AN APPLICATION OF GEOMETRIC PROGRAMMING TO AN INVENTORY

PROBLEM

BY

UMOH, EMMANUEL ALPHONSUS

PG/M.Sc/05/40209

BEING A PROJECT SUBMITTED IN PARTIAL

FULFILLMENT OF THE REQUIREMENT FOR THE

AWARD OF THE DEGREE OF MASTER OF SCIENCE

(M.SC.) STATISTICS

DEPARTMENT OF STATISTICS UNIVERSITY OF NIGERIA

NSUKA

SEPTEMBER, 2008

CERTIFICATION

The work embodied in this project report is original and has not

been in substance for any other degree of this University or other

Universities.

SUPERVISOR:………………………………….

Mr. W. I. E. Chukwu

Department of Statistics, UNN.

SUPERVISOR:………………………………….

Dr. F. I. Ugwuowo

Department of Statistics, UNN.

………………………………….

EXTERNAL EXAMINER

DEDICATION

This project is dedicated to all members of my family.

ACKNOWLEDGMENTS

I am highly indebted to many of my lecturers especially my project

supervisor (Assoc. Prof.) W. I. E. Chukwu whose constructively criticism,

encouragements and corrections have made this work a success. I am

also indebted to my lovely wife (Unyime) and my dear daughter

(Edidiong-Abasi) whose prayers and encouragements have seen me

through this programme.

To my father (late) and dear mother Mrs. Grace A. Umoh, my

brother Peter and my sisters Mrs. Akpan, Veronica, Esther and Felicia, I

owe you people a million of thanks for your daily prayers and the show of

understanding during this period.

Also worthy of commendation are my good friends, the post-

graduate students of the University of Nigeria, Nsukka and Akwa Ibom

State Post-graduate students Association for their cooperation to the

actualization of this work.

ABSTRACT

In this research work, an Economic Production Quality (EPQ)

model with flexibility and reliability consideration of production process

and demand dependent unit production cost was considered. The model

involves one storage space constraint. This inventory problem was

converted into a geometric Programming problem (GPP) because the

problem was a non linear programming problem. The inventory problem

was solved by both Geometric programming and Lagrange Multiplier

techniques. The results of both solution techniques were compared

using the total average cost to find out which one gives a better result.

From the results obtained, it was discovered that geometric

Programming technique gives a better result than the Lagrange

Multiplier technique.

TABLE OF CONTENTS

Title Page … … … … … … … …

Certification … … … … … … …

Dedication … … … … … … … …

Acknowledgment … … … … … … …

Abstract … … … … … … … …

Table of contents … … … … … … …

CHAPTER ONE

Introduction … … … … … … …

1.1 Statement of Problem … … … … …

1.2 Objectives … … … … … … …

1.3 Definition of Terms … … … … …

CHAPTER TWO

Introduction … … … … … … …

2.1 Review of Literature … … … … …

METHODOLOGY

3.1 Optimization of Posynomial … … … …

3.2 The Role of Arithmetic-Geometric Mean Inequality …

3.3 Definition and Theorem … … … … …

3.4 The Inventory Model … … … … … …

3.5 Notations … … … … … … … …

3.6 Assumptions … … … … … … …

3.7 Advantages of Employing Duality Theory

in geometric Programming … … … … …

CHAPTER FOUR

NUMERICAL DEMONSTRATION OF RESULT

4.1 Introduction … … … … … … …

4.2 Numerical Example … … … … … …

CHAPTER FIVE

CONCLUSION

5.1 Summary and Conclusion … … … … …

5.2 Bibliography … … … … … … …

BACKGROUND OF THE STUDY

1.0 INTRODUCTION

Decision makers from a wide variety of job classifications are

concerned with making optimal decisions. By making optimal decisions,

we mean choosing the “best” decision from a number of feasible

alternatives based on some measure of effectiveness such as profits,

cost, time or volume. Obviously, if there is not feasible alternative to

consider, there is no decision to make.

Often, it is possible to build mathematical models which portray

optimization problems and aid in their solutions. Such models normally

consist of a set of decision variables and mathematical representation of

a measure of effectiveness (called the objective function) and a set of

constraints that the decision variables must satisfy. The optimization

problem then is to determine that particular set of decision variables

which optimize the objective functions while simultaneously satisfying all

the constraints. This special set of variable values yields the optimal

decision for the problem at hand.

Mathematical programming is a branch of applied mathematics

that is designed to solve optimization problems. Mathematical

programming can be partitioned into two main areas: Linear

Programming and Non-Linear programming. Linear Programming is

concerned with determining optimal decision variables for problems in

which the objective functions and all the constraints are linear.

