AFM 31130
STDM Linear Programming
By
Isuru Manawadu
B.Sc in Accounting Sp. (USJP), ACA,
1
Learning Outcomes
After studying this session you will be able to:
Why linear programming
Assumptions of linear programming
Graphical method of linear programming
Cost Minimization Problem Using Linear Programming
Shadow price calculation
Why Linear Programming?
Programming formulations gives solutions for
the problems.
The output generated by linear programs
provides useful “what-if” information.
Improves quality of decision.
Utilized to analyze numerous economic,
social, military and industrial problem.
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.
 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
Linear Programming Assumptions:
1. Constant prices for outputs (as in a perfectly
competitive market).
2. There are no interactions between the
décision variables.
3. The parameters are know with certainly.
4. Constant returns to scale for production
processes.
Linear Programming Assumptions:
6. Typically, each decision variable
has a non-negativity constraint.
also
For example, the time spent using a machine
cannot be negative. The décision variable are
continuons.
.
Graphical Analysis – The Feasible
Region
The non-negativity constraints
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.
Solution Methods
Cont….
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.
Solution Methods
 There
Cont….
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.
GRAPHICAL
Corner Points
A, B, and C
X1
CONSTRAINT # 1
A
Feasible
Region OABC
O
B
CONSTRAINT
#2
C
X2
Extreme points and optimal solutions
If a linear programming problem has an
optimal solution, an extreme point is
optimal.
13
Multiple optimal solutions
• For multiple optimal solutions to exist, the
objective function must be parallel to one of
the constraints
•Any weighted average
of optimal solutions is
also an optimal solution.
14
Shadow Prices
l
l
Assuming there are no other changes to the
input parameters, the change to the objective
function value per unit increase to a right hand
side of a constraint is called the “Shadow
Price”
One 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.
Shadow Price – graphical demonstration
The Plastic
constraint
X2
1000
When more plastic becomes available
(the plastic constraint is relaxed), the
right hand side of the plastic constraint
increases.
Maximum profit = 4360
Maximum profit = 4363.4
500
Shadow price =
4363.40 – 4360.00 = 3.40
Production time
constraint
X1
500
16
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.
Cost Minimization Problem Using
Linear Programming
Scheduling of hours per week in each mine
has the objective of minimizing cost, but
achieving the required outputs
Thank you