OPTIMIZATION OF BAGASSE ASH CONTENT IN CEMENT-STABILIZED · PDF fileOPTIMIZATION OF BAGASSE...

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OKONKWO, UGOCHUKWU NNATUANYA

PG/Ph. D/09/52059

OPTIMIZATION OF BAGASSE ASH CONTENT IN CEMENT- STABILIZED LATERITIC SOIL

FACULTY OF ENGINEERING

DEPARTMENT OF CIVIL ENGINEERING

Paul Okeke

Digitally Signed by

Name

DN : CN = Webmaster’s name

O= University of Nigeri

OU = Innovation Centre

OKONKWO, UGOCHUKWU NNATUANYA

PG/Ph. D/09/52059

OPTIMIZATION OF BAGASSE ASH CONTENT IN STABILIZED LATERITIC SOIL

FACULTY OF ENGINEERING

EPARTMENT OF CIVIL ENGINEERING

Digitally Signed by: Content manager’s

Webmaster’s name

O= University of Nigeria, Nsukka

OU = Innovation Centre

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By

OKONKWO, UGOCHUKWU NNATUANYA PG/Ph. D/09/52059

FACULTY OF ENGINEERING DEPARTMENT OF CIVIL ENGINEERING

UNIVERSITY OF NIGERIA NSUKKA.

FEBRUARY, 2015

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By

OKONKWO, UGOCHUKWU NNATUANYA PG/Ph. D/09/52059

A THESIS SUBMITTED IN PARTIAL FULFILMENT OF THE REQUIREMENTS FOR THE AWARD OF DOCTOR OF

PHILOSOPHY (PhD) IN CIVIL ENGINEERING (GEOTECHNICAL ENGINEERING), DEPARTMENT OF CIVIL

ENGINEERING, UNIVERSITY OF NIGERIA, NSUKKA.

SUPERVISOR: ENGR.PROF. J. C. AGUNWAMBA

FEBRUARY, 2015

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CERTIFICATION THIS IS TO CERTIFY THAT OKONKWO, UGOCHUKWU N. A POSTGRADUATE STUDENT IN THE DEPARTMENT OF CIVIL ENGINEERING UNIVERSITY OF NIGERIA NSUKKA WITH REGISTRATION NUMBER PG/PhD/09/ 52059 HAS SATISFACTORILY COMPLETED THE REQUIREMENTS FOR THE AWARD OF DEGREE OF DOCTOR OF PHILOSOPHY (PhD) IN CIVIL ENGINEERNG. THE WORK EMBODIED IN THIS THESIS IS ORIGINAL AND HAS NOT BEEN SUBMITTED IN PART OR WHOLE FOR ANY OTHER DEGREE OR DIPLOMA OF THIS OR ANY OTHER UNIVERSITY.

ENGR.PROF. J. C. AGUNWAMBA DATE SUPERVISOR ENGR. PROF. O. O. UGWU DATE HEAD OF DEPARTMENT

ENGR. PROF. E. S. OBE DATE CHAIRMAN, FACULTY POSTGRADUATE COMMITTEE

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APPROVAL PAGE

OPTIMIZATION OF BAGASSE ASH CONTENT IN CEMENT-

STABILIZED LATERITIC SOIL

BY

OKONKWO UGOCHUKWU NNATUANYA

PG/Ph.D/09/52059

A THESIS SUBMITTED IN PARTIAL FULFILLMENTOF THE

REQUIREMENTS FOR THE AWARD OF THE DEGREE OF

DOCTOR OF PHILOSOPHY (Ph.D) IN CIVIL ENGINEERING,

UNIVERSITY OF NIGERIA, NSUKKA

JUNE, 2014

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OKONKWO, U. N. Signature: Date: (Student) ENGR.PROF. J.C. AGUNWAMBA Signature: Date: (Supervisor) External Examiner Signature: Date: ENGR.PROF. O.O. UGWU Signature: Date: (Head of Department) ENGR. PROF. E. S. OBE Signature: Date: Chairman, Faculty Postgraduate Committee

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DEDICATION

I dedicate this work to the Almighty God who has sustained me throughout this work.

He has never failed even in my weaknesses and will never fail me for ever.

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ACKNOWLEDGEMENTS

I wish to sincerely express my profound gratitude to my supervisor, academic

luminary and an erudite scholar; Engr. Prof. J. C. Agunwamba for his noble

encouragement and guidance throughout this work. I also wish to use this opportunity

to appreciate the other distinguished lecturers of the Civil Engineering Department for

their great contributions and suggestions for this work to be a success. My lovely

wife, Mrs. J. N. Okonkwo and precious daughter, Okonkwo Chinonyelum Awesome

are special gifts from Almighty God to me. They gave me understanding, support and

offered prayers for me even when I often deprived them of the fatherly role in the

course of pursuing this work. I will never fail to remember my father, late Pa Harford

Obiora Okonkwo (of blessed memory) who when he was alive continually

encouraged me at the beginning of this programme to press on in spite of the initial

challenges, may his gentle soul rest in the bosom of the Lord. I also wish to express

my indebtedness to my dear mother, Mrs. R. I. Okonkwo and my siblings for their

encouragement and prayers throughout this work. May the Almighty God who

rewards all good works, recompense all these in mighty folds.

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ABSTRACT The frequent rises in the price of cement and other binders have resulted in the escalation of the cost of construction, rehabilitation and maintenance of roads. One of the possible ways of cost reduction is to convert waste bagasse residue into ash and use it as a supplement/partial replacement for cement. Therefore this study is an attempt to optimize bagasse ash content in cement-stabilized lateritic soil for low-cost roads. The bagasse ash and lateritic soil were characterized by carrying out Atomic Absorption Spectrometer and soil preliminary tests as well as X-ray diffraction respectively. Compaction test, California bearing ratio, unconfined compressive strength and durability tests were carried out on the soil stabilized with 2%, 4%, 6% and 8% cement contents and bagasse ash ranging from 0% to 20% at 2% intervals; all percentages of the bagasse ash and cement were by the weight of dry soil. Cost analysis was carried out for the constituents of the stabilized material and a model was formed for cost evaluation. Also three regression models were developed that involved relationships of cost of bagasse ash, cement content, optimum moisture content, California bearing ratio and unconfined compressive strength at 7 days curing period. The three regression models were used to form a non-linear model which was linearized and solved with the simplex method including sensitivity analysis on the objective function and the constraints. Attempt was also made to apply Scheffe’s regression method from obtained results. It was observed that the increase in bagasse ash content increased the optimum moisture content but reduced maximum dry density. On the other hand higher bagasse ash tremendously improved the strength properties of the stabilized matrix. The optimum contents for bagasse ash, cement and optimum moisture content for an economic mix were 14.03%, 4.52% and 22.46% respectively at a cost of 39.50 kobo for stabilizing 100 grams of the lateritic soil as against 43.52 kobo for stabilizing with only cement.

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TABLE OF CONTENTS PAGE

TITLE PAGE i

CERTIFICATION ii

APPROVAL PAGE iii

DEDICATION iv

ACKNOWLEDGEMENTS v

ABSTRACT vi

TABLE OF CONTENT vii

LIST OF TABLES

LIST OF FIGURES

LIST OF NOTATIONS

CHAPTER ONE INTRODUCTION

1.1 Background of the Study 1

1.2 Statement of Problem 2

1.3 Aim and Objectives of the Study 3

1.4 Scope of the Study 3

1.5 Significance of the Study 4

CHAPTER TWO LITERATURE REVIEW

2.1 Definition of Laterite 6

2.1.1 Formation of Laterite 9

2.1.2 Mineralogical Composition of Laterite 12

2.1.3 Uses and Economic Relevance of Laterites 13

2.1.3.1 Building Blocks 13

2.1.3.2 Road Building 13

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2.1.3.3 Water Supply 14

2.1.3.4 Waste Water Treatment 14

2.1.3.5 Ores 15

2.2 Definition of Soil Stabilization 19

2.2.1 Techniques for Soil Stabilization 19

2.2.1.1 Stabilization by Compaction 20

2.2.1.2 Mechanical Stabilization 21

2.2.1.3 Stabilizing by the Use of Stabilizing Agents 23

2.3 Soil Stabilizing Agents Available 23

2.3.1 Primary Stabilizing Agent 23

2.3.1.1 Portland Cement 23

2.3.1.2 Lime 27

2.3.1.3 Bitumen 28

2.3.2 Secondary Stabilizing Agents 29

2.3.2.1 Blast Furnace Slag 29

2.3.2.2 Iron Fillings 30

2.3.2.3 Rice Husk Ash 30

2.3.2.4 Bagasse Ash 31

2.4 Mechanisms of Stabilization 32

2.5 Mathematical Modeling 33

2.5.1 Mathematical Model-Building Techniques 39

2.6 The Non-linear Programming Modeling 41

2.6.1 Monomial and Polynomial Functions 42

2.6.2 Previous Works on Optimization Techniques for Construction Materials 43

2.7 Classification of Soil 44

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2.7.1 AASHTO Soil Classification System 45

2.7.2 The Unified Classification System 46

CHAPTER THREE: METHODOLOGY

3.1 Introduction 49

3.2 Characterization of the Lateritic Soil 49

3.2.1 Moisture Content Determination 49

3.2.2 Liquid Limit 50

3.2.3 Plastic Limit 50

3.2.4 Linear Shrinkage 51

3.2.5 Particle Size Analysis 51

3.2.6 Identification of Clay Mineral 53

3.2.7 Classification of Soil 53

3.2.8 Compaction Test 53

3.2.9 Specific Gravity of Solids 54

3.2.10 California Bearing Ratio 55

3.2.11 Unconfined Compressive Strength 55

3.3 Characterization of Bagasse Ash 56

3.4 Test Requirements for the Stabilized Lateritic Soil 56

3.4.1 Unconfined Compressive Strength 56

3.4.2 California Bearing Ratio 57

3.4.3 Durability Tests 58

3.5 Method of Formulation of Non-linear Programming Model 58

3.5.1 Objective Function 58

3.5.2 Constraints 60

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3.6 Solution of Non-linear Programming Model 61

3.6.1 Sensitivity Analysis 63

3.7 Scheffe’s Simplex Regression Model 64

3.7.1 Determination of the Coefficients of the Polynomial Function 67

3.7.2 Validation of Optimization Models 69

CHAPTER FOUR RESULTS AND DISCUSSION

4.1 Presentation of Results 71

4.2 Soil Characterization 75

4.3 Characterization of Bagasse Ash 78

4.4 Stabilized Soil Tests 79

4.4.1 Compaction Characteristics 79

4.4.2 Strength Characteristics 81

CHAPTER FIVE MODELING AND OPTIMIZATION OF BAGASSE ASH

CONTENT

4.1 Cost Analysis for the Stabilized Matrix 86

5.1.1 Cement Cost 86

5.1.2 Projected Cost of Bagasse Ash 86

5.1.3 Cost of Water 87

5.1.4 Cost of Lateritic Soil 87

5.2 Regression Models 88

5.2.1 Calibration and Verification of Models 90

5.3 Non-linear Programming Model 92

5.3.1 Sensitivity Analysis 97

5.3.1.1 Sensitivity Analysis on Constraints 98

5.3.1.2 Sensitivity Analysis on Objective Function 103

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5.4 Application of Scheffe’s Simplex Regression Model 109

5.4.1 Determination of Densities of Materials 109

5.4.2 Formulation of Optimization Models 110

5.4.3 Validation and Verification of the Scheffe’s Optimization Models 112

CHAPTER SIX CONCLUSIONS AND RECOMMENDATIONS

6.1 Conclusion 118

6.2 Recommendations 119

REFERENCES

APPENDICES

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LIST OF TABLES PAGE

Table 2.1: Percentages of Main Elements in Laterites and their Corresponding Parent

Rocks 8

Table 2.2: Bogue’s Compounds 24

Table 2.3: Approximate Oxide Composition Limits of Ordinary Portland Cement 25

Table 2.4: AASHTO Soil Classification System 46

Table2.5: Unified Classification System 47

Table 3.1: Format for the Simplex Matrix 62

Table 4.1: Properties of Lateritic Soil 71

Table 4.2: Clay Minerals Characteristics and 2θ Angles at the Peak of X-ray

Diffraction of Soil Minerals 73

Table 4.3: Identification of Soil Minerals Using the Spacing of the Atomic Plane 74

Table 4.4: Properties of Bagasse Ash (Oxide Compositions and Specific Gravity) 78

Table 4.5: Percentage Losses in Unconfined Compressive Strength between 14 Days

Curing and 7 Days Curing + 7 Days Soaking 85

Table 5.1: Bagasse Ash Content and Corresponding Attached Cost 88

Table 5.2: Comparison of Predicted Results to Experimental Results 91

Table 5.4: Change in Constraint with Corresponding Change in Optimal Solution for

Constrained Equation (5.3) 98





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Table 5.7: Change in Coefficient of Unconfined Compressive Strength with

Corresponding Change in Optimal Solution 103

Table 5.8: Change in Coefficient of California Bearing Ratio with Corresponding

Change in Optimal Solution 105

Table 5.9: Change in Coefficient of Cement Content with Corresponding Change

in Optimal Solution 106

Table 5.10: Change in Coefficient of Optimum Moisture Content with Corresponding


Table 5.11: Mix Proportions in Mass with the Corresponding Response Function 110

Table 5.12: Mix Proportions in Volume with the Corresponding Pseudo Mixes 112

Table 5.13: Mix Proportions in Mass with Corresponding Response Functions for the

Validation of Scheffe’s Optimization Models 113

Table 5.14: Mix Proportions in Volume with the Corresponding Pseudo Mixes for the

Validation of Scheffe’s Optimization Models 113

Table 5.15: Statistical Student’s Two-Tailed T-test for Unconfined Compressive

Strength 114

Table 5.16: Statistical Student’s Two-Tailed T-test for California Bearing Ratio 114

Table 4.6: Variations of Maximum Dry Density with Increase in Bagasse Ash Content

at 2%, 4%, 6% and 8% Cement Content

Table 4.7: Variations of California Bearing Ratio with Increase in Bagasse Ash

Content at 2%, 4%, 6%, and 8% Cement Content

Table 4.5: Variations of Optimum Moisture Content with Increase in Bagasse Ash

Content at 2%, 4%, 6% and 8% Cement Contents

Table 4.6: Variations of Unconfined Compressive Strength and Age with Increase in Bagasse Ash Content at 2%, 4%, 6% and 8% Cement Content

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LIST OF FIGURES PAGE

Figure 4.1: Particle Size Curve 72

Figure 4.2: X-ray Diffractometer Chart for Soil Minerals 74

Figure 4.4: Variations of Optimum Moisture Content with Increase in Bagasse Ash

Content at 2%, 4%, 6% and 8% Cement Contents 79

Figure 4.5: Variations of Maximum Dry Density with Increase in Bagasse Ash


Figure 4.6: Variations of California Bearing Ratiowith Increase in Bagasse Ash


Figure 4.7: Variations of Unconfined Compressive Strength with Increase in Bagasse

Ash Content at 2%, 4%, 6% and 8% Cement Contents 82

Figure 5.1: Variations of Change in Constraint with Change in Optimal Solution for






Figure 5.4: Variations of Change in Coefficient of Unconfined Compressive Strength

with Change in Optimal Solution 104

Figure 5.5: Variations of Change in Coefficient of California Bearing Ratio with


Figure 5.6: Variations of Change in Coefficient of Cement Content with Change in

Optimal Solution 107

Figure 5.7: Variations of Change in Coefficient of Optimum Moisture Content with


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LIST OF NOTATIONS Ls Linear shrinkage

�1 Length of oven-dry sample

�� Initial length of specimen

� The percentage finer for any given size

� The specific gravity of the soil

ℛ Corrected hydrometer reading

ℳs Total mass of the soil

60 Particles with diameter 60% finer



Λ The wavelength of a parallel beam of X-rays

Angle parallel to the atomic planes

� Distance between parallel atomic planes

� Cost of bagasse ash in Kobo

Optimum Moisture Content in percentage

� California Bearing Ratio in Percentage

� Unconfined Compressive Strength for 7 days curing period in kN/m2

� Cement content in percentage

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CHAPTER ONE

INTRODUCTION

1.1 Background of the Study

Bagasse-ash is an agricultural material obtained after squeezing out the sweet juice in

sugarcane and incinerating the residue to ash. Bagasse is the fibrous residue obtained

from sugarcane after the extraction of sugar juice at sugarcane mills or sugar

producing factories (Osinubi and Stephen, 2005). The climatic and soil conditions

favourable for the production of sugarcane are present in the Northern part of Nigeria

and consequently, there is abundant production of it in the area. Sequel to the

foregoing is massive generation of sugarcane residue waste which constitutes disposal

problems and requires handling. There is yet no adequate awareness about the

usefulness of the sugarcane residue in the country, in other words very little value has

been attached to it. In some cases, the residue is being utilized as a primary fuel

source for sugar mills and also for paper production. However incinerating it to ash

and adopting it as admixture in stabilized soils because it has been found to be a good

pozzolana, adds to its economic value.

The major part of Nigeria is underlain by basement complex rocks, the weathering of

which had produced lateritic materials spread over most part of the area. It is virtually

impossible to execute any construction work in Nigeria without the use of lateritic soil

because they are virtually non-swelling (Osinubi, 1998a). The climatic and geological

position of Abia state with her alternating humid and dry periods enhanced the rich

deposition and formation of lateritic soils which have been very often utilized as fill

materials in road construction and other civil engineering works. These have shown

promising potentials in the lateritic soils for road pavements in the stabilized form and

prompted for more studies on them. In the past, several admixtures have been used on

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lateritic soil in the south east of Nigeria such as rice husk ash (Okafor and Okonkwo,

2009), and others. However, much has not been done with bagasse-ash on lateritic soil

in the region.

1.2 Statement of Problem

Roads in Nigeria have not received adequate attention with regards to maintenance

and even some rural areas are still inaccessible because of lack of motorable roads.

These roads are classified as Trunks A, B and C which implies that the roads are

managed and controlled by Federal, State and Local Authorities respectively.

However this has not paid off in ensuring that the roads are sufficiently maintained

and kept in good condition such that the users are not endangered in any way. In some

cases they are left in a very deplorable state or a point where maintenance by mere

cutting and patching bad portions cannot bring them to a satisfactory level to the users

but might require total re-building.

One of the plausible reasons for allowing roads to deteriorate so much is that the cost

of construction, maintenance and re-building has remained very high. The cost of

materials is a vital component in the total cost of road work. Thus, if it is substantially

reduced, the total cost of the road work would also be affected and consequently

becomes affordable. Therefore efforts should be geared towards harnessing the

natural potentials in the environment for use as construction materials to reduce the

cost of road work to the most possible minimum.

1.3 Aim and Objectives of the Study

This work used bagasse ash (sugar-cane residue ash) as an admixture in cement-

stabilized Ndoro Oboro lateritic soil for road construction works. The study is aimed

at optimizing bagasse ash content in the cement-stabilized soil.

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The objectives of this work were to:

i. Characterize bagasse ash and lateritic soil.

ii. Examine the effects of bagasse ash on the compaction and strength

characteristics of the cement-stabilized lateritic soil.

iii. Develop relationships comprising cost of bagasse ash content, cement content,

cement-stabilized lateritic soil compaction and strength characteristics.

iv. Calibrate and verify the model using experimental results.

v. Develop a non-linear programming model for predicting the optimum content

of bagasse ash.

vi. Optimize bagasse ash content in the cement stabilized lateritic soil and

compare results with unoptimized solution.

1.4 Scope of Study

Soils have peculiarities, they vary in properties. In other words, no two soils can be

similar in all properties but can behave alike in some cases. For example, peculiarities

of structure may play more important role in cement stabilization than the Atterberg

limits. Lateritic soils with the same and similar plasticity index may have completely

different behaviours in mixing operations (Osinubi, 1998b). Osinubi (1998b) equally

pointed out that one of the major problems confronting geotechnical engineers in the

tropics is the fact that most local soils are not amenable to standard pretest

preparations and testing procedures, resulting in variations of test results. These

variabilities have been discussed by Gidigasu (1988). However, the differences in

opinion are expressed over the understanding of engineering behaviour of residual

soils. According to Vaughan (1985), the development of classical concepts of soil

mechanics has been based largely on the investigation of sedimentary deposits of

unweathered soils. These concepts have been found to be inappropriate in describing

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the behaviour of residual soils and could lead to significant errors if inadvertently

applied. Gidigasu (1988) concludes that classical soil-mechanics principles have

failed in answering some of the geotechnical problems of some soils formed under

subtropical and tropical environments. In other words the results, model and

recommendations are only limited to the lateritic soil deposit in Ndoro in Ikwuano

local government area of Abia State, which is stabilized with ordinary Portland

cement as the binder and bagasse-ash as admixture. The tests that were carried out

include compaction, California bearing ratio, unconfined compressive strength and

durability tests which are the test requirements for stabilized materials.

1.5 Significance of the Study

Soil stabilization techniques for road construction are used in most parts of world

although the circumstances and reasons for resorting to stabilization vary

considerably. In industrialized, densely populated countries, the demand for

aggregates has come into sharp conflict between agricultural and environmental

interests. In less developed countries and in remote areas the availability of good

aggregates of consistent quality at economic prices may be limited. In either case

these factors produce an escalation in aggregate costs and maintenance costs. The

upgrading by stabilization of materials therefore emerges as an attractive proposition

(Sherwood, 1993).

The importance of cement stabilization of lateritic soils has been emphasized by

researchers with soil-cement mixtures being used as sub-base or base courses of low-

cost roads. However, excessive addition of cement becomes uneconomical; therefore

the cheap agricultural waste (bagasse ash) becomes a partial replacement/supplement

for the more expensive cement. Bagasse ash has been globally confirmed to be a good

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pozzolana because of its high silica content which indicates that there is a promising

potential in the agricultural material to serve as an admixture. This would be one of

the ways to guarantee the federal government’s efforts of meeting Millennium

Development Goals of providing low-cost roads. The availability of good road

networks becomes possible which would enhance the symbiotic relationship between

the urban and rural areas for economic development.

The trade-off between cost effectiveness and the strength characteristics of the

stabilized matrix resulting from the partial replacement/supplement of cement with

the bagasse ash for road work should be balanced. Instead of going through a rigorous

laboratory experiments with very many specimens in order to determine the optimum

content of bagasse ash, a predictive model could be developed using relatively fewer

observations. The model could also be useful in predicting other factors like the

compaction and strength characteristics with variation in bagasse ash and cement

contents.

Because of limited resources, there is a need to be very conscious not to be wasteful.

The process of mathematical modeling and prediction puts a check on how effective

limited field data are put to use in decision-making. In other words, it would be

beneficial to predict the optimum amount of bagasse ash required with a certain

amount of cement in the stabilized matrix to achieve the desired result with regards to

the compaction and strength characteristics at minimum cost.

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CHAPTER TWO

LITERATURE REVIEW

2.1 Definition of Laterite

The term laterite has been put into diverse usage and controversially defined, since it

was first coined by Buchanan (1807) from the latin word ‘later’ which means a brick

because it was easily moulded into brick-shaped blocks for building. It was originally

described as a ferruginous vesicular unstratified and porous material with yellow

ochre due to high iron content.

Joachin and Kandiah (1941) categorized laterite, lateritic and non-lateritic soil based

on their silica-sesquioxide ratios, which is represented by SiO2 / (Fe2O3+Al2O3). Ratio

less than 1.33 indicates laterites, those between 1.33 and 2.00 indicate lateritic soils

and above 2.00 indicate non-lateritic soils, which have also been tropically weathered.

A sesquioxide is an oxide with three atoms of oxygen and two metal atoms.

Another definition for laterite was proposed by Little (1969) as igneous rock

tropically, partially or totally weathered with a concentration of iron and aluminium

oxides (sesquioxides) at the expense of silica. Gidigasu (1976) grouped the various

definitions according to the soil-hardening properties, chemistry and morphology.

Madu (1977) while agreeing with the residual nature of the laterites, used the silica

sesquioxide ratio to divide eastern-Nigeria laterite into two main sub-genetic groups

of sandstone laterite and lateritic shales. And also, it was recorded that low iron oxide

content in lateritic shales and comparatively high content in the sandstone laterites

which was explained to be due to modes of formation of laterites.

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Ola (1978) did not agree with Joachin and Kandiah (1941) owing to its inconvenience

from an engineering point of view particularly where there is a lack of adequate

laboratory facilities. Therefore local terminology was adopted which defines lateritic

soils as all products of tropical weathering with red, reddish brown or dark brown

colour, with or without nodules or concretion and generally (but not exclusively)

found below hardened ferruginous crusts or hard pan.

According to Alexander and Cady (1962) laterite is a highly weathered material, rich

in secondary oxides of iron, aluminium or both. It is nearly void of bases and primary

silicates but it may contain large amount of quartz and kaolinite. It is either hard or

capable of hardening on exposure to wetting and drying. Osula (1984) modified the

definition to read “laterite is a highly weathered tropical soil, rich in secondary oxides

of any or a combination of iron, aluminium and manganese”. Manganese has been

reported as a predominant element in combination with iron in some varieties of

laterite, notably those in India (Rastal, 1941). Melfi (1985) defined lateritic soils as

soils belonging to horizon A and B of well-drained profiles kaolinite group and of

iron and aluminium hydrated oxides. Smith (1998) defined laterite as a residual soil

formed from limestone after the leaching out of solid rock material by rainwater to

leave behind the insoluble hydroxides of iron and aluminium.

Makasa (2004) stated that the degree of laterization is estimated by silica-sesquioxide

ratio (SiO2 / Fe2O3 + Al2O3). In later studies by Schellmann (2008), it was found that

intensive chemical decomposition of rocks is a wide spread phenomenon in tropical

regions and affects each kind of rock. Obviously, tropical weathering causes an

increase of iron indicated by reddish-brown colour of laterites. The progress in

chemical analysis of more samples showed that the tropical weathering increases iron

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content and frequently aluminium content and decreases silica content in relation to

the underlying parent rocks. Therefore, attempt was made to define laterites by the

ratio Si: (Al + Fe) but a definite limit was not applicable for laterites on different

parent rocks. Rather laterites are described to be soil types rich in iron and alminium,

formed in hot and wet tropical areas. Nearly all laterites are rusty-red because of iron

oxides. They develop by intensive and long-lasting weathering of the underlying

parent rock. The majority of the land areas with laterites was or is between the tropics

of cancer and capricon which include stable areas of African Shield, the South

American Shield and the Australian Shield. Laterites on mafic (basalt, gabbro) and on

ultramafic rocks (serpentine, peridotite, dunite) are formed from these rocks which are

free of quartz and show lower silica and higher iron content while laterites on acidic

rocks (not only granites and granitic gneisses but also sediments as clays, shales and

sandstone shall be included) are formed from rocks which contain quartz and have

higher silica and lower iron contents. The main element percentages of rocks from

these two groups and their corresponding laterites are shown in Table 2.1 and the

percentages shown are typical average values of numerous laterite samples and their

parent rocks in many tropical countries.

Table 2.1: Percentages of Main Elements in Laterites and Their Corresponding Parent Rocks SiO2 Al2O3 Fe2O3 Fe2O3 : Al2O3 Laterite 46.2 24.5 16.3 0.67 Granite 73.3 16.3 3.1 0.19 Laterite 39.2 26.9 19.7 0.73 Clay 56.5 24.4 5.3 0.22 Laterite 23.7 24.6 28.3 1.15 Basalt 47.9 13.7 14.9 1.09 Laterite 3.0 5.5 67.0 12.2 Serpentinite 38.8 0.7 9.4 14.1

Source: Schellmann (2008)

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Lateritic soils form the uppermost part of the laterite cover which form a thick

weathered layer on top of the basement rocks. Tardy (1997) calculated that laterites

cover about one-third of the Earth’s continental land area. Lateritic soils are also

referred to as the subsoils of equatorial forests, of Savannas of the humid tropical

regions, and of the Sahelian steppes. Engelhardt (2010) also reported that laterite is

mined while it is below the water table, so it is wet and soft and upon exposure to air

it gradually hardens as the moisture between the flat clay particles evaporates and

larger iron salts lock into a rigid lattice structure and become resistant to atmospheric

conditions.

According to the foregoing, it is very obvious that the depth at which most of the soils

obtained from borrow pits or being encountered during construction work in either

case are still in the range of lateritic soil. Therefore in order to remain economical

with facts available, it is rather better to refer to these groups of soils as lateritic soil

than laterites.

2.1.1 Formation of Laterite

Tropical weathering otherwise referred to as laterization is a prolonged process of

chemical weathering which produces a wide variety in the thickness, grade, chemistry

and ore mineralogy of the resulting soils (Dalvi, et al. 2004). The initial products of

weathering are essentially kaolinized rocks called saprolites. A period of active

laterization extended from about the mid-Tertiary to the mid-Quaternary periods [35

to 1.5 million years ago] (Dalvi, et al. 2004).

Laterites are formed from the leaching of parent sedimentary rocks (sandstones, clays,

limestones); metamorphic rocks (schists, gneisses, migmatites); igneous rocks

(grainites, basalts, gabbros, peridotites); and mineralized proto-ores; which leaves the

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more insoluble ions, predominantly iron and aluminium (Tardy, 1997). The

mechanism of leaching involves acid dissolving the host mineral lattice, followed by

hydrolysis and precipitation of insoluble oxides and sulphates of iron, aluminium and

silica under high temperature conditions of humid sub-tropical monsoon climate (Hill,

et al. 2000). An essential feature for the formation of laterite is the repetition of wet

and dry seasons. Rocks are leached by percolating rain water during the wet season;

the resulting solution containing the leached ions is brought to the surface by capillary

action during the dry season. These ions form soluble salt compounds which dry on

the surface; these salts are washed away during the next wet season (Yamaguchi,

2010). Laterite formation is favoured in low topographical reliefs of gentle crests and

plateaus which prevent erosion of the surface cover (Dalvi, et al. 2004). The reaction

zone where rocks are in contact with water from the lowest to the highest water table

levels is progressively depleted of the easily leached ions of sodium, potassium,

calcium and magnesium. A solution of these ions can have the correct pH to

preferentially dissolve silicon oxide rather than the aluminium oxides and iron oxides

(Yamaguchi, 2010).

The transformation of rock into laterite proceeds in general gradually as indicated by

the steady increase of iron and decrease of silica in laterite profiles above the parent

rock. It goes without saying that the initial products of weathering cannot be called

laterites. They also form in moderate climates and are essentially kaolinized rocks still

showing the structure of the rock. They are called saprolites in which iron is not as

strongly concentrated as in laterites. Some saprolites show due to finely disseminated

hematite a deep-red colour and are sometimes erroneously considered as laterite.

