(3) Assignment Method (Special method)

Assignment Problems and Methods to solve such problems

P1 P2P3 J1 J2 J3 Assignment Problem (AP)

Formulation of Assignment Problem as LPP

Jobs (Activities) n 12..m12..m Persons (Resources) Cases Cases m = n m  n Cij Pij Cij = Cost associated with assigning i th resource to j th activity Assignment Problem

A. Balanced Minimization  m = n with Cij B. Unbalanced Minimization  m  n with Cij C. Balanced Maximization  m = n with Pij D. Unbalanced Maximization  m  n with Pij Categories of Assignment Problems

Jobs (Activities) n 12..n12..n Persons (Resources) Xij = assignment of i th resource to j th activity Assignments are made on one to one basis A. Balanced Minimization Problem

Formulation of Assignment Problem as LPP

Methods to solve Assignment Problems 4. Hungarian Method 1. Enumeration Method 2. Integer Programming Method 3. Transportation Method

Enumeration Method For n x n Problem---- n! 2! (=2) No. of Possible solutions = P1J1, P2J2 OR P1J2, P2J1 For 4 x 4 Problem---- 4! = 24 For 5 x 5 Problem----5! = 120

All Xij = 0 or 1  0-1 Integer programming. 2. Integer Programming Method Difficult to solve manually. For n x n Problem  Variables = n x n. Constraints = n+n = 2n For 5 x 5 Problem  Variables = 25 Constraints = 10

Formulate the problem in Transportation Problem format. 3. Transportation Method n 12..n12..n

4. Hungarian Method 1. Row Deduction 2. Column Deduction 3. Assign zeros If all assignments are over, then STOP Else Go To Step 4 4. Adopt Ticking (Marking) Procedure 5. Modify Cij and Go To Step 3 (Mr. D. Konig - A Hungarian Mathematician) Steps :

Problem 1. Minimization Problem Perform Row Deduction

Row Deduction Perform Column Deduction Perform Column Deduction

Column Deduction Minimum uncrossed no = 3. Modify Numbers Assign Zeros Adopt Tick Marking

Hence optimal solution is P1J1, P2J2, P3J3 giving Z = = 49

Procedure of Assigning Zeros Steps to be followed, after getting at-least one zero in each row & each column. 1. Start with 1st row. If there is only one uncrossed, unassigned zero, assign it & cross other zeros in respective column (of assigned zero), if they exits, else go to next row. Repeat this for next all other rows. 2. If still uncrossed, unassigned zeros are available, start with first column. If there is only one uncrossed, unassigned zero, assign it and cross other zeros in respective row of assigned zero, if they exit, else go to next column. Repeat this for next all other columns. 3. Repeat 1 & 2 until single uncrossed, unassigned zeros are available, while going through rows & columns. 4. If still multiple uncrossed, unassigned zeros are available while going through rows & columns, it indicates that multiple (alternative) optimal solutions are possible. Assign any one remaining uncrossed, unassigned zero & cross remaining zeros in respective row & column of assigned zero. Then go to If required assignments are completed then STOP, else perform “Tick Marking Procedure”. Modify numbers & go to 1.

Marking Procedure (To draw minimum number of lines through zeros.) 1. Tick marks row/rows where there is no assigned zero. 2. Tick mark column/columns w.r.t. crossed zero/zeros in marked row/rows. 3. Tick mark row/rows w.r.t. assigned zero in marked column/columns. 4. Go to to step 2 and repeat the procedure until no zero is available for tick marking. Then 5. Draw lines through unmarked rows and marked columns (Check no. of lines = No. of assign zeros)

Marking Procedure (Continue) How to Modify numbers ? Find minimum number out of uncrossed numbers. 1. Add this minimum number to crossings. 2. Deduct this number from all uncrossed numbers one by one. 3. Keep crossed numbers, on horizontal & vertical lines, except on crossing, same

Jobs (Activities) Persons (Resources) Problem 2. Problem 2.

Jobs (Activities) Persons (Resources)

Jobs (Activities) Persons (Resources) Minimum uncrossed Number = 1. Marking or Ticking procedure

Jobs (Activities) Persons (Resources) Hence optimal solution is P1J2, P2J1, P3J5, P4J3, P5J4 giving Z = = 23 Assigment

Jobs (Activities) Persons (Resources) Problem 3. Problem 3.

Jobs (Activities) Persons (Resources) After Row deduction 

Jobs (Activities) Persons (Resources) After Column deduction  Now modified matrix will be :

Jobs (Activities) Persons (Resources) Hence, this is a case of alternative optimal solutions. Assign any one remaining zero and cross existing zeros in respective row and column, then apply assigning procedure. Hence, one of the optimal solutions is P1J2, P2J4, P3J1, P4J3 giving Z = = 18

Jobs (Activities) Persons (Resources) Hence, another optimal solution is P1J3, P2J2, P3J1, P4J4 giving Z = = 18 To get another optimal solution, assign another remaining zero. P1J3 can not be assigned, as it is already assigned before.

