FORE School of Management 2011
PROJECT REPORT ON ASSIGNMENT PROBLEM
IN
DECISION MAKING MODELS
Submitted To
Mr Hitesh Arora
Professor FORE School of Management
Submitted By
Amit Kumar Singh
FMG20 Sec-A
201012
1
FORE School of Management 2011
EXECUTIVE SUMMARY
Assignment problems arise in different situations where we have to find an optimal way to
assign n objects to m other objects in an injective fashion Depending on the objective we
want to optimize we obtain different problems ranging from linear assignment problems to
quadratic and higher dimensional assignment problems The assignment problems are a well
studied topic in combinatorial optimization These problems find numerous applications in
production planning telecommunication VLSI design economics etc They can be
classified into three groups linear assignment problems three and higher dimensional
assignment problems and quadratic assignment problems and problems related to it For each
group of problems we mention some applications show some basic properties and describe
briefly some of the most successful algorithms used to solve these problems Although
assignment problem can be solved using the techniques of Linear Programming or the
transportation method the assignment method is much faster and efficient This method was
developed by D Konig a Hungarian mathematician and is therefore known as the Hungarian
method of assignment problem So we will try to figure out the application of assignment
problem with a case on NASA astronaut assignment to space missions and will see how excel
solver can be applied to solve this kind of problems
2
FORE School of Management 2011
Table of ContentsChapter 14
11 Introduction to Assignment Problems4
Chapter 25
21 Application Areas of Assignment Problem5
22 Formulation Of The Problem6
23 Solution Methods7
24 Hungarian Method7
Chapter 310
31 A Case of Assignment Problem10
32 Solution to the Case11
References17
3
FORE School of Management 2011
Chapter 1
11 Introduction to Assignment Problems
In the world of trade Business Organisations are confronting the conflicting need for
optimal utilization of their limited resources among competing activities When the
information available on resources and relationship between variables is known we can
use LP very reliably The course of action chosen will invariably lead to optimal or nearly
optimal results
The assignment problem is a special case of transportation problem in which the objective
is to assign a number of origins to the equal number of destinations at the minimum cost
(or maximum profit) It involves assignment of people to projects jobs to machines
workers to jobs and teachers to classes etc while minimizing the total assignment costs
One of the important characteristics of assignment problem is that only one job (or
worker) is assigned to one machine (or project) Hence the number of sources are equal
the number of destinations and each requirement and capacity value is exactly one unit
Although assignment problem can be solved using the techniques of Linear
Programming or the transportation method the assignment method is much faster and
efficient This method was developed by D Konig a Hungarian mathematician and is
therefore known as the Hungarian method of assignment problem In order to use this
method one needs to know only the cost of making all the possible assignments Each
assignment problem has a matrix (table) associated with it Normally the objects (or
people) one wishes to assign are expressed in rows whereas the columns represent the
tasks (or things) assigned to them The number in the table would then be the costs
associated with each particular assignment It may be noted that the assignment problem
is a variation of transportation problem with two characteristics(i)the cost matrix is a
square matrix and (ii)the optimum solution for the problem would be such that there
would be only one assignment in a row or column of the cost matrix
4
FORE School of Management 2011
Chapter 2
21 Application Areas of Assignment ProblemThough assignment problem finds applicability in various diverse business situations we discuss some of its main application areas
(i) In assigning machines to factory orders
(ii) In assigning salesmarketing people to sales territories
(iii) In assigning contracts to bidders by systematic bid-evaluation
(iv) In assigning teachers to classes
(v) In assigning accountants to accounts of the clients
22 Formulation Of The ProblemLet there are n jobs and n persons are available with different skills If the cost of doing jth work by ith person is cijThen the cost matrix is given in the table 1 below
JobsPersons
1 2 3 j n
1 C11 C12 C13 C1j C1n
2 C21 C22 C23 C2j C2n
i
Ci1
Ci2
Ci3
Cij
Cin
n
Cn1
Cn2
Cn3
Cnj
Cnn
Now the problem is which work is to be assigned to whom so that the cost of completion of
work will be minimum
5
FORE School of Management 2011
Mathematically we can express the problem as follows
n n
Z = Σ Σ Cij Xij
i=1 j=1Subject to the constraints n
Σ Xij = 1 for all i (resource availability) j=1
n
Σ Xij = 1 for all i (activity requirement) i=1
and Xij = 0 or 1 for all i to activity j
23 Solution Methods The assignment problem can be solved by the following four methods
Enumeration method
Simplex method
Transportation method
Hungarian method
As i am going to use Hungarian Method for solving the case so i will briefly describe about the method here
24 Hungarian Method
Step 1 Determine the cost table from the given problem
(i) If the no of sources is equal to no of destinations go to step 3
(ii) If the no of sources is not equal to the no of destination go to step2
Step 2 Add a dummy source or dummy destination so that the cost table becomes a square
matrix The cost entries of the dummy sourcedestinations are always zero
6
FORE School of Management 2011
Step 3 Locate the smallest element in each row of the given cost matrix and then subtract the
same from each element of the row
Step 4 In the reduced matrix obtained in the step 3 locate the smallest element of each
column and then subtract the same from each element of that column Each column and row
now have at least one zero
Step 5 In the modified matrix obtained in the step 4 search for the optimal assignment as
follows
(a) Examine the rows successively until a row with a single zero is found Enrectangle this
row (1048576)and cross off (X) all other zeros in its column Continue in this manner until
all the rows have been taken care of
(b) Repeat the procedure for each column of the reduced matrix
(c) If a row andor column has two or more zeros and one cannot be chosen by inspection
then assign arbitrary any one of these zeros and cross off all other zeros of that row
column
(d) Repeat (a) through (c) above successively until the chain of assigning (1048576) or cross (X)
ends
Step 6 If the number of assignment (1048576) is equal to n (the order of the cost matrix) an
optimum solution is reached
If the number of assignment is less than n(the order of the matrix) go to the next step
Step7 Draw the minimum number of horizontal andor vertical lines to cover all the zeros
of the reduced matrix
Step 8 Develop the new revised cost matrix as follows
(a)Find the smallest element of the reduced matrix not covered by any of the lines
(b)Subtract this element from all uncovered elements and add the same to all the elements
laying at the intersection of any two lines
Step 9 Go to step 6 and repeat the procedure until an optimum solution is attained
7
FORE School of Management 2011
The flowchart to solve any Assignment problem by Hungarian Method is given below
8
FORE School of Management 2011
Chapter 3
31 A Case of Assignment Problem
NASArsquoS astronaut crew currently includes 10 mission specialists who hold the doctoral
degree in either astrophysics or astromedicine One of these specialists will be assigned to
each of the 10 flights scheduled for the upcoming 9 months Mission specialists are
responsible for carrying out scientific and medical experiments in space or for launching
retrieving or repairing satellites The chief of Astronaut personnel himself a former crew
member with three missions under his belt must decide who should be assigned and trained
for each of the very different missions Clearly astronauts with medical educations are more
suited to other types of missions The chief assigns each astronaut a rating on a scale of 1 to
10 for each possible mission with 10 being a perfect match for the task at hand and a 1 being
a mismatch Only one specialist is assigned to each flight and none is reassigned until all
others have flown at least once
A) Who should be assigned to which flight
B) NASA has just been notified that Anderson is getting married in February and has
been granted a highly sought publicity tour in Europe that month (He intends to take
his wife and let the trip double as a honeymoon) How does this change the final
schedule
C) Creto has complained that he was misrated on his January missions Both ratings
should be 10s he claims to the chief who agrees and recomputes the schedule Do
any changes occur over the schedule set in part (b)
D) What are the strengths and weaknesses of this approach to scheduling
9
FORE School of Management 2011
Table 31 Data for problem
32 Solution to the Case
We can solve this case by two methods one is that we can go for manually solving the
question or else we can use Excel Solver
But before that we have to understand the problem that what is says and how it can be
interpreted
The problem is basically an assignment problem and here chief astronaut has to assign
various astronauts to respective missions keeping in consideration the rating which has been
given to astronauts We will have to handle the case in such a manner that proper assignment
can be done in each of the cases given in question Also we need to calculate the total rating
points when assignment has been done to see how efficient the mission is on a scale of 100
combining together the total ratings of ten astronauts
Using Solver
Setting up the LP in Solver
When all of the LP components have been entered into the worksheet and given names
Bring up Solver using the Tools rarr Solver menu There are four main elements of the
solver
10
FORE School of Management 2011
Solver dialog box
Set Target Cell The Target Cell contains the quantity you wish to optimizendashthe Objective
function value To specify the Target Cell either click on the cell with the mouse or type in
the name of the cell containing the objective function value
Equal To This specifies the direction of the optimization Click on either of the ldquoMaxrdquo or
ldquoMinrdquo radio buttons
By Changing Cells Recall that our goal is to optimize the value of the objective Function by
choosing an appropriate vector of decision variables Therefore we will Allow Excel to
change the decision variables x In the ldquoBy Changing Cellsrdquo
Subject to the Constraints Specify a constraint by clicking on the Add button While it is
possible to add each constraint one at a time it is easier (and more concise) to enter a single
inequality between the constraint function Be sure to include any additional constraints such
as nonnegativity constraints
On the right hand side of the Solver dialog box is a button labelled Options Click on this
button to bring up another dialog box Since we will be dealing primarily with linear
programs option of greatest interest is ldquoAssume Linear Modelrdquo Selecting this option forces
Excel to use a method for solving LPs known as the Simplex algorithm It is important that
ldquoAssume Linear Modelrdquo is selected or else we may end up with inappropriate outputs Once
the LP has been properly set up in the Solver dialog box press the Solve button to run Solver
A) Now we would assign each Astronaut to different Missions
11
FORE School of Management 2011
In Solver we add all the constraints and target cell as well as we set the solver to
Maximization type as we have to take in consideration the Astronauts with maximum rating
points
Solver Output Options
Pressing the Solve button runs Solver Depending on the size of the LP it may take some
time for Solver to get ready If Solver reaches a solution a new dialog box will appear and
prompt you to either accept the solution or restore the original worksheet values At this
point you may also choose to see a number of output reports The Answer report provides a
summary of the optimal decision variable values binding and non-binding constraints and
the optimal objective function value The Sensitivity report provides information describing
the sensitivity of the optimal solution to perturbations in the problem data
Following is the solution obtained from solving the Excel
12
FORE School of Management 2011
Hence assignments for each astronaut can be given as follows
Vincze Mar-26Viet April-12Anderson Feb-26Herbert Feb-05Schatz Jan-12Plane June-09Certo Sep-19Moses May-01Brandon Aug-20Drtina Jan-27
Hence we see that each astronaut has been allocated to a different mission Total rating has
been 96
B) Now in case b Anderson is getting married in February and has been granted a highly
sought publicity tour in Europe that month so he cant be assigned to any mission in
February So we would put 0 rating for him in the month of February
That is the only change in the main table and how it will affect the current solution we
will see in the solution which we will obtain after solving the problem through excel
13
FORE School of Management 2011
Now on solving the problem through solver we get the following Solution
Hence assignments for each astronaut can be given as follows
Vincze Mar-26Viet April-12Anderson Aug-20Herbert Feb-05Schatz Jan-12Plane June-09Certo Sep-19Moses May-01Brandon Feb-26Drtina Jan-27
So we see that after changing data for Anderson the assignments have also changed and rating also come down to 92
C) Now for January missions Certorsquos ratings should be 10s he claims to the chief who
agrees and recomputes the schedule hence we will change the table New table is as
follows
14
FORE School of Management 2011
On solving through excel we get the following solution
So we see that there has been a change in the total score and it has gone up by 1 to 93 But the
assignments of astronauts have remained same
D) The strengths and weaknesses of this approach to scheduling are given as follows
Strengths
Solvers or optimizers are software tools that help users find the best way to allocate
scarce resources The resources may be raw materials machine time or people time
money or anything else in limited supply The best or optimal solution may mean
maximizing profits minimizing costs or achieving the best possible quality
Weaknesses
Sometimes due to technical glitches it may give faulty result which may not be
optimized Also when we are assigning values to cell small error can have big impact
It does not take into consideration the effect of time and uncertainty There may be
cases of infeasibility and un-bounded
15
FORE School of Management 2011
References
1) Quantitative Analysis for Management by Barry Render Ralph M Stair Michael Hanna TN Badri (Pearson 10th Edition) Transportation and assignment models Question No 10-40
2) The Dynamic Hungarian Algorithm for the Assignment Problem with Changing Costs
G Ayorkor Mills Tettey Anthony Stentz M Bernardine Dias CMU-RI-TR-07-27 July 2007
Robotics Institute Carnegie Mellon Date of Access 28th September 2011
3) httpwwwniosacinsrsec311opt-lp5pdf Date of Access 28th September 2011
4) httpmikemccreavycomhungarian-assignment-problempdf Date of Access 28th September 2011
5) httpwwwamsjhuedu~castello362Handoutshungarianpdf Date of Access 28th September 2011
16
FORE School of Management 2011
EXECUTIVE SUMMARY
Assignment problems arise in different situations where we have to find an optimal way to
assign n objects to m other objects in an injective fashion Depending on the objective we
want to optimize we obtain different problems ranging from linear assignment problems to
quadratic and higher dimensional assignment problems The assignment problems are a well
studied topic in combinatorial optimization These problems find numerous applications in
production planning telecommunication VLSI design economics etc They can be
classified into three groups linear assignment problems three and higher dimensional
assignment problems and quadratic assignment problems and problems related to it For each
