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Question:
1. Solve the following problem using the simplex method.
Maximise p = 4 x + 6 y + 3 z
Subjected to the following constraints:
(N+1) x + 4 y + 3 z ≤ 12
4 x + 3 y + 2 z ≤ 10
2 x + 4 y + z ≤ 8
x≥ 0, y≥ 0 and z≥ 0
2. By using the Lagrange multiplier method, find the dimensions of a rectangular box of volume V = 1000 × (N+1)2 cm3 for which the total length of the 12 edges is minimum.
3. Consider the function f(x) = x4 – 14x3 + (N+51) x2 - 70x
Use the golden section search technique to find the value of x that minimises f over the range 0 ≤ x ≤ 2. Locate the x value within an interval of 15% uncertainty of the initial search interval. Present your answer in a tabular format.
4. A manufacturing company produces two types of products, Product A and Product B. The various restrictions, requirements and unit profits are shown below. By using the analytical linear programming method determine how many of Product A and Product B should be produced to maximise the profit. Present your answer in a tabular format.
† Individual assignment
Profit Raw Material Electricity Labour
Product A 3 5 1 1
Product B 4 2 1 2
Amount available ---- 31 + N 10

As given in question,

Maximize p = 4x + 6y + 3z subject to

6x + 4y + 3z <= 12

4x + 3y +2z <= 10

2w + 4y + z <= 8

x>=0

y>=0

z>=0

Suppose s­­1, s2, s3, s4, s5 and s6  are the slack variable, the above equation can be tabulated as follows

 Table #1 x y z s1 s2 s3 s4 s5 s6 p 6 4 3 1 0 0 0 0 0 0 12 4 3 2 0 1 0 0 0 0 0 10 2 4 1 0 0 1 0 0 0 0 8 4 6 3 0 0 0 0 0 0 0

Further row and column operation can be done as follows

 Tableau #1 x y z s1 s2 s3 s4 s5 s6 p 6 4 3 1 0 0 0 0 0 0 12 4 3 2 0 1 0 0 0 0 0 10 0 4 1 0 0 1 0 0 0 0 8 1 0 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 -4 -6 -3 0 0 0 0 0 0 1 0

 Tableau #2 x y z s1 s2 s3 s4 s5 s6 p 6 4 3 1 0 0 0 0 0 0 12 4 3 2 0 1 0 0 0 0 0 10 0 4 1 0 0 1 0 0 0 0 8 -1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 -4 -6 -3 0 0 0 0 0 0 1 0

 Tableau #3 x y z s1 s2 s3 s4 s5 s6 p 6 4 3 1 0 0 0 0 0 0 12 4 3 2 0 1 0 0 0 0 0 10 0 4 1 0 0 1 0 0 0 0 8 -1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 -4 -6 -3 0 0 0 0 0 0 1 0

 Tableau #4 x y z s1 s2 s3 s4 s5 s6 p 6 4 3 1 0 0 0 0 0 0 12 4 3 2 0 1 0 0 0 0 0 10 0 4 1 0 0 1 0 0 0 0 8 -1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 1 0 0 0 0 0 -1 0 0 0 0 0 1 0 0 -4 -6 -3 0 0 0 0 0 0 1 0

 Tableau #5 x y z s1 s2 s3 s4 s5 s6 p 6 0 2 1 0 -1 0 0 0 0 4 4 0 1.3 0 1 -0.75 0 0 0 0 4 0 1 0.25 0 0 0.25 0 0 0 0 2 -1 0 0 0 0 0 1 0 0 0 0 0 0 0.25 0 0 0.25 0 1 0 0 2 0 0 -1 0 0 0 0 0 1 0 0 -4 0 -1.5 0 0 1.5 0 0 0 1 12

 Tableau #6 x y z s1 s2 s3 s4 s5 s6 p 1 0 0.33 0.17 0 -0.17 0 0 0 0 0.67 0 0 -0.083 -0.67 1 -0.083 0 0 0 0 1.3 0 1 0.25 0 0 0.25 0 0 0 0 2 0 0 0.33 0.17 0 -0.17 1 0 0 0 0.67 0 0 0.25 0 0 0.25 0 1 0 0 2 0 0 -1 0 0 0 0 0 1 0 0 0 0 -0.17 0.67 0 0.83 0 0 0 1 15

 Table #7 x y z s1 s2 s3 s4 s5 s6 p 3 0 1 0.5 0 -0.5 0 0 0 0 2 0.25 0 0 -0.63 1 -0.12 0 0 0 0 1.5 -0.75 1 0 -0.13 0 0.38 0 0 0 0 1.5 -1 0 0 0 0 0 1 0 0 0 0 -0.75 0 0 -0.13 0 0.38 0 1 0 0 1.5 3 0 0 0.5 0 -0.5 0 0 1 0 2 0.5 0 0 0.75 0 0.75 0 0 0 1 15

Since, we can see that, no positive unit solution for x variable, in this condition it can be taken as 0, y = 1.5 and z = 2, the maximum value of P whiting the given condition is 15

Solution 2

As given in problem,

We have to minimize total length i.e. f(x,y,z) = 4x+4y+4z, in which x, y and z are length width and height

Which is subjected to the constant g(x,y,z) = xyz = 1000 (5+1)2 = 36000 cm3,

By implementing Lagrange multiplier, we get

f(x­0,y0,z0) = λ f (x­0,y0,z0)

Using this equation in the above minimization problem

4x + 4y =  λxy             ….. (i)

4y + 4z =  λyz             ….. (ii)

4x + 4z =  λzx             ….. (iii)

Since the constraints x  0, y  0, and z  0, in this condition, we have to solve λ for each equation in pairs we get,

….. (iv)

….. (v)

….. (vi)

