System Analysis and Management Techniques
Adama Science and Technology University
School of Civil Engineering and Architecture
Department of civil Engineering
Getachew Tegegne (Ph.D.)
Assistant Professor of Civil Engineering
E-mail: getachewtegegne21@gmail.com
 Optimization approaches for project management
 Classification of Optimization Techniques
 Optimization techniques are also known as mathematical
programming techniques
 The mathematical programming can be classified in several ways,
such as on the nature of the problem, the nature of the equations
involved, a number of objective functions involved, etc.
 These mathematical programming can be classified as:
o Linear Programming (LP)
o Nonlinear Programming (NLP)
o Dynamic Programming (DP), etc.
 Optimization approaches for project management
 Basics of optimization problem:
1) Objective function: it expresses the main aim of the model
which is either to be minimized or maximized.
2) Unknowns or variables: they control the value of the
objective function
3) Constraints: pre-defined search space that they restrict the
range over which the decision variables can change & thus
affect the optimum solution.
 Optimization problem is then to:
o Find values of the variables that minimize or maximize the
objective function while satisfying the constraints.
 Optimization approaches for project management
 An optimization problem can be stated as:
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑜𝑟 𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 𝑓 𝑋
𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜
𝑔𝑗 𝑋 ≥ 0,
𝑗 = 1, 2, … , 𝑚
ℎ𝑗 𝑋 = 0,
𝑗 = 𝑚 + 1, 𝑚 + 2, … , 𝑝
𝑋𝐿 ≤ 𝑋 ≤ 𝑋𝑈
• where
o X is a vector of n-variables which are known as decision
variables,
o g(X) are the inequality constraints, h(X) are the equality
constraints and XL and XU, respectively, are the lower and
upper limits of the decision variable X
 Optimization: Linear vs. Nonlinear
o Are the functions J(x), g(x), and h(x) linear or nonlinear
functions of x?
o Linear programming (LP):
• When the objective function and the constraints are all
linear, the problem is called a LP problem.
o Non-linear programming:
• When either the objective function or any of the
constraints is a non-linear function of the design variables,
the problem is called a nonlinear programming (NP)
problem
 Formulation and Application of Linear Programming
 Optimization using linear programming
– Linear programming is a technique for determining an
optimum schedule of interdependent activities in view of the
available resources and constraints
– Programming is just another word for planning and refers to
the process of determining a particular plan of action from
amongst several alternatives
– The word linear stands for indicating that all relationships
involved in a particular problem are linear
 Formulation and Application of Linear Programming
 Optimization using linear programming
– Linear programming indicates the right combination of the
various decision variables which can be best employed to
achieve the objective taking full account of the practical
limitations with in which the problem must be solved
– Linear programming problems consist of a linear objective
(cost) function (consisting of a certain number of variables)
which is to be minimized or maximized subject to a certain
number of constraints
 Formulation and Application of Linear Programming
 Optimization using linear programming
– Linear programming indicates the right combination of the
various decision variables which can be best employed to
achieve the objective taking full account of the practical
limitations with in which the problem must be solved
– Linear programming problems consist of a linear objective
(cost) function (consisting of a certain number of variables)
which is to be minimized or maximized subject to a certain
number of constraints
 Formulation and Application of Linear Programming
• Formulation of linear programming problems
– The steps to formulate the LP problems are:
• Step1: Understand the problem
• Step2: Identify decision variables
• Step3: State the objective function as a linear combination
of the decision variables
• Step4: State the constraints as linear combinations of the
decision variables
• Step5: Identify the upper or lower bounds on the decision
variables (Additional Constraints)
 Formulation and Application of Linear Programming
 LP optimization problem meets the following conditions:
o The decision variables of the problem are non-negative (i.e.,
positive or zero)
o The objective function is a linear function of the decision
variables (i.e., a function involving only the first powers of the
variables with no cross products)
o The operating rules governing the processes, commonly known
as constraints, are expressed as a set of linear equations or linear
inequalities
Mathematical Representation of a Linear Programming (LP)
 There are many ways to represent a linear programming problem
 A compact way of representing a linear program is a matrix form
 A general LP problem can be written in conventional form as:
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑜𝑟 𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 : 𝑍 = 𝑓 𝑋
𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜:
𝑔𝑗 𝑋 ≥ 0,
𝑗 = 1, 2, … , 𝑚1
ℎ𝑗 𝑋 ≤ 0,
𝑗 = 1, 2, … , 𝑚2
𝑙𝑗 𝑋 = 0,
𝑗 = 1, 2, … , 𝑚3
 where Z is the objective function; x is an n-dimensional decision vector;
gj(x) & hj(x) are inequality constraints; lj(x) are equality constraints; and
m1, m2, and m3 indicate the number of constraints. The objective function
is a linear function of decision variables.
