6. assignment problems
TRANSCRIPT
ADVANCED OPERATIONS
RESEARCH
By: -Hakeem–Ur–Rehman
IQTM–PU
A ROASSIGNMENT PROBLEMS
ASSIGNMENT PROBLEMS The assignment problem refers to the class of LP problems
that involve determining the most efficient assignment ofresources to tasks
The objective is most often to minimize total costs or totaltime to perform the tasks at hand
One important characteristic of assignment problems isthat only one job or worker can be assigned to onemachine or project
An assignment problem can be viewed as a transportationproblem in which the capacity from each source is 1 andthe demand at each destination is 1
3
Let m be the number of jobs as well as the operators, and tij be the processing time of the job i if it is assigned to the operator j. Here the objective is to assign the jobs to the operators such that the total processing time is minimized.
Operators
Job
1 2 … j … m
1 t11 t12 t1j t1m
2
.
i ti1 tij tim
.
m tm1 tm2 tmj tmm
GENERAL FORMAT OF ASSIGNMENT PROBLEM
4
Examples of assignment problem
Row entity Column entity Cell entity
jobs operators Processing time
Programmer program Processing time
operators machine Processing time
Drivers Routes Travel time
Teachers Subjects Students pass
percentage
5
Assignment problem as a zero-one ( Binary) programming problem
Min Z= c11x11++cijXij+.+cmmXmm=
Subject to x11+…………...+x1m =1x21+…………...+x2m =1……..
xm1+…………...+xmm =1x11+…………...+xm1 =1x12+…………...+xm2 =1………………..
x1m+…………...+xmm =1xij.=0 or 1 for i=1,2….m and j=1,2…..m.
THE HUNGARIAN METHOD (FLOOD’S TECHNIQUE)
Test opportunity cost table to seeif optimal assignments arepossible by drawing the minimumpossible lines on columns and/orrows such that all zeros arecovered
Find opportunity cost
(a) Subtract smallest number ineach row from every number inthat row, then
(b) subtract smallest number ineach column from everynumber in that column
Set up cost table for problem Revise opportunity cost table intwo steps:
(a) Subtract the smallestnumber not covered by a linefrom itself and every otheruncovered number
(b) add this number at everyintersection of any two lines
Optimal solution at zerolocations. Systematically makefinal assignments.
(a) Check each row and columnfor a unique zero and make thefirst assignment in that row orcolumn
(b) Eliminate that row andcolumn and search for anotherunique zero. Make thatassignment and proceed in a likemanner.
Step 1
Notoptimal
Step 2
Optimal
SPECIAL CASES IN ASSIGNMENT PROBLEMS
Unbalanced Problem
Maximum Case
Multiple Alternative Solution
Prohibited Assignment
ASSIGNMENT PROBLEMS USING EXCEL SOLVER
The Fix-It Shop has three rush projects to repair. They have three repairpersons with different talents and abilities. The owner has estimates ofwage costs for each worker for each project. The owner’s objective is toassign the three project to the workers in a way that will result in thelowest cost to the shop. Each project will be assigned exclusively to oneworker.Estimated project repair costs for the Fix-It shop assignment problem
PROJECT
PERSON 1 2 3
Adams $11 $14 $6
Brown 8 10 11
Cooper 9 12 7
ASSIGNMENT PROBLEMS USING EXCEL SOLVER
ASSIGNMENT PROBLEMS USING EXCEL SOLVER
QUESTIONS