1. Assignment Model Lecture 21 By Dr Arshad Zaheer

2. RECAP  Transportation model (Maximization)  Illustration (Demand > Supply)  Optimal Solution  Modi Method

3. Assignment Model For making one-to-one assignments Such as: – People to tasks – Classes to classrooms – Machines to jobs

4. Example A shop has 3 workers and 3 repair projects Decision: Which worker to assign to which project? Objective: Minimize cost in wages to get all 3 projects done Assignment problem is solved by Hungarian Method.

5. Minimization Step 1: Identify the minimum of each row and subtract identified minimum number from each entry of corresponding row Step 2: IdentifY the minimum of each column and subtract the identified minimum number from each entry of the column. (Note that step 1 and 2 are interchangeable)

6. Minimization Step 3: Start from first row. If there is single zero in any row, assign it (by squaring) and cross all other zeros in the corresponding column. If initially there are two zeros in any row, leave it and move to next row. Similarly start column wise now. If there is single zero in any column, assign it (by squaring) and cross zeros in the corresponding rows. If initially there are two zeros in any column, leave it and move to next column. This step should be continued till all the zeros are either assigned or crossed. If this step don’t give the feasible solution (all jobs are not assigned) go to step 4.

7. Minimization Step 4: In this step, do following steps sequence wise: 1.Tick an un-assigned row. 2.If a ticked row has a zero then tick the corresponding column. 3.If a ticked column has an assignment then tick the corresponding row. 4.Repeat step 2 and 3 till no more ticking is possible. 5.Draw lines through un-ticked rows and ticked columns. The number of lines represent the maximum number of assignments possible.

8. Minimization Step 5: Find the smallest number through which line is not passed. Add this smallest number to the number at the intersection of lined rows and columns. Subtract this number from those numbers through which line is not passing. Rest numbers will remain same (through which only one line is passed i.e. other than intersection points).

9. Minimization Step 6: Repeat step 3. Now the optimal assignment can be made by putting square on the zero entry of the assigned job which will have no alternatives. If optimal assignment is still not possible, Repeat this process (step 3-6) in similar way until all the jobs are assigned. To calculate the total cost, refer these finally assigned zeros to the original pay off matrix.

10. Assignment Model J1J1 J2J2 J3J3 J4J4 M1M1 C 11 C 12 C 13 C 14 M2M2 C 21 C 22 C 23 C 24 M3M3 C 31 C 32 C 33 C 34 M4M4 C 41 C 42 C 43 C 44 M a c h i n e s Cost Jobs

11. Illustrations

12. Problem J1J1 J2J2 J3J3 J4J4 M1M1 15$13 $ 14 $ 17 $ M2M2 11 $ 12 $ 15 $ 13 $ M3M3 12 $ 10 $ 11 $ M4M4 15 $ 17 $ 14 $ 16 $ Requirement: Which job is to be assigned to any one machine to get the minimum cost.

15. Projects P1P2P3 A11$14$6$ Persons B8$10$11$ C9$12$7$ Requirement: Which project is to be assigned to which person to get the minimum cost Problem