# Linear Programming and Maximization of Contribution Margin – Graphical Method:

Learning Objective of the Article:

1. Define and explain linear programming graphical method.
2. 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.