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MATHEMATICAL PROGRAMMING APPLICATIONS IN SAMPLING
ABSTRACT
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THESIS SUBMITTED FOR THE AWARD OF THE DEGREE OF
Mottov of $I|tIos^opI|p IN
STATISTICS
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BY
NUJHAT JAHAN
XASA^' UNDER THE SUPERVISION OF
DR. MOHAMMAD JAMEEL AHSAN
DEPARTMENT OF STATISTICS & OPERATIONS RESEARCH ALIGARH MUSLIM UNIVERSITY, ALIGARH 202002, INDIA.
1994
ABSTRACT
The present thesis entilled "MATHEMATICAL PROGRAMMING
APPLICATIONS IN SAMPLING " is submitted to Aligarh Muslim
University, Aligarh to supplicate the degree of Doctor of
Philosophy in Statistics. It embodies the research work
carried out by me in the Department of Statistics and
Operations Research, Aligarh Muslim University during 1991-
1994.
In the development of theory underlying statistical
methods, one is frequently faced with optimization
problems. Attempts have therefore been made to find
optimization techniques that have wider applicability and
can be easily implemented with the available computing
power. Mathematical programming is one such technique that
has the potential for increasing the scope for application
of statistical methodology. Some optimization problems from
the area of sampling are considered in this thesis and
mathematical programming techniques like dynamic
programming, separable programming etc. are applied to
obtain a solution.
This thesis consists of four chapters. Chapter I gives
an account of the gradual development in mathematical
programming techniques. Mathematical programming models are
widely used to solve a variety of military, economic,
(i)
industrial, social, engineering etc. problems. Applications
of mathematical programming in various fields are also
presented in brief.
In CHAPTER II, the problem of determining the optimum
number of strata in various situations is considered. The
cases, when the main variable y itself and when an
auxiliary variable x is used as stratification variable are
discussed. The problem is formulated as a mathematical
programming problem and a solution is obtained by using
Kuhn-Tucker (K-T) conditions. The contents of this chapter
constitute my joint paper entilled " Optimum Number of
Strata for surveys with small sample size" is to appear in
the Aligarh Journal of Statistics, Aligarh, India in its
1995 issue.
The problem of optimum allocation of sample sizes in
stratified sample survey is considered in chapter III. The
objective is to find sample sizes that minimizes the total
cost of the survey for a desired precision of the estimated
population mean. Apart from the measurement cost, the total
cost of the survey also includes the travelling cost within
strata. The problem is formulated as a mathematical
programming problem and a solution procedure is indicated
by using Simplex method by approximating the nonlinear
functions to linear functions. The possible extension of
this technique to the multivariate case is also presented.
This work is based on my joint research paper entilled
(ii)
"Optimum Allocation Using Separable Programming" which is
to appear in the Dhaka University Journal of Sciences in
its January 1995 issue.
In multivariate surveys where more than one characters
are under study the optimum allocation becomes complicated
due to the fact that what is optimum for one character is
generally not optimum for others. Thus a suitable overall
optimality criterion is to be worked out. In chapter IV we
consider the situation when the cost of measurement varies
with stratum as well as with various characters under
study. The resulting problem is formulated as a nonlinear
programming problem and dynamic programming technique is
applied to obtain a solution. A graphical representation of
the problem is also presented by a numerical problem. This
chapter is based on my joint research paper entitled "A
Generalized Optimum Allocation" presented in the Fourth
Islamic Countries Conference on Statistical Sciences held
at Lahore (Pakistan) in August 1994. This paper is also
appearing in the Journal of Indian Statistical Association,
Pune, India, in its December 1994 issue.
A comprehensive list of references is presented at the
end of the thesis.
(iii)
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To my parents
DEPARTMENT OF
STATISTICS k OPERATIONS RESEARCH
ALIOARH MUSLIM UNIVERSITY
ALIGARH-202 002, INDIA
CERTIFICATE
I certify that the material contained in this thesis entitled
" MATHEMATICAL PROGRAMMING APPLICATIONS IN SAMPLING " submitted by
Mrs. Nujhat Jahan for the award of the degree of Doctor of
Philosophy in Statistics is original.
The work has been done under my supervision. In my opinion the
work contained in this thesis is sufficient for consideration for
the award of a Ph.D. degree in Statistics.
Dated /. 3.9r ( Dr. MOHAMMAD JAMEEL AHSAN ) Supervisor Reader Department of Statistics & Operations Research Aligarh Muslim University Aligarh - 202002, India.
Telephone : (0571) J5696 Telex : 564-230 A W If
PREFACE
The present thesis entilled "MATHEMATICAL PROGRAMMING
APPLICATIONS IN SAMPLING " is submitted to Aligarh Muslim
University, Aligarh to supplicate the degree of Doctor of
Philosophy in Statistics. It embodies the research work
carried out by me in the Department of Statistics and
Operations Research, Aligarh Muslim University during 1991-
1994.
In the development of theory underlying statistical
methods, one is frequently faced with optimization
problems. Attempts have therefore been made to find
optimization techniques that have wider applicsibility and
can be easily implemented with the available computing
power. Mathematical programming is one such technique that
has the potential for increasing the scope for application
of statistical methodology. Some optimization problems from
the area of sampling are considered in this thesis and
mathematical programming techniques like dynamic
programming, separable programming etc. are applied to
obtain a solution.
This thesis consists of four chapters. Chapter I gives
an account of the gradual development in mathematical
programming techniques. Mathematical programming models are
widely used to solve a variety of military, economic,
industrial, social, engineering etc. problems. Applications
of mathematical programming in various fields are also
presented in brief.
In CHAPTER II, the problem of determining the optimum
number of strata in various situations is considered. The
cases, when the main variable y itself and when an
auxiliary variable x is used as stratification variable are
discussed. The problem is formulated as a mathematical
programming problem and a solution is obtained by using
Kuhn-Tucker (K-T) conditions. The contents of this chapter
constitute my joint paper entilled " Optimum Number of
Strata for surveys with small sample size" is to appear in
the Aligarh Journal of Statistics, Aligarh, India in its
1995 issue.
The problem of optimum allocation of sample sizes in
stratified sample survey is considered in chapter III. The
objective is to find sample sizes that minimizes the total
cost of the survey for a desired precision of the estimated
population mean. Apart from the measurement cost, the total
cost of the survey also includes the travelling cost within
strata. The problem is formulated as a mathematical
programming problem and a solution procedure is indicated
by using Simplex method by approximating the nonlinear
functions to linear functions. The possible extension of
this technique to the multivariate case is also presented.
This work is based on my joint research paper entilled
(ii)
"Optimum Allocation Using Separable Programming" which is
to appear in the Dhaka University Journal of Sciences in
its January 1995 issue.
In multivariate surveys where more than one characters
are under study the optimum allocation becomes complicated
due to the fact that what is optimum for one character is
generally not optimum for others. Thus a suitable overall
optimality criterion is to be worked out. In chapter IV we
consider the situation when the cost of measurement varies
with stratum as well as with various characters under
study. The resulting problem is formulated as a nonlinear
programming problem and dynamic programming technique is
applied to obtain a solution. A graphical representation of
the problem is also presented by a numerical problem. This
chapter is based on my joint research paper entitled "A
Generalized Optimum Allocation" presented in the Fourth
Islamic Countries Conference on Statistical Sciences held
at Lahore (Pakistan) in August 1994. This paper is also
appearing in the Journal of Indian Statistical Association,
Pune, India, in its December 1994 issue.
A comprehensive list of references is presented at the
end of the thesis.
(iii)
ACKNOWLEDGEMENT
I feel great pleasure at the completion of this
research work and would like to show deep indebtedness and
sincere gratitude to all those who helped me in the
completion of this work.
I like to express my indebtedness and sincere
gratitude to ray supervisor Dr. Mohd. Jameel Ahsan, Reader,
Dept. of Statistics and Operations Research, Aligarh Muslim
University, Aligarh, for his invaluable guidance, and
constant encouragement throughout the course of this
research work. His great involvement and sympathetic
behavior enabled me to complete this work within the
stipulated time.
I am extremely grateful to Prof. S. U. Khan, Chairman,
Department of Statistics and Operations Research and Prof.
S. Rahman (Ex-Chairman) for providing me necessary
facilities to complete this work. I am also thankful to
Prof. A.H. Khan and Prof. Zaheeruddin for their constant
encouragement and well wishes during the pursuit of this
work.
I am also thankful to all teaching and non-teaching
members of the department especially to Dr. S.M. Arshad,
for their kind help and co-operation.
(iv)
I shall be failing in my duty if I do not express my
gratitude to Mr. M. G. M. Khan, and Mr. Md. Almech Ali,
research scholars of this department for their
encouragement and moral support, which enabled me to pursue
this research work and write this thesis. Thanks are also
due to all research scholars and friends for their
encouragement and well wishes.
I am grateful to the Govt. of India for awarding me
India- Bangladesh General Cultural Exchange scholarship for
this course of study.
I am also grateful to the University of Dhaka, Dhaka,
Bangladesh for granting me study leave for this course of
study.
Finally, a very special acknowledgement is due to my
husband Mr.Jamil Akhter,for his support, assistance and
encouragement which inspired me all the time, and to our
daughter Mayeesha for giving up the time that belonged to
her.
(NUJHAT JAHAN)
Department of Statistics and Operations Research Aligarh Muslim University Aligarh - 202002, India.
