MB0048 Set 1 & 2

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Spring 2012, MBA-2nd Semester

ASSIGNMENT SET 1

1. A toy company manufactures two types of dolls, a basic version doll-A and a deluxeversion doll-B. Each doll of type B takes twice as long to produce as one of type A, and

the company would have time to make maximum of 1000 per day. The supply of plastic

is sufficient to produce 1000 dolls per day (both A & B combined). The deluxe version

requires a fancy dress of which there are only 500 per day available. If the company

makes a profit of Rs 3.00 and Rs 5.. per doll, respectively on doll A and B, then how

many of each doll should be produced per day in order to maximize the total profit.

Formulate this problem.

Formulation:

Let X1 and X2 be the number of dolls produced per day of type A and B, respectively.

Let the A require t hrs.

So that the doll B require 2t hrs.

So the total time to manufacture X1 and X2 dolls should not exceed 2000t hrs.

Therefore, tX1 + 2tX2 2000t

Other constraints are simple. Then the linear programming problem becomes:

Maximize p = 3 X1 + 5 X2

Subject to restrictions,

X1 + 2X2 2000 (Time constraint)

X1 + X2 1500 (Plastic constraint)

X2 600 (Dress constraint)

And non-negatively restrictions

X1, X2 0

2. What are the advantages of Linear programming techniques?

1. The linear programming technique helps to make the best possible use of available

productive resources (such as time, labor, machines etc.)

2. It improves the quality of decisions. The individual who makes use of linear programming

methods becomes more objective than subjective.

MB0048: Operations Research Roll No. : 541110058Page 1

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3. It also helps in providing better tools for adjustment to meet changing conditions.

4. In a production process, bottle necks may occur. For example, in a factory some machines

may be in great demand while others may lie idle for some time. A significant advantage of

linear programming is highlighting of such bottle necks.

5. Most business problems involve constraints like raw materials availability, market demand

etc. which must be taken into consideration. Just we can produce so many units of product

does not mean that they can be sold. Linear programming can handle such situation also.

3. Solve the following Assignment Problem

Operations M1 M2 M3 M4

O1 10 15 12 11

O2 9 10 9 12

O3 15 16 16 17

Since the number of rows are less than number of columns, adding a dummy row and

applying Hungarian method,

Row reduction matrix


O1 10 15 12 11

O2 9 10 9 12

O3 15 16 16 17

O4 0 0 0 0

Optimum assignment solution


O1 [0] 5 2 1

O2 x 1 [0] 3

O3 1 [0] x x

O4 x x x [0]

Hungarian Method leads to multiple solutions. Selecting (03, M2) arbitrarily.

O1 M1 10


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O2 M3 09

O3 M2 16

O4 M4 00

-------------------------TOTAL 35

Therefore, the optimum assignment schedule is O1 M1, O2 M3, O3 M2 AND O4 M4.

4. What is integer programming?

If the unknown variables are all required to be integers, then the problem is called an integer

programming(IP) orinteger linear programming (ILP) problem. In contrast to linear

programming, which can be solved efficiently in the worst case, integer programming

problems are in many practical situations (those with bounded variables)NP-hard (non-

deterministic polynomial-time hard), in computational complexity theory, is a class of

problems that are, informally, "at least as hard as the hardest problems in NP"). 0-1 integer

programming orbinary integer programming (BIP) is the special case of integer

programming where variables are required to be 0 or 1 (rather than arbitrary integers). This

problem is also classified as NP-hard, and in fact the decision version was one ofKarp's 21

NP-complete problems.

Integer programming is a mathematical method for determining a way to achieve the best

outcome (such as maximum profit or lowest cost) in a given mathematical model for some

list of requirements represented as linear relationships. Linear programming is a specific case

of mathematical programming (mathematical optimization).

More formally, integer programming is a technique for the optimization of a linearobjective

function, subject to linear equality and linear inequalityconstraints. Its feasible region is a

convex polyhedron, which is a set defined as the intersection of finitely many half spaces,each of which is defined by a linear inequality. Its objective function is a real-valued affine

function defined on this polyhedron. A linear programming algorithm finds a point in the

polyhedron where this function has the smallest (or largest) value if such point exists.

