iii) Material planning for subassembly A (level 1):
Since A is dependent on product 1, therefore the projected requirement for A in week 4 is 105
Lot size = PPB
Lead time = 1
Projected Requirement
Week
1
2
10
3
4
105
Scheduled Receipts
On hand at
end of w
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MS-E2140 Linear Programming
Exercise 4
Fri 23.09.2016
Maari-B
Week 2
This weeks homework https:/mycourses.aalto.fi/mod/folder/view.php?id=130923 is due no
later than Sunday 02.10.2016 23:55.
Exercise 4.1 True and False Statements about Simplex
Course book
MS-E2140 Linear Programming
Exercise 7
Thu 06.10.2016
Maari-B
Week 4
This weeks homework https:/mycourses.aalto.fi/mod/folder/view.php?id=130935 is due no
later than Sunday 16.10.2016 23:55.
Exercise 7.1 Unboundedness and duality
Consider the LP:
c0 x
(P
Chap 2. The Geometry of LP
In the text, polyhedron is defined as P = cfw_ x Rn : Ax b . So some
of our earlier results should be taken with modifications.
Thm 2.1:
(a) The intersections of convex sets is convex.
(b) Every polyhedron is a convex set.
(c)
IE 7200
Supply Chain Engineering
Date:02/05/2017
Homework # 2
Problem 1.
Year
Month
Demand
(Units)
1
Jan
Feb
Mar
Apr
May
Jun
Jul
Aug
Sep
Oct
Nov
Dec
Jan
Feb
Mar
100
125
140
160
185
210
225
235
215
200
190
180
205
230
240
Total error
2
3 week
moving
averag
How to calculate seasonal
index
Pick time period (number of years)
Pick season period (month, quarter)
Calculate average price for season
Calculate average price over time
Divide season average by over time
average price x 100
Sioux Falls Cull Cow Prices
Resume Checklist
Accomplishment-Oriented
First Impression
Is the resume inviting to read, with clear sections and
ample white space?
Does the design look professional and not like a simple
typing job?
Is a qualifications summary included so the reader
imm
I FS , A NDS
AND
O RS
CONS
1999/11/6
page 17
17
When we are talking about a language Lang(AForm; Conn) and we do not
particularly care about AForm or Conn, we will simply refer to it as Lang. We
use A, B and other capitals from the beginning of the alpha
CONS
1999/11/6
page 45
I FS , A NDS
AND
O RS
45
Practice
cfw_2.1 In the table below, match each consecution with the structural rule needed to
prove it.
AAA
B A (B C) (A C)
A (A B) B
ABB
A B (B C) (A C)
K
M
Bc
WI
B
cfw_2.2 Show that in any system contai
I FS , A NDS
AND
O RS
CONS
1999/11/6
page 19
19
as we eat up l 1 of the l in the front of the sum to beef up the l-numbered
value of the sum in the interior to nm . This means that the hypothesis holds
of A too, given that it holds of the subformulae of
T HEORIES
CONS
1999/11/6
page 91
91
The t condition is t is true. The implication condition is if A B is true,
then if A is true so is B. The non-triviality conditions mean simply that is
true and is not.
In traditional, classical logics, many of these c
I FS , A NDS
AND
O RS
CONS
1999/11/6
page 27
27
Here is another proof, using C.
A (B C) A (B C) A A
(E)
A (B C); A B C
!
"
A (B C); A ; B C
!
"
A (B C); B ; A C
A (B C); B A C
BB
(E)
[C]
(I)
A (B C) B (A C)
(I)
The structural rule C gives us strong commu
M ODALITIES
CONS
1999/11/6
page 51
51
P ROOF The positive modality and properties are simple to prove:
A
A
AB
A
B
A
B
(I)
(E)
(E)
(E)
For the disjunction property, we have
AA
AB AB
A
A
BB
(I)
A
A
B
B
B
(I)
(A B)
A
B
(I)
B
A
B
(A B)
A
I FS , A NDS
AND
O RS
CONS
1999/11/6
page 29
29
a string of type B. I leave it to you to consider how you should read fusion in
other applications.
Whatever the interpretation of fusion, we have a number of important behaviours. For example, we can prove
CONS
1999/11/6
page 9
Chapter 2
Ifs, Ands and Ors
A syllogism is a form of words
in which when certain assumptions are made,
something other than what has been assumed
necessarily follows
from the fact that the assumptions are such
Aristotle, Prior Anal
I FS , A NDS
AND
O RS
CONS
1999/11/6
page 37
37
It will be shown in Chapters 9 and 1110 that adding the comma with these
structural rules does not alter the properties of any connectives other than
and . This is not obvious, because there just might be
CONS
1999/11/6
page 85
H ILBERT S YSTEMS
85
L EMMA 4.24 (F USION A XIOMS )
In the presence of B and Bc you can replace the fusion rules with (A B C)
(A (B C).
P ROOF It is sufficient to show that these are provable in the natural deduction
system, give
I FS , A NDS
AND
O RS
CONS
1999/11/6
page 21
21
D EFINITION 2.17 (I NFERENCES AND R ULES )
An inference is a pair consisting of a set of consecutions (the premises of the
inference) and a single consecution (the conclusion of the inference). The
premises
I FS , A NDS
AND
O RS
CONS
1999/11/6
page 13
13
numbers to themselves. So any f S is a function which, when given a number
x, returns a number f (x). The function f : x " 3x + 1, when given 5, returns
16, for example.
We will assume that the set S is cou
I NTRODUCTION
CONS
1999/11/6
page 3
3
E XAMPLE 1.2 (R ESOURCE C ONSCIOUSNESS )
This is not the only way to restrict premise combination. Girard [96] introduced
linear logic as a model for processes and resource use. The idea in this account
of deduction
H ILBERT S YSTEMS
CONS
1999/11/6
page 83
83
If X(A) = Y (A) then we wish to prove
(Y (A B)
Y (A)
Y (B).
Y (A), and Y (B) "
Y (B), and
If " is present, then we have Y (A) "
Y (A) "
Y (B), and Y (B) "
Y (A) "
Y (B), which
hence Y (A) "
Y (A)"
Y (B),
M ODALITIES
CONS
1999/11/6
page 63
63
We can also relate our negation operators to conjunction and disjunction.
L EMMA 3.21 (O NE DE M ORGAN LAW )
For any split negation we have A B (A B) in any logic in which and
are present (and similarly for in place
M ODALITIES
CONS
1999/11/6
page 49
49
These conditions are characteristic of possibility-style operators. If A B is
possible (true in some related state) then one of A and B must be possible too.
Similarly, there is no way that can be possible, as it is
CONS
1999/11/6
page 89
Chapter 5
Theories
Dear friend,
theory is all grey,
And the golden tree of life is green.
Goethe, Faust
Logic is not merely about individual propositions and consequence relations between them. Theories are interesting too. Theori
M ODALITIES
CONS
1999/11/6
page 65
65
If we add these rules we have a converse contraposition result.
L EMMA 3.27 (C ONVERSE C ONTRAPOSITION )
In any logic containing both double negation elimination rules, the two-way contraposition rule
A B
=
B A
holds
I FS , A NDS
AND
O RS
CONS
1999/11/6
page 43
43
by Mints and Dunn, though their work was on Gentzen-style consecution systems (with introduction rules for connectives in the antecedent and consequent,
instead of introduction and elimination rules in the
I FS , A NDS
O RS
AND
CONS
1999/11/6
page 23
23
Note that these rules assume that we are in a language with a conditional,
and a system of structures with at least the semicolon. These rules make sense
in any language with at least these resources. We ha