Saprolites as well as laterites are presently classified as residual rocks which in their

part are grouped within the sedimentary rocks. Lateritic weathering is only one

29

relevant process which is active in the superficial zone of tropical regions. Erosion or

denudation contributes equally to an alteration at the surface together with deposition

of material by water and wind. Not each variation in lateritic profiles can be attributed

to chemical weathering. There are ironstone formations in the world which can hardly

be interpreted by normal laterization processes. If they show signs of reworking,

transport and deposition they should not be defined as laterites but as lateritic

sediments. Lateritic sediments of older epochs can be overprinted by younger lateritic

weathering. Complex lateritic occurrences are grouped as exolaterites, false laterites

and laterite derivative facies. They are relevant in regional studies but not for a

general understanding of the laterization process. This is equally true for loose surface

layers above autochthonus laterites, locally separated by a stone line. They commonly

show a saprolitic composition with higher SiO2 contents and are deposited on the

laterite surface. Very often termites carried this material upwards from deeper

horizons. In other instances zirkonium contents in the surface horizons of laterite

(nickel limonite) above ultramafic rocks indicate an admixture from areas with other

parent rocks (Schellmann, 2008). A three-step-model of tropical weathering,

depending on the intensity of the weathering process was suggested by Schellmann

(2008) as follows:

• Weaker tropical weathering gives rise to formation of saprolites which are

the prevailing weathering products in the tropics and are frequently

misinterpreted as laterites.

• Advanced tropical weathering results in the formation of most of the

laterites showing a much stronger residual enrichment of Fe as against Al. A

higher tropical rainfall and a moderate drainage together with the presence of

quartz are generally not sufficient for pronounced incongruent kaolinite

30

dissolution and a pronounced formation of gibbsite. Al- and Si-bearing

compounds of probably colloidal size are thus removed from the weathering

mantle in high quantities. Laterites formed in this way are frequently indurated

and predominate in the tropics above clays, shales, grainites and granitic

gneisses. Friable laterites with high contents of iron oxides and kaolinite form

on basaltic rocks.

• Strong tropical weathering is prompted by a very pronounced rainfall, a

deep ground water level and a high permeability of the weathered rock,

allowing an excellent drainage. These factors cause an incongruent dissolution

of kaolinite. The composition of the laterite is determined by the composition

of the parent rock. The most wide spread acidic rocks with their high Al- and

Si- and their low Fe- content give rise, in favorable cases, to the formation of

high grade-bauxites. Ferruginous bauxites of a relatively poor quality form on

basaltic rocks. Ultramafic rocks are transformed in thick deposits of a very

ferruginous laterite (nickel limonite ore) which frequently covers nickel

silicate ore.

2.1.2 Mineralogical Composition of Laterite

The mineralogical and chemical composition of laterites are dependant on their parent

rocks. Laterites consist mainly of quartz and oxides of titanium, Zircon, iron, tin,

aluminium and manganese, which remain during the course of weathering (Tardy,

1997). Quartz is the most abundant relic mineral from the parent rock. Laterites vary

significantly according to their location, climate and depth. The main host minerals

for nickel and cobalt can be either iron oxides or clay minerals or manganese oxides.

Iron oxides are derived from mafic igneous rocks and other iron-rich rocks, bauxites

are derived from granitic igneous rock and other iron-poor rocks (Yamaguchi, 2010).

31

Nickel laterites occur in zones of the earth which experienced prolonged tropical

weathering of ultramafic rocks containing the ferro-magnesian minerals olivine,

pyroxene and amphibole (Dalvi, et al. 2004).

2.1.3 Uses and Economic Relevance of Laterites

2.1.3.1 Building blocks

After 1000 CE Angkorian construction changed from circular or irregular earthen

walls to rectangular temple enclosures of laterite, brick and stone structures. Geologic

surveys show areas which have laterite stone alignments which may be foundations of

temple sites that have not survived (Welch, 2010). The Khmer people constructed the

Angkor monuments which are widely distributed in Cambodia and Thailand between

the 9th and 13th centuries. The stone materials used were sandstone and laterite, brick

had been used in monuments constructed in the 9th and 10th centuries (Uchinda, et al.

2003). Angkor Wat located in the present day Cambodia is the largest religious

structure built by Suryavarman II, who ruled Khmer Empire from 1112 to 1152. It is a

world heritage site (Waragai, et al. 2006). The sandstone used for the building of

Angkor Wat is Mesozoic quarried in the Phnom Kulen Mountains, about 40 Km (25

Mi) away from the temple. The foundations and internal parts of the temple contain

laterite blocks behind the sandstone surface. The masonry was laid without joint

mortar (Siedell, 2008).

2.1.3.2 Road building

The French surfaced roads in the Cambodia, Thailand and Vietnam area with crushed

laterite, stone or gravel (Sari, 2004). Kenya during the mid- 1970s and Malawi during

the mid-1980s constructed trial sections of bituminous surfaced low volume roads

using laterite in place of stone as a base course. The laterite did not conform to any

32

accepted specifications but performed equally well when compared with adjoining

sections of road using stone or other stabilized materials as base. In 1984 US$40,000

per 1 Km (0.62 Mi) was saved in Malawi by using laterite in this way (Grace, 1991).

2.1.3.3 Water supply

Bedrock in tropical zones is often granite, gneiss, schist or sandstone; the thick laterite

layer is porous and slightly permeable so the laterite layer can function as an aquifer

in rural areas (Tardy, 1997). One example is the Southwestern Laterite (Cabook)

Aquifer in Sri Lanka. This aquifer is on the southwest border of Sri Lanka, with the

narrow Shallow Aquifers on Coastal Sands between it and the ocean. It has

considerable water-holding capacity, depending on the depth of the formation. The

acquifer in this laterite recharges rapidly with the rains of April-May which follow the

dry season of February-March, and continues to fill with the monsoon rains. The

water table recedes slowly and is recharged several times during the rest of the year.

In some high-density suburban areas the wster table could recede to 15 m (50 ft)

below ground level during a prolonged dry period of more than 65 days. The Cabook

Aquifer laterites support relatively shallow acquifers that are accessible to dug wells

(Panabokke, et al. 2005).

2.1.3.4 Waste water treatment

In Northern Ireland phosphorous enrichment of lakes due to agriculture is a

significant problem. Locally available laterite, a low-grade bauxite rich in iron and

aluminium is used in acid solution, followed by precipitation to remove phosphorous

and heavy metals at several sewage treatment facilities. Calcium-, iron-, and

aluminium-rich solid media are recommended for phosphorous removal. A study,

using both laboratory tests and pilot-scale constructed wetlands, reports the

33

effectiveness of granular laterite in removing phosphorous and heavy metals from

landfill leachate. Initial laboratory studies show that laterite is capable of 99%

removal of phosphorous from solution. A pilot-scale experimental facility containing

laterite achieved 96% removal of phosphorous. This removal is greater than reported

in other systems. Initial removals of aluminium and iron by pilot-scale facilities have

been up to 85% and 98% respectively. Percolating columns of laterite removed

enough cadmium, chromium and lead to undetectable concentrations. There is

possible application of this low-cost, low-technology, visually unobtrusive, efficient

system for rural areas with dispersed point sources of pollution (Wood and

McAtamney, 1996).

2.1.3.5 Ores

Ores are concentrated in metalliferous laterites; aluminium is found in bauxites, iron

and manganese are found in iron-rich hard crusts, nickel and copper are found in

disintegrated rocks, and gold is found in mottled clays (Tardy, 1997).

• Bauxite ore is the main source for aluminium (Thurston, 1913). It was named

after the French village Les Baux-de-Provence where it was discovered.

Bauxite is a variety of laterite (residual sedimentary rock), so it has no precise

chemical formular. It is composed mainly of hydrated alumina minerals such

as gibbsite [Al(OH)3 or Al2O3.3H2O] in newer tropical deposits; in older

subtropical, temperate deposits the major minerals are boehmite [�-AlO(OH)

or Al2O3.H2O] and some diaspore [�-AlO(OH) or Al2O3.H2O]. the average

chemical composition of bauxite, by weight is 45 to 60% Al2O3 and 20 to 30%

Fe2O3. The remaining weight consists of silicas (quartz, chalcedony and

kaolinite), carbonates (calcite, magnesite and dolomite), titanium dioxide and

34

water. Bauxites of economical interest must be low in kaolinite. Formation of

lateritic bauxites occurs world-wide in the 145- to 2-million-year-old

Cretaceous and Tertiary coastal plains. The bauxites form elongate belts,

sometimes hundreds of kilometers long, parallel to lower Tertiary shorelines

in india and South America; their distribution is not related to a particular

mineralogical composition of the parent rock. Many high-level bauxites are

formed in coastal plains which were subsequently uplifted to their present

altitude (Valeton, 1983).

In geosciences lateritic bauxites (silicate bauxites) are distinguished from karst

bauxites (carbonate bauxites). The early discovered karst bauxites occur

predominantly in Europe and Jamaica on Karst surfaces of limestone. They are also

formed by lateritic weathering of silicates either from intercalated clay layers or of

clayey dissolution residues of the limestone. These bauxites frequently contain

boehmite and diaspore in addition to gibbsite. The bauxites in Jamaica rest on tertiary

limestone and are exposed at the surface whereas the European bauxites are bound on

older carbonate rocks of Jurassic and Cretaceous age. If they are covered by younger

sediments, they have to be mined underground. Their contribution to the world

bauxite production is today relatively small (Schellmann, 2008). Most dominant are

nowadays the tropical silicate bauxites which are formed at the surface of various

silicate rocks such as granites, gneisses, basalts, syenites, clays and shales. The

formation of bauxites demands a stronger drainage as laterite formation to enable

precipitation of gibbsite which is the prevailing aluminium hydroxide in lateritic

bauxite. Zones with highest aluminium contents are frequently located below a

ferruginous surface layer and are due to downward leaching of aluminium which is

more soluble than iron under oxidizing conditions. Near the parent rock interface

35

gibbsite frequently replaces primary minerals predominantly feldspars which results

in a preservation of the primary rock structure. Large deposits of lateritic bauxites

with a high production are in Australia, Brazil, Guinea and India together with

Guyana, Suriname and Venezuela.

• High grade iron ores on top of tropical deposits of banded iron formations

(BIF) are also attributed to lateritic weathering which causes dissolution and

removal of siliceous constituents in the banded iron core (Schellmann, 2008).

The basaltic laterites of Northern Ireland were formed by extensive chemical

weathering of basalts during a period of volcanic activity. They reach a

maximum thickness of 30m (100ft) and once provided a major source of iron

and aluminium ore. Percolating waters caused degradation of the parent basalt

and preferential precipitation by acidic water through the lattice left the iron

and aluminium ores. Primary olivine, plagioclase feldspar and augite were

successively broken down and replaced by a mineral assemblage consisting of

hematite, gibbsite, goethite, anatase, halloysite and kaolinite (Hill, et al. 2000).

• Nickel ores were obtained from lateritic ores. Rich laterite deposits in New

Caledonia were mined starting from the end of the 19th century to produce

white metal. The discovery of sulfide deposits of Sudbury, Ontario, Canada,

during the early part of the 20th century shifted the focus to sulfides for nickel

extraction. About 70% of the earth’s land-based nickel resources are contained

in laterites; they currently account for about 40% of the world nickel

production. In 1950 laterite-source nickel was less than 10% of the total

production, in 2003 it accounted for 42%, and by 2012 the share of laterite-

source nickel was expected to be 51%. The four main areas in the world with

36

the largest nickel laterite resources are New Caledonia, with 21%; Australia,

with 20%; the Philipines, with 17%; and Indonesia, with 12% (Dalvi, et al.

2004). The ores are bound on ultramafic rocks above all serpentinites which

consist largely of the magnesium silicate serpentine containing approximately

0.3% nickel. This mineral is nearly completely dissolved in the course of

laterization leaving behind a very iron-rich, soft residue in which nickel is

concentrated up to 1-2% nickel. The bulk of this so-called nickel limonite or

nickel oxide ore consists of the iron oxide goethite in which nickel is

incorporated. Therefore it cannot be concentrated physically by ore dressing

methods. Below the nickel limonite another type of nickel oxide ore has

formed in many deposits. This is called nickel silicate ore which consists

predominantly of partially weathered serpentine. It is depleted in magnesium

and forms with 1.5-2.5% nickel the most relevant type of lateritic nickel ores.

In contrast to the relatively enriched limonite ore, the nickel silicate ore owes

its nickel content to a process of absolute nickel enrichment. The nickel is

leached downwards from the overlying limonite zone since not all of the

nickel, which is released from the serpentinite in the course of nickel limonite

formation, can be incorporated in goethite and therefore cannot be fixed in the

limonite zone. The migrated excess nickel is incorporated in the Mg-depleted

serpentine and occasionally in the neo-formed clay minerals predominantly

smectite. Some deposits show admixtures and layers of secondary quartz

which is precipitated from weathering solutions supersaturated in silicon, due

to a rapid dissolution of serpentine. A third type of lateritic nickel ore is

garnierite which is found in pockets and fissures of the weathered ultramafic

rocks. The green garnierite ore containing mostly 20-40% nickel consists of a

37

mixture of the phyllosilicates serpentine, talc, chlorite and smectite in which a

high percentage of magnesium is substituted by nickel. It is also formed by

downward nickel leaching and precipitation in hollow spaces of of the

weathered rock. Presently, nickel silicate ores with high portions of garnierite

are largely exhausted. Important deposits of nickel laterite are located in many

tropical countries above all in New Caledonia (Schellmann, 2008).

2.2 Definition of Soil Stabilization

Stabilization of soil can be seen as the process of blending and mixing materials with

soil to improve certain properties of the soil. The process may include the blending of

soils to achieve a required gradation or the mixing of commercially available

additives that may alter gradation, texture or act as a binder for cementation of the soil

(United States Army, 1994). O’Flaherty (2002) referred to soil stabilization as any

treatment (including technically i.e compaction) applied to a soil to improve its

strength and reduce its vulnerability to water. Sherwood (1993) also defined soil

stabilization as the alteration of properties of an existing soil to meet the specified

engineering requirements especially the strength properties which are taken to mean

the requirements for use in the various layers of road pavements. The main properties

that may be required to be altered by stabilization are:

Strength: To increase the strength and thus stability and bearing capacity.

The volume stability: To control the swell-shrink characteristics caused by moisture

changes.

Durability: To increase the resistance to erosion, weathering or traffic usage and

Permeability: To reduce permeability and hence the passage of water through the

stabilized soil

38

The foregoing definition covered much of the properties of soils that might be desired

for a deficient soil to possess before embarking on soil stabilization for improvement.

However, alteration of properties of soils by the process of soil stabilization is not

limited to those mentioned in the definition by Sherwood (1993). Some other

properties like the Atterberg limits, soil grading among others could also be altered

through the process of soil stabilization.

2.2.1 Techniques for Soil Stabilization

2.2.1.1 Stabilization by Compaction:

Soil compaction is the process whereby soil particles are constrained to pack more

closely together through a reduction in the amount of air contained in the voids of soil

mass. By compacting under controlled conditions, the air voids in well-graded soils

can be almost eliminated and the soil can be brought to a condition where there are

fewer tendencies for subsequent change in volume to take place. Therefore

compaction is a process which gradually induces artificial saturation or a state of

zero-air voids. However, this is a theoretical saturation because it is practically

impossible to expel all the air voids present in a soil. The fact is that loose material

may be made more stable simply by compacting it. In other words, compaction of

soils densifies, stabilizes and increases the strength of them. Compaction plays a

fundamental role in the properties of even stabilized materials. Compaction is

measured quantitatively in terms of the dry density of the soil, which is the mass of

solids per unit volume of the soil in bulk. The moisture content of the soil is the mass

of water it contains expressed as a percentage of the mass of dry soil. The increase in

dry density of soil produced by compaction depends mainly on the amount of

compactive effort and the moisture content of the soil which lubricates the soil

particles. For a given amount of compactive effort there exists for each soil moisture

39

content value termed optimum moisture content at which the maximum dry density is

obtained and further increase in moisture content will cause the dry density to

plummet. Each soil has its own unique optimum moisture content and the value

depends mainly on the amount and type of plastic fines that it contains. However the

optimum moisture content largely depends on the compactive effort and the term

always needs to be considered in relation to the type of compaction test used to define

the property. Several standards exists like the British Standard (standard proctor) in

which a 2.5kg rammer with height drop of 300mm will be used to give 25 blows on

each layer for 3 layers in a mould of about 1000cm3. Nevertheless this standard has

been modified by so many other institutions to suit different circumstances like West

African standards, Indian standards and so on.

2.2.1.2 Mechanical Stabilization:

Mechanical stabilization is the process whereby the grading of a soil is improved by

the incorporation of another material which affects only the physical properties of the

soil. Unlike stabilization by the incorporation of stabilizing agents the proportion of

material added usually exceeds 10 percent and may be as high as 50 percent.

Compaction is always recommended for well graded materials because nearly all the

air voids can be removed in the process but this is hardly achieved with poorly graded

materials. However their stability is improved by adding another material to fill the

voids between the particles. The blending of the materials has two main uses. The

stability of cohesive soils of low strength may be improved by adding coarse material.

The grading of the mixture is important to ensure that all the voids space is filled.

This grading is given by an equation originally derived by Fuller:

40

P = 100 (d-d100)b (2.1)

Where:

P = Percentage by weight of the total sample passing any given sieve size

d = Aperture of that sieve (mm)

d100 = Size of the largest particle in the sample (mm)

b = An exponent = 5 for a well-graded material

In the mechanical stabilization of clay soils by the addition of non-cohesive granular

material, sufficient granular material has to be added to ensure that the granular

fragments are in contact. In blending granular materials with finer-grained materials

to improve the particle size distribution care needs to be taken that the plasticity of the

fines fraction is not increased to such a degree that there is loss in stability. British

specifications for Type 1 (i.e the better quality materials) granular sub-bases require

that the plasticity index of the fraction of the sub-base material which passes the BS

425µm sieve should be zero and for Type 2 granular sub-base materials the plasticity

index is required to be less than 6. These figures obviously apply to relatively wet

conditions found in wet areas. In drier areas higher figures for plasticity index will be

acceptable, for example Ingles and Metcalf (1972) suggest a plasticity index of 8

rising to 15 in arid areas. Mechanical stabilization has limitations particularly in those

countries which have heavy rainfall or where frost is a problem. Although a

mechanically stable material is highly desirable it cannot always be achieved and even

when it can, it is often necessary to add to a stabilizing agent to bring about a further

improvement in the properties of a material.

41

2.2.1.3 Stabilizing by the Use of stabilizing agents:

The incorporation of stabilizing agents such as lime and cement usually in relatively

low amounts, changes both the physical and chemical properties of the stabilized

soils. The most commonly used primary stabilizing agents are cement, lime and

bitumen.

2.3 Soil Stabilizing Agents Available

2.3.1 Primary Stabilizing Agents:

This group includes the stabilizing agents which can be used alone to bring about

stabilizing action required in soils.

2.3.1.1 Portland cement

Portland cement is defined in BS 12:1991 as “a product consisting mostly of calcium

silicate, obtained by heating to partial fusion a predetermined and homogenous

mixture of materials containing principally lime (CaO) and Silica (SiO2) with a small

portion of alumina (Al2O3) and iron oxide (Fe2O3)”. In other words, it is made by

heating limestone (Calcium Carbonate), with small quantities of other materials (such

as clay) to 1450oC in a kiln in a process known as, ‘Calcination’ whereby a molecule

of carbondioxide is liberated from the calcium carbonate to form Calcium Oxide or

quicklime. Calcaeous materials, typically chalk or limestone, provide the CaO and

argillaceous materials, such as clay or shale, provide the SiO2, Al2O3 and Fe2O3.

Marls, composed of a mixture of chalk, clay and shales are also common raw

materials. As mentioned earlier the oxides present in the raw materials when

subjected to high clinkering temperature combine with each other to form complex

compounds. Four major compounds have been identified as constituents of cement

42

and are usually referred to as “Bogue’s Compounds”. The four compounds are listed

in Table 2.2.

Table 2.2: Bogue’s Compounds

Name of Compound Chemical Composition Usual Abbreviation

Tri-Calcium Silicate Ca3SiO4 C3S

Di-Calcium Silicate Ca2SiO5 C2S

Tri-Calcium Aluminate Ca3Al 2O6 C3A

Tetra-Calcium Alumino

Ferrite

Ca4Al 2Fe3O10 C4AF

Source: (Shetty, 2005)

The advancement made in the various spheres of science and technology has helped

to recognize and understand the microstructure of the cement compounds before and

after hydration. The X-ray powder diffraction method, X-ray fluorescence method and

use of powerful electron microscope capable of magnifying 50,000 times or even

more has helped to reveal the crystalline or amorphous structure of the hydrated and

unhydrated cement to have four different kinds of crystals in thin sections of cement

clinkers which are often referred to as Alite, Belite, Celite and Felite. This description

of the minerals in cement was found to be similar to “Bogue’s compounds”. Therefore

“Bogue’s compounds” C3S, C2S, C3A and C4AF are sometimes called in literature as

Alite, Belite, Celite and Felite repectively (Shetty, 2005). In addition to the four major

compounds, there are many minor compounds formed in the kiln. The influence of

these minor compounds on the properties of cement or hydrated compounds is not

significant. Two of the minor oxides namely K2O and Na2O referred to as alkalis in

cement are of some importance. The raw materials used for the manufacture of

43

cement consist mainly of lime, silica, alumina and iron oxide as mentioned earlier.

These oxides interact with one another in the kiln at high temperature to form more

complex compounds. The relative proportions of these oxide compositions are

responsible for influencing the various properties of the cement, in addition to rate of

cooling and fineness of grinding. Table 2.2 shows the approximate oxide composition

limits of ordinary Portland cement.

Table 2.3: Approximate Oxide Composition Limits of Ordinary Portland Cement. Qxide Percent Content

CaO 60-70

SiO2 17-25

Al 2O3 3.0-8.0

Fe2O3 0.5-6.0

MgO 0.1-4.0

Alkalies (K2O, Na2O) 0.4-1.3

SO3 1.3-3.0

Source: (Shetty, 2005)

Anhydrous cement does not bind the aggregates. It acquires adhesive property only

when mixed with water. The chemical reactions that take place between cement and

water is referred to as hydration of cement. The chemistry of concrete is essentially

the chemistry of the reaction between cement and water. On account of hydration

certain products are formed. These products are important because they have

cementing and adhesive value. The quality, quantity, continuity, stability and the rate

of formation of the hydration products are important. Anhydrous cement compounds

when mixed with water react with each other to form hydrated compounds of very

44

low solubility. The hydration of cement can be visualized in two ways. The first is

‘through solution’ mechanism. In this the cement compounds dissolve to produce a

supersaturated solution from which different hydrated products get precipitated. The

second possibility is that water attacks cement compounds in the solid state

converting the compounds into hydrated products starting from the surface and

proceeding to the interior of the compounds with time. It is probable that both

‘through solution’ and ‘solid state’ types of mechanism may occur during the course

of reaction between cement and water. The former mechanism may predominate in

the early stages of hydration in view of large quantities of water being available, and

the latter mechanism may operate during the later stages of hydration. The equations

of hydration of the major cement compounds are shown in equations (2.2) through

(2.5)

Ca2SiO4 + 2H2O → CaO.SiO2.H2O + Ca(OH) (2.2)

Ca3SiO5 + 3H2O → CaO.SiO2.H2O + 2Ca(OH) (2.3)

Ca3Al 2O6 + 3H2O→ CaO.Al2O3.H2O + 2Ca(OH)2 (2.4)

Ca4Al 2Fe2O10 + 4H2O → CaO.Al2O3.Fe2O3.H2O + 3Ca(OH)2 (2.5)

In the presence of water, the calcium silicate aluminates in Portland cement form

hydrated compounds which in time produce a strong, hard matrix. The hydration

reaction is slow as it proceeds from the surface of the cement particles, and the centre

of the particles may even remain unhydrated. The rate of hydration thus decrease

continuously which explains why the rate of gain of strength of stabilized materials

rapidly decrease with increase in time. Whether or not a bond forms between the

hardened cement matrix and the particles of the stabilized material depends on the

chemical composition of the material. In addition to the hydrated calcium silicates and

45

aluminates. Calcium hydroxide is one of the hydration products of Portland cement

and if, as is often the case, pozzolanic materials are present in the stabilized soil, and

these can react to produce further cementitious material.

The reaction of cement with water is exothermic. The reaction librates a considerable

quantity of heat, this libration of heat is called heat of hydration. This is clearly seen if

freshly mixed with cement is put in a vaccum flask and the temperature of the mass is

read at intervals. The study and control of heat of hydration becomes important in

mass concrete construction. It has been observed that the temperature in the interior of

large mass concrete is 500C above the original temperature of the concrete mass at the

time of placing and this high temperature is found to persist for a prolonged period.

2.3.1.2 Lime

Lime stabilization of soils was known to the Romans for more than seventy (70)

years (Williams, 1986). Sherwood (1993) defined lime to be a broad term which is

used to describe calcium oxide (CaO) - quick lime; calcium hydroxide Ca(OH)2 –

slaked or hydrated – lime; and calcium carbonate (CaCO3) – carbonate of lime. The

relation between these three types of lime can be represented by the following

equations

CaCO3 + Heat = CaO + CO2 (2.6)

CaO + H2O = Ca(OH)2 + Heat (2.7)

Ca(OH)2 + CO2 = CaCO3 + H2O (2.8)

The first reaction in Eq. (2.6) which is reversible does not occur much below 500oC

and is the basis for the manufacture of quicklime from chalk or limestone. Hydrated

lime is produced as a result of the reaction of quicklime with water as shown in Eq.

(2.7). Quicklime by a reversal of Eq. (2.7) and hydrated lime by Eq. (2.8) will both

46

revert to calcium carbonate on exposure to air. Only calcium oxide and calcium

hydroxide react with soil. Calcium carbonate is of no value for stabilization in civil

engineering, although it is used in agriculture as a soil additive to adjust the pH.

Consequently, lime stabilization refers to the addition of either quicklime or slaked

lime to soils. In dolomitic limes some of the calcium is substituted by magnesium.

These types of lime can also be used for stabilization. Hydraulic limes, also known as

grey limes, are produced from impure forms of calcium carbonate, which also contain

clay. They therefore contain less “available lime” to initiate the effects on plasticity

and strength. However, to compensate for this they contain reactive silicates and

aluminates similar to those found in Portland cement. Thus whilst their immediate

effect may be less than that of high calcium limes in the long term they may develop

higher strengths.

There are few countries or substantial areas of the world where some form of calcium

or magnesium carbonate suitable for limestone production is not available. As Spence

(1980) also points out that lime can be made locally by age-old and technologically

unsophisticated processes in most countries where limestone is available. Although

the author adds that most of these processes are highly inefficient, because they are

based on intermittent or batch production of lime.

2.3.1.3 Bitumen

Bitumen is a solid or viscous liquid, which occurs in natural asphalt or can be derived

from petroleum. It has strong adhesive properties and consists essentially of

hydrocarbons. In its natural condition, it is too viscous to be used for stabilization and

has to be rendered more fluid either as“cut-back” bitumen or a “bitumen emulsion”.

Cutback bitumen is a solution of bitumen in kerosene and/or diesel fuel, emulsions are

47

suspensions of bitumen particles in water; when the emulsion “breaks” the bitumen is

deposited on the material to be stabilized. It acts as a binding agent which simply

sticks the particles together and prevents the ingress of water unlike cement and lime

that react chemically with the material being stabilized requires suitable conditions for

the chemical reaction. However, it has little use in countries with high rainfall levels.

This means that the moisture content of road making materials and soils is fairly high

during most of the year and the addition of further fluids in the form of bituminous

materials may cause loss in strength.

2.3.2 Secondary Stabilizing Agent

This group could otherwise be referred to as admixture. They include those materials

which in themselves do not produce a significant stabilizing effect but which have to

be used in association with lime or cement. They are often blended before use in

which case the blended mixture assumes the role of the primary agent. The following

are some examples of admixture;

2.3.2.1 Blast Furnace Slag

Blast furnace slag is a by-product of pig iron production and is formed by the

combination of the siliceous constituents of the iron ore with the limestone flux used

for melting iron. In chemical composition it contains the same elements as Portland

cement. It is not in itself cementitious but it possesses latent hydraulic properties

which can be developed by another alkaline material. Slag may take several forms,

itemized below, according to the various means of cooling to which it is subjected

after leaving the blast furnace.

48

(i) It may be left to cool slowly in the open air, giving a crystallized slag suitable

for crushing and use as an aggregate. In this condition it is known as “air

cooled slag”

(ii) it may be subjected to sudden cooling by using water or air, giving vitrified

slag which in the first case is known as “granulated slag” and in the second as

“ pelletised slag” and

(iii)it may be water-cooled under certain conditions where the steam produced

gives rise to what is known as “expanded slag”

2.3.2.2 Iron fillings

The grinding plates of grinding machine are made up of grey cast- iron from which

when sharpening, fine iron filling is obtained as a by-product. The grey cast-iron is

produced by slow cooling and has silicon content of up to 2.5 percent. The wheel used

in sharpening the plates is made up of aluminium oxide and therefore when the

sharpening is taking place, the wheel strikes the grinding plates, giving out sparks of

light due to friction developed. The resulting material is the fine iron fillings,

comprising some aluminium oxide and cast iron (FeC).

2.3.2.3 Rice Husk Ash

Rice husk is an agricultural by-product generated from rice production. When

incinerated into ash, it has been categorized under pozzolana with about 67-70% silica

and about 4.9 and 0.95% aluminium and iron oxide respectively (Oyetola and

Abdullahi, 2006). The silica is substantially contained in amorphous form, which can

react with the CaOH liberated during hardening (hydration reaction) of cement to

further form cementitious compounds.

49

2.3.2.4 Bagasse-Ash

Bagasse is a fibrous residue as a result of the extraction of juice (sugar) from

sugarcane. Sometimes it is used in the industries as fuel in boilers. Previously, it was

burnt as a means of solid waste disposal but now, since it has been found to be very

useful, it is burnt under controlled temperatures to be further used. The sugarcane

bagasse ash contains high amounts of unburned matter; Silicon, Aluminium and

Calcium Oxides.

Although numerous works have been done on stabilization of soil with cement and

other binders as stabilizing agents, only few researchers have attempted to examine

the effect of bagasse ash on soils. Silvo, et al. (2008) used bagasse ash as a potential

quartz replacement in red ceramic and an improvement was discovered in ceramic/ash

properties up to sintering temperatures higher than 1000oC. Osinubi (2004) studied

the effect of up to 12% bagasse ash by weight of the dry soil on the geotechnical

properties of deficient lateritic soil. It was concluded that bagasse ash cannot be used

as a ‘stand-alone’ stabilizer but should be employed in admixture stabilization.

Mohammed (2007) carried out a work to study the influence of compactive effort on

bagasse ash with cement treated lateritic soil. An increase in optimum moisture

content (OMC) and decrease in maximum dry density (MDD) was observed with

increase in the percentage of bagasse ash and cement. Osinubi, et al. (2009) used

bagasse ash as admixture in lime-stabilized black cotton soil and large quantity of

lime was required to achieve sufficient stabilization. Ijimdiya and Osinubi (2011)

looked at the potential use of black cotton soil treated with bagasse ash for the

attenuation of cationic contaminants in municipal solid waste leachate. Higher

bagasse ash content increased the sorption of the contaminant species. Other works on

improvement of geotechnical characteristics of soils using bagasse ash include

50

Osinubi and Stephen (2007), Osinubi and Ijimdiya (2009), Ijimdiya (2010). Most of

these attempts are on soils in the Northern part of Nigeria however much has not been

done on the effect of bagasse ash on the soil of South-Eastern part of Nigeria and in

addition the application of modeling techniques for the purpose of optimization has

been very scanty.

