(User.programs)
Item: 437 by akcs.rmarimon@hpcvbbs.cv.hp.com [Ricardo Jose Marimon]
Subj: Solving Linear Programming Problems with SIMPLEX3
Date: 14 Nov 1992
This solves linear programs using a "pseudo revised" simplex method.
It works very fast. Take a look at it!!!
[Note: "Linear Programming Problems" is mathematical jargon for what is
simply a system of linear inequalities with the goal of maximizing a
particular variable.
Concrete example: Suppose that the Sierra Cement Company has several
cement trucks, drivers, delivery routes, and types of cement. It is
known that the sum of the costs of the delivery routes plus the drivers'
pay must fall below some particular maximum. And the other variables
are
similarly tied to given maximum values. These inequalities are called
"constraints". If Sierra Cement wants to maximize their profit, they
could spend all day doing "what-if" tinkering on a spreadsheet on a
computer, or they could solve it in a few seconds using the Simplex
method, which kicks out the absolute maximum possible profit, and the
exact values of all the variables required to produce that maximum.
Nice,
eh? Every business should use it, but few have even heard about it.
Complete details can be found in any textbook about "Finite
Mathematics". -jkh-]
SIMPLEX3 can solve a problem with 8 variables and 5 constraints in less
than 15 seconds performing 5 iterations on the tableau.
The problem this program is intended to solve is of the form,
min
st
c.x
A.x = b
x >= 0
(* b must be greater than zero *)
The way the problem is entered is as a matrix of the following form,
[ c | 0 | 0 ]
[---+---+---]
[ A | I | b ]
Where I is the matrix formed by adding the slack and excess variables.
I is NOT necessarily the identity matrix.
For example to solve the following problem
max x1 + 2 x2
st
x1 + x2
x1 - x2 +
x2 +
x1 + x2 +
+ x3
<= 3
x3 <= 3
x3 <= 3
x3 >= 3
x2
<= 2
x1, x2, x3 >=0
We convert it to the standard form
min - x1 - 2 x2 - x3
st
x1 + x2
+ x4
x1 - x2 + x3
+ x5
x2 + x3
+ x6
x1 + x2 + x3
- x7
x2
+ x8
x1, x2, x3 >= 0
=
=
=
=
=
3
3
3
3
2
And write it as a matrix to enter it in the program.
[[ -1 -2 -1 0 0 0 0 0
[ 1 1 0 1 0 0 0 0
[ 1 -1 1 0 1 0 0 0
[ 0 1 1 0 0 1 0 0
[ 1 1 1 0 0 0 -1 0
[ 0 1 0 0 0 0 0 1
0
3
3
3
3
2
]
]
]
]
]
]]
This example matrix is included in the SIMPLEX3 directory under the
name "P".
To solve this problem right away, enter the matrix [just press {alpha}
P {enter} -jkh-], press the [INIT] key which initializes some of the
variables used by the program and then press the [SOLVE] key which
iterates and gives the solution in three parts as follows,
3:
2:
1:
:X: [ 2 .999999999998 2 0 0 0 2 1 ]
:\Gl: [ -1 0 -1 0 0 ]
:Z: -6
Where x is the actual point that minimizes the problem, the \Gl (lambda)
is the solution to the dual problem or shadow prices of each of the
constraints, and Z is the objective function value at the point found to
be the minimum.
This is the end of the quick start procedure to solve linear programming
problems with my program. Now lets go to some details...
The solutions of LP (linear programming problems) can be classified into:
unfeasible, when there is no point satisfying the set of constraints
unbounded, when the set of constraints can not restrict the decrease
of the objective function (c.x)
feasible & bounded, when there is a solution to the set of
constraints
which minimizes the objective function value.
The problem can handle all three cases, giving the solution when it is
feasible and bounded, providing a ray (which is a vector through which
you can infinitely decrease the objective function while remaining in
the feasible region) when the solution is unbounded, or flagging that
there is no feasible solution.
To achieve the different solutions, the program goes through the two
phase simplex method automatically (you don't have to include artificial
variables). Phase 1 is used to derive a starting basic feasible solution
by introducing a set of artificial variables whith the objective of
minimizing its sum; when this is achieved, the set of artificial
variables is removed and Phase 2 starts with the original objective
function. (refer to any LP book for more details into the 2 phase
simplex method) One important thing to notice, is that the columns
corresponding to the artificial variables, although not taken into
account on phase 2, are not removed from the problem because it is
sometimes usefull to have these columns at the end of the problem in
order to perform sensitivity analysis.
