracts to bidders , assuming that each bidder can receive only one contract (Answer : Rs.1,55,000) Compiled By : Prof Amit Kumar Bid Amount (‘000s Rs) I II III IV V B I D D E R S 1 41 72 39 52 25 2 22 29 49 65 81 3 27 39 60 51 40 4 45 50 48 52 37 5 29 40 45 26 30
- 10. Problem 3 : In the modification of a plant layout of a factory four new machines M1, M2, M3 and M4 are to be installed in a machine shop. There are five vacant places A,B,C,D and E available. Because of limited space, machine M2 cannot be placed at C and M3 cannot be placed at A. The Cost of locating the machine at a place (in 000’s INR) is as follow : Find the optimal Assignment schedule and which location will remain vacant. (Rs.32000) Compiled By : Prof Amit Kumar Location A B C D E M A C H INE M1 9 11 15 10 11 M2 12 9 --- 10 9 M3 --- 11 14 11 7 M4 14 8 12 7 8
- 11. Steps Involved : Reduce the whole table by the highest value , then we get opportunity loss table. After getting opportunity loss table, if there is dummy required then its must be provided, thereafter we will apply Hungarian method. On assigning the task from the optimal table, pick up the value from the original table Compiled By : Prof Amit Kumar