04-Unit Four - OR.pptx

Unit Four: Assignment Model
Yalew Mamo
The Way Forward!
Introduction
• The basic objective of an assignment problem is to assign
n number of resources to n number of activities so as to
minimize the total cost or to maximize the total profit of
allocation in such a way that the measure of effectiveness
is optimized.
• The assignment model can be applied in many decision-
making processes like determining optimum processing
time in machine operators and jobs, effectiveness of
teachers and subjects, designing of good plant layout, etc.
Formulation of the Problem
• Let there are n jobs and n persons are available with
different skills. If the cost of doing jth work by ith person is
Cij. Then the cost matrix is given in the table 1 below:
Continued
• Now the problem is which work is to be assigned to whom so that the
cost of completion of work will be minimum. Mathematically, we can
express the problem as follows:
•
• To minimize z (cost) = 𝑖=1
𝑛
𝑗=1
𝑛
𝐶𝑖𝑗 𝑋𝑖𝑗; [𝑖 = 1,2, … 𝑛; 𝑗 = 1,2, … 𝑛
•
• Where xij =
1; 𝑖𝑓 𝑖𝑡ℎ 𝑝𝑟𝑠𝑜𝑛 𝑖𝑠 𝑎𝑠𝑠𝑖𝑔𝑛𝑒𝑑 𝑗𝑡ℎ 𝑤𝑜𝑟𝑘
0; 𝑖𝑓 𝑖𝑡ℎ 𝑝𝑒𝑟𝑠𝑜𝑛 𝑖𝑠 𝑛𝑜𝑡 𝑎𝑠𝑠𝑖𝑔𝑛𝑒𝑑 𝑡ℎ𝑒 𝑗𝑡ℎ𝑤𝑜𝑟𝑘
•
• with the restrictions
• 𝑖=1
𝑛
𝑋𝑖𝑗 = 1; 𝑗 = 1,2, . . 𝑛, 𝑖. 𝑒. , 𝑖𝑡ℎ 𝑝𝑒𝑟𝑠𝑜𝑛 𝑤𝑖𝑙𝑙 𝑑𝑜 𝑜𝑛𝑙𝑦 𝑜𝑛𝑒 𝑤𝑜𝑟𝑘
• 𝑗=1
𝑛
𝑋𝑖𝑗 = 1; 𝑖 = 1,2, . . 𝑛, 𝑖. 𝑒. , 𝑗𝑡ℎ 𝑤𝑜𝑟𝑘 𝑤𝑖𝑙𝑙 𝑏𝑒 𝑑𝑜𝑛𝑒 𝑜𝑛𝑙𝑦 𝑏𝑦 𝑜𝑛𝑒 𝑝𝑒𝑟𝑠𝑜𝑛
Comparison between Transportation
Problem and Assignment Problem
• Now let us see what are the similarities and differences
between Transportation problem and Assignment
Problem.
• Similarities
• 1. Both are special types of linear programming problems.
• 2. Both have objective function, structural constraints, and
non-negativity constraints. And the relationship between
variables and constraints are linear.
• 3. Both are basically minimization problems. For
converting them into maximization problem same
procedure is used.
Differences
Transportation Problem
• 1. The problem may have
rectangular matrix or square
matrix.
• 2. The rows and columns
may have any number of
allocations depending on the
rim conditions.
• 3. The basic feasible
solution is obtained by
northwest corner method or
matrix minimum method or
VAM
• 4. The optimality test is
given by stepping stone
method or by MODI method.
Assignment Problem
• 1. The matrix of the problem
must be a square matrix.
• 2. The rows and columns
must have one to one
allocation. Because of this
property, the matrix must be
a square matrix.
• 3. The basic feasible
solution is obtained by
Hungarian method or
Flood's technique or by
Assignment algorithm.
• 4. Optimality test is given by
drawing minimum number of
horizontal and vertical lines
to cover all the zeros in the
matrix.
Continued
Transportation Problem
• 5. The basic feasible
solution must have m + n
– 1 allocation.
• 6. The rim requirement
may have any numbers
(positive numbers).
• 7. In transportation
problem, the problem
deals with one commodity
being moved from various
origins to various
destinations.
Assignment Problem
• 5. Every column and row
must have at least one
zero. And one machine is
assigned to one job and
vice versa.
• 6. The rim requirements
are always 1 each for
every row and one each
for every column.
• 7. Here row represents
jobs or machines and
columns represent
machines or jobs.
Types of Assignment Problem
• The assignment problems are of two types. It can be
either
• (i) Balanced or
• (ii) Unbalanced.
• If the number of rows is equal to the number of columns
or if the given problem is a square matrix, the problem is
termed as a balanced assignment problem. If the given
problem is not a square matrix, the problem is termed as
an unbalanced assignment problem.
• If the problem is an unbalanced one, add dummy rows
/dummy columns as required so that the matrix becomes
a square matrix or a balanced one. The cost or time
values for the dummy cells are assumed as zero.
Approach to Solution
• Let us consider a simple example and try to understand
the approach to solution and then discuss complicated
problems.
• 1. Solution by visual method
• In this method, first allocation is made to the cell having
lowest element. If there is more than one cell having
smallest element, tie exists and allocation may be made
to any one of them first and then second one is selected.
In such cases, there is a possibility of getting alternate
solution to the problem. This method is suitable for a
matrix of size 3 × 3 or 4 × 4. More than that, we may face
difficulty in allocating.
Continued
• There are 3 jobs A, B, and C and three machines X, Y,
and Z. All the jobs can be processed on all machines. The
time required for processing job on a machine is given
below in the form of matrix. Make allocation to minimize
the total processing time.
Machines (time in hours)
• Allocation: A to X, B to Y and C to Z and the total time =
11 + 13 +12 = 36 hours. (Since 11 is least, Allocate A to X,
12 is the next least, Allocate C to Z)
Jobs X Y Z
A 11 16 21
B 20 13 17
C 13 15 12
Continued
• 2. Solving the assignment problem by enumeration
• Let us take the same problem and workout the solution.
Machines (time in hours)
Jobs X Y Z
A 11 16 21
B 20 13 17
C 13 15 12
S.No Assignment Total cost in Birr.
1 AX BY CZ 11 + 13 + 12 = 36
2 AX BZ CY 11 + 17 + 15 = 43
3 AY BX CZ 16 + 20 + 12 = 48
4 AY BZ CX 16 + 17 + 13 = 46
5 AZ BY CX 21 + 13 + 13 = 47
6 AZ BX CY 21 + 20 + 15 = 56
Continued
• 3. Solution by Transportation method
• Let us take the same example and get the solution and
see the difference between transportation problem and
assignment problem. The rim requirements are 1 each
because of one to one allocation.
Machines (Time in hours)
Jobs X Y Z Available
A 11 16 21 1
B 20 13 17 1
C 13 15 12 1
Requirement 1 1 1 3
By using northwest corner method the assignments are:
Machines (Time in hours)
Jobs X Y Z Available
A 1 Ɛ 1
B 1 Ɛ 1
C 1 1
Requirement 1 1 1 3
Continued
• As the basic feasible solution must have m + n – 1