The basic algorithm that I used to solve the problem is a pseudo
revised simplex method, where all that is maintained is the inverse of
the base, the set of basic and non basic variables, and the rest of
the data provided at the initialization process. I think this is
the most effective procedure in terms of time because it requires less
computations...
Here's a brief explanation of what the programs in the directory do.
This first set of programs are the ones intended to be used directly
to solve the problem:
[INIT]
Initializes the set of variables that are needed by the
program.
[TABL] Provides the current tableau.
[SOLU] Extracts the solution from the current tableau.
[SRAY] Extracts a ray from an unbounded LP.
[TRACE] Performs one iteration at a time; it is equivalent to solve,
but you can take a look at the partial tableaus that are
generated by the procedure. [Great for homework or exams in
Finite Mathematics courses, where you must show "your" work!
-jkh-]
[SOLVE] Gives the answer to the LP if it exists.
This second set of programs should not be used directly unless you
REALLY know what you're doing. There are some system-RPL programs
that can crash your system...
[ITER]
Given which variable enters and which variable exits,
this program performs the necessary iterations. Be careful
because the numbers that this programs expects are the
indices into the BVAR and NVAR variables corresponding to
the basic and non-basic variables.
[INVAR] Calculates which variable enters the base. It returns 0
if it is an optimum.
[OUTVAR] Given which variable enters the base, it gives which
variable must exit the base. It returns 0 if the solution
is unbounded.
[PH1]
Performs all the calculations necessary to start phase 1.
[PH2]
Same but for phase 2.
[MCHOP] Given a matrix and the contents of the 'ZERO' variable
if an element of the matrix is less than 'ZERO' in absolute
value it changes it to 0.
[MVIN] Calculates the min element such that it is strictly less
than zero. It returns the index into the array corresponding
to the element. It returns 0 if no negative elements are
found.
[MVOUT] It performs the ratio test among two vectors.
2: v2
1: v1
Returns i s.t. min over all i [ v1[i]/v2[i] for v2[i]>0 ]
Returns 0 if all v2[i] are less or equal to 0.
[MJOC] Given two matrices it joins them in the standard form
2: [ A ]
1: [ B ]
It returns [ A | B ]
[MSBC] It takes the columns of a column as specified by a list
with the numbers of the columns.
2: [ A ]
1: { 3 2 2 1} It returns [ A3 A2 A2 A1 ] where Ai represents
the ith column of A.
[MELM] This program is used to calculate the matrix that when
multiplied by the inverse of the base, gives the new
inverse.
2: [1 2 4]
1: 1
It returns [[ 1 0 0]
[-2 1 0]
[-4 0 1]
Multyplying by this matrix is equivalent to performing
row reduction in row 1 (as specified by element in level 1)
given that the column involved is specified by the element
in level 2.
[MBASE] It quickly calculates wether there is a base in the system
and if there is, which one it is. It returns a list such that
its elements correspond to the elements of the original matrix
that corresponds to the the columns of the identity.
[MSPL] It just separates the original problem form,
[ c | 0 | 0 ]
[---+---+---]
[ A | I | b ]
Into the matrices [ A | I ], [ b ], and [ c ].
[CLR14] Clears the top 14 rows of the display.
Here are the variables used by the above procedures. Do not play with
these variables during a run of the problem because you might get
incorrect answers.
[ZERO]
[BINV]
Threshold after which numbers are considered to be zero.
This in important because due to numeric problems, results
tend to show some rounding errors.
This is the inverse of the base.
[ACST]
[NVAR]
[BVAR]
[COST]
[DATA]
[RSRC]
[ARTI]
[COLS]
[ROWS]
Actual cost function being used, it changes from phase 1
to phase 2.
Non basic variables.
Basic variables, in the order in which they are in the
inverse of the base.
The original objective function coefficients of the problem.
The matrix [ c ]
The matrix [ A | I ]
The matrix [ b ]
Number of artificial variables included.
Number of variables in the problem, excluding artificials.
Number of constraints in the problem.
The custom menu provides a quick access to all of these variables.
I hope that most of the question will be answered by the above brief
description, but you can always contact me, and if I have time I will
get back yo you fairly soon.
Ricardo Marimon
28-D Escondido Village
Stanford CA 94305
Email: rmarimon@leland.stanford.edu
Most of the bugs have already been worked out, but as Murphy says, there
is always one more bug... Please send any bug reports to my email
address...
Of course, I am not responsible for any damage that this program might
cause to your calculator, your job, yout studies, or your life... :-)
But again, I haven't had any troubles for months.