Solve completely using the simplex method. A company is trying to maximize its p

Question

Solve completely using the simplex method.

A company is trying to maximize its profit on the sale of items A, B, and C. They make a profit of 60 dollars on each item of type A, 25 dollars on each item of type B, and $50 on each item of type C. Each product requires polishing and assembling time. Type A items requre 4 minutes of polishing, type B items require 3 minutes of polishing, and type C items require 5 minutes of polishing. Type A items use 2 minutes of assembly time, and types B and C items each use 1 minute of assembly time. There are 890 minutes of polishing time and 600 minutes of assembly time available. The company must produce at least 100 items of type A and 130 items of type B.

(a) The initial simplex tableau has:
5 rows
9 columns

(b) What is the maximum profit?


This profit is obtained by making:
  items of type A
  items of type B
  items of type C

Explanation / Answer

Solution:

A) 4 rows

8 columns including p

B) 4300

C) A=50, B=52,C=0

Explanation:

Maximize p = 60x + 25y + 50z    (Here A=x, B=y, and C=z)

subject to
4x + 3y + 50z <= 356
2x + y + z <= 240
x>=40
y>=52

We use simplex method as shown

Tableau #1
x      y      z      s1     s2     s3     s4     p           
4      3      50     1      0      0      0      0      356  
2      1      1      0      1      0      0      0      240  
1      0      0      0      0      -1     0      0      40   
0      1      0      0      0      0      -1     0      52   
-60    -25    -50    0      0      0      0      1      0    

Tableau #2
x      y      z      s1     s2     s3     s4     p           
0      3      50     1      0      4      0      0      196  
0      1      1      0      1      2      0      0      160  
1      0      0      0      0      -1     0      0      40   
0      1      0      0      0      0      -1     0      52   
0      -25    -50    0      0      -60    0      1      2400

Tableau #3
x      y      z      s1     s2     s3     s4     p           
0      0      50     1      0      4      3      0      40   
0      0      1      0      1      2      1      0      108  
1      0      0      0      0      -1     0      0      40   
0      1      0      0      0      0      -1     0      52   
0      0      -50    0      0      -60    -25    1      3700

Tableau #4
x      y      z      s1     s2     s3     s4     p           
0      0      12.5   0.25   0      1      0.75   0      10   
0      0      -24    -0.5   1      0      -0.5   0      88   
1      0      12.5   0.25   0      0      0.75   0      50   
0      1      0      0      0      0      -1     0      52   
0      0      700    15     0      0      20     1      4300


We get p = 4300, x = 50, y=52 and z = 0

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