(v)
CONTENTS
PREFACE
ACKNOWLEDGEMENT
PAGE
i
iv
CHAPTER I
INTRODUCTION
1.1 OPTIMIZATION AND MATHEMATICAL PROGRAMMING 1
1.2 THE GENERAL MATHEMATICAL PROGRAMMING PROBLEM 3
1.3 ADVANCES IN MATHEMATICAL PROGRAMMING TECHNIQUES 4
1.4 APPLICATIONS OF MATHEMATICAL PROGRAMMING 8
1.5 MATHEMATICAL PROGRAMMING IN SAMPLING 15
CHAPTER II
OPTIMUM NUMBER OF STRATA USING KUHN-TUCKER CONDITIONS
2.1 INTRODUCTION 18
2.2 THE PROBLEM 18
2.3 THE SOLUTION 22
CHAPTER III
OPTIMUM ALLOCATION IN STRATIFIED SAMPLING
USING SEPARABLE PROGRAMMING
3.1 INTRODUCTION 36
3.2 AN OVERVIEW 37
3.3 FORMULATION OF THE PROBLEM 49
3.4 APPROXIMATE SOLUTION USING SEPARABLE PROGRAMMING 53
3.5 A NUMERICAL EXAMPLE 57
3.6 THE MULTIVARIATE CASE 65
CHAPTER IV
COMPROMISE ALLOCATION USING DYNAMIC PROGRAMMING
4.1 INTRODUCTION 68
4.2 FORMULATION OF THE PROBLEM 69
4.3 THE DYNAMIC PROGRAMMING APPROACH 76
4.4 A NUMERICAL EXAMPLE 80
4.5 GRAPHICAL ILLUSTRATION 84
BIBLIOGRAPHY 86
CHAPTER I
INTRODUCTION
1.1 OPTIMIZATION AND MATHEMATICAL PROGRAMMING:
Any problem that requires a positive decision to be
made can be classified as an operations research (OR)
type problem. According to the Journal of Operations
Research Society, U.K. - Operations Research is the
application of the modern methods of mathematical sciences
to complex problems arising from the direction and
management of large systems of man, machines materials
and money in industry, business, government and
defence. The distinctive approach is to develop a
scientific model of system, incorporating measurements
of factors such as chance and risk with which to predict
and compare the outcomes of alternative decisions,
strategies or controls. The purpose is to help
management to determine its policy and direction
scientifically.
Optimization and uncertainty are dominant themes in
operations research. It is the application of mathematical
analysis to managerial problems. The operations researcher
makes a contribution to this problem solving effort by
mathematical techniques to obtain a solution. These
optimization methods are collectively known as mathematical
programming.
Mathematical programming is a term coined by Robert
Dorfman around 1950, and now is a generic term encompassing
linear programming (LP), integer programing (IP), convex
programming (CP), nonlinear programming (NLP), dynamic
programming (DP), programming under uncertainty, etc.
Programming problems in general may either belong to
the deterministic class or probabilistic class. By
deterministic class it is meant that if certain actions are
taken then it can be predicted with certainty that what
will be (a) the requirements to carry out the actions and
(b) the outcome of any actions. Programs involving
uncertainty of a given action may depend on some chance
event such as the weather, traffic delays, government
policy, employment levels, or the rise and fall of customer
demand. Sometimes the distribution of the chance event is
known, sometimes it is unknown or partially known.
Throughout this thesis by mathematical programming the
deterministic type of mathematical programming is meant.
1.2 THE GENERAL MATHEMATICAL PROGRAMMING PROBLEM :
The standard form of the general mathematical
programming problem may be taken as,
Minimize f(x)
Subject to,
gj(x) i 0 ; j = l,2, . . . ,in
Xj 0 ; i = l,2, . . . ,n (1.2.1)
where X' = (Xi,X2, xj is the vector of unknown
decision variables and f(X), gj (X) are real valued
functions of the n real variables Xi,X2, x .
The function f(x) is called the objective
function, the inequalities g (x) ^ 0 are referred to as
the constraints, and the restrictions X 0 are called non-
negativity restrictions.
The mathematical programming problem stated in (1.2.1)
is also known as a constrained optimization problem. An
optimization problem without any constraint is called an
unconstrained optimization problem.
1.3 ADVANCEMENTS IN MATHEMATICAL PROGRAMMING
TECHNIQUES:
Since the end of World War II, mathematical
programming has developed rapidly as a new field of study
dealing with applications of the scientific method to
business operations and management decision making. But we
can trace the existence of optimization methods to the days
of Newton, Lagrange and Cauchy. The differential calculus
methods of optimization was introduced by Newton and
Leibnitz. The foundation of calculus of variation was laid
by Bernoulli, Euler, Lagrange and Weirstress. Lagrange
introduced his famous Lagrange Multiplier Technicjue to
solve the constrained optimization problems. Cauchy made
the first application of the Steepest Descent method to
solve unconstrained minimization problems. In spite of
these early contributions, very little progress was made
until the middle of the twentieth century.
Linear programming was developed in 1947 by George B.
Dantzig, Marshall Wood and their associates, as a tool for
finding optimal solutions to military planning problems for
the United States Air Force. The early applications were
primarily limited to problems involving military
operations, such as military logistics problems, military
transportation problems, procurement problems, and other
related fields. The numerical procedure for solving a
linear programming problem introduced by Dantzig is known
as Simplex Method. But the method was not available until
it was published in the Cowles Commission Monograph No.13
in 1951.
Kuhn H. W. and Tucker A. W. (1952) published their
important paper dealing with necessary conditions,
popularly known as K-T conditions, to be satisfied by an
optimal solution to a mathematical programming problem,
which laid the foundations for a great deal of later work
in nonlinear programming.
Charnes and Lemke (1954) published an approximation
method of treating problems with separable objective
function subject to linear constraints. Later the technique
was generalized by Miller (1963) to include separable
constraints.
A number of papers by various authors dealing with the
quadratic programming began to appear after 1955. Beale
(1959) gave a method for solving a quadratic programming
problem. Wolfe (1959) transformed the quadratic programming
problem into an equivalent linear programming problem using
K-T conditions, which could be solved by Simplex method.
Other authors who gave techniques for solving quadratic
problem were Markowitz (1956), Hilderth (1957), Houthaker
(1960), Lemke (1962), Panne and Whinston (1964), Graves
(1967), Fletcher (1971), Finkbeiner and Kale (1978),
Arshad, Khan and Ahsan (1981), Fukushima (1986), and other
several authors.
Interest in integer solution to linear prograniming
problems arose early in the development of the field.
Dantzig, Fulkerson and Johnson (1954), Markowitz and Manne
(1957), Land and Doig (1960), gave methods appropriate
for integer programming. Lawler and Wood (1966)
applied the Branch and Bound technique of Land and Doig to
various nonlinear programming problems. Gomory (1960,1963)
developed the cutting plane method which is later extended
by Aggarwal (1974a, 1974b) , Wei Xuan Xu (1981) gave a new
bounding technique for the quadratic assignment problem.
Khan, Ahsan and Khan (1983) deals with the concave
quadratic programming problem when some of the variables
are restricted to take only integer values. A solution is
proposed by adding Gomory type cuts in the procedure
developed by Arshad et al (1981) . Saltzman and Hiller
(1988,1991) presented a new algorithm called the exact
ceiling point algorithm for solving the general integer
linear programming problem.
Bellman (1957) made the major original contribution to
the development of the dynamic programming technique.
Dynamic programming problems paved the way for development
of the methods of constrained optimization. The
contributions of Rosen (1960,1961), Zoutendijk (1966), to
nonlinear programming have been very significant. Although
till date no single technique has been found to be
universally applicable to all nonlinear programming
problems. The work of Carrol (1961), Fiacco and McCormick
(1968) made many a difficult problem to be solved by using
the well-known techniques of unconstrained optimization.
Powell (1964) gave an efficient method for finding the
minimum of a function of a several variables without
calculating derivatives. Other authors who made
contribution for unconstrained optimization are Fletcher
and Reeves (1964), Davidon (1968), Fletcher (1970) etc.
Grandinetti (1982) gave an updating formula for quasi-
newton minimization algorithm.
Geometric programming was developed by Duffin,
Peterson and Zener (1967) . Geometric programming provides
a systematic method for formulating and solving the class
of optimization problems that tend to appear mainly in
engineering designs. Ermer (1971) used geometric
programming for optimization the constrained machinery
economics problem. Dembo (1982) applied sensitivity
analysis in geometric programming.
Dantzig (1955), Charnes and Cooper (1959,1960)
developed stochastic programming techniques and solved
problems by assuming design parameters to be independent
and normally distributed. Some other authors who
contributed in stochastic programming are Evers (1967),
Greenberg (1968), etc..
Recent development in multiobjective fractional
programming are due to Singh (1981,1982, 1984, 1986, 1988),
Shaible (1989), Singh and Hanson (1986,1991), Khan (1990)
and several other authors.
Developments of new techniques for solving
mathematical programming are still going on. Kachian (1979)
gave a polynomial algorithm for linear programming.
Karmarkar (1984) gave the polynomial time algorithm which
is an excellent method for solving linear programming
problem. A number of other authors including Anstreicher
(1986), Gay (1987), Tomlin (1987), Shanno and Marsten
(1985) etc. have worked on Karmarkar's algorithm.
1.4 APPLICATIONS OF MATHEMATICAL PROGRAMMING:
Mathematical programming models are widely used to
solve a variety of military, economic, industrial and
social problems. The early applications were primarily
limited to problems involving military operations, such as
military logistics problems, military transportation
problems, procurement problems, and other related fields.
Some of the works are due to, Brackmen and Burnham (1968),
Wollmer (1970), Dellinger (1971), Miercort and Soland
(1971), Eckler and Burr (1972), Furman and Greenberg
(1973), Burr, Folk and Karr (1985) etc.
In addition, mathematical programming was applied to
interindustry economic problems, based on Wassily Leontiefs
input-output analysis - by Koopman (1951), Leontief (1951)
and Morgenstern (1954) with successful military
applications, it was carried over to business uses. A 1976
survey in America to determined the use of mathematical
programming by American companies shows that 74% of them
use mathematical programming techniques to solve their
various problems.
The uses of mathematical programming range from the
government sector to agricultural, business and industrial
sectors. Some of the applications in agriculture are due to
Fisher and Shruben (1953), Fox and Taeber (1955), Boles
(1955), Candler (1956), Dhondyal (1960), Talib and Singh
(1961), Pierson (1962), Heady and Egbert (1964), Balinga
and Tambal (1964), Swanson and Woodruff (1964), Johl and
Kahlon (1966), Qingzhen, Changyu and et al (1991) etc.
The need for optimal decision making arises from the
relative scarcity of productive resources. The allocation
of limited resources among competitive uses is of major
interest to business decision makers. By an efficient
allocation of resources, an attempt is made to achieve a
specific business objective. In recent years, mathematical
programming has received wide acclaim among business
decision makers and industry management as a tool helpful
in achieving their business objectives. Few contributions
in this area are due to Dantzig, Johnson and White (1958),
Smith (1961), Catchpole (1962), Arabeyre, Fearnley and et
al (1969), Lee (1972), Rubin (1973), Belton (1985),
Vijaylakshimi (1987), Silverman, Steuer and Wishman (1988)
etc., etc.
Mathematical programming are widely used in field of
production-scheduling, planning and inventory control. Many
authors applied mathematical programming technique to the
above areas such as, DeBoer and Vandersloot (1962),
Efroymson and Ray (1966), Silver (1967), Von Lanzenauer
(1970), etc.