Integer programming can be applied to various fields of study. It is used in business and

economics, but can also be utilized for some engineering problems. Industries that use linear

programming models include transportation, energy, telecommunications, and

manufacturing. It has proved useful in modeling diverse types of problems in planning,

routing, scheduling, assignment, and design.

http://en.wikipedia.org/wiki/Integer_programminghttp://en.wikipedia.org/wiki/Integer_programminghttp://en.wikipedia.org/wiki/Integer_programminghttp://en.wikipedia.org/wiki/NP-hardhttp://en.wikipedia.org/wiki/NP_(complexity)http://en.wikipedia.org/wiki/NP_(complexity)http://en.wikipedia.org/wiki/Computational_complexity_theoryhttp://en.wikipedia.org/wiki/Karp's_21_NP-complete_problemshttp://en.wikipedia.org/wiki/Karp's_21_NP-complete_problemshttp://en.wikipedia.org/wiki/Mathematical_modelhttp://en.wikipedia.org/wiki/Mathematical_optimizationhttp://en.wikipedia.org/wiki/Mathematical_optimizationhttp://en.wikipedia.org/wiki/Linearhttp://en.wikipedia.org/wiki/Objective_functionhttp://en.wikipedia.org/wiki/Objective_functionhttp://en.wikipedia.org/wiki/Linear_equalityhttp://en.wikipedia.org/wiki/Linear_inequalityhttp://en.wikipedia.org/wiki/Constraint_(mathematics)http://en.wikipedia.org/wiki/Feasible_regionhttp://en.wikipedia.org/wiki/Convex_polyhedronhttp://en.wikipedia.org/wiki/Half_spacehttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Affine_functionhttp://en.wikipedia.org/wiki/Affine_functionhttp://en.wikipedia.org/wiki/Algorithmhttp://en.wikipedia.org/wiki/Assignment_problemhttp://en.wikipedia.org/wiki/Integer_programminghttp://en.wikipedia.org/wiki/Integer_programminghttp://en.wikipedia.org/wiki/NP-hardhttp://en.wikipedia.org/wiki/NP_(complexity)http://en.wikipedia.org/wiki/NP_(complexity)http://en.wikipedia.org/wiki/Computational_complexity_theoryhttp://en.wikipedia.org/wiki/Karp's_21_NP-complete_problemshttp://en.wikipedia.org/wiki/Karp's_21_NP-complete_problemshttp://en.wikipedia.org/wiki/Mathematical_modelhttp://en.wikipedia.org/wiki/Mathematical_optimizationhttp://en.wikipedia.org/wiki/Mathematical_optimizationhttp://en.wikipedia.org/wiki/Linearhttp://en.wikipedia.org/wiki/Objective_functionhttp://en.wikipedia.org/wiki/Objective_functionhttp://en.wikipedia.org/wiki/Linear_equalityhttp://en.wikipedia.org/wiki/Linear_inequalityhttp://en.wikipedia.org/wiki/Constraint_(mathematics)http://en.wikipedia.org/wiki/Feasible_regionhttp://en.wikipedia.org/wiki/Convex_polyhedronhttp://en.wikipedia.org/wiki/Half_spacehttp://en.wikipedia.org/wiki/Real_numberhttp://en.wikipedia.org/wiki/Affine_functionhttp://en.wikipedia.org/wiki/Affine_functionhttp://en.wikipedia.org/wiki/Algorithmhttp://en.wikipedia.org/wiki/Assignment_problem

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If only some of the unknown variables are required to be integers, then the problem is called

a mixed integer programming (MIP) problem. These are generally also NP-hard.

There are however some important subclasses of IP and MIP problems that are efficientlysolvable, most notably problems where the constraint matrix is totally uni-modular and the

right-hand sides of the constraints are integers.

Advanced algorithms for solving integer linear programs include:

cutting-plane method

branch and bound

branch and cut

branch and price

if the problem has some extra structure, it may be possible to apply delayed column

generation.

Such integer-programming algorithms are discussed by Padberg and in Beasley.

5. Explain the different steps involved in simulation methodologies?

Simulationis a way to model random events, such that simulated outcomes closely match

real-world outcomes. By observing simulated outcomes, researchers gain insight on the real

world.Some situations do not lend themselves to precise mathematical treatment. Others may

be difficult, time-consuming, or expensive to analyze. In these situations, simulation may

approximate real-world results; yet, require less time, effort, and/or money than other

approaches.

The methodology developed for simulation process consists of the following seven steps:

Step 1: Identify and clearly define the problem.

Step 2: List the statement of objectives of the problem.

