# Exercise of maximize total contribution to profit of Smith’s, a Niagara, New YorkExample of Linear Programming - Graphical Method

### Problem Data

We want to Maximize the following problem:

- Objective Function
- Z = 4X
_{1}+ 3X_{2} - Subject to the following constraints
X

_{1}+ 3/4X_{2}≤ 2803/2X

_{1}+ 2X_{2}≤ 450X

_{1}, X_{2}≥ 0- Description
**Exercise of maximize total contribution to profit of Smith’s, a Niagara, New York**Smith’s, a Niagara, New York, clothing manufacturer that produces men’s shirts and pajamas, has two primary resources available: sewing-machine time (in the sewing department) and cutting-machine time (in the cutting department). Over the next month, owner Barbara Smith can schedule up to 280 hours of work on sewing machines and up to 450 hours of work on cutting machines. Each shirt produced requires 1.00 hour of sewing time and 1.50 hours of cutting time. Producing each pair of pajamas requires .75 hours of sewing time and 2 hours of cutting time. Smith’s accounting department analyzes cost and sales figures and states that each shirt produced will yield a $4 contribution to profit and that each pair of pajamas will yield a $3 contribution to profit. Solve the problem with graphical method.

## Solution

To solve the problem we will calculate the **feasible region** which is formed by the area satisfying the set of constraints.

Below we present the detailed calculations and graphs to solve the problem:

### Step 1:

**Non-negativity:** X_{1}, X_{2} ≥ 0

The decision variables of the problem must comply with the **non-negativity**constraint; that is, their values can be from 0 to plus.

In our graph, it means that the feasible region will be in the first quadrant:

**Note:** You can zoom the chart using the scroll wheel, as well as move the view by dragging it with the mouse.

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