allocation, we have to add 2 epsilons. Next we have to
apply optimality test by MODI to get the optimal answer.
• This is a time consuming method. Hence it is better to go
for assignment algorithm to get the solution for an
assignment problem.
4. Hungarian Method for Solving
Assignment Problem
Formulation and Solution of the Assignment problem
Assignment Problem
• A machine toll company decides to make four
subassemblies through four contractors. Each contractor
is to receive only one subassembly. The cost of each
subassembly is determined by the bids submitted by each
contractor and is shown in table below in hundreds of birr.
Subassemblies
Contractors
1 2 3 4
1 15 13 14 17
2 11 12 15 13
3 13 12 10 11
4 15 17 14 16
Continued
• (i) Formulate the mathematical model for the problem
• (ii) Show that the assignment model is a special case of
the transportation model
• (iii) Assign the difference subassemblies to contractors so
as to minimize the total cost.
• (i) Formulation of the Model
• Step I
• Key decision is what to whom i.e., which subassembly be
assigned to which contractor or what are the ‘n’ optimum
assignments on 1-1 basis.
• Step II
• Feasible alternatives are n! possible arrangement for n x
n assignment situation. In the given situation there are 4!
Continued
• Step III
• Objective is to minimize the total cost involved
• i.e., minimize
• Z= 𝑖=1
𝑛
𝑗=1
𝑛
𝐶𝑖𝑗 𝑋𝑖𝑗 = 𝑖=1
4
𝑗=1
4
=
15𝑋11 + 13𝑋12 + 14𝑋13 + 17𝑋14 +
11𝑋21 + 12𝑋22 + 15𝑋23 + 13𝑋24
+ 13𝑋31 + 12𝑋32 + 10𝑋33 + 11𝑋34 +
15𝑋41 + 17𝑋42 + 14𝑋43 + 16𝑋44
• Step IV
• Constraints:
• (a) Constraints on subassemblies are (b) Constraints on contractors are
• X11+ X12+ X13+X14 = 1 X11+ X21+ X31+X41 = 1
• X21+ X22+ X23+X24 = 1 X12+ X22+ X32+X42 = 1
• X31+ X32+ X33+X34 = 1 X13+ X23+ X33+X43 = 1
• X41+ X42+ X43+X44 = 1 X14+ X24+ X34+X44 = 1
Continued
• (ii) Comparing this model with the transportation
model, we find that ai = 1, i = 1,2,3,4, and bj = 1, j =
1,2,3,4. Thus, the assignment model can be represented
as in table below. Therefore, the assignment model is a
special case of the transportation model in which
• (a) All right- hand side constraints in the constraints are
unity i.e., ai =1, bj =1.
Subassemblies
(jobs, tasks or
requirements)
Contractors
1 2 3 4 Supply ai
1
1
1
1
1 15 13 14 17
2 11 12 15 13
3 13 12 10 11
4 15 17 14 16
Demand bj 1 1 1 1
(b) All coefficients of Xij in the constraints are unity.
(c) m = n
Continued
• (iii) Solution of the Model
• We shall apply the food’s technique for solving the assignment
problems. This techniques also known as the Hungarian
Method or Reducing Matrix Method consists the following
steps.
• Step I
• Prepare a square Matrix: since the situation involves a square
matrix, this step is not necessary.
• Step II
• Deduce the Matrix: this involves the following sub steps:
• Sub step 1: In the effectiveness matrix, subtract the minimum
element of each raw from all the elements of that row. The
resulting reduced matrix will have at least one zero element in
each row. Check if there is at least one zero element in each
column also. If so, stop here. If not proceed to subset 2.
Continued
• Sub step 2: Mark the columns that do not have zero
element. Now subtract the minimum element of each such
column from all the elements of that column.
Continued
• Step III
• Check if the Optimal assignment can be made in the
current solution or not
• Basis for making this check is that if the minimum number
of lines crossing all zeros in low than n (in our example n
=4), then an optional assignment cannot be made in the
current solution. If it is equal to n (=4), then optimal
assignment can be made in the current solution.
Subassemblies
Contractors
1 2 3 4
1 2 1 3
2 1 4 1
3 3 2 0
4 1 3 1
0
0
0
0
Continued
• The optimal assignment can be made in the current
solution. Thus minimum total cost is
• = birr (13x1 + 11x1 + 11x1 +14x1) x 100 = 4,900 birr.
• Subassembly 1 - Contractor 2,
• Subassembly 2 - Contractor 1,
• Subassembly 1 - Contractor 4,
• Subassembly 1 - Contractor 3
• The minimal cost of 4900 birr can be determined by
summing up all elements that were subtracted during the
solution procedure i.e., [(13+11+10+14) +1] x 100= 4900.
Example
• Four different jobs can be done on four different
machines. The set-up and take-down time costs are
assumed to be prohibitively high for changeovers. The
matrix below gives the costs in birrs of producing job i on
machine j.
• (i) How should the jobs be assigned to the various
machines so that the total cost is minimized?
Machines
Jobs
M1 M2 M3 M4
J1 5 7 11 6
J2 8 5 9 6
J3 4 7 10 7
J4 10 4 8 3
Continued
• Reduce the Matrix: this involves the following sub steps:
• Step III
• Check if the Optimal assignment can be made in the
current solution or not
M1 M2 M3 M4
J1 0 2 6 1
J2 3 0 4 1
J3 0 3 6 3
J4 7 1 5 0
Matrix after sub step 1
(contains no zero in column 3
M1 M2 M3 M4
First
Feasible
Solution
J1 0 2 2 1
J2 3 0 0 1
J3 0 3 2 3
J4 7 1 1 0
Matrix after sub step 2
M1 M2 M3 M4
J1 2 2 1
J2 3 0 1
J3 0 3 2 3
J4 7 1 1
0
0
0
Continued
• Sub step 4 Mark (✓) the rows for which assignment has not
been made. In our problem it is the third row.
• Sub step 5: Mark (✓) columns (not already marked) which
have zeros in marked rows. Thus column 1 is marked (✓)
• Sub step 6: Mark (✓) rows (not already marked) which have
zeros in the marked columns. Thus row 1 is marked (✓)
• Sub step 8: draw liens through all unmarked rows and
through all marked columns.
Continued
• Step IV
• Iterate Towards Optimality
Continued
• Step VI
• Iterate Towards Optimality
Continued
• As there is assignment in each row and in each column,
optimal assignment can be made in the current solution.
Hence optimal assignment policy is
• Job J1 should be assigned to machine M1,
• Job J2 should be assigned to machine M2,
• Job J3 should be assigned to machine M3,
• Job J4 should be assigned to machine M4,
• And optimal cost = (5+5+10+3) = 23 birr.
1 de 27

