LINEAR PROGRAMMING A STEP BY STEP HANDBOOK

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Step 2: Construct the initial tableau Solution

0
7
-500 -300 0 0 0 0 1200
1 1 -1 1 0 0 12

200 100 0 0 1 0
120001

Step 3: Choose the column (entering variable)

Solution

-500 -300 0 0 0 0 0

1 1 -1 1 0 0 7

200 100 0 0 1 0 1200

120001 12

Step 4: Choose the row (leaving variable)


=

Solution Ratio

-500 -300 0 0 0 0 0 -

111100 7 7

200 100 0 0 1 0 1200 6

1 2 0 0 0 1 12 12

Step 5: Choose the pivot

Solution

-500 -300 0 0 0 0 0

111100 7

200 100 0 0 1 0 1200

120001 12

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A Step By Step Handbook

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Step 6: Convert the pivot to 1 if needed (in this case, pivot
divided by two and so does for the whole row)

Solution
-500 -300 0 000
100 0
111 0 0.005 0 7
1 0.5 0 001 6
12
120

Step 7: Change the remaining column pivot to zero.

1 = 1 + 500 3 so does for the whole ROW 1
2 = 2 − 3 so does for the whole ROW 2
4 = 4 − 3 so does for the whole ROW 4

Solution

0 -50 0 0 2.5 0 3000

0 0.5 -1 -1 -0.005 0 1

1 0.5 0 0 0.005 0 6

0 1.5 0 0 -0.005 1 6

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A Step By Step Handbook

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Step 8: Continue steps (3-7) until there is no negative
value in the index row for the maximum objective
function. Since this tableau already contains non negative
values for the index row, so it is already the final tableau.

Solution
0 -100 100 2 0
0 1 -2 2 -0.01 0 3100
0 0 1 -1 0.01 0 2
1 0 3 -3 0.01 1 5
3
0

Solution

0 0 0 0 2.33 33.33 3200
0
1 1 0 0 -0.003 0.667 4
0
0 0 0 0.006 -0.333 4

0 1 -1 0.003 0.333 1

Step 9: Therefore, the answer for linear programming is

= , = , =

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A Step By Step Handbook

50

Exercise 1.5

1. A manufacturer makes wooden desks (X) and tables (Y). Each desk

requires 2.5 hours to assemble, 3 hours for buffing, and 1 hour for a

crate. Each table requires 1 hour to assemble, 3 hours to buff, and 2

hours to crate. The firm can do only up to 20 hours of assembling, 30

hours of buffing, and 16 hours of crating per week. Profit is RM3 per

desk and RM4 per table. Maximise the profit by using the simplex

method. Ans:

Optimal solution = 4, 6

max = 36

2. A chemical company makes two types of small, solid fuel rocket
motors for testing; motor A's profit is RM3.00 per motor, and motor
B's profit is RM4.00 per motor. A total processing time of 80 hours
per week is available to produce both motors. An average of four
hours per motor is for A, but only two hours per motor is for
B. However, due to the hazardous nature of the material in B, the
preparation time is five hours and two hours for motor A.
A total preparation time of 120 hours per week is available to
produce both motors. Determine the number of each motor that
should be produced to maximise profit using the simplex method.

Ans:
Optimal solution = 10, 20

max = 110

3. A manufacturer produces two types of models, M1 and M2. Each

model of the type M1 requires 4 hours of grinding and 2 hours of

polishing, whereas each model of M2 requires 2 hours of grinding and

5 hours of polishing. The manufacturer has one grinder and two

polishers. Each grinder works for 40 hours a week and each polisher

works 70 hours a week. Profit on the M1 model is RM3.00 and on

model M2 is RM4.00. Whatever is produced in a week is sold in the

market. How should the manufacturer allocate his production

capacity to the two models to make maximum profit in a week using

the simplex method? Ans:

Optimal solution = 0, 20

max = 80

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A Step By Step Handbook

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4. A firm is engaged in producing two products. A and B. Each unit of
product A requires 2 kg of raw material and four labour hours for
processing, whereas each unit of B requires 3 kg of raw materials and
three labour hours for the same type. Every week, the firm has an
availability of 60 kg of raw material and 96 labour hours. One unit of
product A sold yields RM40 and one unit of product B sold gives RM35
as profit. Formulate this as a Linear Programming Problem to
determine how many units of each product should be produced per
week so that the firm can earn maximum profit.
Ans:
Optimal solution = 18, 8
max = 1000

5. The agricultural research institute suggested the farmer spread out
at least 4800 kg of special phosphate fertiliser and not less than
7200 kg of a special nitrogen fertiliser to raise the productivity of
crops in his fields. There are two sources for obtaining these –
mixtures A and mixtures B. Both of these are available in bags
weighing 100kg each and they cost RM40 and RM24 respectively.
Mixture A contains phosphate and nitrogen equivalent of 20kg and
80 kg respectively, while mixture B contains the equivalent of 50 kg
each. Write the objective function and constraints to obtain the
fertiliser at minimum cost and solve it using Simplex Method.
Ans:
Optimal solution = 0, 144
min = 3456

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A Step By Step Handbook

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6. The table below shows the hours of general labour, machine time, and
technical labour required to make one bicycle in each plant. For the
two plants combined, the manufacturer can afford to use up to 4000
hours of general labour, 1500 hours of machine time, and 2300 hours
of technical labour per week. Plant A earns a profit of RM 60 per
bicycle and Plant B earns a profit of RM50 per bicycle. How many
bicycles per week should the manufacturer make in each plant to
maximise profit and solve it using Simplex Method.

Resource Hours per bicycle in Plant A Hours per bicycle in Plant B

General labour 10 1
Machine time 1 3
Technical labour 5 2

Ans:
Optimal solution = 300, 400

max = 38000

7. A firm can produce three types of cloth, A, B and C. 3 kinds of wool

are Black, Green and Blue. One unit of length of type A cloth needs 2

meters of black wool and 3 meters of blue wool.1 unit of length of

type B cloth needs 3 meters of black wool, 2 meters of green wool

and 2 meters of blue wool.1 unit type of C cloth needs 5 meters of

green wool and 4 meters of blue wool. The firm has a stock of 8

meters of black, 10 meters of green and 15 meters of blue. It is

assumed that the income obtained from 1 unit of type A is RM3, from

B is RM5 and from C is RM4. Formulate this as Linear Programming

and solve it by using Simplex Method.

Ans:

Optimal solution 89 50 62
= 41 , 41 , 41

max 765
= 41

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A Step By Step Handbook

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REFERENCE

Calculator of Graphical Method of Linear Programming Step by Step.
(n.d.). Retrieved September 14, 2021, from
http://reshmat.ru/graphical_method_lpp.html

Example - A manufacturing company makes two models A and B. (2018).
https://www.teachoo.com/5437/760/Example-8---A-manufacturing-
company-makes-two-models-A-and-B/category/Examples/

Graphing Calculator - GeoGebra. (2019).
https://www.geogebra.org/graphing?lang=en

Linear Programming: Word Problems and Applications. (n.d.). Retrieved
December 10, 2021, from
https://www.analyzemath.com/linear_programming/linear_prog_ap
plications.html

Linear programming | F5 Performance Management | ACCA Qualification
| Students | ACCA Global. (n.d.). Retrieved December 10, 2021,
from https://www.accaglobal.com/uk/en/student/exam-support-
resources/fundamentals-exams-study-resources/f5/technical-
articles/linear-programming.html

Linear Programming I: Maximization Pages 1 - 18 - Flip PDF Download |
FlipHTML5. (2009). https://fliphtml5.com/ijrx/bkjz/basic

Piyush N. Shah, Anesh P. Shah, & P Shah Varsha. (2003). Simplex
Method calculator. http://bit.ly/2ib46FC

SE: LESSON 1. Mathematical Formulation of The Problem. (2014).
http://ecoursesonline.iasri.res.in/mod/page/view.php?id=2920

What is Formulation of Linear Programming- Minimization Case?
definition and meaning - Business Jargons. (n.d.). Retrieved
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of-linear-programming-minimization-

LINEAR PROGRAMMING
A Step By Step Handbook