group of problems we mention some applications show some basic properties and describe
briefly some of the most successful algorithms used to solve these problems Although
assignment problem can be solved using the techniques of Linear Programming or the
transportation method the assignment method is much faster and efficient This method was
developed by D Konig a Hungarian mathematician and is therefore known as the Hungarian
method of assignment problem So we will try to figure out the application of assignment
problem with a case on NASA astronaut assignment to space missions and will see how excel
solver can be applied to solve this kind of problems
2
FORE School of Management 2011
Table of ContentsChapter 14
11 Introduction to Assignment Problems4
Chapter 25
21 Application Areas of Assignment Problem5
22 Formulation Of The Problem6
23 Solution Methods7
24 Hungarian Method7
Chapter 310
31 A Case of Assignment Problem10
32 Solution to the Case11
References17
3
FORE School of Management 2011
Chapter 1
11 Introduction to Assignment Problems
In the world of trade Business Organisations are confronting the conflicting need for
optimal utilization of their limited resources among competing activities When the
information available on resources and relationship between variables is known we can
use LP very reliably The course of action chosen will invariably lead to optimal or nearly
optimal results
The assignment problem is a special case of transportation problem in which the objective
is to assign a number of origins to the equal number of destinations at the minimum cost
(or maximum profit) It involves assignment of people to projects jobs to machines
workers to jobs and teachers to classes etc while minimizing the total assignment costs
One of the important characteristics of assignment problem is that only one job (or
worker) is assigned to one machine (or project) Hence the number of sources are equal
the number of destinations and each requirement and capacity value is exactly one unit
Although assignment problem can be solved using the techniques of Linear
Programming or the transportation method the assignment method is much faster and
efficient This method was developed by D Konig a Hungarian mathematician and is
therefore known as the Hungarian method of assignment problem In order to use this
method one needs to know only the cost of making all the possible assignments Each
assignment problem has a matrix (table) associated with it Normally the objects (or
people) one wishes to assign are expressed in rows whereas the columns represent the
tasks (or things) assigned to them The number in the table would then be the costs
associated with each particular assignment It may be noted that the assignment problem
is a variation of transportation problem with two characteristics(i)the cost matrix is a
square matrix and (ii)the optimum solution for the problem would be such that there
would be only one assignment in a row or column of the cost matrix
4
FORE School of Management 2011
Chapter 2
21 Application Areas of Assignment ProblemThough assignment problem finds applicability in various diverse business situations we discuss some of its main application areas
(i) In assigning machines to factory orders
(ii) In assigning salesmarketing people to sales territories
(iii) In assigning contracts to bidders by systematic bid-evaluation
(iv) In assigning teachers to classes
(v) In assigning accountants to accounts of the clients
22 Formulation Of The ProblemLet there are n jobs and n persons are available with different skills If the cost of doing jth work by ith person is cijThen the cost matrix is given in the table 1 below
JobsPersons
1 2 3 j n
1 C11 C12 C13 C1j C1n
2 C21 C22 C23 C2j C2n
i
Ci1
Ci2
Ci3
Cij
Cin
n
Cn1
Cn2
Cn3
Cnj
Cnn
Now the problem is which work is to be assigned to whom so that the cost of completion of
work will be minimum
5
FORE School of Management 2011
Mathematically we can express the problem as follows
n n
Z = Σ Σ Cij Xij
i=1 j=1Subject to the constraints n
Σ Xij = 1 for all i (resource availability) j=1
n
Σ Xij = 1 for all i (activity requirement) i=1
and Xij = 0 or 1 for all i to activity j
23 Solution Methods The assignment problem can be solved by the following four methods
Enumeration method
Simplex method
Transportation method
Hungarian method
As i am going to use Hungarian Method for solving the case so i will briefly describe about the method here
24 Hungarian Method
Step 1 Determine the cost table from the given problem
(i) If the no of sources is equal to no of destinations go to step 3
(ii) If the no of sources is not equal to the no of destination go to step2
Step 2 Add a dummy source or dummy destination so that the cost table becomes a square
matrix The cost entries of the dummy sourcedestinations are always zero
6
FORE School of Management 2011
Step 3 Locate the smallest element in each row of the given cost matrix and then subtract the
same from each element of the row
Step 4 In the reduced matrix obtained in the step 3 locate the smallest element of each
column and then subtract the same from each element of that column Each column and row
now have at least one zero
Step 5 In the modified matrix obtained in the step 4 search for the optimal assignment as
follows
(a) Examine the rows successively until a row with a single zero is found Enrectangle this
row (1048576)and cross off (X) all other zeros in its column Continue in this manner until
all the rows have been taken care of
(b) Repeat the procedure for each column of the reduced matrix
(c) If a row andor column has two or more zeros and one cannot be chosen by inspection
then assign arbitrary any one of these zeros and cross off all other zeros of that row
column
(d) Repeat (a) through (c) above successively until the chain of assigning (1048576) or cross (X)
ends
Step 6 If the number of assignment (1048576) is equal to n (the order of the cost matrix) an
optimum solution is reached
If the number of assignment is less than n(the order of the matrix) go to the next step
Step7 Draw the minimum number of horizontal andor vertical lines to cover all the zeros
of the reduced matrix
Step 8 Develop the new revised cost matrix as follows
(a)Find the smallest element of the reduced matrix not covered by any of the lines
(b)Subtract this element from all uncovered elements and add the same to all the elements
laying at the intersection of any two lines
Step 9 Go to step 6 and repeat the procedure until an optimum solution is attained
7
FORE School of Management 2011
The flowchart to solve any Assignment problem by Hungarian Method is given below
8
FORE School of Management 2011
Chapter 3
31 A Case of Assignment Problem
NASArsquoS astronaut crew currently includes 10 mission specialists who hold the doctoral
degree in either astrophysics or astromedicine One of these specialists will be assigned to
each of the 10 flights scheduled for the upcoming 9 months Mission specialists are
responsible for carrying out scientific and medical experiments in space or for launching
retrieving or repairing satellites The chief of Astronaut personnel himself a former crew
member with three missions under his belt must decide who should be assigned and trained
for each of the very different missions Clearly astronauts with medical educations are more
suited to other types of missions The chief assigns each astronaut a rating on a scale of 1 to
10 for each possible mission with 10 being a perfect match for the task at hand and a 1 being
a mismatch Only one specialist is assigned to each flight and none is reassigned until all
others have flown at least once
A) Who should be assigned to which flight
B) NASA has just been notified that Anderson is getting married in February and has
been granted a highly sought publicity tour in Europe that month (He intends to take
his wife and let the trip double as a honeymoon) How does this change the final
schedule
C) Creto has complained that he was misrated on his January missions Both ratings
should be 10s he claims to the chief who agrees and recomputes the schedule Do
any changes occur over the schedule set in part (b)
D) What are the strengths and weaknesses of this approach to scheduling
9
FORE School of Management 2011
Table 31 Data for problem
32 Solution to the Case
We can solve this case by two methods one is that we can go for manually solving the
question or else we can use Excel Solver
But before that we have to understand the problem that what is says and how it can be
interpreted
The problem is basically an assignment problem and here chief astronaut has to assign
various astronauts to respective missions keeping in consideration the rating which has been
given to astronauts We will have to handle the case in such a manner that proper assignment
can be done in each of the cases given in question Also we need to calculate the total rating
points when assignment has been done to see how efficient the mission is on a scale of 100
combining together the total ratings of ten astronauts
Using Solver
Setting up the LP in Solver
When all of the LP components have been entered into the worksheet and given names
Bring up Solver using the Tools rarr Solver menu There are four main elements of the
solver
10
FORE School of Management 2011
Solver dialog box
Set Target Cell The Target Cell contains the quantity you wish to optimizendashthe Objective
function value To specify the Target Cell either click on the cell with the mouse or type in
the name of the cell containing the objective function value
Equal To This specifies the direction of the optimization Click on either of the ldquoMaxrdquo or
ldquoMinrdquo radio buttons
By Changing Cells Recall that our goal is to optimize the value of the objective Function by
choosing an appropriate vector of decision variables Therefore we will Allow Excel to
change the decision variables x In the ldquoBy Changing Cellsrdquo
Subject to the Constraints Specify a constraint by clicking on the Add button While it is
possible to add each constraint one at a time it is easier (and more concise) to enter a single
inequality between the constraint function Be sure to include any additional constraints such
as nonnegativity constraints
On the right hand side of the Solver dialog box is a button labelled Options Click on this
button to bring up another dialog box Since we will be dealing primarily with linear
programs option of greatest interest is ldquoAssume Linear Modelrdquo Selecting this option forces
Excel to use a method for solving LPs known as the Simplex algorithm It is important that
ldquoAssume Linear Modelrdquo is selected or else we may end up with inappropriate outputs Once
the LP has been properly set up in the Solver dialog box press the Solve button to run Solver
A) Now we would assign each Astronaut to different Missions
11
FORE School of Management 2011
In Solver we add all the constraints and target cell as well as we set the solver to
Maximization type as we have to take in consideration the Astronauts with maximum rating
points
Solver Output Options
Pressing the Solve button runs Solver Depending on the size of the LP it may take some
time for Solver to get ready If Solver reaches a solution a new dialog box will appear and
prompt you to either accept the solution or restore the original worksheet values At this
point you may also choose to see a number of output reports The Answer report provides a
summary of the optimal decision variable values binding and non-binding constraints and
the optimal objective function value The Sensitivity report provides information describing
the sensitivity of the optimal solution to perturbations in the problem data
Following is the solution obtained from solving the Excel
12
FORE School of Management 2011
Hence assignments for each astronaut can be given as follows
Vincze Mar-26Viet April-12Anderson Feb-26Herbert Feb-05Schatz Jan-12Plane June-09Certo Sep-19Moses May-01Brandon Aug-20Drtina Jan-27
Hence we see that each astronaut has been allocated to a different mission Total rating has
been 96
B) Now in case b Anderson is getting married in February and has been granted a highly
sought publicity tour in Europe that month so he cant be assigned to any mission in
February So we would put 0 rating for him in the month of February
That is the only change in the main table and how it will affect the current solution we
will see in the solution which we will obtain after solving the problem through excel
13
FORE School of Management 2011
Now on solving the problem through solver we get the following Solution
Hence assignments for each astronaut can be given as follows
Vincze Mar-26Viet April-12Anderson Aug-20Herbert Feb-05Schatz Jan-12Plane June-09Certo Sep-19Moses May-01Brandon Feb-26Drtina Jan-27
So we see that after changing data for Anderson the assignments have also changed and rating also come down to 92
C) Now for January missions Certorsquos ratings should be 10s he claims to the chief who
agrees and recomputes the schedule hence we will change the table New table is as
follows
14
FORE School of Management 2011
On solving through excel we get the following solution
So we see that there has been a change in the total score and it has gone up by 1 to 93 But the
assignments of astronauts have remained same
D) The strengths and weaknesses of this approach to scheduling are given as follows
Strengths
Solvers or optimizers are software tools that help users find the best way to allocate
scarce resources The resources may be raw materials machine time or people time
money or anything else in limited supply The best or optimal solution may mean
maximizing profits minimizing costs or achieving the best possible quality
Weaknesses
Sometimes due to technical glitches it may give faulty result which may not be
optimized Also when we are assigning values to cell small error can have big impact
It does not take into consideration the effect of time and uncertainty There may be
cases of infeasibility and un-bounded
15
FORE School of Management 2011
References
1) Quantitative Analysis for Management by Barry Render Ralph M Stair Michael Hanna TN Badri (Pearson 10th Edition) Transportation and assignment models Question No 10-40
2) The Dynamic Hungarian Algorithm for the Assignment Problem with Changing Costs
G Ayorkor Mills Tettey Anthony Stentz M Bernardine Dias CMU-RI-TR-07-27 July 2007
Robotics Institute Carnegie Mellon Date of Access 28th September 2011
3) httpwwwniosacinsrsec311opt-lp5pdf Date of Access 28th September 2011
4) httpmikemccreavycomhungarian-assignment-problempdf Date of Access 28th September 2011
5) httpwwwamsjhuedu~castello362Handoutshungarianpdf Date of Access 28th September 2011
16
FORE School of Management 2011
Table of ContentsChapter 14
11 Introduction to Assignment Problems4
Chapter 25
21 Application Areas of Assignment Problem5
22 Formulation Of The Problem6
23 Solution Methods7
24 Hungarian Method7
Chapter 310
31 A Case of Assignment Problem10
32 Solution to the Case11
References17
3
FORE School of Management 2011
Chapter 1
11 Introduction to Assignment Problems
In the world of trade Business Organisations are confronting the conflicting need for
optimal utilization of their limited resources among competing activities When the
information available on resources and relationship between variables is known we can
use LP very reliably The course of action chosen will invariably lead to optimal or nearly
optimal results
The assignment problem is a special case of transportation problem in which the objective
is to assign a number of origins to the equal number of destinations at the minimum cost
(or maximum profit) It involves assignment of people to projects jobs to machines
workers to jobs and teachers to classes etc while minimizing the total assignment costs
One of the important characteristics of assignment problem is that only one job (or
worker) is assigned to one machine (or project) Hence the number of sources are equal
the number of destinations and each requirement and capacity value is exactly one unit
Although assignment problem can be solved using the techniques of Linear
Programming or the transportation method the assignment method is much faster and
efficient This method was developed by D Konig a Hungarian mathematician and is
