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
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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
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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
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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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Constraint 3 Because of limited demand, at most 40 bicycles should be produced each
week
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Bicycle ( X 1 )
Car ( X 2 )
Total

Fishing
2
1
100

Carpentry
1
1
80

Therefore constraint 1 may be expressed by:
2 x1  x2  100

(2)

and constraint 2 may be written as
x1  x2  80

(3)

Finally, we express the fact that at most 40 bicycles per week can be sold by limiting the
weekly productions of bicycles to at most 40 bicycles
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Can the decision
variable only assume nonnegative values, or is the decision variable allowed to assume
both positive and negative values?

If a decision variable X 1 can only assume nonnegative values, we add the sign
restriction X 1  0
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For Mapulanga
problem, it is clear that
x1  0 and x2  0

Combining the sign restriction X 1  0 and X 2  0 with the objective function (1) and
constraints (2) – (4) yields the following optimization model:
Max   15 000 x1  10 000 x2 subject to s
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t) means that the values of the decision variables X 1 and X 2 must satisfy
all the constraints and all the sign restriction
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Note that the

Mapulanga’s problem is typical of a wide class of linear programming problems in which
a decision maker’s goal is to maximize profit subject to limited resources
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1
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For example, the contribution
to the objective function from making 5 bicycles 5  10 000  = (K50 000) is
exactly five times the contribution to the objective function from making one
bicycle (K10 000)
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The contribution to the objective function for any variable is independent of the
values of the other decision variables
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1
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For example, it takes exactly three times
as many hours 2 3 = 6 finishing hours) to manufacture three bicycles as it takes
to manufacture one bicycle (2 finishing hours)
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The contribution of a variable to the left-hand side of each constraint is
independent of the values of the variable
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The first implication given in each list is called the proportionally assumption of
Linear programming
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For this reason, the second implication in each list is
called the Additivity Assumption of linear programming
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Two other
assumptions must also be satisfied before an LP can appropriately represent a real
situation
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The Divisibility Assumption requires that each decision variable be allowed to assume
fractional values
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Divisibility implies that it is acceptable to produce 2
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2 cars
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If we were not sure of the exact amount of carpentry and finishing hours
required to build a bicycle, the Certainty Assumption would be violated
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Any other point that is not in an LP’s
feasible region is said to be an infeasible point
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Similarly, for a minimization
problem, an optimal solution is a point in the feasible region with the smallest objective
function value
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The Company believes
that its most likely customers are high-income women and men
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Comedy shows and
football games
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Each football commercial’s seen by 2 million high-income
women and 12 million high-income men
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Dorian would like the commercials to be seen by at
least 28 million high-income women and 24 million high income men
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Example (3) An auto Company manufactures cars and trucks
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If the paint shop were only painting
trucks, 40 per day could be painted
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If the body shop wee producing cars, it could process 50 per day
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Each truck
contributes K1500 000 to profit, and each car contributes K100 000 to profit
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Example (4) Suppose that auto dealers require that the auto company in Example 3
produce at least 30 trucks and 20 cars
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Example 5 Graphically solve the following LP:
Max   2 x1  x2
s
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Case 2

The LP has alternative or multiple optimal solutions:

Two or more

extreme points are optimal, and the LP will have an infinite number of
optimal solutions
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Case 4

The LP is untouched: There are points in the feasible region with
arbitrarily large Z-values (max problem) or arbitrarily small Z values (min
problem)
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

define linear programming,



use graphical method to solve simple Linear Programming

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