Math 414 Lecture 6
RECALL. For the canonical case, X is a solution iff AX =
b; it is feasible iff X >0. A basic solution is feasible if all
the basic variables are > 0 (the parameters, being 0, are
automatically > 0).
Different sequences of elementary row
Math 414 Lecture 18
CUTTING PLANE METHOD. Convert the problem to standard
form with integer coefficients. Solve. If a primal variable
is not integral, cut it off with a new integer-coefficient
constraint. This makes it nonfeasible. Use the dual
method to
Math 414 Lecture 19
All linear inequalities can be rewritten in standard form:
ax + by < c.
`y-3 < 2x
is equivalent to 2x y 3.
0 < x+ y < 5 can be written as 2 linear inequalities.
In the problems below, you may use linear inequalities
in any form.
Variab
Math 414 Lecture 20
Exam 3 covers Lectures 15 - 21
Transportation tableaus
v Bottom row: the values w1, w2, . of the dual demand
variables.
v Rightmost column: the values p1, p2, . of the
dual supply variables.
v The square for xij / oij has cij in the up
Math 414 Lecture 16
Constant column sensitivity
Identify a solution with the constraint boundaries on
which it lies (the constraints with 0 slacks). Thus a solution
will move when its constraint lines move.
Let br = the right-hand constant of constraint r
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 optimal (or super-optimal)
(objective row coefficients are >
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/lb
Math 414 Lecture 13
Recall: The objective row and the constant column.
Primal problem Canonical+
Dual
max z = C:X
max z = C:X
min z = B:W
AX ? B
AX+ .= B
ATW > C
X>0
X, . , > 0
W?0
Where ? is some combination of <, >, =.
The . of the canonical+ is some co
Math 414 Lecture 14
Exam 2 covers Lectures 8-14
Phase two case in which an extra variable is basic
Recall: you may delete phase one columns which
belong to extra variables which are parameters. But if an
extra variable is basic, ignore the pick most negat
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 11.
y
DOUBLE DUAL THEOREM. The dual of a dual is the original
primal problem.
PROOF FOR STANDARD CASE.
PRIMAL
DUAL
DUAL OF DUAL
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 tw
Math 414 Lecture 10
The original problem is called the primal problem.
For any constraint s, the value of its dual variable s =
the rate of change of the optimal value with respect to the
constant of the constraint. Put simply
MARGINAL VALUE THEOREM.
If s
Math 414 Lecture 7
Exam 1, Tues. Sept 21 week from today, Lectures 1-7.
For today, assume all basic variables are positive. Later
we consider the degenerate case with a 0 basic variable.
CONTINUING CANONICAL EXAMPLE.
Max z = 2x + 3y.
with v: x - 2y + v
=
Math 414 Lecture 8
`Solve by geometry and then by the simplex algorithm.
Max z = x + y
with r: 2x + y < 6
s: x +2y < 6, x, y > 0.
6
(0,3)
s
x
(0,0)
#1
x
(2,2)
r
y
y
6
(3,0)
r
s
b
q
RECALL FROM LECTURE 7.
In row r, the q-ratio qr = constant/(positive-coeff
Math 414 Lecture 9
Finding an initial feasible basic solution
If the initial matrix has a set of m independent identity
columns and the constant column b is > 0, start with
these. Otherwise make each constant column entry > 0 by
multiplying negative rows
Math 414 Lecture 5
Go to www.math.hawaii.edu/414, click LPSolve. Install
LPSolve. Run it on the simple example given.
Word problems
Translate into general linear programming problems.
Solve using Linsolve.
`A potter makes lamps (the base part) and pots (f
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 2
inv(A) is the inverse of A. The most efficient way to find
inv(A) by hand involves pivoting on the augmented
matrix [A | I ] = [ A, eye(n,n)] to get a matrix
rref([ A, eye(n,n)]) = [ I | B].
If there is an all 0 row, A is singular and h
Math 414 Lecture 3
THEOREM. For linear functions on nonempty closed
bounded convex sets,
(a) absolute maximas and minimas always exist and
(b) local maximas and minimas are absolute maximas
and minimas.
PROOF. (a) Linear functions are continuous and conti
Math 414 Lecture 21
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 big