Service companies such as restaurants, hotels, gas-
station and convenience stores are often composed of many
distributed facilities. Such companies constantly make
decisions about expanding their operations by establishing
10
new facilities and/or expanding existing ones. The use of
optimization models for capacity expansion, both under
deterministic demand and stochastic demand, has been an
area of active investigation. Luss (1983) provide
comprehensive literature surveys in this area. Recent
articles include Ganz and Berman (1992), Bean, Higel and
Smith (1992) etc..
Problems dealing with personnel assignment and
training are another area where mathematical programming
techniques are successfully applied. Such as physician
manpower requirements of most developing countries can
often be met by developing local training facilities,
sending indigenes to train abroad, and by recruiting
expatriate physicians. Ikem and Reisman (1990) developed a
manpower planning model which coordinates physician
manpower requirements of a developing country with its
capacity to train such physicians along with national
objectives to contain costs. Some others who contributed in
this area are Rothstein (1972), Byrne and Potts (1973),
Martel (1973), Segal (1974) etc.
In design, construction and maintenance of any
engineering system, engineers have to take many
technological and managerial decisions at several stages.
The ultimate goal of all such decisions is either to
11
minimize the effort required or to maximize the desired
benefit. Mathematical programming can be applied to solve
many engineering problems also. The application of
optimization methods in the design of thermal systems was
presented by Stoker (1971) . The works of Haug and Arora
(1979), Johnson (1980), Krisch (1981) deal with the optimum
design of machine and structural systems. Hussain and
Gangiah (1976), Ray and Szekely (1973) discuss the
applications of optimization techniques to chemical and
metallurgical engineering problems. Rao (1973) gave the
minimum cost design of concrete beams.
Apart from the applications of mathematical
programming just listed above, it can be used as an adjunct
to or in place of methods that are now associated with the
fields of artificial intelligence (Al) and information
technology. Ten Dyke (1990) identifies these problems. The
analyst seeks to take raw data and transform them into
useful information. In medical diagnosis also mathematical
programming techniques are used where in we seek to
associate a set of input attributes to a specific outcome
or response. Few works are due to, Klepper (1966), Henschke
and Flehinger (1967), Bahr et el (1968), Redpath and et al
(1975,1976,), McDonald and Rubin (1977), Emerson (1975),
Sonderman and Abrahamson (1985) etc., etc..
Other application of mathematical programming are in
12
education - Bruno (1969), Harden and Tcheng (1971), Heckman
and Taylor (1969), Koenigsberg (1968), McNamara (1976) etc.
; in environment protection - Clark and Helms (1970),
Graves, Whinston and Halfield (1970), Loucks, Revelle and
Lynn (1967), Deiniger (1965,1972), McNamara (1976), Smeers
and Tyteca (1984) etc. ; in media selection - Bass and
Lonsdale (1966), Charnes, Cooper and et al (1968) , Frank
(1966) etc.
Statistical technology plays an indespensable role in
almost every possible sphere of human activity in the
modern world. In fact all statistical procedures are,
solutions to suitably formulated optimization problems.
Whether it is designing a scientific experiment or planning
a large-scale survey for collection of data, or choosing a
stochastic model to characterize observed data, or drawing
inference from available data, such as estimation, testing
of hypothesis, and decision making. One has to choose an
objective function and minimize or maximize it subject to
given constraints on unknown parameters and inputs such as
the costs involved. The classical optimization methods
based on differential calculus are too restrictive and are
either inapplicable or difficult to apply in many
situations that arise in statistical work. The lack of
suitable numerical algorithms for solving optimizing
equations, has placed severe limitations on the choice of
13
objective functions and constraints and led to the
development and use of some inefficient statistical
procedures.
Attempts have therefore been made during the last
three decades to find other optimization techniques that
have wider applicability and can easily be implemented with
the available computing power. One such technique that has
potential for increasing the scope for application of
efficient statistical methodology is mathematical
programming, [ C.R. Rao in Arthanari and Dodge (1981)].
The fundamental paper by Charnes, Cooper and Ferguson
(1955) introduced the application of mathematical
programming to statistics. As an alternative to the least-
square approach to linear regression, they choose to
minimize the sum of the absolute deviations (MINMAD), and
showed the equivalence between the MINMAD problem and a
linear programming problem. Wagner (1959) suggested solving
the problem through the dual approach. An efficient
modification of the simplex method by Barrodale and Roberts
(1973) increased the possibility of using MINMAD regression
as an alternative to classical regression.
Other areas of applications of mathematical
programming in statistics developed simultaneously. Lee,
Judge and Zellner (1968) applied it in maximum likelihood
14
and Bayes estimation of transition probabilities in Markov
chains. Jensen (1969), Vinod (1969), Rao (1971) and
Liittschwager and Wang (1978) in cluster analysis; Sedransk
(1967) in designing some raultifactors of survey data ;
Neuhardt and Bradley (1971) in selection of multifactor
experiments with resource constraints ; Foody and Hedayat
(1977), Whitaker, Thriggs and John (1990) in the
construction of BIB designs, Neuhardt, Bradley and Henning
(1973) in optimal design of multifactor experiments ;
Dantzig and Wald (1951), Barankin (1951), Francis and
Wright (1969), Krafft (1970), Meeks and Francis (1973),
Pulkelsheim (1978), etc. in testing statistical hypotheses
; Chakraborthy (1986,1990) in quality control.
Mathematical programming techniques are applicable to
so wide variety of problems emerging from almost every
branch of science, engineering, industry, management
planning, medical science, military, etc. that it is not
possible to even describe all the applications briefly in
a single thesis.
1.5 MATHEMATICAL PROGRAMMING IN SAMPLING:
Sampling theory deals with problems associated with
the selection of samples from a population according to
certain probability mechanism. The problem of deriving
15
statistical information on population characteristics,
based on sample data, can be formulated as an optimization
problem in which we wish to minimize the cost of the
survey, which is a function of the sample size, size of the
sampling unit, the sampling scheme, and the scope of the
survey, subject to the restriction that the loss in
precision arising out of making decisions on of the basis
of the survey results is within a certain prescribed limit.
Or alternatively, we may minimize the loss in precision,
subject to the restriction that the cost of the survey is
within the given budget. Thus we are interested in finding
the optimal sample size and the optimal sampling scheme
which will enable us to obtain estimates of the population
characteristics with prescribed properties. Mathematical
programming techniques are also applied to certain
estimation problems related to sampling.
Raj (1956), Dalenius (1957), Yates (1960), Kokan
(1963) Hartley (1965), Folks and Antle (1965), Pfanzagl
(1966), Kokan and Khan (1967), Chatterjee (1968,1972),
Huddleston, Claypool and Hocking (1970), Bethel (1985) and
Chromy (1987) etc. discussed the use of mathematical
programming in relation to the multivariate optimum
allocation problem.
The subsequent chapters of this thesis are devoted to
16
the use of mathematical programming to some problems
arising in univariate and multivariate stratified sampling.
17
CHAPTER I I
OPTIMUM NUMBER OF STRATA USING
KUHN-TUCKER CONDITIONS
2.1 INTRODUCTION :
If Stratified sampling is to be used for
estimating some population characteristics of the
character y under study, then before drawing a sample,
the sampler must know the following:
i) How many strata should be there?
ii) What should be the stratum boundaries?
iii) What should be the sample sizes from various
strata?
In this chapter the problem of determining the optimum
number of strata is studied under different situations.
The programming problem and solution procedures for two
different situations are discussed.
2.2 THE PROBLEM :
The problem of determining the optimum number of
strata was first discussed by Dalenius (1950), when the
18
main variable y itself is used as stratification variable
under certain assumtions. He postulated that in
stratified sampling with L strata the varince V(y t ) of
the stratified sample mean y^ is inversely proportional
to l/ , that is
L 2
or V{y,,) - -A
where A is the constant of proportionality. The constant
A is approximately equal to SyVn, where Sy is the
population variance of y and n is the total sample size.
Thus
52 (yst) = — ^ (2.2.1)
On the basis of the above result Dalenius (1953)
conjectured that the relationship between the variances
(yst)L ^rid VkYst^L-^ of the stratified sample means based
on L and L-1 strata respectively is of the type
nyst)L =(^f ^(y.e).-i
Later on Cochran (1961) tested various approximations
empirically and confirmed that the above relationship holds
19
approximately even for skewed distributions and apparently
the rate of reduction in the variance is independent of
the skewness of the population.
Dalenius (1957) expressed the total cost C as a
linear function of n and L, that is,
C = c^L + C2/2 (2.2.2)
where c, = cost per stratum, and
C2 = cost per unit within each stratum.
It can easily be varified that the values of L
and n that minimizes V(y'., ) given by (2.2.1) for a fixed
cost C = Co given by (2.2.2) are,
In sample surveys where the cost of per unit
enumeration is very high it is possible that (2.2.3) yield
a solution with n<2L. In such cases there will be some
strata from which only one unit is to be selected
and we cannot obtain an unbiased variance estimator. To
overcome this difficulty an additional constraint n z
2L is introduced. With this added constraint the problem
of determining the optimum number of strata may be
20
expressed as the following mathematical programming
problem (MPP). Let us call this MPP-I.
Minimize V[n,L) = — ^
Subject to
n t 2L
and n,L -i. Q
MPP-I
In case an auxiliary variable x is used as
stratification variable, Cochran (1963) showed that for
linear regression of y on x is linear then
2
V (7 ) r ^ + (l-p2) • (2.2.4) •'St n L r2
where p is the coefficient of correlation between x and y
in the unstratified population. The RHS of (2.2.4) gives
the minimum value of V(yst) for given values of p, n and L.
The problem is to find (n,L) that minimize RHS of (2.2.4)
for a fixed value of p. The problem of determining the
optimum number of strata in this situation becomes:
Minimize V{n,L) = -21 [ -2- + (i-n ) 1 n y L^ ^ ' \
Subject to
and
c^L + c^n ^ Cg
72 21,
n, L -2: Q
MPP-II
21
2.3 THE SOLUTION:
MPP-I: As both S y/nL' and nL^ are positive valued
functions of n > 0 and L > 0 "(7 ) - •''
will be minimum when Z(n,L) = nL will be maximum. Further
it can be seen that the optimal solution of MPP-I will
always be attained at a boundary point of the feasible
set F defined as
F ^ [{n,L) \c^L + c^n <, Cg ; m2L and n2;0,I. 0( .3.1)
To prove this let (n*,L*) e F be the optimal
solution to MPP-I, that is
Z{n*,L*) ^ Z(n,L) for all {n,L)eF . . (2.3.2)
I f (n*,L*) i s n o t a bounda ry p o i n t we can a lways f i n d
a n o t h e r p o i n t (n* + & , L* +9^) e F f o r da and & > 0,
however s m a l l . ^
Now,
Z{n*+ 6n, L'+ 6L) = {n* + 6n) (L'+ 6L) ^
- {n*+6n) {L'" + 6 L 2 + 2L* 6L)
= n*L'' + A positive quantity
^Z{n\L') + A positive quantity
oz, Z{n'+6n, L*+6L} > Z{n*,L*)
22
which contradicts (2.3.2). Hence (n' , L' ) must be a
boundary point of F.
