SMU MB0048 SET 2 SOLUTION

 Master of Business Administration - Semester 2

MB 0048: “Operations Research”

(4 credits)

(Book ID: B1301)

ASSIGNMENT- Set 2

Marks 60

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

A.1 Most operations research studies involve the construction of a mathematical model. The model is a collection of logical and mathematical relationships that represents aspects of the situation under study. Models describe important relationships between variables, include an objective function with which alternative solutions are evaluated, and constraints that restrict solutions to feasible values.

Although the analyst would hope to study the broad implications of the problem using a systems approach, a model cannot include every aspect of a situation. A model is always an abstraction that is of necessity simpler than the real situation. Elements that are irrelevant or unimportant to the problem are to be ignored, hopefully leaving sufficient detail so that the solution obtained with the model has value with regard to the original problem.

Models must be both tractable, capable of being solved, and valid, representative of the original situation. These dual goals are often contradictory and are not always attainable. It is generally true that the most powerful solution methods can be applied to the simplest, or most abstract, model.

The various types of models used in operations research are :-

1.6.1 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:

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 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

A.2  The given problem is in its standard form:

\ Its dual is

Mini W = 15y1 + 10 y2 + 20 y3

Subject to 3y1 + y2 + 5 y3 ≥ 4

y1 + 2y2 + 2y3 ≥ 5

y1, y2, y3, ≥ 0

3. Write a note on Monte-Carlo simulation.

A.3 Simulation is also called experimentation in the management laboratory. While dealing with business problems, simulation is often referred to as ‘Monte Carlo Analysis’. Two American mathematicians, Von Neumann and Ulan, in the late 1940s found a problem in the field of nuclear physics too complex for analytical solution and too dangerous for actual experimentation. They arrived at an approximate solution by sampling. The method they used had resemblance to the gambler’s betting systems on the roulette 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 only a question of one coin; most people know that there is an equal likelihood of a head or a tail occurring, that is the probability of a head is ½. However, without the application of probability theory, it would be difficult to predict the chances of getting various numbers of heads, when five coins are tossed.

Why don’t you take five coins and toss them repeatedly. Note down the outcomes of each toss after every ten tosses, approximate the probabilities of various outcomes. As you know, the values of these probabilities will initially fluctuate, but they would tend to stabilise as the number of tosses are increased. This approach in effect is a method of sampling, but is not very convenient. Instead of actually tossing the coins, you can conduct the experiment using random numbers. Random numbers have the property that any number is equally likely to occur, irrespective of the digit that has already occurred.

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

Table : Random number set

78466	71923
78722	78870
06401	61208
04754	05003
97118	95983

By following a convention that “even” digits signify a head (H) and the “odd” digits represent a tail (T), the tossing of a coin can be simulated. The probability of occurrence of the first set of digits is ½ and that of the other set is also ½ - a condition corresponding to the probability of the occurrence of a head and the probability of occurrence of a tail respectively.

It is immaterial as to which set of five digits should signify a head. The rule could be that the digits 0, 1, 2, 3 and 4 represent a head and the digits 5, 6, 7, 8 and 9 a tail. It is only necessary to take care that the set of random numbers allotted to any event matches with its probability of occurrence. For instance, if you’re interested in allotting random numbers to three events A, B and C with respective probabilities 0.24, 0.36 and 0.40, choose two digit random numbers 00 to 99.

The numbers 00 to 23 signify event A, 24 to 59 signify B and 60 to 99 signify C. The first set of five random digits in the list of random numbers implies that the outcome of the first toss of 5 coins is as follows:

Table : Outcome of first toss of 5 coins

Coin	1	2	3	4	5
Random Number	7	8	4	6	6
Outcome	T	H	H	H	H

Hence it is 4 heads and 1 tail.

4. Explain PERT

A.4  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 characterise 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.

Project scheduling by PERT-CPM

It consists of three basic phases: planning, scheduling and controlling.

1. Project Planning: In the project planning phase, you need to perform the following 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-relationships between the various activities.

2. Project Scheduling: Once the planning phase is over, scheduling of the project 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 resources required for these execution in the most economic manner.
2. Based on the above time estimates, prepare a time chart showing the start and finish times for each activity. Use the time chart for the following exercises.

· To calculate the total project duration by applying network analysis techniques, such as forward (backward) pass and floats calculation

· To identify the critical path

· To carry out resource smoothing (or levelling) exercises for critical or scarce resources including re-costing of the schedule taking into account resource constraints

3. Project Control: Project control refers to comparing the actual progress against the estimated schedule. If significant differences are observed then you need to re-schedule the project to update or revise the uncompleted part of the project.

5. Explain Maximini-minimax principle

A.5

Maximin – Minimax Principle

Solving a two-person zero-sum game

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

Suppose player A plays. A1

Then, his gain would be a11,a12,.....a1n  accordingly B’s choice would be 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 the second row.

You will find corresponding to A’s 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 called maximin.

The maximin is

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 the jth column.

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

The minimax is

If, 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.

Note:

  cannot be greater than 

6. write short notes on the following:

a. Linear Programming

b. transportation

A.6 a. Linear Programming :-

The LPP is a class of mathematical programming where the functions representing the objectives and the constraints are linear. Optimisation refers 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 + - - - - + c_n x_n

Subject to the constraints,

a₁₁ x₁ + a_{12 x2} + —– + a_1n x_n ~ b₁

a₂₁ x₁ + a_{22 x2} + —– + a_2n x_n ~ b₂

——————————————-

a_m1 x₁ + a_{m2 x2} + ——- + a_mn x_n ~ b_m

and x₁ ≥ 0, x₂ ≥ 0, ——————– x_n ≥ 0

Where c_j, b_i and a_ij (i = 1, 2, 3, ….. m, j = 1, 2, 3 ——- n) are constants determined from the technology of the problem and x_j (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 coefficientsc_j, b_i a_ij are interpreted physically as follows. If b_i is the available amount of resources i, where a_ij 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 c_j.

b. transportation :-

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

Let x_ij be number of units of the homogenous product to be transported from source i to the destination j

Then objective is to

Minimise z = 

Subject to

  (2) (2)                         (2)

With all x_ij ³ 0 and integrals

Theorem: A necessary and sufficient condition for the existence of a feasible solution to the transportation problem (2) is:

Transportation Algorithm (MODI Method)

The first approximation to (2) is integral. Therefore, you always need to find a feasible solution. Rather than determining a first approximation by a direct application of the simplex method, it is more efficient to work with the transportation 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 or least cost method c) Vogel’s approximation method. In the second stage, you test the basic feasible solution for its optimality either by MODI method or by stepping stone method.

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