2.4 Mechanisms of Stabilization

Properties of soil such as plasticity, compressibility and permeability can be altered

by the addition of stabilizing agents but the main interest is usually in finding a means

of increasing soil strength and resistance of softening water. Soil stabilization may be

brought in three ways, by bonding the soil particles together, by water proofing them,

or by the combining and waterproofing.

Bonding agents stabilize soils by cementing the particles together so that the effect of

water on the structure is lessened. The effectiveness of this type of stabilizer depends

on the strength of the stabilized matrix, on whether a bond is formed between the soil

and the matrix and on whether individual particles or agglomerations of particles are

bonded together. These stabilizing agents do not waterproof a soil, although a soil that

has been successfully bonded together will absorb less water than untreated material

owing to the reduced ability of the bonded soil to swell (Sherwood, 1993).

The principle of a waterproofing agent is to maintain the soil at a low moisture

content at which it has adequate strength for its purpose. In actual fact the water in the

stabilized soil and the efficiency of the stabilizers in this group depends on how much

the permeability of the soil is reduced. Very slight or no cementing action is obtained

from these materials and, unlike the process of bonding, the degree of stabilization

does not increase the stabilizer content but attains a maximum which is usually

51

reached with less than 2 percent of stabilizer by weight of soil. Stabilizing agents

which display both a bonding and a waterproofing effect are uncommon, although the

two effects can be achieved together by using a mixture of a bonding and a

waterproofing agent (Sherwood, 1993).

2.5 Mathematical Modeling

Derived from its latin root “modus”, the word model is generally understood to stand

for an object that represents a physical entity with a change of state or an abstract

representation of real world processes, systems or sub-systems (Edwards and

Hamson, 1989; Kapoor, 1993). Cheema (2006) defined a model as a representation of

the essential aspects of an existing system or a system to be constructed which

presents the knowledge of that system in a useable form. Thus Box and Draper (1987)

said that a model is simplified depiction used to enhance our ability to understand,

explain, change, preserve, predict and possibly control behaviour of a system which

may be in existence or still awaiting execution. Nwaogazie (2006) referred to model

as an imitation of something on a smaller scale and also that mathematical model

stands as a mathematical representation of a set of relationship between variables and

parameters. In case of an existing system, a model intends to improve on its

performance, while it explores to identity the best structure/properties of a future

system (Agunwamba, 2007). The trend of modeling is to collate existing records

(data), establish relationships via mathematical equation(s), and calibrate the equation

with experimental results and adopting such equation for forecasting and prediction.

Prediction looks into the future for decision-making.

There is nothing mysterious about models; a house wife’s shopping list, photographs,

maps, organization charts, accounting statements and globe for earth planet are

52

examples of ubiquitous use of a model. This is because each of them partially

represents realities, simplifies complexities/uncertainties and portrays essential

features of the represented systems in their own logical structure that is amenable to

formal analysis (Kapoor, 1993), Models are broadly grouped into two; physical and

mathematical models (Agunwamba, 2007).

A physical model is a three dimensional representation of an object which is tangible

and made to look and perform like the system or some aspects of the system under

study (Kapoor, 1993; Cheema 2006). Physical models are sometimes called iconic

model because they are actually constructed and may be larger or smaller or identical

in size to the object they represent (Kapoor, 1993). Physical models are very

important in the development and analysis of complex engineering systems and

processes such as ships, automobiles, aircrafts and complex chemical plants and so

on. This is due to mathematical intractable of the boundary configuration and

characteristics of these systems. Therefore, it is possible to select optimum design

parameters of complex engineering systems by constructing and monitoring the

performance of their physical models (Agunwamba, 2007).

A mathematical model is a simplified representation of a system or certain aspects of

a real system, created using mathematical concept such as functions, graphs,

diagrams/maps and equations to solve problems in the real world (Edwards and

Hamson, 1989; Cheema, 2006), it is usually referred to as process model and can take

many forms, including but not limited to algebraic equations, inequalities, differential

equations, dynamic systems, statistical and game models (Ike and Mughal, 1997).

Kapoor (1993) classified mathematical models as critical, empirical or semi-empirical

models based on how they are derived. Mathematical models can also be classified in

53

several other ways such as linear versus non-linear, static versus dynamic or steady

versus non-steady, deterministic versus stochastic or probabilistic, lumped parameters

versus distributed parameters, empirical versus mechanistic, continuous versus

discrete, black-box versus white- box models (Lin and Segel, 1998; Aris, 1994;

NIST/SEMATECH, 2006).

Critical models are developed using the principle of scientific laws. In empirical

models the output is related mathematically to the input and a mathematical

relationship is established between the two based on observed or experimental data

from the system under study. Semi-empirical models are developed from a

compromise between critical and empirical models with one or more parameters to be

evaluated from the data generated from the system under study (Kapoor, 1993).

In a linear model, the objective function and constraints are in a linear form while in a

non-linear model, part or all of the constraints and/or the objective function are non-

linear. For deterministic models each variable and parameter is assigned a definite

fixed number or a series of fixed numbers for any set of conditions. When variables

and parameters in a model are difficult to define with unique values it is probabilistic.

Static model does not explicitly take a variable time into account while dynamic

models do. In lumped parameter model, the various parameters and dependent

variables are homogeneous throughout the system while a distributed parameter

model takes account of variations in behaviour from point to point throughout the

system (Aris, 1994).

Furthermore, a mathematical model may be simple or complex. Although

representing a real system mathematically is usually a complex process due to the

presence of several variables and uncertainty associated with physical problems,

54

approximations are often used to obtain a more robust and simple models

(Agunwamba, 2007). This is because as the degree of complexity increases, so do the

amount of information, time and cost required to develop a model and interpret its

outcome.

Mathematical model is generally written as (Myer, 1990; Montgomery et al, 2001;

NIST/SEMATECH.2006):

γ = ƒ (� : �) + ε (2.9)

Consequently, there are three main parts to every mathematical model;

i. Response variable usually denoted by γ

ii. Mathematical function usually denoted by ƒ ��: �� iii. Random error usually denoted by ε

It is based on this that NIST/SEMATECH (2006) expressed mathematical modeling

as a concise description of the total variation in one quantity Y, by partitioning it into

deterministic and random components. The response variable simply called “the

response” or “dependent” variable is a quantity that varies in a way that we hope to be

able to summarize and exploit via the modeling process. Generally it is known that

the variation of the response variable is systematically related to the values of one or

more other variables before the modeling process is begun, although testing the

existence and nature of this dependencies part of the modeling process itself (Dean

and Voss, 1999; Wu and Hamanda, 2000).

The mathematical function, sometimes referred to as the “regression function”,

“regression equation”, ”smoothing function” or “smooth” consists of two parts. These

are the predictor variables �1, �2,……�n and the parameters �0, �1,……..�n

(Montgomery, 1991; NIST/SEMATECH, 2006; Nuran, 2007). The predictor variables

55

are input to the mathematical function and usually observed along with response

variable. Other names for the predictor variables include “explanatory variable”,

“independent variable”, “predictors” or “regressors”. The parameters are the

quantities that are usually estimated during the modeling process. Their true values

are unknown and unknowable except in simulation experiments. The parameters and

predictors are combined in different forms to give the function used to describe the

deterministic variation in the response variable. For example, a straight line with an

unknown intercept and slope goes with two parameters and one predictor variable and

its equation is as follows:

ƒ�� ∶ �� = �o + �1� (2.10)

A straight line with an unknown intercept and a known slope of one goes with one

parameter and is represented in equation 2.11

ƒ�� ∶ �� = �o+ �

(2.11)

For quadratic surface with two predictor variables, the full model goes with six

parameters as described in equation (2.12)

ƒ�� = �o + �1�1+ �2�2 + �12�1�2 + �11�1

2 + �22�22 (2.12)

The random errors are simply the difference between the data and the mathematical

function. They are unknown and assumed to follow a particular probability

distribution which is used to describe their aggregate behaviour. The random errors

cannot be characterized individually and the probability distribution that describes the

56

errors has a mean of zero and an unknown standard deviation � which is another

parameter like the �s.

Mathematical models are the most commonly used model for scientific and

engineering applications because they are versatile, and easier to generalize from

model to real life (Cheema, 2006). In many cases, the models are used to explain

known facts and lay a foundation for the theory behind their phenomena and that of

ambiguous processes. It is a booming engineering tool for design and optimization of

systems/processes because it provides an avenue for understanding qualitative and

quantitative aspects of phenomena of interests and also facilitates access to optimum

design/performance parameters of systems/processes (Kapoor, 1993; Cheema, 2006;

Agunwamba, 2007). NIST/SEMATECH (2006), summarized the four main uses of

mathematical models as estimation, prediction, calibration and optimization.

The goal of estimation is to determine the value of regression function (i.e., the

average value of the response variable) for a particular combination of the values of

the predictor variables. Regression function values can be estimated for any

combination of predictor variables, including values for which no data have been

measured or observed. Function values estimated for points within the observed space

of predictor values are sometimes called interpolation. Estimation of regression

function values for points outside the observed space of predictor values called

extrapolation are sometimes necessary but requires caution in any modeling process

(Agunwamba, 2007). Prediction determines either the value of a new observation of

the response variable, or the values of a specified proportion of all future observations

of the response variable for a particular combination of predictor variables. Prediction

57

can be made for any combination of independent variables including values for which

no data have been measured or observed (NIST/SEMATECH, 2006).

The goal of calibration is to quantitatively relate measurements made using one

measurement system to those of another measurement system. This is done so that

measurements can be compared in common units or to tie results from a relative

measurement method to absolute units. Optimization involves determination of

process inputs that should be used to obtain the desired process output. Typical

optimization goals might be to maximize the yield of a process, to minimize the

processing time and cost required to fabricate a product, or to hit a target product

specification with minimum variation in order to maintain specified tolerances (Myers

and Montgomery, 2002).

2.5.1 Mathematical model-building techniques

The bottom line for all data analysis problems is the selection of most appropriate

method to apply which largely depends on the goal of the analysis and the nature of

the data. However, model building is different from most other areas of data analysis

with regard to method selection because there are more general approaches and more

competing techniques available for this than most other data analysis methods. There

is often more than one technique that can be effectively applied to a given modeling

application. The large menu of methods applicable to modeling problems means that

there are more potentials to perform different analysis on a given problem thereby

resulting to more opportunity of obtaining effective and more efficient solutions

(NIST/SEMATECH, 2006).

Most modern mathematical modeling methods vary considerably in their details but,

their essentials fall within one or more of Analytical-Optimization technique,

58

Statistical technique, Probabilistic technique and Simulation-Search /Sampling

technique. Each of these model-fitting methods is not exclusively independent in

application, that’s why all modern mathematical modeling techniques involve other

scientific advances peculiar to them in addition to elements of one or more if not all of

these basic methods. For instance, response surface analysis consists of experimental

strategy for exploring the space of the process or independent variables, empirical

statistical modeling to develop an appropriate approximating relationship between a

response or responses and the process variables and optimization methods (often

simulation-search/sampling or analytical-optimization techniques or both) for finding

the values of the process variables that produce desirable values of the response

(Myers, 1990; Lawson and Madrigal, 1994; Neddermeijer et al., 2000; Nicolai et al.,

2004).

Analytical technique applies classical calculus and Lagrange multiplier as well as

other mathematical programming techniques which may be linear, non-linear and

dynamic in its study. Statistical techniques include such methods as statistical

inference, decision theory and multi-variable analysis like least square regressions

which may be linear, non-linear or weighted. Probabilistic techniques such as queuing

and inventory theory are used for studying stochastic system elements by means of

appropriate statistical parameters.

Simulation-Search/Sampling techniques are the most widely used for scientific and

engineering applications. Simulation is a descriptive technique that incorporates the

quantifiable relationships among variables and describes the outcome of operating a

system under a given set of inputs/operating conditions. If the objective function is

defined, the values of the objective for several runs generate a response surface. The

59

sampling or search process explores the response surface to determine near-optimal

and optimal solutions (Kathleen et al., 2004; Nuran, 2007).

Although, details vary somewhat from one mathematical modeling method to another,

the basic steps used for developing effective mathematical models are the same across

all modeling methods and this provides a framework in which the results from almost

any method can be interpreted and understood (NIST/SEMATECH,2006).

2.6 The Non-linear Programming Modeling

A Non-linear program is a type of mathematical optimization problem characterized

by objective and constraint functions that have a special form (Boyd et al., 2006).

Objective function is the function of which the optimal value (maximum or minimum)

is to determined, subject to a set of stated restrictions, or constraints placed on the

variables concerned (Stroud, 1996). In other words, constraint is a set of inequalities

that the feasible solution must satisfy while the feasible solution is a solution vector

which satisfies the constraints. Therefore optimal solution is a vector which is both

feasible and optimal.

Non-linear programming optimization model is one in which the objective and

constraint functions can be any nonlinear functions.The basic approach is to attempt

to express practical problems, such as engineering analysis and other design problems

in non-linear program format. In the best case, this formulation is exact but in some

more difficult cases, an approximate formulation is obtained. However nonlinear

programming is not just using some software packages or trying out some algorithm;

it involves some knowledge as well as creativity to be done effectively. The major

advantage of non-linear programming is that large-scale practical problems can be

solved reliably. In addition, unlike the linear programming model for optimization

60

which has a stricter limitation on the form of objective and constraint functions that is

they must be linear.

2.6.1 Monomial and Posynomial Functions

Let �1,………, �n denote n real positive variables, and � = ��1,……,�n) a vector with

components �i. A real valued function ƒ of �, with the form

ƒ (�) = k�1

a�2b………�n

c (2.13)

Where c > 0 and a, b, c ∈ R, is called a monomial function, constant k is referred to

as the coefficient of the monomial and the constants a, b, …., c are the exponents of

the monomial. Any positive constant is a monomial, as is any variable. Monomials are

closed under multiplication and division; if ƒ and g are both monomials then so are ƒg

and ƒ∕g. Any monomial raised to any power is also a monomial (Boyd et al., 2006).

A sum of one or more monomials, that is, a function of the form

k

ƒ (�) = ∑ck �1a1k �2

a2k………..�nank (2.14)

k=1

Where ck > 0, is called a posynomial function or, more simply, a Posynomial (with K

terms, in the variables �1,…………,�n). The term ‘posynomial’ is meant to suggest a

combination of ‘positive’ and ‘polynomial’. Any monomial is also a posynomial.

Posynomials are closed under addition, multiplication and positive scaling.

Posynomials can be divided by monomials (with the result also a posynomial); if ƒ is

a posynomial and g is a monomials, then ƒ∕g is a posynomial. If γ is a nonnegative

61

integer and ƒ is a posynomial, then ƒγ always makes sense and is a posynomial (since

it is the product of γ posynomials).

2.6.2 Previous Works on Optimization Techniques for Construction Materials

Lixiaoyong (2011) used orthogonal method to identify the main influencing factors in

mix ratio on compressive strength of concrete using Portland cement and fly ash. It is

based upon a set of tests relating composition and engineering properties of concrete.

The optimal mix ratios for compressive strength of both 7 days and 28 days were

achieved. The optimization technique most commonly used for construction materials

is the Scheffe’s optimization regression method in simplex design.

NIST/SEMATECH (2006) compared Scheffe’s and Tukey’s methods. Tukey was

preferred when only pairwise comparisons are of interest because it gives a narrower

confidence level while in general case Scheffe was preferred when many or all

contrast might be of interest because it tends to give narrower confidence limit.

Researchers have widely used Scheffe’s method in the past for optimization of

construction materials. Arimanwa et al. (2012) applied it for the prediction of the

compressive strength of aluminium waste-cement concrete and found that the

compressive strengths predicted by the model agreed with the corresponding

experimentally obtained values. The method was also applied by Ezeh and

Ibearugbulem (2009) to optimize the compressive strength of river stone aggregate

concrete and the model was found to be adequate for predicting concrete mix ratios,

when the desired compressive strength is known and vice versa. Eze and

Ibearugbulem (2010) in their work on recycled aggregate concrete also used the

method to optimize and predict strength and the predicted compressive strength were

in good agreement with their corresponding experimentally observed values. Onwuka

et al. (2013) also used the Scheffe’s simplex design for prediction and optimization of

62

compressive strength of sawdust ash-cement concrete. The results of the response

function compared favourably with the corresponding experimental results. The

optimum compressive strength of concrete at 28 days was found to be 20N/mm2

which corresponds to the mix ratio of 0.5: 0.95: 0.05: 2.25: 4 for water, cement,

sawdust ash and granites respectively.

However Scheffe’s theory (1963) has some disadvantages associated with it. It

stipulates that materials involved in the mix must be in volume but in soil stabilization

volume batching is not usually recommended because soils are prone to variations in

volume with time as a result of consolidation of the soils with time caused by natural

forces. Another major disadvantage of Scheffe’s theory (1963) is its rigidity in

application. This is because it involves predetermined points or mix ratios obtained

from a simplex mix design which makes it not to be amenable to stabilized soils. It is

difficult to predict results of stabilized soils prior to adequate laboratory experiments

because soils have peculiarities of structure. Osinubi (1998b) also pointed out that

peculiarities of structure may play more important role in cement stabilization than

Atterberg limits and also that lateritic soils with the same and similar plasticity index

may have completely different behaviours in mixing operation. In addition, the

regression method still has the limitation of not covering a wider scope. Thus the

classical optimization technique appears to be a better approach to overcome the

shortcomings of the simplex regression method and it is of wider scope for use in the

optimization of results from stabilized soils which is a novelty approach.

2.7 Classification of Soil

Soils may be classified in a general way as cohesionless or cohesive or a coarse or

fine grained. As the terms are too general and cover too wide a range of physical and

63

engineering properties, additional refinement or means of classification is necessary

to determine the suitability of a soil for a specific engineering purpose and to be able

to convey this information to others in an understandable way. Numerous

classification systems have been proposed in the past several decades, which are

helpful and guide in classifying soils. The most commonly used are the AASHTO and

the Unified Classification Systems (Bowles, 1992).

2.7.1 AASHTO Soil Classification System

American Association of State Highway and Transportation Officials (AASHTO)

formerly the Bureau of Public Road System is used world wide. The AASHTO

classification system started with the then U.S. Bureau of Public Roads in the years

1927-1929 and the system was revised in 1945. It classifies soils into eight groups, A-

1 through A-8, and originally required the following data:

- Grain-size analysis

- Atterberg limits

The table that was used for the classification is shown Table 3.1 and to establish the

relative ranking of a soil within a subgroup, the group index is a function of the

percent of soil passing sieve No. 200 and the Atterberg limits. The group index can be

obtained using Equation (3.7).

Group Index, GI = 0.2# + 0.005#% + 0.01 &� (3.15)

Where,

# = that part of the percent passing the No. 200 sieve greater than 35 and not

exceeding 75, expressed as a whole number (range= 1 '� 40);

& = that part of the percent passing the No. 200 sieve greater than 15 and not

exceeding 55, expressed as a whole number (range= 1 '� 40);

64

% = that part of the liquid limit greater than 40 and not greater than 60, expressed

as a whole number (range= 1 '� 20);

� = that part of the plasticity index greater than 10 and not exceeding 30,

expressed as a whole number (range1 '� 20).

Table 2.4 AASHTO soil classification system Note that A-8, peak or muck, is by visual classification and is not shown in the table

General classification

Granular materials (35% or less passing No. 200)

Silt-clay materials (More than 35% passing No. 200)

A-1 A-3 A-2 A-4 A-5 A-6 A-7 Group classification

A-1a

A-1b

A-2-4

A-2-5

A-2-6

A-2-7

A-7-5; A-7-6

Sieve analysis: Percent passing: No. 10 No. 40 No. 200

50 max 30 max 15 max

50 max 25 max

51 min. 10 max.

35 max

35 max.

35 max.

35 max.

36 min.

36 min.

36 min.

36 min.

Characteristics of fraction passing No: 40: Liquid limit: Plasticity index

6 max.

N.P.

40 max 10 max

41 min. 10 max.

40 max. 11 min.

41 min. 11 min.

40 max. 11 max.

41 min. 10 max.

40 max. 11 min.

41 min. 11 min.

Group index

0

0

0

0

4 max.

8 max.

12 max.

16 max.

20 max

Usual types of significant constituent materials

Stone fragments, gravel and sand

Fine sand

Silty or clayey gravel and sand

Silty Soils

Clayey soils

General rating as subgrade

Excellent to good

Fair to poor

Source: Bowels (1992)

2.7.2 The Unified Classification System

This system was originally developed for use in airfield construction and it had

already been in use since about 1942, but was slightly modified in 1952 to make it

apply to dams and other construction. The principal soil groups of this classification

system are given in Table 3.2. The soils are designated by group symbols consisting

65

of a prefix and suffix. The prefixes indicate the main soil types and the suffixes

indicate the subdivisions within the groups.

Table 2.5 Unified Classification system

Soil type Prefix Subgroup Suffix

Gravel G Well graded W

Sand S Poorly graded P

Silt M Silty M

Clay C Clayey C

Organic O WL < 50 percent L

Peat Pt WL > 50 percent H

Source: Bowles (1992)

A soil is well-graded gravel or nonuniform if there is a wide distribution of grain sizes

present, if there are some grains of each possible size between the upper and the lower

gradation limits. This could be ascertained by plotting the grain-size curve and either

observing the shape and spread of sizes or computing the coefficient of uniformity

and coefficient of curvature as given by equations (3.4) and (3.5). A poorly graded, or

uniform, if the sample is mostly of one grain size or is deficient in certain grain sizes.

The unified classification system defines a soil as:

1. Coarse-grained if more than 50 percent is retained on the No.200 sieve.

2. Fine-grained if more than 50 percent passes the No. 200 sieve.

The coarse-grained soil is either:

1. Gravel if more than half of the coarse fraction is retained on the No. 4 sieve

2. Sand if more than half of the coarse fraction is between the No.4 and No.200

sieve

66

Classification of coarse-grained soils depends primarily on the grain-size analysis and

particle size distribution. Classification of fine-grained soil requires the use of

plasticity chart; each soil is grouped according to the coordinates of the plasticity

index and liquid limit. On this chart an empirical line (the A line) separates the

inorganic clays (C) from silts (M) and organic (O) soils. Although the silty and

organic soils overlapped areas, they are easily differentiated by visual examination

and odour.

67

CHAPTER THREE

MATERIALS AND METHODS

3.1 Introduction

The soil sample was collected from a lateritic soil deposit in Oboro, Ikwuano Local

Government Area of Abia State. It was collected at a depth of not less than 150mm at

15 different points of about 3m apart using the disturbed sampling technique. The

natural moisture content was determined after which it was air-dried. The Ordinary

Portland Cement was used as the binder and bagasse ash as the admixture in the

stabilized soil while clean tap water was used for the mixing. The bagasse residue was

collected from Panyam district, Mangu Local Government Area, Plateau State. It was

incinerated into ash in a furnace at temperature of up to 5000C for about 2 hours after

which it was allowed to cool and thoroughly ground. It was then sieved through 75µm

sieve as required by BS 12 (1990) and was used for this study.

3.2 Characterization of the lateritic Soil

Soils have peculiarities, they vary in properties. In other words, no two soils can be

similar in all properties but can behave alike in some cases. Therefore it is necessary

to identify a soil and properly classify it to the group it belongs. This can be achieved

by conducting preliminary tests on the natural soil. The following tests were

conducted on the lateritic soil:

3.2.1 Moisture Content Determination

Empty aluminium cans which were properly identified with labels were weighed.

Representative samples of wet soil were placed in the cans and weighed after which it

placed in the oven at 1100C for 20 to 24 hours. The moisture content is computed as

follows:

68

Water Content = �)2 – )1� × 100∕ �)1 – )3) (3.1) Where;

)1 = Weight of dry soil + can

)2 = Weight of wet soil + can

)3 = Weight of can

The natural moisture content is shown in Table 4.1.

3.2.2 Liquid Limit

This is water content above which the soil behaves like a viscous liquid ( a soil- water

mixture with no measurable shear strength). The liquid limit testing apparatus

(Cassagrande apparatus) was used for the determination of liquid limit as

recommended in BS 1377: Part 2: 1990. The soil was sieved with 425µm sieve and

water added in successive stages (drier to wetter). A grooving tool of tip width 2mm –

0.25mm and a drop of liquid limit machine cup of 10mm were also used. The liquid

limit is the water content at which 25 bumps close a groove of about 13mm length.

The result is shown in Table 4.1.

3.2.3 Plastic Limit

This is the moisture content below which the soil no longer behaves as a plastic

material. The plastic limit was determined as specified in BS 1377: Part 2: 1990.The

sample was sieved through 425µm sieve and water was added to about 20g of the

filtrate soil in order to mould it. The moulded lump of soil was broken into smaller

samples and each of them rolled on a glass plate using the fingers to obtain a thread of

uniform diameter (3mm). The plastic limit is described as the water content when a

thread of soil being rolled shear at 3mm diameter ( i.e the first crumbling point or

appearance of little cracks). If the plastic limit could not be attained in first rolling, the

69

thread will be broken into several other pieces, reformed into a ball and re-rolled. The

result is shown in Table 4.1.

3.2.3 Linear Shrinkage

The linear shrinkage was determined as specified in BS 1377: Part 2: 1990. A soil

sample of about 150 grams in mass which passed through a 425µm sieve was taken

into a dish. It is mixed with distilled water to form a smooth paste. The sample is

placed in a brass mould, 140 mm long and with a semi-circular section of 25mm

diameter. The sample is allowed to dry slowly initially in air and then in an oven. The

sample is cooled and its final length measured. The linear shrinkage is given by

�, = - 1 − /010234 × 100

(3.2)

Where:

Ls = Linear shrinkage

�1 = Length of oven-dry sample

�� = Initial length of specimen

3.2.4 Particle Size Analysis

The particle size grading was carried out as specified in ASTM 1992. A set of stack

sieve (apertures ranging from 4.75mm – 0.075mm) was used. A pulverized soil

sample was washed on sieve No. 200 (0.075mm) and the residue soil was oven-dried.

The oven-dried soil sample of known weight was put in the set of stack sieves and

then placed on a mechanical shaker to sieve for about 10 minutes. The weight of the

materials remaining on each sieve was noted and the percentage retained computed as

a percentage of the total weight. The percentage passing and cumulative percentage

70

passing were computed for each sieve. The suspension passing the sieve No.200 after

washing into a 1000ml jar was taken for sedimentation test for silt and clay sized

particles quantitative determination. Enough water was added to make 1000ml of

suspension and the deflocculant sodium-metaphosphate was used. The suspension

was mixed thoroughly by placing a bung on the open end of the jar and turning upside

down and back few times. The jar was placed on the table. The hydrometer was

inserted into the suspension to measure the specific gravity and a stop-watch was used

to record time. The percentage settling at any given time was recorded with the

Equation (3.3) . The particle size curve is shown in Figure 4.1.

� = - ��51�4 × [ℛ/ℳs]× 100 (3.3)

Where:

�= The percentage finer for any given size

� = The specific gravity of the soil

ℛ = Corrected hydrometer reading

ℳs = Total mass of the soil

Coefficient of Uniformity, 8u = 60/10 (3.4)

Coefficient of Curvature,8c= �30)2/�60× 10� (3.5)

Where;

60 = Particles with diameter 60% finer



71

3.2.5 Identification of Clay Mineral

The X-ray diffraction was used for clay mineral identification. Because the small size

of most soil particles prevents the study of single crystals, the powder method and the

orientated aggregates of particles is made use of. In the powder method, a small

sample containing particles at all possible orientations is placed in a collimated beam

of parallel X-rays, and diffracted beams of various intensities are scanned by a

Geiger, proportional, scintillation tube and recorded automatically to produce a chart

showing the intensity of diffracted beam as a function of angle 2ϴ which are

converted to d spacings by Bragg’s law in Equation (3.6). The clay minerals present

in the soil are shown in Table 4.3.

nλ = 2� sin (3.6) where,

λ = the wavelength of a parallel beam of X-rays

= angle parallel to the atomic planes

� = distance between parallel atomic planes

3.2.6 Classification of Soil

The lateritic soil was classified using the AASHTO classification system and Unified

Classification System as presented in section 2.6 .

3.2.7 Compaction Test:

This test is to determine the maximum dry density and the optimum moisture content

with a given compactive effort. This test established the optimum moisture content to

be used for some other performance test like california bearing ratio and the

unconfined compressive strength, which requires compaction. As specified by BS

1377: 1990 (Standard Proctor) was adopted. A cylindrical metal mould (Proctor

mould) of about 1000cm3 volume and a rammer of 2.5kg weight with a height drop of

72

300mm was used as the given compactive effort. Twenty-five (25) blows were given

on each layer of three (3) and moisture content samples were taken from the top and

bottom of the mould. The optimum moisture content was taken as the moisture

content at which the maximum dry density was attained. The dry density was

obtained with the expression shown below in Equations (3.8) and (3.9) and the results

obtained are shown in Table 4.1.

� = =/> (3.8) �d= [ ?

1@ ℳABB

] (3.9)

Where,

� = Bulk density

= = Weight of wet soil

> = Volume of wet soil

γd = Dry density of soil

ℳ = Moisture content of soil in decimal fraction

3.2.8 Specific gravity of solids

The specific gravity of soil was determined in accordance to BS 1377 (1990).

A completely dry density bottle with a stopper was weighed and weight recorded as

W1. About 10g of an oven dried soil that passed through 2 mm BS sieve was put into

the density bottle. The weight of the bottle with the dried soil and the stopper was

recorded as W2. De-aired distilled water was then added to cover the soil in the

density bottle and air was completely removed from the bottle by subjecting it to

vaccum in a dessicator for about an hour. More water was added at a constant

temperature of 200C for an hour. The exterior of the bottle was dried and its weight

73

taken as W3, the bottle was then cleaned, filled with de-aired water and allowed to

stand for one hour. The weight of the bottle containing water with a stopper was taken

as W4 and the specific gravity was computed with the following expression

DS = �E2 − E1�/[�E4 − E1) −�E3 − E2�] (3.10)

3.2.9 California Bearing Ratio

The California bearing ratio (CBR) test is an empirical test developed by the

California State Highway Department for the evaluation of subgrade strengths. In the

test as given in BS 1377: Part 2: 1990, a specimen which is 127mm in height and

152mm in diameter is compacted into the CBR mould. The specimens were prepared

in 5 (five) layers and heavy rammer was used to give 56 (fifty-six) blows onto each

layer. The load required to cause a circular, 49.65mm in diameter, to penetrate the

specimen at a specified rate of 1.25mm per minute is then measured. From the test

results, the CBR value is calculated. This is done by expressing the corrected values

of forces on the plunger for a given penetration as a percentage of a standard force.

The 2.5mm and 5.0mm penetration caused by 13.24KN and 19.96KN loads were used

in comparing the loads that caused the same penetration on the specimens. The CBR

value for the lateritic soil is shown in Table 4.1.

3.2.10 Unconfined Compressive Strength

Unconfined Compressive Strength (UCS) shows a drained condition of the soil and

the ability of the soil to withstand failure by compression. The specimens from the

Proctor mould were used as the unconfined compressive strength specimen and a

correction factor of 1.04 was used on the results to conform to cylindrical specimens

with a height/diameter ratio of 2:1or 150 mm cube specimens. The specimens were

tested by crushing and the load that caused the failure of the specimen divided by the

74

cross sectional area of the specimen gave the strength of the soil. The UCS value for

the natural soil is shown in Table 4.1.