Applications of linear programming techniques are abundant throughout

both industries and government.

Non-linear programming is concerned with finding optimal

solutions in those cases in which the objective functions and/or any of

the constraints are non-linear functions of the decision variables. Non-

linear optimization models like linear models are abundant. Many

classes of non-linear optimization models have been singled out due to

special mathematical structure.

In this research work, a relatively general branch of non-linear

programming technique called “Geometric Programming” has been

explored. Geometric programming originated in 1961. Zener (1961)

discovered an Ingenious method of designing equipment at minimum

total cost – a method that is applicable when the component capital

costs and operating costs can be expressed in terms of the design

variables through a certain type of generalized polynomial (one whose

exponents need not be positive integers only). Geometric Programming

(GP) is an optimization technique developed for solving a class of non-

linear optimization problems with some useful theoretical and

computational properties. These optimization problems are not convex in

their natural form, they can however, be transformed into convex

optimization problems by a change of variables and a transformation of

the objective and constraint functions.

Fein (1961) in a seminar paper observed that some engineering

design problems can be formulated as optimization of generalized

polynomials and that if the number of terms exceed the number of

variables by one, the optimal design can be found by solving a system of

linear equations. Geometric programming (GP) was tied with convex

optimization and Lagrange duality and extended to include more general

formulations beyond polynomials.

Geometric programming (GP) addresses optimization programmes

where the objective functions are sums of monomials. The sum of

monomials with positive signs is called “Polynomials”. If some of the

monomials enter the sum with negative sign; the collection is called

“Signomials”. The term Geometric programming (GP) was adopted

because of the crucial role that the arithmetic mean inequality played in

the initial development. The early works by Zener et al (1963) in

geometric programming was for most part, concerned with minimizing

posynomial functions subject to inequality constraint on such functions.

Thus, the name posynomial programming might well have been chosen

instead if geometric programming.

Peterson (1978) in a reviewed article stated that most of the

applications of geometric programming have not been in the context of

the general theory but have instead involved posynomial programs or

slightly more general class of signomial programs. Applications of

geometric programming span many of the classical subsets of non-linear

programming. Geometric programming theory also suggests very

powerful computational techniques which make its study an interesting

one for any one concerned with non-linear optimization.

Specifically, geometric programming is designed to minimize

constrained generalized posynomials where the posynomial coefficients

are required to be positive and the constraints (also posynomials) are

bounded above by unity. These special polynomial functions, which are

called posynomials, are generally given by:

Many engineering design problems as well as problems from

business and economics can be modeled as constrained posynomials

and solved using geometric programming approach. The geometric

programming solution technique for these constrained posynomial

problems consist of defining and solving an associated problem called

the “dual” program. The optimal solution to the original problem (called

the primal program) is then easily computed from the optimal solution of

the dual program using the relationship provided by the duality theory.

Optimization is performed on the dual program rather than the primal

because the dual is always a concave maximization problem constrained

by linear constraints.

The Primal problem on the other hand, is usually considerably

“more non-linear” than its associated dual problem. Geometric

programming (GP) in standard form is apparently non convex

optimization problem; it can be readily turned into a convex optimization

problem by a logarithmic change of the variables and multiplicative

constants. Hence, the local optimum is also a global optimum and the

duality gap is zero under a mild condition. In general, local minima for

signomial problems are not global minima. It is interesting to note that, in

the special case where all the posynomial functions are in standard form,

they are simply monomials, and then the application of geometric

programming in convex form reduces to a linear programming. Hence,

geometric programming can be viewed as an extension of linear

programming. Zener (1963) used the result called “Cauchy’s arithmetic-

geometric inequality” to show that the arithmetic mean of a group terms

always was greater than or equal to the geometric mean of the group. It

is for this reason, that the name “Geometric Programming (GP)” was

used to describe the class of non-linear optimization problems.

Park (1987) examined the Economic Order Quantity (EOQ)

formula in the fuzzy set theoretic approach stating that in inventory

problems, geometric programming has not been much used as a

solution technique.

The determination of the most cost-effective production quantity is

commonly known as classical Economic production Quantity (EOQ)

model.

Cheng 1989) formulated unconstrained single item economic

production quantity (EOQ) problems with an inventory idea and solve by

geometric programming technique. Hwang et al (1993) also developed a

multi-product economic lot-size model with investment costs for set-up

reduction and quality improvement and solve by geometric programming

technique.

