Optimization Techniques
1. Optimization concepts
Abe Alkaff
Ab. Alkaff
Course objectives
Create and solve Mathematical Models for understanding and solving the optimization
problems
Learning Outcomes:
Student can analyze the optimization problem, building mathematical model, and evaluating
the proper optimization methods,
Requirements:
Engineering Mathematics II
Main References:
1. Alkaff, A. Operation Research Course Handbook, ITS
2. Hillier and Lieberman., “Introduction to Operation Research”, 10th Ed, Mc Graw Hill
International Edition, 2015
3. Taha, H.A., “Operation Research: an Introduction”, 8th Ed, Prentice Hall, 2006
Abe Alkaff
Work Methods
Real
Problems
abstracting
Model
implementing
Problem
Solving
Mathematical
interpreting
Sensitivity
Analysis
solving
Model
Solving
- variable
- parameter
Variable
- objective
and parameter
- constraint
function
Value from
Variable
Ab. Alkaff
Optimization problems illustration
Electric power systems generator:
• Three generator with supply capacity of 100, 80, 150
• Three region that require electric power as follows 140, 90, 70
• Cost distribution per unit from each generator to every region is
known:
D
P
1
2
3
1
4
6
2
2
7
3
5
3
1
4
6
Find the distribution pattern that minimizes the distribution cost!
Electric Power Distribution Scheme
Region
Generator
1
100
80
?
1
4
6
?
2
?
?
?
2
140
?
3
3
?
Abe Alkaff
Ab. Alkaff
1. Mathematical Models Building
a. Variable :
- Distributed power to each generator for every region
(𝑋𝑖𝑗 ) : distributed power from generator i to region j
- Total distribution cost (𝑍)
b. Parameter :
- Generator Load : 𝐾1 = 100, 𝐾2 = 80, 𝐾3 = 150
- Power Demand : 𝑃1 = 140, 𝑃2 = 90, 𝑃3 = 70
4 6 2 Where 𝐶𝑖𝑗 is the cost per unit
- Distribution Cost : C = 7 3 5 from generator 𝑖 to region 𝑗
1 4 6
Ab. Alkaff
c. Objective (variable and parameter function):
Minimize the distribution cost
Min 𝑍 = 4𝑋11 + 6𝑋12 + ..... +6𝑋33
d. Constraints (variable and parameter function):
- Total distributed power that distributed from each
generator should not greater or equal than the distribution
power limit
𝑋11 + 𝑋12 + 𝑋13 ≤ 𝐾1
𝑋21 + 𝑋22 + 𝑋23 ≤ 𝐾2
𝑋31 + 𝑋32 + 𝑋33 ≤ 𝐾3
Ab. Alkaff
- Total distributed power to every region should not less
than every region demand
𝑋11 + 𝑋21 + 𝑋31 ≥ 𝑃1
𝑋12 + 𝑋22 + 𝑋32 ≥ 𝑃2
𝑋13 + 𝑋23 + 𝑋33 ≥ 𝑃3
- Distributed power to every region should not be negative
𝑋𝑖𝑗 ≥ 0
Ab. Alkaff
e. Common mathematical models that fit:
3
3
Min Z = ෍ ෍ 𝐶𝑖𝑗 𝑋𝑖𝑗
𝑗=1 𝑖=1
With constraints:
3
෍ 𝑋𝑖𝑗 ≤ 𝐾𝑖
𝑖 = 1,2,3
𝑗=1
3
෍ 𝑋𝑖𝑗 ≥ 𝑃𝑗
𝑗 = 1,2,3
𝑖=1
𝑋𝑖𝑗 ≥ 0
Model (Mathematics) Transportation
Ab. Alkaff
2. Model Solving
Solved by using transportation method, assume the obtained
values:
𝑋13 = 70
𝑋31 = 140
𝑋22 = 80
𝑋32 = 10
And the other 𝑋𝑖𝑗 = 0
The total distribution cost 𝑍 is equal to 560 (more minimum than
the other alternatives)
Electric Power Distribution Solution Scheme
Generator
Region
1
100
80
?
140
4
1
2
3
6
?
?
?
?
2
?
70
3
?