Jobs (Activities) Persons (Resources) D1 and D2 Dummy Jobs are to be introduced to balance the problem B. Unbalanced Minimization Problem

Jobs (Activities) D1 D Persons (Resources) D1 and D2 are Dummy Jobs : Cij = 0

Jobs (Activities) Persons (Resources) Convert Profit Matrix into Relative Cost Matrix C. Balanced Maximization Problem Pij

How to Convert Profit Matrix into Relative Cost Matrix ? 3. (Pij)max - Pij = RCij 1. (Pij) (-1) = RCij 2. 1/Pij = RCij

Jobs (Activities) Persons (Resources) Convert Unbalanced Profit Matrix into Unbalanced Relative Cost Matrix D. Unbalanced Maximization Problem Pij Then Balance the matrix and solve

1. Restriction in Assignment. e.g. Assignment of P3 to J4 is not possible. Then C34 = M (Big Number) 2. Alternative Optimal Solution Possibility -Already considered. 3. Particular assignment is prefixed. e.g. If P3 & J4 prefixed Then Row3 & Column4 are deleted. Typical Cases in Assignment Problems

Recapitulate Recapitulate Methods to solve Assignment Problem Hungerian Method Application Typical cases of Assignment Problems

Traveling Salesman Problem :

[ 1 ]Travelling Salesman Problem : To α α 634 From 316α α α

1 23 Z = 10 (3-5-3, ) Z = 13 (1-2-1, ) Z = 12 ( ) Optimal Solution (No Route) 3 – 5 = M5 – 3 = M (Route)(No Route) Hence, Optimal Route of Travelling Salesman is

[ 2 ]Travelling Salesman Problem : To α α 1054 From 387α α α

1 2 3 Z = 26 (3-5-3, ) Z = 26 (1-4-1, ) Z = 27 ( ) Optimal Solution (No Route) (Route) (No Route) Hence, Optimal Route of Travelling Salesman is Z = 28 ( ) (Route) Z = 28 ( ) (Route) 3 – 5 = M 5 – 3 = M 1 – 4 = M 4 – 1 = M

Home Assignments :

[ 1 ] Three new automatics feed devices (1 – 3) have been made available for existing punch presses. Six presses (A – F) in the plant can be fitted with these devices. The plant superintendent estimates that the increased output, together with the labour saved will result in the following Rupees increase in profits per day. Determine which presses should receive the feed devices so that the benefit to the plant is maximized. A B C D E F

[ 2 ] Consider the problem of assigning four operators (O1–O4) to four machines (M1 to M4). The assignment costs are given in Rupees. Operator 1 cannot be assigned to machine 3. Also operator 3 cannot be assigned to machine 4. Find the optimal assignment. M1 M2 M3 M4 155   If 5th Machine is made available and the respective costs to the four operators are Rs. 2, 1, 2 and 8. Find whether it is economical to replace any of the four existing machines. If so, which ?

[ 3 ]The jobs J1, J2, J3 are to be assigned to three machines M1, M2, M3. The processing costs in Rs. are given in the matrix. Find : ( i ) the best allocation ( ii ) the worst allocation. M1 M2 M3 J J J

[ 4 ]A Head of the Department in a College has the problem of assigning optimally courses to Teachers. He has one Professor, two Assistant Professors and a Teaching Assistant. The expected marks of a common student, for various combinations, are estimated as given in the matrix below. Get the Optimal Solution. C1C1 C2C2 C3C3 C4C4 P AP AP TA

[ 5 ]The personnel manager of ABC Company wants to assign Mr. X, Mr. Y and Mr. Z to regional offices. But the firm also has an opening in its Chennai office and would send one of the three to that branch, if it were more economical than a move to Delhi, Mumbai or Kolkata. It will cost Rs. 2,000 to relocate Mr. X to Chennai, Rs. 1,600 to reallocate Mr. Y there, and Rs. 3,000 to move Mr. Z. What is the optimal assignment of personnel to offices ? Office DelhiMumbaiKolkata Personnel Mr. X1,6002,2002,400 Mr. Y1,0003,2002,600 Mr. Z1,0002,0004,600

[ 6 ]Find optimal assignments for following “Minimization Assignment Problem”. J1J1 J2J2 J3J3 P1P1  10  12 P2P2  18 66  14 P3P3 66 22 22

[ 7 ]The owner of machine shop has 3 persons and 4 jobs were offered with expected profits as given below. Find optimal solution and corresponding total profit. Which job will not be tackled for optimality ? J1J1 J2J2 J3J3 J4J4 P1P P2P P3P

[ 8 ]The owner of machine shop has 3 persons and 4 jobs were offered with expected profits as given below. Find optimal solution and corresponding total profit. Which job will not be tackled for optimality ? J1J1 J2J2 J3J3 J4J4 P1P P2P P3P

For the following problems get all possible Alternative Optimal Solutions.

[ 9 ] [ 10 ]

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[ 1 ]How many “routes“ are feasible for following ‘Minimization Traveling Salesman Problem‘ ? State them. Which route out of these routes is optimal ? To  2116 From217   Travelling Salesman Problems :

[ 2 ]Following is a Minimization Traveling Salesman Problem. Solve this problem considering it as Assignment Problem. Is optimal solution obtained, a solution of Traveling Salesman Problem ? Why ? M M M M M

Thank you For any Query or suggestion : Contact : Dr. D. I. Lalwani Mech. Engg. Dept. S. V. National Institute of Technology (SVNIT), Ichchhanath, Surat – (Gujarat) INDIA. ID Phone No. : (O)