Recomendados

Linear Programing.pptx por
Linear Programing.pptxLinear Programing.pptx
Linear Programing.pptxAdnanHaleem
19 visualizações18 slides
03-Unit Three -OR.pptx por
03-Unit Three -OR.pptx03-Unit Three -OR.pptx
03-Unit Three -OR.pptxAbdiMuceeTube
3 visualizações61 slides
qadm-ppt-150918102124-lva1-app6892.pdf por
qadm-ppt-150918102124-lva1-app6892.pdfqadm-ppt-150918102124-lva1-app6892.pdf
qadm-ppt-150918102124-lva1-app6892.pdfHari31856
9 visualizações20 slides
Assignment Problem por
Assignment ProblemAssignment Problem
Assignment ProblemNakul Bhardwaj
38.2K visualizações20 slides
AMA_Assignment Theory notes por
AMA_Assignment Theory notesAMA_Assignment Theory notes
AMA_Assignment Theory notesCA Niraj Thapa
776 visualizações7 slides
Assignment Chapter - Q & A Compilation by Niraj Thapa por
Assignment Chapter  - Q & A Compilation by Niraj ThapaAssignment Chapter  - Q & A Compilation by Niraj Thapa
Assignment Chapter - Q & A Compilation by Niraj ThapaCA Niraj Thapa
45K visualizações59 slides

Mais conteúdo relacionado

Similar a 04-Unit Four - OR.pptx

transportation-model.ppt por
transportation-model.ppttransportation-model.ppt
transportation-model.pptanubhuti18
115 visualizações40 slides
Linear programming por
Linear programmingLinear programming
Linear programmingKrantee More
4.8K visualizações31 slides
chapter 3.pptx por
chapter 3.pptxchapter 3.pptx
chapter 3.pptxDejeneDay
19 visualizações53 slides
Asssignment problem por
Asssignment problemAsssignment problem
Asssignment problemMamatha Upadhya
129 visualizações17 slides
Assignment problems por
Assignment problemsAssignment problems
Assignment problemsRoshan Mammen
40K visualizações35 slides
Transportation model and assignment model por
Transportation model and assignment modelTransportation model and assignment model
Transportation model and assignment modelpriyanka yadav
31.1K visualizações37 slides