Step 3: Formulate the variables that influence the situation and an extract or probabilistic

description of their possible values or states.

Step 4: Obtain a consistent set of values (or states) for the variables, i.e., a sample of

probabilistic variables, random sampling technique maybe used.

Step 5: Use the sample obtained in step 2 to calculate the values of the decision criterion, by

actually following the relationships among the variables for each of the alternative decisions.

Step 6: Repeat steps 2 and 3 until a sufficient number of samples are available.

http://en.wikipedia.org/wiki/Totally_unimodularhttp://en.wikipedia.org/wiki/Cutting-plane_methodhttp://en.wikipedia.org/wiki/Branch_and_boundhttp://en.wikipedia.org/wiki/Branch_and_cuthttp://en.wikipedia.org/wiki/Branch_and_pricehttp://en.wikipedia.org/wiki/Delayed_column_generationhttp://en.wikipedia.org/wiki/Delayed_column_generationhttp://en.wikipedia.org/wiki/Totally_unimodularhttp://en.wikipedia.org/wiki/Cutting-plane_methodhttp://en.wikipedia.org/wiki/Branch_and_boundhttp://en.wikipedia.org/wiki/Branch_and_cuthttp://en.wikipedia.org/wiki/Branch_and_pricehttp://en.wikipedia.org/wiki/Delayed_column_generationhttp://en.wikipedia.org/wiki/Delayed_column_generation

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Step 7: Tabulate the various values of the decision criterion and choose the best policy.

6. Write down the basic difference between PERT &CPM.

Project management has evolved as a new field with the development of two analytic

techniques for planning, scheduling and controlling projects. These are the Critical Path

Method (CPM) and the Project Evaluation and Review Technique (PERT). PERT and CPM

are basically time-oriented methods in the sense that they both lead to the determination of a

time schedule.

Though there are no essential differences between PERT and CPM as both of them share in

common the determination of a critical path. Both are based on the network representation of

activities and their scheduling that determines the most critical activities to be controlled so

as to meet the completion date of the project.

PERT

Some key points about PERT are as follows:

1. PERT was developed in connection with an R&D work. Therefore, it had to cope with the

uncertainties that are associated with R&D activities. In PERT, the total project duration is

regarded as a random variable. Therefore, associated probabilities are calculated so as to

characterize it.

2. It is an event-oriented network because in the analysis of a network, emphasis is given onthe important stages of completion of a task rather than the activities required to be

performed to reach a particular event or task.

3. PERT is normally used for projects involving activities of non-repetitive nature in which

time estimates are uncertain.

4. It helps in pinpointing critical areas in a project so that necessary adjustment can be made

to meet the scheduled completion date of the project.

CPM

1. CPM was developed in connection with a construction project, which consisted of routine

tasks whose resource requirements and duration were known with certainty. Therefore, it is

basically deterministic.

2. CPM is suitable for establishing a trade-off for optimum balancing between schedule time

and cost of the project.

3. CPM is used for projects involving activities of repetitive nature.


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ASSIGNMENT SET 2

1. What is a model in OR? Discuss different models available in OR.

A model is an idealized representation or abstraction of a real-life system. The objective of a

model is to identify significant factors that affect the real-life system and their

interrelationships. A model aids the decision-making process as it provides a simplified

description of complexities and uncertainties of a problem in a logical structure. The mostsignificant advantage of a model is that it does not interfere with the real-life system.

A broad classification of OR models

You can broadly classify OR models into the following types.

a. Physical Models include all form of diagrams, graphs and charts. They are designed to

tackle specific problems. They bring out significant factors and interrelationships in pictorial

form to facilitate analysis. There are two types of physical models:

a. Iconic models

b. Analog models

Iconic models are primarily images of objects or systems, represented on a smaller scale.

These models can simulate the actual performance of a product. Analog models are small

physical systems having characteristics similar to the objects they represent, such as toys.

b. Mathematical or Symbolic Models employ a set of mathematical symbols to represent

the decision variable of the system. The variables are related by mathematical systems. Some

examples of mathematical models are allocation, sequencing, and replacement models.

c. By nature of Environment: Models can be further classified as follows:

a. Deterministic model in which everything is defined and the results are certain, such as an

EOQ model.

b. Probabilistic Models in which the input and output variables follow a defined probability

distribution, such as the Games Theory.

d. By the extent of Generality Models can be further classified as follows:

a. General Models are the models which you can apply in general to any problem. For

example: Linear programming.


