The Heat Corporation produces three types of rugs, which it sells to large carpe

Question

The Heat Corporation produces three types of rugs, which it sells to large carpet stores. The production of each rug requires two machine operations, weaving and binding, followed by assembly, which includes inspection and packaging. All three rug types require 0.4 hour of assembly time, but the machining operations have different processing times as shown here in the table below, in hours per unit:

Rectangle rug:
Weaver = 0.2
Binder = 0.6

Runner rug:
Weaver = 0.4
Binder = 0.2

Oval rug:
Weaver = 0.6
Binder = 0.5

Each machine is available 150 hours per month, and the current size of the assembly department provides capacity of 600 hours. Each rug produced yields a unit of profit contribution as seen in the table below:

Profit:
Rectangle = $80
Runner = $60
Oval = $100

What are the optimal (maximum profit) production quantities for the company? Please explain methods to finding answers.

Explanation / Answer

Let x, y, z be the production quantities of three types of rugs that are to be made,

then
The objective is to maximize the profit. So, profit contributions by each tyoe of rugs will become the objective function.

LP Formulation of the above problem is

Maximize profit = 80x + 60y + 100z

subject to

Constraint1: Weaving machine is available only for 150 hours and x type of rug takes 0.2 hrs of weaving machine time, y type of rug takes 0.4 hrs of machine time and z type of rug takes 0.6 hrs of machine time

0.2x + 0.4y + 0.6z <= 150,

Constraint 2: Binding machine is available only for 150 hours and x type of rug takes 0.6 hrs of Binding machine time, y type of rug takes 0.2 hrs of machine time and z type of rug takes 0.5 hrs of machine time

0.6x + 0.2y + 0.5z <= 150,

Constraint 3: Assembly of each rug takes 0.4 hrs and avaialble time is 600 hrs

0.4x + 0.4y + 0.4z <= 600

Constraint4: non-negative constraints: rugs cannot be negative and fractions

x,y,z all integer >=0


The following tables are the step by step solution of simplex method


Tableau #1
x      y      z      s1     s2     s3     p            
1      2      3      5      0      0      0      750   
6      2      5      0      10     0      0      1500  
2      2      2      0      0      5      0      3000  
-80    -60    -100   0      0      0      1      0     

Tableau #2
x      y      z      s1     s2     s3     p            
1      2      3      5      0      0      0      750   
13     -4     0      -25    30     0      0      750   
4      2      0      -10    0      15     0      7500  
-140   20     0      500    0      0      3      75000

Tableau #3
x      y      z      s1     s2     s3     p            
0      10     13     30     -10    0      0      3000  
13     -4     0      -25    30     0      0      750   
0      14     0      -10    -40    65     0      31500
0      -100   0      1000   1400   0      13     360000

Tableau #4
x      y      z      s1     s2     s3     p            
0      10     13     30     -10    0      0      3000  
5      0      2      -5     10     0      0      750   
0      0      -7     -20    -10    25     0      10500
0      0      10     100    100    0      1      30000



Optimal solution is

profit = 30000; Production Quantities of each rugs is x = 150, y = 300, z = 0

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