e1605 simplex algorithm

7/28/2019 E1605 Simplex Algorithm

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Simplex Algorithm

Solving linear programming

problems algebraically


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Initial Example: Maximise P = 10x + 12y

subject to the constraints:

x + y 40 (i)

x + 2y 75 (ii)

x 0, y 0.Step 1: Introduce slack variables to convert the

non-trivial inequalities into equalities:

Equation (i): x + y + s = 40 s 0Equation (ii): x + 2y + t = 75 t 0

s, t are slack variables.


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Step 2: Rewrite the objective function so

that the RHS is a number:

P = 10x + 12y P 10x 12y = 0.

Step 3: Write the objective function and the

non-trivial constraints in tableau format:


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Tableau Format

P x y s t l Equationobjective 1 -10 -12 0 0 0 (1)

0 1 1 1 0 40 (2)constraints

0 1 2 0 1 75 (3)

Last

column

The shaded

cells shouldbe non-

negative

The aim is to solve the equations by combining rows

together. The solution is reached when all entries in the

first row (except possibly the value in the last column)

are non-negative.

We begin by identifying the most negative entry in the

objective function row, here -12 in the y column.

P-10x-12y=0

x+y+s=40x+2y+t=75 Piv

otal column


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P x y s t l Equation Ratio1 -10 -12 0 0 0 (1) 0/-12=0

0 1 1 1 0 40 (2) 40/1=40

0 1 0 1 75 (3) 75/2=37.52

pivot

Pivotal

column

We highlight the pivotal column.

We then divide every entry in the l column by the corresponding

value in the highlighted column.Pick the least positive of these. This is the pivotal row.


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P x y s t l Equation Ratio1 -10 -12 0 0 0 (1) 0/-12=0

0 1 1 1 0 40 (2) 40/1=40

0 1 0 1 75 (3) 75/2=37.52

Divide the pivotal row by the pivot value.

P x y s t l Equation Ratio1 -4 0 0 6 450 (4)=(1)+12(6) 450/-4=-112.5

0 0 0 - 2.5 (5) = (2) (6) 2.5/0.5 = 5

0 1 0 37.5 (6) = (3)/2 37.5/0.5 = 75

The aim is to now get 0 entries elsewhere in the pivotal column.

We now repeat the process, first selecting the new pivotal column,

i.e. the one with the most negative value in the objective function

row.


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P x y s t l Equation Ratio1 -4 0 0 6 450 (4)=(1)+12(6) 450/-4=-112.5

0 0 0 - 2.5 (5) = (2)

(6) 2.5/0.5 = 50 1 0 37.5 (6) = (3)/2 37.5/0.5 = 75

Having identified the pivotal row and the pivot value, we now

divide every entry in the pivotal row by the pivot value.

P x y s t l Equation1 0 0 0 2 470 (7)=(4)+4(8)

0 1 0 0 -1 5 (8) = (5) 0 0 1 0 1 35 (9) = (6)(8)

The process is now finished as every entry on the objective

function row is non-negative.


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P x y s t l Equation1 0 0 0 2 470 (7)

0 1 0 0 -1 5 (8)

0 0 1 0 1 35 (9)

The values of x, y and P can be read from

the table:

x = 5, y = 35, P = 470. This is the

optimal solution.


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Interpretation


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Examination Question

A clocksmith makes 3 types of luxury watches. The mechanism for

each watch is assembled by hand by a skilled watchmaker andthen the complete watch is formed, weatherproofed and

packaged for sale by a fitter.

The table below shows the times (in mins) for each stage of the

process. It also gives the profits to be made on each watch.

Watch type Watchmaker Fitter Profit ()

A 54 60 12

B 72 36 24

C 36 48 20

The watchmaker works for a maximum of 30 hours per week; the fitter

for 25 hours per week.

Let x, y, z represent the number of type A, B, C watches to be made

(respectively).


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Setting up the problem

Profit function: P =

Constraint 1:

Constraint 2:

Watch type Watchmaker Fitter Profit ()A 54 60 12

B 72 36 24

C 36 48 20

e1605 simplex algorithm

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