This course introduces students to operations research and its applications. The course covers several optimization techniques including linear programming, network flows, and transportation problems. Students will learn how to formulate mathematical models of real-world systems and solve them to determine optimal resource allocation. The goal is for students to be able to apply operations research methods to decision-making problems in fields like manufacturing, transportation, and public services. Students will be assessed through assignments, tests, and a final exam.

1. MEC4103: OPERATIONS RESEARCH (3CUs)
Course Instructor: Ms. Gloria Faith Atto
Email: attogloriafaith@gmail.com
Tel: 0784167066/0703548650
Brief course description
• This course introduces students to the systematic planning of resource
allocation and utilization
Course objectives/learning outcomes
• At the end of the course students will be able to decide between
alternatives in industrial design and optimum use of available resources
• Effectively communicate ideas, explain procedures and interpret results and
solutions in simulation

2. DETAILED COURSE DESCRIPTION
• Review of linear algebra (Read about this in Pgs. 27-32 of PDF)
• Linear programming: LP formulations, solving LPs by; Graphical method, simplex
methods, duality, and sensitivity analysis.
• Network flows: transportation and assignment problems, shortest paths, minimum
spanning trees, network simplex method, multi-commodity flows.
• Modeling issues in linear programming and network flow applications.
Course Assessment
• Course work assignments (20%), Tests (20%), Final examination (60%)
Recommended Resources
• Hamdy A. Taha (2006): Operations Research: An Introduction, 8th Edition. Prentice Hall;
• Philip M. Morse (2007): Methods of Operations Research. Kormendi Press;
• Frederick S. Hillier, Gerald J. Lieberman, Frederick Hillier, and Gerald Lieberman (2004):
MP Introduction to Operations Research. McGraw-Hill
• The internet and other electronic resource

3. Definitions
• Operations research (OR) is the application of scientific methods to improve the effectiveness
of operations, decisions and management by means such as analyzing data, creating
mathematical models and proposing innovative approaches. OR is often concerned with
determining the maximum (of profit, performance or yield) or minimum (of loss, risk or cost)
of some real world objective
• “O.R. is applied decision theory, which uses any scientific, mathematical or logical means
to attempt to cope with the problems that confront the executive, when he tries to achieve
a thorough-going rationality in dealing with his decision problem”. (Miller and Starr)
• “Operational Research is the attack of modern science on complex problems arising in the
direction and management of large systems of Men, Machines, Materials and Money in
Industry, Business, Government and Defense (Operational Research Society of Great
Britain)
LECTURE ONE: INTRODUCTION

4. Applications of OR
• Military operations
• Manufacturing eg manufacturing of
safety boots
• transportation
• public services
• construction
• telecommunication
• Finance
Principal phases of implementing
operations research
• The principal phases of
implementing an operations
research in practice include:
• Definition of the problem
• Mathematical model
formulation
• Solution of the model
• Validation of the model
• Implementation of the model
INTRODUCTION…

5. PRINCIPLE PHASES IN DETAILS
1. Problem definition
• This involves defining the scope of the problem under investigation. This phase is carried out by
the entire OR team. The aim is to identify three principal elements of the decision problem:
• Description of the decision alternatives
• Determination of the objective of the study
• Specification of the limitations under which the modeled system operates
2. Model formulation
• This entails an attempt to translate the problem definition into mathematical relationships. The
problem is identified with decision variables such as:
 How many units to buy/sell...
 How much time to spend on a task...
• Measure of performance is through the objective function
 What is the goal/objective?
 Usually: Max/min profit/cost/time/units

6. PRINCIPLE PHASES IN DETAILS…
3. Model Solutions
• Mathematical representations are always approximations of the real world. Model solution is by far
the simplest of all OR phases because it entails the use of well-defined optimization algorithms. An
important aspect of the model solution phase is sensitivity analysis. It deals with obtaining
additional information about the behavior of the optimal solution when the model undergoes some
parameter changes. Sensitivity analysis is particularly needed when the parameters of the model
cannot be estimated accurately.
4. Model testing/validation
• The process of testing/improving model is known as model validation. Model validity checks
whether or not the proposed model does what it purports to do (i.e. does it predict adequately the
behavior of the system under study?). Initially the OR team should be convinced that the model’s
output does not include “surprises.” In other words does the output make sense?
• The common method for checking validity is to compare its output with historical output data. The
model is valid if, under similar input conditions, it reasonably duplicates past performance.
However, if the proposed model represents a new (non-existing) system, no historical data would
be available. In such cases, we may use simulation as an independent tool for verifying the output
of the mathematical model.

7. PRINCIPLE PHASES IN DETAILS…
5. Implementation of the model solution
• Implementation of the solution of the validated model involves the translation of
the results into understandable operating instructions to be issued to the people
who will administer the recommended system. The burden of this task lies
primarily with the OR team.
 OR team explains system to management
 OR team develops procedures required to put system into operation
 Management trains personnel

8. LECTURE TWO: LINEAR PROGRAMMING (LP)
• 2.1 Introduction
• 2.2 Requirements of a Linear Programming problem
• 2.3 Formulating small to moderate LP problems
• 2.4 Graphical solution to an LP problem
• 2.5 Special cases in LP problems
Introduction
• Linear programming is a technique that helps in resource allocation decisions. Resources include
machinery, labour, money, time, warehouse space, raw materials, etc.
• Computer programs help much in solving real life LP problems that are too cumbersome to solve
by hand or with a calculator.
• Requirements of a LP Problem
• All LP problems have three properties in common.
a. All LP problems seek to maximize or minimize a linear function of the decision
variables. The function to be minimized or maximized is called the objective function.
That is max / min z = f(x1, x2, …, xn) = c1x1+c2x2+…+cnxn; where xi is decision variable,
ci is the coefficient of the decision variable.

