finite math ( math for business ) write the solution neatly and clearly please .

Question

finite math ( math for business )
write the solution neatly and clearly please . use the simplex method pivoting procedure.

Question 2.(30 points) A food wholesaler has 3 kinds of individual bags of potato chips; Regular, Barbeque, and Salt and Vinegar. She wants to sell the bags of chips in bulk packages The Bronze package consists of 20 bags of regular, 10 bags of barbeque. The silver package consists 20 bags of regular, 10 bags of barbeque, and 10 bags of salt and vinegar The gold package consists 30 bags of regular, 10 bags of barbeque, and 10 bags of salt and vinegar The profit is S 30 on each bronze package, $40 on each silver package, and $60 on each gold package The wholesaler has 8000 bags or regular, 4000 bags or barbeque, and 2000 bags of salt and vinegar. Assuming maximize the profit? What is the maximum profit and how many bags will be left unused? (i) Write the linear programming problem defining the variables (i) Write the system of equations with slack variables. (iii) Write the initial simplex tableau and initial feasible solution. (iv) Perform pivoting to find the maximum profit. (v) State your answer to the question.

Explanation / Answer

a) Let the number of bronze packages be x

Let the number of silver packages be y

Let the number of gold packages be z

Optimization Fuction, Profit (Maximize) Z = 30x + 40y + 60z

Constraint on regular chips:

20x + 20y + 30z <= 8000

Constraint on barbeque chips:

10x + 10y + 10z <= 4000

Constraint on salt and vinegar chips:

10y + 10z <= 2000

b) After adding the slack variables s1,s2 and s3, the constraints become

Constraint on regular chips:

20x + 20y + 30z + s1 = 8000

Constraint on barbeque chips:

10x + 10y + 10z + s2 = 4000

Constraint on salt and vinegar chips:

10y + 10z + s3 = 2000

c)

Initial Tableau Setup

Table #1
==========================================================
x y z s1 s2 s3 z
==========================================================
20 20 30 1 0 0 0 8000
10 10 10 0 1 0 0 4000
0 10 10 0 0 1 0 2000
-30 -40 -60 0 0 0 1 0   

Initial Feasible Solution

Table #2
==========================================================
x y z s1 s2 s3 z
==========================================================
20 -10 0 1 0 -3 0 2000
10 0 0 0 1 -1 0 2000
0 1 1 0 0 0.1 0 200   
-30 20 0 0 0 6 1 12000  

d)

Tableau #1
x y z s1 s2 s3 z
20 20 30 1 0 0 0 8000
10 10 10 0 1 0 0 4000
0 10 10 0 0 1 0 2000
-30 -40 -60 0 0 0 1 0   

Tableau #2
x y z s1 s2 s3 z
20 -10 0 1 0 -3 0 2000
10 0 0 0 1 -1 0 2000
0 1 1 0 0 0.1 0 200   
-30 20 0 0 0 6 1 12000  

Tableau #3
x y z s1 s2 s3 z
1 -0.5 0 0.05 0 -0.15 0 100   
0 5 0 -0.5 1 0.5 0 1000
0 1 1 0 0 0.1 0 200   
0 5 0 1.5 0 1.5 1 15000  

e)

Number of bronze packages = 100

Number of silver packages = 0

Number of gold packages = 200

Maximum Profit = 100(30) + 0(40) + 200(60) = $15000

Number of bags unused regular chips = 8000 - 100(20) - 200(30) = 0

Number of bags unused barbeque chips = 4000 - 100(10) - 200(10) = 1000

Number of bags unused salt and vinegar chips = 2000 - 200(10) = 0

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