This document discusses an introduction to linear programming and formulating a minimization problem. It presents a case study where the objective is to minimize the daily inspection cost of quality control inspectors at a company. There are constraints on the minimum number of pieces to be inspected and limits on the number of each type of inspector. The document formulates the linear programming model to minimize the objective function of total cost, subject to the constraints. It provides the objective function, constraints, and overall linear programming model to solve the optimization problem.

1. MB106
QUANTITATIVE TECHNIQUES
MODULE I
LECTURE 1
Linear Programming: Introduction, Formulating minimization problems
PROF. KRISHNA ROY

2. LINEAR PROGRAMMING
•Linear Programming deals with the
optimization(maximization/minimization) of a
function of variables known as OBJECTIVE
FUNCTION
•Subject to a set of linear equalities and/or
inequalities known as CONSTRAINTS
09-11-2021 Prof. Krishna Roy, Dr. B. C. Roy Engineering College 2

3. LINEAR PROGRAMMING can be used when…
•A well defined objective function which must either
be maximized or minimized exists.
•Objective can be expressed as a linear function of
decision variables.
•Constraints on the extent of achievement of
objectives exists.
•Constraints should be linear equalities or
inequalities
•Decision variables should be non-negative.
09-11-2021 Prof. Krishna Roy, Dr. B. C. Roy Engineering College 3

4. formulation of LINEAR PROGRAMMING model –a minimization
problem
• A company has two grades of inspectors 1 and 2 to
undertake quality control inspection. At least 1500 pieces
must be inspected in an 8 hour day. Grade 1 inspector
can check 20 pieces in an hour with an accuracy of 96%.
Grade 2 inspector checks 14 pieces an hour with an
accuracy of 92%. The daily wages of grade 1 inspector are
Rs. 5/- per hour while those of grade 2 inspector are Rs.
4/- per hour. An error made by an inspector costs Rs. 3/-
to the company. If there are 10 grade 1 inspectors and 15
grade 2 inspectors, in all, find the optimal assignment of
inspectors that minimize the daily inspection cost.
09-11-2021 Prof. Krishna Roy, Dr. B. C. Roy Engineering College 4

5. formulation of LINEAR PROGRAMMING model –the solution(objective
function)
•To find: number of grade 1 and 2 inspectors
• Let x1 be the number of grade 1 inspectors
• Let x2 be the number of grade 2 inspectors
• Therefore x1 ≥ 0, x2 ≥ 0 non-negativity constraints
• The objective-Minimization of daily inspection cost
• Cost of grade 1 inspector per hourRs(5+3x0.04x20)=Rs. 7.40
• Cost of grade 2 inspector per hourRs(4+3x0.08x14)=Rs. 7.36
• Therefore the objective function is
Minimize Z=8(7.40 x1 +7.36 x2)=59.20 x1 +58.88 x2
09-11-2021 Prof. Krishna Roy, Dr. B. C. Roy Engineering College 5

6. formulation of LINEAR PROGRAMMING model –the
solution(constraints)
• x1≤10 number of grade 1 inspectors
• x2≤15 number of grade 2 inspectors
• 20x8 x1 +14x8 x2 ≥ 1500 or 160x1 +112x2 ≥ 1500
pieces inspected daily
09-11-2021 Prof. Krishna Roy, Dr. B. C. Roy Engineering College 6

7. formulation of LINEAR PROGRAMMING model –the model
Minimize Z=59.20 x1 +58.88 x2
Subject to
x1 ≥ 0, x2 ≥ 0 non-negativity constraints
x1≤10 number of grade 1 inspectors
x2≤15 number of grade 2 inspectors
160x1 +112x2 ≥ 1500 pieces inspected daily
09-11-2021 Prof. Krishna Roy, Dr. B. C. Roy Engineering College 7

8. VIDEO LINKS…….
https://www.youtube.com/watch?v=40sCRtM5xkg&
feature=youtu.be
https://www.youtube.com/watch?v=hXbi59tKHbE&f
eature=youtu.be
09-11-2021 Prof. Krishna Roy, Dr. B. C. Roy Engineering College 8
QUIZ LINK…….
https://docs.google.com/forms/d/e/1FAIpQLSdXmUF2OukJkSl2kjTZmri27d_
SR9tktrA4JHvbNAoLqy6CFA/viewform?usp=sf_link

9. • Till we meet again in the next class……….
PROF. KRISHNA ROY, FMS, BCREC 9
09-11-2021