For solving MPP-I, at first instance we consider
only the cost constraint c L + c.n ^ C,, . From the above
discussion it is clear that the optimal solution will be
a boundary point of F. For obvious reasons it cannot be
a point on the boundaries n = 0 o r L = 0 o f F . Clearly it
will be a point on the boundary of F defined by c.L +
C2n = Co in which case the optimal solution is same
as given by (2.2.3). This optimal point will satisfy the
constraint n ^ 2L provided.
C AC
3 C2 3 Ci
or, —^ > 4 (2.3.3) ^2
Thus if the cost ratio -^ obeys (2.3.3) MPP-I is
competely solved by (2.2.3). If — <4 the
constraint n ^ 2L is violated. As before if we
consider only the constraint n ^ 2L ignore
CiL + c n ^ Co and solve the MPP-I, the optimal solution
will lie on the boundary of F defined by n = 2L and we
will have an unbounded solution, that is n and L can be
23
made arbitrarily large. However n cannot go beyond
the population size N in which case
N L = — . But this will be a complete enumeration not
c, a sample survey. Thus if ^ < 4 the optimal point
will lie on the bounr iry of F at the intersection
of CiL + CjU = Co and n 2L which gives
n -
and L
Ci ^ 2C2
- 0
• ( 2 . 3 . 4 )
The Hessian matrix for the function
nvst) -nL'
IS
H =
d^v d^v s. dn^ dndL
^V d^V dLdn ai,2
( 2S''
2S2
2 5 2 \
6 5 2
\ n^L^ nL'^ )
24
The principal minors of H are
2^ - = - 7 ^ > °
and
^ 2 2 ' , , 25" 45''
- 8 ^ ' > 0
H,i > 0 and H22 > 0 shows that the function
5 2
nL'
is a convex function of n and L.
As the objective function of MPP-I is convex
and the constraints of the problem are linear the Kuhn and
Tucker (1952) necessary conditions to be an optimal
solution of any MPP will be sufficient also. It can be
varified easily that the optimal solution (2.2.3) and
(2.3.4) of the MPP-I satisfy Kuhn and Tucker (K-T)
conditions.
MPP-II To solve MPP-II let us consider first the
subproblem
25
Maximize z{n,L) - -— [ -^ + (1-p^) ]
Subject to
a c-^L - c^n i 0
and /3,L 0
(2.3.5)
Note that the minimization is changed into
maximization, the cost constraint is written in ^ form and
the constraint n ^ 2L is dropped.
The Hessian matrix of the function
Z{n,L) = n L 2
is
H
dLdn
d^Z dndL
d^Z
2„2 /- ^^ P
3r 2 n'L
25^(l-p^) ,3
n-
2„2 25^p
2„2 2S-=p
The two principal minors of H are
//, 25 2„2 3r 2 n'L £f _ 2S^(l-p^) ^
n-
26
and
^22 = 1 1
4,>4 4S"p
Hii < 0 and H,; > 0 shows that the function Z (n,L) is
a concave function of n and L. As the objective
function - — [ -^ + (1-p ) n L 1,2
is concave and the constraint CQ - CiL - Cjn i 0 is linear
the K-T necessary conditions are sufficient also. The K-T
conditions for x to be an optimal solution for the
nonlinear programminig problem "Maximize f (x) subject to
gi (x) ^ 0 ; i = 1, 2, . . . .m and x > 0 " are
^*'^^^{xl,ul) = 0
^^^{xl,ul) ^ 0
u' 0
(2.3.6)
where ^^^^iKliUl) ^nd v^ ^{^,ul) are the gradient vectors
27
of the function <J) with respect to the component of
X and u respectively and <{> is defined as,
<^{x,u) = f{x) +1) U; gj{2£) 1=1
As these conditions are sufficient also if we
are able to find x* and u* satisfying them, then x'
will solve the MPP given by (2.3.5) .
The conditions (2.3.6) for MPP(2.3.5) are
'(n,L) 4> = 2„2 2S^p nL- - uc.
^ 0
(•"'- ) (n,L) <J) = -n i;(f^Ml-p= uc. n I]
+ L 2„2 2.g2p
nL- - uc. = 0
1) i 0
u4> = ^70" 1-^ - C2^ ^ 0
(a)
. . (b)
(cl
(d)
(2.3.7)
uv^^ = U(CQ - c^L - 02^2) = 0 (e)
and u a 0 (f)
28
where
(t>(T2,L, u) = - — f -£- + (l-p2) 1 + u(Co -c,L -c^n) n ^ n^
Taking equality sign in (2.3.7 (a)) we get,
u = ^^ \ -£! + (l-p2) 1 > 0 . . (2.3.8)
and u = ^ > 0 (2.3.9)
Thus u given by (2.3.8) and (2.3.9) satisfies K-T
conditions (2.3.7 (e) ) . As u > 0 by (2.3.7(e)) we get,
or n = (2.3.10) ^2
Again from (2 .3 .8) and (2 .3 .9 ) we g e t ,
C2ii^ •• L" J c^nL'
;2.3.10) and (2.3.11) together on simplification give
29
or, L^ +3 aL +±1 = 0 (2.3.12:
where a = 1-p'
and jb = 2C,p'
c,(l-p2)
Using the theory of equations, the roots of the
cubic equation (2.3.12) are given by
L = pi/3 + gi/3 (2.3.13)
where 1 \ - b + y/b^+4,a
1 and ^ "" 1" i3 - v/P"^4a^
Note that, as
a = -£1- > 0 l-p2
the quantity b + 4a^ > 0 hence p and q are real quantities.
Substituting the values of p and q in terms of p, CQ and Ci in
(2.3.13) and on simplification we get the optimum values of n
and L as
30
c„ 2 N^/^
\ C - n 2
c„ ^
1-p
1-p^ ;
( 2 . 3 . 1 4 )
and
n C, C, \ l - n 2 -' I
C
N
Cn
^ cf i-p-
1/3
1-p^
1/3
( 2 . 3 . 1 5 )
The above values of n and L will satisfy the constraint n^2L
if { R.H.S.of (2.3.15) } ^ 2 { R.H.S. of (2.3.14)}
which gives
(Ci + 20^)^ (dCg - Cj 1-p-
CQ C-^ (2.3.16)
Thus in case the given values of CQ , Ci , C2 and p satisfy
(2.3.16) the vulues of L and n given by (2.3.14) and (2.3.15)
will solve the MPP-II completely.
When (2.3.16) is not satisfied, that is
31
(C, + 2C,)M4C2 - Cj l-p2 > ^-C- (2.3.17)
We will rewrite MPP-II as,
Maximize - — [ -^ + (1-p^)
Subject to,
Cg - c^L - c^n i. 0
12 - 2L ^ 0
and n,L ^ 0
The f u n c t i o n <j> i s d e f i n e d a s
(j)(22,L,Ui,U2) = — [ - ^ + ( 1 - p ^ ) 1 + U^{CQ-C^L-C2n) ^ ^ ^ •• . . . . ( 2 . 3 . 1 8 )
+ Uy [n-2L)
32
The K-T c o n d i t i o n s a r e
^ ( n , Z . ) 4 > ( ^ ' - ^ ' " l ' " 2 ) n- L 2
( 1 -
2
V) ]
" l ^ l
+
-
U1C2
2 U 2
+ " 2
[n,L) v,„^^,(t) = n [ | _ {_£_ + ( i -p2) } - u , c , + U2 ;
+ L nL'
n
^(u„u,)4> = /2 - 21,
^ 0
^ g (a)
(b)
( r ) ^ 2 (c)
(d)
( L ' I ' " 2 ) ^ ( U I , U J ) 4 > = Ui(Co - c^L - c^n) + U2(/2 - 2L) = 0 (e)
( " ' ) ^ 2 <«>
( 2 . 3 . 1 9 )
If Ui and U2 are not equal to zero, in order to
satisfy (2.3.19(e)) we must have
Cg - c^L - c^n = 0
and n - 2L = 0
33
which gives,
L = '^ (2.3.20) c, + 2C2
and n = — ^ - ^ ^ (2.3.21) C, + 2C2
With these values of n and L and taking equality sign
in
(2.3.19(a)) and solving for Uj and Uj we get
"> = ^ ^ 4 i ^ f ^ ^ >° ' — >
where d = —^ ^— > 0 (-0
and U3 = ^ [ dp2 - ^ ^ i ^ ( 3dp2 + Syfd ( l -p^ ) ) ] ( 2 . 3 . 2 3 )
I t can be v e r i f i e d t h a t U2 >0 i f and o n l y i f
Q ' C , P '
34
which is (2.3.17).
Thus the values of L and n given by (2.3.20) and
(2.3.21) will solve MPP-II.
It can be seen that if
(Ci + 20,)' (4C2 - cj _ 1 -r.2
qfc, p'
the expressions (2.3.14) and (2.3.20) give the same value of
L, similarly expressions (2.3.15) and (2.3.21) give the same
value of n.
35
CHAPTER III
OPTIMUM ALLOCATION IN STRATIFIED SAMPLING USING
SEPARABLE PROGRAMMING
3.1 INTRODUCTION :
In stratified random sampling with L strata the
sampler has to determine the sample sizes n ; h=l,2,...,L
in advance. The values of n may be determined either to
maximize the precision of the estimate for given cost or to
minimize the cost of the survey for fixed precision. The
allocation of the sample sizes n according to the above
criteria is known as optimum allocation.
The optimum allocation, when the cost of the survey is
expressed as a linear function of n., is well known in
literature. In this chapter it is assumed that the cost
function is quadratic in y^ . The lower and upper bounds
on the values of n, are also taken into account. The problem
of optimum allocation is then formulated as a mathematical
programming problem and its approximate solution is
36
obtained by using separable programming technique.
