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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
2
OPTIMIZATION OF BAGASSE ASH CONTENT IN CEMENT- STABILIZED LATERITIC SOIL
By
OKONKWO, UGOCHUKWU NNATUANYA PG/Ph. D/09/52059
FACULTY OF ENGINEERING DEPARTMENT OF CIVIL ENGINEERING
UNIVERSITY OF NIGERIA NSUKKA.
FEBRUARY, 2015
3
OPTIMIZATION OF BAGASSE ASH CONTENT IN CEMENT- STABILIZED LATERITIC SOIL
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
6
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
11
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
15
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
Table 5.5: Change in Constraint with Corresponding Change in Optimal Solution for
Constrained Equation (5.6) 100
Table 5.6: Change in Constraint with Corresponding Change in Optimal Solution for
Constrained Equation (5.8) 101
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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
Change in Optimal Solution 108
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
17
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
Content at 2%, 4%, 6% and 8% Cement Contents 80
Figure 4.6: Variations of California Bearing Ratiowith Increase in Bagasse Ash
Content at 2%, 4%, 6% and 8% Cement Contents 81
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
Constrained Equation (5.3) 99
Figure 5.2: Variations of Change in Constraint with Change in Optimal Solution for
Constrained Equation (5.6) 100
Figure 5.3: Variations of Change in Constraint with Change in Optimal Solution for
Constrained Equation (5.8) 102
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
Change in Optimal Solution 105
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
Change in Optimal Solution 108
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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
30 Particles with diameter 30% finer
10 Particles with diameter 10% 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
19
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
20
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.
21
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
22
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
23
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.
24
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.
25
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
26
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)
27
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
28
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
30 = Particles with diameter 30% finer
10 = Particles with diameter 10% 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
83
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)
Bagasse Ash Content (%)
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
(%
)
Bagasse Ash Content (%)
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)
Bagasse Ash Content (%)
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%
Using Equation (5.1) to determine the cost of stabilizing 100grams of soil with this
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)
Using Equation (5.1) to determine the cost of stabilizing 100grams of soil with this
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
Solution for Constrained Equation (5.6)
-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
(%
)
Change in Constraint (%)
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
optimal objective values for constrained Equation (5.6). The relationship is almost a
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
Solution for Constrained Equation (5.8)
Figure 5.3 shows the relationship between changes in constraint with changes in
optimal objective values for constrained Equation (5.8). The relationship is almost a
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
(%
)
Change in Constraint (%)
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
Corresponding Change in Optimal Solution
Change in Coefficient Change in Optimal Solution
-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
Corresponding Change in Optimal Solution
Change in Coefficient Change in Optimal Solution
-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
The * shows the pivot number
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
The * shows the pivot number
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
The * shows the pivot number
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
The * shows the pivot number
Basic s t r E3 E7 [ e
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� = [
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
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
Putting in matrix form and solving with the simplex method
First Matrix
The * shows the pivot number
153
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
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
The * shows the pivot number
154
Basic E2 E4 r E6 E7 E8 e
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
The * shows the pivot number
155
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
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
The * shows the pivot number
156
Basic E2 t r E6 E7 [ e
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
The * shows the pivot number
157
Basic s t r E6 E7 [ e
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
The * shows the pivot number
158
Basic s t r E3 E7 [ e
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� = [
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 ≥ 2.864066
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
Putting in matrix form and solving with the simplex method
First Matrix
161
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
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
The * shows the pivot number
162
Basic E2 E4 r E6 E7 E8 e
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
The * shows the pivot number
163
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
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
Third Iteration The * shows the pivot number
Basic E2 t r E6 E7 [ e
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
The * shows the pivot number
Basic s t r E6 E7 [ e
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
The * shows the pivot number
166
Basic s t r E3 E7 [ e
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� = [
Thus the model becomes;
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
Standard form e − 5.71r + 1.64s − 0.203t − 0.824[ = −8.73
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
Putting in matrix form and solving with the simplex method
First Matrix
169
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
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
The * shows the pivot number
170
Basic E2 E4 r E6 E7 E8 e
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
The * shows the pivot number
171
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
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
The * shows the pivot number
172
Basic E2 t r E6 E7 [ e
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
The * shows the pivot number
173
Basic s t r E6 E7 [ e
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
The * shows the pivot number
174
Basic s t r E3 E7 [ e
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� = [
Thus the model becomes;
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
Standard form e − 5.71r + 1.64s − 0.203t − 0.824[ = −8.73
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
Putting in matrix form and solving with the simplex method
First Matrix
177
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
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
The * shows the pivot number
178
Basic E2 E4 r E6 E7 E8 e
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
The * shows the pivot number
179
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
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
The * shows the pivot number
180
Basic E2 t r E6 E7 [ e
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
The * shows the pivot number
181
Basic s t r E6 E7 [ e
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
The * shows the pivot number
182
Basic s t r E3 E7 [ e
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
other constrained equations to remain as they were:
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
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 ≥ 2.896251
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 = 2.793251
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
Putting in matrix form and solving with the simplex method
First Matrix
185
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
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
The * shows the pivot number
186
Basic E2 E4 r E6 E7 E8 e
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
The * shows the pivot number
187
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
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
The * shows the pivot number
188
Basic E2 t r E6 E7 [ e
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
The * shows the pivot number
189
Basic s t r E6 E7 [ e
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
The * shows the pivot number
190
Basic s t r E3 E7 [ e
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)
Decreasing the right hand side of constrained Equation (5.4) 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 ≥ 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� = [
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
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
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
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
The * shows the pivot number
Basic E2 E4 r E6 E7 E8 e
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
The * shows the pivot number
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
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
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
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
The * shows the pivot number
Basic s t r E6 E7 [ e
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
The * shows the pivot number
Basic s t r E3 E7 [ e
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
Decreasing the right hand side of constrained Equation (5.4) by 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 ≥ 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� = [
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
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
