5x+y≤100

x+y≤60

x≥0

y≥0

x+y=60

5x+y=100

Corner Points

Feasible Region

RESTART

gghh

Objective function

a bentech presentation

A furniture dealer deals in only two items–tables and
chairs. He has Rs 50,000 to invest and has storage space
of at most 60 pieces. A table costs Rs 2500 and a chair
Rs 500. He estimates that from the sale of one table, he
can make a profit of Rs 250 and that from the sale of one
chair a profit of Rs 75. He wants to know how many tables 
and chairs he should buy
from the available money so as to maximise his
 total profit, assuming that he can sell all
the items which he buys

Axes and Grid

Such type of problems which seek to maximise (or, minimise) profit (or, cost) form
a general class of problems called optimisation problems.
A special but a very important class of optimisation problems is linear programming
problem

Let x be the number of tables and  y be the number of chairs that the dealer buys.
Obviously, x and y must be non-negative, 
x≥0 and y≥0
The dealer is constrained by the maximum amount he can invest (Here it is
Rs 50,000) and by the maximum number of items he can store (Here it is 60).
Stated mathematically,

2500x + 500y ≤ 50000 (investment constraint)
or 5x + y ≤ 100 

and x + y ≤ 60  (storage constraint) 

x≥0 \space  and  \space 
y≥0

Mathematically, the given problem (LPP) now reduces to:

Maximise Z = 250x + 75y
subject to the constraints:
5x + y ≤ 100
x + y ≤ 60
x ≥ 0,  y ≥ 0

Problem

Defin LPP

Solution

Constraint 1

Constraint2

Non Negative Constarints

Mathematical Formulation

a(x, y) = (5 * x) + y ≤ 100
b(x, y) = x + y ≤ 60
c(x) = x ≥ 0
d(y) = y ≥ 0
A_1 = Intersect(c, d, (16.458502083219802, 43.708178082619185))
B_1 = Intersect(c, d, (7.524077503836332, 62.32156262300138))
C_1 = Intersect(c, d, (2.3123298325293073, 57.295948797098184))
B = Intersect(e, f)
E = Intersect(c, d, (7.151809813028687, 63.43836569542431))
F = Intersect(c, d, (18.319840537258024, 6.667542847258615))
G = Intersect(c, d, (14.783297474585401, 0.33899210352866915))
H = Intersect(c, d, (14.224895938373935, -0.03327558727897473))
I = Intersect(f, g)
J = Intersect(c, d, (-0.29354400312420476, 31.795611976774577))
K = Intersect(c, d, (3.242999059548419, 56.5514134154829))
L = Intersect(c, d, (4.173668286567531, 55.62074418846379))
M = Intersect(d, ((-(x(e) / y(e))) * x) - z(e) / y(e), (-5.877559365238874, 65.67197184027017))
N = Intersect(c, d, (-0.10741015772038244, 66.7887749126931))
O_1 = Intersect(c, d, (-0.10741015772038244, 66.97490875809693))
P = Intersect(g, e)
Q = Intersect(c, d, (1.3816606055101959, 58.41275186952112))
R = Intersect(c, d, (17.575305155642734, 11.879290518565629))
S = Intersect(f, h)
T = Intersect(c, d, (15.900100547008336, 0.33899210352866915))
U = Intersect(c, d, (6.035006740605754, -0.03327558727897473))
W = Intersect(c, d, (-0.29354400312420476, 4.620070547816574))
C = Intersect(e, g)
O = Intersect(g, h)
A = Intersect(f, h)
i = false
j = false
k = false
l = false
m = false
n = false
p = false
poly1, o_1, a_1, b_1, c_1 = Polygon(O, A, B, C)
o = false
q(x, y) = (250 * x) + (75 * y)
s = 0 (slider from 0 to 6250, step 1)
t: (250 * x) + (75 * y) = s
D = Point(t)
text2: "250x+75y = " + s + "
250x+75y = P"
r = false
u = false
v = false
w = false
d_1 = false
e_1 = false
f_1 = false
g_1 = false
h_1 = false