 Standard Form of LP
 An LP problem with m constraints and n variables can be
represented in standard form as follows:
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑜𝑟 𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 : 𝑍 = 𝑐11 𝑋1 + 𝑐2 𝑋2 + ⋯ + 𝑐𝑛 𝑋𝑛
𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜:
𝑎11 𝑋1 + 𝑎12 𝑋2 + ⋯ + 𝑎1𝑛 𝑋𝑛 = 𝑏1
𝑎21 𝑋1 + 𝑎22 𝑋2 + ⋯ + 𝑎2𝑛 𝑋𝑛 = 𝑏2



𝑎𝑚1 𝑋1 + 𝑎𝑚2 𝑋2 + ⋯ + 𝑎𝑚𝑛 𝑋𝑛 = 𝑏𝑚
𝑋𝑖 ≥ 0,
𝑖 = 1, 2, … , 𝑛
𝑏𝑖 ≥ 0,
𝑖 = 1, 2, … , 𝑚
 where Z is the objective function; xi are the decision variables and ci
are the cost (or benefit) coefficients)
Matrix Form of LP
 In matrix notation, the standard LP problem can be expressed in a
compact form as:
𝑀𝑎𝑥𝑖𝑚𝑖𝑧𝑒 𝑜𝑟 𝑚𝑖𝑛𝑖𝑚𝑖𝑧𝑒 : 𝑍 = 𝐶 𝑇 𝑋
𝑠𝑢𝑏𝑗𝑒𝑐𝑡 𝑡𝑜:
𝐴𝑋 = 𝑏
𝑋≥0
𝑏≥0
 where A is a m*n matrix, X is an n*l column vector, b is an m*l column
vector, C is an n*l column vector, and Z indicates the objective function.
Subscript T in the objective function indicates the transpose operation.
Thus, the equation can be written as:
𝑋 = 𝑥1 , 𝑥2 , ⋯ , 𝑥𝑛 𝑇
𝐶 = 𝑐1 , 𝑐2 , ⋯ , 𝑐𝑛 𝑇
𝑏 = 𝑏1 , 𝑏2 , ⋯ , 𝑏𝑚 𝑇
𝑎11
𝐴= ⋮
𝑎𝑚1
⋯ 𝑎1𝑛
⋱
⋮
⋯ 𝑎𝑚𝑛
Systems of linear equations
 A general set of n equations with n unknowns has the form
• where aij are the coefficients
(known), bi are the right hand
sides (known), and Xj are
unknowns.
 The set of equations can be written more compactly as
 The solution of this matrix equation is equally simple to
write, but can be difficult to compute,
Graphical solution of a LP problem
 The graphical method is a simple way to solve LP problems.
 However, it can be used to solve the LP problems involving at most two
decision variables
 Consider the following Figure for illustration of the graphical method for
two decision variables that shows the definition sketch of feasible region
and constraints
Example: Let objective 𝑧1 𝑥 = 5𝑥1 − 2𝑥2 and
objective 𝑧2 𝑥 = −𝑥1 + 4𝑥2
 Both are to be maximized. Assume that the constraints on
variables 𝑥1 and 𝑥2 are:
−𝑥1 + 𝑥2 ≤ 3
𝑥1 + 𝑥2 ≤ 8
𝑥1 ≤ 6
𝑥2 ≤ 4
𝑥1 , 𝑥2 ≥ 0
 Graph the Pareto optimal or non-inferior solutions in decision
space.