Similar a 04-Unit Four - OR.pptx(20)

transportation-model.ppt por anubhuti18
transportation-model.ppttransportation-model.ppt
transportation-model.ppt
anubhuti18115 visualizações
Linear programming por Krantee More
Linear programmingLinear programming
Linear programming
Krantee More4.8K visualizações
chapter 3.pptx por DejeneDay
chapter 3.pptxchapter 3.pptx
chapter 3.pptx
DejeneDay19 visualizações
Asssignment problem por Mamatha Upadhya
Asssignment problemAsssignment problem
Asssignment problem
Mamatha Upadhya129 visualizações
Assignment problems por Roshan Mammen
Assignment problemsAssignment problems
Assignment problems
Roshan Mammen40K visualizações
Transportation model and assignment model por priyanka yadav
Transportation model and assignment modelTransportation model and assignment model
Transportation model and assignment model
priyanka yadav31.1K visualizações
Ap for b.tech. (mechanical) Assignment Problem por Prashant Khandelwal
Ap for b.tech. (mechanical) Assignment Problem Ap for b.tech. (mechanical) Assignment Problem
Ap for b.tech. (mechanical) Assignment Problem
Prashant Khandelwal3.1K visualizações
Transportation model por msn007
Transportation modelTransportation model
Transportation model
msn00745.4K visualizações
A0280115(1) por prabhat k prasad
A0280115(1)A0280115(1)
A0280115(1)
prabhat k prasad976 visualizações
A Comparative Analysis Of Assignment Problem por Jim Webb
A Comparative Analysis Of Assignment ProblemA Comparative Analysis Of Assignment Problem
A Comparative Analysis Of Assignment Problem
Jim Webb8 visualizações
unit-4-dynamic programming por hodcsencet
unit-4-dynamic programmingunit-4-dynamic programming
unit-4-dynamic programming
hodcsencet385 visualizações
daa-unit-3-greedy method por hodcsencet
daa-unit-3-greedy methoddaa-unit-3-greedy method
daa-unit-3-greedy method
hodcsencet2.1K visualizações
Hungarian Assignment Problem por VivekSaurabh7
Hungarian Assignment ProblemHungarian Assignment Problem
Hungarian Assignment Problem
VivekSaurabh716 visualizações
Assignment.pdf#___text=Assignment problem is a special LPP which deals,offs a... por vishwasmahajan7
Assignment.pdf#___text=Assignment problem is a special LPP which deals,offs a...Assignment.pdf#___text=Assignment problem is a special LPP which deals,offs a...
Assignment.pdf#___text=Assignment problem is a special LPP which deals,offs a...
vishwasmahajan789 visualizações
Mb0048 operations research por smumbahelp
Mb0048  operations researchMb0048  operations research
Mb0048 operations research
smumbahelp110 visualizações
(PBL) Transportation Prblm.pdf por AkashKatiyar22
(PBL) Transportation Prblm.pdf(PBL) Transportation Prblm.pdf
(PBL) Transportation Prblm.pdf
AkashKatiyar2219 visualizações
A brief study on linear programming solving methods por MayurjyotiNeog
A brief study on linear programming solving methodsA brief study on linear programming solving methods
A brief study on linear programming solving methods
MayurjyotiNeog34 visualizações
Mb0048 operations research por smumbahelp
Mb0048  operations researchMb0048  operations research
Mb0048 operations research
smumbahelp243 visualizações
Mb0048 operations research por smumbahelp
Mb0048  operations researchMb0048  operations research
Mb0048 operations research
smumbahelp36 visualizações
Simplex Algorithm por Muhammad Kashif
Simplex AlgorithmSimplex Algorithm
Simplex Algorithm
Muhammad Kashif2.7K visualizações

Mais de AbdiMuceeTube

8904314.ppt por
8904314.ppt8904314.ppt
8904314.pptAbdiMuceeTube
12 visualizações86 slides
Chapter-1-Capital Structure Policy and Leverage.pptx por
Chapter-1-Capital Structure Policy and Leverage.pptxChapter-1-Capital Structure Policy and Leverage.pptx
Chapter-1-Capital Structure Policy and Leverage.pptxAbdiMuceeTube
2 visualizações46 slides
16450253.ppt por
16450253.ppt16450253.ppt
16450253.pptAbdiMuceeTube
3 visualizações29 slides
OR CH 3 Transportation and assignment problem.pptx por
OR CH 3 Transportation and assignment problem.pptxOR CH 3 Transportation and assignment problem.pptx
OR CH 3 Transportation and assignment problem.pptxAbdiMuceeTube
5 visualizações70 slides
Operations-Research-2.pdf por
Operations-Research-2.pdfOperations-Research-2.pdf
Operations-Research-2.pdfAbdiMuceeTube
1 visão4 slides
370_13735_EA221_2010_1__1_1_Linear programming 1.ppt por
370_13735_EA221_2010_1__1_1_Linear programming 1.ppt370_13735_EA221_2010_1__1_1_Linear programming 1.ppt
370_13735_EA221_2010_1__1_1_Linear programming 1.pptAbdiMuceeTube
3 visualizações73 slides

Mais de AbdiMuceeTube(13)

