The manufacturing plant has 3 machine types (A, B, C) and wants to maximize profits by determining the optimal product mix to manufacture from 4 product types (P1, P2, P3, P4) given machine time constraints. The problem is formulated as a linear program with the objective of maximizing total profits and constraints on machine time availability. The linear program is solved graphically by plotting the constraints to find the optimal solution at a corner point of the feasible region.

2. Manufacturing Example
A manufacturing plant has 3 types of machines
A,B,C. There are 20 A-machines, 30 B-machines, and
15 C-machines available.
It also has 4 types of products P1,P2,P3,P4.
Various products have different profits and they take different time
on each machine to be built.
Resources are limited and we want to maximize profit while
operating 60 hours per week.
Determine the most profitable product mix to manufacture.

3. Manufacturing Example
Following table shows amount of time (hours) needed for
each product on different machines and unit of product
profit.
Machine/
Product
A B C Profit (Rs.)
P1 2 0.5 1.5 3.50
P2 2 2 1 4.20
P3 0.5 1 3 6.50
P4 1.5 2 1.5 3.80

4. Manufacturing Example
Same as the previous example we formulate the
problem by creating objective function and
problem constraints.
Lets assume the number of manufactured
product P1 is X1, P2 is X2, P3 is X3, and P4 is
X4.
Then our objective function will be:
Max(3.5 X1 + 4.2 X2 + 6.5 X3 + 3.8 X4)

5. Manufacturing Example
Problem constraints:
(2X1 + 2X2 + 0.5X3 + 1.5X4) ≤ 1200 (20*60 hour/week)
(0.5X1 + 2X2 + X3 + 2X4) ≤ 1800 (30*60 hour/week)
(1.5X1 + X2 + 3X3 + 1.5X4) ≤ 900 (15*60 hour/week)
Non-negativity condition
(X1, X2, X3, X4) ≥ 0

6. Solving Linear Programming
Problems
Graphical Technique
 First graph the constraints:
the solution set of the system is that
region (or set of ordered pairs), which
satisfies ALL the constraints. This region
is called the feasible set

7. Solving Linear Programming Problems:
Graphical Technique continued
￭ Locate all the corner points of the graph:
the coordinates of the corners will be
determined algebraically
It is important to note that the optima is
obtained at the boundary of the solution set and
furthermore at the corner points.
For linear programs, it can be shown that the
optima will always be obtained at corner points.

8. Solving Linear Programming Problems:
Graphical Technique continued
￭ Determine the optimal value:
test all the corner points to see which yields
the optimum value for the objective function
Feasible
set Optimum
Objective function

9. Graphical Approach
ICFAI Bakery vt. Ltd. Is manufacturing two
types of products A and B. Each unit of product
A requires 2 kg of raw material and 4 hours of
labour for processing and each unit of B
requires 3 kg of raw material and 3 hours of
labour. Firm has an availability of 60 kg of raw
material and 96 labour hours. One unit of
product A sold for Rs. 40 as profit and B for Rs.
35 as profit. How many units of product
combination the firm should manufacture to
earn maximum profit.

10. Graphical Approach
Maximize Z = 40x1 + 35x2
Subject to 2x1 + 3x2 ≤ 60
4x1 + 3x2 ≤ 96
x1, x2 ≥ 0

11. Graphical Approach
A research institute suggested to a farmer to spread out
at lest 4800 kg of a special phosphate fertilizer and not
less than 7200 kg of a special nitrogen fertilizer to raise
the productivity of crops in this fields. There are two
sources of obtaining these mixtures A and B. Both of
these are available in bags weighing 100 kg. each and
they cost Rs. 40 and Rs. 24 respectively. Mixture A
contains phosphate and nitrogen equivalent of 20 kg
and 80 kg respectively and mixture B contains these
ingredients equivalent 50 kg. each.
How many bags of each type the farmer should buy in
order to obtain the required fertilizer at minimum cost.

12. Graphical Approach
Minimize Z = 40x1 + 24x2
Subject to 20x1 + 50x2 ≥ 4800
80x1 + 50x2 ≥ 7200
x1, x2 ≥ 0