therefore known as the Hungarian method of assignment problem In order to use this
method one needs to know only the cost of making all the possible assignments Each
assignment problem has a matrix (table) associated with it Normally the objects (or
people) one wishes to assign are expressed in rows whereas the columns represent the
tasks (or things) assigned to them The number in the table would then be the costs
associated with each particular assignment It may be noted that the assignment problem
is a variation of transportation problem with two characteristics(i)the cost matrix is a
square matrix and (ii)the optimum solution for the problem would be such that there
would be only one assignment in a row or column of the cost matrix
4
FORE School of Management 2011
Chapter 2
21 Application Areas of Assignment ProblemThough assignment problem finds applicability in various diverse business situations we discuss some of its main application areas
(i) In assigning machines to factory orders
(ii) In assigning salesmarketing people to sales territories
(iii) In assigning contracts to bidders by systematic bid-evaluation
(iv) In assigning teachers to classes
(v) In assigning accountants to accounts of the clients
22 Formulation Of The ProblemLet there are n jobs and n persons are available with different skills If the cost of doing jth work by ith person is cijThen the cost matrix is given in the table 1 below
JobsPersons
1 2 3 j n
1 C11 C12 C13 C1j C1n
2 C21 C22 C23 C2j C2n
i
Ci1
Ci2
Ci3
Cij
Cin
n
Cn1
Cn2
Cn3
Cnj
Cnn
Now the problem is which work is to be assigned to whom so that the cost of completion of
work will be minimum
5
FORE School of Management 2011
Mathematically we can express the problem as follows
n n
Z = Σ Σ Cij Xij
i=1 j=1Subject to the constraints n
Σ Xij = 1 for all i (resource availability) j=1
n
Σ Xij = 1 for all i (activity requirement) i=1
and Xij = 0 or 1 for all i to activity j
23 Solution Methods The assignment problem can be solved by the following four methods
Enumeration method
Simplex method
Transportation method
Hungarian method
As i am going to use Hungarian Method for solving the case so i will briefly describe about the method here
24 Hungarian Method
Step 1 Determine the cost table from the given problem
(i) If the no of sources is equal to no of destinations go to step 3
(ii) If the no of sources is not equal to the no of destination go to step2
Step 2 Add a dummy source or dummy destination so that the cost table becomes a square
matrix The cost entries of the dummy sourcedestinations are always zero
6
FORE School of Management 2011
Step 3 Locate the smallest element in each row of the given cost matrix and then subtract the
same from each element of the row
Step 4 In the reduced matrix obtained in the step 3 locate the smallest element of each
column and then subtract the same from each element of that column Each column and row
now have at least one zero
Step 5 In the modified matrix obtained in the step 4 search for the optimal assignment as
follows
(a) Examine the rows successively until a row with a single zero is found Enrectangle this
row (1048576)and cross off (X) all other zeros in its column Continue in this manner until
all the rows have been taken care of
(b) Repeat the procedure for each column of the reduced matrix
(c) If a row andor column has two or more zeros and one cannot be chosen by inspection
then assign arbitrary any one of these zeros and cross off all other zeros of that row
column
(d) Repeat (a) through (c) above successively until the chain of assigning (1048576) or cross (X)
ends
Step 6 If the number of assignment (1048576) is equal to n (the order of the cost matrix) an
optimum solution is reached
If the number of assignment is less than n(the order of the matrix) go to the next step
Step7 Draw the minimum number of horizontal andor vertical lines to cover all the zeros
of the reduced matrix
Step 8 Develop the new revised cost matrix as follows
(a)Find the smallest element of the reduced matrix not covered by any of the lines
(b)Subtract this element from all uncovered elements and add the same to all the elements
laying at the intersection of any two lines
Step 9 Go to step 6 and repeat the procedure until an optimum solution is attained
7
FORE School of Management 2011
The flowchart to solve any Assignment problem by Hungarian Method is given below
8
FORE School of Management 2011
Chapter 3
31 A Case of Assignment Problem
NASArsquoS astronaut crew currently includes 10 mission specialists who hold the doctoral
degree in either astrophysics or astromedicine One of these specialists will be assigned to
each of the 10 flights scheduled for the upcoming 9 months Mission specialists are
responsible for carrying out scientific and medical experiments in space or for launching
retrieving or repairing satellites The chief of Astronaut personnel himself a former crew
member with three missions under his belt must decide who should be assigned and trained
for each of the very different missions Clearly astronauts with medical educations are more
suited to other types of missions The chief assigns each astronaut a rating on a scale of 1 to
10 for each possible mission with 10 being a perfect match for the task at hand and a 1 being
a mismatch Only one specialist is assigned to each flight and none is reassigned until all
others have flown at least once
A) Who should be assigned to which flight
B) NASA has just been notified that Anderson is getting married in February and has
been granted a highly sought publicity tour in Europe that month (He intends to take
his wife and let the trip double as a honeymoon) How does this change the final
schedule
C) Creto has complained that he was misrated on his January missions Both ratings
should be 10s he claims to the chief who agrees and recomputes the schedule Do
any changes occur over the schedule set in part (b)
D) What are the strengths and weaknesses of this approach to scheduling
9
FORE School of Management 2011
Table 31 Data for problem
32 Solution to the Case
We can solve this case by two methods one is that we can go for manually solving the
question or else we can use Excel Solver
But before that we have to understand the problem that what is says and how it can be
interpreted
The problem is basically an assignment problem and here chief astronaut has to assign
various astronauts to respective missions keeping in consideration the rating which has been
given to astronauts We will have to handle the case in such a manner that proper assignment
can be done in each of the cases given in question Also we need to calculate the total rating
points when assignment has been done to see how efficient the mission is on a scale of 100
combining together the total ratings of ten astronauts
Using Solver
Setting up the LP in Solver
When all of the LP components have been entered into the worksheet and given names
Bring up Solver using the Tools rarr Solver menu There are four main elements of the
solver
10
FORE School of Management 2011
Solver dialog box
Set Target Cell The Target Cell contains the quantity you wish to optimizendashthe Objective
function value To specify the Target Cell either click on the cell with the mouse or type in
the name of the cell containing the objective function value
Equal To This specifies the direction of the optimization Click on either of the ldquoMaxrdquo or
ldquoMinrdquo radio buttons
By Changing Cells Recall that our goal is to optimize the value of the objective Function by
choosing an appropriate vector of decision variables Therefore we will Allow Excel to
change the decision variables x In the ldquoBy Changing Cellsrdquo
Subject to the Constraints Specify a constraint by clicking on the Add button While it is
possible to add each constraint one at a time it is easier (and more concise) to enter a single
inequality between the constraint function Be sure to include any additional constraints such
as nonnegativity constraints
On the right hand side of the Solver dialog box is a button labelled Options Click on this
button to bring up another dialog box Since we will be dealing primarily with linear
programs option of greatest interest is ldquoAssume Linear Modelrdquo Selecting this option forces
Excel to use a method for solving LPs known as the Simplex algorithm It is important that
ldquoAssume Linear Modelrdquo is selected or else we may end up with inappropriate outputs Once
the LP has been properly set up in the Solver dialog box press the Solve button to run Solver
A) Now we would assign each Astronaut to different Missions
11
FORE School of Management 2011
In Solver we add all the constraints and target cell as well as we set the solver to
Maximization type as we have to take in consideration the Astronauts with maximum rating
points
Solver Output Options
Pressing the Solve button runs Solver Depending on the size of the LP it may take some
time for Solver to get ready If Solver reaches a solution a new dialog box will appear and
prompt you to either accept the solution or restore the original worksheet values At this
point you may also choose to see a number of output reports The Answer report provides a
summary of the optimal decision variable values binding and non-binding constraints and
the optimal objective function value The Sensitivity report provides information describing
the sensitivity of the optimal solution to perturbations in the problem data
Following is the solution obtained from solving the Excel
12
FORE School of Management 2011
Hence assignments for each astronaut can be given as follows
Vincze Mar-26Viet April-12Anderson Feb-26Herbert Feb-05Schatz Jan-12Plane June-09Certo Sep-19Moses May-01Brandon Aug-20Drtina Jan-27
Hence we see that each astronaut has been allocated to a different mission Total rating has
been 96
B) Now in case b Anderson is getting married in February and has been granted a highly
sought publicity tour in Europe that month so he cant be assigned to any mission in
February So we would put 0 rating for him in the month of February
That is the only change in the main table and how it will affect the current solution we
will see in the solution which we will obtain after solving the problem through excel
13
FORE School of Management 2011
Now on solving the problem through solver we get the following Solution
Hence assignments for each astronaut can be given as follows
Vincze Mar-26Viet April-12Anderson Aug-20Herbert Feb-05Schatz Jan-12Plane June-09Certo Sep-19Moses May-01Brandon Feb-26Drtina Jan-27
So we see that after changing data for Anderson the assignments have also changed and rating also come down to 92
C) Now for January missions Certorsquos ratings should be 10s he claims to the chief who
agrees and recomputes the schedule hence we will change the table New table is as
follows
14
FORE School of Management 2011
On solving through excel we get the following solution
So we see that there has been a change in the total score and it has gone up by 1 to 93 But the
assignments of astronauts have remained same
D) The strengths and weaknesses of this approach to scheduling are given as follows
Strengths
Solvers or optimizers are software tools that help users find the best way to allocate
scarce resources The resources may be raw materials machine time or people time
money or anything else in limited supply The best or optimal solution may mean
maximizing profits minimizing costs or achieving the best possible quality
Weaknesses
Sometimes due to technical glitches it may give faulty result which may not be
optimized Also when we are assigning values to cell small error can have big impact
It does not take into consideration the effect of time and uncertainty There may be
cases of infeasibility and un-bounded
15
FORE School of Management 2011
References
1) Quantitative Analysis for Management by Barry Render Ralph M Stair Michael Hanna TN Badri (Pearson 10th Edition) Transportation and assignment models Question No 10-40
2) The Dynamic Hungarian Algorithm for the Assignment Problem with Changing Costs
G Ayorkor Mills Tettey Anthony Stentz M Bernardine Dias CMU-RI-TR-07-27 July 2007
Robotics Institute Carnegie Mellon Date of Access 28th September 2011
3) httpwwwniosacinsrsec311opt-lp5pdf Date of Access 28th September 2011
4) httpmikemccreavycomhungarian-assignment-problempdf Date of Access 28th September 2011
5) httpwwwamsjhuedu~castello362Handoutshungarianpdf Date of Access 28th September 2011
16
FORE School of Management 2011
Chapter 1
11 Introduction to Assignment Problems
In the world of trade Business Organisations are confronting the conflicting need for
optimal utilization of their limited resources among competing activities When the
information available on resources and relationship between variables is known we can
use LP very reliably The course of action chosen will invariably lead to optimal or nearly
optimal results
The assignment problem is a special case of transportation problem in which the objective
is to assign a number of origins to the equal number of destinations at the minimum cost
(or maximum profit) It involves assignment of people to projects jobs to machines
workers to jobs and teachers to classes etc while minimizing the total assignment costs
One of the important characteristics of assignment problem is that only one job (or
worker) is assigned to one machine (or project) Hence the number of sources are equal
the number of destinations and each requirement and capacity value is exactly one unit
Although assignment problem can be solved using the techniques of Linear
Programming or the transportation method the assignment method is much faster and
efficient This method was developed by D Konig a Hungarian mathematician and is
therefore known as the Hungarian method of assignment problem In order to use this
method one needs to know only the cost of making all the possible assignments Each
assignment problem has a matrix (table) associated with it Normally the objects (or
people) one wishes to assign are expressed in rows whereas the columns represent the
tasks (or things) assigned to them The number in the table would then be the costs
associated with each particular assignment It may be noted that the assignment problem
is a variation of transportation problem with two characteristics(i)the cost matrix is a
square matrix and (ii)the optimum solution for the problem would be such that there
would be only one assignment in a row or column of the cost matrix
4
FORE School of Management 2011
Chapter 2
21 Application Areas of Assignment ProblemThough assignment problem finds applicability in various diverse business situations we discuss some of its main application areas
(i) In assigning machines to factory orders
(ii) In assigning salesmarketing people to sales territories
(iii) In assigning contracts to bidders by systematic bid-evaluation
(iv) In assigning teachers to classes
(v) In assigning accountants to accounts of the clients
22 Formulation Of The ProblemLet there are n jobs and n persons are available with different skills If the cost of doing jth work by ith person is cijThen the cost matrix is given in the table 1 below
JobsPersons
1 2 3 j n
1 C11 C12 C13 C1j C1n
2 C21 C22 C23 C2j C2n
i
Ci1
Ci2
Ci3
Cij
Cin
n
Cn1
Cn2
Cn3
Cnj
Cnn
Now the problem is which work is to be assigned to whom so that the cost of completion of
work will be minimum
5
FORE School of Management 2011
Mathematically we can express the problem as follows
n n
Z = Σ Σ Cij Xij
i=1 j=1Subject to the constraints n
Σ Xij = 1 for all i (resource availability) j=1
n
Σ Xij = 1 for all i (activity requirement) i=1
and Xij = 0 or 1 for all i to activity j
23 Solution Methods The assignment problem can be solved by the following four methods
Enumeration method
Simplex method
Transportation method
Hungarian method
As i am going to use Hungarian Method for solving the case so i will briefly describe about the method here
24 Hungarian Method
Step 1 Determine the cost table from the given problem
(i) If the no of sources is equal to no of destinations go to step 3
(ii) If the no of sources is not equal to the no of destination go to step2
Step 2 Add a dummy source or dummy destination so that the cost table becomes a square
matrix The cost entries of the dummy sourcedestinations are always zero
6
FORE School of Management 2011
Step 3 Locate the smallest element in each row of the given cost matrix and then subtract the
