Linear Programming and Maximization of Contribution Margin - Graphical
Method:
Learning Objective of
the Article:
- Define and explain linear programming graphical method.
- How profit maximization problem is solved using linear programming
graphical method.
The
contribution margin is one measure of whether management is making
the best use of resources. When the total
contribution margin is maximized,
management's profit objective should be satisfied.
Example of Linear Programming Graphical Method:
To illustrate the application of linear programming
to the problem of maximizing the
contribution margin, assume that a small
machine shop manufactures two models, standard and deluxe. Each standard
model requires two hours of grinding and four hours of polishing; each
deluxe module requires five hours of grinding and two hours of polishing.
The manufacturer has three grinders and two polishers. Therefore in 40 hours
week there are 120 hours of grinding capacity and 80 hours of polishing
capacity. There is a
contribution margin of $3 on each standard model and
$4
on each deluxe model.
To maximize the total contribution margin, the
management must decide on:
-
The allocation of the available production capacity to standard and
deluxe models.
-
The number of units of each model to produce.
To solve this problem, the symbol "x" is
assigned to the number of standard models and "y"
is assigned to the number of deluxe models. The contribution margin from
making x standard models and y deluxe models is then $3x + $4y. The
contribution margin per unit is the sale price per unit less the unit
variable cost that is directly traceable to the product. The total
contribution margin is the per unit contribution margin multiplied by the
number of units. The restrictions on the machine capacity are expressed
in this manner:
To manufacture one standard unit requires two hours of grinding time; so
that making x standard models uses 2x hours.
Similarly, the production of y deluxe models uses 5y hours of grinding time.
With 120 hours of grinding time available, the grinding capacity is written: 2x + 5y ≤ 120 hours of grinding capacity per week. The
limitation on the polishing capacity is expressed: 4x
+ 2y ≤ 80 hours per week.
In
summary, the relevant information is:
|
|
Grinding Time |
Polishing Time |
Contribution Margin |
Standard model
Deluxe model
Plant capacity |
2 hours
5 hours
120 hours |
4 hours
2 hours
80hours |
$3
$4 |
This information is used in illustrating a
basic linear programming technique - the graphic method.
When a linear programming problem involves only two variables, a two
dimensional graph can be used to determine the optimal solution. In this
example, the x-axis represents the number of standard models, and the y axis
represents the number of deluxe models. The maximum number of each model
that can be produced, given the constraints, is as follows:
|
Operation |
Maximum Number of
Models |
|
Standard |
Deluxe |
| Grinding |
(120 / 2) = 60 |
(120 / 5) = 24 |
| Polishing |
(80 / 4) = 20 |
(80 / 2) = 40 |
The lowest number of each of the two columns measures the impact of the
hours limitations. It appears that at best, the company can produce 20
standard models with a contribution margin of $60(20 × $3) or 24 deluxe
models at a contribution margin of $96(24 × $4). However, producing a
combination of standard and deluxe model may be a better solution.
To determine the combination of production
levels in order to maximize the contribution margin, all the
constraints are
plotted on the graph. In this example, the polishing and grinding
constraints are drawn by connecting the points that represent the extremes
of production of each model. These points are:
|
When x = 0:
|
|
| |
y
≤ 24 grinding
constraint |
| |
y
≤ 40 polishing
constraint |
|
When y = 0: |
|
| |
x
≤ 60 grinding
constraint |
|
x
≤ 20 polishing
constraint |
The constraints sketched on the graph define
the solution space, as shown below:
[Graph will
be available soon]
The solution space represents the area of feasible solutions and is
bounded by the lines AB, BC, CD, and AD on the graph. Any combination of
standard and deluxe units that falls within the solution space is a feasible
solution. However, the best feasible solution, according to mathematical
laws, is one of the four corner points. Consequently, all corner point
variables must be examined to find the combination which maximizes the
contribution margin (CM) of $3x + $4y.
The x and y corner points values can be read from the graph or computed.
The x and y values for the corner points B(24) and D(20) are extreme points
that were used in plotting the constraints. Corner point C values can be
computed as follows:
Write the constraints as equalities:
2x + 5y = 120
4x + 2y = 80
To find the value for y, multiply the first equation by two and subtract
the second equation:
|
4x + 10y = 240
|
|
–4x −2y
= 80 |
|
|
8y = 160 |
|
y = 20 |
Substitute the value of y in the first and solve for x:
2x + 5(20) = 120
2x = 20
x = 10
The value for x and y and the resulting contribution margin values at
each of the corner points are:
|
A |
(x = 0, y = 0); $3(0) + $4(0) = $0 CM |
|
B |
(x = 0, y = 24); $3(0) + $4(24) = $96 CM |
|
C |
(x = 10, y = 20); $3(10) + $4(20) = $110 CM |
|
D |
(x = 200, y = 0); $3(20) + $4(0) = $60 CM |
The total
contribution margin is maximized when 10 standard models and 20 deluxe
models are scheduled for production. This solution uses all of the
constraint resources:
|
2(10) + 5(20) = 120 hours
grinding constraint
4(10) + 2(20) = 80 hours polishing constraint
|
Full utilization of all resources will occur,
however, only in cases in which the optimal solution is at a point of common
intersection of all of the constraint equations in this problem―point
C in this example. |