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
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
The * shows the pivot number
Basic E2 E4 r E6 E7 E8 e
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
The * shows the pivot number
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
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
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
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
The * shows the pivot number
Basic s t r E6 E7 [ e
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
The * shows the pivot number
Basic s t r E3 E7 [ e
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
Increasing the right hand side of constrained Equation (5.4) by 2.5% and allowing the
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� = [
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
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
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
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
The * shows the pivot number
Basic E2 E4 r E6 E7 E8 e
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
The * shows the pivot number
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
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
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
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
The * shows the pivot number
Basic s t r E6 E7 [ e
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
The * shows the pivot number
Basic s t r E3 E7 [ e
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
Increasing the right hand side of constrained Equation (5.4) by 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 ≥ 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� = [
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
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
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
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
The * shows the pivot number
Basic E2 E4 r E6 E7 E8 e
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
The * shows the pivot number
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
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
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
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
The * shows the pivot number
Basic s t r E6 E7 [ e
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
The * shows the pivot number
Basic s t r E3 E7 [ e
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
Sensitivity Analysis on Constrained Equation (5.6)
Decreasing the right hand side of constrained Equation (5.6) 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 ≥ 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� = [
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
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
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
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
The * shows the pivot number
Basic E2 E4 r E6 E7 E8 e
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
The * shows the pivot number
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
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
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
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
The * shows the pivot number
Basic s t r E6 E7 [ e
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
The * shows the pivot number
Basic s t r E3 E7 [ e
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
Decreasing the right hand side of constrained Equation (5.6) by 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 ≥ 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� = [
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
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
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
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
The * shows the pivot number
Basic E2 E4 r E6 E7 E8 e
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
The * shows the pivot number
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
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
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
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
The * shows the pivot number
Basic s t r E6 E7 [ e
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
The * shows the pivot number
Basic s t r E3 E7 [ e
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
Increasing the right hand side of constrained Equation (5.6) 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 ≥ 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� = [
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
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
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
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
The * shows the pivot number
Basic E2 E4 r E6 E7 E8 e
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
The * shows the pivot number
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
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
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
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
The * shows the pivot number
Basic s t r E6 E7 [ e
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
The * shows the pivot number
Basic s t r E3 E7 [ e
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
Increasing the right hand side of constrained Equation (5.6) by 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 ≥ 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� = [
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
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
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
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
The * shows the pivot number
Basic E2 E4 r E6 E7 E8 e
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
The * shows the pivot number
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
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
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
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
The * shows the pivot number
Basic s t r E6 E7 [ e
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
The * shows the pivot number
Basic s t r E3 E7 [ e
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
Sensitivity Analysis on Constrained Equation (5.8)
Decreasing the right hand side of constrained Equation (5.8) 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 ≥ 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� = [
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
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
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
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
The * shows the pivot number
Basic E2 E4 r E6 E7 E8 e
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
The * shows the pivot number
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
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
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
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
The * shows the pivot number
Basic s t r E6 E7 [ e
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
The * shows the pivot number
Basic s t r E3 E7 [ e
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
Decreasing the right hand side of constrained Equation (5.8) by 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 ≥ 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� = [
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
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
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
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
The * shows the pivot number
Basic E2 E4 r E6 E7 E8 e
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
The * shows the pivot number
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
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
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
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
The * shows the pivot number
Basic s t r E6 E7 [ e
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
The * shows the pivot number
Basic s t r E3 E7 [ e
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
Increasing the right hand side of constrained Equation (5.8) 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 ≥ 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� = [
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
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
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
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
The * shows the pivot number
Basic E2 E4 r E6 E7 E8 e
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
The * shows the pivot number
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
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
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
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
The * shows the pivot number
Basic s t r E6 E7 [ e
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
The * shows the pivot number
Basic s t r E3 E7 [ e
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
Increasing the right hand side of constrained Equation (5.8) by 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 ≥ 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� = [
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
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
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
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
The * shows the pivot number
Basic E2 E4 r E6 E7 E8 e
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
The * shows the pivot number
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
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
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
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
The * shows the pivot number
Basic s t r E6 E7 [ e
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
The * shows the pivot number
Basic s t r E3 E7 [ e
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 (%)
2% Cement 4% Cement 6% Cement 8% Cement
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 (%)
2% Cement 4% Cement 6% Cement 8% Cement
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)
2% Cement 4% Cement 6% Cement 8% Cement
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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