LinearProgramming Applications

1
Linear-Programming Applications
2
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.

3
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

4
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.

5
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.

6

• 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.

7
GRAPHICAL
Corner Points A, B, and C
X1
CONSTRAINT 1
A
B
Feasible Region OABC
CONSTRAINT 2
C
O
X2
8
GRAPHICAL
X1
CONSTRAINT 1
Optimal Feasible Solution at Point B
Highest Profit Line
A
B
CONSTRAINT 2
C
O
X2
9
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.

10
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

11
Primal

• Maximize p P1Q1 P2Q2 subject to
• cQ1 dQ2 lt R1 The budget
  constraint, for example.
• eQ1 fQ2 lt R2 The machine
  scheduling time constraint.
• where Q1 and Q2 gt 0 Nonnegativity
  constraint.

12
Dual

• Minimize C R1w1 R2w2 subject to
• cW1 eW2 gt P1 Profit Contribution of
  Product 1
• dW1 fW2 gt P2 Profit Contribution
  of Product 2
• where W1 and W2 gt 0 Nonnegativity
  constraint.

13
Complexity and theMethod 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

14
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.

15

• 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.

16
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.

17

• 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.