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