 Graph the Pareto optimal or non-inferior solutions in decision space
 There are several solutions with the combination of Z1 and Z2 as:
SW  z1 ( x)  z2 ( x)
SW  z1 ( x)  z2 ( x)  (5x1  2 x2 )  ( x1  4 x2 )  4 x1  2 x2
 SW is called by social welfare function. The following approach can
be followed to solve the problem:
(𝒙𝟏 , 𝒙𝟐 )
𝒛𝟏
𝒛𝟐
(0, 0)
0
0
(6, 0)
30
-6
(6, 2)
26
2
(4, 4)
12
12
(1, 4)
-3
15
(0, 3)
-6
12
From above, there is the
maximum value in (6, 2)
Quiz: Solve the following problem using graphical method of
LP
𝑴𝒂𝒙𝒊𝒎𝒊𝒛𝒆 𝒁 = 𝟐𝒙𝟏 + 𝒙𝟐
𝑺𝒖𝒃𝒋𝒆𝒄𝒕 𝒕𝒐:
𝟐𝒙𝟏 − 𝒙𝟐 ≤ 𝟖
𝒙𝟏 + 𝒙𝟐 ≤ 𝟏𝟎
𝒙𝟐 ≤ 𝟕
𝒙𝟏 , 𝒙𝟐 ≤ 𝟎
Application of MATLAB for solving sets of linear equations
 Getting the software
o Get MATLAB: http://software.usc.edu
 The file browser
o Moving around through the folder hierarchy
o Command line tools for navigation
cd
ls
change directory
list directory contents
 The workspace and variable editor
 Settings variables:
x=3
 Clearing variables:
clear x
clear all
 Getting help
 help function
 doc function
>> help sin
sin
Sine of argument in radians.
sin(X) is the sine of the elements of X.
See also asin, sind.
Reference page in Help browser
doc sin
 Scripts
 Anything you type into the workspace can also be run
from a script
 “.m” files are just saved lists of Matlab commands
Functions
 Blocks of code that operate on variables in their own
independent workspaces
 Editor
 Comments
 Syntax highlighting
 Code folding
 Variables
 What is a variable?
 Variable names and conventions
 Variable types
 double: floating point number like
3.24
 integer: no decimal places
345
 Vectors and matrices
 Vectors are like lists
a = [1,2,3,4,5]
 Matrices are like lists of lists
a = [ 1,3,5,7; 2,4,6,8 ]
 Creating vectors
Also try:
x = [1:0.1:10]
>> a = [1 2 3 4 5]
a =
1
2
3
4
5
>> a = [1,2,3,4,5]
a =
1
2
3
4
5
>> a = [1:5]
a =
1
2
4
5
3
 Creating matrices
>> a = [1 2 3; 4 5 6]
a =
1
2
3
4
5
6
>> a = [1 2 3; 4 5 6; 7 8 9]
a =
1
2
3
4
5
6
7
8
9
>> rand(3)
ans =
0.8147
0.9058
0.1270
0.9134
0.6324
0.0975
0.2785
0.5469
0.9575
>> nan(4)
ans =
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
NaN
>> ones(3)
ans =
1
1
1
1
1
1
1
1
1
>> ones(2,3)
ans =
1
1
1
1
1
1
>> zeros(3,4)
ans =
0
0
0
0
0
0
0
0
0
0
0
0
 Describing matrices
 size() will tell you the dimensions of a matrix
 length() will tell you the length of a vector
Accessing elements
>> a = [0:4]
a =
0
1
2
3
>> a(end)
ans =
4
>> a(2)
ans =
1
>> a(end-1)
ans =
>> b = [1,2,3;4,5,6;7,8,9]
b =
1
2
3
4
5
6
7
8
9
>> b(2,3)
ans =
6
3
4
 Vector math
■ Adding a constant to each element in a vector
>> a = [1 2 3]
a =
1
2
>> a + 1
ans =
2
3
3
4
■ Adding two vectors
>> b = [5 1 5]
b =
5
1
>> a + b
ans =
6
3
5
8
Vector multiplication
 The .* sign refers to element-wise multiplication:
>> a = [1 2 3]
a =
1
2
3
>> b = [2 2 4]
b =
2
2
4
>> a * b
Error using *
Inner matrix dimensions must agree.
>> a = [1 2 3]
a =
1
2
>> b = [2 2 4]
b =
2
2
>> a .* b
ans =
2
4
>> b = b'
b =
2
2
4
>> a * b
ans =
18
>> a * 4
ans =
4
>> a .* 4
ans =
4
transposing a matrix:
use ‘ to transpose, i.e.
flip rows and columns
3
4
12
8
12
8
12
 Operators
 Element-wise operators:
.*
multiplication
./
division
.^
exponentiation
 Many other functions work element-wise, e.g.:
 Saving variables
 save() to save the workspace to disk
 load() to load a .mat file that contains variables from disk
 clear() to remove a variable from memory
 who, whos to list variables in memory
 Creating a script
create new blank document
 Your first script
% My first script
x = 5;
y = 6;
z = x + y
Save script as “myFirst.m”
>> myFirst
z =
11
 Function declarations
 All functions must be declared, that is, introduced in the proper way.
