Quantitatve methds

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Q.3 The use of Operations Research (OR) especially Linear Programming (LP) is not

popular even in the large scale public and private sector units. It remains to a larger extend

as a subject of academic importance. But the fact is that even a tiny sector unit has much to

gain from the use of the quantitative tools in the decision making process. In the globalized

economyuse of sophisticated tools which guide more practical decisions taking into

account all possible constraints is relevant and becomes more inevitable. Linear

programming in OR is certainly the most suitable and useable tool in this regard, because

of its simplicity and availability in the form of software packages. While using the LP

packages what is required is a fair understanding of the product mix, market for the existing

products, their demand in the market and the product movement. With all these knowledge,

different models can be developed and run until a suitable solution is obtained. This study

demonstrates the use of OR, especially in solving linear programming technique using

(Linear Interactive Discrete Optimizer) LINDO software to maximize the profit of a Khadi

& Village Industries Commission (KVIC) affiliated Sarvodaya sangham run by Non-

Government Organisation (NGO).

Q. 4Transportation Problems:

Operations Research

Transportation Problems

INTRODUCTION

Many practical problems in operations research can be broadly formulated as linear

programming problems, for which the simplex this is a general method and cannot be used

for specific types of problems like,

(i)transportation models,

(ii)transshipment models and

(iii) the assignment models.

The above models are also basically allocation models. We can adopt the simplex technique

to solve them, but easier algorithms have been developed for solution of such problems. The

following sections deal with the transportation problems and their streamlined procedures for

solution.

TRANSPORTATION MODEL
http://free-books-online.org/mathematics/operations-research/transportation-problems/http://free-books-online.org/mathematics/operations-research/transportation-problems/http://free-books-online.org/mathematics/operations-research/transportation-problems/http://free-books-online.org/category/mathematics/operations-research/http://free-books-online.org/category/mathematics/operations-research/http://free-books-online.org/category/mathematics/operations-research/http://free-books-online.org/mathematics/operations-research/transportation-problems/


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In a transportation problem, we have certain origins, which may represent factories wherewe produce items and supply a required quantity of the products to a certain number ofdestinations. This must be done in such a way as to maximize the profit or minimize the cost.

Thus we have the places of production as origins and the places of supply as destinations.

Sometimes the origins and destinations are also termed as sources and sinks.

To illustrate a typical transportation model, suppose m factories supply certain items to n

warehouses. Let factory i (i= 1, 2, , m) produce ai units and the warehousej (j= 1, 2, ,

n) requires bj units. Suppose the

cost of transportation from factory i to warehousej is cij. Let us define the decisionvariablesxij being the amount transported from the factory i to the warehousej. Our objective

is to find the transportation pattern that will minimize the total transportation cost.

The model of a transportation problem can be represented in a concise tabular form with all

the relevant parameters mentioned above. See table 1

Table 1

Origins

(Factories)

Destinations

(Warehouses)

Available

1 2 .. n

1 c11 c12 c1n a1

2 c21 c22 c2n a2

m cm1 cm2 cmn am

Required b1 b2 bn

The pattern of distribution of items in the form of transportation matrix is separately given

below in table 2.

Table 2

Origins Destinations Available

1 2 .. n

1 x11 x12 x1n a12 x21 x22 x2n a2

m xm1 xm2 xmn am

Required b1 b2 bn

TRANSPORTATION PROBLEM AS AN L.P MODEL

The transportation problem can be represented mathematically as a linear programming

model. The objective function in this problem is to minimize the total transportation cost

given by


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Z = c11 x11 + c12x12 + + cmn xmn

The above is a mathematical formulation of a transportation problem and we can adopt the

linear programming technique with equality constraints. Here the algebraic procedure of thesimplex method may not be the best method to solve the problem and hence more efficient

and simpler streamlined procedures have been developed to solve transportation problems.

Example 1 (Formulation of a transportation problem)

A company has plants located at three places where the production pattern is described inthe following table.

Plant location 1 2 3

Production

(units)40 70 90

The potential demand at five places has been estimated by the marketing department and is

presented below.

Distribution

centre

1 2 3 4 5

Potential demand

(units)30 40 60 40 60

The cost of transportation from a plant to the distribution centre has been displayed in the

table .

Table 3 Distribution centers

Plant 1 2 3 4 5

1 20 25 27 20 15

2 18 21 22 24 20

3 19 17 20 18 19

Represent the above data in a table to represent a transportation problem.

Solution:

In this example the total supply and the total demand do not match as supply is less than

demand. Hence a dummy row (dummy plant) is introduced at a unit transportation cost of 0.

The following is the tabular representation of the transportation problem.

Note: If the total demand (requirements) is less than the total supply (availability), a dummy

column (dummy destination) is introduced with a unit transportation cost of 0.