In multivariate stratified random sampling where more
than one characters are to be measured on each sampled
unit, no single optimality criterion is available to work
out the optimum allocations. The section 3.2 of this
chapter presents an overview of the optimum allocation in
multivariate stratified sampling using various criteria.
In the subsequent sections of this chapter the problem
of optimum allocation in stratified sampling for univariate
case is formulated as a separable programming problem. A
solution procedure is indicated by using Simplex method by
approximating the nonlinear functions into linear
functions. Further extension of the technique for the
multivariate case is also indicated. The objective is to
find sample sizes that minimizes the total cost of the
survey for a desired precision of the estimated population
mean. Apart from the measurement cost, the total cost of
the sur-vey also includes the travelling cost within strata.
3.2 AN OVERVIEW :
Consider the problem of optimum allocation in
univariate stratified sampling in estimating the overall
population mean;
37
?=i|:|n, = |: .y.
where y., = value of the jth population unit in the hth
stratum ; h = 1,2, ,L ; j = 1,2, , N
N^ = size of the hth stratum
N Wfj = ^ = stratum weight
Y^, = -^ LVh-! = Stratum mean
N = L,Nf^ = population size
If independent, without replacement simple random
samples of sizes n ; h = 1, 2, . . . ., L are drawn to construct
the unbiased estimate
for the population mean Y where
38
1 ^
then the well known optimum allocation which minimizes the
variance of y\, for fixed cost CQ given by Neyman (1934) is
„^ ._ £^!3,Iiim. : t-1.2 L .... (3.2.1)
/l-l
where
t c -n ii 0 =" '^h h is the total cost of the survey
c = cost of measuring per unit in the hth stratum
.2 _ 1 V^ ,.. ,; . 2 •5 - .r _;- (yii " y^^ = stratum variance in the
hth stratum.
39
The problem of optimum allocation in multivariate
stratified random sampling may be expressed in the
following general way (Zacks (1970)).
Let Qh = QhiYhi' 'YhNj ' h = l,2, . . . . ,L be specified
parametric functions
where N = size of the hth stratum
and y^ = value of the jth population unit of the
hth stratum.
Consider the problem of estimating p linearly
independent functions
LAY) = t a.^ e ; j=l,2, p
from the stratified sample, where Y = (Y , ,Y^) is the
vector of values in the population.
In univariate case, that is when p=l, the Neyraan
allocation given in (3.2.1) gives
^h ^ \ a^h\ ^isl.Cf,) ; h=l,2, . . .,L
where <t)(5j^,c^) i s s p e c i f i e d f u n c t i o n of S ^ and Ch
40
It can be seen that even if S ^ are known the Neyman
allocation will vary for different functions L ; j =
1,2,....,p. The problem is then to select a single
optimality criterion which is suitable for estimating all
Lj. Many criteria are available in literature to choose a
reasonable allocation. Some of them are indicated below.
Dalenius (1953) suggested to minimize the total
relative loss in precision by applying an allocation n =
(ni, n2, ,nL) other than Neyman allocation such that the
cost of the survey L, Cf^nf^ ^ Cg . The problem may be
mathematically expressed as :
" Find the L component vector n = (ni, nj, ,n^)
which Minimize
t {V{Lj\n)-V^^,{Lj)} Q[n) =2Ii i ^
Subject to ,
where V{L^\n) and V^p^(Lj) denote the variances under an
allocation Q = (n,, n , ,nj and the Neyman allocation.
41
Cochran (1963) suggested the use of the average of the
individual optimum allocations for different characters
under study as an alternative to the optimum allocation for
all characters. He showed that due to the flatness of the
variance function the average allocation gives results
almost as precise as the individual optimum allocations.
Dalenius (1957) considered a weighted linear function
of the loss in precision relative to optimum allocation of
the type ^ w. L- where w are weights assigned to
different characteristics according to their importance and
discussed its minimization subject to cost constraint.
Ghosh (1958) worked out the multivariate optimum
allocation by minimizing the generalized variance of the
estimates of the population means for a fixed total sample
size.
Yates (1960) suggested two criteria for working out
optimum allocation in multivariate cases in the situations
where the individual optimum allocations differ so much
that there is no obvious compromise. The first approach
applies to surveys where the loss due to an error in the
estimates can be measured in terms of money or utility. For
42
estimating the population means, with p variates and a
quadratic loss function he expressed the total expected
loss as a linear function of the variances of the
estimates, that is
This expected loss is minimized under the linear cost
constraint Cg + £ Cf^ n^ <, C
In the second approach, tolerance limits are specified
for each variance and the total cost of the survey is
minimized. In this case we have the prcblem of minimizing Cg + l) c^ n^^
under the p constraints
^^Yisc) ^ ^j ; j=l,2, . . . . ,p ,
and the restriction
0 ^ n,, ^ Nf, ; h = l,2, . . . . ,L ,
where v is the pre-fixed tolerance limit of V(y^,J
43
Several authors e.g. Hartely and Hocking (1963), Kokan
(1963), Chatterjee (1966), Zukhovitsky and Ardeyeva (1966),
Huddleston, and et al (1970), etc. gave algorithms for
solving such problems.
Kokan (1963) suggested the minimization of the total
sampling cost
L
E c = Co + 5] c^ft
subject to the constraints on the individual variances
V{Lj\n) , j=l, 2, . . . . ,p. The condition that he imposes on
each variance V [L^\x) are such that all L^ satisfy the
proportional closeness (relative precision) requirement,
namely :
p{\L^-L^\<X\Lj\] ^ l-a for all j = 1, 2, . . . . ,i, ,
0< X <i, and 0< a <i. in Kokan's approach the functionsL -
do
not have to be linearly independent and p can be greater
than L.
Folks and Antle (1965) solved the problem of
allocation by determining the set of efficient points
A{n^,rhi,, ,n^) , where
44
nn^ '='
<^ n2 11/2 ,
Lay Sjt, 1 N^^n
ra,-=l, a-^O J-1
Kokan and Khan (1967) gave an analytical solution
using n-dimensional geometry to the problem of optimum
allocation in multivariate stratified random sampling and
in few other sampling procedures for estimating the
population mean. The optimality criteria is similar to that
of Yates (1960).
Huddleston, Claypool and Hocking (1970) determined the
optimal sample allocation by using convex programming
methods and compared the allocation with other existing
allocation system.
Ahsan (1975-1976) gave a variation of the problem
discussed in Kokan and Khan (1967) and worked out the
solution on the same lines.
Ericson (1965) used prior information for obtaining
optimum allocation in univariate stratified sampling. The
multivariate case was also discussed under the assumption
that the strata are sufficiently similar.
45
Ahsan and Khan (1977), Ahsan (1978), and Ahsan and
Khan (1982) also devolped algorithms to obtain optimum
allocation in multivariate stratified random sampling using
prior information.
Khan and Islam (1980) formulated the problem of
optimum allocation when multiple characters are measured
into a nonlinear programming problem with multiple
objective functions. A solution has been given similar to
the STEP method for linear programming.
Dayal (1982) gave a procedure of determining sample
sizes and its optimum allocation to various strata under
stratified random sampling when stratum level estimates are
also required. He consider the desired precision in terms
of relative variance for the hth stratum which is given
by
{V{Y^\Y^)} ^ bf, ; h = l,2, . . .,L
where b ' s are given constants.
This gives n a,, ; h = 1,2,...,L,
where a ' s are constants depending upon b , N and S^,\ Thus
the problem is to minimize V(ylj subject to the
constraints.
46
t n^ = n /i-i
and a - n^ 0
Liao and Sendranksk (1983) determine the optimal
allocation of a total sample of size n among the strata by
maximizing for fixed n the probability P, that specified
variance constraints are satisfied and by minimizing the
total sample size n for a desired level of P. Both exact
and approximate solutions are presented and compared using
a secfuence of numerical examples.
Rao (1984) gave an alternative method of deriving
Neyman's optimum allocation. In order to obtain this
allocation, the within stratum standard deviation of the
study variable are required which are unknown and are
substituted by known quantities. He also discussed the
effects of deviations from optimum allocation and other
related problems.
Mukherjee and Rao (1985) consider the problem of
judging how good an actual compromise allocation is as
compared to Neyman allocation by obtaining a bound to the
ratio of the corresponding variances.
Dayal (1985) shows how the values of auxiliary
characteristic linearly related to the study variable, can
47
be used in the allocation of the sample. He also found that
proportional allocation can be more efficient than an
approximation to Neyman's allocation by using estimates of
standard deviations of the study variable from a previous
survey or approximations to them from some variable related
to the study variable.
Omule (1985) expressed and solved the procedure of
determining the optimum sample size in each stratum in
stratified sampling for several variables as a multistage
decision process through dynamic programming.
Bethel (1989) using the Kuhn-Tucker theorem, gave an
expression for the optimum allocation in terms of
lagrangian multipliers. Using this representation, the
partial derivative of the cost function with respect to the
jth variance constraint is found to be,
-2aj g{x* )
where g(x* ) is the cost of the optimum allocation
and a* and v, are respectively the jth normalized lagrangian
multiplier and the upper bound on the precision of the jth
variable. A simple computational algorithm and its
convergence properties are also discussed.
48
3.3 FORMULATION OF THE PROBLEM :
Let the population of size N be divided into L
strata of sizes Nf, ; h=l,2, ,L and that the sampling
within each stratum is independent simple random sampling
without replacement. Further, for the hth stratum, let,
Njj = the stratum size
n^j = the sample size
f;j = — p , the sampling fraction.
Yijj = the value of the characteristic under study
for the j th unit, j = l, 2, ,iV),.
lVh=— = stratum weight
'h--fT y^y^j = the stratum mean
_ ^ L Nt
h^l
= over all population mean
- 1 "' ^^"'JT J^^f^J' ^^^ sample mean based on rz units,
49
which is an unbiased estimate of Y .
1 '''' 5 2 —^^-y^ ^Yhj'^h'l^ = the stratum variance
s 2 _ — ^ y] iyhj'Vh^^ = the sample mean square
based on n units, used as an unbiased
estimate of Sh .
An unbiased estimator of V is given by,
tw,:
The variance of Yg^, ignoring the fpc is,
^(y.t)=E-\^ (3.3.1)
If Wft and S^ are known, Vly ) is a function
of n only. If S are not known s may be substituted
for them.
The cost function is assumed to be of the form
L L
^ = ^ -" "" ^ / \/^ (3.3.2)
where,
C = total cost of the survey
c = measurment cost per unit in the hth stratum
t = travelling cost within hth stratum.
