Question

A company manufactures Products A, B, and C. Each product is processed in three departments: I, II, and III. The total available labor-hours per week for Departments I, II, and III are 1020, 1140, and 960, respectively. The time requirements (in hours per unit) and the profit per unit for each product are as follows. Product A Product B Product C Dept. I 2 1 2 Dept. II 3 1 2 Dept. II 2 2 1 Profit $18 $12 $15

How many units of each product should the company produce in order to maximize its profit?
Product A
Product B
Product C
What is the largest profit the company can realize?
$
Are there any resources left over? (If so, enter the amount remaining. If not, enter 0.)
labor in Dept. I
labor in Dept. II
labor in Dept. III

Answer 1

a)

product A = 12O units

Product B = 220 units

product C = 280 units

b) $9000 = max/largest profit

c) No resource left

Explanation:

Available hours

Dept. I = 1020

Dept. II = 1140

Dept. III = 960

Total available hours = 3120 hours

products produced by each department

product A Product B Product C

Dept. I 2 1 2

Dept. II 3 1 2

Dept. III 2 2 1

profits $18 $12 $15

Determine how many units of each product to be produced to attain maximum profit

let each product be represented as : x , y , z

2x + y + 2z = 1020 -------- ( department A ) --- 1

3x + y + 2z = 1140 -------- ( department B ) --- 2

2x + 2y + z = 960 --------- ( department c ) ---- 3

max profit : 18 x + 12y + 15 y

solving equation 1 from 2

x = 120

solve equation 2 from 3 simultaneously

x - y + z = 180

-y + z = 60

solve equation 1 and 3

-y + z = 60

∴ z = 60 + y

back to equation 1

2( 120 ) + y + 2( 60 + y ) = 1020

240 + y + 120 + 2y = 1020

∴ y = (1020 - 360 )/ 3 = 220

therefore ; z = 60 + 220 = 280

amount of each product to be produced to gain maximum profit

product A = 12O units

Product B = 220 units

product C = 280 units

ii) The largest profit

18 ( 120 ) + 12(220) + 15 ( 280 ) = $9000