Math 414 Lecture 3
THEOREM. For linear functions on nonempty closed bounded
convex sets:
(a) Absolute maximas and minimas always exist.
(b) Local maximas and minimas are absolute maximas and
minimas.
PROOF. (a) Linear functions are continuous and continuo
CUTTING PLANE METHOD
Convert the problem to standard form with integer
coefficients. Solve.
If a primal variable is not integral, cut it off with add a new
integer-coefficient constraint.
Cutting it off, makes it nonfeasible. Use the dual method to
restor
Math 414 Lecture 15
One step of the Simplex method makes feasible
(constant-column is > 0) but nonoptimal solutions (objective row
has a negative coefficient) more optimal.
The dual method makes super-optimal (objective row
coefficients are > 0) but nonfe
Math 414 Lecture 14
Exam 2 covers Lectures 8-14
Find an optimal solution for the primal and dual.
PRIMAL
min z = x y2u
r: x u = 0
s: -x y > -3
t: x + y > 1
x, y, u > 0
Write the canonical version with extra variables.
Solve with SciLab, retaining the extr
Math 414 Lecture 13
FROM LAST LECTURE.
DUAL
PRIMAL PROBLEM CANONICAL+EXTRAS
max z = C:X
max z = C:X
min z = B:W
ATW > C
AX ? B
AX + . = B
X >0
X, ., > 0
W?0
Where ? is some combination of <, >, =.
The . are the slack and extra variables.
Initially arrange
Math 414 Lecture 21.31
Exam 3 covers Lectures 15 - 21.
In a transportation problem, suppose the total supply s
produced exceeds the total demand d, d < s.
Solution: cut back on the supply produced.
Should each plant cut back equally? No, the ones with the
Math 414 Lecture 17
Integer Programming
In many problems the variables must range over integers:
the number of people you must hire, the number of chairs
you must rent, a price in cents, . .
DEFINITION. A linear programming problem is -a pure integer prog
Math 414 Lecture 26
LEMMA. The amount I can expect to win by consistently
playing the strategy for a given row is the minimum payoff
for that row.
THEOREM. The strategy for I with the maximum expected
win is the strategy whose row has the maximum minimum.
Math 414 Lecture 24
LEMMA.
v The distance from the origin to another node = the
minimum of the distances to the nodes edges.
v If the estimated distance of a cut edge is < the estimated
distances of other cut edges, then the estimated distance is
the true
Math 414 Lecture 22
Not included in Exam 3.
ASSIGNMENT ALGORITHM
Input: an nxn cost matrix [cij ].
Output: a minimum cost assignment [xij ].
Row step. From each row, subtract the row minimum.
teach row now has a 0.
Column step. From each column, subtract
Math 414 Lecture 23
Maximum Flows
`The amount of water flowing into node x increases by 5.
How much can the flow into adjacent nodes increase? The
upper number = the capacity; the lower = the flow.
4
3
10
8
5
+5
10
7
x
10
2
We are asking for potential ind
Math 414 Lecture 16
Constant column sensitivity
Let B(brp) be the original constant column B with the
constant br replaced by a new variable p.
Let B' be the constant column (and hence solution) of the
final tableau given by the Simplex Method and let T b
Math 414 Lecture 19
All linear inequalities can be rewritten in standard form:
ax + by < c.
`y-3 < 2x
0 < x+y < 5
is equivalent to 2x y 3.
can be written as 2 linear inequalities.
In the problems below, you may use linear inequalities in
any form.
VARIABL
Math 414 Lecture 4
THEOREM. Every general linear programming problem can be
rewritten in standard and in canonical form.
PROOF.
`Rewriting min in terms of max.
x minimizes f (x) iff x maximizes -f (x)
-f(x)
x
f(x)
`Rewriting > with <.
ax > b iff
-ax < -b
Math 414 Lecture 1
Homework assignments are always due at the beginning of the next lecture.
Operations Research studies linear programming and associated
algorithms. Businesses use it to find resource allocations and production
schedules which yield maxi
Math 414 Lecture 6
CONTINUING ASSUMPTION.
AX = b, X > 0 is a canonical problem with
n variables (columns),
m independent equations (rows) and
k= n m = the number of parameters
LEMMA.
Every basic solution has < m positive entries, > k zeros.
PROOF.
All the
Math 414 Lecture 5
Word problems Write as a general linear programming problem.
`Golf carts are made in Detroit and Newark and shipped to
dealers in Miami, Houston and LA.
Production:
v Detroit can make 100/month,
v Newark can make 95/month.
Needs:
v Miam
Math 414 Lecture 2 Everyone have a laptop?
THEOREM. Let v1, .,vk be k vectors in an n-dimensional space
and A = [v1; . ; vk]
v1, ., v k independent
k<n
v1, ., v k span the space
n<k
v1, ., v k a basis
k=n
v1, ., v k independent
rank(A) = k
v1, ., v k spa
Math 414 Lecture 8
`Solve by geometry.
Max z = x + y
with r: 2x + y < 6
s: x +2y < 6,
6
(0,3)
s
x
(0,0)
x, y > 0.
(2,2)
r
y (3,0)
6
Run the simplex algorithm by looking at the picture.
At each stage what are the parameters?
In the parameter columns,
which
Math 414 Lecture 7
Exam 1 a week from today. Practice Exam 1 also due then.
For today, assume all BASIC variables are positive. Later we
consider the degenerate case where a basic variable is 0.
CONTINUING CANONICAL EXAMPLE.
Max
z = 2x + 3y
with x - 2y +
Math 414 Lecture 10
RECALL:
Upward constraints dual variables > 0,
Downward constraints dual variables < 0,
equalities unrestricted dual variables.
The original problem is called the primal problem.
For any constraint s,
the value of its dual variable s
=
Math 414 Lecture 9
TWO PHASE METHOD
If our matrix has a set of m independent identity columns
and the constant column b is > 0, then the matrix is in tableau
form and we can start the simplex algorithm with the basic
solution associated with the m indepen
Math 414 Lecture 11
DOUBLE DUAL THEOREM. The dual of a dual is the original
primal problem.
PROOF FOR STANDARD CASE.
DUAL
DUAL OF DUAL
PRIMAL
max z = C:X
min z = B:W
max z = C:X
with
with
with
W: ATTX < B
W: AX < B
X: ATW > C
X>0
W>0
X>0
Transposing twice
Math 414 Lecture 12
`You have 100 lbs. of flour and 60 lbs. of sugar.
1 lb. of cookies = .6 lbs. of flour + .4 lbs. of sugar.
1 lb. of candy
=
1 lb. of sugar.
1 lb. of crackers = .8 lbs. of flour + .2 lbs. of sugar.
Cookies sell for $3/lb., candy for $1/l
Math 414 Lecture 25
Job scheduling
A project requires completing several jobs.
Some are independent, some must precede others.
Find the minimum time required to complete the project.
We model this problem as a longest-path problem.
The nodes of the graph