Hariri et al (1997) described a multi-item production lot-size

inventory model with varying order cost under a restriction and solve by

geometric programming technique. We have also considered applying

geometric programming technique to an Economic production Quantity

(EOQ) model with a capacity constraint developed by S. Islam and K.

Roy (2005).

Finally, comparison will be made with the results obtained using

the classical Langrage Multiplier method in solving the same problem to

see which one is more optimal.

Geometric programming (GP) unlike competing analytical

methods, which require the solution of a system of non-linear equations

derived from the differential calculations, this method requires the

solution of a system of linear equations derived from both the differential

calculus and certain ingenious transformations. Unlike competing

numeric methods, which minimize the total cost by either “direct search”

of “steepest descent” or “the Newton-Raphson method” (or one of their

numerous descendants), this methods, geometric programming provides

formulas that show how the minimum total cost and associated optimal

design depend on the design parameters. It has been noted that most, if

not all, equipment-component volumes are posynomials or signomial

functions of their various geometric programming dimension.

1.1 STATEMENT OF PROBLEM

There are some methods of solving non-linear programming

problems but still the best is yet to be achieved. In this work, we want to

apply Geometric programming in solving an inventory problem to see

whether a better solution can be obtained when compared to other

existing solution techniques (Langrage Multiplier).

1.2 OBJECTIVES

This research seeks to achieve the following objectives:

1. To convert an inventory problem into a Geometric Programming

Problem.

2. To solve this inventory problem using Geometric Programming

technique.

3. Solving this inventory problem using the usual method of solving

inventory problem (Langrage Multiplier technique).

4. To compare the results (average total cost) obtained from the two

solution techniques to check which method or technique gives a

better solution.

1.3 DEFINITION OF TERMS

POLYNOMIAL: When the power of the independent random variable is

greater then one with positive or negative coefficients.

POSYNOMIAL: Here the power (exponents) of the independent random

variable can be negative, fraction and even positive integer but with

positive coefficients.

MONOMIAL: Each term in the posynomial function is called monomial or

a posynomial with one term is called monomial as illustrated in page 5.

CHAPTER TWO

REVIEW OF RELATED LITERATURE

2.1 INTRODUCTION

Some authors and researchers have made series of investigations

and researches about giving solutions to some existing non-linear

optimization problems while others have also attempted designing

problem that could be solved using geometric programming approach.

Some areas of application as reviewed by some authors include:

Mechanical and civil engineering, chemical engineering, water resources

engineering, probability and statistics, finance and economics, control

theory, inventory analysis, business management, communication

systems, circuit design, information theory, coding and signal

processing, marketing mix problems, biotechnological systems, wireless

networking etc. This is evidence in some scholarly publications in the

form of textbooks, journals, articles etc.

Wilde et al (1967) developed a theory for negative coefficients and

inequality constraints using Langrage methods. The result of this

research showed that geometric programming is now applicable to a

polynomial economic model with polynomial constraints as equalities

and inequalities.

Duffin et al (1967) extended this ”geometric programming duality”

and associated methodology to the minimization of generalized

polynomials.

In essence, that development provided a non-linear generalization

of “linear programming duality”, one that is frequently applicable to the

optimal design of sophisticated equipment and complicated systems

(such as motors, transformers, generators, heat exchangers etc).

Hariri et al (1997) described a multi-item production lot-size

inventory model with varying order cost under a restriction and solved by

geometric programming technique.

Mendal et al (2005) described a multi-objective fuzzy inventory

model with three constraints and solving this by geometric programming

approach.

Thomas et al (1986) reported that the result for batch and semi

continuous optimization showed that the dual problem can be solved

more readily than the primal problem using the reduced gradient

multidimensional search technique.

Cheng (1989) formulated unconstrained single item economic

production quantity (EOQ) problems with an inventory idea and solve by

geometric programming technique.

Duffin et al (1967) used a generalization of the weighted

arithmetic-geometric mean inequality to obtain lower bounds on the

minimum value for posynomial program.

Das et al (2000) studied a multi-item inventory model with quantity

of dependent inventory costs and demand-dependent unit cost under

precise/imprecise objective and restrictions by geometric programming

(GP) approach.

Woolsey et al (1975) discussed the rules of using geometric

programming as a solution technique to ease the learning process for

readers.

Duffin (1966) recognized that minimizing the “dual function” is

similar to minimizing the “primal function. The dual expression is linear

and much easier to solve than the primal expression.

Cao (2006) showed that geometric programming (GP) can be

applied to models that dispose a complicated optimization problem while

in search of an optimum economical radius for power supply in a

substation.

