Application of matrix algebra (Numerical)
A biscuit manufacturing company has two plants located at two different places. Each plant produces three different types of biscuits Tiger, Glucose and Salty. Matrix A shows the number of packets of each type of biscuit produced per day and matrix B shows their cost and selling price.
A =
Plant I
Plant II
Tiger
300
200
Glucose
200
300
Salty
100
200
B =
C.P.
10
11
12
S.P.
13
12
14
If plant I and Plant II operate for 20 and 25 days respectively in a month. Using matrix algebra, find the total :
i) Production of biscuits from each plant per day.
ii) Production of each type of biscuits in the month,
iii) Profit in the month.
A =
Plant I
Plant II
Tiger
300
200
Glucose
200
300
Salty
100
200
B =
C.P.
10
11
12
S.P.
13
12
14
If plant I and Plant II operate for 20 and 25 days respectively in a month. Using matrix algebra, find the total :
i) Production of biscuits from each plant per day.
ii) Production of each type of biscuits in the month,
iii) Profit in the month.
Solution
A =
300
200
200
300
100
200
300
200
200
300
100
200
Column matrix of cost price, C =
10
11
12
10
11
12
Column matrix of selling price, S =
13
12
14
13
12
14
Row matrix of operating days, D = [ 20 25 ]
i)Production of biscuits from each plant per day :
=
300
200
200
300
100
200
×
1
1
1
=
300×1+200×1+100×1
200×1+300×1+200×1
=
300+200+100
200+300+200
=
600
700
300
200
200
300
100
200
×
1
1
1
=
300×1+200×1+100×1
200×1+300×1+200×1
=
300+200+100
200+300+200
=
600
700
Production of biscuits from each plant per day =
Plant I
Plant II
600
700
Plant I
Plant II
600
700
ii)Production of each type of biscuits in the month :
= [ 20 25 ]
×
300
200
200
300
100
200
= [ 20×300+25×200 20×200+25×300 20×100+25×200 ]
= [ 6000+5000 4000+7500 2000+5000 ]
= [ 11000 11500 7000 ]
×
300
200
200
300
100
200
= [ 20×300+25×200 20×200+25×300 20×100+25×200 ]
= [ 6000+5000 4000+7500 2000+5000 ]
= [ 11000 11500 7000 ]
Production of each type of biscuits in the month, P =
Tiger
11000
Glucose
11500
Salty
7000
Tiger
11000
Glucose
11500
Salty
7000
ii)Profit in the month :
Calculating total selling price in a month :
= [ 11000 11500 7000 ]
×
13
12
14
= [ 11000×13+11500×12+7000×14 ]
= [ 143000+138000+98000 ]
= [ 379000 ]
×
13
12
14
= [ 11000×13+11500×12+7000×14 ]
= [ 143000+138000+98000 ]
= [ 379000 ]
Total selling price in a month = Rs. 379,000
Calculating total cost price in a month :
= [ 11000 11500 7000 ]
×
10
11
12
= [ 11000×10+11500×11+7000×12 ]
= [ 110000+126500+84000 ]
= [ 320500 ]
×
10
11
12
= [ 11000×10+11500×11+7000×12 ]
= [ 110000+126500+84000 ]
= [ 320500 ]
Total cost price in a month = Rs. 320,500
Profit in the month = Total selling price in a month - Total cost price in a month = Rs. 379,000 - Rs. 320,500 = Rs. 58500
Alternatively,
i)
Production of biscuits from Plant I = Sum of elements of Row 1 = 300 + 200 + 100 = 600
Production of biscuits from Plant II = Sum of elements of Row 2 = 200 + 300 + 200 = 700
Production of biscuits from Plant I = Sum of elements of Row 1 = 300 + 200 + 100 = 600
Production of biscuits from Plant II = Sum of elements of Row 2 = 200 + 300 + 200 = 700
ii)
Plant I:
Operating days in a month = 20 days
Production of Tiger biscuits from Plant I in a month = (Production of Tiger biscuits from Plant I in a day × Operating days in a month) = 300 × 20 = 6000
Production of Glucose biscuits from Plant I in a month = (Production of Glucose biscuits from Plant I in a day × Operating days in a month) = 200 × 20 = 4000
Production of Salty biscuits from Plant I in a month = (Production of Salty biscuits from Plant I in a day × Operating days in a month) = 100 × 20 = 2000
Plant II:
Operating days in a month = 25 days
Production of Tiger biscuits from Plant II in a month = (Production of Tiger biscuits from Plant II in a day × Operating days in a month) = 200 × 25 = 5000
Production of Glucose biscuits from Plant II in a month = (Production of Glucose biscuits from Plant II in a day × Operating days in a month) = 300 × 25 = 7500
Production of Salty biscuits from Plant II in a month = (Production of Salty biscuits from Plant II in a day × Operating days in a month) = 200 × 25 = 5000
Production of Tiger biscuits in a month = 6000 + 5000 = 11000
Production of Glucose biscuits in a month = 4000 + 7500 = 11500
Production of Salty biscuits in a month = 2000 + 5000 = 7000
Plant I:
Operating days in a month = 20 days
Production of Tiger biscuits from Plant I in a month = (Production of Tiger biscuits from Plant I in a day × Operating days in a month) = 300 × 20 = 6000
Production of Glucose biscuits from Plant I in a month = (Production of Glucose biscuits from Plant I in a day × Operating days in a month) = 200 × 20 = 4000
Production of Salty biscuits from Plant I in a month = (Production of Salty biscuits from Plant I in a day × Operating days in a month) = 100 × 20 = 2000
Plant II:
Operating days in a month = 25 days
Production of Tiger biscuits from Plant II in a month = (Production of Tiger biscuits from Plant II in a day × Operating days in a month) = 200 × 25 = 5000
Production of Glucose biscuits from Plant II in a month = (Production of Glucose biscuits from Plant II in a day × Operating days in a month) = 300 × 25 = 7500
Production of Salty biscuits from Plant II in a month = (Production of Salty biscuits from Plant II in a day × Operating days in a month) = 200 × 25 = 5000
Production of Tiger biscuits in a month = 6000 + 5000 = 11000
Production of Glucose biscuits in a month = 4000 + 7500 = 11500
Production of Salty biscuits in a month = 2000 + 5000 = 7000
iii)
Profit in a month = Production of Tiger biscuits in a month(S.P. -C.P.) + Production of Glucose biscuits in a month(S.P. -C.P.) + Production of Salty biscuits in a month(S.P. -C.P.)
Profit in a month = 11000(13 - 10) + 11500(12 - 11) + 7000(14 - 12)
= 11000 × 3 + 11500 × 1 + 7000 × 2
33000 + 11500 + 14000 = 58500
Profit in a month is found to be Rs. 58,500.
Profit in a month = Production of Tiger biscuits in a month(S.P. -C.P.) + Production of Glucose biscuits in a month(S.P. -C.P.) + Production of Salty biscuits in a month(S.P. -C.P.)
Profit in a month = 11000(13 - 10) + 11500(12 - 11) + 7000(14 - 12)
= 11000 × 3 + 11500 × 1 + 7000 × 2
33000 + 11500 + 14000 = 58500
Profit in a month is found to be Rs. 58,500.
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