If travel costs within strata are substantial Beardwood et
50
al. (1959) suggested that a fairly good approximation to
L
the total travel cost is V t/,yJT
The problem is to find n, that minimizes (3.3.2) for a
fixed tolerance limit on the value of ViYsc) given in
(3.3.1), where
n is bounded as, 2^nf,^Nfj , h=l,2,...,, L. The upper limit
on rifj is put to avoid oversampling whereas the lower
limit gives an opportunity to estimate the stratum variance
if not known.
Consequently our allocation problem may be expressed
as the following nonlinear programming problem (NLPP).
L L
Minimize C(n^) = 5 ^ c ^ ^ + J ) t ^ ^ . . . . (3.3.3)
Subj ect to , L
^^ (3.3.4)
^nd 2 ^ Uf, ^ Nf, (3.3.5)
where v is the desired tolerance limit for Viy^^)
With x = yjj , the nonlinear programming problem
51
{3.3.3)-(3.3.5) becomes.
Minimize C= L,Cf,xJ -^ l) t, x (3.3.6;
Subject to,
^ w ^ ^ P/ 2 S 2 ^—A- iv (3.3.7)
2
/2 x^ y^ (3.3.8)
As Ch > 0, h=l,2,...L, the objective function (3.3.6)
is convex. The nonlinear constraint (3.3.7) is also
convex. Hence any local minimum for NLPP. (3.3.6) - (3.3.8)
will be a global minimum. Both the objective function and
the constraint function are separable in x ,. The
separability of the functions allow us to apply separable
programming technique which will yield an approximate
global minimum for the NLPP. The separable programming
problem may be stated as:
Minimize yff.(x.) (3.3.9) ^ ^ '
52
Subject to,
t Sr.UJ^^ (3.3.10; /)=•!
y2 X (3.3.11)
where f {x^) = c^ x^^ + t^x^
JV 2 5 2
3.4 APPROXIMATE SOLUTION USING SEPARABLE PROGRAMMING:
Any nonlinear function can be approximated by a
piecewise linear function without much difficulty if the
nonlinear function is well behaved. The basic approximation
technique to be used will be that of replacing the
functions -f/, (-</,) and g;,(x ) by linear approximations,
thereby reducing the problem to a form which can be solved
by simplex method with a simple modification (Hadley
(1970), Kambo(1984)).
Let us define the subset H of the set {l,2,....,L} by
H={h:ff,(Xf,) are linear}. For each h$H, let the feasible range
of the variable x^ be given by the interval [a,,, b^] and let
53
us choose a set of n^j grid points a, (r = l, 2, . . . . , n ) such
that,
It should be noted that the grid points need not be
equally spaced and that varying grid lengths can be used
for the different variables. Let us consider the grid
interval [2 , ,3 ^ ,J . Every point x , in this interval can be
expressed uniquely as,
^h = hrahr + h,r.ia|,,r.i (3.4.1)
where \^^ + A ^ .i = l
and \^,^Q . A ,,.i ^0
Each function fhi-^h'i ,h $ H can be approximated in the grid
interval i^hi > ^h,r*i^ by the linear approximation,
4(^h) = Ar-f/,(a/, ) +Vr*i-f (a;,,,.i) . . (3.4.2)
where x^ is given by (3.4.1). It is easily seen that, for
each h$H and any x,, e [afj,b^] , the entire piecewise linear
approximation -?/,(X/,) with breaks at the grid points can be
written as.
54
hi^,) =TV4(a,J ' «' ' S where
^h- 2^ ^hr "^hr r=l
r=»l
A/,2:0 /or r = l,2, ,12 .
provided, for each h, at most two adjacent X /s are
positive. This condition ensures that the linear
approximation occurs only between adjacent grid points.
Replacing the nonlinear function -f/,(y,) , hiH by their
piecewise linear approximation -?h( ) given by (3.4.2) in
the separable programming problem given by
(3.3.9)-(3.3.10), we obtain the following approximated
program in the variables A ^ {h$H) and Xf, (hEH) as,
Minimize Z = T f,(x^) ^ T'^X,, f, {a,,) (3.4.3) Sew hth r-i
55
Subject to,
Esr.(^J ^EEVsr.(a,,) ^v (3.4.4) heH htk r-l
n h
Y^X^ = 1 h€H (3.4.5) f ' l
X ^ 0 [foz r = 1,2, ,nf, ; h^H ) (3.4.6)
and the additional restriction, namely, for each h$H, no
more than two adjacent k^,^ can be positive, that is if X^r
is positive then only Xf,^r*i °^ "^n.r-i can be positive.
The approximated programming problem (3.4.3)-(3.4.6)
is a linear programming problem. Therefore the problem
(3.4.3)-(3.4.6) can be solved by the simplex method by
using the following restricted basis entry rule. A non
basic variable A.^ is introduced into the basis only if it
improves the value of the objective function and if, for
each h^H, the new basis has no more than two adjacent
/jr's that are positive. The optimal values
Xh ihEH) and Xl, ( i = 1,2, . . . ,n^ ; h$H ) obtained by
solving (3.4.3)-(3.4.6) yield an approximate optimal
solution Si to the problem (3 . 3 . 9) - (3 . 3 .11) , where the
components St^ of i? are given by,
56
j? = •
x^ if hEH
"txt .1, a^, if h€H r=l
It may be remarked here that the foregoing procedure
of solving approximated separable programming gaurantees an
approximate local maximum or minimum. It is only when the
functions fh(Xh) andg^lxj have the appropriate convexity
or concavity properties which assures us that a local
optimum is also a global optimum that we can find a global
optimum for the approximating problem, and hence an
approximated optimal solution. The use of finer grid
points improves the approximate solution but at the
expense of more computational efforts. For more accurate
solutions, it is usually advantageous to initially solve a
problem with a rather coarse grid points and then re-solve
it by using a finer grids only in the neighborhood of the
initial approximate solution.
3.5 A NUMERICAL EXAMPLE:
In a survey the population is stratified into
two strata (L=2) . The following information are available,
57
Stratum
1
2
Wh
0 . 4
0 . 6
Sn
10
20
Ch
$4
$9
tn
$5
$10
N,
200
300
The separable programming problem (3.3.9)-(3.3.11)
thus become
Minimize 4x^ + 9X3 + 5x^ + lOXj (3.5.1)
Subject to,
16 . 144 Xi X2
Si (3.5.2)
/J s X2 s v/IM (3.5.3)
It is clear from the constraint (3.5.2) that x, > 4 and
X2 > 12.
Let us take x^ ^ 4.1 and x^ k 12.1 which implies that
the ranges of the variables are,
4.1 s Xj ^ 14.1 , and
12.1 s X; <, 17.3
58
Let us use the grid points for x, as 4.1, 6.1, 8.1,
10.1, 12.1, 14.1, and that for x as 12.1, 14.1, 16.1,
and 17.3. Also a-, = 4.1, ai = 6.1, a,, = 8.1, a^ = 10.1,
a, = 12.1, and a., = 14.1. Similarly, a,-. = 12.1, ajz = 14.1,
a2, = 16.1, a;, = 17.3.
The piecewise linear approximation to the functions
f^ix^) = 9x| + 10x2
9,2i^2) = are
f i ( x j = 8 7 . 7 4 ^ 1 1 + 1 7 9 . 3 4 X 1 2 + 3 0 2 . 9 4 X 1 3
+ 4 5 8 5 4 X i 4 + 6 4 6 . 1 4 X i 5 + 8 6 5 . 7 4 X ^ 6
f 2 ( x 2 ) = 1 4 3 8 . 6 9 X 2 1 + 1 9 3 0 . 2 9 X 2 2 + 2 4 9 3 . 8 9 X 2 3 + 2 8 6 6 . 6 I X 2 4
g i i ( X i ) = 0 . 9 5 X 1 1 + 0 . 4 3 X 1 2 + 0 . 2 4 X 1 3
+ O . I 6 X 1 4 + 0 . 1 1 X 1 5 + O.O8X16
g i2(X2) = 0 . 9 8 X 2 1 + 0 -72X32 + 0 . 5 6 X 2 3 + 0 . 4 8 X 24
Thus the approximated separable program is,
Minimize ^iCxJ + 2( 2) (3.5.4)
Subject to,
gii(Xi) + ^12 (X2) i 1 (3.5.5)
^11 ^^12 ^^13 ^^14 ^^15 - 16 = 1 (3.5.6)
59
^21 ^^22 ^^23 ^^24 " ^ (3.5.7)
(3.5.8)
Introducing the slack variable X3 ^ 0 in the constraint
(3.5.5) and artificial variables X4 ^ 0, X5 ^ 0 in the
constraint (3.5.6) and (3.5.7).The objective function (3.5.4) is
minimized by Charne' s M-technicjue following the restricted basis
entry rule. The following are the simplex tableaus.
TABLE 3.5.1
Basic varia ble
X,
X4
X5
Cs
0
M
M
XB
1
1
1
\ .
.9
1
0
M-87 .7
^2
.43
1
0
M-179 .3
>,3
.24
1
0
M-30 2.
\ ,
.16
1
0
M-45 8.
A,,
.11
1
0
M-64 6.
^.
.08
1
0
M-86 5.
^ i
.93
0
1
M-14 38
\ 2
.72
0
1
M-19 30
\ .
.56
0
1
M-249 3.9
^,
.48
0
1
M-286 6.6
X,
1
0
0
0
X,
0
1
0
0
x
0
0
1
0
60
TABLE 3 . 5 . 2
Bas ic v a r i a
b l e
Xi
K
X5
CB
0
87 .74
M
XB
. 0
1
1
>,>
0
1
0
0
A,,
- . 5 2
1
0
- v e
A,3
- . 7 1
1
0
ve
A,,
- . 7 9
1
0
v e
^ .
- . 8 +
1
0
v e
\ .
- . 8 7
1
0
ve
A,,
. 9 8
0
1
M-14 38
\ ,
.12
0
1
M-19 30
\ .
. 5 6
0
1
M-24 93
\ ,
. 4 3
0
1
M-28 66
X j
1
0
0
0
Xr
0
0
1
0
TABLE 3 . 5 . 3
Bas ic varia
b l e
\ .
A:
X,
CB
143 8.63
87. 74
M
XB
.OF
1
.9$-
A,,
0
1
0
0
A,2
- . 5 3
1
. 5 3
. 5 3 M
A,3
- . 7 2
1
. 7 2
. 7 2M
A,.