Abe Alkaff
Ab. Alkaff
3. Solution Interpretation
• Generator 1 Distribute 70 MW Power to Region 3
• Generator 2 Distribute 80 MW Power to Region 2
• Generator 3 Distribute 140 MW Power to Region 1 and 10 MW to
Region 2
• Power Demand Fulfilled in Every Region
• There are 30MW Generator Capacity Remains in Generator 1
Ab. Alkaff
4. Sensitivity Analysis
• What if generator 1 capacity is reduced to 90 MW?
• What if generator 3 capacity is reduced to 130 MW?
• What happens if the distribution cost from generator 1 to region 2 is
increased from 6 to 7?
• Idem, if it is reduced from 6 to 5?
• Idem, if it is reduced from 6 to 2?
• What if demand of region 3 is reduced to 60 MW?
• What if demand of region 3 is increased to 80 MW?
• What if demand of region 1 is increased from 140 MW to 150 MW?
Exercise 1
Abe Alkaff
A company produces three types of products using three types of raw
materials. To make the first product, it takes 1 unit of raw material 1. To
make the second product, it takes 2 units of raw materials 1, 1 unit of raw
materials 2, and 1 unit of the first product. To make the third product, it
takes 4 units of 1 raw material, 1 unit of 2 raw materials, 1 unit of 3 raw
materials, and 1 unit of the second product. There are 50 raw material 1,
20 units of raw material 2, and 5 units of raw material 3. Find the number
of each product that must be made so that the company gets the maximum
profit. The first product's profit is 1 per unit, the second product is 7 per
unit, and the third product is 15 per unit.
Problem Scheme
Abe Alkaff
Ab. Alkaff
Variable:
Parameter:
Constraint:
Criteria:
Ab. Alkaff
Mathematical Model:
Sensitivity Analysis:
Exercise 2
Abe Alkaff
The company manufactures 2 types of products and draws up a production plan for the
next 3 months to meet all monthly demands. Companies can work regular working hours
(RT) or overtime (OT). The company can produce more than demand and save the results
for next month's use. Production capacity, demand, production costs, monthly storage costs
are given in the table.
Create a mathematical model that can be used to develop a production plan that
minimizes the total cost of production and storage cost!
Problem Scheme
Abe Alkaff
Ab. Alkaff
Variable:
Parameter:
Constraint:
Criteria:
Ab. Alkaff
Mathematical Model:
Sensitivity Analysis:
Exercise 3
Abe Alkaff
Plane manufacturing companies want to fulfill orders for jet planes that meet specific
specifications by their customers. The company received orders from three customers.
However, because the factory is still busy fulfilling previous orders, not all orders could
be done. Thus, a decision had to be made on how many planes to work for each
customer.
Parameter
Initial production costs
Profits per plane
Factory production capacity
Maximum number of planes order
Customer
1
2
3
$ 3 Million $ 2 Million
0
$ 2 Million $ 3 Million $ 0.8 Million
20%
40%
20%
3
2
5
The table above shows the initial costs of manufacturing the plane, the benefits
obtained from each customer, the factory capacity used in the manufacture of the plane,
and the maximum orders from each customer. Find the number of planes that can be
produced for each customer for maximum profit.
Abe Alkaff
Variable:
1. Number of plane made for a particular customer (𝑥𝑖 ; 𝑖𝑛 𝑖𝑛𝑡𝑒𝑔𝑒𝑟)
2. Processing certain customer requests, for start-up costs
(𝑦𝑖 ; 𝑖𝑛 𝑏𝑖𝑛𝑎𝑟𝑦: 0 → not processed, 1→processed)
Parameter:
1. Initial production costs:
• Customer 1 = $3 Million