same from each element of the row
Step 4 In the reduced matrix obtained in the step 3 locate the smallest element of each
column and then subtract the same from each element of that column Each column and row
now have at least one zero
Step 5 In the modified matrix obtained in the step 4 search for the optimal assignment as
follows
(a) Examine the rows successively until a row with a single zero is found Enrectangle this
row (1048576)and cross off (X) all other zeros in its column Continue in this manner until
all the rows have been taken care of
(b) Repeat the procedure for each column of the reduced matrix
(c) If a row andor column has two or more zeros and one cannot be chosen by inspection
then assign arbitrary any one of these zeros and cross off all other zeros of that row
column
(d) Repeat (a) through (c) above successively until the chain of assigning (1048576) or cross (X)
ends
Step 6 If the number of assignment (1048576) is equal to n (the order of the cost matrix) an
optimum solution is reached
If the number of assignment is less than n(the order of the matrix) go to the next step
Step7 Draw the minimum number of horizontal andor vertical lines to cover all the zeros
of the reduced matrix
Step 8 Develop the new revised cost matrix as follows
(a)Find the smallest element of the reduced matrix not covered by any of the lines
(b)Subtract this element from all uncovered elements and add the same to all the elements
laying at the intersection of any two lines
Step 9 Go to step 6 and repeat the procedure until an optimum solution is attained
7
FORE School of Management 2011
The flowchart to solve any Assignment problem by Hungarian Method is given below
8
FORE School of Management 2011
Chapter 3
31 A Case of Assignment Problem
NASArsquoS astronaut crew currently includes 10 mission specialists who hold the doctoral
degree in either astrophysics or astromedicine One of these specialists will be assigned to
each of the 10 flights scheduled for the upcoming 9 months Mission specialists are
responsible for carrying out scientific and medical experiments in space or for launching
retrieving or repairing satellites The chief of Astronaut personnel himself a former crew
member with three missions under his belt must decide who should be assigned and trained
for each of the very different missions Clearly astronauts with medical educations are more
suited to other types of missions The chief assigns each astronaut a rating on a scale of 1 to
10 for each possible mission with 10 being a perfect match for the task at hand and a 1 being
a mismatch Only one specialist is assigned to each flight and none is reassigned until all
others have flown at least once
A) Who should be assigned to which flight
B) NASA has just been notified that Anderson is getting married in February and has
been granted a highly sought publicity tour in Europe that month (He intends to take
his wife and let the trip double as a honeymoon) How does this change the final
schedule
C) Creto has complained that he was misrated on his January missions Both ratings
should be 10s he claims to the chief who agrees and recomputes the schedule Do
any changes occur over the schedule set in part (b)
D) What are the strengths and weaknesses of this approach to scheduling
9
FORE School of Management 2011
Table 31 Data for problem
32 Solution to the Case
We can solve this case by two methods one is that we can go for manually solving the
question or else we can use Excel Solver
But before that we have to understand the problem that what is says and how it can be
interpreted
The problem is basically an assignment problem and here chief astronaut has to assign
various astronauts to respective missions keeping in consideration the rating which has been
given to astronauts We will have to handle the case in such a manner that proper assignment
can be done in each of the cases given in question Also we need to calculate the total rating
points when assignment has been done to see how efficient the mission is on a scale of 100
combining together the total ratings of ten astronauts
Using Solver
Setting up the LP in Solver
When all of the LP components have been entered into the worksheet and given names
Bring up Solver using the Tools rarr Solver menu There are four main elements of the
solver
10
FORE School of Management 2011
Solver dialog box
Set Target Cell The Target Cell contains the quantity you wish to optimizendashthe Objective
function value To specify the Target Cell either click on the cell with the mouse or type in
the name of the cell containing the objective function value
Equal To This specifies the direction of the optimization Click on either of the ldquoMaxrdquo or
ldquoMinrdquo radio buttons
By Changing Cells Recall that our goal is to optimize the value of the objective Function by
choosing an appropriate vector of decision variables Therefore we will Allow Excel to
change the decision variables x In the ldquoBy Changing Cellsrdquo
Subject to the Constraints Specify a constraint by clicking on the Add button While it is
possible to add each constraint one at a time it is easier (and more concise) to enter a single
inequality between the constraint function Be sure to include any additional constraints such
as nonnegativity constraints
On the right hand side of the Solver dialog box is a button labelled Options Click on this
button to bring up another dialog box Since we will be dealing primarily with linear
programs option of greatest interest is ldquoAssume Linear Modelrdquo Selecting this option forces
Excel to use a method for solving LPs known as the Simplex algorithm It is important that
ldquoAssume Linear Modelrdquo is selected or else we may end up with inappropriate outputs Once
the LP has been properly set up in the Solver dialog box press the Solve button to run Solver
A) Now we would assign each Astronaut to different Missions
11
FORE School of Management 2011
In Solver we add all the constraints and target cell as well as we set the solver to
Maximization type as we have to take in consideration the Astronauts with maximum rating
points
Solver Output Options
Pressing the Solve button runs Solver Depending on the size of the LP it may take some
time for Solver to get ready If Solver reaches a solution a new dialog box will appear and
prompt you to either accept the solution or restore the original worksheet values At this
point you may also choose to see a number of output reports The Answer report provides a
summary of the optimal decision variable values binding and non-binding constraints and
the optimal objective function value The Sensitivity report provides information describing
the sensitivity of the optimal solution to perturbations in the problem data
Following is the solution obtained from solving the Excel
12
FORE School of Management 2011
Hence assignments for each astronaut can be given as follows
Vincze Mar-26Viet April-12Anderson Feb-26Herbert Feb-05Schatz Jan-12Plane June-09Certo Sep-19Moses May-01Brandon Aug-20Drtina Jan-27
Hence we see that each astronaut has been allocated to a different mission Total rating has
been 96
B) Now in case b Anderson is getting married in February and has been granted a highly
sought publicity tour in Europe that month so he cant be assigned to any mission in
February So we would put 0 rating for him in the month of February
That is the only change in the main table and how it will affect the current solution we
will see in the solution which we will obtain after solving the problem through excel
13
FORE School of Management 2011
Now on solving the problem through solver we get the following Solution
Hence assignments for each astronaut can be given as follows
Vincze Mar-26Viet April-12Anderson Aug-20Herbert Feb-05Schatz Jan-12Plane June-09Certo Sep-19Moses May-01Brandon Feb-26Drtina Jan-27
So we see that after changing data for Anderson the assignments have also changed and rating also come down to 92
C) Now for January missions Certorsquos ratings should be 10s he claims to the chief who
agrees and recomputes the schedule hence we will change the table New table is as
follows
14
FORE School of Management 2011
On solving through excel we get the following solution
So we see that there has been a change in the total score and it has gone up by 1 to 93 But the
assignments of astronauts have remained same
D) The strengths and weaknesses of this approach to scheduling are given as follows
Strengths
Solvers or optimizers are software tools that help users find the best way to allocate
scarce resources The resources may be raw materials machine time or people time
money or anything else in limited supply The best or optimal solution may mean
maximizing profits minimizing costs or achieving the best possible quality
Weaknesses
Sometimes due to technical glitches it may give faulty result which may not be
optimized Also when we are assigning values to cell small error can have big impact
It does not take into consideration the effect of time and uncertainty There may be
cases of infeasibility and un-bounded
15
FORE School of Management 2011
References
1) Quantitative Analysis for Management by Barry Render Ralph M Stair Michael Hanna TN Badri (Pearson 10th Edition) Transportation and assignment models Question No 10-40
2) The Dynamic Hungarian Algorithm for the Assignment Problem with Changing Costs
G Ayorkor Mills Tettey Anthony Stentz M Bernardine Dias CMU-RI-TR-07-27 July 2007
Robotics Institute Carnegie Mellon Date of Access 28th September 2011
3) httpwwwniosacinsrsec311opt-lp5pdf Date of Access 28th September 2011
4) httpmikemccreavycomhungarian-assignment-problempdf Date of Access 28th September 2011
5) httpwwwamsjhuedu~castello362Handoutshungarianpdf Date of Access 28th September 2011
16
FORE School of Management 2011
Chapter 2
21 Application Areas of Assignment ProblemThough assignment problem finds applicability in various diverse business situations we discuss some of its main application areas
(i) In assigning machines to factory orders
(ii) In assigning salesmarketing people to sales territories
(iii) In assigning contracts to bidders by systematic bid-evaluation
(iv) In assigning teachers to classes
(v) In assigning accountants to accounts of the clients
22 Formulation Of The ProblemLet there are n jobs and n persons are available with different skills If the cost of doing jth work by ith person is cijThen the cost matrix is given in the table 1 below
JobsPersons
1 2 3 j n
1 C11 C12 C13 C1j C1n
2 C21 C22 C23 C2j C2n
i
Ci1
Ci2
Ci3
Cij
Cin
n
Cn1
Cn2
Cn3
Cnj
Cnn
Now the problem is which work is to be assigned to whom so that the cost of completion of
work will be minimum
5
FORE School of Management 2011
Mathematically we can express the problem as follows
n n
Z = Σ Σ Cij Xij
i=1 j=1Subject to the constraints n
Σ Xij = 1 for all i (resource availability) j=1
n
Σ Xij = 1 for all i (activity requirement) i=1
and Xij = 0 or 1 for all i to activity j
23 Solution Methods The assignment problem can be solved by the following four methods
Enumeration method
Simplex method
Transportation method
Hungarian method
As i am going to use Hungarian Method for solving the case so i will briefly describe about the method here
24 Hungarian Method
Step 1 Determine the cost table from the given problem
(i) If the no of sources is equal to no of destinations go to step 3
(ii) If the no of sources is not equal to the no of destination go to step2
Step 2 Add a dummy source or dummy destination so that the cost table becomes a square
matrix The cost entries of the dummy sourcedestinations are always zero
6
FORE School of Management 2011
Step 3 Locate the smallest element in each row of the given cost matrix and then subtract the
same from each element of the row
Step 4 In the reduced matrix obtained in the step 3 locate the smallest element of each
column and then subtract the same from each element of that column Each column and row
now have at least one zero
Step 5 In the modified matrix obtained in the step 4 search for the optimal assignment as
follows
(a) Examine the rows successively until a row with a single zero is found Enrectangle this
row (1048576)and cross off (X) all other zeros in its column Continue in this manner until
all the rows have been taken care of
(b) Repeat the procedure for each column of the reduced matrix
(c) If a row andor column has two or more zeros and one cannot be chosen by inspection
then assign arbitrary any one of these zeros and cross off all other zeros of that row
column
(d) Repeat (a) through (c) above successively until the chain of assigning (1048576) or cross (X)
ends
Step 6 If the number of assignment (1048576) is equal to n (the order of the cost matrix) an
optimum solution is reached
If the number of assignment is less than n(the order of the matrix) go to the next step
Step7 Draw the minimum number of horizontal andor vertical lines to cover all the zeros
of the reduced matrix
Step 8 Develop the new revised cost matrix as follows
(a)Find the smallest element of the reduced matrix not covered by any of the lines
(b)Subtract this element from all uncovered elements and add the same to all the elements
laying at the intersection of any two lines
Step 9 Go to step 6 and repeat the procedure until an optimum solution is attained
7
FORE School of Management 2011
The flowchart to solve any Assignment problem by Hungarian Method is given below
8
FORE School of Management 2011
Chapter 3
31 A Case of Assignment Problem
NASArsquoS astronaut crew currently includes 10 mission specialists who hold the doctoral
degree in either astrophysics or astromedicine One of these specialists will be assigned to
each of the 10 flights scheduled for the upcoming 9 months Mission specialists are
responsible for carrying out scientific and medical experiments in space or for launching
retrieving or repairing satellites The chief of Astronaut personnel himself a former crew
member with three missions under his belt must decide who should be assigned and trained
for each of the very different missions Clearly astronauts with medical educations are more
suited to other types of missions The chief assigns each astronaut a rating on a scale of 1 to
10 for each possible mission with 10 being a perfect match for the task at hand and a 1 being
a mismatch Only one specialist is assigned to each flight and none is reassigned until all
others have flown at least once
A) Who should be assigned to which flight
B) NASA has just been notified that Anderson is getting married in February and has
been granted a highly sought publicity tour in Europe that month (He intends to take
his wife and let the trip double as a honeymoon) How does this change the final
schedule
C) Creto has complained that he was misrated on his January missions Both ratings
should be 10s he claims to the chief who agrees and recomputes the schedule Do
any changes occur over the schedule set in part (b)
D) What are the strengths and weaknesses of this approach to scheduling
9
FORE School of Management 2011
Table 31 Data for problem
32 Solution to the Case
We can solve this case by two methods one is that we can go for manually solving the
question or else we can use Excel Solver
But before that we have to understand the problem that what is says and how it can be
interpreted
The problem is basically an assignment problem and here chief astronaut has to assign
various astronauts to respective missions keeping in consideration the rating which has been
given to astronauts We will have to handle the case in such a manner that proper assignment
can be done in each of the cases given in question Also we need to calculate the total rating
points when assignment has been done to see how efficient the mission is on a scale of 100
combining together the total ratings of ten astronauts
Using Solver
Setting up the LP in Solver
When all of the LP components have been entered into the worksheet and given names
Bring up Solver using the Tools rarr Solver menu There are four main elements of the
solver
10
FORE School of Management 2011
Solver dialog box
Set Target Cell The Target Cell contains the quantity you wish to optimizendashthe Objective
function value To specify the Target Cell either click on the cell with the mouse or type in
the name of the cell containing the objective function value