8904314.ppt por AbdiMuceeTube
8904314.ppt8904314.ppt
8904314.ppt
AbdiMuceeTube12 visualizações
Chapter-1-Capital Structure Policy and Leverage.pptx por AbdiMuceeTube
Chapter-1-Capital Structure Policy and Leverage.pptxChapter-1-Capital Structure Policy and Leverage.pptx
Chapter-1-Capital Structure Policy and Leverage.pptx
AbdiMuceeTube2 visualizações
16450253.ppt por AbdiMuceeTube
16450253.ppt16450253.ppt
16450253.ppt
AbdiMuceeTube3 visualizações
OR CH 3 Transportation and assignment problem.pptx por AbdiMuceeTube
OR CH 3 Transportation and assignment problem.pptxOR CH 3 Transportation and assignment problem.pptx
OR CH 3 Transportation and assignment problem.pptx
AbdiMuceeTube5 visualizações
370_13735_EA221_2010_1__1_1_Linear programming 1.ppt por AbdiMuceeTube
370_13735_EA221_2010_1__1_1_Linear programming 1.ppt370_13735_EA221_2010_1__1_1_Linear programming 1.ppt
370_13735_EA221_2010_1__1_1_Linear programming 1.ppt
AbdiMuceeTube3 visualizações
Chapter 4 Audit of acquisition and payment cycle.pptx por AbdiMuceeTube
Chapter 4 Audit of acquisition and payment cycle.pptxChapter 4 Audit of acquisition and payment cycle.pptx
Chapter 4 Audit of acquisition and payment cycle.pptx
AbdiMuceeTube56 visualizações
PPT-Introduction to Economics last-1.pptx por AbdiMuceeTube
PPT-Introduction to  Economics last-1.pptxPPT-Introduction to  Economics last-1.pptx
PPT-Introduction to Economics last-1.pptx
AbdiMuceeTube23 visualizações
adv ass.pptx por AbdiMuceeTube
adv ass.pptxadv ass.pptx
adv ass.pptx
AbdiMuceeTube8 visualizações
TP AUDITContents.docx por AbdiMuceeTube
TP AUDITContents.docxTP AUDITContents.docx
TP AUDITContents.docx
AbdiMuceeTube4 visualizações
government-accounting-1.docx por AbdiMuceeTube
government-accounting-1.docxgovernment-accounting-1.docx
government-accounting-1.docx
AbdiMuceeTube66 visualizações
CH-1 Advanced FA .pptx por AbdiMuceeTube
CH-1 Advanced FA .pptxCH-1 Advanced FA .pptx
CH-1 Advanced FA .pptx
AbdiMuceeTube36 visualizações

Último

R. S GROUP.pptx por
R. S GROUP.pptxR. S GROUP.pptx
R. S GROUP.pptxitzrajeshsuthar
11 visualizações14 slides
MechMaf Shipping LLC por
MechMaf Shipping LLCMechMaf Shipping LLC
MechMaf Shipping LLCMechMaf Shipping LLC
70 visualizações288 slides
Valuation Quarterly Webinar Dec23.pdf por
Valuation Quarterly Webinar Dec23.pdfValuation Quarterly Webinar Dec23.pdf
Valuation Quarterly Webinar Dec23.pdfFelixPerez547899
70 visualizações12 slides
SplitMetrics at APS Berlin por
SplitMetrics at APS BerlinSplitMetrics at APS Berlin
SplitMetrics at APS BerlinVikaVlasova1
35 visualizações8 slides
Steele_D&O Summit Keynote.pptx por
Steele_D&O Summit Keynote.pptxSteele_D&O Summit Keynote.pptx
Steele_D&O Summit Keynote.pptxbradgallagher6
13 visualizações16 slides
Quandoo - Passion - Matthias M.pptx por
Quandoo - Passion - Matthias M.pptxQuandoo - Passion - Matthias M.pptx
Quandoo - Passion - Matthias M.pptxMatthias Maximilian
25 visualizações5 slides

Último(20)