code folding
result of the function
name of the function
parameters passed to the function
OUT
IN
>> addemup(1,1)
ans =
6
>> addemup2(1,1)
ans =
10
 Accessing elements
>> odds = [1:2:100];
>> odds([26:50, 1:25])
ans =
Columns 1 through 14
73
51
75
53
77
55
57
59
61
63
65
67
69
71
87
89
91
93
95
97
99
15
17
19
21
23
25
27
43
45
47
49
Columns 15 through 28
79
3
1
81
5
83
85
Columns 29 through 42
29
7
31
9
33
11
13
Columns 43 through 50
35
37
39
41
>> [26:50,1:25]
ans =
Columns 1 through 14
34
26
35
27
36
28
29
37
38
30
39
31
32
33
44
45
46
47
9
10
11
23
24
25
Columns 15 through 28
48
40
49
41
50
42
43
1
2
3
7
16
8
17
Columns 29 through 42
12
4
13
5
14
6
15
Columns 43 through 50
18
19
20
21
22
 Commands are executed immediately when you press Return
 Variables that are created appear in the Workspace
Simple functions and plots
 Form a sine function for a seasonal variable using the
following MATLAB statement.
t = 1:5:365;
temp= 15*sin(2**pi*t/365) + 12;
Plot (t, temp, ‘+’);
Xlabel (‘Time, days’);
Ylabel (‘Smoothed temperature, degrees c’)
 The series of commands above actually produce a figure with
default characteristics on line widths, font size, and so forth.
Application of MATLAB for solving sets of linear equations
 If a system of linear equations is expressed in terms of a square
matrix of coefficients, A, the vector b, and the unknowns x such
that Ax = b, then x can be found using either of two MATLAB
commands:
 The backslash operator “\” solves the system of equations using
Gaussian elimination, without calculating the matrix inverse, and
is preferable to “inv” because it is more efficient (in terms of
computer time and memory) and has better error detection
properties.
 Note that in this case, the vector length must equal the number of rows in
the matrix.
 The form of a set of equations is
Ax = b
where
A is a matrix of (known) coefficients,
x is the unknown vector, and
b is a vector of known quantities.
 The system of equations
3x + 3y =9
2x - 2y = -2
is represented in matrix notation as
Example: Consider the set of equations
 For this set of equations
 In MATLAB, we can set A = [13 -8 -3; -8 10 -1; -3 -1 11] and b = [20;-5;0]
 Solving this with the MATLAB command x=A\b produces (in much less
time than it takes to do it by hand!)
Quiz #1:
Solve the set of linear equations
 using the MATLAB command x=A\b and by hand to 2 or 3
decimal places of accuracy. Compare the answers.
 Linear programming in project management
 Linear programming can be used in construction
management to solve many problems such as:
1.
Optimizing use of resources
2.
Determining most economic product mix.
3.
Transportation and routing problems.
4.
Personnel assignment.
5.
Determining Optimum size of bid.
6.
Location of new production plants, offices and
warehouses
 Linear programming in project management
• Production allocation problem:
– A firm manufactures two types of products A and B and sells
them at a profit of 2 on type A and 3 on type B. Each product
is processed on two machines G and H. type A requires 01
minute of processing time on G and 02 minutes on H; type B
requires 01 minute on G and 01 minute on H. The machine G
is available for not more than 6 hours 40 minutes while
machine H is available for 10 hours during any working day.
Formulate the problem as a LPP.
 Linear programming in project management
• Production allocation problem:
– Let x1 be the number of products of type A; x2 be the
number of products of type B
Machine
Time of products (min)
Available time (min)
Type A (x1
units)
Type B (x2 units)
G
1
1
400
H
2
1
600
Profit per
unit
2
3
 Linear programming in project management
• Production allocation problem:
– Let x1 be the number of products of type A; x2 be the number of
products of type B
– Total Profit (objective function):
o P=2*x1+3*x2
– Constraints:
o 1*x1+1*x2 <= 400
o 2*x1+1*x2 <= 600
o x1 >= 0
o x2 >=0
 Linear programming in project management
 Assignment 2: Linear programming in construction management:
o A pre-mixed concrete firm has to supply concrete to three
different projects A, B, and C. The projects require 200, 350,
and 400 cubic meters of concrete in a particular week. The
firm has three plants P1, P2, and P3 which can produce 250,
400 and 350 respectively. The cost is different from each Plant
to each project since distance will vary. It is required to
determine the quantity to be supplied from each plant to
each project such that cost to be incurred is a minimum