Equal To This specifies the direction of the optimization Click on either of the ldquoMaxrdquo or
ldquoMinrdquo radio buttons
By Changing Cells Recall that our goal is to optimize the value of the objective Function by
choosing an appropriate vector of decision variables Therefore we will Allow Excel to
change the decision variables x In the ldquoBy Changing Cellsrdquo
Subject to the Constraints Specify a constraint by clicking on the Add button While it is
possible to add each constraint one at a time it is easier (and more concise) to enter a single
inequality between the constraint function Be sure to include any additional constraints such
as nonnegativity constraints
On the right hand side of the Solver dialog box is a button labelled Options Click on this
button to bring up another dialog box Since we will be dealing primarily with linear
programs option of greatest interest is ldquoAssume Linear Modelrdquo Selecting this option forces
Excel to use a method for solving LPs known as the Simplex algorithm It is important that
ldquoAssume Linear Modelrdquo is selected or else we may end up with inappropriate outputs Once
the LP has been properly set up in the Solver dialog box press the Solve button to run Solver
A) Now we would assign each Astronaut to different Missions
11
FORE School of Management 2011
In Solver we add all the constraints and target cell as well as we set the solver to
Maximization type as we have to take in consideration the Astronauts with maximum rating
points
Solver Output Options
Pressing the Solve button runs Solver Depending on the size of the LP it may take some
time for Solver to get ready If Solver reaches a solution a new dialog box will appear and
prompt you to either accept the solution or restore the original worksheet values At this
point you may also choose to see a number of output reports The Answer report provides a
summary of the optimal decision variable values binding and non-binding constraints and
the optimal objective function value The Sensitivity report provides information describing
the sensitivity of the optimal solution to perturbations in the problem data
Following is the solution obtained from solving the Excel
12
FORE School of Management 2011
Hence assignments for each astronaut can be given as follows
Vincze Mar-26Viet April-12Anderson Feb-26Herbert Feb-05Schatz Jan-12Plane June-09Certo Sep-19Moses May-01Brandon Aug-20Drtina Jan-27
Hence we see that each astronaut has been allocated to a different mission Total rating has
been 96
B) Now in case b Anderson is getting married in February and has been granted a highly
sought publicity tour in Europe that month so he cant be assigned to any mission in
February So we would put 0 rating for him in the month of February
That is the only change in the main table and how it will affect the current solution we
will see in the solution which we will obtain after solving the problem through excel
13
FORE School of Management 2011
Now on solving the problem through solver we get the following Solution
Hence assignments for each astronaut can be given as follows
Vincze Mar-26Viet April-12Anderson Aug-20Herbert Feb-05Schatz Jan-12Plane June-09Certo Sep-19Moses May-01Brandon Feb-26Drtina Jan-27
So we see that after changing data for Anderson the assignments have also changed and rating also come down to 92
C) Now for January missions Certorsquos ratings should be 10s he claims to the chief who
agrees and recomputes the schedule hence we will change the table New table is as
follows
14
FORE School of Management 2011
On solving through excel we get the following solution
So we see that there has been a change in the total score and it has gone up by 1 to 93 But the
assignments of astronauts have remained same
D) The strengths and weaknesses of this approach to scheduling are given as follows
Strengths
Solvers or optimizers are software tools that help users find the best way to allocate
scarce resources The resources may be raw materials machine time or people time
money or anything else in limited supply The best or optimal solution may mean
maximizing profits minimizing costs or achieving the best possible quality
Weaknesses
Sometimes due to technical glitches it may give faulty result which may not be
optimized Also when we are assigning values to cell small error can have big impact
It does not take into consideration the effect of time and uncertainty There may be
cases of infeasibility and un-bounded
15
FORE School of Management 2011
References
1) Quantitative Analysis for Management by Barry Render Ralph M Stair Michael Hanna TN Badri (Pearson 10th Edition) Transportation and assignment models Question No 10-40
2) The Dynamic Hungarian Algorithm for the Assignment Problem with Changing Costs
G Ayorkor Mills Tettey Anthony Stentz M Bernardine Dias CMU-RI-TR-07-27 July 2007
Robotics Institute Carnegie Mellon Date of Access 28th September 2011
3) httpwwwniosacinsrsec311opt-lp5pdf Date of Access 28th September 2011
4) httpmikemccreavycomhungarian-assignment-problempdf Date of Access 28th September 2011
5) httpwwwamsjhuedu~castello362Handoutshungarianpdf Date of Access 28th September 2011
16
FORE School of Management 2011
Mathematically we can express the problem as follows
n n
Z = Σ Σ Cij Xij
i=1 j=1Subject to the constraints n
Σ Xij = 1 for all i (resource availability) j=1
n
Σ Xij = 1 for all i (activity requirement) i=1
and Xij = 0 or 1 for all i to activity j
23 Solution Methods The assignment problem can be solved by the following four methods
Enumeration method
Simplex method
Transportation method
Hungarian method
As i am going to use Hungarian Method for solving the case so i will briefly describe about the method here
24 Hungarian Method
Step 1 Determine the cost table from the given problem
(i) If the no of sources is equal to no of destinations go to step 3
(ii) If the no of sources is not equal to the no of destination go to step2
Step 2 Add a dummy source or dummy destination so that the cost table becomes a square
matrix The cost entries of the dummy sourcedestinations are always zero
6
FORE School of Management 2011
Step 3 Locate the smallest element in each row of the given cost matrix and then subtract the
same from each element of the row
Step 4 In the reduced matrix obtained in the step 3 locate the smallest element of each
column and then subtract the same from each element of that column Each column and row
now have at least one zero
Step 5 In the modified matrix obtained in the step 4 search for the optimal assignment as
follows
(a) Examine the rows successively until a row with a single zero is found Enrectangle this
row (1048576)and cross off (X) all other zeros in its column Continue in this manner until
all the rows have been taken care of
(b) Repeat the procedure for each column of the reduced matrix
(c) If a row andor column has two or more zeros and one cannot be chosen by inspection
then assign arbitrary any one of these zeros and cross off all other zeros of that row
column
(d) Repeat (a) through (c) above successively until the chain of assigning (1048576) or cross (X)
ends
Step 6 If the number of assignment (1048576) is equal to n (the order of the cost matrix) an
optimum solution is reached
If the number of assignment is less than n(the order of the matrix) go to the next step
Step7 Draw the minimum number of horizontal andor vertical lines to cover all the zeros
of the reduced matrix
Step 8 Develop the new revised cost matrix as follows
(a)Find the smallest element of the reduced matrix not covered by any of the lines
(b)Subtract this element from all uncovered elements and add the same to all the elements
laying at the intersection of any two lines
Step 9 Go to step 6 and repeat the procedure until an optimum solution is attained
7
FORE School of Management 2011
The flowchart to solve any Assignment problem by Hungarian Method is given below
8
FORE School of Management 2011
Chapter 3
31 A Case of Assignment Problem
NASArsquoS astronaut crew currently includes 10 mission specialists who hold the doctoral
degree in either astrophysics or astromedicine One of these specialists will be assigned to
each of the 10 flights scheduled for the upcoming 9 months Mission specialists are
responsible for carrying out scientific and medical experiments in space or for launching
retrieving or repairing satellites The chief of Astronaut personnel himself a former crew
member with three missions under his belt must decide who should be assigned and trained
for each of the very different missions Clearly astronauts with medical educations are more
suited to other types of missions The chief assigns each astronaut a rating on a scale of 1 to
10 for each possible mission with 10 being a perfect match for the task at hand and a 1 being
a mismatch Only one specialist is assigned to each flight and none is reassigned until all
others have flown at least once
A) Who should be assigned to which flight
B) NASA has just been notified that Anderson is getting married in February and has
been granted a highly sought publicity tour in Europe that month (He intends to take
his wife and let the trip double as a honeymoon) How does this change the final
schedule
C) Creto has complained that he was misrated on his January missions Both ratings
should be 10s he claims to the chief who agrees and recomputes the schedule Do
any changes occur over the schedule set in part (b)
D) What are the strengths and weaknesses of this approach to scheduling
9
FORE School of Management 2011
Table 31 Data for problem
32 Solution to the Case
We can solve this case by two methods one is that we can go for manually solving the
question or else we can use Excel Solver
But before that we have to understand the problem that what is says and how it can be
interpreted
The problem is basically an assignment problem and here chief astronaut has to assign
various astronauts to respective missions keeping in consideration the rating which has been
given to astronauts We will have to handle the case in such a manner that proper assignment
can be done in each of the cases given in question Also we need to calculate the total rating
points when assignment has been done to see how efficient the mission is on a scale of 100
combining together the total ratings of ten astronauts
Using Solver
Setting up the LP in Solver
When all of the LP components have been entered into the worksheet and given names
Bring up Solver using the Tools rarr Solver menu There are four main elements of the
solver
10
FORE School of Management 2011
Solver dialog box
Set Target Cell The Target Cell contains the quantity you wish to optimizendashthe Objective
function value To specify the Target Cell either click on the cell with the mouse or type in
the name of the cell containing the objective function value
Equal To This specifies the direction of the optimization Click on either of the ldquoMaxrdquo or
ldquoMinrdquo radio buttons
By Changing Cells Recall that our goal is to optimize the value of the objective Function by
choosing an appropriate vector of decision variables Therefore we will Allow Excel to
change the decision variables x In the ldquoBy Changing Cellsrdquo
Subject to the Constraints Specify a constraint by clicking on the Add button While it is
possible to add each constraint one at a time it is easier (and more concise) to enter a single
inequality between the constraint function Be sure to include any additional constraints such
as nonnegativity constraints
On the right hand side of the Solver dialog box is a button labelled Options Click on this
button to bring up another dialog box Since we will be dealing primarily with linear
programs option of greatest interest is ldquoAssume Linear Modelrdquo Selecting this option forces
Excel to use a method for solving LPs known as the Simplex algorithm It is important that
ldquoAssume Linear Modelrdquo is selected or else we may end up with inappropriate outputs Once
the LP has been properly set up in the Solver dialog box press the Solve button to run Solver
A) Now we would assign each Astronaut to different Missions
11
FORE School of Management 2011
In Solver we add all the constraints and target cell as well as we set the solver to
Maximization type as we have to take in consideration the Astronauts with maximum rating
points
Solver Output Options
Pressing the Solve button runs Solver Depending on the size of the LP it may take some
time for Solver to get ready If Solver reaches a solution a new dialog box will appear and
prompt you to either accept the solution or restore the original worksheet values At this
point you may also choose to see a number of output reports The Answer report provides a
summary of the optimal decision variable values binding and non-binding constraints and
the optimal objective function value The Sensitivity report provides information describing
the sensitivity of the optimal solution to perturbations in the problem data
Following is the solution obtained from solving the Excel
12
FORE School of Management 2011
Hence assignments for each astronaut can be given as follows
Vincze Mar-26Viet April-12Anderson Feb-26Herbert Feb-05Schatz Jan-12Plane June-09Certo Sep-19Moses May-01Brandon Aug-20Drtina Jan-27
Hence we see that each astronaut has been allocated to a different mission Total rating has
been 96
B) Now in case b Anderson is getting married in February and has been granted a highly
sought publicity tour in Europe that month so he cant be assigned to any mission in
February So we would put 0 rating for him in the month of February
That is the only change in the main table and how it will affect the current solution we
will see in the solution which we will obtain after solving the problem through excel
13
FORE School of Management 2011
Now on solving the problem through solver we get the following Solution
Hence assignments for each astronaut can be given as follows
Vincze Mar-26Viet April-12Anderson Aug-20Herbert Feb-05Schatz Jan-12Plane June-09Certo Sep-19Moses May-01Brandon Feb-26Drtina Jan-27
So we see that after changing data for Anderson the assignments have also changed and rating also come down to 92
C) Now for January missions Certorsquos ratings should be 10s he claims to the chief who
agrees and recomputes the schedule hence we will change the table New table is as
follows
14
FORE School of Management 2011
On solving through excel we get the following solution
So we see that there has been a change in the total score and it has gone up by 1 to 93 But the
assignments of astronauts have remained same
D) The strengths and weaknesses of this approach to scheduling are given as follows
Strengths
Solvers or optimizers are software tools that help users find the best way to allocate
scarce resources The resources may be raw materials machine time or people time
money or anything else in limited supply The best or optimal solution may mean
maximizing profits minimizing costs or achieving the best possible quality
Weaknesses
Sometimes due to technical glitches it may give faulty result which may not be
optimized Also when we are assigning values to cell small error can have big impact
It does not take into consideration the effect of time and uncertainty There may be
cases of infeasibility and un-bounded
15
FORE School of Management 2011
References
1) Quantitative Analysis for Management by Barry Render Ralph M Stair Michael Hanna TN Badri (Pearson 10th Edition) Transportation and assignment models Question No 10-40
2) The Dynamic Hungarian Algorithm for the Assignment Problem with Changing Costs
G Ayorkor Mills Tettey Anthony Stentz M Bernardine Dias CMU-RI-TR-07-27 July 2007
Robotics Institute Carnegie Mellon Date of Access 28th September 2011
3) httpwwwniosacinsrsec311opt-lp5pdf Date of Access 28th September 2011
4) httpmikemccreavycomhungarian-assignment-problempdf Date of Access 28th September 2011
5) httpwwwamsjhuedu~castello362Handoutshungarianpdf Date of Access 28th September 2011
16
FORE School of Management 2011
Step 3 Locate the smallest element in each row of the given cost matrix and then subtract the
same from each element of the row
Step 4 In the reduced matrix obtained in the step 3 locate the smallest element of each
column and then subtract the same from each element of that column Each column and row
now have at least one zero
