ch 11 resource constraints and linear programming

Ch 11 Resource Constraints
and Linear Programming

The process of finding an optimum
outcome from a set of constrained
resources, where the objective function
and the constraints can be expressed as
linear equations.

Drawing the Linear Model
Standard Graph
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Number of X units
Number of Y units

Adding the Linear Constraints
Standard Graph: Constraints Added
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Number of X units
Number of Y units
Constraint 1
Constraint 2
Constraint 3
Feasible
Region

Adding the Iso-Contribution Line

The iso-contribution line is a ‘slope’ which represents the objective
function. It is drawn as a generic line, then ‘floated’ to an optimum
location within the feasible region.
Partial Graph: Notional Iso-Contribution
Line, and Constraints.
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Number of X units
Number of Y units
Constraint 1
Constraint 2
Constraint 3
Iso
Contribution
Line

Finding the Optimum Point

Float the iso-contribution line to an optimum position.
Finished Graph:
Optimum Iso-Contribution Line Floated
Into Postion Against the Binding
Constraints.
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Number of X units
N u m b er o f Y
u n its
Constraint 1
Constraint 2
Constraint 3
Iso Contribution
Line
Optimum Iso
Contribution Line
Optimum point.

Algebraic Solution to an Example
LP Problem


Define the objective function:

Z = 0.75 X + 1.82 Y

Set up the resource constraints :
32X + 59 Y <= 4312
200X + 15Y <= 1819

Set up any other limit constraints; e.g;
X >= 0
Y >= 0
X <= 19

Solving the Algebraic Problem 1

In this simple case, the set of algebraic
equations can be easily solved by substitution.

As the Simplex method is tedious, and prone
to error, the solution is best found with
computer software such as Excel Solver.

The standard Excel spreadsheet needs to be
specially adapted to run Solver.

In a more complex case, the Simplex method
can be manually applied.

Solving the Algebraic Problem 2


Additions to the standard spreadsheet are:

An ‘Activity Level’ row for output levels.

A ‘Resource Supply’ column for level of supply of
constrained resources.

A ‘Resource Use’ column for amount of each
constrained resource used, and final objective
function value.

A ‘Sign’ column for the inequality signs:- ( for
information only; not for “Solver” solution.)

Solving the Algebraic Problem 3: The
Adjusted Spreadsheet

Spreadsheet ready for solution.

Solving the Algebraic Problem 4:
Using Excel Solver
Inputs to the Solver dialog box.

Solving the Algebraic Problem 5:
Reading the Solver Results

Read the results from the Solved spreadsheet.

Solving the Algebraic Problem 6:
Reading the Solver Reports (a).


The answer report shows the solution.

Solving the Algebraic Problem 6:
Reading the Solver Reports (b).

The sensitivity report shows possible adjustments to the
solution.

Solving the Algebraic Problem 6:
Reading the Solver Reports (c).

The limits report shows the amount of movement
allowed in the cell values within the constraint levels.

Linear Programming: Summary

Use when an optimum solution is required,
from constrained resources.

Express the objective function and the
constraints as linear equations.

Solve using either the graphical method, or a
computerized model.

Interpret the results.

Consider the sensitivity of the results.


Make a decision.
THE END