Park (1987) examined the economic order quantity (EOQ) formula

in the fuzzy set theoretic approach with cost data stating that in solving

inventory problems geometric programming has not been much used as

a solution technique.

Duffin et al used the arithmetic-harmonic mean inequality to

develop an iterative process for solving signomial problems.

Sommer (1981) applied the fuzzy concept to an inventory and

production-scheduling problem.

Dinkel et al (1974) compared the Avriel-Williams method that uses

the arithmetic-geometric mean with the method suggested by Duffin and

Peterson which uses the arithmetic-harmonic mean inequality and

reported that the computational effort for Avriel-Williams (arithmetic-

geometric mean) appears to be less than that of Duffin and Peterson

(arithmetic-geometric mean).

Hwang et al (1993) developed a multi-product economic lot-size

models with investment costs for set-up reduction and quality

improvement and solve by geometric programming technique.

Jefferson et al (1998) stated that the key feature of posynomial

geometric programming is that it deals with problems that are either

convex or can be transformed into convex problems. This leads to the

important result that any local maximum for the problem is also a global

maximum. Geometric programming now furnishes a powerful set of

techniques, provided that the structure of the problem meets the

convexity requirements.

Rockafellar (1970) studied a class of convex programs whose

duals are linearly constrained. The dual of posynomial program is such a

program. Consequence of this equivalence is the computational

methods based on condensed posynomials that can be viewed on the

familiar framework of linearizing convex functions.

Chiang (2004) on the application of geometric programming in

communication systems stated that geometric programming (GP) in

standard form can be used to efficiently optimize network resources

allocations for non-linear objectives under non-linear Quality of Services

(QoS) constraints. He also stated that the key idea is that resources are

often allocated proportional to some parameters; and when resource

allocations are optimized over these parameters, we are maximizing an

inverted posynomial subject to lower bounds on other inverted

posynomials, which are equivalent to geometric programming (GP) in

standard form.

Islam et al (2005) developed an Economic Production Quantity

(EPQ) model with flexibility and reliability consideration and demand

dependent unit production cost under a space constraint. They also

designed a problem using this model and solved by Newton-Raphson

method. From this solution technique, the average total cost (TC*) was

found to be $145.25.

CHAPTER THREE

THEORETICAL METHODOLOGY

3.1 OPTIMIZATION OF POSYNOMIALS

A posynomial is a polynomial whose terms are all positive. In

general form, a posynomial can be written as:

Where Ci’s are the positive

3.2

CHAPTER FOUR

NUMERICAL DEMONSTRATION OF RESULTS

INTRODUCTION

In this chapter, we intend to apply the geometric programming

approach to the model discussed in the last chapter using the illustrative

example below to obtain the optimal solution of the decision variables.

The result obtain (average total cost) will be compared with the result

obtained using our usual method of solving inventory (Langrage

multiplier method).

NUMERICAL EXAMPLE

A manufacturing company produces a machine. It is given that the

inventory carrying cost of the machine is $10.5 per unit year. The

production cost of the machine varies inversely with the demand. From

the past experience, the production cost of the machine is 12000D-3.6,

where D is the demand rate. The total cost of interest and depreciation

per production cycle is 1500S -1.6r, where S and r are set-up cost per

batch and production process reliability respectively. Storage space area

per unit time (W1) and total storage space area (w) are 10sq.m. and

2000sq.m respectively. Determine the demand rate (D), set-up cost (s),

production quantity (q), production process reliability (r), and optimum

total average cost (TC) of the production system.

CHAPTER FIVE

5.0 CONCLUSION

5.1 SUMMARY AND CONCLUSION

In this research, have discussed “Geometric Programming” as an

optimization technique used in solving non-linear programming

problems. This optimization technique “Geometric Programming” could

be seen as a posynomial or function. By posynomial, we mean the

combination of positive and polynomial. This means that the coefficient

of the posynomial functions are all positive.

We have also applied geometric programming technique to an

already developed inventory model (an economic production quantity

(EPQ) model) by Islam et al (2005). This inventory model has been

converted into an optimization problem where we apply both Lagrange

Multiplier and geometric programming techniques to determine the

optimal values of the decision variables of the model. This model

involves one capacity constraint. Assumptions of the model were also

stated.

However, the results of these two techniques were compared

using the average total cost. Geometric programming technique gave an

optimum average total cost (TC*) of $90.97 while Lagrange Multiplier

technique gave an optimum average total cost (TC*) of $134.54 which

showed that Geometric Programming technique gives a better result

than the Lagrange Multiplier technique.

Finally, the technique presented here is quite general and can be

applied to the model in other areas like structural optimization etc.

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