International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 3, Issue 11, November 2013)
Linear and Integer Programming with Sensitivity Analysis
Approach
Azra Siddiqui1, Shumail Ahmad Siddiqui2
1
Assistant Professor
Tata Consultancy Services
2
The constraints arise from the production times and
Congressional mandate on fuel economy. In 8 hour a day
there are 480 minutes, and so the production times for the
vehicles lead to the following constraint:
Abstract- Linear programming is the name of a branch of
applied mathematics that deals with solving optimization
problems of a particular form. Linear programming problems
consist of a linear cost function or sometimes also called as
objective function which is to be minimized or maximized
subject to a certain number of constraints. The constraints are
linear inequalities of the variables used in the cost function.
Linear programming is closely related to linear algebra.It
often uses inequalities in the problem statement rather than
equalities
This paper is concerned with the advantages and
disadvantages of linear and integer programming anh
through an example is provided a pivot based solution
technique of of linear optimaization.
u +2v + 3w ≤ 480
The average fuel economy restriction can be written:
25u+ 15v+ 10w ≥ 18(u+v+w)
Which simplifies to:
7u-3v-8w ≥ 0
There is an additional implicit constraint that the
variables are all non-negative: u,v,w ≥ 0.
I. INTRODUCTION
This production planning problem can now be written
as:
In linear programming (LP),the word linear refers to the
variables of a problem being combined in a linear way, that
is adding or subtracting variables but not multiplying them
or using them as exponents.Therefore it is obvious that
some physical problems, or other such problems for that
matter, cannot be expressed in a linear way, but those who
can,can be solved in more efficient ways.Lets understand it
by taking an example.
Consider a particular plant can build pilot vehichles at
the rate of one per minute, Ranger at the rate of one every 2
minutes, and EcoSports at the rate of one every 3
minutes.The vehicles get 25, 15, and 10 miles per US quart,
respectively, and Congress mandates that the average fuel
economy of vehicles produced be at least 18 miles per US
Quart. Plant loses $1000 on each Pilot vehicle, but makes a
profit of $5000 on each Ranger and $15,000 on each Eco
Sports.Lets find out the maximum profit this plant can
make in one 8-hour day using linear programming.
The cost function or the objective function is the profit
Ford can make by building ‘u’ Pilot vehicle,’ v’ Ranger,
and ‘w’ Eco Sports, and we want to maximize it:
Maximize
-1000u+ 5000v+ 15000w
subject to
u+2v+3w ≤ 480
7u-3v-8w ≥0
u,v,w ≥0
The solution to the above problem is u=132.41, v=0, and
w=115.86, giving a objective function value of
1,605,517.24. It is obvious that for some problems, noninteger values of the variables may not be desiredTherefore
the concept of Integer Linear programming is introduced.
Integer programming is solving a linear programming
problem for integer values of the variables only and so is a
significantly more difficult problem.
It is not necessary for every scenario that the solution of
the problem solved without the integer constraint is close
and same to the same problem solved by integer
programming . In the given example, the optimal solution if
u, v, and w are constrained to be integers is u=132, y=v,
and w=115 with a resulting objective function value
(profit) of $1,598,000.
− 1000u + 5000v + 15000w
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International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 3, Issue 11, November 2013)
4.
All decision variables are constrained to be
nonnegative.
A problem satisfying the structure is said to be in
canonical form[12]. Given formulation appears to be
limited and restrictive; however, any linear programming
problem can be transformed in canonical form.
II. TERMINOLOGY
Here we will discuss and understand
general
characteristics of linear programming problems,including
the various legitimate forms of mathematical model for
linear programming.
There are some commonly used symbols which denote
the various components of linear programming model.
These symbols are given below:
Table 1
Allocation of resources to activities in Linear Programming Model
Z = value of overall measure of performance.
