civil engineering

1. Optimization Technique
2021-2022
2

2. 1
H/R=1
V=5m3
V=5m3 V=5m3
H/R=2 H/R=3
A=17.13m2 A=16.16m2 A=16.5m2
A B C
Why optimization technique?

3. 3
Learning outcomes:
• To determine or select the optimum value for the variables.
• For formulation of design problems as mathematical programming problems .

4. 4
What is the Optimization technique?
• The term Optimize is “to make perfect”.
• Choosing the best element from some set of available alternatives
• find values of the variables that minimize or maximize the objective function while satisfying the
constraints.
F´(x)=
F´(x)=
maximize
maximize

5. 5
Basic components of an optimization problem:
• An objective function expresses the main aim of the model which is either to be minimized or maximized.
• A set of unknowns or variables which control the value of the objective function.
• A set of constraint equations that allow the unknowns to take on certain values but exclude others.
g(x)=(x1,x2) Constraint Equation
X1,x2…. Unknown and Variables.
F(x)=ax1+ bx2…. Minimize or Maximize the objective function.

6. 6
Example-1/
A manufacturer needs to make a cylindrical water tank that will hold 1000 liters of water. Determine the
dimensions of the can that will minimize the amount of material used in its construction.
Solution/
V=1000L=1m3
 Required: h=? & r=? (variables)
 Objective function: 𝐴 = 2 ∗ 𝜋𝑟2 + 2𝜋𝑟 ∗ ℎ
 Constraint function: V=𝜋𝑟2*h=1
ℎ =
1
𝜋𝑟2
To find the optimal value for (h and r) minimize the Objective function (A´=0)
A(r)=2𝜋𝑟2 + 2𝜋𝑟 ∗
1
𝜋𝑟2= 2𝜋𝑟2 +
2
𝑟
A´(r)= 4𝜋𝑟 −
2
𝑟2=0 r = 0.542m & h=1.083m

7. 7
Home work-1.
A manufacturer needs to make a box water tank that will hold 1m of water. Determine the dimensions of the box
that will minimize the amount of material used in its construction and also compare it with the cylindrical water
tank and which of them do you prefer, Why?
Hint: length(l) and width (w) of the box are equal as shown in the figure.

8. 8
Example-2
A farm is 2000m from a river. It needs a rectangular channel to convey water from the river to the farm with a
discharge of 0.1 m/s with a 0.2 m/s velocity. Determine the optimum dimension for the channel.
Hint: Q=A*V
Solution:
𝐴 =
𝑄
𝑉
=
0.1
0.2
= 0.5𝑚2
 Required: b=? & h=?
 Objective function: L= 2h+b
 Constraint function: A=h*b
0.5=h*b b=
0.5
h
Minimize the objective function.
L=2h+
0.5
h
L´=2-
0.5
ℎ2 =0 2ℎ2
=0.5
h=0.5m & b=1m

9. 9
Home work-2(group task).
Write an example of an engineering problem on the basis of the optimization technique, then formulate and solve
it on the basis of the optimization technique.

10. 10
Thank you