R. S GROUP.pptx por itzrajeshsuthar
R. S GROUP.pptxR. S GROUP.pptx
R. S GROUP.pptx
itzrajeshsuthar11 visualizações
Valuation Quarterly Webinar Dec23.pdf por FelixPerez547899
Valuation Quarterly Webinar Dec23.pdfValuation Quarterly Webinar Dec23.pdf
Valuation Quarterly Webinar Dec23.pdf
FelixPerez54789970 visualizações
SplitMetrics at APS Berlin por VikaVlasova1
SplitMetrics at APS BerlinSplitMetrics at APS Berlin
SplitMetrics at APS Berlin
VikaVlasova135 visualizações
Steele_D&O Summit Keynote.pptx por bradgallagher6
Steele_D&O Summit Keynote.pptxSteele_D&O Summit Keynote.pptx
Steele_D&O Summit Keynote.pptx
bradgallagher613 visualizações
Quandoo - Passion - Matthias M.pptx por Matthias Maximilian
Quandoo - Passion - Matthias M.pptxQuandoo - Passion - Matthias M.pptx
Quandoo - Passion - Matthias M.pptx
Matthias Maximilian25 visualizações
On the Concept of Discovery Power of Enterprise Modeling Languages and its Re... por Ilia Bider
On the Concept of Discovery Power of Enterprise Modeling Languages and its Re...On the Concept of Discovery Power of Enterprise Modeling Languages and its Re...
On the Concept of Discovery Power of Enterprise Modeling Languages and its Re...
Ilia Bider16 visualizações
3Q23_EN.pdf por irhcs
3Q23_EN.pdf3Q23_EN.pdf
3Q23_EN.pdf
irhcs18 visualizações
Learning from Failure_ Lessons from Failed Startups.pptx por Codeventures
Learning from Failure_ Lessons from Failed Startups.pptxLearning from Failure_ Lessons from Failed Startups.pptx
Learning from Failure_ Lessons from Failed Startups.pptx
Codeventures19 visualizações
December 2023 - Meat on the Bones por NZSG
December 2023 - Meat on the BonesDecember 2023 - Meat on the Bones
December 2023 - Meat on the Bones
NZSG32 visualizações
Amazing Opportunities: PCD Pharma Franchise in Kerala.pptx por SaphnixMedicure1
Amazing Opportunities: PCD Pharma Franchise in Kerala.pptxAmazing Opportunities: PCD Pharma Franchise in Kerala.pptx
Amazing Opportunities: PCD Pharma Franchise in Kerala.pptx
SaphnixMedicure125 visualizações
[1Slide] Event Report AWS ReInvent 2023 - Q stands out por Holger Mueller
[1Slide] Event Report AWS ReInvent 2023 - Q stands out[1Slide] Event Report AWS ReInvent 2023 - Q stands out
[1Slide] Event Report AWS ReInvent 2023 - Q stands out
Holger Mueller15 visualizações
Promoting the SEO to the C-Suite por Ash Nallawalla
Promoting the SEO to the C-SuitePromoting the SEO to the C-Suite
Promoting the SEO to the C-Suite
Ash Nallawalla17 visualizações
Navigating the Complexity of Derivatives Valuation 📈 por ValAdvisor
Navigating the Complexity of Derivatives Valuation 📈Navigating the Complexity of Derivatives Valuation 📈
Navigating the Complexity of Derivatives Valuation 📈
ValAdvisor18 visualizações
2023 Tracking Volunteers in Bloomerang.pdf por Bloomerang
2023 Tracking Volunteers in Bloomerang.pdf2023 Tracking Volunteers in Bloomerang.pdf
2023 Tracking Volunteers in Bloomerang.pdf
Bloomerang26 visualizações
23.12.07 Bloomerang - 2023-12-06 21.39.56.pdf por Bloomerang
23.12.07 Bloomerang - 2023-12-06 21.39.56.pdf23.12.07 Bloomerang - 2023-12-06 21.39.56.pdf
23.12.07 Bloomerang - 2023-12-06 21.39.56.pdf
Bloomerang122 visualizações
2024-cio-agenda-ebook.pdf por Alex446314
2024-cio-agenda-ebook.pdf2024-cio-agenda-ebook.pdf
2024-cio-agenda-ebook.pdf
Alex44631414 visualizações