2.1 Characterization of Bagasse Ash

The detectable oxide composition of bagasse ash was obtained from the Atomic

Absorption Spectrometer and its specific gravity was also determined with the

procedure presented in section 3.2.8. The results that were obtained are shown in

Table 4.4.

2.1 Test Requirements for the Stabilized lateritic Soil

The quality of cement stabilized materials is usually assessed on the basis of strength

tests made on the material after the stabilizer has been allowed sufficient time to

harden. The strength of stabilized soils can be evaluated in many ways, of which the

most popular are the Unconfined compressive strength (UCS) test for cement

stabilized soils and the California bearing ratio (CBR) test. Both have been criticized

on the grounds that neither reflects the manner in which a stabilized layer is stressed

in the road pavement. They are used most frequently because of the relative ease

which they can be performed. This makes them particularly suited to routine control

where large numbers of tests may be needed on daily basis (Sherwood, 1993). The

durability test was included to put the test specimens to suit the circumstance where

pavements are subjected to high rainfall or in some other cases in a condition of

freezing and thawing.

3.4.1 Unconfined Compressive Strength

The BS 1924: 1990 has recommended the specifications for the apparatus to be used

and procedure which similar to the natural soil. However, for these stabilized-soil

mixtures, specimens were prepared by first thoroughly mixing dry quantities of

75

pulverized soil with bagasse ash and Portland cement in a mixing tray to obtain a

uniform colour. Constant cement contents of 2%, 4%, 6% and 8% with variations of

bagasse ash from 0% to 20% at 2% intervals and all percentages used were by the

weight of dry soil. The required amount of water which was determined from

moisture density relationships for stabilized-soil mixtures was then added to the

mixture. For each mix 3 (three) specimens were prepared as recommended by the

Nigerian General Specification (1997).

For practical purposes an approximately linear relation may be considered to exist

between strength and the logarithm of time, for both lime and cement stabilized

materials, up to an age of several months. In the case of cement stabilized soils which

contain no clay this relation holds over a long period of time but when clays or other

pozzolanic materials are present the relation is no longer linear after a few weeks. The

7-day curing period generally adopted for test purposes is chosen for convenience and

is purely arbitrary. In many cases provided all comparisons are made at the same age

the actual time is of little importance (Sherwood, 1993). The membrane curing was

used for the curing of the specimens.

3.4.2 The California Bearing Ratio

The BS1924:1990 has recommended the specifications for the apparatus and

procedures to be used which is similar to that of the natural soil. The mixing

procedure for the stabilized-soil mixture for the specimens was also similar to the one

recorded in section 3.4.1. However, the test for CBR was modified so as to conform

to the recommendation of the Nigerian General Specification (1997) which stipulates

that the specimens should be cured for six days unsoaked, immersed in water for 24

hours and allowed to drain for 15 minutes before testing.

76

3.4.3 Durability Tests

Neither the unconfined compressive strength test nor the California bearing ratio test

fully reflects the ability of the stabilized materials to withstand the effects of wetting

and drying. The durability test is employed to examine the ability of the laboratory

specimens ssto resist exposure to such conditions. BS 1924: Part 2: 1990 gives details

of procedure to be followed which adopts the unconfined compressive strength. In

this test two identical sets of UCS specimens are prepared both of which are cured in

the normal manner at constant moisture content for seven days. At the end of the

seven-day period, one set is immersed in water for seven days whilst the other

continues to cure at constant moisture content. When both sets are 14 days old, they

are crushed and the strength of the set immersed in water expressed as a percentage of

the strength of the set cured at constant moisture content. This index is a measure of

the resistance to the effect of water on strength.

3.5 Method of Formulation of Non-Linear Programming Model

3.5.1 Objective Function

The multiple regression approach would be used to develop the objective function.

The cost of bagasse ash stands as the independent variable while the other parameters

of strength and compaction characteristics stand for the dependent variable. Consider

a regression model of the form;

F = % �a1 za2 (3.11)

Where; � and z are the dependent variables; c, a1 and a2 are constants.

Equation (3.11) can be linearized or transformed to multiple regression model by

taking the logarithm of both sides

77

G�HF = G�H% + a1G�HI + a2 G�HJ (3.12)

Thus, the estimates of the coefficients, c, a1 and a2 can be obtained by setting

G�HF = y1, G�H% = a0, �1 = G�HI and �2 = G�HJ

Therefore, equation (3.12) could be represented as

y1 = a0 + a1�1 + a2�2 (3.13)

Equation (3.13) is a linear regression equation of y on �1 and �2. The independent

variables �1 and �2 varies partially due to variations in y, respectively, the coefficients

a1 and a2 represent partial regression coefficients of y on �1 with �2 held constant; y

on �2 with �1 held constant; respectively. For a four dimensional space coordinate

system, equation (3.13) represents a hyperplane usually called a regression

hyperplane.

Given n sets of measurements, (y1, �11, �21, �31)……., (yn, y1n, �2n, �3n) the least square

estimates of a0, a1, a2 and a3 can be obtained using the following equations

(Nwaogazie, 2006):

a0n + a1 ∑�1i + a2 ∑�2i + a3 ∑�3i = ∑ yi (3.14)

a0 ∑�1i + a1 ∑�2

1i + a2 ∑�2i�1i + a3 ∑�3i �1i = ∑yi �1i (3.15)

a0 ∑�2i + a1 ∑�1i�2i + a2 ∑�2

2i + a3 ∑�3i�2i = ∑yi �2i (3.16)

a0 ∑�3i + a1 ∑�1i�3i + a2 ∑�2i�3i + a3 ∑�3i

2 = ∑yi�3i (3.17)

78

However, the above Least Square operation could be made less rigorous by the use of

statistical softwares that have been developed in the computation. Minitab Statistical

software was used in this work.

Thus applying it to the compaction and strength characteristics as independent

variables and cost of bagasse ash as dependent variable to form the equation which

stands as the objective function;

�= Cost of bagasse ash in Kobo

= Optimum Moisture Content in percentage

� = California Bearing Ratio in Percentage

� = Unconfined Compressive Strength for 7 days curing period in kN/m2

� = Cement content in percentage

G�H� =ao+a1G�H +a2G�H� +a3 G�H� +a4G�H� (3.18)

Where ao, a1, …..a2, a3 and a4 are constants.

Equation 3.18 could be transformed to:

� = 10ao Ka1 �a2 �a3�a4 (3.19)

Equation (3.19) stands as the objective function.

3.5.2 Constraints

The Nigeria General Specification (1997) has established evaluation criterion for

stabilized materials, California Bearing Ratio of 180% for laboratory mix was

stipulated. Conventionally, the minimum values of Unconfined Compressive Strength

at 7 days for cement stabilized soils are 750-1500, 1500-3000, 3000-6000 KN/m2 for

sub-base, base (lightly trafficked roads) and base (heavily trafficked roads)

respectively. The constraints were generated from these standards. In addition, this

study will only be meaningful provided that the cost of the required bagasse ash does

79

exceed the cost of cement that should have been used. Therefore constraints were

used to ensure that the cost of the bagasse ash used was that which is just satisfactory

for economic application. The constraint equations were developed similar to

equations (3.18) and (3.19) and would have the same form as follows:

, = L� , �, �� (3.20)

� = L��, �, � (3.21)

Where , �, �and � in equations (3.20) to (3.21) are the usual notations as in equation (3.18).

Therefore the non-linear programming model would be formed using Equation (3.19)

as the objective function and Equations (3.20) and (3.21) to form the constraints.

3.6 Solution of Non-Linear Programming Model

There are several ways of solving the geometric programming model, the graphical

solution not usually common because the optimum solution is always associated with

a corner point of the solution space which might be very difficult to have visual

advantage in very complex problems. However, the simplex method is fundamentally

based on this idea without necessarily showing the plots of the equations like the

graphical solution. It employs an iterative process that starts at a feasible corner point,

normally the origin, and systematically moved from one feasible extreme to another

until the optimum was eventually reached. The first step in the simplex method was to

ensure that each constraint was written with a positive right-hand side constant term.

Then the inequalities were all expressed as equations by the introduction of slack

variables.

Example, a� + bY ≤ N1 can be written as a� + bY + W1 = N1

c� + dY ≤ N2 can be written as c� + dY + W2 = N2

where a,b,c and d are coefficients; � and Y are the problem variables; N1 and N2 are

numerical values; and W1 and W2 are positive (or zero) variables with unit

80

coefficients, required to make up the left-hand side to the value of the right hand side

constant term. The new variables, W1 and W2, are called slack variables.

Subsequently, the simplex table (frame work) is formed as shown in table 3.0 and the

coefficients of the problem variables and of the slack variables in the constraints,

together with the right-hand side numerical values in the column headed RHS. The

Check column was included to provide a check on the numerical calculations as the

simplex operation takes place. For each row, the sum of the entries in that row,

including the RHS column would be entered in the check column. It was always

necessary that the columns of the slack variables form a unity matrix.

Table 3.1: Format for the Simplex Matrix

� Y W1 W2 RHS Check

A B 1 0 N1 Algebraic

sum of row1

C D 0 1 N2 Algebraic

sum of row2

The objective function was included in the bottom of table in a similar manner like

the constraints and the row referred to as index row. In computing the simplex table

the following steps were taken:

i. The key column was selected which was the column containing the most

negative entry.

ii. In each row, the values in the right-hand side column were divided by the

corresponding positive entry in the key column; the row with the smallest ratio

obtained became the key row while the number/entry at the intersection of the

key column and key row became the key number or pivot number.

81

iii. All the entries in the key row were divided by the pivot number to reduce the

pivot entry to unity while the rest of the entries in the table remain unchanged.

The new version of the key row was sometimes called the main row.

iv. The main row was used to operate on the remaining rows of the table

including the index row to reduce the other entries in the key column to zero.

It was noted that the main row remained unaltered. The new value in any

position in the other rows, including the right-hand side column and check

column, can be calculated as follows:

New number = Old number - the product of the corresponding entries in the main

row and key column.

v. The new values in the check column were confirmed that they were all equal

to the sums of the entries in the corresponding rows otherwise it was an

indication that there was an error somewhere.

vi. Steps (i) to (v) were repeated until no negative entry remained in the index

row.

3.6.1 Sensitivity Analysis

Sensitivity analysis considered how small changes in constraints affected the optimal

objective function value/optimal solution. This was achieved for any desired

constraint in the non-linear programming model by slightly varying the right-hand

side of the constraint and keeping other constraints as they were to solve. The

sensitivity of the constraint Si gave the (approximate) fractional change in the optimal

value per fractional change in the right-hand side of the inequality. If the inequality

constraint was not tight at the optimum, then Si = 0 which means that a small change

in the right-hand side of the constraint (loosening or tightening) had no effect on the

optimal value of the problem. Roughly speaking, for constraints that were found to be

82

tight on the optimal value, it was also a measure of comparing how much tightly

binding they were to the optimal value. Therefore optimal sensitivity was also most

useful when a problem was infeasible. Assuming a point that minimizes some

measure of infeasibility was found, the sensitivities associated with the constraints

could be very informative. Each one gives (approximate) relative change in the

optimal infeasibility measure, given a relative change in the constraint. The

constraints with large sensitivities are likely candidates for the one to loosen (for

inequality constraints), or modify (for equality constraints) to make the problem

feasible.

3.7 Scheffe’s Simplex Regression model

Scheffe (1963) introduced simplex lattice design and was later transformed to simplex

centroid design. Simplex is a factor space or a polygon which has its simplest simplex

as a straight line. A straight line is a one-dimensional factor space. Other factor spaces

are two-dimensional factor space, three-dimensional factor space; four-dimensional

factor space etc. In geometry a two-dimensional factor space is called a plane figure.

Some of its examples include triangle, square, rectangle, pentagon, hexagon,

heptagon, octagon, nonagon etc. A three-dimensional factor space is called a solid.

Examples of a three-dimensional factor space are sphere, cylinder, cubes, cuboids,

frustums, tetrahedrons, prisms, cones etc.

This work involved a four component mixture (cement, bagasse ash, water and

lateritic soil), thus the second degree (4, 2) simplex model was used which is a three

dimensional factor space (tetrahedron). The number of terms in the response equation

of simplex design depends on the number components of the mixture and the degree

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of polynomial of the simplex. The value could be obtained using the following

equation

O = �P@Q51�!S�P51�!∗Q!U (3.22)

Where;

O = The number of observations required

V = The degree of the polynomial

W = The number of components in the mixture

The four-component mixture the mix proportion which are in volume, Di must satisfy

the condition

Di ≥ 0 (i= 1, 2, 3, 4� (3.23)

The Scheffe’s method involves the pseudo (virtual) mix ratios, Zi which satisfies the

condition that the summation of the ratios at any point must be equal to unity. Thus

∑Zi = 1 (i = 1, 2, 3, 4� (3.24)

Each component is resident at one vertex of the simplex tetrahedron. This means that

no more than one component can exist at one vertex at the same time. In other

words, at any point in time, only pure components of a mixture exist at the vertices of

the simplex tetrahedron. Binary components only exist along the line joining two

vertices and no more than two components of a mixture exist along a single line

joining two vertices at the same time. The quantity of each of the two components

along the line depends on the position of the point on the line.

The actual (real) mix ratios relate with the pseudo (virtual) mix ratios in this way

[[] = [\] [Z] (3.25)

84

Where [[], [\] and [Z] designate the matrix of the real mix ratios, coefficient of

relation matrix and matrix of pseudo mix ratios.

The sought for the parameter or property, ] of interest is presented using equation of a

polynomial form as shown

] = &o + ∑&iZi + ∑&ijZiZj + ∑&ijkZiZjZk + … + _ (3.26)

Where &i, &ij, &ijk are constants, Zi, Zj, Zk are pseudo components and e is a random

error term which represents the comined effect of all variables not included in the

model

For four pseudo component mixture with two degrees the response equation, ]

becomes

] = &o + ∑&iZi + ∑&ijZiZj + _ �0 ≤ ` ≤ a ≤ 4� (3.27)

Expanding Equation (3.27) by substituting the values of i and j transforms to

] = &o + &1Z1 + &2Z2 + &3Z3 + &4Z4 + &12Z1Z2 + &13Z1Z3 + &14Z1Z4 + &23Z2Z3

+ &24Z2Z4 + &34Z3Z4 + &11Z1 2 + &22Z2

2 + &33Z3

2 + &44Z4

2 (3.28)

Multiplying Equation (3.24) by &o it gives

&o = &oZ1 + &oZ2 + &oZ3 + &oZ4 (3.29)

Multiplying Equation (3.24) sucessively by ZI, Z2, Z3 and Z4 and rearranging the

product it gives

Z1 2 = Z1 − Z1Z2 − Z1 Z3 − Z1Z4 (3.30)

Z2 2 = Z2 − Z1Z2 − Z2Z3 − Z2Z4 (3.31)

Z3 2 = Z3 – Z1Z3 − Z2Z3 − Z3Z4 (3.32)

Z42 = Z4 − Z1Z4 − Z2Z4 − Z3Z4 (3.33)

85

Substituting Equations (3.29) through (3.31) into Equation (3.28)

] = �1Z1 + �2Z2 + �3Z3 + �4Z4 + �12Z1Z2 + �13Z1Z3 + �14Z1Z4 + �23Z2Z3

+ �24Z2Z4 + �34Z3Z4 (3.34)

Where �i and Zi are coefficients of response equation and pseudo components of the

mix respectively.

The coefficients �i and �ij are defined as follows; �i = &o + &i + &ii (3.35)

�ij = &ij − &ii + &jj (3.36)

Thus Equation (3.33) could be represented as

] = ∑�iZI + ∑ �ijZiZj (3.37)

Where i ≥ 1 and i ≤ j ≤ 4

3.6.1 Determination of the Coefficients of the Polynomial Function

If the response function is represented by ], the response function for the pure

component, i and that of the binary mixture components, ij, are ]i and ]ij respectively.

Therefore;

]i = ∑�iZi (3.38)

and

]ij = ∑�iZi + ∑�ijZiZij (3.39)

Where i ≥ 1 and i ≤ j ≤ 4

The substitution of the values of the pseudo components Z1, Z2, Z3 and Z4 at the ith on

the lattice into Equation (3.38) gives

]i = ∑�i (3.39)

86

The substitution of the values of the pseudo components Z1, Z2, Z3 and Z4 at the point

ij into Equation (3,39) gives

]ij = 1b �i + 1

b �j + 1c �ij (3.40)

Rearrangement of Equations (3.40) and (3.41) gives

�i = ]I (3.41)

�ij = 4]ij − 2]i − 2]j (3.42)

Let di = ]i and dij = ]ij, thus Equations (3.42) and (3.43) becomes

�i = di (3.44)

�ij = 4dij − 2di − 2dj (3.45)

Substituting Equations (3.44) and (3.45) into Equation (3.34) and rearranging it gives

] = d1�2Z1− 1�Z1+ d2�2Z2 −1�Z2+ d3�2Z3− 1�Z3+ d4�2Z4−1�Z4+ 4d12Z1Z2

+ 4d13Z1Z3 + 4d14Z1Z4 + 4d23Z2Z3 + 4d24Z2Z4 + 4d34Z3Z4 (3.46)

Equation (3.46) is the form for the optimization equation of a four component mixture

and the second degree polynomial (4, 2) simplex model.

However in lateritic soil stabilization, working with predetermined results could be

unreliable and alternatively the laboratory results with their corresponding mix

proportions could be used to walk back to obtain the pseudo mix ratios. Assuming

that the vertices of the simplex tetrahedron where only pure components of a mixture

exist were located at imaginery points. In order to satisfy the boundary condition in

Equation (3.24) such that the summation of the pseudo mixes at any point must be

equal to one and thus were calculated as follows. If � = Cement content converted to volume

e = Bagasse ash content converted to volume

87

= Optimum Moisture Content converted to volume

� = Lateritic soil content converted to volume

Therefore � + e + + � = Z

(3.47)

The pseudo mixes at any point become

fg = Z1,

hg = Z2,

ig = Z3,

0g = Z4

In other words,

Z1 + Z2 + Z3 +Z4 = 1 (3.48)

The reponse for the optimization model would be California Bearing Ratio in

Percentage, � and Unconfined Compressive Strength for 7 days curing period in

kN/m2, �

3.6.1 Validation of Optimization Models

In order to validate the optimization functions, extra ten points selected at random

from the observations. These observations were used to test the validity of the

response function by testing the adequacy using statistical student’s t-test at 95%

accuracy level. The hypothsis is Null if there is no significant difference between the

laboratory results and the predicted values at 95% accuracy level while the hypothsis

is Alternative if there is significant difference between the laboratory results and the

predicted values at 95% accuracy level.

Let

]E = Experimental results

]M = Responses from the optimization model

j = Number of observations

ki = Difference between ]E and ]M

88

kA, Mean difference between ]E and ]M = ∑ki/j (3.49)

�2 , Variance of difference = �kA− ki�2/�j − 1�

' = kA× j0.5/� = Calculated value of t

89

CHAPTER FOUR

RESULTS AND DISCUSSION

4.1 Presentation of Results

The properties of the lateritic soil and particle size curve are shown in Table 4.1 and

Figure 4.1 respectively. The characterisation of the mineral contents of the soil is

shown in Table 4.2 and Figure 4.2 and 4.3.

Table 4.1: Properties of the Lateritic Soil

S/N Properties Results

1 Colour Reddish-brown

2 Percentage passing sieve No 200 45.3

3 Liquid Limit (%) 38

4 Plastic Limit (%) 11

5 Plasticity Index (%) 27

6 Linear Shrinkage (%) 8

7 Specific Gravity 2.75

8 AASHTO Classification A-6 (2)

9 Unified Classification System SC (Clayey Sand)

10 Major Clay Mineral Present Illite (Inorganic Clay of Medium Plasticity)

11 Maximum Dry Density (Kg/m3) 1709

12 Optimum Moisture Content (%) 14

13 California Bearing Ratio (%) 9

14 Unconfined Compressive Strength

(kN/m2)

186

90

Figure 4.1: Particle Size Curve

0

20

40

60

80

100

120

0.001 0.01 0.1 1 10

Pe

rce

nta

ge

Pa

ssin

g

Sieve Sizes (mm)

91

Table 4.2: Clay Minerals Characteristics and lm Angles at the Peak of X-ray

Diffraction of the Soil Minerals.

92

Figure 4.2: X-ray Diffractometer Chart for the Soil Minerals

Table 4.3: Identification of the Soil Minerals using the Spacing of the Atomic

Planes

Spacing of the Atomic Planes (d in

Angstroms)

Mineral Identified

4.48474 Illite (very strong), Sepiolite

3.54289 Vermiculite

3.12797 Feldspar

2.71776 Carbonate

2.45729 Chlorite

1.59366 Chlorite

1.49827 Illite (strong), Kaolinite

1.41975 Kaolinite

93

4.2 Soil Characterization

The presence of aluminium silicate (Al2Si4O10) as shown in Table 4.2 has indicated

that the soil has a lateritic nature and also it is a residual soil. The process of

laterisation is the leaching of lighter minerals like silica and consequent enrichment of

heavier minerals like aluminium oxide. Table 4.3 showed that illite is the predominant

clay mineral present in the soil and in Figure 4.2 it was also observed that

dehydroxylated pyrophyllite is present in the soil. The clay mineral structure is made

up of the basic structural units of silica sheet and octahedral sheet. The silica

tetrahedral are interconnected in a sheet structure, three of the four oxygen ions in

each tetrahedron are shared to form a hexagonal net, the bases of the tetrahedral are

all in the same plane, and the tips all point in the same direction. The structure has the

composition (Si4O10)4- and can repeat indefinitely. Electrical neutrality can be

obtained by replacement of four oxygen ions by the hydroxyl ions or by union with a

sheet of different composition that is positively charged (Mitchell and Soga, 2005);

whereas the octahedral sheet is composed of aluminium in octahedral coordination

with oxygen ions or hydroxyl ions. When combined with silica sheets, an aluminium

octahedral sheet is referred to as a gibbsite sheet. The basic structural unit of illite,

muscovite and pyrophyllite is made up of the three layer silica-gibbsite-silica

sandwich. Illite has less potassium between the layers of basic structural units. This

potassium ions helps to balance the resulting deficiency of charges (net negative

charges) that resulted as a consequence of some silicon positions being filled by

aluminium (isomorphous substitution). The amount of potassium ions more often is

insufficient to neutralize all the resulting negative charges and this would leave the

clay particles with minimal net negative charges. To maintain electrical neutrality,

cations are attracted and held between the layers, the surfaces and the edges of the

94

clay particles. Many of these cations are exchangeable cations because they may be

replaced by cations of another type. The cation exchange capacity (CEC) is the

quantity of these exchangeable cations which is a measure of water the clay mineral

can absorb and held in between the layers to cause swell. This is very undesirable in

construction soil. Illite which is the predominant clay mineral in the soil has a reduced

cation exchange capacity as result a less affinity for water, medium activity and

moderately stable in volume. Pyrophyllites do not absorb water between the unit

layers and swell which is equally a very good attribute for construction soil. This is

because of the absence of interlayer cations to be hydrated by water and also the

surface hydration energy is too small to overcome the Van der Waals forces between

layers, which are greater in this mineral because of smaller interlayer distance

(Mitchell and Soga, 2005).

Table 4.3 also indicated the presence of kaolinite minerals and a typical example of a

triclinic crystal system as captured in Table 4.2. They are composed of alternating

silica and octahedral sheets. The tips of the silica tetrahedral and one of the planes of

the atoms in the octahedral sheet are common. The tips of the tetrahedral all point in

the same direction, toward the centre of the unit layer. In the plane of the atoms

common to both sheets, two-thirds of the atoms are oxygens and are shared by both

silicon and octahedral cations. The remaining atoms in this plane are (OH) located so

that each is directly below the hole in hexagonal net formed by the bases of silica

tetrahedral (Mitchell and Soga, 2005). The kaolinite group has little or no

isomorphous susbstitution in other words the exchangeable cations would be very

low. Besides, the bonding between successive layers is by both van der Waals forces

and strong hydrogen bonds. The bonding is sufficiently strong that in the presence of

water there is no absorption and thus no interlayer swelling. As a matter of fact the

95

kaolinite subgroup clay minerals are known to be one of the least clay minerals in

activity. This is a very good attribute for construction soil.

The lateritic soil was classified to be A-6(2) soil in AASHTO rating system and SC

(clayey sand) in the Unified Classification System. Though the group is far to the

right of the AASHTO table, it is fairly good for road construction works. This is

because it has a group index of 2 and also from the point of view of Atterberg limits

(liquid limit of 38%, plasticity index of 27% and linear shrinkage of 8%), it is

satisfactory. Using the Atterberg limits in the Casagrande’s plasticity chart, the clay

mineral identified to be present in the lateritic soil is inorganic clay of medium

plasticity. This could also be traced to illite because its activity is known to be

moderately satisfactory which agrees with the result presented in the X-ray diffraction

method. In other words the soil would be somewhat stable in volume at moisture

content variations. However, the high percentage of finer particles (45.3% passing

Sieve No 200) in the soil probably caused the soil to have high cement content

requirement. It will be also noted that the coarse particles are almost inert in the

reaction of the soil with cement rather the finer particles play the major role as the

pozzolanic component in the reaction. Therefore higher fines content results to higher

cement content requirement and this also agrees with Sherwood (1993) and Nigerian

General Specification (1997).

96

4.3 Characterization of Bagasse Ash

Table 4.4: Properties of Bagasse Ash (Oxide Compositions and Specific Gravity)

Properties Results

SiO2 72.8(%)

Al 2O3 6.21(%)

Fe2O3 4.41(%)

CaO 1.95(%)

MgO 2.28(%)

Loss on Ignition (LOI) 12.35(%)

Specific gravity 2.25

ASTM (1976) defined pozzolana as a siliceous, or siliceous and aluminous material

which in itself possesses little or no cementious value but will, in finely divided form

and in the presence of moisture, chemically react with calcium hydroxide at ordinary

temperatures to form compounds possessing cementitious properties. It is very

obvious that bagasse ash fits into this definition because from Table 4.4 showed it to

be rich in silica (SiO2) at 72.8% and trace of aluminium oxide at 6.21 (%). Also

Osinubi (2004) had earlier established that it cannot be used as a ‘stand-alone’

stabilizer but should be employed as admixture. This shows that it possesses little or

no cementitious property but would be very useful during cement stabilization in that

it has the ability to react with huge amounts of calcium hydroxide produced as a result

of hydration reaction which could even be a potential source of instability in the

stabilized matrix. In other words, bagasse ash has been found to be a good pozzolanic

material.

97

4.4 Stabilized Soil Tests

4.4.1 Compaction Characteristics

Figure 4.4: Variations of Optimum Moisture Content with Increase in Bagasse

Ash Content at 2%, 4%, 6% and 8% Cement Contents

Figure 4.4 shows the relationship between optimum moisture content and bagasse ash

content at different cement contents. The optimum moisture content increased

progressively from 16.50% to 22.62%, 17.90% to 23.54%, 18.24% to 24.44% and

20.39% to 25.31% at 2%, 4%, 6% and 8% cement contents respectively with addition

of bagasse ash from 0% to 20%. These increments in optimum moisture content with

increase in bagasse ash could be attributed to the increased amount of water required

in the system to adequately lubricate all the particles in the soil-cement and bagasse

ash mixture. Therefore the optimum moisture content continuously increased with

increase in bagasse ash content. It was also observed from the results that the increase

in cement content in the mixture also increased the optimum moisture content of the

0

5

10

15

20

25

30

0 5 10 15 20 25

Op

tim

um

Mo

istu

re C

on

ten

t (%

)

Bagasse Ash Content (%)

2% cement

4% cement

6% cement

8% cement

98

mixture. The reason for this could be that the increase in cement contents steps-up the

hydration reaction of cement and consequently increases the demand for water in the

system. This would result to the increase in optimum moisture content.

Figure 4.5: Variations of Maximum Dry Density with Increase in Bagasse Ash

Content at 2%, 4%, 6% and 8% Cement Content

Figure 4.5 showed the relationship between maximum dry density and bagasse ash

content at different cement contents. In the corollary, the maximum dry density

reduced from 1661Kg/m3 to 1422 Kg/m3, 1777Kg/m3 to 1572Kg/m3, 1891 Kg/m3 to

1586 Kg/m3 and 2199Kg/m3 to 1791Kg/m3 at 2%,4%, 6% and 8% cement contents

respectively with addition of bagasse ash from 0% to 20%. This could be as a result of

the partial replacement of the soil with higher specific gravity (2.75) by bagasse ash

with lower specific gravity (2.25). Also considering the reaction between cement,

bagasse ash and fine fractions of the soil in which they form clusters that occupied

larger spaces and invariably increasing their volume with decreasing the maximum

0

500

1000

1500

2000

2500

0 5 10 15 20 25

Ma

xim

um

Dry

De

nsi

ty (

Kg

/m³)


2% cement

4% cement

6% cement

8% cement

99

dry density. In some cases the clusters formed were not strongly bound and the

disruption was necessary in order to achieve higher level of compaction at a given

compaction energy, part of the compactive effort was lost in dislodging the weak

bonds which resulted to reduced density. As the cement content increased, the bonds

became stronger and the soil-cement-bagasse ash clusters behave more like coarse

aggregates thus became more amenable to compaction. This resulted in the increase

in maximum dry density with increase in cement content.

4.4.2 Strength Characteristics

Figure 4.6: Variations of California Bearing Ratio with Increase in Bagasse

Ash Content at 2%, 4%, 6% and 8% Cement Contents.

0

50

100

150

200

250

300

0 5 10 15 20 25

Ca

lifo

rnia

Be

ari

ng

Ra

tio

(%

)


2% cement

4% cement

6% cement

8% cement

100

Figure 4.7: Variations of Unconfined Compressive Strength with Increase in

Bagasse Ash Content at 2%, 4%, 6% and 8% Cement Contents.

0

500

1000

1500

2000

2500

0 5 10 15 20 25

Un

con

fin

ed

Co

mp

ress

ive

Str

en

gth

(KN

/m²)


2% cement,7d

2% cement,14d

2% cement,7d+7dsk

4% cement,7d

4% cement,14d

4% cement,7d+7dsk

6% cement,7d

6% cement,14d

6% cement,7d+7dsk

8% cement,7d

The California Bearing Ratio, the unconfined compressive strength and the durability

tests are the strength properties of the stabilized matrix. Figure 4.6 presented the

relationship between California Bearing Ratio and bagasse ash content. The California

Bearing Ratio at 2% cement content increased from 22.30% at 0% bagasse ash content

to attain a value of 25.13% at 8% bagasse ash and in a similar trend the California

Bearing Ratio at 4% cement content rose from 57.99% at 0% bagasse ash content to

attain its peak at 163.59% at 16% bagasse ash content. While the California Bearing

Ratio at 6% and 8% cement content increased continuously from 83.34% to 239.16%

and from 147.16% to 276.30% respectively on addition of bagasse ash from 0% to

20%. Figure 4.7 clearly showed that the unconfined compressive strength of 2%, 4%,

6% and 8% cement contents at 7 days, 14 days curing periods and 7days curing +

7days soaking all increased remarkably and consistently with the addition of bagasse

ash from 0% to 20% bagasse ash content.