• Customer 2 = $2 Million
• Customer 3 = 0
3. Factory capacity for each plane:
• Customer 1 = 20%
• Customer 2 = 40%
• Customer 3 = 20%
2. Profits from each plane:
• Customer 1 = $2 Million
• Customer 2 = $3 Million
• Customer 3 = $0.8 Million
4. Maximum order:
• Customer 1 = 3
• Customer 2 = 2
• Customer 3 = 5
Abe Alkaff
Objective:
Maximizing the profits that can be obtained from plane production
(profit per unit minus initial production costs)
𝑀𝑎𝑥 𝑍 = 2𝑥1 + 3𝑥2 + 0,8𝑥3 − 3𝑦1 − 2𝑦2
Constraint:
20𝑥1 + 40𝑥2 + 20𝑥3 ≤ 100 (Factory production capacity)
𝑥1 ≤ 3 (Maximum order for Customer 1)
𝑥2 ≤ 2 (Maximum order for Customer 2)
𝑥3 ≤ 5 (Maximum order for Customer 3)
𝑥1 ≤ 𝑀𝑦1 (Indicating that the work for Customer 1, if the Customer 1 order is done, then
the value 𝑦1 = 1, where the value of M is a very large number)
𝑥2 ≤ 𝑀𝑦2 (Indicating that the work for Customer 2, if the Customer 2 order is done, then
the value 𝑦2 = 1, where the value of M is a very large number)
Abe Alkaff
Mathematical model formulation:
𝑀𝑎𝑥
𝑆𝑇
2𝑥1 + 3𝑥2 + 0,8𝑥3 − 3𝑦1 − 2𝑦2
20𝑥1 + 40𝑥2 + 20𝑥3 ≤ 100
𝑥1 ≤ 3
𝑥2 ≤ 2
𝑥3 ≤ 5
𝑥1 ≤ 𝑀𝑦1
𝑥2 ≤ 𝑀𝑦2
Solution:
𝑥1 = 0; 𝑥2 = 2; 𝑥3 = 1; 𝑦1 = 0; 𝑦2 = 1
Z = 4.8
Interpretation:
Producing 0 plane for customer 1, 2 planes for customer 2, 1 plane for customer 3.
With maximum profit of $4.8 million
Sensitivity Analysis
Abe Alkaff
Exercise 4
Abe Alkaff
A bus company must provide drivers for its buses. Needs every hour varies. There are 3
types of drivers, namely a part-time driver who works for 4 hours with salary of $ 8 /
hour, a full-time driver who works for 8 hours continuously with salary of $ 12 / hour,
then a split-time driver who works for 4 hours, takes a break. 4 hours, and back to work
4 hours with salary of $ 14 / hour.
The number of drivers available is limited. Full-time and split-time drivers come from
the same bus driver service provider, namely as many as 20 people. The number of
people available can be added because the salary is increased by 25% from the initial
salary. Meanwhile, the availability of part-time drivers is 10 people.
Each driver can only work at most 1 shift per day. However, shifts that started at the end
of the day can resume the next day. How many drivers are needed of each type to meet
the daily needs of the least cost (salary)?
Exercise 4
Abe Alkaff
Abe Alkaff
Variable:
1.
Total demanded driver for each of types in specific period of time.
a. ful-time → 𝑓𝑇
b. part-time → 𝑝𝑇
c. split-time → 𝑠𝑇
d. Additional full-time → 𝑛 𝑇
e. Additional split-time → 𝑚 𝑇
Parameter:
1.
Driver salary (part-time = $8/jam, full-time = $12/jam, split-time = $14/jam, additional fulltime = $15/jam, additional split-time = $17.5/jam)
𝑑1 = 4
➢ 𝑑5 = 12
2. ➢Driver
demand for
each time period. (𝑑 𝑇 )
➢ 𝑑2 = 8
=4
➢➢ 𝑑𝑑
3 1= 10
=8
➢➢ 𝑑𝑑
4 2= 7
➢ 𝑑3 = 10
➢ 𝑑4 = 7
3.
4.
➢ 𝑑6 = 4
➢ 𝑑5 = 12
➢ 𝑑6 = 4
Part-time driver availability = 10
Driver availability full-time f dan split-time s = 20
Abe Alkaff
Objective:
The minimum total cost (salary) to be spent by the bus transportation company.
𝑍 = 4 8 𝑝1 + 8 12 𝑓1 + 8 14 𝑠1 + 8 15 𝑛1 + 8 17.5 𝑚1 + ⋯ + 4 8 𝑝6 + 8 12 𝑓6 +
8 14 𝑠6 + 8 15 𝑛6 + 8 17.5 𝑚6
Constraint:
1.
The number of drivers working in a certain period of time must not exceed the number of
drivers available.
𝑓1 + 𝑠1 + 𝑓2 + 𝑠2 + 𝑓3 + 𝑠3 + 𝑓4 + 𝑠4 + 𝑓5 + 𝑠5 + 𝑓6 + 𝑠6 ≤ 20
𝑝1 + 𝑝2 + 𝑝3 + 𝑝4 + 𝑝5 + 𝑝6 ≤ 10
2.
The number of drivers working in a certain period of time must not be less than the driver's
demand for that time period.