Table 4 Distribution center

Plant 1 2 3 4 5 Supply


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1 20 25 27 20 15 40

2 18 21 22 24 20 70

3 19 17 20 18 19 90

4 0 0 0 0 0 30

Demand 30 40 60 40 60 230

FINDING AN INITIAL BASIC FEASIBLE SOLUTION

An initial basic feasible solution to a transportation problem can be found by any one of the

three following methods:

(i)North west corner rule (NWC)

(ii)Least cost method (LCM)

(iii) Vogels approximation method (VAM)

North West Corner Rule

STEP 1 :

Start with the cell in the upper left hand corner (North West Corner).

STEP 2:

Allocate the maximum feasible amount.

STEP 3 :

Move one cell to the right if there is any remaining supply. Otherwise, move one cell down.

If both are

impossible, stop or go to step (2). An example is considered at this juncture to illustrate theapplication of NWC rule.

Example 2 Table 5

Destinations

Origin P Q R Supply

A 5 7 8 70

B 4 4 6 30

C 6 7 7 50

Demand 65 42 43 150

In table 5 supply level, demands at various destinations, and the unit cost of transportation are

given. Use NWC rule to find the initial solution.


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Solution:

The solution obtained by the North West Corner Rule is a basic feasible solution. In this

method we do not consider the unit cost of transportation. Hence the solution obtained may

not be an optimal solution. But this will serve as an initial solution, which can be improved.

We give below the procedure for the solution of the above problem by NWC rule.

The North West Corner cell (AP) is chosen for allocation. The origin A has 70 items and the

destination P requires only 65 items. Hence it is enough to allot 65 items from A to P. The

origin A which is alive with 5 more items can supply to the destination to the right is alive

with 5 more items can supply to the destination to the right of P namely Q whose requirement

is 42. So, we supply 5 items to Q thereby the origin A is exhausted. Q requires 37 items

more. Now consider the origin B that has 30 items to spare. We allot 30 items to the cell

(BQ) so that the origin B is exhausted. Then move to origin C and supply 7 more items to the

destination Q. Now the requirement of the destination Q is complete and C is left with 43

items and the same can be alloted to the destination R. Now the origin C is emptied and therequirement at the destination R is also complete. This completes the initial solution to the

problem.

Q.5 Assignment Problem

A special case of the transportation problem is the assignment problem which occurs when

each supply is 1 and each demand is 1. In this case, the integrality implies that every supplier

will be assigned one destination and every destination will have one supplier. The costs give

the charge for assigning a supplier and destination to each other.

Example. A company has three new machines of different types. There are four different

plants that could receive the machines. Each plant can only receive one machine, and each

machine can only be given to one plant. The expected profit that can be made by each plant if

assigned a machine is as follows:

This is a transportation problem with all supplies and demands equal to 1, so it is an

assignment problem.

Note that a balanced problem must have the same number of supplies and demands, so we

must add a dummy machine (corresponding to receiving no machine) and assign a zero cost

for assigning the dummy machine to a plant.

Q.6 Concept of inventory:


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Inventory is defined as a usable resource which is physical and tangible such as materials. In

this sense, our stock is our inventory, but even then the term inventory is more

comprehensive. Though inventory is a usable resource, it is also an idle resource, unless it is

managed efficiently and effectively.

Inventory management boils down to maintaining an adequate supply of something to meetthe expected demand pattern subject to budgeting considerations. Inventory could be raw

materials, work in progress (WIP), finished products or the spare parts and other indirect

materials. Effectiveness of the materials and production functions depend to large extent

upon inventory management. Inventory turnover ratio annual Demand / Average Inventory is

an index business performance. Sound management gives a higher inventory turnover ratio.

Inventories have to be procured, stored and carried for production system, since a situation

when they can be instantaneously available is difficult to assume in Indian setting.

Inventories are, therefore, a necessary evil to stay.

Importance of inventory:

Inventories constitute the largest component of current assets in many organizations. Poor

management of inventories therefore may result in business failures. A stock out creates an

unpleasant situation for the organization. In case of a manufacturing organization, the

inability to supply an item from inventory could, bring production process to a halt.

Conversely; if a firm carries excessive inventories, the added carrying cost may represent the

different between profit and loss. Efficient inventory control, therefore, can significantly

contribute to the overall profit position of the organization.

What Function does inventory perform?

Functions that inventory:

In any organizations, inventories add an operating flexibility that would otherwise be lost. In

production, work in process inventories are an absolute necessity unless each individual part

is to be carried from machine to machine and those machine set up to produce that single

part. The many functions inventory performs can be summarized as follows:

Regularizing demand and supply:

Harvest of tobacco is carried out during the concluding part of the summer, but themanufacture of tobacco products such as cigarettes and cigars continues throughout the entire

year. In cases like this sufficient raw materials must be purchased during the tobacco

producing period to last the entire year. It compels the manufacturer to carry inventory.