Step 5 In the modified matrix obtained in the step 4 search for the optimal assignment as
follows
(a) Examine the rows successively until a row with a single zero is found Enrectangle this
row (1048576)and cross off (X) all other zeros in its column Continue in this manner until
all the rows have been taken care of
(b) Repeat the procedure for each column of the reduced matrix
(c) If a row andor column has two or more zeros and one cannot be chosen by inspection
then assign arbitrary any one of these zeros and cross off all other zeros of that row
column
(d) Repeat (a) through (c) above successively until the chain of assigning (1048576) or cross (X)
ends
Step 6 If the number of assignment (1048576) is equal to n (the order of the cost matrix) an
optimum solution is reached
If the number of assignment is less than n(the order of the matrix) go to the next step
Step7 Draw the minimum number of horizontal andor vertical lines to cover all the zeros
of the reduced matrix
Step 8 Develop the new revised cost matrix as follows
(a)Find the smallest element of the reduced matrix not covered by any of the lines
(b)Subtract this element from all uncovered elements and add the same to all the elements
laying at the intersection of any two lines
Step 9 Go to step 6 and repeat the procedure until an optimum solution is attained
7
FORE School of Management 2011
The flowchart to solve any Assignment problem by Hungarian Method is given below
8
FORE School of Management 2011
Chapter 3
31 A Case of Assignment Problem
NASArsquoS astronaut crew currently includes 10 mission specialists who hold the doctoral
degree in either astrophysics or astromedicine One of these specialists will be assigned to
each of the 10 flights scheduled for the upcoming 9 months Mission specialists are
responsible for carrying out scientific and medical experiments in space or for launching
retrieving or repairing satellites The chief of Astronaut personnel himself a former crew
member with three missions under his belt must decide who should be assigned and trained
for each of the very different missions Clearly astronauts with medical educations are more
suited to other types of missions The chief assigns each astronaut a rating on a scale of 1 to
10 for each possible mission with 10 being a perfect match for the task at hand and a 1 being
a mismatch Only one specialist is assigned to each flight and none is reassigned until all
others have flown at least once
A) Who should be assigned to which flight
B) NASA has just been notified that Anderson is getting married in February and has
been granted a highly sought publicity tour in Europe that month (He intends to take
his wife and let the trip double as a honeymoon) How does this change the final
schedule
C) Creto has complained that he was misrated on his January missions Both ratings
should be 10s he claims to the chief who agrees and recomputes the schedule Do
any changes occur over the schedule set in part (b)
D) What are the strengths and weaknesses of this approach to scheduling
9
FORE School of Management 2011
Table 31 Data for problem
32 Solution to the Case
We can solve this case by two methods one is that we can go for manually solving the
question or else we can use Excel Solver
But before that we have to understand the problem that what is says and how it can be
interpreted
The problem is basically an assignment problem and here chief astronaut has to assign
various astronauts to respective missions keeping in consideration the rating which has been
given to astronauts We will have to handle the case in such a manner that proper assignment
can be done in each of the cases given in question Also we need to calculate the total rating
points when assignment has been done to see how efficient the mission is on a scale of 100
combining together the total ratings of ten astronauts
Using Solver
Setting up the LP in Solver
When all of the LP components have been entered into the worksheet and given names
Bring up Solver using the Tools rarr Solver menu There are four main elements of the
solver
10
FORE School of Management 2011
Solver dialog box
Set Target Cell The Target Cell contains the quantity you wish to optimizendashthe Objective
function value To specify the Target Cell either click on the cell with the mouse or type in
the name of the cell containing the objective function value
Equal To This specifies the direction of the optimization Click on either of the ldquoMaxrdquo or
ldquoMinrdquo radio buttons
By Changing Cells Recall that our goal is to optimize the value of the objective Function by
choosing an appropriate vector of decision variables Therefore we will Allow Excel to
change the decision variables x In the ldquoBy Changing Cellsrdquo
Subject to the Constraints Specify a constraint by clicking on the Add button While it is
possible to add each constraint one at a time it is easier (and more concise) to enter a single
inequality between the constraint function Be sure to include any additional constraints such
as nonnegativity constraints
On the right hand side of the Solver dialog box is a button labelled Options Click on this
button to bring up another dialog box Since we will be dealing primarily with linear
programs option of greatest interest is ldquoAssume Linear Modelrdquo Selecting this option forces
Excel to use a method for solving LPs known as the Simplex algorithm It is important that
ldquoAssume Linear Modelrdquo is selected or else we may end up with inappropriate outputs Once
the LP has been properly set up in the Solver dialog box press the Solve button to run Solver
A) Now we would assign each Astronaut to different Missions
11
FORE School of Management 2011
In Solver we add all the constraints and target cell as well as we set the solver to
Maximization type as we have to take in consideration the Astronauts with maximum rating
points
Solver Output Options
Pressing the Solve button runs Solver Depending on the size of the LP it may take some
time for Solver to get ready If Solver reaches a solution a new dialog box will appear and
prompt you to either accept the solution or restore the original worksheet values At this
point you may also choose to see a number of output reports The Answer report provides a
summary of the optimal decision variable values binding and non-binding constraints and
the optimal objective function value The Sensitivity report provides information describing
the sensitivity of the optimal solution to perturbations in the problem data
Following is the solution obtained from solving the Excel
12
FORE School of Management 2011
Hence assignments for each astronaut can be given as follows
Vincze Mar-26Viet April-12Anderson Feb-26Herbert Feb-05Schatz Jan-12Plane June-09Certo Sep-19Moses May-01Brandon Aug-20Drtina Jan-27
Hence we see that each astronaut has been allocated to a different mission Total rating has
been 96
B) Now in case b Anderson is getting married in February and has been granted a highly
sought publicity tour in Europe that month so he cant be assigned to any mission in
February So we would put 0 rating for him in the month of February
That is the only change in the main table and how it will affect the current solution we
will see in the solution which we will obtain after solving the problem through excel
13
FORE School of Management 2011
Now on solving the problem through solver we get the following Solution
Hence assignments for each astronaut can be given as follows
Vincze Mar-26Viet April-12Anderson Aug-20Herbert Feb-05Schatz Jan-12Plane June-09Certo Sep-19Moses May-01Brandon Feb-26Drtina Jan-27
So we see that after changing data for Anderson the assignments have also changed and rating also come down to 92
C) Now for January missions Certorsquos ratings should be 10s he claims to the chief who
agrees and recomputes the schedule hence we will change the table New table is as
follows
14
FORE School of Management 2011
On solving through excel we get the following solution
So we see that there has been a change in the total score and it has gone up by 1 to 93 But the
assignments of astronauts have remained same
D) The strengths and weaknesses of this approach to scheduling are given as follows
Strengths
Solvers or optimizers are software tools that help users find the best way to allocate
scarce resources The resources may be raw materials machine time or people time
money or anything else in limited supply The best or optimal solution may mean
maximizing profits minimizing costs or achieving the best possible quality
Weaknesses
Sometimes due to technical glitches it may give faulty result which may not be
optimized Also when we are assigning values to cell small error can have big impact
It does not take into consideration the effect of time and uncertainty There may be
cases of infeasibility and un-bounded
15
FORE School of Management 2011
References
1) Quantitative Analysis for Management by Barry Render Ralph M Stair Michael Hanna TN Badri (Pearson 10th Edition) Transportation and assignment models Question No 10-40
2) The Dynamic Hungarian Algorithm for the Assignment Problem with Changing Costs
G Ayorkor Mills Tettey Anthony Stentz M Bernardine Dias CMU-RI-TR-07-27 July 2007
Robotics Institute Carnegie Mellon Date of Access 28th September 2011
3) httpwwwniosacinsrsec311opt-lp5pdf Date of Access 28th September 2011
4) httpmikemccreavycomhungarian-assignment-problempdf Date of Access 28th September 2011
5) httpwwwamsjhuedu~castello362Handoutshungarianpdf Date of Access 28th September 2011
16
FORE School of Management 2011
The flowchart to solve any Assignment problem by Hungarian Method is given below
8
FORE School of Management 2011
Chapter 3
31 A Case of Assignment Problem
NASArsquoS astronaut crew currently includes 10 mission specialists who hold the doctoral
degree in either astrophysics or astromedicine One of these specialists will be assigned to
each of the 10 flights scheduled for the upcoming 9 months Mission specialists are
responsible for carrying out scientific and medical experiments in space or for launching
retrieving or repairing satellites The chief of Astronaut personnel himself a former crew
member with three missions under his belt must decide who should be assigned and trained
for each of the very different missions Clearly astronauts with medical educations are more
suited to other types of missions The chief assigns each astronaut a rating on a scale of 1 to
10 for each possible mission with 10 being a perfect match for the task at hand and a 1 being
a mismatch Only one specialist is assigned to each flight and none is reassigned until all
others have flown at least once
A) Who should be assigned to which flight
B) NASA has just been notified that Anderson is getting married in February and has
been granted a highly sought publicity tour in Europe that month (He intends to take
his wife and let the trip double as a honeymoon) How does this change the final
schedule
C) Creto has complained that he was misrated on his January missions Both ratings
should be 10s he claims to the chief who agrees and recomputes the schedule Do
any changes occur over the schedule set in part (b)
D) What are the strengths and weaknesses of this approach to scheduling
9
FORE School of Management 2011
Table 31 Data for problem
32 Solution to the Case
We can solve this case by two methods one is that we can go for manually solving the
question or else we can use Excel Solver
But before that we have to understand the problem that what is says and how it can be
interpreted
The problem is basically an assignment problem and here chief astronaut has to assign
various astronauts to respective missions keeping in consideration the rating which has been
given to astronauts We will have to handle the case in such a manner that proper assignment
can be done in each of the cases given in question Also we need to calculate the total rating
points when assignment has been done to see how efficient the mission is on a scale of 100
combining together the total ratings of ten astronauts
Using Solver
Setting up the LP in Solver
When all of the LP components have been entered into the worksheet and given names
Bring up Solver using the Tools rarr Solver menu There are four main elements of the
solver
10
FORE School of Management 2011
Solver dialog box
Set Target Cell The Target Cell contains the quantity you wish to optimizendashthe Objective
function value To specify the Target Cell either click on the cell with the mouse or type in
the name of the cell containing the objective function value
Equal To This specifies the direction of the optimization Click on either of the ldquoMaxrdquo or
ldquoMinrdquo radio buttons
By Changing Cells Recall that our goal is to optimize the value of the objective Function by
choosing an appropriate vector of decision variables Therefore we will Allow Excel to
change the decision variables x In the ldquoBy Changing Cellsrdquo
Subject to the Constraints Specify a constraint by clicking on the Add button While it is
possible to add each constraint one at a time it is easier (and more concise) to enter a single
inequality between the constraint function Be sure to include any additional constraints such
as nonnegativity constraints
On the right hand side of the Solver dialog box is a button labelled Options Click on this
button to bring up another dialog box Since we will be dealing primarily with linear
programs option of greatest interest is ldquoAssume Linear Modelrdquo Selecting this option forces
Excel to use a method for solving LPs known as the Simplex algorithm It is important that
ldquoAssume Linear Modelrdquo is selected or else we may end up with inappropriate outputs Once
the LP has been properly set up in the Solver dialog box press the Solve button to run Solver
A) Now we would assign each Astronaut to different Missions
11
FORE School of Management 2011
In Solver we add all the constraints and target cell as well as we set the solver to
Maximization type as we have to take in consideration the Astronauts with maximum rating
points
Solver Output Options
Pressing the Solve button runs Solver Depending on the size of the LP it may take some
time for Solver to get ready If Solver reaches a solution a new dialog box will appear and
prompt you to either accept the solution or restore the original worksheet values At this
point you may also choose to see a number of output reports The Answer report provides a
summary of the optimal decision variable values binding and non-binding constraints and
the optimal objective function value The Sensitivity report provides information describing
the sensitivity of the optimal solution to perturbations in the problem data
Following is the solution obtained from solving the Excel
12
FORE School of Management 2011
Hence assignments for each astronaut can be given as follows
Vincze Mar-26Viet April-12Anderson Feb-26Herbert Feb-05Schatz Jan-12Plane June-09Certo Sep-19Moses May-01Brandon Aug-20Drtina Jan-27
Hence we see that each astronaut has been allocated to a different mission Total rating has
been 96
B) Now in case b Anderson is getting married in February and has been granted a highly
sought publicity tour in Europe that month so he cant be assigned to any mission in
February So we would put 0 rating for him in the month of February
That is the only change in the main table and how it will affect the current solution we
will see in the solution which we will obtain after solving the problem through excel
13
FORE School of Management 2011
Now on solving the problem through solver we get the following Solution
Hence assignments for each astronaut can be given as follows
Vincze Mar-26Viet April-12Anderson Aug-20Herbert Feb-05Schatz Jan-12Plane June-09Certo Sep-19Moses May-01Brandon Feb-26Drtina Jan-27
So we see that after changing data for Anderson the assignments have also changed and rating also come down to 92
C) Now for January missions Certorsquos ratings should be 10s he claims to the chief who
agrees and recomputes the schedule hence we will change the table New table is as
follows
14
FORE School of Management 2011
On solving through excel we get the following solution
So we see that there has been a change in the total score and it has gone up by 1 to 93 But the
assignments of astronauts have remained same
D) The strengths and weaknesses of this approach to scheduling are given as follows
Strengths
