Linear programming
Management of operations
often complex poses problems in modeled may be even linear functions. The
technical mathematical of contributes linear programming to solve one wide
range of operations management problems.
The linear program structure
Linear styles of
programming of a function are formed and the objective constraints of this
function. A linear programming model takes the following form:
Objective function:
Z = a1x1 + a2x2 + a3X3
+... + anxn
constraints:
+++ b11X1 b13X3 b12X2...
+ B1nXn < c1
+++ b21X1 b23X3 b22X2...
+ B2nXn < c2
.
.
.
+++ bm1X1 bm3X3 bm2X2...
+ BmnXn < cm
In this system of Linear
Equations, Z is the value of the optimized function objective is that, these
are values decision variable optimal Xi which found must be, and ai, bij, ci
and are of of constant calculated characteristics from the problem.
Assumptions of linear
programming
Need linearity of
programming in the linear Equations as in the structure indicated below -
above. Linear in an equation, Variable decision each multiplied together is one
constant coefficient between variables multiplying without any decision and
non-linear function: as logarithms. Requires dubious assumptions linearity
following elements them:
• proportionality - A
change in a Variable - variation even proportional translated is joins the
contribution of varying this to the value of the function.
• additivity - the value
of the function is the sum of the contributions that each term.
• divisibility -
variable decision are can be values you glimpse in non-integer, fractional
values. In programming techniques can be used whole names if assumption of
divisibility does it not.
In assumptions these
more dubious linearity of the linear programming implies certainty, namely
known them are and constant coefficients.
Formulation of the
problem
With the capable of
solving problems with linear facilitated programming computers, will challenge
is in the formulation of the problem - translate the statement of the problem
in a system of Linear Equations by computer to solve. Information commandeer
write for the objective function is derived the problem statement. The problem
is the statement formulated as the problem follows:
1 Identify the purpose
of the problem, that is, quantity of optimized file must be. For example, can
seek to maximize profits.
Two. Identify
constraints are decision variables and on them. For example, production and
volumes production limits will serve as variable may decision and constraints.
3. Write the function
and terms constraints objective in decision variable, using them statement
information of the problem of the appropriate coefficient he determine for each
term for. Throw all useless information.
April. Add constraints
implied, non-negative: as the restrictions.
May. Organize will join
system of Equations under appropriate form consistent resolve computer for. For
example, all variables set on the left side of their list Equations and in the
order of their indices.
The lines to reduce help
following guidelines will risk of error in the formulation of the problem:
• Make - take you into
account initial conditions.
• that make - you in
each Variable function appears objective at least once in the constraints.
• Let them constraints
could explicitly consider specified not be. For example, if there are physical
quantities must be non - negative, these constraints must be included in the
formulation.
The effect of the
constraints
Limitations some because
it there constraints restrict United Nations of possible values of a variable
range. Considered is as a constraint in a cascade of binding change addition,
it optimal modifies the solution. Severe constraints not less impact on the
optimal binding are not.
Constraint joins
tightening cannot value that aggravate the objective function, in loosening and
cannot stress that improve enclose the value of the objective function. As
such, eleven optimal is a solution, managers can pick them up at this in solution
finding improved ways to be relax constraints.
shadow price
The constraint shadow
price of one which is the amount changes the objective function value per unit
of change in the constraint. Since often determined by resources constraints, a
comparison of shadow prices of each precious constraint provides of information
on the place the most effective use of resources to achieve the best additional
akin valeu
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