operations research set2

7/30/2019 Operations Research Set2

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LAUSON TRISTON MONTEIROREG NO: 521115123

MB0048- OPERATIONS RESEARCH

Assignment Set 2

Q1:

A model is an idealised 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 most significant

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:

I. Iconic models

II. 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:

I. Deterministic model in which everything is defined and the results are

certain, such as an EOQ model.

II. 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:


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I. General Models are the models which you can apply in general to any

problem. For example: Linear programming.

II. Specific Models on the other hand are models that you can apply only

under specific conditions. For example: You can use the sales response curveor equation as a function of only in the marketing function.Q2:

The given problem is in its standard form:Its dual isMini W = 15y1 + 10 y2 + 20 y3Subject to 3y1 + y2 + 5 y3 4y1 + 2y2 + 2y3 5y1, y2, y3, 0

Q3:A.3 Simulation is also called experimentation in the management laboratory.While dealing with business problems, simulation is often referred to asMonte Carlo Analysis. Two American mathematicians, Von Neumann andUlan, in the late 1940s found a problem in the field of nuclear physics toocomplex for analytical solution and too dangerous for actualexperimentation. They arrived at an approximate solution by sampling. Themethod they used had resemblance to the gamblers betting systems on theroulette table, hence the name Monte Carlo has stuck.

Imagine a betting game where the stakes are based on correct prediction of

the number of heads, which occur when five coins are tossed. If it were onlya question of one coin; most people know that there is an equal likelihood ofa head or a tail occurring, that is the probability of a head is . However,without the application of probability theory, it would be difficult to predictthe chances of getting various numbers of heads, when five coins are tossed.

Why dont you take five coins and toss them repeatedly. Note down theoutcomes of each toss after every ten tosses, approximate the probabilitiesof various outcomes. As you know, the values of these probabilities willinitially fluctuate, but they would tend to stabilise as the number of tossesare increased. This approach in effect is a method of sampling, but is not

very convenient. Instead of actually tossing the coins, you can conduct theexperiment using random numbers. Random numbers have the property thatany number is equally likely to occur, irrespective of the digit that hasalready occurred.

Let us estimate the probability of tossing of different numbers of heads withfive coins. We start with set random numbers given below:

Table : Random number set


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1. PERT was developed in connection with an R&D work. Therefore, it had tocope 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 on the important stages of completion of a task ratherthan the activities required to be performed to reach a particular event ortask.

3. PERT is normally used for projects involving activities of non-repetitivenature 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 theproject.

Project scheduling by PERT-CPMIt consists of three basic phases: planning, scheduling and controlling.

1. Project Planning: In the project planning phase, you need to perform thefollowing activities:i) Identify various tasks or work elements to be performed in the project.

ii) Determine requirement of resources, such as men, materials, and

machines, for carrying out activities listed above.

iii) Estimate costs and time for various activities.

iv) Specify the inter-relationship among various activities.

v) Develop a network diagram showing the sequential inter-relationshipsbetween the various activities.

2. Project Scheduling: Once the planning phase is over, scheduling of theproject is when each of the activities required to be performed, is taken up.The various steps involved during this phase are listed below:

1. Estimate the durations of activities. Take into account the resourcesrequired for these execution in the most economic manner.2. Based on the above time estimates, prepare a time chart showingthe start and finish times for each activity. Use the time chart for thefollowing exercises.


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-To calculate the total project duration by applying networkanalysis techniques, such as forward (backward) pass and floatscalculation

-To identify the critical path-To carry out resource smoothing (or leveling) exercises forcritical or scarce resources including re-costing of the scheduletaking into account resource constraints

3. Project Control: Project control refers to comparing the actualprogress against the estimated schedule. If significant differences areobserved then you need to re-schedule the project to update or revisethe uncompleted part of the project.

Q5:

Maximin Minimax Principle

Solving a two-person zero-sum game

Player A and player B are to play a game without knowing the other playersstrategy. However, player A would like to maximise his profit and player Bwould like to minimise his loss. Also each player would expect his opponentto be calculative.

Suppose playerA plays. A1Then, his gain would be a11,a12,.....a1n accordingly Bs choice wouldbe b11,b12,.....b1n.Let a1 = min.{a11,a12,....a1n}Then, a1 is the minimum gain of A when he plays A1(a1 is the minimum pay-off in the first row.)Similarly, if A plays A2, his minimum gain is a2, the least pay-off in thesecond row.

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

the row minimums a1,a2,....am.Suppose A chooses the course of action where a1 is maximum.Then the maximum of the row minimum in the pay-off matrix is calledmaximin.

The maximin is


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Similarly, when B plays, he would minimise his maximum loss.

The maximum loss to B is when Bj is.

This is the maximum pay-off in thejth column.The minimum of the column maximums in the pay-off matrix is calledminimax.

The minimax is

Q6:

6. write short notes on the following:a. Linear Programmingb. transportationA.6 a. Linear Programming :-The LPP is a class of mathematical programming where the functionsrepresenting the objectives and the constraints are linear. Optimisationrefers to the maximisation or minimisation of the objective functions.

You can define the general linear programming model as follows:

Maximise or Minimise:

Z = c1 x1 + c2 x2 + - - - - + cn xnSubject to the constraints,

a11 x1 + a12 x2 + + a1n xn ~ b1a21 x1 + a22 x2 + + a2n xn ~ b2-

am1 x1 + am2 x2 + - + amn xn ~ bm
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and x1 0, x2 0, xn 0Where cj, bi and aij (i = 1, 2, 3, .. m, j = 1, 2, 3 - n) are constantsdetermined from the technology of the problem and xj (j = 1, 2, 3 - n) arethe decision variables. Here ~ is either (less than), (greater than) or =

(equal). Note that, in terms of the above formulation the coefficientscj,bi aijare interpreted physically as follows. Ifbi is the available amount ofresources i, where aij is the amount of resource i that must be allocated toeach unit of activity j, the worth per unit of activity is equal to cj.

b. transportation :-Transportation Algorithm (MODI Method)The first approximation to (2) is integral. Therefore, you always need to finda feasible solution. Rather than determining a first approximation by a directapplication of the simplex method, it is more efficient to work with thetransportation table given below. The transportation algorithm is the simplex

method specialised to the format of table involving the following steps:

i) Finding an integral basic feasible solution

ii) Testing the solution for optimality

iii) Improving the solution, when it is not optimal

iv) Repeating steps (ii) and (iii) until the optimal solution is obtained

The solution to TP is obtained in two stages.

In the first stage, you find the basic feasible solution using any of the

following methods a) North-west corner rule b) Matrix Minima Method orleast cost method c) Vogels approximation method. In the second stage, youtest the basic feasible solution for its optimality either by MODI method or bystepping stone method.
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operations research set2

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