9. LINEAR PROGRAMMING
a. The values of the decision variables must satisfy a set of technological constraints. Each
constraint must be a linear equation or linear inequality. That is a1jx1+a2jx2+…+anjxn = / ≥/≤ bj.
b. A sign restriction is associated with each variable. For any variable xi, the sign restriction
specifies that xi must be nonnegative (xi ≥ 0) or that xi may be unrestricted in sign (urs).
The general LP formulation
The general form for a Linear Programming problem is as follows:
Objective Function:
s.t.
Technological Constraints:
Sign Restrictions:
where ``urs'' implies unrestricted in sign.

10. Formulating a linear program
• 1. Choose decision variables
• 2. Choose an objective function – linear function in variables
• 3. Choose constraints – linear inequalities
• 4. Choose sign restrictions

11. LP Formulation, Example1
• A toy company makes two types of toys: toy soldiers and trains. Each toy is produced in two
stages, first it is constructed in a carpentry shop, and then it is sent to a finishing shop, where it is
varnished, vaxed, and polished. To make one toy soldier costs $10 for raw materials and $14 for
labor; it takes 1 hour in the carpentry shop, and 2 hours for finishing. To make one train costs $9
for raw materials and $10 for labor; it takes 1 hour in the carpentry shop, and 1 hour for finishing.
There are 80 hours available each week in the carpentry shop, and 100 hours for finishing. Each
toy soldier is sold for $27 while each train for $21. Due to decreased demand for toy soldiers, the
company plans to make and sell at most 40 toy soldiers; the number of trains is not restriced in any
way. What is the optimum (best) product mix (i.e., what quantities of which products to make) that
maximizes the profit (assuming all toys produced will be sold)?
• Solution
Hrs in Carpentry Hrs in finishing RM Labour S/price
Soldier 1 1 10 14 27
Train 1 2 9 10 21
Avail hrs 80 100

12. LP Formulation…
• Profits: Selling price -costs
Toy soldier, 27-(10+14)$ =3$
Toy train, 21-(9+10)$ =2$
Decision variables
Let x1 be the no. of toy soldiers produced
Let x2 be the no. of toy trains produced
Mathematical model formulated
Main Objective, Max z=3x1+2x2
S.t. (constraints) x1 + x2 ≤ 80
x1 +2x2≤ 100
x1 ≤ 40
Xi≥0, i=1,2

13. LP Formulation, Example 2
• Furniture company manufactures four models of chairs. Each chair
requires certain amount of raw materials (wood/steel) to make. The
company wants to decide on a production that maximizes profit
(assuming all produced chairs are sold). The required and available
amounts of materials are as follows;
Chair 1 Chair 2 Chair 3 Chair 4 Total
available
Wood
Steel
1
4
1
9
3
7
9
2
4,400 (lbs)
6,000 (lbs)
Profit $12 $20 $18 $40 maximize

14. Example 2…..
Decision variables
Let x1 be the no. of chair 1 produced
Let x2 be the no. of chair 2 produced
Let x3 be the no. of chair 3 produced
Let x4 be the no. of chair 4 produced
Mathematical model formulated
Main Objective, Max (Profit) z=12x1+20x2 +18x3 +40x4
S.t. (constraints) x1+x2 +3x3 +9x4 ≤ 4,400
4x1+9x2 +7x3 +2x4 ≤ 6,000
Xi≥0, i=1,2,3,4

15. LP Formulation, Example 3
• A company wants to produce a certain alloy containing 30% lead, 30% zinc, and
40% tin. This is to be done by mixing certain amounts of existing alloys that can
be purchased at certain prices. The company wishes to minimize the cost. There
are 9 available alloys with the following composition and prices
Alloy 1 2 3 4 5 6 7 8 9 Blend
Lead (%)
Zinc(%)
Tin (%)
20
30
50
50
40
10
30
20
50
30
40
30
30
30
40
60
30
10
40
50
10
10
30
60
10
10
80
30
30
40
Cost ($/lb) 7.3 6.9 7.3 7.5 7.6 6.0 5.8 4.3 4.1 Minimize

16. Example 3…
Decision variables
Let x1 be the amount of alloy 1 in a unit of blend
Let x2 be the amount of alloy 2 in a unit of blend
Let x3 be the amount of alloy 3 in a unit of blend
Let x4 4
Let x5 5
Let x6 6
Let x7 7
Let x8 8
Let x9 9
Mathematical model formulated
Main Objcteive, Min (cost) z=7.3x1+6.9x2 +7.3x3 +7.5x4 +7.6 x5+6.0x6 +5.8x7 +4.3x8+4.1x9
S.t. (constraints) 0.2x1+0.5x2 +0.3x3 +0.3x4 +0.3x5+0.6x6 +0.4x7 +0.1x8 +0.1x49= 0.3
0.3x1+0.4x2 +0.2x3 +0.4x4 +0.3x5+0.3x6 +0.5x7 +0.3x8 +0.1x49= 0.3
0.5x1+0.1x2 +0.5x3 +0.3x4 +0.4x5+0.1x6 +0.1x7 +0.6x8 +0.8x49= 0.4
Xi≥0, i=1,2,3,4,5,6,7,8,9

17. Take home
1. Post office requires different numbers of full-time employees on different days.
Each full time employee works 5 consecutive days (e.g. an employee may work
from Monday to Friday or, say from Wednesday to Sunday). Post office wants to
hire minimum number of employees that meet its daily requirements, which are as
follows.
Monday Tuesday Wednesday Thursday Friday Saturday Sunday
17 13 15 19 14 16 11