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b. Specific Models on the other hand are models that you can apply only under specific

conditions. For example: You can use the sales response curve or equation as a function of

only in the marketing function.

2. Write dual of

Max Z = 4X1 + 5X2

Subject to:

3X1 + X2 15

X1 + 2X2 10

5X1 + 2X2 20

X1, X2 0

Soln:

Min W = 15Y1 + 10Y2 + 20Y3

Subject to

3Y1 + Y2 + 5Y3 4

Y1 + 2Y2 + 2Y3 5

Y1, Y2, Y3 0

3. Write a note on Monte-Carlo simulation.

The Monte-Carlo method is a simulation technique in which statistical distribution functions

are created by using a series of random numbers. This approach has the ability to develop

many months or years of data in a matter of few minutes on a digital computer.

The method is generally used to solve the problems which cannot be adequately represented

by mathematical models, or, where solution of the mode, is not possible by analytical

method.

The Monte-Carlo simulation procedure can be summarized in the following steps:

Step 1: Define the problem:


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a) Identify the objectives of the problem, and

b) Identify the main factors which have the greatest effects on the objectives of the problem

Step 2: Construct an appropriate model:

a) Specify the variables and parameters of the model.

b) Formulate the appropriate decision rules, i.e., state the conditions under which the

experiment is to be performed.

c) Identify the type of distribution that will be used Models use either theoretical

distributions or empirical distributions to state the patterns the occurrence associated with

the variables.

d) Specify the manner in which time will change.

e) Define the relationship between the variables and parameters.

Step 3: Prepare the model for experimentation:

a) Define the starting conditions for the simulation, and

b) Specify the number of runs of simulation to be made.

Step 4: Using step 1 to 3, experiment with the model:

a) Define a coding system that will correlate the factors defined in step 1 with the random

numbers to be generated for the simulation.

b) Select a random number generator and create the random numbers to be used in the

simulation.

c) Associate the generated random numbers with the factors identified in step 1 and coded in

step 4 (a).

Step 5: Summarize and examine the results obtained in step 4.

Step 6: Evaluate the results of the simulation.

Step 7: Formulate proposals for advice to management on the course of action to be

adopted and modify the model, if necessary.

4. Explain PERT

Program (Project) Evaluation and Review Technique (PERT) is a project management tool

used to schedule, organize, and coordinate tasks within a project. It is basically a method to


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analyze the tasks involved in completing a given project, especially the time needed to

complete each task, and to identify the minimum time needed to complete the total project.

Some key points about PERT are as follows:

1. PERT was developed in connection with an R&D work. Therefore, it had to cope with theuncertainties that are associated with R&D activities. In PERT, the total project duration is

regarded as a random variable. Therefore, associated probabilities are calculated so as to

characterize it.

2. It is an event-oriented network because in the analysis of a network, emphasis is given on

the important stages of completion of a task rather than the activities required to be

performed to reach a particular event or task.

3. PERT is normally used for projects involving activities of non-repetitive nature in which

time estimates are uncertain.

4. It helps in pinpointing critical areas in a project so that necessary adjustment can be made

to meet the scheduled completion date of the project.

PERT planning involves the following steps:

Identify the specific activities and milestones.

Determine the proper sequence of the activities.

Construct a network diagram.

Estimate the time required for each activity.

Determine the critical path.

Update the PERT chart as the project progresses.

5. Explain Maximini - minimax principle

Solving a two-person zero-sum game

Player A and player B are to play a game without knowing the other players strategy.

However, player A would like to maximize his profit and player B would like to minimize his

loss. Also each player would expect his opponent to be calculative.

Suppose playerAplaysA1.


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Then, his gain would be a11, a12, ... , a1n, accordingly Bs choice would be B1,B2, , Bn. Let

1 = min { a11, a12, , a1n.

Then, 1is the minimum gain of A when he playsA1(1 is the minimum pay-off in the first

row.)

Similarly, if A plays A2, then his minimum gain is 2, the least pay-off in the second row.

You will find corresponding to As play A1, A2, , Am, the minimum gains are the row

minimums 1, 2, , m.

Suppose A chooses the course of action where i is maximum.

Then the maximum of the row minimum in the pay-off matrix is called maximin.

The maximin is

=max

I {min

j (aij) }

Similarly, whenBplays, he would minimise his maximum loss.

The maximum loss toB is whenBjis j =maxi ( aij).

