Notesale: Turn your study into money

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UNIT 1
Reading
Handy A
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Winston Chapter 1
Hillier and Lieberman Chapter 1

THE ORIGINS OF OPERATIONS RESEARCH
The origin of Operations Research can be traced to operations in the Second World War
II
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After its successful application
in the war, it spread into industry, business and civil government
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The Nature of Operations Research
Operations research is concerned with optimal decision making in, and modelling of,
deterministic and probabilistic systems that originate from real life
...
In these
situations, considerable insight can be obtained from scientific analysis such as that
provided by operations research
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The structuring of the real life situation into a mathematical model, abstracting the
essential elements so that a solution relevant to the decision maker’s objectives can
be sought
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2
...


1

3
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The Impact of Operations Research
Operations research has had an increasingly great impact on the management of
organizations in recent years
...
Some of their techniques involve
quite sophisticated ideas in political science, mathematics, economics, probability theory
and statistics
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Also
financial institutions, governmental agencies, and hospitals are rapidly increasing their
use of operations research
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Police patrol officer scheduling in San Francisco
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Reducing fuel costs in the Electric Power industry
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Designing an Ingot Mold Stripping Facility at Bethelem Steel
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Gasoline Blending at Texaco
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Scheduling Trucks at North American Van lines
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Inventory Management at Blue Bell
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Using Linear programming to Determine Bond Portfolios
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Using linear programming to Plan production
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Equipment Replacement at Phillips Petroleum
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Where should a city locate a New Airport
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The number of full-time employees required on each day is in Table 2
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For example, an employee who works Monday to Friday must be off on Saturday and
Sunday
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Formulate an LP that the post office can use to minimize the number of fulltime employees that must be hired
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At present, the following four foods
are available for consumption: brownies, chocolate ice cream, cola, and pineapple
cheesecake
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Each day, I must ingest at least 500 calories, 6g of chocolate, 10g of sugar, and 8g of fat
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Formulate a linear
programming model that can be used to satisfy my daily nutritional requirements at
minimum cost
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1
K 11
3
13

Time 0 cash outflow
Time 1 cash outflow
NPV

Inv
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3
K5
5
16

Inv
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5
K 29
34
39

A BLENDING PROBLEM
Door -2-Door products blends its own all purpose liquid cleanser, composed of three
ingredients; detergents, ammonia, and lemon concentrate
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In addition, each
gallon must contain at least 2 percent but not more than 5 percent lemon concentrate
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The cleanser is produced by blending bulk cleansers, which Door-2-Door
obtains from two different suppliers
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10 per gallon
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35 per gallon
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The firm’s accounts have stated that production costs are

relatively fixed and that the only variable costs are those of the ingredients used in its
cleanser
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4

The objective function Door-2-Door would like to pay as little as possible (minimize the
cost) for the bulk cleansers used in making their own cleanser
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The cost of bulk cleanser A is its cost per gallon times the
number of gallons purchased, $1
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10 x1 
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35 times variable X 2 1
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Therefore, the objective function, to minimize
total cost is expressed:
Min   1
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35 x2
The Constraints: The constraints on this system are (1) the amount that must be produced
to meet demand for the month, (2) consistency requirements for ammonia, (3)
consistency requirements for lemon concentrate
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Assuming that Door-2-Door will produce exactly
200,000 gallons, the constraint, an equality is expressed:
x1  x2  200 ,000

The ammonia requirement:
 10%
 15%

1
...




20 x1 
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05 x2  30 ,000 (maximum of 15% ammonia)

The lemon requirement
 2%
 5%

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5

The final product B were used, the

upper limit on lemon concentrate would still not be exceeded
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The non-negativity constraints indicate that there can be no negative amounts of an
ingredient in the cleanser
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35 x2

Subject to
x1  x2  200 ,000 (product demand)

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65 x2  200 ,000 (minimum ammonia)

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05 x2  4000
x1 , x2  0

Learning Objectives
After working through this Chapter, you should be able to:


know the origin of Operations Research,



the nature of Operations Research,



the impact of Operations Research, and



know career options in Operations Research
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Taha Chapter 2
Wayne L
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(Figure 1)

Step 2
Observe the system
Step 6
Present results to
organisation

Step 1
Formulate
the problem
Step 3
Formulate a
mathematical model
of the problem

Step 4
Verify the
model and use
the model for
prediction

Step 5
Select a
suitable
alternative

Figure 1: The operations Research Methodology

7

Step 7
Implement and
evaluate
recommendation

Example 1
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A bicycle sells for K135,000 and used K50,000 worth of raw

materials
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A car sells for K105,000 and uses K45,000 worth of raw
materials
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the manufacture of wooden bicycles and cars requires two types of skilled labour:
carpentry and finishing
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A car requires 1 hour of finishing and 1 hour of carpentry labour
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Demand for cars is unlimited, but at most 40 bicycles are bought
each week
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Formulate a mathematical model of Mr Mapulanga’s situation that can be used to
maximize Mapulanga’s weekly profit
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Decision Variables: In any linear programming model, the decision variables should
completely describe the decisions to be made
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With this in mind, we
define:

X 1 = number of bicycles produced each week
X 2 = number of cars produced each week

Objective function: In any linear programming problem, the decision maker wants to
maximize (usually revenue or profit) or minimize (usually costs) some functions of the
decision variables
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For Mr Mulenga’s problem, we note that fixed costs (such as rent and

insurance) do not depend on the values of X 1 and X 2
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Weekly revenues = weekly revenues from bicycles + weekly revenues from cars
135 000 x1  105 000 x2
also

Weekly material costs  5 000 x1  45 000 x2
Other weekly variable costs  70 000 x1  50 000 x2

Then Mr Mapulanga wants to maximize

135 000 x1 105 000 x2  50 000 x  45 000 x2  70 000 x  50 000 x21 
15 000 x1  10 000 x2

We use the variable  to denote the objective function value of any LP
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This means that if Mr Mapulanga were free to choose any values for X 1 and X 2 , the
Company could make an arbitrarily large profit by choosing X 1 and X 2 to be very large
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Constraint 2 Each week, no m

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