How Do You Formulate Inequalities from the Constraints?

Solved Problems
📌 Problem 1. Smitty’s, a 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, Smitty 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 hour of sewing time and 2 hours of cutting time.
To express the LP constraints for this problem mathematically, we let

X1= number of shirts produced
X2= number of pajamas produced

✍ Solution:
First constraint: 1X1+.75X2≤280 hours of sewing-machine time available—our first scarce resource
Second constraint: 1.5X1+② X2≤450 hours of cutting-machine time available—our second scarce resource

Note: ② means that each pair of pajamas takes 2 hours of the cutting resource.
Smitty’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.
This information can be used to create the LP objective function for this problem:
Objective function: maximize total contribution to profit =$4X1+$3X2.

📌 Problem 2. You are given a test consisting of two sections. The first section is on algebra and the second section is on geometry. You are not allowed to answer more than 10 questions from any section, but you have to answer at least 4 algebra questions. The time allowed is not more than 30 minutes. An algebra problem will take 2 minutes and a geometry problem will take 3 minutes to solve.

Let x be the number of algebra questions and y be the number of geometry questions.
a) Formulate the equations and inequalities that satisfy the above constraints.
b) The algebra questions carry 5 marks each and the geometry questions carry 10 marks each. If T is the total marks, write down an expression for T.
✍ Solution:
a) You are not allowed to answer more than 10 questions from any section:

x≤10; y≤10.

You have to answer at least 4 algebra questions:

x≥4

The time allowed is not more than 30 minutes. An algebra problem will take 2 minutes and a geometry problem will take 3 minutes to solve:

2x+3y≤30

b)

T=(5 marks) × (algebra questions answered) +
(10 marks) × (geometry questions answered)
T=5x+10y

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