Korea Advanced Institute of Science and Technology
operation reseach, introduction to anthropology
INDUSTRIAL 322

Spring 2015
Chapter 7
~ HOMEWORK ~
By: Prof. Y. Peter Chiu
7 4
The Noname Computer Company builds a computer design
ated modeI ICU2. It imports the motherboard of the comput
er from Taiwan, but the company inserts the sockets for the
chips and boards in its plant in
Korea Advanced Institute of Science and Technology
operation reseach, introduction to anthropology
INDUSTRIAL 322

Spring 2015
P41
Harold Grey owns a small farm in the Salinas Valley that grows apricots. The apricots are
dried on the premises and sold to a number of large supermarket chains. Based on past
experience and committed contracts, he estimates that sales over the next
Korea Advanced Institute of Science and Technology
operation reseach, introduction to anthropology
INDUSTRIAL 322

Fall 2014
LINEAR PROGRAMMING PROBLEM
MAX 9.68X1+5.91X2+15.2X3+11.74X4+7.34X5+16.97X6+15.44X7
S.T.
1) 35.3X1+32.3X2+20.8X3+25.3X4+49.1X5+46.2X6+20.5X7>0
2) 20X1+23.2X2+22.5X3+33.9X4+5.5X5+21.7X6+44X7>0
3) 28.3X10.9X2+6X3+20.5X4+29.7X5+45.7X621.1X7>0
4) 10.4X1
Korea Advanced Institute of Science and Technology
operation reseach, introduction to anthropology
INDUSTRIAL 322

Fall 2014
LINEAR PROGRAMMING PROBLEM
MAX 178X1+268X2+228X3+380X4+456X5+560X6+199X7+249X8+349X9+385X10+444X11+580X12+1
79X13+380X14+224X15+582X16
S.T.
1) 1X1+1X2+1X3+1X4+1X5+1X6<19
2) 1X7+1X8+1X9+1X10+1X11+1X12<18
3) 1X2+1X5+1X8+1X11+1X13+1X14<12
4) 1X3+1X6+1X9
Korea Advanced Institute of Science and Technology
operation reseach, introduction to anthropology
INDUSTRIAL 322

Fall 2014
LINEAR PROGRAMMING PROBLEM
MIN 614X1+660X2+534X3+680X4+590X5+630X6+603X7+639X8+702X9+693X10+693X11+630X12+8
65X13+830X14+775X15+850X16+900X17+930X18+532X19+553X20+511X21+581X22+595X23+553X
24+720X25+648X26+684X27+693X28+657X29+747X30
S.T.
1) 1X1+1X7+1X1
Korea Advanced Institute of Science and Technology
operation reseach, introduction to anthropology
INDUSTRIAL 322

Fall 2014
LINEAR PROGRAMMING PROBLEM
MAX 9.68X1+5.91X2+15.2X3+11.74X4+7.34X5+16.97X6+15.44X7
S.T.
1) 35.3X1+32.3X2+20.8X3+25.3X4+49.1X5+46.2X6+20.5X7>2
2) 20X1+23.2X2+22.5X3+33.9X4+5.5X5+21.7X6+44X7>2
3) 28.3X10.9X2+6X3+20.5X4+29.7X5+45.7X621.1X7>2
4) 10.4X1
Korea Advanced Institute of Science and Technology
operation reseach, introduction to anthropology
INDUSTRIAL 322

Spring 2015
P31
Stationery Supplies is considering installing an inventory control system in its store in Provo,
Utah. The store carries about 1,400 different inventory items and has annual gross sales of about
$80,000. The inventory control system would cost $12,50
Korea Advanced Institute of Science and Technology
operation reseach, introduction to anthropology
INDUSTRIAL 322

Spring 2015
Q1) Below are the number of cars, which are still under warranty, that show up for
repairs with engine problems at authorized car repair shops for XYZ brand model 3.
The XYZ company executives are worried what is going to happen with the number of
cars wi
Korea Advanced Institute of Science and Technology
operation reseach, introduction to anthropology
INDUSTRIAL 322

Spring 2015
LastName_FirstName_ID
_
Operations Management I 73331 Fall 2003
Odette School of Business
University of Windsor
Midterm Exam II Solution
Thursday, November 20
Education Gym
Instructor: Mohammed Fazle Baki
Aids Permitted: Calculator, straightedge, and a o
Korea Advanced Institute of Science and Technology
operation reseach, introduction to anthropology
INDUSTRIAL 322

Spring 2015
ISE 216
Question Hour
Chapter 5
Mar. 30th 2011
Q 5.10
Happy Henrys car dealer sells an imported car called the EX123. once every
three months, a shipment of the cars is made to Happy Henrys.
Emergency shipments can be made between these threemonth interv
Korea Advanced Institute of Science and Technology
operation reseach, introduction to anthropology
INDUSTRIAL 322

Spring 2015
ISE 216
Question Hour
Chapter 5
Mar. 30th 2011
Q 5.10
Happy Henrys car dealer sells an imported car called the EX123. once every
three months, a shipment of the cars is made to Happy Henrys.
Emergency shipments can be made between these threemonth interv
Korea Advanced Institute of Science and Technology
operation reseach, introduction to anthropology
INDUSTRIAL 322