».81
1
.81
. 8 IM
A,s
- . 8 6
1
. 8 6
. 8 6M
A,e
- . 8 9
1
. 8 9
. 8 9M
^ 1
1
0
0
0
^^
. 7 3
0
. 2 7
. 2 7M
A,,
. 5 7
0
. 4 3
.4 3M
A..
. 4 9
0
.51
. 5 IM
X3
1 . 0
0
- l . O Z
- v e
Xs
0
0
1
0
61
TABLE 3.5.4
Basic varia ble
\r
K
X,
CB
143 8.6
865 .74
M
XB
.94-
1
.Ofe
A,,
.8
1
- . 8 9
-ve
A,.
.36
1
-.36
-ve
^.
.17
' 1
-.17
-ve
A,,
.08
' 1
-.08
—ve
A,,
.03
1
-.03
-ve
A,.
0
1
0
0
A,,
1
0
0
0
\:
.73
0
.27
.2 7M
-880
\>
.57
0
.43
.43 M
-1673
A,,
.49
0
.51
.51M
-5161
•"<!
1.02
0
-1.02
-ve
X=r
0
0
1
0
In table 3.5.4, according to the usual simplex criteria, Aj4
should enter the next basis since the corresponding Z - C has the
most positive value (i.e. .51M-5161) and X5 should leave the basis
since the ratio (b3/a35) = .06/. 51= 0.11 is the smallest one. But
according to the restricted basis entry rule the two nonzero ^^ must
have adjacent subscript r. Hence \^ can not enter the enter the basis
as ^1 remains in the basis. The next most positive cost coefficient
corresponds to Aj, but it also can not enter since Aji remains in the
basis. Finally ^^ enters the basis and x leaves the basis. The
remaining simplex tableaus are
62
Thus j?i = £ X i ^ fii^ - 1 X 1 0 . 1 - 1 0 . 1 r- i
and j?2 = ^^2r ^2T ^ 0 - ^ S X 1 2 . 1 + 0 . 5 2 X 1 4 . 1 = 1 3 . 4 r » l
Therefore the required sample size for the two strata
are ri] = 102.01 « 102
and n = 172.6 = 173.
The minimum cost needed for the survey is C(nJ = 2147.
For better approximated result we may take finer grid points
in the neighbourhood of the current optimum. Let the new grid
points for Xi be 8.1, 9.1, 10.1, 11.1 and 12.1 and that for Xj be
12.1, 13.1 and 14.1. Approximating the nonlinear functions again
in the above grid points and applying the method mentioned above
we obtain the following result,
Xi = 9.1 and x = 13.35
Thus the required optimum sample size for the two strata are
n, = 82.8 « 83 and n = 178.22 « 178 with Ch (nj = 2114.
The number of variables increases rapidly if the number of
strata increases. But this problem can be handled easily on a
high speed digital computer. Since the restricted entry
conditions can be coded for a computer, it is possible to modify
64
TABLE 3.5.5
B a s i c v a n a
b l e
\ .
\ .
\ 2
CB
143 8 . 6 9
865 .74
193 0 . 2 9
XB
.7?
1
.22
^>
3 .3
1
- 3 . 3
- v e
\ .
1.33
1
- 1 . 3 3
32.5
A,,
. 6 3
1
- . 6 3
2 5 3
K
. 3
1
- . 3
260
\ s
.11
1
- . 1 1
166
K
0
1
0
0
\ .
1
0
0
0
\ .
0
0
1
0
\^
- . 5 9
0
1.59
- v e
K
- . 8 9
0
1.89
- v e
X,
3 . 7 8
0
- 3 . 7 8
- v e
TABLE 3.5.6
B a s i c v a n a
b l e
\ .
K
\ 2
CB
143 8 .69
458 .54
193 0 .29
XB
.43
1
.52
A,:
3
1
- 3
- ve
\ 2
1.03
1
- 1 . 0 3
- v e
^ 3
. 3 3
1
- . 3 3
- ve
K
0
1
0
0
\ .
- . l a
1
. 1 9
- v e
\ .
- . 3
1
. 3
- v e
\ ,
1
0
0
0
\z
0
0
1
0
\ .
- . 5 9
0
1.59
- v e
\ .
- . 8 9
0
1.89
- v e
X j
3.78
0
- 3 . 7 8
- v e
Since all the Z -C., in the table (3.5.6) are negative, the
optimum solution is obtained. The optimum solution is
and 21 = 0.48, A,22 = 0-52, 23=0, 24 = 0 and x-^^Q
63
a simplex code to solve the problem (3.4.3)-(3.4.6) with
restricted basis entry rule.
3.6 THE MULTIVARIATE CASE:
The separable programming technique discussed in section 3.4
can also be applied when there are multiple characters under
study.
Let the cost function in multivariate stratified sample for
p characters be defined as
L p L P
( />) = Eir^i^i^^ - EE'^^i v/^ (3.6.1) h=l J^ 7i=l j ^
where C j be the cost of measuring the jth character on a unit of
the hth stratum, and t ^ be the travelling cost.
Let V be the variance of the jth characteristic which is given
by
^^y^st) = t ^ ^ ~ ' J-1.2 p (3.6.2)
where S%,, denote the stratum variance in the hth stratum with
respect to the jth character.
65
The problem of optimum allocation of sample sizes to
different strata can be stated as,
L p L p
Minimize C(n,) = g JJ c j -" + g j^ ti y/ (3.6.3)
Subject to,
j . w l ^ ^ ^^ ^ j = l,2,...p (3.6.4)
2 ^ Uf, ^ Nf, (3.6.5;
With the substitution X^ - Jn^ the nonlinear programming problem
becomes
L p L p
Minimize C - V V) c . ^ +y) T) t ,-x (3.6.6)
Subject to,
L
YZU^^v. ; j = l,2, p (3.i.6) ^ ^l Slj
^ ^ ^ ^ ^ V ^ (3.^.7)
66
Both the objective function and the constraints are
separable function, hence we apply the separable prograiraning
technique, The nonlinear objective and constraint functions are
linearized by the method discussed in section 3.4. The
approximated linear programming problem is thus.
Minimize ^ = V) f . (x ) + T,^Krj ^hj i^hrj)
Subject to,
n J _ ^ ^i
heH
and the additional restrictions on \^^ as described elsewhere in
this chapter.
67
CHAPTER IV
COMPROMISE ALLOCATION USING
DYNAMIC PROGRAMMING
4.1 INTRODUCTION :
In multivariate surveys, the optimum allocation for
one variable will not in general be optimum for others,
therefore some compromise is needed. Different criteria of
compromise are suggested by various authors. Peters and
Bucher (undated) proposed a compromise allocation by
maximizing the average of the relative efficiencies as
compared to Neyman allocation for fixed total sample size.
Cochran (1963) used the average of the individual optimum
allocation for different characters as compromise
allocation. Chatterjee (1967) gave a compromise allocation
by minimizing the total proportional increase in the
variances due to the use of a non-optimum allocation for a
fixed budget. He assumed that the measurement cost in
various strata are independent of the characters measured.
68
In this chapter the problem of obtaining the
compromise allocation is formulated as a non-linear
programming problem and a solution is worked out by using
dynamic programming technique. The criterion for working
out the compromise allocation is same as Chatterjee (1967)
but no assumption about independence of the measurement
cost with respect to various characters has been made.
4.2 FORMULATION OF THE PROBLEM :
Let there be L strata of sizes Ni,N2 ,....,NL and
N ^h ~ ~S' •^='1/2, , ,L, denote the stratum weight of
the h-th stratum,
L
where N = ^ Nf, is the population size. Let p
characters be defined on every unit of the population. In
multivariate surveys generally the cost of measurement
differ from character to character. The total available
budget C may be distributed among various characters in
proportion to some measure of their importance. Thus if
C denote the amount allocated to measure the jth character
on sampled units, we have C - T^ ^j
69
and C- = E^^^i^- ^ - - ^ / ] • !
Thus C-T y c , , n , , (4.2.2) E E ^ i ^ i
where C,, = cost of measuring the jth character on a
unit of hth stratum
and n , = size of sample for jth character in the hth
stratum
If n*h, ,• h = 1,2, ,L, j = 1,2, ,p denote the
optimum allocation for a fixed budget Cj given by (4.2.1)
for measuring the j-th character then n'^ are given by
. C, ^. S,,//^ h=l,2 L
/ J = l
where Shj denote the stratum variance in the h-th stratum
with respect to the j-th character.
If V * denote the variance (ignoring the f .p.c) of the
stratified sample mean for the j-th character under optimum
allocation for fixed cost Cj then
{tw,s,,^^f Vj - _ (4.2.4)
70
The variance V (ignoring f.p.c) of the stratified
sample mean for the j-th character under a
non-optimal compromise allocation n, ; h = 1,2,....,L
is given by
^ Wf, , (4.2.5)
The relative increase in the variance for the j-th
character when a non-optimal compromise allocation is used
is given by
E. - -3- i = -J. - 1 (4.2.6) V* V*
(t:, "'• « v ^ ) " « v ^ «. S.J ^
J
or, Wl Slj c!
From (4.2.3) we get
= V. ' ( C, / (4.2.7)
71
Substituting the value of Wh S . given by (4.2.7) in
(4.2.5) we get,
r* L V- ^ C,, n,,-= J:1 y- ±^]2_!^ (4.2.8)
(4.2.6) and (4.2.8) together give
= A r ^J" •" ' - 1 q .
If C 's remain fixed that is if
= E C/,,. n,, ; J =1,2, ,p
(4.2.9)
E given in (4.2.9) may be expressed as
1 X^ C-M ,_. E, --tS^e'""-"»'
72
The total relative increase is
p i p r-
^ = Z: i E S T ^ «• - ^ ^ ) ' (4.2.10)
The problem of obtaining the compromise allocation can
thus be expressed as.