04-Unit Four - OR.pptx

  • 1. Unit Four: Assignment Model Yalew Mamo The Way Forward!
  • 2. Introduction • The basic objective of an assignment problem is to assign n number of resources to n number of activities so as to minimize the total cost or to maximize the total profit of allocation in such a way that the measure of effectiveness is optimized. • The assignment model can be applied in many decision- making processes like determining optimum processing time in machine operators and jobs, effectiveness of teachers and subjects, designing of good plant layout, etc.
  • 3. Formulation of the Problem • Let there are n jobs and n persons are available with different skills. If the cost of doing jth work by ith person is Cij. Then the cost matrix is given in the table 1 below:
  • 4. Continued • Now the problem is which work is to be assigned to whom so that the cost of completion of work will be minimum. Mathematically, we can express the problem as follows: • • To minimize z (cost) = 𝑖=1 𝑛 𝑗=1 𝑛 𝐶𝑖𝑗 𝑋𝑖𝑗; [𝑖 = 1,2, … 𝑛; 𝑗 = 1,2, … 𝑛 • • Where xij = 1; 𝑖𝑓 𝑖𝑡ℎ 𝑝𝑟𝑠𝑜𝑛 𝑖𝑠 𝑎𝑠𝑠𝑖𝑔𝑛𝑒𝑑 𝑗𝑡ℎ 𝑤𝑜𝑟𝑘 0; 𝑖𝑓 𝑖𝑡ℎ 𝑝𝑒𝑟𝑠𝑜𝑛 𝑖𝑠 𝑛𝑜𝑡 𝑎𝑠𝑠𝑖𝑔𝑛𝑒𝑑 𝑡ℎ𝑒 𝑗𝑡ℎ𝑤𝑜𝑟𝑘 • • with the restrictions • 𝑖=1 𝑛 𝑋𝑖𝑗 = 1; 𝑗 = 1,2, . . 𝑛, 𝑖. 𝑒. , 𝑖𝑡ℎ 𝑝𝑒𝑟𝑠𝑜𝑛 𝑤𝑖𝑙𝑙 𝑑𝑜 𝑜𝑛𝑙𝑦 𝑜𝑛𝑒 𝑤𝑜𝑟𝑘 • 𝑗=1 𝑛 𝑋𝑖𝑗 = 1; 𝑖 = 1,2, . . 𝑛, 𝑖. 𝑒. , 𝑗𝑡ℎ 𝑤𝑜𝑟𝑘 𝑤𝑖𝑙𝑙 𝑏𝑒 𝑑𝑜𝑛𝑒 𝑜𝑛𝑙𝑦 𝑏𝑦 𝑜𝑛𝑒 𝑝𝑒𝑟𝑠𝑜𝑛
  • 5. Comparison between Transportation Problem and Assignment Problem • Now let us see what are the similarities and differences between Transportation problem and Assignment Problem. • Similarities • 1. Both are special types of linear programming problems. • 2. Both have objective function, structural constraints, and non-negativity constraints. And the relationship between variables and constraints are linear. • 3. Both are basically minimization problems. For converting them into maximization problem same procedure is used.
  • 6. Differences Transportation Problem • 1. The problem may have rectangular matrix or square matrix. • 2. The rows and columns may have any number of allocations depending on the rim conditions. • 3. The basic feasible solution is obtained by northwest corner method or matrix minimum method or VAM • 4. The optimality test is given by stepping stone method or by MODI method. Assignment Problem • 1. The matrix of the problem must be a square matrix. • 2. The rows and columns must have one to one allocation. Because of this property, the matrix must be a square matrix. • 3. The basic feasible solution is obtained by Hungarian method or Flood's technique or by Assignment algorithm. • 4. Optimality test is given by drawing minimum number of horizontal and vertical lines to cover all the zeros in the matrix.
  • 7. Continued Transportation Problem • 5. The basic feasible solution must have m + n – 1 allocation. • 6. The rim requirement may have any numbers (positive numbers). • 7. In transportation problem, the problem deals with one commodity being moved from various origins to various destinations. Assignment Problem • 5. Every column and row must have at least one zero. And one machine is assigned to one job and vice versa. • 6. The rim requirements are always 1 each for every row and one each for every column. • 7. Here row represents jobs or machines and columns represent machines or jobs.
  • 8. Types of Assignment Problem • The assignment problems are of two types. It can be either • (i) Balanced or • (ii) Unbalanced. • If the number of rows is equal to the number of columns or if the given problem is a square matrix, the problem is termed as a balanced assignment problem. If the given problem is not a square matrix, the problem is termed as an unbalanced assignment problem. • If the problem is an unbalanced one, add dummy rows /dummy columns as required so that the matrix becomes a square matrix or a balanced one. The cost or time values for the dummy cells are assumed as zero.
  • 9. Approach to Solution • Let us consider a simple example and try to understand the approach to solution and then discuss complicated problems. • 1. Solution by visual method • In this method, first allocation is made to the cell having lowest element. If there is more than one cell having smallest element, tie exists and allocation may be made to any one of them first and then second one is selected. In such cases, there is a possibility of getting alternate solution to the problem. This method is suitable for a matrix of size 3 × 3 or 4 × 4. More than that, we may face difficulty in allocating.
  • 10. Continued • There are 3 jobs A, B, and C and three machines X, Y, and Z. All the jobs can be processed on all machines. The time required for processing job on a machine is given below in the form of matrix. Make allocation to minimize the total processing time. Machines (time in hours) • Allocation: A to X, B to Y and C to Z and the total time = 11 + 13 +12 = 36 hours. (Since 11 is least, Allocate A to X, 12 is the next least, Allocate C to Z) Jobs X Y Z A 11 16 21 B 20 13 17 C 13 15 12
  • 11. Continued • 2. Solving the assignment problem by enumeration • Let us take the same problem and workout the solution. Machines (time in hours) Jobs X Y Z A 11 16 21 B 20 13 17 C 13 15 12 S.No Assignment Total cost in Birr. 1 AX BY CZ 11 + 13 + 12 = 36 2 AX BZ CY 11 + 17 + 15 = 43 3 AY BX CZ 16 + 20 + 12 = 48 4 AY BZ CX 16 + 17 + 13 = 46 5 AZ BY CX 21 + 13 + 13 = 47 6 AZ BX CY 21 + 20 + 15 = 56
  • 12. Continued • 3. Solution by Transportation method • Let us take the same example and get the solution and see the difference between transportation problem and assignment problem. The rim requirements are 1 each because of one to one allocation. Machines (Time in hours) Jobs X Y Z Available A 11 16 21 1 B 20 13 17 1 C 13 15 12 1 Requirement 1 1 1 3 By using northwest corner method the assignments are: Machines (Time in hours) Jobs X Y Z Available A 1 Ɛ 1 B 1 Ɛ 1 C 1 1 Requirement 1 1 1 3
