[30 points] A paper recycling machine can produce toilet paper, writing pads, an

Question

[30 points] A paper recycling machine can produce toilet paper, writing pads, and paper towels, which sell for 18, 29 and 25 cents and consume 0.5, 0.22 and 0.75 kilograms of newspaper and 0.2, 0.4, and 0.22 minutes. Each day 10 hours and 1500 kilograms of newspaper are available, and at least 1000 rolls of toilet paper, 200 writing pads and 400 rolls of paper towels are required. (a) [10 points] Please formulate an appropriate LP to maximize revenue and solve it using tableau form of simplex method. (b) [20 points] Suppose a company asked to buy your newspapers, instead. What should be the minimum price this?

Explanation / Answer

Answer:-

(a)

Let x1 be the number of toilet paper, x2 bet he number of writing pads and x3 be the number of paper towels

Maximize, Z = 18x1 + 29x2 + 25x3

Constraints:

0.5x1 + 0.22x2 + 0.75x3 <= 1500 [Newspaper constraints]

0.2x1 + 0.4x2 + 0.22x3 <= 600 [ Constraint on time]

x1 >= 1000

x2 >= 200

x3 >= 400

Solving it using tableau method

Tableau #1
x1 x2 x3 s1 s2 s3 s4 s5 z
0.5 0.22 0.75 1 0 0 0 0 0 1500
0.2 0.4 0.22 0 1 0 0 0 0 600   
1 0 0 0 0 -1 0 0 0 1000
0 1 0 0 0 0 -1 0 0 200   
0 0 1 0 0 0 0 -1 0 400   
-18 -29 -25 0 0 0 0 0 1 0   

Tableau #2
x1 x2 x3 s1 s2 s3 s4 s5 z
0 0.22 0.75 1 0 0.5 0 0 0 1000
0 0.4 0.22 0 1 0.2 0 0 0 400   
1 0 0 0 0 -1 0 0 0 1000
0 1 0 0 0 0 -1 0 0 200   
0 0 1 0 0 0 0 -1 0 400   
0 -29 -25 0 0 -18 0 0 1 18000  

Tableau #3
x1 x2 x3 s1 s2 s3 s4 s5 z
0 0 0.75 1 0 0.5 0.22 0 0 956   
0 0 0.22 0 1 0.2 0.4 0 0 320   
1 0 0 0 0 -1 0 0 0 1000
0 1 0 0 0 0 -1 0 0 200   
0 0 1 0 0 0 0 -1 0 400   
0 0 -25 0 0 -18 -29 0 1 23800  

Tableau #4
x1 x2 x3 s1 s2 s3 s4 s5 z
0 0 0 1 0 0.5 0.22 0.75 0 656   
0 0 0 0 1 0.2 0.4 0.22 0 232   
1 0 0 0 0 -1 0 0 0 1000
0 1 0 0 0 0 -1 0 0 200   
0 0 1 0 0 0 0 -1 0 400   
0 0 0 0 0 -18 -29 -25 1 33800  

Tableau #5
x1 x2 x3 s1 s2 s3 s4 s5 z
0 0 0 1 -0.55 0.39 0 0.629 0 528.4  
0 0 0 0 2.5 0.5 1 0.55 0 580   
1 0 0 0 0 -1 0 0 0 1000
0 1 0 0 2.5 0.5 0 0.55 0 780   
0 0 1 0 0 0 0 -1 0 400   
0 0 0 0 72.5 -3.5 0 -9.05 1 50620  

Tableau #6
x1 x2 x3 s1 s2 s3 s4 s5 z
0 0 0 1.58983 -0.874404 0.620032 0 1 0 840.064
0 0 0 -0.874404 2.98092 0.158983 1 0 0 117.965
1 0 0 0 0 -1 0 0 0 1000   
0 1 0 -0.874404 2.98092 0.158983 0 0 0 317.965
0 0 1 1.58983 -0.874404 0.620032 0 0 0 1240.06
0 0 0 14.3879 64.5866 2.11129 0 0 1 58222.6

The value of x1=1000,x2=318 and x3=1241

Optimal Solution: z = 1000(18) + 318(29) + 1241(25) = 58247 cents

(b)

Minimum Price = 58247 cents/1500 kg = 0.3883 $ per kg of newspaper

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