Other Math Answers

Consider the following product mix problem:

Three machine shops A, B, C produces three types of products X, Y, Z respectively. Each product involves operation of each of the machine shops. The time required for each operation on various products is given as follows:

Machine shops

Products

A

B

C

Profit per unit

X

10

7

2

Birr 12

Y

2

3

4

Birr 3

Z

1

2

1

Birr 1

Available Hours

100

77

80

Make the Mathematical model of the above primal problem to maximise the profit and then make the dual problem

A work shop contains four persons available for work on the four jobs. Only one person can work on any one job. The following table shows the cost of assigning each person to each job. The objective is to assign person to jobs such that the total assignment cost is a minimum.

Jobs

1

2

3

4

A

20

25

22

28

Persons

B

15

18

23

17

C

19

17

21

24

D

25

23

24

24

You will explore the wonderful world of descriptive statistics. You may not have noticed how often you are presented with statistics in the media and in everyday conversations. It is common for people to make statements like “Statistics show that… [insert claim].”

What sorts of follow up questions about the statistics might you ask that person in order to obtain the data needed to make a decision about the validity of that statement?

Q4. Convert the following linear programming problem into dual problem. Maximise Z = 22x1 + 25x2 +19x3 Subject to: 18x1 + 26x2 + 22x3 ≤ 350 14x1 + 18x2 + 20x3 ≥180 17x1 + 19x2 + 18x3 = 205 x1, x2, x3 ≥ 0 Q5. A work shop contains four persons available for work on the four jobs. Only one person can work on any one job. The following table shows the cost of assigning each person to each job. The objective is to assign person to jobs such that the total assignment cost is a minimum. Jobs 1 2 3 4 A 20 25 22 28 Persons B 15 18 23 17 C 19 17 21 24 D 25 23 24 24

Consider a Cobb-Douglas production function Q =ALαKβ where the amount produced (Q)

is given as a function of the labor (L) and capital (K) used and A, α, β are constants. Let

the value of the constants in the above production function be given by A = 1, α = 2/3

, β = 1/3

.

Wage rate (w) and per unit capital rate (r) be Rs 4 and Rs 27, respectively. Suppose that

the firm wishes to produce 1080 units of output Q. What will be the optimal amount of

factors that the firm needs to employ for this? Also calculate the minimum cost of

producing such an output level?

A consumer has the utility function over goods X and Y,

U(X, Y) = √x + √y

Let the price of good X be given by PX, let the price of good Y be given by PY, and let

income be given by M.

(a) Derive the consumer’s Marshalian demand functions for good X and good Y. (5)

(b) Is good Y normal or inferior? (3)

(c) If PX = 2, PY = 1, and M = 12, compute the utility maximizing consumption bundle of

goods X and Y.

A monopolist operates under two plants, 1 and 2. The marginal costs of the two plants are given by MC1 = 20 + 2Q1 and MC2 = 10 + 5Q2 where Q1 and Q2 represent units of output produced by plant 1 and 2 respectively. If the price of this product is given by 20 – 3(Q1 + Q2), how much should the firm plan to produce in each plant, and at what price should it plan to sell the product?

A monopolist operates under two plants, 1 and 2. The marginal costs of the two plants are given by MC1 = 20 + 2Q1 and MC2 = 10 + 5Q2 where Q1 and Q2 represent units of output produced by plant 1 and 2 respectively. If the price of this product is given by 20 – 3(Q1 + Q2), how much should the firm plan to produce in each plant, and at what price should it plan to sell the product?