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This lecture is from Management Science. Key important points are: Linear Programming Problem, Hours Required, Linear Programming, Simplex Techniques, Optimum Solution, Decision Variable, Chairs Produced, Tables Produced, Objective Function, First Constraint
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Machine
Hours Required Available Hour/Week Chairs Table
M 1 6 6 72
M 2 10 4 100
M 3 4 12 120
(iii) Constraints:
First Constraint: On machine M1, each chair requires 6 hours and each table requires 6 hours. The total of working hours is given by 6x1 + 6x2.
Second Constraint: On machine M2, each chair requires 10 hours and each table requires 4 hours. The total of working hours is given by 10x1 + 4x2.
Third constraint: On machine M1, each chair requires 4 hours and each
table requires 12 hours. The total of working hours is given by 4x1 + 12x2.
Further, since the manufacturer does not have more then 72 hours of
working on machine M1, does not have more then 100 hours of working
on machine M2 and does not have more then 120 hours of working on machine M3. So that the constraint are as follows:
06x1 + 06x2 ≤ 72 10x1 + 04x2 ≤ 100 04x1 + 12x2 ≤ 120
Hence the manufacturer’s allocation problem can be put in
the following mathematical form:
Find two real numbers x1 and x2, such that to maximize the
expression (Objective Function)
Z = 200x1 + 300x
Subject to the Constraints,
06x1 + 06x2 ≤ 72
10x1 + 04x2 ≤ 100
04x1 + 12x2 ≤ 120
and x1; x2 ≥ 0
- 4 x Corresponding New Table Key row values.
Compute the new key row values by using the formula
New Table Key row values
= Old Table Key row values / Key Element
New Table Key row values
= 1/4(12 4 0 1 0 -1/2)
= (12/4 4/4 0/4 1/4 0/4 -1/2 x 1/4)
= (3 1 0 1/4 0 -1/8)
Computing all other row values using the formula
New Table Row values=Old row values-Corresponding
Coefficient in Key Column x Corresponding new table key
row value
New Second Row Values
= Old Second Row Values
- 26/3 x Corresponding New Table Key row values.
= (60 26/3 0 0 1 -1/3)
= (34 0 0 -13/6 1 3/4)
New Third Row Values
=Old Third Row Values – 1/3 x Corresponding New
Table Key row values
= (10 1/3 1 0 0 1/12)
= (9 0 1 -1/12 0 1/8)