In a simple sense, because a commercial vehicle can be driven 100 miles without passing a

petrol pump, its tanks must carry enough petrol to avoid run outs.

Economizing purchases or production by lot buying or batch production

If the product does not have the demand sufficient to sustain its continuous production round

the year, it is usually produced in batches or lots, on an intermittent basis. During the timewhen the item is not being produced, sales are made from inventory which is accumulated


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while the item is being produced. Similarly, a super bazaar selling mens clothing dos not

purchase a new shirt from Double Bull, each time they sell one. Rather they choose to carry

with them an inventory of these shirts so that purchasing can be done in larger quantities,

thereby allowing lower cost and less paper work.

Allowing organizations to cope with perishable materials:

Organizations selling canned foods, particularly fruits and vegetables, operate at peak

production capacity only for a few months each year. They must tore up inventory to supply

sufficient to last them through a years anticipated demand until the next season. They give

thoughtful consideration to the rate of inventory accumulation and depletion throughout the

peak production and sales periods each year.

Inventory can store labor:

Though it may sound abstract to talk of Inventorying labo, it has been done as a matter of

routine. The peak demand for the installation of replacement heating units comes in the falls,just after the old units have been operated fro the first time. The manufacturers of heating

units store up excess labor by having their workers produce at a designated arte all year long,

then having converted labor into finished heating units they hold them in inventory until the

point when demand increases rapidly. Even if demand exceeds current productive capacity, a

manufacturer can supply the difference out of inventory at that time.

Inventory Decisions:

Executives decide two basic issues while dealing with inventories. These two decisions are

made for every item in the inventory.

1) How much of an item to order when the inventory of that item is to be replenished.

2) When to replenish the inventory of that item.

There are many commonly used inventory models which managers use to make the two

decisions enumerated above. Inventory models details are beyond the scope of this article.

q.7Queueingtheoryis used to study the phenomenon of waiting in lines. Some people use the

information gathered from queuing theory in order to determine how to best serve customers

and so prevent them from waiting in line longer than they have to. The theory allows

researchers to analyze several things such as arriving in line, waiting in line, and the time ittakes to service customers. This allows them to gather and derive information on a

customers waiting time, the expected amount of customers that will be in a line, the

probability of a customer encountering a line, as well as other data. This information is used

in order to find ways to reduce lines and wait time.

Applications of queueing theory are used in many aspects of business,customer service,

commerce, industry, healthcare, andengineering. Customer service applications can

especially make use of the information gathered by queueing theory. This information can be

used in order to make decisions on the kind of resources needed to provide service for

customers. The data can be also be applied tocall centers,network serverqueuing,

telecommunications, and traffic flow. It can even be used to dictate what type of linecustomers will be standing in while waiting for different types of service.
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There are several queueing disciplines that have been developed because of queueing

theoryfour of which are First In First Out (FIFO), Last In First Out (LIFO), Processor

Sharing, and Priority. FIFO describes the practice of serving customers in the order they

arrive in so that the person waiting longest is served first, while LIFO describes the practice

of serving customers so that the person who comes in last leaves first, such as in the case of

riding an elevator. Processor sharing serves customers at the same time to that the averagewaiting time for all customers is about the same. The Priority discipline serves the customer

with the highest priority first. It is important to note that these disciplines can be applied to

applications other than customer service.

Examples of how queueing theory works is present in many aspects of everyday life. At bank

tellers and credit unions, one may see one line and multiple tellers. This happens to help

ensure that one slow transaction does not hold up the entire line. Some stores open more

registers if there are more than three people waiting in a line. There are also other stores that

have roaming clerks. These clerks ring up purchases and give customers a number so that the

cashiercan complete the transaction quickly, thus reducing waiting times for all.
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Q. 11


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Q. 15 1. delphi method: forecast is developed by a panel of experts who anonymously

answer a series of questions; responses are fed back to panel members who then may change

their original responses

- very time consuming and expensive

- new groupware makes this process much more feasible

Q.18 Game theory is the study of the ways in which strategic interactions among

economic agents produceoutcomes with respect to the preferences (orutilities) of those

agents, where the outcomes in question might have been intended by none of the agents.

