1. BUS107 Quantitative Methods
Answers:
Objective
The purpose of this advertising campaign is to reach maximum number of people through
tv advertisement and social media campaign. Therefore, the objective of this linear
programing is to optimize number of people reached through advertising campaign
following constraints applicable for this program.
Constraints
Following constraints are applicable for this advertising program:
Total available budget is $30k and no additional budget is available which means that this
proposed advertising campaign has to be completed within this allocated budget.
While the purpose is to allocate budget effectively between these two medium for
advertising campaign, at least $10k budget has to be allocated to each of these advertising
mediums as per requirements.
Decision Variables
Decision variables in context of linear programming is those variables that need to be
determined and for this advertising program, decision variables are amount to be allocated
to each advertising medium while maximizing number of people reached through these
advertising programs.
Analysis Of Linear Programming
Following is the mathematical representation of this linear program:
$X1k = budget allocated for advertising through social media campaign
$X2k is budget allocated for tv advertising campaign.
As per given data, with each $1k that spent on social media campaign, 15 people are
2. reached, and for tv advertising campaign, 10 people are reached with each $1k spent for
advertisement.
Objective Function
Maximize: Z = 15X1 + 10X2
Subject to:
Constraint 1: X1 + X2 ≤ 30
Constraint 2: X1 ≥ 10
Constraint 3: X2 ≥ 10
X1, X2 ≥ 0
Point
Coordinates (X1,X2)
Objective Function Value z=15X1+ 10X2
A
(10,20)
15(10)+ 10(20) = 350
3. B
(20,10)
15(20)+ 10(10) = 400
C
(10,10)
15(10)+ 10(10) = 250
For this linear program, feasible solution space is area represented by intersection of these
constraints and this is area ABC.
Optimal solution corresponds to point B (20, 10) as this provide maximum value for z
complying with applicable constraints for this linear program.
Therefore, to reach maximum number, $20k should be allocated to social media campaign
and $10k should be allocated to tv advertising campaign out of $30k allocated for this
program as described in this context.
(b) As per given data, additional $1k is allocated for advertising program. Therefore, total
budget available is $ (30+1)k = $31k
Objective Function
Maximize: Z = 15X1 + 10X2
Subject to:
Constraint 1: X1 + X2 ≤ 31
4. Constraint 2: X1 ≥ 10
Constraint 3: X2 ≥ 10
X1, X2 ≥ 0
Solution for this new linear program
Point
Coordinates
(X1,X2)
Objective Function Value
15X1+ 10X2
A
(10,21)
15(10)+ 10(21) = 360
B
(21,10)
5. 15(21)+ 10(10) = 415
C
(10,10)
15(10)+ 10(10) = 250
The optimal solution is Z = 415 corresponding to X1=21 and X2=10
Therefore, it is possible to reach 15 more people by increasing advertising budget by $1k.
From optimal solution it can be identified that additional $1k of budget has to be allocated
for social media campaign to reach 15 more people. Therefore, the organization needs to
consider investing additional amount in social media campaign as this increases number of
people reached through advertising campaign, an additional $1k budget spent on
advertising, this advertising campaign reach 15 more people as described in this context.
A Seminar On Linear Programing
Techniques to formulate a problem as a linear problem
To apply linear programing tool:
Analyse problem to be solved and determine objective function that is value that needs to
be minimized or maximized as per requirements.
Identify constraints applicable to chosen problem and then construct linear equations
accordingly.
Identify decision variables which refer to variables that need to be determined to obtain
optimum solution for a linear program while considering constraints related to that
problem.
Once equations have been determined following applicable constraints, equations can be
solved either by graphical method or any software such as excel solver. As part of
demonstration, graphical method will be analysed. In this context, an example has been
provided to provide a comprehensive overview of how to apply this technique for solving
6. problems.
Analyse problem from linear programming perspective
Example Of Problem
Two portfolios are available including portfolio A and portfolio B and total budget available
is $20k. It is also required that at least $5k is allocated to each portfolio as investment. With
every $1k invested into portfolio A generates a revenue of $5k and for portfolio B, revenue
for each $1k invested is $2k. Identify how to invest in these portfolios for maximum profit
from investment.
Analysis of steps to solve a problem with linear programing
Step 1: identify objective of linear programing problem
If analysed carefully, it can be identified that the problem is about finding ways to invest
money into portfolios in such a way that it provides maximum profit. Therefore, the
objective of this problem is to maximize profit from investment into portfolios.
Step 2: identify constraints that are applicable to given problem
As per given data, two portfolios are available including portfolio A and portfolio B and total
budget available is $20k. It is also required that at least $5k is allocated to each portfolio as
investment. While formulating linear program to solve this problem, these constraints
needs to be considered and solutions have to be obtained accordingly.
Step 3: identify decision variables and construct linear equations following constraints
Decision variables for this linear program are investments into portfolios A and B.
Formulate linear equations to solve this problem
Objective Function
Maximize: Z = 15X1 + 10X2
Subject to:
Constraint 1: X1 + X2 ≤ 30
Constraint 2: X1 ≥ 10
Constraint 3: X2 ≥ 10
7. X1, X2 ≥ 0
Coordinates (x1,x2)
Lines through Extreme Point
Objective function value z=5x1+2x2
A(5,5)
2→x1≥5
3→x2≥5
5(5)+2(5)=35
B(15,5)
1→x1+x2≤20
3→x2≥5
5(15)+2(5)=85
8. C(5,15)
1→x1+x2≤20
2→x1≥5
5(5)+2(15)=55
Optimal solution is obtained for point B (15, 5) and maximum value of profit is $85k.
Therefore, $15k should be invested into portfolio A and $5k should be invested into
portfolio for maximum profit as described in this context.
