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1. Introduction to Operations
Research
Dr Indra Mahakalanda
PhD (Canterbury), MSc (Moratuwa), BSc Eng (Moratuwa), DipM (CIM, UK)
Senior Lecturer,
Department of Decision Sciences,
Faculty of Business

2. What is Operations Research (OR)?
 OR seeks to improve a problem solution by providing decision makers with information and
insights gained through problem analysis, often involving mathematical models and
computers.
 Some interesting problems that we solve in OR are:
 What is the shortest path/route from one city to the other? What is the lowest cost path/route from one city to
the other? (We call this as a Transportation Network Optimization Problem).
 What is the optimal hydro-reservoir release that gives maximum benefits to the society via hydropower
generation and agriculture? (This is known as Resource Allocation Problem).
 How would you allocate existing lecture halls and laboratories to maximize their utility? (Resource Allocation
Problem).
 Management Science and Operational Research are some other titles used to refer the
same discipline.

3. Classification of Optimization Problems
 Generally, in an optimization problem, our solution seeks to either
maximize (profit, revenue, resource utilization etc.) or minimize (cost,
time, distance, carbon emission etc.).
 We can categorize any optimization problem into two:
 Unconstrained optimization
 Constrained optimization

4. Unconstrained Optimization
 Any optimization problem whether it is a constrained or an unconstrained, there is an objective.
 We call this as an Objective Function.
 Let us consider a simple function 𝑦 = 𝑥2
. Let us sketch this function.
 So we call this as a minimization problem. The objective is to minimize the value of 𝑦 for some value of 𝑥.
 Why do we call this as unconstrained optimization problem?
 Also we call this objective function as Non-Linear because the power / degree is not equal to 1. So we expect
the function to be non-linear.
0
5
10
15
20
25
30
-6 -4 -2 0 2 4 6
y
x
y = x2
Minimum value of the function, y
is 0 when x = 0

5. Constrained Optimization
 Real world problems are rarely unconstrained. That is there are
limitations to the values that 𝑥 can take for a particular objective.
 So a typical constrained optimization problem has:
 An objective function, 𝑓 𝑥
 This can be a maximization objective or minimization objective
 An a set of constraints
 The limitations on the values of 𝑥 can take.

6.  Let us consider the same function 𝑓 𝑥 = 𝑥2. But we will impose a constraint to 𝑥
such that 𝑥 ≥ 2.
 Now the objective function is:
min
𝑥
𝑓 𝑥 = 𝑥2
Subject to:
Constraint: 𝑥 ≥ 2
 Let us sketch the above:
 You may see that 𝑥 is bounded by the constraint
 That is 𝑥 cannot take values less that 2.
 So the new minimum is 𝑓 𝑥 = 4. This is called constrained optimization.
0
5
10
15
20
25
30
-6 -4 -2 0 2 4 6
y
x
y = x2
𝑥 = 2

7. Re-cap
There are two types of optimization problems:
 Constrained and unconstrained.
 Our focus in this lecture series is about constrained optimization problems.
 A typical optimization problem has the following components:
 Objective function
 This can be Linear (eg: 𝑓 𝑥 = 𝑎𝑥 + 𝑏) or Non-linear (eg: 𝑓 𝑥 = 𝑎𝑥2 + 𝑏𝑥 + 𝑐)
 Subject to:
 One or many constraints (eg: 𝑔 𝑥 ≤ 𝑘)

8. Constrained Optimization: Solution Methods
 Linear Programming (LP)
 Integer Linear Programming (ILP)
 Mixed Integer Linear Programming (MILP)
 Non-Linear Programming
 Dynamic Programming Non traditional optimization methods
 Genetic Algorithms
 Differential Evolution
 Simulated Annealing
 Simulated Quenching

9. In-class exercise
Wyndor is a manufacturing company. It has developed the following new products:
• An 8-foot glass door with aluminum framing.
• A 4-foot by 6-foot double-hung, wood-framed window.
The company has three plants and their functions are as follows:
• Plant 1 produces aluminum frames and hardware.
• Plant 2 produces wood frames.
• Plant 3 produces glass and assembles the windows and doors.
The management has to decide the following:
Should they go ahead with launching these two new products? (What would be the
maximum profit that they can earn?)
If so, what should be the product mix? That is how many door and windows should
they produce to earn maximum profit.

10. Production Time Used for
Each Unit Produced
Plant Doors Windows Available per
week
1 1 hour 0 4 hours
2 0 2 hours 12 hours
3 3 hours 2 hours 18hours
Unit Profit $300 $500

11. Decision Variables:
𝑥𝐷 :No. doors to be manufactured
𝑥𝑤 :No. windows to be manufactured
Objective:
To maximize profit
𝑚𝑎𝑥
𝑥𝐷,𝑥𝑊
300𝑥𝐷 + 500𝑥𝑤
Subject to:
Constraint 1: Capacity/time available of Plant – 1
1. 𝑥𝐷 ≤ 4
Constraint 2: Capacity/time available of Plant – 2
2. 𝑥𝑤 ≤ 12
Constraint 3: Capacity/time available of Plant – 3
3. 𝑥𝐷+ 2. 𝑥𝑤 ≤ 18
Non negativity constraints:
𝑥𝐷 ≥ 0,𝑥𝑤 ≥ 0

12. -2
0
2
4
6
8
10
0 1 2 3 4 5 6 7 8
No
of
Windows,
X
W
No. of Doors, XD
XW versus XD
𝟑. 𝒙𝑫+ 2. 𝒙𝒘 = 𝟏𝟖
𝑥𝐷 = 4
𝑥𝑊 = 6

13. In class exercise : The Profit & Gambit Co.
Management has decided to undertake a major advertising campaign that will
focus on the following three key products:
• A spray prewash stain remover.
• A liquid laundry detergent.
• A powder laundry detergent.
The campaign will use both television and print media. The general goal is to
increase sales of these products. Management has set the following goals for the
campaign:
• Sales of the stain remover should increase by at least 3%.
• Sales of the liquid detergent should increase by at least 18%.
• Sales of the powder detergent should increase by at least 4%.
Question: how much should they advertise in each medium to meet the sales
goals at a minimum total cost?

14. Increase in Sales per Unit of
Advertising
Product TV PM Minimum Required
Increase
Stain remover 0% 1% 3%
Liquid detergent 3% 2% 18%
Powder detergent -1% 4% 4%
Unit Cost $1million $2million

15. Decision Variables:
𝑥𝑇𝑉 :No. advertising units in TV
𝑥𝑃𝑀 : No. advertising units in PM
Objective:
To minimize the total advertising cost:
𝑚𝑖𝑛
𝑥𝑇𝑉,𝑥𝑃𝑀
𝟏. 𝑥𝑇𝑉 + 𝟐. 𝑥𝑃𝑀
Subject to:
Stain remover : Total sales to be increased => 3%
1%. 𝑥𝑃𝑀 ≥ 3%
Liquid detergent : Total sales to be increased => 18%
3%. 𝑥𝑇𝑉 + 2%. 𝑥𝑃𝑀 ≥ 18%
Powder detergent : Total sales to be increased => 4%
−1%. 𝑥𝑇𝑉+ 4%. 𝑥𝑃𝑀 ≥ 4%
Non negativity constraints:
𝑥𝑇𝑉 ≥ 0,𝑥𝑃𝑀 ≥ 0