Homework SourseCan i get the conclusion for this below goal programing Obj
Can i get the conclusion for this below goal programing ??
Objective of the Project:
The objective of this report is to identify the three target audiences customers are trying to reach and with having advertising budget of $600,000.
Defining Goals:
Goal 1: Advertisement needs to be seen by at least 40 million high-income men (HIM)
Goal 2: Advertisement needs to be seen by at least 60 million low-income people (LIP)
Goal; 3: Advertisement needs to be seen by at least 35 million high-income women (HIW)
Defining the Decision Variable:
The advertising during the football games and soap operas will cover the target audience and it is viewed by many viewers. The table below indicates the number of viewers from the different categories that will be viewing these types of programming.
HIM
LIP
HIW
COST
Football ad(per min)
7 million
10 million
5 million
$100,000
Soap opera ad (per min)
3 million
5 million
4 million
$60,000
Expressing the goals as equation:
Let X1 as minutes of football ad
Let X2 as minutes of soap opera ad
Goal 1- (HIM) 7x1 +3x2 ³ 40
Goal 2- (LIP) 10x1 +5x2 ³ 60
Goal 3 – HIW 5x1 +4x2 ³ 35
Ad Budget) 100 X1 + 60 X 2 <-600
Defining the Goal Constraints and Calculations:
Formulating the problem as an Linear programing
HIM) 7x1 +3x2 ³ 40
(LIP) 10x1 +5x2 ³ 60
HIW 5x1 +4x2 ³ 35
(Ad Budget) 100 X1 + 60 X 2 <-600
X1, X2 ³ 0
So now we have the above goals, in goal programming we must remove the in equalities and add the slag variables
So, for the soft constrains we added deviation variables
d- i = represents the amount by which each goal’s target value is underachieved,
d+i = represents the amount by which each goal’s target value is overachieved.
by introducing the deviation variables on the LHS
HIM 7x1 +3x2 + d1 d+1 = 40
LIP 10x1 +5x2 + d2 d+2 = 60
HIW 5x1 +4x2 + d3 d+3 = 35
x1, x2, d1 , d+1 ,+ d2 , d+2 , d3, d+3 ³ 0
We don’t want to go below 40 for HIM
We don’t want to go below 60 for LIP
And not below 35 for HIW
So, we need to minimize d1 , d2,d3
Suppose each shortfall of 1,000,000 viewers from the goal translates to a cost of $200,000 for HIM, 100,000 for LIP and for HIW its $50,000
Then the objective function will be
Min Z = 200 d1 +100 d2,+50 d3
Excel Calculations:
Exposures of each ad media to various groups (input data)
Football ads
Soap opera ads
Goal weight
Goal level
High-income men
7
3
1
1
High-income women
5
4
1.2
1
Low-income people
10
5
2
Cost/unit
$1,00,000
$60,000
Cost of ads
Football ads
Soap opera ads
Total
<=
Budget
500000
100000
600000
$6,00,000
Achievement of goals
Total exposures
Under goals
Sum
Goals
High-income men
40
0
40
>=
40
High-income women
31.66666667
3.333333333
35
>=
35
Low-income people
58.33333333
1.666666667
60
>=
60
Deviation fromgoals
Deviation
Dev from goal 1
0
Dev from goal 2
3.333333333
Dev from goal 3
1.666666667
Weighted goals of L1
4
| HIM | LIP | HIW | COST | |
| Football ad(per min) | 7 million | 10 million | 5 million | $100,000 |
| Soap opera ad (per min) | 3 million | 5 million | 4 million | $60,000 |
Solution
Minimize z = 0x + 0y + 0q + 0s + 0u +200p +100r+50t subject to
7x + 3y +p -q = 40
10x+5y +r - s = 60
5x+4y +t - u = 35
100x+60y <= 600
this is the linear programming equations;
The optimum solution is for
x = 6, q = 2, t = 5;
or we can say that;
total of 6 minute of football ad will be shown;
