Other Math Answers

(b) Sate two basic features, which characterize dynamic programming problems. (2 marks)

(c) Define game theory and state three managerial applications of game theory. (4 marks)

2. (a) A Company has factories at A, B and C which supply warehouses at D, E and F. Weekly factory capacities are 200, 160 and 90 units. Weekly warehouse requirements (demand) are 180, 120 and 150 units respectively. Unit shipping costs (in kshs) are as follows

Factory D E F Capacity
A 16 20 12 200
B 14 8 18 160
C 26 24 16 90
Demand 180 120 150 450
Draw the transportation tableau and show whether it is a degenerate or a non-degenerate problem.
(4 marks)
(b) Sate two basic features, which characterize dynamic programming problems. (2 marks)
(c) Define game theory and state three managerial applications of game theory. (4 marks)

A company produces three products, 1 P2 P , and P , from three raw materials A, B 3
and C. One unit of product P1 requires one unit of A, 3 unit of B and 2 units of C. A
unit of product P2 requires two units each of A and B and 3 units of C, while one
unit of P needs 2 units of A, 6 units of B and 3 4 units of C. The company has a daily
availability of 8 units of A, 12 units of B and 12 units of C. It is further known that
the profit per unit is ` 3, 2 and 5 for 1 P2 P , and P , respectively. How many units of 3
product P1 , product P2 and product P should the company manufacture to 3
maximize the profit? Formulate this problem as a Linear programming problem.

2. (a) A Company has factories at A, B and C which supply warehouses at D, E and F. Weekly factory capacities are 200, 160 and 90 units. Weekly warehouse requirements (demand) are 180, 120 and 150 units respectively. Unit shipping costs (in kshs) are as follows

Factory D E F Capacity
A 16 20 12 200
B 14 8 18 160
C 26 24 16 90
Demand 180 120 150 450
Draw the transportation tableau and show whether it is a degenerate or a non-degenerate problem.
(4 marks)
(b) Sate two basic features, which characterize dynamic programming problems. (2 marks)
(c) Define game theory and state three managerial applications of game theory. (4 marks)

A company produces three products, 1 P2 P , and P , from three raw materials A, B 3
and C. One unit of product P1 requires one unit of A, 3 unit of B and 2 units of C. A
unit of product P2 requires two units each of A and B and 3 units of C, while one
unit of P needs 2 units of A, 6 units of B and 3 4 units of C. The company has a daily
availability of 8 units of A, 12 units of B and 12 units of C. It is further known that
the profit per unit is ` 3, 2 and 5 for 1 P2 P , and P , respectively. How many units of 3
product P1 , product P2 and product P should the company manufacture to 3
maximize the profit? Formulate this problem as a Linear programming problem.

1.(p∧q)=(q∨p) and (p∨q)=(q∨p)
implies an _____
a.Associative laws
b.Distributive Laws
c.Commutative Laws
d.Idempotent Laws

2.____is equivalent to (p∨q)
a.(p↔q)
b.(∼p∧q)
c.(∼∼q)
d.∼(∼p∧∼q)

3.The_____ of a matrix is the inter- changing of its row with the column
a.diagonal matrix
b.zero matrix
c.transpose matrix
d.identity matrix

4.A matrix that has elements only on its diagonal is called
a.transpose matrix
b.diagonal matrix
c.identity matrix
d.zero matrix

5.given that A= ⎛1 2 3⎞
⎝4 5 6⎠ and


⎛1 2⎞
B= ⎜3 4⎟
⎝5 6⎠



. Find AB

a.⎛ 0 12 17⎞
⎜ 7 10 31⎟
⎝20 40 45⎠


b.⎛ 0 12 17⎞
⎜ 19 26 31⎟
⎝ 20 40 45⎠

c.⎛ 0 12 17⎞
⎜ 19 26 31⎟
⎝ 29 40 51⎠

d.⎛ 5 12 15⎞
⎜ 19 26 31⎟
⎝ 29 40 51⎠

For a1,......,an € R, a1<a2<....<an, show that
{n/(a1-a0)}+{(n-1)/(a2-a1)}+.......+ {1/(an- an-1)} >= summation of k=1 to n (k^2/ ak).

Which of the following statements are true?Give reasons for your answers in each of the following parts.
(i)The solution x_1=1,x_2=2,x_3=x_4=x_5=0 of the equations:
2x_1+x_2+x_3+x_4=4
2x_1+x_2+2x_3+x_5=4 is a solution.
(ii)The number of basic variables in a feasible solution of a balanced transportation problem with m sources and n destinations is mn.
(iii)A pay-off matrix can posses more than one saddle-point in case of a pure strategy game problem.
(iv)(1,2) is an optimal solution of LPP
"Max. z=2x_1+4x_2
s.t. x_1+2x_2≤5
x_1+x_2≤4
x_1≥0
x_2≥0."
(v)AUB is a convex set,where A={(x_1,x_2)|x_1+x_2≤8,2x_1+x_2≥10,x_1≥0,x_2≥0} and B={(x_1,x_2)|x_1+x_2≥8,2x_1+x_2≤10,x_1≥0,x_2≥0}.

Solve the cost-minimising assignment problem whose cost matrix is given below:
Machines
M_1 M_2 M_3 M_4
J_1 10 12 9 11
Jobs J_2 5 10 7 8
J_3 12 14 13 11
J_4 8 15 11 9

Write the LPP for player B corresponding to the game given by the matrix.
(Player B)
1 2 3
1 8 4 2
(Player A) 2 2 8 4
3 1 2 8
Further,suppose the final optimal table for the problem you get is given by
BasicsA_1 A_2 A_3 A_4 A_5 A_6Solution
y_1 1 0 0 1/7 -1/14 0 1/14
y_2 0 1 0 -3/98 31/196 -1/14 11/196
y_3 0 0 1 -1/98 -3/98 1/7 5/49
ω 0 0 0 5/49 11/196 1/14 45/196
From this table,find the value of the game and the optimal strategies for both players.