This trend of the improvement in the strength properties of the stabilized matrix could

be better explained with the chemistry of the Bogue’s compounds (Tri-Calcium

101

Silicate, Di-Calcium Silicate, Tri-Calcium Aluminate and Tetra-Calcium Alumino

Ferrite) in cement are shown in equations (2.2) through (2.5) respectively. The

calcium silicate hydrates and calcium hydroxide have been described as dominant

products of hydration which are produced at the early stage of hydration mainly by the

selective hydration of dicalcium silicate and tricalcium silicate. Between the two

foregoing, the tricalcium silicate reacts first and dominates the reaction within first

few days of hydration (Neville, 2003; Scrivener, 2004; Escalante-Garcia and Sharp,

2004; Kjellsen and Justnes, 2004 and Shetty, 2005). Tricalcium silicates was

described as the most important phase of cement and the calcium silicate hydrate gel

resulting from this reaction is reported to be principally responsible for the mechanical

properties of hydrated cement (Escalante-Garcia and Sharp, 2004; Scrivener,2004). A

common product of the four equations for the hydration of cement [equations (2.2)

through (2.5) is calcium hydroxide. The high amount silica provided by bagasse ash

reacted with the excess amounts of calcium hydroxide produced after hydration

reaction of cement compound to further produce additional calcium silicate hydrates

which is very vital for strength development. The additional amount of calcium

silicate hydrates produced will depend on the amount of calcium hydroxide given out

from the hydration reaction of cement compounds. The improved strength of the

resulting stabilized matrix could be attributed to the amounts of calcium silicate

hydrates that were produced as shown in equations (4.1) through (4.3). The following

are the proposed equations for the reaction between silica from bagasse ash and

calcium hydroxide, the common product of hydration reaction of cement compounds:

Ca(OH)2 + SiO2 CaO.SiO2.H2O (4.1)

2Ca(OH)2 + 2SiO2 2CaO.SiO2.H2O (4.2)

3Ca(OH)2 + 3SiO2 3CaO.SiO2.H2O (4.3)

102

For unconfined compressive strength specimen, the minimum conventional values at 7

days curing period for cement stabilized soils of 750-1500 KN/m2, 1500-3000 KN/m2

and 3000-6000KN/m2 for sub-base, base (lightly trafficked roads) and base (heavily

trafficked roads) respectively were adopted in evaluating the strength of the stabilized

soil specimens. The 7 days unconfined compressive strength value at 8% cement

content and 20% bagasse ash was 1424 KN/m2 showed that it could only satisfactorily

meet the requirement for sub-base of road works. Judging by the 180% California

Bearing Ratio value criterion of mix in place condition for laboratory mix as

recommended by the Nigerian General Specification for Roads and Bridges, Works

(1997), the foregoing also attained a value of 276.30% which met the requirement.

The durability requirement for the stabilized soil as stipulated by BS 1924: Part 2:

1990 was also satisfied. This is because the resistance to loss in strength in all the

specimens never exceeded the maximum 20% allowable loss in strength by comparing

the unconfined compressive strength of 14 days old cured specimen with the

unconfined compressive strength of 7 days curing and 7days soaking in water.

Table 4.9: Percentage Losses in Unconfined Compressive Strength between

103

14 Days Curing and 7 Days Curing + 7 Days Soaking

Bagasse Ash

Content (%)

% Loss in Unconfined Compressive Strength

2% Cement 4% Cement 6% Cement 8% Cement

0 9.11 2.92 14.35 16.70

2 7.25 7.10 5.20 8.46

4 12.11 5.77 10.95 6.74

6 12.20 9.89 10.78 13.41

8 16.53 13.90 11.89 13.90

10 18.05 14.15 18.16 13.51

12 14.71 12.06 16.36 11.89

14 8.31 15.64 15.24 10.12

16 6.81 11.86 16.10 7.45

18 7.01 12.13 13.60 7.40

20 8.93 9.71 11.36 5.49

104

CHAPTER FIVE

MODELLING AND OPTIMIZATION BAGASSE ASH CONTENT

5.1 Cost Analysis for the Stabilized Matrix

Practically speaking, little or no value has been attached to neither bagasse residue nor

its ash because of its low demand. In the markets where sugar cane is sold and sugar

factories, the bagasse residues are littered around the surroundings without much

value. Definitely as the awareness of its usefulness increases, the demand will rise and

the value/cost shall equally rise. It is therefore necessary to attach cost to the bagasse

ash in order to make this work meaningful. The market prices or cost of the other

materials at the time of this analysis should also be considered for appropriate basis

for comparison. The bagasse ash, cement and the optimum moisture content were all

measured as a proportion of the weight of the dry soil. Therefore using 100 grams of

dry soil as a reference weight, the corresponding weights of cement, bagasse ash and

water could be determined and their unit cost could be determined.

5.1.1 Cement Cost

50 kilograms of cement = N1,400.00

Thus, 50,000 grams = N1,400.00

1 gram = 140,000 kobo/50,000grams = 2.8 kobo

1% = 1g = 2.8 kobo

5.1.2 Projected Cost of Bagasse Ash Cost of 100,000 kilograms of Bagasse Ash

Haulage cost for (100,000 kilograms) = N180,000.00

Processing (energy + labour) for 100,000 kilograms = N180,000.00

Cost of bagasse residue to yield 100,000 kilograms = N 40,000.00 Total = N400,000.00

105

Therefore, 100,000 kilograms = N400,000.00

1 kilogram = 40,000,000 kobo/100,000,000 grams = 0.4kobo

1% = 1g = 0.4kobo

5.1.3 Cost of Water

1 tank = 1000 gallons = N2,500.00

1000 gallons = 4546 litres = 4546 × 103 millilitres

1 millilitre = 250,000 kobo/4546 × 103 millilitres = 0.055 kobo

Density of water = 1000 kilograms/m3 = 1 gram/milliliter

In other words, 0.055 kobo/millilitre

1% = 1gram = 0.055kobo

5.1.4 Cost of Lateritic Soil

1 lorry truck = 5 tonnes

5 tonnes = 5,000 kilogram = N10,000.00

Thus, 5,000,000 grams = N10,000.00

1 gram = 1,000,000 kobo/5,000,000grams = 0.2 kobo

Therefore, 100 gram = 0.2 × 100 = 20 kobo

The expression for the cost of each mix of stabilized soil matrix is given as

2.8 � + 0.4 � + 0.055 + 20 = Cost of stabilizing 100 grams of soil (kobo) (5.1)

Where, � = Cement content (%)

� = Bagasse ash content (%)

= Optimum moisture content (%)

The bagasse ash content would stand as the objective function in the model, therefore

there is need to attach cost to the function while formulating the model.

106

Table 5.1: Bagasse Ash Content and the Corresponding Attached Cost

Bagasse Ash

Content (%)

0 2 4 6 8 10 12 14 16 18 20

Cost (kobo) 0 0.8 1.6 2.4 3.2 4.0 4.8 5.6 6.4 7.2 8.0

5.2 Regression Models

The procedure of formulation of multiple regression models as discussed in section

3.5.1 was used and Minitab statistical software was also used to make it less rigorous.

The output is shown below:

Regression Analysis 1: Relationship of Cost of Bagasse Ash, Optimum Moisture Content, Cement Content, California Bearing Ratio and Unconfined Compressive Strength at 7 days. The regression equation is logZ = - 8.73 + 5.71 logP - 1.64 logE + 0.203 logC + 0.824 logS Predictor Coef StDev T P Constant -8.7348 0.7468 -11.70 0.000 logP 5.712 1.201 4.75 0.000 logE -1.6370 0.5364 -3.05 0.004 logC 0.2031 0.1731 1.17 0.249 logS 0.8244 0.6387 1.29 0.205 S = 0.1215 R-Sq = 85.8% R-Sq(adj) = 84.1% Analysis of Variance Source DF SS MS F P Regression 4 3.02479 0.75620 51.19 0.000 Error 34 0.50227 0.01477 Total 38 3.52706 Source DF Seq SS logP 1 2.23674 logE 1 0.73066 logC 1 0.03278 logS 1 0.02461 Regression Analysis 2: Relationship of Unconfined Compressive Strength at 7 days, Optimum Moisture Content, California Bearing Ratio and Cement Content.

107

The regression equation is logS = 0.103 + 1.59 logP + 0.0590 logC + 0.747 logE Predictor Coef StDev T P Constant 0.1033 0.1969 0.52 0.603 logP 1.5870 0.1707 9.30 0.000 logC 0.05898 0.04472 1.32 0.196 logE 0.74697 0.06491 11.51 0.000 S = 0.03217 R-Sq = 98.2% R-Sq(adj) = 98.1% Analysis of Variance Source DF SS MS F P Regression 3 2.01665 0.67222 649.70 0.000 Error 35 0.03621 0.00103 Total 38 2.05287 Source DF Seq SS logP 1 0.95673 logC 1 0.92289 logE 1 0.13704 Regression Analysis 3: Relationship of Cement Content, Optimum Moisture Content, California Bearing Ratio and Unconfined Compressive Strength at 7 days The regression equation is logE = 0.027 - 1.88 logP + 0.0697 logC + 1.06 logS Predictor Coef StDev T P Constant 0.0274 0.2353 0.12 0.908 logP -1.8797 0.2058 -9.13 0.000 logC 0.06967 0.05326 1.31 0.199 logS 1.05892 0.09201 11.51 0.000 S = 0.03830 R-Sq = 97.4% R-Sq(adj) = 97.2% Analysis of Variance Source DF SS MS F P Regression 3 1.92586 0.64195 437.67 0.000 Error 35 0.05134 0.00147 Total 38 1.97720 Source DF Seq SS logP 1 0.42451 logC 1 1.30708 logS 1 0.19427 5.2.1 Calibration and Verification of Models The regression equation in analysis 1 represents the relationship of cost of bagasse ash

as the dependent variable while optimum moisture content, cement content, California

bearing ratio and unconfined compressive strength at 7 days were the independent

variables. The square of coefficient of correlation R-sq and R-sq adjusted were 85.8%

and 84.1% respectively. The P-values of 0.000, 0.000, 0.004, 0.249 and 0.205 were

108

the level of significance for the constant value, logarithmic values of optimum

moisture content, cement content, California bearing ratio and the unconfined

compressive strength at 7 days respectively. All the P-values are much less than 0.5

which implies 5% level of significance. Therefore they had high level of significance

in the regression equation. The standard deviation ‘S’ of the equation is 0.1215 which

was quite low and consequently the model equation is dependable.

The regression equation in analysis 2 represents the relationship of unconfined

compressive strength at 7 days as the dependent variable while optimum moisture

content, California bearing ratio and cement content were the independent variables.

The square of coefficient of correlation R-sq and R-sq adjusted were 98.2% and

98.1% respectively. The P-values of 0.603, 0.000, 0.196 and 0.000 were the level of

significance for the constant value, logarithmic values of optimum moisture content,

California bearing ratio and cement content respectively. All the P-values were much

less than 0.5 which implies 5% level of significance. It was only that of the constant

that was higher but it is of less importance in the equation. Therefore the variables

had high level of significance in the regression equation. The standard deviation ‘S’ of

the equation was 0.03217 which was quite low and consequently the model equation

is dependable.

The regression equation in analysis 3 represents the relationship of cement content as

the dependent variable while optimum moisture content, California bearing ratio and

unconfined compressive strength at 7 days was the independent variables. The square

of coefficient of correlation R-sq and R-sq adjusted were 97.4% and 97.2%

respectively. The P-values of 0.908, 0.000, 0.199 and 0.000 were the level of

significance for the constant value, logarithmic values of optimum moisture content,

109

California bearing ratio and cement content respectively. All the P-values were much

less than 0.5 except that of the constant that is very high (almost 1). This is even very

satisfactory because the constant had very little importance in the equation. Thus, the

variables had high level of significance in the regression equation. The standard

deviation ‘S’ of the equation was 0.03830 which was quite low and consequently the

model equation is dependable.

The regression models were further verified using the experimental results. The last

two observations at 8% cement content (18% and 20% bagasse ash) were not used for

the regression model. Afterwards, they were used to compare the predicted results and

experimental results to test the conformity of the models’ predicted results to the

actual values of the properties of the stabilized soil.

Table 5.2: Comparison of Predicted Results to Experimental Results

Regression

Analysis

Predicted Results Experimental Results

18% 20% 18% 20%

Model 1 18.05% 19.67% 18% 20%

Model 2 1394.78kN/m2 1421.96kN/m2 1396kN/m2 1424kN/m2

Model 3 7.97% 7.98% 8% 8%

5.3 Non-linear Programming Model

The regression models could be used to form the non-linear programming model as

shown

Minimize:

� = 10-8.73 K5.71 �0.203 �0.824 �-1.64 ( 5.2)

Subject to:

110

100.103 K1.59 �0.059�0.747 ≥ 750 (5.3)

� ≥ 180 .(5.4)

100.027 K-1.88 �0.0697 �1.06 ≤ 5 (5.5)

� ≤ 190 (5.6)

≤ 23.5 (5.7)

� ≤ 760 (5.8)

Linearize the model

G�H� = −8.73G�H10 + 5.71G�H − 1.64G�H� + 0.203G�H� + 0.824G�H�

Subject to:

0.103G�H10 + 1.59 G�H + 0.0590G�H� + 0.747G�H� ≥ G�H750

G�H� ≥ G�H180

0.027G�H10 − 1.88G�H + 0.0697G�H� + 1.06G�H� ≤ G�H5

G�H� ≤ G�H190

G�H ≤ G�H23.5

G�H� ≤ G�H760

Let:

G�H� = e

G�H = r

G�H� = s

G�H� = t

G�H� = [

Thus the model becomes;

e = −8.73 + 5.71r − 1.64s + 0.203t + 0.824[

Subjected to:

0.103 + 1.59r + 0.0590t + 0.747s ≥ 2.875061

111

t ≥ 2.255273

0.027 − 1.88r + 0.0697t + 1.06[ ≤ 0.698970

t ≤ 2.278754

r ≤ 1.371068

[ ≤ 2.880814

Standard form

e − 5.71r + 1.64s − 0.203t − 0.824[ = −8.73

1.59r + 0.0590t + 0.747s − E1 + E2 = 2.772061

t − E3 + E4 = 2.255273

−1.88r + 0.0697t + 1.06[ + E5 = 0.671970

t + E6 = 2.278754

r + E7 = 1.371068

[ + E8 = 2.880814

Putting in matrix form and solving with the simplex method

First Matrix

The * shows the pivot number

Basic E2 E4 E5 E6 E7 E8 e

r 1.59 0 -1.88* 0 1 0 -5.71

t 0.0590 1 0.0697 1 0 0 -0.203

112

s 0.747 0 0 0 0 0 1.64

[ 0 0 1.06 0 0 1 -0.824

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0 0 1 0 0 0 0

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 2.772061 2.255273 0.671970 2.278754 1.371068 2.880814 -8.73

Check 5.168061 3.255273 0.921670 4.278754 3.371068 4.880814 2M-

13.827

The optimal values obtained after the first through the fifth iterations as shown in

Appendix I are given below;

Thus at optimal solution,

s = 0.654616

t = 2.278754

r = 1.351343

[ = 2.880814

e = 0.748975

In other words;

G�H� = s G�H� = t

� = 100.654616 = 4.51457%, � = 102.278754 = 190%,

113

G�H = 1.351343 G�H� = [

= 101.351343 = 22.456548% � = 102.880814 = 760 KN/m2

G�H� = e

� = 100.748975 = 5.610157

� = v.w1x1vyx.c = 14.025392%

Using Equation (5.1) to determine the cost of stabilizing 100grams of soil with this

mix

2.8�4.52� + 0.4�14.03� + 0.055�22.46� + 20 = 39.50 kobo

If the cement content is increased, it is necessary to observe the effect on the optimal

solution and ultimately the resulting cost for stabilizing 100 grams of soil in order to

have an effective comparison with the optimal solution. This could be achieved by

adjusting the right hand side constrained Equation (5.5) only to 7% cement content.

Minimize:

� = 10-8.73 K5.71 �0.203 �0.824 �-1.64 ( 5.2)

Subject to:

100.103 K1.59 �0.059�0.747 ≥ 750 (5.3)

� ≥ 180 .(5.4)

100.027 K-1.88 �0.0697 �1.06 ≤ 7 (5.5)

� ≤ 190 (5.6)

≤ 23.5 (5.7)

� ≤ 760 (5.8)

Using the procedure followed in model1, linearization of model followed by linear

optimization using simplex method as shown in Appendix II, the following results

were obtained;

114

Thus the solution is,

s = 0.820035

t = 2.278754

r = 1.273615

[ = 2.880814

e = 0.033861

In other words;

G�H� = s G�H� = t

� = 100.820035 = 6.61%, � = 102.278754 = 190%,

G�H = r G�H� = [

= 101.273615 = 18.78% , � = 102.880814 = 760 KN/m2

G�H� = e

� = 100.033861 = 1.081088

� = 1.xz1xzzx.c = 2.70%


mix

2.8�6.61� + 0.4�2.70� + 0.055�18.78� + 20 = 40.62 kobo

Considering stabilizing the soil with only cement (without bagasse ash)


mix

2.8�8� + 0.4�0� + 0.055�20.39� + 20 = 43.52 kobo

115

It is evident that the cost of two foregoing mixes (40.62 kobo and 43.52 kobo) for

stabilizing 100grams of soil would be significantly more expensive than the cost of

stabilizing with the optimal solution (39.50 kobo) in the long run when much weight

of the soil is being used for road construction work. This has clearly shown the cost

benefit of using bagasse ash as admixture. Besides, at 8% cement content with no

bagasse ash had California bearing ratio of 147.16% which fell short of the 180%

California bearing ratio value as stipulated by the Nigeria General Specification of

Road works and Bridges though it had a strength of 942 KN/m2.

Also concerning the constrained Equation (5.5), the adjustment from 5% cement

content to 7% cement content which is a 40% increase resulted in the decrease of the

optimal solution from 14.03% to 2.70% which is 80.76% drop. This goes to show that

Equation (5.5) is very sensitive in the linear programming model.

5.3.1 Sensitivity Analysis

Sensitivity analysis would be very necessary to examine how dependable the linear

programming model could be. This is performed by small adjustments of the

constraints and objective function to monitor the effect on the optimal solution. In

addition for purely local roads with very low volume of traffic, lower values of the

combination of California bearing ratio and unconfined compressive strength that

would be suitable not necessarily the standard could be adopted or alternatively a

lower cement content could be selected to determine the resultant optimal solution or

bagasse ash content required.

5.3.1.1 Senstivity Analysis on Constraints

It is necessary to carry out sensitivity analysis on the other constrained equations to

examine whether small changes in the right hand side would have any effect on the

116

optimal solution. These were performed by altering the right hand side of constrained

equations by -5%, -2.5%, +2.5% and +5% then allowing others to remain as they

were. In each case, the linear programming problem was solved to obtain the optimal

solution.

The optimal solutions for Sensitivity analysis on constrained equation 5.3 is shown

below and the linear programming with the iterations are shown in Appendix III.

At 2.5% decrease

e = 0.773158

G�H� = e

� = 100.773158 = 5.931411

v.{|1c11

x.c = 14.8285

% %ℎ#dH_ = 1c.z|51c.x|1c.x| × 100 = 5.70%

Performing the sensitivity analysis at other percentages, the results are

summarized in Table 5.4

Table 5.4: Change in Constraint with Corresponding Change in Optimal Solution for Constrained Equation (5.3) Change in Constraint (RHS) Change in Optimal Solution -2.5% 5.70% -5% 11.90% 0% 0% 2.5% -5.42% 5% -10.12%

117

Figure 5.1: Variations of Change in Constraint with Change in Optimal Solution for Constrained Equation (5.3) Figure 5.1 shows the relationship between changes in constraint with changes in

optimal objective values for constrained Equation (5.3). The relationship is almost a

linear one which has a slope of about 1:2. This implies that the optimal objective

value is differentiable, that is changes in optimal objective value with respect to small

changes in the constraint. This could assist in predicting the changes in the optimal

objective value as the constraint is being loosened or tightened. It is also evident that

constrained Equation (5.3) is sensitive.

The optimal solutions for sensitivity analysis on constrained Equation 5.6 are shown

below and the linear programming with the iterations is shown in Appendix V.

At 2.5% decrease

-15

-10

-5

0

5

10

15

-6 -4 -2 0 2 4 6C

ha

ng

e i

n O

pti

ma

l S

olu

tio

n (

%)

Change in Constraint (%)

118

e = 0.741568

G�H� = e

� = 100.741568 = 5.515286

Table 5.5: Change in Constraint with Corresponding Change in Optimal

Solution for Constrained Equation (5.6)

Change in Constraint (RHS) Change in Optimal Solution

-2.5% -1.71%

-5% -3.42%

0% 0%

2.5% 1.64%

5% 3.35%

Figure 5.2: Variations of Change in Constraint with Change in Optimal


-4

-3

-2

-1

0

1

2

3

4

-6 -4 -2 0 2 4 6

Ch

an

ge

in

Op

tim

al

So

luti

on

(%

)


v.v1vbzw

x.c = 13.79%

% %ℎ#dH_ = 1|.y{51c.x|1c.x| × 100 = −1.71%

119

Figure 5.2 shows the relationship between changes in constraint with changes in


linear one which has a flatter slope of about 1:0.7. This implies that the optimal

objective value changes very slightly with small changes in the right hand side of

constrained Equation (5.6). This also shows that constrained Equation (5.6) is slightly

sensitive.

The optimal solutions for sensitivity analysis on constrained equation 5.8 are shown

below and the linear programming with the iterations is shown in Appendix VI.

At 2.5% decrease

e = 0.682870

G�H� = e

� = 100.682870 = 4.818036

Performing the sensitivity analysis at the other percentages, the results are

summarized in Table 5.6

Table 5.6: Change in Constraint with Corresponding Change in Optimal Solution for Constrained Equation (5.8) Change in Constraint (RHS) Change in Optimal Solution -2.5% -14.11%

-5% -26.51%

0% 0%

2.5% 15.97%

5% 34.07%

c.z1zx|w

x.c = 12.05%

% %ℎ#dH_ = 1b.xv51c.x| 1c.x| × 100 = −14.11%

120

Figure 5.3: Variations of Change in Constraint with Change in Optimal


Figure 5.3 shows the relationship between changes in constraint with changes in


linear one which has a steep slope of about 1:6. This implies that there are great

changes in optimal objective value with respect to small changes in the right hand side

of the constrained equation. This could assist in predicting the changes in the optimal

objective value as the constraint is being loosened or tightened. It is also evident that

constrained Equation (5.8) seems to be the most sensitive of all the constrained

equations.

The sensitivity analysis on constrained Equation 5.4 shows virtually no change in the

optimal solution and the linear programming with the iterations are shown in

Appendix IV. This simply means that it is virtually insensitive in the non-linear

programming model. Similarly, constrained Equation (5.7) also appears to be

-30

-20

-10

0

10

20

30

40

-6 -4 -2 0 2 4 6

Ch

an

ge

in

Op

tim

al

So

luti

on

(%

)


121

insensitive because after the fifth iteration all the basic variables had already been

reduced to zero. Thus there was no point selecting a pivot number from it for further

iteration. Not with standing that they appear to be insensitive, it is still very necessary

not to relax them because they are very relevant in ensuring that the linear

programming model is solvable. The model contains four basic variables with two

surplus variables which required at least six constraints to make it solvable.

5.3.1.2 Senstivity Analysis on the Objective Function

It is necessary to carry out sensitivity analysis on the objective function to examine

what effect small changes in the coefficients of the variables of the objective function

would have on the optimal solution. These were performed by altering the coefficient

of each variable in the objective function by -5%, -2.5%, +2.5% and +5% and then

allowing the other coefficients to remain as they were. In each case, the linear

programming problem was solved to obtain the optimal solution.

The optimal solutions for sensitivity analysis on the coefficient of variable for

unconfined compressive strength in the objective function are shown in Table 5.7 and

Figure 5.4.

Table 5.7: Change in the Coefficient of Unconfined Compressive Strength

with the Corresponding Change in Optimal Solution

Change in Coefficient Change in Optimal Solution -2.5% -12.40%

-5% -23.81%

0% 0%

2.5% 14.97%

5% 32.15%

122

Figure 5.4: Variations of Change in Coefficient of Unconfined Compressive

Strength with Change in Optimal Solution

Figure 5.4 shows the relationship between changes in the coefficient of unconfined

compressive strength with changes in optimal objective values. The relationship is

almost a linear one with a positive slope which is a clear indication that as the

unconfined compressive strength increased, the optimal solution (bagasse ash content)

also increased at the rate of a steep slope of about 1:5. This implies that there were

great changes in optimal objective value with respect to small changes in the

coefficient of the unconfined compressive strength. This also agrees with the

statistical level of significance of 0.205 as presented in regression analysis of the

objective function equation. This could assist in predicting the changes in the optimal

objective value as the coefficient of unconfined compressive strength is being

increased or decreased.

-30

-20

-10

0

10

20

30

40

-6 -4 -2 0 2 4 6

Ch

an

ge

in

Op

tim

al

So

luti

on

(%

)Change in Coefficient (%)

123

The optimal solutions for sensitivity analysis on the coefficient of variable for

California bearing ratio in the objective function are shown in Table 5.8 and Figure

5.5.

Table 5.8: Change in the Coefficient of California Bearing Ratio with the

Corresponding Change in Optimal Solution

Change in Coefficient Change in Optimal Solution

-2.5% -2.57%

-5% -5.14%

0% 0%

2.5% 2.64%

5% 5.49%

Figure 5.5: Variations of Change in Coefficient of California Bearing Ratio

with Change in Optimal Solution

-6

-4

-2

0

2

4

6

8

-6 -4 -2 0 2 4 6

Ch

an

ge

in

Op

tim

al

So

luti

on

(%

)

Change in Coefficient (%)

124

Figure 5.5 shows the relationship between changes in the coefficient of California

bearing ratio with changes in optimal objective values. The relationship is almost a

linear one with a positive slope which presents the optimal solution (bagasse ash

content) to be increased as the California bearing ratio increased with a flatter slope of

about 1:1. This implies that there were small changes but very significant in optimal

objective value with respect to small changes in the coefficient of California bearing

ratio. This also agrees with the statistical level of significance of 0.249 which was the

least as presented in regression analysis of the objective function Equation. This could

assist in predicting the changes in the optimal objective value as the coefficient of

unconfined compressive strength is being increased or decreased.

The optimal solutions for sensitivity analysis on the coefficient of variable for cement

content in the objective function are shown in Table 5.9 and Figure 5.6.

Table 5.9: Change in the Coefficient of Cement Content with the



-2.5% 6.20%

-5% 12.78%

0% 0%

2.5% -5.99%

5% -11.62%

125

Figure 5.6: Variations of Change in Cement Content with Change in

Optimal Solution

Figure 5.6 shows the relationship between changes in the coefficient of cement

content with changes in optimal objective values. The relationship is almost a linear

one with a negative slope which is a clear indication that as the cement content

increased, the optimal solution (bagasse ash content) decreased at the rate of a slope

of about 1:2.5. This implies that there were very significant changes in optimal

objective value with respect to small changes in the coefficient of cement content. The

statistical level of significance was presented in regression analysis as 0.004 in the

objective function equation which is higher relative to the sensitivity. However this

could assist in predicting the changes in the optimal objective value as the coefficient

of cement content is being increased or decreased.

-15

-10

-5

0

5

10

15

-6 -4 -2 0 2 4 6

Ch

an

ge

in

Op

tim

al

So

luti

on

(%

)Change in Coefficient (%)

126

The optimal solutions for sensitivity analysis on the coefficient of variable for cement

content in the objective function are shown in Table 5.10 and Figure 5.7.

Table 5.10: Change in the Coefficient of Optimum Moisture Content with the



-2.5% -34.07%

-5% -56.45%

0% 0%

2.5% 39.26%

5% 64.21%

Figure 5.7: Variations of Change in the Coefficient of Optimum Moisture

Content with Change in Optimal Solution

-80

-60

-40

-20

0

20

40

60

80

-6 -4 -2 0 2 4 6

Ch

an

ge

in

Op

tim

al

So

luti

on

(%

)

Change in Coefficients (%)

127

Figure 5.7 shows the relationship between changes in the coefficient of Optimum

Moisture Content with changes in optimal objective values. The relationship is almost

a linear one with a positive slope which goes to show that as Optimum Mosture

Content increased, the optimal solution (bagasse ash content) also increased at the rate

of a very steep slope of about 1:12. This implies that there were great changes in

optimal objective value with respect to small changes in the coefficient of optimum

moisture content. This is also in conformity with the statistical level of significance of

0.000 as presented in regression analysis of the objective function equation. This

could also assist in predicting the changes in the optimal objective value as the

coefficient of optimum moisture content is being increased or decreased.

5.4 Application of Scheffe’s Simplex Regression Model

This method requires that the proportions of the constituent materials should be in

volume and soil stabilization requires measuring the materials in weight. In other

words, the proportions of these materials should be converted to volume

measurements using their densities.