Resources
Xj =level of activity j (for j =1,2,….,n)
Cj =increase in Z that would result from each
increase in level of activity j
unit
Amount
of
resource
available
Activity
Bi =amount of resource i that is available for allocation to
activities (for i= 1,2,…,m)
1
aij = amount of resource i consumed by each unit of
activity j.
In linear programming model we pose the problem in
terms of making decisions about the levels of the activities ,
so x1,x2,….,xn are called the decision variables. The
values of cj (j=1,2,…,n), bi (i=1,2,…,m) and aij are the
input constants for the model,thus are also reffered to as the
parameters of the model.
III. STANDARD FORM OF MODEL
Maximixe
Resource usage per
unit of activity
Z=c1x1+c2x2+…….+ cnxn
2
…
3
1
A11 a12
…
a1n
B1
2
A21 a22
…
a2n
B2
.
…
…
m
Am1 am2 …
Contribution
to Z per unit
of activity
C1
…
c2
…
…
amn
…
bm
cn
Subject to the restrictions
a11x1+a12x2+......+a1nxn ≤b1
Every linear programming problem falls into one of the
given categories.It is infeasible if a feasible solution
solution to a problem doesnot exist;that is ,there is no
vector x for which all the constraints or limits of the
problem are satisfied.It is unbounded if the constraints
donot sufficiently restrain the objective function so that for
any given feasible solution,another feasible solution can be
found that is better in making further improvement to the
cost function.Linear programming problems that are not
infeasible or unbounded have an optimal solution;which
means that the cost function has a unique minimum or
maximum cost function value.
a21x1+a22x2+…….+a2nxn ≤b2
am1x1+am2x2+…+amnxn ≤ bm
and
x1 ≥0, x2 ≥0,
…. ,xn ≥0.
To be concise, we can state the following
1. All constraints, except for the nonnegativity of
decision variables, are stated as equalities.
2. The righthand-side coefficients are all nonnegative.
3. One decision variable is isolated in each constraint
with a+1 coefficient (x1 in constraint (1) and x2 in
constraint (2)). The variable isolated in a given
constraint does not appear in any other constraint, and
appears with a zero coefficient in the objective
function.
Note: This doesnot mean that the values of the variables that yield
that optimal solution are unique.
Now we will discuss the basic step for generating a
canonical form with an improved value for the objective
function:
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International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 3, Issue 11, November 2013)
IV.
There has been recent developments in implementing
full scale mathe- matical programming systems which are
of great significance [3, 4,9].
Activities like in data management, modeling, run
control, solution access and analysis, and algorithmic
experimentation can be enhanced by interavtive
implementation .One more advantage is that we can
diminish the obstacle to truly interactive problem solutions
in view of the fact that automated solution methods for
most complex problems do not exist now. Thomas et al. [5]
illustrates a practical application which shows the valuable
advantages in context to linear programming.
IMPROVEMENT CRITERIA
Suppose in a maximization problem, some nonbasic
variable has a positive coefficient in the objective function
of a canonical form. If that variable has a positive
coefficient in some constraint, then a new basic feasible
solution may be obtained by pivoting. This observation
applies in general for any number of constraints, so that we
need never compute ratios for nonpositive coefficients of
the variable that is coming into the basis, and we establish
the following criterion:
V. RATIO AND PIVOTING CRITERION
When improving a given canonical form by introducing
variable into the basis, pivot in a constraint that gives the
minimum ratio of righthand-side coefficient to
corresponding that variable coefficient. Compute these
ratios only for constraints that have a positive coefficient
for the variable.
Finally, we point out that a linear program may have
multiple optimal solutions. Suppose that the problem in
canonical form is satisfied with the optimality criterion and
a nonbasic variable has a zero objective-function
coefficient[11]. Since the value of the objective function
remains unchanged for increases in that variable, we obtain
an alternative optimal solution whenever we can increase
the variable by pivoting.
VI.
VIII. FORMULATING AND OPTIMIZATION MODEL
Dofasco said that “we find it useful to formulate the
optimization problem even if we cannot solve it .False
conclusions are obtained if the model is inadequate as they
depend not on the problem but the model itself.Good
formulation depends on validity and tracibility of model.