Solvers or optimizers are software tools that help users find the best way to allocate
scarce resources The resources may be raw materials machine time or people time
money or anything else in limited supply The best or optimal solution may mean
maximizing profits minimizing costs or achieving the best possible quality
Weaknesses
Sometimes due to technical glitches it may give faulty result which may not be
optimized Also when we are assigning values to cell small error can have big impact
It does not take into consideration the effect of time and uncertainty There may be
cases of infeasibility and un-bounded
15
FORE School of Management 2011
References
1) Quantitative Analysis for Management by Barry Render Ralph M Stair Michael Hanna TN Badri (Pearson 10th Edition) Transportation and assignment models Question No 10-40
2) The Dynamic Hungarian Algorithm for the Assignment Problem with Changing Costs
G Ayorkor Mills Tettey Anthony Stentz M Bernardine Dias CMU-RI-TR-07-27 July 2007
Robotics Institute Carnegie Mellon Date of Access 28th September 2011
3) httpwwwniosacinsrsec311opt-lp5pdf Date of Access 28th September 2011
4) httpmikemccreavycomhungarian-assignment-problempdf Date of Access 28th September 2011
5) httpwwwamsjhuedu~castello362Handoutshungarianpdf Date of Access 28th September 2011
16
FORE School of Management 2011
Chapter 3
31 A Case of Assignment Problem
NASArsquoS astronaut crew currently includes 10 mission specialists who hold the doctoral
degree in either astrophysics or astromedicine One of these specialists will be assigned to
each of the 10 flights scheduled for the upcoming 9 months Mission specialists are
responsible for carrying out scientific and medical experiments in space or for launching
retrieving or repairing satellites The chief of Astronaut personnel himself a former crew
member with three missions under his belt must decide who should be assigned and trained
for each of the very different missions Clearly astronauts with medical educations are more
suited to other types of missions The chief assigns each astronaut a rating on a scale of 1 to
10 for each possible mission with 10 being a perfect match for the task at hand and a 1 being
a mismatch Only one specialist is assigned to each flight and none is reassigned until all
others have flown at least once
A) Who should be assigned to which flight
B) NASA has just been notified that Anderson is getting married in February and has
been granted a highly sought publicity tour in Europe that month (He intends to take
his wife and let the trip double as a honeymoon) How does this change the final
schedule
C) Creto has complained that he was misrated on his January missions Both ratings
should be 10s he claims to the chief who agrees and recomputes the schedule Do
any changes occur over the schedule set in part (b)
D) What are the strengths and weaknesses of this approach to scheduling
9
FORE School of Management 2011
Table 31 Data for problem
32 Solution to the Case
We can solve this case by two methods one is that we can go for manually solving the
question or else we can use Excel Solver
But before that we have to understand the problem that what is says and how it can be
interpreted
The problem is basically an assignment problem and here chief astronaut has to assign
various astronauts to respective missions keeping in consideration the rating which has been
given to astronauts We will have to handle the case in such a manner that proper assignment
can be done in each of the cases given in question Also we need to calculate the total rating
points when assignment has been done to see how efficient the mission is on a scale of 100
combining together the total ratings of ten astronauts
Using Solver
Setting up the LP in Solver
When all of the LP components have been entered into the worksheet and given names
Bring up Solver using the Tools rarr Solver menu There are four main elements of the
solver
10
FORE School of Management 2011
Solver dialog box
Set Target Cell The Target Cell contains the quantity you wish to optimizendashthe Objective
function value To specify the Target Cell either click on the cell with the mouse or type in
the name of the cell containing the objective function value
Equal To This specifies the direction of the optimization Click on either of the ldquoMaxrdquo or
ldquoMinrdquo radio buttons
By Changing Cells Recall that our goal is to optimize the value of the objective Function by
choosing an appropriate vector of decision variables Therefore we will Allow Excel to
change the decision variables x In the ldquoBy Changing Cellsrdquo
Subject to the Constraints Specify a constraint by clicking on the Add button While it is
possible to add each constraint one at a time it is easier (and more concise) to enter a single
inequality between the constraint function Be sure to include any additional constraints such
as nonnegativity constraints
On the right hand side of the Solver dialog box is a button labelled Options Click on this
button to bring up another dialog box Since we will be dealing primarily with linear
programs option of greatest interest is ldquoAssume Linear Modelrdquo Selecting this option forces
Excel to use a method for solving LPs known as the Simplex algorithm It is important that
ldquoAssume Linear Modelrdquo is selected or else we may end up with inappropriate outputs Once
the LP has been properly set up in the Solver dialog box press the Solve button to run Solver
A) Now we would assign each Astronaut to different Missions
11
FORE School of Management 2011
In Solver we add all the constraints and target cell as well as we set the solver to
Maximization type as we have to take in consideration the Astronauts with maximum rating
points
Solver Output Options
Pressing the Solve button runs Solver Depending on the size of the LP it may take some
time for Solver to get ready If Solver reaches a solution a new dialog box will appear and
prompt you to either accept the solution or restore the original worksheet values At this
point you may also choose to see a number of output reports The Answer report provides a
summary of the optimal decision variable values binding and non-binding constraints and
the optimal objective function value The Sensitivity report provides information describing
the sensitivity of the optimal solution to perturbations in the problem data
Following is the solution obtained from solving the Excel
12
FORE School of Management 2011
Hence assignments for each astronaut can be given as follows
Vincze Mar-26Viet April-12Anderson Feb-26Herbert Feb-05Schatz Jan-12Plane June-09Certo Sep-19Moses May-01Brandon Aug-20Drtina Jan-27
Hence we see that each astronaut has been allocated to a different mission Total rating has
been 96
B) Now in case b Anderson is getting married in February and has been granted a highly
sought publicity tour in Europe that month so he cant be assigned to any mission in
February So we would put 0 rating for him in the month of February
That is the only change in the main table and how it will affect the current solution we
will see in the solution which we will obtain after solving the problem through excel
13
FORE School of Management 2011
Now on solving the problem through solver we get the following Solution
Hence assignments for each astronaut can be given as follows
Vincze Mar-26Viet April-12Anderson Aug-20Herbert Feb-05Schatz Jan-12Plane June-09Certo Sep-19Moses May-01Brandon Feb-26Drtina Jan-27
So we see that after changing data for Anderson the assignments have also changed and rating also come down to 92
C) Now for January missions Certorsquos ratings should be 10s he claims to the chief who
agrees and recomputes the schedule hence we will change the table New table is as
follows
14
FORE School of Management 2011
On solving through excel we get the following solution
So we see that there has been a change in the total score and it has gone up by 1 to 93 But the
assignments of astronauts have remained same
D) The strengths and weaknesses of this approach to scheduling are given as follows
Strengths
Solvers or optimizers are software tools that help users find the best way to allocate
scarce resources The resources may be raw materials machine time or people time
money or anything else in limited supply The best or optimal solution may mean
maximizing profits minimizing costs or achieving the best possible quality
Weaknesses
Sometimes due to technical glitches it may give faulty result which may not be
optimized Also when we are assigning values to cell small error can have big impact
It does not take into consideration the effect of time and uncertainty There may be
cases of infeasibility and un-bounded
15
FORE School of Management 2011
References
1) Quantitative Analysis for Management by Barry Render Ralph M Stair Michael Hanna TN Badri (Pearson 10th Edition) Transportation and assignment models Question No 10-40
2) The Dynamic Hungarian Algorithm for the Assignment Problem with Changing Costs
G Ayorkor Mills Tettey Anthony Stentz M Bernardine Dias CMU-RI-TR-07-27 July 2007
Robotics Institute Carnegie Mellon Date of Access 28th September 2011
3) httpwwwniosacinsrsec311opt-lp5pdf Date of Access 28th September 2011
4) httpmikemccreavycomhungarian-assignment-problempdf Date of Access 28th September 2011
5) httpwwwamsjhuedu~castello362Handoutshungarianpdf Date of Access 28th September 2011
16
FORE School of Management 2011
Table 31 Data for problem
32 Solution to the Case
We can solve this case by two methods one is that we can go for manually solving the
question or else we can use Excel Solver
But before that we have to understand the problem that what is says and how it can be
interpreted
The problem is basically an assignment problem and here chief astronaut has to assign
various astronauts to respective missions keeping in consideration the rating which has been
given to astronauts We will have to handle the case in such a manner that proper assignment
can be done in each of the cases given in question Also we need to calculate the total rating
points when assignment has been done to see how efficient the mission is on a scale of 100
combining together the total ratings of ten astronauts
Using Solver
Setting up the LP in Solver
When all of the LP components have been entered into the worksheet and given names
Bring up Solver using the Tools rarr Solver menu There are four main elements of the
solver
10
FORE School of Management 2011
Solver dialog box
Set Target Cell The Target Cell contains the quantity you wish to optimizendashthe Objective
function value To specify the Target Cell either click on the cell with the mouse or type in
the name of the cell containing the objective function value
Equal To This specifies the direction of the optimization Click on either of the ldquoMaxrdquo or
ldquoMinrdquo radio buttons
By Changing Cells Recall that our goal is to optimize the value of the objective Function by
choosing an appropriate vector of decision variables Therefore we will Allow Excel to
change the decision variables x In the ldquoBy Changing Cellsrdquo
Subject to the Constraints Specify a constraint by clicking on the Add button While it is
possible to add each constraint one at a time it is easier (and more concise) to enter a single
inequality between the constraint function Be sure to include any additional constraints such
as nonnegativity constraints
On the right hand side of the Solver dialog box is a button labelled Options Click on this
button to bring up another dialog box Since we will be dealing primarily with linear
programs option of greatest interest is ldquoAssume Linear Modelrdquo Selecting this option forces
Excel to use a method for solving LPs known as the Simplex algorithm It is important that
ldquoAssume Linear Modelrdquo is selected or else we may end up with inappropriate outputs Once
the LP has been properly set up in the Solver dialog box press the Solve button to run Solver
A) Now we would assign each Astronaut to different Missions
11
FORE School of Management 2011
In Solver we add all the constraints and target cell as well as we set the solver to
Maximization type as we have to take in consideration the Astronauts with maximum rating
points
Solver Output Options
Pressing the Solve button runs Solver Depending on the size of the LP it may take some
time for Solver to get ready If Solver reaches a solution a new dialog box will appear and
prompt you to either accept the solution or restore the original worksheet values At this
point you may also choose to see a number of output reports The Answer report provides a
summary of the optimal decision variable values binding and non-binding constraints and
the optimal objective function value The Sensitivity report provides information describing
the sensitivity of the optimal solution to perturbations in the problem data
Following is the solution obtained from solving the Excel
12
FORE School of Management 2011
Hence assignments for each astronaut can be given as follows
Vincze Mar-26Viet April-12Anderson Feb-26Herbert Feb-05Schatz Jan-12Plane June-09Certo Sep-19Moses May-01Brandon Aug-20Drtina Jan-27
Hence we see that each astronaut has been allocated to a different mission Total rating has
been 96
B) Now in case b Anderson is getting married in February and has been granted a highly
sought publicity tour in Europe that month so he cant be assigned to any mission in
February So we would put 0 rating for him in the month of February
That is the only change in the main table and how it will affect the current solution we
will see in the solution which we will obtain after solving the problem through excel
13
FORE School of Management 2011
Now on solving the problem through solver we get the following Solution
Hence assignments for each astronaut can be given as follows
Vincze Mar-26Viet April-12Anderson Aug-20Herbert Feb-05Schatz Jan-12Plane June-09Certo Sep-19Moses May-01Brandon Feb-26Drtina Jan-27
So we see that after changing data for Anderson the assignments have also changed and rating also come down to 92
C) Now for January missions Certorsquos ratings should be 10s he claims to the chief who
agrees and recomputes the schedule hence we will change the table New table is as
follows
14
FORE School of Management 2011
On solving through excel we get the following solution
So we see that there has been a change in the total score and it has gone up by 1 to 93 But the
assignments of astronauts have remained same
D) The strengths and weaknesses of this approach to scheduling are given as follows
Strengths
Solvers or optimizers are software tools that help users find the best way to allocate
scarce resources The resources may be raw materials machine time or people time
money or anything else in limited supply The best or optimal solution may mean
maximizing profits minimizing costs or achieving the best possible quality
Weaknesses
Sometimes due to technical glitches it may give faulty result which may not be
optimized Also when we are assigning values to cell small error can have big impact
It does not take into consideration the effect of time and uncertainty There may be
cases of infeasibility and un-bounded
15
FORE School of Management 2011
References
1) Quantitative Analysis for Management by Barry Render Ralph M Stair Michael Hanna TN Badri (Pearson 10th Edition) Transportation and assignment models Question No 10-40
2) The Dynamic Hungarian Algorithm for the Assignment Problem with Changing Costs
G Ayorkor Mills Tettey Anthony Stentz M Bernardine Dias CMU-RI-TR-07-27 July 2007
Robotics Institute Carnegie Mellon Date of Access 28th September 2011
3) httpwwwniosacinsrsec311opt-lp5pdf Date of Access 28th September 2011
4) httpmikemccreavycomhungarian-assignment-problempdf Date of Access 28th September 2011
5) httpwwwamsjhuedu~castello362Handoutshungarianpdf Date of Access 28th September 2011
16
FORE School of Management 2011
Solver dialog box
Set Target Cell The Target Cell contains the quantity you wish to optimizendashthe Objective
function value To specify the Target Cell either click on the cell with the mouse or type in
the name of the cell containing the objective function value
Equal To This specifies the direction of the optimization Click on either of the ldquoMaxrdquo or
ldquoMinrdquo radio buttons
By Changing Cells Recall that our goal is to optimize the value of the objective Function by