5.4.1 Determination of Densities of Materials

Specific gravity, D = ~�� 2� Q��~�� 2� ��

Specific gravity of the lateritic soil = 2.75

Density of the lateritic soil = 2.75 × 1HW/WG = 2.75 HW/WG

Similarly,

Specific gravity of bagasse ash = 2.25

Density of bagasse ash = 2.25 × 1HW/WG = 2.25 HW/WG

128

Shetty (2005) presented the specific volume of Ordinary Portland Cement to be

0.319 WG/HW.

�K_%`L`% ��G�W_ �L W#'_�`#G = 1~�� 2� Q��

k_d,`'F �L %_W_d' = 1x.|1{ = 3.14 HW/WG

5.4.2 Formulation of Optimization Models

The 10 observation points for the formulation of the optimization model were selected

such that the sample space for the data would be covered while the vertices where

only one pure constituent exist would be at some point imaginery. The California

bearing ratio and the unconfined compressive strength at 7 days which are the

evaluation criterions were used as the response function. They are shown as follows;

Table 5.11: Mix Proportions in Mass with the Corresponding Response

Functions

S/N � � � � � �

1 2 2 16.80 100 23.57 228

2 2 10 20.23 100 25.11 308

3 2 20 22.62 100 24.23 364

4 4 2 17.97 100 84.44 454

5 4 20 23.54 100 160.96 733

6 6 2 18,41 100 93.70 642

7 6 20 24.44 100 239.16 1073

8 8 2 20.56 100 175.12 998

9 8 10 22.63 100 230.24 1180

10 8 20 25.31 100 276.30 1424

129

Table 5.12: Mix Proportions in Volume with the Corresponding Pseudo Mixes

s/N � e � Z Z1 Z2 Z3 Z4

1 0.636943 0.888889 16.80 36.363636 54.689468 1.000000 0.000000 0.000000 0.000000

2 0.636943 4.444445 20.23 36.363636 61.675024 0.010328 0.072062 0.328010 0.589601

3 0.636943 8.888889 22.62 36.363636 68.509468 0.000000 1.000000 0.000000 0.000000

4 1.273885 0.888889 17.97 36.363636 56.496410 0.022548 0.015734 0.318073 0.643645

5 1.273885 8.888889 23.54 36.363636 70.066410 0.018181 0.126864 0.335967 0.518988

6 1.910828 0.888889 18.41 36.363636 57.573353 0.033190 0.015439 0.319766 0.631605

7 1.910828 8.888889 24.44 36.363636 71.603353 0.026686 0.124141 0.341325 0.507848

8 2.547771 0.888889 20.56 36.363636 60.360296 0.000000 0.000000 0.000000 1.000000

9 2.547771 4.444445 22.63 36.363636 65.985852 0.038611 0.067355 0.342952 0.551082

10 2.547771 8.888889 25.31 36.363636 73.110296 0.000000 0.000000 1.000000 0.000000

Regression Analysis 4: Optimization Model for Unconfined Compressive Strength at 7 Days The regression equation is S = 228 (2X1-1)X1 + 364 (2X2-1)X2 + 1424 (2X3-1)X3 + 998 (2X4-1)X4 + 229878 X1X2 + 51060 X1X3 - 88.2 X1X4 - 18894 X2X3 + 13872X2X4 - 233 X3X4 Predictor Coef StDev T P Noconstant (2X1-1)X 228.000 0.000 * * (2X2-1)X 364.000 0.000 * * (2X3-1)X 1424.00 0.00 * * (2X4-1)X 998.000 0.000 * * X1X2 229878 0 * * X1X3 51059.6 0.0 * * X1X4 -88.2451 0.0000 * * X2X3 -18894.1 0.0 * * X2X4 13871.6 0.0 * * X3X4 -233.106 0.000 * * S = * Analysis of Variance Source DF SS MS F P Regression 10 7002422 700242 * * Error 0 * * Total 10 7002422 Regression Analysis 5: Optimization Model for Calfornia Bearing Ratio The regression equation is C = 23.6 (2X1-1)X1 + 24.2 (2X2-1)X2 + 276 (2X3-1)X3 + 175 (2X4-1)X4 + 88038 X1X2 + 318 X1X3 - 109 X1X4 - 7256 X2X3 + 3843 X2X4

130

+ 246 X3X4 Predictor Coef StDev T P Noconstant (2X1-1)X 23.5700 0.0000 * * (2X2-1)X 24.2300 0.0000 * * (2X3-1)X 276.300 0.000 * * (2X4-1)X 175.120 0.000 * * X1X2 88038.0 0.0 * * X1X3 317.733 0.000 * * X1X4 -109.055 0.000 * * X2X3 -7256.33 0.00 * * X2X4 3843.24 0.00 * * X3X4 245.985 0.000 * * S = * Analysis of Variance Source DF SS MS F P Regression 10 260807.7 26080.8 * * Error 0 * * Total 10 260807.7 5.4.2 Validation and Verification of the Scheffe’s Optimization Models

For the purpose of validation of the optimization equations, extra ten points of

observations were selected randomly within the sample space and the student’s t-test

was used to test them. The points are shown as follows;

Table 5.13: Mix Proportions in Mass with the Corresponding Response

Functions for the Validation of Scheffe’s Optimization Models

S/N � � � � � �

11 2 6 18.74 100 26.48 273

12 2 16 22.01 100 24.70 349

131

13 4 8 20.48 100 109.13 575

14 4 10 21.29 100 121.03 613

15 4 14 22.17 100 152.10 665

16 6 6 20.85 100 117.07 801

17 6 12 22.71 100 176.12 941

18 6 18 24.23 100 220.08 1057

19 8 4 21.24 100 196.37 1049

20 8 16 24.69 100 265.30 1366

Table 5.14: Mix Proportions in Volume with the Corresponding Pseudo Mixes

for the Validation of Scheffe’s Optimization Models

S/N � � � � � �1 �2 �3 �4 11 0.636943 2.666667 18.74 36.363636 54.407246 0.010905 0.045656 0.320851 0.622588

12 0.636943 7.111111 22.01 36.363636 66.121690 0.009633 0.107546 0.332871 0.549950

13 1.273885 3.555556 20.48 36.363636 61.673077 0.020656 0.057652 0.332074 0.589619

14 1.273885 4.444444 21.29 36.363636 63.371965 0.020102 0.070133 0.335953 0.573813

15 1.273885 6.222222 22.17 36.363636 66.029743 0.019293 0.094234 0.335758 0.550716

16 1.910828 2.666667 20.85 36.363636 61.791131 0.030924 0.043156 0.337427 0.588493

17 1.910828 5.333333 22.71 36.363636 66.317797 0.028813 0.080421 0.342442 0.548324

18 1.910828 8.000000 24.23 36.363636 70.504464 0.027102 0.113468 0.343666 0.515764

19 2.547771 1.777778 21.24 36.363636 61.929185 0.041140 0.028707 0.342972 0.587181

20 2.547771 7.111111 24.69 36.363636 70.712518 0.036030 0.100564 0.349160 0.514246

Table 5.15: Statistical Student’s Two-Tailed T-Test for Unconfined Compressive

Strength

S/N �E �M �i = �E – �M �A – �i (�A – �i�2

11 273 334.63 -61.63 64.11 4110.09

12 349 366.36 -17.36 19.84 393.63

132

13 575 610.71 -35.71 38.19 1458.48

14 613 637.33 -24.33 26.81 718.78

15 665 693.39 -28.39 30.87 952.96

16 801 795.75 5.25 -2.77 7.67

17 941 951.14 -10.14 12.62 159.26

18 1057 1040.42 16.58 -14.10 198.81

19 1049 921.02 127.98 -125.50 15750.25

20 1366 1313.44 52.56 -50.08 2508.01

∑24.81 ∑26,257.94

j = 10

kA= bc.z11x = 2.481

t#�`#d%_ = bwbvy.{c{ = 2917.55

�'#d�#�� k_�`#'`�d = √2917.55 = 54.01

Actual value of total variation in t-test

' = b.cz1×√1xvc.x1 = 0.145 = 0.15

k_H�__ �L ��__��W = 10 − 1 = 9 5% �`Hd`L`%#d%_ L�� ')�-'#`G_� '_,' = 2.5%

100% − 2.5% = 97.5% = 0.975

Allowable total variation in t-test obtained from statistical table is 2.26

Thus, 2.26 > 0.15

Therefore null hypothesis is accepted and alternative hypothesis rejected. Thus there

is no significant difference between the laboratory and the predicted results of

unconfined compressive strength.

133

Table 5.16: Statistical Student’s Two-Tailed T-Test for California Bearing Ratio

S/N �E �M �i = �E – �M �A – �i (�A – �i�2

11 26.48 90.03 -63.55 46.16 2130.75

12 24.70 80.86 -56.16 38.77 1503.11

13 109.13 131.60 -22.47 5.08 25.81

14 121.03 138.63 -17.60 0.21 0.05

15 152.10 153.35 -1.25 -16.14 260.50

16 117.07 145.96 -28.89 11.50 132.25

17 176.12 198.44 -22.32 4.93 24.31

18 220.08 228.18 -8.1 -9.29 86.31

19 196.37 135.35 61.02 -78.41 6148.13

20 265.30 279.83 -14.53 -2.86 8.18

� −173.85 ∑10319.40

j = 10

kA= 1y|.zv1x = −17.39

t#�`#d%_ = 1x|1{.cx{ = 1,146.60

�'#d�#�� k_�`#'`�d = √1146.60 = 33.86

Actual value of total variation in t-test

' = 51y.|{×√1x||.zw = −1.624

k_H�__ �L ��__��W = 10 − 1 = 9 5% �`Hd`L`%#d%_ L�� ')�-'#`G_� '_,' = 2.5%

100% − 2.5% = 97.5% = 0.975

Allowable total variation in t-test obtained from statistical table is 2.26

134

Thus, 2.26 < 1.624

Therefore null hypothesis is accepted and alternative hypothesis rejected. Thus there

is no significant difference between laboratory and predicted results of California

bearing ratio.

The classical optimization has edge over the Scheffe’s simplex regression method

because it can use as many points as possible in formulating equations while the

foregoing uses limited number of points in formulating the optimization model which

is considered to be grossly inadequate to match the complexity of soil stabilization.

Even if the degree of the polynomial is raised in order to increase the number of

points required, the short-coming in model could be more compounded. Obams

(2006) has attempted to verify the accuracy of Scheffe’s third degree over the second

degree polynomials and the diference was not very significant. The Scheffe’s simplex

regression method might be useful in optimization in concrete because concrete is

mostly made of coarse aggregates which are almost inert in reaction with cement.

However in soil stabilization of this nature is more complex in that the minerals

present in the soil and the bagasse ash were all involved in the reaction with cement

because they are pozzolanic nature. Another advantage of the classical optimization

over the Scheffe’s simplex regression method is that it can handle or consider all the

properties invovlved at the same time to predict a more reliable optimum point unlike

the later which can only handle one property at a time for formulating optimization

model. Thus for roadwork that requires more than one property for judgement, the

classical optimization would be preferable. For sub-base of road work using the

lateritic soil 14.03%, 4.52% and 22.46% by weight of the dry soil for bagasse ash,

cement and optimum moisture content respectively were predicted to satisfy

135

California bearing ratio and unconfined compressive strength at the most minimum

cost of 39.50 kobo for stabilizing 100 grams of the lateritic soil.

CHAPTER SIX

CONCLUSIONS AND RECOMMENDATIONS

6.1 Conclusions

After the investigation into the effects of bagasse ash on the properties of a reddish-

brown lateritic soil classified to be A-6(2) in the AASHTO rating and SC (Clayey

Sand) in the Unified Classification System, the following conclusions were drawn:

136

1. The lateritic soil contains non-problematic clay minerals and thus would be

non-swelling.

2. The bagasse ash has been confirmed to be a good pozzolana or admixture.

3. The increase in bagasse ash content increased the optimum moisture content

but reduced the maximum dry density of the cement-stabilized lateritic soil.

4. The increase in cement content increased the optimum moisture content and

maximum dry density of the lateritic soil treated with bagasse ash and cement.

5. The increase in bagasse ash content improved the strength properties of the

cement-stabilized soil.

6. The lateritic soil treated with bagasse ash and cement satisfied the requirement

to be as sub-base of road work.

7. The optimum content for bagasse ash and cement for the lateritic soil to be

used as Sub-base of a roadwork is 14.03% and 4.52% respectively by the

weight of dry soil for an economic mix while the optimum moisture content

for the economic mix is 22.46%. .

8. The cost of material of stabilizing with the economic mix is 39.50 kobo for

stabilizing 100 grams of lateritic soil as against 43.52 kobo for stabilizing with

only cement.

9. The classical optimization was preferred over Scheffe’s simplex regression

method for optimization in soil stabilization.

6.2 Recommendations

The soil treated with bagasse ash and cement could only satisfy the requirement for

sub- base of road work. However, 8% and 20% of cement and bagasse ash

respectively by the weight of the dry soil could be considered for the base of the local

light-trafficked roads.

137

Soils have perculiarities and variations in engineering behaviour with regards to their

response to the addition of cement and other admixtures. In other words the results

and observations are only exclusively recommended for lateritic soil deposit in Ndoro

in Ikwuano local government area of Abia State. This study had in no doubt shown

that the cement requirement for road work could be substantially reduced to optimum

level and partially replaced by bagasse ash for road work to reduce the cost of

materials. Ultimately, the design, construction, maintenance and re-building of low-

cost roads would be very possible in the environs. These have already been in use in

most developing countries especially in Asia; Nigeria could also tap into these

potentials for the provision of road networks. Furthermore, Federal government has

been projecting for vision 2020 and Millennium Development Goals (MDG). This

could be one way of ensuring that they are achieved as projected. This is possible

because if low-cost roads are being provided and adequately maintained to reach most

rural farmers even in the hinter-land that are hitherto cut-off from transporting their

agricultural products to the urban dwellers. It would assist in ensuring the availability

of food and raw materials for the small and medium scale industries which enhance

the socio-economic relationship between the urban and rural.

For further study, this kind of cost benefit analysis should be extended to other soil

deposits and with various admixtures like rice husk ash, sugar-cane straw ash, palm

kernel husk ash and so on because they are all pozzolanic in nature. These admixtures

are readily available in the country in large quantities and could even constitute

environmental problems if they are not properly handled. This study would be with a

view of finding the one that is the most effective admixture and of lowest cost in order

to maintain the cost of road work very low.

138

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APPENDIX I

Iteration for the optimal solution are shown below

First Iteration


Basic E2 E4 r E6 E7 E8 e

r 0 0 1 0 0 0 0

146

t 0.117949 1 -0.037075 1 0.037075 0 -0.414698

s 0.747 0 0 0 0 0 1.64

[ 0.896490 0 -0.563830 0 0.563830 1* -4.043469

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 3.340396 2.255273 -0.357431 2.278754 1.728499 2.880814 -10.770931

Check 5.947559 3.255273 -0.490250 4.278754 3.861318 4.880814 2M-

16.626328

Second Iteration


Basic E2 E4 r E6 E7 [ e

r 0 0 1 0 0 0 0

t 0.117949 1* -0.037075 1 0.037075 0 -0.414698

147

s 0.747 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.757775 2.255273 1.266858 2.278754 0.104210 2.880814 0.877551

Check 1.571958 3.255273 2.261699 4.278754 1.109369 4.880814 2M+

3.109092

Third Iteration The * shows the pivot number

Basic E2 t r E6 E7 [ e

r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

148

s 0.747* 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0.117949 -1 -0.037075 1 0.037075 0 -0.414698

E4 -0.117949 1 0.037075 -1 -0.037075 0 M+

0.414698

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.491768 2.255273 1.350472 0.023481 0.020596 2.880814 1.812808

Check 1.188002 3.255273 2.382388 1.023481 0.988680 4.880814 2M+

4.459047

Fourth Iteration


Basic s t r E6 E7 [ e

r 0 0 1 0 0 0 0

149

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0.157897 -1 -0.037075 1* 0.037075 0 -0.673649

E4 -0.157897 1 0.037075 -1 -0.037075 0 M+

0.673649

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.658324 2.255273 1.350472 0.023481 0.020596 2.880814 0.733157

Check 1.590364 3.255273 2.382388 1.023481 0.988680 4.880814 2M+

1.850850

Fifth Iteration



150

r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0 0 0 1 0 0 0

E4 0 0 0 -1 0 0 M

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 -0.157897 1 0.037075 1 -0.037075 0 0.673649

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.654616 2.278754 1.351343 0.023481 0.019725 2.880814 0.748975

Check 1.428759 4.278754 2.420334 1.023481 0.950734 4.880814 2M+

2.540317

APPENDIX II

Linear programming and iterations when the right hand side of constrained equation

(5.5) was increased by 40% are shown as follows.

151

Linearize the model

G�H� = −8.73G�H10 + 5.71G�H − 1.64G�H� + 0.203G�H� + 0.824G�H�

Subject to:

0.103G�H10 + 1.59 G�H + 0.0590G�H� + 0.747G�H� ≥ G�H750

G�H� ≥ G�H180

0.027G�H10 − 1.88G�H + 0.0697G�H� + 1.06G�H� ≤ G�H7

G�H� ≤ G�H190

G�H ≤ G�H23.5

G�H� ≤ G�H760

Let:

G�H� = e

G�H = r

G�H� = s

G�H� = t

G�H� = [


e = −8.73 + 5.71r − 1.64s + 0.203t + 0.824[

Subjected to:

0.103 + 1.59r + 0.0590t + 0.747s ≥ 2.875061

t ≥ 2.255273

0.027 − 1.88r + 0.0697t + 1.06[ ≤ 0.845098

t ≤ 2.278754

r ≤ 1.371068

[ ≤ 2.880814

Standard form

152

e − 5.71r + 1.64s − 0.203t − 0.824[ = −8.73

1.59r + 0.0590t + 0.747s − E1 + E2 = 2.772061

t − E3 + E4 = 2.255273

−1.88r + 0.0697t + 1.06[ + E5 = 0.818098

t + E6 = 2.278754

r + E7 = 1.371068

[ + E8 = 2.880814


First Matrix


153


r 1.59 0 -1.88* 0 1 0 -5.71

t 0.0590 1 0.0697 1 0 0 -0.203

s 0.747 0 0 0 0 0 1.64

[ 0 0 1.06 0 0 1 -0.824

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0 0 1 0 0 0 0

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 2.772061 2.255273 0.818098 2.278754 1.371068 2.880814 -8.73

Check 5.168061 3.255273 1.067798

4.278754 3.371068 4.880814 2M-13.827

First Iteration


154


r 0 0 1 0 0 0 0

t 0.117949 1 -0.037075 1 0.037075 0 -0.414698

s 0.747 0 0 0 0 0 1.64

[ 0.896490 0 -0.563830 0 0.563830 1* -4.043469

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 3.463964 2.255273 -0.435159 2.278754 1.806227 2.880814 -11.214758

Check 6.071146 3.255273 -0.567978 4.278754 3.939046 4.880814 2M-

17.070154

Second Iteration


155


r 0 0 1 0 0 0 0

t 0.117949 1* -0.037075 1 0.037075 0 -0.414698

s 0.747 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.881343 2.255273 1.189130 2.278754 0.181938 2.880814 0.433724

Check 1.695545 3.255273 2.183971 4.278754 1.187097 4.880814 2M+

2.665266

Third Iteration


156


r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 0.747* 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0.117949 -1 -0.037075 1 0.037075 0 -0.414698

E4 -0.117949 1 0.037075 -1 -0.037075 0 M+

0.414698

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.615336 2.255273 1.272744 0.023481 0.098324 2.880814 1.368981

Check 1.311589 3.255273 2.304660 1.023481 1.066408 4.880814 2M+

4.015221

Fourth Iteration


157


r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0.157897 -1 -0.037075 1* 0.037075 0 -0.673649

E4 -0.157897 1 0.037075 -1 -0.037075 0 M+

0.673649

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.823743 2.255273 1.272744 0.023481 0.098324 2.880814 0.018043

Check 1.755809 3.255273 2.304660 1.023481 1.066408 4.880814 2M+

1.135694

Fifth Iteration


158


r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0 0 0 1 0 0 0

E4 0 0 0 -1 0 0 M

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 -0.157897 1 0.037075 1 -0.037075 0 0.673649

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.820035 2.278754 1.273615 0.023481 0.097453 2.880814 0.033861

Check 1.594204 4.278754 2.342606 1.023481 1.028462 4.880814 2M+

1.825161

APPENDIX III

Sensitivity Analysis on Constrained Equation (5.3)

159

Decreasing the right hand side of constrained Equation (5.3) by 2.5% and allowing

the other constrained equations to remain as they were:

Minimize:

� = 10-8.73 K5.71 �0.203 �0.824 �-1.64 …………………………………………………(5.2)

Subject to:

100.103 K1.59 �0.059�0.747 ≥ 731.25 ……………………………………………….(5.3)

� ≥ 180 ………………………………………………………………………..(5.4)

100.027 K-1.88 �0.0697 �1.06 ≤ 5 ……………………………………………………….(5.5)

� ≤ 190 ………………………………………………………………………….(5.6)

≤ 23.5 …………………………………………………………………………(5.7)

� ≤ 760 …………………………………………………………………………(5.8)

Linearize the model

G�H� = −8.73G�H10 + 5.71G�H − 1.64G�H� + 0.203G�H� + 0.824G�H�

Subject to:

0.103G�H10 + 1.59 G�H + 0.0590G�H� + 0.747G�H� ≥ G�H731.25

G�H� ≥ G�H180

0.027G�H10 − 1.88G�H + 0.0697G�H� + 1.06G�H� ≤ G�H5

G�H� ≤ G�H190

G�H ≤ G�H23.5

G�H� ≤ G�H760

Let:

G�H� = e

G�H = r

G�H� = s

G�H� = t

160

G�H� = [


e = −8.73 + 5.71r − 1.64s + 0.203t + 0.824[

Subjected to:

0.103 + 1.59r + 0.0590t + 0.747s ≥ 2.864066

t ≥ 2.255273

0.027 − 1.88r + 0.0697t + 1.06[ ≤ 0.698970

t ≤ 2.278754

r ≤ 1.371068

[ ≤ 2.880814

Standard form e − 5.71r + 1.64s − 0.203t − 0.824[ = −8.73

1.59r + 0.0590t + 0.747s − E1 + E2 = 2.761066

t − E3 + E4 = 2.255273

−1.88r + 0.0697t + 1.06[ + E5 = 0.671970

t + E6 = 2.278754

r + E7 = 1.371068

[ + E8 = 2.880814


First Matrix

161



r 1.59 0 -1.88* 0 1 0 -5.71

t 0.0590 1 0.0697 1 0 0 -0.203

s 0.747 0 0 0 0 0 1.64

[ 0 0 1.06 0 0 1 -0.824

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0 0 1 0 0 0 0

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 2.761066 2.255273 0.671970 2.278754 1.371068 2.880814 -8.73

Check 5.157066 3.255273 0.921670 4.278754 3.371068 4.880814 2M-13.827

First Iteration


162


r 0 0 1 0 0 0 0

t 0.117949 1 -0.037075 1 0.037075 0 -0.414698

s 0.747 0 0 0 0 0 1.64

[ 0.896490 0 -0.563830 0 0.563830 1* -4.043469

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 3.329381 2.255273 -0.357431 2.278754 1.728499 2.880814 -10.770931

Check 5.936564 3.255273 -0.490250 4.278754 3.861318 4.880814 2M-

16.626328

Second Iteration


163


r 0 0 1 0 0 0 0

t 0.117949 1* -0.037075 1 0.037075 0 -0.414698

s 0.747 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.746760 2.255273 1.266858 2.278754 0.104210 2.880814 0.877551

Check 1.560963 3.255273 2.261699 4.278754 1.109369 4.880814 2M+

3.109092



164

r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 0.747* 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0.117949 -1 -0.037075 1 0.037075 0 -0.414698

E4 -0.117949 1 0.037075 -1 -0.037075 0 M+

0.414698

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.480753 2.255273 1.350472 0.023481 0.020596 2.880814 1.812808

Check 1.177007 3.255273 2.382388 1.023481 0.988680 4.880814 2M+

4.459047

Fourth Iteration



165

r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0.157897 -1 -0.037075 1* 0.037075 0 -0.673649

E4 -0.157897 1 0.037075 -1 -0.037075 0 M+

0.673649

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.643578 2.255273 1.350472 0.023481 0.020596 2.880814 0.757340

Check 1.575645 3.255273 2.382388 1.023481 0.988680 4.880814 2M+

1.874989

Fifth Iteration


166


r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0 0 0 1 0 0 0

E4 0 0 0 -1 0 0 M

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 -0.157897 1 0.037075 1 -0.037075 0 0.673649

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.639870 2.278754 1.351343 0.023481 0.019725 2.880814 0.773158

Check 1.414040 4.278754 2.420334 1.023481 0.950734 4.880814 2M+

2.564456

Decreasing the right hand side of constrained Equation (5.3) by 5% and allowing the

other constrained equations to remain as they were:

167

Minimize:

� = 10-8.73 K5.71 �0.203 �0.824 �-1.64 …………………………………………………...( 5.2)

Subject to:

100.103 K1.59 �0.059�0.747 ≥ 712.5 …………………………………………………...(5.3)

� ≥ 180 …………………………………………………………………………..(5.4)

100.027 K-1.88 �0.0697 �1.06 ≤ 5 ………………………………………………………….(5.5)

� ≤ 190 …………………………………………………………………………….(5.6)

≤ 23.5 ……………………………………………………………………………(5.7)

� ≤ 760 ……………………………………………………………………………(5.8)

Linearize the model

G�H� = −8.73G�H10 + 5.71G�H − 1.64G�H� + 0.203G�H� + 0.824G�H�

Subject to:

0.103G�H10 + 1.59 G�H + 0.0590G�H� + 0.747G�H� ≥ G�H712.5

G�H� ≥ G�H180

0.027G�H10 − 1.88G�H + 0.0697G�H� + 1.06G�H� ≤ G�H5

G�H� ≤ G�H190

G�H ≤ G�H23.5

G�H� ≤ G�H760

Let:

G�H� = e

G�H = r

G�H� = s

G�H� = t

G�H� = [


168

e = −8.73 + 5.71r − 1.64s + 0.203t + 0.824[

Subjected to: e − 5.71r + 1.64s − 0.203t − 0.824[ = −8.73

0.103 + 1.59r + 0.0590t + 0.747s ≥ 2.852785

t ≥ 2.255273

0.027 − 1.88r + 0.0697t + 1.06[ ≤ 0.698970

t ≤ 2.278754

r ≤ 1.371068

[ ≤ 2.880814


1.59r + 0.0590t + 0.747s − E1 + E2 = 2.749785

t − E3 + E4 = 2.255273

−1.88r + 0.0697t + 1.06[ + E5 = 0.671970

t + E6 = 2.278754

r + E7 = 1.371068

[ + E8 = 2.880814


First Matrix

169



r 1.59 0 -1.88* 0 1 0 -5.71

t 0.0590 1 0.0697 1 0 0 -0.203

s 0.747 0 0 0 0 0 1.64

[ 0 0 1.06 0 0 1 -0.824

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0 0 1 0 0 0 0

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 2.749785 2.255273 0.671970 2.278754 1.371068 2.880814 -8.73

Check 5.145785 3.255273 0.921670 4.278754 3.371068 4.880814 2M-13.827

First Iteration


170


r 0 0 1 0 0 0 0

t 0.117949 1 -0.037075 1 0.037075 0 -0.414698

s 0.747 0 0 0 0 0 1.64

[ 0.896490 0 -0.563830 0 0.563830 1* -4.043469

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 3.318100 2.255273 -0.357431 2.278754 1.728499 2.880814 -10.770931

Check 5.925283 3.255273 -0.490250 4.278754 3.861318 4.880814 2M-

16.626328

Second Iteration


171


r 0 0 1 0 0 0 0

t 0.117949 1* -0.037075 1 0.037075 0 -0.414698

s 0.747 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.735479 2.255273 1.266858 2.278754 0.104210 2.880814 0.877551

Check 1.549682 3.255273 2.261699 4.278754 1.109369 4.880814 2M+

3.109092

Third Iteration


172


r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 0.747* 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0.117949 -1 -0.037075 1 0.037075 0 -0.414698

E4 -0.117949 1 0.037075 -1 -0.037075 0 M+

0.414698

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.469472 2.255273 1.350472 0.023481 0.020596 2.880814 1.812808

Check 1.165726 3.255273 2.382388 1.023481 0.988680 4.880814 2M+

4.459047

Fourth Iteration


173


r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0.157897 -1 -0.037075 1* 0.037075 0 -0.673649

E4 -0.157897 1 0.037075 -1 -0.037075 0 M+

0.673649

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.628477 2.255273 1.350472 0.023481 0.020596 2.880814 0.782106

Check 1.560544 3.255273 2.382388 1.023481 0.988680 4.880814 2M+

1.899755

Fifth Iteration


174


r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0 0 0 1 0 0 0

E4 0 0 0 -1 0 0 M

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 -0.157897 1 0.037075 1 -0.037075 0 0.673649

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.624769 2.278754 1.351343 0.023481 0.019725 2.880814 0.797924

Check 1.398939 4.278754 2.420334 1.023481 0.950734 4.880814 2M+

2.589222

Increasing the right hand side of constrained Equation (5.3) by 2.5% and allowing the

other constrained equation to remain as they were:

175

Minimize:

� = 10-8.73 K5.71 �0.203 �0.824 �-1.64 …………………………………………………(5.2)

Subject to:

100.103 K1.59 �0.059�0.747 ≥ 768.75 ……………………………………………….(5.3)

� ≥ 180 ………………………………………………………………………..(5.4)

100.027 K-1.88 �0.0697 �1.06 ≤ 5 ……………………………………………………….(5.5)

� ≤ 190 ………………………………………………………………………….(5.6)

≤ 23.5 ………………………………………………………………………...(5.7)

� ≤ 760 ………………………………………………………………………...(5.8)

Linearize the model

G�H� = −8.73G�H10 + 5.71G�H − 1.64G�H� + 0.203G�H� + 0.824G�H�

Subject to:

0.103G�H10 + 1.59 G�H + 0.0590G�H� + 0.747G�H� ≥ G�H768.75

G�H� ≥ G�H180

0.027G�H10 − 1.88G�H + 0.0697G�H� + 1.06G�H� ≤ G�H5

G�H� ≤ G�H190

G�H ≤ G�H23.5

G�H� ≤ G�H760

Let:

G�H� = e

G�H = r

G�H� = s

G�H� = t

G�H� = [


176

e = −8.73 + 5.71r − 1.64s + 0.203t + 0.824[

Subjected to:

0.103 + 1.59r + 0.0590t + 0.747s ≥ 2.885785

t ≥ 2.255273

0.027 − 1.88r + 0.0697t + 1.06[ ≤ 0.698970

t ≤ 2.278754

r ≤ 1.371068

[ ≤ 2.880814


1.59r + 0.0590t + 0.747s − E1 + E2 = 2.782785

t − E3 + E4 = 2.255273

−1.88r + 0.0697t + 1.06[ + E5 = 0.671970

t + E6 = 2.278754

r + E7 = 1.371068

[ + E8 = 2.880814


First Matrix

177



r 1.59 0 -1.88* 0 1 0 -5.71

t 0.0590 1 0.0697 1 0 0 -0.203

s 0.747 0 0 0 0 0 1.64

[ 0 0 1.06 0 0 1 -0.824

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0 0 1 0 0 0 0

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 2.782785 2.255273 0.671970 2.278754 1.371068 2.880814 -8.73

Check 5.178785 3.255273 0.921670 4.278754 3.371068 4.880814 2M-13.827

First Iteration


178


r 0 0 1 0 0 0 0

t 0.117949 1 -0.037075 1 0.037075 0 -0.414698

s 0.747 0 0 0 0 0 1.64

[ 0.896490 0 -0.563830 0 0.563830 1* -4.043469

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 3.351100 2.255273 -0.357431 2.278754 1.728499 2.880814 -10.770931

Check 5.958283 3.255273 -0.490250 4.278754 3.861318 4.880814 2M-

16.626328

Second Iteration


179


r 0 0 1 0 0 0 0

t 0.117949 1* -0.037075 1 0.037075 0 -0.414698

s 0.747 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.768479 2.255273 1.266858 2.278754 0.104210 2.880814 0.877551

Check 1.582682 3.255273 2.261699 4.278754 1.109369 4.880814 2M+

3.109092

Third Iteration


180


r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 0.747* 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0.117949 -1 -0.037075 1 0.037075 0 -0.414698

E4 -0.117949 1 0.037075 -1 -0.037075 0 M+

0.414698

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.502472 2.255273 1.350472 0.023481 0.020596 2.880814 1.812808

Check 1.198726 3.255273 2.382388 1.023481 0.988680 4.880814 2M+

4.459047

Fourth Iteration


181


r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0.157897 -1 -0.037075 1* 0.037075 0 -0.673649

E4 -0.157897 1 0.037075 -1 -0.037075 0 M+

0.673649

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.672653 2.255273 1.350472 0.023481 0.020596 2.880814 0.709657

Check 1.604720 3.255273 2.382388 1.023481 0.988680 4.880814 2M+

1.827306

Fifth Iteration


182


r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0 0 0 1 0 0 0

E4 0 0 0 -1 0 0 M

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 -0.157897 1 0.037075 1 -0.037075 0 0.673649

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.668945 2.278754 1.351343 0.023481 0.019725 2.880814 0.725475

Check 1.428759 4.278754 2.420334 1.023481 0.950734 4.880814 2M+

2.51677

Increasing the right hand side of constrained Equation (5.3) by 5% and allowing the


183

Minimize:

� = 10-8.73 K5.71 �0.203 �0.824 �-1.64 …………………………………………………...(

5.2)

Subject to:

100.103 K1.59 �0.059�0.747 ≥ 787.5 ….…………………………………………………..(5.3)

� ≥ 180 …………………………………………………………………………..(5.4)

100.027 K-1.88 �0.0697 �1.06 ≤ 5 ………………………………………………………….(5.5)

� ≤ 190 …………………………………………………………………………….(5.6)

≤ 23.5 ……………………………………………………………………………(5.7)

� ≤ 760 …………………………………………………………………………....(5.8)

Linearize the model

G�H� = −8.73G�H10 + 5.71G�H − 1.64G�H� + 0.203G�H� + 0.824G�H�

Subject to:

0.103G�H10 + 1.59 G�H + 0.0590G�H� + 0.747G�H� ≥ G�H787.5

G�H� ≥ G�H180

0.027G�H10 − 1.88G�H + 0.0697G�H� + 1.06G�H� ≤ G�H5

G�H� ≤ G�H190

G�H ≤ G�H23.5

G�H� ≤ G�H760

Let:

G�H� = e

G�H = r

G�H� = s

G�H� = t

G�H� = [

184


e = −8.73 + 5.71r − 1.64s + 0.203t + 0.824[

Subjected to:

0.103 + 1.59r + 0.0590t + 0.747s ≥ 2.896251

t ≥ 2.255273

0.027 − 1.88r + 0.0697t + 1.06[ ≤ 0.698970

t ≤ 2.278754

r ≤ 1.371068

[ ≤ 2.880814


1.59r + 0.0590t + 0.747s − E1 + E2 = 2.793251

t − E3 + E4 = 2.255273

−1.88r + 0.0697t + 1.06[ + E5 = 0.671970

t + E6 = 2.278754

r + E7 = 1.371068

[ + E8 = 2.880814


First Matrix

185



r 1.59 0 -1.88* 0 1 0 -5.71

t 0.0590 1 0.0697 1 0 0 -0.203

s 0.747 0 0 0 0 0 1.64

[ 0 0 1.06 0 0 1 -0.824

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0 0 1 0 0 0 0

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 2.793251 2.255273 0.671970 2.278754 1.371068 2.880814 -8.73

Check 5.189251 3.255273 0.921670 4.278754 3.371068 4.880814 2M-13.827

First Iteration


186


r 0 0 1 0 0 0 0

t 0.117949 1 -0.037075 1 0.037075 0 -0.414698

s 0.747 0 0 0 0 0 1.64

[ 0.896490 0 -0.563830 0 0.563830 1* -4.043469

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 3.361566 2.255273 -0.357431 2.278754 1.728499 2.880814 -10.770931

Check 5.968749 3.255273 -0.490250 4.278754 3.861318 4.880814 2M-

16.626328

Second Iteration


187


r 0 0 1 0 0 0 0

t 0.117949 1* -0.037075 1 0.037075 0 -0.414698

s 0.747 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.778945 2.255273 1.266858 2.278754 0.104210 2.880814 0.877551

Check 1.593148 3.255273 2.261699 4.278754 1.109369 4.880814 2M+

3.109092

Third Iteration


188


r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 0.747* 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0.117949 -1 -0.037075 1 0.037075 0 -0.414698

E4 -0.117949 1 0.037075 -1 -0.037075 0 M+

0.414698

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.512938 2.255273 1.350472 0.023481 0.020596 2.880814 1.812808

Check 1.209192 3.255273 2.382388 1.023481 0.988680 4.880814 2M+

4.459047

Fourth Iteration


189


r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0.157897 -1 -0.037075 1* 0.037075 0 -0.673649

E4 -0.157897 1 0.037075 -1 -0.037075 0 M+

0.673649

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.686664 2.255273 1.350472 0.023481 0.020596 2.880814 0.686679

Check 1.618731 3.255273 2.382388 1.023481 0.988680 4.880814 2M+

1.804328

Fifth Iteration


190


r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0 0 0 1 0 0 0

E4 0 0 0 -1 0 0 M

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 -0.157897 1 0.037075 1 -0.037075 0 0.673649

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.682956 2.278754 1.351343 0.023481 0.019725 2.880814 0.702497

Check 1.457126 4.278754 2.420334 1.023481 0.950734 4.880814 2M+

2.493795

191

At 5% decrease

e = 0.797924

G�H� = e

� = 100.797924 = 6.279485

w.by{czv

x.c = 15.6987%

% %ℎ#dH_ = 1v.yx51c.x|1c.x| × 100 = 11.90%

At 2.5% increase

e = 0.725475

G�H� = e

� = 100.725475 = 5.314654

v.|1cwvc

x.c = 13.2866%

% %ℎ#dH_ = 1|.by51c.x|1c.x| × 100 = −5.42%

e = 0.702497

At 5% increase

G�H� = e

� = 100.702497 = 5.040771

v.xcxyy1

x.c = 12.6019%

% %ℎ#dH_ = 1b.w151c.x|1c.x| × 100 = −10.12%

192

APPENDIX IV

Sensitivity Analysis on Constrain Equation (5.4)



Minimize:

� = 10-8.73 K5.71 �0.203 �0.824 �-1.64 ……………………………………………………( 5.2)

Subject to:

100.103 K1.59 �0.059�0.747 ≥ 750 ……………………………………………………(5.3)

� ≥ 175.5 ……………………………………………………………………….(5.4)

100.027 K-1.88 �0.0697 �1.06 ≤ 5 …………………………………………………………(5.5)

� ≤ 190 ……………………………………………………………………………(5.6)

≤ 23.5 …………………………………………………………………………..(5.7)

� ≤ 760 …………………………………………………………………………..(5.8)

Linearize the model

G�H� = −8.73G�H10 + 5.71G�H − 1.64G�H� + 0.203G�H� + 0.824G�H�

Subject to:

0.103G�H10 + 1.59 G�H + 0.0590G�H� + 0.747G�H� ≥ G�H750

G�H� ≥ G�H175.5

0.027G�H10 − 1.88G�H + 0.0697G�H� + 1.06G�H� ≤ G�H5

G�H� ≤ G�H190

G�H ≤ G�H23.5

G�H� ≤ G�H760

Let:

G�H� = e

G�H = r

193

G�H� = s

G�H� = t

G�H� = [


e = −8.73 + 5.71r − 1.64s + 0.203t + 0.824[

Subjected to:

0.103 + 1.59r + 0.0590t + 0.747s ≥ 2.875061

t ≥ 2.244277

0.027 − 1.88r + 0.0697t + 1.06[ ≤ 0.698970

t ≤ 2.278754

r ≤ 1.371068

[ ≤ 2.880814

Standard form

e − 5.71r + 1.64s − 0.203t − 0.824[ = −8.73

1.59r + 0.0590t + 0.747s − E1 + E2 = 2.772061

t − E3 + E4 = 2.244277

−1.88r + 0.0697t + 1.06[ + E5 = 0.671970

t + E6 = 2.278754

r + E7 = 1.371068

[ + E8 = 2.880814

194


First Matrix



r 1.59 0 -1.88* 0 1 0 -5.71

t 0.0590 1 0.0697 1 0 0 -0.203

s 0.747 0 0 0 0 0 1.64

[ 0 0 1.06 0 0 1 -0.824

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0 0 1 0 0 0 0

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 2.772061 2.244277 0.671970 2.278754 1.371068 2.880814 -8.73

Check 5.168061 3.244277 0.921670 4.278754 3.371068 4.880814 2M-13.827

195

First Iteration



r 0 0 1 0 0 0 0

t 0.117949 1 -0.037075 1 0.037075 0 -0.414698

s 0.747 0 0 0 0 0 1.64

[ 0.896490 0 -0.563830 0 0.563830 1* -4.043469

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 3.340396 2.244277 -0.357431 2.278754 1.728499 2.880814 -10.770931

Check 5.947559 3.244277 -0.490250 4.278754 3.861318 4.880814 2M-

16.626328

196

Second Iteration



r 0 0 1 0 0 0 0

t 0.117949 1* -0.037075 1 0.037075 0 -0.414698

s 0.747 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.757775 2.244277 1.266858 2.278754 0.104210 2.880814 0.877551

Check 1.571958 3.244277 2.261699 4.278754 1.109369 4.880814 2M+

3.109092

197

Third Iteration



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 0.747* 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0.117949 -1 -0.037075 1 0.037075 0 -0.414698

E4 -0.117949 1 0.037075 -1 -0.037075 0 M+

0.414698

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.493065 2.244277 1.350065 0.034477 0.021003 2.880814 1.808248

Check 1.189299 3.244277 2.381981 1.034477 0.989087 4.880814 2M+

4.454487

198

Fourth Iteration



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0.157897 -1 -0.037075 1* 0.037075 0 -0.673649

E4 -0.157897 1 0.037075 -1 -0.037075 0 M+

0.673649

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.660060 2.244277 1.350065 0.034477 0.021003 2.880814 0.725750

Check 1.592100 3.244277 2.381981 1.034477 0.989087 4.880814 2M+

1.843443

199

Fifth Iteration



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0 0 0 1 0 0 0

E4 0 0 0 -1 0 0 M

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 -0.157897 1 0.037075 1 -0.037075 0 0.673649

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.654616 2.278754 1.351343 0.034477 0.019725 2.880814 0.748975

Check 1.428759 4.278754 2.420334 1.034477 0.950734 4.880814 2M+

2.540317

200



Minimize:

� = 10-8.73 K5.71 �0.203 �0.824 �-1.64 ……………………………………………………( 5.2)

Subject to:

100.103 K1.59 �0.059�0.747 ≥ 750 ……………………………………………………(5.3)

� ≥ 171 …………………………………………………………………………(5.4)

100.027 K-1.88 �0.0697 �1.06 ≤ 5 …………………………………………………………(5.5)

� ≤ 190 ……………………………………………………………………………(5.6)

≤ 23.5 …………………………………………………………………………..(5.7)

� ≤ 760 …………………………………………………………………………..(5.8)

Linearize the model

G�H� = −8.73G�H10 + 5.71G�H − 1.64G�H� + 0.203G�H� + 0.824G�H�

Subject to:

0.103G�H10 + 1.59 G�H + 0.0590G�H� + 0.747G�H� ≥ G�H750

G�H� ≥ G�H171

0.027G�H10 − 1.88G�H + 0.0697G�H� + 1.06G�H� ≤ G�H5

G�H� ≤ G�H190

G�H ≤ G�H23.5

G�H� ≤ G�H760

Let:

G�H� = e

G�H = r

G�H� = s

G�H� = t

201

G�H� = [


e = −8.73 + 5.71r − 1.64s + 0.203t + 0.824[

Subjected to:

0.103 + 1.59r + 0.0590t + 0.747s ≥ 2.875061

t ≥ 2.232996

0.027 − 1.88r + 0.0697t + 1.06[ ≤ 0.698970

t ≤ 2.278754

r ≤ 1.371068

[ ≤ 2.880814

Standard form

e − 5.71r + 1.64s − 0.203t − 0.824[ = −8.73

1.59r + 0.0590t + 0.747s − E1 + E2 = 2.772061

t − E3 + E4 = 2.232996

−1.88r + 0.0697t + 1.06[ + E5 = 0.671970

t + E6 = 2.278754

r + E7 = 1.371068

[ + E8 = 2.880814

202


First Matrix



r 1.59 0 -1.88* 0 1 0 -5.71

t 0.0590 1 0.0697 1 0 0 -0.203

s 0.747 0 0 0 0 0 1.64

[ 0 0 1.06 0 0 1 -0.824

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0 0 1 0 0 0 0

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 2.772061 2.232996 0.671970 2.278754 1.371068 2.880814 -8.73

Check 5.168061 3.232996 0.921670 4.278754 3.371068 4.880814 2M-13.827

203

First Iteration



r 0 0 1 0 0 0 0

t 0.117949 1 -0.037075 1 0.037075 0 -0.414698

s 0.747 0 0 0 0 0 1.64

[ 0.896490 0 -0.563830 0 0.563830 1* -4.043469

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 3.340396 2.232996 -0.357431 2.278754 1.728499 2.880814 -10.770931

Check 5.947559 3.232996 -0.490250 4.278754 3.861318 4.880814 2M-

16.626328

204

Second Iteration



r 0 0 1 0 0 0 0

t 0.117949 1* -0.037075 1 0.037075 0 -0.414698

s 0.747 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.757775 2.232996 1.266858 2.278754 0.104210 2.880814 0.877551

Check 1.571958 3.232996 2.261699 4.278754 1.109369 4.880814 2M+

3.109092

205

Third Iteration



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 0.747* 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0.117949 -1 -0.037075 1 0.037075 0 -0.414698

E4 -0.117949 1 0.037075 -1 -0.037075 0 M+

0.414698

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.494395 2.232996 1.349646 0.045758 0.021422 2.880814 1.803570

Check 1.190629 3.232996 2.381562 1.045758 0.989506 4.880814 2M+

4.449809

206

Fourth Iteration



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0.157897 -1 -0.037075 1* 0.037075 0 -0.673649

E4 -0.157897 1 0.037075 -1 -0.037075 0 M+

0.673649

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.661841 2.232996 1.349646 0.045758 0.021422 2.880814 0.718151

Check 1.593881 3.232996 2.381562 1.045758 0.989506 4.880814 2M+

1.835844

207

Fifth Iteration



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0 0 0 1 0 0 0

E4 0 0 0 -1 0 0 M

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 -0.157897 1 0.037075 1 -0.037075 0 0.673649

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.654616 2.278754 1.351343 0.045758 0.019726 2.880814 0.748976

Check 1.428759 4.278754 2.420334 1.045758 0.950735 4.880814 2M+

2.540318

208


other constrained equation to remain as they were:

Minimize:

� = 10-8.73 K5.71 �0.203 �0.824 �-1.64 ……………………………………………………( 5.2)

Subject to:

100.103 K1.59 �0.059�0.747 ≥ 750 ……………………………………………………(5.3)

� ≥ 184.5 ……………………………………………………………………….(5.4)

100.027 K-1.88 �0.0697 �1.06 ≤ 5 …………………………………………………………(5.5)

� ≤ 190 ……………………………………………………………………………(5.6)

≤ 23.5 …………………………………………………………………………..(5.7)

� ≤ 760 …………………………………………………………………………..(5.8)

Linearize the model

G�H� = −8.73G�H10 + 5.71G�H − 1.64G�H� + 0.203G�H� + 0.824G�H�

Subject to:

0.103G�H10 + 1.59 G�H + 0.0590G�H� + 0.747G�H� ≥ G�H750

G�H� ≥ G�H184.5

0.027G�H10 − 1.88G�H + 0.0697G�H� + 1.06G�H� ≤ G�H5

G�H� ≤ G�H190

G�H ≤ G�H23.5

G�H� ≤ G�H760

Let:

G�H� = e

G�H = r

G�H� = s

G�H� = t

209

G�H� = [


e = −8.73 + 5.71r − 1.64s + 0.203t + 0.824[

Subjected to:

0.103 + 1.59r + 0.0590t + 0.747s ≥ 2.875061

t ≥ 2.265996

0.027 − 1.88r + 0.0697t + 1.06[ ≤ 0.698970

t ≤ 2.278754

r ≤ 1.371068

[ ≤ 2.880814

Standard form

e − 5.71r + 1.64s − 0.203t − 0.824[ = −8.73

1.59r + 0.0590t + 0.747s − E1 + E2 = 2.772061

t − E3 + E4 = 2.265996

−1.88r + 0.0697t + 1.06[ + E5 = 0.671970

t + E6 = 2.278754

r + E7 = 1.371068

[ + E8 = 2.880814

210


First Matrix



r 1.59 0 -1.88* 0 1 0 -5.71

t 0.0590 1 0.0697 1 0 0 -0.203

s 0.747 0 0 0 0 0 1.64

[ 0 0 1.06 0 0 1 -0.824

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0 0 1 0 0 0 0

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 2.772061 2.265996 0.671970 2.278754 1.371068 2.880814 -8.73

Check 5.168061 3.265996 0.921670 4.278754 3.371068 4.880814 2M-13.827

211

First Iteration



r 0 0 1 0 0 0 0

t 0.117949 1 -0.037075 1 0.037075 0 -0.414698

s 0.747 0 0 0 0 0 1.64

[ 0.896490 0 -0.563830 0 0.563830 1* -4.043469

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 3.340396 2.265996 -0.357431 2.278754 1.728499 2.880814 -10.770931

Check 5.947559 3.265996 -0.490250 4.278754 3.861318 4.880814 2M-

16.626328

212

Second Iteration



r 0 0 1 0 0 0 0

t 0.117949 1* -0.037075 1 0.037075 0 -0.414698

s 0.747 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.757775 2.265996 1.266858 2.278754 0.104210 2.880814 0.877551

Check 1.571958 3.265996 2.261699 4.278754 1.109369 4.880814 2M+

3.109092

213

Third Iteration



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 0.747* 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0.117949 -1 -0.037075 1 0.037075 0 -0.414698

E4 -0.117949 1 0.037075 -1 -0.037075 0 M+

0.414698

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.490503 2.265996 1.350870 0.012758 0.020918 2.880814 1.817255

Check 1.186737 3.265996 2.382786 1.012758 0.988282 4.880814 2M+

4.463494

214

Fourth Iteration



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0.157897 -1 -0.037075 1* 0.037075 0 -0.673649

E4 -0.157897 1 0.037075 -1 -0.037075 0 M+

0.673649

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.656631 2.265996 1.350870 0.012758 0.020198 2.880814 0.740380

Check 1.588671 3.265996 2.382786 1.012758 0.988282 4.880814 2M+

1.858074

215

Fifth Iteration



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0 0 0 1 0 0 0

E4 0 0 0 -1 0 0 M

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 -0.157897 1 0.037075 1 -0.037075 0 0.673649

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.654617 2.278754 1.351343 0.019725 0.019725 2.880814 0.748974

Check 1.428760 4.278754 2.420334 0.950734 0.950734 4.880814 2M+

2.540317

216



Minimize:

� = 10-8.73 K5.71 �0.203 �0.824 �-1.64 ……………………………………………………( 5.2)

Subject to:

100.103 K1.59 �0.059�0.747 ≥ 750 ……………………………………………………(5.3)

� ≥ 189 …………………………………………………………………………(5.4)

100.027 K-1.88 �0.0697 �1.06 ≤ 5 …………………………………………………………(5.5)

� ≤ 190 ……………………………………………………………………………(5.6)

≤ 23.5 …………………………………………………………………………..(5.7)

� ≤ 760 …………………………………………………………………………..(5.8)

Linearize the model

G�H� = −8.73G�H10 + 5.71G�H − 1.64G�H� + 0.203G�H� + 0.824G�H�

Subject to:

0.103G�H10 + 1.59 G�H + 0.0590G�H� + 0.747G�H� ≥ G�H750

G�H� ≥ G�H189

0.027G�H10 − 1.88G�H + 0.0697G�H� + 1.06G�H� ≤ G�H5

G�H� ≤ G�H190

G�H ≤ G�H23.5

G�H� ≤ G�H760

Let:

G�H� = e

G�H = r

G�H� = s

G�H� = t

217

G�H� = [


e = −8.73 + 5.71r − 1.64s + 0.203t + 0.824[

Subjected to:

0.103 + 1.59r + 0.0590t + 0.747s ≥ 2.875061

t ≥ 2.276462

0.027 − 1.88r + 0.0697t + 1.06[ ≤ 0.698970

t ≤ 2.278754

r ≤ 1.371068

[ ≤ 2.880814

Standard form

e − 5.71r + 1.64s − 0.203t − 0.824[ = −8.73

1.59r + 0.0590t + 0.747s − E1 + E2 = 2.772061

t − E3 + E4 = 2.276462

−1.88r + 0.0697t + 1.06[ + E5 = 0.671970

t + E6 = 2.278754

r + E7 = 1.371068

[ + E8 = 2.880814

218


First Matrix



r 1.59 0 -1.88* 0 1 0 -5.71

t 0.0590 1 0.0697 1 0 0 -0.203

s 0.747 0 0 0 0 0 1.64

[ 0 0 1.06 0 0 1 -0.824

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0 0 1 0 0 0 0

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 2.772061 2.276462 0.671970 2.278754 1.371068 2.880814 -8.73

Check 5.168061 3.276462 0.921670 4.278754 3.371068 4.880814 2M-13.827

219

First Iteration



r 0 0 1 0 0 0 0

t 0.117949 1 -0.037075 1 0.037075 0 -0.414698

s 0.747 0 0 0 0 0 1.64

[ 0.896490 0 -0.563830 0 0.563830 1* -4.043469

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 3.340396 2.276462 -0.357431 2.278754 1.728499 2.880814 -10.770931

Check 5.947559 3.276462 -0.490250 4.278754 3.861318 4.880814 2M-

16.626328

220

Second Iteration



r 0 0 1 0 0 0 0

t 0.117949 1* -0.037075 1 0.037075 0 -0.414698

s 0.747 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.757775 2.276462 1.266858 2.278754 0.104210 2.880814 0.877551

Check 1.571958 3.276462 2.261699 4.278754 1.109369 4.880814 2M+

3.109092

221

Third Iteration



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 0.747* 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0.117949 -1 -0.037075 1 0.037075 0 -0.414698

E4 -0.117949 1 0.037075 -1 -0.037075 0 M+

0.414698

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.489269 2.276462 1.351258 0.002292 0.019810 2.880814 1.821595

Check 1.185503 3.276462 2.383174 1.002292 0.987894 4.880814 2M+

4.467834

222

Fourth Iteration



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0.157897 -1 -0.037075 1* 0.037075 0 -0.673649

E4 -0.157897 1 0.037075 -1 -0.037075 0 M+

0.673649

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.654979 2.276462 1.351258 0.002292 0.019810 2.880814 0.747429

Check 1.587019 3.276462 2.383174 1.002292 0.987894 4.880814 2M+

1.865123

223

Fifth Iteration



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0 0 0 1 0 0 0

E4 0 0 0 -1 0 0 M

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 -0.157897 1 0.037075 1 -0.037075 0 0.673649

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.654617 2.278754 1.351343 0.002292 0.019725 2.880814 0.748973

Check 1.428760 4.278754 2.420334 1.002292 0.950734 4.880814 2M+

2.540316

224

APPENDIX V




Minimize:

� = 10-8.73 K5.71 �0.203 �0.824 �-1.64 ……………………………………………………( 5.2)

Subject to:

100.103 K1.59 �0.059�0.747 ≥ 750 ……………………………………………………(5.3)

� ≥ 180 …………………………………………………………………………(5.4)

100.027 K-1.88 �0.0697 �1.06 ≤ 5 …………………………………………………………(5.5)

� ≤ 185.25 …………………………………………………………………………(5.6)

≤ 23.5 …………………………………………………………………………..(5.7)

� ≤ 760 …………………………………………………………………………..(5.8)

Linearize the model

G�H� = −8.73G�H10 + 5.71G�H − 1.64G�H� + 0.203G�H� + 0.824G�H�

Subject to:

0.103G�H10 + 1.59 G�H + 0.0590G�H� + 0.747G�H� ≥ G�H750

G�H� ≥ G�H180

0.027G�H10 − 1.88G�H + 0.0697G�H� + 1.06G�H� ≤ G�H5

G�H� ≤ G�H185.25

G�H ≤ G�H23.5

G�H� ≤ G�H760

Let:

G�H� = e

G�H = r

225

G�H� = s

G�H� = t

G�H� = [


e = −8.73 + 5.71r − 1.64s + 0.203t + 0.824[

Subjected to:

0.103 + 1.59r + 0.0590t + 0.747s ≥ 2.875061

t ≥ 2.255273

0.027 − 1.88r + 0.0697t + 1.06[ ≤ 0.698970

t ≤ 2.267758

r ≤ 1.371068

[ ≤ 2.880814

Standard form

e − 5.71r + 1.64s − 0.203t − 0.824[ = −8.73

1.59r + 0.0590t + 0.747s − E1 + E2 = 2.772061

t − E3 + E4 = 2.255273

−1.88r + 0.0697t + 1.06[ + E5 = 0.671970

t + E6 = 2.267758

r + E7 = 1.371068

[ + E8 = 2.880814

226


First Matrix



r 1.59 0 -1.88* 0 1 0 -5.71

t 0.0590 1 0.0697 1 0 0 -0.203

s 0.747 0 0 0 0 0 1.64

[ 0 0 1.06 0 0 1 -0.824

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0 0 1 0 0 0 0

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 2.772061 2.255273 0.671970 2.267758 1.371068 2.880814 -8.73

Check 5.168061 3.255273 0.921670 4.267758 3.371068 4.880814 2M-13.827

227

First Iteration



r 0 0 1 0 0 0 0

t 0.117949 1 -0.037075 1 0.037075 0 -0.414698

s 0.747 0 0 0 0 0 1.64

[ 0.896490 0 -0.563830 0 0.563830 1* -4.043469

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 3.340396 2.255273 -0.357431 2.267758 1.728499 2.880814 -10.770931

Check 5.947559 3.255273 -0.490250 4.267758 3.861318 4.880814 2M-

16.626328

228

Second Iteration



r 0 0 1 0 0 0 0

t 0.117949 1* -0.037075 1 0.037075 0 -0.414698

s 0.747 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.757775 2.255273 1.266858 2.267758 0.104210 2.880814 0.877551

Check 1.571958 3.255273 2.261699 4.267758 1.109369 4.880814 2M+

3.109092

229

Third Iteration



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 0.747* 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0.117949 -1 -0.037075 1 0.037075 0 -0.414698

E4 -0.117949 1 0.037075 -1 -0.037075 0 M+

0.414698

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.491768 2.255273 1.350472 0.012485 0.020596 2.880814 1.812808

Check 1.188002 3.255273 2.382388 1.012485 0.988680 4.880814 2M+

4.459047

230

Fourth Iteration



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0.157897 -1 -0.037075 1* 0.037075 0 -0.673649

E4 -0.157897 1 0.037075 -1 -0.037075 0 M+

0.673649

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.658324 2.255273 1.350472 0.012485 0.020596 2.880814 0.733157

Check 1.590364 3.255273 2.382388 1.012485 0.988680 4.880814 2M+

1.850850

231

Fifth Iteration



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0 0 0 1 0 0 0

E4 0 0 0 -1 0 0 M

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 -0.157897 1 0.037075 1 -0.037075 0 0.673649

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.654616 2.278754 1.351343 0.012485 0.019725 2.880814 0.741568

Check 1.428759 4.278754 2.420334 1.012485 0.950734 4.880814 2M+

2.532910

232



Minimize:

� = 10-8.73 K5.71 �0.203 �0.824 �-1.64 ……………………………………………………( 5.2)

Subject to:

100.103 K1.59 �0.059�0.747 ≥ 750 ……………………………………………………(5.3)

� ≥ 180 ………………………………………………………………………….(5.4)

100.027 K-1.88 �0.0697 �1.06 ≤ 5 …………………………………………………………(5.5)

� ≤ 180.5 …………………………………………………………………………(5.6)

≤ 23.5 …………………………………………………………………………..(5.7)

� ≤ 760 …………………………………………………………………………..(5.8)

Linearize the model

G�H� = −8.73G�H10 + 5.71G�H − 1.64G�H� + 0.203G�H� + 0.824G�H�

Subject to:

0.103G�H10 + 1.59 G�H + 0.0590G�H� + 0.747G�H� ≥ G�H750

G�H� ≥ G�H180

0.027G�H10 − 1.88G�H + 0.0697G�H� + 1.06G�H� ≤ G�H5

G�H� ≤ G�H180.5

G�H ≤ G�H23.5

G�H� ≤ G�H760

Let:

G�H� = e

G�H = r

G�H� = s

G�H� = t

233

G�H� = [


e = −8.73 + 5.71r − 1.64s + 0.203t + 0.824[

Subjected to:

0.103 + 1.59r + 0.0590t + 0.747s ≥ 2.875061

t ≥ 2.255273

0.027 − 1.88r + 0.0697t + 1.06[ ≤ 0.698970

t ≤ 2.278754

r ≤ 1.371068

[ ≤ 2.880814

Standard form

e − 5.71r + 1.64s − 0.203t − 0.824[ = −8.73

1.59r + 0.0590t + 0.747s − E1 + E2 = 2.772061

t − E3 + E4 = 2.255273

−1.88r + 0.0697t + 1.06[ + E5 = 0.671970

t + E6 = 2.256477

r + E7 = 1.371068

[ + E8 = 2.880814

234


First Matrix



r 1.59 0 -1.88* 0 1 0 -5.71

t 0.0590 1 0.0697 1 0 0 -0.203

s 0.747 0 0 0 0 0 1.64

[ 0 0 1.06 0 0 1 -0.824

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0 0 1 0 0 0 0

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 2.772061 2.255273 0.671970 2.256477 1.371068 2.880814 -8.73

Check 5.168061 3.255273 0.921670 4.256477 3.371068 4.880814 2M-13.827

235

First Iteration



r 0 0 1 0 0 0 0

t 0.117949 1 -0.037075 1 0.037075 0 -0.414698

s 0.747 0 0 0 0 0 1.64

[ 0.896490 0 -0.563830 0 0.563830 1* -4.043469

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 3.340396 2.255273 -0.357431 2.256477 1.728499 2.880814 -10.770931

Check 5.947559 3.255273 -0.490250 4.256477 3.861318 4.880814 2M-

16.626328

236

Second Iteration



r 0 0 1 0 0 0 0

t 0.117949 1* -0.037075 1 0.037075 0 -0.414698

s 0.747 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.757775 2.255273 1.266858 2.256477 0.104210 2.880814 0.877551

Check 1.571958 3.255273 2.261699 4.256477 1.109369 4.880814 2M+

3.109092

237

Third Iteration



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 0.747* 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0.117949 -1 -0.037075 1 0.037075 0 -0.414698

E4 -0.117949 1 0.037075 -1 -0.037075 0 M+

0.414698

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.491768 2.255273 1.350472 0.001204 0.020596 2.880814 1.812808

Check 1.188002 3.255273 2.382388 1.001204 0.988680 4.880814 2M+

4.459047

238

Fourth Iteration



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0.157897 -1 -0.037075 1* 0.037075 0 -0.673649

E4 -0.157897 1 0.037075 -1 -0.037075 0 M+

0.673649

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.658324 2.255273 1.350472 0.001204 0.020596 2.880814 0.733157

Check 1.590364 3.255273 2.382388 1.001204 0.988680 4.880814 2M+

1.850850

239

Fifth Iteration



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0 0 0 1 0 0 0

E4 0 0 0 -1 0 0 M

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 -0.157897 1 0.037075 1 -0.037075 0 0.673649

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.654616 2.278754 1.351343 0.001204 0.019725 2.880814 0.733968

Check 1.428759 4.278754 2.420334 1.001204 0.950734 4.880814 2M+

2.525310

240



Minimize:

� = 10-8.73 K5.71 �0.203 �0.824 �-1.64 ……………………………………………………( 5.2)

Subject to:

100.103 K1.59 �0.059�0.747 ≥ 750 ……………………………………………………(5.3)

� ≥ 180 …………………………………………………………………………(5.4)

100.027 K-1.88 �0.0697 �1.06 ≤ 5 …………………………………………………………(5.5)

� ≤ 194.75 …………………………………………………………………………(5.6)