The meaning of this statement will not be clear to the non-expert until each of the

italicized words and phrases has been explained and featured in some examples. Doing

this will be the main business of this article. First, however, we provide some historical

and philosophical context in order to motivate the reader for the technical work ahead. Strategy

Qualitative Methods Quantitative Methods

Methods include focus groups, in-depthinterviews, and reviews of documents for

types of themes

Surveys, structured interviews &observations, and reviews of records or

documents for numeric information

Primarily inductive process used toformulate theory or hypotheses

Primarily deductive process used to testpre-specified concepts, constructs, and

hypotheses that make up a theory

More subjective: describes a problem or

condition from the point of view of thoseexperiencing it

More objective: provides observed effects

(interpreted by researchers) of a programon a problem or condition

Text-based Number-based

More in-depth information on a few cases Less in-depth but more breadth ofinformation across a large number of

cases

Unstructured or semi-structured responseoptions

Fixed response options

No statistical tests Statistical tests are used for analysis

Can be valid and reliable: largely depends

on skill and rigor of the researcherCan be valid and reliable: largely depends

on the measurement device orinstrument used

Time expenditure lighter on the planningend and heavier during the analysisphase

Time expenditure heavier on the planningphase and lighter on the analysis phase

Less generalizable More generalizable


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In a game situation, each of the players has a set of strategies available. The strategy for a

player is the set of alternative courses of action that he will take for every payoff

(outcome) that might arise. It is assumed that the players know the rules governing the

choices in advance. The different outcomes resulting from the choices are also known to

the players in advance and are expressed in terms of the numerical values ( e.g. money,

market share percentage etc. )Strategy may be of two types:

(a) Pure strategy

If the players select the same strategy each time, then it is referred as pure strategy. In

this case each player knows exactly what the opponent is

going to do and the objective of the players is to maximize gains or to minimize losses.

(b) Mixed Strategy

When the players use a combination of strategies with some fixed probabilities and each

player kept guessing as to which course of action is to be selected by the other player at aparticular occasion then this is known as mixed strategy. Thus, there is probabilistic

situation and objective of the player is to maximize expected gains or to minimize losses

strategies. Mixed strategy is a selection among pure strategies with fixed probabilities.

Optimal StrategyA strategy which when adopted puts the player in the most preferred position, irrespective

of the strategy of his competitors is called an optimal strategy. The optimal strategy

involves maximal pay-off to the player.

The games can also be classified on the basis of the number of strategies. A game is said to be

finite if each player has the option of choosing from only a finite number of strategies otherwise it

is called infinite

Q.19Dynamic programming typically applies to optimisation problems in which a set of

choices must be made in order to get an optimal solution. As choices are made,subproblems of the same form often arise. Dynamic programming is effective whenthe given subproblem may arise from more than one partial set of choices.The key technique is to store the solution to each such subproblem in case it couldre-appear (programming in this context refers to a tabular method). Therefore,dynamic programming is applicable when the subproblems are not independent.COMP3600/6466: Lecture 9 2009 1

15.2 Four Steps of Developing DP AlgorithmsThe development of a DP algorithm consists of the following four steps.1. Characterise the structure of an optimal solution2. Define the value of an optimal solution recursively3. Compute the value of an optimal solution in a bottom-up fashion4. Construct an optimal solution from the computed informationSteps 1 - 3 form the basis of a DP solution to a problem.COMP3600/

Q.12 11.1 IntroductionMost of our study of probability has dealt with independent trials processes. Theseprocesses are the basis of classical probability theory and much of statistics. Wehave discussed two of the principal theorems for these processes: the Law of Large


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Numbers and the Central Limit Theorem.We have seen that when a sequence of chance experiments forms an independenttrials process, the possible outcomes for each experiment are the same andoccur with the same probability. Further, knowledge of the outcomes of the previousexperiments does not inuence our predictions for the outcomes of the nextexperiment. The distribution for the outcomes of a single experiment is sucient

to construct a tree and a tree measure for a sequence of nexperiments, and wecan answer any probability question about these experiments by using this treemeasure.Modern probability theory studies chance processes for which the knowledgeof previous outcomes inuences predictions for future experiments. In principle,when we observe a sequence of chance experiments, all of the past outcomes couldinuence our predictions for the next experiment. For example, this should be thecase in predicting a student's grades on a sequence of exams in a course. But toallow this much generality would make it very dicult to prove general results.In 1907, A. A. Markov began the study of an important new type of chanceprocess. In this process, the outcome of a given experiment can aect the outcomeof the next experiment. This type of process is called a Markov chain.

Specifying a Markov ChainWe describe a Markov chain as follows: We have a set of states, S= fs1; s2; : : : ; srg.The process starts in one of these states and moves successively from one state toanother. Each move is called a step. If the chain is currently in state si, thenit moves to state sjat the next step with a probability denoted by pij, and thisprobability does not depend upon which states the chain was in before the current405406 CHAPTER 11. MARKOV CHAINSstate.The probabilities pijare called transition probabilities. The process can remainin the state it is in, and this occurs with probability pii. An initial probabilitydistribution, de ned on S, speci es the starting state. Usually this is done byspecifying a particular state as the starting state.R. A. Howard1 provides us with a picturesque description of a Markov chain asa frog jumping on a set of lily pads. The frog starts on one of the pads and thenjumps from lily pad to lily pad with the appropriate transition probabilities.

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