This document presents a linear programming problem involving assigning quality inspectors to minimize total inspection costs. There are two types of inspectors (Grade I and Grade II) with different inspection rates and accuracy. The objective is to minimize total costs based on wages, inspection pieces, and error costs with constraints on minimum inspection pieces and available inspectors.

1. LPP Formulation

2. 24-1-2019 Anurag Srivastava 2
A company has two grades of inspectors I and II who are to be assigned
for a quality control inspection. It is required that at least 2000 pieces be
inspected per 8 hour day. Grade I inspectors can check pieces @ 40 per
hour with an accuracy of 97%. Grade II inspectors check @ 30 pieces per
hour with an accuracy of 95%. The wage rate of Grade I inspectors is Rs.5
per hour while that of a Grade II inspector is Rs.4 per hour. An error made
by an inspector costs Rs.3 to the company. There are only 9 Grade I
inspectors and 11 Grade II inspectors available in the company. The
company wishes to assign work to the available inspectors so as to minimise
the total cost of inspection. Formulate the LP model for the problem and
solve it by the graphical method.

3. 24-1-2019 Anurag Srivastava 3
𝑴𝒊𝒏 𝒛 = 𝟖. 𝟔𝒙 𝟏 + 𝟖. 𝟓𝒙 𝟐
𝒔/𝒕
𝟒𝟎𝒙 𝟏 + 𝟑𝟎𝒙 𝟐 ≥ 𝟐𝟓𝟎
𝟎 ≤ 𝒙 𝟏 ≤ 𝟗
𝟎 ≤ 𝒙 𝟐 ≤ 𝟏𝟏

4. 24-1-2019 Anurag Srivastava 4
A company, makes two types of leather belts. Belt A is a high quality belt
and belt B is of lower quality. The respective profits are Rs.40 and Rs.30.
Each belt of type A requires twice as much time as a belt of type B and if
all the belts were of type B, the company could make 1000 belts per day.
The supply of leather is sufficient for only 800 belts per day, both A and B
combined. Belt A requires a fancy buckle of which only 400 per day are
available. There are 700 buckles per day available for belt B. Set up the LPP
and solve it by the graphical method.

5. 24-1-2019 Anurag Srivastava 5
𝑴𝒂𝒙 𝒛 = 𝟒𝟎𝒙 𝟏 + 𝟑𝟎𝒙 𝟐
𝒔/𝒕
𝟐𝒙 𝟏 + 𝒙 𝟐 ≤ 𝟏𝟎𝟎𝟎
𝒙 𝟏 + 𝒙 𝟐 ≤ 𝟖𝟎𝟎
𝟎 ≤ 𝒙 𝟏 ≤ 𝟒𝟎𝟎
𝟎 ≤ 𝒙 𝟐 ≤ 𝟕𝟎𝟎

6. 24-1-2019 Anurag Srivastava 6
A 24 hour open supermarket has the following minimal requirements for
cashiers. Period 1 follows immediately after period 6, A cashier works eight
consecutive hours, starting at the beginning of one of the six periods.
Determine a daily employee worksheet which satisfies the requirements
with the least number of personnel.
Period: I II III IV V VI
Duration: 3-7am 7-11am 11am-3pm 3-7pm 7-11pm 11pm-3am
Minimum
number
required:
7 20 14 20 10 5

7. 24-1-2019 Anurag Srivastava 7
𝑴𝒊𝒏 𝒛 = 𝒙 𝟏 + 𝒙 𝟐 + 𝒙 𝟑 + 𝒙 𝟒 + 𝒙 𝟓 + 𝒙 𝟔
𝒔/𝒕
𝒙 𝟔 + 𝒙 𝟏 ≥ 𝟕
𝒙 𝟏 + 𝒙 𝟐 ≥ 𝟐𝟎
𝒙 𝟐 + 𝒙 𝟑 ≥ 𝟏𝟒
𝒙 𝟑 + 𝒙 𝟒 ≥ 𝟐𝟎
𝒙 𝟒 + 𝒙 𝟓 ≥ 𝟏𝟎
𝒙 𝟓 + 𝒙 𝟔 ≥ 𝟓
𝒙 𝟏, 𝒙 𝟐, 𝒙 𝟑, 𝒙 𝟒, 𝒙 𝟓, 𝒙 𝟔 ≥ 𝟎