#The Roxy Company
The Roxy Company owns a small paint factory that produces both exterior and interior paints from two raw materials, A and B. The following table provides the daily requirement of the raw materials which produces per ton exterior and interior paints.
Types of materials Tons of raw materials per ton of Maximum availability (ton)
Exterior paint Interior paint
Raw material A 6 4 24
Raw material B 1 2 6
Profit per ton (tk1000) 5 4
A market survey establishes that
1. The daily demand for interior paint cannot exceed that of exterior paint by more than 1 ton.
2. The maximum demand of the interior paint is limit to 2 tons daily.
How much interior and exterior paints should the company produce daily to maximize the total profit?
Mathematical Model
The company wants to determine produce exterior and interior paints ( variable) that will fill the required raw materials (constraints) with the maximum profit daily (objective).
Let us consider the variables of the model as
x1 =Tons produce daily of exterior paint
x2 =Tons produce daily of interior paint
To construct the objective function,
z represent the total profit daily
Total profit from exterior paint = 5 x1 taka
Total profit from interior paint = 4 x2 taka
The objective function is expressed as, Maximize z = 5 x1+ 4 x2
Now we construct the five constraints
1. Usage and availability of raw materials. The raw material restrictions may be expressed verbally as
Usage of raw materials by both paint ≤ Maximum raw materials availability
Usage of raw material A by exterior paint = 6 x1 tons/day
Usage of raw material A by interior paint = 4 x2 tons/day
Usage of raw material A by both paints = 6 x1 + 4 x2 tons/day
Usage of raw material B by both paints = 1 x1 + 2 x2 tons/day
The daily availabilities of raw materials A and B are limit 24 and 6 tons
respectively, the associated restrictions are given as
6 x1 + 4 x2 ≤ 24
x1 + 2 x2 ≤ 6
3. The first demand restriction that excess of daily production of interior over exterior paint, x2 - x1 should not exceed 1 ton, which as x2 - x1 ≤ 1
4. Demand of interior paint i.e. x2 ≤ 2
5. Nonnegativity x1 ≥ 0 , x2 ≥ 0 .
The complete Roxy Company model is
Maximize z = 5 x1+ 4 x2
Subject to
6 x1 + 4 x2 ≤ 24…….(1)
x1 + 2 x2 ≤ 6 …….(2)
x2 - x1 ≤ 1 …….(3)
x2 ≤ 2 …….(4)
x1 ≥ 0 , x2 ≥ 0 . …….(5)
If we draw the above five equations graphically can get this figure. The feasible solution space is OABCDE. O, A, B, C, D, E are the extreme points of the solution space. We can get optimum solution with one of these points.
Solution by feasible points
Feasible points are O (0,0), A (4,0), B (3,3/2), C (2,2), D (1,2), E (0,1).
For O , z = 0 + 0 = 0
For A , z = 5.4 + 0 = 20
For B , z = 5.3 + 4.3/2 = 21
For C , z = 5.2 + 4.2 = 18
For D , z = 5.1 + 4.2 = 13
For E , z = 0 + 4.1 = 4
We can get the optimum solution (z = 21 ) by the above feasible point B.
Solution by objective function
z = 5 x1+ 4 x2 x2 = - x1 +
m = = - = Slope of the objective function
From this slope we can write x1 = 4 and x2 =5 and ratio is x1 : x2= 4 : 5
According this ratio we can draw lines. These lines must be parallel lines.
We can see that maximum value of 20 which does not exist in the point B. The objective function must satisfy all feasible point. The optimum solution is 21 which touch at point B.