This is the maximum pay-off in thej th column.

The minimum of the column maximums in the pay-off matrix is called minimax.

The minimax is

= minj {maxI (aij) }

If = = v (say), the maximin and the minimax are equal and the game is said to have

saddle point. If < , then the game does not have a saddle point.

Saddle point

In a two-person zero-sum game, if the maximin and the minimax are equal, the game has

saddle point.

Saddle point is the position where the maximin (maximum of the row minimums) andminimax (minimum of the column maximums) coincide.

If the maximin occurs in the rth row and if the minimax occurs in thesth column, the position

(r, s) is the saddle point.

Here, v = arsis the common value of the maximin and the minimax. It is called the value of

the game.

The value of a game is the expected gain of player A, when both the players adopt optimal

strategy.

Note: If a game has saddle point, (r, s), the players strategy is pure strategy.


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Solution to a game with saddle point

Consider a two-person zero-sum game with playersA andB. LetA1, A2, ,Ambe the courses

of action for playerA. LetB1, B2, ,Bnbe the courses of action for player B.

The saddle point of the game is as follows:

1. The minimum pay-off in each row of the pay-off matrix is encircled.

2. The maximum pay-off in each column is written within a box.

3. If any pay-off is circled as well as boxed, that pay-off is the value of the game. The

corresponding position is the saddle point.

Let (r, s) be the saddle point. Then, the suggested pure strategy for playerA is Ar. The

suggested pure strategy for playerB isBs. The value of the game is ars.

Note: However, if none of the pay-offs is circled or boxed, the game does not have a saddle

point. Hence, the suggested solution for the players is mixed strategy.

6. Write short notes on the following:

a. Linear Programming

b. Transportation

Linear Programming

Linear programming focuses on obtaining the best possible output (or a set of outputs) from a

given set of limited resources.

The LPP is a class of mathematical programming where the functions representing the

objectives and the constraints are linear. Optimization refers to the maximization or

minimization of the objective functions.

You can define the general linear programming model as follows:

Maximize or Minimize:

Z = c1X1 + c2X2 + --- +cnXn

Subject to the constraints,

a11X1 + a12X2 + --- + a1nXn ~ b1

a21X1 + a22X2 + --- + a2nXn ~ b2

am1X1 + am2xX2 + --- + amnXn ~ bm


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and X1, X2, .., Xn 0

Where, cj, bi and aij (i = 1, 2, 3, .. m, j = 1, 2, 3 ------- n) are constants determined from the

technology of the problem and Xj (j = 1, 2, 3 ---- n) are the decision variables. Here ~ is either

(less than), (greater than) or = (equal). Note that, in terms of the above formulation the

coefficients cj, bi and aij are interpreted physically as follows. If bi is the available amount ofresources i, where aij is the amount of resource i that must be allocated to each unit of activity

j, the worth per unit of activity is equal to cj.

Transportation

Transportation model is an important class of linear programs. For a given supply at each

source and a given demand at each destination, the model studies the minimization of the cost

of transporting a commodity from a number of sources to several destinations.

The transportation problem involves m sources, each of which has available a i (i = 1, 2 m)

units of homogeneous product and n destinations, each of which requires bj (j = 1, 2., n)

units of products. Here ai and bj are positive integers. The cost c ij of transporting one unit of

the product from the ith source to the jth destination is given for each i and j. The objective is

to develop an integral transportation schedule that meets all demands from the inventory at a

minimum total transportation cost.

It is assumed that the total supply and the total demand are equal.

mi=1 ai =nj=1 bj (1)

The condition (1) is guaranteed by creating either a fictitious destination with a demand equal

to the surplus if total demand is less than the total supply or a (dummy) source with a supply

equal to the shortage if total demand exceeds total supply. The cost of transportation from the

fictitious destination to all sources and from all destinations to the fictitious sources are

assumed to be zero so that total cost of transportation will remain the same.

The standard mathematical model for the transportation problem is as follows.

Let Xij be number of units of the homogenous product to be transported from source i to the

destination j.

Then objective is to

Minimize Z = mi=1nj=1 CIJ Xij

Subject to mi=1 ai, i = 1, 2, 3, -------------, m andnj=1 bj, j = 1, 2, 3, -------------, n

(2)

With all XIJ 0

A necessary and sufficient condition for the existence of a feasible solution to the

transportation problem (2) is: mi=1 ai =nj=1 bj


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MB0048 Set 1 & 2

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