QT 4 - Progress Assignments PDF

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Progress Assignments...

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Quantitative Techniques Cat II 1. A calculator Company produces a scientific calculator and a graphing calculator. Longterm projections indicate an expected demand of at least 100 scientific and 80 graphing calculators each day. Because of limitations on production capacity, no more than 200 scientific and 170 graphing calculators can been made daily. To satisfy a shipping contract, a total of at least 200 calculators much be shipped each day. If each scientific calculator sold results in a profit of $ 2, but each graphing calculator produces a 5 profit, how many of each type should be made daily to maximize net profits? Use graphical method.

(8 marks)

The solution for this problem requires the optimal number of calculators Let x be the number of scientific calculators And y be the number of graphing calculators The non negativity constraints can be ignored (i.e. provided with

x , y ≥0 ) since we are already

x ≥ 100∧ y ≥ 80.

The problem also provides maximum constraints as The minimum shipping requirement provides that

x ≤ 200∧ y ≤ 170 x+ y ≥ 200

The optimization problem in this case will be the profit relation The entire system will be P=¿ 2 x+ 5 y max ¿ s .t 100 ≤ x ≤200, 80 ≤ y ≤ 170, x + y ≥ 200

Graph is as attached.

P=2 x+5 y

The region shaded as R is the required region and so we test the corner points of the graph, that is, the points (100, 100), (100, 170), (120, 80), (200, 80), (200, 170) The maximum P is at (200, 170) with a maximum profit of sh. 1250 2. A marketing manager wants to assign four regions to four different salesmen. The salesmen differ in their efficiency and territories also differ in potentiality. An estimated sale (in Ks, 000s) by different sales men in the four territories are given below. Territory Salesman A B C D

X

Y

16 7 10 25

13 12 25 27

Z 17 37 14 18

W 23 18 9 25

Determine an optimal assignment schedule for maximum sales. (8marks) This is an assignment problem for maximization. The columns are reduced by the largest number in each column to yield

Salesman

X

Y

Z

W

A

9

14

20

2

B

18

15

0

7

C

15

2

23

16

D

0

0

19

0

X

Y

A

7

12

18

0

B

18

15

0

7

C

13

0

21

14

D

0

0

19

0

X

Y

A

7

12

18

0

B

18

15

0

7

C

13

0

21

14

D

0

0

19

0

Row reduction Salesman

Z

W

Column reduction Salesman

Allocations A to W B to Z

Z

W

C to Y D to X Total sales=23000+37000+25000+25000=sh. 110, 000

3. Maureen mimi, is a proprietor in Nyeri Town, she is planning to explain her business with an aim of making a profit of shs 160,000 within the next financial year. Analysis of her past records revealed the following: -Selling price per unit has been shs 200 -Variable cost was as follows: for 10000 units it was shs 800,000 30,000 unit it was shs. 2,400,000 -Fixed cost per year in the past was shs 140,000 for Maureen to attain the said profit she has to intensity her advertisement which is estimated to cost shs.60,000. Determine the break-even point of Maureen’s business.(8 marks) Break even point (units)= total¿ cost ÷(selling price−variable cost) Variable cost per unit= sh. 80 Selling price per unit=200 Total fixed cost=140000+60000=200000 BEP (units)= 200000 ÷( 200 −80 ) =1667units BEP (shillings)=1667x200=sh. 333400 4.

Susan took two tests. The probability of her passing both tests is 0.6. The probability of her passing the first test is 0.8. what is the probability of her passing the second test given that she has passed the first test?(6 marks) P ( second / first )=

P (first ∧second ) 0.6 = =0.75 P(first) 0.8...