L p c Minimize E^VT—^ {n^^ - n^)^ (4.2.11)
A-l pl ^J^h
Subject to,
g Aj A Cj ; J=i.2, ,p . . (4.2.12)
HI, i 0 ; h = l,2, ,L (4.2.13)
However, using (4.2.9) the total relative increase in
the variances may also be expressed as
L p
/i-i i ^
, 2
^A
or E' = E.p = Tf; ^ ^ (4.2.,4)
73
As p, the number of characters, is a constant,
minimization of E' is equivalent to minimization of E. Now
L p
= Ej : E' = Cfti -n^j
/i=l ^h f ^ l ^j
^ (4.2.15) ^ ^^
-t ^^ ' ^-^'^' where C = V " ^ ^ ; i2-l,2, L
Thus :the problem (4.2.11) to (4.2.13) can be
expressed in a relatively simpler form as
L C Minimize E ^ V — (4.2.16)
Subject to,
^C7^i ^n ^ Cj ; j=l,2, p (4.2.17)
Hu i 0 (4.2.18)
74
A compromise allocation is advisable only if the
individual n^/'s do not differ very much with respect
to different characters. Otherwise it will result in
large relative increases E . If the costs C ;
j=l,2, ,p are roughly in proportion to the various
individual costs C , (which is a most practical situation)
the variations in nh/ due to variable Chj will not be
very much and if other factors permit, a compromise
allocation may be used. C can be made roughly proportional
to Ch-, by keeping it proportional to L, Cf,j, that is to the /j=i
total cost of measuring the j-th character on one unit of
each stratum.
In fact when C are allocated according to the above
criterion it may happen that only a fewer number of
constraints out of the p constraints given in (4.2.17)
may remain effective and we then have a much easier
problem to solve. The k-th constraint will become
ineffective (redundant) if there exist a q-th constraint
such that
C C —^ ^ -p^ for all h=l,2, . . . . ,L ^hk ^kq
The problem (4.2.11) to (4.2.13) is a nonlinear
programming problem with a strictly convex objective
75
function and linear constraints. When the values of C ,
C and nf,/ are known the problem can besolved by using an
appropriate convex programming technique. Kokan and
Khan(1967) gave an analytical solution for such problems.
In the next section a solution procedure using dynamic
programming technique is presented to solve the above
nonlinear programming problem given in (4.2.16) - (4.2.18).
A graphical solution of the problem is also provided in
section 3.7 for L=2 and p=2 as an illustration.
4.3 THE DYNAMIC PROGRAMMING APPROACH :
Dynamic programming basically takes a multi-stage
decision process containing many interdependent variables
and converts it into a form that can be optimized more
easily using the standard optimization methods. The
technique has been used quite extensively and successfully
in solving a variety of problems that permit its use. For
example, Pnevmaticos and Mann (1972) used dynamic
programming in the determination of an optimum tree bucking
policy. Omule and Williams (1982) used it to determine an
optimum partial replacement policy for sampling on
successive occasions. In stratified sampling the variance
function of the mean is additive and the sampling cost
functions usually adopted are also additive in nature (with
respect to stratum sample size). This means that the
76
T^54. separability and monotonocity conditions of the objective
and constraint functions which are necessary for the
decomposition of an L-stage problem into L single stage
problems (Namhauser (1966)) are always met in stratified
sampling. The nonlinear programming problem (4.2.16)
(4.2.18) can be considered as an L- stage dynamic
programming problem with L decision variables n,, n
, ,nL and p constraints. The L decision variables are
to be determined at stage h and the amount of the
available budget C are to be determined before allocating
them to any particular stage. For example when the
value of the first decision variable n, is determined at
stage l, there must be sufficient amounts C for
allocation to activity 1 i.e. ni .The amount remaining
after allocation to activity 1 is to be determined before
the value of n2 is determined at stage 2, and so on. Thus
the amount remaining for allocation must be known
before making a decision at any stage of the L-stage
system.
Let us define a subproblem of (4.2.16) - (4.2.18)
involving only the first k strata as
Minimize * C
S ^ (^-^-i)
77
Subject to,
k
T Cf^^n^^y^, ; j-1,2 p (4.3.2)
n^iQ;h = 1.2, k (4.3.3)
where y^'s, j=l,2,....,p are the available amount for
allocation to k strata.
Let -fj (YI'Y2' • • • •' Yp) t*® the optimum value of the
objective function using only the first k strata then
^A(YI'Y2' • "Yp) = min [ E -Jl I
Subject to,
k
y, Chj n^ ^ Yj ; J = 1.2, . .,p
n^ ^ 0 ; h=l,2, . . . ,k
Now, r;{Yi,Y2'• • •'Yp) = min ^ ^k h "/> J
78
Subject to , k-l
^ Cf,x n^^^y^ -
k-X
E ^ « A ^ Y2 -
- <^ki n^
- Ck2 n„ h-l
k-1
E /'P * Yp - C^ n^ h'l
n. ^ 0 ; h=l,2,..,(k-l)
For a fixed value of n,,,
Yi Y2 Yp
'fcp
and
C c ic*(Yi/Y2' •wYp)=-^+ {min J] -
J c - 1
Subject to E ^ h i -"/J Yj- - c'jtj- jc ; J=i/
n^ 0 h=l,2, . .,ic-l }
/P (4.3.4)
The recurrence relationship of dynamic programming
technique when applied to the above nonlinear programming
problem yields,
^;(Yi.Y2.-wYp)= min [ -^ + f^_^(y^ - c,, n„
Y 2 <^A-2 •' A:/ ' Yp (^icp ^k^ (4.3.5)
79
where &^ indicates the maximum value that n can take without
violating any of the constraints stated in the problem. The
value of S|j is obtained by,
since any value of nk greater than S would violate at
least one constraints . Thus at the k-th stage the
optimum values n;,* and f/ can be determined as functions of
Yi < Y2' • • • / Yp •
Finally at the L-th stage for the L-th strata,
since the values of Yi / Y2 < • • • < Yp ^^ known to be Ci, C2, . . ., Cp
respectively we can determine nL* and f j*. Once nL* is
known, the remaining values UL-X, 11^-2, . . . . ,n^ can be
determined by retracing the sub optimization steps. The
following numerical example will illustrate the details of
the dynamic programming procedure.
4.4 A NUMERICAL EXAMPLE :
In a two variate survey {p=2) the population is
stratified into two strata (L=2). The following information
is available;
80
Stratum h
1
2
Wh
0 . 4
0 . 6
Shi
4 . 6
2 . 8
Sh2
3 3 2
1 7 3
( ,•)) =
'A 20
^ 5 22
:q) ) = (400 2000)
Using (4.2.3) n j* are worked out as
(in;;,-}} 48 55 38 41
The nonlinear programming problem, given by (4.2.16)
(4.2.18) thus becomes,
„. . . 53.645 ^ 37.885 , ^ ,, Minimize + (4.4.1
Subject to,
4/2j + 5/22 :£ 4 0 0
20n^ + 2 2 n 2 ^ 2 0 0 0 ' ( 4 . 4 . 2 )
a n d rij^, n^ ^ 0 ( 4 . 4 . 3 )
Since h=2 and j=2, this problem can be considered as a
two-stage dynamic programming problem with two state
parameters . The first stage problem for k=l is to find the
81
minimum value of f (Yi/Y?) ^ s ,
• f i ( Y i ' Y 2 ) = n i i n 5 3 . 6 4 5
Qi n^ sPi 17-^
where Yi and Y2 a re t h e c o s t s a v a i l a b l e for a l l o c a t i o n a t
s t age 1 for 1s t s t r a t a and ni i s a non-negat ive va lue which
s a t i s f i e s the s i d e c o n s t r a i n t s 4ni^Yi and 20n;^Y2- Here
Yi = 400 - 5^2 and Y2 = 2000 - 22/23 •
Hence the maximum v a l u e ft t h a t ni can assume i s g iven by
P^ = Hi = min ^11 ^ 1 2
= mm 400-5/2, 2000-22/2,
20 ( 4 . 4 . 4 )
Thus, fi(Yi'Y2) = A* , r 400-5/2, 2000-22/2,
4
53 .645
20
400-5/22 2000-22/22 m m 20
53.645 *
n-i
The second stage problem i.e. for k:=2 is to find the
minimum value f2 as,
^•2(YI/Y2)= f"ii, Oi n, ib
c, », 400-5/2, 2000-22/2, v /2. M 4 20 I
where Yi and Y2 are the costs available for allocation at
stage 2, which are equal to 400 and 2000 respectively. The
maximum value that n^ can assume without violating the
constraints is given by
82
p2 = min [
= mi " [
Yi T2 1
" - 2 1 ^-22 "
400 2000 5 ' 22
= min [ 8 0 , 9 0 . 9 1 ]
= 80
Thus t h e r e c u r r e n c e r e l a t i o n s h i p ( 4 . 3 . 5 ) can be r e s t a t e d
as
Wi/2 2* (400, 2000) = min Oi Oj s80
3 7 . 8 8 5 5 3 . 6 4 5 77, . ,400-5/2, 2000-22/2,>
mxn( ^ ^ ^ )
Since mm Oi n, S80
4 0 0 - 5 / 2 , 2 0 0 0 - 2 2 / 2 ,
20
400-5/22
4 ; 0 ^ / 2 , ^80
Therefore, min Oi n, i80
3 7 . 8 8 5 ^ 5 3 . 6 4 5 / 2 , 4 0 0 - 5 / 2 ,
min Os ^2 s80
37 . M i + 214 . ^^ 72, 400 -5 /2 2 J
= 2 .018068 a t /22 = 3 8 . 8
Thus n / = 38 .8 and f / ( 4 0 0 , 2 0 0 0 ) = 2 .018068
83
From (4.4.4) ,
400-5/2, 2000-22/3, Til - min
20
= min [ 51.5,57.32 ]
= 51.5.
Thus the optimum solution of the problem is given by
n/ = 51.5, n/ =38.8 and f* = 2.018068.
4.5 GRAPHICAL ILLUSTRATION :
A graphical solution will be represented in this section
for the problem solved by dynamic programming method in
section 4.4.
1000 __ 100 ^^^ 1000 ^ 100 20 4 22 5
the constraint 20ni + 22n2 2000 remain ineffective and we
have to solve the nonlinear programming problem (4.4.1)-
4.4.3) with only 4ni + Snj ^ 400 as a constraint. From the
figure 4.5.1 it is clear that at the optimal point the
constraint 4ni+5n2: 400 will be active and the optimal
solution will be the point of contact of the rectangular
hyperbola represented by the objective function and the
84
straight line represented by 4n,+5n^=400. This point of
contact is 51.5 and 38.74. The optimum values of n^ and n
when rounded off to the nearest integer value is n]=52 &
n. = 39. It can also be seen that this solution satisfies
Kuhn-Tucker(1951) necessary conditions (which are also
sufficient for the problem). The maximum relative increase
in using this compromise allocation of n;' =52, n * = 39,
is for j=2 and is given by —^ - 1 is only 6.5%. V2
100
80
60
40
20
O 20 40 60 80 100 120
FIG. 1
85
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