Answer to Question #302898 in Operations Research for fiker

Question #302898

A bed mart company is in the business of manufacturing beds and

pillows. The company has 40 hours for assembly and 32 hours for

finishing work per day. Manufacturing of a bed requires 4 hours for

assembly and 2 hours for finishing. Similarly a pillow requires 2 hours

for assembly and 4 hours for finishing. Profitability analysis indicates

that every bed would contribute Birr 80, while a pillow contribution is

Birr 55 respectively. Find out the daily production of the company to

maximise the contribution (profit). Solve the problem by graphical

method.


1
Expert's answer
2022-03-01T09:24:30-0500

Let x, y denote the number of units of bed and pillow to be produced to maximize the profit.

The data given is tabulated below:


"\\begin{array}{|c|c|c|c|}\n\\hline\n & \\text{Assembly} & \\text{Finish} & \\text{Profit} \\\\\n\\hline\n\\text{Bed} & 4 & 2 & 80\\\\\n\\text{Pillow} & 2 & 4 & 55\\\\\n\\hline\n\\text{Availability (Time)}& 40 & 32 & \\\\\n\\hline\n\\end{array}"


The objective function is "z = 80x + 55y". The constraints are on the available hours given by,

"\\begin{aligned}\n4x+2y &\\le 40\\\\\n2x+4y &\\le 32\\\\\nx,y & \\ge 0\n\\end{aligned}"

The linear programming model is,

"\\text{Maximise } z = 80x+55y \\\\\n\\text{subject to}\\\\\n\\begin{aligned}\n4x+2y &\\le40\\\\\n2x+4y &\\le 32\\\\\nx,y &\\ge 0\n\\end{aligned}"

To solve it graphically, we consider the constraints as equations and draw straight lines. The equations are,

"\\begin{aligned}\n4x+2y &=40 \\qquad(1)\\\\\n2x+4y &= 32 \\qquad(2)\\\\\n\\end{aligned}"

Equation (1) is the line passing through "(0,20)~ \\& ~(10,0)".

Equation (2) is the line passing through "(0,8)~ \\& ~(16,0)".


The graph is given below.


The feasible region is bounded by the constraints. The extreme points are "O(0,0), A(10,0), B(8,4), C(0,8)". The optimal value occurs at one of the extreme points.

The point B is the point of intersection of the two lines which is obtained by solving the equations (1) and (2).

The value of z at these points are,

"z(O) = z(0,0) = 0\\\\\nz(A) = z(10,0) = 800\\\\\nz(B) = z(8,4) = 860\\\\\nz(C) = z(0,8) = 440"


Hence the maximum value occurs at B(8,4) and the maximum value is z = 860.


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