≤ 23.5 …………………………………………………………………………..(5.7)

� ≤ 760 …………………………………………………………………………..(5.8)

Linearize the model

G�H� = −8.73G�H10 + 5.71G�H − 1.64G�H� + 0.203G�H� + 0.824G�H�

Subject to:

0.103G�H10 + 1.59 G�H + 0.0590G�H� + 0.747G�H� ≥ G�H750

G�H� ≥ G�H180

0.027G�H10 − 1.88G�H + 0.0697G�H� + 1.06G�H� ≤ G�H5

G�H� ≤ G�H194.75

G�H ≤ G�H23.5

G�H� ≤ G�H760

Let:

G�H� = e

G�H = r

G�H� = s

G�H� = t

241

G�H� = [


e = −8.73 + 5.71r − 1.64s + 0.203t + 0.824[

Subjected to:

0.103 + 1.59r + 0.0590t + 0.747s ≥ 2.875061

t ≥ 2.255273

0.027 − 1.88r + 0.0697t + 1.06[ ≤ 0.698970

t ≤ 2.289478

r ≤ 1.371068

[ ≤ 2.880814

Standard form

e − 5.71r + 1.64s − 0.203t − 0.824[ = −8.73

1.59r + 0.0590t + 0.747s − E1 + E2 = 2.772061

t − E3 + E4 = 2.255273

−1.88r + 0.0697t + 1.06[ + E5 = 0.671970

t + E6 = 2.289478

r + E7 = 1.371068

[ + E8 = 2.880814

242


First Matrix



r 1.59 0 -1.88* 0 1 0 -5.71

t 0.0590 1 0.0697 1 0 0 -0.203

s 0.747 0 0 0 0 0 1.64

[ 0 0 1.06 0 0 1 -0.824

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0 0 1 0 0 0 0

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 2.772061 2.255273 0.671970 2.289478 1.371068 2.880814 -8.73

Check 5.168061 3.255273 0.921670 4.289478 3.371068 4.880814 2M-13.827

243

First Iteration



r 0 0 1 0 0 0 0

t 0.117949 1 -0.037075 1 0.037075 0 -0.414698

s 0.747 0 0 0 0 0 1.64

[ 0.896490 0 -0.563830 0 0.563830 1* -4.043469

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 3.340396 2.255273 -0.357431 2.289478 1.728499 2.880814 -10.770931

Check 5.947559 3.255273 -0.490250 4.289478 3.861318 4.880814 2M-

16.626328

244

Second Iteration



r 0 0 1 0 0 0 0

t 0.117949 1* -0.037075 1 0.037075 0 -0.414698

s 0.747 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.757775 2.255273 1.266858 2.289754 0.104210 2.880814 0.877551

Check 1.571958 3.255273 2.261699 4.289754 1.109369 4.880814 2M+

3.109092

245



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 0.747* 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0.117949 -1 -0.037075 1 0.037075 0 -0.414698

E4 -0.117949 1 0.037075 -1 -0.037075 0 M+

0.414698

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.491768 2.255273 1.350472 0.034205 0.020596 2.880814 1.812808

Check 1.188002 3.255273 2.382388 1.034205 0.988680 4.880814 2M+

4.459047

246

Fourth Iteration



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0.157897 -1 -0.037075 1* 0.037075 0 -0.673649

E4 -0.157897 1 0.037075 -1 -0.037075 0 M+

0.673649

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.658324 2.255273 1.350472 0.034205 0.020596 2.880814 0.733157

Check 1.590364 3.255273 2.382388 1.034205 0.988680 4.880814 2M+

1.850850

247

Fifth Iteration



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0 0 0 1 0 0 0

E4 0 0 0 -1 0 0 M

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 -0.157897 1 0.037075 1 -0.037075 0 0.673649

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.654616 2.278754 1.351343 0.034205 0.019725 2.880814 0.756199

Check 1.428759 4.278754 2.420334 1.034205 0.950734 4.880814 2M+

2.547541

248



Minimize:

� = 10-8.73 K5.71 �0.203 �0.824 �-1.64 ……………………………………………………( 5.2)

Subject to:

100.103 K1.59 �0.059�0.747 ≥ 750 ……………………………………………………(5.3)

� ≥ 180 …………………………………………………………………………(5.4)

100.027 K-1.88 �0.0697 �1.06 ≤ 5 …………………………………………………………(5.5)

� ≤ 199.5 …………………………………………………………………………(5.6)

≤ 23.5 …………………………………………………………………………..(5.7)

� ≤ 760 …………………………………………………………………………..(5.8)

Linearize the model

G�H� = −8.73G�H10 + 5.71G�H − 1.64G�H� + 0.203G�H� + 0.824G�H�

Subject to:

0.103G�H10 + 1.59 G�H + 0.0590G�H� + 0.747G�H� ≥ G�H750

G�H� ≥ G�H180

0.027G�H10 − 1.88G�H + 0.0697G�H� + 1.06G�H� ≤ G�H5

G�H� ≤ G�H199.5

G�H ≤ G�H23.5

G�H� ≤ G�H760

Let:

G�H� = e

G�H = r

G�H� = s

G�H� = t

249

G�H� = [


e = −8.73 + 5.71r − 1.64s + 0.203t + 0.824[

Subjected to:

0.103 + 1.59r + 0.0590t + 0.747s ≥ 2.875061

t ≥ 2.255273

0.027 − 1.88r + 0.0697t + 1.06[ ≤ 0.698970

t ≤ 2.299943

r ≤ 1.371068

[ ≤ 2.880814

Standard form

e − 5.71r + 1.64s − 0.203t − 0.824[ = −8.73

1.59r + 0.0590t + 0.747s − E1 + E2 = 2.772061

t − E3 + E4 = 2.255273

−1.88r + 0.0697t + 1.06[ + E5 = 0.671970

t + E6 = 2.299943

r + E7 = 1.371068

[ + E8 = 2.880814

250


First Matrix



r 1.59 0 -1.88* 0 1 0 -5.71

t 0.0590 1 0.0697 1 0 0 -0.203

s 0.747 0 0 0 0 0 1.64

[ 0 0 1.06 0 0 1 -0.824

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0 0 1 0 0 0 0

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 2.772061 2.255273 0.671970 2.299943 1.371068 2.880814 -8.73

Check 5.168061 3.255273 0.921670 4.299943 3.371068 4.880814 2M-13.827

251

First Iteration



r 0 0 1 0 0 0 0

t 0.117949 1 -0.037075 1 0.037075 0 -0.414698

s 0.747 0 0 0 0 0 1.64

[ 0.896490 0 -0.563830 0 0.563830 1* -4.043469

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 3.340396 2.255273 -0.357431 2.299943 1.728499 2.880814 -10.770931

Check 5.947559 3.255273 -0.490250 4.299943 3.861318 4.880814 2M-

16.626328

252

Second Iteration



r 0 0 1 0 0 0 0

t 0.117949 1* -0.037075 1 0.037075 0 -0.414698

s 0.747 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.757775 2.255273 1.266858 2.299943 0.104210 2.880814 0.877551

Check 1.571958 3.255273 2.261699 4.299943 1.109369 4.880814 2M+

3.109092

253

Third Iteration



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 0.747* 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0.117949 -1 -0.037075 1 0.037075 0 -0.414698

E4 -0.117949 1 0.037075 -1 -0.037075 0 M+

0.414698

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.491768 2.255273 1.350472 0.04467 0.020596 2.880814 1.812808

Check 1.188002 3.255273 2.382388 1.04467 0.988680 4.880814 2M+

4.459047

254

Fourth Iteration



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0.157897 -1 -0.037075 1* 0.037075 0 -0.673649

E4 -0.157897 1 0.037075 -1 -0.037075 0 M+

0.673649

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.658324 2.255273 1.350472 0.04467 0.020596 2.880814 0.733157

Check 1.590364 3.255273 2.382388 1.04467 0.988680 4.880814 2M+

1.850850

255

Fifth Iteration



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0 0 0 1 0 0 0

E4 0 0 0 -1 0 0 M

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 -0.157897 1 0.037075 1 -0.037075 0 0.673649

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.654616 2.278754 1.351343 0.04467 0.019725 2.880814 0.763267

Check 1.428759 4.278754 2.420334 1.04467 0.950734 4.880814 2M+

2.554591

256

At 5% decrease

e = 0.733968

G�H� = e

� = 100.733968 = 5.419610

v.c1{w1x

x.c = 13.55%

% %ℎ#dH_ = 1|.vv51c.x|1c.x| × 100 = −3.42%

At 2.5% increase

e = 0.756199

G�H� = e

� = 100.756199 = 5.704256

v.yxcbvw

x.c = 14.26%

% %ℎ#dH_ = 1c.bw51c.x|1c.x| × 100 = 1.64%

At 5% increase

e = 0.763267

G�H� = e

� = 100.763267 = 5.797850

v.y{yzvx

x.c = 14.50%

% %ℎ#dH_ = 1c.vx51c.x|1c.x| × 100 = 3.35%

257

APPENDIX VI




Minimize:

� = 10-8.73 K5.71 �0.203 �0.824 �-1.64 ……………………………………………………( 5.2)

Subject to:

100.103 K1.59 �0.059�0.747 ≥ 750 ……………………………………………………(5.3)

� ≥ 180 …………………………………………………………………………(5.4)

100.027 K-1.88 �0.0697 �1.06 ≤ 5 …………………………………………………………(5.5)

� ≤ 190 ……………………………………………………………………………(5.6)

≤ 23.5 …………………………………………………………………………..(5.7)

� ≤ 741 …………………………………………………………………………..(5.8)

Linearize the model

G�H� = −8.73G�H10 + 5.71G�H − 1.64G�H� + 0.203G�H� + 0.824G�H�

Subject to:

0.103G�H10 + 1.59 G�H + 0.0590G�H� + 0.747G�H� ≥ G�H750

G�H� ≥ G�H180

0.027G�H10 − 1.88G�H + 0.0697G�H� + 1.06G�H� ≤ G�H5

G�H� ≤ G�H190

G�H ≤ G�H23.5

G�H� ≤ G�H741

Let:

G�H� = e

G�H = r

258

G�H� = s

G�H� = t

G�H� = [


e = −8.73 + 5.71r − 1.64s + 0.203t + 0.824[

Subjected to:

0.103 + 1.59r + 0.0590t + 0.747s ≥ 2.875061

t ≥ 2.255273

0.027 − 1.88r + 0.0697t + 1.06[ ≤ 0.698970

t ≤ 2.278754

r ≤ 1.371068

[ ≤ 2.869818

Standard form

e − 5.71r + 1.64s − 0.203t − 0.824[ = −8.73

1.59r + 0.0590t + 0.747s − E1 + E2 = 2.772061

t − E3 + E4 = 2.255273

−1.88r + 0.0697t + 1.06[ + E5 = 0.671970

t + E6 = 2.278754

r + E7 = 1.371068

[ + E8 = 2.869818

259


First Matrix



r 1.59 0 -1.88* 0 1 0 -5.71

t 0.0590 1 0.0697 1 0 0 -0.203

s 0.747 0 0 0 0 0 1.64

[ 0 0 1.06 0 0 1 -0.824

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0 0 1 0 0 0 0

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 2.772061 2.255273 0.671970 2.278754 1.371068 2.869818 -8.73

Check 5.168061 3.255273 0.921670 4.278754 3.371068 4.869818 2M-13.827

260

First Iteration



r 0 0 1 0 0 0 0

t 0.117949 1 -0.037075 1 0.037075 0 -0.414698

s 0.747 0 0 0 0 0 1.64

[ 0.896490 0 -0.563830 0 0.563830 1* -4.043469

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 3.340396 2.255273 -0.357431 2.278754 1.728499 2.869818 -10.770931

Check 5.947559 3.255273 -0.490250 4.278754 3.861318 4.869818 2M-

16.626328

261

Second Iteration



r 0 0 1 0 0 0 0

t 0.117949 1* -0.037075 1 0.037075 0 -0.414698

s 0.747 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.767633 2.255273 1.260659 2.278754 0.110410 2.869818 0.833089

Check 1.581816 3.255273 2.255500 4.278754 1.115569 4.869818 2M+

3.064630

262

Third Iteration



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 0.747* 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0.117949 -1 -0.037075 1 0.037075 0 -0.414698

E4 -0.117949 1 0.037075 -1 -0.037075 0 M+

0.414698

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.501626 2.255273 1.344273 0.023481 0.026796 2.869818 1.768346

Check 1.197860 3.255273 2.376189 1.023481 0.994880 4.869818 2M+

4.414585

263

Fourth Iteration



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0.157897 -1 -0.037075 1* 0.037075 0 -0.673649

E4 -0.157897 1 0.037075 -1 -0.037075 0 M+

0.673649

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.671521 2.255273 1.344273 0.023481 0.026796 2.869818 0.667052

Check 1.603561 3.255273 2.376189 1.023481 0.994880 4.869818 2M+

1.784745

264

Fifth Iteration



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0 0 0 1 0 0 0

E4 0 0 0 -1 0 0 M

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 -0.157897 1 0.037075 1 -0.037075 0 0.673649

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.667813 2.278754 1.345144 0.023481 0.025925 2.869818 0.682870

Check 1.441956 4.278754 2.414135 1.023481 0.956934 4.869818 2M+

2.474212

265



Minimize:

� = 10-8.73 K5.71 �0.203 �0.824 �-1.64 ……………………………………………………( 5.2)

Subject to:

100.103 K1.59 �0.059�0.747 ≥ 750 ……………………………………………………(5.3)

� ≥ 180 …………………………………………………………………………(5.4)

100.027 K-1.88 �0.0697 �1.06 ≤ 5 …………………………………………………………(5.5)

� ≤ 190 ……………………………………………………………………………(5.6)

≤ 23.5 …………………………………………………………………………..(5.7)

� ≤ 722 …………………………………………………………………………..(5.8)

Linearize the model

G�H� = −8.73G�H10 + 5.71G�H − 1.64G�H� + 0.203G�H� + 0.824G�H�

Subject to:

0.103G�H10 + 1.59 G�H + 0.0590G�H� + 0.747G�H� ≥ G�H750

G�H� ≥ G�H180

0.027G�H10 − 1.88G�H + 0.0697G�H� + 1.06G�H� ≤ G�H5

G�H� ≤ G�H190

G�H ≤ G�H23.5

G�H� ≤ G�H722

Let:

G�H� = e

G�H = r

G�H� = s

G�H� = t

266

G�H� = [


e = −8.73 + 5.71r − 1.64s + 0.203t + 0.824[

Subjected to:

0.103 + 1.59r + 0.0590t + 0.747s ≥ 2.875061

t ≥ 2.255273

0.027 − 1.88r + 0.0697t + 1.06[ ≤ 0.698970

t ≤ 2.278754

r ≤ 1.371068

[ ≤ 2.858537

Standard form

e − 5.71r + 1.64s − 0.203t − 0.824[ = −8.73

1.59r + 0.0590t + 0.747s − E1 + E2 = 2.772061

t − E3 + E4 = 2.255273

−1.88r + 0.0697t + 1.06[ + E5 = 0.671970

t + E6 = 2.278754

r + E7 = 1.371068

[ + E8 = 2.858537

267


First Matrix



r 1.59 0 -1.88* 0 1 0 -5.71

t 0.0590 1 0.0697 1 0 0 -0.203

s 0.747 0 0 0 0 0 1.64

[ 0 0 1.06 0 0 1 -0.824

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0 0 1 0 0 0 0

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 2.772061 2.255273 0.671970 2.278754 1.371068 2.858537 -8.73

Check 5.168061 3.255273 0.921670 4.278754 3.371068 4.858537 2M-13.827

268

First Iteration



r 0 0 1 0 0 0 0

t 0.117949 1 -0.037075 1 0.037075 0 -0.414698

s 0.747 0 0 0 0 0 1.64

[ 0.896490 0 -0.563830 0 0.563830 1* -4.043469

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 3.340396 2.255273 -0.357431 2.278754 1.728499 2.858537 -10.770931

Check 5.947559 3.255273 -0.490250 4.278754 3.861318 4.858537 2M-

16.626328

269

Second Iteration



r 0 0 1 0 0 0 0

t 0.117949 1* -0.037075 1 0.037075 0 -0.414698

s 0.747 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.777746 2.255273 1.254298 2.278754 0.116670 2.858537 0.787475

Check 1.591929 3.255273 2.249139 4.278754 1.121929 4.858537 2M+

3.019016

270

Third Iteration



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 0.747* 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0.117949 -1 -0.037075 1 0.037075 0 -0.414698

E4 -0.117949 1 0.037075 -1 -0.037075 0 M+

0.414698

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.511739 2.255273 1.337912 0.023481 0.033156 2.858537 1.722732

Check 1.207973 3.255273 2.369828 1.023481 1.001240 4.858537 2M+

4.368971

Fourth Iteration

271



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0.157897 -1 -0.037075 1* 0.037075 0 -0.673649

E4 -0.157897 1 0.037075 -1 -0.037075 0 M+

0.673649

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.685059 2.255273 1.337912 0.023481 0.033156 2.858537 0.599235

Check 1.617099 3.255273 2.369828 1.023481 1.001240 4.858537 2M+

1.716929

Fifth Iteration

272



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0 0 0 1 0 0 0

E4 0 0 0 -1 0 0 M

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 -0.157897 1 0.037075 1 -0.037075 0 0.673649

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.681351 2.278754 1.338783 0.023481 0.032285 2.858537 0.615053

Check 1.455494 4.278754 2.407779 1.023481 0.963294 4.858537 2M+

2.406396

273



Minimize:

� = 10-8.73 K5.71 �0.203 �0.824 �-1.64 ……………………………………………………( 5.2)

Subject to:

100.103 K1.59 �0.059�0.747 ≥ 750 ……………………………………………………(5.3)

� ≥ 180 …………………………………………………………………………(5.4)

100.027 K-1.88 �0.0697 �1.06 ≤ 5 …………………………………………………………(5.5)

� ≤ 190 ……………………………………………………………………………(5.6)

≤ 23.5 …………………………………………………………………………..(5.7)

� ≤ 760 …………………………………………………………………………..(5.8)

Linearize the model

G�H� = −8.73G�H10 + 5.71G�H − 1.64G�H� + 0.203G�H� + 0.824G�H�

Subject to:

0.103G�H10 + 1.59 G�H + 0.0590G�H� + 0.747G�H� ≥ G�H750

G�H� ≥ G�H180

0.027G�H10 − 1.88G�H + 0.0697G�H� + 1.06G�H� ≤ G�H5

G�H� ≤ G�H190

G�H ≤ G�H23.5

G�H� ≤ G�H779

Let:

G�H� = e

G�H = r

G�H� = s

G�H� = t

274

G�H� = [


e = −8.73 + 5.71r − 1.64s + 0.203t + 0.824[

Subjected to:

0.103 + 1.59r + 0.0590t + 0.747s ≥ 2.875061

t ≥ 2.255273

0.027 − 1.88r + 0.0697t + 1.06[ ≤ 0.698970

t ≤ 2.278754

r ≤ 1.371068

[ ≤ 2.891538

Standard form

e − 5.71r + 1.64s − 0.203t − 0.824[ = −8.73

1.59r + 0.0590t + 0.747s − E1 + E2 = 2.772061

t − E3 + E4 = 2.255273

−1.88r + 0.0697t + 1.06[ + E5 = 0.671970

t + E6 = 2.278754

r + E7 = 1.371068

[ + E8 = 2.891538

275


First Matrix



r 1.59 0 -1.88* 0 1 0 -5.71

t 0.0590 1 0.0697 1 0 0 -0.203

s 0.747 0 0 0 0 0 1.64

[ 0 0 1.06 0 0 1 -0.824

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0 0 1 0 0 0 0

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 2.772061 2.255273 0.671970 2.278754 1.371068 2.891538 -8.73

Check 5.168061 3.255273 0.921670 4.278754 3.371068 4.891538 2M-13.827

276

First Iteration



r 0 0 1 0 0 0 0

t 0.117949 1 -0.037075 1 0.037075 0 -0.414698

s 0.747 0 0 0 0 0 1.64

[ 0.896490 0 -0.563830 0 0.563830 1* -4.043469

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 3.340396 2.255273 -0.357431 2.278754 1.728499 2.891538 -10.770931

Check 5.947559 3.255273 -0.490250 4.278754 3.861318 4.891538 2M-

16.626328

277

Second Iteration



r 0 0 1 0 0 0 0

t 0.117949 1* -0.037075 1 0.037075 0 -0.414698

s 0.747 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.748161 2.255273 1.272905 2.278754 0.098163 2.891538 0.920913

Check 1.562344 3.255273 2.267746 4.278754 1.103322 4.891538 2M+

3.152454

278

Third Iteration



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 0.747* 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0.117949 -1 -0.037075 1 0.037075 0 -0.414698

E4 -0.117949 1 0.037075 -1 -0.037075 0 M+

0.414698

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.482154 2.255273 1.356519 0.023481 0.014549 2.891538 1.856170

Check 1.178388 3.255273 2.388435 1.023481 0.982633 4.891538 2M+

4.502409

279

Fourth Iteration



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0.157897 -1 -0.037075 1* 0.037075 0 -0.673649

E4 -0.157897 1 0.037075 -1 -0.037075 0 M+

0.673649

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.645454 2.255273 1.356519 0.023481 0.014549 2.891538 0.797625

Check 1.577494 3.255273 2.388435 1.023481 0.982633 4.891538 2M+

1.915319

280

Fifth Iteration



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0 0 0 1 0 0 0

E4 0 0 0 -1 0 0 M

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 -0.157897 1 0.037075 1 -0.037075 0 0.673649

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.641746 2.278754 1.357390 0.023481 0.013678 2.891538 0.813443

Check 1.415889 4.278754 2.426381 1.023481 0.944687 4.891538 2M+

2.604786

281



Minimize:

� = 10-8.73 K5.71 �0.203 �0.824 �-1.64 ……………………………………………………( 5.2)

Subject to:

100.103 K1.59 �0.059�0.747 ≥ 750 ……………………………………………………(5.3)

� ≥ 180 …………………………………………………………………………(5.4)

100.027 K-1.88 �0.0697 �1.06 ≤ 5 …………………………………………………………(5.5)

� ≤ 190 ……………………………………………………………………………(5.6)

≤ 23.5 …………………………………………………………………………..(5.7)

� ≤ 798 …………………………………………………………………………..(5.8)

Linearize the model

G�H� = −8.73G�H10 + 5.71G�H − 1.64G�H� + 0.203G�H� + 0.824G�H�

Subject to:

0.103G�H10 + 1.59 G�H + 0.0590G�H� + 0.747G�H� ≥ G�H750

G�H� ≥ G�H180

0.027G�H10 − 1.88G�H + 0.0697G�H� + 1.06G�H� ≤ G�H5

G�H� ≤ G�H190

G�H ≤ G�H23.5

G�H� ≤ G�H798

Let:

G�H� = e

G�H = r

G�H� = s

G�H� = t

282

G�H� = [


e = −8.73 + 5.71r − 1.64s + 0.203t + 0.824[

Subjected to:

0.103 + 1.59r + 0.0590t + 0.747s ≥ 2.875061

t ≥ 2.255273

0.027 − 1.88r + 0.0697t + 1.06[ ≤ 0.698970

t ≤ 2.278754

r ≤ 1.371068

[ ≤ 2.902003

Standard form

e − 5.71r + 1.64s − 0.203t − 0.824[ = −8.73

1.59r + 0.0590t + 0.747s − E1 + E2 = 2.772061

t − E3 + E4 = 2.255273

−1.88r + 0.0697t + 1.06[ + E5 = 0.671970

t + E6 = 2.278754

r + E7 = 1.371068

[ + E8 = 2.902003

283


First Matrix



r 1.59 0 -1.88* 0 1 0 -5.71

t 0.0590 1 0.0697 1 0 0 -0.203

s 0.747 0 0 0 0 0 1.64

[ 0 0 1.06 0 0 1 -0.824

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0 0 1 0 0 0 0

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 2.772061 2.255273 0.671970 2.278754 1.371068 2.902003 -8.73

Check 5.168061 3.255273 0.921670 4.278754 3.371068 4.902003 2M-13.827

284

First Iteration



r 0 0 1 0 0 0 0

t 0.117949 1 -0.037075 1 0.037075 0 -0.414698

s 0.747 0 0 0 0 0 1.64

[ 0.896490 0 -0.563830 0 0.563830 1* -4.043469

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 0 0 0 0 0 1 0

RHS 3.340396 2.255273 -0.357431 2.278754 1.728499 2.902003 -10.770931

Check 5.947559 3.255273 -0.490250 4.278754 3.861318 4.902003 2M-

16.626328

285

Second Iteration



r 0 0 1 0 0 0 0

t 0.117949 1* -0.037075 1 0.037075 0 -0.414698

s 0.747 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0 -1 0 0 0 0 0

E4 0 1 0 0 0 0 M

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.738779 2.255273 1.278805 2.278754 0.092263 2.902003 0.963228

Check 1.552962 3.255273 2.273646 4.278754 1.097422 4.902003 2M+

3.194769

286

Third Iteration



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 0.747* 0 0 0 0 0 1.64

[ 0 0 0 0 0 1 0

E1 -1 0 0 0 0 0 0

E2 1 0 0 0 0 0 M

E3 0.117949 -1 -0.037075 1 0.037075 0 -0.414698

E4 -0.117949 1 0.037075 -1 -0.037075 0 M+

0.414698

E5 0.845745 0 -0.531915 0 0.531915 0 -3.037235

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -0.896490 0 0.563830 0 -0.563830 1 4.043469

RHS 0.472772 2.255273 1.362419 0.023481 0.008649 2.902003 1.898485

Check 1.169006 3.255273 2.394335 1.023481 0.976733 4.902003 2M+

4.544724

287

Fourth Iteration



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0.157897 -1 -0.037075 1* 0.037075 0 -0.673649

E4 -0.157897 1 0.037075 -1 -0.037075 0 M+

0.673649

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 0 0 0 1 0 0 0

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.632894 2.255273 1.362419 0.023481 0.008649 2.902003 0.860539

Check 1.564934 3.255273 2.394335 1.023481 0.976733 4.902003 2M+

1.978232

288

Fifth Iteration



r 0 0 1 0 0 0 0

t 0 1 0 0 0 0 0

s 1 0 0 0 0 0 0

[ 0 0 0 0 0 1 0

E1 -1.338688 0 0 0 0 0 2.195448

E2 1.338688 0 0 0 0 0 M-

2.195448

E3 0 0 0 1 0 0 0

E4 0 0 0 -1 0 0 M

E5 1.132189 0 -0.531915 0 0.531915 0 -4.894025

E6 -0.157897 1 0.037075 1 -0.037075 0 0.673649

E7 0 0 0 0 1 0 0

E8 -1.200121 0 0.563830 0 -0.563830 1 6.011667

RHS 0.629186 2.278754 1.363290 0.023481 0.007778 2.902003 0.876357

Check 1.403329 4.278754 2.432281 1.023481 0.938787 4.902003 2M+

2.667699

289

At 5% decrease

e = 0.615053

G�H� = e

� = 100.615053 = 4.121478

c.1b1cyz

x.c = 10.31%

% %ℎ#dH_ = 1x.|151c.x|1c.x| × 100 = −26.51%

At 2.5% increase

e = 0.813443

G�H� = e

� = 100.813443 = 6.507932

w.vxy{|b

x.c = 16.27%

% %ℎ#dH_ = 1w.by51c.x|1c.x| × 100 = 15.97%

e = 0.876357

At 5% increase

G�H� = e

� = 100.876357 = 7.522410

y.vbbc1x

x.c = 18.81%

% %ℎ#dH_ = 1z.z151c.x|1c.x| × 100 = 34.07%

290

APPENDIX VII

Results of Particle Size Analysis

Sieve Sizes (mm) Percentage Passing

2.400 100

1.200 99.2

0.600 85.2

0.425 77.9

0.300 63.8

0.210 59.1

0.150 49.4

0.075 45.3

0.020 25

0.006 18

0.002 11

291

APPENDIX VIII

Table 4.5: Variations of Optimum Moisture Content with Increase in Bagasse

Ash Content at 2%, 4%, 6% and 8% Cement Contents

Bagasse Ash

Content (%)

Optimum Moisture Content (%)


0 16.50 17.90 18.24 20.39

2 16.80 17.97 18.41 20.56

4 17.71 18.30 18.91 21.24

6 18.74 19.69 20.85 21.63

8 19.58 20.48 21.66 22.08

10 20.23 21.29 22.39 22.63

12 20.81 21.71 22.71 23.05

14 21.32 22.17 23.29 23.95

16 22.01 22.85 23.75 24.69

18 22.22 23.21 24.23 25.02

20 22.62 23.54 24.44 25.31

292

APPENDIX IX

Table 4.6: Variations of Maximum Dry Density with Increase in Bagasse Ash

Content at 2%, 4%, 6% and 8% Cement Content

Bagasse Ash

Content (%)

Maximum

Dry Density (Kg/m3)

2% cement 4% cement 6% cement 8% cement

0 1661 1777 1891 2199

2 1634 1771 1875 2132

4 1612 1759 1805 2084

6 1584 1742 1783 2022

8 1551 1708 1724 1996

10 1533 1691 1702 1971

12 1503 1671 1689 1954

14 1489 1630 1644 1933

16 1463 1602 1628 1877

18 1441 1591 1601 1846

20 1422 1572 1586 1791

293

APPENDIX X

Table 4.7: Variations of California Bearing Ratio with Increase Bagasse Ash

Content at 2%, 4%, 6% and 8% Cement Contents

Bagasse Ash

Content (%)

California Bearing Ratio (%)


0 22.30 57.99 83.34 147.16

2 23.57 84.44 93.70 175.12

4 25.42 85.20 104.94 196.37

6 26.48 93.04 117.07 209.09

8 25.13 109.13 123.68 221.03

10 25.11 121.03 135.59 230.24

12 24.98 135.19 176.12 242.05

14 24.92 152.10 196.50 251.31

16 24.70 163.59 207.26 265.30

18 24.31 161.38 220.08 271.80

20 24.23 160.96 239.16 276.30

294

APPENDIX XI

Table 4.8: Variations of Unconfined Compressive Strength and Age with

Increase in Bagasse Ash Content at 2%, 4%, 6% and 8% Cement Contents.

Bagasse

Ash

Content

(%)

Unconfined Compressive Strength (kN/m2)


7d 14d 7d+7s

k

7d 14d 7d+7s

k

7d 14d 7d+7s

k

7d 14d 7d+7sk

0 213 248 225 419 513 498 549 864 740 942 1210 1008

2 228 262 243 454 549 510 642 924 876 998 1241 1136

4 248 289 254 492 589 555 683 1014 903 1049 1320 1231

6 273 328 288 534 647 583 801 1066 951 1087 1492 1292

8 292 375 313 575 698 601 854 1110 978 1132 1662 1431

10 308 399 327 613 749 643 907 1228 1005 1180 1776 1536

12 321 428 365 642 788 693 941 1259 1053 1221 1833 1615

14 335 421 386 665 863 728 985 1312 1112 1298 1868 1679

16 349 426 397 697 902 795 1018 1373 1152 1366 1905 1763

18 353 442 411 717 915 804 1057 1390 1201 1396 1945 1801

20 364 459 418 733 948 856 1073 1435 1272 1424 1986 1877

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