MA3264
Tutorial 2
1. Scheduling Production A manufacturer of an industrial product has to meet the
following shipping schedule:
Month
January
February
March
Required shipment (units)
10,000
40,000
20,000
The monthly production capacity is 30,000 units and
MA3264
Tutorial 1
1. Your grandparents have an annuity. The value of the annuity increases each month
as 1% interest on the previous months balance is deposited. Your grandparents withdraw
$1,000 each month for living expenses. Presently, they have $50,00
Sample solutions to Tutorial 8
Q.1
Let w(t) be Williams feelings for Zelda at time t and and z(t) be Zeldas
feelings for William at time t. The following model for w(t) and v(t) can be
established:
dw
= az(t),
dt
dz
= bw(t),
dt
where a > 0 and b > 0. To s
Sample solutions to Tutorial 7
Q.1
Supposing that pn and qn are the percentage of students who eat at Grease
Dinning Hall and Sweet Dinning Hall respectively at n-th period (n=0,1,2,
), a discrete model can be established as follows:
pn+1 = 0.25pn + (1 0
MA3264
Sample solutions to Tutorial 2
Q1. Let x1 , x2 and x3 be the amounts of unites produced in Jan, Feb and Mar
respectively, then the problem can be formulated as
min :
s.t.
c = 10(x1 + x2 + x3 ) + 3 [(x1 10000) + (x1 10000 + x2 40000)]
x1 + x2 + x3 =
Sample solutions to Tutorial 6
Q.1
(i) Multiplying both sides of the equation
sin t
dy(t)
=
dt
y
by y and integrating the obtained equation with respect to t, we get
1 2
y (t) = cos t + C,
2
where C is some constant to be determined.
(a) If y(0) = 2, we c
Sample solutions to Tutorial 5
Q.1
Because the manger of the reserve is considering allowing controlled hunting of E birds annually, the population of the bird, i.e., N(t) will satisfy
(1 0.01N)N
dN
=
E.
dt
1 + 0.1N
d
dt
Suppose that the limit of N(t) ex
MA3264
Sample solutions to Tutorial 4
Q1.
According to the assumptions, the model can be formulated as
dN(t)
= N 2 ,
dt
t > 0.
Here we use dozen as the unit of population, so N(0) = 1 and N(10) = 2. Solving the
initial-value problem, we have
1
N(t) =
.
1
MA3264
Sample solutions to Tutorial 3
Q1.
(i). From the assumption, we have the following discrete model,
an+1 = an + r(M an )(an m) := f (an ),
n 0,
with a constant r > 0.
(ii). To nd the equilibrium ,
= f () = = M or m.
Since
f () = 1 + r(M + m 2),
f (
Sample solutions to Tutorial 10
Q.1
(a) Let x = x(t) denote the displacement of the body from its equilibrium
at time t. We assume that the body does not sustain any other horizontal
forces except for the force coming from the spring. At time t = 0, the b
Sample solutions to Tutorial 9
Q.1
(a) Let r = r(t) denote the height of the projectile at time t. Because
the projectile undergoes only the gravitational force between the earth and
itself, we get the mathematical model for r(t)
m
dr 2
GMm
= 2 ,
2
dt
r
w
MA3264
Tutorial 3
1. Assume we are considering the survival of whales and that if the number of whales
below a minimum survival level m the species will become extinct. Assume also that the
population is limited by the carrying capacity M of the environme
Chapter 3
SIMPLEX METHOD
In this chapter, we put the theory developed in the last to practice. We develop the simplex method algorithm for LP problems given in feasible canonical form and standard form. We also discuss two methods, the M -Method and the T
MA3264
Tutorial 9
1. Suppose that a projectile is red straight upward from the surface of the earth with
initial velocity v0 .
(a) Assume the projectile is only undergoing the gravitational force between the earth
and itself. Build a mathematical model fo
MA3264
Tutorial 8
1. Consider the model in the romantic relationships with the dierence that the more
William loves Zelda, the more Zelda loves William, and conversely. Establish a mathematical
model.
(a) Suppose that initially Zelda is neutral when Willi
MA3264
Tutorial 6
1. Given the rst order ordinary dierential equation (ODE)
sin t
dy(t)
=
,
dt
y
t > 0.
(i) Solve the above ODE with each of the following initial values: (a) y(0) = 2; (b)
y(0)=1; and (c) y(0) = 3.
(ii) Sketch the solution with the initia
MA3264
Tutorial 5
1. In a wildlife reserve a non-indigenous bird endangers the other birds. To control this
pest a scientic study is made of the population growth of the unwelcome visitors, and it is
found that
dN(t)
(1 0.01N)N
=
, t > 0,
dt
1 + 0.1N
wher
MA3264
Tutorial 4
1. The time rate change of an alligator population N in a swamp is proportional to the
square of N. The swamp contained a dozen alligators in 1988, two dozen in 1998. When
will there be four dozen alligators in the swamp? What happens th
MA3264
Tutorial 7
1. Consider a model for the long-term dining behavior of the students at College USA.
It is found that 25% of students who eat at the colleges Grease Dining Hall return to eat
there again, whereas whose who eat at Sweet Dining Hall have
Sample solutions to Homework 3
Q.1
(i) The mathematical model for the amounts x(t) and y(t) of salt in the
two tanks is
dx
x
= ,
dt
20
x
y
dy
=
,
dt
20 40
with the initial conditions x(0) = y(0) = 50.
(ii) The solution to the above ODE system is
t
x(t) =
Sample solutions to Homework 2
Q.1
(i) If we assume that pn is the amount of digoxin present at n-th day (n =
0, 1, 2, ), then pn+1 pn = kpn because of the change in concentration per
day is proportional to the amount of digoxin present. Here k is some co
Sample solutions to Homework 1
Q.1 (15)
Let S(n) be the amount in the mortgage after n-th month and x be the monthly
payment, the following mathematical model
S(n + 1) = S(n) + rS(n) x,
, n = 0, 1, 2,
can be established where r is the monthly interest ra
Chapter 5 Probability Models
Introduction
Modeling situations that involve an element of chance
Either independent or state variables is probability or random
variable
Markov chain
Random variables
Statistics, system reliability,
Game theory
Casino
Technique of nondimensionalization
Aim:
To remove physical dimensions
To reduce the number of parameters
To balance or distinguish different terms in the equation
To choose proper scale for different variables
Method:
u us u * with us : chosen scale;
u*:
Free fall with air resistance
Forces which resist motion play an important part in
everyday life:
Brakes in cars; friction in many forms, damped vibrations
Laws established by experiments:
For very low speed (<1 meter per second), ignore air resistance.
What is a model
Some notations
Independent variables:
Time variable: t
Space variable: x in one dimension (1D), (x,y) in 2D or (x,y,z) in 3D
State variables or dependent variables: is a scalar or vector
valued variables, generally a function of the in
Chapter 3
Discrete Models
Introduction
Independent variables are chosen discrete values
Interest is accumulated monthly
World population is collected yearly
Number of tables
Number of population
Paradigm for state variables
Future value = present value