Fall 2014
3
Applications of
Differentiation
1
3.3
How Derivatives Affect the Shape of a Graph
What Does f Say About f ?
3
Example 1
Find where the function f (x) = 3x4 4x3 12x2 + 5 is increasing and
where it is decreasing.
Solution:
f (x) = 12x3 12x2 24x
= 12x(x 2)
Korea Advanced Institute of Science and Technology
operation reseach, introduction to anthropology
INDUSTRIAL 322

Fall 2014
1
Functions and Limits
1.6
Calculating Limits Using
the Limit Laws
Calculating Limits Using the Limit Laws
Here we use the following properties of limits, called the Limit Laws,
to calculate limits.
Sum law
Difference law
Constant multiple law
Product law
Korea Advanced Institute of Science and Technology
operation reseach, introduction to anthropology
INDUSTRIAL 322

Fall 2014
1
Functions and Limits
1.2
A Catalog of Essential Functions
Mathematical Models
A mathematical model is a mathematical description
(often by means of a function or an equation) of a realworld phenomenon.
The purpose of the model is to understand the
pheno
Korea Advanced Institute of Science and Technology
operation reseach, introduction to anthropology
INDUSTRIAL 322

Fall 2014
1
Functions and Limits
1.8
Continuity
Continuity
The limit of a function as x approaches a can often be
found simply by calculating the value of the function at a.
Functions with this property are called continuous at a.
3
Continuity
Notice that Definitio
Korea Advanced Institute of Science and Technology
operation reseach, introduction to anthropology
INDUSTRIAL 322

Fall 2014
2
Derivatives
2.1 Derivatives and Rates of Change
Derivatives and Rates of Change
The problem of finding the tangent line to a curve and the
problem of finding the velocity of an object both involve
finding the same type of limit.
This special type of lim
Korea Advanced Institute of Science and Technology
operation reseach, introduction to anthropology
INDUSTRIAL 322

Fall 2014
2
Derivatives
2.2
The Derivative as a Function
The Derivative as a Function
We have considered the derivative of a function f at a fixed number a:
Here we change our point of view and let the number a vary. If we
replace a in Equation 1 by a variable x, w
Korea Advanced Institute of Science and Technology
operation reseach, introduction to anthropology
INDUSTRIAL 322

Fall 2014
2
Derivatives
1
2.5
The Chain Rule
The Chain Rule
Suppose you are asked to differentiate the function
The differentiation formulas you learned in the previous sections of this
chapter do not enable you to calculate F(x).
Observe that F is a composite func
Korea Advanced Institute of Science and Technology
operation reseach, introduction to anthropology
INDUSTRIAL 322

Fall 2014
1
Functions and Limits
1.5
The Limit of a Function
The Limit of a Function
As an example, we investigate the behavior of the function f
defined by f(x) = x2 x + 2 for values of x near 2.
3
The Limit of a Function
From the table and the graph of f (a parab
Korea Advanced Institute of Science and Technology
operation reseach, introduction to anthropology
INDUSTRIAL 322

Fall 2014
2
Derivatives
2.4
Derivatives of Trigonometric
Functions
Derivatives of Trigonometric Functions
In particular, it is important to remember that when we talk about the
function f defined for all real numbers x by
f (x) = sin x
it is understood that sin x m
Korea Advanced Institute of Science and Technology
operation reseach, introduction to anthropology
INDUSTRIAL 322

Fall 2014
2
Derivatives
1
2.3
Differentiation Formulas
Differentiation Formulas
Consider the constant function f (x) = c.
3
Power Functions
4
Power Functions
Proof. First note the identity
(an bn) = (a b) (an 1 + an 2b + an 3b2 + . . . + abn 2 + bn 1)
(xn) = limz x
Korea Advanced Institute of Science and Technology
operation reseach, introduction to anthropology
INDUSTRIAL 322

Fall 2014
2
Derivatives
1
2.6
Implicit Differentiation
Implicit Differentiation
Some times a curve in the plane can be defined implicitly by a relation
between x and y such as
x2 + y2 = 25
or
x3 + y3 = 6xy
3
Implicit Differentiation
In some cases it is possible to
Korea Advanced Institute of Science and Technology
operation reseach, introduction to anthropology
INDUSTRIAL 322

Fall 2014
2
Derivatives
2.8
Related Rates
Related Rates
If we are pumping air into a balloon, both the volume and the radius of
the balloon are increasing and their rates of increase are related to
each other.
But it is much easier to measure directly the rate of i
Korea Advanced Institute of Science and Technology
operation reseach, introduction to anthropology
INDUSTRIAL 322

Fall 2014
2
Derivatives
2.9
Linear Approximations and
Differentials
Linear Approximations and Differentials
We have seen that a curve lies very close to its tangent line near the
point of tangency. In fact, by zooming in toward a point on the graph of
a differentia
Korea Advanced Institute of Science and Technology
operation reseach, introduction to anthropology
INDUSTRIAL 322

Fall 2014
3
Applications of
Differentiation
3.1 Maximum and Minimum Values
Maximum and Minimum Values
In an optimization problem we want to find the maximum or minimum
values of a function.
3
Maximum and Minimum Values
In Definition 2 (and elsewhere), if we say tha