𝑓1 + 𝑓6 + 𝑝1 + 𝑠1 + 𝑠5 + 𝑛1 + 𝑛6 + 𝑚1 + 𝑚5 ≥ 4
𝑓2 + 𝑓1 + 𝑝2 + 𝑠2 + 𝑠6 + 𝑛2 + 𝑛1 + 𝑚2 + 𝑚6 ≥ 8
𝑓3 + 𝑓2 + 𝑝3 + 𝑠3 + 𝑠1 + 𝑛3 + 𝑛2 + 𝑚3 + 𝑚1 ≥ 10
𝑓4 + 𝑓3 + 𝑝4 + 𝑠4 + 𝑠2 + 𝑛4 + 𝑛3 + 𝑚4 + 𝑚2 ≥ 7
𝑓5 + 𝑓4 + 𝑝5 + 𝑠5 + 𝑠3 + 𝑛5 + 𝑛4 + 𝑚5 + 𝑚3 ≥ 12
𝑓6 + 𝑓5 + 𝑝6 + 𝑠6 + 𝑠4 + 𝑛6 + 𝑛5 + 𝑚6 + 𝑚4 ≥ 4
Abe Alkaff
Mathematical Model:
6
𝑀𝑎𝑥
𝑍 = 4 ෍(8𝑝𝑡 + 2(12𝑓𝑡 ) + 2(14𝑠𝑡 ) + 2(15𝑛𝑡 ) + 2(17.5𝑚𝑡 ))
𝑡=1
𝑆𝑇
𝑓𝑡 + 𝑓𝑡−1 + 𝑝𝑡 + 𝑠𝑡 + 𝑠𝑡−2 + 𝑛𝑡 + 𝑛𝑡−1 + 𝑚𝑡 + 𝑚𝑡−2 ≥ 𝑑 𝑇
for 𝑡 = 1,2,3,4,5,6
6
෍ 𝑓𝑡 + 𝑠𝑡 ≤ 20
𝑡=1
6
෍ 𝑝𝑡 ≤ 10
𝑡=1
Solution:
𝑝1 = 1; 𝑓2 = 8; 𝑝3 = 2; 𝑓4 = 7; 𝑝5 = 5; 𝑓6 = 3; 𝑝6 = 1
With value of 𝑍 = 2016
Interpretation:
1.
2.
3.
Provides 8 people full-time driver for period 2, 7 people for period 4, 3 people for period 6.
Provides 1 person part-time driver for period 1 and 6, 2 people for period 3, and 5 people for
period 5.
Spends cost in a total of $2,016 for paying the driver salary in 1 day.
Course Plan
1.
2.
3.
4.
5.
6.
7.
8.
Abe Alkaff
Optimization concepts
Basics of optimization mathematics
Analytical solution of optimization problem
Numerical Solution of optimization problem: bisection, bisection with
derivative, golden section, steepest descent, conjugate gradient,
Newton
Linear programming: standard and non-standard linear programming
Linear programming variations: integer, mixed, quadratic,
transportation, assignment, and transhipment
Dynamic programming: standard, continue, stochastic, optimal
control, and long-term
Heuristics Methods
Grading
Abe Alkaff
• Weekly Assignments (+/- 8 Assignments): 35-40%
• Report
• Presentation
• Mid-term Exam, Final Exam/Project: 50-60%
• Mid-term, Final Exam: Open Book
• Class Participation: 10%
• Weekly Assignments Or Final Project Is Worked in Group of 2
People
• Assignments is graded based on:
• Substance(Complexity, Realistic, Correctness)
• Systematic (Completeness, Order)
• Language aspect
Assignment 1
Abe Alkaff
1. Do the exercise 2
2. Change this slide into article in word format
3. Find an optimization problem from anywhere
• Write the problem description
• Define and symbolize:
• variable value that should be known,
• Parameter that should be used,
• Constraints that should be fulfilled, and
• Criteria that should be optimized
• Create the mathematical model
• Find the solution, which is the variable value that should be known (random or could be taken
from the source that is used)
• Give an interpretation to the solution, which is interpretation from the obtained variable
• Think about some sensitivity analysis that could occurs on that problem
• Submit it in form of softcopy via email to abealkaff@gmail.com with cc to
drahmadi41@gmail.com and to share-its in Saturday no later than 23:59