Validity specify how much the model hold for the real
system on the basis of the inference drawn from
it.Tacibility is the extent to which the model can be
conveniently analysed
Sommer [6] focussed on achieving the eqvivalent
formulations for providing better LP bounds, so does
Williams [7]. It is recommended that the integer
programming modeler should not focus on economizing
on the number of constraints in the representation of a
model as it leads to more disadvantages rather than
benifits. If we minimize on the number of integer variables
we will achieve the desired conditionbinary variables have
always been used in the methods which helps in attaining
transformation of combinatorial problems or nonlinear
integer programs into linear integer programs with more
compactness.
For example, using only n continuous variables and 4n
constraints for linearizing quadratic forms in n binary
variables is given in [8]. But it should be kept in mind that
transforming nonlinear problem into a linear integer one is
not at all efficient.
PITFALLS OF LINEAR PROGRAMMING
Tomlin [1] and White [2] presented significant review
on Linear Programming developments in large-scale ,out
of which White [2] emphasised more on practical approach
of software consideration.
Every linear programming problem where we seek to
maximize the objective function gives rise to a related
problem, called the dual problem, where we seek to
minimize the objective function. There is an interesting
way of interaction between the two problems that if one of
the problem has an optimal solution,the other problem also
has an optimal solution and also the two objective function
values are same.The other case is when original problem is
infeasible or unbounded,so is the dual problem.
IX. SENSITIVITY ANALYSIS
Suppose through linear programming
we have
determined the increase of the the overall production
capacity. But the point is that how confident and satisfied
we are with the result.What will be the impact on the result
if the basic data are slightly changed?
VII. INTEGER PROGRAMMING
An optimal solution if exist, is always achieved by
linear programming ,this is the biggest advantage of it as
an optimization methods.
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International Journal of Emerging Technology and Advanced Engineering
Website: www.ijetae.com (ISSN 2250-2459, ISO 9001:2008 Certified Journal, Volume 3, Issue 11, November 2013)
These doubts and questions are the basis of sensitive
analysis.Formally, it is : is the optimum solution wchich
includes both the variables and the value of objective
function sensitive to one of the original problem
coefficients?
Various approaches are there for sensitive analysis.
Brute force: If the model is small to solve quickly then we
change the initial data and solve the model again and see
the results.this process can be done several times.Classical
sensitivity analysis: if the model is large and takes much
time to solve then this method is applied which shows the
relationship between initial table and optimal table to
quickly update the optimum solution when changes are
made to the coefficient of original table.
Computer based ranging: This is in between the two
extreme approaches.It deals with how much certain
coefficients can change before the current optimal solution
is fundamentally changed. The weekness of simple
computer based sensitivity analysis is that it doesnot deal
with the changes to constraint coefficients[10].
Following information is returned from the LP solver
when use computer based ranging sensitive analysis
approach.
1. Reduced Costs: the objective function coefficients for
the original variables at the optimum.
2. Dual Prices: the objective function coefficients for the
slack and surplus variables at the optimum.
3. The ranges of the original objective function coefficients
of the original variables for which the current basis remains
optimal.
4. The ranges of the right hand side constants for the
constraint for which the current basis remains optimal.
The advantage with this approach is that we can still use
the solution that we already have to obtain what the new
point is and will be after the coefficients have changed.
The integer linear programming can also be restricted
by not being able to properly model non linear
behaviour.The integer linear programming approach gives
a quality measure during the solution process and also
generally gives guarantee to coverage to an optimal
solution of the problem defined. Various problems like
karyotyping where 46 chromosomes are assigned to 24
classes ,aircraft and crew scheduling have been great
improved by the use of linear programming
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X. CONCLUSION AND FUTURE WORKS
The integer linear programming models do provide a
successful framework to solve both the problem of
maximizing production and the problem of minimizing
costs. It is an important branch of applied mathematics and
solves optimization problems.
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