  • 13. Continued • As the basic feasible solution must have m + n – 1 allocation, we have to add 2 epsilons. Next we have to apply optimality test by MODI to get the optimal answer. • This is a time consuming method. Hence it is better to go for assignment algorithm to get the solution for an assignment problem.
  • 14. 4. Hungarian Method for Solving Assignment Problem Formulation and Solution of the Assignment problem Assignment Problem • A machine toll company decides to make four subassemblies through four contractors. Each contractor is to receive only one subassembly. The cost of each subassembly is determined by the bids submitted by each contractor and is shown in table below in hundreds of birr. Subassemblies Contractors 1 2 3 4 1 15 13 14 17 2 11 12 15 13 3 13 12 10 11 4 15 17 14 16
  • 15. Continued • (i) Formulate the mathematical model for the problem • (ii) Show that the assignment model is a special case of the transportation model • (iii) Assign the difference subassemblies to contractors so as to minimize the total cost. • (i) Formulation of the Model • Step I • Key decision is what to whom i.e., which subassembly be assigned to which contractor or what are the ‘n’ optimum assignments on 1-1 basis. • Step II • Feasible alternatives are n! possible arrangement for n x n assignment situation. In the given situation there are 4!
  • 16. Continued • Step III • Objective is to minimize the total cost involved • i.e., minimize • Z= 𝑖=1 𝑛 𝑗=1 𝑛 𝐶𝑖𝑗 𝑋𝑖𝑗 = 𝑖=1 4 𝑗=1 4 = 15𝑋11 + 13𝑋12 + 14𝑋13 + 17𝑋14 + 11𝑋21 + 12𝑋22 + 15𝑋23 + 13𝑋24 + 13𝑋31 + 12𝑋32 + 10𝑋33 + 11𝑋34 + 15𝑋41 + 17𝑋42 + 14𝑋43 + 16𝑋44 • Step IV • Constraints: • (a) Constraints on subassemblies are (b) Constraints on contractors are • X11+ X12+ X13+X14 = 1 X11+ X21+ X31+X41 = 1 • X21+ X22+ X23+X24 = 1 X12+ X22+ X32+X42 = 1 • X31+ X32+ X33+X34 = 1 X13+ X23+ X33+X43 = 1 • X41+ X42+ X43+X44 = 1 X14+ X24+ X34+X44 = 1
  • 17. Continued • (ii) Comparing this model with the transportation model, we find that ai = 1, i = 1,2,3,4, and bj = 1, j = 1,2,3,4. Thus, the assignment model can be represented as in table below. Therefore, the assignment model is a special case of the transportation model in which • (a) All right- hand side constraints in the constraints are unity i.e., ai =1, bj =1. Subassemblies (jobs, tasks or requirements) Contractors 1 2 3 4 Supply ai 1 1 1 1 1 15 13 14 17 2 11 12 15 13 3 13 12 10 11 4 15 17 14 16 Demand bj 1 1 1 1 (b) All coefficients of Xij in the constraints are unity. (c) m = n
  • 18. Continued • (iii) Solution of the Model • We shall apply the food’s technique for solving the assignment problems. This techniques also known as the Hungarian Method or Reducing Matrix Method consists the following steps. • Step I • Prepare a square Matrix: since the situation involves a square matrix, this step is not necessary. • Step II • Deduce the Matrix: this involves the following sub steps: • Sub step 1: In the effectiveness matrix, subtract the minimum element of each raw from all the elements of that row. The resulting reduced matrix will have at least one zero element in each row. Check if there is at least one zero element in each column also. If so, stop here. If not proceed to subset 2.
  • 19. Continued • Sub step 2: Mark the columns that do not have zero element. Now subtract the minimum element of each such column from all the elements of that column.
  • 20. Continued • Step III • Check if the Optimal assignment can be made in the current solution or not • Basis for making this check is that if the minimum number of lines crossing all zeros in low than n (in our example n =4), then an optional assignment cannot be made in the current solution. If it is equal to n (=4), then optimal assignment can be made in the current solution. Subassemblies Contractors 1 2 3 4 1 2 1 3 2 1 4 1 3 3 2 0 4 1 3 1 0 0 0 0
  • 21. Continued • The optimal assignment can be made in the current solution. Thus minimum total cost is • = birr (13x1 + 11x1 + 11x1 +14x1) x 100 = 4,900 birr. • Subassembly 1 - Contractor 2, • Subassembly 2 - Contractor 1, • Subassembly 1 - Contractor 4, • Subassembly 1 - Contractor 3 • The minimal cost of 4900 birr can be determined by summing up all elements that were subtracted during the solution procedure i.e., [(13+11+10+14) +1] x 100= 4900.
  • 22. Example • Four different jobs can be done on four different machines. The set-up and take-down time costs are assumed to be prohibitively high for changeovers. The matrix below gives the costs in birrs of producing job i on machine j. • (i) How should the jobs be assigned to the various machines so that the total cost is minimized? Machines Jobs M1 M2 M3 M4 J1 5 7 11 6 J2 8 5 9 6 J3 4 7 10 7 J4 10 4 8 3
  • 23. Continued • Reduce the Matrix: this involves the following sub steps: • Step III • Check if the Optimal assignment can be made in the current solution or not M1 M2 M3 M4 J1 0 2 6 1 J2 3 0 4 1 J3 0 3 6 3 J4 7 1 5 0 Matrix after sub step 1 (contains no zero in column 3 M1 M2 M3 M4 First Feasible Solution J1 0 2 2 1 J2 3 0 0 1 J3 0 3 2 3 J4 7 1 1 0 Matrix after sub step 2 M1 M2 M3 M4 J1 2 2 1 J2 3 0 1 J3 0 3 2 3 J4 7 1 1 0 0 0
  • 24. Continued • Sub step 4 Mark (✓) the rows for which assignment has not been made. In our problem it is the third row. • Sub step 5: Mark (✓) columns (not already marked) which have zeros in marked rows. Thus column 1 is marked (✓) • Sub step 6: Mark (✓) rows (not already marked) which have zeros in the marked columns. Thus row 1 is marked (✓) • Sub step 8: draw liens through all unmarked rows and through all marked columns.
  • 25. Continued • Step IV • Iterate Towards Optimality
  • 26. Continued • Step VI • Iterate Towards Optimality
  • 27. Continued • As there is assignment in each row and in each column, optimal assignment can be made in the current solution. Hence optimal assignment policy is • Job J1 should be assigned to machine M1, • Job J2 should be assigned to machine M2, • Job J3 should be assigned to machine M3, • Job J4 should be assigned to machine M4, • And optimal cost = (5+5+10+3) = 23 birr.