Project Report on Assignment Problem

FORE School of Management 201 1

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


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.

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Table of Contents Table of Contents................................................................................................... 3 2.1 Application Areas of Assignment Problem..............................................................................5 2.2 Formulation Of The Problem...................................................................................................5 2.3 Solution Methods ...................................................................................................................... 6 2.4 Hungarian Method.................................................................................................................... 6 3.1 A Case of Assignment Problem............................................................................................... 11 3.2 Solution to the Case................................................................................................................. 12

References:.......................................................................................................... 18

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Chapter 1 1.1 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 .

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Chapter 2 2.1 Application Areas of Assignment Problem Though 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 sales/marketing 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.

2.2 Formulation Of The Problem Let there are n jobs and n persons are available with different skills. If the cost of doing jth work by ith person is cij.Then the cost matrix is given in the table 1 below: Jobs/ Persons

1

2

3

.......j

........n

1

C11

C12

C13

.....C1j

......C1n

2

C21

C22

C23

......C2j

......C2n

.. .. .. ..i

.. .. .. Ci1

.. .. .. Ci2

.. .. .. Ci3

.. .. .. Cij

.. .. .. Cin

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FORE School of Management 201 1 .. .. .. ...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.

Mathematically, we can express the problem as follows: nn

Z = Σ Σ Cij Xij i=1 j=1 Subject 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.

2.3 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.

2.4 Hungarian Method Step 1. Determine the cost table from the given problem. 6

FORE School of Management 201 1 (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 source/destinations are always zero. 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 ()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 and/or 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 ( ) or cross (X) ends. Step 6. If the number of assignment ( ) 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 and/or vertical lines to cover all the zeros of

the reduced matrix.

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FORE School of Management 201 1 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.

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FORE School of Management 201 1 The flowchart to solve any Assignment problem by Hungarian Method is given below:

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Chapter 3 3.1 A Case of Assignment Problem NASA’S 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?

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FORE School of Management 201 1 Table 3.1 Data for problem

3.2 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 → Solver... menu. There are four main elements of the solver:

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FORE School of Management 201 1 Solver dialog box: Set Target Cell: The Target Cell contains the quantity you wish to optimize–the 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 “Max” or

“Min” 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 “By Changing Cells:” 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 “Assume Linear Model.” Selecting this option forces Excel to use a method for solving LPs known as the Simplex algorithm. It is important that “Assume Linear Model” 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.

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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 14

FORE School of Management 201 1 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.

Hence assignments for each astronaut can be given as follows: Vincze Viet Anderson Herbert Schatz Plane Certo Moses Brandon Drtina

Mar-26 April-12 Feb-26 Feb-05 Jan-12 June-09 Sep-19 May-01 Aug-20 Jan-27

Hence, we see that each astronaut has been allocated to a different mission. Total rating has been 96

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FORE School of Management 201 1 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.

Now on solving the problem through solver we get the following Solution:

Hence assignments for each astronaut can be given as follows:

Vincze Viet Anderson Herbert Schatz Plane Certo Moses Brandon Drtina 16

Mar-26 April-12 Aug-20 Feb-05 Jan-12 June-09 Sep-19 May-01 Feb-26 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 Certo’s 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:

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,

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FORE School of Management 201 1 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.

References: 1) Quantitative Analysis for Management by Barry Render, Ralph M. Stair, Michael Hanna, T.N. 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-0727 July 2007 Robotics Institute, Carnegie Mellon Date of Access: 28 th September 2011 3) http://www.nios.ac.in/srsec311/opt-lp5.pdf Date of Access: 28th September 2011 4) http://mike.mccreavy.com/hungarian-assignment-problem.pdf Date of Access: 28th September 2011 5) http://www.ams.jhu.edu/~castello/362/Handouts/hungarian.pdf Date of Access: 28th September 2011

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Project Report on Assignment Problem

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