In the previous section, we provided the approach to formulate the **goal programming model**. Here we provide an example of **model formulation**.

The Japan Life Company produces two products- A and B. According to the past experience, production of either product A or product B requires an average of one hour in the plant. The plant has a normal production capacity of 300 hours a month. The marketing department of the firm reports that because of limited market, the maximum number of product A and product B that can be sold in a month are 140 and 200 respectively. The net profit from the sale of product A and product B are Rs. 600 and Rs. 200 respectively. The manager has set the following goals.

P_{1}: The first goal is to avoid any underutilization of normal
production capacity.

P_{2}: He wants to sell maximum possible units of product A
and B. Since the net profit from the sale of product A is thrice the
amount from Product B, therefore, the manager has thrice as much desire
to achieve sales for product A as for Product B.

P_{3}: He wants to minimize the overtime operation of the plant
as much as possible.

**Solution.**

Production capacity

Let x_{1 } = number of units of product A

x_{2} = number of units of product B

Since overtime operation of the plant is allowed to a certain extent,
the constraint can be written as

x_{1} + x_{2} + d_{1}^{−} - d_{1}^{+} = 300

Where

d_{1}^{−} = underutilization
(idle) of production capacity and

d_{1}+ = overtime operation
of the normal production capacity.

Sales constraints

In this case, the maximum sales for product A and product B are set at 140 and 200 respectively. Hence, it is assumed that overachievement of sales beyond the maximum limits is impossible. Then, the sales (market) constraints can be expressed as

x_{1} + d_{2}^{−} = 140

x_{2} + d_{3}^{−} = 200

Where d_{2}^{−} and
d_{3}^{−} are the underachievement
of the sales goal for product A and B respectively.

Therefore, the complete goal programming model can be written as

Minimize z = P_{1}d_{1}^{−} + 3P_{2}d_{2}^{−} + P_{2}d_{3}^{−} + P_{3}d_{1}^{+}

subject to

x_{1} + x_{2} + d_{1}^{−} - d_{1}^{+} = 300

x_{1} + d_{2}^{−} = 140

x_{2} + d_{3}^{−} = 200

and x_{1}, x_{2}, d_{1}^{−},
d_{2}^{−}, d_{3}^{−},
d_{1}^{+} ≥ 0