Linear-Programming
Applications
Linear-Programming
Applications
Constrained Optimization problems occur
frequently in economics:
» maximizing output from a given budget;
» or minimizing cost of a set of required
outputs.
Lagrangian multiplier problems required
binding constraints.
A number of business problems have
inequality constraints.
Profit Maximization Problem
Using Linear Programming
Constraints of production capacity, time,
money, raw materials, budget, space,
and other restrictions on choices.
These constraints can be viewed as
inequality constraints
 A "linear" programming problem
assumes a linear objective function, and
a series of linear inequality constraints

Linearity implies:
1.
constant prices for outputs (as in a
perfectly competitive market).
2.
constant returns to scale for
production processes.
3.
Typically, each decision variable
also has a non-negativity constraint.
For example, the time spent using a machine
cannot be negative.
Solution Methods

Linear programming problems can be
solved using graphical techniques,
SIMPLEX algorithms using matrices, or
using software, such as ForeProfit software.

In the graphical technique, each inequality
constraint is graphed as an equality constraint.
The Feasible Solution Space is the area which
satisfies all of the inequality constraints.
The Optimal Feasible Solution occurs along the
boundary of the Feasible Solution Space, at the
extreme points or corner points.


The corner point that maximize the objective
function is the Optimal Feasible Solution.

There may be several optimal solutions.
Examination of the slope of the objective function
and the slopes of the constraints is useful in
determining which is the optimal corner point.
One or more of the constraints may be slack, which
means it is not binding.
Each constraint has an implicit price, the shadow
price of the constraint. If a constraint is slack, its
shadow price is zero.
Each shadow price has much the same meaning as
a Lagrangian multiplier.



GRAPHICAL
Corner Points
A, B, and C
X1
CONSTRAINT # 1
A
Feasible
Region OABC
O
B
CONSTRAINT
#2
C
X2
GRAPHICAL
X1
Highest
Profit
Line
A
CONSTRAINT # 1 Optimal Feasible
Solution at
Point B
B
CONSTRAINT
#2
O
C
X2
The Dual Problem


Each linear programming problem (the
primal problem) has an associated dual
problem.
EXAMPLE: A maximization of profit objective
function, subject to resource constraints has
an associated dual problem
» The dual is a minimization of the total
costs of the resources subject to
constraints that the value of the resources
used in producing one unit of each output
be at least as great as the profit received
from the sale of that output.
Duality Theorem
THEOREM: the maximum value of the
primal (profit max problem) equals the
minimum value of the dual (cost
minimization) problem.
 The resource constraints of the primal
problem appear in the objective function
of the dual problem

Primal:
Maximize p = P1·Q1 + P2·Q2
c·Q1 + d·Q2 < R1
e·Q1 + f·Q2 < R2
where Q1 and Q2 > 0
subject to:
The budget
constraint,
for example.
The machine
scheduling
time constraint.
Nonnegativity
constraint.
Dual:
Minimize C = R1·w1 + R2·w2
c·W1 + e·W2 > P1
d·W1 + f·W2 > P2
Contribution
Product 2
where W1 and W2 > 0
subject to:
Profit Contribution
of Product 1
Profit
of
Nonnegativity
constraint.
Complexity and the
Method of Solution

The solutions to primal and dual
problems may be solved graphically, so
long as this involves two dimensions.
 With
many products, the solution
involves the SIMPLEX algorithm, or
software available in FOREPROFIT
Cost Minimization Problem Using
Linear Programming



Multi-plant firms want to produce with the lowest
cost across their disparate facilities. Sometimes,
the relative efficiencies of the different plants can
be exploited to reduce costs.
A firm may have two mines that produces different
qualities of ore. The firm has output requirements
in each ore quality.
Scheduling of hours per week in each mine has
the objective of minimizing cost, but achieving the
required outputs.
 If
one mine is more efficient in all
categories of ore, and is less costly
to operate, the optimal solution may
involve shutting one mine down.
 The dual of this problem involves
the shadow prices of the ore
constraints. It tells the implicit
value of each quality of ore.
Capital Rationing Problem



Financial decisions sometimes may be
viewed as a linear programming problem.
EXAMPLE: A financial officer may want to
maximize the return on investments available,
given a limited amount of money to invest.
The usual problem in finance is to accept all
projects with positive net present values, but
sometimes the capital budgets are fixed or
limited to create "capital rationing"
among projects.
 The
solution involves determining
what fraction of money allotted
should be invested in each of the
possible projects or investments.
 In some problems, projects cannot
be broken into small parts.
 When this is the case, integer
programming can be added to the
problem.