choosing an appropriate vector of decision variables Therefore we will Allow Excel to
change the decision variables x In the ldquoBy Changing Cellsrdquo
Subject to the Constraints Specify a constraint by clicking on the Add button While it is
possible to add each constraint one at a time it is easier (and more concise) to enter a single
inequality between the constraint function Be sure to include any additional constraints such
as nonnegativity constraints
On the right hand side of the Solver dialog box is a button labelled Options Click on this
button to bring up another dialog box Since we will be dealing primarily with linear
programs option of greatest interest is ldquoAssume Linear Modelrdquo Selecting this option forces
Excel to use a method for solving LPs known as the Simplex algorithm It is important that
ldquoAssume Linear Modelrdquo is selected or else we may end up with inappropriate outputs Once
the LP has been properly set up in the Solver dialog box press the Solve button to run Solver
A) Now we would assign each Astronaut to different Missions
11
FORE School of Management 2011
In Solver we add all the constraints and target cell as well as we set the solver to
Maximization type as we have to take in consideration the Astronauts with maximum rating
points
Solver Output Options
Pressing the Solve button runs Solver Depending on the size of the LP it may take some
time for Solver to get ready If Solver reaches a solution a new dialog box will appear and
prompt you to either accept the solution or restore the original worksheet values At this
point you may also choose to see a number of output reports The Answer report provides a
summary of the optimal decision variable values binding and non-binding constraints and
the optimal objective function value The Sensitivity report provides information describing
the sensitivity of the optimal solution to perturbations in the problem data
Following is the solution obtained from solving the Excel
12
FORE School of Management 2011
Hence assignments for each astronaut can be given as follows
Vincze Mar-26Viet April-12Anderson Feb-26Herbert Feb-05Schatz Jan-12Plane June-09Certo Sep-19Moses May-01Brandon Aug-20Drtina Jan-27
Hence we see that each astronaut has been allocated to a different mission Total rating has
been 96
B) Now in case b Anderson is getting married in February and has been granted a highly
sought publicity tour in Europe that month so he cant be assigned to any mission in
February So we would put 0 rating for him in the month of February
That is the only change in the main table and how it will affect the current solution we
will see in the solution which we will obtain after solving the problem through excel
13
FORE School of Management 2011
Now on solving the problem through solver we get the following Solution
Hence assignments for each astronaut can be given as follows
Vincze Mar-26Viet April-12Anderson Aug-20Herbert Feb-05Schatz Jan-12Plane June-09Certo Sep-19Moses May-01Brandon Feb-26Drtina Jan-27
So we see that after changing data for Anderson the assignments have also changed and rating also come down to 92
C) Now for January missions Certorsquos ratings should be 10s he claims to the chief who
agrees and recomputes the schedule hence we will change the table New table is as
follows
14
FORE School of Management 2011
On solving through excel we get the following solution
So we see that there has been a change in the total score and it has gone up by 1 to 93 But the
assignments of astronauts have remained same
D) The strengths and weaknesses of this approach to scheduling are given as follows
Strengths
Solvers or optimizers are software tools that help users find the best way to allocate
scarce resources The resources may be raw materials machine time or people time
money or anything else in limited supply The best or optimal solution may mean
maximizing profits minimizing costs or achieving the best possible quality
Weaknesses
Sometimes due to technical glitches it may give faulty result which may not be
optimized Also when we are assigning values to cell small error can have big impact
It does not take into consideration the effect of time and uncertainty There may be
cases of infeasibility and un-bounded
15
FORE School of Management 2011
References
1) Quantitative Analysis for Management by Barry Render Ralph M Stair Michael Hanna TN Badri (Pearson 10th Edition) Transportation and assignment models Question No 10-40
2) The Dynamic Hungarian Algorithm for the Assignment Problem with Changing Costs
G Ayorkor Mills Tettey Anthony Stentz M Bernardine Dias CMU-RI-TR-07-27 July 2007
Robotics Institute Carnegie Mellon Date of Access 28th September 2011
3) httpwwwniosacinsrsec311opt-lp5pdf Date of Access 28th September 2011
4) httpmikemccreavycomhungarian-assignment-problempdf Date of Access 28th September 2011
5) httpwwwamsjhuedu~castello362Handoutshungarianpdf Date of Access 28th September 2011
16
FORE School of Management 2011
In Solver we add all the constraints and target cell as well as we set the solver to
Maximization type as we have to take in consideration the Astronauts with maximum rating
points
Solver Output Options
Pressing the Solve button runs Solver Depending on the size of the LP it may take some
time for Solver to get ready If Solver reaches a solution a new dialog box will appear and
prompt you to either accept the solution or restore the original worksheet values At this
point you may also choose to see a number of output reports The Answer report provides a
summary of the optimal decision variable values binding and non-binding constraints and
the optimal objective function value The Sensitivity report provides information describing
the sensitivity of the optimal solution to perturbations in the problem data
Following is the solution obtained from solving the Excel
12
FORE School of Management 2011
Hence assignments for each astronaut can be given as follows
Vincze Mar-26Viet April-12Anderson Feb-26Herbert Feb-05Schatz Jan-12Plane June-09Certo Sep-19Moses May-01Brandon Aug-20Drtina Jan-27
Hence we see that each astronaut has been allocated to a different mission Total rating has
been 96
B) Now in case b Anderson is getting married in February and has been granted a highly
sought publicity tour in Europe that month so he cant be assigned to any mission in
February So we would put 0 rating for him in the month of February
That is the only change in the main table and how it will affect the current solution we
will see in the solution which we will obtain after solving the problem through excel
13
FORE School of Management 2011
Now on solving the problem through solver we get the following Solution
Hence assignments for each astronaut can be given as follows
Vincze Mar-26Viet April-12Anderson Aug-20Herbert Feb-05Schatz Jan-12Plane June-09Certo Sep-19Moses May-01Brandon Feb-26Drtina Jan-27
So we see that after changing data for Anderson the assignments have also changed and rating also come down to 92
C) Now for January missions Certorsquos ratings should be 10s he claims to the chief who
agrees and recomputes the schedule hence we will change the table New table is as
follows
14
FORE School of Management 2011
On solving through excel we get the following solution
So we see that there has been a change in the total score and it has gone up by 1 to 93 But the
assignments of astronauts have remained same
D) The strengths and weaknesses of this approach to scheduling are given as follows
Strengths
Solvers or optimizers are software tools that help users find the best way to allocate
scarce resources The resources may be raw materials machine time or people time
money or anything else in limited supply The best or optimal solution may mean
maximizing profits minimizing costs or achieving the best possible quality
Weaknesses
Sometimes due to technical glitches it may give faulty result which may not be
optimized Also when we are assigning values to cell small error can have big impact
It does not take into consideration the effect of time and uncertainty There may be
cases of infeasibility and un-bounded
15
FORE School of Management 2011
References
1) Quantitative Analysis for Management by Barry Render Ralph M Stair Michael Hanna TN Badri (Pearson 10th Edition) Transportation and assignment models Question No 10-40
2) The Dynamic Hungarian Algorithm for the Assignment Problem with Changing Costs
G Ayorkor Mills Tettey Anthony Stentz M Bernardine Dias CMU-RI-TR-07-27 July 2007
Robotics Institute Carnegie Mellon Date of Access 28th September 2011
3) httpwwwniosacinsrsec311opt-lp5pdf Date of Access 28th September 2011
4) httpmikemccreavycomhungarian-assignment-problempdf Date of Access 28th September 2011
5) httpwwwamsjhuedu~castello362Handoutshungarianpdf Date of Access 28th September 2011
16
FORE School of Management 2011
Hence assignments for each astronaut can be given as follows
Vincze Mar-26Viet April-12Anderson Feb-26Herbert Feb-05Schatz Jan-12Plane June-09Certo Sep-19Moses May-01Brandon Aug-20Drtina Jan-27
Hence we see that each astronaut has been allocated to a different mission Total rating has
been 96
B) Now in case b Anderson is getting married in February and has been granted a highly
sought publicity tour in Europe that month so he cant be assigned to any mission in
February So we would put 0 rating for him in the month of February
That is the only change in the main table and how it will affect the current solution we
will see in the solution which we will obtain after solving the problem through excel
13
FORE School of Management 2011
Now on solving the problem through solver we get the following Solution
Hence assignments for each astronaut can be given as follows
Vincze Mar-26Viet April-12Anderson Aug-20Herbert Feb-05Schatz Jan-12Plane June-09Certo Sep-19Moses May-01Brandon Feb-26Drtina Jan-27
So we see that after changing data for Anderson the assignments have also changed and rating also come down to 92
C) Now for January missions Certorsquos ratings should be 10s he claims to the chief who
agrees and recomputes the schedule hence we will change the table New table is as
follows
14
FORE School of Management 2011
On solving through excel we get the following solution
So we see that there has been a change in the total score and it has gone up by 1 to 93 But the
assignments of astronauts have remained same
D) The strengths and weaknesses of this approach to scheduling are given as follows
Strengths
Solvers or optimizers are software tools that help users find the best way to allocate
scarce resources The resources may be raw materials machine time or people time
money or anything else in limited supply The best or optimal solution may mean
maximizing profits minimizing costs or achieving the best possible quality
Weaknesses
Sometimes due to technical glitches it may give faulty result which may not be
optimized Also when we are assigning values to cell small error can have big impact
It does not take into consideration the effect of time and uncertainty There may be
cases of infeasibility and un-bounded
15
FORE School of Management 2011
References
1) Quantitative Analysis for Management by Barry Render Ralph M Stair Michael Hanna TN Badri (Pearson 10th Edition) Transportation and assignment models Question No 10-40
2) The Dynamic Hungarian Algorithm for the Assignment Problem with Changing Costs
G Ayorkor Mills Tettey Anthony Stentz M Bernardine Dias CMU-RI-TR-07-27 July 2007
Robotics Institute Carnegie Mellon Date of Access 28th September 2011
3) httpwwwniosacinsrsec311opt-lp5pdf Date of Access 28th September 2011
4) httpmikemccreavycomhungarian-assignment-problempdf Date of Access 28th September 2011
5) httpwwwamsjhuedu~castello362Handoutshungarianpdf Date of Access 28th September 2011
16
FORE School of Management 2011
Now on solving the problem through solver we get the following Solution
Hence assignments for each astronaut can be given as follows
Vincze Mar-26Viet April-12Anderson Aug-20Herbert Feb-05Schatz Jan-12Plane June-09Certo Sep-19Moses May-01Brandon Feb-26Drtina Jan-27
So we see that after changing data for Anderson the assignments have also changed and rating also come down to 92
C) Now for January missions Certorsquos ratings should be 10s he claims to the chief who
agrees and recomputes the schedule hence we will change the table New table is as
follows
14
FORE School of Management 2011
On solving through excel we get the following solution
So we see that there has been a change in the total score and it has gone up by 1 to 93 But the
assignments of astronauts have remained same
D) The strengths and weaknesses of this approach to scheduling are given as follows
Strengths
Solvers or optimizers are software tools that help users find the best way to allocate
scarce resources The resources may be raw materials machine time or people time
money or anything else in limited supply The best or optimal solution may mean
maximizing profits minimizing costs or achieving the best possible quality
Weaknesses
Sometimes due to technical glitches it may give faulty result which may not be
optimized Also when we are assigning values to cell small error can have big impact
It does not take into consideration the effect of time and uncertainty There may be
cases of infeasibility and un-bounded
15
FORE School of Management 2011
References
1) Quantitative Analysis for Management by Barry Render Ralph M Stair Michael Hanna TN Badri (Pearson 10th Edition) Transportation and assignment models Question No 10-40
2) The Dynamic Hungarian Algorithm for the Assignment Problem with Changing Costs
G Ayorkor Mills Tettey Anthony Stentz M Bernardine Dias CMU-RI-TR-07-27 July 2007
Robotics Institute Carnegie Mellon Date of Access 28th September 2011
3) httpwwwniosacinsrsec311opt-lp5pdf Date of Access 28th September 2011
4) httpmikemccreavycomhungarian-assignment-problempdf Date of Access 28th September 2011
5) httpwwwamsjhuedu~castello362Handoutshungarianpdf Date of Access 28th September 2011
16
FORE School of Management 2011
On solving through excel we get the following solution
So we see that there has been a change in the total score and it has gone up by 1 to 93 But the
assignments of astronauts have remained same
D) The strengths and weaknesses of this approach to scheduling are given as follows
Strengths
Solvers or optimizers are software tools that help users find the best way to allocate
scarce resources The resources may be raw materials machine time or people time
money or anything else in limited supply The best or optimal solution may mean
maximizing profits minimizing costs or achieving the best possible quality
Weaknesses
Sometimes due to technical glitches it may give faulty result which may not be
optimized Also when we are assigning values to cell small error can have big impact
It does not take into consideration the effect of time and uncertainty There may be
cases of infeasibility and un-bounded
15
FORE School of Management 2011
References
1) Quantitative Analysis for Management by Barry Render Ralph M Stair Michael Hanna TN Badri (Pearson 10th Edition) Transportation and assignment models Question No 10-40
2) The Dynamic Hungarian Algorithm for the Assignment Problem with Changing Costs
G Ayorkor Mills Tettey Anthony Stentz M Bernardine Dias CMU-RI-TR-07-27 July 2007
Robotics Institute Carnegie Mellon Date of Access 28th September 2011
3) httpwwwniosacinsrsec311opt-lp5pdf Date of Access 28th September 2011
4) httpmikemccreavycomhungarian-assignment-problempdf Date of Access 28th September 2011
5) httpwwwamsjhuedu~castello362Handoutshungarianpdf Date of Access 28th September 2011
16
FORE School of Management 2011
References
1) Quantitative Analysis for Management by Barry Render Ralph M Stair Michael Hanna TN Badri (Pearson 10th Edition) Transportation and assignment models Question No 10-40
2) The Dynamic Hungarian Algorithm for the Assignment Problem with Changing Costs
G Ayorkor Mills Tettey Anthony Stentz M Bernardine Dias CMU-RI-TR-07-27 July 2007
Robotics Institute Carnegie Mellon Date of Access 28th September 2011
3) httpwwwniosacinsrsec311opt-lp5pdf Date of Access 28th September 2011
4) httpmikemccreavycomhungarian-assignment-problempdf Date of Access 28th September 2011
5) httpwwwamsjhuedu~castello362Handoutshungarianpdf Date of Access 28th September 2011
16
Top Related