we will get 42 million HIM
60 million LIP
30 million HIW;
total cost incurred will be 600,000 on advert;
and total loss will be 5 million due to HIW, which is 50*5 250,000 in loss; which is minimum;
Using simplex we solved;
********************************/*********************************/*******************************/******************************
Tableau #1
x y q s u p r t s1 s2 s3 s4 s5 s6 s7 -z
7 3 -1 0 0 1 0 0 1 0 0 0 0 0 0 0 40
10 5 0 -1 0 0 1 0 0 1 0 0 0 0 0 0 60
5 4 0 0 -1 0 0 1 0 0 1 0 0 0 0 0 35
100 60 0 0 0 0 0 0 0 0 0 1 0 0 0 0 600
7 3 -1 0 0 1 0 0 0 0 0 0 -1 0 0 0 40
10 5 0 -1 0 0 1 0 0 0 0 0 0 -1 0 0 60
5 4 0 0 -1 0 0 1 0 0 0 0 0 0 -1 0 35
0 0 0 0 0 200 100 50 0 0 0 0 0 0 0 1 0
Tableau #2
x y q s u p r t s1 s2 s3 s4 s5 s6 s7 -z
0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0
0 0.714286 1.42857 -1 0 -1.42857 1 0 0 1 0 0 1.42857 0 0 0 2.85714
0 1.85714 0.714286 0 -1 -0.714286 0 1 0 0 1 0 0.714286 0 0 0 6.42857
0 17.1429 14.2857 0 0 -14.2857 0 0 0 0 0 1 14.2857 0 0 0 28.5714
1 0.428571 -0.142857 0 0 0.142857 0 0 0 0 0 0 -0.142857 0 0 0 5.71429
0 0.714286 1.42857 -1 0 -1.42857 1 0 0 0 0 0 1.42857 -1 0 0 2.85714
0 1.85714 0.714286 0 -1 -0.714286 0 1 0 0 0 0 0.714286 0 -1 0 6.42857
0 0 0 0 0 200 100 50 0 0 0 0 0 0 0 1 0
Tableau #3
x y q s u p r t s1 s2 s3 s4 s5 s6 s7 -z
0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0
0 1.5 0 0.5 -1 0 -0.5 1 0 0 1 0 0 0.5 0 0 5
0 10 0 10 0 0 -10 0 0 0 0 1 0 10 0 0 0
1 0.5 0 -0.1 0 0 0.1 0 0 0 0 0 0 -0.1 0 0 6
0 0.5 1 -0.7 0 -1 0.7 0 0 0 0 0 1 -0.7 0 0 2
0 1.5 0 0.5 -1 0 -0.5 1 0 0 0 0 0 0.5 -1 0 5
0 0 0 0 0 200 100 50 0 0 0 0 0 0 0 1 0
Tableau #4
x y q s u p r t s1 s2 s3 s4 s5 s6 s7 -z
0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0
0 0 0 -1 -1 0 1 1 0 0 1 -0.15 0 -1 0 0 5
0 1 0 1 0 0 -1 0 0 0 0 0.1 0 1 0 0 0
1 0 0 -0.6 0 0 0.6 0 0 0 0 -0.05 0 -0.6 0 0 6
0 0 1 -1.2 0 -1 1.2 0 0 0 0 -0.05 1 -1.2 0 0 2
0 0 0 -1 -1 0 1 1 0 0 0 -0.15 0 -1 -1 0 5
0 0 0 0 0 200 100 50 0 0 0 0 0 0 0 1 0
Tableau #5
x y q s u p r t s1 s2 s3 s4 s5 s6 s7 -z
0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0
0 0 -0.833333 0 -1 0.833333 0 1 0 0 1 -0.108333 -0.833333 0 0 0 3.33333
0 1 0.833333 0 0 -0.833333 0 0 0 0 0 0.0583333 0.833333 0 0 0 1.66667
1 0 -0.5 0 0 0.5 0 0 0 0 0 -0.025 -0.5 0 0 0 5
0 0 0.833333 -1 0 -0.833333 1 0 0 0 0 -0.0416667 0.833333 -1 0 0 1.66667
0 0 -0.833333 0 -1 0.833333 0 1 0 0 0 -0.108333 -0.833333 0 -1 0 3.33333
0 0 -83.3333 100 0 283.333 0 50 0 0 0 4.16667 -83.3333 100 0 1 -166.667
Tableau #6
x y q s u p r t s1 s2 s3 s4 s5 s6 s7 -z
0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0
0 1 0.833333 0 0 -0.833333 0 0 0 0 0 0.0583333 0.833333 0 0 0 1.66667
1 0 -0.5 0 0 0.5 0 0 0 0 0 -0.025 -0.5 0 0 0 5
0 0 0.833333 -1 0 -0.833333 1 0 0 0 0 -0.0416667 0.833333 -1 0 0 1.66667
0 0 -0.833333 0 -1 0.833333 0 1 0 0 0 -0.108333 -0.833333 0 -1 0 3.33333
0 0 -41.6667 100 50 241.667 0 0 0 0 0 9.58333 -41.6667 100 50 1 -333.333
Tableau #7
x y q s u p r t s1 s2 s3 s4 s5 s6 s7 -z
0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 0
0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0
0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0
0 1 0 1 0 0 -1 0 0 0 0 0.1 0 1 0 0 0
1 0 0 -0.6 0 0 0.6 0 0 0 0 -0.05 0 -0.6 0 0 6
0 0 1 -1.2 0 -1 1.2 0 0 0 0 -0.05 1 -1.2 0 0 2
0 0 0 -1 -1 0 1 1 0 0 0 -0.15 0 -1 -1 0 5
0 0 0 50 50 200 50 0 0 0 0 7.5 0 50 50 1 -250