From equation (iv) and (v) we get,

=   or x =z

Similarly, equation (v) and (vi) we get

X =z, i.e. x =y=z

From the above solution it’s clear that the sum will be minimum when all dimension will be equal, i.e. the box will be in the form of cuboid

Putting g the value in constrain

Xyz = x3 = 36000

Or,   x = , therefore all side will be x = =            Ans

Solution 3

As given in question,

The value of x range as given =  = [0, 2]

The steps to be followed is as given

1

Now applying the gold section method at  (Suppose)

2  Now putting the value in function,

We have to minimise the problem

F(0) = 4

Now f(0) = 0, f(0.7639) = -24.361,  f(1.2631) = -18.958

Value less than f(2) = 4

X3 0.4722 = (0+L3*)

F(0) = 0 à  f(x1) = -24.361

F(0) = 0

 Function L1 L­2 (0, 1.2361) L3 (0.4722, 1.2361) f(x1) (0,2) -24.361 f(x2) (0,2) -18.958 f(x3) (0,2) -21.1
Solution 4
As given in question,

The constraints can be tabulated as follows

 Profit Raw Material Electricity Labor Product A 3 5 1 1 Product B 4 2 1 2 Amount available ---- 36 10 16

Suppose the number of product A =x

And No of product B = y

Then, Profit (Z) = 3x +4y, we must maximize Z to solve the problem

The constraints can be written as

5x+2y  36       ……… (i)

x+y  10            ……… (ii)

x+2y  16           ……... (iii)

Suppose s1 , s2, s3, s4 , s5  are the slack variable for the given above equation.

After performing different operation in row and column

 Table 1 x y s1 s2 s3 s4 s5 p 5 2 1 0 0 0 0 0 36 1 1 0 1 0 0 0 0 10 1 2 0 0 1 0 0 0 16 1 0 0 0 0 -1 0 0 0 0 1 0 0 0 0 -1 0 0 -3 -4 0 0 0 0 0 1 0

 Table 2 x y s1 s2 s3 s4 s5 p 5 2 1 0 0 0 0 0 36 1 1 0 1 0 0 0 0 10 1 2 0 0 1 0 0 0 16 -1 0 0 0 0 1 0 0 0 0 1 0 0 0 0 -1 0 0 -3 -4 0 0 0 0 0 1 0

 Table 3 x y s1 s2 s3 s4 s5 p 5 2 1 0 0 0 0 0 36 1 1 0 1 0 0 0 0 10 1 2 0 0 1 0 0 0 16 -1 0 0 0 0 1 0 0 0 0 -1 0 0 0 0 1 0 0 -3 -4 0 0 0 0 0 1 0

 Table 4 x y s1 s2 s3 s4 s5 p 4 0 1 0 -1 0 0 0 20 0.5 0 0 1 -0.5 0 0 0 2 0.5 1 0 0 0.5 0 0 0 8 -1 0 0 0 0 1 0 0 0 0.5 0 0 0 0.5 0 1 0 8 -1 0 0 0 2 0 0 1 32

 Table 5 x y s1 s2 s3 s4 s5 p 0 0 1 -8 3 0 0 0 4 1 0 0 2 -1 0 0 0 4 0 1 0 -1 1 0 0 0 6 0 0 0 2 -1 1 0 0 4 0 0 0 -1 1 0 1 0 6 0 0 0 2 1 0 0 1 36

After making positive in all bottom line It was found that the value of x and are 4 and 6 respectively, the maximum profit that can be made is 36. This is tabulated below.

 Profit Raw Material Electricity Labor Product A 12 20 4 4 Product B 24 12 6 12 Amount available ---- 32 <=36 10 16
References

Aczel Amir D.  Complete business statistics. NJ :: Morristown, , 2012.

Shuqin Yang. Applications of Excel in teaching Statistics. 2005.

Berenson Mark. 2 Basic Business Statistics. Sydney :: Pearson. 2012.

Broadman Bart.  DBS Group Holdings Ltd | Annual Report 2015. Singapore :: DBS Pub. 2014.

Barron's.  Business Statistics. New York :: 2010.

Krehbiel Heather Haskin & Timothy. Business statistics at the top 50 US business programmes. Dubline:: NSK. 2011.

Chng Pamela.  Bettr Barista Coffee Academy. Singapore : :Singa publisher. 2015.

Douglas Lind. William Marchal. Samuel Wathen. Basic Statistics for Business and Economics. New York : :McGraw-Hill Higher Education. 2012.

Cite This Work

My Assignment Help. (2020). Optimization Problems With Simplex And Lagrange Methods. Retrieved from https://myassignmenthelp.com/free-samples/mcen4011-engineering-design-methodology/dimensions-of-a-rectangular.html.

"Optimization Problems With Simplex And Lagrange Methods." My Assignment Help, 2020, https://myassignmenthelp.com/free-samples/mcen4011-engineering-design-methodology/dimensions-of-a-rectangular.html.

My Assignment Help (2020) Optimization Problems With Simplex And Lagrange Methods [Online]. Available from: https://myassignmenthelp.com/free-samples/mcen4011-engineering-design-methodology/dimensions-of-a-rectangular.html
[Accessed 24 July 2024].

My Assignment Help. 'Optimization Problems With Simplex And Lagrange Methods' (My Assignment Help, 2020) <https://myassignmenthelp.com/free-samples/mcen4011-engineering-design-methodology/dimensions-of-a-rectangular.html> accessed 24 July 2024.

My Assignment Help. Optimization Problems With Simplex And Lagrange Methods [Internet]. My Assignment Help. 2020 [cited 24 July 2024]. Available from: https://myassignmenthelp.com/free-samples/mcen4011-engineering-design-methodology/dimensions-of-a-rectangular.html.

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